Research ArticleA Modified Hybrid Genetic Algorithm forSolving Nonlinear Optimal Control Problems
Saeed Nezhadhosein1 Aghileh Heydari1 and Reza Ghanbari2
1Department of Applied Mathematics Payame Noor University Tehran 193953697 Iran2Department of Applied Mathematics Faculty of Mathematical Science Ferdowsi University of MashhadMashhad 9177948953 Iran
Correspondence should be addressed to Saeed Nezhadhosein s nejhadhoseinpnuacir
Received 4 September 2014 Accepted 30 January 2015
Academic Editor Alain Vande Wouwer
Copyright copy 2015 Saeed Nezhadhosein et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Here a two-phase algorithm is proposed for solving bounded continuous-time nonlinear optimal control problems (NOCP) Ineach phase of the algorithm a modified hybrid genetic algorithm (MHGA) is applied which performs a local search on offspringsIn first phase a random initial population of control input values in time nodes is constructed Next MHGA starts with thispopulation After phase 1 to achieve more accurate solutions the number of time nodes is increased The values of the associatednew control inputs are estimated by Linear interpolation (LI) or Spline interpolation (SI) using the curves obtained from the phase1 In addition to maintain the diversity in the population some additional individuals are added randomly Next in the secondphase MHGA restarts with the new population constructed by above procedure and tries to improve the obtained solutions at theend of phase 1 We implement our proposed algorithm on 20 well-known benchmark and real world problems then the results arecompared with some recently proposed algorithms Moreover two statistical approaches are considered for the comparison of theLI and SI methods and investigation of sensitivity analysis for the MHGA parameters
1 Introduction
NOCPs are dynamic optimization problemswithmany appli-cations in industrial processes such as airplane robotic armbio-process system biomedicine electric power systems andplasma physics [1]
High-quality solutions and the less required computa-tional time aremain issues for solvingNOCPsThenumericalmethods direct [2] or indirect [3] usually have two maindeficiencies less accuracy and convergence to a local solu-tion In direct methods the quality of solution depends ondiscretization resolution Since these methods using controlparametrization convert the continuous problem to discreteproblem they have less accuracy However the adaptivestrategies [4 5] can overcome these defects but they maybe trapped by a local optimal yet In indirect approachesthe problem through the use of the Pontryagins minimumprinciple (PMP) is converted into a two-boundary valueproblem (TBVP) that can be solved by numerical methods
such as shooting method [6] These methods need the goodinitial guesses that lie within the domain of convergenceTherefore the numerical methods usually are not suitablefor solving NOCPs especially for large-scale andmultimodalmodels
Metaheuristics as the global optimization methods canovercome these problems but they usually need morecomputational time though they do not really need goodinitial guesses and deterministic rules Several researchersused metaheuristics to solve optimal control problems ForinstanceMichalewicz et al [7] applied floating-point Geneticalgorithms (GA) to solve discrete time optimal control prob-lems Yamashita and Shima [8] used the classical GAs to solvethe free final time optimal control problems with terminalconstraints Abo-Hammour et al [9] used continuous GA forsolvingNOCPsMoreover the other usages ofGA for optimalcontrol problems can be found in [6 10] Lopez Cruz et al[11] applied differential evolution (DE) algorithms for solvingthe multimodal optimal control problems Recently Ghosh
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 139036 21 pageshttpdxdoiorg1011552015139036
2 Mathematical Problems in Engineering
et al [12] developed an ecologically inspired optimizationtechnique called invasive weed optimization (IWO) forsolving optimal control problems The other well-knownmetaheuristic algorithms used for solving NOCPs are geneticprogramming (GP) [13] particle swarm optimization (PSO)[14 15] ant colony optimization (ACO) [16] and DE [17 18]
To increase the quality of solutions and decrease therunning time hybrid methods were introduced which useda local search in the implementation of a population-basedmetaheuristics [19] Modares and Naghibi-Sistani [20] pro-posed a hybrid algorithm by integrating an improved PSOwith successive quadratic programming (SQP) for solvingNOCPs Recently Sun et al [21] proposed a hybrid improvedGA which used simplex method (SM) to perform a localsearch for solving NOCPs and applied it for chemicalprocesses
Based on the success of the hybrid methods for solvingNOCPs mentioned above we here use a modified hybridgenetic algorithm (MHGA) which combines GA with SQPsee [22] as a local search SQP is an iterative algorithm forsolving nonlinear programming (NLP) problems which usesgradient information It can moreover be used for solvingNOCPs see [23ndash25] For decreasing the running time in theearly generations (iterations) of MHGA a less number ofiterations for SQP was used and then when the promisingregion of search space was found we increase the number ofiterations of SQP gradually
To perform MHGA for solving an NOCP the timeinterval is uniformly divided by using a constant number oftime nodes Next in each of these time nodes the controlvariable is approximated by a scaler vector of control inputvalues Thus an infinite dimensional NOCP is changed toa finite dimensional NLP Now we encounter two conflictsituations the quality of the global solution and the neededcomputational time In other words when the number oftime nodes is increased then we expect that the quality ofthe global solution is also increased but we know that in thissituation the computational time is increased dramatically Inother situation if we consider less number of time nodesthen the computational time is decreased but we may finda poor local solution To conquer these problems MHGAperforms in two phases In the first phase (explorationphase) to decrease the computational time and to find apromising region of search space MHGA uses a less numberof time nodes After phase 1 to increase the quality ofsolutions obtained from phase 1 the number of time nodesis increased Using the population obtained in phase 1 thevalues of the new control inputs are estimated by Linear orSpline interpolations Next in the second phase (exploitationphase) MHGA uses the solutions constructed by the aboveprocedure as an initial population
The paper is organized as follows in Section 2 theformulation of the problem is introduced In Section 3 theproposed MHGA is presented In Section 4 we introduceour algorithm for solving NOCP In Section 5 we provide20 numerical benchmark examples to compare the proposedalgorithm with the other recently proposed algorithmsIn Section 6 we consider two statistical approaches for
the comparison of the LI and SI methods and investigation ofsensitivity analysis of the algorithm parameters The impactof SQP as local search in the proposed algorithm is surveyedin Section 7 We conclude in Section 8
2 Formulation of Problem
The bounded continuous-time NOCP is considered as find-ing the control input 119906(119905) isin R119898 over the planning horizon[1199050 119905119891] which minimizes the cost functional
119869 = 120601 (119909 (119905119891) 119905119891) + int
119905119891
1199050
119892 (119909 (119905) 119906 (119905) 119905) 119889119905 (1)
subject to
(119905) = 119891 (119909 (119905) 119906 (119905) 119905) (2)
119888 (119909 119906 119905) = 0 (3)
119889 (119909 119906 119905) le 0 (4)
120595 (119909 (119905119891) 119905119891) = 0 (5)
119909 (1199050) = 1199090 (6)
where 119909(119905) isin R119899 denotes the state vector for the system and1199090isin R119899 is the initial state The functions 119891 R119899 timesR119898 timesR rarr
R119899 119892 R119899 timesR119898 timesR rarr R 119888 R119899 timesR119898 timesR rarr R119899119888 119889 R119899 times
R119898timesR rarr R119899119889 120595 R119899timesR rarr R119899120595 and 120601 R119899timesR rarr R areassumed to be sufficiently smooth on appropriate open setsCost function (1) must be minimized subject to dynamic (2)control and state equality constraints (3) and control and stateinequality constraints (4) the final state constraints (5) andthe initial condition (6) A special case of the NOCPs is thelinear quadratic regulator (LQR) problemwhere the dynamicequations are linear and the objective function is a quadraticfunction of 119909 and 119906 The minimum time problems trackingproblem terminal control problem andminimum energy areanother special case of NOCPs
3 Modified Hybrid Genetic Algorithm
In this section first MHGA as a subprocedure for the mainalgorithm is introduced To perform MHGA the controlvariables are discretized Next NOCP is changed into a finitedimensional NLP see [21 26] Now we can imply a GA tofind the global solution of the corresponding NLP In thefollowing we introduce GA operators
31 Underlying GA GAs introduced by Holland in 1975 areheuristics and probabilistic methods [27] These algorithmsstart with an initial population of solutions This populationis evaluated by using genetic operators that include selectioncrossover andmutation Here inMHGA the underlying GAhas the following steps
InitializationThe time interval is divided into119873119905minus1 subinter-
vals using time nodes 1199050 119905
119873119905minus1and then the control input
Mathematical Problems in Engineering 3
Initialization Input the the size of tournament119873tour and select119873tour
individuals from population randomly Let 119896119894 119894 = 1 2 119873tour be the indexes of them
repeatif 119873tour = 2 thenif 119868(1198801198961 ) lt 119868(119880
1198962) then
Let 119896 = 1198961
elseLet 119896 = 119896
2
end ifend if
Perform Tournament algorithm with size119873tour2 with output 1198961
Perform Tournament algorithm with size119873tour2 with output 1198962
until 1198961
= 1198962
Return the 119896th individual of population
Algorithm 1 Tournament algorithm
values are computed (or selected randomly)This can be doneby the following stages
(1) Let 119905119895= 1199050+ 119895ℎ where ℎ = (119905
119891minus 1199050)(119873119905minus 1) 119895 =
0 1 119873119905minus1 be time nodes where 119905
0and 119905119891are the
initial and final times respectively
(2) The corresponding control input value at each timenode 119905
119895is an119898times1 vector 119906
119895 which can be calculated
randomly with the following components
119906119894119895= 119906119895(119894) = 119906left (119894) + (119906right (119894) minus 119906left (119894)) sdot 119903119894119895
119894 = 1 2 119898 119895 = 0 1 119873119905minus 1
(7)
where 119903119894119895is a random number in [0 1]with a uniform
distribution and 119906left 119906right isin R119898 are the lower and theupper bound vectors of control input values whichcan be given by the problemrsquos definition or the user(eg see the NOCPs numbers (6) and (5) in theAppendix resp) So each individual of the popula-tion is an 119898 times 119873
119905matrix as 119880 = (119906
119894119895)119898times119873119905
= [119906119895]119873119905minus1
119895=0
Next we let 119880(119896) = (119906119894119895)119898times119873119905
119896 = 1 2 119873119901as 119896th
individual of the population which 119873119901is the size of
the population
Evaluation For each control input matrix 119880(119896) 119896 =
1 2 119873119901 the corresponding state variable is an 119899 times 119873
119905
matrix119883(119896) and it can be computed by the forthRunge-Kuttamethod on dynamic system (2) with the initial condition(6) approximately Then the performance index 119869(119880(119896))is approximated by a numerical method (denoted by 119869) IfNOCP includes equality or inequality constraints (3) or (4)then we add some penalty terms to the corresponding fitness
value of the solution Finally we assign 119868(119880(119896)
) to 119880(119896) as the
fitness value as follows
119868 (119880(119896)
) = 119869 +
119899119889
sum
119897=1
119873119905minus1
sum
119895=0
1198721119897max 0 119889
119897(119909119895 119906119895 119905119895)
+
119899119888
sum
ℎ=1
119873119905minus1
sum
119895=0
1198722ℎ1198882
ℎ(119909119895 119906119895 119905119895)
+
119899120595
sum
119894=119901
11987231199011205952
119901(119909119873119905minus1
119905119873119905minus1
)
(8)
where 1198721= [119872
11 119872
1119899119889]119879 1198722= [119872
21 119872
2119899119888]119879 and
1198723
= [11987231 119872
3119899120595]119879 are big numbers as the penalty
parameters 119888ℎ(sdot sdot) ℎ = 1 2 119899
119888 119889119897(sdot sdot) 119897 = 1 2 119899
119889
and 120595119901(sdot sdot) 119901 = 1 2 119899
120595are defined in (3) (4) and (5)
respectively
Selection To select two parents we use a tournament selec-tion [27] It can be applied for parallel GA The tournamentoperator applies competition among the same individualsand the best of them is selected for next generation At firstwe select a specified number of individuals from populationrandomly This number is tournament selection parameterwhich is denoted by119873tourThe tournament algorithm is givenin Algorithm 1
Crossover When two parents 119880(1) and 119880(2) are selected we
use the following stages to construct an offspring(1) Select the following numbers
1205821isin [0 1] 120582
2isin [minus120582max 0] 120582
3isin [1 1 + 120582max]
(9)randomly where 120582max is a random number in [0 1]
(2) Let
of119896 = 120582119896119880(1)
+ (1 minus 120582119896) 119880(2)
119896 = 1 2 3 (10)
4 Mathematical Problems in Engineering
where 120582119896 119896 = 1 2 3 is defined in (9) For 119894 = 1 119898
and 119895 = 1 119873119905 if (of119896)
119894119895gt 119906right(119894) then let (of
119896
)119894119895=
119906right(119894) and if (of119896
)119894119895lt 119906left(119894) then let (of
119896
)119894119895= 119906left(119894)
(3) Let of = oflowast where oflowast is the best of119896 119896 = 1 2 3
constructed by (10)
MutationWe apply a perturbation on each component of theoffspring as follows
(of)119894119895= (of)
119894119895+ 119903119894119895sdot 120572 119894 = 1 2 119898 119895 = 1 2 119873
119905
(11)
where 119903119894119895is selected randomly in minus1 1 and 120572 is a random
number with a uniform distribution in [0 1] If (of)119894119895
gt
119906right(119894) then let (of)119894119895= 119906right(119894) and if (of)
119894119895lt 119906left(119894) then
let (of)119894119895= 119906left(119894)
ReplacementHere in the underling GA we use a traditionalreplacement strategy The replacement is done if the newoffspring has two properties first it is better than the worstperson in the population 119868(of) lt max
1le119894le119873119901119868(119880(119894)
) secondit is not very similar to a person in the population that is foreach 119894 = 1 119873
119901 at least one of the following conditions is
satisfied10038161003816100381610038161003816119868 (of) minus 119868 (119880
(119894)
)
10038161003816100381610038161003816gt 120576
10038171003817100381710038171003817of minus 119880
(119894)10038171003817100381710038171003817gt 120576
(12)
where 120576 is the machine epsilon
Termination Conditions Underlying GA is terminated whenat least one of the following conditions is occurred
(1) Themaximumnumber of generations119873119892 is reached
(2) Over a specified number of generations 119873119894 we do
not have any improvement (the best individual is notchanged) or the two-norm or error of final stateconstraints will reach a small number as the desiredprecision 120576 that is
120593119891=100381710038171003817100381712059510038171003817100381710038172
lt 120576 (13)
where120595 = [1205951 1205952 120595
119899120595]119879 is the vector of final state
constraints defined in (5)
32 SQP The fitness value in (8) can be viewed as anonlinear objective function with the decision variable as119906 = [119906
0 1199061 119906
119873119905minus1] This cost function with upper and
lower bounds of input signals construct a finite dimensionalNLP problem as following
min 119868 (119906) = 119868 (1199060 1199061 119906
119873119905minus1) (14)
st 119906left le 119906119895le 119906right 119895 = 0 1 119873
119905minus 1 (15)
SQP algorithm [22 28] is performed on the NLP (14)-(15)using 1199060 = of and constructed in (11) as the initial solutionwhen the maximum number of iteration is 119904119902119901119898119886119909119894119905119890119903
SQP is an effective and iterative algorithm for thenumerical solution of the constrained NLP problem Thistechnique is based on finding a solution to the system ofnonlinear equations that arise from the first-order necessaryconditions for an extremum of the NLP problem Usingan initial solution of NLP 119906119896 119896 = 0 1 a sequence ofsolutions as 119906119896+1 = 119906
119896
+ 119889119896 is constructed which 119889
119896 is theoptimal solution of the constructed quadratic programming(QP) that approximates NLP in the iteration 119896 based on119906119896 as the search direction in the line search procedure
For the NLP (15) the principal idea is the formulation of aQP subproblem based on a quadratic approximation of theLagrangian function as 119871(119906 120582) = 119868(119906) + 120582
119879
ℎ(119906) where thevector 120582 is Lagrangian multiplier and ℎ(119906) return the vectorof inequality constraints evaluated at 119906 The QP is obtainedby linearizing the nonlinear functions as follows
min 1
2
119889119879
119867(119906119896
) 119889 + nabla119868 (119906119896
)
119879
119889 (16)
nablaℎ (119906119896
)
119879
119889 + ℎ (119906119896
) le 0 (17)
Similar to [26] here a finite difference approximation isapplied to compute the gradient of the cost function and theconstraints with the following components
120597119868
120597119906119895
=
119868 (sdot sdot sdot 119906119895+ 120575 sdot sdot sdot ) minus 119868 (119906)
120575
119895 = 0 1 119873119905minus 1 (18)
where 120575 is the double precision of machine So the gradientvector is nabla119868 = [120597119868120597119906
0 120597119868120597119906
119873119905minus1]119879 Also at each major
iteration a positive definite quasi-Newton approximation oftheHessian of the Lagrangian function119867 is calculated usingthe BFGS method [28] where 120582
119894 119894 = 1 119898 is an estimate
of the Lagrange multipliers The general procedure for SQPfor NLP (14)-(15) is as follows
(1) Given an initial solution 1199060 Let 119896 = 0
(2) Construct the QP subproblem (16)-(17) based on 1199060
using the approximations of the gradient and theHessian of the the Lagrangian function
(3) Compute the new point as 119906119896+1 = 119906119896
+ 119889119896 where 119889119896
is the optimal solution of the current QP(4) Let 119896 = 119896 + 1 and go to step (2)
Here in MHGA SQP is used as the local search and weuse the maximum number of iterations as the main criterionfor stopping SQP In other words we terminate SQP when itconverges either to local solution or themaximumnumber ofSQPrsquos iterations is reached
33 MHGA In MHGA GA uses a local search method toimprove solutions Here we use SQP as a local search UsingSQP as a local search in the hybrid metaheuristic is commonfor example see [20]
MHGA can be seen as a multi start local search whereinitial solutions are constructed by GA From another per-spective MHGA can be seen as a GA that the quality of
Mathematical Problems in Engineering 5
Initialization
sqpmaxiter
an initial population
EvaluationEvaluate the fitnessof each individual
using (8)
Local search(Section 32)
ReturnThe best individual inthe final populationas an approximate
of the global solution of NOCP
Stoppingconditions
Selection(Algorithm 1)
Crossover(using Equation (10))
Replacement(using Equations
(12))
Mutation(using Equation(11))
Local search(Section 32)
Input NtNpNiNg Pm
120576Mi i = 1 2 3 and
=sqpmaxitersqpmaxiter + 1
Figure 1 Flowchart of the MHGA algorithm
its population is intensified by SQP In the beginning ofMHGA a less number of iterations for SQP was used Thenwhen the promising regions of search space were found byGA operators we increase the number of iterations of SQPgradually Using this approach we may decrease the neededrunning time (in [19] the philosophy of this approach isdiscussed)
Finally we give our modified MHGA to find the globalsolution by the flowchart in Figure 1
4 Proposed Algorithm
Here we give a new algorithm which is a direct approachbased on MHGA for solving NOCPs The proposed algo-rithm has two main phases In the first phase we performMHGA with a completely random initial population con-structed by (7) In the first phase to find the promisingregions of the search space in a less running time we usea few numbers of time nodes In addition to have a fasterMHGA the size of the population in the first phase is usuallyless than the size of the population in the second phase
After phase 1 to maintain the property of individuals inthe last population of phase 1 and to increase the accuratelyof solutions we add some additional time nodes Whenthe number of time nodes is increased it is estimated thatthe quality of solution obtained by numerical methods (eg
Runge-Kutta and Simpson) is increased Thus we increasetime nodes from 119873
1199051in phase 1 to 119873
1199052in phase 2 The
corresponding control input values of the new time nodesare added to individuals To use the information of theobtained solutions from phase 1 in the construction of theinitial population of phase 2 we use either Linear or Splineinterpolation to estimate the value of the control inputs inthe new time nodes in each individual of the last populationof phase 1 Moreover to maintain the diversity in the initialpopulation of phase 2 we add new random individuals tothe population using (7) In the second phase MHGA startswith this population and new value of parameters Finally theproposed algorithm is given in Algorithm 2
5 Numerical Experiments
In this section to investigate the efficiency of the proposedalgorithm 20 well-known and real world NOCPs as bench-mark problems are considered which are presented in termsof (1)ndash(6) in the Appendix These NOCPs are selected withsingle control signal and multi control signals which will bedemonstrated in a general manner
The numerical behaviour of algorithms can be studiedfrom two view of points the relative error of the performanceindex and the status of the final state constraints Let 119869 be theobtained performance index by an algorithm 120593
119891 defined in
6 Mathematical Problems in Engineering
Initialization Input the desired precision 120576 in (13) the penalty parameters119872119894 119894 = 1 2 3 in (8) the bounds of control input values in (7) 119906left and 119906right
Phase 1 Perform MHGA with a random population and1198731199051 1198731199011 1198731198941 1198731198921 1198751198981
and 1199041199021199011198981198861199091198941199051198901199031
Construction of the initial population of the phase 2 Increase time nodes uniformly to1198731199052
and estimate the corresponding control input values of the new time nodes in eachindividual obtained from phase 1 using either Linear or Spline interpolationCreate119873
1199012minus 1198731199011new different individuals with119873
1199052time nodes randomly
Phase 2 Perform MHGA with the constructed population and1198731199052 1198731199012 1198731198942 1198731198922 1198751198982
and 1199041199021199011198981198861199091198941199051198901199032
Algorithm 2 The proposed algorithm
(13) be the error of final state constraints and 119869lowast be the best
obtained solution among all implementations or the exactsolution (when exists) Now the relative error of 119869 119864
119869 of the
algorithm can be defined as
119864119869=
10038161003816100381610038161003816100381610038161003816
119869 minus 119869lowast
119869lowast
10038161003816100381610038161003816100381610038161003816
(19)
Tomore accurate study we now define a new criterion calledfactor to compare the algorithms as follows
119870120595= 119864119869+ 120593119891 (20)
Note that 119870120595shows the summation of two important errors
Thus based on 119870120595
we can study the behaviour of thealgorithms on the quality and feasibility of given solutionssimultaneously
To solve any NOCP described in the Appendix wemust know the algorithmrsquos parameters including MHGArsquosparameters including 119873
119905 119873119901 119873119894 119873119892 119875119898and 119904119902119901119898119886119909119894119905119890119903
in both phases in Algorithm 2 and the problemrsquos parametersincluding 120576 in (13) 119872
119894 119894 = 1 2 3 in (8) 119906left and 119906right in
(7) To estimate the best value of the algorithmrsquos parameterswe ran the proposed algorithm with different values ofparameters and then select the best However the sensitivityof MHGA parameters are studied in the next section In bothof phases of Algorithm 2 in MHGA we let 119904119902119901119898119886119909119894119905119890119903
119894= 4
and 119875119898119894
= 08 for 119894 = 1 2 Also we consider 1198731198921
= 1198731198922
1198731198941
= 1198731198942 1198731199011
= 9 and 1198731199012
= 12 The other MHGAparameters are given in the associated subsection and theproblemrsquos parameters in Table 2 For each NOCP 12 differentruns were done and the best results are reported in Table 1which the best value of each column is seen in the bold
The reported numerical results of the proposed algo-rithm for each NOCP include the value of performanceindex 119869 the relative error of 119869119864
119869 defined in (19) the required
computational time Time the norm of final state constraints120593119891 defined in (13) and the factor 119870
120595 defined in (20)
The algorithm was implemented in Matlab R2011a envi-ronment on a Notebook with Windows 7 Ultimate CPU253GHz and 400GB RAM Also to implement SQP in ourproposed algorithm we used ldquofminconrdquo in Matlab when theldquoAlgorithmrdquo was set to ldquoSQPrdquo Moreover we use compositeSimpsonrsquos method [29] to approximate integrations
Remark 1 We use the following abbreviations to show theused interpolation method in our proposed algorithm
(1) LI linear interpolation(2) SI spline interpolation
For comparing the numerical results of the proposedalgorithm two subsections are considered comparison withsome metaheuristic algorithms in Section 51 and com-parison with some numerical methods in Section 52 Wegive more details of these comparisons in the followingsubsections
51 Comparison with Metaheuristic Algorithms The numeri-cal results for the NOCPs numbers (1)ndash(3) in the Appendixare compared with a continuous GA CGA as a meta-heuristic proposed in [9] which gave better solutions thanshooting method and gradient algorithm from the indirectmethods category [2 30] and SUMT from the directmethodscategory [26] For NOCPs numbers (4) and (5) the resultsare compared with another metaheuristic which is a hybridimproved PSO called IPSO proposed in [20]
511 VDP Problem [9] The first NOCP in the Appendix isVanDer Pol Problem VDP which has two state variables andone control variable VDPproblemhas a final state constraintwhich is 120595 = 119909
1(119905119891) minus 1199092(119905119891) + 1 = 0 The results of the
proposed algorithm with the MHGArsquos parameters as 1198731199051
=
311198731199052= 71119873
119892= 300 and119873
119894= 200 are reported in Table 1
FromTable 1 it is obvious that the numerical results of LI andSI methods are more accurate than CGA with less amount of119870120595
512 CRP Problem [9] The second NOCP in the Appendixis Chemical Reactor Problem CRP which has two statevariables and one control variableThe results of the proposedalgorithm with the MHGArsquos parameters as 119873
1199051= 31 119873
1199052=
71 119873119892
= 300 and 119873119894= 200 are shown in the second
row of Table 1 CRP problem has two final state constraints120595 = [119909
1 1199092]119879 Although from Table 1 the norm of final state
constraints 120593119891 for the CGA equals 120593lowast
119891= 757 times 10
minus10 isless than 120593
119891rsquos of LI and SI methods which equals 115 times 10
minus9
and 599 times 10minus9 respectively but the performance index
Mathematical Problems in Engineering 7
Table 1 The best numerical results in 12 different runs of the SI and LI methods for NOCPs in the Appendix
Problem Algorithm 119869 119864119869
120593119891
119870120595
Time
VDPCGA 17404 00912 267 times 10
minus11
00912 50128LI 15949 0 108 times 10
minus13 108 times 10minus13 28282SI 15950 627 times 10
minus5 999 times 10minus14 627 times 10minus5 23914
CRPCGA 163 times 10
minus2 02835 757 times 10minus10 02835 50128LI 127 times 10minus2 0 115 times 10
minus9 115 times 10minus9 17194
SI 127 times 10minus2 0 599 times 10minus9
599 times 10minus9
12405
FFRPCGA 8363 13256 465 times 10
minus3
13302 1413LI 3596 0 122 times 10
minus5 122 times 10minus5 126858
SI 3599 834 times 10minus4 901 times 10minus6 843 times 10
minus4 134340
MSNICIPSO 01727 00165 mdash mdash 12626LI 01699 0 mdash mdash 13882
SI 01700 589 times 10minus4 mdash mdash 13868
CSTCRIPSO 01355 02098 mdash mdash 3134
LI 01120 0 mdash mdash 10358
SI 01120 0 mdash mdash 10673
No 6Bezier minus53898 00251 mdash mdash NRa
LI minus54309 00177 mdash mdash 13838
SI minus54309 00177 mdash mdash 10734
Number 7HPM 02353 01677 420 times 10
minus6 01677 NRLI 02015 0 235 times 10
minus9
235 times 10minus9 21392
SI 02015 0 282 times 10minus10 282 times 10minus10 20696
Number 8
SQP 636 times 10minus6
220 times 109 mdash mdash NR
SUMT 515 times 10minus6
178 times 109 mdash mdash NR
LI 289 times 10minus15 0 mdash mdash 7282SI 364 times 10
minus15
02595 mdash mdash 6868
Number 9
SQP 17950 00873 mdash mdash NRSUMT 17980 00891 mdash mdash NRLI 16509 0 mdash mdash 63910SI 16509 0 mdash mdash 5950
Number 10
SQP 02163 03964 mdash mdash NRSUMT 01703 00994 mdash mdash NRLI 01549 0 mdash mdash 62570
SI 01549 0 mdash mdash 67662
Number 11
SQP 325 01648 0 01648 NRSUMT 325 01648 0 01648 NRLI 27901 0 162 times 10
minus9
162 times 10minus9 14778
SI 27901 0 586 times 10minus10 586 times 10minus10 21946
Number 12
SQP minus02490 40 times 10minus3 0 40 times 10
minus3 NRSUMT minus02490 40 times 10
minus3 0 40 times 10minus3 NR
LI minus02500 0 285 times 10minus8
285 times 10minus8
54771
SI minus02500 0 390 times 10minus10 390 times 10minus10 54548
Number 13
SQP 168 times 10minus2 01748 0 01748 NR
SUMT 167 times 10minus2 01678 0 01678 NR
LI 143 times 10minus2 0 118 times 10minus9 118 times 10minus9 42599SI 144 times 10
minus2
70 times 10minus3
632 times 10minus9
70 times 10minus3
49027
Number 14
SQP 37220 00961 0 00961 NRSUMT 37700 01103 0 01103 NRLI 33956 0 786 times 10minus7 786 times 10minus7 104124SI 33965 265 times 10
minus4
276 times 10minus6
267 times 10minus4
115144
8 Mathematical Problems in Engineering
Table 1 Continued
Problem Algorithm 119869 119864119869
120593119891
119870120595
Time
Number 15
SQP 103 times 10minus3 43368 0 43368 NR
SUMT 929 times 10minus4 38135 0 38135 NR
LI 193 times 10minus4 0 810 times 10minus10 810 times 10minus10 33354SI 194 times 10
minus4
520 times 10minus3
164 times 10minus9
520 times 10minus3
34014
Number 16
SQP 22120 00753 0 00753 NRSUMT 22080 00734 0 00734 NRLI 20571 0 394 times 10
minus12
394 times 10minus12
46982
SI 20571 0 560 times 10minus13 560 times 10minus13 70423
Number 17
SQP minus88690 225 times 10minus5 0 225 times 10
minus5 NRSUMT minus88690 225 times 10
minus5 0 225 times 10minus5 NR
LI minus88692 0 145 times 10minus10 145 times 10minus10 32183SI minus88692 0 279 times 10
minus10
279 times 10minus10
30959
Number 18
SQP 00368 01288 mdash mdash NRSUMT 00386 01840 mdash mdash NRLI 00326 0 mdash mdash 55163
SI 00326 0 mdash mdash 60671
Number 19
SQP 03439 49293 0 49293 NRSUMT 03428 49103 0 49103 NRLI 00689 01879 410 times 10
minus3
01919 164509
SI 00580 0 405 times 10minus4 405 times 10minus4 178783
Number 20
SQP 7752 12554 0 12554 NRSUMT 7683 12353 0 12353 NRLI 343716 232 times 10
minus5
225 times 10minus4
248 times 10minus4 81519
SI 343708 0 157 times 10minus4 157 times 10minus4 80420
aNot reported
the relative error of 119869 and the factor of the proposed algorithmis better and so the proposed algorithm is more robust thanCGA
513 FFRP Problem [9] The third NOCP in the Appendixis Free Floating Robot Problem FFRP which has six statevariables and four control variables FFRP has been solvedby CGA and the proposed algorithm with the MHGArsquosparameters as 119873
1199051= 31 119873
1199052= 71 119873
119892= 300 and
119873119894
= 200 This problem has six final state constraints120595 = [119909
1minus 4 119909
2 1199093minus 4 119909
4 1199095 1199096]119879 The numerical results
are shown in Table 1 The values of 119869 119864119869 120593119891and 119870
120595for LI
and SI methods separately are less than CGA Thereforethe proposed algorithm can achieve much better qualitysolutions than the CGA with reasonable computational time
514 MSNIC Problem [20] For the forth NOCP in theAppendix which is a Mathematical System with NonlinearInequality Constraint NSNIC the numerical results are com-pared with IPSO MSNIC contains an inequality constraint119889(119909 119905) = 119909
2(119905) + 05 minus 8(119905 minus 05)
2
le 0 The problem solvedby several numerical methods as [24 31] From [20] IPSOmethod could achieved more accurate results than men-tioned numericalmethods AlsoMSNIC can be solved by theproposed algorithm with the MHGArsquos parameters as 119873
1199051=
31 1198731199052
= 91 119873119892= 100 and 119873
119894= 60 From the forth row
of Table 1 the absolute error of 119869 119864119869 for LI and SI methods
equal 0 and 589 times 10minus4 respectively which are less than
IPSOrsquos 00165Subplots (a) and (b) in Figure 2 show the graphs of
the convergence rate for the performance index and theinequality constraint respect to the number of iterationrespectively
515 CSTCR Problem [20] The fifth NOCP in the Appendixis a model of a nonlinear Continuous Stirred-tank ChemicalReactor CSTCR It has two state variables 119909
1(119905) and 119909
2(119905)
as the deviation from the steady-state temperature and con-centration and one control variable 119906(119905) which represent theeffect of the flow rate of cooling fluid on chemical reactorTheobjective is to maintain the temperature and concentrationclose to steady-state values without expending large amountof control effort Also this is a benchmark problem in thehandbook of test problems in local and global optimization[32] which is a multimodal optimal control problem [33]It involves two different local minima The values of theperformance indexes for these solutions equal 0244 and0133 Similarly to the MSNIC the numerical results of theproposed algorithm with the MHGArsquos parameters as 119873
1199051=
31 1198731199052
= 51 119873119892= 100 and 119873
119894= 50 are compared with
IPSO From Table 1 the performance index 119869 for LI and SImethods is equal to 119869
lowast
= 01120 which is less than IPSOs01355
Mathematical Problems in Engineering 9
Table 2 The problem parameters for NOCPs in the Appendix
Parameters Problem number1 2 3 4 5 6 7 8 9 10
119906left minus05 minus15 minus15 minus20 0 minus2 0 minus2 minus1 minus20
119906right 2 2 10 20 5 2 1 2 1 20119872119894
103
102 70 1 mdash 1 mdash 10
2 mdash 102
120576 10minus12
10minus10
10minus3 mdash mdash mdash mdash mdash mdash 10
minus9
Parameters Problem no11 12 13 14 15 16 17 18 19 20
119906left minus5 minus1 minus2 minus120587 minus1 minus3 minus30 minus1 minus15 minus15
119906right 5 1 2 120587 1 3 30 1 10 10119872119894
10 102 mdash 10
2
102
102 mdash 10 70 10
minus3
120576 10minus9
10minus11
10minus11 mdash 10
minus10
10minus10
10minus10 mdash 10
minus3
10minus4
0 5 10 15 20 25 30 35 40 45 50017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 2 Graphical results of MSNIC problem using SI method (a) convergence rate of the performance index and (b) the inequalityconstraint 119889(119909 119905) le 0 in respect to the number of iterations
52 Comparison with Numerical Methods For NOCPs num-bers (6)ndash(20) the comparison are done with some numericalmethods Unfortunately for these methods usually the finalstate constraints and the required computational time arenot reported which are shown with NR in Table 1 but thesevalues are reported for both LI and SI methods in Table 1For all NOCPs in this section the MHGArsquos parameters areconsidered as 119873
1199051= 31 119873
1199052= 51 119873
119892= 100 119873
119894= 50 with
the problem parameters in Table 2
521 Compared with Bezier [34] The NOCP number (6) inthe Appendix has exact solution which has an inequalityconstraint as 119889(119909
1 119905) = minus6 minus 119909
1(119905) le 0 The exact value of
performance index equals 119869lowast = minus55285 [35] This problemhas been solved by a numerical method proposed in [34]called Bezier From sixth row of Table 1 the absolute error
of the LI and SI methods equal 00177 which is less thanBezierrsquos 00251
Figure 3 shows the graphs of the convergence rate of theperformance index subplot (a) and inequality constraintsubplot (b) respect to the number of iteration using SImethod
522 Compared with HPM [36] For NOCP number (7) inthe Appendix which is a constraint nonlinear model thenumerical results of the proposed algorithm are comparedwith HPM proposed in [36] This NOCP has a final stateconstraint as 120595 = 119909 minus 05 = 0 From [36] the norm offinal state constraint for HPM equals 42times10minus6 however thiscriterion for the LI and SI methods equals 235 times 10
minus9 and120593lowast
119891= 282 times 10
minus10 respectively From Table 1 it is obviousthat the obtained values of the performance index the norm
10 Mathematical Problems in Engineering
2 4 6 8 10 12 14minus5353
minus5352
minus5351
minus535
minus5349
minus5348
minus5347
minus5346
minus5345
minus5344
minus5343
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 3 Graphical results for NOCP number (6) using SI method (a) convergence rate for the performance index and (b) the inequalityconstraint 119889(119909
1 119905) le 0 in respect to the number of iterations
of final state constraint and 119870120595for the SI method are more
accurate than LI than HPMmethods
523 Compared with SQP and SUMT [26] For the NOCPsnumbers (8)ndash(20) in the Appendix the numerical resultsof LI and SI methods are compared with two numericalmethods contain SQP and SUMT proposed in [26] Amongthese NOCPs only two problems numbers (8) and (18) areunconstrained and the others have at least one constraintfinal state constraint or inequality constraint All of theseNOCPs solved by the proposed algorithm with the problemparameters in Table 2 and their results are summarized inTable 1 Because the final state constraints in these methodsare not reported we let 120593
119891= 0 to calculate the factor119870
120595
Figures 4ndash16 show the graphical results of NOCPsnumbers (8)ndash(20) in the Appendix using SI method Forunconstrained NOCPs numbers (8) and (18) only the graphof convergent rate of performance index is shown see Figures4 and 14 For constraint NOCPs and NOCPs numbers(11)ndash(17) and numbers (19)-(20) with final state constraintthe graphs of convergent rates of performance index and theerror of final state constraint are shown see Figures 7ndash13 and15-16 For the constraint NOCPs with inequality constraintsNOCPs numbers (9) and (10) the graphs of convergent rateof performance index and inequality constraint are shownsee Figures 5 and 6
Table 1 shows that the proposed algorithm LI and SImethods was 100 percent successful in point of views theperformance index 119869 and the factor119870
120595 numerically So the
proposed algorithm provides robust solutions with respect tothe other mentioned numerical or metaheuristic methodsTo compare 119869 in LI and SI methods in 35 percent of NOCPsLI is more accurate than SI in 10 percent SI is more accurate
1 15 2 25 3 35 40
002
004
006
008
01
012
Iter
Perfo
rman
ce in
dex
Figure 4 Convergence rate for the performance index of SI methodfor NOCP number (8)
than LI and in 55 percent are same In point of view the finalconditions 65 percent of NOCPs in the Appendix have finalstate constraints In all of them 120593
119891in the proposed algorithm
is improved except CRP problem however the factor of theproposed algorithm is the best yet In 54 percent of theseNOCPs LI is better performance and in 46 percent SI isbetter Also in point of the running time Time LI is same asSI that is in 50 percent of NOCPs LI and in 50 percent SI isbetter than another So the proposed algorithmcould providevery suitable solutions in a reasonable computational timeAlso for more accurate comparison of LI and SI methods astatistical approach will done in next section
To compare with CGA the mean of relative error of 119869 119864119869
for CGA LI and SImethods inNOCPs (1)ndash(3) equal 056680 and 298 times 10
minus4 respectively Also the mean for the error
Mathematical Problems in Engineering 11
5 10 15 20 25165
17
175
18
185
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3 35 4 45 50
005
01
015
02
025
TimeIn
equa
lity
cons
trai
nt(b)
Figure 5 (a) Convergence rate for the performance index and (b) the inequality constraint of SI method for NOCP number (9)
5 10 15 20 25 30 35 40 45 500155
016
0165
017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 6 (a) Convergence rate of the performance index and (b) the inequality constraint for SI method for NOCP number (10)
of the final state constraints 120593119891 are 16 times 10
minus3 407 times 10minus6
and 301 times 10minus6 respectively and for 119870
120595 these values equal
05883 407 times 10minus6 and 302 times 10
minus4 Thus we can say that thefeasibility of the solutions given by the proposed algorithmand CGA are competitive
From Table 1 to compare with numerical methods SQPand SUMT in NOCPs (8)ndash(20) the mean of 119864
119869for LI SI
SQP and SUMT equals 00145 00209 169 times 108 and 136 times
108 respectively Also the mean for the error of final state
constraints for these NOCPs equal 334 times 10minus4 441 times 10
minus50 and 0 respectively For 119870
120595 these values are 00213 00014
12263 and 11644 Therefore the performance index 119869 andthe factor 119870
120595 for the LI and SI methods are more accurate
than SQP and SUMT So the proposed algorithm gave morebetter solution in comparison with the numerical methods
12 Mathematical Problems in Engineering
5 10 15 20 25minus025
minus0245
minus024
minus0235
minus023
minus0225
Iter
Perfo
rman
ce in
dex
(a)
1 15 2 25 3 35 4 45 5 55 60
02
04
06
08
1
12
14
16
IterFi
nal c
ondi
tion
times10minus5
(b)
Figure 7 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (11)
2 4 6 8 10 12 14minus025
minus0245
minus024
minus0235
minus023
minus0225
minus022
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
02
04
06
08
1
12
14
16
18
2
Iter
Fina
l con
ditio
n
times10minus5
(b)
Figure 8 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (12)
Therefore based on this numerical studywe can concludethat the proposed algorithm outperform well-known numer-ical method Since the algorithms were not implementedon the same PC the computational times of them are notcompetitive Therefore we did not give the computationaltimes in bold in Table 1
6 Sensitivity Analysis andComparing LI and SI
In this section two statistical analysis based on the one-way analysis of variance (ANOVA) used for investigatingthe sensitivity of MHGA parameters and Mann-Whitney
Mathematical Problems in Engineering 13
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
07
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 9 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (13)
5 10 15 20 25 30
35
4
45
5
55
6
65
7
75
8
85
9
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
Iter
Fina
l con
ditio
n
(b)
Figure 10 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (14)
applied comparing LI and SI methods are done by thestatistical software IBM SPSS version 21
61 Sensitivity Analysis In order to survey the sensitivityof the MHGA parameters from the proposed algorithmthe VDP problem is selected for example The influenceof these parameters are investigated for this NOCP on
the dependent outputs consist of the performance index119869 the relative error of 119869 119864
119869 the required computational
time Time the error of final state constraints 120593119891and the
factor 119870120595 The independent parameters are consist of the
number of time nodes in both two phases 1198731199051and 119873
1199052 the
size of population in both two phases 1198731199011
and 1198731199012 the
maximum number of generations without improvement119873119894
14 Mathematical Problems in Engineering
5 10 15 20 25 30 35 40 45 50
00955
0096
00965
0097
00975
0098
00985
0099
00995
Iter
Perfo
rman
ce in
dex
(a)
2 4 6 8 10 12 14 16 18 200
05
1
15
2
25
3
35
4
45
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 11 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (15)
5 10 15 20 25
21
22
23
24
25
26
27
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
1
2
3
4
5
6
Iter
Fina
l con
ditio
n
times10minus3
(b)
Figure 12 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (16)
and the maximum number of generations 119873119892 Because the
mutation implementation probability119875119898 has a less influence
in numerical results it is not considered Also 119904119902119901119898119886119909119894119905119890119903 ischanged in each iteration of the proposed algorithm so it isnot considered too
At first we selected at least four constant value foreach of parameters and then in each pose 35 differentruns were made independently The statistical analysis isdone based on ANOVA The descriptive statistics whichcontains the number of different runs (119873) the mean of each
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7minus887
minus886
minus885
minus884
minus883
minus882
minus881
minus88
minus879
minus878
minus877
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 70
01
02
03
04
05
06
07
08
09
1
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 13 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (17)
1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333
Iter
Perfo
rman
ce in
dex
Figure 14 Convergence rate for the performance index of SImethod for NOCP numbers (18)
output in119873 different runs (Mean) standard deviation (Sd)maximum (Max) and minimum (Min) of each outputcould be achieved Among of the MHGA parameters wepresent the descriptive statistics of the ANOVA only for the1198731199051parameter which is reported in Table 3Table 4 summarized the statistical data contain the test
statistics (119865) and 119875-values of ANOVA tests Sensitivityanalysis for each parameter separately are done based on thistable as follows
1198731199051 From Table 4 the significant level or 119875-value for119870120595
is equal to 0279 which is greater than 005So from ANOVA 119873
1199051parameter has no significant
effect on the factor 119870120595 With similar analysis this
parameter has no effect on the other outputs except
required computational time Time because its 119875-value is equal to 0 which is less than 005 that is thisoutput sensitive to the parameter119873
1199051
1198731199052 From the second row of Table 4 the outputs 119869 119864
119869119870120595
and Time are sensitive to the parameter 1198731199052 and the
only output 120593119891is independent
1198731199011 From the third row of Table 4 only the computationalrequired time Time is sensitive because the its 119875-value is equal to 0006 which is less than 005 and theother outputs contain 119869 119864
119869 120593119891119870120595 are independent
1198731199012 From the forth row of Table 4 the output 120593
119891is
independent respect to the parameter 1198731199012 but other
outputs contain 119869 119864119869 Time 119870
120601are sensitive to this
parameter119873119892 From the fifth row of Table 4 the outputs 119869 119864
119869and
119870120595are independent to the parameter 119873
119892 and other
outputs contain 120593119891and Time are sensitive respect to
this parameter119873119894 The sensitivity analysis is similar to119873
1199011
From above cases it is obvious that all parameters canbe effect on the required computational time except 119873
119892
Moreover intuitions shows the normof final state constraints120593119891is independent output with respect to all parameters that
is any of the parameters could not effect on this output and itis not sensitive with respect to any of the MHGA parameters
62 Comparison of LI and SI To compare the efficiency ofthe LI and SI methods for NOCPs in the Appendix weused the Mann-Whitney nonparametric statistical test [37]
16 Mathematical Problems in Engineering
1 15 2 25 3 35 40
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
200
400
600
800
1000
1200
1400
1600
IterFi
nal c
ondi
tion
(b)
Figure 15 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (19)
2 4 6 8 10 12 14 16 18 20
50
100
150
200
250
300
350
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 10 11 120
05
1
15
2
25
3
Iter
Fina
l con
ditio
n
(b)
Figure 16 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (20)
Using the nonparametric statistical tests for comparing theperformance of several algorithms presented in [38] Forthis purpose At first 15 different runs for each of NOCPwere made Table 5 shows the mean of numerical resultsseparately Here we apply the fixMHGAparameters as119873
1199011=
91198731199012
= 121198731199051= 31119873
1199052= 91119873
119892= 100 and119873
119894= 50 with
the previous problem parameters in Table 2 Comparison
criterions contain 119869 120593119891 119870120595and Time The results of Mann-
Whitney test are shown in Tables 6ndash9 From Table 6 since119885 = 0 gt 119885
001= minus234 then the average relative error of
the LI and SI methods are same with the significant level 001In other words with probability 99 we can say that LI andSI methods have same effectiveness form the perspective oferror for 119869 Similarly the results of Tables 7ndash9 indicate that LI
Mathematical Problems in Engineering 17
Table 3 Descriptive statistics for the sensitivity analysis of1198731199051 for VDP problem
Output 1198731199051
119873 Mean Sd Min Max
119869
11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449
Total 140 15357 00294 15285 18449
120593119891
11 35 281 times 10minus8
952 times 10minus8
1 times 10minus12
452 times 10minus7
21 35 467 times 10minus8
235 times 10minus7
1 times 10minus12
141 times 10minus6
31 35 163 times 10minus8
491 times 10minus8
1 times 10minus12
221 times 10minus7
41 35 817 times 10minus8
285 times 10minus7
1 times 10minus12
126 times 10minus6
Total 140 426 times 10minus8 191 times 10minus7 1 times 10minus12 141 times 10minus6
Time
11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514
Total 140 1142131 410891 456615 2849514
119864119869
11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070
Total 140 00047 00192 0 02070
119870120595
11 35 00031 00045 475 times 10minus9
00204
21 35 00027 00064 550 times 10minus10
00369
31 35 00029 00050 154 times 10minus13
00271
41 35 00111 00397 193 times 10minus10
02070
Total 140 00047 00192 154 times 10minus13 02070
Table 4 Summary statistical data of ANOVA test for the parameters1198731199051 1198731199052 1198731199011 1198731199012 119873119892 119873119894
Parameters 119869 119864119869
120593119891
Time 119870120595
Test statistic (119865)
1198731199051
1294 1296 0624 1079 12961198731199052
110572 393 0868 5939 3931198731199011
1945 1835 1271 4317 18351198731199012
2740 2478 1011 2302 2478119873119892
3541 3591 0491 0890 3591119873119894
0495 0309 0301 5849 1046
119875 value
1198731199051
0280 0279 0601 0 02791198731199052
0 0005 0489 0 00051198731199011
0125 0143 0286 0006 01431198731199012
0021 0034 0413 0 0034119873119892
0009 0008 0743 0472 0008119873119894
0739 0819 0877 0 0386
and SI methods from the perspective of errors for 120593119891 Time
and119870120595 have same behaviour
7 The Impact of SQP
In this section we investigation the impact of the SQP on theproposed algorithm For this purpose we remove the SQPfrom Algorithm 2 as without SQP algorithm For comparingthe numerical results with the previous results the required
running time in each NOCP is considered fixed which is themaximum of running time in 12 different runs were done inSection 5 Also the all of the parameters for each problemwasset as same as Section 5 The numerical results of the withoutSQP algorithm is summarized in Table 10 By comparing theresults of Tables 10 and 1 it is obvious that for all NOCPsin the Appendix the obtained values of the performanceindex and the norm of final state constraint for the proposedalgorithm (Algorithm 2) are more accurate than the withoutSQP algorithm
18 Mathematical Problems in Engineering
Table 5 The mean of numerical results of 15 different runs for NOCPs in the Appendix using LI and SI methods
Problem LI SI119869 120593
119891119870120595
Time 119869 120593119891
119870120595
TimeVDP 15977 157 times 10
minus9
170 times 10minus3 17720 16002 667 times 10
minus8
326 times 10minus3 15757
CRP 00126 307 times 10minus8
307 times 10minus8 19714 00119 161 times 10
minus8
161 times 10minus8 19532
FFPP 4774 805 times 10minus5 03277 133280 5351 592 times 10
minus5 04868 137989MSNIC 01702 mdash mdash 13678 01703 mdash mdash 14092CSTCR 01120 mdash mdash 17385 01120 mdash mdash 17606Number 6 minus54307 mdash mdash 10541 minus54308 mdash mdash 10372Number 7 02015 326 times 10
minus9
695 times 10minus6 20807 02020 313 times 10
minus9
662 times 10minus6 21019
Number 8 630 times 10minus15 mdash mdash 6376 124 times 10
minus14 mdash mdash 5837Number 9 16512 mdash mdash 59239 16512 mdash mdash 58125Number 10 01550 mdash mdash 63153 01550 mdash mdash 65597Number 11 29701 270 times 10
minus9
272 times 10minus7 22860 29701 337 times 10
minus9
811 times 10minus7 23276
Number 12 minus02499 201 times 10minus8
464 times 10minus5 54718 minus02499 115 times 10
minus9
827 times 10minus5 52765
Number 13 00147 115 times 10minus8
00252 57797 00151 139 times 10minus8
00472 66098Number 14 34761 672 times 10
minus6
00237 104585 34290 757 times 10minus6
950 times 10minus3 108767
Number 15 195 times 10minus4
111 times 10minus8
569 times 10minus3 363 194 times 10
minus4
704 times 10minus9
486 times 10minus3 365
Number 16 20571 166 times 10minus10
166 times 10minus10 54391 20571 115 times 10
minus10
115 times 10minus10 54431
Number 17 minus88692 803 times 10minus8
803 times 10minus8 22744 minus88692 661 times 10
minus8
661 times 10minus8 23254
Number 18 00326 mdash mdash 65957 00326 mdash mdash 70897Number 19 011588 655 times 10
minus4 06828 161813 01091 884 times 10minus4 08808 159990
Number 20 5242 00176 05429 84165 4464 453 times 10minus4 03765 86891
Table 6 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 119869
Method Mean rank Sum of ranks Test statistics ValueLI 2050 4100 Mann-Whitney 119880 200SI 2050 4100 Wilcoxon119882 410mdash mdash mdash 119885 0
Table 7 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 120593119891
Method Mean rank Sum of ranks Test statistics ValueLI 1362 177 Mann-Whitney 119880 83SI 1338 174 Wilcoxon119882 174mdash mdash mdash 119885 minus0077
Table 8 Results of Mann-Whitney test on relative errors of the pair (LI SI) for Time
Method Mean rank Sum of ranks Test statistics ValueLI 2025 4050 Mann-Whitney 119880 195SI 2075 4150 Wilcoxon119882 405mdash mdash mdash 119885 minus0135
Table 9 Results of Mann-Whitney test on relative errors of the pair (LI SI) for119870120595
Method Mean rank Sum of ranks Test statistics ValueLI 1354 1760 Mann-Whitney 119880 84SI 1346 1750 Wilcoxon119882 175mdash mdash mdash 119885 minus0026
Mathematical Problems in Engineering 19
Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix
Problem no1 2 3 4 5 6 7 8 9 10
119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048
120601119891
330 times 10minus9
135 times 10minus5 11445 mdash mdash mdash 758 times 10
minus9 mdash mdash mdashProblem no
11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10
minus4 23182 minus90882 00336 2513 20274120601119891
167 times 10minus5
468 times 10minus9 00161 433 times 10
minus4
594 times 10minus3
156 times 10minus5 164 mdash 1706 25283
8 Conclusions
Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters
Appendix
Test Problems
The following problems are presented using notation given in(1)ndash(6)
(1) (VDP) [9] 119892 = (12)(1199092
1+ 1199092
2+ 1199062
) 1199050= 0 119905119891= 5 119891 =
[1199092 minus1199092+(1minus119909
2
1)1199092+119906]119879 1199090= [1 0]
119879120595 = 1199091minus1199092+1
(2) (CRP) [9] 119892 = (12)(1199092
1+1199092
2+01119906
2
) 1199050= 0 119905119891= 078
119891 = [1199091minus2(1199091+025)+(119909
2+05) exp(25119909
1(1199091+2))minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(3) (FFRP) [9] 119892 = (12)(1199062
1+ 1199062
2+ 1199062
3+ 1199062
4) 1199050
=
0 119905119891
= 5 119891 = [1199092 (110)((119906
1+ 1199062) cos119909
5minus
(1199062+ 1199064) sin119909
5) 1199094 (110)((119906
1+ 1199063) sin119909
5+ (1199062+
1199064) cos119909
5) 1199096 (512)((119906
1+ 1199063) minus (119906
2+ 1199064))]119879 1199090=
[0 0 0 0 0 0]119879 120595 = [119909
1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879
(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909
2 minus1199092+
119906 1199092
1+ 1199092
2+ 0005119906
2
]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909
2+
05 minus 8(119905 minus 05)2
]119879 1199090= [0 minus1 0]
119879
(5) (CSTCR) [20] 119892 = 1199092
1+ 1199092
2+ 01119906
2 1199050= 0 119905119891= 078
119891 = [minus(2 + 119906)(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+
2)) 05 minus 1199092minus (1199092+ 05) exp(25119909
1(1199091+ 2))]
119879 1199090=
[009 009]119879
(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909
2 119906]119879 119889 = [minus(2 +
119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]
119879(7) [36] 119892 = 119906
2 1199050= 0 119905
119891= 1 119891 = (12)119909
2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05
(8) [26] 119892 = 1199092cos2119906 119905
0= 0 119905119891= 120587 119891 = sin(1199062) 119909
0=
1205872(9) [26] 119892 = (12)(119909
2
1+ 1199092
2+ 1199062
) 1199050= 0 119905
119891= 5 119891 =
[1199092 minus1199091+ (1 minus 119909
2
1)1199092+ 119906]119879 119889 = minus(119909
2+ 025) 119909
0=
[1 0]119879
(10) [26] 119892 = 1199092
1+ 1199092
2+ 0005119906
2
1199050= 0 119905
119891= 1 119891 =
[1199092 minus1199092+119906]119879
119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus
05) minus 05 minus 1199092)]119879
1199090= [0 minus1]
119879(11) [26] 119892 = (12)119906
2 1199050= 0 119905
119891= 2 119891 = [119909
2 119906]119879 1199090=
[1 1]119879 120595 = [119909
1 1199092]119879
(12) [26] 119892 = minus1199092 1199050= 0 119905
119891= 1 119891 = [119909
2 119906]119879 119889 =
minus(1 minus 119906)(1 + 119906) 1199090= [0 0]
119879 120595 = 1199092
(13) [26] 119892 = (12)(1199092
1+ 1199092
2+ 01119906
2
) 1199050= 0 119905
119891= 078
119891 = [minus2(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+ 2)) minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(14) [26] 119892 = (12)1199062 1199050= 0 119905
119891= 10 119891 = [cos 119906 minus
1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909
0= [366 minus186]
119879120595 = [119909
1 1199092]119879
(15) [26]119892 = (12)(1199092
1+1199092
2) 1199050= 0 119905119891= 078119891 = [minus2(119909
1+
025)+(1199092+05) exp(25119909
1(1199091+2))minus(119909
1+025)119906 05minus
1199092minus(1199092+05) exp(25119909
1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(16) [26] 120601 = 1199093 1199050= 0 119905
119891= 1 119891 = [119909
2 119906 (12)119906
2
]119879
119889 = 1199091minus 19 119909
0= [0 0 0]
119879 120595 = [1199091 1199092+ 1]119879
(17) [26] 120601 = minus1199093 1199050= 0 119905
119891= 5 119891 = [119909
2 minus2 + 119906119909
3
minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909
0= [10 minus2 10]
119879120595 = [119909
1 1199092]119879
(18) [26] 120601 = (1199091minus1)2
+1199092
2+1199092
3 119892 = (12)119906
2 1199050= 0 119905119891= 5
119891 = [1199093cos 119906 119909
3sin 119906 sin 119906]119879 119909
0= [0 0 0]
119879
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
et al [12] developed an ecologically inspired optimizationtechnique called invasive weed optimization (IWO) forsolving optimal control problems The other well-knownmetaheuristic algorithms used for solving NOCPs are geneticprogramming (GP) [13] particle swarm optimization (PSO)[14 15] ant colony optimization (ACO) [16] and DE [17 18]
To increase the quality of solutions and decrease therunning time hybrid methods were introduced which useda local search in the implementation of a population-basedmetaheuristics [19] Modares and Naghibi-Sistani [20] pro-posed a hybrid algorithm by integrating an improved PSOwith successive quadratic programming (SQP) for solvingNOCPs Recently Sun et al [21] proposed a hybrid improvedGA which used simplex method (SM) to perform a localsearch for solving NOCPs and applied it for chemicalprocesses
Based on the success of the hybrid methods for solvingNOCPs mentioned above we here use a modified hybridgenetic algorithm (MHGA) which combines GA with SQPsee [22] as a local search SQP is an iterative algorithm forsolving nonlinear programming (NLP) problems which usesgradient information It can moreover be used for solvingNOCPs see [23ndash25] For decreasing the running time in theearly generations (iterations) of MHGA a less number ofiterations for SQP was used and then when the promisingregion of search space was found we increase the number ofiterations of SQP gradually
To perform MHGA for solving an NOCP the timeinterval is uniformly divided by using a constant number oftime nodes Next in each of these time nodes the controlvariable is approximated by a scaler vector of control inputvalues Thus an infinite dimensional NOCP is changed toa finite dimensional NLP Now we encounter two conflictsituations the quality of the global solution and the neededcomputational time In other words when the number oftime nodes is increased then we expect that the quality ofthe global solution is also increased but we know that in thissituation the computational time is increased dramatically Inother situation if we consider less number of time nodesthen the computational time is decreased but we may finda poor local solution To conquer these problems MHGAperforms in two phases In the first phase (explorationphase) to decrease the computational time and to find apromising region of search space MHGA uses a less numberof time nodes After phase 1 to increase the quality ofsolutions obtained from phase 1 the number of time nodesis increased Using the population obtained in phase 1 thevalues of the new control inputs are estimated by Linear orSpline interpolations Next in the second phase (exploitationphase) MHGA uses the solutions constructed by the aboveprocedure as an initial population
The paper is organized as follows in Section 2 theformulation of the problem is introduced In Section 3 theproposed MHGA is presented In Section 4 we introduceour algorithm for solving NOCP In Section 5 we provide20 numerical benchmark examples to compare the proposedalgorithm with the other recently proposed algorithmsIn Section 6 we consider two statistical approaches for
the comparison of the LI and SI methods and investigation ofsensitivity analysis of the algorithm parameters The impactof SQP as local search in the proposed algorithm is surveyedin Section 7 We conclude in Section 8
2 Formulation of Problem
The bounded continuous-time NOCP is considered as find-ing the control input 119906(119905) isin R119898 over the planning horizon[1199050 119905119891] which minimizes the cost functional
119869 = 120601 (119909 (119905119891) 119905119891) + int
119905119891
1199050
119892 (119909 (119905) 119906 (119905) 119905) 119889119905 (1)
subject to
(119905) = 119891 (119909 (119905) 119906 (119905) 119905) (2)
119888 (119909 119906 119905) = 0 (3)
119889 (119909 119906 119905) le 0 (4)
120595 (119909 (119905119891) 119905119891) = 0 (5)
119909 (1199050) = 1199090 (6)
where 119909(119905) isin R119899 denotes the state vector for the system and1199090isin R119899 is the initial state The functions 119891 R119899 timesR119898 timesR rarr
R119899 119892 R119899 timesR119898 timesR rarr R 119888 R119899 timesR119898 timesR rarr R119899119888 119889 R119899 times
R119898timesR rarr R119899119889 120595 R119899timesR rarr R119899120595 and 120601 R119899timesR rarr R areassumed to be sufficiently smooth on appropriate open setsCost function (1) must be minimized subject to dynamic (2)control and state equality constraints (3) and control and stateinequality constraints (4) the final state constraints (5) andthe initial condition (6) A special case of the NOCPs is thelinear quadratic regulator (LQR) problemwhere the dynamicequations are linear and the objective function is a quadraticfunction of 119909 and 119906 The minimum time problems trackingproblem terminal control problem andminimum energy areanother special case of NOCPs
3 Modified Hybrid Genetic Algorithm
