Research ArticleImproved Quantum Genetic Algorithm in Application ofScheduling Engineering Personnel

Huaixiao Wang Ling Li Jianyong Liu Yong Wang and Chengqun Fu

College of Field Engineering PLA University of Science and Technology Nanjing 210007 China

Correspondence should be addressed to Huaixiao Wang 753048237qqcom

Received 22 January 2014 Revised 5 June 2014 Accepted 13 July 2014 Published 23 July 2014

Academic Editor Mohammad T Darvishi

Copyright copy 2014 Huaixiao Wang et alThis 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

To verify the availability of the improved quantum genetic algorithm in solving the scheduling engineering personnel problemthe following work has been carried out the characteristics of the scheduling engineering personnel problem are analyzed thequantum encoding method is proposed and an improved quantum genetic algorithm is applied to address the issue Taking thelow efficiency and the bad performance of the conventional quantumgenetic algorithm into account a universal improved quantumgenetic algorithm is introduced to solve the scheduling engineering personnel problem Finally the examples are applied to verifythe effectiveness and superiority of the improved quantum genetic algorithm and the rationality of the encoding method

1 Introduction

In engineering work the number of participants is large andthe qualities of the personnel are uneven so the time of eachgroup carrying out an engineering work is different the pro-cesses of engineering work are somany that the scheduling ofthe engineering personnel is complex [1]There is parallelismin some processes of engineering work Any work groupcan be selected for any process of engineering work and theoperating times of different work groups are different Thesefactors increase the complexity and flexibility of schedulingDifferent scheduling schemes cause different total operationtime Proper scheduling scheme can improve efficiency andsave time [2] So solving the personnel scheduling problemof engineering work is one of the key aspects to improve theefficiency of engineering work

Scheduling engineering personnel is a complex NP-hard problem [3] Currently the main methods to solvesuch problems contain branch-bound method heuristicalgorithms and so on but these algorithms can only solvethe scheduling engineering personnel problem with a smallscale and there are imperfections in coding method of thesealgorithms which may lead to illegal solutions [4] Theimproved quantum genetic algorithm has better robustness

versatility excellent computing power implicit parallelismand global search capability and can be successful in solvingthe scheduling engineering operating personnel problem

2 Description of Scheduling EngineeringOperating Personnel Problem

Scheduling engineering operating personnel problem is theextension of standard scheduling personnel problem

21 Standard Scheduling Personnel Problem The standardpersonnel scheduling problem can be described as followssuppose there are 119898 tasks which need to be finished all ofwhich can be divided into two stages119860 and 119861The two stagesmust be finished in order as in Figure 1 Every stage of thetask is finished by only one work group The object is to findan order to make the time from the beginning of the first taskto the end of the last task shortest

The rule of sorting of the two-stage engineering work hasbeen put forward by Johnson in 1954 which is as follows

min (119886119894 119887119895) le min (119886

119895 119887119894) (1)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 164961 10 pageshttpdxdoiorg1011552014164961

2 Abstract and Applied Analysis

t

Aai

Task i

t

Task i minus 1

B

Task i minus 1

When t le ai

When t ge ai

bi

bi

t minus ai + bi

Figure 1 Diagram of standard scheduling personnel problem

J1 J2 J3 JSmiddot middot middot

middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middot

middot middot middot

middot middot middot

M12

M1m M2n M3t MSn

MS2

MS1M11M21

M22 M32

M31

Figure 2 Distribution of engineering personnel

Equation (1) is the condition that task 119894 is ordered beforethe task 119895 where 119886

119894is the time needed by the first stage of task

119894 and 119887119895is the time needed by the second stage of task 119895

22 Extension of Scheduling Engineering Personnel ProblemExtension of scheduling engineering personnel problem isbased on the standard personnel scheduling problem Thestages of the task needed to be done are more than two thework groups for finishing every stage of the task aremore thanone and the time in which different work teams finish thetask is different There is not an authentic rule of sorting ofthe extension of scheduling engineering personnel problem

Extension of scheduling engineering operating personnelproblem can be described as follows There are a numberof engineering works which can be decomposed into acertain number of steps and at least one of all steps can becompleted in parallel by multiple work groups The problemis to determine the distribution of work groups in parallelThat is there are 119899 work groups and 119873 engineering works119882111988221198823 119882

119873 Decompose these engineering works

into 119878 processes uniformly 1198691 1198692 1198693 119869

119878 Each process

119869119894has 119885

119894parallel work group (1 le 119894 le 119878 1 le 119885

119894le

119899sum119885119894gt 119878) The figure of scheduling engineering operating

personnel problem is shown in Figure 2 The target is tominimize the maximum flow time through conducting anoptimal allocation [5]

Each process of each engineering work can be completedby any parallel team of corresponding process but the work

times are different [6] The specific example is shown inTable 1 119869

119894stands for the 119894th process 119872119894

119895stands for the

119895th parallel work group of the 119894th process and the dataof 119872119894

119895stands for the work time that the 119895th parallel

work group needs to complete the 119894th process Table 1 isan operational timetable which includes four engineeringworks (119882

1119882211988231198824) three processes (119869

1 1198692 1198693) and nine

engineering work groupsThe engineering personnel scheduling problem in this

paper satisfies the following constraints (1) different engi-neering works have the same priority (2) there is no orderconstraint between different engineering works but theprocesses of the same engineering work are successive [7](3) any engineering work can start at the beginning (4) onework group only can carry out one engineering work at thesame time (5) the same process of the same engineeringworkcan only be completed by one work group (6) Once startedeach process of the engineering work cannot be interrupted(7) The only performance is the maximum completion time119879max that is the purpose of scheduling is to minimize themaximum completion time

min119879max = min (max119879119896) 1 le 119896 le 119898 (2)

where 119879119896is the completion time of the work group 119872119904

119901

1 le 119901 le 119885119904 (119885119904 is the total number of work groups of eachprocess) and119872119904

119901is the 119901th parallel work group of the final

process

3 Calculation of Engineering Work Time

We need to follow certain principles to calculate the worktime of the determined scheme to minimize the work timewhen the scheduling scheme of engineering work groupsis determined During the whole engineering work theconsuming time is divided into two parts the working timeand the waiting time [8]

Suppose the 119900119896th work group of the work groups of the

first process 119872(1)(119900119896)

needs cost 119883(2)(119900119896)119896

to complete the firstprocess of the 119896th engineering work the 119901

119896th work group of

the work groups of the second process119872(2)(119901119896)

needs cost119883(2)(119901119896)119896

to complete the second process and the 119902119896th work group of

Abstract and Applied Analysis 3

Table 1 Example of scheduling engineering personnel problem

Engineering worksProcesses

1198691

1198692

1198693

11987211

11987212

11987213

11987214

11987221

11987222

11987231

11987232

11987233

1198821

3 2 4 3 3 4 5 3 41198822

4 5 3 2 6 5 5 6 31198823

4 3 5 6 4 2 3 4 51198824

2 4 1 3 5 3 4 6 2

the work groups of the third process119872(3)(119902119896)

needs cost 119883(2)(119901119896)119896

to complete the third process The total time the first processof the 119896th engineering work costs is119872(1)

(119900119896)119896 the total time the

119901119896minus1

th work group119872(2)(119901119896minus1)

costs on the second process of the1198961th engineering work is119872(2)

(119901119896minus1)119896minus1 (119896 gt 1) Calculation of

119872(2)

(119901119896)119896is divided into two situations

(1) when119872(1)(119900119896)119896

ge 119872(2)

(119901119896minus1)119896minus1119872(2)(119901119896)119896

= 119872(1)

(119900119896)119896+ 119883(2)

(119901119896)119896

(2) when119872(1)(119900119896)119896

lt 119872(2)

(119901119896minus1)119896minus1119872(2)(119901119896)119896

= 119872(2)

(119901119896minus1)119896minus1+119883(2)

(119901119896)119896

Take the two situations together the formula of calcu-lating the total time 119872(2)

(119901119896)119896of the 119901

119896th work group 119872(2)

(119901119896)

completing the second process of the 119896th engineering workis

119872(2)

(119901119896)119896= max (119872(1)

(119900119896)119896119872(2)

(119901119896minus1)119896minus1) + 119883

(2)

(119901119896)119896 (3)

Similarly similar formula of the third step can beobtained The specific formulas are in Table 2 The formulasin Table 2 indicate the total time119872119894

119895of the 119896th engineering

work cost from the beginning to the 119894th process

4 The Improved Quantum GeneticAlgorithm Used to Solve the SchedulingEngineering Personnel Problem

Quantum genetic algorithm (QGA) is the product of thecombination of quantum computation and genetic algo-rithms and it is a new evolutionary algorithm of probability[9] It is successfully used to solve the TSP problem [10] Thequantum state vector is introduced in the genetic algorithmto express genetic code and quantum logic gates are usedto realize the chromosome evolution By these means betterresults are achieved [11] Genetic algorithm has been widelyapplied in scheduling engineering personnel [12ndash14] Theimproved quantum genetic algorithm is applied to solvethe scheduling engineering personnel problem in order toachieve the optimization of scheduling engineering person-nel

41 The Qubit Encoding of Scheduling Engineering PersonnelProblem Ensuring that the encoding method is the mostimportant for applying improved quantum genetic algorithm

to solve the scheduling engineering personnel problem Thequbit encoding needs to combine the optimization problemwith the improved quantum genetic algorithmThe engineer-ing works which can be divided into the same number ofprocesses need to be completed and there is at least oneprocess that can be completed by multiple parallel workgroups It needs to ensure the distribution of parallel workgroups in order to minimize the maximum completion time

The features of scheduling engineering personnel prob-lemare as follows for example there are119873 engineeringworksthat need to be completed every work must be completedthrough 119878 processes The number of parallel work groups ofevery process is 119885

119894(1 le 119894 le 119878) There is at least one process

that can be completed by multiple parallel work groups thatis there is at least one119885

119895which is greater than one Construct

a119873 times 119878 dimensional matrix

119860119873times119878

=

[

[

[

[

[

1198861111988612sdot sdot sdot 1198861119878

1198862111988622sdot sdot sdot 1198862119878

11988611198781198862119878sdot sdot sdot 119886119873119878

]

]

]

]

]

(4)

where the element 119886119894119895of matrix is a random real number

between the ranges [1 119885119895+ 1) which means that the 119895th

process of the 119894th engineering work is completed by theint 119886119894119895th work group (int means getting the greatest integer

which is not greater than 119886119894119895)

If int 1198861198951198961

= int 1198861198951198962(1198961

= 1198962) it indicates that the same

processes of multiple engineering works are completed bythe same work group Use variable 119899

119894(119894 = 1 sum119885

119895) to

record the number of engineering works every work groupcompletes

Accord the structure of (2) to design the chromosome ofquantum genetic algorithm

chromosome119896= [119873 119878 (

11986911198692sdot sdot sdot 119869119904

11988511198852sdot sdot sdot 119885

119904

)

11988611 11988612 119886

1119878 119886

1198731 119886

119873119878]

(5)

4 Abstract and Applied Analysis

Table 2 Total time of completion of every process

Work ProcessProcess 1 Process 2 Process 3 sdot sdot sdot Process 119894

1 0 + 119883(1)

(1199001)1119872(1)

(1199001)1+ 119883(2)

(1199011)1119872(2)

(1199011)1+ 119883(3)

(1199021)1sdot sdot sdot 119872

(119894minus1)

(1199101)1+ 119883(119894)

(1199111)1

2 119872(1)

(1199001)1+ 119883(1)

(1199002)2max (119872(1)

(1199002)2119872(2)

(1199011)1) + 119883

(2)

(1199012)2max (119872(2)

(1199012)2119872(3)

(1199021)1) + 119883

(3)

(1199022)2sdot sdot sdot max (119872(119894minus1)

(1199102)2119872(119894)

(1199111)1) + 119883

(119894)

(1199112)2

3 119872(1)

(1199002)2+ 119883(1)

(1199003)3max (119872(1)

(1199003)3119872(2)

(1199012)2) + 119883

(2)

(1199012)3max (119872(2)

(1199013)3119872(3)

(1199022)2) + 119883

(3)

(1199023)3sdot sdot sdot max (119872(119894minus1)

(1199103)3119872(119894)

(1199112)2) + 119883

(119894)

(1199113)3

119896 119872(1)

(119900119896minus1)119896minus1+ 119883(1)

(119900119896)119896max (119872(1)

(119900119896)119896119872(2)

(119901119896minus1)119896minus1) + 119883

(2)

(119901119896)119896max (119872(2)

(119901119896)119896119872(3)

(119902119896minus1)119896minus1) + 119883

(3)

(119902119896)119896sdot sdot sdot max (119872(119894minus1)

(119910119896)119896119872(119894)

(119911119896minus1)119896minus1) + 119883

(119894)

(1199112)119896

Themathematical model ismin = max(119872(2)(119901119896)119896119872(3)

(119902119896minus1)119896minus1) + 119883(3)

(119902119896)119896 where 119896 is the number of engineering works

The improved quantum genetic algorithm adopted in thispaper is the algorithm based on the algorithm in document

[11] The multiqubits encoding which has 119898 parameters isdefined as follows

119902119905

119895=[

[

120572119905

11

120573119905

11

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

12

120573119905

12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572119905

1119896

120573119905

1119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

21

120573119905

21

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

22

120573119905

22

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572119905

2119896

120573119905

2119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

1198981

120573119905

1198981

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

1198982

120573119905

1198982

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572119905

119898119896

120573119905

119898119905

]

]

(6)

where 119902119905119895represents the 119895th individual chromosome of the tth

generation 119896 represents the number of qubit encoding of eachgene 119898 represents the number of genes in the chromosome

[15] Initialize the quantum encoding (120572 120573) of each individualin the population with (1radic2 1radic2) which indicates thatwhen 119905 = 0 the possibility of each state expressed by achromosome is equal [16]

The allocation coding matrix is encoded by qubit as

119876 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120572111

120573111

10038161003816100381610038161003816100381610038161003816

120572112

120573112

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572111198761

120573111198761

120572121

120573121

10038161003816100381610038161003816100381610038161003816

120572122

120573122

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572121198762

120573121198762

sdot sdot sdot

12057211198781

12057311198781

10038161003816100381610038161003816100381610038161003816

12057211198782

12057311198782

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

1205721119878119876119878

1205731119878119876119878

120572211

120573211

10038161003816100381610038161003816100381610038161003816

120572212

120573212

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572211198761

120573211198761

120572221

120573221

10038161003816100381610038161003816100381610038161003816

120572222

120573222

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572221198762

120573221198762

sdot sdot sdot

12057221198781

12057321198781

10038161003816100381610038161003816100381610038161003816

12057221198782

12057321198782

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

1205722119878119876119878

1205732119878119876119878

d

12057211987311

12057311987311

10038161003816100381610038161003816100381610038161003816

12057211987312

12057311987312

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

12057211987311198761

12057311987311198761

12057211987321

12057311987321

10038161003816100381610038161003816100381610038161003816

12057211987322

12057311987322

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

12057211987321198762

12057311987321198762

sdot sdot sdot

1205721198731198781

1205731198731198781

10038161003816100381610038161003816100381610038161003816

1205721198731198782

1205731198731198782

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572119873119878119876119878

120573119873119878119876119878

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(7)

The element in (7) is1205721198941198951

1205731198941198951

100381610038161003816100381610038161003816

1205721198941198952

1205731198941198952

100381610038161003816100381610038161003816

sdotsdotsdot

sdotsdotsdot

100381610038161003816100381610038161003816

120572119894119895119876119895

120573119894119895119876119895which represents

the qubit encoding of the ordinal number of work teamwhere 119894 (119894 = 1 2 119873) represents the ordinal number ofengineering work 119895 (119894 = 1 2 119878) represents the ordinalnumber of process and 119876

119896(119896 = 1 119878) represents the

length of the qubit encoding usually the more work groupsthere are for the process 119896 the longer the length is Thenumber 119886

119894119895generated by qubit encoding which represents the

119895th process of the 119894th work is finished by the 119886119894119895th work group

42 The Solution Based on Improved Quantum Genetic Algo-rithm The solution should be solved based on the qubitcoding

421 Quantum Rotating Gates Quantum genetic algorithmapplies the probability amplitude of qubits to encode chro-mosome and uses quantum rotating gates to realize chromo-somal updated operation

Abstract and Applied Analysis 5

Quantum rotating gates can be designed according to thepractical problems and usually be defined as

119880 (120579119894) = [

cos 120579119894minus sin 120579

119894

sin 120579119894

cos 120579119894

] (8)

The updated process is

[

[

1205721015840

119894

1205731015840

119894

]

]

= 119880 (120579119894) [

120572119894

120573119894

] = [

cos 120579119894minus sin 120579

119894

sin 120579119894

cos 120579119894

] [

120572119894

120573119894

] (9)

where (120572119894 120573119894)119879 and (1205721015840

119894 1205731015840

119894)

119879 are the probability amplitudes ofthe 119894th qubit in chromosome before and after the quantumrotating gates updating respectively 120579

119894is the rotating angle

and the value and the sign of 120579119894are determined by the adjust-

ment strategy [17 18] 120579119894is static in conventional quantum

genetic algorithmThevalue of 120579119894can be dynamically adjusted

based on the evolutionary process in the improved quantumgenetic algorithm Adjust the value of the rotating angleaccording to the number of generation the regulator is asfollows

120579119894= 120579max minus (

120579max minus 120579min119894119905max

) lowast iter (10)

where 120579119894is the value of the rotating angle of the 119894th generation

120579max is the maximal rotating angle 120579min is the minimalrotating angle 119894119905max is the maximal generation and iter isthe current generation 120579max = 001120587 120579min = 0001120587

The adjustment strategy is as follows comparing thefitness of the currently measured value 119891(119909) of the individual119902119905

119895with the fitness of the current optimal individual 119891(best

119894)

if 119891(119909) gt 119891(best) then adjust the corresponding qubits of119902119905

119895 making the probability amplitude (120572

119894 120573119894) evolve toward

the direction that is propitious to the emergence of 119909119894

Conversely if 119891(119909) lt 119891(best) then adjust the correspondingqubits of 119902119905

119895 making the probability amplitude (120572

119894 120573119894) evolve

toward the direction that is propitious to the emergence ofbest

Construct a determinant119860 = 10038161003816100381610038161003816

120572119887 120572119894

120573119887 120573119894

10038161003816100381610038161003816 where (120572

119887 120573119887) is the

probability amplitude of some qubit of the optimal solutionwhich has been searched (120572

119894 120573119894) is probability amplitude of

corresponding state of qubit in the current solution When119909119894= best119894 if 119860 = 0 then the direction of the rotating angle is

minussgn(119860) If 119860 = 0 the direction of the rotating angle can beeither positive of negative

422 Quantum Mutation Add quantum mutation opera-tion in conventional quantum genetic algorithm Quantummutation enables some individual to deviate slightly fromthe current evolutionary direction and prevent the evolutionof individual into a local optimal solution [19] Quantummutation can completely reverse the individualrsquos evolutionarydirection by swapping the value of probability amplitude ofqubits (120572 120573) Quantum NOT gates are adopted to realizechromosomal variation

Table 3 One result of the whole interference crossover

Chromosomenumber New chromosome

1 A(1) D(2) C(3) B(4) A(5) D(6) C(7) B(8)

2 B(1) A(2) D(3) C(4) B(5) A(6) D(7) C(8)

3 C(1) B(2) A(3) D(4) C(5) B(6) A(7) D(8)

4 D(1) C(2) B(3) A(4) D(5) C(6) B(7) A(8)

423 QuantumDisaster Add quantum disaster operation inthe conventional quantum genetic algorithm The methodis to apply a large disturbance to some individuals inthe population and regenerate some other new randomindividuals [20]

424 Whole Interference Crossover Whole interferencecrossover operator is utilized to realize evolution insteadof the traditional crossover operator All chromosomes areinvolved in the whole interference crossover Set the numberof population to be 119898 and length of chromosome is 119899(119898 lt 119899) 119860

119894119895is the 119895th bit of the 119894th individual Then the

chromosome of the 119894th individual of the new family throughthewhole interference crossover is119860

(119894 mod 119898)1minus119860 (119894+1 mod 119898)2 minussdot sdot sdot minus 119860

(119894+119896 mod 119898)(119896+1) minus sdot sdot sdot minus 119860 (119894+119899minus1 mod 119898)119899 The new wholeinterference crossover operator is mainly used when thealgorithm converges to local optimal solution to generate newindividuals to overcome the prematurity in later evolutionSuppose that there are 4 populations and the lengths of themare all 8 One result of the whole interference crossover is asin Table 3

It is important to note that quantum rotating gatesquantummutation and quantum disaster all act on quantumchromosome However whole interference crossover acts onthe chromosome generated by quantum chromosome Theevolution performance will not be introduced if the wholeinterference crossover acts on quantum chromosome

The improved strategy of quantum genetic algorithmmainly contains self-adaptive rotating angle strategy theimprovement of program structure of quantum evolutionadded quantum mutation operation quantum disaster oper-ation and whole interference crossover operation Theflowchart of improved quantum genetic algorithm is shownin Figure 3

43 The Determining Flow of Scheduling Engineering Per-sonnel The steps to determine the scheme of schedulingengineering personnel are as follows

Step 1 (initialize distribution scheme) Initialize allocationscheme to form the initial population according to the needsof the evolution of quantum genetic algorithm and theassignment matrix

Step 2 (time calculation) Calculate the time of all theallocated schemes in the population

6 Abstract and Applied Analysis

Start

Satisfy final condition

Memorize the best individual and its fitness

End

Yes

No

Colony disaster

Yes

No

Initialize the population Q(t0) and every parameter

Test every individual of Q(t0)

Evaluate the fitness of every individual of Q(t0)

Set the best individual as the evolutional goal of next generation

Test every individual of Q(t0)

Quantum rotating door update population get Q(t + 1)

Quantum mutation whole interference crossover

Whether or not disaster

t = t + 1

Figure 3 Flowchart of improved quantum genetic algorithm

Start

Initialize distribution scheme

Satisfy final condition

No

Yes

End

Calculate the time of the allocated schemes in the population

Record the shortest work time of the corresponding scheme

Optimize the allocated scheme

Calculate the shortest time of the optimal allocated scheme

Figure 4 Flowchart of acquiring the optimal allocated scheme and the shortest work time

Abstract and Applied Analysis 7

Step 3 (optimize the allocation scheme) Apply the improvedquantumgenetic algorithm to optimize the allocation schemein order to get the optimal allocation scheme Go to Step 2according to the termination condition to do the judgmentif the termination condition is satisfied then go to the End orelse do the loop to Step 3

Step 4 (determination of the optimal allocation schemeand the shortest work time) Choose the optimal allocationscheme form the population satisfied the termination con-dition Calculate the work time of the optimal allocationscheme

The flowchart is shown in Figure 4

5 Simulation Analyses

Take some road building project as an example There are12 roads which need to be built The building work canbe divided into three procedures prospecting drawing andlocation designing and constructionThere are 3 prospectinggroups 2 drawing and location designing groups and 3 con-struction groups The capacities of the groups are differentThe specific work times are shown in Table 4

The specific steps of acquiring the optimal allocatedscheme and the shortest work time are as follows

(1) Construct 12 times 3 dimensional matrix

11986012times3

=

[

[

[

[

[

11988611

11988612

11988613

11988621

11988622

11988623

119886121

119886122

119886123

]

]

]

]

]

(11)

where 1 le 11988611 11988621 11988631 119886

121lt 4 1 le

11988612 11988622 11988632 119886

122lt 3 1 le 119886

13 11988623 11988633 119886

123lt 4

The corresponding qubit chromosome is

11987612times3

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120572111

120573111

10038161003816100381610038161003816100381610038161003816

120572112

120573112

120572121

120573121

120572131

120573131

10038161003816100381610038161003816100381610038161003816

120572132

120573132

120572211

120573211

10038161003816100381610038161003816100381610038161003816

120572212

120573212

120572221

120573221

120572231

120573231

10038161003816100381610038161003816100381610038161003816

120572232

120573232

1205721211

1205731211

10038161003816100381610038161003816100381610038161003816

1205721212

1205731212

1205721221

1205731221

1205721231

1205731231

10038161003816100381610038161003816100381610038161003816

1205721232

1205731232

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(12)

Suppose the generated matrix is

11986012times3

=

[

[

[

[

[

11988611

11988612

11988613

11988621

11988622

11988623

119886121

119886122

119886123

]

]

]

]

]

=[

[

12 23 1 36 21 18 13 32 25 27

29 12 16 23 11 25 27 14 22 13

37 15 24 16 33 28 25 36 19 21

34 17

16 28

19 22

]

]

119879

1198601015840

12times3=[

[

1 2 1 3 2 1 1 3 2 2

2 1 1 2 1 2 2 1 2 1

3 1 2 1 3 2 2 3 1 2

3 1

1 2

1 2

]

]

119879

(13)

Then the encoding section of the assigned chromosome is

[1 2 1 3 2 1 1 3 2 2 3 1 2 1 1 2 1 2 2 1 2

1 1 2 3 1 2 1 3 2 2 3 1 2 1 2]

(14)

