Research Article Research on the Production Scheduling ...downloads.hindawi.com/journals/mpe/2013/492158.pdf · MinHuang, 1 RuixianHuang, 1 BoSun, 2 andLinrongLi 1 School of Soware

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 492158 9 pageshttpdxdoiorg1011552013492158

Research ArticleResearch on the Production Scheduling Optimization forVirtual Enterprises

Min Huang1 Ruixian Huang1 Bo Sun2 and Linrong Li1

1 School of Software Engineering South China University of Technology Guangzhou 510006 China2Department of Research Guangdong University of Foreign Studies Guangzhou 510420 China

Correspondence should be addressed to Min Huang minhscuteducn

Received 12 March 2013 Accepted 2 July 2013

Academic Editor Orwa Jaber Housheya

Copyright copy 2013 Min Huang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Production scheduling is a rather difficult problem in virtual enterprises (VE) for the tasks of production which would be executedby some distributed and independentmembers Factors such as the timing constraints of task and ability restrictions of themembersare considered comprehensibly to solve the global scheduling optimization problem This paper establishes a partner selectionmodel based on an improved ant colony algorithm at first then presents a production scheduling framework with two layers asglobal scheduling and local scheduling for virtual enterprise and gives a global scheduling mathematical model with the smallesttotal production time based on it An improved genetic algorithm is proposed in the model to solve the time complexity of virtualenterprise production scheduling The presented experimental results validate the optimization of the model and the efficiency ofthe algorithm

1 Introduction

Virtual enterprise (VE) is a dynamic alliance consisting ofindependent enterprises Production scheduling is one of themost important aspects in VE to seize the market opportuni-ties with the shortest production cycle and maximum profitsHence production planning has become an important areain VE

As Helaakoski et al defined VE is defined as a temporarycollaborative network consisting of independent enterprisesformed to exploit a particular business opportunity [1] Goelet al look into both the paradigms and Enterprise Architec-ture viewpoint of virtual enterprise Enterprise Architecturedeals with the structure of an enterprise relationships andinteractions of its units [2] Capuano et al propose anoverview of the Knowledge Virtual Enterprise model wherethe virtual enterprise vision is extended with Knowledge-based assets in order to provide an agreement model tosupport the interoperability among organizations [3] Daneshet al propose a 6-layer framework with multiple componentswithin each layer and present a distributed SOA infras-tructure that facilitates peer-to-peer collaboration between

enterprises in a virtual enterprise [4] Esparcia et al presentan extension of the EnvironmentDimension of theVEmodelwhich is an Organization Modeling Language to defineOrganization-Centered Multiagent Systems [5]

Partner selection of virtual enterprise is important toVE and production scheduling of VE Crispim and Sousamainly talk about the procedure of partner selection in VEand give a model but cannot solve it absolutely [6] Jarimoand Salo create a mixed-integer linear programming (MILP)model to solve partner selection in virtual organization[7] In additionally to fixed and variable costs we presentextensions that accommodate transportation costs capacityrisk measures and interorganizational dependencies suchas the success of past collaboration Nayak et al proposea variant of swarm optimization to handle combinatorialproblems efficiently compared to its continuous counterpart[8] Simona and Raluca propose partner selection as adifficult task and involve important decision-making becauseit includes many factors quality cost trust delivery timelimitations of geography and communicate abilities [9] Atlast they solve the problem by genetic algorithm Mohamedfocuses on the solution procedure of the multiobjective

2 Mathematical Problems in Engineering

partner selection problem in virtual enterprise where the costcoefficients are expressed as interval by the decision makerand uses a multiobjective algorithm (PSA) to solve [10]

Yalaoui et al solve a hybrid flow shop scheduling problemand create a new method to solve the problem based onthe nature which is the particle swarm optimization methodunder fuzzy logic controller (FLCPSO) [11] Dugardin et alcreate a model to solve multiobjective scheduling of a reen-trant hybrid flow shop problem and can be used for partnerselection inVE if improved [12] Gao and Jiang andDing et alestablish a mathematical model solved using a hybrid geneticalgorithm (GA) to acquire the shortest operating cycle basedon the characteristics of the production scheduling of VE[13 14] However this model caused premature convergencebecause that it only used a single population Song combinesthe widely applied ldquocloud computingrdquo theory to present thecloud of VE production planning and control model [15]However the model has not yet been tested in practicalapplications Gao and Ding and Li and Liu discuss an orderrarely multiple orders established a multiorder productionschedulingmodel and introduce various population genetic-simulated annealing algorithms to solve this model [16 17]Although the problem of population diversity is solvedthe execution time is very slow Some researchers such asCamarinha-Matos and Afsarmanesh and Zhao and Zhouapply multiagent technology to solve VE production plan-ning and control [18 19] They mainly attempt to establisha framework of the production scheduling system The VEproduction planningmodel and the corresponding algorithmneed to be further studied According to the two-tier schedul-ing model of VE Tao and Xie establish a mathematicalmodel with the smallest total operating time and used theant colony algorithm to solve the model [20] Howeverpremature convergence and slow execution time were thedisadvantages of this model

The structure of the paper is as follows Section 2 statesthe problems of partner selection and production planningin VE and creates twomodels Section 3 introduces improvedant colony algorithm (IACO) to solve partner selectionmodeland improved multipopulation genetic algorithm (IMGA) tosolve the production scheduling model Section 4 presents anumerical simulation to show the feasibility of the algorithmFinally our solution to the problem and the performanceof IMGA are discussed Section 5 concludes this paper andongoing works

2 Description of the Problem and Model

21 Partner Selection Model

211 Three-Stage Model of Partner Selection According tothe actual characteristics of the virtual enterprise partnerselection and the cycle of the selection the virtual enterprisepartner selection can be roughly divided into three stages asthe primary selection fine selection and optimized combi-nation as shown in Figure 1

Primary and fine selection stage is relatively simplegenerally the dominated enterprise establishes a specific

assessment team and filters based on certain indicatorsexcluding the enterprises which do not have the requiredcore resources Combinatorial optimization phase occupiesan important position in the entire virtual enterprise partnerselection and it is related to the success or failure ofthe partner selection Therefore this paper focuses on theestablishment of the stages of the model and use of improvedant colony algorithm to solve it

212 Partner Selection Combinatorial Optimization Model

Description of the Problem Assume that the dominant enter-prise has decomposed the task into 119869 different subtasksbased on the decomposition of business processes After theprimaries of the first phase and the fine selection of the secondphase we can get the combination of the candidate partnerenterprises set 119864119868times119869 that need to optimize where 119869 representsthe task number and 119868 is the number of candidate enterprisesfor each task 119906

119894119895is task 119895 which selects its candidate partners

in the corporate collection of the 119894th enterprise as the final im-plementation of the enterprise 119894 = 1 2 119868 119895 = 1 2 119869

Due to the difference of the factor of the characteristicsof the market opportunities and core enterprise defectsconsideration of core enterprise during the partner selectionis also different Overall the time costs and risk are factorsto be considered basically in every virtual enterprise [21]Therefore this paper considers the impact of the virtualenterprise partner selection factors from three aspects oftime cost and risk

Determination of the Objective Function According to thecharacteristics of the virtual enterprise partner selection setthe time 119879 the cost 119862 and the risk 119877 the three objectivefunctions as follows

min119879 = min[

[

119869

sum

119895=1

119868

sum

119894=1

(119879119894119895119867119894119895)]

]

(1)

min119862 = min[

[

119869

sum

119895=1

119868

sum

119894=1

(119862119894119895119867119894119895)]