In this section first MHGA as a subprocedure for the mainalgorithm is introduced To perform MHGA the controlvariables are discretized Next NOCP is changed into a finitedimensional NLP see [21 26] Now we can imply a GA tofind the global solution of the corresponding NLP In thefollowing we introduce GA operators
31 Underlying GA GAs introduced by Holland in 1975 areheuristics and probabilistic methods [27] These algorithmsstart with an initial population of solutions This populationis evaluated by using genetic operators that include selectioncrossover andmutation Here inMHGA the underlying GAhas the following steps
InitializationThe time interval is divided into119873119905minus1 subinter-
vals using time nodes 1199050 119905
119873119905minus1and then the control input
Mathematical Problems in Engineering 3
Initialization Input the the size of tournament119873tour and select119873tour
individuals from population randomly Let 119896119894 119894 = 1 2 119873tour be the indexes of them
repeatif 119873tour = 2 thenif 119868(1198801198961 ) lt 119868(119880
1198962) then
Let 119896 = 1198961
elseLet 119896 = 119896
2
end ifend if
Perform Tournament algorithm with size119873tour2 with output 1198961
Perform Tournament algorithm with size119873tour2 with output 1198962
until 1198961
= 1198962
Return the 119896th individual of population
Algorithm 1 Tournament algorithm
values are computed (or selected randomly)This can be doneby the following stages
(1) Let 119905119895= 1199050+ 119895ℎ where ℎ = (119905
119891minus 1199050)(119873119905minus 1) 119895 =
0 1 119873119905minus1 be time nodes where 119905
0and 119905119891are the
initial and final times respectively
(2) The corresponding control input value at each timenode 119905
119895is an119898times1 vector 119906
119895 which can be calculated
randomly with the following components
119906119894119895= 119906119895(119894) = 119906left (119894) + (119906right (119894) minus 119906left (119894)) sdot 119903119894119895
119894 = 1 2 119898 119895 = 0 1 119873119905minus 1
(7)
where 119903119894119895is a random number in [0 1]with a uniform
distribution and 119906left 119906right isin R119898 are the lower and theupper bound vectors of control input values whichcan be given by the problemrsquos definition or the user(eg see the NOCPs numbers (6) and (5) in theAppendix resp) So each individual of the popula-tion is an 119898 times 119873
119905matrix as 119880 = (119906
119894119895)119898times119873119905
= [119906119895]119873119905minus1
119895=0
Next we let 119880(119896) = (119906119894119895)119898times119873119905
119896 = 1 2 119873119901as 119896th
individual of the population which 119873119901is the size of
the population
Evaluation For each control input matrix 119880(119896) 119896 =
1 2 119873119901 the corresponding state variable is an 119899 times 119873
119905
matrix119883(119896) and it can be computed by the forthRunge-Kuttamethod on dynamic system (2) with the initial condition(6) approximately Then the performance index 119869(119880(119896))is approximated by a numerical method (denoted by 119869) IfNOCP includes equality or inequality constraints (3) or (4)then we add some penalty terms to the corresponding fitness
value of the solution Finally we assign 119868(119880(119896)
) to 119880(119896) as the
fitness value as follows
119868 (119880(119896)
) = 119869 +
119899119889
sum
119897=1
119873119905minus1
sum
119895=0
1198721119897max 0 119889
119897(119909119895 119906119895 119905119895)
+
119899119888
sum
ℎ=1
119873119905minus1
sum
119895=0
1198722ℎ1198882
ℎ(119909119895 119906119895 119905119895)
+
119899120595
sum
119894=119901
11987231199011205952
119901(119909119873119905minus1
119905119873119905minus1
)
(8)
where 1198721= [119872
11 119872
1119899119889]119879 1198722= [119872
21 119872
2119899119888]119879 and
1198723
= [11987231 119872
3119899120595]119879 are big numbers as the penalty
parameters 119888ℎ(sdot sdot) ℎ = 1 2 119899
119888 119889119897(sdot sdot) 119897 = 1 2 119899
119889
and 120595119901(sdot sdot) 119901 = 1 2 119899
120595are defined in (3) (4) and (5)
respectively
Selection To select two parents we use a tournament selec-tion [27] It can be applied for parallel GA The tournamentoperator applies competition among the same individualsand the best of them is selected for next generation At firstwe select a specified number of individuals from populationrandomly This number is tournament selection parameterwhich is denoted by119873tourThe tournament algorithm is givenin Algorithm 1
Crossover When two parents 119880(1) and 119880(2) are selected we
use the following stages to construct an offspring(1) Select the following numbers
1205821isin [0 1] 120582
2isin [minus120582max 0] 120582
3isin [1 1 + 120582max]
(9)randomly where 120582max is a random number in [0 1]
(2) Let
of119896 = 120582119896119880(1)
+ (1 minus 120582119896) 119880(2)
119896 = 1 2 3 (10)
4 Mathematical Problems in Engineering
where 120582119896 119896 = 1 2 3 is defined in (9) For 119894 = 1 119898
and 119895 = 1 119873119905 if (of119896)
119894119895gt 119906right(119894) then let (of
119896
)119894119895=
119906right(119894) and if (of119896
)119894119895lt 119906left(119894) then let (of
119896
)119894119895= 119906left(119894)
(3) Let of = oflowast where oflowast is the best of119896 119896 = 1 2 3
constructed by (10)
MutationWe apply a perturbation on each component of theoffspring as follows
(of)119894119895= (of)
119894119895+ 119903119894119895sdot 120572 119894 = 1 2 119898 119895 = 1 2 119873
119905
(11)
where 119903119894119895is selected randomly in minus1 1 and 120572 is a random
number with a uniform distribution in [0 1] If (of)119894119895
gt
119906right(119894) then let (of)119894119895= 119906right(119894) and if (of)
119894119895lt 119906left(119894) then
let (of)119894119895= 119906left(119894)
ReplacementHere in the underling GA we use a traditionalreplacement strategy The replacement is done if the newoffspring has two properties first it is better than the worstperson in the population 119868(of) lt max
1le119894le119873119901119868(119880(119894)
) secondit is not very similar to a person in the population that is foreach 119894 = 1 119873
119901 at least one of the following conditions is
satisfied10038161003816100381610038161003816119868 (of) minus 119868 (119880
(119894)
)
10038161003816100381610038161003816gt 120576
10038171003817100381710038171003817of minus 119880
(119894)10038171003817100381710038171003817gt 120576
(12)
where 120576 is the machine epsilon
Termination Conditions Underlying GA is terminated whenat least one of the following conditions is occurred
(1) Themaximumnumber of generations119873119892 is reached
(2) Over a specified number of generations 119873119894 we do
not have any improvement (the best individual is notchanged) or the two-norm or error of final stateconstraints will reach a small number as the desiredprecision 120576 that is
120593119891=100381710038171003817100381712059510038171003817100381710038172
lt 120576 (13)
where120595 = [1205951 1205952 120595
119899120595]119879 is the vector of final state
constraints defined in (5)
32 SQP The fitness value in (8) can be viewed as anonlinear objective function with the decision variable as119906 = [119906
0 1199061 119906
119873119905minus1] This cost function with upper and
lower bounds of input signals construct a finite dimensionalNLP problem as following
min 119868 (119906) = 119868 (1199060 1199061 119906
119873119905minus1) (14)
st 119906left le 119906119895le 119906right 119895 = 0 1 119873
119905minus 1 (15)
SQP algorithm [22 28] is performed on the NLP (14)-(15)using 1199060 = of and constructed in (11) as the initial solutionwhen the maximum number of iteration is 119904119902119901119898119886119909119894119905119890119903
SQP is an effective and iterative algorithm for thenumerical solution of the constrained NLP problem Thistechnique is based on finding a solution to the system ofnonlinear equations that arise from the first-order necessaryconditions for an extremum of the NLP problem Usingan initial solution of NLP 119906119896 119896 = 0 1 a sequence ofsolutions as 119906119896+1 = 119906
119896
+ 119889119896 is constructed which 119889
119896 is theoptimal solution of the constructed quadratic programming(QP) that approximates NLP in the iteration 119896 based on119906119896 as the search direction in the line search procedure
For the NLP (15) the principal idea is the formulation of aQP subproblem based on a quadratic approximation of theLagrangian function as 119871(119906 120582) = 119868(119906) + 120582
119879
ℎ(119906) where thevector 120582 is Lagrangian multiplier and ℎ(119906) return the vectorof inequality constraints evaluated at 119906 The QP is obtainedby linearizing the nonlinear functions as follows
min 1
2
119889119879
119867(119906119896
) 119889 + nabla119868 (119906119896
)
119879
119889 (16)
nablaℎ (119906119896
)
119879
119889 + ℎ (119906119896
) le 0 (17)
Similar to [26] here a finite difference approximation isapplied to compute the gradient of the cost function and theconstraints with the following components
120597119868
120597119906119895
=
119868 (sdot sdot sdot 119906119895+ 120575 sdot sdot sdot ) minus 119868 (119906)
120575
119895 = 0 1 119873119905minus 1 (18)
where 120575 is the double precision of machine So the gradientvector is nabla119868 = [120597119868120597119906
0 120597119868120597119906
119873119905minus1]119879 Also at each major
iteration a positive definite quasi-Newton approximation oftheHessian of the Lagrangian function119867 is calculated usingthe BFGS method [28] where 120582
119894 119894 = 1 119898 is an estimate
of the Lagrange multipliers The general procedure for SQPfor NLP (14)-(15) is as follows
(1) Given an initial solution 1199060 Let 119896 = 0
(2) Construct the QP subproblem (16)-(17) based on 1199060
using the approximations of the gradient and theHessian of the the Lagrangian function
(3) Compute the new point as 119906119896+1 = 119906119896
+ 119889119896 where 119889119896
is the optimal solution of the current QP(4) Let 119896 = 119896 + 1 and go to step (2)
Here in MHGA SQP is used as the local search and weuse the maximum number of iterations as the main criterionfor stopping SQP In other words we terminate SQP when itconverges either to local solution or themaximumnumber ofSQPrsquos iterations is reached
33 MHGA In MHGA GA uses a local search method toimprove solutions Here we use SQP as a local search UsingSQP as a local search in the hybrid metaheuristic is commonfor example see [20]
MHGA can be seen as a multi start local search whereinitial solutions are constructed by GA From another per-spective MHGA can be seen as a GA that the quality of
Mathematical Problems in Engineering 5
Initialization
sqpmaxiter
an initial population
EvaluationEvaluate the fitnessof each individual
using (8)
Local search(Section 32)
ReturnThe best individual inthe final populationas an approximate
of the global solution of NOCP
Stoppingconditions
Selection(Algorithm 1)
Crossover(using Equation (10))
Replacement(using Equations
(12))
Mutation(using Equation(11))
Local search(Section 32)
Input NtNpNiNg Pm
120576Mi i = 1 2 3 and
=sqpmaxitersqpmaxiter + 1
Figure 1 Flowchart of the MHGA algorithm
its population is intensified by SQP In the beginning ofMHGA a less number of iterations for SQP was used Thenwhen the promising regions of search space were found byGA operators we increase the number of iterations of SQPgradually Using this approach we may decrease the neededrunning time (in [19] the philosophy of this approach isdiscussed)
Finally we give our modified MHGA to find the globalsolution by the flowchart in Figure 1
4 Proposed Algorithm
Here we give a new algorithm which is a direct approachbased on MHGA for solving NOCPs The proposed algo-rithm has two main phases In the first phase we performMHGA with a completely random initial population con-structed by (7) In the first phase to find the promisingregions of the search space in a less running time we usea few numbers of time nodes In addition to have a fasterMHGA the size of the population in the first phase is usuallyless than the size of the population in the second phase
After phase 1 to maintain the property of individuals inthe last population of phase 1 and to increase the accuratelyof solutions we add some additional time nodes Whenthe number of time nodes is increased it is estimated thatthe quality of solution obtained by numerical methods (eg
Runge-Kutta and Simpson) is increased Thus we increasetime nodes from 119873
1199051in phase 1 to 119873
1199052in phase 2 The
corresponding control input values of the new time nodesare added to individuals To use the information of theobtained solutions from phase 1 in the construction of theinitial population of phase 2 we use either Linear or Splineinterpolation to estimate the value of the control inputs inthe new time nodes in each individual of the last populationof phase 1 Moreover to maintain the diversity in the initialpopulation of phase 2 we add new random individuals tothe population using (7) In the second phase MHGA startswith this population and new value of parameters Finally theproposed algorithm is given in Algorithm 2
5 Numerical Experiments
In this section to investigate the efficiency of the proposedalgorithm 20 well-known and real world NOCPs as bench-mark problems are considered which are presented in termsof (1)ndash(6) in the Appendix These NOCPs are selected withsingle control signal and multi control signals which will bedemonstrated in a general manner
The numerical behaviour of algorithms can be studiedfrom two view of points the relative error of the performanceindex and the status of the final state constraints Let 119869 be theobtained performance index by an algorithm 120593
119891 defined in
6 Mathematical Problems in Engineering
Initialization Input the desired precision 120576 in (13) the penalty parameters119872119894 119894 = 1 2 3 in (8) the bounds of control input values in (7) 119906left and 119906right
Phase 1 Perform MHGA with a random population and1198731199051 1198731199011 1198731198941 1198731198921 1198751198981
and 1199041199021199011198981198861199091198941199051198901199031
Construction of the initial population of the phase 2 Increase time nodes uniformly to1198731199052
and estimate the corresponding control input values of the new time nodes in eachindividual obtained from phase 1 using either Linear or Spline interpolationCreate119873
1199012minus 1198731199011new different individuals with119873
1199052time nodes randomly
Phase 2 Perform MHGA with the constructed population and1198731199052 1198731199012 1198731198942 1198731198922 1198751198982
and 1199041199021199011198981198861199091198941199051198901199032
Algorithm 2 The proposed algorithm
(13) be the error of final state constraints and 119869lowast be the best
obtained solution among all implementations or the exactsolution (when exists) Now the relative error of 119869 119864
119869 of the
algorithm can be defined as
119864119869=
10038161003816100381610038161003816100381610038161003816
119869 minus 119869lowast
119869lowast
10038161003816100381610038161003816100381610038161003816
(19)
Tomore accurate study we now define a new criterion calledfactor to compare the algorithms as follows
119870120595= 119864119869+ 120593119891 (20)
Note that 119870120595shows the summation of two important errors
Thus based on 119870120595
we can study the behaviour of thealgorithms on the quality and feasibility of given solutionssimultaneously
To solve any NOCP described in the Appendix wemust know the algorithmrsquos parameters including MHGArsquosparameters including 119873
119905 119873119901 119873119894 119873119892 119875119898and 119904119902119901119898119886119909119894119905119890119903
in both phases in Algorithm 2 and the problemrsquos parametersincluding 120576 in (13) 119872
119894 119894 = 1 2 3 in (8) 119906left and 119906right in
(7) To estimate the best value of the algorithmrsquos parameterswe ran the proposed algorithm with different values ofparameters and then select the best However the sensitivityof MHGA parameters are studied in the next section In bothof phases of Algorithm 2 in MHGA we let 119904119902119901119898119886119909119894119905119890119903
119894= 4
and 119875119898119894
= 08 for 119894 = 1 2 Also we consider 1198731198921
= 1198731198922
1198731198941
= 1198731198942 1198731199011
= 9 and 1198731199012
= 12 The other MHGAparameters are given in the associated subsection and theproblemrsquos parameters in Table 2 For each NOCP 12 differentruns were done and the best results are reported in Table 1which the best value of each column is seen in the bold
The reported numerical results of the proposed algo-rithm for each NOCP include the value of performanceindex 119869 the relative error of 119869119864
119869 defined in (19) the required
computational time Time the norm of final state constraints120593119891 defined in (13) and the factor 119870
120595 defined in (20)
The algorithm was implemented in Matlab R2011a envi-ronment on a Notebook with Windows 7 Ultimate CPU253GHz and 400GB RAM Also to implement SQP in ourproposed algorithm we used ldquofminconrdquo in Matlab when theldquoAlgorithmrdquo was set to ldquoSQPrdquo Moreover we use compositeSimpsonrsquos method [29] to approximate integrations
Remark 1 We use the following abbreviations to show theused interpolation method in our proposed algorithm
(1) LI linear interpolation(2) SI spline interpolation
For comparing the numerical results of the proposedalgorithm two subsections are considered comparison withsome metaheuristic algorithms in Section 51 and com-parison with some numerical methods in Section 52 Wegive more details of these comparisons in the followingsubsections
51 Comparison with Metaheuristic Algorithms The numeri-cal results for the NOCPs numbers (1)ndash(3) in the Appendixare compared with a continuous GA CGA as a meta-heuristic proposed in [9] which gave better solutions thanshooting method and gradient algorithm from the indirectmethods category [2 30] and SUMT from the directmethodscategory [26] For NOCPs numbers (4) and (5) the resultsare compared with another metaheuristic which is a hybridimproved PSO called IPSO proposed in [20]
511 VDP Problem [9] The first NOCP in the Appendix isVanDer Pol Problem VDP which has two state variables andone control variable VDPproblemhas a final state constraintwhich is 120595 = 119909
1(119905119891) minus 1199092(119905119891) + 1 = 0 The results of the
proposed algorithm with the MHGArsquos parameters as 1198731199051
=
311198731199052= 71119873
119892= 300 and119873
119894= 200 are reported in Table 1
FromTable 1 it is obvious that the numerical results of LI andSI methods are more accurate than CGA with less amount of119870120595
512 CRP Problem [9] The second NOCP in the Appendixis Chemical Reactor Problem CRP which has two statevariables and one control variableThe results of the proposedalgorithm with the MHGArsquos parameters as 119873
1199051= 31 119873
1199052=
71 119873119892
= 300 and 119873119894= 200 are shown in the second
row of Table 1 CRP problem has two final state constraints120595 = [119909
1 1199092]119879 Although from Table 1 the norm of final state
constraints 120593119891 for the CGA equals 120593lowast
119891= 757 times 10
minus10 isless than 120593
119891rsquos of LI and SI methods which equals 115 times 10
minus9
and 599 times 10minus9 respectively but the performance index
Mathematical Problems in Engineering 7
Table 1 The best numerical results in 12 different runs of the SI and LI methods for NOCPs in the Appendix
Problem Algorithm 119869 119864119869
120593119891
119870120595
Time
VDPCGA 17404 00912 267 times 10
minus11
00912 50128LI 15949 0 108 times 10
minus13 108 times 10minus13 28282SI 15950 627 times 10
minus5 999 times 10minus14 627 times 10minus5 23914
CRPCGA 163 times 10
minus2 02835 757 times 10minus10 02835 50128LI 127 times 10minus2 0 115 times 10
minus9 115 times 10minus9 17194
SI 127 times 10minus2 0 599 times 10minus9
599 times 10minus9
12405
FFRPCGA 8363 13256 465 times 10
minus3
13302 1413LI 3596 0 122 times 10
minus5 122 times 10minus5 126858
SI 3599 834 times 10minus4 901 times 10minus6 843 times 10
minus4 134340
MSNICIPSO 01727 00165 mdash mdash 12626LI 01699 0 mdash mdash 13882
SI 01700 589 times 10minus4 mdash mdash 13868
CSTCRIPSO 01355 02098 mdash mdash 3134
LI 01120 0 mdash mdash 10358
SI 01120 0 mdash mdash 10673
No 6Bezier minus53898 00251 mdash mdash NRa
LI minus54309 00177 mdash mdash 13838
SI minus54309 00177 mdash mdash 10734
Number 7HPM 02353 01677 420 times 10
minus6 01677 NRLI 02015 0 235 times 10
minus9
235 times 10minus9 21392
SI 02015 0 282 times 10minus10 282 times 10minus10 20696
Number 8
SQP 636 times 10minus6
220 times 109 mdash mdash NR
SUMT 515 times 10minus6
178 times 109 mdash mdash NR
LI 289 times 10minus15 0 mdash mdash 7282SI 364 times 10
minus15
02595 mdash mdash 6868
Number 9
SQP 17950 00873 mdash mdash NRSUMT 17980 00891 mdash mdash NRLI 16509 0 mdash mdash 63910SI 16509 0 mdash mdash 5950
Number 10
SQP 02163 03964 mdash mdash NRSUMT 01703 00994 mdash mdash NRLI 01549 0 mdash mdash 62570
SI 01549 0 mdash mdash 67662
Number 11
SQP 325 01648 0 01648 NRSUMT 325 01648 0 01648 NRLI 27901 0 162 times 10
minus9
162 times 10minus9 14778
SI 27901 0 586 times 10minus10 586 times 10minus10 21946
Number 12
SQP minus02490 40 times 10minus3 0 40 times 10
minus3 NRSUMT minus02490 40 times 10
minus3 0 40 times 10minus3 NR
LI minus02500 0 285 times 10minus8
285 times 10minus8
54771
SI minus02500 0 390 times 10minus10 390 times 10minus10 54548
Number 13
SQP 168 times 10minus2 01748 0 01748 NR
SUMT 167 times 10minus2 01678 0 01678 NR
LI 143 times 10minus2 0 118 times 10minus9 118 times 10minus9 42599SI 144 times 10
minus2
70 times 10minus3
632 times 10minus9
70 times 10minus3
49027
Number 14
SQP 37220 00961 0 00961 NRSUMT 37700 01103 0 01103 NRLI 33956 0 786 times 10minus7 786 times 10minus7 104124SI 33965 265 times 10
minus4
276 times 10minus6
267 times 10minus4
115144
8 Mathematical Problems in Engineering
Table 1 Continued
Problem Algorithm 119869 119864119869
120593119891
119870120595
Time
Number 15
SQP 103 times 10minus3 43368 0 43368 NR
SUMT 929 times 10minus4 38135 0 38135 NR
LI 193 times 10minus4 0 810 times 10minus10 810 times 10minus10 33354SI 194 times 10
minus4
520 times 10minus3
164 times 10minus9
520 times 10minus3
34014
Number 16
SQP 22120 00753 0 00753 NRSUMT 22080 00734 0 00734 NRLI 20571 0 394 times 10
minus12
394 times 10minus12
46982
SI 20571 0 560 times 10minus13 560 times 10minus13 70423
Number 17
SQP minus88690 225 times 10minus5 0 225 times 10
minus5 NRSUMT minus88690 225 times 10
minus5 0 225 times 10minus5 NR
LI minus88692 0 145 times 10minus10 145 times 10minus10 32183SI minus88692 0 279 times 10
minus10
279 times 10minus10
30959
Number 18
SQP 00368 01288 mdash mdash NRSUMT 00386 01840 mdash mdash NRLI 00326 0 mdash mdash 55163
SI 00326 0 mdash mdash 60671
Number 19
SQP 03439 49293 0 49293 NRSUMT 03428 49103 0 49103 NRLI 00689 01879 410 times 10
minus3
01919 164509
SI 00580 0 405 times 10minus4 405 times 10minus4 178783
Number 20
SQP 7752 12554 0 12554 NRSUMT 7683 12353 0 12353 NRLI 343716 232 times 10
minus5
225 times 10minus4
248 times 10minus4 81519
SI 343708 0 157 times 10minus4 157 times 10minus4 80420
aNot reported
the relative error of 119869 and the factor of the proposed algorithmis better and so the proposed algorithm is more robust thanCGA
513 FFRP Problem [9] The third NOCP in the Appendixis Free Floating Robot Problem FFRP which has six statevariables and four control variables FFRP has been solvedby CGA and the proposed algorithm with the MHGArsquosparameters as 119873
1199051= 31 119873
1199052= 71 119873
119892= 300 and
119873119894
= 200 This problem has six final state constraints120595 = [119909
1minus 4 119909
2 1199093minus 4 119909
4 1199095 1199096]119879 The numerical results
are shown in Table 1 The values of 119869 119864119869 120593119891and 119870
120595for LI
and SI methods separately are less than CGA Thereforethe proposed algorithm can achieve much better qualitysolutions than the CGA with reasonable computational time
514 MSNIC Problem [20] For the forth NOCP in theAppendix which is a Mathematical System with NonlinearInequality Constraint NSNIC the numerical results are com-pared with IPSO MSNIC contains an inequality constraint119889(119909 119905) = 119909
2(119905) + 05 minus 8(119905 minus 05)
2
le 0 The problem solvedby several numerical methods as [24 31] From [20] IPSOmethod could achieved more accurate results than men-tioned numericalmethods AlsoMSNIC can be solved by theproposed algorithm with the MHGArsquos parameters as 119873
1199051=
31 1198731199052
= 91 119873119892= 100 and 119873
119894= 60 From the forth row
of Table 1 the absolute error of 119869 119864119869 for LI and SI methods
equal 0 and 589 times 10minus4 respectively which are less than
IPSOrsquos 00165Subplots (a) and (b) in Figure 2 show the graphs of
the convergence rate for the performance index and theinequality constraint respect to the number of iterationrespectively
515 CSTCR Problem [20] The fifth NOCP in the Appendixis a model of a nonlinear Continuous Stirred-tank ChemicalReactor CSTCR It has two state variables 119909
1(119905) and 119909
2(119905)
as the deviation from the steady-state temperature and con-centration and one control variable 119906(119905) which represent theeffect of the flow rate of cooling fluid on chemical reactorTheobjective is to maintain the temperature and concentrationclose to steady-state values without expending large amountof control effort Also this is a benchmark problem in thehandbook of test problems in local and global optimization[32] which is a multimodal optimal control problem [33]It involves two different local minima The values of theperformance indexes for these solutions equal 0244 and0133 Similarly to the MSNIC the numerical results of theproposed algorithm with the MHGArsquos parameters as 119873
1199051=
31 1198731199052
= 51 119873119892= 100 and 119873
119894= 50 are compared with
IPSO From Table 1 the performance index 119869 for LI and SImethods is equal to 119869
lowast
= 01120 which is less than IPSOs01355
Mathematical Problems in Engineering 9
Table 2 The problem parameters for NOCPs in the Appendix
Parameters Problem number1 2 3 4 5 6 7 8 9 10
119906left minus05 minus15 minus15 minus20 0 minus2 0 minus2 minus1 minus20
119906right 2 2 10 20 5 2 1 2 1 20119872119894
103
102 70 1 mdash 1 mdash 10
2 mdash 102
120576 10minus12
10minus10
10minus3 mdash mdash mdash mdash mdash mdash 10
minus9
Parameters Problem no11 12 13 14 15 16 17 18 19 20
119906left minus5 minus1 minus2 minus120587 minus1 minus3 minus30 minus1 minus15 minus15
119906right 5 1 2 120587 1 3 30 1 10 10119872119894
10 102 mdash 10
2
102
102 mdash 10 70 10
minus3
120576 10minus9
10minus11
10minus11 mdash 10
minus10
10minus10
10minus10 mdash 10
minus3
10minus4
0 5 10 15 20 25 30 35 40 45 50017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 2 Graphical results of MSNIC problem using SI method (a) convergence rate of the performance index and (b) the inequalityconstraint 119889(119909 119905) le 0 in respect to the number of iterations
52 Comparison with Numerical Methods For NOCPs num-bers (6)ndash(20) the comparison are done with some numericalmethods Unfortunately for these methods usually the finalstate constraints and the required computational time arenot reported which are shown with NR in Table 1 but thesevalues are reported for both LI and SI methods in Table 1For all NOCPs in this section the MHGArsquos parameters areconsidered as 119873
1199051= 31 119873
1199052= 51 119873
119892= 100 119873
119894= 50 with
the problem parameters in Table 2
521 Compared with Bezier [34] The NOCP number (6) inthe Appendix has exact solution which has an inequalityconstraint as 119889(119909
1 119905) = minus6 minus 119909
1(119905) le 0 The exact value of
performance index equals 119869lowast = minus55285 [35] This problemhas been solved by a numerical method proposed in [34]called Bezier From sixth row of Table 1 the absolute error
of the LI and SI methods equal 00177 which is less thanBezierrsquos 00251
Figure 3 shows the graphs of the convergence rate of theperformance index subplot (a) and inequality constraintsubplot (b) respect to the number of iteration using SImethod
522 Compared with HPM [36] For NOCP number (7) inthe Appendix which is a constraint nonlinear model thenumerical results of the proposed algorithm are comparedwith HPM proposed in [36] This NOCP has a final stateconstraint as 120595 = 119909 minus 05 = 0 From [36] the norm offinal state constraint for HPM equals 42times10minus6 however thiscriterion for the LI and SI methods equals 235 times 10
minus9 and120593lowast
119891= 282 times 10
minus10 respectively From Table 1 it is obviousthat the obtained values of the performance index the norm
10 Mathematical Problems in Engineering
2 4 6 8 10 12 14minus5353
minus5352
minus5351
minus535
minus5349
minus5348
minus5347
minus5346
minus5345
minus5344
minus5343
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 3 Graphical results for NOCP number (6) using SI method (a) convergence rate for the performance index and (b) the inequalityconstraint 119889(119909
1 119905) le 0 in respect to the number of iterations
of final state constraint and 119870120595for the SI method are more
accurate than LI than HPMmethods
523 Compared with SQP and SUMT [26] For the NOCPsnumbers (8)ndash(20) in the Appendix the numerical resultsof LI and SI methods are compared with two numericalmethods contain SQP and SUMT proposed in [26] Amongthese NOCPs only two problems numbers (8) and (18) areunconstrained and the others have at least one constraintfinal state constraint or inequality constraint All of theseNOCPs solved by the proposed algorithm with the problemparameters in Table 2 and their results are summarized inTable 1 Because the final state constraints in these methodsare not reported we let 120593
119891= 0 to calculate the factor119870
120595
Figures 4ndash16 show the graphical results of NOCPsnumbers (8)ndash(20) in the Appendix using SI method Forunconstrained NOCPs numbers (8) and (18) only the graphof convergent rate of performance index is shown see Figures4 and 14 For constraint NOCPs and NOCPs numbers(11)ndash(17) and numbers (19)-(20) with final state constraintthe graphs of convergent rates of performance index and theerror of final state constraint are shown see Figures 7ndash13 and15-16 For the constraint NOCPs with inequality constraintsNOCPs numbers (9) and (10) the graphs of convergent rateof performance index and inequality constraint are shownsee Figures 5 and 6
Table 1 shows that the proposed algorithm LI and SImethods was 100 percent successful in point of views theperformance index 119869 and the factor119870
120595 numerically So the
proposed algorithm provides robust solutions with respect tothe other mentioned numerical or metaheuristic methodsTo compare 119869 in LI and SI methods in 35 percent of NOCPsLI is more accurate than SI in 10 percent SI is more accurate
1 15 2 25 3 35 40
002
004
006
008
01
012
Iter
Perfo
rman
ce in
dex
Figure 4 Convergence rate for the performance index of SI methodfor NOCP number (8)
than LI and in 55 percent are same In point of view the finalconditions 65 percent of NOCPs in the Appendix have finalstate constraints In all of them 120593
119891in the proposed algorithm
is improved except CRP problem however the factor of theproposed algorithm is the best yet In 54 percent of theseNOCPs LI is better performance and in 46 percent SI isbetter Also in point of the running time Time LI is same asSI that is in 50 percent of NOCPs LI and in 50 percent SI isbetter than another So the proposed algorithmcould providevery suitable solutions in a reasonable computational timeAlso for more accurate comparison of LI and SI methods astatistical approach will done in next section
To compare with CGA the mean of relative error of 119869 119864119869
for CGA LI and SImethods inNOCPs (1)ndash(3) equal 056680 and 298 times 10
minus4 respectively Also the mean for the error
Mathematical Problems in Engineering 11
5 10 15 20 25165
17
175
18
185
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3 35 4 45 50
005
01
015
02
025
TimeIn
equa
lity
cons
trai
nt(b)
Figure 5 (a) Convergence rate for the performance index and (b) the inequality constraint of SI method for NOCP number (9)
5 10 15 20 25 30 35 40 45 500155
016
0165
017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 6 (a) Convergence rate of the performance index and (b) the inequality constraint for SI method for NOCP number (10)
of the final state constraints 120593119891 are 16 times 10
minus3 407 times 10minus6
and 301 times 10minus6 respectively and for 119870
120595 these values equal
05883 407 times 10minus6 and 302 times 10
minus4 Thus we can say that thefeasibility of the solutions given by the proposed algorithmand CGA are competitive
From Table 1 to compare with numerical methods SQPand SUMT in NOCPs (8)ndash(20) the mean of 119864
119869for LI SI
SQP and SUMT equals 00145 00209 169 times 108 and 136 times
108 respectively Also the mean for the error of final state
constraints for these NOCPs equal 334 times 10minus4 441 times 10
minus50 and 0 respectively For 119870
120595 these values are 00213 00014
12263 and 11644 Therefore the performance index 119869 andthe factor 119870
120595 for the LI and SI methods are more accurate
than SQP and SUMT So the proposed algorithm gave morebetter solution in comparison with the numerical methods
12 Mathematical Problems in Engineering
5 10 15 20 25minus025
minus0245
minus024
minus0235
minus023
minus0225
Iter
Perfo
rman
ce in
dex
(a)
1 15 2 25 3 35 4 45 5 55 60
02
04
06
08
1
12
14
16
IterFi
nal c
ondi
tion
times10minus5
(b)
Figure 7 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (11)
2 4 6 8 10 12 14minus025
minus0245
minus024
minus0235
minus023
minus0225
minus022
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
02
04
06
08
1
12
14
16
18
2
Iter
Fina
l con
ditio
n
times10minus5
(b)
Figure 8 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (12)
Therefore based on this numerical studywe can concludethat the proposed algorithm outperform well-known numer-ical method Since the algorithms were not implementedon the same PC the computational times of them are notcompetitive Therefore we did not give the computationaltimes in bold in Table 1
6 Sensitivity Analysis andComparing LI and SI
In this section two statistical analysis based on the one-way analysis of variance (ANOVA) used for investigatingthe sensitivity of MHGA parameters and Mann-Whitney
Mathematical Problems in Engineering 13
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
07
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 9 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (13)
5 10 15 20 25 30
35
4
45
5
55
6
65
7
75
8
85
9
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
Iter
Fina
l con
ditio
n
(b)
Figure 10 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (14)
applied comparing LI and SI methods are done by thestatistical software IBM SPSS version 21
61 Sensitivity Analysis In order to survey the sensitivityof the MHGA parameters from the proposed algorithmthe VDP problem is selected for example The influenceof these parameters are investigated for this NOCP on
the dependent outputs consist of the performance index119869 the relative error of 119869 119864
119869 the required computational
time Time the error of final state constraints 120593119891and the
factor 119870120595 The independent parameters are consist of the
number of time nodes in both two phases 1198731199051and 119873
1199052 the
size of population in both two phases 1198731199011
and 1198731199012 the
maximum number of generations without improvement119873119894
14 Mathematical Problems in Engineering
5 10 15 20 25 30 35 40 45 50
00955
0096
00965
0097
00975
0098
00985
0099
00995
Iter
Perfo
rman
ce in
dex
(a)
2 4 6 8 10 12 14 16 18 200
05
1
15
2
25
3
35
4
45
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 11 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (15)
5 10 15 20 25
21
22
23
24
25
26
27
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
1
2
3
4
5
6
Iter
Fina
l con
ditio
n
times10minus3
(b)
Figure 12 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (16)
and the maximum number of generations 119873119892 Because the
mutation implementation probability119875119898 has a less influence
in numerical results it is not considered Also 119904119902119901119898119886119909119894119905119890119903 ischanged in each iteration of the proposed algorithm so it isnot considered too
At first we selected at least four constant value foreach of parameters and then in each pose 35 differentruns were made independently The statistical analysis isdone based on ANOVA The descriptive statistics whichcontains the number of different runs (119873) the mean of each
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7minus887
minus886
minus885
minus884
minus883
minus882
minus881
minus88
minus879
minus878
minus877
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 70
01
02
03
04
05
06
07
08
09
1
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 13 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (17)
1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333
Iter
Perfo
rman
ce in
dex
Figure 14 Convergence rate for the performance index of SImethod for NOCP numbers (18)
output in119873 different runs (Mean) standard deviation (Sd)maximum (Max) and minimum (Min) of each outputcould be achieved Among of the MHGA parameters wepresent the descriptive statistics of the ANOVA only for the1198731199051parameter which is reported in Table 3Table 4 summarized the statistical data contain the test
statistics (119865) and 119875-values of ANOVA tests Sensitivityanalysis for each parameter separately are done based on thistable as follows
1198731199051 From Table 4 the significant level or 119875-value for119870120595
is equal to 0279 which is greater than 005So from ANOVA 119873
1199051parameter has no significant
effect on the factor 119870120595 With similar analysis this
parameter has no effect on the other outputs except
required computational time Time because its 119875-value is equal to 0 which is less than 005 that is thisoutput sensitive to the parameter119873
1199051
1198731199052 From the second row of Table 4 the outputs 119869 119864
119869119870120595
and Time are sensitive to the parameter 1198731199052 and the
only output 120593119891is independent
1198731199011 From the third row of Table 4 only the computationalrequired time Time is sensitive because the its 119875-value is equal to 0006 which is less than 005 and theother outputs contain 119869 119864
119869 120593119891119870120595 are independent
1198731199012 From the forth row of Table 4 the output 120593
119891is
independent respect to the parameter 1198731199012 but other
outputs contain 119869 119864119869 Time 119870
120601are sensitive to this
parameter119873119892 From the fifth row of Table 4 the outputs 119869 119864
119869and
119870120595are independent to the parameter 119873
119892 and other
outputs contain 120593119891and Time are sensitive respect to
this parameter119873119894 The sensitivity analysis is similar to119873
1199011
From above cases it is obvious that all parameters canbe effect on the required computational time except 119873
119892
Moreover intuitions shows the normof final state constraints120593119891is independent output with respect to all parameters that
is any of the parameters could not effect on this output and itis not sensitive with respect to any of the MHGA parameters
62 Comparison of LI and SI To compare the efficiency ofthe LI and SI methods for NOCPs in the Appendix weused the Mann-Whitney nonparametric statistical test [37]
16 Mathematical Problems in Engineering
1 15 2 25 3 35 40
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
200
400
600
800
1000
1200
1400
1600
IterFi
nal c
ondi
tion
(b)
Figure 15 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (19)
2 4 6 8 10 12 14 16 18 20
50
100
150
200
250
300
350
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 10 11 120
05
1
15
2
25
3
Iter
Fina
l con
ditio
n
(b)
Figure 16 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (20)
Using the nonparametric statistical tests for comparing theperformance of several algorithms presented in [38] Forthis purpose At first 15 different runs for each of NOCPwere made Table 5 shows the mean of numerical resultsseparately Here we apply the fixMHGAparameters as119873
1199011=
91198731199012
= 121198731199051= 31119873
1199052= 91119873
119892= 100 and119873
119894= 50 with
the previous problem parameters in Table 2 Comparison
criterions contain 119869 120593119891 119870120595and Time The results of Mann-
Whitney test are shown in Tables 6ndash9 From Table 6 since119885 = 0 gt 119885
001= minus234 then the average relative error of
the LI and SI methods are same with the significant level 001In other words with probability 99 we can say that LI andSI methods have same effectiveness form the perspective oferror for 119869 Similarly the results of Tables 7ndash9 indicate that LI
Mathematical Problems in Engineering 17
Table 3 Descriptive statistics for the sensitivity analysis of1198731199051 for VDP problem
Output 1198731199051
119873 Mean Sd Min Max
119869
11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449
Total 140 15357 00294 15285 18449
120593119891
11 35 281 times 10minus8
952 times 10minus8
1 times 10minus12
452 times 10minus7
21 35 467 times 10minus8
235 times 10minus7
1 times 10minus12
141 times 10minus6
31 35 163 times 10minus8
491 times 10minus8
1 times 10minus12
221 times 10minus7
41 35 817 times 10minus8
285 times 10minus7
1 times 10minus12
126 times 10minus6
Total 140 426 times 10minus8 191 times 10minus7 1 times 10minus12 141 times 10minus6
Time
11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514
Total 140 1142131 410891 456615 2849514
119864119869
11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070
Total 140 00047 00192 0 02070
119870120595
11 35 00031 00045 475 times 10minus9
00204
21 35 00027 00064 550 times 10minus10
00369
31 35 00029 00050 154 times 10minus13
00271
41 35 00111 00397 193 times 10minus10
02070
Total 140 00047 00192 154 times 10minus13 02070
Table 4 Summary statistical data of ANOVA test for the parameters1198731199051 1198731199052 1198731199011 1198731199012 119873119892 119873119894
Parameters 119869 119864119869
120593119891
Time 119870120595
Test statistic (119865)
1198731199051
1294 1296 0624 1079 12961198731199052
110572 393 0868 5939 3931198731199011
1945 1835 1271 4317 18351198731199012
2740 2478 1011 2302 2478119873119892
3541 3591 0491 0890 3591119873119894
0495 0309 0301 5849 1046
119875 value
1198731199051
0280 0279 0601 0 02791198731199052
0 0005 0489 0 00051198731199011
0125 0143 0286 0006 01431198731199012
0021 0034 0413 0 0034119873119892
0009 0008 0743 0472 0008119873119894
0739 0819 0877 0 0386
and SI methods from the perspective of errors for 120593119891 Time
and119870120595 have same behaviour
7 The Impact of SQP
In this section we investigation the impact of the SQP on theproposed algorithm For this purpose we remove the SQPfrom Algorithm 2 as without SQP algorithm For comparingthe numerical results with the previous results the required
running time in each NOCP is considered fixed which is themaximum of running time in 12 different runs were done inSection 5 Also the all of the parameters for each problemwasset as same as Section 5 The numerical results of the withoutSQP algorithm is summarized in Table 10 By comparing theresults of Tables 10 and 1 it is obvious that for all NOCPsin the Appendix the obtained values of the performanceindex and the norm of final state constraint for the proposedalgorithm (Algorithm 2) are more accurate than the withoutSQP algorithm
18 Mathematical Problems in Engineering
Table 5 The mean of numerical results of 15 different runs for NOCPs in the Appendix using LI and SI methods
Problem LI SI119869 120593
119891119870120595
Time 119869 120593119891
119870120595
TimeVDP 15977 157 times 10
minus9
170 times 10minus3 17720 16002 667 times 10
minus8
326 times 10minus3 15757
CRP 00126 307 times 10minus8
307 times 10minus8 19714 00119 161 times 10
minus8
161 times 10minus8 19532
FFPP 4774 805 times 10minus5 03277 133280 5351 592 times 10
minus5 04868 137989MSNIC 01702 mdash mdash 13678 01703 mdash mdash 14092CSTCR 01120 mdash mdash 17385 01120 mdash mdash 17606Number 6 minus54307 mdash mdash 10541 minus54308 mdash mdash 10372Number 7 02015 326 times 10
minus9
695 times 10minus6 20807 02020 313 times 10
minus9
662 times 10minus6 21019
Number 8 630 times 10minus15 mdash mdash 6376 124 times 10
minus14 mdash mdash 5837Number 9 16512 mdash mdash 59239 16512 mdash mdash 58125Number 10 01550 mdash mdash 63153 01550 mdash mdash 65597Number 11 29701 270 times 10
minus9
272 times 10minus7 22860 29701 337 times 10
minus9
811 times 10minus7 23276
Number 12 minus02499 201 times 10minus8
464 times 10minus5 54718 minus02499 115 times 10
minus9
827 times 10minus5 52765
Number 13 00147 115 times 10minus8
00252 57797 00151 139 times 10minus8
00472 66098Number 14 34761 672 times 10
minus6
00237 104585 34290 757 times 10minus6
950 times 10minus3 108767
Number 15 195 times 10minus4
111 times 10minus8
569 times 10minus3 363 194 times 10
minus4
704 times 10minus9
486 times 10minus3 365
Number 16 20571 166 times 10minus10
166 times 10minus10 54391 20571 115 times 10
minus10
115 times 10minus10 54431
Number 17 minus88692 803 times 10minus8
803 times 10minus8 22744 minus88692 661 times 10
minus8
661 times 10minus8 23254
Number 18 00326 mdash mdash 65957 00326 mdash mdash 70897Number 19 011588 655 times 10
minus4 06828 161813 01091 884 times 10minus4 08808 159990
Number 20 5242 00176 05429 84165 4464 453 times 10minus4 03765 86891
Table 6 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 119869
Method Mean rank Sum of ranks Test statistics ValueLI 2050 4100 Mann-Whitney 119880 200SI 2050 4100 Wilcoxon119882 410mdash mdash mdash 119885 0
Table 7 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 120593119891
Method Mean rank Sum of ranks Test statistics ValueLI 1362 177 Mann-Whitney 119880 83SI 1338 174 Wilcoxon119882 174mdash mdash mdash 119885 minus0077
Table 8 Results of Mann-Whitney test on relative errors of the pair (LI SI) for Time
Method Mean rank Sum of ranks Test statistics ValueLI 2025 4050 Mann-Whitney 119880 195SI 2075 4150 Wilcoxon119882 405mdash mdash mdash 119885 minus0135
Table 9 Results of Mann-Whitney test on relative errors of the pair (LI SI) for119870120595
Method Mean rank Sum of ranks Test statistics ValueLI 1354 1760 Mann-Whitney 119880 84SI 1346 1750 Wilcoxon119882 175mdash mdash mdash 119885 minus0026
Mathematical Problems in Engineering 19
Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix
Problem no1 2 3 4 5 6 7 8 9 10
119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048
120601119891
330 times 10minus9
135 times 10minus5 11445 mdash mdash mdash 758 times 10
minus9 mdash mdash mdashProblem no
11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10
minus4 23182 minus90882 00336 2513 20274120601119891
167 times 10minus5
468 times 10minus9 00161 433 times 10
minus4
594 times 10minus3
156 times 10minus5 164 mdash 1706 25283
8 Conclusions
Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters
Appendix
Test Problems
The following problems are presented using notation given in(1)ndash(6)
(1) (VDP) [9] 119892 = (12)(1199092
1+ 1199092
2+ 1199062
) 1199050= 0 119905119891= 5 119891 =
[1199092 minus1199092+(1minus119909
2
1)1199092+119906]119879 1199090= [1 0]
119879120595 = 1199091minus1199092+1
(2) (CRP) [9] 119892 = (12)(1199092
1+1199092
2+01119906
2
) 1199050= 0 119905119891= 078
119891 = [1199091minus2(1199091+025)+(119909
2+05) exp(25119909
1(1199091+2))minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(3) (FFRP) [9] 119892 = (12)(1199062
1+ 1199062
2+ 1199062
3+ 1199062
4) 1199050
=
0 119905119891
= 5 119891 = [1199092 (110)((119906
1+ 1199062) cos119909
5minus
(1199062+ 1199064) sin119909
5) 1199094 (110)((119906
1+ 1199063) sin119909
5+ (1199062+
1199064) cos119909
5) 1199096 (512)((119906
1+ 1199063) minus (119906
2+ 1199064))]119879 1199090=
[0 0 0 0 0 0]119879 120595 = [119909
1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879
(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909
2 minus1199092+
119906 1199092
1+ 1199092
2+ 0005119906
2
]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909
2+
05 minus 8(119905 minus 05)2
]119879 1199090= [0 minus1 0]
119879
(5) (CSTCR) [20] 119892 = 1199092
1+ 1199092
2+ 01119906
2 1199050= 0 119905119891= 078
119891 = [minus(2 + 119906)(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+
2)) 05 minus 1199092minus (1199092+ 05) exp(25119909
1(1199091+ 2))]
119879 1199090=
[009 009]119879
(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909
2 119906]119879 119889 = [minus(2 +
119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]
119879(7) [36] 119892 = 119906
2 1199050= 0 119905
119891= 1 119891 = (12)119909
2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05
(8) [26] 119892 = 1199092cos2119906 119905
0= 0 119905119891= 120587 119891 = sin(1199062) 119909
0=
1205872(9) [26] 119892 = (12)(119909
2
1+ 1199092
2+ 1199062
) 1199050= 0 119905
119891= 5 119891 =
[1199092 minus1199091+ (1 minus 119909
2
1)1199092+ 119906]119879 119889 = minus(119909
2+ 025) 119909
0=
[1 0]119879
(10) [26] 119892 = 1199092
1+ 1199092
2+ 0005119906
2
1199050= 0 119905
119891= 1 119891 =
[1199092 minus1199092+119906]119879
119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus
05) minus 05 minus 1199092)]119879
1199090= [0 minus1]
119879(11) [26] 119892 = (12)119906
2 1199050= 0 119905
119891= 2 119891 = [119909
2 119906]119879 1199090=
[1 1]119879 120595 = [119909
1 1199092]119879
(12) [26] 119892 = minus1199092 1199050= 0 119905
119891= 1 119891 = [119909
2 119906]119879 119889 =
minus(1 minus 119906)(1 + 119906) 1199090= [0 0]
119879 120595 = 1199092
(13) [26] 119892 = (12)(1199092
1+ 1199092
2+ 01119906
2
) 1199050= 0 119905
119891= 078
119891 = [minus2(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+ 2)) minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(14) [26] 119892 = (12)1199062 1199050= 0 119905
119891= 10 119891 = [cos 119906 minus
1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909
0= [366 minus186]
119879120595 = [119909
1 1199092]119879
(15) [26]119892 = (12)(1199092
1+1199092
2) 1199050= 0 119905119891= 078119891 = [minus2(119909
1+
025)+(1199092+05) exp(25119909
1(1199091+2))minus(119909
1+025)119906 05minus
1199092minus(1199092+05) exp(25119909
1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(16) [26] 120601 = 1199093 1199050= 0 119905
119891= 1 119891 = [119909
2 119906 (12)119906
2
]119879
119889 = 1199091minus 19 119909
0= [0 0 0]
119879 120595 = [1199091 1199092+ 1]119879
(17) [26] 120601 = minus1199093 1199050= 0 119905
119891= 5 119891 = [119909
2 minus2 + 119906119909
3
minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909
0= [10 minus2 10]
119879120595 = [119909
1 1199092]119879
(18) [26] 120601 = (1199091minus1)2
+1199092
2+1199092
3 119892 = (12)119906
2 1199050= 0 119905119891= 5
119891 = [1199093cos 119906 119909
3sin 119906 sin 119906]119879 119909
0= [0 0 0]
119879
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Initialization Input the the size of tournament119873tour and select119873tour
individuals from population randomly Let 119896119894 119894 = 1 2 119873tour be the indexes of them
repeatif 119873tour = 2 thenif 119868(1198801198961 ) lt 119868(119880
1198962) then
Let 119896 = 1198961
elseLet 119896 = 119896
2
end ifend if
Perform Tournament algorithm with size119873tour2 with output 1198961
Perform Tournament algorithm with size119873tour2 with output 1198962
until 1198961
= 1198962
Return the 119896th individual of population
Algorithm 1 Tournament algorithm
values are computed (or selected randomly)This can be doneby the following stages
(1) Let 119905119895= 1199050+ 119895ℎ where ℎ = (119905
119891minus 1199050)(119873119905minus 1) 119895 =
0 1 119873119905minus1 be time nodes where 119905
0and 119905119891are the
initial and final times respectively
(2) The corresponding control input value at each timenode 119905
119895is an119898times1 vector 119906
119895 which can be calculated
randomly with the following components
119906119894119895= 119906119895(119894) = 119906left (119894) + (119906right (119894) minus 119906left (119894)) sdot 119903119894119895
119894 = 1 2 119898 119895 = 0 1 119873119905minus 1
(7)
where 119903119894119895is a random number in [0 1]with a uniform
distribution and 119906left 119906right isin R119898 are the lower and theupper bound vectors of control input values whichcan be given by the problemrsquos definition or the user(eg see the NOCPs numbers (6) and (5) in theAppendix resp) So each individual of the popula-tion is an 119898 times 119873
119905matrix as 119880 = (119906
119894119895)119898times119873119905
= [119906119895]119873119905minus1
119895=0
Next we let 119880(119896) = (119906119894119895)119898times119873119905
119896 = 1 2 119873119901as 119896th
individual of the population which 119873119901is the size of
the population
Evaluation For each control input matrix 119880(119896) 119896 =
1 2 119873119901 the corresponding state variable is an 119899 times 119873
119905
matrix119883(119896) and it can be computed by the forthRunge-Kuttamethod on dynamic system (2) with the initial condition(6) approximately Then the performance index 119869(119880(119896))is approximated by a numerical method (denoted by 119869) IfNOCP includes equality or inequality constraints (3) or (4)then we add some penalty terms to the corresponding fitness
value of the solution Finally we assign 119868(119880(119896)
) to 119880(119896) as the
fitness value as follows
119868 (119880(119896)
) = 119869 +
119899119889
sum
119897=1
119873119905minus1
sum
119895=0
1198721119897max 0 119889
119897(119909119895 119906119895 119905119895)
+
119899119888
sum
ℎ=1
119873119905minus1
sum
119895=0
1198722ℎ1198882
ℎ(119909119895 119906119895 119905119895)
+
119899120595
sum
119894=119901
11987231199011205952
119901(119909119873119905minus1
119905119873119905minus1
)
(8)
where 1198721= [119872
11 119872
1119899119889]119879 1198722= [119872
21 119872
2119899119888]119879 and
1198723
= [11987231 119872
3119899120595]119879 are big numbers as the penalty
parameters 119888ℎ(sdot sdot) ℎ = 1 2 119899
119888 119889119897(sdot sdot) 119897 = 1 2 119899
119889
and 120595119901(sdot sdot) 119901 = 1 2 119899
120595are defined in (3) (4) and (5)
respectively
Selection To select two parents we use a tournament selec-tion [27] It can be applied for parallel GA The tournamentoperator applies competition among the same individualsand the best of them is selected for next generation At firstwe select a specified number of individuals from populationrandomly This number is tournament selection parameterwhich is denoted by119873tourThe tournament algorithm is givenin Algorithm 1
Crossover When two parents 119880(1) and 119880(2) are selected we
use the following stages to construct an offspring(1) Select the following numbers
1205821isin [0 1] 120582
2isin [minus120582max 0] 120582
3isin [1 1 + 120582max]
(9)randomly where 120582max is a random number in [0 1]
(2) Let
of119896 = 120582119896119880(1)
+ (1 minus 120582119896) 119880(2)
119896 = 1 2 3 (10)
4 Mathematical Problems in Engineering
where 120582119896 119896 = 1 2 3 is defined in (9) For 119894 = 1 119898
and 119895 = 1 119873119905 if (of119896)
119894119895gt 119906right(119894) then let (of
119896
)119894119895=
119906right(119894) and if (of119896
)119894119895lt 119906left(119894) then let (of
119896
)119894119895= 119906left(119894)
(3) Let of = oflowast where oflowast is the best of119896 119896 = 1 2 3
constructed by (10)
MutationWe apply a perturbation on each component of theoffspring as follows
(of)119894119895= (of)
119894119895+ 119903119894119895sdot 120572 119894 = 1 2 119898 119895 = 1 2 119873
119905
(11)
where 119903119894119895is selected randomly in minus1 1 and 120572 is a random
number with a uniform distribution in [0 1] If (of)119894119895
gt
119906right(119894) then let (of)119894119895= 119906right(119894) and if (of)
119894119895lt 119906left(119894) then
let (of)119894119895= 119906left(119894)
ReplacementHere in the underling GA we use a traditionalreplacement strategy The replacement is done if the newoffspring has two properties first it is better than the worstperson in the population 119868(of) lt max
1le119894le119873119901119868(119880(119894)
) secondit is not very similar to a person in the population that is foreach 119894 = 1 119873
119901 at least one of the following conditions is
satisfied10038161003816100381610038161003816119868 (of) minus 119868 (119880
(119894)
)
10038161003816100381610038161003816gt 120576
10038171003817100381710038171003817of minus 119880
(119894)10038171003817100381710038171003817gt 120576
(12)
where 120576 is the machine epsilon
Termination Conditions Underlying GA is terminated whenat least one of the following conditions is occurred
(1) Themaximumnumber of generations119873119892 is reached
(2) Over a specified number of generations 119873119894 we do
not have any improvement (the best individual is notchanged) or the two-norm or error of final stateconstraints will reach a small number as the desiredprecision 120576 that is
120593119891=100381710038171003817100381712059510038171003817100381710038172
lt 120576 (13)
where120595 = [1205951 1205952 120595
119899120595]119879 is the vector of final state
constraints defined in (5)
32 SQP The fitness value in (8) can be viewed as anonlinear objective function with the decision variable as119906 = [119906
0 1199061 119906
119873119905minus1] This cost function with upper and
lower bounds of input signals construct a finite dimensionalNLP problem as following
min 119868 (119906) = 119868 (1199060 1199061 119906
119873119905minus1) (14)
st 119906left le 119906119895le 119906right 119895 = 0 1 119873
119905minus 1 (15)
SQP algorithm [22 28] is performed on the NLP (14)-(15)using 1199060 = of and constructed in (11) as the initial solutionwhen the maximum number of iteration is 119904119902119901119898119886119909119894119905119890119903
SQP is an effective and iterative algorithm for thenumerical solution of the constrained NLP problem Thistechnique is based on finding a solution to the system ofnonlinear equations that arise from the first-order necessaryconditions for an extremum of the NLP problem Usingan initial solution of NLP 119906119896 119896 = 0 1 a sequence ofsolutions as 119906119896+1 = 119906
119896
+ 119889119896 is constructed which 119889
119896 is theoptimal solution of the constructed quadratic programming(QP) that approximates NLP in the iteration 119896 based on119906119896 as the search direction in the line search procedure
For the NLP (15) the principal idea is the formulation of aQP subproblem based on a quadratic approximation of theLagrangian function as 119871(119906 120582) = 119868(119906) + 120582
119879
ℎ(119906) where thevector 120582 is Lagrangian multiplier and ℎ(119906) return the vectorof inequality constraints evaluated at 119906 The QP is obtainedby linearizing the nonlinear functions as follows
min 1
2
119889119879
119867(119906119896
) 119889 + nabla119868 (119906119896
)
119879
119889 (16)
nablaℎ (119906119896
)
119879
119889 + ℎ (119906119896
) le 0 (17)
Similar to [26] here a finite difference approximation isapplied to compute the gradient of the cost function and theconstraints with the following components
120597119868
120597119906119895
=
119868 (sdot sdot sdot 119906119895+ 120575 sdot sdot sdot ) minus 119868 (119906)
120575
119895 = 0 1 119873119905minus 1 (18)
where 120575 is the double precision of machine So the gradientvector is nabla119868 = [120597119868120597119906
0 120597119868120597119906
119873119905minus1]119879 Also at each major
iteration a positive definite quasi-Newton approximation oftheHessian of the Lagrangian function119867 is calculated usingthe BFGS method [28] where 120582
119894 119894 = 1 119898 is an estimate
of the Lagrange multipliers The general procedure for SQPfor NLP (14)-(15) is as follows
(1) Given an initial solution 1199060 Let 119896 = 0
(2) Construct the QP subproblem (16)-(17) based on 1199060
using the approximations of the gradient and theHessian of the the Lagrangian function
(3) Compute the new point as 119906119896+1 = 119906119896
+ 119889119896 where 119889119896
is the optimal solution of the current QP(4) Let 119896 = 119896 + 1 and go to step (2)
Here in MHGA SQP is used as the local search and weuse the maximum number of iterations as the main criterionfor stopping SQP In other words we terminate SQP when itconverges either to local solution or themaximumnumber ofSQPrsquos iterations is reached
33 MHGA In MHGA GA uses a local search method toimprove solutions Here we use SQP as a local search UsingSQP as a local search in the hybrid metaheuristic is commonfor example see [20]
MHGA can be seen as a multi start local search whereinitial solutions are constructed by GA From another per-spective MHGA can be seen as a GA that the quality of
Mathematical Problems in Engineering 5
Initialization
sqpmaxiter
an initial population
EvaluationEvaluate the fitnessof each individual
using (8)
Local search(Section 32)
ReturnThe best individual inthe final populationas an approximate
of the global solution of NOCP
Stoppingconditions
Selection(Algorithm 1)
Crossover(using Equation (10))
Replacement(using Equations
(12))
Mutation(using Equation(11))
Local search(Section 32)
Input NtNpNiNg Pm
120576Mi i = 1 2 3 and
=sqpmaxitersqpmaxiter + 1
Figure 1 Flowchart of the MHGA algorithm
its population is intensified by SQP In the beginning ofMHGA a less number of iterations for SQP was used Thenwhen the promising regions of search space were found byGA operators we increase the number of iterations of SQPgradually Using this approach we may decrease the neededrunning time (in [19] the philosophy of this approach isdiscussed)
Finally we give our modified MHGA to find the globalsolution by the flowchart in Figure 1
4 Proposed Algorithm
Here we give a new algorithm which is a direct approachbased on MHGA for solving NOCPs The proposed algo-rithm has two main phases In the first phase we performMHGA with a completely random initial population con-structed by (7) In the first phase to find the promisingregions of the search space in a less running time we usea few numbers of time nodes In addition to have a fasterMHGA the size of the population in the first phase is usuallyless than the size of the population in the second phase
After phase 1 to maintain the property of individuals inthe last population of phase 1 and to increase the accuratelyof solutions we add some additional time nodes Whenthe number of time nodes is increased it is estimated thatthe quality of solution obtained by numerical methods (eg
Runge-Kutta and Simpson) is increased Thus we increasetime nodes from 119873
1199051in phase 1 to 119873
1199052in phase 2 The
corresponding control input values of the new time nodesare added to individuals To use the information of theobtained solutions from phase 1 in the construction of theinitial population of phase 2 we use either Linear or Splineinterpolation to estimate the value of the control inputs inthe new time nodes in each individual of the last populationof phase 1 Moreover to maintain the diversity in the initialpopulation of phase 2 we add new random individuals tothe population using (7) In the second phase MHGA startswith this population and new value of parameters Finally theproposed algorithm is given in Algorithm 2
5 Numerical Experiments
In this section to investigate the efficiency of the proposedalgorithm 20 well-known and real world NOCPs as bench-mark problems are considered which are presented in termsof (1)ndash(6) in the Appendix These NOCPs are selected withsingle control signal and multi control signals which will bedemonstrated in a general manner
The numerical behaviour of algorithms can be studiedfrom two view of points the relative error of the performanceindex and the status of the final state constraints Let 119869 be theobtained performance index by an algorithm 120593
119891 defined in
6 Mathematical Problems in Engineering
Initialization Input the desired precision 120576 in (13) the penalty parameters119872119894 119894 = 1 2 3 in (8) the bounds of control input values in (7) 119906left and 119906right
Phase 1 Perform MHGA with a random population and1198731199051 1198731199011 1198731198941 1198731198921 1198751198981
and 1199041199021199011198981198861199091198941199051198901199031
Construction of the initial population of the phase 2 Increase time nodes uniformly to1198731199052
and estimate the corresponding control input values of the new time nodes in eachindividual obtained from phase 1 using either Linear or Spline interpolationCreate119873
1199012minus 1198731199011new different individuals with119873
1199052time nodes randomly