(2) Seek the values of time 1198871198961198941198861015840

119894119895

corresponding with1198821198961198721198941198861015840

119896119894

(where 1 le 119896 le 12 1 le 119894 le 3 119894 119895 119896are integers) in Table 2 Then the work timetable ofevery engineeringwork is gotten InTable 5 the figuredenotes the work time and the subscript denotes thework group

(3) Apply the method of Section 3 to calculate the worktimes of every work group according to the orderdetermined by Step (2) and the time determined byStep (3) Then find the maximum value as the totalengineering work time

Apply GA (genetic algorithm) QGA (quantum geneticalgorithm) and IQGA (improved quantum genetic algo-rithm) to solve the problem of scheduling engineering per-sonnelThe three kinds of algorithmarewritten and compiledin MATLAB The simulation test in this paper is based onthe AMD CPU 18GHz 15 GB RAM The parameters of GAare set as follows GA is encoded by real number the totalevolution generation is 80 the size of population is 40 and theprobability ofmutation is 001The parameters ofQGAare setas follows QGA is encoded by the binary the total evolutiongeneration is 80 the size of population is 40 the length ofthe variable expressing 3 work groups is 2 and the length ofthe variable expressing 2 work groups is 1 The parametersof IQGA are set as follows IQGA is encoded by the binarythe total evolution generation is 80 the size of population is40 the length of the variable expressing 3 work groups is 2the length of the variable expressing 2 work groups is 1 andthe probability of mutation is 001 The fitness function is theobjective function The fitness function tends to the optimal

8 Abstract and Applied Analysis

Table 4 Work time of every work group

1198691

1198692

1198693

11987211

11987212

11987213

11987221

11987222

11987231

11987232

11987233

1198821

3 2 4 3 4 5 3 41198822

4 5 3 6 5 5 6 31198823

4 3 5 4 2 3 4 51198824

2 5 3 5 4 1 5 41198825

4 2 6 3 1 3 4 61198826

1 5 3 2 3 5 2 71198827

7 6 8 4 6 6 7 61198828

6 4 5 7 5 4 8 51198829

2 5 4 1 2 3 6 311988210

3 6 4 3 4 4 3 911988211

5 2 3 3 5 8 5 411988212

4 3 4 6 5 4 7 5

Table 5 Work time of every engineering work and correspondingwork group

Engineering work Process1198691

1198692

1198693

1198821

31

42

43

1198822

52

61

51

1198823

41

41

42

1198824

33

42

11

1198825

22

31

63

1198826

11

32

22

1198827

71

62

72

1198828

53

71

53

1198829

52

22

31

11988210

62

31

32

11988211

33

31

81

11988212

41

52

72

value while the evolution increases Simulate the process ofevolution Ten rounds of experiments were presented in thispaper The simulation results of the 3 kinds of algorithm arein Tables 6 7 and 8 The result of some time of the IQGA isshown in Figures 5 6 and 7

The statistical results of the experiment are shown inTable 9

Table 9 shows that compared with GA QGA can get itssmallest value in a short time however QGA can hardly getthe optimal result Though the evolution time of the IQGA islonger and the evolution generation of the IQGA is more theIQGA can get the optimal result in a bigger probability

In Figure 5 x-axis represents the evolution generationsand y-axis represents the work timeThe work time graduallybecomes shorter with the increase of the generation

In Figure 6 x-axis represents the evolution algebra andy-axis represents the work time The average work time of allindividuals has a decreasing trend but there is a gap betweenthe optimal value and the average work time

0 10 20 30 40 50 60 70 8026

27

28

29

30

31

32

Evolutionary generations

The b

est fi

tnes

s of e

very

gen

erat

ion

Figure 5The changing curve of work time of the best individual inthe population

0 10 20 30 40 50 60 70 8036

37

38

39

40

41

42

43

44

45

Figure 6 The changing curve of average work time of all individu-als

Abstract and Applied Analysis 9

Table 6 The simulation results of ten rounds of experiments of GA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 27 26 26 28 27 29 28 28 28 28Simulation time (second) 3007 2905 2894 3102 2732 2940 2946 2846 3468 2937Evolution generation 22 20 23 28 19 19 18 29 18 29Evolution time (second) 08269 07263 08320 10857 06488 06983 06629 10317 07803 10647

Table 7 The simulation results of ten rounds of experiments of QGA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 30 27 29 30 28 29 31 30 29 28Simulation time (second) 5679 5337 5547 5576 5766 6150 5813 5828 6024 5802Evolution generation 12 13 10 4 4 15 3 4 8 21Evolution time (second) 08519 08673 06934 02788 02883 11531 02180 02914 06024 15230

Table 8 The simulation results of ten rounds of experiments of IQGA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 27 28 26 27 28 27 27 26 28 26Simulation time (second) 8847 8580 8527 8459 8505 8364 8422 8712 8342 8524Evolution generation 18 10 12 12 11 9 13 22 21 16Evolution time (second) 19906 10725 12791 12689 11694 09410 13686 23958 21898 17048

Table 9 The statistical results of the experiment

Algorithm The optimalresult

The worstresult

The averageresult

The fastestevolution time (s)

The slowestevolution time (s)

The averageevolution time (s)

Times ofconvergence

GA 26 29 275 06488 10857 08358 2QGA 27 31 291 02788 15230 06768 0IQGA 26 28 27 09410 23958 15381 3

0 5 10 15 20 25 300

2

4

6

8

10

122 1 23 2 1

2 2 13 1 1

2 2 21 2 2

2 1 21 2 1

3 1 31 1 2

1 1 32 2 1

Work time

The n

umbe

r of t

he en

gine

erin

g w

ork

Figure 7 Gantt chart of the optimal scheme

In Figure 7 x-axis represents time y-axis represents thenumber of the engineering workThe length of the rectanglesrepresents thework time of the engineeringwork the numberin the rectangle represents the number of the work group ofthe corresponding process

Engineering personnel can be scheduled based on theGantt chart of the optimal schemeThe role of the Gantt chartis to aid in allocation decisions of scheduling engineeringpersonnel

6 Conclusions

Compared with genetic algorithm and standard quantumevolutionary algorithm the advantages of improved quantumgenetic algorithm are as follows (1) IQGA adopts directedevolution which can find the optimal solution more quicklythan undirected evolution (2) the self-adaptive quantumrotation angle with larger rotation angle at the early stageof the evolution and with smaller rotation angle in the lateevolution is advantageous for exploration and exploitation(3) quantum mutation is good to generate new quantumchromosome to increase the diversity of population to extendthe search space (4) quantum disaster can make evolutionjump out of local optimal solution to increase the searchprobability of the global optimal solution (5) the whole inter-ference crossover takes effect on the level of chromosomecoding generated by quantum chromosome to generate newchromosome codingThe search space is effectively expandedand the probability to find optimal solution is increased

10 Abstract and Applied Analysis

So IQGA can search the optimal solution faster and moreefficiently based on the above-mentioned advantages

Application of improved quantum genetic algorithmand appropriate mathematical model can solve the complexscheduling engineering personnel problem Through thismethod we can obtain the optimal allocation scheme andthe work time of the allocation scheme is also determinedThe Gantt chart of the result clearly describes the work timeand work order of each engineering work The method inthis paper can provide powerful support for the schedulingengineering personnel The proposed method in this papernot only can be applied in the scheduling engineeringpersonnel but also can be used for similar types of problemwith multistage multiple projects and multiple subjects andthe abilities of subjects are different such as other schedulingproblems and flexible flow shop scheduling problem

Conflict of Interests

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

References

[1] P Brucker R Qu and E Burke ldquoPersonnel scheduling modelsand complexityrdquo European Journal of Operational Research vol210 no 3 pp 467ndash473 2011

[2] T Lapegue O Bellenguez-Morineau and D Prot ldquoAconstraint-based approach for the shift design personnel taskscheduling problem with equityrdquo Computers and OperationsResearch vol 40 no 10 pp 2450ndash2465 2013

[3] S-W Lin and K-C Ying ldquoMinimizing shifts for personneltask scheduling problems a three-phase algorithmrdquo EuropeanJournal ofOperational Research vol 237 no 1 pp 323ndash334 2014

[4] M Krishnamoorthy A T Ernst and D Baatar ldquoAlgorithmsfor large scale shift minimisation personnel task schedulingproblemsrdquo European Journal of Operational Research vol 219no 1 pp 34ndash48 2012

[5] G DuCote and E M Malstrom ldquoDesign of personnel schedul-ing software for manufacturingrdquo Computers and IndustrialEngineering vol 37 no 1 pp 473ndash476 1999

[6] P M Koeleman S Bhulai and M van Meersbergen ldquoOptimalpatient and personnel scheduling policies for care-at-homeservice facilitiesrdquo European Journal of Operational Research vol219 no 3 pp 557ndash563 2012

[7] J van den Bergh J Belien P de Bruecker E Demeulemeesterand L de Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013

[8] M Sabar B Montreuil and J-M Frayret ldquoA multi-agent-based approach for personnel scheduling in assembly centersrdquoEngineering Applications of Artificial Intelligence vol 22 no 7pp 1080ndash1088 2009

[9] K-H Han and J-H Kim ldquoGenetic quantum algorithm andits application to combinatorial optimization problemrdquo inProceedings of the Congress on Evolutionary Computation pp1354ndash1360 Piscataway NJ USA July 2000

[10] A Narayanan and M Moore ldquoQuantum-inspired genetic algo-rithmsrdquo in Proceedings of the IEEE International Conference on

Evolutionary Computatio (ICEC 96) pp 61ndash66 Nagoya JapanMay 1996

[11] HWang J Liu J Zhi andC Fu ldquoThe improvement of quantumgenetic algorithm and its application on function optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID730749 10 pages 2013

[12] J Tang G Zhang B Lin and B Zhang ldquoHybrid genetic algo-rithm for flow shop scheduling problemrdquo in Proceedings of theInternational Conference on Intelligent Computation Technologyand Automation (ICICTA 10) vol 2 pp 449ndash452 ChangshaChina May 2010

[13] R ZhangWWei Z Jiang andXMa ldquoResearch and simulationon flow-shop scheduling problem based on improved geneticalgorithmrdquo in Proceedings of the 7th International Conference onComputer Science and Education (ICCSE rsquo12) pp 916ndash919 July2012

[14] F Jin and Y Fu ldquoMaster-slave genetic algorithm for flowshop scheduling with resource flexibilityrdquo in Proceedings ofthe IEEE International Conference on Advanced ManagementScience (ICAMS rsquo10) vol 1 pp 341ndash346 Chengdu China July2010

[15] G Zhang and H Rong ldquoQuantum-inspired genetic algorithmbased time-frequency atom decompositionrdquo in Proceedings ofthe 7th international conference on Computational Science (ICCSrsquo07) 2007

[16] J-A Yang and Z-Q Zhuang ldquoActuality of research on quantumgenetic algorithmrdquo Journal of Computer Science amp Technologyvol 30 no 11 pp 13ndash15 2003

[17] W Shu and B He ldquoA quantum genetic simulated annealingalgorithm for task schedulingrdquo inAdvances in Computation andIntelligence L Kang Y Liu and S Zeng Eds vol 4683 ofLecture Notes in Computer Science pp 169ndash176 Springer BerlinGermany 2007

[18] Y Li Y Zhang Y Cheng and X Jiang ldquoA novel immunequantum-inspired genetic algorithmrdquo in Advances in NaturalComputation vol 3612 of Lecture Notes in Computer Science pp215ndash218 2005

[19] KHan and J Kim ldquoGenetic quantum algorithm and its applica-tion to combinatorial optimization problemrdquo in Proceedings ofthe Congress on Evolutionary Computation pp 1354ndash1360 SanDiego Calif USA July 2000

[20] H Miao H Wang and Z Deng ldquoQuantum genetic algo-rithm and its application in power system reactive poweroptimizationrdquo in Proceedings of the International Conference onComputational Intelligence and Security (CIS 09) vol 1 pp 107ndash111 Beijing China 2009

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

t

Aai

Task i

t

Task i minus 1

B

Task i minus 1

When t le ai

When t ge ai

bi

bi

t minus ai + bi

Figure 1 Diagram of standard scheduling personnel problem

J1 J2 J3 JSmiddot middot middot

middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middot

middot middot middot

middot middot middot

M12

M1m M2n M3t MSn

MS2

MS1M11M21

M22 M32

M31

Figure 2 Distribution of engineering personnel

Equation (1) is the condition that task 119894 is ordered beforethe task 119895 where 119886

119894is the time needed by the first stage of task

119894 and 119887119895is the time needed by the second stage of task 119895

22 Extension of Scheduling Engineering Personnel ProblemExtension of scheduling engineering personnel problem isbased on the standard personnel scheduling problem Thestages of the task needed to be done are more than two thework groups for finishing every stage of the task aremore thanone and the time in which different work teams finish thetask is different There is not an authentic rule of sorting ofthe extension of scheduling engineering personnel problem

Extension of scheduling engineering operating personnelproblem can be described as follows There are a numberof engineering works which can be decomposed into acertain number of steps and at least one of all steps can becompleted in parallel by multiple work groups The problemis to determine the distribution of work groups in parallelThat is there are 119899 work groups and 119873 engineering works119882111988221198823 119882

119873 Decompose these engineering works

into 119878 processes uniformly 1198691 1198692 1198693 119869

119878 Each process

119869119894has 119885

119894parallel work group (1 le 119894 le 119878 1 le 119885

119894le

119899sum119885119894gt 119878) The figure of scheduling engineering operating

personnel problem is shown in Figure 2 The target is tominimize the maximum flow time through conducting anoptimal allocation [5]

Each process of each engineering work can be completedby any parallel team of corresponding process but the work

times are different [6] The specific example is shown inTable 1 119869

119894stands for the 119894th process 119872119894

119895stands for the

119895th parallel work group of the 119894th process and the dataof 119872119894

119895stands for the work time that the 119895th parallel

work group needs to complete the 119894th process Table 1 isan operational timetable which includes four engineeringworks (119882

1119882211988231198824) three processes (119869

1 1198692 1198693) and nine

engineering work groupsThe engineering personnel scheduling problem in this

paper satisfies the following constraints (1) different engi-neering works have the same priority (2) there is no orderconstraint between different engineering works but theprocesses of the same engineering work are successive [7](3) any engineering work can start at the beginning (4) onework group only can carry out one engineering work at thesame time (5) the same process of the same engineeringworkcan only be completed by one work group (6) Once startedeach process of the engineering work cannot be interrupted(7) The only performance is the maximum completion time119879max that is the purpose of scheduling is to minimize themaximum completion time

min119879max = min (max119879119896) 1 le 119896 le 119898 (2)

where 119879119896is the completion time of the work group 119872119904

119901

1 le 119901 le 119885119904 (119885119904 is the total number of work groups of eachprocess) and119872119904

119901is the 119901th parallel work group of the final

process

3 Calculation of Engineering Work Time

We need to follow certain principles to calculate the worktime of the determined scheme to minimize the work timewhen the scheduling scheme of engineering work groupsis determined During the whole engineering work theconsuming time is divided into two parts the working timeand the waiting time [8]

Suppose the 119900119896th work group of the work groups of the

first process 119872(1)(119900119896)

needs cost 119883(2)(119900119896)119896

to complete the firstprocess of the 119896th engineering work the 119901

119896th work group of

the work groups of the second process119872(2)(119901119896)

needs cost119883(2)(119901119896)119896

to complete the second process and the 119902119896th work group of

Abstract and Applied Analysis 3

Table 1 Example of scheduling engineering personnel problem

Engineering worksProcesses

1198691

1198692

1198693

11987211

11987212

11987213

11987214

11987221

11987222

11987231

11987232

11987233

1198821

3 2 4 3 3 4 5 3 41198822

4 5 3 2 6 5 5 6 31198823

4 3 5 6 4 2 3 4 51198824

2 4 1 3 5 3 4 6 2

the work groups of the third process119872(3)(119902119896)

needs cost 119883(2)(119901119896)119896

to complete the third process The total time the first processof the 119896th engineering work costs is119872(1)

(119900119896)119896 the total time the

119901119896minus1

th work group119872(2)(119901119896minus1)

costs on the second process of the1198961th engineering work is119872(2)

(119901119896minus1)119896minus1 (119896 gt 1) Calculation of

119872(2)

(119901119896)119896is divided into two situations

(1) when119872(1)(119900119896)119896

ge 119872(2)

(119901119896minus1)119896minus1119872(2)(119901119896)119896

= 119872(1)

(119900119896)119896+ 119883(2)

(119901119896)119896

(2) when119872(1)(119900119896)119896

lt 119872(2)

(119901119896minus1)119896minus1119872(2)(119901119896)119896

= 119872(2)

(119901119896minus1)119896minus1+119883(2)

(119901119896)119896

Take the two situations together the formula of calcu-lating the total time 119872(2)

(119901119896)119896of the 119901

119896th work group 119872(2)

(119901119896)

completing the second process of the 119896th engineering workis

119872(2)

(119901119896)119896= max (119872(1)

(119900119896)119896119872(2)

(119901119896minus1)119896minus1) + 119883

(2)

(119901119896)119896 (3)

Similarly similar formula of the third step can beobtained The specific formulas are in Table 2 The formulasin Table 2 indicate the total time119872119894

119895of the 119896th engineering

work cost from the beginning to the 119894th process

4 The Improved Quantum GeneticAlgorithm Used to Solve the SchedulingEngineering Personnel Problem

Quantum genetic algorithm (QGA) is the product of thecombination of quantum computation and genetic algo-rithms and it is a new evolutionary algorithm of probability[9] It is successfully used to solve the TSP problem [10] Thequantum state vector is introduced in the genetic algorithmto express genetic code and quantum logic gates are usedto realize the chromosome evolution By these means betterresults are achieved [11] Genetic algorithm has been widelyapplied in scheduling engineering personnel [12ndash14] Theimproved quantum genetic algorithm is applied to solvethe scheduling engineering personnel problem in order toachieve the optimization of scheduling engineering person-nel

41 The Qubit Encoding of Scheduling Engineering PersonnelProblem Ensuring that the encoding method is the mostimportant for applying improved quantum genetic algorithm

to solve the scheduling engineering personnel problem Thequbit encoding needs to combine the optimization problemwith the improved quantum genetic algorithmThe engineer-ing works which can be divided into the same number ofprocesses need to be completed and there is at least oneprocess that can be completed by multiple parallel workgroups It needs to ensure the distribution of parallel workgroups in order to minimize the maximum completion time

The features of scheduling engineering personnel prob-lemare as follows for example there are119873 engineeringworksthat need to be completed every work must be completedthrough 119878 processes The number of parallel work groups ofevery process is 119885

119894(1 le 119894 le 119878) There is at least one process

that can be completed by multiple parallel work groups thatis there is at least one119885

119895which is greater than one Construct

a119873 times 119878 dimensional matrix

119860119873times119878

=

[

[

[

[

[

1198861111988612sdot sdot sdot 1198861119878

1198862111988622sdot sdot sdot 1198862119878

11988611198781198862119878sdot sdot sdot 119886119873119878

]

]

]

]

]

(4)

where the element 119886119894119895of matrix is a random real number

between the ranges [1 119885119895+ 1) which means that the 119895th

process of the 119894th engineering work is completed by theint 119886119894119895th work group (int means getting the greatest integer

which is not greater than 119886119894119895)

If int 1198861198951198961

= int 1198861198951198962(1198961

= 1198962) it indicates that the same

processes of multiple engineering works are completed bythe same work group Use variable 119899

119894(119894 = 1 sum119885

119895) to

record the number of engineering works every work groupcompletes

Accord the structure of (2) to design the chromosome ofquantum genetic algorithm

chromosome119896= [119873 119878 (

11986911198692sdot sdot sdot 119869119904

11988511198852sdot sdot sdot 119885

119904

)

11988611 11988612 119886

1119878 119886

1198731 119886

119873119878]

(5)

4 Abstract and Applied Analysis

Table 2 Total time of completion of every process

Work ProcessProcess 1 Process 2 Process 3 sdot sdot sdot Process 119894

1 0 + 119883(1)

(1199001)1119872(1)

(1199001)1+ 119883(2)

(1199011)1119872(2)

(1199011)1+ 119883(3)

(1199021)1sdot sdot sdot 119872

(119894minus1)

(1199101)1+ 119883(119894)

(1199111)1

2 119872(1)

(1199001)1+ 119883(1)

(1199002)2max (119872(1)

(1199002)2119872(2)

(1199011)1) + 119883

(2)

(1199012)2max (119872(2)

(1199012)2119872(3)

(1199021)1) + 119883

(3)

(1199022)2sdot sdot sdot max (119872(119894minus1)

(1199102)2119872(119894)

(1199111)1) + 119883

(119894)

(1199112)2

3 119872(1)

(1199002)2+ 119883(1)

(1199003)3max (119872(1)

(1199003)3119872(2)

(1199012)2) + 119883

(2)

(1199012)3max (119872(2)

(1199013)3119872(3)

(1199022)2) + 119883

(3)

(1199023)3sdot sdot sdot max (119872(119894minus1)

(1199103)3119872(119894)

(1199112)2) + 119883

(119894)

(1199113)3

119896 119872(1)

(119900119896minus1)119896minus1+ 119883(1)

(119900119896)119896max (119872(1)

(119900119896)119896119872(2)

(119901119896minus1)119896minus1) + 119883

(2)

(119901119896)119896max (119872(2)

(119901119896)119896119872(3)

(119902119896minus1)119896minus1) + 119883

(3)

(119902119896)119896sdot sdot sdot max (119872(119894minus1)

(119910119896)119896119872(119894)

(119911119896minus1)119896minus1) + 119883

(119894)

(1199112)119896

Themathematical model ismin = max(119872(2)(119901119896)119896119872(3)

(119902119896minus1)119896minus1) + 119883(3)

(119902119896)119896 where 119896 is the number of engineering works

The improved quantum genetic algorithm adopted in thispaper is the algorithm based on the algorithm in document

[11] The multiqubits encoding which has 119898 parameters isdefined as follows

119902119905

119895=[

[

120572119905

11

120573119905

11

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

12

120573119905

12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572119905

1119896

120573119905

1119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

21

120573119905

21

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

22

120573119905

22

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572119905

2119896

120573119905

2119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

1198981

120573119905

1198981

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

1198982

120573119905

1198982

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572119905

119898119896

120573119905

119898119905

]

]

(6)

where 119902119905119895represents the 119895th individual chromosome of the tth

generation 119896 represents the number of qubit encoding of eachgene 119898 represents the number of genes in the chromosome

[15] Initialize the quantum encoding (120572 120573) of each individualin the population with (1radic2 1radic2) which indicates thatwhen 119905 = 0 the possibility of each state expressed by achromosome is equal [16]

The allocation coding matrix is encoded by qubit as

119876 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120572111

120573111

10038161003816100381610038161003816100381610038161003816

120572112

120573112

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572111198761

120573111198761

120572121

120573121

10038161003816100381610038161003816100381610038161003816

120572122

120573122

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572121198762

120573121198762

sdot sdot sdot

12057211198781

12057311198781

10038161003816100381610038161003816100381610038161003816

12057211198782

12057311198782

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

1205721119878119876119878

1205731119878119876119878

120572211

120573211

10038161003816100381610038161003816100381610038161003816

120572212

120573212

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572211198761

120573211198761

120572221

120573221

10038161003816100381610038161003816100381610038161003816

120572222

120573222

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572221198762

120573221198762

sdot sdot sdot

12057221198781

12057321198781

10038161003816100381610038161003816100381610038161003816

12057221198782

12057321198782

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

1205722119878119876119878

1205732119878119876119878

d

12057211987311

12057311987311

10038161003816100381610038161003816100381610038161003816

12057211987312

12057311987312

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

12057211987311198761

12057311987311198761

12057211987321

12057311987321

10038161003816100381610038161003816100381610038161003816

12057211987322

12057311987322

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

12057211987321198762

12057311987321198762

sdot sdot sdot

1205721198731198781

1205731198731198781

10038161003816100381610038161003816100381610038161003816

1205721198731198782

1205731198731198782

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572119873119878119876119878

120573119873119878119876119878

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(7)

The element in (7) is1205721198941198951

1205731198941198951

100381610038161003816100381610038161003816

1205721198941198952

1205731198941198952

100381610038161003816100381610038161003816

sdotsdotsdot

sdotsdotsdot

100381610038161003816100381610038161003816

120572119894119895119876119895

120573119894119895119876119895which represents

the qubit encoding of the ordinal number of work teamwhere 119894 (119894 = 1 2 119873) represents the ordinal number ofengineering work 119895 (119894 = 1 2 119878) represents the ordinalnumber of process and 119876

119896(119896 = 1 119878) represents the

length of the qubit encoding usually the more work groupsthere are for the process 119896 the longer the length is Thenumber 119886

119894119895generated by qubit encoding which represents the

119895th process of the 119894th work is finished by the 119886119894119895th work group

42 The Solution Based on Improved Quantum Genetic Algo-rithm The solution should be solved based on the qubitcoding

421 Quantum Rotating Gates Quantum genetic algorithmapplies the probability amplitude of qubits to encode chro-mosome and uses quantum rotating gates to realize chromo-somal updated operation

Abstract and Applied Analysis 5

Quantum rotating gates can be designed according to thepractical problems and usually be defined as

119880 (120579119894) = [

cos 120579119894minus sin 120579

119894

sin 120579119894

cos 120579119894

] (8)

The updated process is

[

[

1205721015840

119894

1205731015840

119894

]