]

(2)

min119877 = min[

[

119869

sum

119895=1

119868

sum

119894=1

(119877119894119895119867119894119895)]

]

(3)

119867119894119895=

1 the candidate partner 119906119894119895is selected

0 the candidate partner 119906119894119895is not selected

(4)

Wherein119879119894119895indicates the required time when the task 119895 is

the 119894th candidate corporate executive119862119894119895represents required

costs when task 119895 is the 119894th candidate corporate executive 119877119894119895

is the risk that chooses the 119894th candidate enterprises to executetask 119895 119869 represents the total number of tasks 119868 is the numberof candidate enterprises for the 119895th task Ultimate goal is aminimum time cost and risk

For multiobjective decision-making problems it is dif-ficult for the decision makers to prepare give the specificmeans and consider the polarity inconsistency of goal

Mathematical Problems in Engineering 3

Business process divided

Using core competitiveness index

Experts assess

Optimized combination by IACO

Virtual enterprise construction

Selection start

Primary selection

Fine selection

Optimized combination

The selection process end

middot middot middot




Figure 1 Three-stage model of partner selection

Ideal point method (TOPSIS) is a good method for solv-ing multiobjective decision problems Guo and Jin appliedTOPSIS to solve the problem of multiple index decision greyrelation [22] Ideal point method was first proposed by C LHwang and K Yoon in 1981 [23] sort ideal point methodsort the closeness e based a finite number of evaluationobjects with idealized goal and the relative merits of existingobject evaluation ideal point method is an effective methodmultiobjective decision analysis also known as pros and consof solutions of distance method Define a measure in space tomeasure the degree of a program close to the ideal solutionand away from negative ideal solution pros and cons of sortof candidate programs based on the value of this measure toprovide a basis for decision-makingThis method is based onobtaining a good application

Use ideal point method and construct the fitness func-tion of themultiobjective decision problems decisionmakersneed to give positive ideal value (positive ideal point) ofeach goal and negative ideal value (negative ideal point) Thepositive ideal point of the time cost and risk in this paper is(119879+ 119862+ 119877+) negative ideal point is (119879minus 119862minus 119877minus) This can be

constructed out the objective function 119891(119905)

119891 (119905) =119889minus

119889minus + 119889+ (5)

119889minus(119905) =

100381710038171003817100381710038171003817100381710038171003817

(119879 (119905) minus 119879minus)

119879+(119862 (119905) minus 119862

minus)

119862+(119877 (119905) minus 119877

minus)

119877+

100381710038171003817100381710038171003817100381710038171003817

(6)

119889+(119905) =

100381710038171003817100381710038171003817100381710038171003817

(119879 (119905) minus 119879+)

119879+(119862 (119905) minus 119862

+)

119862+(119877 (119905) minus 119877

+)

119877+

100381710038171003817100381710038171003817100381710038171003817

(7)

Task0Task2

Task1

Task3 Task6

Task7

Task8 Task10Task4

Task5

Figure 2 The figure of task decomposition

where means to take norm 119905 is the number of ants119889+ is thepositive ideal distance measure and 119889minus point is the negativeideal distance measure The Euclidean norm can be used tosolve it and the values 119889+ and 119889minus can be calculated using

119889+(119905) = radic(119879 (119905) minus 119879+)

2+ (119862 (119905) minus 119862+)

2+ (119877 (119905) minus 119877+)

2

119889minus(119905) = radic(119879 (119905) minus 119879minus)

2+ (119862 (119905) minus 119862minus)

2+ (119877 (119905) minus 119877minus)

2

(8)

213 Numerical Example As shown in Figure 2 it is assumedthat the dominated enterprise decomposition market oppor-tunities objectives into Task1 Task8 Task0 and Task9 arethe starting point and end point

Through the first phase and the secondphase of the select-ing the number of candidate enterprises which can enter intothe optimum combination of stage has been narrowed intoan appropriate range The information set of Task1 Task8in optimized combination stage corresponding to candidatecompanies is shown in Table 1

According to the information provided in Table 1 cal-culate each candidate set using the ideal point methodFor example positive ideal points of the candidate setP11P12P13P14 is (90 55 035) and the negative ideal

point is (110 75 045) The calculated ant path hunt to thediagram shown in Figure 3 is carried out for all candidatesets Calculating the collection of all candidates can get thedigraph of antsrsquo path finding

22 Production Scheduling Model

221 Description of the Problem After the dominant enter-prise establishes VE the 119873 subtasks are assigned to the119872 member enterprises The production plan is developedbased on the timing relationships between the tasks andthe constraints of the overall scheduling This model aimsat obtaining the maximum profits and deliver customer-specified products with the shortest production cycle [24] Inthis paper we consider that the VE production planning hasthe following characteristics [25]

(a) The dominant enterprise specifies the operating timeof the decomposition of the task

(b) Subtasks in the implementation process withoutinterruption for authors of only one affiliation

(c) Subtasks with timing constraints(d) When member enterprises are assigned to multiple

tasks considering the member enterprises of their


Start

7107

56 56 71

352125

58

48

54

3635

42

65

097868

84 End

P11

P21

P31P41

P42

P43

P54

P53

P52

P51

P61

P62

P63P73

P72

P71

P81

P82

P83

P84

P32

P33

P22

P12

P13

P14

2316

42

131426

24

Figure 3 Digraph with weight

Table 1 Time cost and risk of candidate enterprises

Task Candidateenterprise

Cost(10000$)

Time(month) Risk

Task1

11987511

100 65 041P12 99 71 040P13 110 55 035P14 90 75 045

Task2 P21 56 27 022P22 77 33 033

Task3P31 59 53 050P32 67 55 045P33 50 45 035

Task4P41 80 5 033P42 88 32 051P43 79 34 052

Task5

P51 130 52 022P52 139 4 033P53 122 62 025P54 140 4 020

Task6P61 80 5 033P62 79 34 052P63 88 32 051

Task7P71 56 27 022P72 77 33 022P73 103 41 025

Task8

P81 59 53 050P82 67 55 045P83 50 45 035P84 66 4 025

own resource constraints a moment can only per-form one task

(e) Initial time of the production scheduling is 0 and thegoal is the shortest operating cycle

222 Symbol Description This paper uses the followingsymbols to describe the mathematical model of VE

119872 the total number of tasks

119873 the total number of enterprises

119875119894 task number 119894 isin [119894119873]

119864119894 the number of enterprises 119894 isin [119894119872]

119877119894 enterprise 119864

119894assuming the task set

119870119894 the number of tasks to be undertaken by the

enterprise 119864119894

119904119905119894119890 the initial production time of 119875

119894in the enterprise

119864119890

119888119905119894119890 production time of 119875

119894in the enterprise 119864

119890

119890119905119894119890 final time of 119875


119890

119905119905119898119899 transport time from enterprise 119864

119898to 119864119899

119909119894119890=

1 task 119875119894in the enterprise 119864

119894

0 otherwise

119891119894119895=

1 task 119875119894is the preamble of the task of 119875

119895

0 otherwise

(9)

119871119909 extending task and 119909 isin [119894119873]

119878 delaying the task 119871119909as a preorder or in connection

with tasks performed in the same enterprise

1198781015840 completed tasks including delayed tasks

223 Model Under the constraints with the initial time ofthe production scheduling set to 0 the entire productioncycle is ensured to be the shortest for each production taskin the VE production planning problems by determining theshortest completion time [26] Delayed by the duration ofthe task prior to the completion of this task the subsequent


sequence of tasks and the same enterprise to complete the taskto a collection of tasks in 119878 cannot be executed The tasks in119878 can be reallocated by the dominant enterprise or adjusteddynamically in accordance with the original distribution VEcan be dynamically adjusted according to the sign constraintsThe production scheduling model is expressed as follow

min (119879) = min max1le119894le119873

max1le119890le119872

119890119905119894119890 (10)