Phase 2 Perform MHGA with the constructed population and1198731199052 1198731199012 1198731198942 1198731198922 1198751198982
and 1199041199021199011198981198861199091198941199051198901199032
Algorithm 2 The proposed algorithm
(13) be the error of final state constraints and 119869lowast be the best
obtained solution among all implementations or the exactsolution (when exists) Now the relative error of 119869 119864
119869 of the
algorithm can be defined as
119864119869=
10038161003816100381610038161003816100381610038161003816
119869 minus 119869lowast
119869lowast
10038161003816100381610038161003816100381610038161003816
(19)
Tomore accurate study we now define a new criterion calledfactor to compare the algorithms as follows
119870120595= 119864119869+ 120593119891 (20)
Note that 119870120595shows the summation of two important errors
Thus based on 119870120595
we can study the behaviour of thealgorithms on the quality and feasibility of given solutionssimultaneously
To solve any NOCP described in the Appendix wemust know the algorithmrsquos parameters including MHGArsquosparameters including 119873
119905 119873119901 119873119894 119873119892 119875119898and 119904119902119901119898119886119909119894119905119890119903
in both phases in Algorithm 2 and the problemrsquos parametersincluding 120576 in (13) 119872
119894 119894 = 1 2 3 in (8) 119906left and 119906right in
(7) To estimate the best value of the algorithmrsquos parameterswe ran the proposed algorithm with different values ofparameters and then select the best However the sensitivityof MHGA parameters are studied in the next section In bothof phases of Algorithm 2 in MHGA we let 119904119902119901119898119886119909119894119905119890119903
119894= 4
and 119875119898119894
= 08 for 119894 = 1 2 Also we consider 1198731198921
= 1198731198922
1198731198941
= 1198731198942 1198731199011
= 9 and 1198731199012
= 12 The other MHGAparameters are given in the associated subsection and theproblemrsquos parameters in Table 2 For each NOCP 12 differentruns were done and the best results are reported in Table 1which the best value of each column is seen in the bold
The reported numerical results of the proposed algo-rithm for each NOCP include the value of performanceindex 119869 the relative error of 119869119864
119869 defined in (19) the required
computational time Time the norm of final state constraints120593119891 defined in (13) and the factor 119870
120595 defined in (20)
The algorithm was implemented in Matlab R2011a envi-ronment on a Notebook with Windows 7 Ultimate CPU253GHz and 400GB RAM Also to implement SQP in ourproposed algorithm we used ldquofminconrdquo in Matlab when theldquoAlgorithmrdquo was set to ldquoSQPrdquo Moreover we use compositeSimpsonrsquos method [29] to approximate integrations
Remark 1 We use the following abbreviations to show theused interpolation method in our proposed algorithm
(1) LI linear interpolation(2) SI spline interpolation
For comparing the numerical results of the proposedalgorithm two subsections are considered comparison withsome metaheuristic algorithms in Section 51 and com-parison with some numerical methods in Section 52 Wegive more details of these comparisons in the followingsubsections
51 Comparison with Metaheuristic Algorithms The numeri-cal results for the NOCPs numbers (1)ndash(3) in the Appendixare compared with a continuous GA CGA as a meta-heuristic proposed in [9] which gave better solutions thanshooting method and gradient algorithm from the indirectmethods category [2 30] and SUMT from the directmethodscategory [26] For NOCPs numbers (4) and (5) the resultsare compared with another metaheuristic which is a hybridimproved PSO called IPSO proposed in [20]
511 VDP Problem [9] The first NOCP in the Appendix isVanDer Pol Problem VDP which has two state variables andone control variable VDPproblemhas a final state constraintwhich is 120595 = 119909
1(119905119891) minus 1199092(119905119891) + 1 = 0 The results of the
proposed algorithm with the MHGArsquos parameters as 1198731199051
=
311198731199052= 71119873
119892= 300 and119873
119894= 200 are reported in Table 1
FromTable 1 it is obvious that the numerical results of LI andSI methods are more accurate than CGA with less amount of119870120595
512 CRP Problem [9] The second NOCP in the Appendixis Chemical Reactor Problem CRP which has two statevariables and one control variableThe results of the proposedalgorithm with the MHGArsquos parameters as 119873
1199051= 31 119873
1199052=
71 119873119892
= 300 and 119873119894= 200 are shown in the second
row of Table 1 CRP problem has two final state constraints120595 = [119909
1 1199092]119879 Although from Table 1 the norm of final state
constraints 120593119891 for the CGA equals 120593lowast
119891= 757 times 10
minus10 isless than 120593
119891rsquos of LI and SI methods which equals 115 times 10
minus9
and 599 times 10minus9 respectively but the performance index
Mathematical Problems in Engineering 7
Table 1 The best numerical results in 12 different runs of the SI and LI methods for NOCPs in the Appendix
Problem Algorithm 119869 119864119869
120593119891
119870120595
Time
VDPCGA 17404 00912 267 times 10
minus11
00912 50128LI 15949 0 108 times 10
minus13 108 times 10minus13 28282SI 15950 627 times 10
minus5 999 times 10minus14 627 times 10minus5 23914
CRPCGA 163 times 10
minus2 02835 757 times 10minus10 02835 50128LI 127 times 10minus2 0 115 times 10
minus9 115 times 10minus9 17194
SI 127 times 10minus2 0 599 times 10minus9
599 times 10minus9
12405
FFRPCGA 8363 13256 465 times 10
minus3
13302 1413LI 3596 0 122 times 10
minus5 122 times 10minus5 126858
SI 3599 834 times 10minus4 901 times 10minus6 843 times 10
minus4 134340
MSNICIPSO 01727 00165 mdash mdash 12626LI 01699 0 mdash mdash 13882
SI 01700 589 times 10minus4 mdash mdash 13868
CSTCRIPSO 01355 02098 mdash mdash 3134
LI 01120 0 mdash mdash 10358
SI 01120 0 mdash mdash 10673
No 6Bezier minus53898 00251 mdash mdash NRa
LI minus54309 00177 mdash mdash 13838
SI minus54309 00177 mdash mdash 10734
Number 7HPM 02353 01677 420 times 10
minus6 01677 NRLI 02015 0 235 times 10
minus9
235 times 10minus9 21392
SI 02015 0 282 times 10minus10 282 times 10minus10 20696
Number 8
SQP 636 times 10minus6
220 times 109 mdash mdash NR
SUMT 515 times 10minus6
178 times 109 mdash mdash NR
LI 289 times 10minus15 0 mdash mdash 7282SI 364 times 10
minus15
02595 mdash mdash 6868
Number 9
SQP 17950 00873 mdash mdash NRSUMT 17980 00891 mdash mdash NRLI 16509 0 mdash mdash 63910SI 16509 0 mdash mdash 5950
Number 10
SQP 02163 03964 mdash mdash NRSUMT 01703 00994 mdash mdash NRLI 01549 0 mdash mdash 62570
SI 01549 0 mdash mdash 67662
Number 11
SQP 325 01648 0 01648 NRSUMT 325 01648 0 01648 NRLI 27901 0 162 times 10
minus9
162 times 10minus9 14778
SI 27901 0 586 times 10minus10 586 times 10minus10 21946
Number 12
SQP minus02490 40 times 10minus3 0 40 times 10
minus3 NRSUMT minus02490 40 times 10
minus3 0 40 times 10minus3 NR
LI minus02500 0 285 times 10minus8
285 times 10minus8
54771
SI minus02500 0 390 times 10minus10 390 times 10minus10 54548
Number 13
SQP 168 times 10minus2 01748 0 01748 NR
SUMT 167 times 10minus2 01678 0 01678 NR
LI 143 times 10minus2 0 118 times 10minus9 118 times 10minus9 42599SI 144 times 10
minus2
70 times 10minus3
632 times 10minus9
70 times 10minus3
49027
Number 14
SQP 37220 00961 0 00961 NRSUMT 37700 01103 0 01103 NRLI 33956 0 786 times 10minus7 786 times 10minus7 104124SI 33965 265 times 10
minus4
276 times 10minus6
267 times 10minus4
115144
8 Mathematical Problems in Engineering
Table 1 Continued
Problem Algorithm 119869 119864119869
120593119891
119870120595
Time
Number 15
SQP 103 times 10minus3 43368 0 43368 NR
SUMT 929 times 10minus4 38135 0 38135 NR
LI 193 times 10minus4 0 810 times 10minus10 810 times 10minus10 33354SI 194 times 10
minus4
520 times 10minus3
164 times 10minus9
520 times 10minus3
34014
Number 16
SQP 22120 00753 0 00753 NRSUMT 22080 00734 0 00734 NRLI 20571 0 394 times 10
minus12
394 times 10minus12
46982
SI 20571 0 560 times 10minus13 560 times 10minus13 70423
Number 17
SQP minus88690 225 times 10minus5 0 225 times 10
minus5 NRSUMT minus88690 225 times 10
minus5 0 225 times 10minus5 NR
LI minus88692 0 145 times 10minus10 145 times 10minus10 32183SI minus88692 0 279 times 10
minus10
279 times 10minus10
30959
Number 18
SQP 00368 01288 mdash mdash NRSUMT 00386 01840 mdash mdash NRLI 00326 0 mdash mdash 55163
SI 00326 0 mdash mdash 60671
Number 19
SQP 03439 49293 0 49293 NRSUMT 03428 49103 0 49103 NRLI 00689 01879 410 times 10
minus3
01919 164509
SI 00580 0 405 times 10minus4 405 times 10minus4 178783
Number 20
SQP 7752 12554 0 12554 NRSUMT 7683 12353 0 12353 NRLI 343716 232 times 10
minus5
225 times 10minus4
248 times 10minus4 81519
SI 343708 0 157 times 10minus4 157 times 10minus4 80420
aNot reported
the relative error of 119869 and the factor of the proposed algorithmis better and so the proposed algorithm is more robust thanCGA
513 FFRP Problem [9] The third NOCP in the Appendixis Free Floating Robot Problem FFRP which has six statevariables and four control variables FFRP has been solvedby CGA and the proposed algorithm with the MHGArsquosparameters as 119873
1199051= 31 119873
1199052= 71 119873
119892= 300 and
119873119894
= 200 This problem has six final state constraints120595 = [119909
1minus 4 119909
2 1199093minus 4 119909
4 1199095 1199096]119879 The numerical results
are shown in Table 1 The values of 119869 119864119869 120593119891and 119870
120595for LI
and SI methods separately are less than CGA Thereforethe proposed algorithm can achieve much better qualitysolutions than the CGA with reasonable computational time
514 MSNIC Problem [20] For the forth NOCP in theAppendix which is a Mathematical System with NonlinearInequality Constraint NSNIC the numerical results are com-pared with IPSO MSNIC contains an inequality constraint119889(119909 119905) = 119909
2(119905) + 05 minus 8(119905 minus 05)
2
le 0 The problem solvedby several numerical methods as [24 31] From [20] IPSOmethod could achieved more accurate results than men-tioned numericalmethods AlsoMSNIC can be solved by theproposed algorithm with the MHGArsquos parameters as 119873
1199051=
31 1198731199052
= 91 119873119892= 100 and 119873
119894= 60 From the forth row
of Table 1 the absolute error of 119869 119864119869 for LI and SI methods
equal 0 and 589 times 10minus4 respectively which are less than
IPSOrsquos 00165Subplots (a) and (b) in Figure 2 show the graphs of
the convergence rate for the performance index and theinequality constraint respect to the number of iterationrespectively
515 CSTCR Problem [20] The fifth NOCP in the Appendixis a model of a nonlinear Continuous Stirred-tank ChemicalReactor CSTCR It has two state variables 119909
1(119905) and 119909
2(119905)
as the deviation from the steady-state temperature and con-centration and one control variable 119906(119905) which represent theeffect of the flow rate of cooling fluid on chemical reactorTheobjective is to maintain the temperature and concentrationclose to steady-state values without expending large amountof control effort Also this is a benchmark problem in thehandbook of test problems in local and global optimization[32] which is a multimodal optimal control problem [33]It involves two different local minima The values of theperformance indexes for these solutions equal 0244 and0133 Similarly to the MSNIC the numerical results of theproposed algorithm with the MHGArsquos parameters as 119873
1199051=
31 1198731199052
= 51 119873119892= 100 and 119873
119894= 50 are compared with
IPSO From Table 1 the performance index 119869 for LI and SImethods is equal to 119869
lowast
= 01120 which is less than IPSOs01355
Mathematical Problems in Engineering 9
Table 2 The problem parameters for NOCPs in the Appendix
Parameters Problem number1 2 3 4 5 6 7 8 9 10
119906left minus05 minus15 minus15 minus20 0 minus2 0 minus2 minus1 minus20
119906right 2 2 10 20 5 2 1 2 1 20119872119894
103
102 70 1 mdash 1 mdash 10
2 mdash 102
120576 10minus12
10minus10
10minus3 mdash mdash mdash mdash mdash mdash 10
minus9
Parameters Problem no11 12 13 14 15 16 17 18 19 20
119906left minus5 minus1 minus2 minus120587 minus1 minus3 minus30 minus1 minus15 minus15
119906right 5 1 2 120587 1 3 30 1 10 10119872119894
10 102 mdash 10
2
102
102 mdash 10 70 10
minus3
120576 10minus9
10minus11
10minus11 mdash 10
minus10
10minus10
10minus10 mdash 10
minus3
10minus4
0 5 10 15 20 25 30 35 40 45 50017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 2 Graphical results of MSNIC problem using SI method (a) convergence rate of the performance index and (b) the inequalityconstraint 119889(119909 119905) le 0 in respect to the number of iterations
52 Comparison with Numerical Methods For NOCPs num-bers (6)ndash(20) the comparison are done with some numericalmethods Unfortunately for these methods usually the finalstate constraints and the required computational time arenot reported which are shown with NR in Table 1 but thesevalues are reported for both LI and SI methods in Table 1For all NOCPs in this section the MHGArsquos parameters areconsidered as 119873
1199051= 31 119873
1199052= 51 119873
119892= 100 119873
119894= 50 with
the problem parameters in Table 2
521 Compared with Bezier [34] The NOCP number (6) inthe Appendix has exact solution which has an inequalityconstraint as 119889(119909
1 119905) = minus6 minus 119909
1(119905) le 0 The exact value of
performance index equals 119869lowast = minus55285 [35] This problemhas been solved by a numerical method proposed in [34]called Bezier From sixth row of Table 1 the absolute error
of the LI and SI methods equal 00177 which is less thanBezierrsquos 00251
Figure 3 shows the graphs of the convergence rate of theperformance index subplot (a) and inequality constraintsubplot (b) respect to the number of iteration using SImethod
522 Compared with HPM [36] For NOCP number (7) inthe Appendix which is a constraint nonlinear model thenumerical results of the proposed algorithm are comparedwith HPM proposed in [36] This NOCP has a final stateconstraint as 120595 = 119909 minus 05 = 0 From [36] the norm offinal state constraint for HPM equals 42times10minus6 however thiscriterion for the LI and SI methods equals 235 times 10
minus9 and120593lowast
119891= 282 times 10
minus10 respectively From Table 1 it is obviousthat the obtained values of the performance index the norm
10 Mathematical Problems in Engineering
2 4 6 8 10 12 14minus5353
minus5352
minus5351
minus535
minus5349
minus5348
minus5347
minus5346
minus5345
minus5344
minus5343
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 3 Graphical results for NOCP number (6) using SI method (a) convergence rate for the performance index and (b) the inequalityconstraint 119889(119909
1 119905) le 0 in respect to the number of iterations
of final state constraint and 119870120595for the SI method are more
accurate than LI than HPMmethods
523 Compared with SQP and SUMT [26] For the NOCPsnumbers (8)ndash(20) in the Appendix the numerical resultsof LI and SI methods are compared with two numericalmethods contain SQP and SUMT proposed in [26] Amongthese NOCPs only two problems numbers (8) and (18) areunconstrained and the others have at least one constraintfinal state constraint or inequality constraint All of theseNOCPs solved by the proposed algorithm with the problemparameters in Table 2 and their results are summarized inTable 1 Because the final state constraints in these methodsare not reported we let 120593
119891= 0 to calculate the factor119870
120595
Figures 4ndash16 show the graphical results of NOCPsnumbers (8)ndash(20) in the Appendix using SI method Forunconstrained NOCPs numbers (8) and (18) only the graphof convergent rate of performance index is shown see Figures4 and 14 For constraint NOCPs and NOCPs numbers(11)ndash(17) and numbers (19)-(20) with final state constraintthe graphs of convergent rates of performance index and theerror of final state constraint are shown see Figures 7ndash13 and15-16 For the constraint NOCPs with inequality constraintsNOCPs numbers (9) and (10) the graphs of convergent rateof performance index and inequality constraint are shownsee Figures 5 and 6
Table 1 shows that the proposed algorithm LI and SImethods was 100 percent successful in point of views theperformance index 119869 and the factor119870
120595 numerically So the
proposed algorithm provides robust solutions with respect tothe other mentioned numerical or metaheuristic methodsTo compare 119869 in LI and SI methods in 35 percent of NOCPsLI is more accurate than SI in 10 percent SI is more accurate
1 15 2 25 3 35 40
002
004
006
008
01
012
Iter
Perfo
rman
ce in
dex
Figure 4 Convergence rate for the performance index of SI methodfor NOCP number (8)
than LI and in 55 percent are same In point of view the finalconditions 65 percent of NOCPs in the Appendix have finalstate constraints In all of them 120593
119891in the proposed algorithm
is improved except CRP problem however the factor of theproposed algorithm is the best yet In 54 percent of theseNOCPs LI is better performance and in 46 percent SI isbetter Also in point of the running time Time LI is same asSI that is in 50 percent of NOCPs LI and in 50 percent SI isbetter than another So the proposed algorithmcould providevery suitable solutions in a reasonable computational timeAlso for more accurate comparison of LI and SI methods astatistical approach will done in next section
To compare with CGA the mean of relative error of 119869 119864119869
for CGA LI and SImethods inNOCPs (1)ndash(3) equal 056680 and 298 times 10
minus4 respectively Also the mean for the error
Mathematical Problems in Engineering 11
5 10 15 20 25165
17
175
18
185
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3 35 4 45 50
005
01
015
02
025
TimeIn
equa
lity
cons
trai
nt(b)
Figure 5 (a) Convergence rate for the performance index and (b) the inequality constraint of SI method for NOCP number (9)
5 10 15 20 25 30 35 40 45 500155
016
0165
017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 6 (a) Convergence rate of the performance index and (b) the inequality constraint for SI method for NOCP number (10)
of the final state constraints 120593119891 are 16 times 10
minus3 407 times 10minus6
and 301 times 10minus6 respectively and for 119870
120595 these values equal
05883 407 times 10minus6 and 302 times 10
minus4 Thus we can say that thefeasibility of the solutions given by the proposed algorithmand CGA are competitive
From Table 1 to compare with numerical methods SQPand SUMT in NOCPs (8)ndash(20) the mean of 119864
119869for LI SI
SQP and SUMT equals 00145 00209 169 times 108 and 136 times
108 respectively Also the mean for the error of final state
constraints for these NOCPs equal 334 times 10minus4 441 times 10
minus50 and 0 respectively For 119870
120595 these values are 00213 00014
12263 and 11644 Therefore the performance index 119869 andthe factor 119870
120595 for the LI and SI methods are more accurate
than SQP and SUMT So the proposed algorithm gave morebetter solution in comparison with the numerical methods
12 Mathematical Problems in Engineering
5 10 15 20 25minus025
minus0245
minus024
minus0235
minus023
minus0225
Iter
Perfo
rman
ce in
dex
(a)
1 15 2 25 3 35 4 45 5 55 60
02
04
06
08
1
12
14
16
IterFi
nal c
ondi
tion
times10minus5
(b)
Figure 7 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (11)
2 4 6 8 10 12 14minus025
minus0245
minus024
minus0235
minus023
minus0225
minus022
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
02
04
06
08
1
12
14
16
18
2
Iter
Fina
l con
ditio
n
times10minus5
(b)
Figure 8 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (12)
Therefore based on this numerical studywe can concludethat the proposed algorithm outperform well-known numer-ical method Since the algorithms were not implementedon the same PC the computational times of them are notcompetitive Therefore we did not give the computationaltimes in bold in Table 1
6 Sensitivity Analysis andComparing LI and SI
In this section two statistical analysis based on the one-way analysis of variance (ANOVA) used for investigatingthe sensitivity of MHGA parameters and Mann-Whitney
Mathematical Problems in Engineering 13
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
07
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 9 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (13)
5 10 15 20 25 30
35
4
45
5
55
6
65
7
75
8
85
9
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
Iter
Fina
l con
ditio
n
(b)
Figure 10 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (14)
applied comparing LI and SI methods are done by thestatistical software IBM SPSS version 21
61 Sensitivity Analysis In order to survey the sensitivityof the MHGA parameters from the proposed algorithmthe VDP problem is selected for example The influenceof these parameters are investigated for this NOCP on
the dependent outputs consist of the performance index119869 the relative error of 119869 119864
119869 the required computational
time Time the error of final state constraints 120593119891and the
factor 119870120595 The independent parameters are consist of the
number of time nodes in both two phases 1198731199051and 119873
1199052 the
size of population in both two phases 1198731199011
and 1198731199012 the
maximum number of generations without improvement119873119894
14 Mathematical Problems in Engineering
5 10 15 20 25 30 35 40 45 50
00955
0096
00965
0097
00975
0098
00985
0099
00995
Iter
Perfo
rman
ce in
dex
(a)
2 4 6 8 10 12 14 16 18 200
05
1
15
2
25
3
35
4
45
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 11 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (15)
5 10 15 20 25
21
22
23
24
25
26
27
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
1
2
3
4
5
6
Iter
Fina
l con
ditio
n
times10minus3
(b)
Figure 12 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (16)
and the maximum number of generations 119873119892 Because the
mutation implementation probability119875119898 has a less influence
in numerical results it is not considered Also 119904119902119901119898119886119909119894119905119890119903 ischanged in each iteration of the proposed algorithm so it isnot considered too
At first we selected at least four constant value foreach of parameters and then in each pose 35 differentruns were made independently The statistical analysis isdone based on ANOVA The descriptive statistics whichcontains the number of different runs (119873) the mean of each
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7minus887
minus886
minus885
minus884
minus883
minus882
minus881
minus88
minus879
minus878
minus877
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 70
01
02
03
04
05
06
07
08
09
1
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 13 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (17)
1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333
Iter
Perfo
rman
ce in
dex
Figure 14 Convergence rate for the performance index of SImethod for NOCP numbers (18)
output in119873 different runs (Mean) standard deviation (Sd)maximum (Max) and minimum (Min) of each outputcould be achieved Among of the MHGA parameters wepresent the descriptive statistics of the ANOVA only for the1198731199051parameter which is reported in Table 3Table 4 summarized the statistical data contain the test
statistics (119865) and 119875-values of ANOVA tests Sensitivityanalysis for each parameter separately are done based on thistable as follows
1198731199051 From Table 4 the significant level or 119875-value for119870120595
is equal to 0279 which is greater than 005So from ANOVA 119873
1199051parameter has no significant
effect on the factor 119870120595 With similar analysis this
parameter has no effect on the other outputs except
required computational time Time because its 119875-value is equal to 0 which is less than 005 that is thisoutput sensitive to the parameter119873
1199051
1198731199052 From the second row of Table 4 the outputs 119869 119864
119869119870120595
and Time are sensitive to the parameter 1198731199052 and the
only output 120593119891is independent
1198731199011 From the third row of Table 4 only the computationalrequired time Time is sensitive because the its 119875-value is equal to 0006 which is less than 005 and theother outputs contain 119869 119864
119869 120593119891119870120595 are independent
1198731199012 From the forth row of Table 4 the output 120593
119891is
independent respect to the parameter 1198731199012 but other
outputs contain 119869 119864119869 Time 119870
120601are sensitive to this
parameter119873119892 From the fifth row of Table 4 the outputs 119869 119864
119869and
119870120595are independent to the parameter 119873
119892 and other
outputs contain 120593119891and Time are sensitive respect to
this parameter119873119894 The sensitivity analysis is similar to119873
1199011
From above cases it is obvious that all parameters canbe effect on the required computational time except 119873
119892
Moreover intuitions shows the normof final state constraints120593119891is independent output with respect to all parameters that
is any of the parameters could not effect on this output and itis not sensitive with respect to any of the MHGA parameters
62 Comparison of LI and SI To compare the efficiency ofthe LI and SI methods for NOCPs in the Appendix weused the Mann-Whitney nonparametric statistical test [37]
16 Mathematical Problems in Engineering
1 15 2 25 3 35 40
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
200
400
600
800
1000
1200
1400
1600
IterFi
nal c
ondi
tion
(b)
Figure 15 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (19)
2 4 6 8 10 12 14 16 18 20
50
100
150
200
250
300
350
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 10 11 120
05
1
15
2
25
3
Iter
Fina
l con
ditio
n
(b)
Figure 16 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (20)
Using the nonparametric statistical tests for comparing theperformance of several algorithms presented in [38] Forthis purpose At first 15 different runs for each of NOCPwere made Table 5 shows the mean of numerical resultsseparately Here we apply the fixMHGAparameters as119873
1199011=
91198731199012
= 121198731199051= 31119873
1199052= 91119873
119892= 100 and119873
119894= 50 with
the previous problem parameters in Table 2 Comparison
criterions contain 119869 120593119891 119870120595and Time The results of Mann-
Whitney test are shown in Tables 6ndash9 From Table 6 since119885 = 0 gt 119885
001= minus234 then the average relative error of
the LI and SI methods are same with the significant level 001In other words with probability 99 we can say that LI andSI methods have same effectiveness form the perspective oferror for 119869 Similarly the results of Tables 7ndash9 indicate that LI
Mathematical Problems in Engineering 17
Table 3 Descriptive statistics for the sensitivity analysis of1198731199051 for VDP problem
Output 1198731199051
119873 Mean Sd Min Max
119869
11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449
Total 140 15357 00294 15285 18449
120593119891
11 35 281 times 10minus8
952 times 10minus8
1 times 10minus12
452 times 10minus7
21 35 467 times 10minus8
235 times 10minus7
1 times 10minus12
141 times 10minus6
31 35 163 times 10minus8
491 times 10minus8
1 times 10minus12
221 times 10minus7
41 35 817 times 10minus8
285 times 10minus7
1 times 10minus12
126 times 10minus6
Total 140 426 times 10minus8 191 times 10minus7 1 times 10minus12 141 times 10minus6
Time
11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514
Total 140 1142131 410891 456615 2849514
119864119869
11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070
Total 140 00047 00192 0 02070
119870120595
11 35 00031 00045 475 times 10minus9
00204
21 35 00027 00064 550 times 10minus10
00369
31 35 00029 00050 154 times 10minus13
00271
41 35 00111 00397 193 times 10minus10
02070
Total 140 00047 00192 154 times 10minus13 02070
Table 4 Summary statistical data of ANOVA test for the parameters1198731199051 1198731199052 1198731199011 1198731199012 119873119892 119873119894
Parameters 119869 119864119869
120593119891
Time 119870120595
Test statistic (119865)
1198731199051
1294 1296 0624 1079 12961198731199052
110572 393 0868 5939 3931198731199011
1945 1835 1271 4317 18351198731199012
2740 2478 1011 2302 2478119873119892
3541 3591 0491 0890 3591119873119894
0495 0309 0301 5849 1046
119875 value
1198731199051
0280 0279 0601 0 02791198731199052
0 0005 0489 0 00051198731199011
0125 0143 0286 0006 01431198731199012
0021 0034 0413 0 0034119873119892
0009 0008 0743 0472 0008119873119894
0739 0819 0877 0 0386
and SI methods from the perspective of errors for 120593119891 Time
and119870120595 have same behaviour
7 The Impact of SQP
In this section we investigation the impact of the SQP on theproposed algorithm For this purpose we remove the SQPfrom Algorithm 2 as without SQP algorithm For comparingthe numerical results with the previous results the required
running time in each NOCP is considered fixed which is themaximum of running time in 12 different runs were done inSection 5 Also the all of the parameters for each problemwasset as same as Section 5 The numerical results of the withoutSQP algorithm is summarized in Table 10 By comparing theresults of Tables 10 and 1 it is obvious that for all NOCPsin the Appendix the obtained values of the performanceindex and the norm of final state constraint for the proposedalgorithm (Algorithm 2) are more accurate than the withoutSQP algorithm
18 Mathematical Problems in Engineering
Table 5 The mean of numerical results of 15 different runs for NOCPs in the Appendix using LI and SI methods
Problem LI SI119869 120593
119891119870120595
Time 119869 120593119891
119870120595
TimeVDP 15977 157 times 10
minus9
170 times 10minus3 17720 16002 667 times 10
minus8
326 times 10minus3 15757
CRP 00126 307 times 10minus8
307 times 10minus8 19714 00119 161 times 10
minus8
161 times 10minus8 19532
FFPP 4774 805 times 10minus5 03277 133280 5351 592 times 10
minus5 04868 137989MSNIC 01702 mdash mdash 13678 01703 mdash mdash 14092CSTCR 01120 mdash mdash 17385 01120 mdash mdash 17606Number 6 minus54307 mdash mdash 10541 minus54308 mdash mdash 10372Number 7 02015 326 times 10
minus9
695 times 10minus6 20807 02020 313 times 10
minus9
662 times 10minus6 21019
Number 8 630 times 10minus15 mdash mdash 6376 124 times 10
minus14 mdash mdash 5837Number 9 16512 mdash mdash 59239 16512 mdash mdash 58125Number 10 01550 mdash mdash 63153 01550 mdash mdash 65597Number 11 29701 270 times 10
minus9
272 times 10minus7 22860 29701 337 times 10
minus9
811 times 10minus7 23276
Number 12 minus02499 201 times 10minus8
464 times 10minus5 54718 minus02499 115 times 10
minus9
827 times 10minus5 52765
Number 13 00147 115 times 10minus8
00252 57797 00151 139 times 10minus8
00472 66098Number 14 34761 672 times 10
minus6
00237 104585 34290 757 times 10minus6
950 times 10minus3 108767
Number 15 195 times 10minus4
111 times 10minus8
569 times 10minus3 363 194 times 10
minus4
704 times 10minus9
486 times 10minus3 365
Number 16 20571 166 times 10minus10
166 times 10minus10 54391 20571 115 times 10
minus10
115 times 10minus10 54431
Number 17 minus88692 803 times 10minus8
803 times 10minus8 22744 minus88692 661 times 10
minus8
661 times 10minus8 23254
Number 18 00326 mdash mdash 65957 00326 mdash mdash 70897Number 19 011588 655 times 10
minus4 06828 161813 01091 884 times 10minus4 08808 159990
Number 20 5242 00176 05429 84165 4464 453 times 10minus4 03765 86891
Table 6 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 119869
Method Mean rank Sum of ranks Test statistics ValueLI 2050 4100 Mann-Whitney 119880 200SI 2050 4100 Wilcoxon119882 410mdash mdash mdash 119885 0
Table 7 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 120593119891
Method Mean rank Sum of ranks Test statistics ValueLI 1362 177 Mann-Whitney 119880 83SI 1338 174 Wilcoxon119882 174mdash mdash mdash 119885 minus0077
Table 8 Results of Mann-Whitney test on relative errors of the pair (LI SI) for Time
Method Mean rank Sum of ranks Test statistics ValueLI 2025 4050 Mann-Whitney 119880 195SI 2075 4150 Wilcoxon119882 405mdash mdash mdash 119885 minus0135
Table 9 Results of Mann-Whitney test on relative errors of the pair (LI SI) for119870120595
Method Mean rank Sum of ranks Test statistics ValueLI 1354 1760 Mann-Whitney 119880 84SI 1346 1750 Wilcoxon119882 175mdash mdash mdash 119885 minus0026
Mathematical Problems in Engineering 19
Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix
Problem no1 2 3 4 5 6 7 8 9 10
119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048
120601119891
330 times 10minus9
135 times 10minus5 11445 mdash mdash mdash 758 times 10
minus9 mdash mdash mdashProblem no
11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10
minus4 23182 minus90882 00336 2513 20274120601119891
167 times 10minus5
468 times 10minus9 00161 433 times 10
minus4
594 times 10minus3
156 times 10minus5 164 mdash 1706 25283
8 Conclusions
Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters
Appendix
Test Problems
The following problems are presented using notation given in(1)ndash(6)
(1) (VDP) [9] 119892 = (12)(1199092
1+ 1199092
2+ 1199062
) 1199050= 0 119905119891= 5 119891 =
[1199092 minus1199092+(1minus119909
2
1)1199092+119906]119879 1199090= [1 0]
119879120595 = 1199091minus1199092+1
(2) (CRP) [9] 119892 = (12)(1199092
1+1199092
2+01119906
2
) 1199050= 0 119905119891= 078
119891 = [1199091minus2(1199091+025)+(119909
2+05) exp(25119909
1(1199091+2))minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(3) (FFRP) [9] 119892 = (12)(1199062
1+ 1199062
2+ 1199062
3+ 1199062
4) 1199050
=
0 119905119891
= 5 119891 = [1199092 (110)((119906
1+ 1199062) cos119909
5minus
(1199062+ 1199064) sin119909
5) 1199094 (110)((119906
1+ 1199063) sin119909
5+ (1199062+
1199064) cos119909
5) 1199096 (512)((119906
1+ 1199063) minus (119906
2+ 1199064))]119879 1199090=
[0 0 0 0 0 0]119879 120595 = [119909
1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879
(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909
2 minus1199092+
119906 1199092
1+ 1199092
2+ 0005119906
2
]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909
2+
05 minus 8(119905 minus 05)2
]119879 1199090= [0 minus1 0]
119879
(5) (CSTCR) [20] 119892 = 1199092
1+ 1199092
2+ 01119906
2 1199050= 0 119905119891= 078
119891 = [minus(2 + 119906)(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+
2)) 05 minus 1199092minus (1199092+ 05) exp(25119909
1(1199091+ 2))]
119879 1199090=
[009 009]119879
(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909
2 119906]119879 119889 = [minus(2 +
119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]
119879(7) [36] 119892 = 119906
2 1199050= 0 119905
119891= 1 119891 = (12)119909
2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05
(8) [26] 119892 = 1199092cos2119906 119905
0= 0 119905119891= 120587 119891 = sin(1199062) 119909
0=
1205872(9) [26] 119892 = (12)(119909
2
1+ 1199092
2+ 1199062
) 1199050= 0 119905
119891= 5 119891 =
[1199092 minus1199091+ (1 minus 119909
2
1)1199092+ 119906]119879 119889 = minus(119909
2+ 025) 119909
0=
[1 0]119879
(10) [26] 119892 = 1199092
1+ 1199092
2+ 0005119906
2
1199050= 0 119905
119891= 1 119891 =
[1199092 minus1199092+119906]119879
119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus
05) minus 05 minus 1199092)]119879
1199090= [0 minus1]
119879(11) [26] 119892 = (12)119906
2 1199050= 0 119905
119891= 2 119891 = [119909
2 119906]119879 1199090=
[1 1]119879 120595 = [119909
1 1199092]119879
(12) [26] 119892 = minus1199092 1199050= 0 119905
119891= 1 119891 = [119909
2 119906]119879 119889 =
minus(1 minus 119906)(1 + 119906) 1199090= [0 0]
119879 120595 = 1199092
(13) [26] 119892 = (12)(1199092
1+ 1199092
2+ 01119906
2
) 1199050= 0 119905
119891= 078
119891 = [minus2(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+ 2)) minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(14) [26] 119892 = (12)1199062 1199050= 0 119905
119891= 10 119891 = [cos 119906 minus
1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909
0= [366 minus186]
119879120595 = [119909
1 1199092]119879
(15) [26]119892 = (12)(1199092
1+1199092
2) 1199050= 0 119905119891= 078119891 = [minus2(119909
1+
025)+(1199092+05) exp(25119909
1(1199091+2))minus(119909
1+025)119906 05minus
1199092minus(1199092+05) exp(25119909
1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(16) [26] 120601 = 1199093 1199050= 0 119905
119891= 1 119891 = [119909
2 119906 (12)119906
2
]119879
119889 = 1199091minus 19 119909
0= [0 0 0]
119879 120595 = [1199091 1199092+ 1]119879
(17) [26] 120601 = minus1199093 1199050= 0 119905
119891= 5 119891 = [119909
2 minus2 + 119906119909
3
minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909
0= [10 minus2 10]
119879120595 = [119909
1 1199092]119879
(18) [26] 120601 = (1199091minus1)2
+1199092
2+1199092
3 119892 = (12)119906
2 1199050= 0 119905119891= 5
119891 = [1199093cos 119906 119909
3sin 119906 sin 119906]119879 119909
0= [0 0 0]
119879
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
where 120582119896 119896 = 1 2 3 is defined in (9) For 119894 = 1 119898
and 119895 = 1 119873119905 if (of119896)
119894119895gt 119906right(119894) then let (of
119896
)119894119895=
119906right(119894) and if (of119896
)119894119895lt 119906left(119894) then let (of
119896
)119894119895= 119906left(119894)
(3) Let of = oflowast where oflowast is the best of119896 119896 = 1 2 3
constructed by (10)
MutationWe apply a perturbation on each component of theoffspring as follows
(of)119894119895= (of)
119894119895+ 119903119894119895sdot 120572 119894 = 1 2 119898 119895 = 1 2 119873
119905
(11)
where 119903119894119895is selected randomly in minus1 1 and 120572 is a random
number with a uniform distribution in [0 1] If (of)119894119895
gt
119906right(119894) then let (of)119894119895= 119906right(119894) and if (of)
119894119895lt 119906left(119894) then
let (of)119894119895= 119906left(119894)
ReplacementHere in the underling GA we use a traditionalreplacement strategy The replacement is done if the newoffspring has two properties first it is better than the worstperson in the population 119868(of) lt max
1le119894le119873119901119868(119880(119894)
) secondit is not very similar to a person in the population that is foreach 119894 = 1 119873
119901 at least one of the following conditions is
satisfied10038161003816100381610038161003816119868 (of) minus 119868 (119880
(119894)
)
10038161003816100381610038161003816gt 120576
10038171003817100381710038171003817of minus 119880
(119894)10038171003817100381710038171003817gt 120576
(12)
where 120576 is the machine epsilon
Termination Conditions Underlying GA is terminated whenat least one of the following conditions is occurred
(1) Themaximumnumber of generations119873119892 is reached
(2) Over a specified number of generations 119873119894 we do
not have any improvement (the best individual is notchanged) or the two-norm or error of final stateconstraints will reach a small number as the desiredprecision 120576 that is
120593119891=100381710038171003817100381712059510038171003817100381710038172
lt 120576 (13)
where120595 = [1205951 1205952 120595
119899120595]119879 is the vector of final state
constraints defined in (5)
32 SQP The fitness value in (8) can be viewed as anonlinear objective function with the decision variable as119906 = [119906
0 1199061 119906
119873119905minus1] This cost function with upper and
lower bounds of input signals construct a finite dimensionalNLP problem as following
min 119868 (119906) = 119868 (1199060 1199061 119906
119873119905minus1) (14)
st 119906left le 119906119895le 119906right 119895 = 0 1 119873
119905minus 1 (15)
SQP algorithm [22 28] is performed on the NLP (14)-(15)using 1199060 = of and constructed in (11) as the initial solutionwhen the maximum number of iteration is 119904119902119901119898119886119909119894119905119890119903
SQP is an effective and iterative algorithm for thenumerical solution of the constrained NLP problem Thistechnique is based on finding a solution to the system ofnonlinear equations that arise from the first-order necessaryconditions for an extremum of the NLP problem Usingan initial solution of NLP 119906119896 119896 = 0 1 a sequence ofsolutions as 119906119896+1 = 119906
119896
+ 119889119896 is constructed which 119889
119896 is theoptimal solution of the constructed quadratic programming(QP) that approximates NLP in the iteration 119896 based on119906119896 as the search direction in the line search procedure
For the NLP (15) the principal idea is the formulation of aQP subproblem based on a quadratic approximation of theLagrangian function as 119871(119906 120582) = 119868(119906) + 120582
119879
ℎ(119906) where thevector 120582 is Lagrangian multiplier and ℎ(119906) return the vectorof inequality constraints evaluated at 119906 The QP is obtainedby linearizing the nonlinear functions as follows
min 1
2
119889119879
119867(119906119896
) 119889 + nabla119868 (119906119896
)
119879
119889 (16)
nablaℎ (119906119896
)
119879
119889 + ℎ (119906119896
) le 0 (17)
Similar to [26] here a finite difference approximation isapplied to compute the gradient of the cost function and theconstraints with the following components
120597119868
120597119906119895
=
119868 (sdot sdot sdot 119906119895+ 120575 sdot sdot sdot ) minus 119868 (119906)
120575
119895 = 0 1 119873119905minus 1 (18)
where 120575 is the double precision of machine So the gradientvector is nabla119868 = [120597119868120597119906
0 120597119868120597119906
119873119905minus1]119879 Also at each major
iteration a positive definite quasi-Newton approximation oftheHessian of the Lagrangian function119867 is calculated usingthe BFGS method [28] where 120582
119894 119894 = 1 119898 is an estimate
of the Lagrange multipliers The general procedure for SQPfor NLP (14)-(15) is as follows
(1) Given an initial solution 1199060 Let 119896 = 0
(2) Construct the QP subproblem (16)-(17) based on 1199060
using the approximations of the gradient and theHessian of the the Lagrangian function
(3) Compute the new point as 119906119896+1 = 119906119896
+ 119889119896 where 119889119896
is the optimal solution of the current QP(4) Let 119896 = 119896 + 1 and go to step (2)
Here in MHGA SQP is used as the local search and weuse the maximum number of iterations as the main criterionfor stopping SQP In other words we terminate SQP when itconverges either to local solution or themaximumnumber ofSQPrsquos iterations is reached
33 MHGA In MHGA GA uses a local search method toimprove solutions Here we use SQP as a local search UsingSQP as a local search in the hybrid metaheuristic is commonfor example see [20]
MHGA can be seen as a multi start local search whereinitial solutions are constructed by GA From another per-spective MHGA can be seen as a GA that the quality of
Mathematical Problems in Engineering 5
Initialization
sqpmaxiter
an initial population
EvaluationEvaluate the fitnessof each individual
using (8)
Local search(Section 32)
ReturnThe best individual inthe final populationas an approximate
of the global solution of NOCP
Stoppingconditions
Selection(Algorithm 1)
Crossover(using Equation (10))
Replacement(using Equations
(12))
Mutation(using Equation(11))
Local search(Section 32)
Input NtNpNiNg Pm
120576Mi i = 1 2 3 and
=sqpmaxitersqpmaxiter + 1
Figure 1 Flowchart of the MHGA algorithm
its population is intensified by SQP In the beginning ofMHGA a less number of iterations for SQP was used Thenwhen the promising regions of search space were found byGA operators we increase the number of iterations of SQPgradually Using this approach we may decrease the neededrunning time (in [19] the philosophy of this approach isdiscussed)
Finally we give our modified MHGA to find the globalsolution by the flowchart in Figure 1
4 Proposed Algorithm
Here we give a new algorithm which is a direct approachbased on MHGA for solving NOCPs The proposed algo-rithm has two main phases In the first phase we performMHGA with a completely random initial population con-structed by (7) In the first phase to find the promisingregions of the search space in a less running time we usea few numbers of time nodes In addition to have a fasterMHGA the size of the population in the first phase is usuallyless than the size of the population in the second phase
After phase 1 to maintain the property of individuals inthe last population of phase 1 and to increase the accuratelyof solutions we add some additional time nodes Whenthe number of time nodes is increased it is estimated thatthe quality of solution obtained by numerical methods (eg
Runge-Kutta and Simpson) is increased Thus we increasetime nodes from 119873
1199051in phase 1 to 119873
1199052in phase 2 The
corresponding control input values of the new time nodesare added to individuals To use the information of theobtained solutions from phase 1 in the construction of theinitial population of phase 2 we use either Linear or Splineinterpolation to estimate the value of the control inputs inthe new time nodes in each individual of the last populationof phase 1 Moreover to maintain the diversity in the initialpopulation of phase 2 we add new random individuals tothe population using (7) In the second phase MHGA startswith this population and new value of parameters Finally theproposed algorithm is given in Algorithm 2
5 Numerical Experiments
In this section to investigate the efficiency of the proposedalgorithm 20 well-known and real world NOCPs as bench-mark problems are considered which are presented in termsof (1)ndash(6) in the Appendix These NOCPs are selected withsingle control signal and multi control signals which will bedemonstrated in a general manner
The numerical behaviour of algorithms can be studiedfrom two view of points the relative error of the performanceindex and the status of the final state constraints Let 119869 be theobtained performance index by an algorithm 120593
119891 defined in
6 Mathematical Problems in Engineering
Initialization Input the desired precision 120576 in (13) the penalty parameters119872119894 119894 = 1 2 3 in (8) the bounds of control input values in (7) 119906left and 119906right
Phase 1 Perform MHGA with a random population and1198731199051 1198731199011 1198731198941 1198731198921 1198751198981
and 1199041199021199011198981198861199091198941199051198901199031
Construction of the initial population of the phase 2 Increase time nodes uniformly to1198731199052
and estimate the corresponding control input values of the new time nodes in eachindividual obtained from phase 1 using either Linear or Spline interpolationCreate119873
1199012minus 1198731199011new different individuals with119873
1199052time nodes randomly
Phase 2 Perform MHGA with the constructed population and1198731199052 1198731199012 1198731198942 1198731198922 1198751198982
and 1199041199021199011198981198861199091198941199051198901199032
Algorithm 2 The proposed algorithm
(13) be the error of final state constraints and 119869lowast be the best
obtained solution among all implementations or the exactsolution (when exists) Now the relative error of 119869 119864
119869 of the
algorithm can be defined as
119864119869=
10038161003816100381610038161003816100381610038161003816
119869 minus 119869lowast
119869lowast
10038161003816100381610038161003816100381610038161003816
(19)
Tomore accurate study we now define a new criterion calledfactor to compare the algorithms as follows
119870120595= 119864119869+ 120593119891 (20)
Note that 119870120595shows the summation of two important errors
Thus based on 119870120595
we can study the behaviour of thealgorithms on the quality and feasibility of given solutionssimultaneously
To solve any NOCP described in the Appendix wemust know the algorithmrsquos parameters including MHGArsquosparameters including 119873
119905 119873119901 119873119894 119873119892 119875119898and 119904119902119901119898119886119909119894119905119890119903
in both phases in Algorithm 2 and the problemrsquos parametersincluding 120576 in (13) 119872
119894 119894 = 1 2 3 in (8) 119906left and 119906right in
(7) To estimate the best value of the algorithmrsquos parameterswe ran the proposed algorithm with different values ofparameters and then select the best However the sensitivityof MHGA parameters are studied in the next section In bothof phases of Algorithm 2 in MHGA we let 119904119902119901119898119886119909119894119905119890119903
119894= 4
and 119875119898119894
= 08 for 119894 = 1 2 Also we consider 1198731198921
= 1198731198922
1198731198941
= 1198731198942 1198731199011
= 9 and 1198731199012
= 12 The other MHGAparameters are given in the associated subsection and theproblemrsquos parameters in Table 2 For each NOCP 12 differentruns were done and the best results are reported in Table 1which the best value of each column is seen in the bold
The reported numerical results of the proposed algo-rithm for each NOCP include the value of performanceindex 119869 the relative error of 119869119864
119869 defined in (19) the required
computational time Time the norm of final state constraints120593119891 defined in (13) and the factor 119870
120595 defined in (20)
The algorithm was implemented in Matlab R2011a envi-ronment on a Notebook with Windows 7 Ultimate CPU253GHz and 400GB RAM Also to implement SQP in ourproposed algorithm we used ldquofminconrdquo in Matlab when theldquoAlgorithmrdquo was set to ldquoSQPrdquo Moreover we use compositeSimpsonrsquos method [29] to approximate integrations
Remark 1 We use the following abbreviations to show theused interpolation method in our proposed algorithm
(1) LI linear interpolation(2) SI spline interpolation
For comparing the numerical results of the proposedalgorithm two subsections are considered comparison withsome metaheuristic algorithms in Section 51 and com-parison with some numerical methods in Section 52 Wegive more details of these comparisons in the followingsubsections
51 Comparison with Metaheuristic Algorithms The numeri-cal results for the NOCPs numbers (1)ndash(3) in the Appendixare compared with a continuous GA CGA as a meta-heuristic proposed in [9] which gave better solutions thanshooting method and gradient algorithm from the indirectmethods category [2 30] and SUMT from the directmethodscategory [26] For NOCPs numbers (4) and (5) the resultsare compared with another metaheuristic which is a hybridimproved PSO called IPSO proposed in [20]
511 VDP Problem [9] The first NOCP in the Appendix isVanDer Pol Problem VDP which has two state variables andone control variable VDPproblemhas a final state constraintwhich is 120595 = 119909
1(119905119891) minus 1199092(119905119891) + 1 = 0 The results of the
proposed algorithm with the MHGArsquos parameters as 1198731199051
=
311198731199052= 71119873
119892= 300 and119873
119894= 200 are reported in Table 1
FromTable 1 it is obvious that the numerical results of LI andSI methods are more accurate than CGA with less amount of119870120595
512 CRP Problem [9] The second NOCP in the Appendixis Chemical Reactor Problem CRP which has two statevariables and one control variableThe results of the proposedalgorithm with the MHGArsquos parameters as 119873
1199051= 31 119873
1199052=
71 119873119892
= 300 and 119873119894= 200 are shown in the second
row of Table 1 CRP problem has two final state constraints120595 = [119909
1 1199092]119879 Although from Table 1 the norm of final state
constraints 120593119891 for the CGA equals 120593lowast
119891= 757 times 10
minus10 isless than 120593
119891rsquos of LI and SI methods which equals 115 times 10
minus9
and 599 times 10minus9 respectively but the performance index
Mathematical Problems in Engineering 7
Table 1 The best numerical results in 12 different runs of the SI and LI methods for NOCPs in the Appendix
Problem Algorithm 119869 119864119869
120593119891
119870120595
Time
VDPCGA 17404 00912 267 times 10
minus11
00912 50128LI 15949 0 108 times 10
minus13 108 times 10minus13 28282SI 15950 627 times 10
minus5 999 times 10minus14 627 times 10minus5 23914
CRPCGA 163 times 10
minus2 02835 757 times 10minus10 02835 50128LI 127 times 10minus2 0 115 times 10
minus9 115 times 10minus9 17194
SI 127 times 10minus2 0 599 times 10minus9
599 times 10minus9
12405
FFRPCGA 8363 13256 465 times 10
minus3
13302 1413LI 3596 0 122 times 10
minus5 122 times 10minus5 126858
SI 3599 834 times 10minus4 901 times 10minus6 843 times 10
minus4 134340
MSNICIPSO 01727 00165 mdash mdash 12626LI 01699 0 mdash mdash 13882
SI 01700 589 times 10minus4 mdash mdash 13868
CSTCRIPSO 01355 02098 mdash mdash 3134
LI 01120 0 mdash mdash 10358
SI 01120 0 mdash mdash 10673
No 6Bezier minus53898 00251 mdash mdash NRa
LI minus54309 00177 mdash mdash 13838
SI minus54309 00177 mdash mdash 10734
Number 7HPM 02353 01677 420 times 10
minus6 01677 NRLI 02015 0 235 times 10
minus9
235 times 10minus9 21392
SI 02015 0 282 times 10minus10 282 times 10minus10 20696
Number 8
SQP 636 times 10minus6
220 times 109 mdash mdash NR
SUMT 515 times 10minus6
178 times 109 mdash mdash NR
LI 289 times 10minus15 0 mdash mdash 7282SI 364 times 10
minus15
02595 mdash mdash 6868
Number 9
SQP 17950 00873 mdash mdash NRSUMT 17980 00891 mdash mdash NRLI 16509 0 mdash mdash 63910SI 16509 0 mdash mdash 5950
Number 10
SQP 02163 03964 mdash mdash NRSUMT 01703 00994 mdash mdash NRLI 01549 0 mdash mdash 62570
SI 01549 0 mdash mdash 67662
Number 11
SQP 325 01648 0 01648 NRSUMT 325 01648 0 01648 NRLI 27901 0 162 times 10
minus9
162 times 10minus9 14778
SI 27901 0 586 times 10minus10 586 times 10minus10 21946
Number 12
SQP minus02490 40 times 10minus3 0 40 times 10
minus3 NRSUMT minus02490 40 times 10
minus3 0 40 times 10minus3 NR
LI minus02500 0 285 times 10minus8
285 times 10minus8
54771
SI minus02500 0 390 times 10minus10 390 times 10minus10 54548
Number 13
SQP 168 times 10minus2 01748 0 01748 NR
SUMT 167 times 10minus2 01678 0 01678 NR
LI 143 times 10minus2 0 118 times 10minus9 118 times 10minus9 42599SI 144 times 10
minus2
70 times 10minus3
632 times 10minus9
70 times 10minus3
49027
Number 14
SQP 37220 00961 0 00961 NRSUMT 37700 01103 0 01103 NRLI 33956 0 786 times 10minus7 786 times 10minus7 104124SI 33965 265 times 10
minus4
276 times 10minus6
267 times 10minus4
115144
8 Mathematical Problems in Engineering
Table 1 Continued
Problem Algorithm 119869 119864119869
120593119891
119870120595
Time
Number 15
SQP 103 times 10minus3 43368 0 43368 NR
SUMT 929 times 10minus4 38135 0 38135 NR
LI 193 times 10minus4 0 810 times 10minus10 810 times 10minus10 33354SI 194 times 10
minus4
520 times 10minus3
164 times 10minus9
520 times 10minus3
34014
Number 16
SQP 22120 00753 0 00753 NRSUMT 22080 00734 0 00734 NRLI 20571 0 394 times 10
minus12
394 times 10minus12
46982
SI 20571 0 560 times 10minus13 560 times 10minus13 70423
Number 17
SQP minus88690 225 times 10minus5 0 225 times 10
minus5 NRSUMT minus88690 225 times 10
minus5 0 225 times 10minus5 NR
LI minus88692 0 145 times 10minus10 145 times 10minus10 32183SI minus88692 0 279 times 10
minus10
279 times 10minus10
30959
Number 18
SQP 00368 01288 mdash mdash NRSUMT 00386 01840 mdash mdash NRLI 00326 0 mdash mdash 55163
SI 00326 0 mdash mdash 60671
Number 19
SQP 03439 49293 0 49293 NRSUMT 03428 49103 0 49103 NRLI 00689 01879 410 times 10
minus3
01919 164509
SI 00580 0 405 times 10minus4 405 times 10minus4 178783
Number 20
SQP 7752 12554 0 12554 NRSUMT 7683 12353 0 12353 NRLI 343716 232 times 10
minus5
225 times 10minus4
248 times 10minus4 81519
SI 343708 0 157 times 10minus4 157 times 10minus4 80420
aNot reported
the relative error of 119869 and the factor of the proposed algorithmis better and so the proposed algorithm is more robust thanCGA
513 FFRP Problem [9] The third NOCP in the Appendixis Free Floating Robot Problem FFRP which has six statevariables and four control variables FFRP has been solvedby CGA and the proposed algorithm with the MHGArsquosparameters as 119873
1199051= 31 119873
1199052= 71 119873
119892= 300 and
119873119894
= 200 This problem has six final state constraints120595 = [119909
1minus 4 119909
2 1199093minus 4 119909
4 1199095 1199096]119879 The numerical results
are shown in Table 1 The values of 119869 119864119869 120593119891and 119870
120595for LI
and SI methods separately are less than CGA Thereforethe proposed algorithm can achieve much better qualitysolutions than the CGA with reasonable computational time
514 MSNIC Problem [20] For the forth NOCP in theAppendix which is a Mathematical System with NonlinearInequality Constraint NSNIC the numerical results are com-pared with IPSO MSNIC contains an inequality constraint119889(119909 119905) = 119909
2(119905) + 05 minus 8(119905 minus 05)
2
le 0 The problem solvedby several numerical methods as [24 31] From [20] IPSOmethod could achieved more accurate results than men-tioned numericalmethods AlsoMSNIC can be solved by theproposed algorithm with the MHGArsquos parameters as 119873
1199051=
31 1198731199052
= 91 119873119892= 100 and 119873
119894= 60 From the forth row
of Table 1 the absolute error of 119869 119864119869 for LI and SI methods
equal 0 and 589 times 10minus4 respectively which are less than
IPSOrsquos 00165Subplots (a) and (b) in Figure 2 show the graphs of
the convergence rate for the performance index and theinequality constraint respect to the number of iterationrespectively
515 CSTCR Problem [20] The fifth NOCP in the Appendixis a model of a nonlinear Continuous Stirred-tank ChemicalReactor CSTCR It has two state variables 119909
1(119905) and 119909
2(119905)
as the deviation from the steady-state temperature and con-centration and one control variable 119906(119905) which represent theeffect of the flow rate of cooling fluid on chemical reactorTheobjective is to maintain the temperature and concentrationclose to steady-state values without expending large amountof control effort Also this is a benchmark problem in thehandbook of test problems in local and global optimization[32] which is a multimodal optimal control problem [33]It involves two different local minima The values of theperformance indexes for these solutions equal 0244 and0133 Similarly to the MSNIC the numerical results of theproposed algorithm with the MHGArsquos parameters as 119873
1199051=
31 1198731199052
= 51 119873119892= 100 and 119873
119894= 50 are compared with
IPSO From Table 1 the performance index 119869 for LI and SImethods is equal to 119869
lowast
= 01120 which is less than IPSOs01355
Mathematical Problems in Engineering 9
Table 2 The problem parameters for NOCPs in the Appendix
Parameters Problem number1 2 3 4 5 6 7 8 9 10
119906left minus05 minus15 minus15 minus20 0 minus2 0 minus2 minus1 minus20
119906right 2 2 10 20 5 2 1 2 1 20119872119894
103
102 70 1 mdash 1 mdash 10
2 mdash 102
120576 10minus12
10minus10
10minus3 mdash mdash mdash mdash mdash mdash 10
minus9
Parameters Problem no11 12 13 14 15 16 17 18 19 20
119906left minus5 minus1 minus2 minus120587 minus1 minus3 minus30 minus1 minus15 minus15
119906right 5 1 2 120587 1 3 30 1 10 10119872119894
10 102 mdash 10
2
102
102 mdash 10 70 10
minus3
120576 10minus9
10minus11
10minus11 mdash 10
minus10
10minus10
10minus10 mdash 10
minus3
10minus4
0 5 10 15 20 25 30 35 40 45 50017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 2 Graphical results of MSNIC problem using SI method (a) convergence rate of the performance index and (b) the inequalityconstraint 119889(119909 119905) le 0 in respect to the number of iterations
52 Comparison with Numerical Methods For NOCPs num-bers (6)ndash(20) the comparison are done with some numericalmethods Unfortunately for these methods usually the finalstate constraints and the required computational time arenot reported which are shown with NR in Table 1 but thesevalues are reported for both LI and SI methods in Table 1For all NOCPs in this section the MHGArsquos parameters areconsidered as 119873
1199051= 31 119873
1199052= 51 119873
119892= 100 119873
119894= 50 with
the problem parameters in Table 2
521 Compared with Bezier [34] The NOCP number (6) inthe Appendix has exact solution which has an inequalityconstraint as 119889(119909
1 119905) = minus6 minus 119909
1(119905) le 0 The exact value of
performance index equals 119869lowast = minus55285 [35] This problemhas been solved by a numerical method proposed in [34]called Bezier From sixth row of Table 1 the absolute error
of the LI and SI methods equal 00177 which is less thanBezierrsquos 00251
Figure 3 shows the graphs of the convergence rate of theperformance index subplot (a) and inequality constraintsubplot (b) respect to the number of iteration using SImethod
522 Compared with HPM [36] For NOCP number (7) inthe Appendix which is a constraint nonlinear model thenumerical results of the proposed algorithm are comparedwith HPM proposed in [36] This NOCP has a final stateconstraint as 120595 = 119909 minus 05 = 0 From [36] the norm offinal state constraint for HPM equals 42times10minus6 however thiscriterion for the LI and SI methods equals 235 times 10
minus9 and120593lowast
119891= 282 times 10
minus10 respectively From Table 1 it is obviousthat the obtained values of the performance index the norm
10 Mathematical Problems in Engineering
2 4 6 8 10 12 14minus5353
minus5352
minus5351
minus535
minus5349
minus5348
minus5347
minus5346
minus5345
minus5344
minus5343
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 3 Graphical results for NOCP number (6) using SI method (a) convergence rate for the performance index and (b) the inequalityconstraint 119889(119909
1 119905) le 0 in respect to the number of iterations
of final state constraint and 119870120595for the SI method are more
accurate than LI than HPMmethods
523 Compared with SQP and SUMT [26] For the NOCPsnumbers (8)ndash(20) in the Appendix the numerical resultsof LI and SI methods are compared with two numericalmethods contain SQP and SUMT proposed in [26] Amongthese NOCPs only two problems numbers (8) and (18) areunconstrained and the others have at least one constraintfinal state constraint or inequality constraint All of theseNOCPs solved by the proposed algorithm with the problemparameters in Table 2 and their results are summarized inTable 1 Because the final state constraints in these methodsare not reported we let 120593
119891= 0 to calculate the factor119870
120595
Figures 4ndash16 show the graphical results of NOCPsnumbers (8)ndash(20) in the Appendix using SI method Forunconstrained NOCPs numbers (8) and (18) only the graphof convergent rate of performance index is shown see Figures4 and 14 For constraint NOCPs and NOCPs numbers(11)ndash(17) and numbers (19)-(20) with final state constraintthe graphs of convergent rates of performance index and theerror of final state constraint are shown see Figures 7ndash13 and15-16 For the constraint NOCPs with inequality constraintsNOCPs numbers (9) and (10) the graphs of convergent rateof performance index and inequality constraint are shownsee Figures 5 and 6
Table 1 shows that the proposed algorithm LI and SImethods was 100 percent successful in point of views theperformance index 119869 and the factor119870
120595 numerically So the
proposed algorithm provides robust solutions with respect tothe other mentioned numerical or metaheuristic methodsTo compare 119869 in LI and SI methods in 35 percent of NOCPsLI is more accurate than SI in 10 percent SI is more accurate
1 15 2 25 3 35 40
002
004
006
008
01
012
Iter
Perfo
rman
ce in
dex
Figure 4 Convergence rate for the performance index of SI methodfor NOCP number (8)
than LI and in 55 percent are same In point of view the finalconditions 65 percent of NOCPs in the Appendix have finalstate constraints In all of them 120593
119891in the proposed algorithm
is improved except CRP problem however the factor of theproposed algorithm is the best yet In 54 percent of theseNOCPs LI is better performance and in 46 percent SI isbetter Also in point of the running time Time LI is same asSI that is in 50 percent of NOCPs LI and in 50 percent SI isbetter than another So the proposed algorithmcould providevery suitable solutions in a reasonable computational timeAlso for more accurate comparison of LI and SI methods astatistical approach will done in next section
To compare with CGA the mean of relative error of 119869 119864119869
for CGA LI and SImethods inNOCPs (1)ndash(3) equal 056680 and 298 times 10
minus4 respectively Also the mean for the error
Mathematical Problems in Engineering 11
5 10 15 20 25165
17
175
18
185
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3 35 4 45 50
005
01
015
02
025
TimeIn
equa
lity
cons
trai
nt(b)
Figure 5 (a) Convergence rate for the performance index and (b) the inequality constraint of SI method for NOCP number (9)
5 10 15 20 25 30 35 40 45 500155
016
0165
017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 6 (a) Convergence rate of the performance index and (b) the inequality constraint for SI method for NOCP number (10)
of the final state constraints 120593119891 are 16 times 10
minus3 407 times 10minus6
and 301 times 10minus6 respectively and for 119870
120595 these values equal
05883 407 times 10minus6 and 302 times 10
minus4 Thus we can say that thefeasibility of the solutions given by the proposed algorithmand CGA are competitive
From Table 1 to compare with numerical methods SQPand SUMT in NOCPs (8)ndash(20) the mean of 119864
119869for LI SI
SQP and SUMT equals 00145 00209 169 times 108 and 136 times
108 respectively Also the mean for the error of final state
constraints for these NOCPs equal 334 times 10minus4 441 times 10
