research article steady modeling for an ammonia synthesis

Research ArticleSteady Modeling for an Ammonia Synthesis Reactor Based ona Novel CDEAS-LS-SVM Model

Zhuoqian Liu1 Lingbo Zhang1 Wei Xu2 and Xingsheng Gu1

1 Key Laboratory of Advanced Control and Optimization for Chemical Process Ministry of Education Shanghai 200237 China2 Shanghai Electric Group Co Ltd Central Academe Shanghai 200070 China

Correspondence should be addressed to Xingsheng Gu xsguecusteducn

Received 6 December 2013 Accepted 5 February 2014 Published 18 March 2014

Academic Editor Huaicheng Yan

Copyright copy 2014 Zhuoqian Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A steady-state mathematical model is built in order to represent plant behavior under stationary operating conditions A novelmodeling using LS-SVR based on Cultural Differential Evolution with Ant Search is proposed LS-SVM is adopted to establish themodel of the net value of ammoniaThemodelingmethod has fast convergence speed and good global adaptability for identificationof the ammonia synthesis processThe LS-SVRmodel was established using the above-mentionedmethod Simulation results verifythe validity of the method

1 Introduction

Ammonia is one of the important chemicals that has innu-merable uses in a wide range of areas that is explosivematerials pharmaceuticals polymers acids and coolersparticularly in synthetic fertilizers It is produced worldwideon a large scale with capacities extending to about 159milliontons at 2010 Generally the average energy consumption ofammonia production per ton is 1900KG of standard coal inChina which is much higher than the advanced standard of1570KG around the world At the same time the haze andparticulate matter 25 has been serious exceeded in big citiesin China at recent years and one of the important reasons isthe emission of coal chemical factories Thus an economicpotential exists in energy consumption of the ammoniasynthesis as prices of energy rise and reduce the ammoniasynthesis pollution to protect the environment Ammoniasynthesis process has the characteristics of nonlinearitystrong coupling large time-delay and great inertia load andso forth Steady-state operation-optimization can be a reliabletechnique for output improvement and energy reductionwithout changing any devices

The optimization of ammonia synthesis process highlyrelies on the accurate system model To establish an appro-priate mathematical model of ammonia synthesis process is a

principal problem of operation optimization It has receivedconsiderable attention since last century Heterogeneoussimulation models imitating different types of ammonia syn-thesis reactors have been developed for design optimizationand control [1] Elnashaie et al [2] studied the optimizationof an ammonia synthesis reactor which has three adiabaticbedsThe optimal temperature profile was obtained using theorthogonal collocation method in the paper Pedemera et al[3] studied the steady state analysis and optimization of aradial-flow ammonia synthesis reactor

The above study indicated that both the productive capac-ity and the stability of the ammonia reactor are influenced bythe cold quench and the feed temperature significantly Babuand Angira [4] described the simulation and optimizationdesign of an auto-thermal ammonia synthesis reactor usingQuasi-Newton and NAG subroutine method The optimaltemperature trajectory along the reactor and optimal flowsthroughput 33 additional ammonia production Sadeghiand Kavianiboroujeni [1] evaluated the process behavior ofan industrial ammonia synthesis reactorby one-dimensionalmodel and two-dimensional model genetic algorithm (GA)was applied to optimize the reactor performance in varyingits quench flows FromThe above literatures we can find thatmost models are built based on thermodynamic kinetic andmass equilibria calculations It is very difficult to simulate the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 168371 18 pageshttpdxdoiorg1011552014168371

2 Mathematical Problems in Engineering

specific internal mechanism because a lot of parameters areunknown in real industrial process

In order to achieve the required accuracy of the modelsome researches focus on the novel modeling methodscombining some heuristic methods such as ANN (ArtificialNeural Network) LS-SVM (Least Squares Support VectorMachine) with Evolutionary Algorithm for example geneticalgorithm ant colony optimization (ACO) particle swarmoptimization (PSO) differential evolution (DE) and so forthDE is one of themost popular algorithms for this problemandhas been applied in many fields Sacco and Hendersonb [5]introduced a variant of the differential evolution algorithmwith a new mutation operator based on a topographicalheuristic and used it to solve the nuclear reactor core designoptimization problem Rout et al [6] proposed a simple butpromising hybrid prediction model by suitably combiningan adaptive autoregressive moving average architecture anddifferential evolution for forecasting of exchange rates Ozcanet al [7] carried out the cost optimization of an air coolingsystem by using Lagrange multipliers method differentialevolution algorithm and particle swarm optimization forvarious temperatures andmass flow ratesThe results showedthat the method gives high accuracy results within a shorttime interval Zhang et al [8] proposed a hybrid differentialevolution algorithm for the job shop scheduling problemwithrandom processing times under the objective of minimizingthe expected total tardiness Arya and Choube [9] describeda methodology for allocating repair time and failure rates tosegments of a meshed distribution system using differentialevolution technique Xu et al [10] proposed a model ofammonia conversion rate by LS-SVM and a hybrid algorithmof PSO and DE is described to identify the hyper-parametersof LS-SVM

To describe the relationship between net value of ammo-nia in ammonia synthesis reactor and the key operationalparameters least squares support vectormachine is employedto build the structure of the relationship model in which anovel algorithm called CDEAS is proposed to identify theparametersThe experiment results showed that the proposedCDEAS-LS-SVM optimizing model is very effective of beingused to obtain the optimal operational parameters of ammo-nia synthesis converter

The remaining of the paper is organized as followsSection 2 describes the ammonia synthesis production pro-cess Section 3 proposes a novel Cultural Differential Evolu-tion with Ant Colony Search (CDEAS) algorithm Section 4constructs a model using LS-SVM based on the proposedCDEAS algorithm Section 5 presents the experiments andcomputational results and discussion Finally Section 6 sum-marizes the above results and presents several problemswhich remain to be solved

2 Ammonia Synthesis Production Process

A normal ammonia production flow chart includes thesynthesis gas production purification gas compression andammonia synthesis Ammonia synthesis loop is one of themost critical units in the entire process The system has

been realized by LuHua Inc a medium fertilizers factory ofYanKuang Group China

Figure 1 represents a flow sheet for the ammonia syn-thesis process The ammonia synthesis reactor is a one-axialflow and two-radial flow three-bed quench-type unit [11]Hydrogen-nitrogen mixture is reacted in the catalyst bedunder high temperature and pressure The temperature inthe reactor is sustained by the heat of reaction because thereaction is exothermic [1]The reaction of ammonia synthesisprocess contains

3

2H2+1

2N2999445999468 NH

3+Q (1)

The reaction is limited by the unfavorable position ofthe chemical equilibrium and by the low activity of thepromoted iron catalysts with high pressure and temperature[12] In general no more than 20 of the synthesis gas isconverted into ammonia per pass even at high pressure of30MPa [12] As the ammonia reaction is exothermic it isnecessary for removing the heat generated in the catalyst bedby the progress of the reaction to obtain a reasonable overallconversion rate as same as to protect the life of the catalyst[13] The mixture gas from the condenser is divided into twoparts Q1 and Q2 to go to the converter The first cold shotQ1 is recirculated to the annular space between the outershell reactor and catalyst bed from the top to the bottomto refrigerate the shell and remove the heat released by thereaction Then the gas Q1 from the bottom of reactor goesthrough the preheater and is heated by the counter-currentflowing reacted gas from waste heat boiler Q1 gas is dividedinto 4 cold quench gas (q1 q2 q3 and q4) and Q2 gas formixing with the gas between consecutive catalyst beds toquench the hot spots before entry to the subsequent catalystbeds The hot spot temperatures (TIRA705 TIRA712N andTIRA714) represent the highest reaction temperatures at eachstage of the catalyst bed

Figure 2 represents the ammonia synthesis unit Thereacted gas including N

2 H2 NH3 and inert gas after reactor

passes through the waste heat boiler Then it goes throughthe preheater and the water cooler to be further cooled Partof the ammonia is condensed and separated by ammoniaseparator I Inert gas from the ammonia synthesis loop areejected by purge gas from separator to prevent accumulationof inert gas in the system The fresh feed gas is producedby the Texaco coal gasification air separation section aprocess that converts the Coal Water Slurry into synthesisgas for ammonia The fresh gas consists of hydrogen andnitrogen in stoichiometric proportions of 3 1 approximatelyand mixes with small amounts of argon and methane Thefresh gas which passes compressor is compounded with therecycle gas which comes from the circulator and then themixture goes through oil separator and condenser Mixturegas is further cooled by liquid ammonia and goes throughammonia separator II to separate the partial liquid ammoniaand then it goes out with very few ammonia The liquidammonia from ammonia separator I and separator II flowsto the liquid ammonia jar Mixture is heated in ammoniacondenser to about 25∘C and flows to the reactor and thewhole cycle starts again

Mathematical Problems in Engineering 3

Ammoniareactor

CirculatorWaste heat

boiler Preheater Water cooler Ammonia

separator IOil

separator

Condenser

Ammonia cooler

Synthesis gas

Evaporator

Ammonia separator II

Hydrogen recovery unit

Ammonia recovery unit

TIRA705

TIR712N

TIRA714

AR701

AR701-4

FIR705 703

PI

725TI

Liquid ammonia

Compressor

Figure 1 Ammonia synthesis system

Preheater

Q2

Preheaterq1

q4

q3

q2

Waste heat boiler

Q1

Q 1

I radial bed

II radial bed

Inter-changer

Axial bedFIR704

703

702

FIR705

FIR

FIR

Figure 2 The ammonia synthesis unit

3 Proposed Cultural Differential Evolutionwith Ant Search Algorithm

31 Differential Evolution Algorithm Evolutionary Algo-rithms which are inspired by the evolution of species havebeen adopted to solve a wide range of optimization problemssuccessfully in different fields The primary advantage ofEvolutionaryAlgorithms is that they just require the objectivefunction values while properties such as differentiability andcontinuity are not necessary [14]

Differential evolution proposed by Storn and Price is afast and simple population based stochastic search technique[15] DE employs mutation crossover and selection opera-tions It focuses on differential vectors of individuals with thecharacteristics of simple structure and rapid convergenceThedetailed procedure of DE is presented below

(1) Initialization In a 119863-dimension space NP parametervectors so-called individuals cover the entire search spaceby uniformly randomizing the initial individuals within thesearch space constrained by the minimum and maximumparameter bounds119883min and119883max

1199090

119894119889= 1199090

min119889 + rand (0 1) (1199090

max119889 minus 1199090

min119889) 119895 = 1 2 119863

(2)


(2)MutationDE employs themutation operation to produceamutant vector 119906119905

119894119889called target vector corresponding to each

individual 119909119905119894119889after initialization In iteration 119905 the mutant

vector 119906119905119894119889

of individual 119909119905119894119889

can be generated according tocertain mutation strategies Equations (3)ndash(7) indicate themost frequent mutation strategies version respectively

DErand1 119906119905

119894119889= 119883119905

1199031119894119889+ 119865 (119883

119905

1199032119894119889minus 119883119905

1199033119894119889) (3)

DErand2 119906119905

119894119889= 119883119905

1199031119894119889+ 119865 (119883

119905

1199032119894119889minus 119883119905

1199033119894119889)

+ 119865 (119883119905

1199034119894119889minus 119883119905

1199035119894119889)

(4)

DEbest1 119906119905

119894119889= 119883119905

best119889 + 119865 (119883119905

1199031119894119889minus 119883119905

1199032119894119889) (5)

DEbest2 119906119905

119894119889= 119883119905

best119889 + 119865 (119883119905

1199031119894119889minus 119883119905

1199032119894119889)

+ 119865 (119883119905

1199033119894119889minus 119883119905

1199034119894119889)

(6)

DErand-to-best1 119906119905

119894119889= 119883119905

119894119889+ 119865 (119883

119905

best119889 minus 119883119905

119894119889)

+ 119865 (119883119905

1199031119894119889minus 119883119905

1199032119894119889)

(7)

where 1199031119894 1199032119894 1199033119894 1199034119894 and 1199035119894are mutually exclusive integersrandomly generated within the range [1NP] which shouldnot be 119894 119865 is the mutation factor for scaling the differencevector usually bounded in [0 2] 119883119905best is the best individualwith the best fitness value at generation 119905 in the population

(3) Crossover The individual 119883119905119894and mutant vector 119906119905

119894are

hybridized to compose the trial vector 119910119905119894after mutation

operation The binomial crossover is adopted by the DE inthe paper which is defined as

119910119905

119894119889= 119906119905

119894119889if rand le 119862

119877or 119894 = 119894rand

119883119905

119894119889otherwise

(8)

where rand is a random number between in 0 and 1 dis-tributed uniformly The crossover factor 119862

119877is a probability

rate within the range 0 and 1 which influences the tradeoffbetween the ability of exploration and exploitation 119894rand is aninteger chosen randomly in [1 119863] To ensure that the trialvector (119910119905

119894) differs from its corresponding individual (119883119905

119894) by

at least one dimension 119894 = 119894rand is recommended

(4) Selection When a newly generated trial vector exceedsits corresponding upper and lower bounds it is reinitializedwithin the presetting range uniformly and randomly Thenthe trial individual119910119905

119894is comparedwith the individual119883119905

119894 and

the one with better fitness is selected as the new individual inthe next iteration

119883119905+1

119894119889= 119910119905

119894119889if 119891 (119910119905

119894119889) le 119891 (119883

119905

119894119889)

119883119905


(9)

(5) Termination All above three evolutionary operationscontinue until termination criterion is achieved such asthe evolution reaching the maximumminimum of functionevaluations

As an effective and powerful random optimizationmethod DE has been successfully used to solve real worldproblems in diverse fields both unconstrained and con-strained optimization problems

32 Cultural Differential Evolution with Ant Search As wementioned in Section 31 mutation factor 119865 mutation strate-gies and crossover factor 119862

119877have great influence on the bal-

ance ofDErsquos exploration and exploitation ability119865decides theamplification of differential variation 119862

119877is used to control

the possibility of the crossover operation mutation strategieshave great influence on the results of mutation operation Insome literatures 119865 119862

119877 and mutation strategies are defined in

advance or varied by some specific regulations But the factors119865119862119877 and strategies are very difficult to choose since the prior

knowledge is absent Therefore Ant Colony Search is usedto search the suitable combination of 119865 119862

119877 and mutation

strategies adaptively to accelerate the global search Someresearchers have found an inevitable relationship betweenthe parameters (119865 119862

119877 and mutation strategies) and the

optimization results of DE [16ndash18] However the approachesabove are not applying the most suitable 119865 119862

119877 and mutation

strategies simultaneouslyIn this paper based on the theory of Cultural Algorithm

and Ant Colony Optimization (ACO) an improved Cul-tural Differential Algorithm incorporation with Ant ColonySearch is presented In order to accelerate searching out theglobal solution the Ant Colony Search is used to searchthe optimal combination of 119865 and 119862

119877in subpopulation 1 as

well as mutation strategy in subpopulation 2 The frameworkof Cultural Differential Evolution with Ant Search is brieflydescribed in Figure 3

321 Population Space The population space is divided intotwo parts subpopulation 1 and subpopulation 2 The twosubpopulations contain equal number of the individuals

In subpopulation 1 the individual is set as ant at each gen-eration 119865 and 119862

119877are defined to be the values between [0 1]

119865 isin 01times119894 119894 = 1 2 10 and119862119877isin 01times119894 119894 = 1 2 10

Each of the ants chooses a combination of119865 and119862119877according

to the information which is calculated by the fitness functionof ants During search process the information gathered bythe ants is preserved in the pheromone trails 120591 By exchanginginformation according to pheromone the ants cooperatewitheach other to choose appropriate combination of 119865 and 119862

119877

Then ant colony renews the pheromone trails of all antsThen the pheromone trail 120591

119898119899is updated in the following

equation

120591119898119899(119905 + 1) = (1 minus 120588

1) 120591119898119899(119905) +

subpopulation1

sum

119894=1

Δ120591119894

119898119899(119905)

(10)

where 0 le 1205881lt 1 means the pheromone trail evaporation

rate 119899 = 1 2 10119898 = 1 2 1st parameter represents 119865 and


Subpopulation 1 Subpopulation 2

Belief space

Influencefunction

Acceptancefunction

Select Performancefunction

Ant Search of mutation strategy

Knowledge exchange

Population space

Ant Search ofF and CR

(situational knowledge and normative knowledge)

Figure 3 The framework of CDEAS algorithm

01 02 03 10

01 02 03 10

12059111 12059112 12059113 120591110

F

F

CR

CR

12059121 12059122 12059123 120591210

01

10

02

03

01

10

02

03

middot middot middot




Figure 4 Relationship between pheromone and ant paths of 119865 119862119877

2nd parameter represents 119862119877 Δ120591119894119898119899(119905) is the quantity of the

pheromone trail of ant 119894

Δ120591119894

119898119899(119905)

=

1 if 119894 isin 119883119898119899

and fitness (119910119905119894) lt fitness (119909best119905)

05 if 119894 isin 119883119898119899

and fitness (119909best119905) lt fitness (119910119905119894)

and fitness (119910119905119894) lt fitness (119909119905

119894)

0 otherwise(11)

where 119883119898119899

is the ant group that chooses 119899th value as theselection of119898th parameter119909best119905 denotes the best individualof ant colony till 119905th generation