The constraints of the initial and final times of a task areshown

119890119905119894119890= 119904119905119894119890+ 119888119905119894119890 (11)

The task execution time constraints with direct sequencerelationship are expressed in

(119890119905119894119898minus 119904119905119895119899minus 119905119905119898119899) sdot 119891119894119895sdot 119909119894119898sdot 119909119895119899ge 0 (12)

As shown in (13) the same task cannot be performed withindifferent enterprises

119877119898cap 119877119899= 0 forall119898 119899 isin [1119872] (13)

As shown in (14) all subtasks are allocated to the enterprise

1198771cup 1198772cup sdot sdot sdot cup 119877

119872= 1198751cup 1198752cup sdot sdot sdot cup 119875

119873 (14)

The time constraints between tasks before and after in thesame enterprise are expressed in

(119904119905119895119890minus 119904119905119894119890) 119891119894119895ge 0 (15)

As shown in (16) the initial time of each task cannot benegative

119904119905119894119890ge 0 forall119894 isin [1119872] forall119890 isin [1119872] (16)

As shown in (17) each enterprise can only perform one taskat a time

119870119899

sum

119894=1

119909119894119890= 1 forall119890 isin [1119872] (17)

As shown in (18) the finished and unfinished tasks are thesum of all the tasks

119878 cup 1198781015840= 1198751cup 1198752cup sdot sdot sdot cup 119875

119873 (18)

3 Solving the Models by IACO and IMGA

31 Solving Partner Selection Model by IACO Through theabove analysis the use of improved ant colony algorithm(IACO) for virtual enterprise partner selection problem isto use the ants traverse set of candidate companies for eachtask each time after only a candidate for a collection of tasksin a candidate enterprises Improved ant colony algorithmsfor virtual enterprise partner selection concrete steps are asfollows

311 Initialization Assume that the core enterprise decom-poses the market opportunity goal into 119898 tasks and thereis a collection of selected partners for each task The initialnumber of ants is 119899 ants carry the initial pheromone as120591119894119895(0) = 120591

0 (1205910is a constant)The initial size of the pheromone

between the enterprise node is obtained by the formula (5)The required value of the reciprocal is called the degree ofattraction in this article For the initial placed node of eachant is set randomly so the starting point of each ant is notnecessarily the same

312 Ants Jump Ants exist in discrete states They useprobability transfer rules to move from one node to anothernode The path selection probability of ant 119905 from node 119894moving to node 119895 is

119875119896

119894119895(119905) =

120591120572

119894119895(119905) 120593120573

119894119896(119905)

sum119904isinallowed

119896

[120591119894119904 (119905)]120572sdot [120593119894119904 (119905)]120573 if in allowed

119896

0 else(19)

where 120591119894119895(119905) represents the concentration of pheromone of

the edge (119894 119895) at the 119905th search cycle 119896 is the attraction of 119895point (obtained by the inverse of the value of (5)) parameters120572 120573 represent relative weights for adjusting the relativeimportance of 119896 and 119895 and the larger 120572 the more ants tendto choose the road section that other ants use reflecting thecollaboration among ants The greater 120573 represents the moreinfluence to the degree of attraction of jump probabilitiesand the probabilities are close to the greedy rule The 119895 (119895 isinallowed

119896) denotes the set of ant 119896 that is allowed to choose

the next node that taboo [119895] = 1 If you simply transfer themaximumprobability it will soon fall into local optimum theoptimal solution cannot be found Therefore we choose theroulette algorithm which combines the probability to guidethe transfer of ants

313 Update Taboo Table For each ant can only access thenodes which havenrsquot been visited before so a taboo table[119898][119899] is set to mark it in which the value of taboo [119896][119905]refers the point accessed by ant 119896 at the time 119905 In thisalgorithm one task can only be assigned one partner sowhen a node is accessed by ants all of the other nodes in thecollection of the corresponding candidate enterprises wouldbe marked as accessed

314 Update Pheromone Ants traverse a candidate enter-prise collection to construct a feasible solution that needsto update the global pheromone The pheromone updateincludes two aspects the pheromone that ants leave andvolatile pheromone over time Ants will leave a certainamount of pheromone on the edge of the path and the size ofthe pheromone left by ant 119896 is calculated by the formula (20)as follows

Δ120591119896

119894119895(119905) =

119876

119871 [119896] (20)


where 119876 is a positive constant number according to thespecific circumstances 119871[119896] is the length of the path that ant119896 traveled through in this paper the length is calculated bythe degrees of the attraction of each other node size plus 1At the end of the 119905th search cycle the pheromone on theconcentration increment remains in the edge (119894 119895) as shown

Δ120591119894119895 (119905) =

119898

sum

119896=1

Δ120591119896

119894119895(119905) (21)

Pheromonewill disappear gradually over time Assumingthat the pheromone retention factor is 119896 and the volatilesratio is 119895 which represents the degree of disappearance ofthe pheromone So the edge (119894 119895) on the global pheromoneupdate is carried out in accordance with

120591119894119895 (119905 + 1) = 120588 sdot 120591119894119895 (119905) + Δ120591119894119895 (119905) 120588 isin (0 1) (22)

After (2) (3) and (4) the ants completed a full traver-sal after getting an optimal solution update the globalpheromone Repeated (2) (3) and (4) operations in accor-dance with the initial number of ants and finally choose onepath that ant go through most as the optimal solution of thealgorithm execution

Using example of part II the IAOC is run in MATLAB70 the number of ants in initialization is 30 the constants119876 = 20 and 119862 = 01 and the weight coefficients 120572 120573 are 07and 09 After all ants finished 27 ants converge to the path(P14 P22 P31 P43 P51 P61 P73 and P83) So we can selectthe candidate as the ultimate corporate partners on this pathAnd the candidate corporate on this path can be selected asthe ultimate partners

In the case of large solution space the algorithm will notconverge and thus cannot get the optimal combination of theoptimal solution We solve this problem through adjust thevalues of 120572 120573 and antsrsquo initial phenomenon

32 Solving Scheduling Model by IMGA

321 Encoding and Decoding For simple operation geneticencoding is based on task numbers For tasks numbered as1 minus 119873 genetic sequences are the permutations of the 119873numbers Decoding is the sequence of topological sortingTherefore genetic sequences are transformed topologicallyfollowing the constraint graph of the task

322 Genetic Operator

Population Initialization A population size of pop size israndomly generated and then randomly allocated to 119899 sub-populations The size of sub-population is pop size119899

Improved Multipopulation Genetic Algorithm The multi-population GAimproves the performance of GA This paperdesigns an IMGA [27] based on the existingmulti-populationGA The entire population consists of a more general popu-lation and an excellent population with a good populationof the ordinary population evolution of a certain algebraiccross to guide the evolution of the general population and

improve the quality of the general population The excellentpopulation continues to absorb the best individual from thegeneral population to improve its quality and hasten theconvergence rate of GA

Fitness FunctionThe productionmodel of this paper is basedon the shortest production time for the target Thus thefitness function is defined as 119891

119894= 119865max 119895 minus 119865119894119895 where 119865max 119895

is the maximum execution time of all the genes in the sub-population 119895 and 119865

