research article multiobjective joint optimization...

Research ArticleMultiobjective Joint Optimization of Production Scheduling andMaintenance Planning in the Flexible Job-Shop Problem

Jianfei Ye1 and Huimin Ma2

1Business School University of Shanghai for Science and Technology Shanghai 200093 China2Business School Shanghai Dianji University Shanghai 201306 China

Correspondence should be addressed to Huimin Ma mahm81163com

Received 8 June 2015 Revised 26 September 2015 Accepted 28 September 2015

Academic Editor Thomas Hanne

Copyright copy 2015 J Ye and H Ma 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

In order to solve the joint optimization of production scheduling and maintenance planning problem in the flexible job-shopa multiobjective joint optimization model considering the maximum completion time and maintenance costs per unit time isestablished based on the concept of flexible job-shop and preventive maintenance A weighted summethod is adopted to eliminatethe index dimension In addition a double-coded genetic algorithm is designed according to the problem characteristics The bestresult under the circumstances of joint decision-making is obtained through multiple simulation experiments which proves thevalidity of the algorithmWe can prove the superiority of joint optimizationmodel by comparing the result of joint decision-makingproject with the result of independent decision-making project under fixed preventive maintenance period This study will enrichand expand the theoretical framework and analytical methods of this problem it provides a scientific decision analysis method forenterprise to make production plan and maintenance plan

1 Introduction

Production scheduling and maintenance activities are thetwo processes which are correlated with each other in anenterprise Production department and maintenance depart-ment contribute to enterprisersquos profit target together Unco-ordinated arrangement of production plan and maintenanceplan will result in losses in enterprisersquos business opera-tions But in actual production production and mainte-nance department have different goals Production depart-ment pursues the shortest completion time which aimsto avoid delay in delivery Maintenance department pur-sues minimum maintenance cost The difference in twodepartmentsrsquo objective will create conflicts Therefore inorder to realize the best interests of the whole enterpriseit is necessary to study the optimal joint maintenanceand scheduling decision scheme We should consider therequirement of production and maintenance departmentjointly In this paper we continue our study on the basisof previous studies conducted by Ma and Ye [1] We expectto make a further study on the multi-objective optimization

problems of flexible job-shop scheduling and maintenanceplanning

Many experts and scholars have done some research inthe integrated optimization of production scheduling andequipment maintenance Sortrakul et al [2] proposed aclassical model for production scheduling and preventivemaintenance planning Da et al [3] established the joint opti-mizationmodel of the batch scheduling and the maintenanceunder the flow shop in order to minimize the maximumcompletion time the corresponding genetic algorithm tosolve this problem was put forward Zhang et al [4] didsome research on flow shop scheduling problem with peri-odical maintenance The authors took minimum makespanas the optimized objective and designed a kind of hybridgenetic algorithm to solve the problem Chung et al [5] alsoaimed to minimize the makespan of the jobs the authorsfound an approach for scheduling of perfect maintenancein distributed production scheduling With the objective ofmakespanminimizationWang and Yu [6] studied joint opti-mization problem in job-shop schedulingMor andMosheiov[7] focused on a single machine batch scheduling problem

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 725460 9 pageshttpdxdoiorg1011552015725460

2 Mathematical Problems in Engineering

Pan et al [8] built a joint model for minimizing job tardinessAll the references above established the single objectiveoptimization model but some other researchers establishedthe multiobjective optimizing model Berrichi et al [9]presented an algorithm based on Ant Colony Optimizationparadigm to solve the joint production and maintenancescheduling problem the goal is to simultaneously minimizethe system unavailability andmakespan After thatMoradi etal [10] attempted to simultaneously optimize two objectivesfor scheduling flexible job-shop problem theminimization ofthe makespan for the production part and the minimizationof the system unavailability for the maintenance part Cuiet al [11] proposed an integrated production schedulingand maintenance policy in a single machine the objectivefunction is to minimize the weighted sum of two subgoalsone is the expectation of total deviation between the jobsrsquoplanned start times and actual start times and the other is theexpectation of the maximum tardiness Lu et al [12] appliedthe model to a single machine with failure uncertainty Allof the articles mentioned above are aimed at minimizing themaximum completion time As research continues it is notenough to consider the time factor only It needs to be quickin production scheduling but when maintenance activitiesare not sooner that is better The influence of maintenancecosts should also be considered

In the question about integrated optimization of makes-pan and maintenance costs the following scholars have doneuseful research Zammori et al [13] focused on the singlemachine scheduling problem with planned maintenancethe objective is to minimize total earliness and tardinesspenalties Dhouib et al [14] proposed a joint optimal pro-duction preventive maintenance policy for imperfect processmanufacturing cell the main objective is to minimize theoverall cost comprised of setup maintenance inventoryholding and shortage costs as well as the cost incurredby producing nonconforming items Rivera-Gomez et al[15] extended this model to a production system subjectto quality and reliability deteriorations the objective is tominimize the total cost comprising the inventory backlogrepair and overhaul cost Fitouhi and Nourelfath [16 17]dealt with the problem of integrating noncyclical preventivemaintenance and tactical production planning for a singlemachine andmultistate systemsThe objective is to minimizethe sum of costs associated with maintenance productionand backorder Nourelfath and Chatelet [18] extended thismodel to a parallel system they dealt with the problem ofintegrating preventive maintenance and tactical productionplanning for a parallel system Although the references abovetake the cost factor into consideration they consider costfactor only all these references neglect the time factor

Pandey et al [19] proposed a joint optimization policyfor maintenance scheduling quality control and productionscheduling the objective is to minimize expected cost perunit time Gan et al [20] studied joint optimization ofmaintenance buffer and spare parts for a production systemthe objective is to achieve the minimization of the long-termexpected cost rateThe studies above simultaneously considerthe two factors completion time andmaintenance costsTheyadopt a cost per unit time method to deal with the objective

function Moghaddam [21] presented a new multiobjectivemodel to determine an optimal preventive maintenanceand replacement schedule Total operational costs overallreliability and the system availability are incorporated as theobjective functions This literature considers the cost factorthe reliability and availability factor is also in considerationresearch on reliability is similar to Berrichi et alrsquos andMoradiet alrsquos research

Cui et al [22 23] studied a multiobjective optimizationproblem aimed at minimizing the makespan and main-tenance cost simultaneously on a single machine systemA modified genetic algorithm was proposed to optimizethe Pareto front Yulan et al [24] studied a multiobjectiveoptimization problem jointly determining preventive main-tenance planning and production scheduling for a singlemachine Although some researchers have taken operationtime and maintenance cost into consideration they mainlyfocused on the single machine area few researchers focuson multiobjective optimization study in flexible job-shopscheduling At present most of the research that referredto flexible job-shop scheduling is concentrated on the fieldof scheduling In the aspect of maintenance Xiong et al[25] conducted a study about flexible job-shop schedulingstability considering machine fault Li and Pan [26] studieda flexible job-shop scheduling problem considering mainte-nance activities it was solved by a hybrid chemical-reactionoptimization (HCRO) algorithmwith the goal of minimizingthe makespan The research about maintenance activitiesin the flexible job-shop scheduling area above still aims tominimize the completion timeThe purpose of this paper is tostudy integrated optimization problem of maintenance costsand total completion time under flexible shop schedulingthe corresponding joint optimization model is establisheda double-coded genetic algorithm is designed to solve thisproblem Finally a simulation experiment is conducted by aset of data