minus50 and 0 respectively For 119870
120595 these values are 00213 00014
12263 and 11644 Therefore the performance index 119869 andthe factor 119870
120595 for the LI and SI methods are more accurate
than SQP and SUMT So the proposed algorithm gave morebetter solution in comparison with the numerical methods
12 Mathematical Problems in Engineering
5 10 15 20 25minus025
minus0245
minus024
minus0235
minus023
minus0225
Iter
Perfo
rman
ce in
dex
(a)
1 15 2 25 3 35 4 45 5 55 60
02
04
06
08
1
12
14
16
IterFi
nal c
ondi
tion
times10minus5
(b)
Figure 7 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (11)
2 4 6 8 10 12 14minus025
minus0245
minus024
minus0235
minus023
minus0225
minus022
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
02
04
06
08
1
12
14
16
18
2
Iter
Fina
l con
ditio
n
times10minus5
(b)
Figure 8 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (12)
Therefore based on this numerical studywe can concludethat the proposed algorithm outperform well-known numer-ical method Since the algorithms were not implementedon the same PC the computational times of them are notcompetitive Therefore we did not give the computationaltimes in bold in Table 1
6 Sensitivity Analysis andComparing LI and SI
In this section two statistical analysis based on the one-way analysis of variance (ANOVA) used for investigatingthe sensitivity of MHGA parameters and Mann-Whitney
Mathematical Problems in Engineering 13
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
07
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 9 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (13)
5 10 15 20 25 30
35
4
45
5
55
6
65
7
75
8
85
9
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
Iter
Fina
l con
ditio
n
(b)
Figure 10 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (14)
applied comparing LI and SI methods are done by thestatistical software IBM SPSS version 21
61 Sensitivity Analysis In order to survey the sensitivityof the MHGA parameters from the proposed algorithmthe VDP problem is selected for example The influenceof these parameters are investigated for this NOCP on
the dependent outputs consist of the performance index119869 the relative error of 119869 119864
119869 the required computational
time Time the error of final state constraints 120593119891and the
factor 119870120595 The independent parameters are consist of the
number of time nodes in both two phases 1198731199051and 119873
1199052 the
size of population in both two phases 1198731199011
and 1198731199012 the
maximum number of generations without improvement119873119894
14 Mathematical Problems in Engineering
5 10 15 20 25 30 35 40 45 50
00955
0096
00965
0097
00975
0098
00985
0099
00995
Iter
Perfo
rman
ce in
dex
(a)
2 4 6 8 10 12 14 16 18 200
05
1
15
2
25
3
35
4
45
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 11 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (15)
5 10 15 20 25
21
22
23
24
25
26
27
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
1
2
3
4
5
6
Iter
Fina
l con
ditio
n
times10minus3
(b)
Figure 12 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (16)
and the maximum number of generations 119873119892 Because the
mutation implementation probability119875119898 has a less influence
in numerical results it is not considered Also 119904119902119901119898119886119909119894119905119890119903 ischanged in each iteration of the proposed algorithm so it isnot considered too
At first we selected at least four constant value foreach of parameters and then in each pose 35 differentruns were made independently The statistical analysis isdone based on ANOVA The descriptive statistics whichcontains the number of different runs (119873) the mean of each
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7minus887
minus886
minus885
minus884
minus883
minus882
minus881
minus88
minus879
minus878
minus877
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 70
01
02
03
04
05
06
07
08
09
1
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 13 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (17)
1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333
Iter
Perfo
rman
ce in
dex
Figure 14 Convergence rate for the performance index of SImethod for NOCP numbers (18)
output in119873 different runs (Mean) standard deviation (Sd)maximum (Max) and minimum (Min) of each outputcould be achieved Among of the MHGA parameters wepresent the descriptive statistics of the ANOVA only for the1198731199051parameter which is reported in Table 3Table 4 summarized the statistical data contain the test
statistics (119865) and 119875-values of ANOVA tests Sensitivityanalysis for each parameter separately are done based on thistable as follows
1198731199051 From Table 4 the significant level or 119875-value for119870120595
is equal to 0279 which is greater than 005So from ANOVA 119873
1199051parameter has no significant
effect on the factor 119870120595 With similar analysis this
parameter has no effect on the other outputs except
required computational time Time because its 119875-value is equal to 0 which is less than 005 that is thisoutput sensitive to the parameter119873
1199051
1198731199052 From the second row of Table 4 the outputs 119869 119864
119869119870120595
and Time are sensitive to the parameter 1198731199052 and the
only output 120593119891is independent
1198731199011 From the third row of Table 4 only the computationalrequired time Time is sensitive because the its 119875-value is equal to 0006 which is less than 005 and theother outputs contain 119869 119864
119869 120593119891119870120595 are independent
1198731199012 From the forth row of Table 4 the output 120593
119891is
independent respect to the parameter 1198731199012 but other
outputs contain 119869 119864119869 Time 119870
120601are sensitive to this
parameter119873119892 From the fifth row of Table 4 the outputs 119869 119864
119869and
119870120595are independent to the parameter 119873
119892 and other
outputs contain 120593119891and Time are sensitive respect to
this parameter119873119894 The sensitivity analysis is similar to119873
1199011
From above cases it is obvious that all parameters canbe effect on the required computational time except 119873
119892
Moreover intuitions shows the normof final state constraints120593119891is independent output with respect to all parameters that
is any of the parameters could not effect on this output and itis not sensitive with respect to any of the MHGA parameters
62 Comparison of LI and SI To compare the efficiency ofthe LI and SI methods for NOCPs in the Appendix weused the Mann-Whitney nonparametric statistical test [37]
16 Mathematical Problems in Engineering
1 15 2 25 3 35 40
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
200
400
600
800
1000
1200
1400
1600
IterFi
nal c
ondi
tion
(b)
Figure 15 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (19)
2 4 6 8 10 12 14 16 18 20
50
100
150
200
250
300
350
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 10 11 120
05
1
15
2
25
3
Iter
Fina
l con
ditio
n
(b)
Figure 16 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (20)
Using the nonparametric statistical tests for comparing theperformance of several algorithms presented in [38] Forthis purpose At first 15 different runs for each of NOCPwere made Table 5 shows the mean of numerical resultsseparately Here we apply the fixMHGAparameters as119873
1199011=
91198731199012
= 121198731199051= 31119873
1199052= 91119873
119892= 100 and119873
119894= 50 with
the previous problem parameters in Table 2 Comparison
criterions contain 119869 120593119891 119870120595and Time The results of Mann-
Whitney test are shown in Tables 6ndash9 From Table 6 since119885 = 0 gt 119885
001= minus234 then the average relative error of
the LI and SI methods are same with the significant level 001In other words with probability 99 we can say that LI andSI methods have same effectiveness form the perspective oferror for 119869 Similarly the results of Tables 7ndash9 indicate that LI
Mathematical Problems in Engineering 17
Table 3 Descriptive statistics for the sensitivity analysis of1198731199051 for VDP problem
Output 1198731199051
119873 Mean Sd Min Max
119869
11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449
Total 140 15357 00294 15285 18449
120593119891
11 35 281 times 10minus8
952 times 10minus8
1 times 10minus12
452 times 10minus7
21 35 467 times 10minus8
235 times 10minus7
1 times 10minus12
141 times 10minus6
31 35 163 times 10minus8
491 times 10minus8
1 times 10minus12
221 times 10minus7
41 35 817 times 10minus8
285 times 10minus7
1 times 10minus12
126 times 10minus6
Total 140 426 times 10minus8 191 times 10minus7 1 times 10minus12 141 times 10minus6
Time
11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514
Total 140 1142131 410891 456615 2849514
119864119869
11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070
Total 140 00047 00192 0 02070
119870120595
11 35 00031 00045 475 times 10minus9
00204
21 35 00027 00064 550 times 10minus10
00369
31 35 00029 00050 154 times 10minus13
00271
41 35 00111 00397 193 times 10minus10
02070
Total 140 00047 00192 154 times 10minus13 02070
Table 4 Summary statistical data of ANOVA test for the parameters1198731199051 1198731199052 1198731199011 1198731199012 119873119892 119873119894
Parameters 119869 119864119869
120593119891
Time 119870120595
Test statistic (119865)
1198731199051
1294 1296 0624 1079 12961198731199052
110572 393 0868 5939 3931198731199011
1945 1835 1271 4317 18351198731199012
2740 2478 1011 2302 2478119873119892
3541 3591 0491 0890 3591119873119894
0495 0309 0301 5849 1046
119875 value
1198731199051
0280 0279 0601 0 02791198731199052
0 0005 0489 0 00051198731199011
0125 0143 0286 0006 01431198731199012
0021 0034 0413 0 0034119873119892
0009 0008 0743 0472 0008119873119894
0739 0819 0877 0 0386
and SI methods from the perspective of errors for 120593119891 Time
and119870120595 have same behaviour
7 The Impact of SQP
In this section we investigation the impact of the SQP on theproposed algorithm For this purpose we remove the SQPfrom Algorithm 2 as without SQP algorithm For comparingthe numerical results with the previous results the required
running time in each NOCP is considered fixed which is themaximum of running time in 12 different runs were done inSection 5 Also the all of the parameters for each problemwasset as same as Section 5 The numerical results of the withoutSQP algorithm is summarized in Table 10 By comparing theresults of Tables 10 and 1 it is obvious that for all NOCPsin the Appendix the obtained values of the performanceindex and the norm of final state constraint for the proposedalgorithm (Algorithm 2) are more accurate than the withoutSQP algorithm
18 Mathematical Problems in Engineering
Table 5 The mean of numerical results of 15 different runs for NOCPs in the Appendix using LI and SI methods
Problem LI SI119869 120593
119891119870120595
Time 119869 120593119891
119870120595
TimeVDP 15977 157 times 10
minus9
170 times 10minus3 17720 16002 667 times 10
minus8
326 times 10minus3 15757
CRP 00126 307 times 10minus8
307 times 10minus8 19714 00119 161 times 10
minus8
161 times 10minus8 19532
FFPP 4774 805 times 10minus5 03277 133280 5351 592 times 10
minus5 04868 137989MSNIC 01702 mdash mdash 13678 01703 mdash mdash 14092CSTCR 01120 mdash mdash 17385 01120 mdash mdash 17606Number 6 minus54307 mdash mdash 10541 minus54308 mdash mdash 10372Number 7 02015 326 times 10
minus9
695 times 10minus6 20807 02020 313 times 10
minus9
662 times 10minus6 21019
Number 8 630 times 10minus15 mdash mdash 6376 124 times 10
minus14 mdash mdash 5837Number 9 16512 mdash mdash 59239 16512 mdash mdash 58125Number 10 01550 mdash mdash 63153 01550 mdash mdash 65597Number 11 29701 270 times 10
minus9
272 times 10minus7 22860 29701 337 times 10
minus9
811 times 10minus7 23276
Number 12 minus02499 201 times 10minus8
464 times 10minus5 54718 minus02499 115 times 10
minus9
827 times 10minus5 52765
Number 13 00147 115 times 10minus8
00252 57797 00151 139 times 10minus8
00472 66098Number 14 34761 672 times 10
minus6
00237 104585 34290 757 times 10minus6
950 times 10minus3 108767
Number 15 195 times 10minus4
111 times 10minus8
569 times 10minus3 363 194 times 10
minus4
704 times 10minus9
486 times 10minus3 365
Number 16 20571 166 times 10minus10
166 times 10minus10 54391 20571 115 times 10
minus10
115 times 10minus10 54431
Number 17 minus88692 803 times 10minus8
803 times 10minus8 22744 minus88692 661 times 10
minus8
661 times 10minus8 23254
Number 18 00326 mdash mdash 65957 00326 mdash mdash 70897Number 19 011588 655 times 10
minus4 06828 161813 01091 884 times 10minus4 08808 159990
Number 20 5242 00176 05429 84165 4464 453 times 10minus4 03765 86891
Table 6 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 119869
Method Mean rank Sum of ranks Test statistics ValueLI 2050 4100 Mann-Whitney 119880 200SI 2050 4100 Wilcoxon119882 410mdash mdash mdash 119885 0
Table 7 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 120593119891
Method Mean rank Sum of ranks Test statistics ValueLI 1362 177 Mann-Whitney 119880 83SI 1338 174 Wilcoxon119882 174mdash mdash mdash 119885 minus0077
Table 8 Results of Mann-Whitney test on relative errors of the pair (LI SI) for Time
Method Mean rank Sum of ranks Test statistics ValueLI 2025 4050 Mann-Whitney 119880 195SI 2075 4150 Wilcoxon119882 405mdash mdash mdash 119885 minus0135
Table 9 Results of Mann-Whitney test on relative errors of the pair (LI SI) for119870120595
Method Mean rank Sum of ranks Test statistics ValueLI 1354 1760 Mann-Whitney 119880 84SI 1346 1750 Wilcoxon119882 175mdash mdash mdash 119885 minus0026
Mathematical Problems in Engineering 19
Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix
Problem no1 2 3 4 5 6 7 8 9 10
119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048
120601119891
330 times 10minus9
135 times 10minus5 11445 mdash mdash mdash 758 times 10
minus9 mdash mdash mdashProblem no
11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10
minus4 23182 minus90882 00336 2513 20274120601119891
167 times 10minus5
468 times 10minus9 00161 433 times 10
minus4
594 times 10minus3
156 times 10minus5 164 mdash 1706 25283
8 Conclusions
Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters
Appendix
Test Problems
The following problems are presented using notation given in(1)ndash(6)
(1) (VDP) [9] 119892 = (12)(1199092
1+ 1199092
2+ 1199062
) 1199050= 0 119905119891= 5 119891 =
[1199092 minus1199092+(1minus119909
2
1)1199092+119906]119879 1199090= [1 0]
119879120595 = 1199091minus1199092+1
(2) (CRP) [9] 119892 = (12)(1199092
1+1199092
2+01119906
2
) 1199050= 0 119905119891= 078
119891 = [1199091minus2(1199091+025)+(119909
2+05) exp(25119909
1(1199091+2))minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(3) (FFRP) [9] 119892 = (12)(1199062
1+ 1199062
2+ 1199062
3+ 1199062
4) 1199050
=
0 119905119891
= 5 119891 = [1199092 (110)((119906
1+ 1199062) cos119909
5minus
(1199062+ 1199064) sin119909
5) 1199094 (110)((119906
1+ 1199063) sin119909
5+ (1199062+
1199064) cos119909
5) 1199096 (512)((119906
1+ 1199063) minus (119906
2+ 1199064))]119879 1199090=
[0 0 0 0 0 0]119879 120595 = [119909
1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879
(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909
2 minus1199092+
119906 1199092
1+ 1199092
2+ 0005119906
2
]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909
2+
05 minus 8(119905 minus 05)2
]119879 1199090= [0 minus1 0]
119879
(5) (CSTCR) [20] 119892 = 1199092
1+ 1199092
2+ 01119906
2 1199050= 0 119905119891= 078
119891 = [minus(2 + 119906)(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+
2)) 05 minus 1199092minus (1199092+ 05) exp(25119909
1(1199091+ 2))]
119879 1199090=
[009 009]119879
(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909
2 119906]119879 119889 = [minus(2 +
119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]
119879(7) [36] 119892 = 119906
2 1199050= 0 119905
119891= 1 119891 = (12)119909
2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05
(8) [26] 119892 = 1199092cos2119906 119905
0= 0 119905119891= 120587 119891 = sin(1199062) 119909
0=
1205872(9) [26] 119892 = (12)(119909
2
1+ 1199092
2+ 1199062
) 1199050= 0 119905
119891= 5 119891 =
[1199092 minus1199091+ (1 minus 119909
2
1)1199092+ 119906]119879 119889 = minus(119909
2+ 025) 119909
0=
[1 0]119879
(10) [26] 119892 = 1199092
1+ 1199092
2+ 0005119906
2
1199050= 0 119905
119891= 1 119891 =
[1199092 minus1199092+119906]119879
119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus
05) minus 05 minus 1199092)]119879
1199090= [0 minus1]
119879(11) [26] 119892 = (12)119906
2 1199050= 0 119905
119891= 2 119891 = [119909
2 119906]119879 1199090=
[1 1]119879 120595 = [119909
1 1199092]119879
(12) [26] 119892 = minus1199092 1199050= 0 119905
119891= 1 119891 = [119909
2 119906]119879 119889 =
minus(1 minus 119906)(1 + 119906) 1199090= [0 0]
119879 120595 = 1199092
(13) [26] 119892 = (12)(1199092
1+ 1199092
2+ 01119906
2
) 1199050= 0 119905
119891= 078
119891 = [minus2(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+ 2)) minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(14) [26] 119892 = (12)1199062 1199050= 0 119905
119891= 10 119891 = [cos 119906 minus
1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909
0= [366 minus186]
119879120595 = [119909
1 1199092]119879
(15) [26]119892 = (12)(1199092
1+1199092
2) 1199050= 0 119905119891= 078119891 = [minus2(119909
1+
025)+(1199092+05) exp(25119909
1(1199091+2))minus(119909
1+025)119906 05minus
1199092minus(1199092+05) exp(25119909
1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(16) [26] 120601 = 1199093 1199050= 0 119905
119891= 1 119891 = [119909
2 119906 (12)119906
2
]119879
119889 = 1199091minus 19 119909
0= [0 0 0]
119879 120595 = [1199091 1199092+ 1]119879
(17) [26] 120601 = minus1199093 1199050= 0 119905
119891= 5 119891 = [119909
2 minus2 + 119906119909
3
minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909
0= [10 minus2 10]
119879120595 = [119909
1 1199092]119879
(18) [26] 120601 = (1199091minus1)2
+1199092
2+1199092
3 119892 = (12)119906
2 1199050= 0 119905119891= 5
119891 = [1199093cos 119906 119909
3sin 119906 sin 119906]119879 119909
0= [0 0 0]
119879
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Initialization
sqpmaxiter
an initial population
EvaluationEvaluate the fitnessof each individual
using (8)
Local search(Section 32)
ReturnThe best individual inthe final populationas an approximate
of the global solution of NOCP
Stoppingconditions
Selection(Algorithm 1)
Crossover(using Equation (10))
Replacement(using Equations
(12))
Mutation(using Equation(11))
Local search(Section 32)
Input NtNpNiNg Pm
120576Mi i = 1 2 3 and
=sqpmaxitersqpmaxiter + 1
Figure 1 Flowchart of the MHGA algorithm
its population is intensified by SQP In the beginning ofMHGA a less number of iterations for SQP was used Thenwhen the promising regions of search space were found byGA operators we increase the number of iterations of SQPgradually Using this approach we may decrease the neededrunning time (in [19] the philosophy of this approach isdiscussed)
Finally we give our modified MHGA to find the globalsolution by the flowchart in Figure 1
4 Proposed Algorithm
Here we give a new algorithm which is a direct approachbased on MHGA for solving NOCPs The proposed algo-rithm has two main phases In the first phase we performMHGA with a completely random initial population con-structed by (7) In the first phase to find the promisingregions of the search space in a less running time we usea few numbers of time nodes In addition to have a fasterMHGA the size of the population in the first phase is usuallyless than the size of the population in the second phase
After phase 1 to maintain the property of individuals inthe last population of phase 1 and to increase the accuratelyof solutions we add some additional time nodes Whenthe number of time nodes is increased it is estimated thatthe quality of solution obtained by numerical methods (eg
Runge-Kutta and Simpson) is increased Thus we increasetime nodes from 119873
1199051in phase 1 to 119873
1199052in phase 2 The
corresponding control input values of the new time nodesare added to individuals To use the information of theobtained solutions from phase 1 in the construction of theinitial population of phase 2 we use either Linear or Splineinterpolation to estimate the value of the control inputs inthe new time nodes in each individual of the last populationof phase 1 Moreover to maintain the diversity in the initialpopulation of phase 2 we add new random individuals tothe population using (7) In the second phase MHGA startswith this population and new value of parameters Finally theproposed algorithm is given in Algorithm 2
5 Numerical Experiments
In this section to investigate the efficiency of the proposedalgorithm 20 well-known and real world NOCPs as bench-mark problems are considered which are presented in termsof (1)ndash(6) in the Appendix These NOCPs are selected withsingle control signal and multi control signals which will bedemonstrated in a general manner
The numerical behaviour of algorithms can be studiedfrom two view of points the relative error of the performanceindex and the status of the final state constraints Let 119869 be theobtained performance index by an algorithm 120593
119891 defined in
6 Mathematical Problems in Engineering
Initialization Input the desired precision 120576 in (13) the penalty parameters119872119894 119894 = 1 2 3 in (8) the bounds of control input values in (7) 119906left and 119906right
Phase 1 Perform MHGA with a random population and1198731199051 1198731199011 1198731198941 1198731198921 1198751198981
and 1199041199021199011198981198861199091198941199051198901199031
Construction of the initial population of the phase 2 Increase time nodes uniformly to1198731199052
and estimate the corresponding control input values of the new time nodes in eachindividual obtained from phase 1 using either Linear or Spline interpolationCreate119873
1199012minus 1198731199011new different individuals with119873
1199052time nodes randomly
Phase 2 Perform MHGA with the constructed population and1198731199052 1198731199012 1198731198942 1198731198922 1198751198982
and 1199041199021199011198981198861199091198941199051198901199032
Algorithm 2 The proposed algorithm
(13) be the error of final state constraints and 119869lowast be the best
obtained solution among all implementations or the exactsolution (when exists) Now the relative error of 119869 119864
119869 of the
algorithm can be defined as
119864119869=
10038161003816100381610038161003816100381610038161003816
119869 minus 119869lowast
119869lowast
10038161003816100381610038161003816100381610038161003816
(19)
Tomore accurate study we now define a new criterion calledfactor to compare the algorithms as follows
119870120595= 119864119869+ 120593119891 (20)
Note that 119870120595shows the summation of two important errors
Thus based on 119870120595
we can study the behaviour of thealgorithms on the quality and feasibility of given solutionssimultaneously
To solve any NOCP described in the Appendix wemust know the algorithmrsquos parameters including MHGArsquosparameters including 119873
119905 119873119901 119873119894 119873119892 119875119898and 119904119902119901119898119886119909119894119905119890119903
in both phases in Algorithm 2 and the problemrsquos parametersincluding 120576 in (13) 119872
119894 119894 = 1 2 3 in (8) 119906left and 119906right in
(7) To estimate the best value of the algorithmrsquos parameterswe ran the proposed algorithm with different values ofparameters and then select the best However the sensitivityof MHGA parameters are studied in the next section In bothof phases of Algorithm 2 in MHGA we let 119904119902119901119898119886119909119894119905119890119903
119894= 4
and 119875119898119894
= 08 for 119894 = 1 2 Also we consider 1198731198921
= 1198731198922
1198731198941
= 1198731198942 1198731199011
= 9 and 1198731199012
= 12 The other MHGAparameters are given in the associated subsection and theproblemrsquos parameters in Table 2 For each NOCP 12 differentruns were done and the best results are reported in Table 1which the best value of each column is seen in the bold
The reported numerical results of the proposed algo-rithm for each NOCP include the value of performanceindex 119869 the relative error of 119869119864
119869 defined in (19) the required
computational time Time the norm of final state constraints120593119891 defined in (13) and the factor 119870
120595 defined in (20)
The algorithm was implemented in Matlab R2011a envi-ronment on a Notebook with Windows 7 Ultimate CPU253GHz and 400GB RAM Also to implement SQP in ourproposed algorithm we used ldquofminconrdquo in Matlab when theldquoAlgorithmrdquo was set to ldquoSQPrdquo Moreover we use compositeSimpsonrsquos method [29] to approximate integrations
Remark 1 We use the following abbreviations to show theused interpolation method in our proposed algorithm
(1) LI linear interpolation(2) SI spline interpolation
For comparing the numerical results of the proposedalgorithm two subsections are considered comparison withsome metaheuristic algorithms in Section 51 and com-parison with some numerical methods in Section 52 Wegive more details of these comparisons in the followingsubsections
51 Comparison with Metaheuristic Algorithms The numeri-cal results for the NOCPs numbers (1)ndash(3) in the Appendixare compared with a continuous GA CGA as a meta-heuristic proposed in [9] which gave better solutions thanshooting method and gradient algorithm from the indirectmethods category [2 30] and SUMT from the directmethodscategory [26] For NOCPs numbers (4) and (5) the resultsare compared with another metaheuristic which is a hybridimproved PSO called IPSO proposed in [20]
511 VDP Problem [9] The first NOCP in the Appendix isVanDer Pol Problem VDP which has two state variables andone control variable VDPproblemhas a final state constraintwhich is 120595 = 119909
1(119905119891) minus 1199092(119905119891) + 1 = 0 The results of the
proposed algorithm with the MHGArsquos parameters as 1198731199051
=
311198731199052= 71119873
119892= 300 and119873
119894= 200 are reported in Table 1
FromTable 1 it is obvious that the numerical results of LI andSI methods are more accurate than CGA with less amount of119870120595
512 CRP Problem [9] The second NOCP in the Appendixis Chemical Reactor Problem CRP which has two statevariables and one control variableThe results of the proposedalgorithm with the MHGArsquos parameters as 119873
1199051= 31 119873
1199052=
71 119873119892
= 300 and 119873119894= 200 are shown in the second
row of Table 1 CRP problem has two final state constraints120595 = [119909
1 1199092]119879 Although from Table 1 the norm of final state
constraints 120593119891 for the CGA equals 120593lowast
119891= 757 times 10
minus10 isless than 120593
119891rsquos of LI and SI methods which equals 115 times 10
minus9
and 599 times 10minus9 respectively but the performance index
Mathematical Problems in Engineering 7
Table 1 The best numerical results in 12 different runs of the SI and LI methods for NOCPs in the Appendix
Problem Algorithm 119869 119864119869
120593119891
119870120595
Time
VDPCGA 17404 00912 267 times 10
minus11
00912 50128LI 15949 0 108 times 10
minus13 108 times 10minus13 28282SI 15950 627 times 10
minus5 999 times 10minus14 627 times 10minus5 23914
CRPCGA 163 times 10
minus2 02835 757 times 10minus10 02835 50128LI 127 times 10minus2 0 115 times 10
minus9 115 times 10minus9 17194
SI 127 times 10minus2 0 599 times 10minus9
599 times 10minus9
12405
FFRPCGA 8363 13256 465 times 10
minus3
13302 1413LI 3596 0 122 times 10
minus5 122 times 10minus5 126858
SI 3599 834 times 10minus4 901 times 10minus6 843 times 10
minus4 134340
MSNICIPSO 01727 00165 mdash mdash 12626LI 01699 0 mdash mdash 13882
SI 01700 589 times 10minus4 mdash mdash 13868
CSTCRIPSO 01355 02098 mdash mdash 3134
LI 01120 0 mdash mdash 10358
SI 01120 0 mdash mdash 10673
No 6Bezier minus53898 00251 mdash mdash NRa
LI minus54309 00177 mdash mdash 13838
SI minus54309 00177 mdash mdash 10734
Number 7HPM 02353 01677 420 times 10
minus6 01677 NRLI 02015 0 235 times 10
minus9
235 times 10minus9 21392
SI 02015 0 282 times 10minus10 282 times 10minus10 20696
Number 8
SQP 636 times 10minus6
220 times 109 mdash mdash NR
SUMT 515 times 10minus6
178 times 109 mdash mdash NR
LI 289 times 10minus15 0 mdash mdash 7282SI 364 times 10
minus15
02595 mdash mdash 6868
Number 9
SQP 17950 00873 mdash mdash NRSUMT 17980 00891 mdash mdash NRLI 16509 0 mdash mdash 63910SI 16509 0 mdash mdash 5950
Number 10
SQP 02163 03964 mdash mdash NRSUMT 01703 00994 mdash mdash NRLI 01549 0 mdash mdash 62570
SI 01549 0 mdash mdash 67662
Number 11
SQP 325 01648 0 01648 NRSUMT 325 01648 0 01648 NRLI 27901 0 162 times 10
minus9
162 times 10minus9 14778
SI 27901 0 586 times 10minus10 586 times 10minus10 21946
Number 12
SQP minus02490 40 times 10minus3 0 40 times 10
minus3 NRSUMT minus02490 40 times 10
minus3 0 40 times 10minus3 NR
LI minus02500 0 285 times 10minus8
285 times 10minus8
54771
SI minus02500 0 390 times 10minus10 390 times 10minus10 54548
Number 13
SQP 168 times 10minus2 01748 0 01748 NR
SUMT 167 times 10minus2 01678 0 01678 NR
LI 143 times 10minus2 0 118 times 10minus9 118 times 10minus9 42599SI 144 times 10
minus2
70 times 10minus3
632 times 10minus9
70 times 10minus3
49027
Number 14
SQP 37220 00961 0 00961 NRSUMT 37700 01103 0 01103 NRLI 33956 0 786 times 10minus7 786 times 10minus7 104124SI 33965 265 times 10
minus4
276 times 10minus6
267 times 10minus4
115144
8 Mathematical Problems in Engineering
Table 1 Continued
Problem Algorithm 119869 119864119869
120593119891
119870120595
Time
Number 15
SQP 103 times 10minus3 43368 0 43368 NR
SUMT 929 times 10minus4 38135 0 38135 NR
LI 193 times 10minus4 0 810 times 10minus10 810 times 10minus10 33354SI 194 times 10
minus4
520 times 10minus3
164 times 10minus9
520 times 10minus3
34014
Number 16
SQP 22120 00753 0 00753 NRSUMT 22080 00734 0 00734 NRLI 20571 0 394 times 10
minus12
394 times 10minus12
46982
SI 20571 0 560 times 10minus13 560 times 10minus13 70423
Number 17
SQP minus88690 225 times 10minus5 0 225 times 10
minus5 NRSUMT minus88690 225 times 10
minus5 0 225 times 10minus5 NR
LI minus88692 0 145 times 10minus10 145 times 10minus10 32183SI minus88692 0 279 times 10
minus10
279 times 10minus10
30959
Number 18
SQP 00368 01288 mdash mdash NRSUMT 00386 01840 mdash mdash NRLI 00326 0 mdash mdash 55163
SI 00326 0 mdash mdash 60671
Number 19
SQP 03439 49293 0 49293 NRSUMT 03428 49103 0 49103 NRLI 00689 01879 410 times 10
minus3
01919 164509
SI 00580 0 405 times 10minus4 405 times 10minus4 178783
Number 20
SQP 7752 12554 0 12554 NRSUMT 7683 12353 0 12353 NRLI 343716 232 times 10
minus5
225 times 10minus4
248 times 10minus4 81519
SI 343708 0 157 times 10minus4 157 times 10minus4 80420
aNot reported
the relative error of 119869 and the factor of the proposed algorithmis better and so the proposed algorithm is more robust thanCGA
513 FFRP Problem [9] The third NOCP in the Appendixis Free Floating Robot Problem FFRP which has six statevariables and four control variables FFRP has been solvedby CGA and the proposed algorithm with the MHGArsquosparameters as 119873
1199051= 31 119873
1199052= 71 119873
119892= 300 and
119873119894
= 200 This problem has six final state constraints120595 = [119909
1minus 4 119909
2 1199093minus 4 119909
4 1199095 1199096]119879 The numerical results
are shown in Table 1 The values of 119869 119864119869 120593119891and 119870
120595for LI
and SI methods separately are less than CGA Thereforethe proposed algorithm can achieve much better qualitysolutions than the CGA with reasonable computational time
514 MSNIC Problem [20] For the forth NOCP in theAppendix which is a Mathematical System with NonlinearInequality Constraint NSNIC the numerical results are com-pared with IPSO MSNIC contains an inequality constraint119889(119909 119905) = 119909
2(119905) + 05 minus 8(119905 minus 05)
2
le 0 The problem solvedby several numerical methods as [24 31] From [20] IPSOmethod could achieved more accurate results than men-tioned numericalmethods AlsoMSNIC can be solved by theproposed algorithm with the MHGArsquos parameters as 119873
1199051=
31 1198731199052
= 91 119873119892= 100 and 119873
119894= 60 From the forth row
of Table 1 the absolute error of 119869 119864119869 for LI and SI methods
equal 0 and 589 times 10minus4 respectively which are less than
IPSOrsquos 00165Subplots (a) and (b) in Figure 2 show the graphs of
the convergence rate for the performance index and theinequality constraint respect to the number of iterationrespectively
515 CSTCR Problem [20] The fifth NOCP in the Appendixis a model of a nonlinear Continuous Stirred-tank ChemicalReactor CSTCR It has two state variables 119909
1(119905) and 119909
2(119905)
as the deviation from the steady-state temperature and con-centration and one control variable 119906(119905) which represent theeffect of the flow rate of cooling fluid on chemical reactorTheobjective is to maintain the temperature and concentrationclose to steady-state values without expending large amountof control effort Also this is a benchmark problem in thehandbook of test problems in local and global optimization[32] which is a multimodal optimal control problem [33]It involves two different local minima The values of theperformance indexes for these solutions equal 0244 and0133 Similarly to the MSNIC the numerical results of theproposed algorithm with the MHGArsquos parameters as 119873
1199051=
31 1198731199052
= 51 119873119892= 100 and 119873
119894= 50 are compared with
IPSO From Table 1 the performance index 119869 for LI and SImethods is equal to 119869
lowast
= 01120 which is less than IPSOs01355
Mathematical Problems in Engineering 9
Table 2 The problem parameters for NOCPs in the Appendix
Parameters Problem number1 2 3 4 5 6 7 8 9 10
119906left minus05 minus15 minus15 minus20 0 minus2 0 minus2 minus1 minus20
119906right 2 2 10 20 5 2 1 2 1 20119872119894
103
102 70 1 mdash 1 mdash 10
2 mdash 102
120576 10minus12
10minus10
10minus3 mdash mdash mdash mdash mdash mdash 10
minus9
Parameters Problem no11 12 13 14 15 16 17 18 19 20
119906left minus5 minus1 minus2 minus120587 minus1 minus3 minus30 minus1 minus15 minus15
119906right 5 1 2 120587 1 3 30 1 10 10119872119894
10 102 mdash 10
2
102
102 mdash 10 70 10
minus3
120576 10minus9
10minus11
10minus11 mdash 10
minus10
10minus10
10minus10 mdash 10
minus3
10minus4
0 5 10 15 20 25 30 35 40 45 50017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 2 Graphical results of MSNIC problem using SI method (a) convergence rate of the performance index and (b) the inequalityconstraint 119889(119909 119905) le 0 in respect to the number of iterations
52 Comparison with Numerical Methods For NOCPs num-bers (6)ndash(20) the comparison are done with some numericalmethods Unfortunately for these methods usually the finalstate constraints and the required computational time arenot reported which are shown with NR in Table 1 but thesevalues are reported for both LI and SI methods in Table 1For all NOCPs in this section the MHGArsquos parameters areconsidered as 119873
1199051= 31 119873
1199052= 51 119873
119892= 100 119873
119894= 50 with
the problem parameters in Table 2
521 Compared with Bezier [34] The NOCP number (6) inthe Appendix has exact solution which has an inequalityconstraint as 119889(119909
1 119905) = minus6 minus 119909
1(119905) le 0 The exact value of
performance index equals 119869lowast = minus55285 [35] This problemhas been solved by a numerical method proposed in [34]called Bezier From sixth row of Table 1 the absolute error
of the LI and SI methods equal 00177 which is less thanBezierrsquos 00251
Figure 3 shows the graphs of the convergence rate of theperformance index subplot (a) and inequality constraintsubplot (b) respect to the number of iteration using SImethod
522 Compared with HPM [36] For NOCP number (7) inthe Appendix which is a constraint nonlinear model thenumerical results of the proposed algorithm are comparedwith HPM proposed in [36] This NOCP has a final stateconstraint as 120595 = 119909 minus 05 = 0 From [36] the norm offinal state constraint for HPM equals 42times10minus6 however thiscriterion for the LI and SI methods equals 235 times 10
minus9 and120593lowast
119891= 282 times 10
minus10 respectively From Table 1 it is obviousthat the obtained values of the performance index the norm
10 Mathematical Problems in Engineering
2 4 6 8 10 12 14minus5353
minus5352
minus5351
minus535
minus5349
minus5348
minus5347
minus5346
minus5345
minus5344
minus5343
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 3 Graphical results for NOCP number (6) using SI method (a) convergence rate for the performance index and (b) the inequalityconstraint 119889(119909
1 119905) le 0 in respect to the number of iterations
of final state constraint and 119870120595for the SI method are more
accurate than LI than HPMmethods
523 Compared with SQP and SUMT [26] For the NOCPsnumbers (8)ndash(20) in the Appendix the numerical resultsof LI and SI methods are compared with two numericalmethods contain SQP and SUMT proposed in [26] Amongthese NOCPs only two problems numbers (8) and (18) areunconstrained and the others have at least one constraintfinal state constraint or inequality constraint All of theseNOCPs solved by the proposed algorithm with the problemparameters in Table 2 and their results are summarized inTable 1 Because the final state constraints in these methodsare not reported we let 120593
119891= 0 to calculate the factor119870
120595
Figures 4ndash16 show the graphical results of NOCPsnumbers (8)ndash(20) in the Appendix using SI method Forunconstrained NOCPs numbers (8) and (18) only the graphof convergent rate of performance index is shown see Figures4 and 14 For constraint NOCPs and NOCPs numbers(11)ndash(17) and numbers (19)-(20) with final state constraintthe graphs of convergent rates of performance index and theerror of final state constraint are shown see Figures 7ndash13 and15-16 For the constraint NOCPs with inequality constraintsNOCPs numbers (9) and (10) the graphs of convergent rateof performance index and inequality constraint are shownsee Figures 5 and 6
Table 1 shows that the proposed algorithm LI and SImethods was 100 percent successful in point of views theperformance index 119869 and the factor119870
120595 numerically So the
proposed algorithm provides robust solutions with respect tothe other mentioned numerical or metaheuristic methodsTo compare 119869 in LI and SI methods in 35 percent of NOCPsLI is more accurate than SI in 10 percent SI is more accurate
1 15 2 25 3 35 40
002
004
006
008
01
012
Iter
Perfo
rman
ce in
dex
Figure 4 Convergence rate for the performance index of SI methodfor NOCP number (8)
than LI and in 55 percent are same In point of view the finalconditions 65 percent of NOCPs in the Appendix have finalstate constraints In all of them 120593
119891in the proposed algorithm
is improved except CRP problem however the factor of theproposed algorithm is the best yet In 54 percent of theseNOCPs LI is better performance and in 46 percent SI isbetter Also in point of the running time Time LI is same asSI that is in 50 percent of NOCPs LI and in 50 percent SI isbetter than another So the proposed algorithmcould providevery suitable solutions in a reasonable computational timeAlso for more accurate comparison of LI and SI methods astatistical approach will done in next section
To compare with CGA the mean of relative error of 119869 119864119869
for CGA LI and SImethods inNOCPs (1)ndash(3) equal 056680 and 298 times 10
minus4 respectively Also the mean for the error
Mathematical Problems in Engineering 11
5 10 15 20 25165
17
175
18
185
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3 35 4 45 50
005
01
015
02
025
TimeIn
equa
lity
cons
trai
nt(b)
Figure 5 (a) Convergence rate for the performance index and (b) the inequality constraint of SI method for NOCP number (9)
5 10 15 20 25 30 35 40 45 500155
016
0165
017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 6 (a) Convergence rate of the performance index and (b) the inequality constraint for SI method for NOCP number (10)
of the final state constraints 120593119891 are 16 times 10
minus3 407 times 10minus6
and 301 times 10minus6 respectively and for 119870
120595 these values equal
05883 407 times 10minus6 and 302 times 10
minus4 Thus we can say that thefeasibility of the solutions given by the proposed algorithmand CGA are competitive
From Table 1 to compare with numerical methods SQPand SUMT in NOCPs (8)ndash(20) the mean of 119864
119869for LI SI
SQP and SUMT equals 00145 00209 169 times 108 and 136 times
108 respectively Also the mean for the error of final state
constraints for these NOCPs equal 334 times 10minus4 441 times 10
minus50 and 0 respectively For 119870
120595 these values are 00213 00014
12263 and 11644 Therefore the performance index 119869 andthe factor 119870
120595 for the LI and SI methods are more accurate
than SQP and SUMT So the proposed algorithm gave morebetter solution in comparison with the numerical methods
12 Mathematical Problems in Engineering
5 10 15 20 25minus025
minus0245
minus024
minus0235
minus023
minus0225
Iter
Perfo
rman
ce in
dex
(a)
1 15 2 25 3 35 4 45 5 55 60
02
04
06
08
1
12
14
16
IterFi
nal c
ondi
tion
times10minus5
(b)
Figure 7 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (11)
2 4 6 8 10 12 14minus025
minus0245
minus024
minus0235
minus023
minus0225
minus022
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
02
04
06
08
1
12
14
16
18
2
Iter
Fina
l con
ditio
n
times10minus5
(b)
Figure 8 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (12)
Therefore based on this numerical studywe can concludethat the proposed algorithm outperform well-known numer-ical method Since the algorithms were not implementedon the same PC the computational times of them are notcompetitive Therefore we did not give the computationaltimes in bold in Table 1
6 Sensitivity Analysis andComparing LI and SI
In this section two statistical analysis based on the one-way analysis of variance (ANOVA) used for investigatingthe sensitivity of MHGA parameters and Mann-Whitney
Mathematical Problems in Engineering 13
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
07
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 9 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (13)
5 10 15 20 25 30
35
4
45
5
55
6
65
7
75
8
85
9
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
Iter
Fina
l con
ditio
n
(b)
Figure 10 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (14)
applied comparing LI and SI methods are done by thestatistical software IBM SPSS version 21
61 Sensitivity Analysis In order to survey the sensitivityof the MHGA parameters from the proposed algorithmthe VDP problem is selected for example The influenceof these parameters are investigated for this NOCP on
the dependent outputs consist of the performance index119869 the relative error of 119869 119864
119869 the required computational
time Time the error of final state constraints 120593119891and the
factor 119870120595 The independent parameters are consist of the
number of time nodes in both two phases 1198731199051and 119873
1199052 the
size of population in both two phases 1198731199011
and 1198731199012 the
maximum number of generations without improvement119873119894
14 Mathematical Problems in Engineering
5 10 15 20 25 30 35 40 45 50
00955
0096
00965
0097
00975
0098
00985
0099
00995
Iter
Perfo
rman
ce in
dex
(a)
2 4 6 8 10 12 14 16 18 200
05
1
15
2
25
3
35
4
45
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 11 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (15)
5 10 15 20 25
21
22
23
24
25
26
27
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
1
2
3
4
5
6
Iter
Fina
l con
ditio
n
times10minus3
(b)
Figure 12 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (16)
and the maximum number of generations 119873119892 Because the
mutation implementation probability119875119898 has a less influence
in numerical results it is not considered Also 119904119902119901119898119886119909119894119905119890119903 ischanged in each iteration of the proposed algorithm so it isnot considered too
At first we selected at least four constant value foreach of parameters and then in each pose 35 differentruns were made independently The statistical analysis isdone based on ANOVA The descriptive statistics whichcontains the number of different runs (119873) the mean of each
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7minus887
minus886
minus885
minus884
minus883
minus882
minus881
minus88
minus879
minus878
minus877
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 70
01
02
03
04
05
06
07
08
09
1
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 13 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (17)
1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333
Iter
Perfo
rman
ce in
dex
Figure 14 Convergence rate for the performance index of SImethod for NOCP numbers (18)
output in119873 different runs (Mean) standard deviation (Sd)maximum (Max) and minimum (Min) of each outputcould be achieved Among of the MHGA parameters wepresent the descriptive statistics of the ANOVA only for the1198731199051parameter which is reported in Table 3Table 4 summarized the statistical data contain the test
statistics (119865) and 119875-values of ANOVA tests Sensitivityanalysis for each parameter separately are done based on thistable as follows
1198731199051 From Table 4 the significant level or 119875-value for119870120595
is equal to 0279 which is greater than 005So from ANOVA 119873
1199051parameter has no significant
effect on the factor 119870120595 With similar analysis this
parameter has no effect on the other outputs except
required computational time Time because its 119875-value is equal to 0 which is less than 005 that is thisoutput sensitive to the parameter119873
1199051
1198731199052 From the second row of Table 4 the outputs 119869 119864
119869119870120595
and Time are sensitive to the parameter 1198731199052 and the
only output 120593119891is independent
1198731199011 From the third row of Table 4 only the computationalrequired time Time is sensitive because the its 119875-value is equal to 0006 which is less than 005 and theother outputs contain 119869 119864
119869 120593119891119870120595 are independent
1198731199012 From the forth row of Table 4 the output 120593
119891is
independent respect to the parameter 1198731199012 but other
outputs contain 119869 119864119869 Time 119870
120601are sensitive to this
parameter119873119892 From the fifth row of Table 4 the outputs 119869 119864
119869and
119870120595are independent to the parameter 119873
119892 and other
outputs contain 120593119891and Time are sensitive respect to
this parameter119873119894 The sensitivity analysis is similar to119873
1199011
From above cases it is obvious that all parameters canbe effect on the required computational time except 119873
119892
Moreover intuitions shows the normof final state constraints120593119891is independent output with respect to all parameters that
is any of the parameters could not effect on this output and itis not sensitive with respect to any of the MHGA parameters
62 Comparison of LI and SI To compare the efficiency ofthe LI and SI methods for NOCPs in the Appendix weused the Mann-Whitney nonparametric statistical test [37]
16 Mathematical Problems in Engineering
1 15 2 25 3 35 40
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
200
400
600
800
1000
1200
1400
1600
IterFi
nal c
ondi
tion
(b)
Figure 15 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (19)
2 4 6 8 10 12 14 16 18 20
50
100
150
200
250
300
350
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 10 11 120
05
1
15
2
25
3
Iter
Fina
l con
ditio
n
(b)
Figure 16 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (20)
Using the nonparametric statistical tests for comparing theperformance of several algorithms presented in [38] Forthis purpose At first 15 different runs for each of NOCPwere made Table 5 shows the mean of numerical resultsseparately Here we apply the fixMHGAparameters as119873
1199011=
91198731199012
= 121198731199051= 31119873
1199052= 91119873
119892= 100 and119873
119894= 50 with
the previous problem parameters in Table 2 Comparison
criterions contain 119869 120593119891 119870120595and Time The results of Mann-
Whitney test are shown in Tables 6ndash9 From Table 6 since119885 = 0 gt 119885
001= minus234 then the average relative error of
the LI and SI methods are same with the significant level 001In other words with probability 99 we can say that LI andSI methods have same effectiveness form the perspective oferror for 119869 Similarly the results of Tables 7ndash9 indicate that LI
Mathematical Problems in Engineering 17
Table 3 Descriptive statistics for the sensitivity analysis of1198731199051 for VDP problem
Output 1198731199051
119873 Mean Sd Min Max
119869
11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449
Total 140 15357 00294 15285 18449
120593119891
11 35 281 times 10minus8
952 times 10minus8
1 times 10minus12
452 times 10minus7
21 35 467 times 10minus8
235 times 10minus7
1 times 10minus12
141 times 10minus6
31 35 163 times 10minus8
491 times 10minus8
1 times 10minus12
221 times 10minus7
41 35 817 times 10minus8
285 times 10minus7
1 times 10minus12
126 times 10minus6
Total 140 426 times 10minus8 191 times 10minus7 1 times 10minus12 141 times 10minus6
Time
11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514
Total 140 1142131 410891 456615 2849514
119864119869
11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070
Total 140 00047 00192 0 02070
119870120595
11 35 00031 00045 475 times 10minus9
00204
21 35 00027 00064 550 times 10minus10
00369
31 35 00029 00050 154 times 10minus13
00271
41 35 00111 00397 193 times 10minus10
02070
Total 140 00047 00192 154 times 10minus13 02070
Table 4 Summary statistical data of ANOVA test for the parameters1198731199051 1198731199052 1198731199011 1198731199012 119873119892 119873119894
Parameters 119869 119864119869
120593119891
Time 119870120595
Test statistic (119865)
1198731199051
1294 1296 0624 1079 12961198731199052
110572 393 0868 5939 3931198731199011
1945 1835 1271 4317 18351198731199012
2740 2478 1011 2302 2478119873119892
3541 3591 0491 0890 3591119873119894
0495 0309 0301 5849 1046
119875 value
1198731199051
0280 0279 0601 0 02791198731199052
0 0005 0489 0 00051198731199011
0125 0143 0286 0006 01431198731199012
0021 0034 0413 0 0034119873119892
0009 0008 0743 0472 0008119873119894
0739 0819 0877 0 0386
and SI methods from the perspective of errors for 120593119891 Time
and119870120595 have same behaviour
7 The Impact of SQP
In this section we investigation the impact of the SQP on theproposed algorithm For this purpose we remove the SQPfrom Algorithm 2 as without SQP algorithm For comparingthe numerical results with the previous results the required
running time in each NOCP is considered fixed which is themaximum of running time in 12 different runs were done inSection 5 Also the all of the parameters for each problemwasset as same as Section 5 The numerical results of the withoutSQP algorithm is summarized in Table 10 By comparing theresults of Tables 10 and 1 it is obvious that for all NOCPsin the Appendix the obtained values of the performanceindex and the norm of final state constraint for the proposedalgorithm (Algorithm 2) are more accurate than the withoutSQP algorithm
18 Mathematical Problems in Engineering
Table 5 The mean of numerical results of 15 different runs for NOCPs in the Appendix using LI and SI methods
Problem LI SI119869 120593
119891119870120595
Time 119869 120593119891
119870120595
TimeVDP 15977 157 times 10
minus9
170 times 10minus3 17720 16002 667 times 10
minus8
326 times 10minus3 15757
CRP 00126 307 times 10minus8
307 times 10minus8 19714 00119 161 times 10
minus8
161 times 10minus8 19532
FFPP 4774 805 times 10minus5 03277 133280 5351 592 times 10
minus5 04868 137989MSNIC 01702 mdash mdash 13678 01703 mdash mdash 14092CSTCR 01120 mdash mdash 17385 01120 mdash mdash 17606Number 6 minus54307 mdash mdash 10541 minus54308 mdash mdash 10372Number 7 02015 326 times 10
minus9
695 times 10minus6 20807 02020 313 times 10
minus9
662 times 10minus6 21019
Number 8 630 times 10minus15 mdash mdash 6376 124 times 10
minus14 mdash mdash 5837Number 9 16512 mdash mdash 59239 16512 mdash mdash 58125Number 10 01550 mdash mdash 63153 01550 mdash mdash 65597Number 11 29701 270 times 10
minus9
272 times 10minus7 22860 29701 337 times 10
minus9
811 times 10minus7 23276
Number 12 minus02499 201 times 10minus8
464 times 10minus5 54718 minus02499 115 times 10
minus9
827 times 10minus5 52765
Number 13 00147 115 times 10minus8
00252 57797 00151 139 times 10minus8
00472 66098Number 14 34761 672 times 10
minus6
00237 104585 34290 757 times 10minus6
950 times 10minus3 108767
Number 15 195 times 10minus4
111 times 10minus8
569 times 10minus3 363 194 times 10
minus4
704 times 10minus9
486 times 10minus3 365
Number 16 20571 166 times 10minus10
166 times 10minus10 54391 20571 115 times 10
minus10
115 times 10minus10 54431
Number 17 minus88692 803 times 10minus8
803 times 10minus8 22744 minus88692 661 times 10
minus8
661 times 10minus8 23254
Number 18 00326 mdash mdash 65957 00326 mdash mdash 70897Number 19 011588 655 times 10
minus4 06828 161813 01091 884 times 10minus4 08808 159990
Number 20 5242 00176 05429 84165 4464 453 times 10minus4 03765 86891
Table 6 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 119869
Method Mean rank Sum of ranks Test statistics ValueLI 2050 4100 Mann-Whitney 119880 200SI 2050 4100 Wilcoxon119882 410mdash mdash mdash 119885 0
Table 7 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 120593119891
Method Mean rank Sum of ranks Test statistics ValueLI 1362 177 Mann-Whitney 119880 83SI 1338 174 Wilcoxon119882 174mdash mdash mdash 119885 minus0077
Table 8 Results of Mann-Whitney test on relative errors of the pair (LI SI) for Time
Method Mean rank Sum of ranks Test statistics ValueLI 2025 4050 Mann-Whitney 119880 195SI 2075 4150 Wilcoxon119882 405mdash mdash mdash 119885 minus0135
Table 9 Results of Mann-Whitney test on relative errors of the pair (LI SI) for119870120595
Method Mean rank Sum of ranks Test statistics ValueLI 1354 1760 Mann-Whitney 119880 84SI 1346 1750 Wilcoxon119882 175mdash mdash mdash 119885 minus0026
Mathematical Problems in Engineering 19
Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix
Problem no1 2 3 4 5 6 7 8 9 10
119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048
120601119891
330 times 10minus9
135 times 10minus5 11445 mdash mdash mdash 758 times 10
minus9 mdash mdash mdashProblem no
11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10
minus4 23182 minus90882 00336 2513 20274120601119891
167 times 10minus5
468 times 10minus9 00161 433 times 10
minus4
594 times 10minus3
156 times 10minus5 164 mdash 1706 25283
8 Conclusions
Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters
Appendix
Test Problems
The following problems are presented using notation given in(1)ndash(6)
(1) (VDP) [9] 119892 = (12)(1199092
1+ 1199092
2+ 1199062
) 1199050= 0 119905119891= 5 119891 =
[1199092 minus1199092+(1minus119909
2
1)1199092+119906]119879 1199090= [1 0]
119879120595 = 1199091minus1199092+1
(2) (CRP) [9] 119892 = (12)(1199092
1+1199092
2+01119906
2
) 1199050= 0 119905119891= 078
119891 = [1199091minus2(1199091+025)+(119909
2+05) exp(25119909
1(1199091+2))minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(3) (FFRP) [9] 119892 = (12)(1199062
1+ 1199062
2+ 1199062
3+ 1199062
4) 1199050
=
0 119905119891
= 5 119891 = [1199092 (110)((119906
1+ 1199062) cos119909
5minus
(1199062+ 1199064) sin119909
5) 1199094 (110)((119906
1+ 1199063) sin119909
5+ (1199062+
1199064) cos119909
5) 1199096 (512)((119906
1+ 1199063) minus (119906
2+ 1199064))]119879 1199090=
[0 0 0 0 0 0]119879 120595 = [119909
1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879
(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909
2 minus1199092+
119906 1199092
1+ 1199092
2+ 0005119906
2
]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909
2+
05 minus 8(119905 minus 05)2
]119879 1199090= [0 minus1 0]
119879
(5) (CSTCR) [20] 119892 = 1199092
1+ 1199092
2+ 01119906
2 1199050= 0 119905119891= 078
119891 = [minus(2 + 119906)(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+
2)) 05 minus 1199092minus (1199092+ 05) exp(25119909
1(1199091+ 2))]
119879 1199090=
[009 009]119879
(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909
2 119906]119879 119889 = [minus(2 +
119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]
119879(7) [36] 119892 = 119906
2 1199050= 0 119905
119891= 1 119891 = (12)119909
2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05
(8) [26] 119892 = 1199092cos2119906 119905
0= 0 119905119891= 120587 119891 = sin(1199062) 119909
0=
1205872(9) [26] 119892 = (12)(119909
2
1+ 1199092
2+ 1199062
) 1199050= 0 119905
119891= 5 119891 =
[1199092 minus1199091+ (1 minus 119909
2
1)1199092+ 119906]119879 119889 = minus(119909
2+ 025) 119909
0=
[1 0]119879
(10) [26] 119892 = 1199092
1+ 1199092
2+ 0005119906
2
1199050= 0 119905
119891= 1 119891 =
[1199092 minus1199092+119906]119879
119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus
05) minus 05 minus 1199092)]119879
1199090= [0 minus1]
119879(11) [26] 119892 = (12)119906
2 1199050= 0 119905
119891= 2 119891 = [119909
2 119906]119879 1199090=
[1 1]119879 120595 = [119909
1 1199092]119879
(12) [26] 119892 = minus1199092 1199050= 0 119905
119891= 1 119891 = [119909
2 119906]119879 119889 =
minus(1 minus 119906)(1 + 119906) 1199090= [0 0]
119879 120595 = 1199092
(13) [26] 119892 = (12)(1199092
1+ 1199092
2+ 01119906
2
) 1199050= 0 119905
119891= 078
119891 = [minus2(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+ 2)) minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(14) [26] 119892 = (12)1199062 1199050= 0 119905
119891= 10 119891 = [cos 119906 minus
1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909
0= [366 minus186]
119879120595 = [119909
1 1199092]119879
(15) [26]119892 = (12)(1199092
1+1199092
2) 1199050= 0 119905119891= 078119891 = [minus2(119909
1+
025)+(1199092+05) exp(25119909
1(1199091+2))minus(119909
1+025)119906 05minus
1199092minus(1199092+05) exp(25119909
1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(16) [26] 120601 = 1199093 1199050= 0 119905
119891= 1 119891 = [119909
2 119906 (12)119906
2
]119879
119889 = 1199091minus 19 119909
0= [0 0 0]
119879 120595 = [1199091 1199092+ 1]119879
(17) [26] 120601 = minus1199093 1199050= 0 119905
119891= 5 119891 = [119909
2 minus2 + 119906119909
3
minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909
0= [10 minus2 10]
119879120595 = [119909
1 1199092]119879
(18) [26] 120601 = (1199091minus1)2
+1199092
2+1199092
3 119892 = (12)119906
2 1199050= 0 119905119891= 5
119891 = [1199093cos 119906 119909
3sin 119906 sin 119906]119879 119909
0= [0 0 0]
119879
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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OptimizationJournal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
Initialization Input the desired precision 120576 in (13) the penalty parameters119872119894 119894 = 1 2 3 in (8) the bounds of control input values in (7) 119906left and 119906right
Phase 1 Perform MHGA with a random population and1198731199051 1198731199011 1198731198941 1198731198921 1198751198981
and 1199041199021199011198981198861199091198941199051198901199031
Construction of the initial population of the phase 2 Increase time nodes uniformly to1198731199052
and estimate the corresponding control input values of the new time nodes in eachindividual obtained from phase 1 using either Linear or Spline interpolationCreate119873
1199012minus 1198731199011new different individuals with119873
1199052time nodes randomly
Phase 2 Perform MHGA with the constructed population and1198731199052 1198731199012 1198731198942 1198731198922 1198751198982
and 1199041199021199011198981198861199091198941199051198901199032
Algorithm 2 The proposed algorithm
(13) be the error of final state constraints and 119869lowast be the best
obtained solution among all implementations or the exactsolution (when exists) Now the relative error of 119869 119864
119869 of the
algorithm can be defined as
119864119869=
10038161003816100381610038161003816100381610038161003816
119869 minus 119869lowast
119869lowast
10038161003816100381610038161003816100381610038161003816
(19)
Tomore accurate study we now define a new criterion calledfactor to compare the algorithms as follows
119870120595= 119864119869+ 120593119891 (20)
Note that 119870120595shows the summation of two important errors
Thus based on 119870120595
we can study the behaviour of thealgorithms on the quality and feasibility of given solutionssimultaneously
To solve any NOCP described in the Appendix wemust know the algorithmrsquos parameters including MHGArsquosparameters including 119873
119905 119873119901 119873119894 119873119892 119875119898and 119904119902119901119898119886119909119894119905119890119903
in both phases in Algorithm 2 and the problemrsquos parametersincluding 120576 in (13) 119872
119894 119894 = 1 2 3 in (8) 119906left and 119906right in
(7) To estimate the best value of the algorithmrsquos parameterswe ran the proposed algorithm with different values ofparameters and then select the best However the sensitivityof MHGA parameters are studied in the next section In bothof phases of Algorithm 2 in MHGA we let 119904119902119901119898119886119909119894119905119890119903
119894= 4
and 119875119898119894
= 08 for 119894 = 1 2 Also we consider 1198731198921
= 1198731198922
1198731198941
= 1198731198942 1198731199011
= 9 and 1198731199012
= 12 The other MHGAparameters are given in the associated subsection and theproblemrsquos parameters in Table 2 For each NOCP 12 differentruns were done and the best results are reported in Table 1which the best value of each column is seen in the bold
The reported numerical results of the proposed algo-rithm for each NOCP include the value of performanceindex 119869 the relative error of 119869119864
119869 defined in (19) the required
computational time Time the norm of final state constraints120593119891 defined in (13) and the factor 119870
120595 defined in (20)
The algorithm was implemented in Matlab R2011a envi-ronment on a Notebook with Windows 7 Ultimate CPU253GHz and 400GB RAM Also to implement SQP in ourproposed algorithm we used ldquofminconrdquo in Matlab when theldquoAlgorithmrdquo was set to ldquoSQPrdquo Moreover we use compositeSimpsonrsquos method [29] to approximate integrations
Remark 1 We use the following abbreviations to show theused interpolation method in our proposed algorithm
(1) LI linear interpolation(2) SI spline interpolation
For comparing the numerical results of the proposedalgorithm two subsections are considered comparison withsome metaheuristic algorithms in Section 51 and com-parison with some numerical methods in Section 52 Wegive more details of these comparisons in the followingsubsections
51 Comparison with Metaheuristic Algorithms The numeri-cal results for the NOCPs numbers (1)ndash(3) in the Appendixare compared with a continuous GA CGA as a meta-heuristic proposed in [9] which gave better solutions thanshooting method and gradient algorithm from the indirectmethods category [2 30] and SUMT from the directmethodscategory [26] For NOCPs numbers (4) and (5) the resultsare compared with another metaheuristic which is a hybridimproved PSO called IPSO proposed in [20]
511 VDP Problem [9] The first NOCP in the Appendix isVanDer Pol Problem VDP which has two state variables andone control variable VDPproblemhas a final state constraintwhich is 120595 = 119909
1(119905119891) minus 1199092(119905119891) + 1 = 0 The results of the
proposed algorithm with the MHGArsquos parameters as 1198731199051
=
311198731199052= 71119873
119892= 300 and119873
119894= 200 are reported in Table 1
FromTable 1 it is obvious that the numerical results of LI andSI methods are more accurate than CGA with less amount of119870120595
512 CRP Problem [9] The second NOCP in the Appendixis Chemical Reactor Problem CRP which has two statevariables and one control variableThe results of the proposedalgorithm with the MHGArsquos parameters as 119873
1199051= 31 119873
1199052=
71 119873119892
= 300 and 119873119894= 200 are shown in the second
row of Table 1 CRP problem has two final state constraints120595 = [119909
1 1199092]119879 Although from Table 1 the norm of final state
constraints 120593119891 for the CGA equals 120593lowast
119891= 757 times 10
minus10 isless than 120593
119891rsquos of LI and SI methods which equals 115 times 10
minus9
and 599 times 10minus9 respectively but the performance index
Mathematical Problems in Engineering 7
Table 1 The best numerical results in 12 different runs of the SI and LI methods for NOCPs in the Appendix
Problem Algorithm 119869 119864119869
120593119891
119870120595
Time
VDPCGA 17404 00912 267 times 10
minus11
00912 50128LI 15949 0 108 times 10
minus13 108 times 10minus13 28282SI 15950 627 times 10
minus5 999 times 10minus14 627 times 10minus5 23914
CRPCGA 163 times 10
minus2 02835 757 times 10minus10 02835 50128LI 127 times 10minus2 0 115 times 10
minus9 115 times 10minus9 17194
SI 127 times 10minus2 0 599 times 10minus9
599 times 10minus9