119894119895is the execution time of gene 119894 in the

sub-population 119895

Select Operator In this paper the general populations areselected using the roulette wheel selection operator and theelitist strategy to ensure that the highest fitness individualsare copied to the next generation

Crossover Operator The partially mapped crossover is a goodmethod for increasing the diversity of the offspring throughthe intersection between the parent individuals Two cross-points are randomly selected and the fragments betweenthe cross-points of the parent individuals are exchangedA conflict occurs if the gene outside the cross-point doesnot crossover the gene fragment Otherwise the genes aredetermined through gene mapping

Mutation Operator Mutation in combinatorial optimizationproblems is commonly used in reverse swap and insertoperation The so-called reversion is reversing two differentrandomposition genes in the chromosome [28] In this paperthe reverse operation is used to ensure that the largest possiblechromosomal variation is obtained

Simulated Annealing Mechanism In simulated annealingalgorithm [29] an initial solution is provided Anothersolution is randomly generated from the fieldThe acceptancecriteria allows the target function to deteriorate within a finiterange and it is decided by a parameter similar to the tem-perature control parameter in physical process Combinedwith genetic algorithm the acceptance criteria can preventpremature convergence

Termination The algorithm terminates when the global opti-mal solution does not change in a continuous 119870-generationand when the per-set maximum number of iterations iscompleted

323 Steps of the Improved Genetic Algorithm

Step 1 The GA parameters are initialized Population sizeis pop size the largest breeding algebra is max gen thenumber of subpopulations is 119899 the cross rate and variationrate of sub-population are 119865

119894119888and 119865119894119898 respectively the global

optimal solution unchanged termination of algebraic 119870 thesubpopulation independent evolution of the119885 algebra globalevolution of algebra loop1 = 1 general sub-population evolu-tionary algebra loop2 = 1 the initial annealing temperatureis 119879 the minimum temperature is 119905 and the annealingcoefficients are 119888


P25

P26

P21 P16

P17

P22 P18 P13 P8

P6 P4

P2 P1

P19 P14 P9

P12

P11 P7 P5 P3

P20 P15 P10

P24

P23

18 6

10 5

2

1

8 4 5

12 18

13

4 5

6 2

825

12

14165

768

3 4

4

0 2

05

1

0

1

2 1

1 1

1

05

0 0 2

1

1

10

05

05

Figure 4 Timing constraints of tasks

Step 2 The following operations are implemented for eachgeneral sub-population 119894

(a) The sub-populations in the fitness of each individualare evaluated

(b) 119909119894119895 119909119894119896

are randomly selected from the sub-population Two new individuals 119909

1015840

119894119895 1199091015840119894119896

arecrossed over and generated and their fitness valuesare 119891(1199091015840

119894119895) and 119891(1199091015840

119894119896) If 119891(1199091015840

119894119895) le 119891(119909

119894119895) accept 119909

119894119895 if

min 1 exp(minus(minus119891(1199091015840119894119895) minus 119891(119909

119894119895)))119879 gt random and

119891(1199091015840

119894119895) gt 119891(119909

119894119895) accept 1199091015840

119894119895 The acceptance of 119909

119894119896and

1199091015840

119894119896are the same as before

Step 3 Mutation Whether the new individual is acceptedaccording to Step 2 is determined

Step 4 If loop1 lt 119885 loop1 = loop + 1 Step 2 is repeatedotherwise Step 5 is followed

Step 5 The best individual of each of the general population(20) is used to establish an excellent population Crossoverand mutation according to Steps 2 and 3 in the excellentpopulation are performed

Step 6 Form a mixed population by several general popu-lations and replace the individuals of the mixed populationby of mixed population Excellent populations before 20individual replacement of the mixed population are thendivided into new general populations

Step 7 If the optimal solution is the same algebra 119870 thissolution is obtained and the algorithm terminates otherwiseStep 8 is followed

Step 8 If loop2 lt max gen 119879 = 119888119879 119888 isin (0 1) and loop2= loop2 + 1 Step 2 is repeated otherwise the algorithm isterminated

4 Experiment

41 Examples of Production Scheduling Enterprise 1198641could

not independently complete a device manufacturing taskIt divides the task into 26 subtasks (119875

1 11987526) Finally

1198641establishes a VE with four partners (119864

2 1198643 1198644 and

1198645) through a tender 119864

1implementation of the task

set 1198751 1198755 1198757 11987514 11987523 11987526 1198642implementation of the task

set 1198754 1198759 11987515 11987520 11987524 1198643implementation of the task set

1198752 11987510 11987512 11987516 11987519 1198644implementation of the task set

1198756 1198758 11987511 11987517 11987525 and 119864

5implementation of the task set

1198753 11987513 11987518 11987521 11987522The task execution order constraints are

shown in Figure 4 The top of the rectangular box representsthe task number the bottom represents themission time andthe arrow line numbers indicate the transit time (when theproduction tasks are completed by the same enterprise thetransit time is 0)

42 Scheduling Optimization All parameters are initializedand MATLAB programming is used to achieve the improvedmultipopulation GAThe initial population size is 100 whichis then divided into three general populations (with sizes of30 30 and 40) The crossover and mutation probabilitiesof the three populations are 08 01 085 015 and 0902 respectively The length of the chromosome is 26 Thesub-population evolution algebra is 10 and the maximumevolution generation of the GA is 100 The initial annealingtemperature annealing coefficient and lowest temperatureare 100 ∘C 097 and 10 ∘C respectively After implementation


E1

E2

E3

E4

E5

0 10 20 30 40 50 60 70 80Time (day)

Ente

rpris

e

P26 P23 P14 P7 P5 P1

TaskTransport

P22 P18 P13 P3

P6P8P11P17P25

P21 P19 P16 P12 P10

P24 P20 P15 P9 P4

P14

Figure 5 Optimal plan program by IMGSA

125

120

115

110

105

100

95

90

85

80

Aver

age t

ime MGSA

IMGSA

10 20 30 40 50 60 70 80 90 100Hereditary algebra

Figure 6The average time comparison chart ofMSGAand IMGAS

of the algorithm the optimal manufacturing cycle is 775days The optimal scheme using the Gantt chart is shown inFigure 5

Using the above algorithm parameters the unimprovedmulti-population (UMGSA) and improved multi-populationalgorithm (IMGSA) are run for 100 generations The averagecompletion time of the task with the convergence of geneticalgebraic diagram is shown in Figure 6 The improved GAconvergence rate is fast

5 Conclusion

This paper presents a model for solving production planningThe improved GA is used to solve the model It providesa good method for solving the problem of the productionplanning of VE The convergence rate of the improved GAis faster than that of the unimproved GA Thus an optimalproduction plan is obtained quickly to guide the productionof VE During the production operation the production plan

may change any time because companies are subjected tocertain external factorsTherefore the unfinished task of con-centration can be based on the information of the completedtask and this algorithm can be used for adjustments

Acknowledgments

This work was supported by some grants from GuangdongProvince Production Education and Scientific Study Pro-grams China (no 2012B091100490 and no 2011B090400056)and the Fundamental Research Funds for the Central Univer-sities SCUT

References

[1] H Helaakoski P Iskanius and I Peltomaa ldquoAgent-basedarchitecture for virtual enterprises to support agilityrdquo IFIPInternational Federation for Information Processing vol 243 pp299ndash306 2007

[2] A Goel H Schmidt and D Gilbert ldquoTowards formalizing vir-tual enterprise architecturerdquo inProceedings of the 13th EnterpriseDistributed Object Computing Conference Workshops (EDOCWrsquo09) pp 238ndash242 September 2009