The main features of the papers mentioned above aresummarized in Table 1

2 Problem Formulation

In this paper the flexible job-shop scheduling problem isstudied the problem can be described as follows Thereare 119868 independent jobs that are indexed by 119894 There are 119869machines indexed by 119895 Each job 119894 consists of a sequence of119870operationsThe execution of operation requires one machineout of a set of givenmachines represented as 119869The processingtime of an operation onmachine 119895 is predefined and given by119905119894119895119896Theobjective is to find a schedule that has the lowest value

of makespanThe model is based on the assumptions as follows jobs

are independent from each other and machines are indepen-dent from each other production preparation time on eachmachine and moving time between jobs in the machine arenegligible a job could be processed on one machine at thesame time the processing of job cannot be interrupted onceit is started all jobs share the same processing priority

Machine failure rate rises with the increasing ofmachinersquosservice ageThe failure rate of themachine is assumed to obey

Mathematical Problems in Engineering 3

Table 1 Literature review

Data ofpublication Type of production environment Objective function Heuristic approach

2005 Single machine Minimum makespan Genetic algorithms2013 Flow shop Minimum makespan Genetic algorithms2014 Flow shop Minimum makespan Genetic algorithms

2009 Distributed production scheduling Minimum makespan A modified geneticalgorithm

2010 Flexible job-shop Minimum makespan A filtered beam searchbased heuristic algorithm

2014 Single machine Minimize total flow time Polynomial optimalalgorithm

2012 Single machine Minimizing job tardiness

2010 Parallel machine Minimize unavailability and makespan Ant Colony Optimizationparadigm algorithm

2011 Flexible job-shop Minimize unavailability and makespan Ant Colony Optimizationparadigm algorithm

2014 Single machine Minimize solution and qualityrobustness

Three-phase heuristicalgorithm

2015 Single machine System robustness and stability Genetic algorithms

2014 Single machine Minimize total earliness and tardinesspenalties Harmony search algorithm

2012 Imperfect process manufacturing cell Minimizes overall cost Extensive sensitivityanalysis

2013 Unreliable manufacturing system Minimizes overall cost Extensive sensitivityanalysis

2014 Multistate systems Minimize maintenance costs The mixed integer solver2012 A single machine Minimize maintenance costs The mixed integer solver2012 Parallel system Minimize maintenance costs The mixed integer solver2011 Single machine Minimize expected cost per unit time2015 Two serial machines Minimize long-term expected cost rate Genetic algorithm

2013 Multiworkstation manufacturing system Pareto-optimal maintenance andreplacement schedules Pareto-optimal solutions

2014 Single machine Minimize the makespan andmaintenance cost Genetic algorithm


2008 Single machine Minimize maintenance costmakespan and so forth Genetic algorithm

2013 Flexible job-shop Minimum makespan A multiobjectiveevolutionary algorithm

2013 Flexible job-shop Minimum makespan Chemical-reactionoptimization

Weibull distributionWhenmachine failure happens we takeminimal maintenancemeasures to deal with failure Minimalmaintenance will not change machinersquos service age but onlymake themachine condition return to the level before failureminimal maintenance will not interrupt production Themachine needs to decide whether to perform preventivemaintenance after each processing operation preventivemaintenance is perfect maintenance it means the machinerecovers as new after each maintenance operation

We assume that 119862max denotes makespan of all jobs119862lowast

max denotes the reference value of makespan 119888 denotes

maintenance cost per unit time in the whole process 119888lowastdenotes the reference value of maintenance cost per unittime 120572

1denotes the production plan weight factor 120572

2

denotes the maintenance plan weight factor 119905119901

denotespreventive maintenance time 119905

119903denotes minimal repair

time 119888119901denotes preventive maintenance cost 119888

119903denotes

minimal repair cost 120582(119905) denotes failure rate function andfollows Weibull distribution 119902

0denotes the initial service

age of each machine and 120573 120578 denote shape parame-ter and scale parameter in Weibull distribution respec-tively


Some intermediate variables involved in this model areshown as follows119862

119894119895119896denotes the completion time of the 119896th

operation of job 119894 on machine 119895 119905119894119895119896

denotes the processingtime of the 119896th operation of job 119894 on machine 119895119872

ℎ119895denotes

the time required to repair the machine in the ℎth job ofmachine 119895 and 119909

ℎ119895denotes preventive maintenance decision

variables if PM (preventive maintenance) on machine 119895 isperformed after the ℎth job 119909

ℎ119895= 1 otherwise 119909

ℎ119895= 0

119873119903(ℎ)(119895)

denotes the expected number of failures in the ℎth jobof machine 119895 119888

119905(119895)denotes total maintenance cost of machine

119895 119905ℎ119895denotes the processing time of the ℎth job on machine

119895119873119903(119895)

denotes the expected number of failures onmachine 119895during the whole planning period119873

119901(119895)denotes the number

of preventive maintenance operations on machine 119895 duringthe whole planning period and 119902

ℎ119905denotes the service age

before the ℎth job on machine 119895The notation used in this paper is summarized in Nota-

tionsWe establish the model as follows

min119885 = 1205721

119862max119862lowastmax

+ 1205722

119888

119888lowast (1)

Equation (1) is the target function The reason whywe built it in this form is explained as follows In orderto consider production and maintenance department goalssynthetically we propose weighted sum method to deal withmaintenance cost and completion time But the units usedto measure the maintenance cost and completion time aredifferent we adopt a method mentioned in Huang et al [27]to eliminate the differences between two units

119862max = max 119862119894119895119896 (2)

119862119894119895119896= 119878119894119895119896+ 119905119894119895119896+119872ℎ119895 (3)

119878119894119895119896= max 119862

119894119896minus1 119862ℎminus1119895

+ 119909ℎminus1119895119905119901 (4)

119872ℎ119895= 119905119903119873119903(ℎ)(119895)

(5)

Equation (2) denotes the makespan of all operations ofjob Equation (3) denotes the completion time of the 119896thoperation of job 119894 onmachine 119895 Equation (4) denotes the starttime of the 119896th operation of job 119894 on machine 119895 Equation (5)denotes repair time of the ℎth job on machine 119895 Consider

119888 =

119869

sum

119895=1

119867

sum

ℎ=1

119888119905(119895)

119905ℎ119895

(6)