In order to prevent the ants from being limited to oneant path and improve the possibility of choosing other paths

considerably the probability of each ant chooses 119899th value of119898th parameter (119865 and 119862

119877) in Figure 4 is set by

119901119898119899(119905) =

120591119894

119898119899(119905)

sum119899120591119898119899(119905)

if rand1lt 119875119904

rand2

otherwise(12)

Figure 4 illustrates the relationship between pheromonematrix and ant path of 119865 and 119862

119877 where 119875

119904is a constant

which is defined as selection parameter and rand1and rand

2

are two random values which are uniformly distributed in[0 1] Selection of the values of 119865 and 119862

119877depends on the

pheromone of each path According to the performance ofall the individuals the individual is chosen by the mostappropriate combination of 119865 and 119862

119877in each generation

In subpopulation 2 the individual is set as ant at eachgeneration Mutation strategies which are listed at (3)ndash(7) are


Mutation strategy

Mutation strategy

DErand1 DErand2 DEbest1 DEbest2 DErand-to-best1

02 04 06 08 1

1205761 1205762 1205763 1205764 1205765

04

06

02

10

Figure 5 Relationship between and ant paths of mutation strategy

defined to be of the values 02 04 06 08 10 respectivelyFor example 02 means the first mutation strategy equation(3) is selected Each of the ants chooses a mutation strategyaccording to the informationwhich is calculated by the fitnessfunction of ants During search process the informationgathered by the ants is preserved in the pheromone trails 120576By exchanging information according to pheromone the antscooperate with each other to choose appropriate mutationstrategy Then ant colony renews the pheromone trails of allants

Then the pheromone trail 120576 is updated in the followingequation

120576119896(119905 + 1) = (1 minus 120588

2) 120576119896(119905) +

subpopulation2

sum

119894=1

Δ120576119894

119896(119905) (13)


rate and Δ120576119894119896(119905) is the quantity of the pheromone trail of ant 119894

120576119894

119896(119905)

=

1 if 119894 isin 119883119896and fitness (119910119905

119894) lt fitness (119909best119905)

05 if 119894 isin 119883119896and fitness (119909best119905) lt fitness (119910119905

119894)


119894)

0 otherwise(14)

where 119883119896is the ant group that chooses 119896th value as the

selection of parameter 119909best119905 denotes the best individual ofant colony till 119905th generation


considerably the probability of each ant choosing 119899th valueof119898th parameter (mutation strategies) is set by

119901119896(119905) =

120576119894

119896(119905)

sum119899120576119896(119905)

if rand3lt 1198751015840

119904

rand4

otherwise(15)

where1198751015840119904is a constant which is defined as selection parameter

and rand3and rand

4are two random values which are

uniformly distributed in [0 1] Selection of the values ofmutation strategies depends on the pheromone of each pathAccording to the performance of all the individuals theindividual is chosen by the most appropriate combination ofmutation strategies in each generation

Figure 5 illustrates the relationship between pheromonematrix and ant path of mutation strategies

322 Belief Space In our approach the belief space isdivided into two knowledge sources situational knowledgeand normative knowledge

Situational knowledge consists of the global best exem-plar 119864 which is found along the searching process andprovides guidance for individuals of population space Theupdate of the situational knowledge is done if the bestindividual found in the current populations space is betterthan 119864

The normative knowledge contains the intervals thatdecide the individuals of population space where to move 119897

119894

and 119906119894are the lower and upper bounds of the search range

in population space 119871119894and 119880

119894are the value of the fitness

function associated with that bound If the 119897119894and 119906

119894are

updated the 119871119894and 119880

119894must be updated too


119897119894and 119906

119894are set by

119897119894= 119909119894min if 119909

119894min lt 119897119894 or 119891 (119909119894min) lt 119871 119894119897119894

otherwise

119906119894= 119909119894max if 119909

119894max gt 119906119894 or 119891 (119909119894max) gt 119880119894119906119894

otherwise

(16)

323 Acceptance Function Acceptance function controls theamount of good individuals which impact on the update ofbelief space [19] In this paper 30 of the individuals inthe belief space are replaced by the good ones in populationspace

324 Influence Function In the CDEAS situational knowl-edge and normative knowledge are involved to influence eachindividual in the population space and then population spaceis updated

The individuals in population space are updated in thefollowing equation

119909119905+1

119894119889=

119909119905

119894119889+10038161003816100381610038161003816119873 (05 03) lowast (119883

119905

1199032119894119889minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889le 119864119895 119909119905

119894119889ge 119906119894

119909119905

119894119889minus10038161003816100381610038161003816119873 (05 03) lowast (119883

119905

1199032119894119889minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889gt 119864119895 119909119905

119894119889lt 119906119894

119883119905

1199031119894119889+10038161003816100381610038161003816119873 (05 03) lowast (119906

119894minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889le 119864119895 119909119905

119894119889ge 119897119894

119883119905

1199031119894119889minus10038161003816100381610038161003816119873 (05 03) lowast (119897

119894minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889gt 119864119895 119909119905

119894119889gt 119897119894

119909119905+1

119894119889=

119883119905

1199031119894119889+ 119865 lowast (119906

119894minus 119883119905

1199031119894119889) times rand if 119909

119894119889gt 119897119894

119883119905

1199031119894119889minus 119865 lowast (119883

119905

1199031119894119889minus 119897119894) times rand if 119909

119894119889lt 119906119894

119883119905

1199031119894119889+ 119865 lowast (119906


119894lt 119909119894119889lt 119906119894

(17)where 119865 is a constant of 02

325 Knowledge Exchange After 119905 steps the 119865 and 119862119877

of subpopulation 2 are replaced by the suitable 119865 and 119862119877

calculated by subpopulation 1 and the mutation strategyof subpopulation 1 is displaced by the suitable mutationstrategy calculated by subpopulation 2 simultaneously Sothe 119865 and 119862

119877and mutation strategy are varying in the two

subpopulations to enable the individuals to converge globallyand fast

326 Procedure of CDEAS The procedure of CDEAS isproposed as follows

Step 1 Initialize the population spaces and the belief spacesthe population space is divided into subpopulation 1 andsubpopulation 2

Step 2 Evaluate each individualrsquos fitness

Step 3 To find the proper 119865 119862119877 and mutation strategy the

Ant Colony Search strategy is used in subpopulation 1 andsubpopulation 2 respectively

Step 4 According to acceptance function choose good indi-viduals from subpopulation 1 and subpopulation 2 and thenupdate the normative knowledge and situational knowledge

Step 5 Adopt the normative knowledge and situationalknowledge to influence each individual in population spacethrough the influence functions and generate two corre-sponding subpopulations

Step 6 Select individuals from subpopulation 1 and subpop-ulation 2 and update the belief spaces including the twoknowledge sources for the next generation

Step 7 If the algorithm reaches the given times exchangethe knowledge of 119865 119862

119877 and mutation strategy between

subpopulation 1 and subpopulation 2 otherwise go to Step 8

Step 8 If the stop criteria are achieved terminate the itera-tion otherwise go back to Step 2

33 Simulation Results of CDEAS The proposed CDEASalgorithm is compared with original DE algorithm To getthe average performance of the CDEAS algorithm 30 runson each problem instance were performed and the solutionquality was averagedThe parameters of CDEAS and originalDE algorithm are set as follows the maximum evolutiongeneration is 2000 the size of the population is 50 for originalDE algorithm 119865 = 03 and 119862

119877= 05 for CDEAS the size

of both two subpopulations is 25 the initial 119865 and 119862119877are

randomly selected in (0 1) and the initial mutation strategyis DErand1 the interval information exchanges between thetwo subpopulations 119905 is 50 generations the thresholds 119875

119904=

1198751015840

119904= 05 and 120588

1= 1205882= 01

To illustrate the effectiveness and performance of CDEASalgorithm for optimization problems a set of 18 representa-tive benchmark functions which were listed in the appendixwere employed to evaluate them in comparison with originalDE The test problems are heterogeneous nonlinear andnumerical benchmark functions and the global optimum for1198912 1198914 1198917 1198919 11989111 11989113 and 119891

15is shifted Functions 119891

1sim1198917are

unimodal and functions1198918sim11989118are multimodalThe detailed

principle of functions is presented in [11] The comparisonsresults of CDEAS and original DE algorithm are shown inTable 4 of the appendix The experimental results of originalDE and CDEAS algorithm on each function are listed inTable 1 Mean best worst std success rate time representthe mean minimum best minimum worst minimum thestandard deviation of minimum the success rate and theaverage computing time in 30 trials respectively

From simulation results of Table 1 we can obtain thatCDEAS reached the global optimum of 119891

2and 1198917in all trials

and the success rate reached 100 of functions 1198911 1198912 1198913 1198914

1198916 1198917 and 119891

18 For most of the test functions the success


Table 1 The comparison results of the CDEAS algorithm and original DE algorithm

Original DE CDEASSphere function 119891

1

Best 11746 times 10minus65 50147 times 10minus79

Worst 10815 times 10minus23 93244 times 10minus75

Mean 36052 times 10minus25 16390 times 10minus75

Std 19746 times 10minus24 22315 times 10minus75

Success rate () 100 100Times (s) 18803 146017

Shifted sphere function 1198912

Best 0 0Worst 80779 times 10

minus28 0Mean 33658 times 10

minus29 0Std 15078 times 10

minus28 0Success rate () 100 100Times (s) 21788 181117

Schwefelrsquos Problem 12 1198913






Shifted Schwefelrsquos Problem 12 1198914


minus27 34331 times 10minus27




Rosenbrockrsquos function 1198915

Best 130060 52659Worst 1661159 1391358Mean 709399 394936Std 400052 312897Success rate () 8667 9667Times (s) 19594 167233

Schwefelrsquos Problem 12 with noise in fitness 1198916






Shifted Schwefelrsquos Problem 12 with noise in fitness 1198917

Best 0 0Worst 0 0Mean 0 0


Table 1 Continued

Original DE CDEASStd 0 0Success rate () 100 100Times (s) 33374 285638

Ackleyrsquos function 1198918

Best 71054 times 10minus15

35527 times 10minus15

Worst 48999 times 10minus7 13404Mean 16332 times 10minus8 01763Std 89457 times 10minus8 04068Success rate () 100 8333Times (s) 24820 209353

Shifted Ackleyrsquos function 1198919


Worst 09313 09313Mean 00310 00620Std 01700 02362Success rate () 9667 9333Times (s) 27337 216841

Griewankrsquos function 11989110


Shifted Griewankrsquos function 11989111


Rastriginrsquos function 11989112


Shifted Rastriginrsquos function 11989113


Noncontiguous Rastriginrsquos function 11989114

Best 207617 39949Worst 299112 119899Mean 254556 81947


Table 1 Continued


Shifted noncontiguous Rastriginrsquos function 11989115


Schwefelrsquos function 11989116










rate of CDEAS is higher in comparison with original DEMoreover CDEAS gets very close to the global optimum insome other functions 119891

1 1198913 1198914 1198916 and 119891

18 It also presents

that the mean minimum best minimum worst minimumthe standard deviation of minimum and the success rate ofCDEAS algorithm are clearly better than the original DE forfunctions 119891

1 1198913 1198914 1198915 1198916 11989112 11989113 11989114 11989115 and 119891

18although

the computing time of CDEAS is longer than that of originalDE because of its complexity

The convergence figures of CDEAS comparing withoriginal DE for 18 instances are listed as Figure 6

From Figure 6 one can observe that the convergencespeed of CDEAS is faster than original DE for 119891

1 1198912 1198913 1198914

1198916 1198917 11989111 11989112 11989113 11989114 11989115 and 119891

18

All these comparisons of CDEAS with original DE algo-rithm have shown that CDEAS is a competitive algorithmto solve all the unimodal function problems and most ofthe multimodal function optimization problems listed aboveAs shown in the descriptions and all the illustrations beforeCDEAS is efficacious on those typical function optimizations

4 Model of Net Value of Ammonia UsingCDEAS-LS-SVM

41 Auxiliary Variables Selection of theModel There are someprocess variables which have the greatest influence on the netvalue of ammina such as system pressure recycle gas flowrate feed composition (HN ratio) ammonia and inert gascencetration in the gas of reactor inlet hot spot temperaturesand so forth The relations between the process variablesare coupling and the operational variables interact with eachother

The inlet ammonia concentration is an important processvariable which is beneficial to operation-optimization but thedevice of online catharometer is very expensive Accordingto the mechanism and soft sensor model a IIO-BP modelwas built to get the more accurate value of the inlet ammoniaconcentration [20]

Δ (NH3) = 119860NH3OUT minus 119860NH3IN (18)


0 400 800 1200 1600 2000Evolution generation

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)

Convergence figure of originalDE and CDEAS for f1


log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5



0

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

10

log(fi

tnes

s val

ue)



10

log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5


123456789

1011


log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus35

minus40

minus45

minus25

minus20

minus15

minus10

minus5

0

5


10

Original DECDEAS


log(fi

tnes

s val

ue)

minus30

minus35

minus25

minus20

minus15

minus10

minus5

0

5

Convergence fgure of originalDE and CDEAS for f7

Original DECDEAS


minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2

log(fi

tnes

s val

ue)


Figure 6 Continued





081

12141618

222242628


081

12141618

222242628


081

12141618

222242628


0

05

1

15

2

25

3


Original DECDEAS

22242628

332343638

442


Original DECDEAS

02

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2 024

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

1

2

3

minus3

minus2

minus1









log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

Figure 6 Continued


0

05

1

15

2

Original DECDEAS


Original DECDEAS


minus50 minus05

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)



Figure 6 Convergence figure of CDEAS comparing with original DE for 1198911sim11989118

Table 2 Auxiliary variables of model of net value of Ammonia

List Symbols Name Unit1 119877HN HN ratio 2 119860CH4 Methane concentration in recycled synthesis gas at the reactor inlet Mole ratio3 ANH3 Ammonia concentration in recycled synthesis gas at the reactor inlet Mole ratio4 119875

119878System pressure Mpa

5 119865119876

Recycle gas flow rate Nm3h6 119865

1198761Quench gas flows of axial layer Nm3h

7 1198651199022

Cold quench gas flows of 1st radial layers Nm3h8 119865

1199023Quench gas flows of 2nd radial layers Nm3h

9 1198651199024

Hot quench gas flows of 1st radial layers Nm3h10 119879

119860Hot-spot temperatures of axial bed ∘C

11 1198791198771 Hot-spot temperatures of radial bed I ∘C

12 1198791198772 Hot-spot temperatures of radial bed II ∘C

13 119879EO Outlet gas temperature of evaporator ∘C

From the analysis discussed above some important vari-ables have significant effects on the net value of ammoniaBy discussion with experienced engineers and taking intoconsideration a priori knowledge about the process thesystem pressure recycle gas flow rate the HN ratio hot-spottemperatures in the catalyst bed and ammonia and methaneconcentration in the recycle gas are identified as the keyauxiliary variables to model net value of ammonia which islisted in Table 2

42 Modeling the Net Value of Ammonia Using CDEAS-LS-SVM LS-SVM is an alternate formulation of SVMwhich is proposed by Suykens The e-insensitive loss func-tion is replaced by a squared loss function which con-structs the Lagrange function by solving the problem linearKarush Kuhn Tucker (KKT)

[0 119868

119879

119899

119868119899119870 + 120574

minus1119868] [1198870

119887] = [

0

119910] (19)

where 119868119899is a [119899 times 1] vector of ones 119879 is the transpose of a

matrix or vector 120574 is a weight vector 1198870means the model

offset and 119887 is regression vector119870 is Mercer kernel matrix which is defined as

119870 = (

11989611

sdot sdot sdot 1198961119899

d

1198961198991

sdot sdot sdot 119896119899119899

) (20)

where 119896119894119895is defined by kernel function

There are several kinds of kernel functions such ashyperbolic tangent polynomial and Gaussian radial basisfunction (RBF) which are commonly used Literatures haveproved that RBF kernel function has strong generalizationso in this study RBF kernel was used

119896119894119895= 119890minus|119909119894minus119909119895|

221205902

(21)

where 119909119894and 119909

119895indicated different training samples 120590 is the

kernel width parameter


Table 3 The comparisons of training error and testing error of LS-SVM

Method Type of error RElowast MAElowast MSElowast

BP-NN Training error 94422 times 10minus04 10544 times 10minus04 13970 times 10minus04

Testing error 0008085 89666 times 10minus04

0001188

LS-SVM Training error 0002231 24785 times 10minus04




DE-LS-SVM Training error 0002739 304286 times 10minus04




CDEAS-LS-SVM Training error 0002830 31415 times 10minus04


Testing error 0004661 51752 times 10minus04 68952 times 10minus04lowastRE relative error MAE mean absolute error MSE mean square error

As we can see from (19)sim(21) only two parameters(120574 120590) are needed for LS-SVM It makes LS-SVM problemcomputationally easier than SVR problem

Grid search is a commonly used method to select theparameters of LS-SVM but it is time-consuming and inef-ficient CDEAS algorithm has strong search capabilities andthe algorithm is simple and easy to implementTherefore thispaper proposes the CDEAS algorithm to calculate the bestparameters (120574 120590) of LS-SVM