12405
FFRPCGA 8363 13256 465 times 10
minus3
13302 1413LI 3596 0 122 times 10
minus5 122 times 10minus5 126858
SI 3599 834 times 10minus4 901 times 10minus6 843 times 10
minus4 134340
MSNICIPSO 01727 00165 mdash mdash 12626LI 01699 0 mdash mdash 13882
SI 01700 589 times 10minus4 mdash mdash 13868
CSTCRIPSO 01355 02098 mdash mdash 3134
LI 01120 0 mdash mdash 10358
SI 01120 0 mdash mdash 10673
No 6Bezier minus53898 00251 mdash mdash NRa
LI minus54309 00177 mdash mdash 13838
SI minus54309 00177 mdash mdash 10734
Number 7HPM 02353 01677 420 times 10
minus6 01677 NRLI 02015 0 235 times 10
minus9
235 times 10minus9 21392
SI 02015 0 282 times 10minus10 282 times 10minus10 20696
Number 8
SQP 636 times 10minus6
220 times 109 mdash mdash NR
SUMT 515 times 10minus6
178 times 109 mdash mdash NR
LI 289 times 10minus15 0 mdash mdash 7282SI 364 times 10
minus15
02595 mdash mdash 6868
Number 9
SQP 17950 00873 mdash mdash NRSUMT 17980 00891 mdash mdash NRLI 16509 0 mdash mdash 63910SI 16509 0 mdash mdash 5950
Number 10
SQP 02163 03964 mdash mdash NRSUMT 01703 00994 mdash mdash NRLI 01549 0 mdash mdash 62570
SI 01549 0 mdash mdash 67662
Number 11
SQP 325 01648 0 01648 NRSUMT 325 01648 0 01648 NRLI 27901 0 162 times 10
minus9
162 times 10minus9 14778
SI 27901 0 586 times 10minus10 586 times 10minus10 21946
Number 12
SQP minus02490 40 times 10minus3 0 40 times 10
minus3 NRSUMT minus02490 40 times 10
minus3 0 40 times 10minus3 NR
LI minus02500 0 285 times 10minus8
285 times 10minus8
54771
SI minus02500 0 390 times 10minus10 390 times 10minus10 54548
Number 13
SQP 168 times 10minus2 01748 0 01748 NR
SUMT 167 times 10minus2 01678 0 01678 NR
LI 143 times 10minus2 0 118 times 10minus9 118 times 10minus9 42599SI 144 times 10
minus2
70 times 10minus3
632 times 10minus9
70 times 10minus3
49027
Number 14
SQP 37220 00961 0 00961 NRSUMT 37700 01103 0 01103 NRLI 33956 0 786 times 10minus7 786 times 10minus7 104124SI 33965 265 times 10
minus4
276 times 10minus6
267 times 10minus4
115144
8 Mathematical Problems in Engineering
Table 1 Continued
Problem Algorithm 119869 119864119869
120593119891
119870120595
Time
Number 15
SQP 103 times 10minus3 43368 0 43368 NR
SUMT 929 times 10minus4 38135 0 38135 NR
LI 193 times 10minus4 0 810 times 10minus10 810 times 10minus10 33354SI 194 times 10
minus4
520 times 10minus3
164 times 10minus9
520 times 10minus3
34014
Number 16
SQP 22120 00753 0 00753 NRSUMT 22080 00734 0 00734 NRLI 20571 0 394 times 10
minus12
394 times 10minus12
46982
SI 20571 0 560 times 10minus13 560 times 10minus13 70423
Number 17
SQP minus88690 225 times 10minus5 0 225 times 10
minus5 NRSUMT minus88690 225 times 10
minus5 0 225 times 10minus5 NR
LI minus88692 0 145 times 10minus10 145 times 10minus10 32183SI minus88692 0 279 times 10
minus10
279 times 10minus10
30959
Number 18
SQP 00368 01288 mdash mdash NRSUMT 00386 01840 mdash mdash NRLI 00326 0 mdash mdash 55163
SI 00326 0 mdash mdash 60671
Number 19
SQP 03439 49293 0 49293 NRSUMT 03428 49103 0 49103 NRLI 00689 01879 410 times 10
minus3
01919 164509
SI 00580 0 405 times 10minus4 405 times 10minus4 178783
Number 20
SQP 7752 12554 0 12554 NRSUMT 7683 12353 0 12353 NRLI 343716 232 times 10
minus5
225 times 10minus4
248 times 10minus4 81519
SI 343708 0 157 times 10minus4 157 times 10minus4 80420
aNot reported
the relative error of 119869 and the factor of the proposed algorithmis better and so the proposed algorithm is more robust thanCGA
513 FFRP Problem [9] The third NOCP in the Appendixis Free Floating Robot Problem FFRP which has six statevariables and four control variables FFRP has been solvedby CGA and the proposed algorithm with the MHGArsquosparameters as 119873
1199051= 31 119873
1199052= 71 119873
119892= 300 and
119873119894
= 200 This problem has six final state constraints120595 = [119909
1minus 4 119909
2 1199093minus 4 119909
4 1199095 1199096]119879 The numerical results
are shown in Table 1 The values of 119869 119864119869 120593119891and 119870
120595for LI
and SI methods separately are less than CGA Thereforethe proposed algorithm can achieve much better qualitysolutions than the CGA with reasonable computational time
514 MSNIC Problem [20] For the forth NOCP in theAppendix which is a Mathematical System with NonlinearInequality Constraint NSNIC the numerical results are com-pared with IPSO MSNIC contains an inequality constraint119889(119909 119905) = 119909
2(119905) + 05 minus 8(119905 minus 05)
2
le 0 The problem solvedby several numerical methods as [24 31] From [20] IPSOmethod could achieved more accurate results than men-tioned numericalmethods AlsoMSNIC can be solved by theproposed algorithm with the MHGArsquos parameters as 119873
1199051=
31 1198731199052
= 91 119873119892= 100 and 119873
119894= 60 From the forth row
of Table 1 the absolute error of 119869 119864119869 for LI and SI methods
equal 0 and 589 times 10minus4 respectively which are less than
IPSOrsquos 00165Subplots (a) and (b) in Figure 2 show the graphs of
the convergence rate for the performance index and theinequality constraint respect to the number of iterationrespectively
515 CSTCR Problem [20] The fifth NOCP in the Appendixis a model of a nonlinear Continuous Stirred-tank ChemicalReactor CSTCR It has two state variables 119909
1(119905) and 119909
2(119905)
as the deviation from the steady-state temperature and con-centration and one control variable 119906(119905) which represent theeffect of the flow rate of cooling fluid on chemical reactorTheobjective is to maintain the temperature and concentrationclose to steady-state values without expending large amountof control effort Also this is a benchmark problem in thehandbook of test problems in local and global optimization[32] which is a multimodal optimal control problem [33]It involves two different local minima The values of theperformance indexes for these solutions equal 0244 and0133 Similarly to the MSNIC the numerical results of theproposed algorithm with the MHGArsquos parameters as 119873
1199051=
31 1198731199052
= 51 119873119892= 100 and 119873
119894= 50 are compared with
IPSO From Table 1 the performance index 119869 for LI and SImethods is equal to 119869
lowast
= 01120 which is less than IPSOs01355
Mathematical Problems in Engineering 9
Table 2 The problem parameters for NOCPs in the Appendix
Parameters Problem number1 2 3 4 5 6 7 8 9 10
119906left minus05 minus15 minus15 minus20 0 minus2 0 minus2 minus1 minus20
119906right 2 2 10 20 5 2 1 2 1 20119872119894
103
102 70 1 mdash 1 mdash 10
2 mdash 102
120576 10minus12
10minus10
10minus3 mdash mdash mdash mdash mdash mdash 10
minus9
Parameters Problem no11 12 13 14 15 16 17 18 19 20
119906left minus5 minus1 minus2 minus120587 minus1 minus3 minus30 minus1 minus15 minus15
119906right 5 1 2 120587 1 3 30 1 10 10119872119894
10 102 mdash 10
2
102
102 mdash 10 70 10
minus3
120576 10minus9
10minus11
10minus11 mdash 10
minus10
10minus10
10minus10 mdash 10
minus3
10minus4
0 5 10 15 20 25 30 35 40 45 50017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 2 Graphical results of MSNIC problem using SI method (a) convergence rate of the performance index and (b) the inequalityconstraint 119889(119909 119905) le 0 in respect to the number of iterations
52 Comparison with Numerical Methods For NOCPs num-bers (6)ndash(20) the comparison are done with some numericalmethods Unfortunately for these methods usually the finalstate constraints and the required computational time arenot reported which are shown with NR in Table 1 but thesevalues are reported for both LI and SI methods in Table 1For all NOCPs in this section the MHGArsquos parameters areconsidered as 119873
1199051= 31 119873
1199052= 51 119873
119892= 100 119873
119894= 50 with
the problem parameters in Table 2
521 Compared with Bezier [34] The NOCP number (6) inthe Appendix has exact solution which has an inequalityconstraint as 119889(119909
1 119905) = minus6 minus 119909
1(119905) le 0 The exact value of
performance index equals 119869lowast = minus55285 [35] This problemhas been solved by a numerical method proposed in [34]called Bezier From sixth row of Table 1 the absolute error
of the LI and SI methods equal 00177 which is less thanBezierrsquos 00251
Figure 3 shows the graphs of the convergence rate of theperformance index subplot (a) and inequality constraintsubplot (b) respect to the number of iteration using SImethod
522 Compared with HPM [36] For NOCP number (7) inthe Appendix which is a constraint nonlinear model thenumerical results of the proposed algorithm are comparedwith HPM proposed in [36] This NOCP has a final stateconstraint as 120595 = 119909 minus 05 = 0 From [36] the norm offinal state constraint for HPM equals 42times10minus6 however thiscriterion for the LI and SI methods equals 235 times 10
minus9 and120593lowast
119891= 282 times 10
minus10 respectively From Table 1 it is obviousthat the obtained values of the performance index the norm
10 Mathematical Problems in Engineering
2 4 6 8 10 12 14minus5353
minus5352
minus5351
minus535
minus5349
minus5348
minus5347
minus5346
minus5345
minus5344
minus5343
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 3 Graphical results for NOCP number (6) using SI method (a) convergence rate for the performance index and (b) the inequalityconstraint 119889(119909
1 119905) le 0 in respect to the number of iterations
of final state constraint and 119870120595for the SI method are more
accurate than LI than HPMmethods
523 Compared with SQP and SUMT [26] For the NOCPsnumbers (8)ndash(20) in the Appendix the numerical resultsof LI and SI methods are compared with two numericalmethods contain SQP and SUMT proposed in [26] Amongthese NOCPs only two problems numbers (8) and (18) areunconstrained and the others have at least one constraintfinal state constraint or inequality constraint All of theseNOCPs solved by the proposed algorithm with the problemparameters in Table 2 and their results are summarized inTable 1 Because the final state constraints in these methodsare not reported we let 120593
119891= 0 to calculate the factor119870
120595
Figures 4ndash16 show the graphical results of NOCPsnumbers (8)ndash(20) in the Appendix using SI method Forunconstrained NOCPs numbers (8) and (18) only the graphof convergent rate of performance index is shown see Figures4 and 14 For constraint NOCPs and NOCPs numbers(11)ndash(17) and numbers (19)-(20) with final state constraintthe graphs of convergent rates of performance index and theerror of final state constraint are shown see Figures 7ndash13 and15-16 For the constraint NOCPs with inequality constraintsNOCPs numbers (9) and (10) the graphs of convergent rateof performance index and inequality constraint are shownsee Figures 5 and 6
Table 1 shows that the proposed algorithm LI and SImethods was 100 percent successful in point of views theperformance index 119869 and the factor119870
120595 numerically So the
proposed algorithm provides robust solutions with respect tothe other mentioned numerical or metaheuristic methodsTo compare 119869 in LI and SI methods in 35 percent of NOCPsLI is more accurate than SI in 10 percent SI is more accurate
1 15 2 25 3 35 40
002
004
006
008
01
012
Iter
Perfo
rman
ce in
dex
Figure 4 Convergence rate for the performance index of SI methodfor NOCP number (8)
than LI and in 55 percent are same In point of view the finalconditions 65 percent of NOCPs in the Appendix have finalstate constraints In all of them 120593
119891in the proposed algorithm
is improved except CRP problem however the factor of theproposed algorithm is the best yet In 54 percent of theseNOCPs LI is better performance and in 46 percent SI isbetter Also in point of the running time Time LI is same asSI that is in 50 percent of NOCPs LI and in 50 percent SI isbetter than another So the proposed algorithmcould providevery suitable solutions in a reasonable computational timeAlso for more accurate comparison of LI and SI methods astatistical approach will done in next section
To compare with CGA the mean of relative error of 119869 119864119869
for CGA LI and SImethods inNOCPs (1)ndash(3) equal 056680 and 298 times 10
minus4 respectively Also the mean for the error
Mathematical Problems in Engineering 11
5 10 15 20 25165
17
175
18
185
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3 35 4 45 50
005
01
015
02
025
TimeIn
equa
lity
cons
trai
nt(b)
Figure 5 (a) Convergence rate for the performance index and (b) the inequality constraint of SI method for NOCP number (9)
5 10 15 20 25 30 35 40 45 500155
016
0165
017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 6 (a) Convergence rate of the performance index and (b) the inequality constraint for SI method for NOCP number (10)
of the final state constraints 120593119891 are 16 times 10
minus3 407 times 10minus6
and 301 times 10minus6 respectively and for 119870
120595 these values equal
05883 407 times 10minus6 and 302 times 10
minus4 Thus we can say that thefeasibility of the solutions given by the proposed algorithmand CGA are competitive
From Table 1 to compare with numerical methods SQPand SUMT in NOCPs (8)ndash(20) the mean of 119864
119869for LI SI
SQP and SUMT equals 00145 00209 169 times 108 and 136 times
108 respectively Also the mean for the error of final state
constraints for these NOCPs equal 334 times 10minus4 441 times 10
minus50 and 0 respectively For 119870
120595 these values are 00213 00014
12263 and 11644 Therefore the performance index 119869 andthe factor 119870
120595 for the LI and SI methods are more accurate
than SQP and SUMT So the proposed algorithm gave morebetter solution in comparison with the numerical methods
12 Mathematical Problems in Engineering
5 10 15 20 25minus025
minus0245
minus024
minus0235
minus023
minus0225
Iter
Perfo
rman
ce in
dex
(a)
1 15 2 25 3 35 4 45 5 55 60
02
04
06
08
1
12
14
16
IterFi
nal c
ondi
tion
times10minus5
(b)
Figure 7 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (11)
2 4 6 8 10 12 14minus025
minus0245
minus024
minus0235
minus023
minus0225
minus022
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
02
04
06
08
1
12
14
16
18
2
Iter
Fina
l con
ditio
n
times10minus5
(b)
Figure 8 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (12)
Therefore based on this numerical studywe can concludethat the proposed algorithm outperform well-known numer-ical method Since the algorithms were not implementedon the same PC the computational times of them are notcompetitive Therefore we did not give the computationaltimes in bold in Table 1
6 Sensitivity Analysis andComparing LI and SI
In this section two statistical analysis based on the one-way analysis of variance (ANOVA) used for investigatingthe sensitivity of MHGA parameters and Mann-Whitney
Mathematical Problems in Engineering 13
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
07
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 9 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (13)
5 10 15 20 25 30
35
4
45
5
55
6
65
7
75
8
85
9
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
Iter
Fina
l con
ditio
n
(b)
Figure 10 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (14)
applied comparing LI and SI methods are done by thestatistical software IBM SPSS version 21
61 Sensitivity Analysis In order to survey the sensitivityof the MHGA parameters from the proposed algorithmthe VDP problem is selected for example The influenceof these parameters are investigated for this NOCP on
the dependent outputs consist of the performance index119869 the relative error of 119869 119864
119869 the required computational
time Time the error of final state constraints 120593119891and the
factor 119870120595 The independent parameters are consist of the
number of time nodes in both two phases 1198731199051and 119873
1199052 the
size of population in both two phases 1198731199011
and 1198731199012 the
maximum number of generations without improvement119873119894
14 Mathematical Problems in Engineering
5 10 15 20 25 30 35 40 45 50
00955
0096
00965
0097
00975
0098
00985
0099
00995
Iter
Perfo
rman
ce in
dex
(a)
2 4 6 8 10 12 14 16 18 200
05
1
15
2
25
3
35
4
45
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 11 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (15)
5 10 15 20 25
21
22
23
24
25
26
27
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
1
2
3
4
5
6
Iter
Fina
l con
ditio
n
times10minus3
(b)
Figure 12 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (16)
and the maximum number of generations 119873119892 Because the
mutation implementation probability119875119898 has a less influence
in numerical results it is not considered Also 119904119902119901119898119886119909119894119905119890119903 ischanged in each iteration of the proposed algorithm so it isnot considered too
At first we selected at least four constant value foreach of parameters and then in each pose 35 differentruns were made independently The statistical analysis isdone based on ANOVA The descriptive statistics whichcontains the number of different runs (119873) the mean of each
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7minus887
minus886
minus885
minus884
minus883
minus882
minus881
minus88
minus879
minus878
minus877
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 70
01
02
03
04
05
06
07
08
09
1
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 13 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (17)
1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333
Iter
Perfo
rman
ce in
dex
Figure 14 Convergence rate for the performance index of SImethod for NOCP numbers (18)
output in119873 different runs (Mean) standard deviation (Sd)maximum (Max) and minimum (Min) of each outputcould be achieved Among of the MHGA parameters wepresent the descriptive statistics of the ANOVA only for the1198731199051parameter which is reported in Table 3Table 4 summarized the statistical data contain the test
statistics (119865) and 119875-values of ANOVA tests Sensitivityanalysis for each parameter separately are done based on thistable as follows
1198731199051 From Table 4 the significant level or 119875-value for119870120595
is equal to 0279 which is greater than 005So from ANOVA 119873
1199051parameter has no significant
effect on the factor 119870120595 With similar analysis this
parameter has no effect on the other outputs except
required computational time Time because its 119875-value is equal to 0 which is less than 005 that is thisoutput sensitive to the parameter119873
1199051
1198731199052 From the second row of Table 4 the outputs 119869 119864
119869119870120595
and Time are sensitive to the parameter 1198731199052 and the
only output 120593119891is independent
1198731199011 From the third row of Table 4 only the computationalrequired time Time is sensitive because the its 119875-value is equal to 0006 which is less than 005 and theother outputs contain 119869 119864
119869 120593119891119870120595 are independent
1198731199012 From the forth row of Table 4 the output 120593
119891is
independent respect to the parameter 1198731199012 but other
outputs contain 119869 119864119869 Time 119870
120601are sensitive to this
parameter119873119892 From the fifth row of Table 4 the outputs 119869 119864
119869and
119870120595are independent to the parameter 119873
119892 and other
outputs contain 120593119891and Time are sensitive respect to
this parameter119873119894 The sensitivity analysis is similar to119873
1199011
From above cases it is obvious that all parameters canbe effect on the required computational time except 119873
119892
Moreover intuitions shows the normof final state constraints120593119891is independent output with respect to all parameters that
is any of the parameters could not effect on this output and itis not sensitive with respect to any of the MHGA parameters
62 Comparison of LI and SI To compare the efficiency ofthe LI and SI methods for NOCPs in the Appendix weused the Mann-Whitney nonparametric statistical test [37]
16 Mathematical Problems in Engineering
1 15 2 25 3 35 40
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
200
400
600
800
1000
1200
1400
1600
IterFi
nal c
ondi
tion
(b)
Figure 15 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (19)
2 4 6 8 10 12 14 16 18 20
50
100
150
200
250
300
350
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 10 11 120
05
1
15
2
25
3
Iter
Fina
l con
ditio
n
(b)
Figure 16 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (20)
Using the nonparametric statistical tests for comparing theperformance of several algorithms presented in [38] Forthis purpose At first 15 different runs for each of NOCPwere made Table 5 shows the mean of numerical resultsseparately Here we apply the fixMHGAparameters as119873
1199011=
91198731199012
= 121198731199051= 31119873
1199052= 91119873
119892= 100 and119873
119894= 50 with
the previous problem parameters in Table 2 Comparison
criterions contain 119869 120593119891 119870120595and Time The results of Mann-
Whitney test are shown in Tables 6ndash9 From Table 6 since119885 = 0 gt 119885
001= minus234 then the average relative error of
the LI and SI methods are same with the significant level 001In other words with probability 99 we can say that LI andSI methods have same effectiveness form the perspective oferror for 119869 Similarly the results of Tables 7ndash9 indicate that LI
Mathematical Problems in Engineering 17
Table 3 Descriptive statistics for the sensitivity analysis of1198731199051 for VDP problem
Output 1198731199051
119873 Mean Sd Min Max
119869
11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449
Total 140 15357 00294 15285 18449
120593119891
11 35 281 times 10minus8
952 times 10minus8
1 times 10minus12
452 times 10minus7
21 35 467 times 10minus8
235 times 10minus7
1 times 10minus12
141 times 10minus6
31 35 163 times 10minus8
491 times 10minus8
1 times 10minus12
221 times 10minus7
41 35 817 times 10minus8
285 times 10minus7
1 times 10minus12
126 times 10minus6
Total 140 426 times 10minus8 191 times 10minus7 1 times 10minus12 141 times 10minus6
Time
11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514
Total 140 1142131 410891 456615 2849514
119864119869
11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070
Total 140 00047 00192 0 02070
119870120595
11 35 00031 00045 475 times 10minus9
00204
21 35 00027 00064 550 times 10minus10
00369
31 35 00029 00050 154 times 10minus13
00271
41 35 00111 00397 193 times 10minus10
02070
Total 140 00047 00192 154 times 10minus13 02070
Table 4 Summary statistical data of ANOVA test for the parameters1198731199051 1198731199052 1198731199011 1198731199012 119873119892 119873119894
Parameters 119869 119864119869
120593119891
Time 119870120595
Test statistic (119865)
1198731199051
1294 1296 0624 1079 12961198731199052
110572 393 0868 5939 3931198731199011
1945 1835 1271 4317 18351198731199012
2740 2478 1011 2302 2478119873119892
3541 3591 0491 0890 3591119873119894
0495 0309 0301 5849 1046
119875 value
1198731199051
0280 0279 0601 0 02791198731199052
0 0005 0489 0 00051198731199011
0125 0143 0286 0006 01431198731199012
0021 0034 0413 0 0034119873119892
0009 0008 0743 0472 0008119873119894
0739 0819 0877 0 0386
and SI methods from the perspective of errors for 120593119891 Time
and119870120595 have same behaviour
7 The Impact of SQP
In this section we investigation the impact of the SQP on theproposed algorithm For this purpose we remove the SQPfrom Algorithm 2 as without SQP algorithm For comparingthe numerical results with the previous results the required
running time in each NOCP is considered fixed which is themaximum of running time in 12 different runs were done inSection 5 Also the all of the parameters for each problemwasset as same as Section 5 The numerical results of the withoutSQP algorithm is summarized in Table 10 By comparing theresults of Tables 10 and 1 it is obvious that for all NOCPsin the Appendix the obtained values of the performanceindex and the norm of final state constraint for the proposedalgorithm (Algorithm 2) are more accurate than the withoutSQP algorithm
18 Mathematical Problems in Engineering
Table 5 The mean of numerical results of 15 different runs for NOCPs in the Appendix using LI and SI methods
Problem LI SI119869 120593
119891119870120595
Time 119869 120593119891
119870120595
TimeVDP 15977 157 times 10
minus9
170 times 10minus3 17720 16002 667 times 10
minus8
326 times 10minus3 15757
CRP 00126 307 times 10minus8
307 times 10minus8 19714 00119 161 times 10
minus8
161 times 10minus8 19532
FFPP 4774 805 times 10minus5 03277 133280 5351 592 times 10
minus5 04868 137989MSNIC 01702 mdash mdash 13678 01703 mdash mdash 14092CSTCR 01120 mdash mdash 17385 01120 mdash mdash 17606Number 6 minus54307 mdash mdash 10541 minus54308 mdash mdash 10372Number 7 02015 326 times 10
minus9
695 times 10minus6 20807 02020 313 times 10
minus9
662 times 10minus6 21019
Number 8 630 times 10minus15 mdash mdash 6376 124 times 10
minus14 mdash mdash 5837Number 9 16512 mdash mdash 59239 16512 mdash mdash 58125Number 10 01550 mdash mdash 63153 01550 mdash mdash 65597Number 11 29701 270 times 10
minus9
272 times 10minus7 22860 29701 337 times 10
minus9
811 times 10minus7 23276
Number 12 minus02499 201 times 10minus8
464 times 10minus5 54718 minus02499 115 times 10
minus9
827 times 10minus5 52765
Number 13 00147 115 times 10minus8
00252 57797 00151 139 times 10minus8
00472 66098Number 14 34761 672 times 10
minus6
00237 104585 34290 757 times 10minus6
950 times 10minus3 108767
Number 15 195 times 10minus4
111 times 10minus8
569 times 10minus3 363 194 times 10
minus4
704 times 10minus9
486 times 10minus3 365
Number 16 20571 166 times 10minus10
166 times 10minus10 54391 20571 115 times 10
minus10
115 times 10minus10 54431
Number 17 minus88692 803 times 10minus8
803 times 10minus8 22744 minus88692 661 times 10
minus8
661 times 10minus8 23254
Number 18 00326 mdash mdash 65957 00326 mdash mdash 70897Number 19 011588 655 times 10
minus4 06828 161813 01091 884 times 10minus4 08808 159990
Number 20 5242 00176 05429 84165 4464 453 times 10minus4 03765 86891
Table 6 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 119869
Method Mean rank Sum of ranks Test statistics ValueLI 2050 4100 Mann-Whitney 119880 200SI 2050 4100 Wilcoxon119882 410mdash mdash mdash 119885 0
Table 7 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 120593119891
Method Mean rank Sum of ranks Test statistics ValueLI 1362 177 Mann-Whitney 119880 83SI 1338 174 Wilcoxon119882 174mdash mdash mdash 119885 minus0077
Table 8 Results of Mann-Whitney test on relative errors of the pair (LI SI) for Time
Method Mean rank Sum of ranks Test statistics ValueLI 2025 4050 Mann-Whitney 119880 195SI 2075 4150 Wilcoxon119882 405mdash mdash mdash 119885 minus0135
Table 9 Results of Mann-Whitney test on relative errors of the pair (LI SI) for119870120595
Method Mean rank Sum of ranks Test statistics ValueLI 1354 1760 Mann-Whitney 119880 84SI 1346 1750 Wilcoxon119882 175mdash mdash mdash 119885 minus0026
Mathematical Problems in Engineering 19
Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix
Problem no1 2 3 4 5 6 7 8 9 10
119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048
120601119891
330 times 10minus9
135 times 10minus5 11445 mdash mdash mdash 758 times 10
minus9 mdash mdash mdashProblem no
11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10
minus4 23182 minus90882 00336 2513 20274120601119891
167 times 10minus5
468 times 10minus9 00161 433 times 10
minus4
594 times 10minus3
156 times 10minus5 164 mdash 1706 25283
8 Conclusions
Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters
Appendix
Test Problems
The following problems are presented using notation given in(1)ndash(6)
(1) (VDP) [9] 119892 = (12)(1199092
1+ 1199092
2+ 1199062
) 1199050= 0 119905119891= 5 119891 =
[1199092 minus1199092+(1minus119909
2
1)1199092+119906]119879 1199090= [1 0]
119879120595 = 1199091minus1199092+1
(2) (CRP) [9] 119892 = (12)(1199092
1+1199092
2+01119906
2
) 1199050= 0 119905119891= 078
119891 = [1199091minus2(1199091+025)+(119909
2+05) exp(25119909
1(1199091+2))minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(3) (FFRP) [9] 119892 = (12)(1199062
1+ 1199062
2+ 1199062
3+ 1199062
4) 1199050
=
0 119905119891
= 5 119891 = [1199092 (110)((119906
1+ 1199062) cos119909
5minus
(1199062+ 1199064) sin119909
5) 1199094 (110)((119906
1+ 1199063) sin119909
5+ (1199062+
1199064) cos119909
5) 1199096 (512)((119906
1+ 1199063) minus (119906
2+ 1199064))]119879 1199090=
[0 0 0 0 0 0]119879 120595 = [119909
1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879
(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909
2 minus1199092+
119906 1199092
1+ 1199092
2+ 0005119906
2
]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909
2+
05 minus 8(119905 minus 05)2
]119879 1199090= [0 minus1 0]
119879
(5) (CSTCR) [20] 119892 = 1199092
1+ 1199092
2+ 01119906
2 1199050= 0 119905119891= 078
119891 = [minus(2 + 119906)(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+
2)) 05 minus 1199092minus (1199092+ 05) exp(25119909
1(1199091+ 2))]
119879 1199090=
[009 009]119879
(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909
2 119906]119879 119889 = [minus(2 +
119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]
119879(7) [36] 119892 = 119906
2 1199050= 0 119905
119891= 1 119891 = (12)119909
2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05
(8) [26] 119892 = 1199092cos2119906 119905
0= 0 119905119891= 120587 119891 = sin(1199062) 119909
0=
1205872(9) [26] 119892 = (12)(119909
2
1+ 1199092
2+ 1199062
) 1199050= 0 119905
119891= 5 119891 =
[1199092 minus1199091+ (1 minus 119909
2
1)1199092+ 119906]119879 119889 = minus(119909
2+ 025) 119909
0=
[1 0]119879
(10) [26] 119892 = 1199092
1+ 1199092
2+ 0005119906
2
1199050= 0 119905
119891= 1 119891 =
[1199092 minus1199092+119906]119879
119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus
05) minus 05 minus 1199092)]119879
1199090= [0 minus1]
119879(11) [26] 119892 = (12)119906
2 1199050= 0 119905
119891= 2 119891 = [119909
2 119906]119879 1199090=
[1 1]119879 120595 = [119909
1 1199092]119879
(12) [26] 119892 = minus1199092 1199050= 0 119905
119891= 1 119891 = [119909
2 119906]119879 119889 =
minus(1 minus 119906)(1 + 119906) 1199090= [0 0]
119879 120595 = 1199092
(13) [26] 119892 = (12)(1199092
1+ 1199092
2+ 01119906
2
) 1199050= 0 119905
119891= 078
119891 = [minus2(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+ 2)) minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(14) [26] 119892 = (12)1199062 1199050= 0 119905
119891= 10 119891 = [cos 119906 minus
1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909
0= [366 minus186]
119879120595 = [119909
1 1199092]119879
(15) [26]119892 = (12)(1199092
1+1199092
2) 1199050= 0 119905119891= 078119891 = [minus2(119909
1+
025)+(1199092+05) exp(25119909
1(1199091+2))minus(119909
1+025)119906 05minus
1199092minus(1199092+05) exp(25119909
1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(16) [26] 120601 = 1199093 1199050= 0 119905
119891= 1 119891 = [119909
2 119906 (12)119906
2
]119879
119889 = 1199091minus 19 119909
0= [0 0 0]
119879 120595 = [1199091 1199092+ 1]119879
(17) [26] 120601 = minus1199093 1199050= 0 119905
119891= 5 119891 = [119909
2 minus2 + 119906119909
3
minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909
0= [10 minus2 10]
119879120595 = [119909
1 1199092]119879
(18) [26] 120601 = (1199091minus1)2
+1199092
2+1199092
3 119892 = (12)119906
2 1199050= 0 119905119891= 5
119891 = [1199093cos 119906 119909
3sin 119906 sin 119906]119879 119909
0= [0 0 0]
119879
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
Table 1 The best numerical results in 12 different runs of the SI and LI methods for NOCPs in the Appendix
Problem Algorithm 119869 119864119869
120593119891
119870120595
Time
VDPCGA 17404 00912 267 times 10
minus11
00912 50128LI 15949 0 108 times 10
minus13 108 times 10minus13 28282SI 15950 627 times 10
minus5 999 times 10minus14 627 times 10minus5 23914
CRPCGA 163 times 10
minus2 02835 757 times 10minus10 02835 50128LI 127 times 10minus2 0 115 times 10
minus9 115 times 10minus9 17194
SI 127 times 10minus2 0 599 times 10minus9
599 times 10minus9
12405
FFRPCGA 8363 13256 465 times 10
minus3
13302 1413LI 3596 0 122 times 10
minus5 122 times 10minus5 126858
SI 3599 834 times 10minus4 901 times 10minus6 843 times 10
minus4 134340
MSNICIPSO 01727 00165 mdash mdash 12626LI 01699 0 mdash mdash 13882
SI 01700 589 times 10minus4 mdash mdash 13868
CSTCRIPSO 01355 02098 mdash mdash 3134
LI 01120 0 mdash mdash 10358
SI 01120 0 mdash mdash 10673
No 6Bezier minus53898 00251 mdash mdash NRa
LI minus54309 00177 mdash mdash 13838
SI minus54309 00177 mdash mdash 10734
Number 7HPM 02353 01677 420 times 10
minus6 01677 NRLI 02015 0 235 times 10
minus9
235 times 10minus9 21392
SI 02015 0 282 times 10minus10 282 times 10minus10 20696
Number 8
SQP 636 times 10minus6
220 times 109 mdash mdash NR
SUMT 515 times 10minus6
178 times 109 mdash mdash NR
LI 289 times 10minus15 0 mdash mdash 7282SI 364 times 10
minus15
02595 mdash mdash 6868
Number 9
SQP 17950 00873 mdash mdash NRSUMT 17980 00891 mdash mdash NRLI 16509 0 mdash mdash 63910SI 16509 0 mdash mdash 5950
Number 10
SQP 02163 03964 mdash mdash NRSUMT 01703 00994 mdash mdash NRLI 01549 0 mdash mdash 62570
SI 01549 0 mdash mdash 67662
Number 11
SQP 325 01648 0 01648 NRSUMT 325 01648 0 01648 NRLI 27901 0 162 times 10
minus9
162 times 10minus9 14778
SI 27901 0 586 times 10minus10 586 times 10minus10 21946
Number 12
SQP minus02490 40 times 10minus3 0 40 times 10
minus3 NRSUMT minus02490 40 times 10
minus3 0 40 times 10minus3 NR
LI minus02500 0 285 times 10minus8
285 times 10minus8
54771
SI minus02500 0 390 times 10minus10 390 times 10minus10 54548
Number 13
SQP 168 times 10minus2 01748 0 01748 NR
SUMT 167 times 10minus2 01678 0 01678 NR
LI 143 times 10minus2 0 118 times 10minus9 118 times 10minus9 42599SI 144 times 10
minus2
70 times 10minus3
632 times 10minus9
70 times 10minus3
49027
Number 14
SQP 37220 00961 0 00961 NRSUMT 37700 01103 0 01103 NRLI 33956 0 786 times 10minus7 786 times 10minus7 104124SI 33965 265 times 10
minus4
276 times 10minus6
267 times 10minus4
115144
8 Mathematical Problems in Engineering
Table 1 Continued
Problem Algorithm 119869 119864119869
120593119891
119870120595
Time
Number 15
SQP 103 times 10minus3 43368 0 43368 NR
SUMT 929 times 10minus4 38135 0 38135 NR
LI 193 times 10minus4 0 810 times 10minus10 810 times 10minus10 33354SI 194 times 10
minus4
520 times 10minus3
164 times 10minus9
520 times 10minus3
34014
Number 16
SQP 22120 00753 0 00753 NRSUMT 22080 00734 0 00734 NRLI 20571 0 394 times 10
minus12
394 times 10minus12
46982
SI 20571 0 560 times 10minus13 560 times 10minus13 70423
Number 17
SQP minus88690 225 times 10minus5 0 225 times 10
minus5 NRSUMT minus88690 225 times 10
minus5 0 225 times 10minus5 NR
LI minus88692 0 145 times 10minus10 145 times 10minus10 32183SI minus88692 0 279 times 10
minus10
279 times 10minus10
30959
Number 18
SQP 00368 01288 mdash mdash NRSUMT 00386 01840 mdash mdash NRLI 00326 0 mdash mdash 55163
SI 00326 0 mdash mdash 60671
Number 19
SQP 03439 49293 0 49293 NRSUMT 03428 49103 0 49103 NRLI 00689 01879 410 times 10
minus3
01919 164509
SI 00580 0 405 times 10minus4 405 times 10minus4 178783
Number 20
SQP 7752 12554 0 12554 NRSUMT 7683 12353 0 12353 NRLI 343716 232 times 10
minus5
225 times 10minus4
248 times 10minus4 81519
SI 343708 0 157 times 10minus4 157 times 10minus4 80420
aNot reported
the relative error of 119869 and the factor of the proposed algorithmis better and so the proposed algorithm is more robust thanCGA
513 FFRP Problem [9] The third NOCP in the Appendixis Free Floating Robot Problem FFRP which has six statevariables and four control variables FFRP has been solvedby CGA and the proposed algorithm with the MHGArsquosparameters as 119873
1199051= 31 119873
1199052= 71 119873
119892= 300 and
119873119894
= 200 This problem has six final state constraints120595 = [119909
1minus 4 119909
2 1199093minus 4 119909
4 1199095 1199096]119879 The numerical results
are shown in Table 1 The values of 119869 119864119869 120593119891and 119870
120595for LI
and SI methods separately are less than CGA Thereforethe proposed algorithm can achieve much better qualitysolutions than the CGA with reasonable computational time
514 MSNIC Problem [20] For the forth NOCP in theAppendix which is a Mathematical System with NonlinearInequality Constraint NSNIC the numerical results are com-pared with IPSO MSNIC contains an inequality constraint119889(119909 119905) = 119909
2(119905) + 05 minus 8(119905 minus 05)
2
le 0 The problem solvedby several numerical methods as [24 31] From [20] IPSOmethod could achieved more accurate results than men-tioned numericalmethods AlsoMSNIC can be solved by theproposed algorithm with the MHGArsquos parameters as 119873
1199051=
31 1198731199052
= 91 119873119892= 100 and 119873
119894= 60 From the forth row
of Table 1 the absolute error of 119869 119864119869 for LI and SI methods
equal 0 and 589 times 10minus4 respectively which are less than
IPSOrsquos 00165Subplots (a) and (b) in Figure 2 show the graphs of
the convergence rate for the performance index and theinequality constraint respect to the number of iterationrespectively
515 CSTCR Problem [20] The fifth NOCP in the Appendixis a model of a nonlinear Continuous Stirred-tank ChemicalReactor CSTCR It has two state variables 119909
1(119905) and 119909
2(119905)
as the deviation from the steady-state temperature and con-centration and one control variable 119906(119905) which represent theeffect of the flow rate of cooling fluid on chemical reactorTheobjective is to maintain the temperature and concentrationclose to steady-state values without expending large amountof control effort Also this is a benchmark problem in thehandbook of test problems in local and global optimization[32] which is a multimodal optimal control problem [33]It involves two different local minima The values of theperformance indexes for these solutions equal 0244 and0133 Similarly to the MSNIC the numerical results of theproposed algorithm with the MHGArsquos parameters as 119873
1199051=
31 1198731199052
= 51 119873119892= 100 and 119873
119894= 50 are compared with
IPSO From Table 1 the performance index 119869 for LI and SImethods is equal to 119869
lowast
= 01120 which is less than IPSOs01355
Mathematical Problems in Engineering 9
Table 2 The problem parameters for NOCPs in the Appendix
Parameters Problem number1 2 3 4 5 6 7 8 9 10
119906left minus05 minus15 minus15 minus20 0 minus2 0 minus2 minus1 minus20
119906right 2 2 10 20 5 2 1 2 1 20119872119894
103
102 70 1 mdash 1 mdash 10
2 mdash 102
120576 10minus12
10minus10
10minus3 mdash mdash mdash mdash mdash mdash 10
minus9
Parameters Problem no11 12 13 14 15 16 17 18 19 20
119906left minus5 minus1 minus2 minus120587 minus1 minus3 minus30 minus1 minus15 minus15
119906right 5 1 2 120587 1 3 30 1 10 10119872119894
10 102 mdash 10
2
102
102 mdash 10 70 10
minus3
120576 10minus9
10minus11
10minus11 mdash 10
minus10
10minus10
10minus10 mdash 10
minus3
10minus4
0 5 10 15 20 25 30 35 40 45 50017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 2 Graphical results of MSNIC problem using SI method (a) convergence rate of the performance index and (b) the inequalityconstraint 119889(119909 119905) le 0 in respect to the number of iterations
52 Comparison with Numerical Methods For NOCPs num-bers (6)ndash(20) the comparison are done with some numericalmethods Unfortunately for these methods usually the finalstate constraints and the required computational time arenot reported which are shown with NR in Table 1 but thesevalues are reported for both LI and SI methods in Table 1For all NOCPs in this section the MHGArsquos parameters areconsidered as 119873
1199051= 31 119873
1199052= 51 119873
119892= 100 119873
119894= 50 with
the problem parameters in Table 2
521 Compared with Bezier [34] The NOCP number (6) inthe Appendix has exact solution which has an inequalityconstraint as 119889(119909
1 119905) = minus6 minus 119909
1(119905) le 0 The exact value of
performance index equals 119869lowast = minus55285 [35] This problemhas been solved by a numerical method proposed in [34]called Bezier From sixth row of Table 1 the absolute error
of the LI and SI methods equal 00177 which is less thanBezierrsquos 00251
Figure 3 shows the graphs of the convergence rate of theperformance index subplot (a) and inequality constraintsubplot (b) respect to the number of iteration using SImethod
522 Compared with HPM [36] For NOCP number (7) inthe Appendix which is a constraint nonlinear model thenumerical results of the proposed algorithm are comparedwith HPM proposed in [36] This NOCP has a final stateconstraint as 120595 = 119909 minus 05 = 0 From [36] the norm offinal state constraint for HPM equals 42times10minus6 however thiscriterion for the LI and SI methods equals 235 times 10
minus9 and120593lowast
119891= 282 times 10
minus10 respectively From Table 1 it is obviousthat the obtained values of the performance index the norm
10 Mathematical Problems in Engineering
2 4 6 8 10 12 14minus5353
minus5352
minus5351
minus535
minus5349
minus5348
minus5347
minus5346
minus5345
minus5344
minus5343
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 3 Graphical results for NOCP number (6) using SI method (a) convergence rate for the performance index and (b) the inequalityconstraint 119889(119909
1 119905) le 0 in respect to the number of iterations
of final state constraint and 119870120595for the SI method are more
accurate than LI than HPMmethods
523 Compared with SQP and SUMT [26] For the NOCPsnumbers (8)ndash(20) in the Appendix the numerical resultsof LI and SI methods are compared with two numericalmethods contain SQP and SUMT proposed in [26] Amongthese NOCPs only two problems numbers (8) and (18) areunconstrained and the others have at least one constraintfinal state constraint or inequality constraint All of theseNOCPs solved by the proposed algorithm with the problemparameters in Table 2 and their results are summarized inTable 1 Because the final state constraints in these methodsare not reported we let 120593
119891= 0 to calculate the factor119870
120595
Figures 4ndash16 show the graphical results of NOCPsnumbers (8)ndash(20) in the Appendix using SI method Forunconstrained NOCPs numbers (8) and (18) only the graphof convergent rate of performance index is shown see Figures4 and 14 For constraint NOCPs and NOCPs numbers(11)ndash(17) and numbers (19)-(20) with final state constraintthe graphs of convergent rates of performance index and theerror of final state constraint are shown see Figures 7ndash13 and15-16 For the constraint NOCPs with inequality constraintsNOCPs numbers (9) and (10) the graphs of convergent rateof performance index and inequality constraint are shownsee Figures 5 and 6
Table 1 shows that the proposed algorithm LI and SImethods was 100 percent successful in point of views theperformance index 119869 and the factor119870
120595 numerically So the
proposed algorithm provides robust solutions with respect tothe other mentioned numerical or metaheuristic methodsTo compare 119869 in LI and SI methods in 35 percent of NOCPsLI is more accurate than SI in 10 percent SI is more accurate
1 15 2 25 3 35 40
002
004
006
008
01
012
Iter
Perfo
rman
ce in
dex
Figure 4 Convergence rate for the performance index of SI methodfor NOCP number (8)
than LI and in 55 percent are same In point of view the finalconditions 65 percent of NOCPs in the Appendix have finalstate constraints In all of them 120593
119891in the proposed algorithm
is improved except CRP problem however the factor of theproposed algorithm is the best yet In 54 percent of theseNOCPs LI is better performance and in 46 percent SI isbetter Also in point of the running time Time LI is same asSI that is in 50 percent of NOCPs LI and in 50 percent SI isbetter than another So the proposed algorithmcould providevery suitable solutions in a reasonable computational timeAlso for more accurate comparison of LI and SI methods astatistical approach will done in next section
To compare with CGA the mean of relative error of 119869 119864119869
for CGA LI and SImethods inNOCPs (1)ndash(3) equal 056680 and 298 times 10
minus4 respectively Also the mean for the error
Mathematical Problems in Engineering 11
5 10 15 20 25165
17
175
18
185
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3 35 4 45 50
005
01
015
02
025
TimeIn
equa
lity
cons
trai
nt(b)
Figure 5 (a) Convergence rate for the performance index and (b) the inequality constraint of SI method for NOCP number (9)
5 10 15 20 25 30 35 40 45 500155
016
0165
017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 6 (a) Convergence rate of the performance index and (b) the inequality constraint for SI method for NOCP number (10)
of the final state constraints 120593119891 are 16 times 10
minus3 407 times 10minus6
and 301 times 10minus6 respectively and for 119870
120595 these values equal
05883 407 times 10minus6 and 302 times 10
minus4 Thus we can say that thefeasibility of the solutions given by the proposed algorithmand CGA are competitive
From Table 1 to compare with numerical methods SQPand SUMT in NOCPs (8)ndash(20) the mean of 119864
119869for LI SI
SQP and SUMT equals 00145 00209 169 times 108 and 136 times
108 respectively Also the mean for the error of final state
constraints for these NOCPs equal 334 times 10minus4 441 times 10
minus50 and 0 respectively For 119870
120595 these values are 00213 00014
12263 and 11644 Therefore the performance index 119869 andthe factor 119870
120595 for the LI and SI methods are more accurate
than SQP and SUMT So the proposed algorithm gave morebetter solution in comparison with the numerical methods
12 Mathematical Problems in Engineering
5 10 15 20 25minus025
minus0245
minus024
minus0235
minus023
minus0225
Iter
Perfo
rman
ce in
dex
(a)
1 15 2 25 3 35 4 45 5 55 60
02
04
06
08
1
12
14
16
IterFi
nal c
ondi
tion
times10minus5
(b)
Figure 7 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (11)
2 4 6 8 10 12 14minus025
minus0245
minus024
minus0235
minus023
minus0225
minus022
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
02
04
06
08
1
12
14
16
18
2
Iter
Fina
l con
ditio
n
times10minus5
(b)
Figure 8 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (12)
Therefore based on this numerical studywe can concludethat the proposed algorithm outperform well-known numer-ical method Since the algorithms were not implementedon the same PC the computational times of them are notcompetitive Therefore we did not give the computationaltimes in bold in Table 1
6 Sensitivity Analysis andComparing LI and SI
In this section two statistical analysis based on the one-way analysis of variance (ANOVA) used for investigatingthe sensitivity of MHGA parameters and Mann-Whitney
Mathematical Problems in Engineering 13
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
07
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 9 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (13)
5 10 15 20 25 30
35
4
45
5
55
6
65
7
75
8
85
9
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
Iter
Fina
l con
ditio
n
(b)
Figure 10 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (14)
applied comparing LI and SI methods are done by thestatistical software IBM SPSS version 21
61 Sensitivity Analysis In order to survey the sensitivityof the MHGA parameters from the proposed algorithmthe VDP problem is selected for example The influenceof these parameters are investigated for this NOCP on
the dependent outputs consist of the performance index119869 the relative error of 119869 119864
119869 the required computational
time Time the error of final state constraints 120593119891and the
factor 119870120595 The independent parameters are consist of the
number of time nodes in both two phases 1198731199051and 119873
1199052 the
size of population in both two phases 1198731199011
and 1198731199012 the
maximum number of generations without improvement119873119894
14 Mathematical Problems in Engineering
5 10 15 20 25 30 35 40 45 50
00955
0096
00965
0097
00975
0098
00985
0099
00995
Iter
Perfo
rman
ce in
dex
(a)
2 4 6 8 10 12 14 16 18 200
05
1
15
2
25
3
35
4
45
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 11 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (15)
5 10 15 20 25
21
22
23
24
25
26
27
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
1
2
3
4
5
6
Iter
Fina
l con
ditio
n
times10minus3
(b)
Figure 12 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (16)
and the maximum number of generations 119873119892 Because the
mutation implementation probability119875119898 has a less influence
in numerical results it is not considered Also 119904119902119901119898119886119909119894119905119890119903 ischanged in each iteration of the proposed algorithm so it isnot considered too
At first we selected at least four constant value foreach of parameters and then in each pose 35 differentruns were made independently The statistical analysis isdone based on ANOVA The descriptive statistics whichcontains the number of different runs (119873) the mean of each
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7minus887
minus886
minus885
minus884
minus883
minus882
minus881
minus88
minus879
minus878
minus877
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 70
01
02
03
04
05
06
07
08
09
1
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 13 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (17)
1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333
Iter
Perfo
rman
ce in
dex
Figure 14 Convergence rate for the performance index of SImethod for NOCP numbers (18)
output in119873 different runs (Mean) standard deviation (Sd)maximum (Max) and minimum (Min) of each outputcould be achieved Among of the MHGA parameters wepresent the descriptive statistics of the ANOVA only for the1198731199051parameter which is reported in Table 3Table 4 summarized the statistical data contain the test
statistics (119865) and 119875-values of ANOVA tests Sensitivityanalysis for each parameter separately are done based on thistable as follows
1198731199051 From Table 4 the significant level or 119875-value for119870120595
is equal to 0279 which is greater than 005So from ANOVA 119873
1199051parameter has no significant
effect on the factor 119870120595 With similar analysis this
parameter has no effect on the other outputs except
required computational time Time because its 119875-value is equal to 0 which is less than 005 that is thisoutput sensitive to the parameter119873
1199051
1198731199052 From the second row of Table 4 the outputs 119869 119864
119869119870120595
and Time are sensitive to the parameter 1198731199052 and the
only output 120593119891is independent
1198731199011 From the third row of Table 4 only the computationalrequired time Time is sensitive because the its 119875-value is equal to 0006 which is less than 005 and theother outputs contain 119869 119864
119869 120593119891119870120595 are independent
1198731199012 From the forth row of Table 4 the output 120593
119891is
independent respect to the parameter 1198731199012 but other
outputs contain 119869 119864119869 Time 119870
120601are sensitive to this
parameter119873119892 From the fifth row of Table 4 the outputs 119869 119864
119869and
119870120595are independent to the parameter 119873
119892 and other
outputs contain 120593119891and Time are sensitive respect to
this parameter119873119894 The sensitivity analysis is similar to119873
1199011
From above cases it is obvious that all parameters canbe effect on the required computational time except 119873
119892
Moreover intuitions shows the normof final state constraints120593119891is independent output with respect to all parameters that
is any of the parameters could not effect on this output and itis not sensitive with respect to any of the MHGA parameters
62 Comparison of LI and SI To compare the efficiency ofthe LI and SI methods for NOCPs in the Appendix weused the Mann-Whitney nonparametric statistical test [37]
16 Mathematical Problems in Engineering
1 15 2 25 3 35 40
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
200
400
600
800
1000
1200
1400
1600
IterFi
nal c
ondi
tion
(b)
Figure 15 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (19)
2 4 6 8 10 12 14 16 18 20
50
100
150
200
250
300
350
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 10 11 120
05
1
15
2
25
3
Iter
Fina
l con
ditio
n
(b)
Figure 16 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (20)
Using the nonparametric statistical tests for comparing theperformance of several algorithms presented in [38] Forthis purpose At first 15 different runs for each of NOCPwere made Table 5 shows the mean of numerical resultsseparately Here we apply the fixMHGAparameters as119873
1199011=
91198731199012
= 121198731199051= 31119873
1199052= 91119873
119892= 100 and119873
119894= 50 with
the previous problem parameters in Table 2 Comparison
criterions contain 119869 120593119891 119870120595and Time The results of Mann-
Whitney test are shown in Tables 6ndash9 From Table 6 since119885 = 0 gt 119885
001= minus234 then the average relative error of
the LI and SI methods are same with the significant level 001In other words with probability 99 we can say that LI andSI methods have same effectiveness form the perspective oferror for 119869 Similarly the results of Tables 7ndash9 indicate that LI
Mathematical Problems in Engineering 17
Table 3 Descriptive statistics for the sensitivity analysis of1198731199051 for VDP problem
Output 1198731199051
119873 Mean Sd Min Max
119869
11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449
Total 140 15357 00294 15285 18449
120593119891
11 35 281 times 10minus8
952 times 10minus8
1 times 10minus12
452 times 10minus7
21 35 467 times 10minus8
235 times 10minus7
1 times 10minus12
141 times 10minus6
31 35 163 times 10minus8
491 times 10minus8
1 times 10minus12
221 times 10minus7
41 35 817 times 10minus8
285 times 10minus7
1 times 10minus12
126 times 10minus6
Total 140 426 times 10minus8 191 times 10minus7 1 times 10minus12 141 times 10minus6
Time
11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514
Total 140 1142131 410891 456615 2849514
119864119869
11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070
Total 140 00047 00192 0 02070
119870120595
11 35 00031 00045 475 times 10minus9
00204
21 35 00027 00064 550 times 10minus10
00369
31 35 00029 00050 154 times 10minus13
00271
41 35 00111 00397 193 times 10minus10
02070
Total 140 00047 00192 154 times 10minus13 02070
Table 4 Summary statistical data of ANOVA test for the parameters1198731199051 1198731199052 1198731199011 1198731199012 119873119892 119873119894
Parameters 119869 119864119869
120593119891
Time 119870120595
Test statistic (119865)
1198731199051
1294 1296 0624 1079 12961198731199052
110572 393 0868 5939 3931198731199011
1945 1835 1271 4317 18351198731199012
2740 2478 1011 2302 2478119873119892
3541 3591 0491 0890 3591119873119894
0495 0309 0301 5849 1046
119875 value
1198731199051
0280 0279 0601 0 02791198731199052
0 0005 0489 0 00051198731199011
0125 0143 0286 0006 01431198731199012
0021 0034 0413 0 0034119873119892
0009 0008 0743 0472 0008119873119894
0739 0819 0877 0 0386
and SI methods from the perspective of errors for 120593119891 Time
and119870120595 have same behaviour
7 The Impact of SQP
In this section we investigation the impact of the SQP on theproposed algorithm For this purpose we remove the SQPfrom Algorithm 2 as without SQP algorithm For comparingthe numerical results with the previous results the required
running time in each NOCP is considered fixed which is themaximum of running time in 12 different runs were done inSection 5 Also the all of the parameters for each problemwasset as same as Section 5 The numerical results of the withoutSQP algorithm is summarized in Table 10 By comparing theresults of Tables 10 and 1 it is obvious that for all NOCPsin the Appendix the obtained values of the performanceindex and the norm of final state constraint for the proposedalgorithm (Algorithm 2) are more accurate than the withoutSQP algorithm
18 Mathematical Problems in Engineering
Table 5 The mean of numerical results of 15 different runs for NOCPs in the Appendix using LI and SI methods
Problem LI SI119869 120593
119891119870120595
Time 119869 120593119891
119870120595
TimeVDP 15977 157 times 10
minus9
170 times 10minus3 17720 16002 667 times 10
minus8
326 times 10minus3 15757
CRP 00126 307 times 10minus8
307 times 10minus8 19714 00119 161 times 10
minus8
161 times 10minus8 19532
FFPP 4774 805 times 10minus5 03277 133280 5351 592 times 10
minus5 04868 137989MSNIC 01702 mdash mdash 13678 01703 mdash mdash 14092CSTCR 01120 mdash mdash 17385 01120 mdash mdash 17606Number 6 minus54307 mdash mdash 10541 minus54308 mdash mdash 10372Number 7 02015 326 times 10
minus9
695 times 10minus6 20807 02020 313 times 10
minus9
662 times 10minus6 21019
Number 8 630 times 10minus15 mdash mdash 6376 124 times 10
minus14 mdash mdash 5837Number 9 16512 mdash mdash 59239 16512 mdash mdash 58125Number 10 01550 mdash mdash 63153 01550 mdash mdash 65597Number 11 29701 270 times 10
minus9
272 times 10minus7 22860 29701 337 times 10
minus9
811 times 10minus7 23276
Number 12 minus02499 201 times 10minus8
464 times 10minus5 54718 minus02499 115 times 10
minus9
827 times 10minus5 52765
Number 13 00147 115 times 10minus8
00252 57797 00151 139 times 10minus8
00472 66098Number 14 34761 672 times 10
minus6
00237 104585 34290 757 times 10minus6
950 times 10minus3 108767
Number 15 195 times 10minus4
111 times 10minus8
569 times 10minus3 363 194 times 10
minus4
704 times 10minus9
486 times 10minus3 365
Number 16 20571 166 times 10minus10
166 times 10minus10 54391 20571 115 times 10
minus10
115 times 10minus10 54431
Number 17 minus88692 803 times 10minus8
803 times 10minus8 22744 minus88692 661 times 10
minus8
661 times 10minus8 23254
Number 18 00326 mdash mdash 65957 00326 mdash mdash 70897Number 19 011588 655 times 10
minus4 06828 161813 01091 884 times 10minus4 08808 159990
Number 20 5242 00176 05429 84165 4464 453 times 10minus4 03765 86891
Table 6 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 119869
Method Mean rank Sum of ranks Test statistics ValueLI 2050 4100 Mann-Whitney 119880 200SI 2050 4100 Wilcoxon119882 410mdash mdash mdash 119885 0
Table 7 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 120593119891
Method Mean rank Sum of ranks Test statistics ValueLI 1362 177 Mann-Whitney 119880 83SI 1338 174 Wilcoxon119882 174mdash mdash mdash 119885 minus0077
Table 8 Results of Mann-Whitney test on relative errors of the pair (LI SI) for Time
Method Mean rank Sum of ranks Test statistics ValueLI 2025 4050 Mann-Whitney 119880 195SI 2075 4150 Wilcoxon119882 405mdash mdash mdash 119885 minus0135
Table 9 Results of Mann-Whitney test on relative errors of the pair (LI SI) for119870120595
Method Mean rank Sum of ranks Test statistics ValueLI 1354 1760 Mann-Whitney 119880 84SI 1346 1750 Wilcoxon119882 175mdash mdash mdash 119885 minus0026
Mathematical Problems in Engineering 19
Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix
Problem no1 2 3 4 5 6 7 8 9 10
119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048
120601119891
330 times 10minus9
135 times 10minus5 11445 mdash mdash mdash 758 times 10
minus9 mdash mdash mdashProblem no
11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10
minus4 23182 minus90882 00336 2513 20274120601119891
167 times 10minus5
468 times 10minus9 00161 433 times 10
minus4
594 times 10minus3
156 times 10minus5 164 mdash 1706 25283
8 Conclusions
Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters
Appendix
Test Problems
The following problems are presented using notation given in(1)ndash(6)
(1) (VDP) [9] 119892 = (12)(1199092
1+ 1199092
2+ 1199062
) 1199050= 0 119905119891= 5 119891 =
[1199092 minus1199092+(1minus119909
2
1)1199092+119906]119879 1199090= [1 0]
119879120595 = 1199091minus1199092+1
(2) (CRP) [9] 119892 = (12)(1199092
1+1199092
2+01119906
2
) 1199050= 0 119905119891= 078
119891 = [1199091minus2(1199091+025)+(119909
2+05) exp(25119909
1(1199091+2))minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(3) (FFRP) [9] 119892 = (12)(1199062
1+ 1199062
2+ 1199062
3+ 1199062
4) 1199050
=
0 119905119891
= 5 119891 = [1199092 (110)((119906
1+ 1199062) cos119909
5minus
(1199062+ 1199064) sin119909
5) 1199094 (110)((119906
1+ 1199063) sin119909
5+ (1199062+
1199064) cos119909
5) 1199096 (512)((119906
1+ 1199063) minus (119906
2+ 1199064))]119879 1199090=
[0 0 0 0 0 0]119879 120595 = [119909
1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879
(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909
2 minus1199092+
119906 1199092
1+ 1199092
2+ 0005119906
2
]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909
2+
05 minus 8(119905 minus 05)2
]119879 1199090= [0 minus1 0]
119879
(5) (CSTCR) [20] 119892 = 1199092
1+ 1199092
2+ 01119906
2 1199050= 0 119905119891= 078
119891 = [minus(2 + 119906)(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+
2)) 05 minus 1199092minus (1199092+ 05) exp(25119909
1(1199091+ 2))]
119879 1199090=
[009 009]119879
(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909
2 119906]119879 119889 = [minus(2 +
119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]
119879(7) [36] 119892 = 119906
2 1199050= 0 119905
119891= 1 119891 = (12)119909
2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05
(8) [26] 119892 = 1199092cos2119906 119905
0= 0 119905119891= 120587 119891 = sin(1199062) 119909
0=
1205872(9) [26] 119892 = (12)(119909
2
1+ 1199092
2+ 1199062
) 1199050= 0 119905
119891= 5 119891 =
[1199092 minus1199091+ (1 minus 119909
2
1)1199092+ 119906]119879 119889 = minus(119909
2+ 025) 119909
0=
[1 0]119879
(10) [26] 119892 = 1199092
1+ 1199092
2+ 0005119906
2
1199050= 0 119905
119891= 1 119891 =
[1199092 minus1199092+119906]119879
119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus
05) minus 05 minus 1199092)]119879
1199090= [0 minus1]
119879(11) [26] 119892 = (12)119906
2 1199050= 0 119905
119891= 2 119891 = [119909
2 119906]119879 1199090=
[1 1]119879 120595 = [119909
1 1199092]119879
(12) [26] 119892 = minus1199092 1199050= 0 119905
119891= 1 119891 = [119909
2 119906]119879 119889 =
minus(1 minus 119906)(1 + 119906) 1199090= [0 0]
119879 120595 = 1199092
(13) [26] 119892 = (12)(1199092
1+ 1199092
2+ 01119906
2
) 1199050= 0 119905
119891= 078
119891 = [minus2(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+ 2)) minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(14) [26] 119892 = (12)1199062 1199050= 0 119905
119891= 10 119891 = [cos 119906 minus
1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909
0= [366 minus186]
119879120595 = [119909
1 1199092]119879
(15) [26]119892 = (12)(1199092
1+1199092
2) 1199050= 0 119905119891= 078119891 = [minus2(119909
1+
025)+(1199092+05) exp(25119909
1(1199091+2))minus(119909
1+025)119906 05minus
1199092minus(1199092+05) exp(25119909
1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(16) [26] 120601 = 1199093 1199050= 0 119905
119891= 1 119891 = [119909
2 119906 (12)119906
2
]119879
119889 = 1199091minus 19 119909
0= [0 0 0]
119879 120595 = [1199091 1199092+ 1]119879
(17) [26] 120601 = minus1199093 1199050= 0 119905
119891= 5 119891 = [119909
2 minus2 + 119906119909
3
minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909
0= [10 minus2 10]
119879120595 = [119909
1 1199092]119879
(18) [26] 120601 = (1199091minus1)2
+1199092
2+1199092
3 119892 = (12)119906
2 1199050= 0 119905119891= 5
119891 = [1199093cos 119906 119909
3sin 119906 sin 119906]119879 119909
0= [0 0 0]
119879
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Table 1 Continued
Problem Algorithm 119869 119864119869
120593119891
119870120595
Time
Number 15
SQP 103 times 10minus3 43368 0 43368 NR
SUMT 929 times 10minus4 38135 0 38135 NR
LI 193 times 10minus4 0 810 times 10minus10 810 times 10minus10 33354SI 194 times 10
minus4
520 times 10minus3
164 times 10minus9
520 times 10minus3
34014
Number 16
SQP 22120 00753 0 00753 NRSUMT 22080 00734 0 00734 NRLI 20571 0 394 times 10
minus12
394 times 10minus12
46982
SI 20571 0 560 times 10minus13 560 times 10minus13 70423
Number 17
SQP minus88690 225 times 10minus5 0 225 times 10
minus5 NRSUMT minus88690 225 times 10
minus5 0 225 times 10minus5 NR
LI minus88692 0 145 times 10minus10 145 times 10minus10 32183SI minus88692 0 279 times 10
minus10
279 times 10minus10
30959
Number 18
SQP 00368 01288 mdash mdash NRSUMT 00386 01840 mdash mdash NRLI 00326 0 mdash mdash 55163
SI 00326 0 mdash mdash 60671
Number 19
SQP 03439 49293 0 49293 NRSUMT 03428 49103 0 49103 NRLI 00689 01879 410 times 10
minus3
01919 164509
SI 00580 0 405 times 10minus4 405 times 10minus4 178783
Number 20
SQP 7752 12554 0 12554 NRSUMT 7683 12353 0 12353 NRLI 343716 232 times 10
minus5
225 times 10minus4
248 times 10minus4 81519
SI 343708 0 157 times 10minus4 157 times 10minus4 80420
aNot reported
the relative error of 119869 and the factor of the proposed algorithmis better and so the proposed algorithm is more robust thanCGA
513 FFRP Problem [9] The third NOCP in the Appendixis Free Floating Robot Problem FFRP which has six statevariables and four control variables FFRP has been solvedby CGA and the proposed algorithm with the MHGArsquosparameters as 119873
1199051= 31 119873
1199052= 71 119873
119892= 300 and
119873119894
= 200 This problem has six final state constraints120595 = [119909
1minus 4 119909
2 1199093minus 4 119909
4 1199095 1199096]119879 The numerical results
are shown in Table 1 The values of 119869 119864119869 120593119891and 119870
120595for LI