]

= 119880 (120579119894) [

120572119894

120573119894

] = [

cos 120579119894minus sin 120579

119894

sin 120579119894

cos 120579119894

] [

120572119894

120573119894

] (9)

where (120572119894 120573119894)119879 and (1205721015840

119894 1205731015840

119894)

119879 are the probability amplitudes ofthe 119894th qubit in chromosome before and after the quantumrotating gates updating respectively 120579

119894is the rotating angle

and the value and the sign of 120579119894are determined by the adjust-

ment strategy [17 18] 120579119894is static in conventional quantum

genetic algorithmThevalue of 120579119894can be dynamically adjusted

based on the evolutionary process in the improved quantumgenetic algorithm Adjust the value of the rotating angleaccording to the number of generation the regulator is asfollows

120579119894= 120579max minus (

120579max minus 120579min119894119905max

) lowast iter (10)

where 120579119894is the value of the rotating angle of the 119894th generation

120579max is the maximal rotating angle 120579min is the minimalrotating angle 119894119905max is the maximal generation and iter isthe current generation 120579max = 001120587 120579min = 0001120587

The adjustment strategy is as follows comparing thefitness of the currently measured value 119891(119909) of the individual119902119905

119895with the fitness of the current optimal individual 119891(best

119894)

if 119891(119909) gt 119891(best) then adjust the corresponding qubits of119902119905

119895 making the probability amplitude (120572

119894 120573119894) evolve toward

the direction that is propitious to the emergence of 119909119894

Conversely if 119891(119909) lt 119891(best) then adjust the correspondingqubits of 119902119905

119895 making the probability amplitude (120572

119894 120573119894) evolve

toward the direction that is propitious to the emergence ofbest

Construct a determinant119860 = 10038161003816100381610038161003816

120572119887 120572119894

120573119887 120573119894

10038161003816100381610038161003816 where (120572

119887 120573119887) is the

probability amplitude of some qubit of the optimal solutionwhich has been searched (120572

119894 120573119894) is probability amplitude of

corresponding state of qubit in the current solution When119909119894= best119894 if 119860 = 0 then the direction of the rotating angle is

minussgn(119860) If 119860 = 0 the direction of the rotating angle can beeither positive of negative

422 Quantum Mutation Add quantum mutation opera-tion in conventional quantum genetic algorithm Quantummutation enables some individual to deviate slightly fromthe current evolutionary direction and prevent the evolutionof individual into a local optimal solution [19] Quantummutation can completely reverse the individualrsquos evolutionarydirection by swapping the value of probability amplitude ofqubits (120572 120573) Quantum NOT gates are adopted to realizechromosomal variation

Table 3 One result of the whole interference crossover

Chromosomenumber New chromosome

1 A(1) D(2) C(3) B(4) A(5) D(6) C(7) B(8)

2 B(1) A(2) D(3) C(4) B(5) A(6) D(7) C(8)

3 C(1) B(2) A(3) D(4) C(5) B(6) A(7) D(8)

4 D(1) C(2) B(3) A(4) D(5) C(6) B(7) A(8)

423 QuantumDisaster Add quantum disaster operation inthe conventional quantum genetic algorithm The methodis to apply a large disturbance to some individuals inthe population and regenerate some other new randomindividuals [20]

424 Whole Interference Crossover Whole interferencecrossover operator is utilized to realize evolution insteadof the traditional crossover operator All chromosomes areinvolved in the whole interference crossover Set the numberof population to be 119898 and length of chromosome is 119899(119898 lt 119899) 119860

119894119895is the 119895th bit of the 119894th individual Then the

chromosome of the 119894th individual of the new family throughthewhole interference crossover is119860

(119894 mod 119898)1minus119860 (119894+1 mod 119898)2 minussdot sdot sdot minus 119860

(119894+119896 mod 119898)(119896+1) minus sdot sdot sdot minus 119860 (119894+119899minus1 mod 119898)119899 The new wholeinterference crossover operator is mainly used when thealgorithm converges to local optimal solution to generate newindividuals to overcome the prematurity in later evolutionSuppose that there are 4 populations and the lengths of themare all 8 One result of the whole interference crossover is asin Table 3

It is important to note that quantum rotating gatesquantummutation and quantum disaster all act on quantumchromosome However whole interference crossover acts onthe chromosome generated by quantum chromosome Theevolution performance will not be introduced if the wholeinterference crossover acts on quantum chromosome

The improved strategy of quantum genetic algorithmmainly contains self-adaptive rotating angle strategy theimprovement of program structure of quantum evolutionadded quantum mutation operation quantum disaster oper-ation and whole interference crossover operation Theflowchart of improved quantum genetic algorithm is shownin Figure 3

43 The Determining Flow of Scheduling Engineering Per-sonnel The steps to determine the scheme of schedulingengineering personnel are as follows

Step 1 (initialize distribution scheme) Initialize allocationscheme to form the initial population according to the needsof the evolution of quantum genetic algorithm and theassignment matrix

Step 2 (time calculation) Calculate the time of all theallocated schemes in the population

6 Abstract and Applied Analysis

Start

Satisfy final condition

Memorize the best individual and its fitness

End

Yes

No

Colony disaster

Yes

No

Initialize the population Q(t0) and every parameter

Test every individual of Q(t0)

Evaluate the fitness of every individual of Q(t0)

Set the best individual as the evolutional goal of next generation

Test every individual of Q(t0)

Quantum rotating door update population get Q(t + 1)

Quantum mutation whole interference crossover

Whether or not disaster

t = t + 1

Figure 3 Flowchart of improved quantum genetic algorithm

Start

Initialize distribution scheme

Satisfy final condition

No

Yes

End

Calculate the time of the allocated schemes in the population

Record the shortest work time of the corresponding scheme

Optimize the allocated scheme

Calculate the shortest time of the optimal allocated scheme

Figure 4 Flowchart of acquiring the optimal allocated scheme and the shortest work time

Abstract and Applied Analysis 7

Step 3 (optimize the allocation scheme) Apply the improvedquantumgenetic algorithm to optimize the allocation schemein order to get the optimal allocation scheme Go to Step 2according to the termination condition to do the judgmentif the termination condition is satisfied then go to the End orelse do the loop to Step 3

Step 4 (determination of the optimal allocation schemeand the shortest work time) Choose the optimal allocationscheme form the population satisfied the termination con-dition Calculate the work time of the optimal allocationscheme

The flowchart is shown in Figure 4

5 Simulation Analyses

Take some road building project as an example There are12 roads which need to be built The building work canbe divided into three procedures prospecting drawing andlocation designing and constructionThere are 3 prospectinggroups 2 drawing and location designing groups and 3 con-struction groups The capacities of the groups are differentThe specific work times are shown in Table 4

The specific steps of acquiring the optimal allocatedscheme and the shortest work time are as follows

(1) Construct 12 times 3 dimensional matrix

11986012times3

=

[

[

[

[

[

11988611

11988612

11988613

11988621

11988622

11988623

119886121

119886122

119886123

]

]

]

]

]

(11)

where 1 le 11988611 11988621 11988631 119886

121lt 4 1 le

11988612 11988622 11988632 119886

122lt 3 1 le 119886

13 11988623 11988633 119886

123lt 4

The corresponding qubit chromosome is

11987612times3

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120572111

120573111

10038161003816100381610038161003816100381610038161003816

120572112

120573112

120572121

120573121

120572131

120573131

10038161003816100381610038161003816100381610038161003816

120572132

120573132

120572211

120573211

10038161003816100381610038161003816100381610038161003816

120572212

120573212

120572221

120573221

120572231

120573231

10038161003816100381610038161003816100381610038161003816

120572232

120573232

1205721211

1205731211

10038161003816100381610038161003816100381610038161003816

1205721212

1205731212

1205721221

1205731221

1205721231

1205731231

10038161003816100381610038161003816100381610038161003816

1205721232

1205731232

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(12)

Suppose the generated matrix is

11986012times3

=

[

[

[

[

[

11988611

11988612

11988613

11988621

11988622

11988623

119886121

119886122

119886123

]

]

]

]

]

=[

[

12 23 1 36 21 18 13 32 25 27

29 12 16 23 11 25 27 14 22 13

37 15 24 16 33 28 25 36 19 21

34 17

16 28

19 22

]

]

119879

1198601015840

12times3=[

[

1 2 1 3 2 1 1 3 2 2

2 1 1 2 1 2 2 1 2 1

3 1 2 1 3 2 2 3 1 2

3 1

1 2

1 2

]

]

119879

(13)

Then the encoding section of the assigned chromosome is

[1 2 1 3 2 1 1 3 2 2 3 1 2 1 1 2 1 2 2 1 2

1 1 2 3 1 2 1 3 2 2 3 1 2 1 2]

(14)

(2) Seek the values of time 1198871198961198941198861015840

119894119895

corresponding with1198821198961198721198941198861015840

119896119894

(where 1 le 119896 le 12 1 le 119894 le 3 119894 119895 119896are integers) in Table 2 Then the work timetable ofevery engineeringwork is gotten InTable 5 the figuredenotes the work time and the subscript denotes thework group

(3) Apply the method of Section 3 to calculate the worktimes of every work group according to the orderdetermined by Step (2) and the time determined byStep (3) Then find the maximum value as the totalengineering work time

Apply GA (genetic algorithm) QGA (quantum geneticalgorithm) and IQGA (improved quantum genetic algo-rithm) to solve the problem of scheduling engineering per-sonnelThe three kinds of algorithmarewritten and compiledin MATLAB The simulation test in this paper is based onthe AMD CPU 18GHz 15 GB RAM The parameters of GAare set as follows GA is encoded by real number the totalevolution generation is 80 the size of population is 40 and theprobability ofmutation is 001The parameters ofQGAare setas follows QGA is encoded by the binary the total evolutiongeneration is 80 the size of population is 40 the length ofthe variable expressing 3 work groups is 2 and the length ofthe variable expressing 2 work groups is 1 The parametersof IQGA are set as follows IQGA is encoded by the binarythe total evolution generation is 80 the size of population is40 the length of the variable expressing 3 work groups is 2the length of the variable expressing 2 work groups is 1 andthe probability of mutation is 001 The fitness function is theobjective function The fitness function tends to the optimal

8 Abstract and Applied Analysis

Table 4 Work time of every work group

1198691

1198692

1198693

11987211

11987212

11987213

11987221

11987222

11987231

11987232

11987233

1198821

3 2 4 3 4 5 3 41198822

4 5 3 6 5 5 6 31198823

4 3 5 4 2 3 4 51198824

2 5 3 5 4 1 5 41198825

4 2 6 3 1 3 4 61198826

1 5 3 2 3 5 2 71198827

7 6 8 4 6 6 7 61198828

6 4 5 7 5 4 8 51198829

2 5 4 1 2 3 6 311988210

3 6 4 3 4 4 3 911988211

5 2 3 3 5 8 5 411988212

4 3 4 6 5 4 7 5

Table 5 Work time of every engineering work and correspondingwork group

Engineering work Process1198691

1198692

1198693

1198821

31

42

43

1198822

52

61

51

1198823

41

41

42

1198824

33

42

11

1198825

22

31

63

1198826

11

32

22

1198827

71

62

72

1198828

53

71

53

1198829

52

22

31

11988210

62

31

32

11988211

33

31

81

11988212

41

52

72

value while the evolution increases Simulate the process ofevolution Ten rounds of experiments were presented in thispaper The simulation results of the 3 kinds of algorithm arein Tables 6 7 and 8 The result of some time of the IQGA isshown in Figures 5 6 and 7

The statistical results of the experiment are shown inTable 9

Table 9 shows that compared with GA QGA can get itssmallest value in a short time however QGA can hardly getthe optimal result Though the evolution time of the IQGA islonger and the evolution generation of the IQGA is more theIQGA can get the optimal result in a bigger probability

In Figure 5 x-axis represents the evolution generationsand y-axis represents the work timeThe work time graduallybecomes shorter with the increase of the generation

In Figure 6 x-axis represents the evolution algebra andy-axis represents the work time The average work time of allindividuals has a decreasing trend but there is a gap betweenthe optimal value and the average work time

0 10 20 30 40 50 60 70 8026

27

28

29

30

31

32

Evolutionary generations

The b

est fi

tnes

s of e

very

gen

erat

ion

Figure 5The changing curve of work time of the best individual inthe population

0 10 20 30 40 50 60 70 8036

37

38

39

40

41

42

43

44

45

Figure 6 The changing curve of average work time of all individu-als

Abstract and Applied Analysis 9

Table 6 The simulation results of ten rounds of experiments of GA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 27 26 26 28 27 29 28 28 28 28Simulation time (second) 3007 2905 2894 3102 2732 2940 2946 2846 3468 2937Evolution generation 22 20 23 28 19 19 18 29 18 29Evolution time (second) 08269 07263 08320 10857 06488 06983 06629 10317 07803 10647

Table 7 The simulation results of ten rounds of experiments of QGA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 30 27 29 30 28 29 31 30 29 28Simulation time (second) 5679 5337 5547 5576 5766 6150 5813 5828 6024 5802Evolution generation 12 13 10 4 4 15 3 4 8 21Evolution time (second) 08519 08673 06934 02788 02883 11531 02180 02914 06024 15230

Table 8 The simulation results of ten rounds of experiments of IQGA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 27 28 26 27 28 27 27 26 28 26Simulation time (second) 8847 8580 8527 8459 8505 8364 8422 8712 8342 8524Evolution generation 18 10 12 12 11 9 13 22 21 16Evolution time (second) 19906 10725 12791 12689 11694 09410 13686 23958 21898 17048

Table 9 The statistical results of the experiment

Algorithm The optimalresult

The worstresult

The averageresult

The fastestevolution time (s)

The slowestevolution time (s)

The averageevolution time (s)

Times ofconvergence

GA 26 29 275 06488 10857 08358 2QGA 27 31 291 02788 15230 06768 0IQGA 26 28 27 09410 23958 15381 3

0 5 10 15 20 25 300

2

4

6

8

10

122 1 23 2 1

2 2 13 1 1

2 2 21 2 2

2 1 21 2 1

3 1 31 1 2

1 1 32 2 1

Work time

The n

umbe

r of t

he en

gine

erin

g w

ork

Figure 7 Gantt chart of the optimal scheme

In Figure 7 x-axis represents time y-axis represents thenumber of the engineering workThe length of the rectanglesrepresents thework time of the engineeringwork the numberin the rectangle represents the number of the work group ofthe corresponding process

Engineering personnel can be scheduled based on theGantt chart of the optimal schemeThe role of the Gantt chartis to aid in allocation decisions of scheduling engineeringpersonnel

6 Conclusions

Compared with genetic algorithm and standard quantumevolutionary algorithm the advantages of improved quantumgenetic algorithm are as follows (1) IQGA adopts directedevolution which can find the optimal solution more quicklythan undirected evolution (2) the self-adaptive quantumrotation angle with larger rotation angle at the early stageof the evolution and with smaller rotation angle in the lateevolution is advantageous for exploration and exploitation(3) quantum mutation is good to generate new quantumchromosome to increase the diversity of population to extendthe search space (4) quantum disaster can make evolutionjump out of local optimal solution to increase the searchprobability of the global optimal solution (5) the whole inter-ference crossover takes effect on the level of chromosomecoding generated by quantum chromosome to generate newchromosome codingThe search space is effectively expandedand the probability to find optimal solution is increased

10 Abstract and Applied Analysis

So IQGA can search the optimal solution faster and moreefficiently based on the above-mentioned advantages

Application of improved quantum genetic algorithmand appropriate mathematical model can solve the complexscheduling engineering personnel problem Through thismethod we can obtain the optimal allocation scheme andthe work time of the allocation scheme is also determinedThe Gantt chart of the result clearly describes the work timeand work order of each engineering work The method inthis paper can provide powerful support for the schedulingengineering personnel The proposed method in this papernot only can be applied in the scheduling engineeringpersonnel but also can be used for similar types of problemwith multistage multiple projects and multiple subjects andthe abilities of subjects are different such as other schedulingproblems and flexible flow shop scheduling problem

Conflict of Interests

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

References

[1] P Brucker R Qu and E Burke ldquoPersonnel scheduling modelsand complexityrdquo European Journal of Operational Research vol210 no 3 pp 467ndash473 2011

[2] T Lapegue O Bellenguez-Morineau and D Prot ldquoAconstraint-based approach for the shift design personnel taskscheduling problem with equityrdquo Computers and OperationsResearch vol 40 no 10 pp 2450ndash2465 2013

[3] S-W Lin and K-C Ying ldquoMinimizing shifts for personneltask scheduling problems a three-phase algorithmrdquo EuropeanJournal ofOperational Research vol 237 no 1 pp 323ndash334 2014

[4] M Krishnamoorthy A T Ernst and D Baatar ldquoAlgorithmsfor large scale shift minimisation personnel task schedulingproblemsrdquo European Journal of Operational Research vol 219no 1 pp 34ndash48 2012

[5] G DuCote and E M Malstrom ldquoDesign of personnel schedul-ing software for manufacturingrdquo Computers and IndustrialEngineering vol 37 no 1 pp 473ndash476 1999

[6] P M Koeleman S Bhulai and M van Meersbergen ldquoOptimalpatient and personnel scheduling policies for care-at-homeservice facilitiesrdquo European Journal of Operational Research vol219 no 3 pp 557ndash563 2012

[7] J van den Bergh J Belien P de Bruecker E Demeulemeesterand L de Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013

[8] M Sabar B Montreuil and J-M Frayret ldquoA multi-agent-based approach for personnel scheduling in assembly centersrdquoEngineering Applications of Artificial Intelligence vol 22 no 7pp 1080ndash1088 2009

[9] K-H Han and J-H Kim ldquoGenetic quantum algorithm andits application to combinatorial optimization problemrdquo inProceedings of the Congress on Evolutionary Computation pp1354ndash1360 Piscataway NJ USA July 2000

[10] A Narayanan and M Moore ldquoQuantum-inspired genetic algo-rithmsrdquo in Proceedings of the IEEE International Conference on

Evolutionary Computatio (ICEC 96) pp 61ndash66 Nagoya JapanMay 1996

[11] HWang J Liu J Zhi andC Fu ldquoThe improvement of quantumgenetic algorithm and its application on function optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID730749 10 pages 2013

[12] J Tang G Zhang B Lin and B Zhang ldquoHybrid genetic algo-rithm for flow shop scheduling problemrdquo in Proceedings of theInternational Conference on Intelligent Computation Technologyand Automation (ICICTA 10) vol 2 pp 449ndash452 ChangshaChina May 2010

[13] R ZhangWWei Z Jiang andXMa ldquoResearch and simulationon flow-shop scheduling problem based on improved geneticalgorithmrdquo in Proceedings of the 7th International Conference onComputer Science and Education (ICCSE rsquo12) pp 916ndash919 July2012

[14] F Jin and Y Fu ldquoMaster-slave genetic algorithm for flowshop scheduling with resource flexibilityrdquo in Proceedings ofthe IEEE International Conference on Advanced ManagementScience (ICAMS rsquo10) vol 1 pp 341ndash346 Chengdu China July2010

[15] G Zhang and H Rong ldquoQuantum-inspired genetic algorithmbased time-frequency atom decompositionrdquo in Proceedings ofthe 7th international conference on Computational Science (ICCSrsquo07) 2007

[16] J-A Yang and Z-Q Zhuang ldquoActuality of research on quantumgenetic algorithmrdquo Journal of Computer Science amp Technologyvol 30 no 11 pp 13ndash15 2003

[17] W Shu and B He ldquoA quantum genetic simulated annealingalgorithm for task schedulingrdquo inAdvances in Computation andIntelligence L Kang Y Liu and S Zeng Eds vol 4683 ofLecture Notes in Computer Science pp 169ndash176 Springer BerlinGermany 2007

[18] Y Li Y Zhang Y Cheng and X Jiang ldquoA novel immunequantum-inspired genetic algorithmrdquo in Advances in NaturalComputation vol 3612 of Lecture Notes in Computer Science pp215ndash218 2005

[19] KHan and J Kim ldquoGenetic quantum algorithm and its applica-tion to combinatorial optimization problemrdquo in Proceedings ofthe Congress on Evolutionary Computation pp 1354ndash1360 SanDiego Calif USA July 2000

[20] H Miao H Wang and Z Deng ldquoQuantum genetic algo-rithm and its application in power system reactive poweroptimizationrdquo in Proceedings of the International Conference onComputational Intelligence and Security (CIS 09) vol 1 pp 107ndash111 Beijing China 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Table 1 Example of scheduling engineering personnel problem

Engineering worksProcesses

1198691

1198692

1198693

11987211

11987212

11987213

11987214

11987221

11987222

11987231

11987232

11987233

1198821

3 2 4 3 3 4 5 3 41198822

4 5 3 2 6 5 5 6 31198823

4 3 5 6 4 2 3 4 51198824

2 4 1 3 5 3 4 6 2

the work groups of the third process119872(3)(119902119896)

needs cost 119883(2)(119901119896)119896

to complete the third process The total time the first processof the 119896th engineering work costs is119872(1)

(119900119896)119896 the total time the

119901119896minus1

th work group119872(2)(119901119896minus1)

costs on the second process of the1198961th engineering work is119872(2)

(119901119896minus1)119896minus1 (119896 gt 1) Calculation of

119872(2)

(119901119896)119896is divided into two situations

(1) when119872(1)(119900119896)119896

ge 119872(2)

(119901119896minus1)119896minus1119872(2)(119901119896)119896

= 119872(1)

(119900119896)119896+ 119883(2)

(119901119896)119896

(2) when119872(1)(119900119896)119896

lt 119872(2)

(119901119896minus1)119896minus1119872(2)(119901119896)119896

= 119872(2)

(119901119896minus1)119896minus1+119883(2)

(119901119896)119896

Take the two situations together the formula of calcu-lating the total time 119872(2)

(119901119896)119896of the 119901

119896th work group 119872(2)

(119901119896)

completing the second process of the 119896th engineering workis

119872(2)

(119901119896)119896= max (119872(1)

(119900119896)119896119872(2)

(119901119896minus1)119896minus1) + 119883

(2)

(119901119896)119896 (3)

Similarly similar formula of the third step can beobtained The specific formulas are in Table 2 The formulasin Table 2 indicate the total time119872119894

119895of the 119896th engineering

work cost from the beginning to the 119894th process

4 The Improved Quantum GeneticAlgorithm Used to Solve the SchedulingEngineering Personnel Problem

Quantum genetic algorithm (QGA) is the product of thecombination of quantum computation and genetic algo-rithms and it is a new evolutionary algorithm of probability[9] It is successfully used to solve the TSP problem [10] Thequantum state vector is introduced in the genetic algorithmto express genetic code and quantum logic gates are usedto realize the chromosome evolution By these means betterresults are achieved [11] Genetic algorithm has been widelyapplied in scheduling engineering personnel [12ndash14] Theimproved quantum genetic algorithm is applied to solvethe scheduling engineering personnel problem in order toachieve the optimization of scheduling engineering person-nel

41 The Qubit Encoding of Scheduling Engineering PersonnelProblem Ensuring that the encoding method is the mostimportant for applying improved quantum genetic algorithm

to solve the scheduling engineering personnel problem Thequbit encoding needs to combine the optimization problemwith the improved quantum genetic algorithmThe engineer-ing works which can be divided into the same number ofprocesses need to be completed and there is at least oneprocess that can be completed by multiple parallel workgroups It needs to ensure the distribution of parallel workgroups in order to minimize the maximum completion time

The features of scheduling engineering personnel prob-lemare as follows for example there are119873 engineeringworksthat need to be completed every work must be completedthrough 119878 processes The number of parallel work groups ofevery process is 119885

119894(1 le 119894 le 119878) There is at least one process

that can be completed by multiple parallel work groups thatis there is at least one119885

119895which is greater than one Construct

a119873 times 119878 dimensional matrix

119860119873times119878

=

[

[

[

[

[

1198861111988612sdot sdot sdot 1198861119878

1198862111988622sdot sdot sdot 1198862119878

11988611198781198862119878sdot sdot sdot 119886119873119878

]

]

]

]

]

(4)

where the element 119886119894119895of matrix is a random real number

between the ranges [1 119885119895+ 1) which means that the 119895th

process of the 119894th engineering work is completed by theint 119886119894119895th work group (int means getting the greatest integer

which is not greater than 119886119894119895)

If int 1198861198951198961

= int 1198861198951198962(1198961

= 1198962) it indicates that the same

processes of multiple engineering works are completed bythe same work group Use variable 119899

119894(119894 = 1 sum119885

119895) to

record the number of engineering works every work groupcompletes

Accord the structure of (2) to design the chromosome ofquantum genetic algorithm

chromosome119896= [119873 119878 (

11986911198692sdot sdot sdot 119869119904

11988511198852sdot sdot sdot 119885

119904

)

11988611 11988612 119886

1119878 119886

1198731 119886

119873119878]

(5)

4 Abstract and Applied Analysis

Table 2 Total time of completion of every process

Work ProcessProcess 1 Process 2 Process 3 sdot sdot sdot Process 119894

1 0 + 119883(1)

(1199001)1119872(1)

(1199001)1+ 119883(2)

(1199011)1119872(2)

(1199011)1+ 119883(3)

(1199021)1sdot sdot sdot 119872

(119894minus1)