[3] N Capuano S Miranda F Orciuoli and S Vassallo ldquoE-Learning at work in the knowledge virtual enterpriserdquo inProceedings of the 2nd International Conference on ComplexIntelligent and Software Intensive Systems (CISIS rsquo08) pp 507ndash512 March 2008

[4] MH Danesh B Raahemi andM A Kamali ldquoA framework forprocess management in service oriented virtual organizationsrdquoin Proceedings of the 7th International Conference on Next Gen-eration Web Services Practices (NWeSP rsquo11) pp 12ndash17 October2011

[5] S Esparcia R Centeno R Hermoso and E Argente ldquoArtifact-ing and regulating the environment of a virtual organizationrdquo inProceedings of the 23rd IEEE International Conference on Toolswith Artificial Intelligence (ICTAI rsquo11) pp 547ndash554 November2011

[6] J A Crispim and J P De Sousa ldquoPartner selection in virtualenterprisesrdquo International Journal of Production Research vol48 no 3 pp 683ndash707 2010

[7] T Jarimo and A Salo ldquoMulticriteria partner selection in virtualorganizations with transportation costs and other networkinterdependenciesrdquo IEEE Transactions on Systems Man andCybernetics C vol 39 no 1 pp 124ndash129 2009

[8] N Nayak K Prasanna S Datta S S Mahapatra and S SahuldquoA novel swarm optimization technique for partner selectionin virtual enterpriserdquo in IEEE International Conference onIndustrial Engineering and EngineeringManagement (IEEM rsquo10)pp 1118ndash1122 December 2010

[9] D Simona and P Raluca ldquoIntellegent modeling method basedon genetic algorithm for partner selection in virtual organiza-tionsrdquo Business and Economic Horizons vol 5 no 2 pp 23ndash242011

[10] A M Mohamed ldquoOptimal composition of virtual enterpriseswith interval cost parametersrdquo in Proceedings of the 8th Inter-national Conference on Informatics and Systems (INFOS rsquo12) ppBIO-188ndashBIO-194 2012

[11] N Yalaoui L Amodeo F Yalaoui and H Mahdi ldquoParticleswarm optimization under fuzzy logic controller for solving ahybrid reentrant flow shop problemrdquo in Proceedings of the IEEE


International Symposium on Parallel and Distributed ProcessingWorkshops and Phd Forum (IPDPSW rsquo10) pp 1ndash6 April 2010

[12] F Dugardin L Amodeo and F Yalaoui ldquoMultiobjectivescheduling of a reentrant hybrid flowshoprdquo in Proceedings ofthe 39th International Conference on Computers and IndustrialEngineering (CIE rsquo09) pp 193ndash198 July 2009

[13] Y Gao and Z B Jiang ldquoHybrid genetic algorithm for virtualproduction planningrdquoControl andDecision-Making vol 22 no8 2007

[14] Y Ding Y Gao and G Luo ldquoVirtual enterprise global produc-tion planning based on cost optimizationwith time constraintsrdquoin Proceedings of the International Conference on Managementand Service Science (MASS rsquo09) September 2009

[15] Q J SongThe Cloud of Virtual Enterprise Production Planningand Control Silicon Valley 2008

[16] Y Gao and Y S Ding Virtual Enterprise Production PlanningModel vol 22 Enterprise Management 2009

[17] Q S Li and J G Liu ldquoResearch on VE production planningbased on multi-agentrdquo Research and Design vol 22 no 1 2008

[18] L M Camarinha-Matos and H Afsarmanesh VE Model-ing and Support Infrastructures Applying Multi-Agent SystemApproaches Springer New York NY USA 2001

[19] Q Zhao and M Zhou ldquoVirtual production tasks based onant colony optimization schedulingrdquo Wuhan University ofTechnology vol 34 no 3 2011

[20] Z Tao and L Y Xie ldquoBased on hybrid genetic algorithmfor job shop scheduling problemrdquo Computer Engineering andApplications vol 18 2005

[21] M Huang XWang F-Q Lu and H Bi ldquoA coordination of riskmanagement for supply chains organized as virtual enterprisesrdquoMathematical Problems in Engineering vol 2013 Article ID931690 11 pages 2013

[22] X Y Guo and L Jin ldquoThe gray relational MCDM TOPSISrdquoTechnology and Management vol 12 no 5 2010

[23] J Singh and A Madhukar ldquoNew method for calculating non-ideal point defect induced electronic structurerdquo Solid StateCommunications vol 41 no 12 pp 947ndash950 1982

[24] W Zhou and Y Bu ldquoCultural algorithm based on particleswarm optimization for partner selection of virtual enterpriserdquoin Proceedings of the 31st Chinese Control Conference 2012

[25] S-L YangH-WKang andH Zhou ldquoResearch onweb service-based virtual enterprise integration frameworkrdquo in Proceedingsof the 3rd International Conference on System Science Engineer-ing Design and Manufacturing Informatization 2012

[26] M A O Pessoa F Junqueira and D J S Filho ldquoVirtualenterprise planning system using time windows and capacityconstraint conceptsrdquo in Proceedings of the 38th Annual Con-ference on IEEE Industrial Electronics Society (IECON rsquo12) pp2851ndash22856 2012

[27] C F M Toledo and J M G Lima ldquoA multi-population geneticalgorithm approach for PID controller auto-tuningrdquo in Proceed-ings of the IEEE 17th Conference on Emerging Technologies ampFactory Automation (ETFA rsquo12) pp 1ndash8 2012

[28] B Xiaojun and L Guangxin ldquoThe improvement of ant colonyalgorithm based on the inver-over operatorrdquo in Proceedingsof the IEEE International Conference on Mechatronics andAutomation (ICMA rsquo07) pp 2383ndash2387 August 2007

[29] Q Xu JMao andZ Jin ldquoSimulated annealing-based ant colonyalgorithm for tugboat scheduling optimizationrdquo MathematicalProblems in Engineering vol 2012 Article ID 246978 22 pages2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


partner selection problem in virtual enterprise where the costcoefficients are expressed as interval by the decision makerand uses a multiobjective algorithm (PSA) to solve [10]

Yalaoui et al solve a hybrid flow shop scheduling problemand create a new method to solve the problem based onthe nature which is the particle swarm optimization methodunder fuzzy logic controller (FLCPSO) [11] Dugardin et alcreate a model to solve multiobjective scheduling of a reen-trant hybrid flow shop problem and can be used for partnerselection inVE if improved [12] Gao and Jiang andDing et alestablish a mathematical model solved using a hybrid geneticalgorithm (GA) to acquire the shortest operating cycle basedon the characteristics of the production scheduling of VE[13 14] However this model caused premature convergencebecause that it only used a single population Song combinesthe widely applied ldquocloud computingrdquo theory to present thecloud of VE production planning and control model [15]However the model has not yet been tested in practicalapplications Gao and Ding and Li and Liu discuss an orderrarely multiple orders established a multiorder productionschedulingmodel and introduce various population genetic-simulated annealing algorithms to solve this model [16 17]Although the problem of population diversity is solvedthe execution time is very slow Some researchers such asCamarinha-Matos and Afsarmanesh and Zhao and Zhouapply multiagent technology to solve VE production plan-ning and control [18 19] They mainly attempt to establisha framework of the production scheduling system The VEproduction planningmodel and the corresponding algorithmneed to be further studied According to the two-tier schedul-ing model of VE Tao and Xie establish a mathematicalmodel with the smallest total operating time and used theant colony algorithm to solve the model [20] Howeverpremature convergence and slow execution time were thedisadvantages of this model