and SI methods separately are less than CGA Thereforethe proposed algorithm can achieve much better qualitysolutions than the CGA with reasonable computational time
514 MSNIC Problem [20] For the forth NOCP in theAppendix which is a Mathematical System with NonlinearInequality Constraint NSNIC the numerical results are com-pared with IPSO MSNIC contains an inequality constraint119889(119909 119905) = 119909
2(119905) + 05 minus 8(119905 minus 05)
2
le 0 The problem solvedby several numerical methods as [24 31] From [20] IPSOmethod could achieved more accurate results than men-tioned numericalmethods AlsoMSNIC can be solved by theproposed algorithm with the MHGArsquos parameters as 119873
1199051=
31 1198731199052
= 91 119873119892= 100 and 119873
119894= 60 From the forth row
of Table 1 the absolute error of 119869 119864119869 for LI and SI methods
equal 0 and 589 times 10minus4 respectively which are less than
IPSOrsquos 00165Subplots (a) and (b) in Figure 2 show the graphs of
the convergence rate for the performance index and theinequality constraint respect to the number of iterationrespectively
515 CSTCR Problem [20] The fifth NOCP in the Appendixis a model of a nonlinear Continuous Stirred-tank ChemicalReactor CSTCR It has two state variables 119909
1(119905) and 119909
2(119905)
as the deviation from the steady-state temperature and con-centration and one control variable 119906(119905) which represent theeffect of the flow rate of cooling fluid on chemical reactorTheobjective is to maintain the temperature and concentrationclose to steady-state values without expending large amountof control effort Also this is a benchmark problem in thehandbook of test problems in local and global optimization[32] which is a multimodal optimal control problem [33]It involves two different local minima The values of theperformance indexes for these solutions equal 0244 and0133 Similarly to the MSNIC the numerical results of theproposed algorithm with the MHGArsquos parameters as 119873
1199051=
31 1198731199052
= 51 119873119892= 100 and 119873
119894= 50 are compared with
IPSO From Table 1 the performance index 119869 for LI and SImethods is equal to 119869
lowast
= 01120 which is less than IPSOs01355
Mathematical Problems in Engineering 9
Table 2 The problem parameters for NOCPs in the Appendix
Parameters Problem number1 2 3 4 5 6 7 8 9 10
119906left minus05 minus15 minus15 minus20 0 minus2 0 minus2 minus1 minus20
119906right 2 2 10 20 5 2 1 2 1 20119872119894
103
102 70 1 mdash 1 mdash 10
2 mdash 102
120576 10minus12
10minus10
10minus3 mdash mdash mdash mdash mdash mdash 10
minus9
Parameters Problem no11 12 13 14 15 16 17 18 19 20
119906left minus5 minus1 minus2 minus120587 minus1 minus3 minus30 minus1 minus15 minus15
119906right 5 1 2 120587 1 3 30 1 10 10119872119894
10 102 mdash 10
2
102
102 mdash 10 70 10
minus3
120576 10minus9
10minus11
10minus11 mdash 10
minus10
10minus10
10minus10 mdash 10
minus3
10minus4
0 5 10 15 20 25 30 35 40 45 50017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 2 Graphical results of MSNIC problem using SI method (a) convergence rate of the performance index and (b) the inequalityconstraint 119889(119909 119905) le 0 in respect to the number of iterations
52 Comparison with Numerical Methods For NOCPs num-bers (6)ndash(20) the comparison are done with some numericalmethods Unfortunately for these methods usually the finalstate constraints and the required computational time arenot reported which are shown with NR in Table 1 but thesevalues are reported for both LI and SI methods in Table 1For all NOCPs in this section the MHGArsquos parameters areconsidered as 119873
1199051= 31 119873
1199052= 51 119873
119892= 100 119873
119894= 50 with
the problem parameters in Table 2
521 Compared with Bezier [34] The NOCP number (6) inthe Appendix has exact solution which has an inequalityconstraint as 119889(119909
1 119905) = minus6 minus 119909
1(119905) le 0 The exact value of
performance index equals 119869lowast = minus55285 [35] This problemhas been solved by a numerical method proposed in [34]called Bezier From sixth row of Table 1 the absolute error
of the LI and SI methods equal 00177 which is less thanBezierrsquos 00251
Figure 3 shows the graphs of the convergence rate of theperformance index subplot (a) and inequality constraintsubplot (b) respect to the number of iteration using SImethod
522 Compared with HPM [36] For NOCP number (7) inthe Appendix which is a constraint nonlinear model thenumerical results of the proposed algorithm are comparedwith HPM proposed in [36] This NOCP has a final stateconstraint as 120595 = 119909 minus 05 = 0 From [36] the norm offinal state constraint for HPM equals 42times10minus6 however thiscriterion for the LI and SI methods equals 235 times 10
minus9 and120593lowast
119891= 282 times 10
minus10 respectively From Table 1 it is obviousthat the obtained values of the performance index the norm
10 Mathematical Problems in Engineering
2 4 6 8 10 12 14minus5353
minus5352
minus5351
minus535
minus5349
minus5348
minus5347
minus5346
minus5345
minus5344
minus5343
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 3 Graphical results for NOCP number (6) using SI method (a) convergence rate for the performance index and (b) the inequalityconstraint 119889(119909
1 119905) le 0 in respect to the number of iterations
of final state constraint and 119870120595for the SI method are more
accurate than LI than HPMmethods
523 Compared with SQP and SUMT [26] For the NOCPsnumbers (8)ndash(20) in the Appendix the numerical resultsof LI and SI methods are compared with two numericalmethods contain SQP and SUMT proposed in [26] Amongthese NOCPs only two problems numbers (8) and (18) areunconstrained and the others have at least one constraintfinal state constraint or inequality constraint All of theseNOCPs solved by the proposed algorithm with the problemparameters in Table 2 and their results are summarized inTable 1 Because the final state constraints in these methodsare not reported we let 120593
119891= 0 to calculate the factor119870
120595
Figures 4ndash16 show the graphical results of NOCPsnumbers (8)ndash(20) in the Appendix using SI method Forunconstrained NOCPs numbers (8) and (18) only the graphof convergent rate of performance index is shown see Figures4 and 14 For constraint NOCPs and NOCPs numbers(11)ndash(17) and numbers (19)-(20) with final state constraintthe graphs of convergent rates of performance index and theerror of final state constraint are shown see Figures 7ndash13 and15-16 For the constraint NOCPs with inequality constraintsNOCPs numbers (9) and (10) the graphs of convergent rateof performance index and inequality constraint are shownsee Figures 5 and 6
Table 1 shows that the proposed algorithm LI and SImethods was 100 percent successful in point of views theperformance index 119869 and the factor119870
120595 numerically So the
proposed algorithm provides robust solutions with respect tothe other mentioned numerical or metaheuristic methodsTo compare 119869 in LI and SI methods in 35 percent of NOCPsLI is more accurate than SI in 10 percent SI is more accurate
1 15 2 25 3 35 40
002
004
006
008
01
012
Iter
Perfo
rman
ce in
dex
Figure 4 Convergence rate for the performance index of SI methodfor NOCP number (8)
than LI and in 55 percent are same In point of view the finalconditions 65 percent of NOCPs in the Appendix have finalstate constraints In all of them 120593
119891in the proposed algorithm
is improved except CRP problem however the factor of theproposed algorithm is the best yet In 54 percent of theseNOCPs LI is better performance and in 46 percent SI isbetter Also in point of the running time Time LI is same asSI that is in 50 percent of NOCPs LI and in 50 percent SI isbetter than another So the proposed algorithmcould providevery suitable solutions in a reasonable computational timeAlso for more accurate comparison of LI and SI methods astatistical approach will done in next section
To compare with CGA the mean of relative error of 119869 119864119869
for CGA LI and SImethods inNOCPs (1)ndash(3) equal 056680 and 298 times 10
minus4 respectively Also the mean for the error
Mathematical Problems in Engineering 11
5 10 15 20 25165
17
175
18
185
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3 35 4 45 50
005
01
015
02
025
TimeIn
equa
lity
cons
trai
nt(b)
Figure 5 (a) Convergence rate for the performance index and (b) the inequality constraint of SI method for NOCP number (9)
5 10 15 20 25 30 35 40 45 500155
016
0165
017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 6 (a) Convergence rate of the performance index and (b) the inequality constraint for SI method for NOCP number (10)
of the final state constraints 120593119891 are 16 times 10
minus3 407 times 10minus6
and 301 times 10minus6 respectively and for 119870
120595 these values equal
05883 407 times 10minus6 and 302 times 10
minus4 Thus we can say that thefeasibility of the solutions given by the proposed algorithmand CGA are competitive
From Table 1 to compare with numerical methods SQPand SUMT in NOCPs (8)ndash(20) the mean of 119864
119869for LI SI
SQP and SUMT equals 00145 00209 169 times 108 and 136 times
108 respectively Also the mean for the error of final state
constraints for these NOCPs equal 334 times 10minus4 441 times 10
minus50 and 0 respectively For 119870
120595 these values are 00213 00014
12263 and 11644 Therefore the performance index 119869 andthe factor 119870
120595 for the LI and SI methods are more accurate
than SQP and SUMT So the proposed algorithm gave morebetter solution in comparison with the numerical methods
12 Mathematical Problems in Engineering
5 10 15 20 25minus025
minus0245
minus024
minus0235
minus023
minus0225
Iter
Perfo
rman
ce in
dex
(a)
1 15 2 25 3 35 4 45 5 55 60
02
04
06
08
1
12
14
16
IterFi
nal c
ondi
tion
times10minus5
(b)
Figure 7 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (11)
2 4 6 8 10 12 14minus025
minus0245
minus024
minus0235
minus023
minus0225
minus022
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
02
04
06
08
1
12
14
16
18
2
Iter
Fina
l con
ditio
n
times10minus5
(b)
Figure 8 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (12)
Therefore based on this numerical studywe can concludethat the proposed algorithm outperform well-known numer-ical method Since the algorithms were not implementedon the same PC the computational times of them are notcompetitive Therefore we did not give the computationaltimes in bold in Table 1
6 Sensitivity Analysis andComparing LI and SI
In this section two statistical analysis based on the one-way analysis of variance (ANOVA) used for investigatingthe sensitivity of MHGA parameters and Mann-Whitney
Mathematical Problems in Engineering 13
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
07
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 9 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (13)
5 10 15 20 25 30
35
4
45
5
55
6
65
7
75
8
85
9
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
Iter
Fina
l con
ditio
n
(b)
Figure 10 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (14)
applied comparing LI and SI methods are done by thestatistical software IBM SPSS version 21
61 Sensitivity Analysis In order to survey the sensitivityof the MHGA parameters from the proposed algorithmthe VDP problem is selected for example The influenceof these parameters are investigated for this NOCP on
the dependent outputs consist of the performance index119869 the relative error of 119869 119864
119869 the required computational
time Time the error of final state constraints 120593119891and the
factor 119870120595 The independent parameters are consist of the
number of time nodes in both two phases 1198731199051and 119873
1199052 the
size of population in both two phases 1198731199011
and 1198731199012 the
maximum number of generations without improvement119873119894
14 Mathematical Problems in Engineering
5 10 15 20 25 30 35 40 45 50
00955
0096
00965
0097
00975
0098
00985
0099
00995
Iter
Perfo
rman
ce in
dex
(a)
2 4 6 8 10 12 14 16 18 200
05
1
15
2
25
3
35
4
45
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 11 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (15)
5 10 15 20 25
21
22
23
24
25
26
27
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
1
2
3
4
5
6
Iter
Fina
l con
ditio
n
times10minus3
(b)
Figure 12 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (16)
and the maximum number of generations 119873119892 Because the
mutation implementation probability119875119898 has a less influence
in numerical results it is not considered Also 119904119902119901119898119886119909119894119905119890119903 ischanged in each iteration of the proposed algorithm so it isnot considered too
At first we selected at least four constant value foreach of parameters and then in each pose 35 differentruns were made independently The statistical analysis isdone based on ANOVA The descriptive statistics whichcontains the number of different runs (119873) the mean of each
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7minus887
minus886
minus885
minus884
minus883
minus882
minus881
minus88
minus879
minus878
minus877
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 70
01
02
03
04
05
06
07
08
09
1
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 13 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (17)
1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333
Iter
Perfo
rman
ce in
dex
Figure 14 Convergence rate for the performance index of SImethod for NOCP numbers (18)
output in119873 different runs (Mean) standard deviation (Sd)maximum (Max) and minimum (Min) of each outputcould be achieved Among of the MHGA parameters wepresent the descriptive statistics of the ANOVA only for the1198731199051parameter which is reported in Table 3Table 4 summarized the statistical data contain the test
statistics (119865) and 119875-values of ANOVA tests Sensitivityanalysis for each parameter separately are done based on thistable as follows
1198731199051 From Table 4 the significant level or 119875-value for119870120595
is equal to 0279 which is greater than 005So from ANOVA 119873
1199051parameter has no significant
effect on the factor 119870120595 With similar analysis this
parameter has no effect on the other outputs except
required computational time Time because its 119875-value is equal to 0 which is less than 005 that is thisoutput sensitive to the parameter119873
1199051
1198731199052 From the second row of Table 4 the outputs 119869 119864
119869119870120595
and Time are sensitive to the parameter 1198731199052 and the
only output 120593119891is independent
1198731199011 From the third row of Table 4 only the computationalrequired time Time is sensitive because the its 119875-value is equal to 0006 which is less than 005 and theother outputs contain 119869 119864
119869 120593119891119870120595 are independent
1198731199012 From the forth row of Table 4 the output 120593
119891is
independent respect to the parameter 1198731199012 but other
outputs contain 119869 119864119869 Time 119870
120601are sensitive to this
parameter119873119892 From the fifth row of Table 4 the outputs 119869 119864
119869and
119870120595are independent to the parameter 119873
119892 and other
outputs contain 120593119891and Time are sensitive respect to
this parameter119873119894 The sensitivity analysis is similar to119873
1199011
From above cases it is obvious that all parameters canbe effect on the required computational time except 119873
119892
Moreover intuitions shows the normof final state constraints120593119891is independent output with respect to all parameters that
is any of the parameters could not effect on this output and itis not sensitive with respect to any of the MHGA parameters
62 Comparison of LI and SI To compare the efficiency ofthe LI and SI methods for NOCPs in the Appendix weused the Mann-Whitney nonparametric statistical test [37]
16 Mathematical Problems in Engineering
1 15 2 25 3 35 40
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
200
400
600
800
1000
1200
1400
1600
IterFi
nal c
ondi
tion
(b)
Figure 15 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (19)
2 4 6 8 10 12 14 16 18 20
50
100
150
200
250
300
350
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 10 11 120
05
1
15
2
25
3
Iter
Fina
l con
ditio
n
(b)
Figure 16 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (20)
Using the nonparametric statistical tests for comparing theperformance of several algorithms presented in [38] Forthis purpose At first 15 different runs for each of NOCPwere made Table 5 shows the mean of numerical resultsseparately Here we apply the fixMHGAparameters as119873
1199011=
91198731199012
= 121198731199051= 31119873
1199052= 91119873
119892= 100 and119873
119894= 50 with
the previous problem parameters in Table 2 Comparison
criterions contain 119869 120593119891 119870120595and Time The results of Mann-
Whitney test are shown in Tables 6ndash9 From Table 6 since119885 = 0 gt 119885
001= minus234 then the average relative error of
the LI and SI methods are same with the significant level 001In other words with probability 99 we can say that LI andSI methods have same effectiveness form the perspective oferror for 119869 Similarly the results of Tables 7ndash9 indicate that LI
Mathematical Problems in Engineering 17
Table 3 Descriptive statistics for the sensitivity analysis of1198731199051 for VDP problem
Output 1198731199051
119873 Mean Sd Min Max
119869
11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449
Total 140 15357 00294 15285 18449
120593119891
11 35 281 times 10minus8
952 times 10minus8
1 times 10minus12
452 times 10minus7
21 35 467 times 10minus8
235 times 10minus7
1 times 10minus12
141 times 10minus6
31 35 163 times 10minus8
491 times 10minus8
1 times 10minus12
221 times 10minus7
41 35 817 times 10minus8
285 times 10minus7
1 times 10minus12
126 times 10minus6
Total 140 426 times 10minus8 191 times 10minus7 1 times 10minus12 141 times 10minus6
Time
11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514
Total 140 1142131 410891 456615 2849514
119864119869
11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070
Total 140 00047 00192 0 02070
119870120595
11 35 00031 00045 475 times 10minus9
00204
21 35 00027 00064 550 times 10minus10
00369
31 35 00029 00050 154 times 10minus13
00271
41 35 00111 00397 193 times 10minus10
02070
Total 140 00047 00192 154 times 10minus13 02070
Table 4 Summary statistical data of ANOVA test for the parameters1198731199051 1198731199052 1198731199011 1198731199012 119873119892 119873119894
Parameters 119869 119864119869
120593119891
Time 119870120595
Test statistic (119865)
1198731199051
1294 1296 0624 1079 12961198731199052
110572 393 0868 5939 3931198731199011
1945 1835 1271 4317 18351198731199012
2740 2478 1011 2302 2478119873119892
3541 3591 0491 0890 3591119873119894
0495 0309 0301 5849 1046
119875 value
1198731199051
0280 0279 0601 0 02791198731199052
0 0005 0489 0 00051198731199011
0125 0143 0286 0006 01431198731199012
0021 0034 0413 0 0034119873119892
0009 0008 0743 0472 0008119873119894
0739 0819 0877 0 0386
and SI methods from the perspective of errors for 120593119891 Time
and119870120595 have same behaviour
7 The Impact of SQP
In this section we investigation the impact of the SQP on theproposed algorithm For this purpose we remove the SQPfrom Algorithm 2 as without SQP algorithm For comparingthe numerical results with the previous results the required
running time in each NOCP is considered fixed which is themaximum of running time in 12 different runs were done inSection 5 Also the all of the parameters for each problemwasset as same as Section 5 The numerical results of the withoutSQP algorithm is summarized in Table 10 By comparing theresults of Tables 10 and 1 it is obvious that for all NOCPsin the Appendix the obtained values of the performanceindex and the norm of final state constraint for the proposedalgorithm (Algorithm 2) are more accurate than the withoutSQP algorithm
18 Mathematical Problems in Engineering
Table 5 The mean of numerical results of 15 different runs for NOCPs in the Appendix using LI and SI methods
Problem LI SI119869 120593
119891119870120595
Time 119869 120593119891
119870120595
TimeVDP 15977 157 times 10
minus9
170 times 10minus3 17720 16002 667 times 10
minus8
326 times 10minus3 15757
CRP 00126 307 times 10minus8
307 times 10minus8 19714 00119 161 times 10
minus8
161 times 10minus8 19532
FFPP 4774 805 times 10minus5 03277 133280 5351 592 times 10
minus5 04868 137989MSNIC 01702 mdash mdash 13678 01703 mdash mdash 14092CSTCR 01120 mdash mdash 17385 01120 mdash mdash 17606Number 6 minus54307 mdash mdash 10541 minus54308 mdash mdash 10372Number 7 02015 326 times 10
minus9
695 times 10minus6 20807 02020 313 times 10
minus9
662 times 10minus6 21019
Number 8 630 times 10minus15 mdash mdash 6376 124 times 10
minus14 mdash mdash 5837Number 9 16512 mdash mdash 59239 16512 mdash mdash 58125Number 10 01550 mdash mdash 63153 01550 mdash mdash 65597Number 11 29701 270 times 10
minus9
272 times 10minus7 22860 29701 337 times 10
minus9
811 times 10minus7 23276
Number 12 minus02499 201 times 10minus8
464 times 10minus5 54718 minus02499 115 times 10
minus9
827 times 10minus5 52765
Number 13 00147 115 times 10minus8
00252 57797 00151 139 times 10minus8
00472 66098Number 14 34761 672 times 10
minus6
00237 104585 34290 757 times 10minus6
950 times 10minus3 108767
Number 15 195 times 10minus4
111 times 10minus8
569 times 10minus3 363 194 times 10
minus4
704 times 10minus9
486 times 10minus3 365
Number 16 20571 166 times 10minus10
166 times 10minus10 54391 20571 115 times 10
minus10
115 times 10minus10 54431
Number 17 minus88692 803 times 10minus8
803 times 10minus8 22744 minus88692 661 times 10
minus8
661 times 10minus8 23254
Number 18 00326 mdash mdash 65957 00326 mdash mdash 70897Number 19 011588 655 times 10
minus4 06828 161813 01091 884 times 10minus4 08808 159990
Number 20 5242 00176 05429 84165 4464 453 times 10minus4 03765 86891
Table 6 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 119869
Method Mean rank Sum of ranks Test statistics ValueLI 2050 4100 Mann-Whitney 119880 200SI 2050 4100 Wilcoxon119882 410mdash mdash mdash 119885 0
Table 7 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 120593119891
Method Mean rank Sum of ranks Test statistics ValueLI 1362 177 Mann-Whitney 119880 83SI 1338 174 Wilcoxon119882 174mdash mdash mdash 119885 minus0077
Table 8 Results of Mann-Whitney test on relative errors of the pair (LI SI) for Time
Method Mean rank Sum of ranks Test statistics ValueLI 2025 4050 Mann-Whitney 119880 195SI 2075 4150 Wilcoxon119882 405mdash mdash mdash 119885 minus0135
Table 9 Results of Mann-Whitney test on relative errors of the pair (LI SI) for119870120595
Method Mean rank Sum of ranks Test statistics ValueLI 1354 1760 Mann-Whitney 119880 84SI 1346 1750 Wilcoxon119882 175mdash mdash mdash 119885 minus0026
Mathematical Problems in Engineering 19
Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix
Problem no1 2 3 4 5 6 7 8 9 10
119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048
120601119891
330 times 10minus9
135 times 10minus5 11445 mdash mdash mdash 758 times 10
minus9 mdash mdash mdashProblem no
11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10
minus4 23182 minus90882 00336 2513 20274120601119891
167 times 10minus5
468 times 10minus9 00161 433 times 10
minus4
594 times 10minus3
156 times 10minus5 164 mdash 1706 25283
8 Conclusions
Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters
Appendix
Test Problems
The following problems are presented using notation given in(1)ndash(6)
(1) (VDP) [9] 119892 = (12)(1199092
1+ 1199092
2+ 1199062
) 1199050= 0 119905119891= 5 119891 =
[1199092 minus1199092+(1minus119909
2
1)1199092+119906]119879 1199090= [1 0]
119879120595 = 1199091minus1199092+1
(2) (CRP) [9] 119892 = (12)(1199092
1+1199092
2+01119906
2
) 1199050= 0 119905119891= 078
119891 = [1199091minus2(1199091+025)+(119909
2+05) exp(25119909
1(1199091+2))minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(3) (FFRP) [9] 119892 = (12)(1199062
1+ 1199062
2+ 1199062
3+ 1199062
4) 1199050
=
0 119905119891
= 5 119891 = [1199092 (110)((119906
1+ 1199062) cos119909
5minus
(1199062+ 1199064) sin119909
5) 1199094 (110)((119906
1+ 1199063) sin119909
5+ (1199062+
1199064) cos119909
5) 1199096 (512)((119906
1+ 1199063) minus (119906
2+ 1199064))]119879 1199090=
[0 0 0 0 0 0]119879 120595 = [119909
1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879
(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909
2 minus1199092+
119906 1199092
1+ 1199092
2+ 0005119906
2
]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909
2+
05 minus 8(119905 minus 05)2
]119879 1199090= [0 minus1 0]
119879
(5) (CSTCR) [20] 119892 = 1199092
1+ 1199092
2+ 01119906
2 1199050= 0 119905119891= 078
119891 = [minus(2 + 119906)(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+
2)) 05 minus 1199092minus (1199092+ 05) exp(25119909
1(1199091+ 2))]
119879 1199090=
[009 009]119879
(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909
2 119906]119879 119889 = [minus(2 +
119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]
119879(7) [36] 119892 = 119906
2 1199050= 0 119905
119891= 1 119891 = (12)119909
2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05
(8) [26] 119892 = 1199092cos2119906 119905
0= 0 119905119891= 120587 119891 = sin(1199062) 119909
0=
1205872(9) [26] 119892 = (12)(119909
2
1+ 1199092
2+ 1199062
) 1199050= 0 119905
119891= 5 119891 =
[1199092 minus1199091+ (1 minus 119909
2
1)1199092+ 119906]119879 119889 = minus(119909
2+ 025) 119909
0=
[1 0]119879
(10) [26] 119892 = 1199092
1+ 1199092
2+ 0005119906
2
1199050= 0 119905
119891= 1 119891 =
[1199092 minus1199092+119906]119879
119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus
05) minus 05 minus 1199092)]119879
1199090= [0 minus1]
119879(11) [26] 119892 = (12)119906
2 1199050= 0 119905
119891= 2 119891 = [119909
2 119906]119879 1199090=
[1 1]119879 120595 = [119909
1 1199092]119879
(12) [26] 119892 = minus1199092 1199050= 0 119905
119891= 1 119891 = [119909
2 119906]119879 119889 =
minus(1 minus 119906)(1 + 119906) 1199090= [0 0]
119879 120595 = 1199092
(13) [26] 119892 = (12)(1199092
1+ 1199092
2+ 01119906
2
) 1199050= 0 119905
119891= 078
119891 = [minus2(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+ 2)) minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(14) [26] 119892 = (12)1199062 1199050= 0 119905
119891= 10 119891 = [cos 119906 minus
1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909
0= [366 minus186]
119879120595 = [119909
1 1199092]119879
(15) [26]119892 = (12)(1199092
1+1199092
2) 1199050= 0 119905119891= 078119891 = [minus2(119909
1+
025)+(1199092+05) exp(25119909
1(1199091+2))minus(119909
1+025)119906 05minus
1199092minus(1199092+05) exp(25119909
1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(16) [26] 120601 = 1199093 1199050= 0 119905
119891= 1 119891 = [119909
2 119906 (12)119906
2
]119879
119889 = 1199091minus 19 119909
0= [0 0 0]
119879 120595 = [1199091 1199092+ 1]119879
(17) [26] 120601 = minus1199093 1199050= 0 119905
119891= 5 119891 = [119909
2 minus2 + 119906119909
3
minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909
0= [10 minus2 10]
119879120595 = [119909
1 1199092]119879
(18) [26] 120601 = (1199091minus1)2
+1199092
2+1199092
3 119892 = (12)119906
2 1199050= 0 119905119891= 5
119891 = [1199093cos 119906 119909
3sin 119906 sin 119906]119879 119909
0= [0 0 0]
119879
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Table 2 The problem parameters for NOCPs in the Appendix
Parameters Problem number1 2 3 4 5 6 7 8 9 10
119906left minus05 minus15 minus15 minus20 0 minus2 0 minus2 minus1 minus20
119906right 2 2 10 20 5 2 1 2 1 20119872119894
103
102 70 1 mdash 1 mdash 10
2 mdash 102
120576 10minus12
10minus10
10minus3 mdash mdash mdash mdash mdash mdash 10
minus9
Parameters Problem no11 12 13 14 15 16 17 18 19 20
119906left minus5 minus1 minus2 minus120587 minus1 minus3 minus30 minus1 minus15 minus15
119906right 5 1 2 120587 1 3 30 1 10 10119872119894
10 102 mdash 10
2
102
102 mdash 10 70 10
minus3
120576 10minus9
10minus11
10minus11 mdash 10
minus10
10minus10
10minus10 mdash 10
minus3
10minus4
0 5 10 15 20 25 30 35 40 45 50017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 2 Graphical results of MSNIC problem using SI method (a) convergence rate of the performance index and (b) the inequalityconstraint 119889(119909 119905) le 0 in respect to the number of iterations
52 Comparison with Numerical Methods For NOCPs num-bers (6)ndash(20) the comparison are done with some numericalmethods Unfortunately for these methods usually the finalstate constraints and the required computational time arenot reported which are shown with NR in Table 1 but thesevalues are reported for both LI and SI methods in Table 1For all NOCPs in this section the MHGArsquos parameters areconsidered as 119873
1199051= 31 119873
1199052= 51 119873
119892= 100 119873
119894= 50 with
the problem parameters in Table 2
521 Compared with Bezier [34] The NOCP number (6) inthe Appendix has exact solution which has an inequalityconstraint as 119889(119909
1 119905) = minus6 minus 119909
1(119905) le 0 The exact value of
performance index equals 119869lowast = minus55285 [35] This problemhas been solved by a numerical method proposed in [34]called Bezier From sixth row of Table 1 the absolute error
of the LI and SI methods equal 00177 which is less thanBezierrsquos 00251
Figure 3 shows the graphs of the convergence rate of theperformance index subplot (a) and inequality constraintsubplot (b) respect to the number of iteration using SImethod
522 Compared with HPM [36] For NOCP number (7) inthe Appendix which is a constraint nonlinear model thenumerical results of the proposed algorithm are comparedwith HPM proposed in [36] This NOCP has a final stateconstraint as 120595 = 119909 minus 05 = 0 From [36] the norm offinal state constraint for HPM equals 42times10minus6 however thiscriterion for the LI and SI methods equals 235 times 10
minus9 and120593lowast
119891= 282 times 10
minus10 respectively From Table 1 it is obviousthat the obtained values of the performance index the norm
10 Mathematical Problems in Engineering
2 4 6 8 10 12 14minus5353
minus5352
minus5351
minus535
minus5349
minus5348
minus5347
minus5346
minus5345
minus5344
minus5343
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 3 Graphical results for NOCP number (6) using SI method (a) convergence rate for the performance index and (b) the inequalityconstraint 119889(119909
1 119905) le 0 in respect to the number of iterations
of final state constraint and 119870120595for the SI method are more
accurate than LI than HPMmethods
523 Compared with SQP and SUMT [26] For the NOCPsnumbers (8)ndash(20) in the Appendix the numerical resultsof LI and SI methods are compared with two numericalmethods contain SQP and SUMT proposed in [26] Amongthese NOCPs only two problems numbers (8) and (18) areunconstrained and the others have at least one constraintfinal state constraint or inequality constraint All of theseNOCPs solved by the proposed algorithm with the problemparameters in Table 2 and their results are summarized inTable 1 Because the final state constraints in these methodsare not reported we let 120593
119891= 0 to calculate the factor119870
120595
Figures 4ndash16 show the graphical results of NOCPsnumbers (8)ndash(20) in the Appendix using SI method Forunconstrained NOCPs numbers (8) and (18) only the graphof convergent rate of performance index is shown see Figures4 and 14 For constraint NOCPs and NOCPs numbers(11)ndash(17) and numbers (19)-(20) with final state constraintthe graphs of convergent rates of performance index and theerror of final state constraint are shown see Figures 7ndash13 and15-16 For the constraint NOCPs with inequality constraintsNOCPs numbers (9) and (10) the graphs of convergent rateof performance index and inequality constraint are shownsee Figures 5 and 6
Table 1 shows that the proposed algorithm LI and SImethods was 100 percent successful in point of views theperformance index 119869 and the factor119870
120595 numerically So the
proposed algorithm provides robust solutions with respect tothe other mentioned numerical or metaheuristic methodsTo compare 119869 in LI and SI methods in 35 percent of NOCPsLI is more accurate than SI in 10 percent SI is more accurate
1 15 2 25 3 35 40
002
004
006
008
01
012
Iter
Perfo
rman
ce in
dex
Figure 4 Convergence rate for the performance index of SI methodfor NOCP number (8)
than LI and in 55 percent are same In point of view the finalconditions 65 percent of NOCPs in the Appendix have finalstate constraints In all of them 120593
119891in the proposed algorithm
is improved except CRP problem however the factor of theproposed algorithm is the best yet In 54 percent of theseNOCPs LI is better performance and in 46 percent SI isbetter Also in point of the running time Time LI is same asSI that is in 50 percent of NOCPs LI and in 50 percent SI isbetter than another So the proposed algorithmcould providevery suitable solutions in a reasonable computational timeAlso for more accurate comparison of LI and SI methods astatistical approach will done in next section
To compare with CGA the mean of relative error of 119869 119864119869
for CGA LI and SImethods inNOCPs (1)ndash(3) equal 056680 and 298 times 10
minus4 respectively Also the mean for the error
Mathematical Problems in Engineering 11
5 10 15 20 25165
17
175
18
185
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3 35 4 45 50
005
01
015
02
025
TimeIn
equa
lity
cons
trai
nt(b)
Figure 5 (a) Convergence rate for the performance index and (b) the inequality constraint of SI method for NOCP number (9)
5 10 15 20 25 30 35 40 45 500155
016
0165
017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 6 (a) Convergence rate of the performance index and (b) the inequality constraint for SI method for NOCP number (10)
of the final state constraints 120593119891 are 16 times 10
minus3 407 times 10minus6
and 301 times 10minus6 respectively and for 119870
120595 these values equal
05883 407 times 10minus6 and 302 times 10
minus4 Thus we can say that thefeasibility of the solutions given by the proposed algorithmand CGA are competitive
From Table 1 to compare with numerical methods SQPand SUMT in NOCPs (8)ndash(20) the mean of 119864
119869for LI SI
SQP and SUMT equals 00145 00209 169 times 108 and 136 times
108 respectively Also the mean for the error of final state
constraints for these NOCPs equal 334 times 10minus4 441 times 10
minus50 and 0 respectively For 119870
120595 these values are 00213 00014
12263 and 11644 Therefore the performance index 119869 andthe factor 119870
120595 for the LI and SI methods are more accurate
than SQP and SUMT So the proposed algorithm gave morebetter solution in comparison with the numerical methods
12 Mathematical Problems in Engineering
5 10 15 20 25minus025
minus0245
minus024
minus0235
minus023
minus0225
Iter
Perfo
rman
ce in
dex
(a)
1 15 2 25 3 35 4 45 5 55 60
02
04
06
08
1
12
14
16
IterFi
nal c
ondi
tion
times10minus5
(b)
Figure 7 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (11)
2 4 6 8 10 12 14minus025
minus0245
minus024
minus0235
minus023
minus0225
minus022
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
02
04
06
08
1
12
14
16
18
2
Iter
Fina
l con
ditio
n
times10minus5
(b)
Figure 8 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (12)
Therefore based on this numerical studywe can concludethat the proposed algorithm outperform well-known numer-ical method Since the algorithms were not implementedon the same PC the computational times of them are notcompetitive Therefore we did not give the computationaltimes in bold in Table 1
6 Sensitivity Analysis andComparing LI and SI
In this section two statistical analysis based on the one-way analysis of variance (ANOVA) used for investigatingthe sensitivity of MHGA parameters and Mann-Whitney
Mathematical Problems in Engineering 13
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
07
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 9 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (13)
5 10 15 20 25 30
35
4
45
5
55
6
65
7
75
8
85
9
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
Iter
Fina
l con
ditio
n
(b)
Figure 10 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (14)
applied comparing LI and SI methods are done by thestatistical software IBM SPSS version 21
61 Sensitivity Analysis In order to survey the sensitivityof the MHGA parameters from the proposed algorithmthe VDP problem is selected for example The influenceof these parameters are investigated for this NOCP on
the dependent outputs consist of the performance index119869 the relative error of 119869 119864
119869 the required computational
time Time the error of final state constraints 120593119891and the
factor 119870120595 The independent parameters are consist of the
number of time nodes in both two phases 1198731199051and 119873
1199052 the
size of population in both two phases 1198731199011
and 1198731199012 the
maximum number of generations without improvement119873119894
14 Mathematical Problems in Engineering
5 10 15 20 25 30 35 40 45 50
00955
0096
00965
0097
00975
0098
00985
0099
00995
Iter
Perfo
rman
ce in
dex
(a)
2 4 6 8 10 12 14 16 18 200
05
1
15
2
25
3
35
4
45
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 11 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (15)
5 10 15 20 25
21
22
23
24
25
26
27
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
1
2
3
4
5
6
Iter
Fina
l con
ditio
n
times10minus3
(b)
Figure 12 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (16)
and the maximum number of generations 119873119892 Because the
mutation implementation probability119875119898 has a less influence
in numerical results it is not considered Also 119904119902119901119898119886119909119894119905119890119903 ischanged in each iteration of the proposed algorithm so it isnot considered too
At first we selected at least four constant value foreach of parameters and then in each pose 35 differentruns were made independently The statistical analysis isdone based on ANOVA The descriptive statistics whichcontains the number of different runs (119873) the mean of each
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7minus887
minus886
minus885
minus884
minus883
minus882
minus881
minus88
minus879
minus878
minus877
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 70
01
02
03
04
05
06
07
08
09
1
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 13 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (17)
1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333
Iter
Perfo
rman
ce in
dex
Figure 14 Convergence rate for the performance index of SImethod for NOCP numbers (18)
output in119873 different runs (Mean) standard deviation (Sd)maximum (Max) and minimum (Min) of each outputcould be achieved Among of the MHGA parameters wepresent the descriptive statistics of the ANOVA only for the1198731199051parameter which is reported in Table 3Table 4 summarized the statistical data contain the test
statistics (119865) and 119875-values of ANOVA tests Sensitivityanalysis for each parameter separately are done based on thistable as follows
1198731199051 From Table 4 the significant level or 119875-value for119870120595
is equal to 0279 which is greater than 005So from ANOVA 119873
1199051parameter has no significant
effect on the factor 119870120595 With similar analysis this
parameter has no effect on the other outputs except
required computational time Time because its 119875-value is equal to 0 which is less than 005 that is thisoutput sensitive to the parameter119873
1199051
1198731199052 From the second row of Table 4 the outputs 119869 119864
119869119870120595
and Time are sensitive to the parameter 1198731199052 and the
only output 120593119891is independent
1198731199011 From the third row of Table 4 only the computationalrequired time Time is sensitive because the its 119875-value is equal to 0006 which is less than 005 and theother outputs contain 119869 119864
119869 120593119891119870120595 are independent
1198731199012 From the forth row of Table 4 the output 120593
119891is
independent respect to the parameter 1198731199012 but other
outputs contain 119869 119864119869 Time 119870
120601are sensitive to this
parameter119873119892 From the fifth row of Table 4 the outputs 119869 119864
119869and
119870120595are independent to the parameter 119873
119892 and other
outputs contain 120593119891and Time are sensitive respect to
this parameter119873119894 The sensitivity analysis is similar to119873
1199011
From above cases it is obvious that all parameters canbe effect on the required computational time except 119873
119892
Moreover intuitions shows the normof final state constraints120593119891is independent output with respect to all parameters that
is any of the parameters could not effect on this output and itis not sensitive with respect to any of the MHGA parameters
62 Comparison of LI and SI To compare the efficiency ofthe LI and SI methods for NOCPs in the Appendix weused the Mann-Whitney nonparametric statistical test [37]
16 Mathematical Problems in Engineering
1 15 2 25 3 35 40
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
200
400
600
800
1000
1200
1400
1600
IterFi
nal c
ondi
tion
(b)
Figure 15 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (19)
2 4 6 8 10 12 14 16 18 20
50
100
150
200
250
300
350
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 10 11 120
05
1
15
2
25
3
Iter
Fina
l con
ditio
n
(b)
Figure 16 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (20)
Using the nonparametric statistical tests for comparing theperformance of several algorithms presented in [38] Forthis purpose At first 15 different runs for each of NOCPwere made Table 5 shows the mean of numerical resultsseparately Here we apply the fixMHGAparameters as119873
1199011=
91198731199012
= 121198731199051= 31119873
1199052= 91119873
119892= 100 and119873
119894= 50 with
the previous problem parameters in Table 2 Comparison
criterions contain 119869 120593119891 119870120595and Time The results of Mann-
Whitney test are shown in Tables 6ndash9 From Table 6 since119885 = 0 gt 119885
001= minus234 then the average relative error of
the LI and SI methods are same with the significant level 001In other words with probability 99 we can say that LI andSI methods have same effectiveness form the perspective oferror for 119869 Similarly the results of Tables 7ndash9 indicate that LI
Mathematical Problems in Engineering 17
Table 3 Descriptive statistics for the sensitivity analysis of1198731199051 for VDP problem
Output 1198731199051
119873 Mean Sd Min Max
119869
11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449
Total 140 15357 00294 15285 18449
120593119891
11 35 281 times 10minus8
952 times 10minus8
1 times 10minus12
452 times 10minus7
21 35 467 times 10minus8
235 times 10minus7
1 times 10minus12
141 times 10minus6
31 35 163 times 10minus8
491 times 10minus8
1 times 10minus12
221 times 10minus7
41 35 817 times 10minus8
285 times 10minus7
1 times 10minus12
126 times 10minus6
Total 140 426 times 10minus8 191 times 10minus7 1 times 10minus12 141 times 10minus6
Time
11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514
Total 140 1142131 410891 456615 2849514
119864119869
11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070
Total 140 00047 00192 0 02070
119870120595
11 35 00031 00045 475 times 10minus9
00204
21 35 00027 00064 550 times 10minus10
00369
31 35 00029 00050 154 times 10minus13
00271
41 35 00111 00397 193 times 10minus10
02070
Total 140 00047 00192 154 times 10minus13 02070
Table 4 Summary statistical data of ANOVA test for the parameters1198731199051 1198731199052 1198731199011 1198731199012 119873119892 119873119894
Parameters 119869 119864119869
120593119891
Time 119870120595
Test statistic (119865)
1198731199051
1294 1296 0624 1079 12961198731199052
110572 393 0868 5939 3931198731199011
1945 1835 1271 4317 18351198731199012
2740 2478 1011 2302 2478119873119892
3541 3591 0491 0890 3591119873119894
0495 0309 0301 5849 1046
119875 value
1198731199051
0280 0279 0601 0 02791198731199052
0 0005 0489 0 00051198731199011
0125 0143 0286 0006 01431198731199012
0021 0034 0413 0 0034119873119892
0009 0008 0743 0472 0008119873119894
0739 0819 0877 0 0386
and SI methods from the perspective of errors for 120593119891 Time
and119870120595 have same behaviour
7 The Impact of SQP
In this section we investigation the impact of the SQP on theproposed algorithm For this purpose we remove the SQPfrom Algorithm 2 as without SQP algorithm For comparingthe numerical results with the previous results the required
running time in each NOCP is considered fixed which is themaximum of running time in 12 different runs were done inSection 5 Also the all of the parameters for each problemwasset as same as Section 5 The numerical results of the withoutSQP algorithm is summarized in Table 10 By comparing theresults of Tables 10 and 1 it is obvious that for all NOCPsin the Appendix the obtained values of the performanceindex and the norm of final state constraint for the proposedalgorithm (Algorithm 2) are more accurate than the withoutSQP algorithm
18 Mathematical Problems in Engineering
Table 5 The mean of numerical results of 15 different runs for NOCPs in the Appendix using LI and SI methods
Problem LI SI119869 120593
119891119870120595
Time 119869 120593119891
119870120595
TimeVDP 15977 157 times 10
minus9
170 times 10minus3 17720 16002 667 times 10
minus8
326 times 10minus3 15757
CRP 00126 307 times 10minus8
307 times 10minus8 19714 00119 161 times 10
minus8
161 times 10minus8 19532
FFPP 4774 805 times 10minus5 03277 133280 5351 592 times 10
minus5 04868 137989MSNIC 01702 mdash mdash 13678 01703 mdash mdash 14092CSTCR 01120 mdash mdash 17385 01120 mdash mdash 17606Number 6 minus54307 mdash mdash 10541 minus54308 mdash mdash 10372Number 7 02015 326 times 10
minus9
695 times 10minus6 20807 02020 313 times 10
minus9
662 times 10minus6 21019
Number 8 630 times 10minus15 mdash mdash 6376 124 times 10
minus14 mdash mdash 5837Number 9 16512 mdash mdash 59239 16512 mdash mdash 58125Number 10 01550 mdash mdash 63153 01550 mdash mdash 65597Number 11 29701 270 times 10
minus9
272 times 10minus7 22860 29701 337 times 10
minus9
811 times 10minus7 23276
Number 12 minus02499 201 times 10minus8
464 times 10minus5 54718 minus02499 115 times 10
minus9
827 times 10minus5 52765
Number 13 00147 115 times 10minus8
00252 57797 00151 139 times 10minus8
00472 66098Number 14 34761 672 times 10
minus6
00237 104585 34290 757 times 10minus6
950 times 10minus3 108767
Number 15 195 times 10minus4
111 times 10minus8
569 times 10minus3 363 194 times 10
minus4
704 times 10minus9
486 times 10minus3 365
Number 16 20571 166 times 10minus10
166 times 10minus10 54391 20571 115 times 10
minus10
115 times 10minus10 54431
Number 17 minus88692 803 times 10minus8
803 times 10minus8 22744 minus88692 661 times 10
minus8
661 times 10minus8 23254
Number 18 00326 mdash mdash 65957 00326 mdash mdash 70897Number 19 011588 655 times 10
minus4 06828 161813 01091 884 times 10minus4 08808 159990
Number 20 5242 00176 05429 84165 4464 453 times 10minus4 03765 86891
Table 6 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 119869
Method Mean rank Sum of ranks Test statistics ValueLI 2050 4100 Mann-Whitney 119880 200SI 2050 4100 Wilcoxon119882 410mdash mdash mdash 119885 0
Table 7 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 120593119891
Method Mean rank Sum of ranks Test statistics ValueLI 1362 177 Mann-Whitney 119880 83SI 1338 174 Wilcoxon119882 174mdash mdash mdash 119885 minus0077
Table 8 Results of Mann-Whitney test on relative errors of the pair (LI SI) for Time
Method Mean rank Sum of ranks Test statistics ValueLI 2025 4050 Mann-Whitney 119880 195SI 2075 4150 Wilcoxon119882 405mdash mdash mdash 119885 minus0135
Table 9 Results of Mann-Whitney test on relative errors of the pair (LI SI) for119870120595
Method Mean rank Sum of ranks Test statistics ValueLI 1354 1760 Mann-Whitney 119880 84SI 1346 1750 Wilcoxon119882 175mdash mdash mdash 119885 minus0026
Mathematical Problems in Engineering 19
Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix
Problem no1 2 3 4 5 6 7 8 9 10
119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048
120601119891
330 times 10minus9
135 times 10minus5 11445 mdash mdash mdash 758 times 10
minus9 mdash mdash mdashProblem no
11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10
minus4 23182 minus90882 00336 2513 20274120601119891
167 times 10minus5
468 times 10minus9 00161 433 times 10
minus4
594 times 10minus3
156 times 10minus5 164 mdash 1706 25283
8 Conclusions
Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters
Appendix
Test Problems
The following problems are presented using notation given in(1)ndash(6)
(1) (VDP) [9] 119892 = (12)(1199092
1+ 1199092
2+ 1199062
) 1199050= 0 119905119891= 5 119891 =
[1199092 minus1199092+(1minus119909
2
1)1199092+119906]119879 1199090= [1 0]
119879120595 = 1199091minus1199092+1
(2) (CRP) [9] 119892 = (12)(1199092
1+1199092
2+01119906
2
) 1199050= 0 119905119891= 078
119891 = [1199091minus2(1199091+025)+(119909
2+05) exp(25119909
1(1199091+2))minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(3) (FFRP) [9] 119892 = (12)(1199062
1+ 1199062
2+ 1199062
3+ 1199062
4) 1199050
=
0 119905119891
= 5 119891 = [1199092 (110)((119906
1+ 1199062) cos119909
5minus
(1199062+ 1199064) sin119909
5) 1199094 (110)((119906
1+ 1199063) sin119909
5+ (1199062+
1199064) cos119909
5) 1199096 (512)((119906
1+ 1199063) minus (119906
2+ 1199064))]119879 1199090=
[0 0 0 0 0 0]119879 120595 = [119909
1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879
(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909
2 minus1199092+
119906 1199092
1+ 1199092
2+ 0005119906
2
]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909
2+
05 minus 8(119905 minus 05)2
]119879 1199090= [0 minus1 0]
119879
(5) (CSTCR) [20] 119892 = 1199092
1+ 1199092
2+ 01119906
2 1199050= 0 119905119891= 078
119891 = [minus(2 + 119906)(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+
2)) 05 minus 1199092minus (1199092+ 05) exp(25119909
1(1199091+ 2))]
119879 1199090=
[009 009]119879
(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909
2 119906]119879 119889 = [minus(2 +
119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]
119879(7) [36] 119892 = 119906
2 1199050= 0 119905
119891= 1 119891 = (12)119909
2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05
(8) [26] 119892 = 1199092cos2119906 119905
0= 0 119905119891= 120587 119891 = sin(1199062) 119909
0=
1205872(9) [26] 119892 = (12)(119909
2
1+ 1199092
2+ 1199062
) 1199050= 0 119905
119891= 5 119891 =
[1199092 minus1199091+ (1 minus 119909
2
1)1199092+ 119906]119879 119889 = minus(119909
2+ 025) 119909
0=
[1 0]119879
(10) [26] 119892 = 1199092
1+ 1199092
2+ 0005119906
2
1199050= 0 119905
119891= 1 119891 =
[1199092 minus1199092+119906]119879
119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus
05) minus 05 minus 1199092)]119879
1199090= [0 minus1]
119879(11) [26] 119892 = (12)119906
2 1199050= 0 119905
119891= 2 119891 = [119909
2 119906]119879 1199090=
[1 1]119879 120595 = [119909
1 1199092]119879
(12) [26] 119892 = minus1199092 1199050= 0 119905
119891= 1 119891 = [119909
2 119906]119879 119889 =
minus(1 minus 119906)(1 + 119906) 1199090= [0 0]
119879 120595 = 1199092
(13) [26] 119892 = (12)(1199092
1+ 1199092
2+ 01119906
2
) 1199050= 0 119905
119891= 078
119891 = [minus2(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+ 2)) minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(14) [26] 119892 = (12)1199062 1199050= 0 119905
119891= 10 119891 = [cos 119906 minus
1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909
0= [366 minus186]
119879120595 = [119909
1 1199092]119879
(15) [26]119892 = (12)(1199092
1+1199092
2) 1199050= 0 119905119891= 078119891 = [minus2(119909
1+
025)+(1199092+05) exp(25119909
1(1199091+2))minus(119909
1+025)119906 05minus
1199092minus(1199092+05) exp(25119909
1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(16) [26] 120601 = 1199093 1199050= 0 119905
119891= 1 119891 = [119909
2 119906 (12)119906
2
]119879
119889 = 1199091minus 19 119909
0= [0 0 0]
119879 120595 = [1199091 1199092+ 1]119879
(17) [26] 120601 = minus1199093 1199050= 0 119905
119891= 5 119891 = [119909
2 minus2 + 119906119909
3
minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909
0= [10 minus2 10]
119879120595 = [119909
1 1199092]119879
(18) [26] 120601 = (1199091minus1)2
+1199092
2+1199092
3 119892 = (12)119906
2 1199050= 0 119905119891= 5
119891 = [1199093cos 119906 119909
3sin 119906 sin 119906]119879 119909
0= [0 0 0]
119879
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
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10 Mathematical Problems in Engineering
2 4 6 8 10 12 14minus5353
minus5352
minus5351
minus535
minus5349
minus5348
minus5347
minus5346
minus5345
minus5344
minus5343
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 3 Graphical results for NOCP number (6) using SI method (a) convergence rate for the performance index and (b) the inequalityconstraint 119889(119909
1 119905) le 0 in respect to the number of iterations
of final state constraint and 119870120595for the SI method are more
accurate than LI than HPMmethods
523 Compared with SQP and SUMT [26] For the NOCPsnumbers (8)ndash(20) in the Appendix the numerical resultsof LI and SI methods are compared with two numericalmethods contain SQP and SUMT proposed in [26] Amongthese NOCPs only two problems numbers (8) and (18) areunconstrained and the others have at least one constraintfinal state constraint or inequality constraint All of theseNOCPs solved by the proposed algorithm with the problemparameters in Table 2 and their results are summarized inTable 1 Because the final state constraints in these methodsare not reported we let 120593
119891= 0 to calculate the factor119870
120595
Figures 4ndash16 show the graphical results of NOCPsnumbers (8)ndash(20) in the Appendix using SI method Forunconstrained NOCPs numbers (8) and (18) only the graphof convergent rate of performance index is shown see Figures4 and 14 For constraint NOCPs and NOCPs numbers(11)ndash(17) and numbers (19)-(20) with final state constraintthe graphs of convergent rates of performance index and theerror of final state constraint are shown see Figures 7ndash13 and15-16 For the constraint NOCPs with inequality constraintsNOCPs numbers (9) and (10) the graphs of convergent rateof performance index and inequality constraint are shownsee Figures 5 and 6
Table 1 shows that the proposed algorithm LI and SImethods was 100 percent successful in point of views theperformance index 119869 and the factor119870
120595 numerically So the
proposed algorithm provides robust solutions with respect tothe other mentioned numerical or metaheuristic methodsTo compare 119869 in LI and SI methods in 35 percent of NOCPsLI is more accurate than SI in 10 percent SI is more accurate
1 15 2 25 3 35 40
002
004
006
008
01
012
Iter
Perfo
rman
ce in
dex
Figure 4 Convergence rate for the performance index of SI methodfor NOCP number (8)
than LI and in 55 percent are same In point of view the finalconditions 65 percent of NOCPs in the Appendix have finalstate constraints In all of them 120593
119891in the proposed algorithm
is improved except CRP problem however the factor of theproposed algorithm is the best yet In 54 percent of theseNOCPs LI is better performance and in 46 percent SI isbetter Also in point of the running time Time LI is same asSI that is in 50 percent of NOCPs LI and in 50 percent SI isbetter than another So the proposed algorithmcould providevery suitable solutions in a reasonable computational timeAlso for more accurate comparison of LI and SI methods astatistical approach will done in next section
To compare with CGA the mean of relative error of 119869 119864119869
for CGA LI and SImethods inNOCPs (1)ndash(3) equal 056680 and 298 times 10
minus4 respectively Also the mean for the error
Mathematical Problems in Engineering 11
5 10 15 20 25165
17
175
18
185
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3 35 4 45 50
005
01
015
02
025
TimeIn
equa
lity
cons
trai
nt(b)
Figure 5 (a) Convergence rate for the performance index and (b) the inequality constraint of SI method for NOCP number (9)
5 10 15 20 25 30 35 40 45 500155
016
0165
017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 6 (a) Convergence rate of the performance index and (b) the inequality constraint for SI method for NOCP number (10)
of the final state constraints 120593119891 are 16 times 10
minus3 407 times 10minus6
and 301 times 10minus6 respectively and for 119870
120595 these values equal
05883 407 times 10minus6 and 302 times 10
minus4 Thus we can say that thefeasibility of the solutions given by the proposed algorithmand CGA are competitive
From Table 1 to compare with numerical methods SQPand SUMT in NOCPs (8)ndash(20) the mean of 119864
119869for LI SI
SQP and SUMT equals 00145 00209 169 times 108 and 136 times
108 respectively Also the mean for the error of final state
constraints for these NOCPs equal 334 times 10minus4 441 times 10
minus50 and 0 respectively For 119870
120595 these values are 00213 00014
12263 and 11644 Therefore the performance index 119869 andthe factor 119870
120595 for the LI and SI methods are more accurate
than SQP and SUMT So the proposed algorithm gave morebetter solution in comparison with the numerical methods
12 Mathematical Problems in Engineering
5 10 15 20 25minus025
minus0245
minus024
minus0235
minus023
minus0225
Iter
Perfo
rman
ce in
dex
(a)
1 15 2 25 3 35 4 45 5 55 60
02
04
06
08
1
12
14
16
IterFi
nal c
ondi
tion
times10minus5
(b)
Figure 7 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (11)
2 4 6 8 10 12 14minus025
minus0245
minus024
minus0235
minus023
minus0225
minus022
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
02
04
06
08
1
12
14
16
18
2
Iter
Fina
l con
ditio
n
times10minus5
(b)
Figure 8 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (12)
Therefore based on this numerical studywe can concludethat the proposed algorithm outperform well-known numer-ical method Since the algorithms were not implementedon the same PC the computational times of them are notcompetitive Therefore we did not give the computationaltimes in bold in Table 1
6 Sensitivity Analysis andComparing LI and SI
In this section two statistical analysis based on the one-way analysis of variance (ANOVA) used for investigatingthe sensitivity of MHGA parameters and Mann-Whitney
Mathematical Problems in Engineering 13
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
07
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 9 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (13)
5 10 15 20 25 30
35
4
45
5
55
6
65
7
75
8
85
9
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
Iter
Fina
l con
ditio
n
(b)
Figure 10 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (14)
applied comparing LI and SI methods are done by thestatistical software IBM SPSS version 21
61 Sensitivity Analysis In order to survey the sensitivityof the MHGA parameters from the proposed algorithmthe VDP problem is selected for example The influenceof these parameters are investigated for this NOCP on
the dependent outputs consist of the performance index119869 the relative error of 119869 119864
119869 the required computational
time Time the error of final state constraints 120593119891and the
factor 119870120595 The independent parameters are consist of the
number of time nodes in both two phases 1198731199051and 119873
1199052 the
size of population in both two phases 1198731199011
and 1198731199012 the
maximum number of generations without improvement119873119894
14 Mathematical Problems in Engineering
5 10 15 20 25 30 35 40 45 50
00955
0096
00965
0097
00975
0098
00985
0099
00995
Iter
Perfo
rman
ce in
dex
(a)
2 4 6 8 10 12 14 16 18 200
05
1
15
2
25
3
35
4
45
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 11 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (15)
5 10 15 20 25
21
22
23
24
25
26
27
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
1
2
3
4
5
6
Iter
Fina
l con
ditio
n
times10minus3
(b)
Figure 12 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (16)
and the maximum number of generations 119873119892 Because the
mutation implementation probability119875119898 has a less influence
in numerical results it is not considered Also 119904119902119901119898119886119909119894119905119890119903 ischanged in each iteration of the proposed algorithm so it isnot considered too
At first we selected at least four constant value foreach of parameters and then in each pose 35 differentruns were made independently The statistical analysis isdone based on ANOVA The descriptive statistics whichcontains the number of different runs (119873) the mean of each
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7minus887
minus886
minus885
minus884
minus883
minus882
minus881
minus88
minus879
minus878
minus877
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 70
01
02
03
04
05
06
07
08
09
1
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 13 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (17)
1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333
Iter
Perfo
rman
ce in
dex
Figure 14 Convergence rate for the performance index of SImethod for NOCP numbers (18)
output in119873 different runs (Mean) standard deviation (Sd)maximum (Max) and minimum (Min) of each outputcould be achieved Among of the MHGA parameters wepresent the descriptive statistics of the ANOVA only for the1198731199051parameter which is reported in Table 3Table 4 summarized the statistical data contain the test
statistics (119865) and 119875-values of ANOVA tests Sensitivityanalysis for each parameter separately are done based on thistable as follows
1198731199051 From Table 4 the significant level or 119875-value for119870120595
is equal to 0279 which is greater than 005So from ANOVA 119873
1199051parameter has no significant
effect on the factor 119870120595 With similar analysis this
parameter has no effect on the other outputs except
required computational time Time because its 119875-value is equal to 0 which is less than 005 that is thisoutput sensitive to the parameter119873
1199051
1198731199052 From the second row of Table 4 the outputs 119869 119864
119869119870120595
and Time are sensitive to the parameter 1198731199052 and the
only output 120593119891is independent
1198731199011 From the third row of Table 4 only the computationalrequired time Time is sensitive because the its 119875-value is equal to 0006 which is less than 005 and theother outputs contain 119869 119864
119869 120593119891119870120595 are independent
1198731199012 From the forth row of Table 4 the output 120593
119891is
independent respect to the parameter 1198731199012 but other
outputs contain 119869 119864119869 Time 119870
120601are sensitive to this
parameter119873119892 From the fifth row of Table 4 the outputs 119869 119864
119869and
119870120595are independent to the parameter 119873
119892 and other
outputs contain 120593119891and Time are sensitive respect to
this parameter119873119894 The sensitivity analysis is similar to119873
1199011
From above cases it is obvious that all parameters canbe effect on the required computational time except 119873
119892
Moreover intuitions shows the normof final state constraints120593119891is independent output with respect to all parameters that
is any of the parameters could not effect on this output and itis not sensitive with respect to any of the MHGA parameters
62 Comparison of LI and SI To compare the efficiency ofthe LI and SI methods for NOCPs in the Appendix weused the Mann-Whitney nonparametric statistical test [37]
16 Mathematical Problems in Engineering
1 15 2 25 3 35 40
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
200
400
600
800
1000
1200
1400
1600
IterFi
nal c
ondi
tion
(b)
Figure 15 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (19)
2 4 6 8 10 12 14 16 18 20
50
100
150
200
250
300
350
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 10 11 120
05
1
15
2
25
3
Iter
Fina
l con
ditio
n
(b)
Figure 16 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (20)
Using the nonparametric statistical tests for comparing theperformance of several algorithms presented in [38] Forthis purpose At first 15 different runs for each of NOCPwere made Table 5 shows the mean of numerical resultsseparately Here we apply the fixMHGAparameters as119873
1199011=
91198731199012
= 121198731199051= 31119873
1199052= 91119873
119892= 100 and119873
119894= 50 with
the previous problem parameters in Table 2 Comparison
criterions contain 119869 120593119891 119870120595and Time The results of Mann-
Whitney test are shown in Tables 6ndash9 From Table 6 since119885 = 0 gt 119885
001= minus234 then the average relative error of
the LI and SI methods are same with the significant level 001In other words with probability 99 we can say that LI andSI methods have same effectiveness form the perspective oferror for 119869 Similarly the results of Tables 7ndash9 indicate that LI
Mathematical Problems in Engineering 17
Table 3 Descriptive statistics for the sensitivity analysis of1198731199051 for VDP problem
Output 1198731199051
119873 Mean Sd Min Max
119869
11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449