119888119905(119895)= 119888119903119873119903(119895)+ 119888119901119873119901(119895) (7)

119873119901(119895)

=

119867

sum

ℎ=1

119909ℎ119895 (8)

119873119903(119895)=

119867

sum

ℎ=1

119873119903(ℎ)(119895)

(9)

119873119903(ℎ)(119895)

= int

119902ℎ119895+119905ℎ119895

119902ℎ119895

120582 (119905) 119889119905 (10)

119902ℎ119895= 1199020119909ℎminus1119895

+ (1 minus 119909ℎminus1119895) (119902ℎminus1119895

+ 119905ℎ119895) (11)

119909ℎ119895isin 0 1 (12)

Equation (6) denotes maintenance cost per unit time inthe whole process Equation (7) denotes total maintenancecost of machine 119895 in the whole process Equation (8) denotestotal number of preventative maintenance operations onmachine 119895 during the whole planning period Equation (9)denotes the total number of minimal repairs on machine 119895during the whole planning period Equation (10) denotes thenumber of expected failures in the ℎth job on machine 119895Equation (11) denotes the service age before the ℎth job onmachine 119895 Inclusion (12) denotes preventive maintenancedecision variables which could only be set to 0 or 1

The objective function is aimed at balancing the objectiveof the production and maintenance department and min-imizing the deviation between the actual scheme and theplanned scheme of each department We set the maximumcompletion time reference value 119862lowastmax equal to the totalcompletion time when only production scheduling is con-sidered namely the maximum completion time in a flexiblejob-shop scheduling problem regardless of the equipmentmaintenance We set the maintenance cost per unit timereference value 119888lowast equal to themost economical maintenancecost per unit timeThe calculating process is shown as followsWe assume that during a preventive maintenance period 119879maintenance cost per unit time 119888 = (119888

119903int119879

0120582(119905)119889119905 + 119888

119901)119879 we

set the derivative of the equation equal to 0 we can get theoptimal preventive maintenance cycle 119879lowast = 120573radic119888

119901120578120573119888119903(120573 minus 1)

Then by plugging 119879lowast into these formulas we can get the mosteconomical maintenance cost per unit time 119888lowast = 119888

119901120573(120573 minus

1)119879lowast

3 Genetic Algorithm

Flexible job-shop scheduling is a complex problem multiplejobs are processed onmultiple machines each job is assignedto a unique processing route Each chromosome of geneticalgorithm denotes a potential solution to the problem eachgene on a chromosome can express the information of prob-lem So we can transform the difficult problem into geneticcode and represent job and machine with genetic coderespectively production and maintenance should be takeninto consideration in the fitness function This paper adoptsa double-coded genetic algorithm to solve this problem thealgorithm process is represented as follows

31 Chromosome Representation Each chromosome is divid-ed into two parts dispatching part and machine schedulingpart The first half of chromosome is the dispatching part itdenotes the order of all jobs processed on the machine thesecond part (half) of chromosome is the machine schedul-ing part it denotes that the machine corresponds to eachworking operation both parts are represented with real-coded method When the total number of jobs is119898 the totalnumber of working operations is 119899 chromosome individualsare expressed as an integer string that is 2 lowast 119898119899 in length


For example [213132 121212] denotes 3 jobs that need tobe processed 2 times on their machines the first half denotesworking operation they are in order job 2 job 1 job 3 job1 job 3 and job 2 the second part denotes the machine theyare in order the 1st machine in the set of machines the 2ndmachine in the set of machines the 1st machine in the setof machines the 2nd machine in the set of machines the 1stmachine in the set of machines the 2nd machine in the set ofmachines The machine scheduling part adopts 0-1 encodingin subroutine where 0 means maintenance and 1 means nomaintenance

32 Calculate the Objective Function Fitness value is thebasis of chromosome evaluation fitness should reflect therequirements of the objective function so we establish thefitness function of this algorithm according to the objectivefunction The objective function in this algorithm is theweighted sumof the deviation between the actual scheme andthe planned scheme of each department the lower this valueis the better the chromosome is The target value of the 119899thchromosome calculation formula is shown as follows

fitness (119899) = 1205721119862max119862lowastmax

+ 1205722

119888

119888lowast (13)

33 Selection Operation Selection operation is to selectexcellent individuals from the old population at a certainprobability to form new population the purpose is to breedthe next generation chromosome The probability of theindividual being selected is related to fitness value the oneswith better fitness have more chances to be selected Lowertarget value is better in this paper so ranking functionis adopted to sort the target value from low to high andreturn the individual fitness value Lower objective functionvalue equates to higher fitness value Then we can select thebetter individuals according to the fitness values through theroulette method Let 119865

119894be the fitness value of individual 119894 and

119873 the number of populationThen the probability of selectingthe individual 119894 is

119875119894=

119865119894

sum119873

119895=1119865119895

(14)

34 Crossover Operation Crossover operation is to randomlyselect two chromosomes from the population and take outthe first half of each chromosome and then randomly selectcrossover position in it to perform crossover operationTheremay be some jobs extra or missing after crossover operationwe need to add or delete some working operation to ensurethe number of processing operations we will also adjustthe corresponding machine such as the following examplecrossover operation is only for the first half cross point is 4

parent 1- [112323213 112121222] offspring 1- [221323213 112121222]Cross997888997888997888997888rarr

parent 2- [221331213 122122211] offspring 2- [112331213 122122211]

(15)

After crossing redundant operation needs to be removedfrom certain job (such as job 2 in offspring 1) missingoperation needs to be added (such as job 3 in offspring 1)that is to make up the missing operation with the redundantoperation Then we will adjust the machine located in thelatter half of the chromosome according to the machinebefore the crossover operation

offspring 1- [221323213 112121222]adjust997888997888997888997888rarr [221323213 122121222]

(16)

35 Variation In variation operation we will select a chro-mosome from the population first and then randomly selecttwo positions from the top 119898 lowast 119899 of the chromosomethe values of the two positions are then inverted theircorrespondingmachine in the latter119898lowast119899 of the chromosomeis also inverted Variation operation guarantees the diversityof population As shown in the following example variationposition is 3 and 6

individual - [112323213 112121222]

997888rarr individual - [113322213 111122222] (17)

4 Numerical Experiments

In order to verify the validity of the model and algorithmwe designed an example in this paper The machine for eachworking operation is given in Table 2 the processing time foreach working operation is presented in Table 3

Our algorithm was implemented in MATLAB 2010b theparameter values are given as follows the failure rate functionof machine 119895 follows 120573 = 2 120578 = 100 in Weibull distributionPreventive maintenance time is 119905