The structure of the paper is as follows Section 2 statesthe problems of partner selection and production planningin VE and creates twomodels Section 3 introduces improvedant colony algorithm (IACO) to solve partner selectionmodeland improved multipopulation genetic algorithm (IMGA) tosolve the production scheduling model Section 4 presents anumerical simulation to show the feasibility of the algorithmFinally our solution to the problem and the performanceof IMGA are discussed Section 5 concludes this paper andongoing works

2 Description of the Problem and Model

21 Partner Selection Model

211 Three-Stage Model of Partner Selection According tothe actual characteristics of the virtual enterprise partnerselection and the cycle of the selection the virtual enterprisepartner selection can be roughly divided into three stages asthe primary selection fine selection and optimized combi-nation as shown in Figure 1

Primary and fine selection stage is relatively simplegenerally the dominated enterprise establishes a specific

assessment team and filters based on certain indicatorsexcluding the enterprises which do not have the requiredcore resources Combinatorial optimization phase occupiesan important position in the entire virtual enterprise partnerselection and it is related to the success or failure ofthe partner selection Therefore this paper focuses on theestablishment of the stages of the model and use of improvedant colony algorithm to solve it

212 Partner Selection Combinatorial Optimization Model

Description of the Problem Assume that the dominant enter-prise has decomposed the task into 119869 different subtasksbased on the decomposition of business processes After theprimaries of the first phase and the fine selection of the secondphase we can get the combination of the candidate partnerenterprises set 119864119868times119869 that need to optimize where 119869 representsthe task number and 119868 is the number of candidate enterprisesfor each task 119906

119894119895is task 119895 which selects its candidate partners

in the corporate collection of the 119894th enterprise as the final im-plementation of the enterprise 119894 = 1 2 119868 119895 = 1 2 119869

Due to the difference of the factor of the characteristicsof the market opportunities and core enterprise defectsconsideration of core enterprise during the partner selectionis also different Overall the time costs and risk are factorsto be considered basically in every virtual enterprise [21]Therefore this paper considers the impact of the virtualenterprise partner selection factors from three aspects oftime cost and risk

Determination of the Objective Function According to thecharacteristics of the virtual enterprise partner selection setthe time 119879 the cost 119862 and the risk 119877 the three objectivefunctions as follows

min119879 = min[

[

119869

sum

119895=1

119868

sum

119894=1

(119879119894119895119867119894119895)]

]

(1)

min119862 = min[

[

119869

sum

119895=1

119868

sum

119894=1

(119862119894119895119867119894119895)]

]

(2)

min119877 = min[

[

119869

sum

119895=1

119868

sum

119894=1

(119877119894119895119867119894119895)]

]

(3)

119867119894119895=

1 the candidate partner 119906119894119895is selected

0 the candidate partner 119906119894119895is not selected

(4)

Wherein119879119894119895indicates the required time when the task 119895 is

the 119894th candidate corporate executive119862119894119895represents required

costs when task 119895 is the 119894th candidate corporate executive 119877119894119895

is the risk that chooses the 119894th candidate enterprises to executetask 119895 119869 represents the total number of tasks 119868 is the numberof candidate enterprises for the 119895th task Ultimate goal is aminimum time cost and risk

For multiobjective decision-making problems it is dif-ficult for the decision makers to prepare give the specificmeans and consider the polarity inconsistency of goal




Experts assess



Selection start

Primary selection

Fine selection











119891 (119905) =119889minus

119889minus + 119889+ (5)

119889minus(119905) =

100381710038171003817100381710038171003817100381710038171003817

(119879 (119905) minus 119879minus)

119879+(119862 (119905) minus 119862

minus)

119862+(119877 (119905) minus 119877

minus)

119877+

100381710038171003817100381710038171003817100381710038171003817

(6)

119889+(119905) =

100381710038171003817100381710038171003817100381710038171003817

(119879 (119905) minus 119879+)

119879+(119862 (119905) minus 119862

+)

119862+(119877 (119905) minus 119877

+)

119877+

100381710038171003817100381710038171003817100381710038171003817

(7)

Task0Task2

Task1

Task3 Task6

Task7

Task8 Task10Task4

Task5



119889+(119905) = radic(119879 (119905) minus 119879+)

2+ (119862 (119905) minus 119862+)

2+ (119877 (119905) minus 119877+)

2


2+ (119862 (119905) minus 119862minus)

2+ (119877 (119905) minus 119877minus)

2

(8)












Start

7107

56 56 71

352125

58

48

54

3635

42

65

097868

84 End

P11

P21

P31P41

P42

P43

P54

P53

P52

P51

P61

P62

P63P73

P72

P71

P81

P82

P83

P84

P32

P33

P22

P12

P13

P14

2316

42

131426

24




Cost(10000$)

Time(month) Risk

Task1

11987511

100 65 041P12 99 71 040P13 110 55 035P14 90 75 045

Task2 P21 56 27 022P22 77 33 033

Task3P31 59 53 050P32 67 55 045P33 50 45 035

Task4P41 80 5 033P42 88 32 051P43 79 34 052

Task5

P51 130 52 022P52 139 4 033P53 122 62 025P54 140 4 020

Task6P61 80 5 033P62 79 34 052P63 88 32 051

Task7P71 56 27 022P72 77 33 022P73 103 41 025

Task8

P81 59 53 050P82 67 55 045P83 50 45 035P84 66 4 025






119875119894 task number 119894 isin [119894119873]


119877119894 enterprise 119864






119864119890



119890

119890119905119894119890 final time of 119875


119890


119898to 119864119899

119909119894119890=


119894

0 otherwise

119891119894119895=


119895

0 otherwise

(9)








min (119879) = min max1le119894le119873

max1le119890le119872

119890119905119894119890 (10)


119890119905119894119890= 119904119905119894119890+ 119888119905119894119890 (11)




119877119898cap 119877119899= 0 forall119898 119899 isin [1119872] (13)




119873 (14)


(119904119905119895119890minus 119904119905119894119890) 119891119894119895ge 0 (15)




119870119899

sum

119894=1

119909119894119890= 1 forall119890 isin [1119872] (17)



119873 (18)







119875119896

119894119895(119905) =

120591120572

119894119895(119905) 120593120573

119894119896(119905)


119896


119896

0 else(19)







Δ120591119896

119894119895(119905) =

119876

119871 [119896] (20)



Δ120591119894119895 (119905) =

119898

sum

119896=1

Δ120591119896

119894119895(119905) (21)


120591119894119895 (119905 + 1) = 120588 sdot 120591119894119895 (119905) + Δ120591119894119895 (119905) 120588 isin (0 1) (22)

























P25

P26

P21 P16

P17

P22 P18 P13 P8

P6 P4

P2 P1

P19 P14 P9

P12

P11 P7 P5 P3

P20 P15 P10

P24

P23

18 6

10 5

2

1

8 4 5

12 18

13

4 5

6 2

825

12

14165

768

3 4

4

0 2

05

1

0

1

2 1

1 1

1

05

0 0 2

1

1

10

05

05




(b) 119909119894119895 119909119894119896


1015840

119894119895 1199091015840119894119896


119894119895) and 119891(1199091015840

119894119896) If 119891(1199091015840

119894119895) le 119891(119909

119894119895) accept 119909

119894119895 if


119894119895)))119879 gt random and

119891(1199091015840

119894119895) gt 119891(119909

119894119895) accept 1199091015840


119894119896and

1199091015840








4 Experiment



1 11987526) Finally


2 1198643 1198644 and






1198756 1198758 11987511 11987517 11987525 and 119864






E1

E2

E3

E4

E5

0 10 20 30 40 50 60 70 80Time (day)