5 Results and Discussion

Operational parameters such as 119860H2 119860CH4 and 119875119878 werecollected and acquired from plant DCS from the year 2011-2012 In addition data on the inlet ammonia concentrationof recycle gas 119860NH3 were simulated by mechanism and softsensor model [20]

The extreme values are eliminated from the data using the3120590 criterion After the smoothing and normalization eachdata group is divided into 2 parts 223 groups of trainingsamples which are used to train model while 90 groups oftesting samples which are valuing the generalization of themodel for identifying the parameters of the LS-SVM thekernel width parameter and the weight vector

BP-NN LS-SVM and DE-LS-SVM are also used tomodel the net value of ammonia respectively BP-NN isa 13-15-1 three-layer network with back-propagation algo-rithm LS-SVM gains the (120574 120590) with grid-search and cross-validation The parameter settings of CDEAS-LS-SVM arethe same as those in the benchmark tests Each model is run30 times and the best value is shown in Table 3 Descriptivestatistics of training results and testing results of modelinclude the relative error absolute error and mean squareerror The performance of the four models is compared asshown in Table 3 The training and testing results of fourmodels are illustrated in Figure 7

Despite the fact that the training error using BP-NN issmaller than that using CDEAS-LS-SVM which is becauseBP-NN is overfitting to the training data the mean squareerror (MSE) on training data using CDEAS-LS-SVM isreduced by 256 and 232 compared with LS-SVM andDE-LS-SVM respectively In comparison with the othermodels (BP-NN LS-SVM and DE-LS-SVM) testing errorusing CDEAS-LS-SVMmodel is reduced by 141 and 112

respectively The results indicate that the proposed CDEAS-LS-SVM model has a good tracking precision performanceand guides production better

6 Conclusion

In this paper an optimizing model which describes therelationship between net value of ammonia and key opera-tional parameters in ammonia synthesis has been proposedSome representative benchmark functions were employed toevaluate the performance of a novel algorithm CDEAS Theobtained results show that CDEAS algorithm is efficaciousfor solving most of the optimization problems comparisonswith original DE Least squares support vector machineis used to build the model while CDEAS algorithm isemployed to identify the parameters of LS-SVM The sim-ulation results indicated that CDEAS-LS-SVM is superiorto other models (BP-NN LS-SVM and DE-LS-SVM) andmeets the requirements of ammonia synthesis process TheCDEAS-LS-SVM optimizing model makes it a promisingcandidate for obtaining the optimal operational parametersof ammonia synthesis process and meets the maximumbenefit of ammonia synthesis production

Appendix

(1) Sphere function

1198911(119909) =

119863

sum

119894=1

1199092

119894

119900 = [0 0 0] the global optimum

(A1)

(2) Shifted sphere function

1198912(119909) =

119863

sum

119894=1

1199112

119894

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579

119863] the shifted global optimum

(A2)


0 50 100 150 200 2500106010701080109

011011101120113011401150116

Sample number

Net

val

ue o

f am

mon

ia (

)

0 10 20 30 40 50 60 70 80 9001050106010701080109

01101110112011301140115

Sample number

Net

val

ue o

f am

mon

ia (

)

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

Net

val

ue o

f am

mon

ia (

)

Sample number0 50 100 150 200 250

0106010701080109

011011101120113011401150116

Actual valuesTraining results of CDEAS-LS-SVM

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115


Figure 7 The analyzed results training results and testing results of BP-NN LS-SVM DE-LS-SVM and CDEAS-LS-SVM


Table 4 Global optimum search ranges and initialization ranges of the test functions

119891 Dimension Global optimum 997888119909 119891(997888119909) Search range Target

1198911

30

0 0 [minus100 100]119863

10minus5

1198912

120579 0 [minus100 100]119863

10minus5

1198913

0 0 [minus100 100]119863

10minus5

1198914

120579 0 [minus100 100]119863

10minus5

1198915

1 1 [minus100 100]119863 100

1198916

0 0 [minus32 32]119863

10minus5

1198917

120579 0 [minus32 32]119863

10minus5

1198918

0 0 [minus32 32]119863

10minus5

1198919

120579 0 [minus32 32]119863 01

11989110

0 0 [0 600]119863 0001

11989111

120579 0 [minus600 600]119863 001

11989112

0 0 [minus5 5]119863 10

11989113

120579 0 [minus5 5]119863 10

11989114

0 0 [minus5 5]119863 10

11989115

120579 0 [minus5 5]119863 5

11989116

4189829 0 [minus500 500]119863 500

11989117

0 0 [minus100 100]119863 1

11989118

0 0 [minus10 10]119863

10minus5

120579 is the shifted vector

(3) Schwefelrsquos Problem 12

1198913(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2


(A3)

(4) Shifted Schwefelrsquos Problem 12

1198914(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A4)

(5) Rosenbrockrsquos function

1198915(119909) =

119863minus1

sum

119894=1

(100(1199092

119894minus 1199092

119894+1)2

+ (119909119894minus 1)2

)


(A5)

(6) Schwefelrsquos Problem 12 with noise in fitness

1198916(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2

) lowast (1 + 04 |119873 (0 1)|)


(A6)

(7) Shifted Schwefelrsquos Problem 12 with noise in fitness

1198917(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

) lowast (1 + 04 |119873 (0 1)|)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A7)

(8) Ackleyrsquos function

1198918(119909) = minus 20 exp(minus02radic 1

119863

119863

sum

119894=1

1199092

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119909119894)) + 20 + 119890


(A8)

(9) Shifted Ackleyrsquos function


119863

119863

sum

119894=1

1199112

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119911119894)) + 20 + 119890

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A9)


(10) Griewankrsquos function

11989110(119909) =

119863

sum

119894=1

1199092

119894

4000minus

119863

prod

119894=1

cos119909119894

radic119894

+ 1


(A10)

(11) Shifted Griewankrsquos function

11989111(119909) =

119863

sum

119894=1

1199112

119894

4000minus

119863

prod

119894=1

cos119911119894

radic119894

+ 1

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A11)

(12) Rastriginrsquos function

11989112(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10)


(A12)

(13) Shifted Rastriginrsquos function

11989113(119909) =

119863

sum

119894=1

(1199112

119894minus 10 cos (2120587119911

119894) + 10)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A13)

(14) Noncontiguous Rastriginrsquos function

11989114(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt1

2round (2119909

119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863

119900 = [0 0 0] the global optimum(A14)

(15) Shifted noncontiguous Rastriginrsquos function

11989115(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119911119894

10038161003816100381610038161199111198941003816100381610038161003816 lt1

2round (2119911

119894)

2

10038161003816100381610038161199111198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A15)

(16) Schwefelrsquos function

11989116(119909) = 4189829 times 119863 minus

119863

sum

119894=1

119909119894sin (1003816100381610038161003816119909119894

1003816100381610038161003816

12

)

119900=[4189829 4189829 4189829] the global optimum(A16)


11989118(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863



11989117(119909) =

119863

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119863

prod

119894=1

119909119894


(A18)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers forgiving us helpful suggestions This work is supported byNational Natural Science Foundation of China (Grant nos61174040 and 61104178) and Fundamental Research Funds forthe Central Universities Shanghai Commission of Scienceand Technology (Grant no 12JC1403400)

References

[1] M T Sadeghi and A Kavianiboroujeni ldquoThe optimizationof an ammonia synthesis reactor using genetic algorithmrdquoInternational Journal of Chemical Reactor Engineering vol 6 no1 article A113 2009

[2] S S E H Elnashaie A T Mahfouz and S S ElshishinildquoDigital simulation of an industrial ammonia reactorrdquoChemicalEngineering and Processing vol 23 no 3 pp 165ndash177 1988

[3] M N Pedemera D O Borio and N S Schbib ldquoSteady-Stateanalysis and optimization of a radial-flow ammonia synthesisreactorrdquo Computers amp Chemical Engineering vol 23 no 1 ppS783ndashS786 1999

[4] B V Babu and R Angira ldquoOptimal design of an auto-thermalammonia synthesis reactorrdquo Computers amp Chemical Engineer-ing vol 29 no 5 pp 1041ndash1045 2005

[5] W F Sacco and N Hendersonb ldquoDifferential evolution withtopographical mutation applied to nuclear reactor core designrdquoProgress in Nuclear Energy vol 70 pp 140ndash148 2014

[6] M Rout B Majhi R Majhi and G Panda ldquoForecasting ofcurrency exchange rates using an adaptive ARMA model withdifferential evolution based trainingrdquo Journal of King SaudUniversity vol 26 no 1 pp 7ndash18 2014

[7] H Ozcan K Ozdemir and H Ciloglu ldquoOptimum cost of anair cooling system by using differential evolution and particleswarm algorithmsrdquo Energy and Buildings vol 65 pp 93ndash1002013


[8] R Zhang S Song and C Wu ldquoA hybrid differential evolutionalgorithm for job shop scheduling problems with expected totaltardiness criterionrdquo Applied Soft Computing Journal vol 13 no3 pp 1448ndash1458 2013

[9] R Arya and S C Choube ldquoDifferential evolution based tech-nique for reliability design of meshed electrical distributionsystemsrdquo International Journal of Electrical Power amp EnergySystems vol 48 pp 10ndash20 2013

[10] W Xu L Zhang and X Gu ldquoSoft sensor for ammoniaconcentration at the ammonia converter outlet based on animproved particle swarm optimization and BP neural networkrdquoChemical Engineering Research and Design vol 89 no 10 pp2102ndash2109 2011

[11] M R Sawant K V Patwardhan A W Patwardhan V GGaikar and M Bhaskaran ldquoOptimization of primary enrich-ment section of mono-thermal ammonia-hydrogen chemicalexchange processrdquo Chemical Engineering Journal vol 142 no3 pp 285ndash300 2008

[12] Z Kirova-Yordanova ldquoExergy analysis of industrial ammoniasynthesisrdquo Energy vol 29 no 12ndash15 pp 2373ndash2384 2004

[13] B Mansson and B Andresen ldquoOptimal temperature profilefor an ammonia reactorrdquo Industrial amp Engineering ChemistryProcess Design and Development vol 25 no 1 pp 59ndash65 1986

[14] R Mallipeddi P N Suganthan Q K Pan andM F TasgetirenldquoDifferential evolution algorithm with ensemble of parametersand mutation strategiesrdquo Applied Soft Computing Journal vol11 no 2 pp 1679ndash1696 2011

[15] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[16] C Hu and X Yan ldquoA hybrid differential evolution algorithmintegrated with an ant system and its applicationrdquo Computers ampMathematics with Applications vol 62 no 1 pp 32ndash43 2011

[17] K Vaisakh and L R Srinivas ldquoGenetic evolving ant directionHDE for OPF with non-smooth cost functions and statisticalanalysisrdquo Expert Systems with Applications vol 38 no 3 pp2046ndash2062 2011

[18] R Wang J Zhang Y Zhang and X Wang ldquoAssessment ofhuman operator functional state using a novel differentialevolution optimization based adaptive fuzzymodelrdquoBiomedicalSignal Processing and Control vol 7 no 5 pp 490ndash498 2012

[19] W Xu L Zhang and X Gu ldquoA novel cultural algorithm and itsapplication to the constrained optimization in ammonia syn-thesisrdquo Communications in Computer and Information Sciencevol 98 no 2 pp 52ndash58 2010

[20] Z Liu and X Gu ldquoSoft-sensor modelling of inlet ammoniacontent of synthetic tower based on integrated intelligentoptimizationrdquo Huagong Xuebao vol 61 no 8 pp 2051ndash20552010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


specific internal mechanism because a lot of parameters areunknown in real industrial process

In order to achieve the required accuracy of the modelsome researches focus on the novel modeling methodscombining some heuristic methods such as ANN (ArtificialNeural Network) LS-SVM (Least Squares Support VectorMachine) with Evolutionary Algorithm for example geneticalgorithm ant colony optimization (ACO) particle swarmoptimization (PSO) differential evolution (DE) and so forthDE is one of themost popular algorithms for this problemandhas been applied in many fields Sacco and Hendersonb [5]introduced a variant of the differential evolution algorithmwith a new mutation operator based on a topographicalheuristic and used it to solve the nuclear reactor core designoptimization problem Rout et al [6] proposed a simple butpromising hybrid prediction model by suitably combiningan adaptive autoregressive moving average architecture anddifferential evolution for forecasting of exchange rates Ozcanet al [7] carried out the cost optimization of an air coolingsystem by using Lagrange multipliers method differentialevolution algorithm and particle swarm optimization forvarious temperatures andmass flow ratesThe results showedthat the method gives high accuracy results within a shorttime interval Zhang et al [8] proposed a hybrid differentialevolution algorithm for the job shop scheduling problemwithrandom processing times under the objective of minimizingthe expected total tardiness Arya and Choube [9] describeda methodology for allocating repair time and failure rates tosegments of a meshed distribution system using differentialevolution technique Xu et al [10] proposed a model ofammonia conversion rate by LS-SVM and a hybrid algorithmof PSO and DE is described to identify the hyper-parametersof LS-SVM

To describe the relationship between net value of ammo-nia in ammonia synthesis reactor and the key operationalparameters least squares support vectormachine is employedto build the structure of the relationship model in which anovel algorithm called CDEAS is proposed to identify theparametersThe experiment results showed that the proposedCDEAS-LS-SVM optimizing model is very effective of beingused to obtain the optimal operational parameters of ammo-nia synthesis converter

The remaining of the paper is organized as followsSection 2 describes the ammonia synthesis production pro-cess Section 3 proposes a novel Cultural Differential Evolu-tion with Ant Colony Search (CDEAS) algorithm Section 4constructs a model using LS-SVM based on the proposedCDEAS algorithm Section 5 presents the experiments andcomputational results and discussion Finally Section 6 sum-marizes the above results and presents several problemswhich remain to be solved

2 Ammonia Synthesis Production Process

A normal ammonia production flow chart includes thesynthesis gas production purification gas compression andammonia synthesis Ammonia synthesis loop is one of themost critical units in the entire process The system has

been realized by LuHua Inc a medium fertilizers factory ofYanKuang Group China

Figure 1 represents a flow sheet for the ammonia syn-thesis process The ammonia synthesis reactor is a one-axialflow and two-radial flow three-bed quench-type unit [11]Hydrogen-nitrogen mixture is reacted in the catalyst bedunder high temperature and pressure The temperature inthe reactor is sustained by the heat of reaction because thereaction is exothermic [1]The reaction of ammonia synthesisprocess contains

3

2H2+1

2N2999445999468 NH

3+Q (1)

The reaction is limited by the unfavorable position ofthe chemical equilibrium and by the low activity of thepromoted iron catalysts with high pressure and temperature[12] In general no more than 20 of the synthesis gas isconverted into ammonia per pass even at high pressure of30MPa [12] As the ammonia reaction is exothermic it isnecessary for removing the heat generated in the catalyst bedby the progress of the reaction to obtain a reasonable overallconversion rate as same as to protect the life of the catalyst[13] The mixture gas from the condenser is divided into twoparts Q1 and Q2 to go to the converter The first cold shotQ1 is recirculated to the annular space between the outershell reactor and catalyst bed from the top to the bottomto refrigerate the shell and remove the heat released by thereaction Then the gas Q1 from the bottom of reactor goesthrough the preheater and is heated by the counter-currentflowing reacted gas from waste heat boiler Q1 gas is dividedinto 4 cold quench gas (q1 q2 q3 and q4) and Q2 gas formixing with the gas between consecutive catalyst beds toquench the hot spots before entry to the subsequent catalystbeds The hot spot temperatures (TIRA705 TIRA712N andTIRA714) represent the highest reaction temperatures at eachstage of the catalyst bed

Figure 2 represents the ammonia synthesis unit Thereacted gas including N

2 H2 NH3 and inert gas after reactor

passes through the waste heat boiler Then it goes throughthe preheater and the water cooler to be further cooled Partof the ammonia is condensed and separated by ammoniaseparator I Inert gas from the ammonia synthesis loop areejected by purge gas from separator to prevent accumulationof inert gas in the system The fresh feed gas is producedby the Texaco coal gasification air separation section aprocess that converts the Coal Water Slurry into synthesisgas for ammonia The fresh gas consists of hydrogen andnitrogen in stoichiometric proportions of 3 1 approximatelyand mixes with small amounts of argon and methane Thefresh gas which passes compressor is compounded with therecycle gas which comes from the circulator and then themixture goes through oil separator and condenser Mixturegas is further cooled by liquid ammonia and goes throughammonia separator II to separate the partial liquid ammoniaand then it goes out with very few ammonia The liquidammonia from ammonia separator I and separator II flowsto the liquid ammonia jar Mixture is heated in ammoniacondenser to about 25∘C and flows to the reactor and thewhole cycle starts again


Ammoniareactor



separator IOil

separator

Condenser

Ammonia cooler

Synthesis gas

Evaporator




TIRA705

TIR712N

TIRA714

AR701

AR701-4

FIR705 703

PI

725TI

Liquid ammonia

Compressor


Preheater

Q2

Preheaterq1

q4

q3

q2

Waste heat boiler

Q1

Q 1

I radial bed

II radial bed

Inter-changer

Axial bedFIR704

703

702

FIR705

FIR

FIR






1199090

119894119889= 1199090

min119889 + rand (0 1) (1199090

max119889 minus 1199090

min119889) 119895 = 1 2 119863

(2)





vector 119906119905119894119889

of individual 119909119905119894119889


DErand1 119906119905

119894119889= 119883119905

1199031119894119889+ 119865 (119883

119905

1199032119894119889minus 119883119905

1199033119894119889) (3)