119901= 10 minimal repair

time is 119905119903= 15 preventive maintenance cost is 119888

119901= 500

minimal repair cost is 119888119903= 300 the initial service age of

each machine is 1199020= 0 and weight coefficients are 120572

1= 05

1205722= 05 We have worked out reference values 119862lowastmax = 398

119888lowast= 775 Parameters of genetic algorithm are given as

follows population size is 100 maximum iterations are 1000crossover rate is 08 and variation rate is 05 The programwas run 15 times independently on an Intel Core2 22GHzPC with 2GB memory The optimal results were chosen outThe best objective function value in 15 experiments is 10221the corresponding preventive maintenance cost per unit timeis 60246 and the makespan is 5042130


Table 2 Optional machine of process

Job Operation 1 Operation 2 Operation 3 Operation 4 Operation 5 Operation 6Job 1 5 6 4 [2 9] [3 7] 5Job 2 4 [2 9] 8 [6 7] 5 [1 10]Job 3 3 [6 8] 7 [2 1] [4 10] 5Job 4 5 2 [4 7] 10 [2 5] [3 6]Job 5 [4 5] 5 [9 10] 6 2 [3 8]Job 6 [2 6] 4 [6 9] 7 8 [3 9]

Table 3 Work time of process


The result above shows that the algorithm proposed inthis paper can solve the problem In order to verify thevalidity of the joint optimization model we contrast the jointdecision results with the independent decision results Inthe traditional workshop production process the productiondepartment and maintenance department make their planaccording to their own target respectively then they submittheir plan to the decision-making department It is hardto avoid conflict between the two departments becauseproduction department aims to minimize the completiontime whilemaintenance department aims tominimizemain-tenance cost Faced with this situation the decision-makingdepartment tends to make plan according to maintenancedepartment fixed period preventive maintenance is adoptedmaintenance cycle 119879 is the preventive maintenance cycleunder the most economical maintenance costs unit time 119879lowastIn this case we have worked out that 119879lowast = 129 We adopt themethod used in the joint decision scheme asmentioned in theprevious paragraph the same parameter is set the programis run for 15 times independently on the same computer thenwe get the results The best objective function value in 15experiments is 10718 the corresponding maintenance costper unit time is 65717 and the makespan is 5267905 Thecomparison result of joint decision-making and independentdecision-making is shown in Table 4

The searching process of genetic algorithm is random sothe result fluctuates in a small range As it is shown in Table 4when we contrast the overall levels between joint decision-making and independent decision-making the target valueof joint decision-making distributed in 102sim107 the targetvalue of independent decision-making distributed in 107sim12 the target value of joint decision-making is better thanindependent decision-making By comparison with the bestsolution of two schemes in the 15 times simulation exper-iments joint decision-making is better than independentdecision-making in all three indicators target value 119862max

and 119888 We can conclude that joint decision-making is betterthan independent decision-making

In the previous calculation we assume that the produc-tion and maintenance weight coefficients are 120572

1= 05

1205722= 05 It shows that we give the same emphasis to

production department andmaintenance department But inthe practical industrial production two weight coefficientsare not necessarily the same Next we will study the influenceof the weight coefficient on the objective function value Wewill give 120572

1 12057225 arrays of values then a comparative study of

the different 119862max and 119888 value is conducted when 1205721 1205722are

given to the 5 values The result is shown in Figures 1 and 2We can conclude from Figure 1 that different weight coef-

ficient will affect the objective function value In the optimaljoint decision scheme the higher the productionweight valuecoefficient is the lower the maximum completion time willbe and the higher the maintenance cost per unit time will beWe can conclude from Figure 2 that joint decision-makingis superior to independent decision-making no matter whenthe weight coefficient is given to any value the superiordegree did not change with the change in weight coefficientIt shows that the joint decision-making scheme we proposedhas wide applicability

5 Conclusions

We study joint optimization of production scheduling andmaintenance planning in the flexible job-shop problem inthis paper the model considering two factors is built Theyare completion time and maintenance costs The objectivefunction is aimed at finding the best scheme which will min-imize the deviation between production and maintenancedepartment double-coded genetic algorithm is adopted tosolve this problem the result is compared with the indepen-dent decision scheme under fixed preventive maintenancecycles It turned out from the simulation experiment that


Table 4 The comparison of joint decision-making and independent decision-making result

Number Joint decision-making Independent decision-makingTarget value 119888 119862max Target value 119888 119862max

1 10461 62652 5109310 11390 68067 55710502 10603 63023 5203310 10738 62649 53301653 10450 61176 5176450 11464 66676 52009204 10566 63133 5168490 10980 63327 54882355 10255 61017 5029205 11123 60239 57605506 10306 63731 4930624 11395 61229 56176807 10594 64205 5135695 11048 65184 54467808 10595 61168 5292235 11219 59664 58664409 10742 62172 5352540 11211 70432 530681510 10615 62732 5227675 11246 68517 543291511 10598 61750 5264900 10822 65737 523855012 10995 62278 5553895 11912 74115 567619513 10458 58153 5338415 10718 65717 515699014 10506 62337 5161150 10941 67001 526790515 10221 60246 5042130 11478 70268 5521755

c

Cmax

1205721= 09

1205722= 01

1205721= 07

1205722= 03

1205721= 05

1205722= 05

1205721= 03

1205722= 07

1205721= 01

1205722= 09

5

55

6

65

7

75

8

480

530

580

630

680

730

Figure 1 The influence of different weight coefficient on the resultof joint decision-making

we can get better objective function value from joint deci-sion of production scheduling and maintenance activitiesMoreover the corresponding completion time is shorterand maintenance cost per unit time is lower Thereforejoint decision of production and maintenance activities canshorten the completion time of the enterprise and reducerunning cost it is better than the independent decisionFinally we study the performance of the objective functionvalues of the independent decision and joint decision underdifferent weight coefficient we can conclude that differentemphasis on production and maintenance has an impacton their own results but joint decision-making schemeis superior to independent decision-making scheme andthe superior degree did not change with the change inweight coefficient The validity and wide applicability of themodel and algorithm are proved through the simulationexperiments

In the future we will try to adopt a new algorithm tosolve this problem In this paper we adopted small sample

1205721= 09

1205722= 01

1205721= 07

1205722= 03

1205721= 05

1205722= 05

1205721= 03

1205722= 07

1205721= 01

1205722= 09

Joint decision-makingIndependent decision-making

08

085

09

095

1

105

11

115

12

125

13

Figure 2 The influence of different weight coefficient on the resultof two kinds of decision-making

data to conduct simulation experiments we will add testswith large databases having a public repository of standarddata for different types of tests We will collect data fromthe production line of an enterprise The results comingfrom our research will be more convincing and more ableto solve practical problems In this mathematical modelcompletion time and cost are considered but it is not enoughIn the future we will consider more factors in the objectivefunction such as backlog of orders and the delivery delaypenalty We will research joint optimization of more goals

Notations

119894 Index of jobs 119894 = 1 2 119868119895 Index of machines 119895 = 1 2 119869119896 Index of operations 119896 = 1 2 119870ℎ Index of jobs on one machine