Ente

rpris

e

P26 P23 P14 P7 P5 P1

TaskTransport

P22 P18 P13 P3

P6P8P11P17P25

P21 P19 P16 P12 P10

P24 P20 P15 P9 P4

P14


125

120

115

110

105

100

95

90

85

80

Aver

age t

ime MGSA

IMGSA





5 Conclusion



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Experts assess



Selection start

Primary selection

Fine selection











119891 (119905) =119889minus

119889minus + 119889+ (5)

119889minus(119905) =

100381710038171003817100381710038171003817100381710038171003817

(119879 (119905) minus 119879minus)

119879+(119862 (119905) minus 119862

minus)

119862+(119877 (119905) minus 119877

minus)

119877+

100381710038171003817100381710038171003817100381710038171003817

(6)

119889+(119905) =

100381710038171003817100381710038171003817100381710038171003817

(119879 (119905) minus 119879+)

119879+(119862 (119905) minus 119862

+)

119862+(119877 (119905) minus 119877

+)

119877+

100381710038171003817100381710038171003817100381710038171003817

(7)

Task0Task2

Task1

Task3 Task6

Task7

Task8 Task10Task4

Task5



119889+(119905) = radic(119879 (119905) minus 119879+)

2+ (119862 (119905) minus 119862+)

2+ (119877 (119905) minus 119877+)

2


2+ (119862 (119905) minus 119862minus)

2+ (119877 (119905) minus 119877minus)

2

(8)












Start

7107

56 56 71

352125

58

48

54

3635

42

65

097868

84 End

P11

P21

P31P41

P42

P43

P54

P53

P52

P51

P61

P62

P63P73

P72

P71

P81

P82

P83

P84

P32

P33

P22

P12

P13

P14

2316

42

131426

24




Cost(10000$)

Time(month) Risk

Task1

11987511

100 65 041P12 99 71 040P13 110 55 035P14 90 75 045

Task2 P21 56 27 022P22 77 33 033

Task3P31 59 53 050P32 67 55 045P33 50 45 035

Task4P41 80 5 033P42 88 32 051P43 79 34 052

Task5

P51 130 52 022P52 139 4 033P53 122 62 025P54 140 4 020

Task6P61 80 5 033P62 79 34 052P63 88 32 051

Task7P71 56 27 022P72 77 33 022P73 103 41 025

Task8

P81 59 53 050P82 67 55 045P83 50 45 035P84 66 4 025






119875119894 task number 119894 isin [119894119873]


119877119894 enterprise 119864






119864119890



119890

119890119905119894119890 final time of 119875


119890


119898to 119864119899

119909119894119890=


119894

0 otherwise

119891119894119895=


119895

0 otherwise

(9)








min (119879) = min max1le119894le119873

max1le119890le119872

119890119905119894119890 (10)


119890119905119894119890= 119904119905119894119890+ 119888119905119894119890 (11)




119877119898cap 119877119899= 0 forall119898 119899 isin [1119872] (13)




119873 (14)


(119904119905119895119890minus 119904119905119894119890) 119891119894119895ge 0 (15)




119870119899

sum

119894=1

119909119894119890= 1 forall119890 isin [1119872] (17)



119873 (18)







119875119896

119894119895(119905) =

120591120572

119894119895(119905) 120593120573

119894119896(119905)


119896


119896

0 else(19)







Δ120591119896

119894119895(119905) =

119876

119871 [119896] (20)



Δ120591119894119895 (119905) =

119898

sum

119896=1

Δ120591119896

119894119895(119905) (21)


120591119894119895 (119905 + 1) = 120588 sdot 120591119894119895 (119905) + Δ120591119894119895 (119905) 120588 isin (0 1) (22)

























P25

P26

P21 P16

P17

P22 P18 P13 P8

P6 P4

P2 P1

P19 P14 P9

P12

P11 P7 P5 P3

P20 P15 P10

P24

P23

18 6

10 5

2

1

8 4 5

12 18

13

4 5

6 2

825

12

14165

768

3 4

4

0 2

05

1

0

1

2 1

1 1

1

05

0 0 2

1

1

10

05

05




(b) 119909119894119895 119909119894119896


1015840

119894119895 1199091015840119894119896


119894119895) and 119891(1199091015840

119894119896) If 119891(1199091015840

119894119895) le 119891(119909

119894119895) accept 119909

119894119895 if


119894119895)))119879 gt random and

119891(1199091015840

119894119895) gt 119891(119909

119894119895) accept 1199091015840


119894119896and

1199091015840








4 Experiment



1 11987526) Finally


2 1198643 1198644 and






1198756 1198758 11987511 11987517 11987525 and 119864






E1

E2

E3

E4

E5

0 10 20 30 40 50 60 70 80Time (day)

Ente

rpris

e

P26 P23 P14 P7 P5 P1

TaskTransport

P22 P18 P13 P3

P6P8P11P17P25

P21 P19 P16 P12 P10

P24 P20 P15 P9 P4

P14


125

120

115

110

105

100

95

90

85

80

Aver

age t

ime MGSA

IMGSA





5 Conclusion



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start

7107

56 56 71

352125

58

48

54

3635

42

65

097868

84 End

P11

P21

P31P41

P42

P43

P54

P53

P52

P51

P61

P62

P63P73

P72

P71

P81

P82

P83

P84

P32

P33

P22

P12

P13

P14

2316

42

131426

24




Cost(10000$)

Time(month) Risk

Task1

11987511

100 65 041P12 99 71 040P13 110 55 035P14 90 75 045

Task2 P21 56 27 022P22 77 33 033

Task3P31 59 53 050P32 67 55 045P33 50 45 035

Task4P41 80 5 033P42 88 32 051P43 79 34 052

Task5

P51 130 52 022P52 139 4 033P53 122 62 025P54 140 4 020

Task6P61 80 5 033P62 79 34 052P63 88 32 051

Task7P71 56 27 022P72 77 33 022P73 103 41 025

Task8

P81 59 53 050P82 67 55 045P83 50 45 035P84 66 4 025






119875119894 task number 119894 isin [119894119873]


119877119894 enterprise 119864






119864119890



119890

119890119905119894119890 final time of 119875


119890


119898to 119864119899

119909119894119890=


119894

0 otherwise

119891119894119895=


119895

0 otherwise

(9)








min (119879) = min max1le119894le119873

max1le119890le119872

119890119905119894119890 (10)


119890119905119894119890= 119904119905119894119890+ 119888119905119894119890 (11)




119877119898cap 119877119899= 0 forall119898 119899 isin [1119872] (13)




119873 (14)


(119904119905119895119890minus 119904119905119894119890) 119891119894119895ge 0 (15)




119870119899

sum

119894=1

119909119894119890= 1 forall119890 isin [1119872] (17)



119873 (18)







119875119896

119894119895(119905) =

120591120572

119894119895(119905) 120593120573

119894119896(119905)


119896


119896

0 else(19)







Δ120591119896

119894119895(119905) =

119876

119871 [119896] (20)



Δ120591119894119895 (119905) =

119898

sum

119896=1

Δ120591119896

119894119895(119905) (21)


120591119894119895 (119905 + 1) = 120588 sdot 120591119894119895 (119905) + Δ120591119894119895 (119905) 120588 isin (0 1) (22)

