Total 140 15357 00294 15285 18449
120593119891
11 35 281 times 10minus8
952 times 10minus8
1 times 10minus12
452 times 10minus7
21 35 467 times 10minus8
235 times 10minus7
1 times 10minus12
141 times 10minus6
31 35 163 times 10minus8
491 times 10minus8
1 times 10minus12
221 times 10minus7
41 35 817 times 10minus8
285 times 10minus7
1 times 10minus12
126 times 10minus6
Total 140 426 times 10minus8 191 times 10minus7 1 times 10minus12 141 times 10minus6
Time
11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514
Total 140 1142131 410891 456615 2849514
119864119869
11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070
Total 140 00047 00192 0 02070
119870120595
11 35 00031 00045 475 times 10minus9
00204
21 35 00027 00064 550 times 10minus10
00369
31 35 00029 00050 154 times 10minus13
00271
41 35 00111 00397 193 times 10minus10
02070
Total 140 00047 00192 154 times 10minus13 02070
Table 4 Summary statistical data of ANOVA test for the parameters1198731199051 1198731199052 1198731199011 1198731199012 119873119892 119873119894
Parameters 119869 119864119869
120593119891
Time 119870120595
Test statistic (119865)
1198731199051
1294 1296 0624 1079 12961198731199052
110572 393 0868 5939 3931198731199011
1945 1835 1271 4317 18351198731199012
2740 2478 1011 2302 2478119873119892
3541 3591 0491 0890 3591119873119894
0495 0309 0301 5849 1046
119875 value
1198731199051
0280 0279 0601 0 02791198731199052
0 0005 0489 0 00051198731199011
0125 0143 0286 0006 01431198731199012
0021 0034 0413 0 0034119873119892
0009 0008 0743 0472 0008119873119894
0739 0819 0877 0 0386
and SI methods from the perspective of errors for 120593119891 Time
and119870120595 have same behaviour
7 The Impact of SQP
In this section we investigation the impact of the SQP on theproposed algorithm For this purpose we remove the SQPfrom Algorithm 2 as without SQP algorithm For comparingthe numerical results with the previous results the required
running time in each NOCP is considered fixed which is themaximum of running time in 12 different runs were done inSection 5 Also the all of the parameters for each problemwasset as same as Section 5 The numerical results of the withoutSQP algorithm is summarized in Table 10 By comparing theresults of Tables 10 and 1 it is obvious that for all NOCPsin the Appendix the obtained values of the performanceindex and the norm of final state constraint for the proposedalgorithm (Algorithm 2) are more accurate than the withoutSQP algorithm
18 Mathematical Problems in Engineering
Table 5 The mean of numerical results of 15 different runs for NOCPs in the Appendix using LI and SI methods
Problem LI SI119869 120593
119891119870120595
Time 119869 120593119891
119870120595
TimeVDP 15977 157 times 10
minus9
170 times 10minus3 17720 16002 667 times 10
minus8
326 times 10minus3 15757
CRP 00126 307 times 10minus8
307 times 10minus8 19714 00119 161 times 10
minus8
161 times 10minus8 19532
FFPP 4774 805 times 10minus5 03277 133280 5351 592 times 10
minus5 04868 137989MSNIC 01702 mdash mdash 13678 01703 mdash mdash 14092CSTCR 01120 mdash mdash 17385 01120 mdash mdash 17606Number 6 minus54307 mdash mdash 10541 minus54308 mdash mdash 10372Number 7 02015 326 times 10
minus9
695 times 10minus6 20807 02020 313 times 10
minus9
662 times 10minus6 21019
Number 8 630 times 10minus15 mdash mdash 6376 124 times 10
minus14 mdash mdash 5837Number 9 16512 mdash mdash 59239 16512 mdash mdash 58125Number 10 01550 mdash mdash 63153 01550 mdash mdash 65597Number 11 29701 270 times 10
minus9
272 times 10minus7 22860 29701 337 times 10
minus9
811 times 10minus7 23276
Number 12 minus02499 201 times 10minus8
464 times 10minus5 54718 minus02499 115 times 10
minus9
827 times 10minus5 52765
Number 13 00147 115 times 10minus8
00252 57797 00151 139 times 10minus8
00472 66098Number 14 34761 672 times 10
minus6
00237 104585 34290 757 times 10minus6
950 times 10minus3 108767
Number 15 195 times 10minus4
111 times 10minus8
569 times 10minus3 363 194 times 10
minus4
704 times 10minus9
486 times 10minus3 365
Number 16 20571 166 times 10minus10
166 times 10minus10 54391 20571 115 times 10
minus10
115 times 10minus10 54431
Number 17 minus88692 803 times 10minus8
803 times 10minus8 22744 minus88692 661 times 10
minus8
661 times 10minus8 23254
Number 18 00326 mdash mdash 65957 00326 mdash mdash 70897Number 19 011588 655 times 10
minus4 06828 161813 01091 884 times 10minus4 08808 159990
Number 20 5242 00176 05429 84165 4464 453 times 10minus4 03765 86891
Table 6 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 119869
Method Mean rank Sum of ranks Test statistics ValueLI 2050 4100 Mann-Whitney 119880 200SI 2050 4100 Wilcoxon119882 410mdash mdash mdash 119885 0
Table 7 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 120593119891
Method Mean rank Sum of ranks Test statistics ValueLI 1362 177 Mann-Whitney 119880 83SI 1338 174 Wilcoxon119882 174mdash mdash mdash 119885 minus0077
Table 8 Results of Mann-Whitney test on relative errors of the pair (LI SI) for Time
Method Mean rank Sum of ranks Test statistics ValueLI 2025 4050 Mann-Whitney 119880 195SI 2075 4150 Wilcoxon119882 405mdash mdash mdash 119885 minus0135
Table 9 Results of Mann-Whitney test on relative errors of the pair (LI SI) for119870120595
Method Mean rank Sum of ranks Test statistics ValueLI 1354 1760 Mann-Whitney 119880 84SI 1346 1750 Wilcoxon119882 175mdash mdash mdash 119885 minus0026
Mathematical Problems in Engineering 19
Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix
Problem no1 2 3 4 5 6 7 8 9 10
119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048
120601119891
330 times 10minus9
135 times 10minus5 11445 mdash mdash mdash 758 times 10
minus9 mdash mdash mdashProblem no
11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10
minus4 23182 minus90882 00336 2513 20274120601119891
167 times 10minus5
468 times 10minus9 00161 433 times 10
minus4
594 times 10minus3
156 times 10minus5 164 mdash 1706 25283
8 Conclusions
Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters
Appendix
Test Problems
The following problems are presented using notation given in(1)ndash(6)
(1) (VDP) [9] 119892 = (12)(1199092
1+ 1199092
2+ 1199062
) 1199050= 0 119905119891= 5 119891 =
[1199092 minus1199092+(1minus119909
2
1)1199092+119906]119879 1199090= [1 0]
119879120595 = 1199091minus1199092+1
(2) (CRP) [9] 119892 = (12)(1199092
1+1199092
2+01119906
2
) 1199050= 0 119905119891= 078
119891 = [1199091minus2(1199091+025)+(119909
2+05) exp(25119909
1(1199091+2))minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(3) (FFRP) [9] 119892 = (12)(1199062
1+ 1199062
2+ 1199062
3+ 1199062
4) 1199050
=
0 119905119891
= 5 119891 = [1199092 (110)((119906
1+ 1199062) cos119909
5minus
(1199062+ 1199064) sin119909
5) 1199094 (110)((119906
1+ 1199063) sin119909
5+ (1199062+
1199064) cos119909
5) 1199096 (512)((119906
1+ 1199063) minus (119906
2+ 1199064))]119879 1199090=
[0 0 0 0 0 0]119879 120595 = [119909
1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879
(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909
2 minus1199092+
119906 1199092
1+ 1199092
2+ 0005119906
2
]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909
2+
05 minus 8(119905 minus 05)2
]119879 1199090= [0 minus1 0]
119879
(5) (CSTCR) [20] 119892 = 1199092
1+ 1199092
2+ 01119906
2 1199050= 0 119905119891= 078
119891 = [minus(2 + 119906)(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+
2)) 05 minus 1199092minus (1199092+ 05) exp(25119909
1(1199091+ 2))]
119879 1199090=
[009 009]119879
(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909
2 119906]119879 119889 = [minus(2 +
119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]
119879(7) [36] 119892 = 119906
2 1199050= 0 119905
119891= 1 119891 = (12)119909
2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05
(8) [26] 119892 = 1199092cos2119906 119905
0= 0 119905119891= 120587 119891 = sin(1199062) 119909
0=
1205872(9) [26] 119892 = (12)(119909
2
1+ 1199092
2+ 1199062
) 1199050= 0 119905
119891= 5 119891 =
[1199092 minus1199091+ (1 minus 119909
2
1)1199092+ 119906]119879 119889 = minus(119909
2+ 025) 119909
0=
[1 0]119879
(10) [26] 119892 = 1199092
1+ 1199092
2+ 0005119906
2
1199050= 0 119905
119891= 1 119891 =
[1199092 minus1199092+119906]119879
119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus
05) minus 05 minus 1199092)]119879
1199090= [0 minus1]
119879(11) [26] 119892 = (12)119906
2 1199050= 0 119905
119891= 2 119891 = [119909
2 119906]119879 1199090=
[1 1]119879 120595 = [119909
1 1199092]119879
(12) [26] 119892 = minus1199092 1199050= 0 119905
119891= 1 119891 = [119909
2 119906]119879 119889 =
minus(1 minus 119906)(1 + 119906) 1199090= [0 0]
119879 120595 = 1199092
(13) [26] 119892 = (12)(1199092
1+ 1199092
2+ 01119906
2
) 1199050= 0 119905
119891= 078
119891 = [minus2(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+ 2)) minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(14) [26] 119892 = (12)1199062 1199050= 0 119905
119891= 10 119891 = [cos 119906 minus
1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909
0= [366 minus186]
119879120595 = [119909
1 1199092]119879
(15) [26]119892 = (12)(1199092
1+1199092
2) 1199050= 0 119905119891= 078119891 = [minus2(119909
1+
025)+(1199092+05) exp(25119909
1(1199091+2))minus(119909
1+025)119906 05minus
1199092minus(1199092+05) exp(25119909
1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(16) [26] 120601 = 1199093 1199050= 0 119905
119891= 1 119891 = [119909
2 119906 (12)119906
2
]119879
119889 = 1199091minus 19 119909
0= [0 0 0]
119879 120595 = [1199091 1199092+ 1]119879
(17) [26] 120601 = minus1199093 1199050= 0 119905
119891= 5 119891 = [119909
2 minus2 + 119906119909
3
minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909
0= [10 minus2 10]
119879120595 = [119909
1 1199092]119879
(18) [26] 120601 = (1199091minus1)2
+1199092
2+1199092
3 119892 = (12)119906
2 1199050= 0 119905119891= 5
119891 = [1199093cos 119906 119909
3sin 119906 sin 119906]119879 119909
0= [0 0 0]
119879
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
5 10 15 20 25165
17
175
18
185
Iter
Perfo
rman
ce in
dex
(a)
0 05 1 15 2 25 3 35 4 45 50
005
01
015
02
025
TimeIn
equa
lity
cons
trai
nt(b)
Figure 5 (a) Convergence rate for the performance index and (b) the inequality constraint of SI method for NOCP number (9)
5 10 15 20 25 30 35 40 45 500155
016
0165
017
0175
018
0185
019
0195
02
0205
Iter
Perfo
rman
ce in
dex
(a)
0 01 02 03 04 05 06 07 08 09 1minus25
minus2
minus15
minus1
minus05
0
Time
Ineq
ualit
y co
nstr
aint
(b)
Figure 6 (a) Convergence rate of the performance index and (b) the inequality constraint for SI method for NOCP number (10)
of the final state constraints 120593119891 are 16 times 10
minus3 407 times 10minus6
and 301 times 10minus6 respectively and for 119870
120595 these values equal
05883 407 times 10minus6 and 302 times 10
minus4 Thus we can say that thefeasibility of the solutions given by the proposed algorithmand CGA are competitive
From Table 1 to compare with numerical methods SQPand SUMT in NOCPs (8)ndash(20) the mean of 119864
119869for LI SI
SQP and SUMT equals 00145 00209 169 times 108 and 136 times
108 respectively Also the mean for the error of final state
constraints for these NOCPs equal 334 times 10minus4 441 times 10
minus50 and 0 respectively For 119870
120595 these values are 00213 00014
12263 and 11644 Therefore the performance index 119869 andthe factor 119870
120595 for the LI and SI methods are more accurate
than SQP and SUMT So the proposed algorithm gave morebetter solution in comparison with the numerical methods
12 Mathematical Problems in Engineering
5 10 15 20 25minus025
minus0245
minus024
minus0235
minus023
minus0225
Iter
Perfo
rman
ce in
dex
(a)
1 15 2 25 3 35 4 45 5 55 60
02
04
06
08
1
12
14
16
IterFi
nal c
ondi
tion
times10minus5
(b)
Figure 7 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (11)
2 4 6 8 10 12 14minus025
minus0245
minus024
minus0235
minus023
minus0225
minus022
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
02
04
06
08
1
12
14
16
18
2
Iter
Fina
l con
ditio
n
times10minus5
(b)
Figure 8 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (12)
Therefore based on this numerical studywe can concludethat the proposed algorithm outperform well-known numer-ical method Since the algorithms were not implementedon the same PC the computational times of them are notcompetitive Therefore we did not give the computationaltimes in bold in Table 1
6 Sensitivity Analysis andComparing LI and SI
In this section two statistical analysis based on the one-way analysis of variance (ANOVA) used for investigatingthe sensitivity of MHGA parameters and Mann-Whitney
Mathematical Problems in Engineering 13
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
07
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 9 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (13)
5 10 15 20 25 30
35
4
45
5
55
6
65
7
75
8
85
9
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
Iter
Fina
l con
ditio
n
(b)
Figure 10 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (14)
applied comparing LI and SI methods are done by thestatistical software IBM SPSS version 21
61 Sensitivity Analysis In order to survey the sensitivityof the MHGA parameters from the proposed algorithmthe VDP problem is selected for example The influenceof these parameters are investigated for this NOCP on
the dependent outputs consist of the performance index119869 the relative error of 119869 119864
119869 the required computational
time Time the error of final state constraints 120593119891and the
factor 119870120595 The independent parameters are consist of the
number of time nodes in both two phases 1198731199051and 119873
1199052 the
size of population in both two phases 1198731199011
and 1198731199012 the
maximum number of generations without improvement119873119894
14 Mathematical Problems in Engineering
5 10 15 20 25 30 35 40 45 50
00955
0096
00965
0097
00975
0098
00985
0099
00995
Iter
Perfo
rman
ce in
dex
(a)
2 4 6 8 10 12 14 16 18 200
05
1
15
2
25
3
35
4
45
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 11 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (15)
5 10 15 20 25
21
22
23
24
25
26
27
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
1
2
3
4
5
6
Iter
Fina
l con
ditio
n
times10minus3
(b)
Figure 12 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (16)
and the maximum number of generations 119873119892 Because the
mutation implementation probability119875119898 has a less influence
in numerical results it is not considered Also 119904119902119901119898119886119909119894119905119890119903 ischanged in each iteration of the proposed algorithm so it isnot considered too
At first we selected at least four constant value foreach of parameters and then in each pose 35 differentruns were made independently The statistical analysis isdone based on ANOVA The descriptive statistics whichcontains the number of different runs (119873) the mean of each
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7minus887
minus886
minus885
minus884
minus883
minus882
minus881
minus88
minus879
minus878
minus877
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 70
01
02
03
04
05
06
07
08
09
1
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 13 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (17)
1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333
Iter
Perfo
rman
ce in
dex
Figure 14 Convergence rate for the performance index of SImethod for NOCP numbers (18)
output in119873 different runs (Mean) standard deviation (Sd)maximum (Max) and minimum (Min) of each outputcould be achieved Among of the MHGA parameters wepresent the descriptive statistics of the ANOVA only for the1198731199051parameter which is reported in Table 3Table 4 summarized the statistical data contain the test
statistics (119865) and 119875-values of ANOVA tests Sensitivityanalysis for each parameter separately are done based on thistable as follows
1198731199051 From Table 4 the significant level or 119875-value for119870120595
is equal to 0279 which is greater than 005So from ANOVA 119873
1199051parameter has no significant
effect on the factor 119870120595 With similar analysis this
parameter has no effect on the other outputs except
required computational time Time because its 119875-value is equal to 0 which is less than 005 that is thisoutput sensitive to the parameter119873
1199051
1198731199052 From the second row of Table 4 the outputs 119869 119864
119869119870120595
and Time are sensitive to the parameter 1198731199052 and the
only output 120593119891is independent
1198731199011 From the third row of Table 4 only the computationalrequired time Time is sensitive because the its 119875-value is equal to 0006 which is less than 005 and theother outputs contain 119869 119864
119869 120593119891119870120595 are independent
1198731199012 From the forth row of Table 4 the output 120593
119891is
independent respect to the parameter 1198731199012 but other
outputs contain 119869 119864119869 Time 119870
120601are sensitive to this
parameter119873119892 From the fifth row of Table 4 the outputs 119869 119864
119869and
119870120595are independent to the parameter 119873
119892 and other
outputs contain 120593119891and Time are sensitive respect to
this parameter119873119894 The sensitivity analysis is similar to119873
1199011
From above cases it is obvious that all parameters canbe effect on the required computational time except 119873
119892
Moreover intuitions shows the normof final state constraints120593119891is independent output with respect to all parameters that
is any of the parameters could not effect on this output and itis not sensitive with respect to any of the MHGA parameters
62 Comparison of LI and SI To compare the efficiency ofthe LI and SI methods for NOCPs in the Appendix weused the Mann-Whitney nonparametric statistical test [37]
16 Mathematical Problems in Engineering
1 15 2 25 3 35 40
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
200
400
600
800
1000
1200
1400
1600
IterFi
nal c
ondi
tion
(b)
Figure 15 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (19)
2 4 6 8 10 12 14 16 18 20
50
100
150
200
250
300
350
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 10 11 120
05
1
15
2
25
3
Iter
Fina
l con
ditio
n
(b)
Figure 16 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (20)
Using the nonparametric statistical tests for comparing theperformance of several algorithms presented in [38] Forthis purpose At first 15 different runs for each of NOCPwere made Table 5 shows the mean of numerical resultsseparately Here we apply the fixMHGAparameters as119873
1199011=
91198731199012
= 121198731199051= 31119873
1199052= 91119873
119892= 100 and119873
119894= 50 with
the previous problem parameters in Table 2 Comparison
criterions contain 119869 120593119891 119870120595and Time The results of Mann-
Whitney test are shown in Tables 6ndash9 From Table 6 since119885 = 0 gt 119885
001= minus234 then the average relative error of
the LI and SI methods are same with the significant level 001In other words with probability 99 we can say that LI andSI methods have same effectiveness form the perspective oferror for 119869 Similarly the results of Tables 7ndash9 indicate that LI
Mathematical Problems in Engineering 17
Table 3 Descriptive statistics for the sensitivity analysis of1198731199051 for VDP problem
Output 1198731199051
119873 Mean Sd Min Max
119869
11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449
Total 140 15357 00294 15285 18449
120593119891
11 35 281 times 10minus8
952 times 10minus8
1 times 10minus12
452 times 10minus7
21 35 467 times 10minus8
235 times 10minus7
1 times 10minus12
141 times 10minus6
31 35 163 times 10minus8
491 times 10minus8
1 times 10minus12
221 times 10minus7
41 35 817 times 10minus8
285 times 10minus7
1 times 10minus12
126 times 10minus6
Total 140 426 times 10minus8 191 times 10minus7 1 times 10minus12 141 times 10minus6
Time
11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514
Total 140 1142131 410891 456615 2849514
119864119869
11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070
Total 140 00047 00192 0 02070
119870120595
11 35 00031 00045 475 times 10minus9
00204
21 35 00027 00064 550 times 10minus10
00369
31 35 00029 00050 154 times 10minus13
00271
41 35 00111 00397 193 times 10minus10
02070
Total 140 00047 00192 154 times 10minus13 02070
Table 4 Summary statistical data of ANOVA test for the parameters1198731199051 1198731199052 1198731199011 1198731199012 119873119892 119873119894
Parameters 119869 119864119869
120593119891
Time 119870120595
Test statistic (119865)
1198731199051
1294 1296 0624 1079 12961198731199052
110572 393 0868 5939 3931198731199011
1945 1835 1271 4317 18351198731199012
2740 2478 1011 2302 2478119873119892
3541 3591 0491 0890 3591119873119894
0495 0309 0301 5849 1046
119875 value
1198731199051
0280 0279 0601 0 02791198731199052
0 0005 0489 0 00051198731199011
0125 0143 0286 0006 01431198731199012
0021 0034 0413 0 0034119873119892
0009 0008 0743 0472 0008119873119894
0739 0819 0877 0 0386
and SI methods from the perspective of errors for 120593119891 Time
and119870120595 have same behaviour
7 The Impact of SQP
In this section we investigation the impact of the SQP on theproposed algorithm For this purpose we remove the SQPfrom Algorithm 2 as without SQP algorithm For comparingthe numerical results with the previous results the required
running time in each NOCP is considered fixed which is themaximum of running time in 12 different runs were done inSection 5 Also the all of the parameters for each problemwasset as same as Section 5 The numerical results of the withoutSQP algorithm is summarized in Table 10 By comparing theresults of Tables 10 and 1 it is obvious that for all NOCPsin the Appendix the obtained values of the performanceindex and the norm of final state constraint for the proposedalgorithm (Algorithm 2) are more accurate than the withoutSQP algorithm
18 Mathematical Problems in Engineering
Table 5 The mean of numerical results of 15 different runs for NOCPs in the Appendix using LI and SI methods
Problem LI SI119869 120593
119891119870120595
Time 119869 120593119891
119870120595
TimeVDP 15977 157 times 10
minus9
170 times 10minus3 17720 16002 667 times 10
minus8
326 times 10minus3 15757
CRP 00126 307 times 10minus8
307 times 10minus8 19714 00119 161 times 10
minus8
161 times 10minus8 19532
FFPP 4774 805 times 10minus5 03277 133280 5351 592 times 10
minus5 04868 137989MSNIC 01702 mdash mdash 13678 01703 mdash mdash 14092CSTCR 01120 mdash mdash 17385 01120 mdash mdash 17606Number 6 minus54307 mdash mdash 10541 minus54308 mdash mdash 10372Number 7 02015 326 times 10
minus9
695 times 10minus6 20807 02020 313 times 10
minus9
662 times 10minus6 21019
Number 8 630 times 10minus15 mdash mdash 6376 124 times 10
minus14 mdash mdash 5837Number 9 16512 mdash mdash 59239 16512 mdash mdash 58125Number 10 01550 mdash mdash 63153 01550 mdash mdash 65597Number 11 29701 270 times 10
minus9
272 times 10minus7 22860 29701 337 times 10
minus9
811 times 10minus7 23276
Number 12 minus02499 201 times 10minus8
464 times 10minus5 54718 minus02499 115 times 10
minus9
827 times 10minus5 52765
Number 13 00147 115 times 10minus8
00252 57797 00151 139 times 10minus8
00472 66098Number 14 34761 672 times 10
minus6
00237 104585 34290 757 times 10minus6
950 times 10minus3 108767
Number 15 195 times 10minus4
111 times 10minus8
569 times 10minus3 363 194 times 10
minus4
704 times 10minus9
486 times 10minus3 365
Number 16 20571 166 times 10minus10
166 times 10minus10 54391 20571 115 times 10
minus10
115 times 10minus10 54431
Number 17 minus88692 803 times 10minus8
803 times 10minus8 22744 minus88692 661 times 10
minus8
661 times 10minus8 23254
Number 18 00326 mdash mdash 65957 00326 mdash mdash 70897Number 19 011588 655 times 10
minus4 06828 161813 01091 884 times 10minus4 08808 159990
Number 20 5242 00176 05429 84165 4464 453 times 10minus4 03765 86891
Table 6 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 119869
Method Mean rank Sum of ranks Test statistics ValueLI 2050 4100 Mann-Whitney 119880 200SI 2050 4100 Wilcoxon119882 410mdash mdash mdash 119885 0
Table 7 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 120593119891
Method Mean rank Sum of ranks Test statistics ValueLI 1362 177 Mann-Whitney 119880 83SI 1338 174 Wilcoxon119882 174mdash mdash mdash 119885 minus0077
Table 8 Results of Mann-Whitney test on relative errors of the pair (LI SI) for Time
Method Mean rank Sum of ranks Test statistics ValueLI 2025 4050 Mann-Whitney 119880 195SI 2075 4150 Wilcoxon119882 405mdash mdash mdash 119885 minus0135
Table 9 Results of Mann-Whitney test on relative errors of the pair (LI SI) for119870120595
Method Mean rank Sum of ranks Test statistics ValueLI 1354 1760 Mann-Whitney 119880 84SI 1346 1750 Wilcoxon119882 175mdash mdash mdash 119885 minus0026
Mathematical Problems in Engineering 19
Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix
Problem no1 2 3 4 5 6 7 8 9 10
119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048
120601119891
330 times 10minus9
135 times 10minus5 11445 mdash mdash mdash 758 times 10
minus9 mdash mdash mdashProblem no
11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10
minus4 23182 minus90882 00336 2513 20274120601119891
167 times 10minus5
468 times 10minus9 00161 433 times 10
minus4
594 times 10minus3
156 times 10minus5 164 mdash 1706 25283
8 Conclusions
Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters
Appendix
Test Problems
The following problems are presented using notation given in(1)ndash(6)
(1) (VDP) [9] 119892 = (12)(1199092
1+ 1199092
2+ 1199062
) 1199050= 0 119905119891= 5 119891 =
[1199092 minus1199092+(1minus119909
2
1)1199092+119906]119879 1199090= [1 0]
119879120595 = 1199091minus1199092+1
(2) (CRP) [9] 119892 = (12)(1199092
1+1199092
2+01119906
2
) 1199050= 0 119905119891= 078
119891 = [1199091minus2(1199091+025)+(119909
2+05) exp(25119909
1(1199091+2))minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(3) (FFRP) [9] 119892 = (12)(1199062
1+ 1199062
2+ 1199062
3+ 1199062
4) 1199050
=
0 119905119891
= 5 119891 = [1199092 (110)((119906
1+ 1199062) cos119909
5minus
(1199062+ 1199064) sin119909
5) 1199094 (110)((119906
1+ 1199063) sin119909
5+ (1199062+
1199064) cos119909
5) 1199096 (512)((119906
1+ 1199063) minus (119906
2+ 1199064))]119879 1199090=
[0 0 0 0 0 0]119879 120595 = [119909
1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879
(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909
2 minus1199092+
119906 1199092
1+ 1199092
2+ 0005119906
2
]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909
2+
05 minus 8(119905 minus 05)2
]119879 1199090= [0 minus1 0]
119879
(5) (CSTCR) [20] 119892 = 1199092
1+ 1199092
2+ 01119906
2 1199050= 0 119905119891= 078
119891 = [minus(2 + 119906)(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+
2)) 05 minus 1199092minus (1199092+ 05) exp(25119909
1(1199091+ 2))]
119879 1199090=
[009 009]119879
(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909
2 119906]119879 119889 = [minus(2 +
119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]
119879(7) [36] 119892 = 119906
2 1199050= 0 119905
119891= 1 119891 = (12)119909
2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05
(8) [26] 119892 = 1199092cos2119906 119905
0= 0 119905119891= 120587 119891 = sin(1199062) 119909
0=
1205872(9) [26] 119892 = (12)(119909
2
1+ 1199092
2+ 1199062
) 1199050= 0 119905
119891= 5 119891 =
[1199092 minus1199091+ (1 minus 119909
2
1)1199092+ 119906]119879 119889 = minus(119909
2+ 025) 119909
0=
[1 0]119879
(10) [26] 119892 = 1199092
1+ 1199092
2+ 0005119906
2
1199050= 0 119905
119891= 1 119891 =
[1199092 minus1199092+119906]119879
119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus
05) minus 05 minus 1199092)]119879
1199090= [0 minus1]
119879(11) [26] 119892 = (12)119906
2 1199050= 0 119905
119891= 2 119891 = [119909
2 119906]119879 1199090=
[1 1]119879 120595 = [119909
1 1199092]119879
(12) [26] 119892 = minus1199092 1199050= 0 119905
119891= 1 119891 = [119909
2 119906]119879 119889 =
minus(1 minus 119906)(1 + 119906) 1199090= [0 0]
119879 120595 = 1199092
(13) [26] 119892 = (12)(1199092
1+ 1199092
2+ 01119906
2
) 1199050= 0 119905
119891= 078
119891 = [minus2(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+ 2)) minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(14) [26] 119892 = (12)1199062 1199050= 0 119905
119891= 10 119891 = [cos 119906 minus
1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909
0= [366 minus186]
119879120595 = [119909
1 1199092]119879
(15) [26]119892 = (12)(1199092
1+1199092
2) 1199050= 0 119905119891= 078119891 = [minus2(119909
1+
025)+(1199092+05) exp(25119909
1(1199091+2))minus(119909
1+025)119906 05minus
1199092minus(1199092+05) exp(25119909
1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(16) [26] 120601 = 1199093 1199050= 0 119905
119891= 1 119891 = [119909
2 119906 (12)119906
2
]119879
119889 = 1199091minus 19 119909
0= [0 0 0]
119879 120595 = [1199091 1199092+ 1]119879
(17) [26] 120601 = minus1199093 1199050= 0 119905
119891= 5 119891 = [119909
2 minus2 + 119906119909
3
minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909
0= [10 minus2 10]
119879120595 = [119909
1 1199092]119879
(18) [26] 120601 = (1199091minus1)2
+1199092
2+1199092
3 119892 = (12)119906
2 1199050= 0 119905119891= 5
119891 = [1199093cos 119906 119909
3sin 119906 sin 119906]119879 119909
0= [0 0 0]
119879
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
5 10 15 20 25minus025
minus0245
minus024
minus0235
minus023
minus0225
Iter
Perfo
rman
ce in
dex
(a)
1 15 2 25 3 35 4 45 5 55 60
02
04
06
08
1
12
14
16
IterFi
nal c
ondi
tion
times10minus5
(b)
Figure 7 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (11)
2 4 6 8 10 12 14minus025
minus0245
minus024
minus0235
minus023
minus0225
minus022
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
02
04
06
08
1
12
14
16
18
2
Iter
Fina
l con
ditio
n
times10minus5
(b)
Figure 8 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (12)
Therefore based on this numerical studywe can concludethat the proposed algorithm outperform well-known numer-ical method Since the algorithms were not implementedon the same PC the computational times of them are notcompetitive Therefore we did not give the computationaltimes in bold in Table 1
6 Sensitivity Analysis andComparing LI and SI
In this section two statistical analysis based on the one-way analysis of variance (ANOVA) used for investigatingthe sensitivity of MHGA parameters and Mann-Whitney
Mathematical Problems in Engineering 13
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
07
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 9 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (13)
5 10 15 20 25 30
35
4
45
5
55
6
65
7
75
8
85
9
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
Iter
Fina
l con
ditio
n
(b)
Figure 10 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (14)
applied comparing LI and SI methods are done by thestatistical software IBM SPSS version 21
61 Sensitivity Analysis In order to survey the sensitivityof the MHGA parameters from the proposed algorithmthe VDP problem is selected for example The influenceof these parameters are investigated for this NOCP on
the dependent outputs consist of the performance index119869 the relative error of 119869 119864
119869 the required computational
time Time the error of final state constraints 120593119891and the
factor 119870120595 The independent parameters are consist of the
number of time nodes in both two phases 1198731199051and 119873
1199052 the
size of population in both two phases 1198731199011
and 1198731199012 the
maximum number of generations without improvement119873119894
14 Mathematical Problems in Engineering
5 10 15 20 25 30 35 40 45 50
00955
0096
00965
0097
00975
0098
00985
0099
00995
Iter
Perfo
rman
ce in
dex
(a)
2 4 6 8 10 12 14 16 18 200
05
1
15
2
25
3
35
4
45
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 11 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (15)
5 10 15 20 25
21
22
23
24
25
26
27
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
1
2
3
4
5
6
Iter
Fina
l con
ditio
n
times10minus3
(b)
Figure 12 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (16)
and the maximum number of generations 119873119892 Because the
mutation implementation probability119875119898 has a less influence
in numerical results it is not considered Also 119904119902119901119898119886119909119894119905119890119903 ischanged in each iteration of the proposed algorithm so it isnot considered too
At first we selected at least four constant value foreach of parameters and then in each pose 35 differentruns were made independently The statistical analysis isdone based on ANOVA The descriptive statistics whichcontains the number of different runs (119873) the mean of each
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7minus887
minus886
minus885
minus884
minus883
minus882
minus881
minus88
minus879
minus878
minus877
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 70
01
02
03
04
05
06
07
08
09
1
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 13 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (17)
1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333
Iter
Perfo
rman
ce in
dex
Figure 14 Convergence rate for the performance index of SImethod for NOCP numbers (18)
output in119873 different runs (Mean) standard deviation (Sd)maximum (Max) and minimum (Min) of each outputcould be achieved Among of the MHGA parameters wepresent the descriptive statistics of the ANOVA only for the1198731199051parameter which is reported in Table 3Table 4 summarized the statistical data contain the test
statistics (119865) and 119875-values of ANOVA tests Sensitivityanalysis for each parameter separately are done based on thistable as follows
1198731199051 From Table 4 the significant level or 119875-value for119870120595
is equal to 0279 which is greater than 005So from ANOVA 119873
1199051parameter has no significant
effect on the factor 119870120595 With similar analysis this
parameter has no effect on the other outputs except
required computational time Time because its 119875-value is equal to 0 which is less than 005 that is thisoutput sensitive to the parameter119873
1199051
1198731199052 From the second row of Table 4 the outputs 119869 119864
119869119870120595
and Time are sensitive to the parameter 1198731199052 and the
only output 120593119891is independent
1198731199011 From the third row of Table 4 only the computationalrequired time Time is sensitive because the its 119875-value is equal to 0006 which is less than 005 and theother outputs contain 119869 119864
119869 120593119891119870120595 are independent
1198731199012 From the forth row of Table 4 the output 120593
119891is
independent respect to the parameter 1198731199012 but other
outputs contain 119869 119864119869 Time 119870
120601are sensitive to this
parameter119873119892 From the fifth row of Table 4 the outputs 119869 119864
119869and
119870120595are independent to the parameter 119873
119892 and other
outputs contain 120593119891and Time are sensitive respect to
this parameter119873119894 The sensitivity analysis is similar to119873
1199011
From above cases it is obvious that all parameters canbe effect on the required computational time except 119873
119892
Moreover intuitions shows the normof final state constraints120593119891is independent output with respect to all parameters that
is any of the parameters could not effect on this output and itis not sensitive with respect to any of the MHGA parameters
62 Comparison of LI and SI To compare the efficiency ofthe LI and SI methods for NOCPs in the Appendix weused the Mann-Whitney nonparametric statistical test [37]
16 Mathematical Problems in Engineering
1 15 2 25 3 35 40
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
200
400
600
800
1000
1200
1400
1600
IterFi
nal c
ondi
tion
(b)
Figure 15 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (19)
2 4 6 8 10 12 14 16 18 20
50
100
150
200
250
300
350
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 10 11 120
05
1
15
2
25
3
Iter
Fina
l con
ditio
n
(b)
Figure 16 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (20)
Using the nonparametric statistical tests for comparing theperformance of several algorithms presented in [38] Forthis purpose At first 15 different runs for each of NOCPwere made Table 5 shows the mean of numerical resultsseparately Here we apply the fixMHGAparameters as119873
1199011=
91198731199012
= 121198731199051= 31119873
1199052= 91119873
119892= 100 and119873
119894= 50 with
the previous problem parameters in Table 2 Comparison
criterions contain 119869 120593119891 119870120595and Time The results of Mann-
Whitney test are shown in Tables 6ndash9 From Table 6 since119885 = 0 gt 119885
001= minus234 then the average relative error of
the LI and SI methods are same with the significant level 001In other words with probability 99 we can say that LI andSI methods have same effectiveness form the perspective oferror for 119869 Similarly the results of Tables 7ndash9 indicate that LI
Mathematical Problems in Engineering 17
Table 3 Descriptive statistics for the sensitivity analysis of1198731199051 for VDP problem
Output 1198731199051
119873 Mean Sd Min Max
119869
11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449
Total 140 15357 00294 15285 18449
120593119891
11 35 281 times 10minus8
952 times 10minus8
1 times 10minus12
452 times 10minus7
21 35 467 times 10minus8
235 times 10minus7
1 times 10minus12
141 times 10minus6
31 35 163 times 10minus8
491 times 10minus8
1 times 10minus12
221 times 10minus7
41 35 817 times 10minus8
285 times 10minus7
1 times 10minus12
126 times 10minus6
Total 140 426 times 10minus8 191 times 10minus7 1 times 10minus12 141 times 10minus6
Time
11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514
Total 140 1142131 410891 456615 2849514
119864119869
11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070
Total 140 00047 00192 0 02070
119870120595
11 35 00031 00045 475 times 10minus9
00204
21 35 00027 00064 550 times 10minus10
00369
31 35 00029 00050 154 times 10minus13
00271
41 35 00111 00397 193 times 10minus10
02070
Total 140 00047 00192 154 times 10minus13 02070
Table 4 Summary statistical data of ANOVA test for the parameters1198731199051 1198731199052 1198731199011 1198731199012 119873119892 119873119894
Parameters 119869 119864119869
120593119891
Time 119870120595
Test statistic (119865)
1198731199051
1294 1296 0624 1079 12961198731199052
110572 393 0868 5939 3931198731199011
1945 1835 1271 4317 18351198731199012
2740 2478 1011 2302 2478119873119892
3541 3591 0491 0890 3591119873119894
0495 0309 0301 5849 1046
119875 value
1198731199051
0280 0279 0601 0 02791198731199052
0 0005 0489 0 00051198731199011
0125 0143 0286 0006 01431198731199012
0021 0034 0413 0 0034119873119892
0009 0008 0743 0472 0008119873119894
0739 0819 0877 0 0386
and SI methods from the perspective of errors for 120593119891 Time
and119870120595 have same behaviour
7 The Impact of SQP
In this section we investigation the impact of the SQP on theproposed algorithm For this purpose we remove the SQPfrom Algorithm 2 as without SQP algorithm For comparingthe numerical results with the previous results the required
running time in each NOCP is considered fixed which is themaximum of running time in 12 different runs were done inSection 5 Also the all of the parameters for each problemwasset as same as Section 5 The numerical results of the withoutSQP algorithm is summarized in Table 10 By comparing theresults of Tables 10 and 1 it is obvious that for all NOCPsin the Appendix the obtained values of the performanceindex and the norm of final state constraint for the proposedalgorithm (Algorithm 2) are more accurate than the withoutSQP algorithm
18 Mathematical Problems in Engineering
Table 5 The mean of numerical results of 15 different runs for NOCPs in the Appendix using LI and SI methods
Problem LI SI119869 120593
119891119870120595
Time 119869 120593119891
119870120595
TimeVDP 15977 157 times 10
minus9
170 times 10minus3 17720 16002 667 times 10
minus8
326 times 10minus3 15757
CRP 00126 307 times 10minus8
307 times 10minus8 19714 00119 161 times 10
minus8
161 times 10minus8 19532
FFPP 4774 805 times 10minus5 03277 133280 5351 592 times 10
minus5 04868 137989MSNIC 01702 mdash mdash 13678 01703 mdash mdash 14092CSTCR 01120 mdash mdash 17385 01120 mdash mdash 17606Number 6 minus54307 mdash mdash 10541 minus54308 mdash mdash 10372Number 7 02015 326 times 10
minus9
695 times 10minus6 20807 02020 313 times 10
minus9
662 times 10minus6 21019
Number 8 630 times 10minus15 mdash mdash 6376 124 times 10
minus14 mdash mdash 5837Number 9 16512 mdash mdash 59239 16512 mdash mdash 58125Number 10 01550 mdash mdash 63153 01550 mdash mdash 65597Number 11 29701 270 times 10
minus9
272 times 10minus7 22860 29701 337 times 10
minus9
811 times 10minus7 23276
Number 12 minus02499 201 times 10minus8
464 times 10minus5 54718 minus02499 115 times 10
minus9
827 times 10minus5 52765
Number 13 00147 115 times 10minus8
00252 57797 00151 139 times 10minus8
00472 66098Number 14 34761 672 times 10
minus6
00237 104585 34290 757 times 10minus6
950 times 10minus3 108767
Number 15 195 times 10minus4
111 times 10minus8
569 times 10minus3 363 194 times 10
minus4
704 times 10minus9
486 times 10minus3 365
Number 16 20571 166 times 10minus10
166 times 10minus10 54391 20571 115 times 10
minus10
115 times 10minus10 54431
Number 17 minus88692 803 times 10minus8
803 times 10minus8 22744 minus88692 661 times 10
minus8
661 times 10minus8 23254
Number 18 00326 mdash mdash 65957 00326 mdash mdash 70897Number 19 011588 655 times 10
minus4 06828 161813 01091 884 times 10minus4 08808 159990
Number 20 5242 00176 05429 84165 4464 453 times 10minus4 03765 86891
Table 6 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 119869
Method Mean rank Sum of ranks Test statistics ValueLI 2050 4100 Mann-Whitney 119880 200SI 2050 4100 Wilcoxon119882 410mdash mdash mdash 119885 0
Table 7 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 120593119891
Method Mean rank Sum of ranks Test statistics ValueLI 1362 177 Mann-Whitney 119880 83SI 1338 174 Wilcoxon119882 174mdash mdash mdash 119885 minus0077
Table 8 Results of Mann-Whitney test on relative errors of the pair (LI SI) for Time
Method Mean rank Sum of ranks Test statistics ValueLI 2025 4050 Mann-Whitney 119880 195SI 2075 4150 Wilcoxon119882 405mdash mdash mdash 119885 minus0135
Table 9 Results of Mann-Whitney test on relative errors of the pair (LI SI) for119870120595
Method Mean rank Sum of ranks Test statistics ValueLI 1354 1760 Mann-Whitney 119880 84SI 1346 1750 Wilcoxon119882 175mdash mdash mdash 119885 minus0026
Mathematical Problems in Engineering 19
Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix
Problem no1 2 3 4 5 6 7 8 9 10
119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048
120601119891
330 times 10minus9
135 times 10minus5 11445 mdash mdash mdash 758 times 10
minus9 mdash mdash mdashProblem no
11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10
minus4 23182 minus90882 00336 2513 20274120601119891
167 times 10minus5
468 times 10minus9 00161 433 times 10
minus4
594 times 10minus3
156 times 10minus5 164 mdash 1706 25283
8 Conclusions
Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters
Appendix
Test Problems
The following problems are presented using notation given in(1)ndash(6)
(1) (VDP) [9] 119892 = (12)(1199092
1+ 1199092
2+ 1199062
) 1199050= 0 119905119891= 5 119891 =
[1199092 minus1199092+(1minus119909
2
1)1199092+119906]119879 1199090= [1 0]
119879120595 = 1199091minus1199092+1
(2) (CRP) [9] 119892 = (12)(1199092
1+1199092
2+01119906
2
) 1199050= 0 119905119891= 078
119891 = [1199091minus2(1199091+025)+(119909
2+05) exp(25119909
1(1199091+2))minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(3) (FFRP) [9] 119892 = (12)(1199062
1+ 1199062
2+ 1199062
3+ 1199062
4) 1199050
=
0 119905119891
= 5 119891 = [1199092 (110)((119906
1+ 1199062) cos119909
5minus
(1199062+ 1199064) sin119909
5) 1199094 (110)((119906
1+ 1199063) sin119909
5+ (1199062+
1199064) cos119909
5) 1199096 (512)((119906
1+ 1199063) minus (119906
2+ 1199064))]119879 1199090=
[0 0 0 0 0 0]119879 120595 = [119909
1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879
(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909
2 minus1199092+
119906 1199092
1+ 1199092
2+ 0005119906
2
]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909
2+
05 minus 8(119905 minus 05)2
]119879 1199090= [0 minus1 0]
119879
(5) (CSTCR) [20] 119892 = 1199092
1+ 1199092
2+ 01119906
2 1199050= 0 119905119891= 078
119891 = [minus(2 + 119906)(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+
2)) 05 minus 1199092minus (1199092+ 05) exp(25119909
1(1199091+ 2))]
119879 1199090=
[009 009]119879
(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909
2 119906]119879 119889 = [minus(2 +
119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]
119879(7) [36] 119892 = 119906
2 1199050= 0 119905
119891= 1 119891 = (12)119909
2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05
(8) [26] 119892 = 1199092cos2119906 119905
0= 0 119905119891= 120587 119891 = sin(1199062) 119909
0=
1205872(9) [26] 119892 = (12)(119909
2
1+ 1199092
2+ 1199062
) 1199050= 0 119905
119891= 5 119891 =
[1199092 minus1199091+ (1 minus 119909
2
1)1199092+ 119906]119879 119889 = minus(119909
2+ 025) 119909
0=
[1 0]119879
(10) [26] 119892 = 1199092
1+ 1199092
2+ 0005119906
2
1199050= 0 119905
119891= 1 119891 =
[1199092 minus1199092+119906]119879
119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus
05) minus 05 minus 1199092)]119879
1199090= [0 minus1]
119879(11) [26] 119892 = (12)119906
2 1199050= 0 119905
119891= 2 119891 = [119909
2 119906]119879 1199090=
[1 1]119879 120595 = [119909
1 1199092]119879
(12) [26] 119892 = minus1199092 1199050= 0 119905
119891= 1 119891 = [119909
2 119906]119879 119889 =
minus(1 minus 119906)(1 + 119906) 1199090= [0 0]
119879 120595 = 1199092
(13) [26] 119892 = (12)(1199092
1+ 1199092
2+ 01119906
2
) 1199050= 0 119905
119891= 078
119891 = [minus2(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+ 2)) minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(14) [26] 119892 = (12)1199062 1199050= 0 119905
119891= 10 119891 = [cos 119906 minus
1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909
0= [366 minus186]
119879120595 = [119909
1 1199092]119879
(15) [26]119892 = (12)(1199092
1+1199092
2) 1199050= 0 119905119891= 078119891 = [minus2(119909
1+
025)+(1199092+05) exp(25119909
1(1199091+2))minus(119909
1+025)119906 05minus
1199092minus(1199092+05) exp(25119909
1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(16) [26] 120601 = 1199093 1199050= 0 119905
119891= 1 119891 = [119909
2 119906 (12)119906
2
]119879
119889 = 1199091minus 19 119909
0= [0 0 0]
119879 120595 = [1199091 1199092+ 1]119879
(17) [26] 120601 = minus1199093 1199050= 0 119905
119891= 5 119891 = [119909
2 minus2 + 119906119909
3
minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909
0= [10 minus2 10]
119879120595 = [119909
1 1199092]119879
(18) [26] 120601 = (1199091minus1)2
+1199092
2+1199092
3 119892 = (12)119906
2 1199050= 0 119905119891= 5
119891 = [1199093cos 119906 119909
3sin 119906 sin 119906]119879 119909
0= [0 0 0]
119879
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
07
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 9 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (13)
5 10 15 20 25 30
35
4
45
5
55
6
65
7
75
8
85
9
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 100
01
02
03
04
05
06
Iter
Fina
l con
ditio
n
(b)
Figure 10 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (14)
applied comparing LI and SI methods are done by thestatistical software IBM SPSS version 21
61 Sensitivity Analysis In order to survey the sensitivityof the MHGA parameters from the proposed algorithmthe VDP problem is selected for example The influenceof these parameters are investigated for this NOCP on
the dependent outputs consist of the performance index119869 the relative error of 119869 119864
119869 the required computational
time Time the error of final state constraints 120593119891and the
factor 119870120595 The independent parameters are consist of the
number of time nodes in both two phases 1198731199051and 119873
1199052 the
size of population in both two phases 1198731199011
and 1198731199012 the
maximum number of generations without improvement119873119894
14 Mathematical Problems in Engineering
5 10 15 20 25 30 35 40 45 50
00955
0096
00965
0097
00975
0098
00985
0099
00995
Iter
Perfo
rman
ce in
dex
(a)
2 4 6 8 10 12 14 16 18 200
05
1
15
2
25
3
35
4
45
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 11 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (15)
5 10 15 20 25
21
22
23
24
25
26
27
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
1
2
3
4
5
6
Iter
Fina
l con
ditio
n
times10minus3
(b)
Figure 12 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (16)
and the maximum number of generations 119873119892 Because the
mutation implementation probability119875119898 has a less influence
in numerical results it is not considered Also 119904119902119901119898119886119909119894119905119890119903 ischanged in each iteration of the proposed algorithm so it isnot considered too
At first we selected at least four constant value foreach of parameters and then in each pose 35 differentruns were made independently The statistical analysis isdone based on ANOVA The descriptive statistics whichcontains the number of different runs (119873) the mean of each
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7minus887
minus886
minus885
minus884
minus883
minus882
minus881
minus88
minus879
minus878
minus877
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 70
01
02
03
04
05
06
07
08
09
1
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 13 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (17)
1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333
Iter
Perfo
rman
ce in
dex
Figure 14 Convergence rate for the performance index of SImethod for NOCP numbers (18)
output in119873 different runs (Mean) standard deviation (Sd)maximum (Max) and minimum (Min) of each outputcould be achieved Among of the MHGA parameters wepresent the descriptive statistics of the ANOVA only for the1198731199051parameter which is reported in Table 3Table 4 summarized the statistical data contain the test
statistics (119865) and 119875-values of ANOVA tests Sensitivityanalysis for each parameter separately are done based on thistable as follows
1198731199051 From Table 4 the significant level or 119875-value for119870120595
is equal to 0279 which is greater than 005So from ANOVA 119873
1199051parameter has no significant
effect on the factor 119870120595 With similar analysis this
parameter has no effect on the other outputs except
required computational time Time because its 119875-value is equal to 0 which is less than 005 that is thisoutput sensitive to the parameter119873
1199051
1198731199052 From the second row of Table 4 the outputs 119869 119864
119869119870120595
and Time are sensitive to the parameter 1198731199052 and the
only output 120593119891is independent
1198731199011 From the third row of Table 4 only the computationalrequired time Time is sensitive because the its 119875-value is equal to 0006 which is less than 005 and theother outputs contain 119869 119864
119869 120593119891119870120595 are independent
1198731199012 From the forth row of Table 4 the output 120593
119891is
independent respect to the parameter 1198731199012 but other
outputs contain 119869 119864119869 Time 119870
120601are sensitive to this
parameter119873119892 From the fifth row of Table 4 the outputs 119869 119864
119869and
119870120595are independent to the parameter 119873
119892 and other
outputs contain 120593119891and Time are sensitive respect to
this parameter119873119894 The sensitivity analysis is similar to119873
1199011
From above cases it is obvious that all parameters canbe effect on the required computational time except 119873
119892
Moreover intuitions shows the normof final state constraints120593119891is independent output with respect to all parameters that
is any of the parameters could not effect on this output and itis not sensitive with respect to any of the MHGA parameters
62 Comparison of LI and SI To compare the efficiency ofthe LI and SI methods for NOCPs in the Appendix weused the Mann-Whitney nonparametric statistical test [37]
16 Mathematical Problems in Engineering
1 15 2 25 3 35 40
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
200
400
600
800
1000
1200
1400
1600
IterFi
nal c
ondi
tion
(b)
Figure 15 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (19)
2 4 6 8 10 12 14 16 18 20
50
100
150
200
250
300
350
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 10 11 120
05
1
15
2
25
3
Iter
Fina
l con
ditio
n
(b)
Figure 16 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (20)
Using the nonparametric statistical tests for comparing theperformance of several algorithms presented in [38] Forthis purpose At first 15 different runs for each of NOCPwere made Table 5 shows the mean of numerical resultsseparately Here we apply the fixMHGAparameters as119873
1199011=
91198731199012
= 121198731199051= 31119873
1199052= 91119873
119892= 100 and119873
119894= 50 with
the previous problem parameters in Table 2 Comparison
criterions contain 119869 120593119891 119870120595and Time The results of Mann-
Whitney test are shown in Tables 6ndash9 From Table 6 since119885 = 0 gt 119885
001= minus234 then the average relative error of
the LI and SI methods are same with the significant level 001In other words with probability 99 we can say that LI andSI methods have same effectiveness form the perspective oferror for 119869 Similarly the results of Tables 7ndash9 indicate that LI
Mathematical Problems in Engineering 17
Table 3 Descriptive statistics for the sensitivity analysis of1198731199051 for VDP problem
Output 1198731199051
119873 Mean Sd Min Max
119869
11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449
Total 140 15357 00294 15285 18449
120593119891
11 35 281 times 10minus8
952 times 10minus8
1 times 10minus12
452 times 10minus7
21 35 467 times 10minus8
235 times 10minus7
1 times 10minus12
141 times 10minus6
31 35 163 times 10minus8
491 times 10minus8
1 times 10minus12
221 times 10minus7
41 35 817 times 10minus8
285 times 10minus7
1 times 10minus12
126 times 10minus6
Total 140 426 times 10minus8 191 times 10minus7 1 times 10minus12 141 times 10minus6
Time
11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514
Total 140 1142131 410891 456615 2849514
119864119869
11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070
Total 140 00047 00192 0 02070
119870120595
11 35 00031 00045 475 times 10minus9
00204
21 35 00027 00064 550 times 10minus10
00369
31 35 00029 00050 154 times 10minus13
00271
41 35 00111 00397 193 times 10minus10
02070
Total 140 00047 00192 154 times 10minus13 02070
Table 4 Summary statistical data of ANOVA test for the parameters1198731199051 1198731199052 1198731199011 1198731199012 119873119892 119873119894
Parameters 119869 119864119869
120593119891
Time 119870120595
Test statistic (119865)
1198731199051
1294 1296 0624 1079 12961198731199052
110572 393 0868 5939 3931198731199011
1945 1835 1271 4317 18351198731199012
2740 2478 1011 2302 2478119873119892
3541 3591 0491 0890 3591119873119894
0495 0309 0301 5849 1046
119875 value
1198731199051
0280 0279 0601 0 02791198731199052
0 0005 0489 0 00051198731199011
0125 0143 0286 0006 01431198731199012
0021 0034 0413 0 0034119873119892
0009 0008 0743 0472 0008119873119894
0739 0819 0877 0 0386
and SI methods from the perspective of errors for 120593119891 Time
and119870120595 have same behaviour
7 The Impact of SQP
In this section we investigation the impact of the SQP on theproposed algorithm For this purpose we remove the SQPfrom Algorithm 2 as without SQP algorithm For comparingthe numerical results with the previous results the required
running time in each NOCP is considered fixed which is themaximum of running time in 12 different runs were done inSection 5 Also the all of the parameters for each problemwasset as same as Section 5 The numerical results of the withoutSQP algorithm is summarized in Table 10 By comparing theresults of Tables 10 and 1 it is obvious that for all NOCPsin the Appendix the obtained values of the performanceindex and the norm of final state constraint for the proposedalgorithm (Algorithm 2) are more accurate than the withoutSQP algorithm
18 Mathematical Problems in Engineering
Table 5 The mean of numerical results of 15 different runs for NOCPs in the Appendix using LI and SI methods
Problem LI SI119869 120593
119891119870120595
Time 119869 120593119891
119870120595
TimeVDP 15977 157 times 10
minus9
170 times 10minus3 17720 16002 667 times 10
minus8
326 times 10minus3 15757
CRP 00126 307 times 10minus8
307 times 10minus8 19714 00119 161 times 10
minus8
161 times 10minus8 19532
FFPP 4774 805 times 10minus5 03277 133280 5351 592 times 10
minus5 04868 137989MSNIC 01702 mdash mdash 13678 01703 mdash mdash 14092CSTCR 01120 mdash mdash 17385 01120 mdash mdash 17606Number 6 minus54307 mdash mdash 10541 minus54308 mdash mdash 10372Number 7 02015 326 times 10
minus9
695 times 10minus6 20807 02020 313 times 10
minus9
662 times 10minus6 21019
Number 8 630 times 10minus15 mdash mdash 6376 124 times 10
minus14 mdash mdash 5837Number 9 16512 mdash mdash 59239 16512 mdash mdash 58125Number 10 01550 mdash mdash 63153 01550 mdash mdash 65597Number 11 29701 270 times 10
minus9
272 times 10minus7 22860 29701 337 times 10
minus9
811 times 10minus7 23276
Number 12 minus02499 201 times 10minus8
464 times 10minus5 54718 minus02499 115 times 10
minus9
827 times 10minus5 52765
Number 13 00147 115 times 10minus8
00252 57797 00151 139 times 10minus8
00472 66098Number 14 34761 672 times 10
minus6
00237 104585 34290 757 times 10minus6
950 times 10minus3 108767
Number 15 195 times 10minus4
111 times 10minus8
569 times 10minus3 363 194 times 10
minus4
704 times 10minus9
486 times 10minus3 365
Number 16 20571 166 times 10minus10
166 times 10minus10 54391 20571 115 times 10
minus10
115 times 10minus10 54431
Number 17 minus88692 803 times 10minus8
803 times 10minus8 22744 minus88692 661 times 10
minus8
661 times 10minus8 23254
Number 18 00326 mdash mdash 65957 00326 mdash mdash 70897Number 19 011588 655 times 10
minus4 06828 161813 01091 884 times 10minus4 08808 159990
Number 20 5242 00176 05429 84165 4464 453 times 10minus4 03765 86891
Table 6 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 119869
Method Mean rank Sum of ranks Test statistics ValueLI 2050 4100 Mann-Whitney 119880 200SI 2050 4100 Wilcoxon119882 410mdash mdash mdash 119885 0
Table 7 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 120593119891
Method Mean rank Sum of ranks Test statistics ValueLI 1362 177 Mann-Whitney 119880 83SI 1338 174 Wilcoxon119882 174mdash mdash mdash 119885 minus0077
Table 8 Results of Mann-Whitney test on relative errors of the pair (LI SI) for Time
Method Mean rank Sum of ranks Test statistics ValueLI 2025 4050 Mann-Whitney 119880 195SI 2075 4150 Wilcoxon119882 405mdash mdash mdash 119885 minus0135
Table 9 Results of Mann-Whitney test on relative errors of the pair (LI SI) for119870120595
Method Mean rank Sum of ranks Test statistics ValueLI 1354 1760 Mann-Whitney 119880 84SI 1346 1750 Wilcoxon119882 175mdash mdash mdash 119885 minus0026
Mathematical Problems in Engineering 19
Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix
Problem no1 2 3 4 5 6 7 8 9 10
119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048
120601119891
330 times 10minus9
135 times 10minus5 11445 mdash mdash mdash 758 times 10
minus9 mdash mdash mdashProblem no
11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10
minus4 23182 minus90882 00336 2513 20274120601119891
167 times 10minus5
468 times 10minus9 00161 433 times 10
minus4
594 times 10minus3
156 times 10minus5 164 mdash 1706 25283
8 Conclusions
Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters
Appendix
Test Problems
The following problems are presented using notation given in(1)ndash(6)
(1) (VDP) [9] 119892 = (12)(1199092
1+ 1199092
2+ 1199062
) 1199050= 0 119905119891= 5 119891 =
[1199092 minus1199092+(1minus119909
2
1)1199092+119906]119879 1199090= [1 0]
119879120595 = 1199091minus1199092+1
(2) (CRP) [9] 119892 = (12)(1199092
1+1199092
2+01119906
2
) 1199050= 0 119905119891= 078
119891 = [1199091minus2(1199091+025)+(119909
2+05) exp(25119909
1(1199091+2))minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(3) (FFRP) [9] 119892 = (12)(1199062
1+ 1199062
2+ 1199062
3+ 1199062
4) 1199050
=
0 119905119891
= 5 119891 = [1199092 (110)((119906
1+ 1199062) cos119909
5minus
(1199062+ 1199064) sin119909
5) 1199094 (110)((119906
1+ 1199063) sin119909
5+ (1199062+
1199064) cos119909
5) 1199096 (512)((119906
1+ 1199063) minus (119906
2+ 1199064))]119879 1199090=
[0 0 0 0 0 0]119879 120595 = [119909
1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879
(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909
2 minus1199092+
119906 1199092
1+ 1199092
2+ 0005119906
2
]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909
2+
05 minus 8(119905 minus 05)2
]119879 1199090= [0 minus1 0]
119879
(5) (CSTCR) [20] 119892 = 1199092
1+ 1199092
2+ 01119906
2 1199050= 0 119905119891= 078
119891 = [minus(2 + 119906)(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+
2)) 05 minus 1199092minus (1199092+ 05) exp(25119909
1(1199091+ 2))]
119879 1199090=
[009 009]119879
(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909
2 119906]119879 119889 = [minus(2 +
119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]
119879(7) [36] 119892 = 119906
2 1199050= 0 119905
119891= 1 119891 = (12)119909
2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05
(8) [26] 119892 = 1199092cos2119906 119905
0= 0 119905119891= 120587 119891 = sin(1199062) 119909
0=
1205872(9) [26] 119892 = (12)(119909
2
1+ 1199092
2+ 1199062
) 1199050= 0 119905
119891= 5 119891 =
[1199092 minus1199091+ (1 minus 119909
2
1)1199092+ 119906]119879 119889 = minus(119909
2+ 025) 119909
0=
[1 0]119879
(10) [26] 119892 = 1199092
1+ 1199092
2+ 0005119906
2
1199050= 0 119905
119891= 1 119891 =
[1199092 minus1199092+119906]119879
119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus
05) minus 05 minus 1199092)]119879
1199090= [0 minus1]
119879(11) [26] 119892 = (12)119906
2 1199050= 0 119905
119891= 2 119891 = [119909
2 119906]119879 1199090=
[1 1]119879 120595 = [119909
1 1199092]119879
(12) [26] 119892 = minus1199092 1199050= 0 119905
119891= 1 119891 = [119909
2 119906]119879 119889 =
minus(1 minus 119906)(1 + 119906) 1199090= [0 0]
119879 120595 = 1199092
(13) [26] 119892 = (12)(1199092
1+ 1199092
2+ 01119906
2
) 1199050= 0 119905
119891= 078
119891 = [minus2(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+ 2)) minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(14) [26] 119892 = (12)1199062 1199050= 0 119905
119891= 10 119891 = [cos 119906 minus
1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909
0= [366 minus186]
119879120595 = [119909
1 1199092]119879
(15) [26]119892 = (12)(1199092
1+1199092
2) 1199050= 0 119905119891= 078119891 = [minus2(119909
1+
025)+(1199092+05) exp(25119909
1(1199091+2))minus(119909
1+025)119906 05minus
1199092minus(1199092+05) exp(25119909
1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(16) [26] 120601 = 1199093 1199050= 0 119905
119891= 1 119891 = [119909
2 119906 (12)119906
2
]119879
119889 = 1199091minus 19 119909
0= [0 0 0]
119879 120595 = [1199091 1199092+ 1]119879
(17) [26] 120601 = minus1199093 1199050= 0 119905
119891= 5 119891 = [119909
2 minus2 + 119906119909
3
minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909
0= [10 minus2 10]
119879120595 = [119909
1 1199092]119879
(18) [26] 120601 = (1199091minus1)2
+1199092
2+1199092
3 119892 = (12)119906
2 1199050= 0 119905119891= 5
119891 = [1199093cos 119906 119909
3sin 119906 sin 119906]119879 119909
0= [0 0 0]
119879
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
5 10 15 20 25 30 35 40 45 50
00955
0096
00965
0097
00975
0098
00985
0099
00995
Iter
Perfo
rman
ce in
dex
(a)
2 4 6 8 10 12 14 16 18 200
05
1
15
2
25
3
35
4
45
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 11 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (15)
5 10 15 20 25
21
22
23
24
25
26
27
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
1
2
3
4
5
6
Iter
Fina
l con
ditio
n
times10minus3
(b)
Figure 12 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (16)
and the maximum number of generations 119873119892 Because the
mutation implementation probability119875119898 has a less influence
in numerical results it is not considered Also 119904119902119901119898119886119909119894119905119890119903 ischanged in each iteration of the proposed algorithm so it isnot considered too
At first we selected at least four constant value foreach of parameters and then in each pose 35 differentruns were made independently The statistical analysis isdone based on ANOVA The descriptive statistics whichcontains the number of different runs (119873) the mean of each
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7minus887
minus886
minus885
minus884
minus883
minus882
minus881
minus88
minus879
minus878
minus877
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 70
01
02
03
04
05
06
07
08
09
1
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 13 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (17)