ℎ = 1 2 119867

119862max Makespan of all jobs119862lowast

max Reference value of makespan


119888 Maintenance cost per unit time119888lowast Reference value of maintenance cost per

unit time1205721 Production plan weight factor

1205722 Maintenance plan weight factor

119905119901 Preventive maintenance time

119905119903 Minimal repair time119888119901 Preventive maintenance cost

119888119903 Minimal repair cost120582(119905) Failure rate function1199020 Initial service age of each machine

120573 120578 Shape parameter and scale parameter inWeibull distribution

119862119894119895119896 Completion time of the 119896th operation of

job 119894 on machine 119895119905119894119895119896 Processing time of the 119896th operation of job

119894 on machine 119895119872ℎ119895 Time required to repair the machine in the

ℎth job of machine 119895119909ℎ119895 Preventive maintenance decision variables

119873119903(ℎ)(119895)

Expected number of failures in the ℎth jobof machine 119895

119888119905(119895)

Total maintenance cost of machine 119895119905ℎ119895 Processing time of the ℎth job on machine

119895

119873119903(119895)

Expected number of failures on machine 119895during the whole planning period

119873119901(119895)

Number of preventive maintenanceoperations on machine 119895 during the wholeplanning period

119902ℎ119905 Service age before the ℎth job on machine

119895

Conflict of Interests

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

Acknowledgment

This work is supported by the National Natural ScienceFoundation of China (no 71472125)

References

[1] H M Ma and J F Ye ldquoJoint optimization of productionscheduling and maintenance planning in the flexible job-shopproblemrdquoMachinery Design ampManufacture no 7 pp 248ndash2502015

[2] N Sortrakul H L Nachtmann and C R Cassady ldquoGeneticalgorithms for integrated preventive maintenance planning andproduction scheduling for a single machinerdquo Computers inIndustry vol 56 no 2 pp 161ndash168 2005

[3] JMDa L YWang S L Xu and S X Zhao ldquoJoint optimizationof production lot planning and preventive maintenancerdquo Engi-neering Science and Technology vol 10 pp 2700ndash2709 2013

[4] S-Y Zhang Z-Q Lu and W-W Cui ldquoFlow shop schedulingalgorithm with periodical maintenancerdquo Computer IntegratedManufacturing Systems vol 20 no 6 pp 1379ndash1387 2014

[5] S H Chung F T S Chan and H K Chan ldquoAmodified geneticalgorithm approach for scheduling of perfect maintenance indistributed production schedulingrdquo Engineering Applications ofArtificial Intelligence vol 22 no 7 pp 1005ndash1014 2009

[6] S Wang and J Yu ldquoAn effective heuristic for flexible job-shopscheduling problem with maintenance activitiesrdquo Computers ampIndustrial Engineering vol 59 no 3 pp 436ndash447 2010

[7] B Mor and G Mosheiov ldquoBatch scheduling with a rate-modifying maintenance activity to minimize total flowtimerdquoInternational Journal of Production Economics vol 153 pp 238ndash242 2014

[8] E Pan W Liao and L Xi ldquoA joint model of productionscheduling and predictive maintenance for minimizing jobtardinessrdquo International Journal of Advanced ManufacturingTechnology vol 60 no 9ndash12 pp 1049ndash1061 2012

[9] A Berrichi F Yalaoui L Amodeo and M Mezghiche ldquoBi-Objective Ant Colony Optimization approach to optimize pro-duction andmaintenance schedulingrdquoComputers ampOperationsResearch vol 37 no 9 pp 1584ndash1596 2010

[10] E Moradi S M T Fatemi Ghomi and M Zandieh ldquoBi-objective optimization research on integrated fixed time inter-val preventive maintenance and production for schedulingflexible job-shop problemrdquo Expert Systems with Applicationsvol 38 no 6 pp 7169ndash7178 2011

[11] W-W Cui Z Q Lu and E S Pan ldquoIntegrated productionscheduling and maintenance policy for robustness in a singlemachinerdquo Computers amp Operations Research vol 47 no 7 pp81ndash91 2014

[12] Z Q Lu W W Cui and X Han ldquoIntegrated production andpreventive maintenance scheduling for a single machine withfailure uncertaintyrdquo Computers amp Industrial Engineering vol80 pp 236ndash244 2015

[13] F Zammori M Braglia and D Castellano ldquoHarmony searchalgorithm for single-machine scheduling problemwith plannedmaintenancerdquoComputers and Industrial Engineering vol 76 pp333ndash346 2014

[14] K Dhouib A Gharbi andM N Ben Aziza ldquoJoint optimal pro-duction controlpreventive maintenance policy for imperfectprocess manufacturing cellrdquo International Journal of ProductionEconomics vol 137 no 1 pp 126ndash136 2012

[15] H Rivera-Gomez A Gharbi and J P Kenne ldquoJoint control ofproduction overhaul and preventive maintenance for a pro-duction system subject to quality and reliability deteriorationsrdquoInternational Journal of Advanced Manufacturing Technologyvol 69 no 9ndash12 pp 2111ndash2130 2013

[16] M-C Fitouhi and M Nourelfath ldquoIntegrating noncyclicalpreventive maintenance scheduling and production planningfor multi-state systemsrdquo Reliability Engineering amp System Safetyvol 121 pp 175ndash186 2014

[17] M-C Fitouhi and M Nourelfath ldquoIntegrating noncyclical pre-ventive maintenance scheduling and production planning for asingle machinerdquo International Journal of Production Economicsvol 136 no 2 pp 344ndash351 2012

[18] M Nourelfath and E Chatelet ldquoIntegrating production inven-tory and maintenance planning for a parallel system withdependent componentsrdquo Reliability Engineering amp SystemSafety vol 101 pp 59ndash66 2012

[19] D PandeyM S Kulkarni and P Vrat ldquoAmethodology for jointoptimization for maintenance planning process quality and


production schedulingrdquo Computers amp Industrial Engineeringvol 61 no 4 pp 1098ndash1106 2011

[20] S Gan Z Zhang Y Zhou and J Shi ldquoJoint optimization ofmaintenance buffer and spare parts for a production systemrdquoApplied Mathematical Modelling vol 39 no 19 pp 6032ndash60422015

[21] K S Moghaddam ldquoMulti-objective preventive maintenanceand replacement scheduling in a manufacturing system usinggoal programmingrdquo International Journal of Production Eco-nomics vol 146 no 2 pp 704ndash716 2013

[22] W-W Cui Z-Q Lu and E-S Pan ldquoProduction scheduling andpreventive maintenance integration based on multi-objectiveoptimizationrdquoComputer IntegratedManufacturing Systems vol20 no 6 pp 1398ndash1404 2014

[23] WW Cui and Z Q Lu ldquoIntegrating production scheduling andpreventive maintenance planning for a single machinerdquo Journalof Shanghai Jiao Tong University vol 46 no 12 pp 2009ndash20132012