(1199101)1+ 119883(119894)

(1199111)1

2 119872(1)

(1199001)1+ 119883(1)

(1199002)2max (119872(1)

(1199002)2119872(2)

(1199011)1) + 119883

(2)

(1199012)2max (119872(2)

(1199012)2119872(3)

(1199021)1) + 119883

(3)

(1199022)2sdot sdot sdot max (119872(119894minus1)

(1199102)2119872(119894)

(1199111)1) + 119883

(119894)

(1199112)2

3 119872(1)

(1199002)2+ 119883(1)

(1199003)3max (119872(1)

(1199003)3119872(2)

(1199012)2) + 119883

(2)

(1199012)3max (119872(2)

(1199013)3119872(3)

(1199022)2) + 119883

(3)

(1199023)3sdot sdot sdot max (119872(119894minus1)

(1199103)3119872(119894)

(1199112)2) + 119883

(119894)

(1199113)3

119896 119872(1)

(119900119896minus1)119896minus1+ 119883(1)

(119900119896)119896max (119872(1)

(119900119896)119896119872(2)

(119901119896minus1)119896minus1) + 119883

(2)

(119901119896)119896max (119872(2)

(119901119896)119896119872(3)

(119902119896minus1)119896minus1) + 119883

(3)

(119902119896)119896sdot sdot sdot max (119872(119894minus1)

(119910119896)119896119872(119894)

(119911119896minus1)119896minus1) + 119883

(119894)

(1199112)119896

Themathematical model ismin = max(119872(2)(119901119896)119896119872(3)

(119902119896minus1)119896minus1) + 119883(3)

(119902119896)119896 where 119896 is the number of engineering works

The improved quantum genetic algorithm adopted in thispaper is the algorithm based on the algorithm in document

[11] The multiqubits encoding which has 119898 parameters isdefined as follows

119902119905

119895=[

[

120572119905

11

120573119905

11

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

12

120573119905

12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572119905

1119896

120573119905

1119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

21

120573119905

21

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

22

120573119905

22

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572119905

2119896

120573119905

2119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

1198981

120573119905

1198981

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

1198982

120573119905

1198982

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572119905

119898119896

120573119905

119898119905

]

]

(6)

where 119902119905119895represents the 119895th individual chromosome of the tth

generation 119896 represents the number of qubit encoding of eachgene 119898 represents the number of genes in the chromosome

[15] Initialize the quantum encoding (120572 120573) of each individualin the population with (1radic2 1radic2) which indicates thatwhen 119905 = 0 the possibility of each state expressed by achromosome is equal [16]

The allocation coding matrix is encoded by qubit as

119876 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120572111

120573111

10038161003816100381610038161003816100381610038161003816

120572112

120573112

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572111198761

120573111198761

120572121

120573121

10038161003816100381610038161003816100381610038161003816

120572122

120573122

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572121198762

120573121198762

sdot sdot sdot

12057211198781

12057311198781

10038161003816100381610038161003816100381610038161003816

12057211198782

12057311198782

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

1205721119878119876119878

1205731119878119876119878

120572211

120573211

10038161003816100381610038161003816100381610038161003816

120572212

120573212

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572211198761

120573211198761

120572221

120573221

10038161003816100381610038161003816100381610038161003816

120572222

120573222

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572221198762

120573221198762

sdot sdot sdot

12057221198781

12057321198781

10038161003816100381610038161003816100381610038161003816

12057221198782

12057321198782

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

1205722119878119876119878

1205732119878119876119878

d

12057211987311

12057311987311

10038161003816100381610038161003816100381610038161003816

12057211987312

12057311987312

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

12057211987311198761

12057311987311198761

12057211987321

12057311987321

10038161003816100381610038161003816100381610038161003816

12057211987322

12057311987322

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

12057211987321198762

12057311987321198762

sdot sdot sdot

1205721198731198781

1205731198731198781

10038161003816100381610038161003816100381610038161003816

1205721198731198782

1205731198731198782

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572119873119878119876119878

120573119873119878119876119878

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(7)

The element in (7) is1205721198941198951

1205731198941198951

100381610038161003816100381610038161003816

1205721198941198952

1205731198941198952

100381610038161003816100381610038161003816

sdotsdotsdot

sdotsdotsdot

100381610038161003816100381610038161003816

120572119894119895119876119895

120573119894119895119876119895which represents

the qubit encoding of the ordinal number of work teamwhere 119894 (119894 = 1 2 119873) represents the ordinal number ofengineering work 119895 (119894 = 1 2 119878) represents the ordinalnumber of process and 119876

119896(119896 = 1 119878) represents the

length of the qubit encoding usually the more work groupsthere are for the process 119896 the longer the length is Thenumber 119886

119894119895generated by qubit encoding which represents the

119895th process of the 119894th work is finished by the 119886119894119895th work group

42 The Solution Based on Improved Quantum Genetic Algo-rithm The solution should be solved based on the qubitcoding

421 Quantum Rotating Gates Quantum genetic algorithmapplies the probability amplitude of qubits to encode chro-mosome and uses quantum rotating gates to realize chromo-somal updated operation

Abstract and Applied Analysis 5

Quantum rotating gates can be designed according to thepractical problems and usually be defined as

119880 (120579119894) = [

cos 120579119894minus sin 120579

119894

sin 120579119894

cos 120579119894

] (8)

The updated process is

[

[

1205721015840

119894

1205731015840

119894

]

]

= 119880 (120579119894) [

120572119894

120573119894

] = [

cos 120579119894minus sin 120579

119894

sin 120579119894

cos 120579119894

] [

120572119894

120573119894

] (9)

where (120572119894 120573119894)119879 and (1205721015840

119894 1205731015840

119894)

119879 are the probability amplitudes ofthe 119894th qubit in chromosome before and after the quantumrotating gates updating respectively 120579

119894is the rotating angle

and the value and the sign of 120579119894are determined by the adjust-

ment strategy [17 18] 120579119894is static in conventional quantum

genetic algorithmThevalue of 120579119894can be dynamically adjusted

based on the evolutionary process in the improved quantumgenetic algorithm Adjust the value of the rotating angleaccording to the number of generation the regulator is asfollows

120579119894= 120579max minus (

120579max minus 120579min119894119905max

) lowast iter (10)

where 120579119894is the value of the rotating angle of the 119894th generation

120579max is the maximal rotating angle 120579min is the minimalrotating angle 119894119905max is the maximal generation and iter isthe current generation 120579max = 001120587 120579min = 0001120587

The adjustment strategy is as follows comparing thefitness of the currently measured value 119891(119909) of the individual119902119905

119895with the fitness of the current optimal individual 119891(best

119894)

if 119891(119909) gt 119891(best) then adjust the corresponding qubits of119902119905

119895 making the probability amplitude (120572

119894 120573119894) evolve toward

the direction that is propitious to the emergence of 119909119894

Conversely if 119891(119909) lt 119891(best) then adjust the correspondingqubits of 119902119905

119895 making the probability amplitude (120572

119894 120573119894) evolve

toward the direction that is propitious to the emergence ofbest

Construct a determinant119860 = 10038161003816100381610038161003816

120572119887 120572119894

120573119887 120573119894

10038161003816100381610038161003816 where (120572

119887 120573119887) is the

probability amplitude of some qubit of the optimal solutionwhich has been searched (120572

119894 120573119894) is probability amplitude of

corresponding state of qubit in the current solution When119909119894= best119894 if 119860 = 0 then the direction of the rotating angle is

minussgn(119860) If 119860 = 0 the direction of the rotating angle can beeither positive of negative

422 Quantum Mutation Add quantum mutation opera-tion in conventional quantum genetic algorithm Quantummutation enables some individual to deviate slightly fromthe current evolutionary direction and prevent the evolutionof individual into a local optimal solution [19] Quantummutation can completely reverse the individualrsquos evolutionarydirection by swapping the value of probability amplitude ofqubits (120572 120573) Quantum NOT gates are adopted to realizechromosomal variation

Table 3 One result of the whole interference crossover

Chromosomenumber New chromosome

1 A(1) D(2) C(3) B(4) A(5) D(6) C(7) B(8)

2 B(1) A(2) D(3) C(4) B(5) A(6) D(7) C(8)

3 C(1) B(2) A(3) D(4) C(5) B(6) A(7) D(8)

4 D(1) C(2) B(3) A(4) D(5) C(6) B(7) A(8)

423 QuantumDisaster Add quantum disaster operation inthe conventional quantum genetic algorithm The methodis to apply a large disturbance to some individuals inthe population and regenerate some other new randomindividuals [20]

424 Whole Interference Crossover Whole interferencecrossover operator is utilized to realize evolution insteadof the traditional crossover operator All chromosomes areinvolved in the whole interference crossover Set the numberof population to be 119898 and length of chromosome is 119899(119898 lt 119899) 119860

119894119895is the 119895th bit of the 119894th individual Then the

chromosome of the 119894th individual of the new family throughthewhole interference crossover is119860

(119894 mod 119898)1minus119860 (119894+1 mod 119898)2 minussdot sdot sdot minus 119860

(119894+119896 mod 119898)(119896+1) minus sdot sdot sdot minus 119860 (119894+119899minus1 mod 119898)119899 The new wholeinterference crossover operator is mainly used when thealgorithm converges to local optimal solution to generate newindividuals to overcome the prematurity in later evolutionSuppose that there are 4 populations and the lengths of themare all 8 One result of the whole interference crossover is asin Table 3

It is important to note that quantum rotating gatesquantummutation and quantum disaster all act on quantumchromosome However whole interference crossover acts onthe chromosome generated by quantum chromosome Theevolution performance will not be introduced if the wholeinterference crossover acts on quantum chromosome

The improved strategy of quantum genetic algorithmmainly contains self-adaptive rotating angle strategy theimprovement of program structure of quantum evolutionadded quantum mutation operation quantum disaster oper-ation and whole interference crossover operation Theflowchart of improved quantum genetic algorithm is shownin Figure 3

43 The Determining Flow of Scheduling Engineering Per-sonnel The steps to determine the scheme of schedulingengineering personnel are as follows

Step 1 (initialize distribution scheme) Initialize allocationscheme to form the initial population according to the needsof the evolution of quantum genetic algorithm and theassignment matrix

Step 2 (time calculation) Calculate the time of all theallocated schemes in the population

6 Abstract and Applied Analysis

Start

Satisfy final condition

Memorize the best individual and its fitness

End

Yes

No

Colony disaster

Yes

No

Initialize the population Q(t0) and every parameter

Test every individual of Q(t0)

Evaluate the fitness of every individual of Q(t0)

Set the best individual as the evolutional goal of next generation

Test every individual of Q(t0)

Quantum rotating door update population get Q(t + 1)

Quantum mutation whole interference crossover

Whether or not disaster

t = t + 1

Figure 3 Flowchart of improved quantum genetic algorithm

Start

Initialize distribution scheme

Satisfy final condition

No

Yes

End

Calculate the time of the allocated schemes in the population

Record the shortest work time of the corresponding scheme

Optimize the allocated scheme

Calculate the shortest time of the optimal allocated scheme

Figure 4 Flowchart of acquiring the optimal allocated scheme and the shortest work time

Abstract and Applied Analysis 7

Step 3 (optimize the allocation scheme) Apply the improvedquantumgenetic algorithm to optimize the allocation schemein order to get the optimal allocation scheme Go to Step 2according to the termination condition to do the judgmentif the termination condition is satisfied then go to the End orelse do the loop to Step 3

Step 4 (determination of the optimal allocation schemeand the shortest work time) Choose the optimal allocationscheme form the population satisfied the termination con-dition Calculate the work time of the optimal allocationscheme

The flowchart is shown in Figure 4

5 Simulation Analyses

Take some road building project as an example There are12 roads which need to be built The building work canbe divided into three procedures prospecting drawing andlocation designing and constructionThere are 3 prospectinggroups 2 drawing and location designing groups and 3 con-struction groups The capacities of the groups are differentThe specific work times are shown in Table 4

The specific steps of acquiring the optimal allocatedscheme and the shortest work time are as follows

(1) Construct 12 times 3 dimensional matrix

11986012times3

=

[

[

[

[

[

11988611

11988612

11988613

11988621

11988622

11988623

119886121

119886122

119886123

]

]

]

]

]

(11)

where 1 le 11988611 11988621 11988631 119886

121lt 4 1 le

11988612 11988622 11988632 119886

122lt 3 1 le 119886

13 11988623 11988633 119886

123lt 4

The corresponding qubit chromosome is

11987612times3

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120572111

120573111

10038161003816100381610038161003816100381610038161003816

120572112

120573112

120572121

120573121

120572131

120573131

10038161003816100381610038161003816100381610038161003816

120572132

120573132

120572211

120573211

10038161003816100381610038161003816100381610038161003816

120572212

120573212

120572221

120573221

120572231

120573231

10038161003816100381610038161003816100381610038161003816

120572232

120573232

1205721211

1205731211

10038161003816100381610038161003816100381610038161003816

1205721212

1205731212

1205721221

1205731221

1205721231

1205731231

10038161003816100381610038161003816100381610038161003816

1205721232

1205731232

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(12)

Suppose the generated matrix is

11986012times3

=

[

[

[

[

[

11988611

11988612

11988613

11988621

11988622

11988623

119886121

119886122

119886123

]

]

]

]

]

=[

[

12 23 1 36 21 18 13 32 25 27

29 12 16 23 11 25 27 14 22 13

37 15 24 16 33 28 25 36 19 21

34 17

16 28

19 22

]

]

119879

1198601015840

12times3=[

[

1 2 1 3 2 1 1 3 2 2

2 1 1 2 1 2 2 1 2 1

3 1 2 1 3 2 2 3 1 2

3 1

1 2

1 2

]

]

119879

(13)

Then the encoding section of the assigned chromosome is

[1 2 1 3 2 1 1 3 2 2 3 1 2 1 1 2 1 2 2 1 2

1 1 2 3 1 2 1 3 2 2 3 1 2 1 2]

(14)

(2) Seek the values of time 1198871198961198941198861015840

119894119895

corresponding with1198821198961198721198941198861015840

119896119894

(where 1 le 119896 le 12 1 le 119894 le 3 119894 119895 119896are integers) in Table 2 Then the work timetable ofevery engineeringwork is gotten InTable 5 the figuredenotes the work time and the subscript denotes thework group

(3) Apply the method of Section 3 to calculate the worktimes of every work group according to the orderdetermined by Step (2) and the time determined byStep (3) Then find the maximum value as the totalengineering work time

Apply GA (genetic algorithm) QGA (quantum geneticalgorithm) and IQGA (improved quantum genetic algo-rithm) to solve the problem of scheduling engineering per-sonnelThe three kinds of algorithmarewritten and compiledin MATLAB The simulation test in this paper is based onthe AMD CPU 18GHz 15 GB RAM The parameters of GAare set as follows GA is encoded by real number the totalevolution generation is 80 the size of population is 40 and theprobability ofmutation is 001The parameters ofQGAare setas follows QGA is encoded by the binary the total evolutiongeneration is 80 the size of population is 40 the length ofthe variable expressing 3 work groups is 2 and the length ofthe variable expressing 2 work groups is 1 The parametersof IQGA are set as follows IQGA is encoded by the binarythe total evolution generation is 80 the size of population is40 the length of the variable expressing 3 work groups is 2the length of the variable expressing 2 work groups is 1 andthe probability of mutation is 001 The fitness function is theobjective function The fitness function tends to the optimal

8 Abstract and Applied Analysis

Table 4 Work time of every work group

1198691

1198692

1198693

11987211

11987212

11987213

11987221

11987222

11987231

11987232

11987233

1198821

3 2 4 3 4 5 3 41198822

4 5 3 6 5 5 6 31198823

4 3 5 4 2 3 4 51198824

2 5 3 5 4 1 5 41198825

4 2 6 3 1 3 4 61198826

1 5 3 2 3 5 2 71198827

7 6 8 4 6 6 7 61198828

6 4 5 7 5 4 8 51198829

2 5 4 1 2 3 6 311988210

3 6 4 3 4 4 3 911988211

5 2 3 3 5 8 5 411988212

4 3 4 6 5 4 7 5

Table 5 Work time of every engineering work and correspondingwork group

Engineering work Process1198691

1198692

1198693

1198821

31

42

43

1198822

52

61

51

1198823

41

41

42

1198824

33

42

11

1198825

22

31

63

1198826

11

32

22

1198827

71

62

72

1198828

53

71

53

1198829

52

22

31

11988210

62

31

32

11988211

33

31

81

11988212

41

52

72

value while the evolution increases Simulate the process ofevolution Ten rounds of experiments were presented in thispaper The simulation results of the 3 kinds of algorithm arein Tables 6 7 and 8 The result of some time of the IQGA isshown in Figures 5 6 and 7

The statistical results of the experiment are shown inTable 9

Table 9 shows that compared with GA QGA can get itssmallest value in a short time however QGA can hardly getthe optimal result Though the evolution time of the IQGA islonger and the evolution generation of the IQGA is more theIQGA can get the optimal result in a bigger probability

In Figure 5 x-axis represents the evolution generationsand y-axis represents the work timeThe work time graduallybecomes shorter with the increase of the generation

In Figure 6 x-axis represents the evolution algebra andy-axis represents the work time The average work time of allindividuals has a decreasing trend but there is a gap betweenthe optimal value and the average work time

0 10 20 30 40 50 60 70 8026

27

28

29

30

31

32

Evolutionary generations

The b

est fi

tnes

s of e

very

gen

erat

ion

Figure 5The changing curve of work time of the best individual inthe population

0 10 20 30 40 50 60 70 8036

37

38

39

40

41

42

43

44

45

Figure 6 The changing curve of average work time of all individu-als

Abstract and Applied Analysis 9

Table 6 The simulation results of ten rounds of experiments of GA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 27 26 26 28 27 29 28 28 28 28Simulation time (second) 3007 2905 2894 3102 2732 2940 2946 2846 3468 2937Evolution generation 22 20 23 28 19 19 18 29 18 29Evolution time (second) 08269 07263 08320 10857 06488 06983 06629 10317 07803 10647

Table 7 The simulation results of ten rounds of experiments of QGA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 30 27 29 30 28 29 31 30 29 28Simulation time (second) 5679 5337 5547 5576 5766 6150 5813 5828 6024 5802Evolution generation 12 13 10 4 4 15 3 4 8 21Evolution time (second) 08519 08673 06934 02788 02883 11531 02180 02914 06024 15230

Table 8 The simulation results of ten rounds of experiments of IQGA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 27 28 26 27 28 27 27 26 28 26Simulation time (second) 8847 8580 8527 8459 8505 8364 8422 8712 8342 8524Evolution generation 18 10 12 12 11 9 13 22 21 16Evolution time (second) 19906 10725 12791 12689 11694 09410 13686 23958 21898 17048

Table 9 The statistical results of the experiment

Algorithm The optimalresult

The worstresult

The averageresult

The fastestevolution time (s)

The slowestevolution time (s)

The averageevolution time (s)

Times ofconvergence

GA 26 29 275 06488 10857 08358 2QGA 27 31 291 02788 15230 06768 0IQGA 26 28 27 09410 23958 15381 3

0 5 10 15 20 25 300

2

4

6

8

10

122 1 23 2 1

2 2 13 1 1

2 2 21 2 2

2 1 21 2 1

3 1 31 1 2

1 1 32 2 1

Work time

The n

umbe

r of t

he en

gine

erin

g w

ork

Figure 7 Gantt chart of the optimal scheme

In Figure 7 x-axis represents time y-axis represents thenumber of the engineering workThe length of the rectanglesrepresents thework time of the engineeringwork the numberin the rectangle represents the number of the work group ofthe corresponding process

Engineering personnel can be scheduled based on theGantt chart of the optimal schemeThe role of the Gantt chartis to aid in allocation decisions of scheduling engineeringpersonnel

6 Conclusions

Compared with genetic algorithm and standard quantumevolutionary algorithm the advantages of improved quantumgenetic algorithm are as follows (1) IQGA adopts directedevolution which can find the optimal solution more quicklythan undirected evolution (2) the self-adaptive quantumrotation angle with larger rotation angle at the early stageof the evolution and with smaller rotation angle in the lateevolution is advantageous for exploration and exploitation(3) quantum mutation is good to generate new quantumchromosome to increase the diversity of population to extendthe search space (4) quantum disaster can make evolutionjump out of local optimal solution to increase the searchprobability of the global optimal solution (5) the whole inter-ference crossover takes effect on the level of chromosomecoding generated by quantum chromosome to generate newchromosome codingThe search space is effectively expandedand the probability to find optimal solution is increased

10 Abstract and Applied Analysis

So IQGA can search the optimal solution faster and moreefficiently based on the above-mentioned advantages

Application of improved quantum genetic algorithmand appropriate mathematical model can solve the complexscheduling engineering personnel problem Through thismethod we can obtain the optimal allocation scheme andthe work time of the allocation scheme is also determinedThe Gantt chart of the result clearly describes the work timeand work order of each engineering work The method inthis paper can provide powerful support for the schedulingengineering personnel The proposed method in this papernot only can be applied in the scheduling engineeringpersonnel but also can be used for similar types of problemwith multistage multiple projects and multiple subjects andthe abilities of subjects are different such as other schedulingproblems and flexible flow shop scheduling problem

Conflict of Interests

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

References

[1] P Brucker R Qu and E Burke ldquoPersonnel scheduling modelsand complexityrdquo European Journal of Operational Research vol210 no 3 pp 467ndash473 2011

[2] T Lapegue O Bellenguez-Morineau and D Prot ldquoAconstraint-based approach for the shift design personnel taskscheduling problem with equityrdquo Computers and OperationsResearch vol 40 no 10 pp 2450ndash2465 2013

[3] S-W Lin and K-C Ying ldquoMinimizing shifts for personneltask scheduling problems a three-phase algorithmrdquo EuropeanJournal ofOperational Research vol 237 no 1 pp 323ndash334 2014

[4] M Krishnamoorthy A T Ernst and D Baatar ldquoAlgorithmsfor large scale shift minimisation personnel task schedulingproblemsrdquo European Journal of Operational Research vol 219no 1 pp 34ndash48 2012

[5] G DuCote and E M Malstrom ldquoDesign of personnel schedul-ing software for manufacturingrdquo Computers and IndustrialEngineering vol 37 no 1 pp 473ndash476 1999

[6] P M Koeleman S Bhulai and M van Meersbergen ldquoOptimalpatient and personnel scheduling policies for care-at-homeservice facilitiesrdquo European Journal of Operational Research vol219 no 3 pp 557ndash563 2012

[7] J van den Bergh J Belien P de Bruecker E Demeulemeesterand L de Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013

[8] M Sabar B Montreuil and J-M Frayret ldquoA multi-agent-based approach for personnel scheduling in assembly centersrdquoEngineering Applications of Artificial Intelligence vol 22 no 7pp 1080ndash1088 2009

[9] K-H Han and J-H Kim ldquoGenetic quantum algorithm andits application to combinatorial optimization problemrdquo inProceedings of the Congress on Evolutionary Computation pp1354ndash1360 Piscataway NJ USA July 2000

[10] A Narayanan and M Moore ldquoQuantum-inspired genetic algo-rithmsrdquo in Proceedings of the IEEE International Conference on

Evolutionary Computatio (ICEC 96) pp 61ndash66 Nagoya JapanMay 1996

[11] HWang J Liu J Zhi andC Fu ldquoThe improvement of quantumgenetic algorithm and its application on function optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID730749 10 pages 2013

[12] J Tang G Zhang B Lin and B Zhang ldquoHybrid genetic algo-rithm for flow shop scheduling problemrdquo in Proceedings of theInternational Conference on Intelligent Computation Technologyand Automation (ICICTA 10) vol 2 pp 449ndash452 ChangshaChina May 2010

[13] R ZhangWWei Z Jiang andXMa ldquoResearch and simulationon flow-shop scheduling problem based on improved geneticalgorithmrdquo in Proceedings of the 7th International Conference onComputer Science and Education (ICCSE rsquo12) pp 916ndash919 July2012

[14] F Jin and Y Fu ldquoMaster-slave genetic algorithm for flowshop scheduling with resource flexibilityrdquo in Proceedings ofthe IEEE International Conference on Advanced ManagementScience (ICAMS rsquo10) vol 1 pp 341ndash346 Chengdu China July2010

[15] G Zhang and H Rong ldquoQuantum-inspired genetic algorithmbased time-frequency atom decompositionrdquo in Proceedings ofthe 7th international conference on Computational Science (ICCSrsquo07) 2007

[16] J-A Yang and Z-Q Zhuang ldquoActuality of research on quantumgenetic algorithmrdquo Journal of Computer Science amp Technologyvol 30 no 11 pp 13ndash15 2003

[17] W Shu and B He ldquoA quantum genetic simulated annealingalgorithm for task schedulingrdquo inAdvances in Computation andIntelligence L Kang Y Liu and S Zeng Eds vol 4683 ofLecture Notes in Computer Science pp 169ndash176 Springer BerlinGermany 2007

[18] Y Li Y Zhang Y Cheng and X Jiang ldquoA novel immunequantum-inspired genetic algorithmrdquo in Advances in NaturalComputation vol 3612 of Lecture Notes in Computer Science pp215ndash218 2005