P25

P26

P21 P16

P17

P22 P18 P13 P8

P6 P4

P2 P1

P19 P14 P9

P12

P11 P7 P5 P3

P20 P15 P10

P24

P23

18 6

10 5

2

1

8 4 5

12 18

13

4 5

6 2

825

12

14165

768

3 4

4

0 2

05

1

0

1

2 1

1 1

1

05

0 0 2

1

1

10

05

05




(b) 119909119894119895 119909119894119896


1015840

119894119895 1199091015840119894119896


119894119895) and 119891(1199091015840

119894119896) If 119891(1199091015840

119894119895) le 119891(119909

119894119895) accept 119909

119894119895 if


119894119895)))119879 gt random and

119891(1199091015840

119894119895) gt 119891(119909

119894119895) accept 1199091015840


119894119896and

1199091015840








4 Experiment



1 11987526) Finally


2 1198643 1198644 and






1198756 1198758 11987511 11987517 11987525 and 119864






E1

E2

E3

E4

E5

0 10 20 30 40 50 60 70 80Time (day)

Ente

rpris

e

P26 P23 P14 P7 P5 P1

TaskTransport

P22 P18 P13 P3

P6P8P11P17P25

P21 P19 P16 P12 P10

P24 P20 P15 P9 P4

P14


125

120

115

110

105

100

95

90

85

80

Aver

age t

ime MGSA

IMGSA





5 Conclusion



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











min (119879) = min max1le119894le119873

max1le119890le119872

119890119905119894119890 (10)


119890119905119894119890= 119904119905119894119890+ 119888119905119894119890 (11)




119877119898cap 119877119899= 0 forall119898 119899 isin [1119872] (13)




119873 (14)


(119904119905119895119890minus 119904119905119894119890) 119891119894119895ge 0 (15)




119870119899

sum

119894=1

119909119894119890= 1 forall119890 isin [1119872] (17)



119873 (18)







119875119896

119894119895(119905) =

120591120572

119894119895(119905) 120593120573

119894119896(119905)


119896


119896

0 else(19)







Δ120591119896

119894119895(119905) =

119876

119871 [119896] (20)



Δ120591119894119895 (119905) =

119898

sum

119896=1

Δ120591119896

119894119895(119905) (21)


120591119894119895 (119905 + 1) = 120588 sdot 120591119894119895 (119905) + Δ120591119894119895 (119905) 120588 isin (0 1) (22)

























P25

P26

P21 P16

P17

P22 P18 P13 P8

P6 P4

P2 P1

P19 P14 P9

P12

P11 P7 P5 P3

P20 P15 P10

P24

P23

18 6

10 5

2

1

8 4 5

12 18

13

4 5

6 2

825

12

14165

768

3 4

4

0 2

05

1

0

1

2 1

1 1

1

05

0 0 2

1

1

10

05

05




(b) 119909119894119895 119909119894119896


1015840

119894119895 1199091015840119894119896


119894119895) and 119891(1199091015840

119894119896) If 119891(1199091015840

119894119895) le 119891(119909

119894119895) accept 119909

119894119895 if


119894119895)))119879 gt random and

119891(1199091015840

119894119895) gt 119891(119909

119894119895) accept 1199091015840


119894119896and

1199091015840








4 Experiment



1 11987526) Finally


2 1198643 1198644 and






1198756 1198758 11987511 11987517 11987525 and 119864






E1

E2

E3

E4

E5

0 10 20 30 40 50 60 70 80Time (day)

Ente

rpris

e

P26 P23 P14 P7 P5 P1

TaskTransport

P22 P18 P13 P3

P6P8P11P17P25

P21 P19 P16 P12 P10

P24 P20 P15 P9 P4

P14


125

120

115

110

105

100

95

90

85

80

Aver

age t

ime MGSA

IMGSA





5 Conclusion



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Δ120591119894119895 (119905) =

119898

sum

119896=1

Δ120591119896

119894119895(119905) (21)


120591119894119895 (119905 + 1) = 120588 sdot 120591119894119895 (119905) + Δ120591119894119895 (119905) 120588 isin (0 1) (22)

























P25

P26

P21 P16

P17

P22 P18 P13 P8

P6 P4

P2 P1

P19 P14 P9

P12

P11 P7 P5 P3

P20 P15 P10

P24

P23

18 6

10 5

2

1

8 4 5

12 18

13

4 5

6 2

825

12

14165

768

3 4

4

0 2

05

1

0

1

2 1

1 1

1

05

0 0 2

1

1

10

05

05




(b) 119909119894119895 119909119894119896


1015840

119894119895 1199091015840119894119896


119894119895) and 119891(1199091015840

119894119896) If 119891(1199091015840

119894119895) le 119891(119909

119894119895) accept 119909

119894119895 if


119894119895)))119879 gt random and

119891(1199091015840

119894119895) gt 119891(119909

119894119895) accept 1199091015840


119894119896and

1199091015840








4 Experiment



1 11987526) Finally


2 1198643 1198644 and






1198756 1198758 11987511 11987517 11987525 and 119864






E1

E2

E3

E4

E5

0 10 20 30 40 50 60 70 80Time (day)

Ente

rpris

e

P26 P23 P14 P7 P5 P1

TaskTransport

P22 P18 P13 P3

P6P8P11P17P25

P21 P19 P16 P12 P10

P24 P20 P15 P9 P4

P14


125

120

115

110

105

100

95

90

85

80

Aver

age t

ime MGSA

IMGSA





5 Conclusion



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










P25

P26

P21 P16

P17

P22 P18 P13 P8

P6 P4

P2 P1

P19 P14 P9

P12

P11 P7 P5 P3

P20 P15 P10

P24

P23

18 6

10 5

2

1

8 4 5

12 18

13

4 5

6 2

825

12

14165

768

3 4

4

0 2

05

1

0

1

2 1

1 1

1

05

0 0 2

1

1

10

05

05




(b) 119909119894119895 119909119894119896


1015840

119894119895 1199091015840119894119896


119894119895) and 119891(1199091015840

119894119896) If 119891(1199091015840

119894119895) le 119891(119909

119894119895) accept 119909

119894119895 if


119894119895)))119879 gt random and

119891(1199091015840

119894119895) gt 119891(119909

119894119895) accept 1199091015840


119894119896and

1199091015840








4 Experiment



1 11987526) Finally


2 1198643 1198644 and






1198756 1198758 11987511 11987517 11987525 and 119864






E1

E2

E3

E4

E5

0 10 20 30 40 50 60 70 80Time (day)

Ente

rpris

e

P26 P23 P14 P7 P5 P1

TaskTransport

P22 P18 P13 P3

P6P8P11P17P25

P21 P19 P16 P12 P10

P24 P20 P15 P9 P4

P14


125

120

115

110

105

100

95

90

85

80

Aver

age t

ime MGSA

IMGSA





5 Conclusion



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










E1

E2

E3

E4

E5

0 10 20 30 40 50 60 70 80Time (day)

Ente

rpris

e

P26 P23 P14 P7 P5 P1

TaskTransport

P22 P18 P13 P3

P6P8P11P17P25

P21 P19 P16 P12 P10

P24 P20 P15 P9 P4

P14


125

120

115

110

105

100

95

90

85

80

Aver

age t

ime MGSA

IMGSA





5 Conclusion



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Research on the Production Scheduling ...downloads.hindawi.com/journals/mpe/2013/492158.pdf · MinHuang, 1 RuixianHuang, 1 BoSun, 2 andLinrongLi 1 School of Soware