[24] J Yulan J Zuhua and H Wenrui ldquoMulti-objective integratedoptimization research on preventive maintenance planningand production scheduling for a single machinerdquo InternationalJournal of AdvancedManufacturing Technology vol 39 no 9-10pp 954ndash964 2008

[25] J Xiong L-N Xing and Y-W Chen ldquoRobust schedulingfor multi-objective flexible job-shop problems with randommachine breakdownsrdquo International Journal of Production Eco-nomics vol 141 no 1 pp 112ndash126 2013

[26] J-Q Li and Q-K Pan ldquoChemical-reaction optimization forsolving fuzzy job-shop scheduling problem with flexible main-tenance activitiesrdquo International Journal of Production Eco-nomics vol 145 no 1 pp 4ndash17 2013

[27] F M Huang Z Q Lu and W W Cui ldquoIntegrated preventivemaintenance planning and production scheduling on parallelmachinerdquo Industrial Engineering and Management vol 4 pp49ndash55 2013

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


Pan et al [8] built a joint model for minimizing job tardinessAll the references above established the single objectiveoptimization model but some other researchers establishedthe multiobjective optimizing model Berrichi et al [9]presented an algorithm based on Ant Colony Optimizationparadigm to solve the joint production and maintenancescheduling problem the goal is to simultaneously minimizethe system unavailability andmakespan After thatMoradi etal [10] attempted to simultaneously optimize two objectivesfor scheduling flexible job-shop problem theminimization ofthe makespan for the production part and the minimizationof the system unavailability for the maintenance part Cuiet al [11] proposed an integrated production schedulingand maintenance policy in a single machine the objectivefunction is to minimize the weighted sum of two subgoalsone is the expectation of total deviation between the jobsrsquoplanned start times and actual start times and the other is theexpectation of the maximum tardiness Lu et al [12] appliedthe model to a single machine with failure uncertainty Allof the articles mentioned above are aimed at minimizing themaximum completion time As research continues it is notenough to consider the time factor only It needs to be quickin production scheduling but when maintenance activitiesare not sooner that is better The influence of maintenancecosts should also be considered

In the question about integrated optimization of makes-pan and maintenance costs the following scholars have doneuseful research Zammori et al [13] focused on the singlemachine scheduling problem with planned maintenancethe objective is to minimize total earliness and tardinesspenalties Dhouib et al [14] proposed a joint optimal pro-duction preventive maintenance policy for imperfect processmanufacturing cell the main objective is to minimize theoverall cost comprised of setup maintenance inventoryholding and shortage costs as well as the cost incurredby producing nonconforming items Rivera-Gomez et al[15] extended this model to a production system subjectto quality and reliability deteriorations the objective is tominimize the total cost comprising the inventory backlogrepair and overhaul cost Fitouhi and Nourelfath [16 17]dealt with the problem of integrating noncyclical preventivemaintenance and tactical production planning for a singlemachine andmultistate systemsThe objective is to minimizethe sum of costs associated with maintenance productionand backorder Nourelfath and Chatelet [18] extended thismodel to a parallel system they dealt with the problem ofintegrating preventive maintenance and tactical productionplanning for a parallel system Although the references abovetake the cost factor into consideration they consider costfactor only all these references neglect the time factor

Pandey et al [19] proposed a joint optimization policyfor maintenance scheduling quality control and productionscheduling the objective is to minimize expected cost perunit time Gan et al [20] studied joint optimization ofmaintenance buffer and spare parts for a production systemthe objective is to achieve the minimization of the long-termexpected cost rateThe studies above simultaneously considerthe two factors completion time andmaintenance costsTheyadopt a cost per unit time method to deal with the objective

function Moghaddam [21] presented a new multiobjectivemodel to determine an optimal preventive maintenanceand replacement schedule Total operational costs overallreliability and the system availability are incorporated as theobjective functions This literature considers the cost factorthe reliability and availability factor is also in considerationresearch on reliability is similar to Berrichi et alrsquos andMoradiet alrsquos research

Cui et al [22 23] studied a multiobjective optimizationproblem aimed at minimizing the makespan and main-tenance cost simultaneously on a single machine systemA modified genetic algorithm was proposed to optimizethe Pareto front Yulan et al [24] studied a multiobjectiveoptimization problem jointly determining preventive main-tenance planning and production scheduling for a singlemachine Although some researchers have taken operationtime and maintenance cost into consideration they mainlyfocused on the single machine area few researchers focuson multiobjective optimization study in flexible job-shopscheduling At present most of the research that referredto flexible job-shop scheduling is concentrated on the fieldof scheduling In the aspect of maintenance Xiong et al[25] conducted a study about flexible job-shop schedulingstability considering machine fault Li and Pan [26] studieda flexible job-shop scheduling problem considering mainte-nance activities it was solved by a hybrid chemical-reactionoptimization (HCRO) algorithmwith the goal of minimizingthe makespan The research about maintenance activitiesin the flexible job-shop scheduling area above still aims tominimize the completion timeThe purpose of this paper is tostudy integrated optimization problem of maintenance costsand total completion time under flexible shop schedulingthe corresponding joint optimization model is establisheda double-coded genetic algorithm is designed to solve thisproblem Finally a simulation experiment is conducted by aset of data

The main features of the papers mentioned above aresummarized in Table 1

2 Problem Formulation

In this paper the flexible job-shop scheduling problem isstudied the problem can be described as follows Thereare 119868 independent jobs that are indexed by 119894 There are 119869machines indexed by 119895 Each job 119894 consists of a sequence of119870operationsThe execution of operation requires one machineout of a set of givenmachines represented as 119869The processingtime of an operation onmachine 119895 is predefined and given by119905119894119895119896Theobjective is to find a schedule that has the lowest value

of makespanThe model is based on the assumptions as follows jobs

are independent from each other and machines are indepen-dent from each other production preparation time on eachmachine and moving time between jobs in the machine arenegligible a job could be processed on one machine at thesame time the processing of job cannot be interrupted onceit is started all jobs share the same processing priority

Machine failure rate rises with the increasing ofmachinersquosservice ageThe failure rate of themachine is assumed to obey





























2





119903denotes









ℎ119895denotes




ℎ119895= 1 otherwise 119909

ℎ119895= 0

119873119903(ℎ)(119895)




119895119873119903(119895)







min119885 = 1205721


+ 1205722

119888

119888lowast (1)


119862max = max 119862119894119895119896 (2)

119862119894119895119896= 119878119894119895119896+ 119905119894119895119896+119872ℎ119895 (3)

119878119894119895119896= max 119862

119894119896minus1 119862ℎminus1119895

+ 119909ℎminus1119895119905119901 (4)

119872ℎ119895= 119905119903119873119903(ℎ)(119895)

(5)


119888 =

119869

sum

119895=1

119867

sum

ℎ=1

119888119905(119895)

119905ℎ119895

(6)