[19] KHan and J Kim ldquoGenetic quantum algorithm and its applica-tion to combinatorial optimization problemrdquo in Proceedings ofthe Congress on Evolutionary Computation pp 1354ndash1360 SanDiego Calif USA July 2000

[20] H Miao H Wang and Z Deng ldquoQuantum genetic algo-rithm and its application in power system reactive poweroptimizationrdquo in Proceedings of the International Conference onComputational Intelligence and Security (CIS 09) vol 1 pp 107ndash111 Beijing China 2009

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Table 2 Total time of completion of every process

Work ProcessProcess 1 Process 2 Process 3 sdot sdot sdot Process 119894

1 0 + 119883(1)

(1199001)1119872(1)

(1199001)1+ 119883(2)

(1199011)1119872(2)

(1199011)1+ 119883(3)

(1199021)1sdot sdot sdot 119872

(119894minus1)

(1199101)1+ 119883(119894)

(1199111)1

2 119872(1)

(1199001)1+ 119883(1)

(1199002)2max (119872(1)

(1199002)2119872(2)

(1199011)1) + 119883

(2)

(1199012)2max (119872(2)

(1199012)2119872(3)

(1199021)1) + 119883

(3)

(1199022)2sdot sdot sdot max (119872(119894minus1)

(1199102)2119872(119894)

(1199111)1) + 119883

(119894)

(1199112)2

3 119872(1)

(1199002)2+ 119883(1)

(1199003)3max (119872(1)

(1199003)3119872(2)

(1199012)2) + 119883

(2)

(1199012)3max (119872(2)

(1199013)3119872(3)

(1199022)2) + 119883

(3)

(1199023)3sdot sdot sdot max (119872(119894minus1)

(1199103)3119872(119894)

(1199112)2) + 119883

(119894)

(1199113)3

119896 119872(1)

(119900119896minus1)119896minus1+ 119883(1)

(119900119896)119896max (119872(1)

(119900119896)119896119872(2)

(119901119896minus1)119896minus1) + 119883

(2)

(119901119896)119896max (119872(2)

(119901119896)119896119872(3)

(119902119896minus1)119896minus1) + 119883

(3)

(119902119896)119896sdot sdot sdot max (119872(119894minus1)

(119910119896)119896119872(119894)

(119911119896minus1)119896minus1) + 119883

(119894)

(1199112)119896

Themathematical model ismin = max(119872(2)(119901119896)119896119872(3)

(119902119896minus1)119896minus1) + 119883(3)

(119902119896)119896 where 119896 is the number of engineering works

The improved quantum genetic algorithm adopted in thispaper is the algorithm based on the algorithm in document

[11] The multiqubits encoding which has 119898 parameters isdefined as follows

119902119905

119895=[

[

120572119905

11

120573119905

11

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

12

120573119905

12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572119905

1119896

120573119905

1119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

21

120573119905

21

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

22

120573119905

22

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572119905

2119896

120573119905

2119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

1198981

120573119905

1198981

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572119905

1198982

120573119905

1198982

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572119905

119898119896

120573119905

119898119905

]

]

(6)

where 119902119905119895represents the 119895th individual chromosome of the tth

generation 119896 represents the number of qubit encoding of eachgene 119898 represents the number of genes in the chromosome

[15] Initialize the quantum encoding (120572 120573) of each individualin the population with (1radic2 1radic2) which indicates thatwhen 119905 = 0 the possibility of each state expressed by achromosome is equal [16]

The allocation coding matrix is encoded by qubit as

119876 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120572111

120573111

10038161003816100381610038161003816100381610038161003816

120572112

120573112

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572111198761

120573111198761

120572121

120573121

10038161003816100381610038161003816100381610038161003816

120572122

120573122

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572121198762

120573121198762

sdot sdot sdot

12057211198781

12057311198781

10038161003816100381610038161003816100381610038161003816

12057211198782

12057311198782

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

1205721119878119876119878

1205731119878119876119878

120572211

120573211

10038161003816100381610038161003816100381610038161003816

120572212

120573212

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572211198761

120573211198761

120572221

120573221

10038161003816100381610038161003816100381610038161003816

120572222

120573222

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572221198762

120573221198762

sdot sdot sdot

12057221198781

12057321198781

10038161003816100381610038161003816100381610038161003816

12057221198782

12057321198782

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

1205722119878119876119878

1205732119878119876119878

d

12057211987311

12057311987311

10038161003816100381610038161003816100381610038161003816

12057211987312

12057311987312

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

12057211987311198761

12057311987311198761

12057211987321

12057311987321

10038161003816100381610038161003816100381610038161003816

12057211987322

12057311987322

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

12057211987321198762

12057311987321198762

sdot sdot sdot

1205721198731198781

1205731198731198781

10038161003816100381610038161003816100381610038161003816

1205721198731198782

1205731198731198782

10038161003816100381610038161003816100381610038161003816

sdot sdot sdot

sdot sdot sdot

10038161003816100381610038161003816100381610038161003816

120572119873119878119876119878

120573119873119878119876119878

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(7)

The element in (7) is1205721198941198951

1205731198941198951

100381610038161003816100381610038161003816

1205721198941198952

1205731198941198952

100381610038161003816100381610038161003816

sdotsdotsdot

sdotsdotsdot

100381610038161003816100381610038161003816

120572119894119895119876119895

120573119894119895119876119895which represents

the qubit encoding of the ordinal number of work teamwhere 119894 (119894 = 1 2 119873) represents the ordinal number ofengineering work 119895 (119894 = 1 2 119878) represents the ordinalnumber of process and 119876

119896(119896 = 1 119878) represents the

length of the qubit encoding usually the more work groupsthere are for the process 119896 the longer the length is Thenumber 119886

119894119895generated by qubit encoding which represents the

119895th process of the 119894th work is finished by the 119886119894119895th work group

42 The Solution Based on Improved Quantum Genetic Algo-rithm The solution should be solved based on the qubitcoding

421 Quantum Rotating Gates Quantum genetic algorithmapplies the probability amplitude of qubits to encode chro-mosome and uses quantum rotating gates to realize chromo-somal updated operation

Abstract and Applied Analysis 5

Quantum rotating gates can be designed according to thepractical problems and usually be defined as

119880 (120579119894) = [

cos 120579119894minus sin 120579

119894

sin 120579119894

cos 120579119894

] (8)

The updated process is

[

[

1205721015840

119894

1205731015840

119894

]

]

= 119880 (120579119894) [

120572119894

120573119894

] = [

cos 120579119894minus sin 120579

119894

sin 120579119894

cos 120579119894

] [

120572119894

120573119894

] (9)

where (120572119894 120573119894)119879 and (1205721015840

119894 1205731015840

119894)

119879 are the probability amplitudes ofthe 119894th qubit in chromosome before and after the quantumrotating gates updating respectively 120579

119894is the rotating angle

and the value and the sign of 120579119894are determined by the adjust-

ment strategy [17 18] 120579119894is static in conventional quantum

genetic algorithmThevalue of 120579119894can be dynamically adjusted

based on the evolutionary process in the improved quantumgenetic algorithm Adjust the value of the rotating angleaccording to the number of generation the regulator is asfollows

120579119894= 120579max minus (

120579max minus 120579min119894119905max

) lowast iter (10)

where 120579119894is the value of the rotating angle of the 119894th generation

120579max is the maximal rotating angle 120579min is the minimalrotating angle 119894119905max is the maximal generation and iter isthe current generation 120579max = 001120587 120579min = 0001120587

The adjustment strategy is as follows comparing thefitness of the currently measured value 119891(119909) of the individual119902119905

119895with the fitness of the current optimal individual 119891(best

119894)

if 119891(119909) gt 119891(best) then adjust the corresponding qubits of119902119905

119895 making the probability amplitude (120572

119894 120573119894) evolve toward

the direction that is propitious to the emergence of 119909119894

Conversely if 119891(119909) lt 119891(best) then adjust the correspondingqubits of 119902119905

119895 making the probability amplitude (120572

119894 120573119894) evolve

toward the direction that is propitious to the emergence ofbest

Construct a determinant119860 = 10038161003816100381610038161003816

120572119887 120572119894

120573119887 120573119894

10038161003816100381610038161003816 where (120572

119887 120573119887) is the

probability amplitude of some qubit of the optimal solutionwhich has been searched (120572

119894 120573119894) is probability amplitude of

corresponding state of qubit in the current solution When119909119894= best119894 if 119860 = 0 then the direction of the rotating angle is

minussgn(119860) If 119860 = 0 the direction of the rotating angle can beeither positive of negative

422 Quantum Mutation Add quantum mutation opera-tion in conventional quantum genetic algorithm Quantummutation enables some individual to deviate slightly fromthe current evolutionary direction and prevent the evolutionof individual into a local optimal solution [19] Quantummutation can completely reverse the individualrsquos evolutionarydirection by swapping the value of probability amplitude ofqubits (120572 120573) Quantum NOT gates are adopted to realizechromosomal variation

Table 3 One result of the whole interference crossover

Chromosomenumber New chromosome

1 A(1) D(2) C(3) B(4) A(5) D(6) C(7) B(8)

2 B(1) A(2) D(3) C(4) B(5) A(6) D(7) C(8)

3 C(1) B(2) A(3) D(4) C(5) B(6) A(7) D(8)

4 D(1) C(2) B(3) A(4) D(5) C(6) B(7) A(8)

423 QuantumDisaster Add quantum disaster operation inthe conventional quantum genetic algorithm The methodis to apply a large disturbance to some individuals inthe population and regenerate some other new randomindividuals [20]

424 Whole Interference Crossover Whole interferencecrossover operator is utilized to realize evolution insteadof the traditional crossover operator All chromosomes areinvolved in the whole interference crossover Set the numberof population to be 119898 and length of chromosome is 119899(119898 lt 119899) 119860

119894119895is the 119895th bit of the 119894th individual Then the

chromosome of the 119894th individual of the new family throughthewhole interference crossover is119860

(119894 mod 119898)1minus119860 (119894+1 mod 119898)2 minussdot sdot sdot minus 119860

(119894+119896 mod 119898)(119896+1) minus sdot sdot sdot minus 119860 (119894+119899minus1 mod 119898)119899 The new wholeinterference crossover operator is mainly used when thealgorithm converges to local optimal solution to generate newindividuals to overcome the prematurity in later evolutionSuppose that there are 4 populations and the lengths of themare all 8 One result of the whole interference crossover is asin Table 3

It is important to note that quantum rotating gatesquantummutation and quantum disaster all act on quantumchromosome However whole interference crossover acts onthe chromosome generated by quantum chromosome Theevolution performance will not be introduced if the wholeinterference crossover acts on quantum chromosome

The improved strategy of quantum genetic algorithmmainly contains self-adaptive rotating angle strategy theimprovement of program structure of quantum evolutionadded quantum mutation operation quantum disaster oper-ation and whole interference crossover operation Theflowchart of improved quantum genetic algorithm is shownin Figure 3

43 The Determining Flow of Scheduling Engineering Per-sonnel The steps to determine the scheme of schedulingengineering personnel are as follows

Step 1 (initialize distribution scheme) Initialize allocationscheme to form the initial population according to the needsof the evolution of quantum genetic algorithm and theassignment matrix

Step 2 (time calculation) Calculate the time of all theallocated schemes in the population

6 Abstract and Applied Analysis

Start

Satisfy final condition

Memorize the best individual and its fitness

End

Yes

No

Colony disaster

Yes

No

Initialize the population Q(t0) and every parameter

Test every individual of Q(t0)

Evaluate the fitness of every individual of Q(t0)

Set the best individual as the evolutional goal of next generation

Test every individual of Q(t0)

Quantum rotating door update population get Q(t + 1)

Quantum mutation whole interference crossover

Whether or not disaster

t = t + 1

Figure 3 Flowchart of improved quantum genetic algorithm

Start

Initialize distribution scheme

Satisfy final condition

No

Yes

End

Calculate the time of the allocated schemes in the population

Record the shortest work time of the corresponding scheme

Optimize the allocated scheme

Calculate the shortest time of the optimal allocated scheme

Figure 4 Flowchart of acquiring the optimal allocated scheme and the shortest work time

Abstract and Applied Analysis 7

Step 3 (optimize the allocation scheme) Apply the improvedquantumgenetic algorithm to optimize the allocation schemein order to get the optimal allocation scheme Go to Step 2according to the termination condition to do the judgmentif the termination condition is satisfied then go to the End orelse do the loop to Step 3

Step 4 (determination of the optimal allocation schemeand the shortest work time) Choose the optimal allocationscheme form the population satisfied the termination con-dition Calculate the work time of the optimal allocationscheme

The flowchart is shown in Figure 4

5 Simulation Analyses

Take some road building project as an example There are12 roads which need to be built The building work canbe divided into three procedures prospecting drawing andlocation designing and constructionThere are 3 prospectinggroups 2 drawing and location designing groups and 3 con-struction groups The capacities of the groups are differentThe specific work times are shown in Table 4

The specific steps of acquiring the optimal allocatedscheme and the shortest work time are as follows

(1) Construct 12 times 3 dimensional matrix

11986012times3

=

[

[

[

[

[

11988611

11988612

11988613

11988621

11988622

11988623

119886121

119886122

119886123

]

]

]

]

]

(11)

where 1 le 11988611 11988621 11988631 119886

121lt 4 1 le

11988612 11988622 11988632 119886

122lt 3 1 le 119886

13 11988623 11988633 119886

123lt 4

The corresponding qubit chromosome is

11987612times3

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120572111

120573111

10038161003816100381610038161003816100381610038161003816

120572112

120573112

120572121

120573121

120572131

120573131

10038161003816100381610038161003816100381610038161003816

120572132

120573132

120572211

120573211

10038161003816100381610038161003816100381610038161003816

120572212

120573212

120572221

120573221

120572231

120573231

10038161003816100381610038161003816100381610038161003816

120572232

120573232

1205721211

1205731211

10038161003816100381610038161003816100381610038161003816

1205721212

1205731212

1205721221

1205731221

1205721231

1205731231

10038161003816100381610038161003816100381610038161003816

1205721232

1205731232

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(12)

Suppose the generated matrix is

11986012times3

=

[

[

[

[

[

11988611

11988612

11988613

11988621

11988622

11988623

119886121

119886122

119886123

]

]

]

]

]

=[

[

12 23 1 36 21 18 13 32 25 27

29 12 16 23 11 25 27 14 22 13

37 15 24 16 33 28 25 36 19 21

34 17

16 28

19 22

]

]

119879

1198601015840

12times3=[

[

1 2 1 3 2 1 1 3 2 2

2 1 1 2 1 2 2 1 2 1

3 1 2 1 3 2 2 3 1 2

3 1

1 2

1 2

]

]

119879

(13)

Then the encoding section of the assigned chromosome is

[1 2 1 3 2 1 1 3 2 2 3 1 2 1 1 2 1 2 2 1 2

1 1 2 3 1 2 1 3 2 2 3 1 2 1 2]

(14)

(2) Seek the values of time 1198871198961198941198861015840

119894119895

corresponding with1198821198961198721198941198861015840

119896119894

(where 1 le 119896 le 12 1 le 119894 le 3 119894 119895 119896are integers) in Table 2 Then the work timetable ofevery engineeringwork is gotten InTable 5 the figuredenotes the work time and the subscript denotes thework group

(3) Apply the method of Section 3 to calculate the worktimes of every work group according to the orderdetermined by Step (2) and the time determined byStep (3) Then find the maximum value as the totalengineering work time

Apply GA (genetic algorithm) QGA (quantum geneticalgorithm) and IQGA (improved quantum genetic algo-rithm) to solve the problem of scheduling engineering per-sonnelThe three kinds of algorithmarewritten and compiledin MATLAB The simulation test in this paper is based onthe AMD CPU 18GHz 15 GB RAM The parameters of GAare set as follows GA is encoded by real number the totalevolution generation is 80 the size of population is 40 and theprobability ofmutation is 001The parameters ofQGAare setas follows QGA is encoded by the binary the total evolutiongeneration is 80 the size of population is 40 the length ofthe variable expressing 3 work groups is 2 and the length ofthe variable expressing 2 work groups is 1 The parametersof IQGA are set as follows IQGA is encoded by the binarythe total evolution generation is 80 the size of population is40 the length of the variable expressing 3 work groups is 2the length of the variable expressing 2 work groups is 1 andthe probability of mutation is 001 The fitness function is theobjective function The fitness function tends to the optimal

8 Abstract and Applied Analysis

Table 4 Work time of every work group

1198691

1198692

1198693

11987211

11987212

11987213

11987221

11987222

11987231

11987232

11987233

1198821

3 2 4 3 4 5 3 41198822

4 5 3 6 5 5 6 31198823

4 3 5 4 2 3 4 51198824

2 5 3 5 4 1 5 41198825

4 2 6 3 1 3 4 61198826

1 5 3 2 3 5 2 71198827

7 6 8 4 6 6 7 61198828

6 4 5 7 5 4 8 51198829

2 5 4 1 2 3 6 311988210

3 6 4 3 4 4 3 911988211

5 2 3 3 5 8 5 411988212

4 3 4 6 5 4 7 5

Table 5 Work time of every engineering work and correspondingwork group

Engineering work Process1198691

1198692

1198693

1198821

31

42

43

1198822

52

61

51

1198823

41

41

42

1198824

33

42

11

1198825

22

31

63

1198826

11

32

22

1198827

71

62

72

1198828

53

71

53

1198829

52

22

31

11988210

62

31

32

11988211

33

31

81

11988212

41

52

72

value while the evolution increases Simulate the process ofevolution Ten rounds of experiments were presented in thispaper The simulation results of the 3 kinds of algorithm arein Tables 6 7 and 8 The result of some time of the IQGA isshown in Figures 5 6 and 7

The statistical results of the experiment are shown inTable 9

Table 9 shows that compared with GA QGA can get itssmallest value in a short time however QGA can hardly getthe optimal result Though the evolution time of the IQGA islonger and the evolution generation of the IQGA is more theIQGA can get the optimal result in a bigger probability

In Figure 5 x-axis represents the evolution generationsand y-axis represents the work timeThe work time graduallybecomes shorter with the increase of the generation

In Figure 6 x-axis represents the evolution algebra andy-axis represents the work time The average work time of allindividuals has a decreasing trend but there is a gap betweenthe optimal value and the average work time

0 10 20 30 40 50 60 70 8026

27

28

29

30

31

32

Evolutionary generations

The b

est fi

tnes

s of e

very

gen

erat

ion

Figure 5The changing curve of work time of the best individual inthe population

0 10 20 30 40 50 60 70 8036

37

38

39

40

41

42

43

44

45

Figure 6 The changing curve of average work time of all individu-als

Abstract and Applied Analysis 9

Table 6 The simulation results of ten rounds of experiments of GA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 27 26 26 28 27 29 28 28 28 28Simulation time (second) 3007 2905 2894 3102 2732 2940 2946 2846 3468 2937Evolution generation 22 20 23 28 19 19 18 29 18 29Evolution time (second) 08269 07263 08320 10857 06488 06983 06629 10317 07803 10647

Table 7 The simulation results of ten rounds of experiments of QGA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 30 27 29 30 28 29 31 30 29 28Simulation time (second) 5679 5337 5547 5576 5766 6150 5813 5828 6024 5802Evolution generation 12 13 10 4 4 15 3 4 8 21Evolution time (second) 08519 08673 06934 02788 02883 11531 02180 02914 06024 15230

Table 8 The simulation results of ten rounds of experiments of IQGA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 27 28 26 27 28 27 27 26 28 26Simulation time (second) 8847 8580 8527 8459 8505 8364 8422 8712 8342 8524Evolution generation 18 10 12 12 11 9 13 22 21 16Evolution time (second) 19906 10725 12791 12689 11694 09410 13686 23958 21898 17048

Table 9 The statistical results of the experiment

Algorithm The optimalresult

The worstresult

The averageresult

The fastestevolution time (s)

The slowestevolution time (s)

The averageevolution time (s)

Times ofconvergence

GA 26 29 275 06488 10857 08358 2QGA 27 31 291 02788 15230 06768 0IQGA 26 28 27 09410 23958 15381 3

0 5 10 15 20 25 300

2

4

6

8

10

122 1 23 2 1

2 2 13 1 1

2 2 21 2 2

2 1 21 2 1

3 1 31 1 2

1 1 32 2 1

Work time

The n

umbe

r of t

he en

gine

erin

g w

ork

Figure 7 Gantt chart of the optimal scheme

In Figure 7 x-axis represents time y-axis represents thenumber of the engineering workThe length of the rectanglesrepresents thework time of the engineeringwork the numberin the rectangle represents the number of the work group ofthe corresponding process

Engineering personnel can be scheduled based on theGantt chart of the optimal schemeThe role of the Gantt chartis to aid in allocation decisions of scheduling engineeringpersonnel

6 Conclusions

Compared with genetic algorithm and standard quantumevolutionary algorithm the advantages of improved quantumgenetic algorithm are as follows (1) IQGA adopts directedevolution which can find the optimal solution more quicklythan undirected evolution (2) the self-adaptive quantumrotation angle with larger rotation angle at the early stageof the evolution and with smaller rotation angle in the lateevolution is advantageous for exploration and exploitation(3) quantum mutation is good to generate new quantumchromosome to increase the diversity of population to extendthe search space (4) quantum disaster can make evolutionjump out of local optimal solution to increase the searchprobability of the global optimal solution (5) the whole inter-ference crossover takes effect on the level of chromosomecoding generated by quantum chromosome to generate newchromosome codingThe search space is effectively expandedand the probability to find optimal solution is increased

10 Abstract and Applied Analysis

So IQGA can search the optimal solution faster and moreefficiently based on the above-mentioned advantages

Application of improved quantum genetic algorithmand appropriate mathematical model can solve the complexscheduling engineering personnel problem Through thismethod we can obtain the optimal allocation scheme andthe work time of the allocation scheme is also determinedThe Gantt chart of the result clearly describes the work timeand work order of each engineering work The method inthis paper can provide powerful support for the schedulingengineering personnel The proposed method in this papernot only can be applied in the scheduling engineeringpersonnel but also can be used for similar types of problemwith multistage multiple projects and multiple subjects andthe abilities of subjects are different such as other schedulingproblems and flexible flow shop scheduling problem

Conflict of Interests

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

References

[1] P Brucker R Qu and E Burke ldquoPersonnel scheduling modelsand complexityrdquo European Journal of Operational Research vol210 no 3 pp 467ndash473 2011

[2] T Lapegue O Bellenguez-Morineau and D Prot ldquoAconstraint-based approach for the shift design personnel taskscheduling problem with equityrdquo Computers and OperationsResearch vol 40 no 10 pp 2450ndash2465 2013

[3] S-W Lin and K-C Ying ldquoMinimizing shifts for personneltask scheduling problems a three-phase algorithmrdquo EuropeanJournal ofOperational Research vol 237 no 1 pp 323ndash334 2014

[4] M Krishnamoorthy A T Ernst and D Baatar ldquoAlgorithmsfor large scale shift minimisation personnel task schedulingproblemsrdquo European Journal of Operational Research vol 219no 1 pp 34ndash48 2012

[5] G DuCote and E M Malstrom ldquoDesign of personnel schedul-ing software for manufacturingrdquo Computers and IndustrialEngineering vol 37 no 1 pp 473ndash476 1999

[6] P M Koeleman S Bhulai and M van Meersbergen ldquoOptimalpatient and personnel scheduling policies for care-at-homeservice facilitiesrdquo European Journal of Operational Research vol219 no 3 pp 557ndash563 2012

[7] J van den Bergh J Belien P de Bruecker E Demeulemeesterand L de Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013

[8] M Sabar B Montreuil and J-M Frayret ldquoA multi-agent-based approach for personnel scheduling in assembly centersrdquoEngineering Applications of Artificial Intelligence vol 22 no 7pp 1080ndash1088 2009

[9] K-H Han and J-H Kim ldquoGenetic quantum algorithm andits application to combinatorial optimization problemrdquo inProceedings of the Congress on Evolutionary Computation pp1354ndash1360 Piscataway NJ USA July 2000

[10] A Narayanan and M Moore ldquoQuantum-inspired genetic algo-rithmsrdquo in Proceedings of the IEEE International Conference on

Evolutionary Computatio (ICEC 96) pp 61ndash66 Nagoya JapanMay 1996

[11] HWang J Liu J Zhi andC Fu ldquoThe improvement of quantumgenetic algorithm and its application on function optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID730749 10 pages 2013

[12] J Tang G Zhang B Lin and B Zhang ldquoHybrid genetic algo-rithm for flow shop scheduling problemrdquo in Proceedings of theInternational Conference on Intelligent Computation Technologyand Automation (ICICTA 10) vol 2 pp 449ndash452 ChangshaChina May 2010

[13] R ZhangWWei Z Jiang andXMa ldquoResearch and simulationon flow-shop scheduling problem based on improved geneticalgorithmrdquo in Proceedings of the 7th International Conference onComputer Science and Education (ICCSE rsquo12) pp 916ndash919 July2012

[14] F Jin and Y Fu ldquoMaster-slave genetic algorithm for flowshop scheduling with resource flexibilityrdquo in Proceedings ofthe IEEE International Conference on Advanced ManagementScience (ICAMS rsquo10) vol 1 pp 341ndash346 Chengdu China July2010