DErand2 119906119905

119894119889= 119883119905

1199031119894119889+ 119865 (119883

119905

1199032119894119889minus 119883119905

1199033119894119889)

+ 119865 (119883119905

1199034119894119889minus 119883119905

1199035119894119889)

(4)

DEbest1 119906119905

119894119889= 119883119905

best119889 + 119865 (119883119905

1199031119894119889minus 119883119905

1199032119894119889) (5)

DEbest2 119906119905

119894119889= 119883119905

best119889 + 119865 (119883119905

1199031119894119889minus 119883119905

1199032119894119889)

+ 119865 (119883119905

1199033119894119889minus 119883119905

1199034119894119889)

(6)


119894119889= 119883119905

119894119889+ 119865 (119883

119905

best119889 minus 119883119905

119894119889)

+ 119865 (119883119905

1199031119894119889minus 119883119905

1199032119894119889)

(7)



119894are



119910119905

119894119889= 119906119905

119894119889if rand le 119862

119877or 119894 = 119894rand

119883119905


(8)





119894) by




119894 and


119883119905+1

119894119889= 119910119905

119894119889if 119891 (119910119905

119894119889) le 119891 (119883

119905

119894119889)

119883119905


(9)











119877 and mutation




119877 and mutation











119877



equation

120591119898119899(119905 + 1) = (1 minus 120588

1) 120591119898119899(119905) +

subpopulation1

sum

119894=1

Δ120591119894

119898119899(119905)

(10)





Belief space

Influencefunction

Acceptancefunction



Knowledge exchange

Population space




01 02 03 10

01 02 03 10

12059111 12059112 12059113 120591110

F

F

CR

CR

12059121 12059122 12059123 120591210

01

10

02

03

01

10

02

03








Δ120591119894

119898119899(119905)

=

1 if 119894 isin 119883119898119899


05 if 119894 isin 119883119898119899



119894)

0 otherwise(11)

where 119883119898119899





119901119898119899(119905) =

120591119894

119898119899(119905)

sum119899120591119898119899(119905)

if rand1lt 119875119904

rand2

otherwise(12)


119877 where 119875

119904is a constant


2







Mutation strategy

Mutation strategy


02 04 06 08 1

1205761 1205762 1205763 1205764 1205765

04

06

02

10




120576119896(119905 + 1) = (1 minus 120588

2) 120576119896(119905) +

subpopulation2

sum

119894=1

Δ120576119894

119896(119905) (13)



120576119894

119896(119905)

=


119894) lt fitness (119909best119905)


119894)


119894)

0 otherwise(14)





119901119896(119905) =

120576119894

119896(119905)

sum119899120576119896(119905)

if rand3lt 1198751015840

119904

rand4

otherwise(15)


and rand3and rand







119894





119894are

updated the 119871119894and 119880



119897119894and 119906

119894are set by

119897119894= 119909119894min if 119909

119894min lt 119897119894 or 119891 (119909119894min) lt 119871 119894119897119894

otherwise

119906119894= 119909119894max if 119909

119894max gt 119906119894 or 119891 (119909119894max) gt 119880119894119906119894

otherwise

(16)




119909119905+1

119894119889=

119909119905

119894119889+10038161003816100381610038161003816119873 (05 03) lowast (119883

119905

1199032119894119889minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889le 119864119895 119909119905

119894119889ge 119906119894

119909119905

119894119889minus10038161003816100381610038161003816119873 (05 03) lowast (119883

119905

1199032119894119889minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889gt 119864119895 119909119905

119894119889lt 119906119894

119883119905

1199031119894119889+10038161003816100381610038161003816119873 (05 03) lowast (119906

119894minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889le 119864119895 119909119905

119894119889ge 119897119894

119883119905

1199031119894119889minus10038161003816100381610038161003816119873 (05 03) lowast (119897

119894minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889gt 119864119895 119909119905

119894119889gt 119897119894

119909119905+1

119894119889=

119883119905

1199031119894119889+ 119865 lowast (119906

119894minus 119883119905

1199031119894119889) times rand if 119909

119894119889gt 119897119894

119883119905

1199031119894119889minus 119865 lowast (119883

119905


119894119889lt 119906119894

119883119905

1199031119894119889+ 119865 lowast (119906


119894lt 119909119894119889lt 119906119894























119904=

1198751015840

119904= 05 and 120588

1= 1205882= 01



1sim1198917are






1198916 1198917 and 119891





1


































Table 1 Continued


















Best 207617 39949Worst 299112 119899Mean 254556 81947


Table 1 Continued















1 1198913 1198914 1198916 and 119891

18 It also presents


1 1198913 1198914 1198915 1198916 11989112 11989113 11989114 11989115 and 119891

18although




1 1198912 1198913 1198914

1198916 1198917 11989111 11989112 11989113 11989114 11989115 and 119891

18








minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5



0

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

10

log(fi

tnes

s val

ue)



10

log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5


123456789

1011


log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus35

minus40

minus45

minus25

minus20

minus15

minus10

minus5

0

5


10

Original DECDEAS


log(fi

tnes

s val

ue)

minus30

minus35

minus25

minus20

minus15

minus10

minus5

0

5


Original DECDEAS


minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2

log(fi

tnes

s val

ue)


Figure 6 Continued





081

12141618

222242628


081

12141618

222242628


081

12141618

222242628


0

05

1

15

2

25

3


Original DECDEAS

22242628

332343638

442


Original DECDEAS

02

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2 024

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

1

2

3

minus3

minus2

minus1









log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

Figure 6 Continued


0

05

1

15

2

Original DECDEAS


Original DECDEAS


minus50 minus05

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)







5 119865119876



7 1198651199022



9 1198651199024








[0 119868

119879

119899

119868119899119870 + 120574

minus1119868] [1198870

119887] = [

0

119910] (19)




119870 = (

11989611


d

1198961198991

sdot sdot sdot 119896119899119899

) (20)



119896119894119895= 119890minus|119909119894minus119909119895|

221205902

(21)

where 119909119894and 119909








0001188




















6 Conclusion


Appendix

(1) Sphere function

1198911(119909) =

119863

sum

119894=1

1199092

119894


(A1)


1198912(119909) =

119863

sum

119894=1

1199112

119894

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A2)


0 50 100 150 200 2500106010701080109

011011101120113011401150116

Sample number

Net

val

ue o

f am

mon

ia (

)

0 10 20 30 40 50 60 70 80 9001050106010701080109

01101110112011301140115

Sample number

Net

val

ue o

f am

mon

ia (

)

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

Net

val

ue o

f am

mon

ia (

)

Sample number0 50 100 150 200 250

0106010701080109

011011101120113011401150116


Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115






1198911

30

0 0 [minus100 100]119863

10minus5

1198912

120579 0 [minus100 100]119863

10minus5

1198913

0 0 [minus100 100]119863

10minus5

1198914

120579 0 [minus100 100]119863

10minus5

1198915

1 1 [minus100 100]119863 100

1198916

0 0 [minus32 32]119863

10minus5

1198917

120579 0 [minus32 32]119863

10minus5

1198918

0 0 [minus32 32]119863

10minus5

1198919

120579 0 [minus32 32]119863 01

11989110

0 0 [0 600]119863 0001

11989111

120579 0 [minus600 600]119863 001

11989112

0 0 [minus5 5]119863 10

11989113

120579 0 [minus5 5]119863 10

11989114

0 0 [minus5 5]119863 10

11989115

120579 0 [minus5 5]119863 5

11989116

4189829 0 [minus500 500]119863 500

11989117

0 0 [minus100 100]119863 1

11989118

0 0 [minus10 10]119863

10minus5



1198913(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2


(A3)


1198914(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A4)


1198915(119909) =

119863minus1

sum

119894=1

(100(1199092

119894minus 1199092

119894+1)2

+ (119909119894minus 1)2

)


(A5)


1198916(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2

) lowast (1 + 04 |119873 (0 1)|)


(A6)


1198917(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

) lowast (1 + 04 |119873 (0 1)|)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A7)



119863

119863

sum

119894=1

1199092

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119909119894)) + 20 + 119890


(A8)



119863

119863

sum

119894=1

1199112

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119911119894)) + 20 + 119890

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A9)



11989110(119909) =

119863

sum

119894=1

1199092

119894

4000minus

119863

prod

119894=1

cos119909119894

radic119894

+ 1


(A10)


11989111(119909) =

119863

sum

119894=1

1199112

119894

4000minus

119863

prod

119894=1

cos119911119894

radic119894

+ 1

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A11)


11989112(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10)


(A12)


11989113(119909) =

119863

sum

119894=1

(1199112

119894minus 10 cos (2120587119911

119894) + 10)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A13)


11989114(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt1

2round (2119909

119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863



11989115(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119911119894

10038161003816100381610038161199111198941003816100381610038161003816 lt1

2round (2119911

119894)

2

10038161003816100381610038161199111198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A15)


11989116(119909) = 4189829 times 119863 minus

119863

sum

119894=1

119909119894sin (1003816100381610038161003816119909119894

1003816100381610038161003816

12

)



11989118(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863



11989117(119909) =

119863

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119863

prod

119894=1

119909119894


(A18)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Ammoniareactor



separator IOil

separator

Condenser

Ammonia cooler

Synthesis gas

Evaporator




TIRA705

TIR712N

TIRA714

AR701

AR701-4

FIR705 703

PI

725TI

Liquid ammonia

Compressor


Preheater

Q2

Preheaterq1

q4

q3

q2

Waste heat boiler

Q1

Q 1

I radial bed

II radial bed

Inter-changer

Axial bedFIR704

703

702

FIR705

FIR

FIR






1199090

119894119889= 1199090

min119889 + rand (0 1) (1199090

max119889 minus 1199090

min119889) 119895 = 1 2 119863

(2)





vector 119906119905119894119889

of individual 119909119905119894119889


DErand1 119906119905

119894119889= 119883119905

1199031119894119889+ 119865 (119883

119905

1199032119894119889minus 119883119905

1199033119894119889) (3)

DErand2 119906119905

119894119889= 119883119905

1199031119894119889+ 119865 (119883

119905

1199032119894119889minus 119883119905

1199033119894119889)

+ 119865 (119883119905

1199034119894119889minus 119883119905

1199035119894119889)

(4)

DEbest1 119906119905

119894119889= 119883119905

best119889 + 119865 (119883119905

1199031119894119889minus 119883119905

1199032119894119889) (5)

DEbest2 119906119905

119894119889= 119883119905

best119889 + 119865 (119883119905

1199031119894119889minus 119883119905

1199032119894119889)

+ 119865 (119883119905

1199033119894119889minus 119883119905

1199034119894119889)

(6)


119894119889= 119883119905

119894119889+ 119865 (119883

119905

best119889 minus 119883119905

119894119889)

+ 119865 (119883119905

1199031119894119889minus 119883119905

1199032119894119889)

(7)



119894are



119910119905

119894119889= 119906119905

119894119889if rand le 119862

119877or 119894 = 119894rand

119883119905


(8)





119894) by




119894 and


119883119905+1

119894119889= 119910119905

119894119889if 119891 (119910119905

119894119889) le 119891 (119883

119905

119894119889)

119883119905


(9)











119877 and mutation




119877 and mutation











119877



equation

120591119898119899(119905 + 1) = (1 minus 120588

1) 120591119898119899(119905) +

subpopulation1

sum

119894=1

Δ120591119894

119898119899(119905)

(10)





Belief space

Influencefunction

Acceptancefunction



Knowledge exchange

Population space




01 02 03 10

01 02 03 10

12059111 12059112 12059113 120591110

F

F

CR

CR

12059121 12059122 12059123 120591210

01

10

02

03

01

10

02

03








Δ120591119894

119898119899(119905)

=

1 if 119894 isin 119883119898119899


05 if 119894 isin 119883119898119899



119894)

0 otherwise(11)

where 119883119898119899





119901119898119899(119905) =

120591119894

119898119899(119905)

sum119899120591119898119899(119905)

if rand1lt 119875119904

rand2

otherwise(12)


119877 where 119875

119904is a constant


2







Mutation strategy

Mutation strategy


02 04 06 08 1

1205761 1205762 1205763 1205764 1205765

04

06

02

10




120576119896(119905 + 1) = (1 minus 120588

2) 120576119896(119905) +

subpopulation2

sum

119894=1

Δ120576119894

119896(119905) (13)



120576119894

119896(119905)

=


119894) lt fitness (119909best119905)


119894)


119894)

0 otherwise(14)





119901119896(119905) =

120576119894

119896(119905)

sum119899120576119896(119905)

if rand3lt 1198751015840

119904

rand4

otherwise(15)


and rand3and rand







119894





119894are

updated the 119871119894and 119880



119897119894and 119906

119894are set by

119897119894= 119909119894min if 119909

119894min lt 119897119894 or 119891 (119909119894min) lt 119871 119894119897119894

otherwise

119906119894= 119909119894max if 119909

119894max gt 119906119894 or 119891 (119909119894max) gt 119880119894119906119894

otherwise

(16)




119909119905+1

119894119889=

119909119905

119894119889+10038161003816100381610038161003816119873 (05 03) lowast (119883

119905

1199032119894119889minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889le 119864119895 119909119905

119894119889ge 119906119894

119909119905

119894119889minus10038161003816100381610038161003816119873 (05 03) lowast (119883

119905

1199032119894119889minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889gt 119864119895 119909119905

119894119889lt 119906119894

119883119905

1199031119894119889+10038161003816100381610038161003816119873 (05 03) lowast (119906

119894minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889le 119864119895 119909119905

119894119889ge 119897119894

119883119905

1199031119894119889minus10038161003816100381610038161003816119873 (05 03) lowast (119897

119894minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889gt 119864119895 119909119905

119894119889gt 119897119894

119909119905+1

119894119889=

119883119905

1199031119894119889+ 119865 lowast (119906

119894minus 119883119905

1199031119894119889) times rand if 119909

119894119889gt 119897119894

119883119905

1199031119894119889minus 119865 lowast (119883

119905


119894119889lt 119906119894

119883119905

1199031119894119889+ 119865 lowast (119906


119894lt 119909119894119889lt 119906119894























119904=

1198751015840

119904= 05 and 120588

1= 1205882= 01



1sim1198917are






1198916 1198917 and 119891





1


































Table 1 Continued


















Best 207617 39949Worst 299112 119899Mean 254556 81947


Table 1 Continued















1 1198913 1198914 1198916 and 119891

18 It also presents


1 1198913 1198914 1198915 1198916 11989112 11989113 11989114 11989115 and 119891

18although




1 1198912 1198913 1198914

1198916 1198917 11989111 11989112 11989113 11989114 11989115 and 119891

18








minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5



0

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

10

log(fi

tnes

s val

ue)



10

log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5


123456789

1011


log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus35

minus40

minus45

minus25

minus20

minus15

minus10

minus5

0

5


10

Original DECDEAS


log(fi

tnes

s val

ue)

minus30

minus35

minus25

minus20

minus15

minus10

minus5

0

5


Original DECDEAS


minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2

log(fi

tnes

s val

ue)


Figure 6 Continued





081

12141618

222242628


081

12141618

222242628


081

12141618

222242628


0

05

1

15

2

25

3


Original DECDEAS

22242628

332343638

442


Original DECDEAS

02

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2 024

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

1

2

3

minus3

minus2

minus1









log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

Figure 6 Continued


0

05

1

15

2

Original DECDEAS


Original DECDEAS


minus50 minus05

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)







5 119865119876



7 1198651199022



9 1198651199024








[0 119868

119879

119899

119868119899119870 + 120574

minus1119868] [1198870

119887] = [

0

119910] (19)




119870 = (

11989611


d

1198961198991

sdot sdot sdot 119896119899119899

) (20)



119896119894119895= 119890minus|119909119894minus119909119895|

221205902

(21)

where 119909119894and 119909








0001188




















6 Conclusion


Appendix

(1) Sphere function

1198911(119909) =

119863

sum

119894=1

1199092

119894


(A1)


1198912(119909) =

119863

sum

119894=1

1199112

119894

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A2)


0 50 100 150 200 2500106010701080109

011011101120113011401150116

Sample number

Net

val

ue o

f am

mon

ia (

)

0 10 20 30 40 50 60 70 80 9001050106010701080109

01101110112011301140115

Sample number

Net

val

ue o

f am

mon

ia (

)

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

Net

val

ue o

f am

mon

ia (

)