119888119905(119895)= 119888119903119873119903(119895)+ 119888119901119873119901(119895) (7)

119873119901(119895)

=

119867

sum

ℎ=1

119909ℎ119895 (8)

119873119903(119895)=

119867

sum

ℎ=1

119873119903(ℎ)(119895)

(9)

119873119903(ℎ)(119895)

= int

119902ℎ119895+119905ℎ119895

119902ℎ119895

120582 (119905) 119889119905 (10)

119902ℎ119895= 1199020119909ℎminus1119895


+ 119905ℎ119895) (11)

119909ℎ119895isin 0 1 (12)



119903int119879

0120582(119905)119889119905 + 119888

119901)119879 we


119901120578120573119888119903(120573 minus 1)


119901120573(120573 minus

1)119879lowast

3 Genetic Algorithm







+ 1205722

119888

119888lowast (13)




119875119894=

119865119894

sum119873

119895=1119865119895

(14)




(15)



(16)


individual - [112323213 112121222]

997888rarr individual - [113322213 111122222] (17)






119901= 500



1= 05













1= 05







5 Conclusions





1 10461 62652 5109310 11390 68067 55710502 10603 63023 5203310 10738 62649 53301653 10450 61176 5176450 11464 66676 52009204 10566 63133 5168490 10980 63327 54882355 10255 61017 5029205 11123 60239 57605506 10306 63731 4930624 11395 61229 56176807 10594 64205 5135695 11048 65184 54467808 10595 61168 5292235 11219 59664 58664409 10742 62172 5352540 11211 70432 530681510 10615 62732 5227675 11246 68517 543291511 10598 61750 5264900 10822 65737 523855012 10995 62278 5553895 11912 74115 567619513 10458 58153 5338415 10718 65717 515699014 10506 62337 5161150 10941 67001 526790515 10221 60246 5042130 11478 70268 5521755

c

Cmax

1205721= 09

1205722= 01

1205721= 07

1205722= 03

1205721= 05

1205722= 05

1205721= 03

1205722= 07

1205721= 01

1205722= 09

5

55

6

65

7

75

8

480

530

580

630

680

730




1205721= 09

1205722= 01

1205721= 07

1205722= 03

1205721= 05

1205722= 05

1205721= 03

1205722= 07

1205721= 01

1205722= 09


08

085

09

095

1

105

11

115

12

125

13



Notations


ℎ = 1 2 119867















119873119903(ℎ)(119895)


119888119905(119895)


119895

119873119903(119895)


119873119901(119895)



119895



Acknowledgment


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































2





119903denotes









ℎ119895denotes




ℎ119895= 1 otherwise 119909

ℎ119895= 0

119873119903(ℎ)(119895)




119895119873119903(119895)







min119885 = 1205721


+ 1205722

119888

119888lowast (1)


119862max = max 119862119894119895119896 (2)

119862119894119895119896= 119878119894119895119896+ 119905119894119895119896+119872ℎ119895 (3)

119878119894119895119896= max 119862

119894119896minus1 119862ℎminus1119895

+ 119909ℎminus1119895119905119901 (4)

119872ℎ119895= 119905119903119873119903(ℎ)(119895)

(5)


119888 =

119869

sum

119895=1

119867

sum

ℎ=1

119888119905(119895)

119905ℎ119895

(6)

119888119905(119895)= 119888119903119873119903(119895)+ 119888119901119873119901(119895) (7)

119873119901(119895)

=

119867

sum

ℎ=1

119909ℎ119895 (8)

119873119903(119895)=

119867

sum

ℎ=1

119873119903(ℎ)(119895)

(9)

119873119903(ℎ)(119895)

= int

119902ℎ119895+119905ℎ119895

119902ℎ119895

120582 (119905) 119889119905 (10)

119902ℎ119895= 1199020119909ℎminus1119895


+ 119905ℎ119895) (11)

119909ℎ119895isin 0 1 (12)



119903int119879

0120582(119905)119889119905 + 119888

119901)119879 we


119901120578120573119888119903(120573 minus 1)


119901120573(120573 minus

1)119879lowast

3 Genetic Algorithm







+ 1205722

119888

119888lowast (13)




119875119894=

119865119894

sum119873

119895=1119865119895

(14)




(15)



(16)


individual - [112323213 112121222]

997888rarr individual - [113322213 111122222] (17)






119901= 500



1= 05













1= 05







5 Conclusions





1 10461 62652 5109310 11390 68067 55710502 10603 63023 5203310 10738 62649 53301653 10450 61176 5176450 11464 66676 52009204 10566 63133 5168490 10980 63327 54882355 10255 61017 5029205 11123 60239 57605506 10306 63731 4930624 11395 61229 56176807 10594 64205 5135695 11048 65184 54467808 10595 61168 5292235 11219 59664 58664409 10742 62172 5352540 11211 70432 530681510 10615 62732 5227675 11246 68517 543291511 10598 61750 5264900 10822 65737 523855012 10995 62278 5553895 11912 74115 567619513 10458 58153 5338415 10718 65717 515699014 10506 62337 5161150 10941 67001 526790515 10221 60246 5042130 11478 70268 5521755

c

Cmax

1205721= 09

1205722= 01

1205721= 07

1205722= 03

1205721= 05

1205722= 05

1205721= 03

1205722= 07

1205721= 01

1205722= 09

5

55

6

65

7

75

8

480

530

580

630

680

730




1205721= 09

1205722= 01

1205721= 07

1205722= 03

1205721= 05

1205722= 05

1205721= 03

1205722= 07

1205721= 01

1205722= 09


08

085

09

095

1

105

11

115

12

125

13



Notations


ℎ = 1 2 119867















119873119903(ℎ)(119895)


119888119905(119895)


119895

119873119903(119895)


119873119901(119895)



119895



Acknowledgment


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














ℎ119895denotes




ℎ119895= 1 otherwise 119909

ℎ119895= 0

119873119903(ℎ)(119895)




119895119873119903(119895)







min119885 = 1205721


+ 1205722

119888

119888lowast (1)


119862max = max 119862119894119895119896 (2)

119862119894119895119896= 119878119894119895119896+ 119905119894119895119896+119872ℎ119895 (3)

119878119894119895119896= max 119862

119894119896minus1 119862ℎminus1119895

+ 119909ℎminus1119895119905119901 (4)

119872ℎ119895= 119905119903119873119903(ℎ)(119895)

(5)


119888 =

119869

sum

119895=1

119867

sum

ℎ=1

119888119905(119895)

119905ℎ119895

(6)

119888119905(119895)= 119888119903119873119903(119895)+ 119888119901119873119901(119895) (7)

119873119901(119895)

=

119867

sum

ℎ=1

119909ℎ119895 (8)

119873119903(119895)=

119867

sum

ℎ=1

119873119903(ℎ)(119895)

(9)

119873119903(ℎ)(119895)