1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333
Iter
Perfo
rman
ce in
dex
Figure 14 Convergence rate for the performance index of SImethod for NOCP numbers (18)
output in119873 different runs (Mean) standard deviation (Sd)maximum (Max) and minimum (Min) of each outputcould be achieved Among of the MHGA parameters wepresent the descriptive statistics of the ANOVA only for the1198731199051parameter which is reported in Table 3Table 4 summarized the statistical data contain the test
statistics (119865) and 119875-values of ANOVA tests Sensitivityanalysis for each parameter separately are done based on thistable as follows
1198731199051 From Table 4 the significant level or 119875-value for119870120595
is equal to 0279 which is greater than 005So from ANOVA 119873
1199051parameter has no significant
effect on the factor 119870120595 With similar analysis this
parameter has no effect on the other outputs except
required computational time Time because its 119875-value is equal to 0 which is less than 005 that is thisoutput sensitive to the parameter119873
1199051
1198731199052 From the second row of Table 4 the outputs 119869 119864
119869119870120595
and Time are sensitive to the parameter 1198731199052 and the
only output 120593119891is independent
1198731199011 From the third row of Table 4 only the computationalrequired time Time is sensitive because the its 119875-value is equal to 0006 which is less than 005 and theother outputs contain 119869 119864
119869 120593119891119870120595 are independent
1198731199012 From the forth row of Table 4 the output 120593
119891is
independent respect to the parameter 1198731199012 but other
outputs contain 119869 119864119869 Time 119870
120601are sensitive to this
parameter119873119892 From the fifth row of Table 4 the outputs 119869 119864
119869and
119870120595are independent to the parameter 119873
119892 and other
outputs contain 120593119891and Time are sensitive respect to
this parameter119873119894 The sensitivity analysis is similar to119873
1199011
From above cases it is obvious that all parameters canbe effect on the required computational time except 119873
119892
Moreover intuitions shows the normof final state constraints120593119891is independent output with respect to all parameters that
is any of the parameters could not effect on this output and itis not sensitive with respect to any of the MHGA parameters
62 Comparison of LI and SI To compare the efficiency ofthe LI and SI methods for NOCPs in the Appendix weused the Mann-Whitney nonparametric statistical test [37]
16 Mathematical Problems in Engineering
1 15 2 25 3 35 40
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
200
400
600
800
1000
1200
1400
1600
IterFi
nal c
ondi
tion
(b)
Figure 15 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (19)
2 4 6 8 10 12 14 16 18 20
50
100
150
200
250
300
350
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 10 11 120
05
1
15
2
25
3
Iter
Fina
l con
ditio
n
(b)
Figure 16 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (20)
Using the nonparametric statistical tests for comparing theperformance of several algorithms presented in [38] Forthis purpose At first 15 different runs for each of NOCPwere made Table 5 shows the mean of numerical resultsseparately Here we apply the fixMHGAparameters as119873
1199011=
91198731199012
= 121198731199051= 31119873
1199052= 91119873
119892= 100 and119873
119894= 50 with
the previous problem parameters in Table 2 Comparison
criterions contain 119869 120593119891 119870120595and Time The results of Mann-
Whitney test are shown in Tables 6ndash9 From Table 6 since119885 = 0 gt 119885
001= minus234 then the average relative error of
the LI and SI methods are same with the significant level 001In other words with probability 99 we can say that LI andSI methods have same effectiveness form the perspective oferror for 119869 Similarly the results of Tables 7ndash9 indicate that LI
Mathematical Problems in Engineering 17
Table 3 Descriptive statistics for the sensitivity analysis of1198731199051 for VDP problem
Output 1198731199051
119873 Mean Sd Min Max
119869
11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449
Total 140 15357 00294 15285 18449
120593119891
11 35 281 times 10minus8
952 times 10minus8
1 times 10minus12
452 times 10minus7
21 35 467 times 10minus8
235 times 10minus7
1 times 10minus12
141 times 10minus6
31 35 163 times 10minus8
491 times 10minus8
1 times 10minus12
221 times 10minus7
41 35 817 times 10minus8
285 times 10minus7
1 times 10minus12
126 times 10minus6
Total 140 426 times 10minus8 191 times 10minus7 1 times 10minus12 141 times 10minus6
Time
11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514
Total 140 1142131 410891 456615 2849514
119864119869
11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070
Total 140 00047 00192 0 02070
119870120595
11 35 00031 00045 475 times 10minus9
00204
21 35 00027 00064 550 times 10minus10
00369
31 35 00029 00050 154 times 10minus13
00271
41 35 00111 00397 193 times 10minus10
02070
Total 140 00047 00192 154 times 10minus13 02070
Table 4 Summary statistical data of ANOVA test for the parameters1198731199051 1198731199052 1198731199011 1198731199012 119873119892 119873119894
Parameters 119869 119864119869
120593119891
Time 119870120595
Test statistic (119865)
1198731199051
1294 1296 0624 1079 12961198731199052
110572 393 0868 5939 3931198731199011
1945 1835 1271 4317 18351198731199012
2740 2478 1011 2302 2478119873119892
3541 3591 0491 0890 3591119873119894
0495 0309 0301 5849 1046
119875 value
1198731199051
0280 0279 0601 0 02791198731199052
0 0005 0489 0 00051198731199011
0125 0143 0286 0006 01431198731199012
0021 0034 0413 0 0034119873119892
0009 0008 0743 0472 0008119873119894
0739 0819 0877 0 0386
and SI methods from the perspective of errors for 120593119891 Time
and119870120595 have same behaviour
7 The Impact of SQP
In this section we investigation the impact of the SQP on theproposed algorithm For this purpose we remove the SQPfrom Algorithm 2 as without SQP algorithm For comparingthe numerical results with the previous results the required
running time in each NOCP is considered fixed which is themaximum of running time in 12 different runs were done inSection 5 Also the all of the parameters for each problemwasset as same as Section 5 The numerical results of the withoutSQP algorithm is summarized in Table 10 By comparing theresults of Tables 10 and 1 it is obvious that for all NOCPsin the Appendix the obtained values of the performanceindex and the norm of final state constraint for the proposedalgorithm (Algorithm 2) are more accurate than the withoutSQP algorithm
18 Mathematical Problems in Engineering
Table 5 The mean of numerical results of 15 different runs for NOCPs in the Appendix using LI and SI methods
Problem LI SI119869 120593
119891119870120595
Time 119869 120593119891
119870120595
TimeVDP 15977 157 times 10
minus9
170 times 10minus3 17720 16002 667 times 10
minus8
326 times 10minus3 15757
CRP 00126 307 times 10minus8
307 times 10minus8 19714 00119 161 times 10
minus8
161 times 10minus8 19532
FFPP 4774 805 times 10minus5 03277 133280 5351 592 times 10
minus5 04868 137989MSNIC 01702 mdash mdash 13678 01703 mdash mdash 14092CSTCR 01120 mdash mdash 17385 01120 mdash mdash 17606Number 6 minus54307 mdash mdash 10541 minus54308 mdash mdash 10372Number 7 02015 326 times 10
minus9
695 times 10minus6 20807 02020 313 times 10
minus9
662 times 10minus6 21019
Number 8 630 times 10minus15 mdash mdash 6376 124 times 10
minus14 mdash mdash 5837Number 9 16512 mdash mdash 59239 16512 mdash mdash 58125Number 10 01550 mdash mdash 63153 01550 mdash mdash 65597Number 11 29701 270 times 10
minus9
272 times 10minus7 22860 29701 337 times 10
minus9
811 times 10minus7 23276
Number 12 minus02499 201 times 10minus8
464 times 10minus5 54718 minus02499 115 times 10
minus9
827 times 10minus5 52765
Number 13 00147 115 times 10minus8
00252 57797 00151 139 times 10minus8
00472 66098Number 14 34761 672 times 10
minus6
00237 104585 34290 757 times 10minus6
950 times 10minus3 108767
Number 15 195 times 10minus4
111 times 10minus8
569 times 10minus3 363 194 times 10
minus4
704 times 10minus9
486 times 10minus3 365
Number 16 20571 166 times 10minus10
166 times 10minus10 54391 20571 115 times 10
minus10
115 times 10minus10 54431
Number 17 minus88692 803 times 10minus8
803 times 10minus8 22744 minus88692 661 times 10
minus8
661 times 10minus8 23254
Number 18 00326 mdash mdash 65957 00326 mdash mdash 70897Number 19 011588 655 times 10
minus4 06828 161813 01091 884 times 10minus4 08808 159990
Number 20 5242 00176 05429 84165 4464 453 times 10minus4 03765 86891
Table 6 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 119869
Method Mean rank Sum of ranks Test statistics ValueLI 2050 4100 Mann-Whitney 119880 200SI 2050 4100 Wilcoxon119882 410mdash mdash mdash 119885 0
Table 7 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 120593119891
Method Mean rank Sum of ranks Test statistics ValueLI 1362 177 Mann-Whitney 119880 83SI 1338 174 Wilcoxon119882 174mdash mdash mdash 119885 minus0077
Table 8 Results of Mann-Whitney test on relative errors of the pair (LI SI) for Time
Method Mean rank Sum of ranks Test statistics ValueLI 2025 4050 Mann-Whitney 119880 195SI 2075 4150 Wilcoxon119882 405mdash mdash mdash 119885 minus0135
Table 9 Results of Mann-Whitney test on relative errors of the pair (LI SI) for119870120595
Method Mean rank Sum of ranks Test statistics ValueLI 1354 1760 Mann-Whitney 119880 84SI 1346 1750 Wilcoxon119882 175mdash mdash mdash 119885 minus0026
Mathematical Problems in Engineering 19
Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix
Problem no1 2 3 4 5 6 7 8 9 10
119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048
120601119891
330 times 10minus9
135 times 10minus5 11445 mdash mdash mdash 758 times 10
minus9 mdash mdash mdashProblem no
11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10
minus4 23182 minus90882 00336 2513 20274120601119891
167 times 10minus5
468 times 10minus9 00161 433 times 10
minus4
594 times 10minus3
156 times 10minus5 164 mdash 1706 25283
8 Conclusions
Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters
Appendix
Test Problems
The following problems are presented using notation given in(1)ndash(6)
(1) (VDP) [9] 119892 = (12)(1199092
1+ 1199092
2+ 1199062
) 1199050= 0 119905119891= 5 119891 =
[1199092 minus1199092+(1minus119909
2
1)1199092+119906]119879 1199090= [1 0]
119879120595 = 1199091minus1199092+1
(2) (CRP) [9] 119892 = (12)(1199092
1+1199092
2+01119906
2
) 1199050= 0 119905119891= 078
119891 = [1199091minus2(1199091+025)+(119909
2+05) exp(25119909
1(1199091+2))minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(3) (FFRP) [9] 119892 = (12)(1199062
1+ 1199062
2+ 1199062
3+ 1199062
4) 1199050
=
0 119905119891
= 5 119891 = [1199092 (110)((119906
1+ 1199062) cos119909
5minus
(1199062+ 1199064) sin119909
5) 1199094 (110)((119906
1+ 1199063) sin119909
5+ (1199062+
1199064) cos119909
5) 1199096 (512)((119906
1+ 1199063) minus (119906
2+ 1199064))]119879 1199090=
[0 0 0 0 0 0]119879 120595 = [119909
1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879
(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909
2 minus1199092+
119906 1199092
1+ 1199092
2+ 0005119906
2
]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909
2+
05 minus 8(119905 minus 05)2
]119879 1199090= [0 minus1 0]
119879
(5) (CSTCR) [20] 119892 = 1199092
1+ 1199092
2+ 01119906
2 1199050= 0 119905119891= 078
119891 = [minus(2 + 119906)(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+
2)) 05 minus 1199092minus (1199092+ 05) exp(25119909
1(1199091+ 2))]
119879 1199090=
[009 009]119879
(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909
2 119906]119879 119889 = [minus(2 +
119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]
119879(7) [36] 119892 = 119906
2 1199050= 0 119905
119891= 1 119891 = (12)119909
2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05
(8) [26] 119892 = 1199092cos2119906 119905
0= 0 119905119891= 120587 119891 = sin(1199062) 119909
0=
1205872(9) [26] 119892 = (12)(119909
2
1+ 1199092
2+ 1199062
) 1199050= 0 119905
119891= 5 119891 =
[1199092 minus1199091+ (1 minus 119909
2
1)1199092+ 119906]119879 119889 = minus(119909
2+ 025) 119909
0=
[1 0]119879
(10) [26] 119892 = 1199092
1+ 1199092
2+ 0005119906
2
1199050= 0 119905
119891= 1 119891 =
[1199092 minus1199092+119906]119879
119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus
05) minus 05 minus 1199092)]119879
1199090= [0 minus1]
119879(11) [26] 119892 = (12)119906
2 1199050= 0 119905
119891= 2 119891 = [119909
2 119906]119879 1199090=
[1 1]119879 120595 = [119909
1 1199092]119879
(12) [26] 119892 = minus1199092 1199050= 0 119905
119891= 1 119891 = [119909
2 119906]119879 119889 =
minus(1 minus 119906)(1 + 119906) 1199090= [0 0]
119879 120595 = 1199092
(13) [26] 119892 = (12)(1199092
1+ 1199092
2+ 01119906
2
) 1199050= 0 119905
119891= 078
119891 = [minus2(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+ 2)) minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(14) [26] 119892 = (12)1199062 1199050= 0 119905
119891= 10 119891 = [cos 119906 minus
1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909
0= [366 minus186]
119879120595 = [119909
1 1199092]119879
(15) [26]119892 = (12)(1199092
1+1199092
2) 1199050= 0 119905119891= 078119891 = [minus2(119909
1+
025)+(1199092+05) exp(25119909
1(1199091+2))minus(119909
1+025)119906 05minus
1199092minus(1199092+05) exp(25119909
1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(16) [26] 120601 = 1199093 1199050= 0 119905
119891= 1 119891 = [119909
2 119906 (12)119906
2
]119879
119889 = 1199091minus 19 119909
0= [0 0 0]
119879 120595 = [1199091 1199092+ 1]119879
(17) [26] 120601 = minus1199093 1199050= 0 119905
119891= 5 119891 = [119909
2 minus2 + 119906119909
3
minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909
0= [10 minus2 10]
119879120595 = [119909
1 1199092]119879
(18) [26] 120601 = (1199091minus1)2
+1199092
2+1199092
3 119892 = (12)119906
2 1199050= 0 119905119891= 5
119891 = [1199093cos 119906 119909
3sin 119906 sin 119906]119879 119909
0= [0 0 0]
119879
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Function Spaces
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7minus887
minus886
minus885
minus884
minus883
minus882
minus881
minus88
minus879
minus878
minus877
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 70
01
02
03
04
05
06
07
08
09
1
IterFi
nal c
ondi
tion
times10minus3
(b)
Figure 13 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (17)
1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333
Iter
Perfo
rman
ce in
dex
Figure 14 Convergence rate for the performance index of SImethod for NOCP numbers (18)
output in119873 different runs (Mean) standard deviation (Sd)maximum (Max) and minimum (Min) of each outputcould be achieved Among of the MHGA parameters wepresent the descriptive statistics of the ANOVA only for the1198731199051parameter which is reported in Table 3Table 4 summarized the statistical data contain the test
statistics (119865) and 119875-values of ANOVA tests Sensitivityanalysis for each parameter separately are done based on thistable as follows
1198731199051 From Table 4 the significant level or 119875-value for119870120595
is equal to 0279 which is greater than 005So from ANOVA 119873
1199051parameter has no significant
effect on the factor 119870120595 With similar analysis this
parameter has no effect on the other outputs except
required computational time Time because its 119875-value is equal to 0 which is less than 005 that is thisoutput sensitive to the parameter119873
1199051
1198731199052 From the second row of Table 4 the outputs 119869 119864
119869119870120595
and Time are sensitive to the parameter 1198731199052 and the
only output 120593119891is independent
1198731199011 From the third row of Table 4 only the computationalrequired time Time is sensitive because the its 119875-value is equal to 0006 which is less than 005 and theother outputs contain 119869 119864
119869 120593119891119870120595 are independent
1198731199012 From the forth row of Table 4 the output 120593
119891is
independent respect to the parameter 1198731199012 but other
outputs contain 119869 119864119869 Time 119870
120601are sensitive to this
parameter119873119892 From the fifth row of Table 4 the outputs 119869 119864
119869and
119870120595are independent to the parameter 119873
119892 and other
outputs contain 120593119891and Time are sensitive respect to
this parameter119873119894 The sensitivity analysis is similar to119873
1199011
From above cases it is obvious that all parameters canbe effect on the required computational time except 119873
119892
Moreover intuitions shows the normof final state constraints120593119891is independent output with respect to all parameters that
is any of the parameters could not effect on this output and itis not sensitive with respect to any of the MHGA parameters
62 Comparison of LI and SI To compare the efficiency ofthe LI and SI methods for NOCPs in the Appendix weused the Mann-Whitney nonparametric statistical test [37]
16 Mathematical Problems in Engineering
1 15 2 25 3 35 40
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
200
400
600
800
1000
1200
1400
1600
IterFi
nal c
ondi
tion
(b)
Figure 15 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (19)
2 4 6 8 10 12 14 16 18 20
50
100
150
200
250
300
350
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 10 11 120
05
1
15
2
25
3
Iter
Fina
l con
ditio
n
(b)
Figure 16 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (20)
Using the nonparametric statistical tests for comparing theperformance of several algorithms presented in [38] Forthis purpose At first 15 different runs for each of NOCPwere made Table 5 shows the mean of numerical resultsseparately Here we apply the fixMHGAparameters as119873
1199011=
91198731199012
= 121198731199051= 31119873
1199052= 91119873
119892= 100 and119873
119894= 50 with
the previous problem parameters in Table 2 Comparison
criterions contain 119869 120593119891 119870120595and Time The results of Mann-
Whitney test are shown in Tables 6ndash9 From Table 6 since119885 = 0 gt 119885
001= minus234 then the average relative error of
the LI and SI methods are same with the significant level 001In other words with probability 99 we can say that LI andSI methods have same effectiveness form the perspective oferror for 119869 Similarly the results of Tables 7ndash9 indicate that LI
Mathematical Problems in Engineering 17
Table 3 Descriptive statistics for the sensitivity analysis of1198731199051 for VDP problem
Output 1198731199051
119873 Mean Sd Min Max
119869
11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449
Total 140 15357 00294 15285 18449
120593119891
11 35 281 times 10minus8
952 times 10minus8
1 times 10minus12
452 times 10minus7
21 35 467 times 10minus8
235 times 10minus7
1 times 10minus12
141 times 10minus6
31 35 163 times 10minus8
491 times 10minus8
1 times 10minus12
221 times 10minus7
41 35 817 times 10minus8
285 times 10minus7
1 times 10minus12
126 times 10minus6
Total 140 426 times 10minus8 191 times 10minus7 1 times 10minus12 141 times 10minus6
Time
11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514
Total 140 1142131 410891 456615 2849514
119864119869
11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070
Total 140 00047 00192 0 02070
119870120595
11 35 00031 00045 475 times 10minus9
00204
21 35 00027 00064 550 times 10minus10
00369
31 35 00029 00050 154 times 10minus13
00271
41 35 00111 00397 193 times 10minus10
02070
Total 140 00047 00192 154 times 10minus13 02070
Table 4 Summary statistical data of ANOVA test for the parameters1198731199051 1198731199052 1198731199011 1198731199012 119873119892 119873119894
Parameters 119869 119864119869
120593119891
Time 119870120595
Test statistic (119865)
1198731199051
1294 1296 0624 1079 12961198731199052
110572 393 0868 5939 3931198731199011
1945 1835 1271 4317 18351198731199012
2740 2478 1011 2302 2478119873119892
3541 3591 0491 0890 3591119873119894
0495 0309 0301 5849 1046
119875 value
1198731199051
0280 0279 0601 0 02791198731199052
0 0005 0489 0 00051198731199011
0125 0143 0286 0006 01431198731199012
0021 0034 0413 0 0034119873119892
0009 0008 0743 0472 0008119873119894
0739 0819 0877 0 0386
and SI methods from the perspective of errors for 120593119891 Time
and119870120595 have same behaviour
7 The Impact of SQP
In this section we investigation the impact of the SQP on theproposed algorithm For this purpose we remove the SQPfrom Algorithm 2 as without SQP algorithm For comparingthe numerical results with the previous results the required
running time in each NOCP is considered fixed which is themaximum of running time in 12 different runs were done inSection 5 Also the all of the parameters for each problemwasset as same as Section 5 The numerical results of the withoutSQP algorithm is summarized in Table 10 By comparing theresults of Tables 10 and 1 it is obvious that for all NOCPsin the Appendix the obtained values of the performanceindex and the norm of final state constraint for the proposedalgorithm (Algorithm 2) are more accurate than the withoutSQP algorithm
18 Mathematical Problems in Engineering
Table 5 The mean of numerical results of 15 different runs for NOCPs in the Appendix using LI and SI methods
Problem LI SI119869 120593
119891119870120595
Time 119869 120593119891
119870120595
TimeVDP 15977 157 times 10
minus9
170 times 10minus3 17720 16002 667 times 10
minus8
326 times 10minus3 15757
CRP 00126 307 times 10minus8
307 times 10minus8 19714 00119 161 times 10
minus8
161 times 10minus8 19532
FFPP 4774 805 times 10minus5 03277 133280 5351 592 times 10
minus5 04868 137989MSNIC 01702 mdash mdash 13678 01703 mdash mdash 14092CSTCR 01120 mdash mdash 17385 01120 mdash mdash 17606Number 6 minus54307 mdash mdash 10541 minus54308 mdash mdash 10372Number 7 02015 326 times 10
minus9
695 times 10minus6 20807 02020 313 times 10
minus9
662 times 10minus6 21019
Number 8 630 times 10minus15 mdash mdash 6376 124 times 10
minus14 mdash mdash 5837Number 9 16512 mdash mdash 59239 16512 mdash mdash 58125Number 10 01550 mdash mdash 63153 01550 mdash mdash 65597Number 11 29701 270 times 10
minus9
272 times 10minus7 22860 29701 337 times 10
minus9
811 times 10minus7 23276
Number 12 minus02499 201 times 10minus8
464 times 10minus5 54718 minus02499 115 times 10
minus9
827 times 10minus5 52765
Number 13 00147 115 times 10minus8
00252 57797 00151 139 times 10minus8
00472 66098Number 14 34761 672 times 10
minus6
00237 104585 34290 757 times 10minus6
950 times 10minus3 108767
Number 15 195 times 10minus4
111 times 10minus8
569 times 10minus3 363 194 times 10
minus4
704 times 10minus9
486 times 10minus3 365
Number 16 20571 166 times 10minus10
166 times 10minus10 54391 20571 115 times 10
minus10
115 times 10minus10 54431
Number 17 minus88692 803 times 10minus8
803 times 10minus8 22744 minus88692 661 times 10
minus8
661 times 10minus8 23254
Number 18 00326 mdash mdash 65957 00326 mdash mdash 70897Number 19 011588 655 times 10
minus4 06828 161813 01091 884 times 10minus4 08808 159990
Number 20 5242 00176 05429 84165 4464 453 times 10minus4 03765 86891
Table 6 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 119869
Method Mean rank Sum of ranks Test statistics ValueLI 2050 4100 Mann-Whitney 119880 200SI 2050 4100 Wilcoxon119882 410mdash mdash mdash 119885 0
Table 7 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 120593119891
Method Mean rank Sum of ranks Test statistics ValueLI 1362 177 Mann-Whitney 119880 83SI 1338 174 Wilcoxon119882 174mdash mdash mdash 119885 minus0077
Table 8 Results of Mann-Whitney test on relative errors of the pair (LI SI) for Time
Method Mean rank Sum of ranks Test statistics ValueLI 2025 4050 Mann-Whitney 119880 195SI 2075 4150 Wilcoxon119882 405mdash mdash mdash 119885 minus0135
Table 9 Results of Mann-Whitney test on relative errors of the pair (LI SI) for119870120595
Method Mean rank Sum of ranks Test statistics ValueLI 1354 1760 Mann-Whitney 119880 84SI 1346 1750 Wilcoxon119882 175mdash mdash mdash 119885 minus0026
Mathematical Problems in Engineering 19
Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix
Problem no1 2 3 4 5 6 7 8 9 10
119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048
120601119891
330 times 10minus9
135 times 10minus5 11445 mdash mdash mdash 758 times 10
minus9 mdash mdash mdashProblem no
11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10
minus4 23182 minus90882 00336 2513 20274120601119891
167 times 10minus5
468 times 10minus9 00161 433 times 10
minus4
594 times 10minus3
156 times 10minus5 164 mdash 1706 25283
8 Conclusions
Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters
Appendix
Test Problems
The following problems are presented using notation given in(1)ndash(6)
(1) (VDP) [9] 119892 = (12)(1199092
1+ 1199092
2+ 1199062
) 1199050= 0 119905119891= 5 119891 =
[1199092 minus1199092+(1minus119909
2
1)1199092+119906]119879 1199090= [1 0]
119879120595 = 1199091minus1199092+1
(2) (CRP) [9] 119892 = (12)(1199092
1+1199092
2+01119906
2
) 1199050= 0 119905119891= 078
119891 = [1199091minus2(1199091+025)+(119909
2+05) exp(25119909
1(1199091+2))minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(3) (FFRP) [9] 119892 = (12)(1199062
1+ 1199062
2+ 1199062
3+ 1199062
4) 1199050
=
0 119905119891
= 5 119891 = [1199092 (110)((119906
1+ 1199062) cos119909
5minus
(1199062+ 1199064) sin119909
5) 1199094 (110)((119906
1+ 1199063) sin119909
5+ (1199062+
1199064) cos119909
5) 1199096 (512)((119906
1+ 1199063) minus (119906
2+ 1199064))]119879 1199090=
[0 0 0 0 0 0]119879 120595 = [119909
1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879
(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909
2 minus1199092+
119906 1199092
1+ 1199092
2+ 0005119906
2
]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909
2+
05 minus 8(119905 minus 05)2
]119879 1199090= [0 minus1 0]
119879
(5) (CSTCR) [20] 119892 = 1199092
1+ 1199092
2+ 01119906
2 1199050= 0 119905119891= 078
119891 = [minus(2 + 119906)(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+
2)) 05 minus 1199092minus (1199092+ 05) exp(25119909
1(1199091+ 2))]
119879 1199090=
[009 009]119879
(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909
2 119906]119879 119889 = [minus(2 +
119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]
119879(7) [36] 119892 = 119906
2 1199050= 0 119905
119891= 1 119891 = (12)119909
2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05
(8) [26] 119892 = 1199092cos2119906 119905
0= 0 119905119891= 120587 119891 = sin(1199062) 119909
0=
1205872(9) [26] 119892 = (12)(119909
2
1+ 1199092
2+ 1199062
) 1199050= 0 119905
119891= 5 119891 =
[1199092 minus1199091+ (1 minus 119909
2
1)1199092+ 119906]119879 119889 = minus(119909
2+ 025) 119909
0=
[1 0]119879
(10) [26] 119892 = 1199092
1+ 1199092
2+ 0005119906
2
1199050= 0 119905
119891= 1 119891 =
[1199092 minus1199092+119906]119879
119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus
05) minus 05 minus 1199092)]119879
1199090= [0 minus1]
119879(11) [26] 119892 = (12)119906
2 1199050= 0 119905
119891= 2 119891 = [119909
2 119906]119879 1199090=
[1 1]119879 120595 = [119909
1 1199092]119879
(12) [26] 119892 = minus1199092 1199050= 0 119905
119891= 1 119891 = [119909
2 119906]119879 119889 =
minus(1 minus 119906)(1 + 119906) 1199090= [0 0]
119879 120595 = 1199092
(13) [26] 119892 = (12)(1199092
1+ 1199092
2+ 01119906
2
) 1199050= 0 119905
119891= 078
119891 = [minus2(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+ 2)) minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(14) [26] 119892 = (12)1199062 1199050= 0 119905
119891= 10 119891 = [cos 119906 minus
1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909
0= [366 minus186]
119879120595 = [119909
1 1199092]119879
(15) [26]119892 = (12)(1199092
1+1199092
2) 1199050= 0 119905119891= 078119891 = [minus2(119909
1+
025)+(1199092+05) exp(25119909
1(1199091+2))minus(119909
1+025)119906 05minus
1199092minus(1199092+05) exp(25119909
1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(16) [26] 120601 = 1199093 1199050= 0 119905
119891= 1 119891 = [119909
2 119906 (12)119906
2
]119879
119889 = 1199091minus 19 119909
0= [0 0 0]
119879 120595 = [1199091 1199092+ 1]119879
(17) [26] 120601 = minus1199093 1199050= 0 119905
119891= 5 119891 = [119909
2 minus2 + 119906119909
3
minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909
0= [10 minus2 10]
119879120595 = [119909
1 1199092]119879
(18) [26] 120601 = (1199091minus1)2
+1199092
2+1199092
3 119892 = (12)119906
2 1199050= 0 119905119891= 5
119891 = [1199093cos 119906 119909
3sin 119906 sin 119906]119879 119909
0= [0 0 0]
119879
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Mathematical Problems in Engineering
1 15 2 25 3 35 40
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 80
200
400
600
800
1000
1200
1400
1600
IterFi
nal c
ondi
tion
(b)
Figure 15 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (19)
2 4 6 8 10 12 14 16 18 20
50
100
150
200
250
300
350
Iter
Perfo
rman
ce in
dex
(a)
1 2 3 4 5 6 7 8 9 10 11 120
05
1
15
2
25
3
Iter
Fina
l con
ditio
n
(b)
Figure 16 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (20)
Using the nonparametric statistical tests for comparing theperformance of several algorithms presented in [38] Forthis purpose At first 15 different runs for each of NOCPwere made Table 5 shows the mean of numerical resultsseparately Here we apply the fixMHGAparameters as119873
1199011=
91198731199012
= 121198731199051= 31119873
1199052= 91119873
119892= 100 and119873
119894= 50 with
the previous problem parameters in Table 2 Comparison
criterions contain 119869 120593119891 119870120595and Time The results of Mann-
Whitney test are shown in Tables 6ndash9 From Table 6 since119885 = 0 gt 119885
001= minus234 then the average relative error of
the LI and SI methods are same with the significant level 001In other words with probability 99 we can say that LI andSI methods have same effectiveness form the perspective oferror for 119869 Similarly the results of Tables 7ndash9 indicate that LI
Mathematical Problems in Engineering 17
Table 3 Descriptive statistics for the sensitivity analysis of1198731199051 for VDP problem
Output 1198731199051
119873 Mean Sd Min Max
119869
11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449
Total 140 15357 00294 15285 18449
120593119891
11 35 281 times 10minus8
952 times 10minus8
1 times 10minus12
452 times 10minus7
21 35 467 times 10minus8
235 times 10minus7
1 times 10minus12
141 times 10minus6
31 35 163 times 10minus8
491 times 10minus8
1 times 10minus12
221 times 10minus7
41 35 817 times 10minus8
285 times 10minus7
1 times 10minus12
126 times 10minus6
Total 140 426 times 10minus8 191 times 10minus7 1 times 10minus12 141 times 10minus6
Time
11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514
Total 140 1142131 410891 456615 2849514
119864119869
11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070
Total 140 00047 00192 0 02070
119870120595
11 35 00031 00045 475 times 10minus9
00204
21 35 00027 00064 550 times 10minus10
00369
31 35 00029 00050 154 times 10minus13
00271
41 35 00111 00397 193 times 10minus10
02070
Total 140 00047 00192 154 times 10minus13 02070
Table 4 Summary statistical data of ANOVA test for the parameters1198731199051 1198731199052 1198731199011 1198731199012 119873119892 119873119894
Parameters 119869 119864119869
120593119891
Time 119870120595
Test statistic (119865)
1198731199051
1294 1296 0624 1079 12961198731199052
110572 393 0868 5939 3931198731199011
1945 1835 1271 4317 18351198731199012
2740 2478 1011 2302 2478119873119892
3541 3591 0491 0890 3591119873119894
0495 0309 0301 5849 1046
119875 value
1198731199051
0280 0279 0601 0 02791198731199052
0 0005 0489 0 00051198731199011
0125 0143 0286 0006 01431198731199012
0021 0034 0413 0 0034119873119892
0009 0008 0743 0472 0008119873119894
0739 0819 0877 0 0386
and SI methods from the perspective of errors for 120593119891 Time
and119870120595 have same behaviour
7 The Impact of SQP
In this section we investigation the impact of the SQP on theproposed algorithm For this purpose we remove the SQPfrom Algorithm 2 as without SQP algorithm For comparingthe numerical results with the previous results the required
running time in each NOCP is considered fixed which is themaximum of running time in 12 different runs were done inSection 5 Also the all of the parameters for each problemwasset as same as Section 5 The numerical results of the withoutSQP algorithm is summarized in Table 10 By comparing theresults of Tables 10 and 1 it is obvious that for all NOCPsin the Appendix the obtained values of the performanceindex and the norm of final state constraint for the proposedalgorithm (Algorithm 2) are more accurate than the withoutSQP algorithm
18 Mathematical Problems in Engineering
Table 5 The mean of numerical results of 15 different runs for NOCPs in the Appendix using LI and SI methods
Problem LI SI119869 120593
119891119870120595
Time 119869 120593119891
119870120595
TimeVDP 15977 157 times 10
minus9
170 times 10minus3 17720 16002 667 times 10
minus8
326 times 10minus3 15757
CRP 00126 307 times 10minus8
307 times 10minus8 19714 00119 161 times 10
minus8
161 times 10minus8 19532
FFPP 4774 805 times 10minus5 03277 133280 5351 592 times 10
minus5 04868 137989MSNIC 01702 mdash mdash 13678 01703 mdash mdash 14092CSTCR 01120 mdash mdash 17385 01120 mdash mdash 17606Number 6 minus54307 mdash mdash 10541 minus54308 mdash mdash 10372Number 7 02015 326 times 10
minus9
695 times 10minus6 20807 02020 313 times 10
minus9
662 times 10minus6 21019
Number 8 630 times 10minus15 mdash mdash 6376 124 times 10
minus14 mdash mdash 5837Number 9 16512 mdash mdash 59239 16512 mdash mdash 58125Number 10 01550 mdash mdash 63153 01550 mdash mdash 65597Number 11 29701 270 times 10
minus9
272 times 10minus7 22860 29701 337 times 10
minus9
811 times 10minus7 23276
Number 12 minus02499 201 times 10minus8
464 times 10minus5 54718 minus02499 115 times 10
minus9
827 times 10minus5 52765
Number 13 00147 115 times 10minus8
00252 57797 00151 139 times 10minus8
00472 66098Number 14 34761 672 times 10
minus6
00237 104585 34290 757 times 10minus6
950 times 10minus3 108767
Number 15 195 times 10minus4
111 times 10minus8
569 times 10minus3 363 194 times 10
minus4
704 times 10minus9
486 times 10minus3 365
Number 16 20571 166 times 10minus10
166 times 10minus10 54391 20571 115 times 10
minus10
115 times 10minus10 54431
Number 17 minus88692 803 times 10minus8
803 times 10minus8 22744 minus88692 661 times 10
minus8
661 times 10minus8 23254
Number 18 00326 mdash mdash 65957 00326 mdash mdash 70897Number 19 011588 655 times 10
minus4 06828 161813 01091 884 times 10minus4 08808 159990
Number 20 5242 00176 05429 84165 4464 453 times 10minus4 03765 86891
Table 6 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 119869
Method Mean rank Sum of ranks Test statistics ValueLI 2050 4100 Mann-Whitney 119880 200SI 2050 4100 Wilcoxon119882 410mdash mdash mdash 119885 0
Table 7 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 120593119891
Method Mean rank Sum of ranks Test statistics ValueLI 1362 177 Mann-Whitney 119880 83SI 1338 174 Wilcoxon119882 174mdash mdash mdash 119885 minus0077
Table 8 Results of Mann-Whitney test on relative errors of the pair (LI SI) for Time
Method Mean rank Sum of ranks Test statistics ValueLI 2025 4050 Mann-Whitney 119880 195SI 2075 4150 Wilcoxon119882 405mdash mdash mdash 119885 minus0135
Table 9 Results of Mann-Whitney test on relative errors of the pair (LI SI) for119870120595
Method Mean rank Sum of ranks Test statistics ValueLI 1354 1760 Mann-Whitney 119880 84SI 1346 1750 Wilcoxon119882 175mdash mdash mdash 119885 minus0026
Mathematical Problems in Engineering 19
Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix
Problem no1 2 3 4 5 6 7 8 9 10
119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048
120601119891
330 times 10minus9
135 times 10minus5 11445 mdash mdash mdash 758 times 10
minus9 mdash mdash mdashProblem no
11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10
minus4 23182 minus90882 00336 2513 20274120601119891
167 times 10minus5
468 times 10minus9 00161 433 times 10
minus4
594 times 10minus3
156 times 10minus5 164 mdash 1706 25283
8 Conclusions
Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters
Appendix
Test Problems
The following problems are presented using notation given in(1)ndash(6)
(1) (VDP) [9] 119892 = (12)(1199092
1+ 1199092
2+ 1199062
) 1199050= 0 119905119891= 5 119891 =
[1199092 minus1199092+(1minus119909
2
1)1199092+119906]119879 1199090= [1 0]
119879120595 = 1199091minus1199092+1
(2) (CRP) [9] 119892 = (12)(1199092
1+1199092
2+01119906
2
) 1199050= 0 119905119891= 078
119891 = [1199091minus2(1199091+025)+(119909
2+05) exp(25119909
1(1199091+2))minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(3) (FFRP) [9] 119892 = (12)(1199062
1+ 1199062
2+ 1199062
3+ 1199062
4) 1199050
=
0 119905119891
= 5 119891 = [1199092 (110)((119906
1+ 1199062) cos119909
5minus
(1199062+ 1199064) sin119909
5) 1199094 (110)((119906
1+ 1199063) sin119909
5+ (1199062+
1199064) cos119909
5) 1199096 (512)((119906
1+ 1199063) minus (119906
2+ 1199064))]119879 1199090=
[0 0 0 0 0 0]119879 120595 = [119909
1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879
(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909
2 minus1199092+
119906 1199092
1+ 1199092
2+ 0005119906
2
]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909
2+
05 minus 8(119905 minus 05)2
]119879 1199090= [0 minus1 0]
119879
(5) (CSTCR) [20] 119892 = 1199092
1+ 1199092
2+ 01119906
2 1199050= 0 119905119891= 078
119891 = [minus(2 + 119906)(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+
2)) 05 minus 1199092minus (1199092+ 05) exp(25119909
1(1199091+ 2))]
119879 1199090=
[009 009]119879
(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909
2 119906]119879 119889 = [minus(2 +
119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]
119879(7) [36] 119892 = 119906
2 1199050= 0 119905
119891= 1 119891 = (12)119909
2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05
(8) [26] 119892 = 1199092cos2119906 119905
0= 0 119905119891= 120587 119891 = sin(1199062) 119909
0=
1205872(9) [26] 119892 = (12)(119909
2
1+ 1199092
2+ 1199062
) 1199050= 0 119905
119891= 5 119891 =
[1199092 minus1199091+ (1 minus 119909
2
1)1199092+ 119906]119879 119889 = minus(119909
2+ 025) 119909
0=
[1 0]119879
(10) [26] 119892 = 1199092
1+ 1199092
2+ 0005119906
2
1199050= 0 119905
119891= 1 119891 =
[1199092 minus1199092+119906]119879
119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus
05) minus 05 minus 1199092)]119879
1199090= [0 minus1]
119879(11) [26] 119892 = (12)119906
2 1199050= 0 119905
119891= 2 119891 = [119909
2 119906]119879 1199090=
[1 1]119879 120595 = [119909
1 1199092]119879
(12) [26] 119892 = minus1199092 1199050= 0 119905
119891= 1 119891 = [119909
2 119906]119879 119889 =
minus(1 minus 119906)(1 + 119906) 1199090= [0 0]
119879 120595 = 1199092
(13) [26] 119892 = (12)(1199092
1+ 1199092
2+ 01119906
2
) 1199050= 0 119905
119891= 078
119891 = [minus2(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+ 2)) minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(14) [26] 119892 = (12)1199062 1199050= 0 119905
119891= 10 119891 = [cos 119906 minus
1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909
0= [366 minus186]
119879120595 = [119909
1 1199092]119879
(15) [26]119892 = (12)(1199092
1+1199092
2) 1199050= 0 119905119891= 078119891 = [minus2(119909
1+
025)+(1199092+05) exp(25119909
1(1199091+2))minus(119909
1+025)119906 05minus
1199092minus(1199092+05) exp(25119909
1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(16) [26] 120601 = 1199093 1199050= 0 119905
119891= 1 119891 = [119909
2 119906 (12)119906
2
]119879
119889 = 1199091minus 19 119909
0= [0 0 0]
119879 120595 = [1199091 1199092+ 1]119879
(17) [26] 120601 = minus1199093 1199050= 0 119905
119891= 5 119891 = [119909
2 minus2 + 119906119909
3
minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909
0= [10 minus2 10]
119879120595 = [119909
1 1199092]119879
(18) [26] 120601 = (1199091minus1)2
+1199092
2+1199092
3 119892 = (12)119906
2 1199050= 0 119905119891= 5
119891 = [1199093cos 119906 119909
3sin 119906 sin 119906]119879 119909
0= [0 0 0]
119879
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 17
Table 3 Descriptive statistics for the sensitivity analysis of1198731199051 for VDP problem
Output 1198731199051
119873 Mean Sd Min Max
119869
11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449
Total 140 15357 00294 15285 18449
120593119891
11 35 281 times 10minus8
952 times 10minus8
1 times 10minus12
452 times 10minus7
21 35 467 times 10minus8
235 times 10minus7
1 times 10minus12
141 times 10minus6
31 35 163 times 10minus8
491 times 10minus8
1 times 10minus12
221 times 10minus7
41 35 817 times 10minus8
285 times 10minus7
1 times 10minus12
126 times 10minus6
Total 140 426 times 10minus8 191 times 10minus7 1 times 10minus12 141 times 10minus6
Time
11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514
Total 140 1142131 410891 456615 2849514
119864119869
11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070
Total 140 00047 00192 0 02070
119870120595
11 35 00031 00045 475 times 10minus9
00204
21 35 00027 00064 550 times 10minus10
00369
31 35 00029 00050 154 times 10minus13
00271
41 35 00111 00397 193 times 10minus10
02070
Total 140 00047 00192 154 times 10minus13 02070
Table 4 Summary statistical data of ANOVA test for the parameters1198731199051 1198731199052 1198731199011 1198731199012 119873119892 119873119894
Parameters 119869 119864119869
120593119891
Time 119870120595
Test statistic (119865)
1198731199051
1294 1296 0624 1079 12961198731199052
110572 393 0868 5939 3931198731199011
1945 1835 1271 4317 18351198731199012
2740 2478 1011 2302 2478119873119892
3541 3591 0491 0890 3591119873119894
0495 0309 0301 5849 1046
119875 value
1198731199051
0280 0279 0601 0 02791198731199052
0 0005 0489 0 00051198731199011
0125 0143 0286 0006 01431198731199012
0021 0034 0413 0 0034119873119892
0009 0008 0743 0472 0008119873119894
0739 0819 0877 0 0386
and SI methods from the perspective of errors for 120593119891 Time
and119870120595 have same behaviour
7 The Impact of SQP
In this section we investigation the impact of the SQP on theproposed algorithm For this purpose we remove the SQPfrom Algorithm 2 as without SQP algorithm For comparingthe numerical results with the previous results the required
running time in each NOCP is considered fixed which is themaximum of running time in 12 different runs were done inSection 5 Also the all of the parameters for each problemwasset as same as Section 5 The numerical results of the withoutSQP algorithm is summarized in Table 10 By comparing theresults of Tables 10 and 1 it is obvious that for all NOCPsin the Appendix the obtained values of the performanceindex and the norm of final state constraint for the proposedalgorithm (Algorithm 2) are more accurate than the withoutSQP algorithm
18 Mathematical Problems in Engineering
Table 5 The mean of numerical results of 15 different runs for NOCPs in the Appendix using LI and SI methods
Problem LI SI119869 120593
119891119870120595
Time 119869 120593119891
119870120595
TimeVDP 15977 157 times 10
minus9
170 times 10minus3 17720 16002 667 times 10
minus8
326 times 10minus3 15757
CRP 00126 307 times 10minus8
307 times 10minus8 19714 00119 161 times 10
minus8
161 times 10minus8 19532
FFPP 4774 805 times 10minus5 03277 133280 5351 592 times 10
minus5 04868 137989MSNIC 01702 mdash mdash 13678 01703 mdash mdash 14092CSTCR 01120 mdash mdash 17385 01120 mdash mdash 17606Number 6 minus54307 mdash mdash 10541 minus54308 mdash mdash 10372Number 7 02015 326 times 10
minus9
695 times 10minus6 20807 02020 313 times 10
minus9
662 times 10minus6 21019
Number 8 630 times 10minus15 mdash mdash 6376 124 times 10
minus14 mdash mdash 5837Number 9 16512 mdash mdash 59239 16512 mdash mdash 58125Number 10 01550 mdash mdash 63153 01550 mdash mdash 65597Number 11 29701 270 times 10
minus9
272 times 10minus7 22860 29701 337 times 10
minus9
811 times 10minus7 23276
Number 12 minus02499 201 times 10minus8
464 times 10minus5 54718 minus02499 115 times 10
minus9
827 times 10minus5 52765
Number 13 00147 115 times 10minus8
00252 57797 00151 139 times 10minus8
00472 66098Number 14 34761 672 times 10
minus6
00237 104585 34290 757 times 10minus6
950 times 10minus3 108767
Number 15 195 times 10minus4
111 times 10minus8
569 times 10minus3 363 194 times 10
minus4
704 times 10minus9
486 times 10minus3 365
Number 16 20571 166 times 10minus10
166 times 10minus10 54391 20571 115 times 10
minus10
115 times 10minus10 54431
Number 17 minus88692 803 times 10minus8
803 times 10minus8 22744 minus88692 661 times 10
minus8
661 times 10minus8 23254
Number 18 00326 mdash mdash 65957 00326 mdash mdash 70897Number 19 011588 655 times 10
minus4 06828 161813 01091 884 times 10minus4 08808 159990
Number 20 5242 00176 05429 84165 4464 453 times 10minus4 03765 86891
Table 6 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 119869
Method Mean rank Sum of ranks Test statistics ValueLI 2050 4100 Mann-Whitney 119880 200SI 2050 4100 Wilcoxon119882 410mdash mdash mdash 119885 0
Table 7 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 120593119891
Method Mean rank Sum of ranks Test statistics ValueLI 1362 177 Mann-Whitney 119880 83SI 1338 174 Wilcoxon119882 174mdash mdash mdash 119885 minus0077
Table 8 Results of Mann-Whitney test on relative errors of the pair (LI SI) for Time
Method Mean rank Sum of ranks Test statistics ValueLI 2025 4050 Mann-Whitney 119880 195SI 2075 4150 Wilcoxon119882 405mdash mdash mdash 119885 minus0135
Table 9 Results of Mann-Whitney test on relative errors of the pair (LI SI) for119870120595
Method Mean rank Sum of ranks Test statistics ValueLI 1354 1760 Mann-Whitney 119880 84SI 1346 1750 Wilcoxon119882 175mdash mdash mdash 119885 minus0026
Mathematical Problems in Engineering 19
Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix
Problem no1 2 3 4 5 6 7 8 9 10
119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048
120601119891
330 times 10minus9
135 times 10minus5 11445 mdash mdash mdash 758 times 10
minus9 mdash mdash mdashProblem no
11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10
minus4 23182 minus90882 00336 2513 20274120601119891
167 times 10minus5
468 times 10minus9 00161 433 times 10
minus4
594 times 10minus3
156 times 10minus5 164 mdash 1706 25283
8 Conclusions
Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters
Appendix
Test Problems
The following problems are presented using notation given in(1)ndash(6)
(1) (VDP) [9] 119892 = (12)(1199092
1+ 1199092
2+ 1199062
) 1199050= 0 119905119891= 5 119891 =
[1199092 minus1199092+(1minus119909
2
1)1199092+119906]119879 1199090= [1 0]
119879120595 = 1199091minus1199092+1
(2) (CRP) [9] 119892 = (12)(1199092
1+1199092
2+01119906
2
) 1199050= 0 119905119891= 078
119891 = [1199091minus2(1199091+025)+(119909
2+05) exp(25119909
1(1199091+2))minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(3) (FFRP) [9] 119892 = (12)(1199062
1+ 1199062
2+ 1199062
3+ 1199062
4) 1199050
=
0 119905119891
= 5 119891 = [1199092 (110)((119906
1+ 1199062) cos119909
5minus
(1199062+ 1199064) sin119909
5) 1199094 (110)((119906
1+ 1199063) sin119909
5+ (1199062+
1199064) cos119909
5) 1199096 (512)((119906
1+ 1199063) minus (119906
2+ 1199064))]119879 1199090=
[0 0 0 0 0 0]119879 120595 = [119909
1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879
(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909
2 minus1199092+
119906 1199092
1+ 1199092
2+ 0005119906
2
]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909
2+
05 minus 8(119905 minus 05)2
]119879 1199090= [0 minus1 0]
119879
(5) (CSTCR) [20] 119892 = 1199092
1+ 1199092
2+ 01119906
2 1199050= 0 119905119891= 078
119891 = [minus(2 + 119906)(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+
2)) 05 minus 1199092minus (1199092+ 05) exp(25119909
1(1199091+ 2))]
119879 1199090=
[009 009]119879
(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909
2 119906]119879 119889 = [minus(2 +
119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]
119879(7) [36] 119892 = 119906
2 1199050= 0 119905
119891= 1 119891 = (12)119909
2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05
(8) [26] 119892 = 1199092cos2119906 119905
0= 0 119905119891= 120587 119891 = sin(1199062) 119909
0=
1205872(9) [26] 119892 = (12)(119909
2
1+ 1199092
2+ 1199062
) 1199050= 0 119905
119891= 5 119891 =
[1199092 minus1199091+ (1 minus 119909
2
1)1199092+ 119906]119879 119889 = minus(119909
2+ 025) 119909
0=
[1 0]119879
(10) [26] 119892 = 1199092
1+ 1199092
2+ 0005119906
2
1199050= 0 119905
119891= 1 119891 =
[1199092 minus1199092+119906]119879
119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus
05) minus 05 minus 1199092)]119879
1199090= [0 minus1]
119879(11) [26] 119892 = (12)119906
2 1199050= 0 119905
119891= 2 119891 = [119909
2 119906]119879 1199090=
[1 1]119879 120595 = [119909
1 1199092]119879
(12) [26] 119892 = minus1199092 1199050= 0 119905
119891= 1 119891 = [119909
2 119906]119879 119889 =
minus(1 minus 119906)(1 + 119906) 1199090= [0 0]
119879 120595 = 1199092
(13) [26] 119892 = (12)(1199092
1+ 1199092
2+ 01119906
2
) 1199050= 0 119905
119891= 078
119891 = [minus2(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+ 2)) minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(14) [26] 119892 = (12)1199062 1199050= 0 119905
119891= 10 119891 = [cos 119906 minus
1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909
0= [366 minus186]
119879120595 = [119909
1 1199092]119879
(15) [26]119892 = (12)(1199092
1+1199092
2) 1199050= 0 119905119891= 078119891 = [minus2(119909
1+
025)+(1199092+05) exp(25119909
1(1199091+2))minus(119909
1+025)119906 05minus
1199092minus(1199092+05) exp(25119909
1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(16) [26] 120601 = 1199093 1199050= 0 119905
119891= 1 119891 = [119909
2 119906 (12)119906
2
]119879
119889 = 1199091minus 19 119909
0= [0 0 0]
119879 120595 = [1199091 1199092+ 1]119879
(17) [26] 120601 = minus1199093 1199050= 0 119905
119891= 5 119891 = [119909
2 minus2 + 119906119909
3
minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909
0= [10 minus2 10]
119879120595 = [119909
1 1199092]119879
(18) [26] 120601 = (1199091minus1)2
+1199092
2+1199092
3 119892 = (12)119906
2 1199050= 0 119905119891= 5
119891 = [1199093cos 119906 119909
3sin 119906 sin 119906]119879 119909
0= [0 0 0]
119879
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
18 Mathematical Problems in Engineering
Table 5 The mean of numerical results of 15 different runs for NOCPs in the Appendix using LI and SI methods
Problem LI SI119869 120593
119891119870120595
Time 119869 120593119891
119870120595
TimeVDP 15977 157 times 10
minus9
170 times 10minus3 17720 16002 667 times 10
minus8
326 times 10minus3 15757
CRP 00126 307 times 10minus8
307 times 10minus8 19714 00119 161 times 10
minus8
161 times 10minus8 19532
FFPP 4774 805 times 10minus5 03277 133280 5351 592 times 10
minus5 04868 137989MSNIC 01702 mdash mdash 13678 01703 mdash mdash 14092CSTCR 01120 mdash mdash 17385 01120 mdash mdash 17606Number 6 minus54307 mdash mdash 10541 minus54308 mdash mdash 10372Number 7 02015 326 times 10
minus9
695 times 10minus6 20807 02020 313 times 10
minus9
662 times 10minus6 21019
Number 8 630 times 10minus15 mdash mdash 6376 124 times 10
minus14 mdash mdash 5837Number 9 16512 mdash mdash 59239 16512 mdash mdash 58125Number 10 01550 mdash mdash 63153 01550 mdash mdash 65597Number 11 29701 270 times 10
minus9
272 times 10minus7 22860 29701 337 times 10
minus9
811 times 10minus7 23276
Number 12 minus02499 201 times 10minus8
464 times 10minus5 54718 minus02499 115 times 10
minus9
827 times 10minus5 52765
Number 13 00147 115 times 10minus8
00252 57797 00151 139 times 10minus8
00472 66098Number 14 34761 672 times 10
minus6
00237 104585 34290 757 times 10minus6
950 times 10minus3 108767
Number 15 195 times 10minus4
111 times 10minus8
569 times 10minus3 363 194 times 10
minus4
704 times 10minus9
486 times 10minus3 365
Number 16 20571 166 times 10minus10
166 times 10minus10 54391 20571 115 times 10
minus10
115 times 10minus10 54431
Number 17 minus88692 803 times 10minus8
803 times 10minus8 22744 minus88692 661 times 10
minus8
661 times 10minus8 23254
Number 18 00326 mdash mdash 65957 00326 mdash mdash 70897Number 19 011588 655 times 10
minus4 06828 161813 01091 884 times 10minus4 08808 159990
Number 20 5242 00176 05429 84165 4464 453 times 10minus4 03765 86891
Table 6 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 119869
Method Mean rank Sum of ranks Test statistics ValueLI 2050 4100 Mann-Whitney 119880 200SI 2050 4100 Wilcoxon119882 410mdash mdash mdash 119885 0
Table 7 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 120593119891
Method Mean rank Sum of ranks Test statistics ValueLI 1362 177 Mann-Whitney 119880 83SI 1338 174 Wilcoxon119882 174mdash mdash mdash 119885 minus0077
Table 8 Results of Mann-Whitney test on relative errors of the pair (LI SI) for Time
Method Mean rank Sum of ranks Test statistics ValueLI 2025 4050 Mann-Whitney 119880 195SI 2075 4150 Wilcoxon119882 405mdash mdash mdash 119885 minus0135
Table 9 Results of Mann-Whitney test on relative errors of the pair (LI SI) for119870120595
Method Mean rank Sum of ranks Test statistics ValueLI 1354 1760 Mann-Whitney 119880 84SI 1346 1750 Wilcoxon119882 175mdash mdash mdash 119885 minus0026
Mathematical Problems in Engineering 19
Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix
Problem no1 2 3 4 5 6 7 8 9 10
119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048
120601119891
330 times 10minus9
135 times 10minus5 11445 mdash mdash mdash 758 times 10
minus9 mdash mdash mdashProblem no
11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10
minus4 23182 minus90882 00336 2513 20274120601119891
167 times 10minus5
468 times 10minus9 00161 433 times 10
minus4
594 times 10minus3
156 times 10minus5 164 mdash 1706 25283
8 Conclusions
Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters
Appendix
Test Problems
The following problems are presented using notation given in(1)ndash(6)
(1) (VDP) [9] 119892 = (12)(1199092
1+ 1199092
2+ 1199062
) 1199050= 0 119905119891= 5 119891 =
[1199092 minus1199092+(1minus119909
2
1)1199092+119906]119879 1199090= [1 0]
119879120595 = 1199091minus1199092+1
(2) (CRP) [9] 119892 = (12)(1199092
1+1199092
2+01119906
2
) 1199050= 0 119905119891= 078
119891 = [1199091minus2(1199091+025)+(119909
2+05) exp(25119909
1(1199091+2))minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(3) (FFRP) [9] 119892 = (12)(1199062
1+ 1199062
2+ 1199062
3+ 1199062
4) 1199050
=
0 119905119891
= 5 119891 = [1199092 (110)((119906
1+ 1199062) cos119909
5minus
(1199062+ 1199064) sin119909
5) 1199094 (110)((119906
1+ 1199063) sin119909
5+ (1199062+
1199064) cos119909
5) 1199096 (512)((119906
1+ 1199063) minus (119906
2+ 1199064))]119879 1199090=
[0 0 0 0 0 0]119879 120595 = [119909
1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879
(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909
2 minus1199092+
119906 1199092
1+ 1199092
2+ 0005119906
2
]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909
2+
05 minus 8(119905 minus 05)2
]119879 1199090= [0 minus1 0]
119879
(5) (CSTCR) [20] 119892 = 1199092
1+ 1199092
2+ 01119906
2 1199050= 0 119905119891= 078
119891 = [minus(2 + 119906)(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+
2)) 05 minus 1199092minus (1199092+ 05) exp(25119909
1(1199091+ 2))]
119879 1199090=
[009 009]119879
(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909
2 119906]119879 119889 = [minus(2 +
119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]
119879(7) [36] 119892 = 119906
2 1199050= 0 119905
119891= 1 119891 = (12)119909
2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05
(8) [26] 119892 = 1199092cos2119906 119905
0= 0 119905119891= 120587 119891 = sin(1199062) 119909
0=
1205872(9) [26] 119892 = (12)(119909
2
1+ 1199092
2+ 1199062
) 1199050= 0 119905
119891= 5 119891 =
[1199092 minus1199091+ (1 minus 119909
2
1)1199092+ 119906]119879 119889 = minus(119909
2+ 025) 119909
0=
[1 0]119879
(10) [26] 119892 = 1199092
1+ 1199092
2+ 0005119906
2
1199050= 0 119905
119891= 1 119891 =
[1199092 minus1199092+119906]119879
119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus
05) minus 05 minus 1199092)]119879
1199090= [0 minus1]
119879(11) [26] 119892 = (12)119906
2 1199050= 0 119905
119891= 2 119891 = [119909
2 119906]119879 1199090=
[1 1]119879 120595 = [119909
1 1199092]119879
(12) [26] 119892 = minus1199092 1199050= 0 119905
119891= 1 119891 = [119909
2 119906]119879 119889 =
minus(1 minus 119906)(1 + 119906) 1199090= [0 0]
119879 120595 = 1199092
(13) [26] 119892 = (12)(1199092
1+ 1199092
2+ 01119906
2
) 1199050= 0 119905
119891= 078
119891 = [minus2(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+ 2)) minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(14) [26] 119892 = (12)1199062 1199050= 0 119905
119891= 10 119891 = [cos 119906 minus
1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909
0= [366 minus186]
119879120595 = [119909
1 1199092]119879
(15) [26]119892 = (12)(1199092
1+1199092
2) 1199050= 0 119905119891= 078119891 = [minus2(119909
1+
025)+(1199092+05) exp(25119909
1(1199091+2))minus(119909
1+025)119906 05minus
1199092minus(1199092+05) exp(25119909
1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(16) [26] 120601 = 1199093 1199050= 0 119905
119891= 1 119891 = [119909
2 119906 (12)119906
2
]119879
119889 = 1199091minus 19 119909
0= [0 0 0]
119879 120595 = [1199091 1199092+ 1]119879
(17) [26] 120601 = minus1199093 1199050= 0 119905
119891= 5 119891 = [119909
2 minus2 + 119906119909
3
minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909
0= [10 minus2 10]
119879120595 = [119909
1 1199092]119879
(18) [26] 120601 = (1199091minus1)2
+1199092
2+1199092
3 119892 = (12)119906
2 1199050= 0 119905119891= 5
119891 = [1199093cos 119906 119909
3sin 119906 sin 119906]119879 119909
0= [0 0 0]
119879
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 19
Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix
Problem no1 2 3 4 5 6 7 8 9 10
119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048
120601119891
330 times 10minus9
135 times 10minus5 11445 mdash mdash mdash 758 times 10
minus9 mdash mdash mdashProblem no
11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10
minus4 23182 minus90882 00336 2513 20274120601119891
167 times 10minus5
468 times 10minus9 00161 433 times 10
minus4
594 times 10minus3
156 times 10minus5 164 mdash 1706 25283
8 Conclusions
Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters
Appendix
Test Problems
The following problems are presented using notation given in(1)ndash(6)
(1) (VDP) [9] 119892 = (12)(1199092
1+ 1199092
2+ 1199062
) 1199050= 0 119905119891= 5 119891 =
[1199092 minus1199092+(1minus119909
2
1)1199092+119906]119879 1199090= [1 0]
119879120595 = 1199091minus1199092+1
(2) (CRP) [9] 119892 = (12)(1199092
1+1199092
2+01119906
2
) 1199050= 0 119905119891= 078
119891 = [1199091minus2(1199091+025)+(119909
2+05) exp(25119909
1(1199091+2))minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(3) (FFRP) [9] 119892 = (12)(1199062
1+ 1199062
2+ 1199062
3+ 1199062
4) 1199050
=
0 119905119891
= 5 119891 = [1199092 (110)((119906
1+ 1199062) cos119909
5minus
(1199062+ 1199064) sin119909
5) 1199094 (110)((119906
1+ 1199063) sin119909
5+ (1199062+
1199064) cos119909
5) 1199096 (512)((119906
1+ 1199063) minus (119906
2+ 1199064))]119879 1199090=
[0 0 0 0 0 0]119879 120595 = [119909
1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879
(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909
2 minus1199092+
119906 1199092
1+ 1199092
2+ 0005119906
2
]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909
2+
05 minus 8(119905 minus 05)2
]119879 1199090= [0 minus1 0]
119879
(5) (CSTCR) [20] 119892 = 1199092
1+ 1199092
2+ 01119906
2 1199050= 0 119905119891= 078
119891 = [minus(2 + 119906)(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+
2)) 05 minus 1199092minus (1199092+ 05) exp(25119909
1(1199091+ 2))]
119879 1199090=
[009 009]119879
(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909
2 119906]119879 119889 = [minus(2 +
119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]
119879(7) [36] 119892 = 119906
2 1199050= 0 119905
119891= 1 119891 = (12)119909
2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05
(8) [26] 119892 = 1199092cos2119906 119905
0= 0 119905119891= 120587 119891 = sin(1199062) 119909
0=
1205872(9) [26] 119892 = (12)(119909
2
1+ 1199092
2+ 1199062
) 1199050= 0 119905
119891= 5 119891 =
[1199092 minus1199091+ (1 minus 119909
2
1)1199092+ 119906]119879 119889 = minus(119909
2+ 025) 119909
0=
[1 0]119879
(10) [26] 119892 = 1199092
1+ 1199092
2+ 0005119906
2
1199050= 0 119905
119891= 1 119891 =
[1199092 minus1199092+119906]119879
119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus
05) minus 05 minus 1199092)]119879
1199090= [0 minus1]
119879(11) [26] 119892 = (12)119906
2 1199050= 0 119905
119891= 2 119891 = [119909
2 119906]119879 1199090=
[1 1]119879 120595 = [119909
1 1199092]119879
(12) [26] 119892 = minus1199092 1199050= 0 119905
119891= 1 119891 = [119909
2 119906]119879 119889 =
minus(1 minus 119906)(1 + 119906) 1199090= [0 0]
119879 120595 = 1199092
(13) [26] 119892 = (12)(1199092
1+ 1199092
2+ 01119906
2
) 1199050= 0 119905
119891= 078
119891 = [minus2(1199091+ 025) + (119909
2+ 05) exp(25119909
1(1199091+ 2)) minus
(1199091+025)119906 05minus119909
2minus(1199092+05) exp(25119909
1(1199091+2))]119879
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(14) [26] 119892 = (12)1199062 1199050= 0 119905
119891= 10 119891 = [cos 119906 minus
1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909
0= [366 minus186]
119879120595 = [119909
1 1199092]119879
(15) [26]119892 = (12)(1199092
1+1199092
2) 1199050= 0 119905119891= 078119891 = [minus2(119909
1+
025)+(1199092+05) exp(25119909
1(1199091+2))minus(119909
1+025)119906 05minus
1199092minus(1199092+05) exp(25119909
1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)
1199090= [005 0]
119879 120595 = [1199091 1199092]119879
(16) [26] 120601 = 1199093 1199050= 0 119905
119891= 1 119891 = [119909
2 119906 (12)119906
2
]119879
119889 = 1199091minus 19 119909
0= [0 0 0]
119879 120595 = [1199091 1199092+ 1]119879
(17) [26] 120601 = minus1199093 1199050= 0 119905
119891= 5 119891 = [119909
2 minus2 + 119906119909
3
minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909
0= [10 minus2 10]
119879120595 = [119909
1 1199092]119879
(18) [26] 120601 = (1199091minus1)2
+1199092
2+1199092
3 119892 = (12)119906
2 1199050= 0 119905119891= 5
119891 = [1199093cos 119906 119909
3sin 119906 sin 119906]119879 119909
0= [0 0 0]
119879
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
20 Mathematical Problems in Engineering
(19) [26] 119892 = 45(1199092
3+ 1199092
6) + 05(119906
2
1+ 1199062
2) 1199050= 0 119905
119891= 1
119891 = [91199094 91199095 91199096 9(1199061+ 1725119909
3) 91199062 minus(9119909
2)(1199061minus
270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]
119879 120595 =
[1199091minus 10 119909
2minus 14 119909
3 1199094minus 25 119909
5 1199096]119879
(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909
2 1199093minus
4 1199094 1199095minus 1205874 119909
6]119879
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010
[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004
[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003
[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001
[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005
[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000
[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992
[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997
[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011
[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003
[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003
[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011
[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009
[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008
[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009
[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009
[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999
[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997
[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011
[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011
[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013
[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006
[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991
[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988
[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996
[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998
[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007
[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 21
[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009
[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975
[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997
[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999
[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997
[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012
[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988
[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011
[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996
[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of