Sample number0 50 100 150 200 250

0106010701080109

011011101120113011401150116


Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115






1198911

30

0 0 [minus100 100]119863

10minus5

1198912

120579 0 [minus100 100]119863

10minus5

1198913

0 0 [minus100 100]119863

10minus5

1198914

120579 0 [minus100 100]119863

10minus5

1198915

1 1 [minus100 100]119863 100

1198916

0 0 [minus32 32]119863

10minus5

1198917

120579 0 [minus32 32]119863

10minus5

1198918

0 0 [minus32 32]119863

10minus5

1198919

120579 0 [minus32 32]119863 01

11989110

0 0 [0 600]119863 0001

11989111

120579 0 [minus600 600]119863 001

11989112

0 0 [minus5 5]119863 10

11989113

120579 0 [minus5 5]119863 10

11989114

0 0 [minus5 5]119863 10

11989115

120579 0 [minus5 5]119863 5

11989116

4189829 0 [minus500 500]119863 500

11989117

0 0 [minus100 100]119863 1

11989118

0 0 [minus10 10]119863

10minus5



1198913(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2


(A3)


1198914(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A4)


1198915(119909) =

119863minus1

sum

119894=1

(100(1199092

119894minus 1199092

119894+1)2

+ (119909119894minus 1)2

)


(A5)


1198916(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2

) lowast (1 + 04 |119873 (0 1)|)


(A6)


1198917(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

) lowast (1 + 04 |119873 (0 1)|)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A7)



119863

119863

sum

119894=1

1199092

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119909119894)) + 20 + 119890


(A8)



119863

119863

sum

119894=1

1199112

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119911119894)) + 20 + 119890

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A9)



11989110(119909) =

119863

sum

119894=1

1199092

119894

4000minus

119863

prod

119894=1

cos119909119894

radic119894

+ 1


(A10)


11989111(119909) =

119863

sum

119894=1

1199112

119894

4000minus

119863

prod

119894=1

cos119911119894

radic119894

+ 1

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A11)


11989112(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10)


(A12)


11989113(119909) =

119863

sum

119894=1

(1199112

119894minus 10 cos (2120587119911

119894) + 10)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A13)


11989114(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt1

2round (2119909

119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863



11989115(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119911119894

10038161003816100381610038161199111198941003816100381610038161003816 lt1

2round (2119911

119894)

2

10038161003816100381610038161199111198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A15)


11989116(119909) = 4189829 times 119863 minus

119863

sum

119894=1

119909119894sin (1003816100381610038161003816119909119894

1003816100381610038161003816

12

)



11989118(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863



11989117(119909) =

119863

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119863

prod

119894=1

119909119894


(A18)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













vector 119906119905119894119889

of individual 119909119905119894119889


DErand1 119906119905

119894119889= 119883119905

1199031119894119889+ 119865 (119883

119905

1199032119894119889minus 119883119905

1199033119894119889) (3)

DErand2 119906119905

119894119889= 119883119905

1199031119894119889+ 119865 (119883

119905

1199032119894119889minus 119883119905

1199033119894119889)

+ 119865 (119883119905

1199034119894119889minus 119883119905

1199035119894119889)

(4)

DEbest1 119906119905

119894119889= 119883119905

best119889 + 119865 (119883119905

1199031119894119889minus 119883119905

1199032119894119889) (5)

DEbest2 119906119905

119894119889= 119883119905

best119889 + 119865 (119883119905

1199031119894119889minus 119883119905

1199032119894119889)

+ 119865 (119883119905

1199033119894119889minus 119883119905

1199034119894119889)

(6)


119894119889= 119883119905

119894119889+ 119865 (119883

119905

best119889 minus 119883119905

119894119889)

+ 119865 (119883119905

1199031119894119889minus 119883119905

1199032119894119889)

(7)



119894are



119910119905

119894119889= 119906119905

119894119889if rand le 119862

119877or 119894 = 119894rand

119883119905


(8)





119894) by




119894 and


119883119905+1

119894119889= 119910119905

119894119889if 119891 (119910119905

119894119889) le 119891 (119883

119905

119894119889)

119883119905


(9)











119877 and mutation




119877 and mutation











119877



equation

120591119898119899(119905 + 1) = (1 minus 120588

1) 120591119898119899(119905) +

subpopulation1

sum

119894=1

Δ120591119894

119898119899(119905)

(10)





Belief space

Influencefunction

Acceptancefunction



Knowledge exchange

Population space




01 02 03 10

01 02 03 10

12059111 12059112 12059113 120591110

F

F

CR

CR

12059121 12059122 12059123 120591210

01

10

02

03

01

10

02

03








Δ120591119894

119898119899(119905)

=

1 if 119894 isin 119883119898119899


05 if 119894 isin 119883119898119899



119894)

0 otherwise(11)

where 119883119898119899





119901119898119899(119905) =

120591119894

119898119899(119905)

sum119899120591119898119899(119905)

if rand1lt 119875119904

rand2

otherwise(12)


119877 where 119875

119904is a constant


2







Mutation strategy

Mutation strategy


02 04 06 08 1

1205761 1205762 1205763 1205764 1205765

04

06

02

10




120576119896(119905 + 1) = (1 minus 120588

2) 120576119896(119905) +

subpopulation2

sum

119894=1

Δ120576119894

119896(119905) (13)



120576119894

119896(119905)

=


119894) lt fitness (119909best119905)


119894)


119894)

0 otherwise(14)





119901119896(119905) =

120576119894

119896(119905)

sum119899120576119896(119905)

if rand3lt 1198751015840

119904

rand4

otherwise(15)


and rand3and rand







119894





119894are

updated the 119871119894and 119880



119897119894and 119906

119894are set by

119897119894= 119909119894min if 119909

119894min lt 119897119894 or 119891 (119909119894min) lt 119871 119894119897119894

otherwise

119906119894= 119909119894max if 119909

119894max gt 119906119894 or 119891 (119909119894max) gt 119880119894119906119894

otherwise

(16)




119909119905+1

119894119889=

119909119905

119894119889+10038161003816100381610038161003816119873 (05 03) lowast (119883

119905

1199032119894119889minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889le 119864119895 119909119905

119894119889ge 119906119894

119909119905

119894119889minus10038161003816100381610038161003816119873 (05 03) lowast (119883

119905

1199032119894119889minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889gt 119864119895 119909119905

119894119889lt 119906119894

119883119905

1199031119894119889+10038161003816100381610038161003816119873 (05 03) lowast (119906

119894minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889le 119864119895 119909119905

119894119889ge 119897119894

119883119905

1199031119894119889minus10038161003816100381610038161003816119873 (05 03) lowast (119897

119894minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889gt 119864119895 119909119905

119894119889gt 119897119894

119909119905+1

119894119889=

119883119905

1199031119894119889+ 119865 lowast (119906

119894minus 119883119905

1199031119894119889) times rand if 119909

119894119889gt 119897119894

119883119905

1199031119894119889minus 119865 lowast (119883

119905


119894119889lt 119906119894

119883119905

1199031119894119889+ 119865 lowast (119906


119894lt 119909119894119889lt 119906119894























119904=

1198751015840

119904= 05 and 120588

1= 1205882= 01



1sim1198917are






1198916 1198917 and 119891





1


































Table 1 Continued


















Best 207617 39949Worst 299112 119899Mean 254556 81947


Table 1 Continued















1 1198913 1198914 1198916 and 119891

18 It also presents


1 1198913 1198914 1198915 1198916 11989112 11989113 11989114 11989115 and 119891

18although




1 1198912 1198913 1198914

1198916 1198917 11989111 11989112 11989113 11989114 11989115 and 119891

18








minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5



0

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

10

log(fi

tnes

s val

ue)



10

log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5


123456789

1011


log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus35

minus40

minus45

minus25

minus20

minus15

minus10

minus5

0

5


10

Original DECDEAS


log(fi

tnes

s val

ue)

minus30

minus35

minus25

minus20

minus15

minus10

minus5

0

5


Original DECDEAS


minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2

log(fi

tnes

s val

ue)


Figure 6 Continued





081

12141618

222242628


081

12141618

222242628


081

12141618

222242628


0

05

1

15

2

25

3


Original DECDEAS

22242628

332343638

442


Original DECDEAS

02

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2 024

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

1

2

3

minus3

minus2

minus1









log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

Figure 6 Continued


0

05

1

15

2

Original DECDEAS


Original DECDEAS


minus50 minus05

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)







5 119865119876



7 1198651199022



9 1198651199024








[0 119868

119879

119899

119868119899119870 + 120574

minus1119868] [1198870

119887] = [

0

119910] (19)




119870 = (

11989611


d

1198961198991

sdot sdot sdot 119896119899119899

) (20)



119896119894119895= 119890minus|119909119894minus119909119895|

221205902

(21)

where 119909119894and 119909








0001188




















6 Conclusion


Appendix

(1) Sphere function

1198911(119909) =

119863

sum

119894=1

1199092

119894


(A1)


1198912(119909) =

119863

sum

119894=1

1199112

119894

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A2)


0 50 100 150 200 2500106010701080109

011011101120113011401150116

Sample number

Net

val

ue o

f am

mon

ia (

)

0 10 20 30 40 50 60 70 80 9001050106010701080109

01101110112011301140115

Sample number

Net

val

ue o

f am

mon

ia (

)

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

Net

val

ue o

f am

mon

ia (

)

Sample number0 50 100 150 200 250

0106010701080109

011011101120113011401150116


Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115






1198911

30

0 0 [minus100 100]119863

10minus5

1198912

120579 0 [minus100 100]119863

10minus5

1198913

0 0 [minus100 100]119863

10minus5

1198914

120579 0 [minus100 100]119863

10minus5

1198915

1 1 [minus100 100]119863 100

1198916

0 0 [minus32 32]119863

10minus5

1198917

120579 0 [minus32 32]119863

10minus5

1198918

0 0 [minus32 32]119863

10minus5

1198919

120579 0 [minus32 32]119863 01

11989110

0 0 [0 600]119863 0001

11989111

120579 0 [minus600 600]119863 001

11989112

0 0 [minus5 5]119863 10

11989113

120579 0 [minus5 5]119863 10

11989114

0 0 [minus5 5]119863 10

11989115

120579 0 [minus5 5]119863 5

11989116

4189829 0 [minus500 500]119863 500

11989117

0 0 [minus100 100]119863 1

11989118

0 0 [minus10 10]119863

10minus5



1198913(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2


(A3)


1198914(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A4)


1198915(119909) =

119863minus1

sum

119894=1

(100(1199092

119894minus 1199092

119894+1)2

+ (119909119894minus 1)2

)


(A5)


1198916(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2

) lowast (1 + 04 |119873 (0 1)|)


(A6)


1198917(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

) lowast (1 + 04 |119873 (0 1)|)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A7)



119863

119863

sum

119894=1

1199092

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119909119894)) + 20 + 119890


(A8)



119863

119863

sum

119894=1

1199112

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119911119894)) + 20 + 119890

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A9)



11989110(119909) =

119863

sum

119894=1

1199092

119894

4000minus

119863

prod

119894=1

cos119909119894

radic119894

+ 1


(A10)


11989111(119909) =

119863

sum

119894=1

1199112

119894

4000minus

119863

prod

119894=1

cos119911119894

radic119894

+ 1

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A11)


11989112(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10)


(A12)


11989113(119909) =

119863

sum

119894=1

(1199112

119894minus 10 cos (2120587119911

119894) + 10)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A13)


11989114(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt1

2round (2119909

119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863



11989115(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119911119894

10038161003816100381610038161199111198941003816100381610038161003816 lt1

2round (2119911

119894)

2

10038161003816100381610038161199111198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A15)


11989116(119909) = 4189829 times 119863 minus

119863

sum

119894=1

119909119894sin (1003816100381610038161003816119909119894

1003816100381610038161003816

12

)



11989118(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863



11989117(119909) =

119863

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119863

prod

119894=1

119909119894


(A18)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Belief space

Influencefunction

Acceptancefunction



Knowledge exchange

Population space




01 02 03 10

01 02 03 10

12059111 12059112 12059113 120591110

F

F

CR

CR

12059121 12059122 12059123 120591210

01

10

02

03

01

10

02

03








Δ120591119894

119898119899(119905)

=

1 if 119894 isin 119883119898119899


05 if 119894 isin 119883119898119899



119894)

0 otherwise(11)

where 119883119898119899





119901119898119899(119905) =

120591119894

119898119899(119905)

sum119899120591119898119899(119905)

if rand1lt 119875119904

rand2

otherwise(12)


119877 where 119875

119904is a constant


2







Mutation strategy

Mutation strategy


02 04 06 08 1

1205761 1205762 1205763 1205764 1205765

04

06

02

10




120576119896(119905 + 1) = (1 minus 120588

2) 120576119896(119905) +

subpopulation2

sum

119894=1

Δ120576119894

119896(119905) (13)



120576119894

119896(119905)

=


119894) lt fitness (119909best119905)


119894)


119894)

0 otherwise(14)





119901119896(119905) =

120576119894

119896(119905)

sum119899120576119896(119905)

if rand3lt 1198751015840

119904

rand4

otherwise(15)


and rand3and rand







119894





119894are

updated the 119871119894and 119880



119897119894and 119906

119894are set by

119897119894= 119909119894min if 119909

119894min lt 119897119894 or 119891 (119909119894min) lt 119871 119894119897119894

otherwise

119906119894= 119909119894max if 119909

119894max gt 119906119894 or 119891 (119909119894max) gt 119880119894119906119894

otherwise

(16)




119909119905+1

119894119889=

119909119905

119894119889+10038161003816100381610038161003816119873 (05 03) lowast (119883

119905

1199032119894119889minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889le 119864119895 119909119905

119894119889ge 119906119894

119909119905

119894119889minus10038161003816100381610038161003816119873 (05 03) lowast (119883

119905

1199032119894119889minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889gt 119864119895 119909119905

119894119889lt 119906119894

119883119905

1199031119894119889+10038161003816100381610038161003816119873 (05 03) lowast (119906

119894minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889le 119864119895 119909119905

119894119889ge 119897119894

119883119905

1199031119894119889minus10038161003816100381610038161003816119873 (05 03) lowast (119897

119894minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889gt 119864119895 119909119905

119894119889gt 119897119894

119909119905+1

119894119889=

119883119905

1199031119894119889+ 119865 lowast (119906

119894minus 119883119905

1199031119894119889) times rand if 119909

119894119889gt 119897119894

119883119905

1199031119894119889minus 119865 lowast (119883

119905


119894119889lt 119906119894

119883119905

1199031119894119889+ 119865 lowast (119906


119894lt 119909119894119889lt 119906119894























119904=

1198751015840

119904= 05 and 120588

1= 1205882= 01



1sim1198917are






1198916 1198917 and 119891





1


































Table 1 Continued


















Best 207617 39949Worst 299112 119899Mean 254556 81947


Table 1 Continued















1 1198913 1198914 1198916 and 119891

18 It also presents


1 1198913 1198914 1198915 1198916 11989112 11989113 11989114 11989115 and 119891

18although




1 1198912 1198913 1198914

1198916 1198917 11989111 11989112 11989113 11989114 11989115 and 119891

18








minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5



0

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

10

log(fi

tnes

s val

ue)



10

log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5


123456789

1011


log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus35

minus40

minus45

minus25

minus20

minus15

minus10

minus5

0

5


10

Original DECDEAS


log(fi

tnes

s val

ue)

minus30

minus35

minus25

minus20

minus15

minus10

minus5

0

5


Original DECDEAS


minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2

log(fi

tnes

s val

ue)


Figure 6 Continued





081

12141618

222242628


081

12141618

222242628


081

12141618

222242628


0

05

1

15

2

25

3


Original DECDEAS

22242628

332343638

442


Original DECDEAS

02

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2 024

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

1

2

3

minus3

minus2

minus1









log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

Figure 6 Continued


0

05

1

15

2

Original DECDEAS


Original DECDEAS


minus50 minus05

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)







5 119865119876



7 1198651199022



9 1198651199024








[0 119868

119879

119899

119868119899119870 + 120574

minus1119868] [1198870

119887] = [

0

119910] (19)




119870 = (

11989611


d

1198961198991

sdot sdot sdot 119896119899119899

) (20)



119896119894119895= 119890minus|119909119894minus119909119895|

221205902

(21)

where 119909119894and 119909








0001188




















6 Conclusion


Appendix

(1) Sphere function

1198911(119909) =

119863

sum

119894=1

1199092

119894


(A1)


1198912(119909) =

119863

sum

119894=1

1199112

119894

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A2)


0 50 100 150 200 2500106010701080109

011011101120113011401150116

Sample number

Net

val

ue o

f am

mon

ia (

)

0 10 20 30 40 50 60 70 80 9001050106010701080109

01101110112011301140115

Sample number

Net

val

ue o

f am

mon

ia (

)

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

Net

val

ue o

f am

mon

ia (

)

Sample number0 50 100 150 200 250

0106010701080109

011011101120113011401150116


Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115






1198911

30

0 0 [minus100 100]119863

10minus5

1198912

120579 0 [minus100 100]119863

10minus5

1198913

0 0 [minus100 100]119863

10minus5

1198914

120579 0 [minus100 100]119863

10minus5

1198915

1 1 [minus100 100]119863 100

1198916

0 0 [minus32 32]119863

10minus5

1198917

120579 0 [minus32 32]119863

10minus5

1198918

0 0 [minus32 32]119863

10minus5

1198919

120579 0 [minus32 32]119863 01

11989110

0 0 [0 600]119863 0001

11989111

120579 0 [minus600 600]119863 001

11989112

0 0 [minus5 5]119863 10

11989113

120579 0 [minus5 5]119863 10

11989114

0 0 [minus5 5]119863 10

11989115

120579 0 [minus5 5]119863 5

11989116

4189829 0 [minus500 500]119863 500

11989117

0 0 [minus100 100]119863 1

11989118

0 0 [minus10 10]119863

10minus5



1198913(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2


(A3)