[15] G Zhang and H Rong ldquoQuantum-inspired genetic algorithmbased time-frequency atom decompositionrdquo in Proceedings ofthe 7th international conference on Computational Science (ICCSrsquo07) 2007

[16] J-A Yang and Z-Q Zhuang ldquoActuality of research on quantumgenetic algorithmrdquo Journal of Computer Science amp Technologyvol 30 no 11 pp 13ndash15 2003

[17] W Shu and B He ldquoA quantum genetic simulated annealingalgorithm for task schedulingrdquo inAdvances in Computation andIntelligence L Kang Y Liu and S Zeng Eds vol 4683 ofLecture Notes in Computer Science pp 169ndash176 Springer BerlinGermany 2007

[18] Y Li Y Zhang Y Cheng and X Jiang ldquoA novel immunequantum-inspired genetic algorithmrdquo in Advances in NaturalComputation vol 3612 of Lecture Notes in Computer Science pp215ndash218 2005

[19] KHan and J Kim ldquoGenetic quantum algorithm and its applica-tion to combinatorial optimization problemrdquo in Proceedings ofthe Congress on Evolutionary Computation pp 1354ndash1360 SanDiego Calif USA July 2000

[20] H Miao H Wang and Z Deng ldquoQuantum genetic algo-rithm and its application in power system reactive poweroptimizationrdquo in Proceedings of the International Conference onComputational Intelligence and Security (CIS 09) vol 1 pp 107ndash111 Beijing China 2009

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

Quantum rotating gates can be designed according to thepractical problems and usually be defined as

119880 (120579119894) = [

cos 120579119894minus sin 120579

119894

sin 120579119894

cos 120579119894

] (8)

The updated process is

[

[

1205721015840

119894

1205731015840

119894

]

]

= 119880 (120579119894) [

120572119894

120573119894

] = [

cos 120579119894minus sin 120579

119894

sin 120579119894

cos 120579119894

] [

120572119894

120573119894

] (9)

where (120572119894 120573119894)119879 and (1205721015840

119894 1205731015840

119894)

119879 are the probability amplitudes ofthe 119894th qubit in chromosome before and after the quantumrotating gates updating respectively 120579

119894is the rotating angle

and the value and the sign of 120579119894are determined by the adjust-

ment strategy [17 18] 120579119894is static in conventional quantum

genetic algorithmThevalue of 120579119894can be dynamically adjusted

based on the evolutionary process in the improved quantumgenetic algorithm Adjust the value of the rotating angleaccording to the number of generation the regulator is asfollows

120579119894= 120579max minus (

120579max minus 120579min119894119905max

) lowast iter (10)

where 120579119894is the value of the rotating angle of the 119894th generation

120579max is the maximal rotating angle 120579min is the minimalrotating angle 119894119905max is the maximal generation and iter isthe current generation 120579max = 001120587 120579min = 0001120587

The adjustment strategy is as follows comparing thefitness of the currently measured value 119891(119909) of the individual119902119905

119895with the fitness of the current optimal individual 119891(best

119894)

if 119891(119909) gt 119891(best) then adjust the corresponding qubits of119902119905

119895 making the probability amplitude (120572

119894 120573119894) evolve toward

the direction that is propitious to the emergence of 119909119894

Conversely if 119891(119909) lt 119891(best) then adjust the correspondingqubits of 119902119905

119895 making the probability amplitude (120572

119894 120573119894) evolve

toward the direction that is propitious to the emergence ofbest

Construct a determinant119860 = 10038161003816100381610038161003816

120572119887 120572119894

120573119887 120573119894

10038161003816100381610038161003816 where (120572

119887 120573119887) is the

probability amplitude of some qubit of the optimal solutionwhich has been searched (120572

119894 120573119894) is probability amplitude of

corresponding state of qubit in the current solution When119909119894= best119894 if 119860 = 0 then the direction of the rotating angle is

minussgn(119860) If 119860 = 0 the direction of the rotating angle can beeither positive of negative

422 Quantum Mutation Add quantum mutation opera-tion in conventional quantum genetic algorithm Quantummutation enables some individual to deviate slightly fromthe current evolutionary direction and prevent the evolutionof individual into a local optimal solution [19] Quantummutation can completely reverse the individualrsquos evolutionarydirection by swapping the value of probability amplitude ofqubits (120572 120573) Quantum NOT gates are adopted to realizechromosomal variation

Table 3 One result of the whole interference crossover

Chromosomenumber New chromosome

1 A(1) D(2) C(3) B(4) A(5) D(6) C(7) B(8)

2 B(1) A(2) D(3) C(4) B(5) A(6) D(7) C(8)

3 C(1) B(2) A(3) D(4) C(5) B(6) A(7) D(8)

4 D(1) C(2) B(3) A(4) D(5) C(6) B(7) A(8)

423 QuantumDisaster Add quantum disaster operation inthe conventional quantum genetic algorithm The methodis to apply a large disturbance to some individuals inthe population and regenerate some other new randomindividuals [20]

424 Whole Interference Crossover Whole interferencecrossover operator is utilized to realize evolution insteadof the traditional crossover operator All chromosomes areinvolved in the whole interference crossover Set the numberof population to be 119898 and length of chromosome is 119899(119898 lt 119899) 119860

119894119895is the 119895th bit of the 119894th individual Then the

chromosome of the 119894th individual of the new family throughthewhole interference crossover is119860

(119894 mod 119898)1minus119860 (119894+1 mod 119898)2 minussdot sdot sdot minus 119860

(119894+119896 mod 119898)(119896+1) minus sdot sdot sdot minus 119860 (119894+119899minus1 mod 119898)119899 The new wholeinterference crossover operator is mainly used when thealgorithm converges to local optimal solution to generate newindividuals to overcome the prematurity in later evolutionSuppose that there are 4 populations and the lengths of themare all 8 One result of the whole interference crossover is asin Table 3

It is important to note that quantum rotating gatesquantummutation and quantum disaster all act on quantumchromosome However whole interference crossover acts onthe chromosome generated by quantum chromosome Theevolution performance will not be introduced if the wholeinterference crossover acts on quantum chromosome

The improved strategy of quantum genetic algorithmmainly contains self-adaptive rotating angle strategy theimprovement of program structure of quantum evolutionadded quantum mutation operation quantum disaster oper-ation and whole interference crossover operation Theflowchart of improved quantum genetic algorithm is shownin Figure 3

43 The Determining Flow of Scheduling Engineering Per-sonnel The steps to determine the scheme of schedulingengineering personnel are as follows

Step 1 (initialize distribution scheme) Initialize allocationscheme to form the initial population according to the needsof the evolution of quantum genetic algorithm and theassignment matrix

Step 2 (time calculation) Calculate the time of all theallocated schemes in the population

6 Abstract and Applied Analysis

Start

Satisfy final condition

Memorize the best individual and its fitness

End

Yes

No

Colony disaster

Yes

No

Initialize the population Q(t0) and every parameter

Test every individual of Q(t0)

Evaluate the fitness of every individual of Q(t0)

Set the best individual as the evolutional goal of next generation

Test every individual of Q(t0)

Quantum rotating door update population get Q(t + 1)

Quantum mutation whole interference crossover

Whether or not disaster

t = t + 1

Figure 3 Flowchart of improved quantum genetic algorithm

Start

Initialize distribution scheme

Satisfy final condition

No

Yes

End

Calculate the time of the allocated schemes in the population

Record the shortest work time of the corresponding scheme

Optimize the allocated scheme

Calculate the shortest time of the optimal allocated scheme

Figure 4 Flowchart of acquiring the optimal allocated scheme and the shortest work time

Abstract and Applied Analysis 7

Step 3 (optimize the allocation scheme) Apply the improvedquantumgenetic algorithm to optimize the allocation schemein order to get the optimal allocation scheme Go to Step 2according to the termination condition to do the judgmentif the termination condition is satisfied then go to the End orelse do the loop to Step 3

Step 4 (determination of the optimal allocation schemeand the shortest work time) Choose the optimal allocationscheme form the population satisfied the termination con-dition Calculate the work time of the optimal allocationscheme

The flowchart is shown in Figure 4

5 Simulation Analyses

Take some road building project as an example There are12 roads which need to be built The building work canbe divided into three procedures prospecting drawing andlocation designing and constructionThere are 3 prospectinggroups 2 drawing and location designing groups and 3 con-struction groups The capacities of the groups are differentThe specific work times are shown in Table 4

The specific steps of acquiring the optimal allocatedscheme and the shortest work time are as follows

(1) Construct 12 times 3 dimensional matrix

11986012times3

=

[

[

[

[

[

11988611

11988612

11988613

11988621

11988622

11988623

119886121

119886122

119886123

]

]

]

]

]

(11)

where 1 le 11988611 11988621 11988631 119886

121lt 4 1 le

11988612 11988622 11988632 119886

122lt 3 1 le 119886

13 11988623 11988633 119886

123lt 4

The corresponding qubit chromosome is

11987612times3

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120572111

120573111

10038161003816100381610038161003816100381610038161003816

120572112

120573112

120572121

120573121

120572131

120573131

10038161003816100381610038161003816100381610038161003816

120572132

120573132

120572211

120573211

10038161003816100381610038161003816100381610038161003816

120572212

120573212

120572221

120573221

120572231

120573231

10038161003816100381610038161003816100381610038161003816

120572232

120573232

1205721211

1205731211

10038161003816100381610038161003816100381610038161003816

1205721212

1205731212

1205721221

1205731221

1205721231

1205731231

10038161003816100381610038161003816100381610038161003816

1205721232

1205731232

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(12)

Suppose the generated matrix is

11986012times3

=

[

[

[

[

[

11988611

11988612

11988613

11988621

11988622

11988623

119886121

119886122

119886123

]

]

]

]

]

=[

[

12 23 1 36 21 18 13 32 25 27

29 12 16 23 11 25 27 14 22 13

37 15 24 16 33 28 25 36 19 21

34 17

16 28

19 22

]

]

119879

1198601015840

12times3=[

[

1 2 1 3 2 1 1 3 2 2

2 1 1 2 1 2 2 1 2 1

3 1 2 1 3 2 2 3 1 2

3 1

1 2

1 2

]

]

119879

(13)

Then the encoding section of the assigned chromosome is

[1 2 1 3 2 1 1 3 2 2 3 1 2 1 1 2 1 2 2 1 2

1 1 2 3 1 2 1 3 2 2 3 1 2 1 2]

(14)

(2) Seek the values of time 1198871198961198941198861015840

119894119895

corresponding with1198821198961198721198941198861015840

119896119894

(where 1 le 119896 le 12 1 le 119894 le 3 119894 119895 119896are integers) in Table 2 Then the work timetable ofevery engineeringwork is gotten InTable 5 the figuredenotes the work time and the subscript denotes thework group

(3) Apply the method of Section 3 to calculate the worktimes of every work group according to the orderdetermined by Step (2) and the time determined byStep (3) Then find the maximum value as the totalengineering work time

Apply GA (genetic algorithm) QGA (quantum geneticalgorithm) and IQGA (improved quantum genetic algo-rithm) to solve the problem of scheduling engineering per-sonnelThe three kinds of algorithmarewritten and compiledin MATLAB The simulation test in this paper is based onthe AMD CPU 18GHz 15 GB RAM The parameters of GAare set as follows GA is encoded by real number the totalevolution generation is 80 the size of population is 40 and theprobability ofmutation is 001The parameters ofQGAare setas follows QGA is encoded by the binary the total evolutiongeneration is 80 the size of population is 40 the length ofthe variable expressing 3 work groups is 2 and the length ofthe variable expressing 2 work groups is 1 The parametersof IQGA are set as follows IQGA is encoded by the binarythe total evolution generation is 80 the size of population is40 the length of the variable expressing 3 work groups is 2the length of the variable expressing 2 work groups is 1 andthe probability of mutation is 001 The fitness function is theobjective function The fitness function tends to the optimal

8 Abstract and Applied Analysis

Table 4 Work time of every work group

1198691

1198692

1198693

11987211

11987212

11987213

11987221

11987222

11987231

11987232

11987233

1198821

3 2 4 3 4 5 3 41198822

4 5 3 6 5 5 6 31198823

4 3 5 4 2 3 4 51198824

2 5 3 5 4 1 5 41198825

4 2 6 3 1 3 4 61198826

1 5 3 2 3 5 2 71198827

7 6 8 4 6 6 7 61198828

6 4 5 7 5 4 8 51198829

2 5 4 1 2 3 6 311988210

3 6 4 3 4 4 3 911988211

5 2 3 3 5 8 5 411988212

4 3 4 6 5 4 7 5

Table 5 Work time of every engineering work and correspondingwork group

Engineering work Process1198691

1198692

1198693

1198821

31

42

43

1198822

52

61

51

1198823

41

41

42

1198824

33

42

11

1198825

22

31

63

1198826

11

32

22

1198827

71

62

72

1198828

53

71

53

1198829

52

22

31

11988210

62

31

32

11988211

33

31

81

11988212

41

52

72

value while the evolution increases Simulate the process ofevolution Ten rounds of experiments were presented in thispaper The simulation results of the 3 kinds of algorithm arein Tables 6 7 and 8 The result of some time of the IQGA isshown in Figures 5 6 and 7

The statistical results of the experiment are shown inTable 9

Table 9 shows that compared with GA QGA can get itssmallest value in a short time however QGA can hardly getthe optimal result Though the evolution time of the IQGA islonger and the evolution generation of the IQGA is more theIQGA can get the optimal result in a bigger probability

In Figure 5 x-axis represents the evolution generationsand y-axis represents the work timeThe work time graduallybecomes shorter with the increase of the generation

In Figure 6 x-axis represents the evolution algebra andy-axis represents the work time The average work time of allindividuals has a decreasing trend but there is a gap betweenthe optimal value and the average work time

0 10 20 30 40 50 60 70 8026

27

28

29

30

31

32

Evolutionary generations

The b

est fi

tnes

s of e

very

gen

erat

ion

Figure 5The changing curve of work time of the best individual inthe population

0 10 20 30 40 50 60 70 8036

37

38

39

40

41

42

43

44

45

Figure 6 The changing curve of average work time of all individu-als

Abstract and Applied Analysis 9

Table 6 The simulation results of ten rounds of experiments of GA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 27 26 26 28 27 29 28 28 28 28Simulation time (second) 3007 2905 2894 3102 2732 2940 2946 2846 3468 2937Evolution generation 22 20 23 28 19 19 18 29 18 29Evolution time (second) 08269 07263 08320 10857 06488 06983 06629 10317 07803 10647

Table 7 The simulation results of ten rounds of experiments of QGA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 30 27 29 30 28 29 31 30 29 28Simulation time (second) 5679 5337 5547 5576 5766 6150 5813 5828 6024 5802Evolution generation 12 13 10 4 4 15 3 4 8 21Evolution time (second) 08519 08673 06934 02788 02883 11531 02180 02914 06024 15230

Table 8 The simulation results of ten rounds of experiments of IQGA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 27 28 26 27 28 27 27 26 28 26Simulation time (second) 8847 8580 8527 8459 8505 8364 8422 8712 8342 8524Evolution generation 18 10 12 12 11 9 13 22 21 16Evolution time (second) 19906 10725 12791 12689 11694 09410 13686 23958 21898 17048

Table 9 The statistical results of the experiment

Algorithm The optimalresult

The worstresult

The averageresult

The fastestevolution time (s)

The slowestevolution time (s)

The averageevolution time (s)

Times ofconvergence

GA 26 29 275 06488 10857 08358 2QGA 27 31 291 02788 15230 06768 0IQGA 26 28 27 09410 23958 15381 3

0 5 10 15 20 25 300

2

4

6

8

10

122 1 23 2 1

2 2 13 1 1

2 2 21 2 2

2 1 21 2 1

3 1 31 1 2

1 1 32 2 1

Work time

The n

umbe

r of t

he en

gine

erin

g w

ork

Figure 7 Gantt chart of the optimal scheme

In Figure 7 x-axis represents time y-axis represents thenumber of the engineering workThe length of the rectanglesrepresents thework time of the engineeringwork the numberin the rectangle represents the number of the work group ofthe corresponding process

Engineering personnel can be scheduled based on theGantt chart of the optimal schemeThe role of the Gantt chartis to aid in allocation decisions of scheduling engineeringpersonnel

6 Conclusions

Compared with genetic algorithm and standard quantumevolutionary algorithm the advantages of improved quantumgenetic algorithm are as follows (1) IQGA adopts directedevolution which can find the optimal solution more quicklythan undirected evolution (2) the self-adaptive quantumrotation angle with larger rotation angle at the early stageof the evolution and with smaller rotation angle in the lateevolution is advantageous for exploration and exploitation(3) quantum mutation is good to generate new quantumchromosome to increase the diversity of population to extendthe search space (4) quantum disaster can make evolutionjump out of local optimal solution to increase the searchprobability of the global optimal solution (5) the whole inter-ference crossover takes effect on the level of chromosomecoding generated by quantum chromosome to generate newchromosome codingThe search space is effectively expandedand the probability to find optimal solution is increased

10 Abstract and Applied Analysis

So IQGA can search the optimal solution faster and moreefficiently based on the above-mentioned advantages

Application of improved quantum genetic algorithmand appropriate mathematical model can solve the complexscheduling engineering personnel problem Through thismethod we can obtain the optimal allocation scheme andthe work time of the allocation scheme is also determinedThe Gantt chart of the result clearly describes the work timeand work order of each engineering work The method inthis paper can provide powerful support for the schedulingengineering personnel The proposed method in this papernot only can be applied in the scheduling engineeringpersonnel but also can be used for similar types of problemwith multistage multiple projects and multiple subjects andthe abilities of subjects are different such as other schedulingproblems and flexible flow shop scheduling problem

Conflict of Interests

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

References

[1] P Brucker R Qu and E Burke ldquoPersonnel scheduling modelsand complexityrdquo European Journal of Operational Research vol210 no 3 pp 467ndash473 2011

[2] T Lapegue O Bellenguez-Morineau and D Prot ldquoAconstraint-based approach for the shift design personnel taskscheduling problem with equityrdquo Computers and OperationsResearch vol 40 no 10 pp 2450ndash2465 2013

[3] S-W Lin and K-C Ying ldquoMinimizing shifts for personneltask scheduling problems a three-phase algorithmrdquo EuropeanJournal ofOperational Research vol 237 no 1 pp 323ndash334 2014

[4] M Krishnamoorthy A T Ernst and D Baatar ldquoAlgorithmsfor large scale shift minimisation personnel task schedulingproblemsrdquo European Journal of Operational Research vol 219no 1 pp 34ndash48 2012

[5] G DuCote and E M Malstrom ldquoDesign of personnel schedul-ing software for manufacturingrdquo Computers and IndustrialEngineering vol 37 no 1 pp 473ndash476 1999

[6] P M Koeleman S Bhulai and M van Meersbergen ldquoOptimalpatient and personnel scheduling policies for care-at-homeservice facilitiesrdquo European Journal of Operational Research vol219 no 3 pp 557ndash563 2012

[7] J van den Bergh J Belien P de Bruecker E Demeulemeesterand L de Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013

[8] M Sabar B Montreuil and J-M Frayret ldquoA multi-agent-based approach for personnel scheduling in assembly centersrdquoEngineering Applications of Artificial Intelligence vol 22 no 7pp 1080ndash1088 2009

[9] K-H Han and J-H Kim ldquoGenetic quantum algorithm andits application to combinatorial optimization problemrdquo inProceedings of the Congress on Evolutionary Computation pp1354ndash1360 Piscataway NJ USA July 2000

[10] A Narayanan and M Moore ldquoQuantum-inspired genetic algo-rithmsrdquo in Proceedings of the IEEE International Conference on

Evolutionary Computatio (ICEC 96) pp 61ndash66 Nagoya JapanMay 1996

[11] HWang J Liu J Zhi andC Fu ldquoThe improvement of quantumgenetic algorithm and its application on function optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID730749 10 pages 2013

[12] J Tang G Zhang B Lin and B Zhang ldquoHybrid genetic algo-rithm for flow shop scheduling problemrdquo in Proceedings of theInternational Conference on Intelligent Computation Technologyand Automation (ICICTA 10) vol 2 pp 449ndash452 ChangshaChina May 2010

[13] R ZhangWWei Z Jiang andXMa ldquoResearch and simulationon flow-shop scheduling problem based on improved geneticalgorithmrdquo in Proceedings of the 7th International Conference onComputer Science and Education (ICCSE rsquo12) pp 916ndash919 July2012

[14] F Jin and Y Fu ldquoMaster-slave genetic algorithm for flowshop scheduling with resource flexibilityrdquo in Proceedings ofthe IEEE International Conference on Advanced ManagementScience (ICAMS rsquo10) vol 1 pp 341ndash346 Chengdu China July2010

[15] G Zhang and H Rong ldquoQuantum-inspired genetic algorithmbased time-frequency atom decompositionrdquo in Proceedings ofthe 7th international conference on Computational Science (ICCSrsquo07) 2007

[16] J-A Yang and Z-Q Zhuang ldquoActuality of research on quantumgenetic algorithmrdquo Journal of Computer Science amp Technologyvol 30 no 11 pp 13ndash15 2003

[17] W Shu and B He ldquoA quantum genetic simulated annealingalgorithm for task schedulingrdquo inAdvances in Computation andIntelligence L Kang Y Liu and S Zeng Eds vol 4683 ofLecture Notes in Computer Science pp 169ndash176 Springer BerlinGermany 2007

[18] Y Li Y Zhang Y Cheng and X Jiang ldquoA novel immunequantum-inspired genetic algorithmrdquo in Advances in NaturalComputation vol 3612 of Lecture Notes in Computer Science pp215ndash218 2005

[19] KHan and J Kim ldquoGenetic quantum algorithm and its applica-tion to combinatorial optimization problemrdquo in Proceedings ofthe Congress on Evolutionary Computation pp 1354ndash1360 SanDiego Calif USA July 2000

[20] H Miao H Wang and Z Deng ldquoQuantum genetic algo-rithm and its application in power system reactive poweroptimizationrdquo in Proceedings of the International Conference onComputational Intelligence and Security (CIS 09) vol 1 pp 107ndash111 Beijing China 2009

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Start

Satisfy final condition

Memorize the best individual and its fitness

End

Yes

No

Colony disaster

Yes

No

Initialize the population Q(t0) and every parameter

Test every individual of Q(t0)

Evaluate the fitness of every individual of Q(t0)

Set the best individual as the evolutional goal of next generation

Test every individual of Q(t0)

Quantum rotating door update population get Q(t + 1)

Quantum mutation whole interference crossover

Whether or not disaster

t = t + 1

Figure 3 Flowchart of improved quantum genetic algorithm

Start

Initialize distribution scheme

Satisfy final condition

No

Yes

End

Calculate the time of the allocated schemes in the population

Record the shortest work time of the corresponding scheme

Optimize the allocated scheme

Calculate the shortest time of the optimal allocated scheme

Figure 4 Flowchart of acquiring the optimal allocated scheme and the shortest work time

Abstract and Applied Analysis 7

Step 3 (optimize the allocation scheme) Apply the improvedquantumgenetic algorithm to optimize the allocation schemein order to get the optimal allocation scheme Go to Step 2according to the termination condition to do the judgmentif the termination condition is satisfied then go to the End orelse do the loop to Step 3

Step 4 (determination of the optimal allocation schemeand the shortest work time) Choose the optimal allocationscheme form the population satisfied the termination con-dition Calculate the work time of the optimal allocationscheme

The flowchart is shown in Figure 4

5 Simulation Analyses

Take some road building project as an example There are12 roads which need to be built The building work canbe divided into three procedures prospecting drawing andlocation designing and constructionThere are 3 prospectinggroups 2 drawing and location designing groups and 3 con-struction groups The capacities of the groups are differentThe specific work times are shown in Table 4

The specific steps of acquiring the optimal allocatedscheme and the shortest work time are as follows

(1) Construct 12 times 3 dimensional matrix

11986012times3

=

[

[

[

[

[

11988611

11988612

11988613

11988621

11988622

11988623

119886121

119886122

119886123

]

]

]

]

]

(11)

where 1 le 11988611 11988621 11988631 119886

121lt 4 1 le

11988612 11988622 11988632 119886

122lt 3 1 le 119886

13 11988623 11988633 119886

123lt 4

The corresponding qubit chromosome is

11987612times3

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120572111

120573111

10038161003816100381610038161003816100381610038161003816

120572112

120573112

120572121

120573121

120572131

120573131

10038161003816100381610038161003816100381610038161003816

120572132

120573132

120572211

120573211

10038161003816100381610038161003816100381610038161003816

120572212

120573212

120572221

120573221

120572231

120573231

10038161003816100381610038161003816100381610038161003816

120572232

120573232

1205721211

1205731211

10038161003816100381610038161003816100381610038161003816

1205721212

1205731212

1205721221

1205731221

1205721231

1205731231

10038161003816100381610038161003816100381610038161003816

1205721232

1205731232

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(12)

Suppose the generated matrix is

11986012times3

=

[

[

[

[

[

11988611

11988612

11988613

11988621

11988622

11988623

119886121

119886122

119886123

]

]

]

]

]

=[

[

12 23 1 36 21 18 13 32 25 27

29 12 16 23 11 25 27 14 22 13

37 15 24 16 33 28 25 36 19 21

34 17

16 28

19 22

]

]