= int

119902ℎ119895+119905ℎ119895

119902ℎ119895

120582 (119905) 119889119905 (10)

119902ℎ119895= 1199020119909ℎminus1119895


+ 119905ℎ119895) (11)

119909ℎ119895isin 0 1 (12)



119903int119879

0120582(119905)119889119905 + 119888

119901)119879 we


119901120578120573119888119903(120573 minus 1)


119901120573(120573 minus

1)119879lowast

3 Genetic Algorithm







+ 1205722

119888

119888lowast (13)




119875119894=

119865119894

sum119873

119895=1119865119895

(14)




(15)



(16)


individual - [112323213 112121222]

997888rarr individual - [113322213 111122222] (17)






119901= 500



1= 05













1= 05







5 Conclusions





1 10461 62652 5109310 11390 68067 55710502 10603 63023 5203310 10738 62649 53301653 10450 61176 5176450 11464 66676 52009204 10566 63133 5168490 10980 63327 54882355 10255 61017 5029205 11123 60239 57605506 10306 63731 4930624 11395 61229 56176807 10594 64205 5135695 11048 65184 54467808 10595 61168 5292235 11219 59664 58664409 10742 62172 5352540 11211 70432 530681510 10615 62732 5227675 11246 68517 543291511 10598 61750 5264900 10822 65737 523855012 10995 62278 5553895 11912 74115 567619513 10458 58153 5338415 10718 65717 515699014 10506 62337 5161150 10941 67001 526790515 10221 60246 5042130 11478 70268 5521755

c

Cmax

1205721= 09

1205722= 01

1205721= 07

1205722= 03

1205721= 05

1205722= 05

1205721= 03

1205722= 07

1205721= 01

1205722= 09

5

55

6

65

7

75

8

480

530

580

630

680

730




1205721= 09

1205722= 01

1205721= 07

1205722= 03

1205721= 05

1205722= 05

1205721= 03

1205722= 07

1205721= 01

1205722= 09


08

085

09

095

1

105

11

115

12

125

13



Notations


ℎ = 1 2 119867















119873119903(ℎ)(119895)


119888119905(119895)


119895

119873119903(119895)


119873119901(119895)



119895



Acknowledgment


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













+ 1205722

119888

119888lowast (13)




119875119894=

119865119894

sum119873

119895=1119865119895

(14)




(15)



(16)


individual - [112323213 112121222]

997888rarr individual - [113322213 111122222] (17)






119901= 500



1= 05













1= 05







5 Conclusions





1 10461 62652 5109310 11390 68067 55710502 10603 63023 5203310 10738 62649 53301653 10450 61176 5176450 11464 66676 52009204 10566 63133 5168490 10980 63327 54882355 10255 61017 5029205 11123 60239 57605506 10306 63731 4930624 11395 61229 56176807 10594 64205 5135695 11048 65184 54467808 10595 61168 5292235 11219 59664 58664409 10742 62172 5352540 11211 70432 530681510 10615 62732 5227675 11246 68517 543291511 10598 61750 5264900 10822 65737 523855012 10995 62278 5553895 11912 74115 567619513 10458 58153 5338415 10718 65717 515699014 10506 62337 5161150 10941 67001 526790515 10221 60246 5042130 11478 70268 5521755

c

Cmax

1205721= 09

1205722= 01

1205721= 07

1205722= 03

1205721= 05

1205722= 05

1205721= 03

1205722= 07

1205721= 01

1205722= 09

5

55

6

65

7

75

8

480

530

580

630

680

730




1205721= 09

1205722= 01

1205721= 07

1205722= 03

1205721= 05

1205722= 05

1205721= 03

1205722= 07

1205721= 01

1205722= 09


08

085

09

095

1

105

11

115

12

125

13



Notations


ℎ = 1 2 119867















119873119903(ℎ)(119895)


119888119905(119895)


119895

119873119903(119895)


119873119901(119895)



119895



Acknowledgment


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















1= 05







5 Conclusions





1 10461 62652 5109310 11390 68067 55710502 10603 63023 5203310 10738 62649 53301653 10450 61176 5176450 11464 66676 52009204 10566 63133 5168490 10980 63327 54882355 10255 61017 5029205 11123 60239 57605506 10306 63731 4930624 11395 61229 56176807 10594 64205 5135695 11048 65184 54467808 10595 61168 5292235 11219 59664 58664409 10742 62172 5352540 11211 70432 530681510 10615 62732 5227675 11246 68517 543291511 10598 61750 5264900 10822 65737 523855012 10995 62278 5553895 11912 74115 567619513 10458 58153 5338415 10718 65717 515699014 10506 62337 5161150 10941 67001 526790515 10221 60246 5042130 11478 70268 5521755

c

Cmax

1205721= 09

1205722= 01

1205721= 07

1205722= 03

1205721= 05

1205722= 05

1205721= 03

1205722= 07

1205721= 01

1205722= 09

5

55

6

65

7

75

8

480

530

580

630

680

730




1205721= 09

1205722= 01

1205721= 07

1205722= 03

1205721= 05

1205722= 05

1205721= 03

1205722= 07

1205721= 01

1205722= 09


08

085

09

095

1

105

11

115

12

125

13



Notations


ℎ = 1 2 119867















119873119903(ℎ)(119895)


119888119905(119895)


119895

119873119903(119895)


119873119901(119895)



119895



Acknowledgment


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1 10461 62652 5109310 11390 68067 55710502 10603 63023 5203310 10738 62649 53301653 10450 61176 5176450 11464 66676 52009204 10566 63133 5168490 10980 63327 54882355 10255 61017 5029205 11123 60239 57605506 10306 63731 4930624 11395 61229 56176807 10594 64205 5135695 11048 65184 54467808 10595 61168 5292235 11219 59664 58664409 10742 62172 5352540 11211 70432 530681510 10615 62732 5227675 11246 68517 543291511 10598 61750 5264900 10822 65737 523855012 10995 62278 5553895 11912 74115 567619513 10458 58153 5338415 10718 65717 515699014 10506 62337 5161150 10941 67001 526790515 10221 60246 5042130 11478 70268 5521755

c

Cmax

1205721= 09

1205722= 01

1205721= 07

1205722= 03

1205721= 05

1205722= 05

1205721= 03

1205722= 07

1205721= 01

1205722= 09

5

55

6

65

7

75

8

480

530

580

630

680

730




1205721= 09

1205722= 01

1205721= 07

1205722= 03

1205721= 05

1205722= 05

1205721= 03

1205722= 07

1205721= 01

1205722= 09


08

085

09

095

1

105

11

115

12

125

13



Notations


ℎ = 1 2 119867















119873119903(ℎ)(119895)


119888119905(119895)


119895

119873119903(119895)


119873119901(119895)



119895



Acknowledgment


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















119873119903(ℎ)(119895)


119888119905(119895)


119895

119873119903(119895)


119873119901(119895)



119895



Acknowledgment


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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