1198914(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A4)


1198915(119909) =

119863minus1

sum

119894=1

(100(1199092

119894minus 1199092

119894+1)2

+ (119909119894minus 1)2

)


(A5)


1198916(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2

) lowast (1 + 04 |119873 (0 1)|)


(A6)


1198917(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

) lowast (1 + 04 |119873 (0 1)|)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A7)



119863

119863

sum

119894=1

1199092

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119909119894)) + 20 + 119890


(A8)



119863

119863

sum

119894=1

1199112

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119911119894)) + 20 + 119890

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A9)



11989110(119909) =

119863

sum

119894=1

1199092

119894

4000minus

119863

prod

119894=1

cos119909119894

radic119894

+ 1


(A10)


11989111(119909) =

119863

sum

119894=1

1199112

119894

4000minus

119863

prod

119894=1

cos119911119894

radic119894

+ 1

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A11)


11989112(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10)


(A12)


11989113(119909) =

119863

sum

119894=1

(1199112

119894minus 10 cos (2120587119911

119894) + 10)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A13)


11989114(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt1

2round (2119909

119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863



11989115(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119911119894

10038161003816100381610038161199111198941003816100381610038161003816 lt1

2round (2119911

119894)

2

10038161003816100381610038161199111198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A15)


11989116(119909) = 4189829 times 119863 minus

119863

sum

119894=1

119909119894sin (1003816100381610038161003816119909119894

1003816100381610038161003816

12

)



11989118(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863



11989117(119909) =

119863

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119863

prod

119894=1

119909119894


(A18)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Mutation strategy

Mutation strategy


02 04 06 08 1

1205761 1205762 1205763 1205764 1205765

04

06

02

10




120576119896(119905 + 1) = (1 minus 120588

2) 120576119896(119905) +

subpopulation2

sum

119894=1

Δ120576119894

119896(119905) (13)



120576119894

119896(119905)

=


119894) lt fitness (119909best119905)


119894)


119894)

0 otherwise(14)





119901119896(119905) =

120576119894

119896(119905)

sum119899120576119896(119905)

if rand3lt 1198751015840

119904

rand4

otherwise(15)


and rand3and rand







119894





119894are

updated the 119871119894and 119880



119897119894and 119906

119894are set by

119897119894= 119909119894min if 119909

119894min lt 119897119894 or 119891 (119909119894min) lt 119871 119894119897119894

otherwise

119906119894= 119909119894max if 119909

119894max gt 119906119894 or 119891 (119909119894max) gt 119880119894119906119894

otherwise

(16)




119909119905+1

119894119889=

119909119905

119894119889+10038161003816100381610038161003816119873 (05 03) lowast (119883

119905

1199032119894119889minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889le 119864119895 119909119905

119894119889ge 119906119894

119909119905

119894119889minus10038161003816100381610038161003816119873 (05 03) lowast (119883

119905

1199032119894119889minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889gt 119864119895 119909119905

119894119889lt 119906119894

119883119905

1199031119894119889+10038161003816100381610038161003816119873 (05 03) lowast (119906

119894minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889le 119864119895 119909119905

119894119889ge 119897119894

119883119905

1199031119894119889minus10038161003816100381610038161003816119873 (05 03) lowast (119897

119894minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889gt 119864119895 119909119905

119894119889gt 119897119894

119909119905+1

119894119889=

119883119905

1199031119894119889+ 119865 lowast (119906

119894minus 119883119905

1199031119894119889) times rand if 119909

119894119889gt 119897119894

119883119905

1199031119894119889minus 119865 lowast (119883

119905


119894119889lt 119906119894

119883119905

1199031119894119889+ 119865 lowast (119906


119894lt 119909119894119889lt 119906119894























119904=

1198751015840

119904= 05 and 120588

1= 1205882= 01



1sim1198917are






1198916 1198917 and 119891





1


































Table 1 Continued


















Best 207617 39949Worst 299112 119899Mean 254556 81947


Table 1 Continued















1 1198913 1198914 1198916 and 119891

18 It also presents


1 1198913 1198914 1198915 1198916 11989112 11989113 11989114 11989115 and 119891

18although




1 1198912 1198913 1198914

1198916 1198917 11989111 11989112 11989113 11989114 11989115 and 119891

18








minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5



0

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

10

log(fi

tnes

s val

ue)



10

log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5


123456789

1011


log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus35

minus40

minus45

minus25

minus20

minus15

minus10

minus5

0

5


10

Original DECDEAS


log(fi

tnes

s val

ue)

minus30

minus35

minus25

minus20

minus15

minus10

minus5

0

5


Original DECDEAS


minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2

log(fi

tnes

s val

ue)


Figure 6 Continued





081

12141618

222242628


081

12141618

222242628


081

12141618

222242628


0

05

1

15

2

25

3


Original DECDEAS

22242628

332343638

442


Original DECDEAS

02

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2 024

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

1

2

3

minus3

minus2

minus1









log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

Figure 6 Continued


0

05

1

15

2

Original DECDEAS


Original DECDEAS


minus50 minus05

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)







5 119865119876



7 1198651199022



9 1198651199024








[0 119868

119879

119899

119868119899119870 + 120574

minus1119868] [1198870

119887] = [

0

119910] (19)




119870 = (

11989611


d

1198961198991

sdot sdot sdot 119896119899119899

) (20)



119896119894119895= 119890minus|119909119894minus119909119895|

221205902

(21)

where 119909119894and 119909








0001188




















6 Conclusion


Appendix

(1) Sphere function

1198911(119909) =

119863

sum

119894=1

1199092

119894


(A1)


1198912(119909) =

119863

sum

119894=1

1199112

119894

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A2)


0 50 100 150 200 2500106010701080109

011011101120113011401150116

Sample number

Net

val

ue o

f am

mon

ia (

)

0 10 20 30 40 50 60 70 80 9001050106010701080109

01101110112011301140115

Sample number

Net

val

ue o

f am

mon

ia (

)

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

Net

val

ue o

f am

mon

ia (

)

Sample number0 50 100 150 200 250

0106010701080109

011011101120113011401150116


Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115






1198911

30

0 0 [minus100 100]119863

10minus5

1198912

120579 0 [minus100 100]119863

10minus5

1198913

0 0 [minus100 100]119863

10minus5

1198914

120579 0 [minus100 100]119863

10minus5

1198915

1 1 [minus100 100]119863 100

1198916

0 0 [minus32 32]119863

10minus5

1198917

120579 0 [minus32 32]119863

10minus5

1198918

0 0 [minus32 32]119863

10minus5

1198919

120579 0 [minus32 32]119863 01

11989110

0 0 [0 600]119863 0001

11989111

120579 0 [minus600 600]119863 001

11989112

0 0 [minus5 5]119863 10

11989113

120579 0 [minus5 5]119863 10

11989114

0 0 [minus5 5]119863 10

11989115

120579 0 [minus5 5]119863 5

11989116

4189829 0 [minus500 500]119863 500

11989117

0 0 [minus100 100]119863 1

11989118

0 0 [minus10 10]119863

10minus5



1198913(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2


(A3)


1198914(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A4)


1198915(119909) =

119863minus1

sum

119894=1

(100(1199092

119894minus 1199092

119894+1)2

+ (119909119894minus 1)2

)


(A5)


1198916(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2

) lowast (1 + 04 |119873 (0 1)|)


(A6)


1198917(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

) lowast (1 + 04 |119873 (0 1)|)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A7)



119863

119863

sum

119894=1

1199092

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119909119894)) + 20 + 119890


(A8)



119863

119863

sum

119894=1

1199112

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119911119894)) + 20 + 119890

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A9)



11989110(119909) =

119863

sum

119894=1

1199092

119894

4000minus

119863

prod

119894=1

cos119909119894

radic119894

+ 1


(A10)


11989111(119909) =

119863

sum

119894=1

1199112

119894

4000minus

119863

prod

119894=1

cos119911119894

radic119894

+ 1

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A11)


11989112(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10)


(A12)


11989113(119909) =

119863

sum

119894=1

(1199112

119894minus 10 cos (2120587119911

119894) + 10)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A13)


11989114(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt1

2round (2119909

119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863



11989115(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119911119894

10038161003816100381610038161199111198941003816100381610038161003816 lt1

2round (2119911

119894)

2

10038161003816100381610038161199111198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A15)


11989116(119909) = 4189829 times 119863 minus

119863

sum

119894=1

119909119894sin (1003816100381610038161003816119909119894

1003816100381610038161003816

12

)



11989118(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863



11989117(119909) =

119863

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119863

prod

119894=1

119909119894


(A18)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119897119894and 119906

119894are set by

119897119894= 119909119894min if 119909

119894min lt 119897119894 or 119891 (119909119894min) lt 119871 119894119897119894

otherwise

119906119894= 119909119894max if 119909

119894max gt 119906119894 or 119891 (119909119894max) gt 119880119894119906119894

otherwise

(16)




119909119905+1

119894119889=

119909119905

119894119889+10038161003816100381610038161003816119873 (05 03) lowast (119883

119905

1199032119894119889minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889le 119864119895 119909119905

119894119889ge 119906119894

119909119905

119894119889minus10038161003816100381610038161003816119873 (05 03) lowast (119883

119905

1199032119894119889minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889gt 119864119895 119909119905

119894119889lt 119906119894

119883119905

1199031119894119889+10038161003816100381610038161003816119873 (05 03) lowast (119906

119894minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889le 119864119895 119909119905

119894119889ge 119897119894

119883119905

1199031119894119889minus10038161003816100381610038161003816119873 (05 03) lowast (119897

119894minus 119883119905

1199033119894119889)10038161003816100381610038161003816times rand

if 119909119905119894119889gt 119864119895 119909119905

119894119889gt 119897119894

119909119905+1

119894119889=

119883119905

1199031119894119889+ 119865 lowast (119906

119894minus 119883119905

1199031119894119889) times rand if 119909

119894119889gt 119897119894

119883119905

1199031119894119889minus 119865 lowast (119883

119905


119894119889lt 119906119894

119883119905

1199031119894119889+ 119865 lowast (119906


119894lt 119909119894119889lt 119906119894























119904=

1198751015840

119904= 05 and 120588

1= 1205882= 01



1sim1198917are






1198916 1198917 and 119891





1


































Table 1 Continued


















Best 207617 39949Worst 299112 119899Mean 254556 81947


Table 1 Continued















1 1198913 1198914 1198916 and 119891

18 It also presents


1 1198913 1198914 1198915 1198916 11989112 11989113 11989114 11989115 and 119891

18although




1 1198912 1198913 1198914

1198916 1198917 11989111 11989112 11989113 11989114 11989115 and 119891

18








minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5



0

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

10

log(fi

tnes

s val

ue)



10

log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5


123456789

1011


log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus35

minus40

minus45

minus25

minus20

minus15

minus10

minus5

0

5


10

Original DECDEAS


log(fi

tnes

s val

ue)

minus30

minus35

minus25

minus20

minus15

minus10

minus5

0

5


Original DECDEAS


minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2

log(fi

tnes

s val

ue)


Figure 6 Continued





081

12141618

222242628


081

12141618

222242628


081

12141618

222242628


0

05

1

15

2

25

3


Original DECDEAS

22242628

332343638

442


Original DECDEAS

02

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2 024

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

1

2

3

minus3

minus2

minus1









log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

Figure 6 Continued


0

05

1

15

2

Original DECDEAS


Original DECDEAS


minus50 minus05

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)







5 119865119876



7 1198651199022



9 1198651199024








[0 119868

119879

119899

119868119899119870 + 120574

minus1119868] [1198870

119887] = [

0

119910] (19)




119870 = (

11989611


d

1198961198991

sdot sdot sdot 119896119899119899

) (20)



119896119894119895= 119890minus|119909119894minus119909119895|

221205902

(21)

where 119909119894and 119909








0001188




















6 Conclusion


Appendix

(1) Sphere function

1198911(119909) =

119863

sum

119894=1

1199092

119894


(A1)


1198912(119909) =

119863

sum

119894=1

1199112

119894

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A2)


0 50 100 150 200 2500106010701080109

011011101120113011401150116

Sample number

Net

val

ue o

f am

mon

ia (

)

0 10 20 30 40 50 60 70 80 9001050106010701080109

01101110112011301140115

Sample number

Net

val

ue o

f am

mon

ia (

)

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

Net

val

ue o

f am

mon

ia (

)

Sample number0 50 100 150 200 250

0106010701080109

011011101120113011401150116


Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115






1198911

30

0 0 [minus100 100]119863

10minus5

1198912

120579 0 [minus100 100]119863

10minus5

1198913

0 0 [minus100 100]119863

10minus5

1198914

120579 0 [minus100 100]119863

10minus5

1198915

1 1 [minus100 100]119863 100

1198916

0 0 [minus32 32]119863

10minus5

1198917

120579 0 [minus32 32]119863

10minus5

1198918

0 0 [minus32 32]119863

10minus5

1198919

120579 0 [minus32 32]119863 01

11989110

0 0 [0 600]119863 0001

11989111

120579 0 [minus600 600]119863 001

11989112

0 0 [minus5 5]119863 10

11989113

120579 0 [minus5 5]119863 10

11989114

0 0 [minus5 5]119863 10

11989115

120579 0 [minus5 5]119863 5

11989116

4189829 0 [minus500 500]119863 500

11989117

0 0 [minus100 100]119863 1

11989118

0 0 [minus10 10]119863

10minus5



1198913(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2


(A3)


1198914(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A4)


1198915(119909) =

119863minus1

sum

119894=1

(100(1199092

119894minus 1199092

119894+1)2

+ (119909119894minus 1)2

)


(A5)


1198916(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2

) lowast (1 + 04 |119873 (0 1)|)


(A6)


1198917(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

) lowast (1 + 04 |119873 (0 1)|)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A7)



119863

119863

sum

119894=1

1199092

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119909119894)) + 20 + 119890


(A8)



119863

119863

sum

119894=1

1199112

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119911119894)) + 20 + 119890

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A9)



11989110(119909) =

119863

sum

119894=1

1199092

119894

4000minus

119863

prod

119894=1

cos119909119894

radic119894

+ 1


(A10)


11989111(119909) =

119863

sum

119894=1

1199112

119894

4000minus

119863

prod

119894=1

cos119911119894

radic119894

+ 1

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A11)


11989112(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10)


(A12)


11989113(119909) =

119863

sum

119894=1

(1199112

119894minus 10 cos (2120587119911

119894) + 10)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A13)


11989114(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt1

2round (2119909

119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863



11989115(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119911119894

10038161003816100381610038161199111198941003816100381610038161003816 lt1

2round (2119911

119894)

2

10038161003816100381610038161199111198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A15)


11989116(119909) = 4189829 times 119863 minus

119863

sum

119894=1

119909119894sin (1003816100381610038161003816119909119894

1003816100381610038161003816

12

)



11989118(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863



11989117(119909) =

119863

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119863

prod

119894=1

119909119894


(A18)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1


































Table 1 Continued


















Best 207617 39949Worst 299112 119899Mean 254556 81947


Table 1 Continued















1 1198913 1198914 1198916 and 119891

18 It also presents


1 1198913 1198914 1198915 1198916 11989112 11989113 11989114 11989115 and 119891

18although




1 1198912 1198913 1198914

1198916 1198917 11989111 11989112 11989113 11989114 11989115 and 119891

18








minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5



0

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

10

log(fi

tnes

s val

ue)



10

log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5


123456789

1011


log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus35

minus40

minus45

minus25

minus20

minus15

minus10

minus5

0

5


10

Original DECDEAS


log(fi

tnes

s val

ue)

minus30

minus35

minus25

minus20

minus15

minus10

minus5

0

5


Original DECDEAS


minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2

log(fi

tnes

s val

ue)


Figure 6 Continued





081

12141618

222242628


081

12141618

222242628


081

12141618

222242628


0

05

1

15

2

25

3


Original DECDEAS

22242628

332343638

442


Original DECDEAS

02

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2 024

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

1

2

3

minus3

minus2

minus1









log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

Figure 6 Continued


0

05

1

15

2

Original DECDEAS


Original DECDEAS


minus50 minus05

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)







5 119865119876



7 1198651199022



9 1198651199024








[0 119868

119879

119899

119868119899119870 + 120574

minus1119868] [1198870

119887] = [

0

119910] (19)




119870 = (

11989611


d

1198961198991

sdot sdot sdot 119896119899119899

) (20)



119896119894119895= 119890minus|119909119894minus119909119895|

221205902

(21)

where 119909119894and 119909








0001188




















6 Conclusion


Appendix

(1) Sphere function

1198911(119909) =

119863

sum

119894=1

1199092

119894


(A1)


1198912(119909) =

119863

sum

119894=1

1199112

119894

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A2)


0 50 100 150 200 2500106010701080109

011011101120113011401150116

Sample number

Net

val

ue o

f am

mon

ia (

)