119879

1198601015840

12times3=[

[

1 2 1 3 2 1 1 3 2 2

2 1 1 2 1 2 2 1 2 1

3 1 2 1 3 2 2 3 1 2

3 1

1 2

1 2

]

]

119879

(13)

Then the encoding section of the assigned chromosome is

[1 2 1 3 2 1 1 3 2 2 3 1 2 1 1 2 1 2 2 1 2

1 1 2 3 1 2 1 3 2 2 3 1 2 1 2]

(14)

(2) Seek the values of time 1198871198961198941198861015840

119894119895

corresponding with1198821198961198721198941198861015840

119896119894

(where 1 le 119896 le 12 1 le 119894 le 3 119894 119895 119896are integers) in Table 2 Then the work timetable ofevery engineeringwork is gotten InTable 5 the figuredenotes the work time and the subscript denotes thework group

(3) Apply the method of Section 3 to calculate the worktimes of every work group according to the orderdetermined by Step (2) and the time determined byStep (3) Then find the maximum value as the totalengineering work time

Apply GA (genetic algorithm) QGA (quantum geneticalgorithm) and IQGA (improved quantum genetic algo-rithm) to solve the problem of scheduling engineering per-sonnelThe three kinds of algorithmarewritten and compiledin MATLAB The simulation test in this paper is based onthe AMD CPU 18GHz 15 GB RAM The parameters of GAare set as follows GA is encoded by real number the totalevolution generation is 80 the size of population is 40 and theprobability ofmutation is 001The parameters ofQGAare setas follows QGA is encoded by the binary the total evolutiongeneration is 80 the size of population is 40 the length ofthe variable expressing 3 work groups is 2 and the length ofthe variable expressing 2 work groups is 1 The parametersof IQGA are set as follows IQGA is encoded by the binarythe total evolution generation is 80 the size of population is40 the length of the variable expressing 3 work groups is 2the length of the variable expressing 2 work groups is 1 andthe probability of mutation is 001 The fitness function is theobjective function The fitness function tends to the optimal

8 Abstract and Applied Analysis

Table 4 Work time of every work group

1198691

1198692

1198693

11987211

11987212

11987213

11987221

11987222

11987231

11987232

11987233

1198821

3 2 4 3 4 5 3 41198822

4 5 3 6 5 5 6 31198823

4 3 5 4 2 3 4 51198824

2 5 3 5 4 1 5 41198825

4 2 6 3 1 3 4 61198826

1 5 3 2 3 5 2 71198827

7 6 8 4 6 6 7 61198828

6 4 5 7 5 4 8 51198829

2 5 4 1 2 3 6 311988210

3 6 4 3 4 4 3 911988211

5 2 3 3 5 8 5 411988212

4 3 4 6 5 4 7 5

Table 5 Work time of every engineering work and correspondingwork group

Engineering work Process1198691

1198692

1198693

1198821

31

42

43

1198822

52

61

51

1198823

41

41

42

1198824

33

42

11

1198825

22

31

63

1198826

11

32

22

1198827

71

62

72

1198828

53

71

53

1198829

52

22

31

11988210

62

31

32

11988211

33

31

81

11988212

41

52

72

value while the evolution increases Simulate the process ofevolution Ten rounds of experiments were presented in thispaper The simulation results of the 3 kinds of algorithm arein Tables 6 7 and 8 The result of some time of the IQGA isshown in Figures 5 6 and 7

The statistical results of the experiment are shown inTable 9

Table 9 shows that compared with GA QGA can get itssmallest value in a short time however QGA can hardly getthe optimal result Though the evolution time of the IQGA islonger and the evolution generation of the IQGA is more theIQGA can get the optimal result in a bigger probability

In Figure 5 x-axis represents the evolution generationsand y-axis represents the work timeThe work time graduallybecomes shorter with the increase of the generation

In Figure 6 x-axis represents the evolution algebra andy-axis represents the work time The average work time of allindividuals has a decreasing trend but there is a gap betweenthe optimal value and the average work time

0 10 20 30 40 50 60 70 8026

27

28

29

30

31

32

Evolutionary generations

The b

est fi

tnes

s of e

very

gen

erat

ion

Figure 5The changing curve of work time of the best individual inthe population

0 10 20 30 40 50 60 70 8036

37

38

39

40

41

42

43

44

45

Figure 6 The changing curve of average work time of all individu-als

Abstract and Applied Analysis 9

Table 6 The simulation results of ten rounds of experiments of GA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 27 26 26 28 27 29 28 28 28 28Simulation time (second) 3007 2905 2894 3102 2732 2940 2946 2846 3468 2937Evolution generation 22 20 23 28 19 19 18 29 18 29Evolution time (second) 08269 07263 08320 10857 06488 06983 06629 10317 07803 10647

Table 7 The simulation results of ten rounds of experiments of QGA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 30 27 29 30 28 29 31 30 29 28Simulation time (second) 5679 5337 5547 5576 5766 6150 5813 5828 6024 5802Evolution generation 12 13 10 4 4 15 3 4 8 21Evolution time (second) 08519 08673 06934 02788 02883 11531 02180 02914 06024 15230

Table 8 The simulation results of ten rounds of experiments of IQGA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 27 28 26 27 28 27 27 26 28 26Simulation time (second) 8847 8580 8527 8459 8505 8364 8422 8712 8342 8524Evolution generation 18 10 12 12 11 9 13 22 21 16Evolution time (second) 19906 10725 12791 12689 11694 09410 13686 23958 21898 17048

Table 9 The statistical results of the experiment

Algorithm The optimalresult

The worstresult

The averageresult

The fastestevolution time (s)

The slowestevolution time (s)

The averageevolution time (s)

Times ofconvergence

GA 26 29 275 06488 10857 08358 2QGA 27 31 291 02788 15230 06768 0IQGA 26 28 27 09410 23958 15381 3

0 5 10 15 20 25 300

2

4

6

8

10

122 1 23 2 1

2 2 13 1 1

2 2 21 2 2

2 1 21 2 1

3 1 31 1 2

1 1 32 2 1

Work time

The n

umbe

r of t

he en

gine

erin

g w

ork

Figure 7 Gantt chart of the optimal scheme

In Figure 7 x-axis represents time y-axis represents thenumber of the engineering workThe length of the rectanglesrepresents thework time of the engineeringwork the numberin the rectangle represents the number of the work group ofthe corresponding process

Engineering personnel can be scheduled based on theGantt chart of the optimal schemeThe role of the Gantt chartis to aid in allocation decisions of scheduling engineeringpersonnel

6 Conclusions

Compared with genetic algorithm and standard quantumevolutionary algorithm the advantages of improved quantumgenetic algorithm are as follows (1) IQGA adopts directedevolution which can find the optimal solution more quicklythan undirected evolution (2) the self-adaptive quantumrotation angle with larger rotation angle at the early stageof the evolution and with smaller rotation angle in the lateevolution is advantageous for exploration and exploitation(3) quantum mutation is good to generate new quantumchromosome to increase the diversity of population to extendthe search space (4) quantum disaster can make evolutionjump out of local optimal solution to increase the searchprobability of the global optimal solution (5) the whole inter-ference crossover takes effect on the level of chromosomecoding generated by quantum chromosome to generate newchromosome codingThe search space is effectively expandedand the probability to find optimal solution is increased

10 Abstract and Applied Analysis

So IQGA can search the optimal solution faster and moreefficiently based on the above-mentioned advantages

Application of improved quantum genetic algorithmand appropriate mathematical model can solve the complexscheduling engineering personnel problem Through thismethod we can obtain the optimal allocation scheme andthe work time of the allocation scheme is also determinedThe Gantt chart of the result clearly describes the work timeand work order of each engineering work The method inthis paper can provide powerful support for the schedulingengineering personnel The proposed method in this papernot only can be applied in the scheduling engineeringpersonnel but also can be used for similar types of problemwith multistage multiple projects and multiple subjects andthe abilities of subjects are different such as other schedulingproblems and flexible flow shop scheduling problem

Conflict of Interests

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

References

[1] P Brucker R Qu and E Burke ldquoPersonnel scheduling modelsand complexityrdquo European Journal of Operational Research vol210 no 3 pp 467ndash473 2011

[2] T Lapegue O Bellenguez-Morineau and D Prot ldquoAconstraint-based approach for the shift design personnel taskscheduling problem with equityrdquo Computers and OperationsResearch vol 40 no 10 pp 2450ndash2465 2013

[3] S-W Lin and K-C Ying ldquoMinimizing shifts for personneltask scheduling problems a three-phase algorithmrdquo EuropeanJournal ofOperational Research vol 237 no 1 pp 323ndash334 2014

[4] M Krishnamoorthy A T Ernst and D Baatar ldquoAlgorithmsfor large scale shift minimisation personnel task schedulingproblemsrdquo European Journal of Operational Research vol 219no 1 pp 34ndash48 2012

[5] G DuCote and E M Malstrom ldquoDesign of personnel schedul-ing software for manufacturingrdquo Computers and IndustrialEngineering vol 37 no 1 pp 473ndash476 1999

[6] P M Koeleman S Bhulai and M van Meersbergen ldquoOptimalpatient and personnel scheduling policies for care-at-homeservice facilitiesrdquo European Journal of Operational Research vol219 no 3 pp 557ndash563 2012

[7] J van den Bergh J Belien P de Bruecker E Demeulemeesterand L de Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013

[8] M Sabar B Montreuil and J-M Frayret ldquoA multi-agent-based approach for personnel scheduling in assembly centersrdquoEngineering Applications of Artificial Intelligence vol 22 no 7pp 1080ndash1088 2009

[9] K-H Han and J-H Kim ldquoGenetic quantum algorithm andits application to combinatorial optimization problemrdquo inProceedings of the Congress on Evolutionary Computation pp1354ndash1360 Piscataway NJ USA July 2000

[10] A Narayanan and M Moore ldquoQuantum-inspired genetic algo-rithmsrdquo in Proceedings of the IEEE International Conference on

Evolutionary Computatio (ICEC 96) pp 61ndash66 Nagoya JapanMay 1996

[11] HWang J Liu J Zhi andC Fu ldquoThe improvement of quantumgenetic algorithm and its application on function optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID730749 10 pages 2013

[12] J Tang G Zhang B Lin and B Zhang ldquoHybrid genetic algo-rithm for flow shop scheduling problemrdquo in Proceedings of theInternational Conference on Intelligent Computation Technologyand Automation (ICICTA 10) vol 2 pp 449ndash452 ChangshaChina May 2010

[13] R ZhangWWei Z Jiang andXMa ldquoResearch and simulationon flow-shop scheduling problem based on improved geneticalgorithmrdquo in Proceedings of the 7th International Conference onComputer Science and Education (ICCSE rsquo12) pp 916ndash919 July2012

[14] F Jin and Y Fu ldquoMaster-slave genetic algorithm for flowshop scheduling with resource flexibilityrdquo in Proceedings ofthe IEEE International Conference on Advanced ManagementScience (ICAMS rsquo10) vol 1 pp 341ndash346 Chengdu China July2010

[15] G Zhang and H Rong ldquoQuantum-inspired genetic algorithmbased time-frequency atom decompositionrdquo in Proceedings ofthe 7th international conference on Computational Science (ICCSrsquo07) 2007

[16] J-A Yang and Z-Q Zhuang ldquoActuality of research on quantumgenetic algorithmrdquo Journal of Computer Science amp Technologyvol 30 no 11 pp 13ndash15 2003

[17] W Shu and B He ldquoA quantum genetic simulated annealingalgorithm for task schedulingrdquo inAdvances in Computation andIntelligence L Kang Y Liu and S Zeng Eds vol 4683 ofLecture Notes in Computer Science pp 169ndash176 Springer BerlinGermany 2007

[18] Y Li Y Zhang Y Cheng and X Jiang ldquoA novel immunequantum-inspired genetic algorithmrdquo in Advances in NaturalComputation vol 3612 of Lecture Notes in Computer Science pp215ndash218 2005

[19] KHan and J Kim ldquoGenetic quantum algorithm and its applica-tion to combinatorial optimization problemrdquo in Proceedings ofthe Congress on Evolutionary Computation pp 1354ndash1360 SanDiego Calif USA July 2000

[20] H Miao H Wang and Z Deng ldquoQuantum genetic algo-rithm and its application in power system reactive poweroptimizationrdquo in Proceedings of the International Conference onComputational Intelligence and Security (CIS 09) vol 1 pp 107ndash111 Beijing China 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

Step 3 (optimize the allocation scheme) Apply the improvedquantumgenetic algorithm to optimize the allocation schemein order to get the optimal allocation scheme Go to Step 2according to the termination condition to do the judgmentif the termination condition is satisfied then go to the End orelse do the loop to Step 3

Step 4 (determination of the optimal allocation schemeand the shortest work time) Choose the optimal allocationscheme form the population satisfied the termination con-dition Calculate the work time of the optimal allocationscheme

The flowchart is shown in Figure 4

5 Simulation Analyses

Take some road building project as an example There are12 roads which need to be built The building work canbe divided into three procedures prospecting drawing andlocation designing and constructionThere are 3 prospectinggroups 2 drawing and location designing groups and 3 con-struction groups The capacities of the groups are differentThe specific work times are shown in Table 4

The specific steps of acquiring the optimal allocatedscheme and the shortest work time are as follows

(1) Construct 12 times 3 dimensional matrix

11986012times3

=

[

[

[

[

[

11988611

11988612

11988613

11988621

11988622

11988623

119886121

119886122

119886123

]

]

]

]

]

(11)

where 1 le 11988611 11988621 11988631 119886

121lt 4 1 le

11988612 11988622 11988632 119886

122lt 3 1 le 119886

13 11988623 11988633 119886

123lt 4

The corresponding qubit chromosome is

11987612times3

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120572111

120573111

10038161003816100381610038161003816100381610038161003816

120572112

120573112

120572121

120573121

120572131

120573131

10038161003816100381610038161003816100381610038161003816

120572132

120573132

120572211

120573211

10038161003816100381610038161003816100381610038161003816

120572212

120573212

120572221

120573221

120572231

120573231

10038161003816100381610038161003816100381610038161003816

120572232

120573232

1205721211

1205731211

10038161003816100381610038161003816100381610038161003816

1205721212

1205731212

1205721221

1205731221

1205721231

1205731231

10038161003816100381610038161003816100381610038161003816

1205721232

1205731232

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(12)

Suppose the generated matrix is

11986012times3

=

[

[

[

[

[

11988611

11988612

11988613

11988621

11988622

11988623

119886121

119886122

119886123

]

]

]

]

]

=[

[

12 23 1 36 21 18 13 32 25 27

29 12 16 23 11 25 27 14 22 13

37 15 24 16 33 28 25 36 19 21

34 17

16 28

19 22

]

]

119879

1198601015840

12times3=[

[

1 2 1 3 2 1 1 3 2 2

2 1 1 2 1 2 2 1 2 1

3 1 2 1 3 2 2 3 1 2

3 1

1 2

1 2

]

]

119879

(13)

Then the encoding section of the assigned chromosome is

[1 2 1 3 2 1 1 3 2 2 3 1 2 1 1 2 1 2 2 1 2

1 1 2 3 1 2 1 3 2 2 3 1 2 1 2]

(14)

(2) Seek the values of time 1198871198961198941198861015840

119894119895

corresponding with1198821198961198721198941198861015840

119896119894

(where 1 le 119896 le 12 1 le 119894 le 3 119894 119895 119896are integers) in Table 2 Then the work timetable ofevery engineeringwork is gotten InTable 5 the figuredenotes the work time and the subscript denotes thework group

(3) Apply the method of Section 3 to calculate the worktimes of every work group according to the orderdetermined by Step (2) and the time determined byStep (3) Then find the maximum value as the totalengineering work time

Apply GA (genetic algorithm) QGA (quantum geneticalgorithm) and IQGA (improved quantum genetic algo-rithm) to solve the problem of scheduling engineering per-sonnelThe three kinds of algorithmarewritten and compiledin MATLAB The simulation test in this paper is based onthe AMD CPU 18GHz 15 GB RAM The parameters of GAare set as follows GA is encoded by real number the totalevolution generation is 80 the size of population is 40 and theprobability ofmutation is 001The parameters ofQGAare setas follows QGA is encoded by the binary the total evolutiongeneration is 80 the size of population is 40 the length ofthe variable expressing 3 work groups is 2 and the length ofthe variable expressing 2 work groups is 1 The parametersof IQGA are set as follows IQGA is encoded by the binarythe total evolution generation is 80 the size of population is40 the length of the variable expressing 3 work groups is 2the length of the variable expressing 2 work groups is 1 andthe probability of mutation is 001 The fitness function is theobjective function The fitness function tends to the optimal

8 Abstract and Applied Analysis

Table 4 Work time of every work group

1198691

1198692

1198693

11987211

11987212

11987213

11987221

11987222

11987231

11987232

11987233

1198821

3 2 4 3 4 5 3 41198822

4 5 3 6 5 5 6 31198823

4 3 5 4 2 3 4 51198824

2 5 3 5 4 1 5 41198825

4 2 6 3 1 3 4 61198826

1 5 3 2 3 5 2 71198827

7 6 8 4 6 6 7 61198828

6 4 5 7 5 4 8 51198829

2 5 4 1 2 3 6 311988210

3 6 4 3 4 4 3 911988211

5 2 3 3 5 8 5 411988212

4 3 4 6 5 4 7 5

Table 5 Work time of every engineering work and correspondingwork group

Engineering work Process1198691

1198692

1198693

1198821

31

42

43

1198822

52

61

51

1198823

41

41

42

1198824

33

42

11

1198825

22

31

63

1198826

11

32

22

1198827

71

62

72

1198828

53

71

53

1198829

52

22

31

11988210

62

31

32

11988211

33

31

81

11988212

41

52

72

value while the evolution increases Simulate the process ofevolution Ten rounds of experiments were presented in thispaper The simulation results of the 3 kinds of algorithm arein Tables 6 7 and 8 The result of some time of the IQGA isshown in Figures 5 6 and 7

The statistical results of the experiment are shown inTable 9

Table 9 shows that compared with GA QGA can get itssmallest value in a short time however QGA can hardly getthe optimal result Though the evolution time of the IQGA islonger and the evolution generation of the IQGA is more theIQGA can get the optimal result in a bigger probability

In Figure 5 x-axis represents the evolution generationsand y-axis represents the work timeThe work time graduallybecomes shorter with the increase of the generation

In Figure 6 x-axis represents the evolution algebra andy-axis represents the work time The average work time of allindividuals has a decreasing trend but there is a gap betweenthe optimal value and the average work time

0 10 20 30 40 50 60 70 8026

27

28

29

30

31

32

Evolutionary generations

The b

est fi

tnes

s of e

very

gen

erat

ion

Figure 5The changing curve of work time of the best individual inthe population

0 10 20 30 40 50 60 70 8036

37

38

39

40

41

42

43

44

45

Figure 6 The changing curve of average work time of all individu-als

Abstract and Applied Analysis 9

Table 6 The simulation results of ten rounds of experiments of GA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 27 26 26 28 27 29 28 28 28 28Simulation time (second) 3007 2905 2894 3102 2732 2940 2946 2846 3468 2937Evolution generation 22 20 23 28 19 19 18 29 18 29Evolution time (second) 08269 07263 08320 10857 06488 06983 06629 10317 07803 10647

Table 7 The simulation results of ten rounds of experiments of QGA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 30 27 29 30 28 29 31 30 29 28Simulation time (second) 5679 5337 5547 5576 5766 6150 5813 5828 6024 5802Evolution generation 12 13 10 4 4 15 3 4 8 21Evolution time (second) 08519 08673 06934 02788 02883 11531 02180 02914 06024 15230

Table 8 The simulation results of ten rounds of experiments of IQGA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 27 28 26 27 28 27 27 26 28 26Simulation time (second) 8847 8580 8527 8459 8505 8364 8422 8712 8342 8524Evolution generation 18 10 12 12 11 9 13 22 21 16Evolution time (second) 19906 10725 12791 12689 11694 09410 13686 23958 21898 17048

Table 9 The statistical results of the experiment

Algorithm The optimalresult

The worstresult

The averageresult

The fastestevolution time (s)

The slowestevolution time (s)

The averageevolution time (s)

Times ofconvergence

GA 26 29 275 06488 10857 08358 2QGA 27 31 291 02788 15230 06768 0IQGA 26 28 27 09410 23958 15381 3

0 5 10 15 20 25 300

2

4

6

8

10

122 1 23 2 1

2 2 13 1 1

2 2 21 2 2

2 1 21 2 1

3 1 31 1 2

1 1 32 2 1

Work time

The n

umbe

r of t

he en

gine

erin

g w

ork

Figure 7 Gantt chart of the optimal scheme

In Figure 7 x-axis represents time y-axis represents thenumber of the engineering workThe length of the rectanglesrepresents thework time of the engineeringwork the numberin the rectangle represents the number of the work group ofthe corresponding process

Engineering personnel can be scheduled based on theGantt chart of the optimal schemeThe role of the Gantt chartis to aid in allocation decisions of scheduling engineeringpersonnel

6 Conclusions

Compared with genetic algorithm and standard quantumevolutionary algorithm the advantages of improved quantumgenetic algorithm are as follows (1) IQGA adopts directedevolution which can find the optimal solution more quicklythan undirected evolution (2) the self-adaptive quantumrotation angle with larger rotation angle at the early stageof the evolution and with smaller rotation angle in the lateevolution is advantageous for exploration and exploitation(3) quantum mutation is good to generate new quantumchromosome to increase the diversity of population to extendthe search space (4) quantum disaster can make evolutionjump out of local optimal solution to increase the searchprobability of the global optimal solution (5) the whole inter-ference crossover takes effect on the level of chromosomecoding generated by quantum chromosome to generate newchromosome codingThe search space is effectively expandedand the probability to find optimal solution is increased

10 Abstract and Applied Analysis

So IQGA can search the optimal solution faster and moreefficiently based on the above-mentioned advantages

Application of improved quantum genetic algorithmand appropriate mathematical model can solve the complexscheduling engineering personnel problem Through thismethod we can obtain the optimal allocation scheme andthe work time of the allocation scheme is also determinedThe Gantt chart of the result clearly describes the work timeand work order of each engineering work The method inthis paper can provide powerful support for the schedulingengineering personnel The proposed method in this papernot only can be applied in the scheduling engineeringpersonnel but also can be used for similar types of problemwith multistage multiple projects and multiple subjects andthe abilities of subjects are different such as other schedulingproblems and flexible flow shop scheduling problem

Conflict of Interests

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

References

[1] P Brucker R Qu and E Burke ldquoPersonnel scheduling modelsand complexityrdquo European Journal of Operational Research vol210 no 3 pp 467ndash473 2011

[2] T Lapegue O Bellenguez-Morineau and D Prot ldquoAconstraint-based approach for the shift design personnel taskscheduling problem with equityrdquo Computers and OperationsResearch vol 40 no 10 pp 2450ndash2465 2013

[3] S-W Lin and K-C Ying ldquoMinimizing shifts for personneltask scheduling problems a three-phase algorithmrdquo EuropeanJournal ofOperational Research vol 237 no 1 pp 323ndash334 2014

[4] M Krishnamoorthy A T Ernst and D Baatar ldquoAlgorithmsfor large scale shift minimisation personnel task schedulingproblemsrdquo European Journal of Operational Research vol 219no 1 pp 34ndash48 2012

[5] G DuCote and E M Malstrom ldquoDesign of personnel schedul-ing software for manufacturingrdquo Computers and IndustrialEngineering vol 37 no 1 pp 473ndash476 1999

[6] P M Koeleman S Bhulai and M van Meersbergen ldquoOptimalpatient and personnel scheduling policies for care-at-homeservice facilitiesrdquo European Journal of Operational Research vol219 no 3 pp 557ndash563 2012

[7] J van den Bergh J Belien P de Bruecker E Demeulemeesterand L de Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013

[8] M Sabar B Montreuil and J-M Frayret ldquoA multi-agent-based approach for personnel scheduling in assembly centersrdquoEngineering Applications of Artificial Intelligence vol 22 no 7pp 1080ndash1088 2009

[9] K-H Han and J-H Kim ldquoGenetic quantum algorithm andits application to combinatorial optimization problemrdquo inProceedings of the Congress on Evolutionary Computation pp1354ndash1360 Piscataway NJ USA July 2000

[10] A Narayanan and M Moore ldquoQuantum-inspired genetic algo-rithmsrdquo in Proceedings of the IEEE International Conference on

Evolutionary Computatio (ICEC 96) pp 61ndash66 Nagoya JapanMay 1996

[11] HWang J Liu J Zhi andC Fu ldquoThe improvement of quantumgenetic algorithm and its application on function optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID730749 10 pages 2013

[12] J Tang G Zhang B Lin and B Zhang ldquoHybrid genetic algo-rithm for flow shop scheduling problemrdquo in Proceedings of theInternational Conference on Intelligent Computation Technologyand Automation (ICICTA 10) vol 2 pp 449ndash452 ChangshaChina May 2010

[13] R ZhangWWei Z Jiang andXMa ldquoResearch and simulationon flow-shop scheduling problem based on improved geneticalgorithmrdquo in Proceedings of the 7th International Conference onComputer Science and Education (ICCSE rsquo12) pp 916ndash919 July2012