0 10 20 30 40 50 60 70 80 9001050106010701080109

01101110112011301140115

Sample number

Net

val

ue o

f am

mon

ia (

)

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

Net

val

ue o

f am

mon

ia (

)

Sample number0 50 100 150 200 250

0106010701080109

011011101120113011401150116


Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115






1198911

30

0 0 [minus100 100]119863

10minus5

1198912

120579 0 [minus100 100]119863

10minus5

1198913

0 0 [minus100 100]119863

10minus5

1198914

120579 0 [minus100 100]119863

10minus5

1198915

1 1 [minus100 100]119863 100

1198916

0 0 [minus32 32]119863

10minus5

1198917

120579 0 [minus32 32]119863

10minus5

1198918

0 0 [minus32 32]119863

10minus5

1198919

120579 0 [minus32 32]119863 01

11989110

0 0 [0 600]119863 0001

11989111

120579 0 [minus600 600]119863 001

11989112

0 0 [minus5 5]119863 10

11989113

120579 0 [minus5 5]119863 10

11989114

0 0 [minus5 5]119863 10

11989115

120579 0 [minus5 5]119863 5

11989116

4189829 0 [minus500 500]119863 500

11989117

0 0 [minus100 100]119863 1

11989118

0 0 [minus10 10]119863

10minus5



1198913(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2


(A3)


1198914(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A4)


1198915(119909) =

119863minus1

sum

119894=1

(100(1199092

119894minus 1199092

119894+1)2

+ (119909119894minus 1)2

)


(A5)


1198916(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2

) lowast (1 + 04 |119873 (0 1)|)


(A6)


1198917(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

) lowast (1 + 04 |119873 (0 1)|)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A7)



119863

119863

sum

119894=1

1199092

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119909119894)) + 20 + 119890


(A8)



119863

119863

sum

119894=1

1199112

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119911119894)) + 20 + 119890

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A9)



11989110(119909) =

119863

sum

119894=1

1199092

119894

4000minus

119863

prod

119894=1

cos119909119894

radic119894

+ 1


(A10)


11989111(119909) =

119863

sum

119894=1

1199112

119894

4000minus

119863

prod

119894=1

cos119911119894

radic119894

+ 1

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A11)


11989112(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10)


(A12)


11989113(119909) =

119863

sum

119894=1

(1199112

119894minus 10 cos (2120587119911

119894) + 10)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A13)


11989114(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt1

2round (2119909

119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863



11989115(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119911119894

10038161003816100381610038161199111198941003816100381610038161003816 lt1

2round (2119911

119894)

2

10038161003816100381610038161199111198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A15)


11989116(119909) = 4189829 times 119863 minus

119863

sum

119894=1

119909119894sin (1003816100381610038161003816119909119894

1003816100381610038161003816

12

)



11989118(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863



11989117(119909) =

119863

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119863

prod

119894=1

119909119894


(A18)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 1 Continued


















Best 207617 39949Worst 299112 119899Mean 254556 81947


Table 1 Continued















1 1198913 1198914 1198916 and 119891

18 It also presents


1 1198913 1198914 1198915 1198916 11989112 11989113 11989114 11989115 and 119891

18although




1 1198912 1198913 1198914

1198916 1198917 11989111 11989112 11989113 11989114 11989115 and 119891

18








minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5



0

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

10

log(fi

tnes

s val

ue)



10

log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5


123456789

1011


log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus35

minus40

minus45

minus25

minus20

minus15

minus10

minus5

0

5


10

Original DECDEAS


log(fi

tnes

s val

ue)

minus30

minus35

minus25

minus20

minus15

minus10

minus5

0

5


Original DECDEAS


minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2

log(fi

tnes

s val

ue)


Figure 6 Continued





081

12141618

222242628


081

12141618

222242628


081

12141618

222242628


0

05

1

15

2

25

3


Original DECDEAS

22242628

332343638

442


Original DECDEAS

02

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2 024

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

1

2

3

minus3

minus2

minus1









log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

Figure 6 Continued


0

05

1

15

2

Original DECDEAS


Original DECDEAS


minus50 minus05

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)







5 119865119876



7 1198651199022



9 1198651199024








[0 119868

119879

119899

119868119899119870 + 120574

minus1119868] [1198870

119887] = [

0

119910] (19)




119870 = (

11989611


d

1198961198991

sdot sdot sdot 119896119899119899

) (20)



119896119894119895= 119890minus|119909119894minus119909119895|

221205902

(21)

where 119909119894and 119909








0001188




















6 Conclusion


Appendix

(1) Sphere function

1198911(119909) =

119863

sum

119894=1

1199092

119894


(A1)


1198912(119909) =

119863

sum

119894=1

1199112

119894

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A2)


0 50 100 150 200 2500106010701080109

011011101120113011401150116

Sample number

Net

val

ue o

f am

mon

ia (

)

0 10 20 30 40 50 60 70 80 9001050106010701080109

01101110112011301140115

Sample number

Net

val

ue o

f am

mon

ia (

)

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

Net

val

ue o

f am

mon

ia (

)

Sample number0 50 100 150 200 250

0106010701080109

011011101120113011401150116


Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115






1198911

30

0 0 [minus100 100]119863

10minus5

1198912

120579 0 [minus100 100]119863

10minus5

1198913

0 0 [minus100 100]119863

10minus5

1198914

120579 0 [minus100 100]119863

10minus5

1198915

1 1 [minus100 100]119863 100

1198916

0 0 [minus32 32]119863

10minus5

1198917

120579 0 [minus32 32]119863

10minus5

1198918

0 0 [minus32 32]119863

10minus5

1198919

120579 0 [minus32 32]119863 01

11989110

0 0 [0 600]119863 0001

11989111

120579 0 [minus600 600]119863 001

11989112

0 0 [minus5 5]119863 10

11989113

120579 0 [minus5 5]119863 10

11989114

0 0 [minus5 5]119863 10

11989115

120579 0 [minus5 5]119863 5

11989116

4189829 0 [minus500 500]119863 500

11989117

0 0 [minus100 100]119863 1

11989118

0 0 [minus10 10]119863

10minus5



1198913(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2


(A3)


1198914(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A4)


1198915(119909) =

119863minus1

sum

119894=1

(100(1199092

119894minus 1199092

119894+1)2

+ (119909119894minus 1)2

)


(A5)


1198916(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2

) lowast (1 + 04 |119873 (0 1)|)


(A6)


1198917(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

) lowast (1 + 04 |119873 (0 1)|)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A7)



119863

119863

sum

119894=1

1199092

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119909119894)) + 20 + 119890


(A8)



119863

119863

sum

119894=1

1199112

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119911119894)) + 20 + 119890

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A9)



11989110(119909) =

119863

sum

119894=1

1199092

119894

4000minus

119863

prod

119894=1

cos119909119894

radic119894

+ 1


(A10)


11989111(119909) =

119863

sum

119894=1

1199112

119894

4000minus

119863

prod

119894=1

cos119911119894

radic119894

+ 1

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A11)


11989112(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10)


(A12)


11989113(119909) =

119863

sum

119894=1

(1199112

119894minus 10 cos (2120587119911

119894) + 10)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A13)


11989114(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt1

2round (2119909

119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863



11989115(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119911119894

10038161003816100381610038161199111198941003816100381610038161003816 lt1

2round (2119911

119894)

2

10038161003816100381610038161199111198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A15)


11989116(119909) = 4189829 times 119863 minus

119863

sum

119894=1

119909119894sin (1003816100381610038161003816119909119894

1003816100381610038161003816

12

)



11989118(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863



11989117(119909) =

119863

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119863

prod

119894=1

119909119894


(A18)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 1 Continued















1 1198913 1198914 1198916 and 119891

18 It also presents


1 1198913 1198914 1198915 1198916 11989112 11989113 11989114 11989115 and 119891

18although




1 1198912 1198913 1198914

1198916 1198917 11989111 11989112 11989113 11989114 11989115 and 119891

18








minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5



0

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

10

log(fi

tnes

s val

ue)



10

log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5


123456789

1011


log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus35

minus40

minus45

minus25

minus20

minus15

minus10

minus5

0

5


10

Original DECDEAS


log(fi

tnes

s val

ue)

minus30

minus35

minus25

minus20

minus15

minus10

minus5

0

5


Original DECDEAS


minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2

log(fi

tnes

s val

ue)


Figure 6 Continued





081

12141618

222242628


081

12141618

222242628


081

12141618

222242628


0

05

1

15

2

25

3


Original DECDEAS

22242628

332343638

442


Original DECDEAS

02

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2 024

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

1

2

3

minus3

minus2

minus1









log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

Figure 6 Continued


0

05

1

15

2

Original DECDEAS


Original DECDEAS


minus50 minus05

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)







5 119865119876



7 1198651199022



9 1198651199024








[0 119868

119879

119899

119868119899119870 + 120574

minus1119868] [1198870

119887] = [

0

119910] (19)




119870 = (

11989611


d

1198961198991

sdot sdot sdot 119896119899119899

) (20)



119896119894119895= 119890minus|119909119894minus119909119895|

221205902

(21)

where 119909119894and 119909








0001188




















6 Conclusion


Appendix

(1) Sphere function

1198911(119909) =

119863

sum

119894=1

1199092

119894


(A1)


1198912(119909) =

119863

sum

119894=1

1199112

119894

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A2)


0 50 100 150 200 2500106010701080109

011011101120113011401150116

Sample number

Net

val

ue o

f am

mon

ia (

)

0 10 20 30 40 50 60 70 80 9001050106010701080109

01101110112011301140115

Sample number

Net

val

ue o

f am

mon

ia (

)

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

Net

val

ue o

f am

mon

ia (

)

Sample number0 50 100 150 200 250

0106010701080109

011011101120113011401150116


Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115






1198911

30

0 0 [minus100 100]119863

10minus5

1198912

120579 0 [minus100 100]119863

10minus5

1198913

0 0 [minus100 100]119863

10minus5

1198914

120579 0 [minus100 100]119863

10minus5

1198915

1 1 [minus100 100]119863 100

1198916

0 0 [minus32 32]119863

10minus5

1198917

120579 0 [minus32 32]119863

10minus5

1198918

0 0 [minus32 32]119863

10minus5

1198919

120579 0 [minus32 32]119863 01

11989110

0 0 [0 600]119863 0001

11989111

120579 0 [minus600 600]119863 001

11989112

0 0 [minus5 5]119863 10

11989113

120579 0 [minus5 5]119863 10

11989114

0 0 [minus5 5]119863 10

11989115

120579 0 [minus5 5]119863 5

11989116

4189829 0 [minus500 500]119863 500

11989117

0 0 [minus100 100]119863 1

11989118

0 0 [minus10 10]119863

10minus5



1198913(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2


(A3)


1198914(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A4)


1198915(119909) =

119863minus1

sum

119894=1

(100(1199092

119894minus 1199092

119894+1)2

+ (119909119894minus 1)2

)


(A5)


1198916(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2

) lowast (1 + 04 |119873 (0 1)|)


(A6)


1198917(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

) lowast (1 + 04 |119873 (0 1)|)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A7)



119863

119863

sum

119894=1

1199092

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119909119894)) + 20 + 119890


(A8)



119863

119863

sum

119894=1

1199112

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119911119894)) + 20 + 119890

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A9)



11989110(119909) =

119863

sum

119894=1

1199092

119894

4000minus

119863

prod

119894=1

cos119909119894

radic119894

+ 1


(A10)


11989111(119909) =

119863

sum

119894=1

1199112

119894

4000minus

119863

prod

119894=1

cos119911119894

radic119894

+ 1

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A11)


11989112(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10)


(A12)


11989113(119909) =

119863

sum

119894=1

(1199112

119894minus 10 cos (2120587119911

119894) + 10)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A13)


11989114(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt1

2round (2119909

119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863



11989115(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119911119894

10038161003816100381610038161199111198941003816100381610038161003816 lt1

2round (2119911

119894)

2

10038161003816100381610038161199111198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A15)


11989116(119909) = 4189829 times 119863 minus

119863

sum

119894=1

119909119894sin (1003816100381610038161003816119909119894

1003816100381610038161003816

12

)



11989118(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863



11989117(119909) =

119863

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119863

prod

119894=1

119909119894


(A18)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5



0

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

10

log(fi

tnes

s val

ue)



10

log(fi

tnes

s val

ue)

minus30

minus25

minus20

minus15

minus10

minus5

0

5


123456789

1011


log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

minus30

minus35

minus40

minus45

minus25

minus20

minus15

minus10

minus5

0

5


10

Original DECDEAS


log(fi

tnes

s val

ue)

minus30

minus35

minus25

minus20

minus15

minus10

minus5

0

5


Original DECDEAS


minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2

log(fi

tnes

s val

ue)


Figure 6 Continued





081

12141618

222242628


081

12141618

222242628


081

12141618

222242628


0

05

1

15

2

25

3


Original DECDEAS

22242628

332343638

442


Original DECDEAS

02

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2 024

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

1

2

3

minus3

minus2

minus1









log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

Figure 6 Continued


0

05

1

15

2

Original DECDEAS


Original DECDEAS


minus50 minus05

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)







5 119865119876



7 1198651199022



9 1198651199024








[0 119868

119879

119899

119868119899119870 + 120574

minus1119868] [1198870

119887] = [

0

119910] (19)




119870 = (

11989611


d

1198961198991

sdot sdot sdot 119896119899119899

) (20)



119896119894119895= 119890minus|119909119894minus119909119895|

221205902

(21)

where 119909119894and 119909








0001188




















6 Conclusion


Appendix

(1) Sphere function

1198911(119909) =

119863

sum

119894=1

1199092

119894


(A1)


1198912(119909) =

119863

sum

119894=1

1199112

119894

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A2)


0 50 100 150 200 2500106010701080109

011011101120113011401150116

Sample number

Net

val

ue o

f am

mon

ia (

)

0 10 20 30 40 50 60 70 80 9001050106010701080109

01101110112011301140115

Sample number

Net

val

ue o

f am

mon

ia (

)

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

Net

val

ue o

f am

mon

ia (

)

Sample number0 50 100 150 200 250

0106010701080109

011011101120113011401150116


Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115






1198911

30

0 0 [minus100 100]119863

10minus5

1198912

120579 0 [minus100 100]119863

10minus5

1198913

0 0 [minus100 100]119863

10minus5

1198914

120579 0 [minus100 100]119863

10minus5

1198915

1 1 [minus100 100]119863 100

1198916

0 0 [minus32 32]119863

10minus5

1198917

120579 0 [minus32 32]119863

10minus5

1198918

0 0 [minus32 32]119863

10minus5

1198919

120579 0 [minus32 32]119863 01

11989110

0 0 [0 600]119863 0001

11989111

120579 0 [minus600 600]119863 001

11989112

0 0 [minus5 5]119863 10

11989113

120579 0 [minus5 5]119863 10

11989114

0 0 [minus5 5]119863 10

11989115

120579 0 [minus5 5]119863 5

11989116

4189829 0 [minus500 500]119863 500

11989117

0 0 [minus100 100]119863 1

11989118

0 0 [minus10 10]119863

10minus5



1198913(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2


(A3)


1198914(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A4)


1198915(119909) =

119863minus1

sum

119894=1

(100(1199092

119894minus 1199092

119894+1)2

+ (119909119894minus 1)2

)


(A5)


1198916(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2

) lowast (1 + 04 |119873 (0 1)|)


(A6)


1198917(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

) lowast (1 + 04 |119873 (0 1)|)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A7)



119863

119863

sum

119894=1

1199092

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119909119894)) + 20 + 119890


(A8)



119863

119863

sum

119894=1

1199112

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119911119894)) + 20 + 119890

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A9)



11989110(119909) =

119863

sum

119894=1

1199092

119894

4000minus

119863

prod

119894=1

cos119909119894

radic119894

+ 1


(A10)


11989111(119909) =

119863

sum

119894=1

1199112

119894

4000minus

119863

prod

119894=1

cos119911119894

radic119894

+ 1

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A11)


11989112(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10)


(A12)


11989113(119909) =

119863

sum

119894=1

(1199112

119894minus 10 cos (2120587119911

119894) + 10)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A13)


11989114(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt1

2round (2119909

119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863



11989115(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119911119894

10038161003816100381610038161199111198941003816100381610038161003816 lt1

2round (2119911

119894)

2

10038161003816100381610038161199111198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A15)


11989116(119909) = 4189829 times 119863 minus

119863

sum

119894=1

119909119894sin (1003816100381610038161003816119909119894

1003816100381610038161003816

12

)



11989118(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863



11989117(119909) =

119863

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119863

prod

119894=1

119909119894


(A18)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













081

12141618

222242628


081

12141618

222242628


081

12141618

222242628


0

05

1

15

2

25

3


Original DECDEAS

22242628

332343638

442


Original DECDEAS

02

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2 024

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

1

2

3

minus3

minus2

minus1









log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

Figure 6 Continued


0

05

1

15

2

Original DECDEAS


Original DECDEAS


minus50 minus05

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)







5 119865119876



7 1198651199022



9 1198651199024








[0 119868

119879

119899

119868119899119870 + 120574

minus1119868] [1198870

119887] = [

0

119910] (19)




119870 = (

11989611


d

1198961198991

sdot sdot sdot 119896119899119899

) (20)



119896119894119895= 119890minus|119909119894minus119909119895|

221205902

(21)

where 119909119894and 119909








0001188




















6 Conclusion


Appendix

(1) Sphere function

1198911(119909) =

119863

sum

119894=1

1199092

119894


(A1)


1198912(119909) =

119863

sum

119894=1

1199112

119894

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A2)


0 50 100 150 200 2500106010701080109

011011101120113011401150116

Sample number

Net

val

ue o

f am

mon

ia (

)

0 10 20 30 40 50 60 70 80 9001050106010701080109

01101110112011301140115

Sample number

Net

val

ue o

f am

mon

ia (

)

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

Net

val

ue o

f am

mon

ia (

)

Sample number0 50 100 150 200 250

0106010701080109

011011101120113011401150116


Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115






1198911

30

0 0 [minus100 100]119863

10minus5

1198912

120579 0 [minus100 100]119863

10minus5

1198913

0 0 [minus100 100]119863

10minus5

1198914

120579 0 [minus100 100]119863

10minus5

1198915

1 1 [minus100 100]119863 100

1198916

0 0 [minus32 32]119863

10minus5

1198917

120579 0 [minus32 32]119863

10minus5

1198918

0 0 [minus32 32]119863

10minus5

1198919

120579 0 [minus32 32]119863 01

11989110

0 0 [0 600]119863 0001

11989111

120579 0 [minus600 600]119863 001

11989112

0 0 [minus5 5]119863 10

11989113

120579 0 [minus5 5]119863 10

11989114

0 0 [minus5 5]119863 10

11989115

120579 0 [minus5 5]119863 5

11989116

4189829 0 [minus500 500]119863 500

11989117

0 0 [minus100 100]119863 1

11989118

0 0 [minus10 10]119863

10minus5



1198913(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2


(A3)


1198914(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A4)


1198915(119909) =

119863minus1

sum

119894=1

(100(1199092

119894minus 1199092

119894+1)2

+ (119909119894minus 1)2

)


(A5)


1198916(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2

) lowast (1 + 04 |119873 (0 1)|)


(A6)


1198917(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

) lowast (1 + 04 |119873 (0 1)|)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A7)



119863

119863

sum

119894=1

1199092

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119909119894)) + 20 + 119890


(A8)



119863

119863

sum

119894=1

1199112

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119911119894)) + 20 + 119890

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A9)



11989110(119909) =

119863

sum

119894=1

1199092

119894

4000minus

119863

prod

119894=1

cos119909119894

radic119894

+ 1


(A10)


11989111(119909) =

119863

sum

119894=1

1199112

119894

4000minus

119863

prod

119894=1

cos119911119894

radic119894

+ 1

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A11)


11989112(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10)


(A12)


11989113(119909) =

119863

sum

119894=1

(1199112

119894minus 10 cos (2120587119911

119894) + 10)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A13)


11989114(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt1

2round (2119909

119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863



11989115(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119911119894

10038161003816100381610038161199111198941003816100381610038161003816 lt1

2round (2119911

119894)

2

10038161003816100381610038161199111198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A15)


11989116(119909) = 4189829 times 119863 minus

119863

sum

119894=1

119909119894sin (1003816100381610038161003816119909119894

1003816100381610038161003816

12

)



11989118(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863



11989117(119909) =

119863

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119863

prod

119894=1

119909119894


(A18)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

05

1

15

2

Original DECDEAS


Original DECDEAS


minus50 minus05

minus40

minus30

minus20

minus10

0

10

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)







5 119865119876



7 1198651199022



9 1198651199024








[0 119868

119879

119899

119868119899119870 + 120574

minus1119868] [1198870

119887] = [

0

119910] (19)




119870 = (

11989611


d

1198961198991

sdot sdot sdot 119896119899119899

) (20)



119896119894119895= 119890minus|119909119894minus119909119895|

221205902

(21)

where 119909119894and 119909








0001188




















6 Conclusion


Appendix

(1) Sphere function

1198911(119909) =

119863

sum

119894=1

1199092

119894


(A1)


1198912(119909) =

119863

sum

119894=1

1199112

119894

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A2)


0 50 100 150 200 2500106010701080109

011011101120113011401150116

Sample number

Net

val

ue o

f am

mon

ia (

)

0 10 20 30 40 50 60 70 80 9001050106010701080109

01101110112011301140115

Sample number

Net

val

ue o

f am

mon

ia (

)

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

Net

val

ue o

f am

mon

ia (

)

Sample number0 50 100 150 200 250

0106010701080109

011011101120113011401150116


Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115






1198911

30

0 0 [minus100 100]119863

10minus5

1198912

120579 0 [minus100 100]119863

10minus5

1198913

0 0 [minus100 100]119863

10minus5

1198914

120579 0 [minus100 100]119863

10minus5

1198915

1 1 [minus100 100]119863 100

1198916

0 0 [minus32 32]119863

10minus5

1198917

120579 0 [minus32 32]119863

10minus5

1198918

0 0 [minus32 32]119863

10minus5

1198919

120579 0 [minus32 32]119863 01

11989110

0 0 [0 600]119863 0001

11989111

120579 0 [minus600 600]119863 001

11989112

0 0 [minus5 5]119863 10

11989113

120579 0 [minus5 5]119863 10

11989114

0 0 [minus5 5]119863 10

11989115

120579 0 [minus5 5]119863 5

11989116

4189829 0 [minus500 500]119863 500

11989117

0 0 [minus100 100]119863 1

11989118

0 0 [minus10 10]119863

10minus5



1198913(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2


(A3)


1198914(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A4)


1198915(119909) =

119863minus1

sum

119894=1

(100(1199092

119894minus 1199092

119894+1)2

+ (119909119894minus 1)2

)


(A5)


1198916(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2

) lowast (1 + 04 |119873 (0 1)|)


(A6)


1198917(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

) lowast (1 + 04 |119873 (0 1)|)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A7)



119863

119863

sum

119894=1

1199092

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119909119894)) + 20 + 119890


(A8)



119863

119863

sum

119894=1

1199112

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119911119894)) + 20 + 119890

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A9)



11989110(119909) =

119863

sum

119894=1

1199092

119894

4000minus

119863

prod

119894=1

cos119909119894

radic119894

+ 1


(A10)


11989111(119909) =

119863

sum

119894=1

1199112

119894

4000minus

119863

prod

119894=1

cos119911119894

radic119894

+ 1

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A11)


11989112(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10)


(A12)


11989113(119909) =

119863

sum

119894=1

(1199112

119894minus 10 cos (2120587119911

119894) + 10)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A13)


11989114(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt1

2round (2119909

119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863



11989115(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119911119894

10038161003816100381610038161199111198941003816100381610038161003816 lt1

2round (2119911

119894)

2

10038161003816100381610038161199111198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A15)


11989116(119909) = 4189829 times 119863 minus

119863

sum

119894=1

119909119894sin (1003816100381610038161003816119909119894

1003816100381610038161003816

12

)



11989118(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863



11989117(119909) =

119863

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119863

prod

119894=1

119909119894


(A18)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














0001188




















6 Conclusion


Appendix

(1) Sphere function

1198911(119909) =

119863

sum

119894=1

1199092

119894


(A1)


1198912(119909) =

119863

sum

119894=1

1199112

119894

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A2)


0 50 100 150 200 2500106010701080109

011011101120113011401150116

Sample number

Net

val

ue o

f am

mon

ia (

)

0 10 20 30 40 50 60 70 80 9001050106010701080109

01101110112011301140115

Sample number

Net

val

ue o

f am

mon

ia (

)

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

Net

val

ue o

f am

mon

ia (

)

Sample number0 50 100 150 200 250

0106010701080109

011011101120113011401150116


Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115






1198911

30

0 0 [minus100 100]119863

10minus5

1198912

120579 0 [minus100 100]119863

10minus5

1198913

0 0 [minus100 100]119863

10minus5

1198914

120579 0 [minus100 100]119863

10minus5

1198915

1 1 [minus100 100]119863 100

1198916

0 0 [minus32 32]119863

10minus5

1198917

120579 0 [minus32 32]119863

10minus5

1198918

0 0 [minus32 32]119863

10minus5

1198919

120579 0 [minus32 32]119863 01

11989110

0 0 [0 600]119863 0001

11989111

120579 0 [minus600 600]119863 001

11989112

0 0 [minus5 5]119863 10

11989113

120579 0 [minus5 5]119863 10

11989114

0 0 [minus5 5]119863 10

11989115

120579 0 [minus5 5]119863 5

11989116

4189829 0 [minus500 500]119863 500

11989117

0 0 [minus100 100]119863 1

11989118

0 0 [minus10 10]119863

10minus5



1198913(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2


(A3)


1198914(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A4)


1198915(119909) =

119863minus1

sum

119894=1

(100(1199092

119894minus 1199092

119894+1)2

+ (119909119894minus 1)2

)


(A5)


1198916(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2

) lowast (1 + 04 |119873 (0 1)|)


(A6)


1198917(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

) lowast (1 + 04 |119873 (0 1)|)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A7)



119863

119863

sum

119894=1

1199092

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119909119894)) + 20 + 119890


(A8)



119863

119863

sum

119894=1

1199112

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119911119894)) + 20 + 119890

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A9)



11989110(119909) =

119863

sum

119894=1

1199092

119894

4000minus

119863

prod

119894=1

cos119909119894

radic119894

+ 1


(A10)


11989111(119909) =

119863

sum

119894=1

1199112

119894

4000minus

119863

prod

119894=1

cos119911119894

radic119894

+ 1

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A11)


11989112(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10)


(A12)


11989113(119909) =

119863

sum

119894=1

(1199112

119894minus 10 cos (2120587119911

119894) + 10)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A13)


11989114(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt1

2round (2119909

119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863



11989115(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119911119894

10038161003816100381610038161199111198941003816100381610038161003816 lt1

2round (2119911

119894)

2

10038161003816100381610038161199111198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A15)


11989116(119909) = 4189829 times 119863 minus

119863

sum

119894=1

119909119894sin (1003816100381610038161003816119909119894

1003816100381610038161003816

12

)



11989118(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863



11989117(119909) =

119863

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119863

prod

119894=1

119909119894


(A18)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 2500106010701080109

011011101120113011401150116

Sample number

Net

val

ue o

f am

mon

ia (

)

0 10 20 30 40 50 60 70 80 9001050106010701080109

01101110112011301140115

Sample number

Net

val

ue o

f am

mon

ia (

)

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (

)

0106010701080109

011011101120113011401150116

Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115

Net

val

ue o

f am

mon

ia (

)

Sample number0 50 100 150 200 250

0106010701080109

011011101120113011401150116


Net

val

ue o

f am

mon

ia (

)

Sample number0 10 20 30 40 50 60 70 80 90

0107

0108

0109

011

0111

0112

0113

0114

0115






1198911

30

0 0 [minus100 100]119863

10minus5

1198912

120579 0 [minus100 100]119863

10minus5

1198913

0 0 [minus100 100]119863

10minus5

1198914

120579 0 [minus100 100]119863

10minus5

1198915

1 1 [minus100 100]119863 100

1198916

0 0 [minus32 32]119863

10minus5

1198917

120579 0 [minus32 32]119863

10minus5

1198918

0 0 [minus32 32]119863

10minus5

1198919

120579 0 [minus32 32]119863 01

11989110

0 0 [0 600]119863 0001

11989111

120579 0 [minus600 600]119863 001

11989112

0 0 [minus5 5]119863 10

11989113

120579 0 [minus5 5]119863 10

11989114

0 0 [minus5 5]119863 10

11989115

120579 0 [minus5 5]119863 5

11989116

4189829 0 [minus500 500]119863 500

11989117

0 0 [minus100 100]119863 1

11989118

0 0 [minus10 10]119863

10minus5



1198913(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2


(A3)


1198914(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A4)


1198915(119909) =

119863minus1

sum

119894=1

(100(1199092

119894minus 1199092

119894+1)2

+ (119909119894minus 1)2

)


(A5)


1198916(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2

) lowast (1 + 04 |119873 (0 1)|)


(A6)


1198917(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

) lowast (1 + 04 |119873 (0 1)|)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A7)



119863

119863

sum

119894=1

1199092

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119909119894)) + 20 + 119890


(A8)



119863

119863

sum

119894=1

1199112

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119911119894)) + 20 + 119890

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A9)



11989110(119909) =

119863

sum

119894=1

1199092

119894

4000minus

119863

prod

119894=1

cos119909119894

radic119894

+ 1


(A10)


11989111(119909) =

119863

sum

119894=1

1199112

119894

4000minus

119863

prod

119894=1

cos119911119894

radic119894

+ 1

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A11)


11989112(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10)


(A12)


11989113(119909) =

119863

sum

119894=1

(1199112

119894minus 10 cos (2120587119911

119894) + 10)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A13)


11989114(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt1

2round (2119909

119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863



11989115(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119911119894

10038161003816100381610038161199111198941003816100381610038161003816 lt1

2round (2119911

119894)

2

10038161003816100381610038161199111198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A15)


11989116(119909) = 4189829 times 119863 minus

119863

sum

119894=1

119909119894sin (1003816100381610038161003816119909119894

1003816100381610038161003816

12

)



11989118(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863



11989117(119909) =

119863

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119863

prod

119894=1

119909119894


(A18)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198911

30

0 0 [minus100 100]119863

10minus5

1198912

120579 0 [minus100 100]119863

10minus5

1198913

0 0 [minus100 100]119863

10minus5

1198914

120579 0 [minus100 100]119863

10minus5

1198915

1 1 [minus100 100]119863 100

1198916

0 0 [minus32 32]119863

10minus5

1198917

120579 0 [minus32 32]119863

10minus5

1198918

0 0 [minus32 32]119863

10minus5

1198919

120579 0 [minus32 32]119863 01

11989110

0 0 [0 600]119863 0001

11989111

120579 0 [minus600 600]119863 001

11989112

0 0 [minus5 5]119863 10

11989113

120579 0 [minus5 5]119863 10

11989114

0 0 [minus5 5]119863 10

11989115

120579 0 [minus5 5]119863 5

11989116

4189829 0 [minus500 500]119863 500

11989117

0 0 [minus100 100]119863 1

11989118

0 0 [minus10 10]119863

10minus5



1198913(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2


(A3)


1198914(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A4)


1198915(119909) =

119863minus1

sum

119894=1

(100(1199092

119894minus 1199092

119894+1)2

+ (119909119894minus 1)2

)


(A5)


1198916(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119909119894)

2

) lowast (1 + 04 |119873 (0 1)|)


(A6)


1198917(119909) = (

119863

sum

119894=1

(

119894

sum

119895=1

119911119894)

2

) lowast (1 + 04 |119873 (0 1)|)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A7)



119863

119863

sum

119894=1

1199092

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119909119894)) + 20 + 119890


(A8)



119863

119863

sum

119894=1

1199112

119894)

minus exp( 1119863

119863

sum

119894=1

cos (2120587119911119894)) + 20 + 119890

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A9)



11989110(119909) =

119863

sum

119894=1

1199092

119894

4000minus

119863

prod

119894=1

cos119909119894

radic119894

+ 1


(A10)


11989111(119909) =

119863

sum

119894=1

1199112

119894

4000minus

119863

prod

119894=1

cos119911119894

radic119894

+ 1

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A11)


11989112(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10)


(A12)


11989113(119909) =

119863

sum

119894=1

(1199112

119894minus 10 cos (2120587119911

119894) + 10)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A13)


11989114(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt1

2round (2119909

119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863



11989115(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119911119894

10038161003816100381610038161199111198941003816100381610038161003816 lt1

2round (2119911

119894)

2

10038161003816100381610038161199111198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A15)


11989116(119909) = 4189829 times 119863 minus

119863

sum

119894=1

119909119894sin (1003816100381610038161003816119909119894

1003816100381610038161003816

12

)



11989118(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863



11989117(119909) =

119863

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119863

prod

119894=1

119909119894


(A18)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











11989110(119909) =

119863

sum

119894=1

1199092

119894

4000minus

119863

prod

119894=1

cos119909119894

radic119894

+ 1


(A10)


11989111(119909) =

119863

sum

119894=1

1199112

119894

4000minus

119863

prod

119894=1

cos119911119894

radic119894

+ 1

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A11)


11989112(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10)


(A12)


11989113(119909) =

119863

sum

119894=1

(1199112

119894minus 10 cos (2120587119911

119894) + 10)

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A13)


11989114(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt1

2round (2119909

119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863



11989115(119909) =

119863

sum

119894=1

(1199102

119894minus 10 cos (2120587119910

119894) + 10)

119910119894=

119911119894

10038161003816100381610038161199111198941003816100381610038161003816 lt1

2round (2119911

119894)

2

10038161003816100381610038161199111198941003816100381610038161003816 ge1

2

for 119894 = 1 2 119863

119911 = 119909 minus 120579

120579 = [1205791 1205792 120579


(A15)


11989116(119909) = 4189829 times 119863 minus

119863

sum

119894=1

119909119894sin (1003816100381610038161003816119909119894

1003816100381610038161003816

12

)



11989118(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863



11989117(119909) =

119863

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119863

prod

119894=1

119909119894


(A18)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article steady modeling for an ammonia synthesis

Documents