[14] F Jin and Y Fu ldquoMaster-slave genetic algorithm for flowshop scheduling with resource flexibilityrdquo in Proceedings ofthe IEEE International Conference on Advanced ManagementScience (ICAMS rsquo10) vol 1 pp 341ndash346 Chengdu China July2010

[15] G Zhang and H Rong ldquoQuantum-inspired genetic algorithmbased time-frequency atom decompositionrdquo in Proceedings ofthe 7th international conference on Computational Science (ICCSrsquo07) 2007

[16] J-A Yang and Z-Q Zhuang ldquoActuality of research on quantumgenetic algorithmrdquo Journal of Computer Science amp Technologyvol 30 no 11 pp 13ndash15 2003

[17] W Shu and B He ldquoA quantum genetic simulated annealingalgorithm for task schedulingrdquo inAdvances in Computation andIntelligence L Kang Y Liu and S Zeng Eds vol 4683 ofLecture Notes in Computer Science pp 169ndash176 Springer BerlinGermany 2007

[18] Y Li Y Zhang Y Cheng and X Jiang ldquoA novel immunequantum-inspired genetic algorithmrdquo in Advances in NaturalComputation vol 3612 of Lecture Notes in Computer Science pp215ndash218 2005

[19] KHan and J Kim ldquoGenetic quantum algorithm and its applica-tion to combinatorial optimization problemrdquo in Proceedings ofthe Congress on Evolutionary Computation pp 1354ndash1360 SanDiego Calif USA July 2000

[20] H Miao H Wang and Z Deng ldquoQuantum genetic algo-rithm and its application in power system reactive poweroptimizationrdquo in Proceedings of the International Conference onComputational Intelligence and Security (CIS 09) vol 1 pp 107ndash111 Beijing China 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

Table 4 Work time of every work group

1198691

1198692

1198693

11987211

11987212

11987213

11987221

11987222

11987231

11987232

11987233

1198821

3 2 4 3 4 5 3 41198822

4 5 3 6 5 5 6 31198823

4 3 5 4 2 3 4 51198824

2 5 3 5 4 1 5 41198825

4 2 6 3 1 3 4 61198826

1 5 3 2 3 5 2 71198827

7 6 8 4 6 6 7 61198828

6 4 5 7 5 4 8 51198829

2 5 4 1 2 3 6 311988210

3 6 4 3 4 4 3 911988211

5 2 3 3 5 8 5 411988212

4 3 4 6 5 4 7 5

Table 5 Work time of every engineering work and correspondingwork group

Engineering work Process1198691

1198692

1198693

1198821

31

42

43

1198822

52

61

51

1198823

41

41

42

1198824

33

42

11

1198825

22

31

63

1198826

11

32

22

1198827

71

62

72

1198828

53

71

53

1198829

52

22

31

11988210

62

31

32

11988211

33

31

81

11988212

41

52

72

value while the evolution increases Simulate the process ofevolution Ten rounds of experiments were presented in thispaper The simulation results of the 3 kinds of algorithm arein Tables 6 7 and 8 The result of some time of the IQGA isshown in Figures 5 6 and 7

The statistical results of the experiment are shown inTable 9

Table 9 shows that compared with GA QGA can get itssmallest value in a short time however QGA can hardly getthe optimal result Though the evolution time of the IQGA islonger and the evolution generation of the IQGA is more theIQGA can get the optimal result in a bigger probability

In Figure 5 x-axis represents the evolution generationsand y-axis represents the work timeThe work time graduallybecomes shorter with the increase of the generation

In Figure 6 x-axis represents the evolution algebra andy-axis represents the work time The average work time of allindividuals has a decreasing trend but there is a gap betweenthe optimal value and the average work time

0 10 20 30 40 50 60 70 8026

27

28

29

30

31

32

Evolutionary generations

The b

est fi

tnes

s of e

very

gen

erat

ion

Figure 5The changing curve of work time of the best individual inthe population

0 10 20 30 40 50 60 70 8036

37

38

39

40

41

42

43

44

45

Figure 6 The changing curve of average work time of all individu-als

Abstract and Applied Analysis 9

Table 6 The simulation results of ten rounds of experiments of GA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 27 26 26 28 27 29 28 28 28 28Simulation time (second) 3007 2905 2894 3102 2732 2940 2946 2846 3468 2937Evolution generation 22 20 23 28 19 19 18 29 18 29Evolution time (second) 08269 07263 08320 10857 06488 06983 06629 10317 07803 10647

Table 7 The simulation results of ten rounds of experiments of QGA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 30 27 29 30 28 29 31 30 29 28Simulation time (second) 5679 5337 5547 5576 5766 6150 5813 5828 6024 5802Evolution generation 12 13 10 4 4 15 3 4 8 21Evolution time (second) 08519 08673 06934 02788 02883 11531 02180 02914 06024 15230

Table 8 The simulation results of ten rounds of experiments of IQGA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 27 28 26 27 28 27 27 26 28 26Simulation time (second) 8847 8580 8527 8459 8505 8364 8422 8712 8342 8524Evolution generation 18 10 12 12 11 9 13 22 21 16Evolution time (second) 19906 10725 12791 12689 11694 09410 13686 23958 21898 17048

Table 9 The statistical results of the experiment

Algorithm The optimalresult

The worstresult

The averageresult

The fastestevolution time (s)

The slowestevolution time (s)

The averageevolution time (s)

Times ofconvergence

GA 26 29 275 06488 10857 08358 2QGA 27 31 291 02788 15230 06768 0IQGA 26 28 27 09410 23958 15381 3

0 5 10 15 20 25 300

2

4

6

8

10

122 1 23 2 1

2 2 13 1 1

2 2 21 2 2

2 1 21 2 1

3 1 31 1 2

1 1 32 2 1

Work time

The n

umbe

r of t

he en

gine

erin

g w

ork

Figure 7 Gantt chart of the optimal scheme

In Figure 7 x-axis represents time y-axis represents thenumber of the engineering workThe length of the rectanglesrepresents thework time of the engineeringwork the numberin the rectangle represents the number of the work group ofthe corresponding process

Engineering personnel can be scheduled based on theGantt chart of the optimal schemeThe role of the Gantt chartis to aid in allocation decisions of scheduling engineeringpersonnel

6 Conclusions

Compared with genetic algorithm and standard quantumevolutionary algorithm the advantages of improved quantumgenetic algorithm are as follows (1) IQGA adopts directedevolution which can find the optimal solution more quicklythan undirected evolution (2) the self-adaptive quantumrotation angle with larger rotation angle at the early stageof the evolution and with smaller rotation angle in the lateevolution is advantageous for exploration and exploitation(3) quantum mutation is good to generate new quantumchromosome to increase the diversity of population to extendthe search space (4) quantum disaster can make evolutionjump out of local optimal solution to increase the searchprobability of the global optimal solution (5) the whole inter-ference crossover takes effect on the level of chromosomecoding generated by quantum chromosome to generate newchromosome codingThe search space is effectively expandedand the probability to find optimal solution is increased

10 Abstract and Applied Analysis

So IQGA can search the optimal solution faster and moreefficiently based on the above-mentioned advantages

Application of improved quantum genetic algorithmand appropriate mathematical model can solve the complexscheduling engineering personnel problem Through thismethod we can obtain the optimal allocation scheme andthe work time of the allocation scheme is also determinedThe Gantt chart of the result clearly describes the work timeand work order of each engineering work The method inthis paper can provide powerful support for the schedulingengineering personnel The proposed method in this papernot only can be applied in the scheduling engineeringpersonnel but also can be used for similar types of problemwith multistage multiple projects and multiple subjects andthe abilities of subjects are different such as other schedulingproblems and flexible flow shop scheduling problem

Conflict of Interests

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

References

[1] P Brucker R Qu and E Burke ldquoPersonnel scheduling modelsand complexityrdquo European Journal of Operational Research vol210 no 3 pp 467ndash473 2011

[2] T Lapegue O Bellenguez-Morineau and D Prot ldquoAconstraint-based approach for the shift design personnel taskscheduling problem with equityrdquo Computers and OperationsResearch vol 40 no 10 pp 2450ndash2465 2013

[3] S-W Lin and K-C Ying ldquoMinimizing shifts for personneltask scheduling problems a three-phase algorithmrdquo EuropeanJournal ofOperational Research vol 237 no 1 pp 323ndash334 2014

[4] M Krishnamoorthy A T Ernst and D Baatar ldquoAlgorithmsfor large scale shift minimisation personnel task schedulingproblemsrdquo European Journal of Operational Research vol 219no 1 pp 34ndash48 2012

[5] G DuCote and E M Malstrom ldquoDesign of personnel schedul-ing software for manufacturingrdquo Computers and IndustrialEngineering vol 37 no 1 pp 473ndash476 1999

[6] P M Koeleman S Bhulai and M van Meersbergen ldquoOptimalpatient and personnel scheduling policies for care-at-homeservice facilitiesrdquo European Journal of Operational Research vol219 no 3 pp 557ndash563 2012

[7] J van den Bergh J Belien P de Bruecker E Demeulemeesterand L de Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013

[8] M Sabar B Montreuil and J-M Frayret ldquoA multi-agent-based approach for personnel scheduling in assembly centersrdquoEngineering Applications of Artificial Intelligence vol 22 no 7pp 1080ndash1088 2009

[9] K-H Han and J-H Kim ldquoGenetic quantum algorithm andits application to combinatorial optimization problemrdquo inProceedings of the Congress on Evolutionary Computation pp1354ndash1360 Piscataway NJ USA July 2000

[10] A Narayanan and M Moore ldquoQuantum-inspired genetic algo-rithmsrdquo in Proceedings of the IEEE International Conference on

Evolutionary Computatio (ICEC 96) pp 61ndash66 Nagoya JapanMay 1996

[11] HWang J Liu J Zhi andC Fu ldquoThe improvement of quantumgenetic algorithm and its application on function optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID730749 10 pages 2013

[12] J Tang G Zhang B Lin and B Zhang ldquoHybrid genetic algo-rithm for flow shop scheduling problemrdquo in Proceedings of theInternational Conference on Intelligent Computation Technologyand Automation (ICICTA 10) vol 2 pp 449ndash452 ChangshaChina May 2010

[13] R ZhangWWei Z Jiang andXMa ldquoResearch and simulationon flow-shop scheduling problem based on improved geneticalgorithmrdquo in Proceedings of the 7th International Conference onComputer Science and Education (ICCSE rsquo12) pp 916ndash919 July2012

[14] F Jin and Y Fu ldquoMaster-slave genetic algorithm for flowshop scheduling with resource flexibilityrdquo in Proceedings ofthe IEEE International Conference on Advanced ManagementScience (ICAMS rsquo10) vol 1 pp 341ndash346 Chengdu China July2010

[15] G Zhang and H Rong ldquoQuantum-inspired genetic algorithmbased time-frequency atom decompositionrdquo in Proceedings ofthe 7th international conference on Computational Science (ICCSrsquo07) 2007

[16] J-A Yang and Z-Q Zhuang ldquoActuality of research on quantumgenetic algorithmrdquo Journal of Computer Science amp Technologyvol 30 no 11 pp 13ndash15 2003

[17] W Shu and B He ldquoA quantum genetic simulated annealingalgorithm for task schedulingrdquo inAdvances in Computation andIntelligence L Kang Y Liu and S Zeng Eds vol 4683 ofLecture Notes in Computer Science pp 169ndash176 Springer BerlinGermany 2007

[18] Y Li Y Zhang Y Cheng and X Jiang ldquoA novel immunequantum-inspired genetic algorithmrdquo in Advances in NaturalComputation vol 3612 of Lecture Notes in Computer Science pp215ndash218 2005

[19] KHan and J Kim ldquoGenetic quantum algorithm and its applica-tion to combinatorial optimization problemrdquo in Proceedings ofthe Congress on Evolutionary Computation pp 1354ndash1360 SanDiego Calif USA July 2000

[20] H Miao H Wang and Z Deng ldquoQuantum genetic algo-rithm and its application in power system reactive poweroptimizationrdquo in Proceedings of the International Conference onComputational Intelligence and Security (CIS 09) vol 1 pp 107ndash111 Beijing China 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

Table 6 The simulation results of ten rounds of experiments of GA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 27 26 26 28 27 29 28 28 28 28Simulation time (second) 3007 2905 2894 3102 2732 2940 2946 2846 3468 2937Evolution generation 22 20 23 28 19 19 18 29 18 29Evolution time (second) 08269 07263 08320 10857 06488 06983 06629 10317 07803 10647

Table 7 The simulation results of ten rounds of experiments of QGA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 30 27 29 30 28 29 31 30 29 28Simulation time (second) 5679 5337 5547 5576 5766 6150 5813 5828 6024 5802Evolution generation 12 13 10 4 4 15 3 4 8 21Evolution time (second) 08519 08673 06934 02788 02883 11531 02180 02914 06024 15230

Table 8 The simulation results of ten rounds of experiments of IQGA

Round 1 2 3 4 5 6 7 8 9 10Simulation result 27 28 26 27 28 27 27 26 28 26Simulation time (second) 8847 8580 8527 8459 8505 8364 8422 8712 8342 8524Evolution generation 18 10 12 12 11 9 13 22 21 16Evolution time (second) 19906 10725 12791 12689 11694 09410 13686 23958 21898 17048

Table 9 The statistical results of the experiment

Algorithm The optimalresult

The worstresult

The averageresult

The fastestevolution time (s)

The slowestevolution time (s)

The averageevolution time (s)

Times ofconvergence

GA 26 29 275 06488 10857 08358 2QGA 27 31 291 02788 15230 06768 0IQGA 26 28 27 09410 23958 15381 3

0 5 10 15 20 25 300

2

4

6

8

10

122 1 23 2 1

2 2 13 1 1

2 2 21 2 2

2 1 21 2 1

3 1 31 1 2

1 1 32 2 1

Work time

The n

umbe

r of t

he en

gine

erin

g w

ork

Figure 7 Gantt chart of the optimal scheme

In Figure 7 x-axis represents time y-axis represents thenumber of the engineering workThe length of the rectanglesrepresents thework time of the engineeringwork the numberin the rectangle represents the number of the work group ofthe corresponding process

Engineering personnel can be scheduled based on theGantt chart of the optimal schemeThe role of the Gantt chartis to aid in allocation decisions of scheduling engineeringpersonnel

6 Conclusions

Compared with genetic algorithm and standard quantumevolutionary algorithm the advantages of improved quantumgenetic algorithm are as follows (1) IQGA adopts directedevolution which can find the optimal solution more quicklythan undirected evolution (2) the self-adaptive quantumrotation angle with larger rotation angle at the early stageof the evolution and with smaller rotation angle in the lateevolution is advantageous for exploration and exploitation(3) quantum mutation is good to generate new quantumchromosome to increase the diversity of population to extendthe search space (4) quantum disaster can make evolutionjump out of local optimal solution to increase the searchprobability of the global optimal solution (5) the whole inter-ference crossover takes effect on the level of chromosomecoding generated by quantum chromosome to generate newchromosome codingThe search space is effectively expandedand the probability to find optimal solution is increased

10 Abstract and Applied Analysis

So IQGA can search the optimal solution faster and moreefficiently based on the above-mentioned advantages

Application of improved quantum genetic algorithmand appropriate mathematical model can solve the complexscheduling engineering personnel problem Through thismethod we can obtain the optimal allocation scheme andthe work time of the allocation scheme is also determinedThe Gantt chart of the result clearly describes the work timeand work order of each engineering work The method inthis paper can provide powerful support for the schedulingengineering personnel The proposed method in this papernot only can be applied in the scheduling engineeringpersonnel but also can be used for similar types of problemwith multistage multiple projects and multiple subjects andthe abilities of subjects are different such as other schedulingproblems and flexible flow shop scheduling problem

Conflict of Interests

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

References

[1] P Brucker R Qu and E Burke ldquoPersonnel scheduling modelsand complexityrdquo European Journal of Operational Research vol210 no 3 pp 467ndash473 2011

[2] T Lapegue O Bellenguez-Morineau and D Prot ldquoAconstraint-based approach for the shift design personnel taskscheduling problem with equityrdquo Computers and OperationsResearch vol 40 no 10 pp 2450ndash2465 2013

[3] S-W Lin and K-C Ying ldquoMinimizing shifts for personneltask scheduling problems a three-phase algorithmrdquo EuropeanJournal ofOperational Research vol 237 no 1 pp 323ndash334 2014

[4] M Krishnamoorthy A T Ernst and D Baatar ldquoAlgorithmsfor large scale shift minimisation personnel task schedulingproblemsrdquo European Journal of Operational Research vol 219no 1 pp 34ndash48 2012

[5] G DuCote and E M Malstrom ldquoDesign of personnel schedul-ing software for manufacturingrdquo Computers and IndustrialEngineering vol 37 no 1 pp 473ndash476 1999

[6] P M Koeleman S Bhulai and M van Meersbergen ldquoOptimalpatient and personnel scheduling policies for care-at-homeservice facilitiesrdquo European Journal of Operational Research vol219 no 3 pp 557ndash563 2012

[7] J van den Bergh J Belien P de Bruecker E Demeulemeesterand L de Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013

[8] M Sabar B Montreuil and J-M Frayret ldquoA multi-agent-based approach for personnel scheduling in assembly centersrdquoEngineering Applications of Artificial Intelligence vol 22 no 7pp 1080ndash1088 2009

[9] K-H Han and J-H Kim ldquoGenetic quantum algorithm andits application to combinatorial optimization problemrdquo inProceedings of the Congress on Evolutionary Computation pp1354ndash1360 Piscataway NJ USA July 2000

[10] A Narayanan and M Moore ldquoQuantum-inspired genetic algo-rithmsrdquo in Proceedings of the IEEE International Conference on

Evolutionary Computatio (ICEC 96) pp 61ndash66 Nagoya JapanMay 1996

[11] HWang J Liu J Zhi andC Fu ldquoThe improvement of quantumgenetic algorithm and its application on function optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID730749 10 pages 2013

[12] J Tang G Zhang B Lin and B Zhang ldquoHybrid genetic algo-rithm for flow shop scheduling problemrdquo in Proceedings of theInternational Conference on Intelligent Computation Technologyand Automation (ICICTA 10) vol 2 pp 449ndash452 ChangshaChina May 2010

[13] R ZhangWWei Z Jiang andXMa ldquoResearch and simulationon flow-shop scheduling problem based on improved geneticalgorithmrdquo in Proceedings of the 7th International Conference onComputer Science and Education (ICCSE rsquo12) pp 916ndash919 July2012

[14] F Jin and Y Fu ldquoMaster-slave genetic algorithm for flowshop scheduling with resource flexibilityrdquo in Proceedings ofthe IEEE International Conference on Advanced ManagementScience (ICAMS rsquo10) vol 1 pp 341ndash346 Chengdu China July2010

[15] G Zhang and H Rong ldquoQuantum-inspired genetic algorithmbased time-frequency atom decompositionrdquo in Proceedings ofthe 7th international conference on Computational Science (ICCSrsquo07) 2007

[16] J-A Yang and Z-Q Zhuang ldquoActuality of research on quantumgenetic algorithmrdquo Journal of Computer Science amp Technologyvol 30 no 11 pp 13ndash15 2003

[17] W Shu and B He ldquoA quantum genetic simulated annealingalgorithm for task schedulingrdquo inAdvances in Computation andIntelligence L Kang Y Liu and S Zeng Eds vol 4683 ofLecture Notes in Computer Science pp 169ndash176 Springer BerlinGermany 2007

[18] Y Li Y Zhang Y Cheng and X Jiang ldquoA novel immunequantum-inspired genetic algorithmrdquo in Advances in NaturalComputation vol 3612 of Lecture Notes in Computer Science pp215ndash218 2005

[19] KHan and J Kim ldquoGenetic quantum algorithm and its applica-tion to combinatorial optimization problemrdquo in Proceedings ofthe Congress on Evolutionary Computation pp 1354ndash1360 SanDiego Calif USA July 2000

[20] H Miao H Wang and Z Deng ldquoQuantum genetic algo-rithm and its application in power system reactive poweroptimizationrdquo in Proceedings of the International Conference onComputational Intelligence and Security (CIS 09) vol 1 pp 107ndash111 Beijing China 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

So IQGA can search the optimal solution faster and moreefficiently based on the above-mentioned advantages

Application of improved quantum genetic algorithmand appropriate mathematical model can solve the complexscheduling engineering personnel problem Through thismethod we can obtain the optimal allocation scheme andthe work time of the allocation scheme is also determinedThe Gantt chart of the result clearly describes the work timeand work order of each engineering work The method inthis paper can provide powerful support for the schedulingengineering personnel The proposed method in this papernot only can be applied in the scheduling engineeringpersonnel but also can be used for similar types of problemwith multistage multiple projects and multiple subjects andthe abilities of subjects are different such as other schedulingproblems and flexible flow shop scheduling problem

Conflict of Interests

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

References

[1] P Brucker R Qu and E Burke ldquoPersonnel scheduling modelsand complexityrdquo European Journal of Operational Research vol210 no 3 pp 467ndash473 2011

[2] T Lapegue O Bellenguez-Morineau and D Prot ldquoAconstraint-based approach for the shift design personnel taskscheduling problem with equityrdquo Computers and OperationsResearch vol 40 no 10 pp 2450ndash2465 2013

[3] S-W Lin and K-C Ying ldquoMinimizing shifts for personneltask scheduling problems a three-phase algorithmrdquo EuropeanJournal ofOperational Research vol 237 no 1 pp 323ndash334 2014

[4] M Krishnamoorthy A T Ernst and D Baatar ldquoAlgorithmsfor large scale shift minimisation personnel task schedulingproblemsrdquo European Journal of Operational Research vol 219no 1 pp 34ndash48 2012

[5] G DuCote and E M Malstrom ldquoDesign of personnel schedul-ing software for manufacturingrdquo Computers and IndustrialEngineering vol 37 no 1 pp 473ndash476 1999

[6] P M Koeleman S Bhulai and M van Meersbergen ldquoOptimalpatient and personnel scheduling policies for care-at-homeservice facilitiesrdquo European Journal of Operational Research vol219 no 3 pp 557ndash563 2012

[7] J van den Bergh J Belien P de Bruecker E Demeulemeesterand L de Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013

[8] M Sabar B Montreuil and J-M Frayret ldquoA multi-agent-based approach for personnel scheduling in assembly centersrdquoEngineering Applications of Artificial Intelligence vol 22 no 7pp 1080ndash1088 2009

[9] K-H Han and J-H Kim ldquoGenetic quantum algorithm andits application to combinatorial optimization problemrdquo inProceedings of the Congress on Evolutionary Computation pp1354ndash1360 Piscataway NJ USA July 2000

[10] A Narayanan and M Moore ldquoQuantum-inspired genetic algo-rithmsrdquo in Proceedings of the IEEE International Conference on

Evolutionary Computatio (ICEC 96) pp 61ndash66 Nagoya JapanMay 1996

[11] HWang J Liu J Zhi andC Fu ldquoThe improvement of quantumgenetic algorithm and its application on function optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID730749 10 pages 2013

[12] J Tang G Zhang B Lin and B Zhang ldquoHybrid genetic algo-rithm for flow shop scheduling problemrdquo in Proceedings of theInternational Conference on Intelligent Computation Technologyand Automation (ICICTA 10) vol 2 pp 449ndash452 ChangshaChina May 2010

[13] R ZhangWWei Z Jiang andXMa ldquoResearch and simulationon flow-shop scheduling problem based on improved geneticalgorithmrdquo in Proceedings of the 7th International Conference onComputer Science and Education (ICCSE rsquo12) pp 916ndash919 July2012

[14] F Jin and Y Fu ldquoMaster-slave genetic algorithm for flowshop scheduling with resource flexibilityrdquo in Proceedings ofthe IEEE International Conference on Advanced ManagementScience (ICAMS rsquo10) vol 1 pp 341ndash346 Chengdu China July2010

[15] G Zhang and H Rong ldquoQuantum-inspired genetic algorithmbased time-frequency atom decompositionrdquo in Proceedings ofthe 7th international conference on Computational Science (ICCSrsquo07) 2007

[16] J-A Yang and Z-Q Zhuang ldquoActuality of research on quantumgenetic algorithmrdquo Journal of Computer Science amp Technologyvol 30 no 11 pp 13ndash15 2003

[17] W Shu and B He ldquoA quantum genetic simulated annealingalgorithm for task schedulingrdquo inAdvances in Computation andIntelligence L Kang Y Liu and S Zeng Eds vol 4683 ofLecture Notes in Computer Science pp 169ndash176 Springer BerlinGermany 2007

[18] Y Li Y Zhang Y Cheng and X Jiang ldquoA novel immunequantum-inspired genetic algorithmrdquo in Advances in NaturalComputation vol 3612 of Lecture Notes in Computer Science pp215ndash218 2005

[19] KHan and J Kim ldquoGenetic quantum algorithm and its applica-tion to combinatorial optimization problemrdquo in Proceedings ofthe Congress on Evolutionary Computation pp 1354ndash1360 SanDiego Calif USA July 2000

[20] H Miao H Wang and Z Deng ldquoQuantum genetic algo-rithm and its application in power system reactive poweroptimizationrdquo in Proceedings of the International Conference onComputational Intelligence and Security (CIS 09) vol 1 pp 107ndash111 Beijing China 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of