Research ArticleScheduling Batch Processing Machine Using MaxndashMin AntSystem Algorithm Improved by a Local Search Method

XiaoLin Li 1 and YuWang 2

1School of Mines China University of Mining and Technology Xuzhou 221116 China2School of Management University of Science and Technology of China Hefei 230026 China

Correspondence should be addressed to Yu Wang YuWangUSTC126com

Received 4 September 2017 Revised 13 December 2017 Accepted 1 January 2018 Published 28 January 2018

Academic Editor Fiorenzo A Fazzolari

Copyright copy 2018 XiaoLin Li and Yu Wang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The problem ofminimizing themakespan on single batch processingmachine is studied in this paper Both job sizes and processingtime are nonidentical and the processing time of each batch is determined by the job with the longest processing time in the batchMaxndashMin Ant System (MMAS) algorithm is developed to solve the problem A local search methodMJE (Multiple Jobs Exchange)is proposed to improve the performance of the algorithmby adjusting jobs between batches Preliminary experiment is conducted todetermine the parameters ofMMASTheperformance of the proposedMMAS algorithm is comparedwithCPLEX aswell as severalother algorithms including ant cycle (AC) algorithm genetic algorithm (GA) and two heuristics First Fit Longest Processing Time(FFLPT) and Best Fit Longest Processing Time (BFLPT) through numerical experimentThe experiment results show that MMASoutperformed others especially for large population size

1 Introduction

Scheduling of batch processing machine is a typical com-binatorial optimization problem Different from traditionalscheduling problems batch processing machine can processseveral jobs simultaneously as a batch Scheduling batchprocessing machine is usually encountered in manufacturingindustry such as heat treatment in metal industry and envi-ronment stress-screening in integrated circuit production Asthese operations often tends to be the bottleneck of manufac-turing sequence scheduling batch processing machine willeffectively promote the completion time of jobs

The problem was first proposed by Ikura and Gimple [1]who studied the problemwith identical job sizes and constantbatch processing time and machine capacity was defined bythe number of jobs processed simultaneously Consideringthe burning operation in semiconductor manufacturing Leeet al [2] presented an efficient dynamic programming basedalgorithms for minimizing a number of different perfor-mancemeasuresThe same problemwas studied by Sung andChoung [3] who proposed branch and bound algorithm andseveral heuristics to minimize the objective of makespan

The problem is muchmore complicated when nonidenti-cal job sizes are considered Uzsoy [4] studied the problemof scheduling single batch processing machine under theobjectives of minimizing makespan (119862max) and the totalprocessing time (sumCi) with nonidentical job sizes Bothproblems were proved to be NP-hard and several heuris-tics were proposed including First Fit Longest ProcessingTime (FFLPT) first fit shortest processing time (FFSPT) etal Dupont and Jolai Ghazvini [5] proposed two effectiveheuristicsrsquo successive knapsack problem (SKP) and best fitlongest processing time (BFLPT) and later one outperformedFFLPT To solve the problem optimally exact algorithm likebranch and bound was proposed by Dupont and Dhaenens-Flipo [6] they present some dominance properties for ageneral enumeration scheme for the makespan criterionEnumeration scheme [7] was also developed combined withexisting heuristics to solve large scale problems

Since the batch processing machine scheduling problemis NP-hard [4] various metaheuristic algorithms have beendeveloped to solve the problem Melouk et al [8] studiedthe problem using Simulated Annealing (SA) and randominstances were generated to evaluate the effectiveness of the

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 3124182 10 pageshttpsdoiorg10115520183124182

2 Mathematical Problems in Engineering

algorithmThe same problem was considered by Damodaranet al [9] with genetic algorithm (GA) The experimentshowed that GA outperformed SA in run time and solu-tion quality Husseinzadeh Kashan et al [10] proposedthe grouping version of the particle swarm optimization(PSO) algorithm and the application of which was madeto the single batch-machine scheduling problem A GRASPapproach developed by Damodaran et al [11] was used tominimize the makespan of a capacitated batch processingmachine and the experimental study concluded that GRASPoutperformed other solution approaches For the problemsconsidering multimachines Zhou et al [12] proposed aneffective differential evolution-based hybrid algorithm tominimize makespan on uniform parallel batch processingmachines and the algorithm was evaluated by comparingwith a random keys genetic algorithm (RKGA) and a particleswarm optimization (PSO) algorithm Similar problem wasstudied by Jiang et al [13] considering batch transportationA hybrid algorithm combining the merits of discrete particleswarm optimization (DPSO) and genetic algorithm (GA)is proposed to solve this problem The performance of theproposed algorithms was improved by using a local searchstrategy as well All of these studies show the effectivenessof solving batch processing machines problems by usingmetaheuristic algorithms

The studies reviewed above mainly solve the problemby sequencing the jobs into job list and then groupingthe jobs into batches Different from the existing studiesa metaheuristic algorithm MMAS (MaxndashMin Ant System)was designed in a constructive way by combining these twostages of decisions together That is to say the batches areconstructed directly without considering job sequences andthen the batches process on a batch processing machine Inthe process of batch construction jobs to be added to theexisting batches can be selected elaborately by consideringbatch utilization and batch processing time To improve theglobal searching ability of MMAS local searchmethod basedon multiple jobs iterative exchange was proposed

The remaining part of this paper is organized as followsThe mathematic model of the problem studied in this paperis presented in Section 2 In Section 3 we show the detailedMMAS algorithm used to solve the problem under the studyThe parameters tuning and the numeric experimentationare given in Section 4 The paper is finally concluded inSection 5

2 Mathematic Model

The problem of scheduling single batch processing machineis studied and the objective is to minimize makespan Batchprocessing machine can process several jobs as a batch andall the jobs in that batch have the same start and completiontime The process cannot be interrupted once the processbegins and no jobs can be added or removed from themachine until all jobs have been finished Batch processingtime is determined by the job with longest processing time inthe batch Jobs are all available at the time of zero

Symbols and notations used in this paper are listed asfollows

(1) There are 119899 jobs 119869 = 1 2 119899 to be processed andeach job 119895 has nonidentical processing time 119901119895 andsize 119904119895

(2) The capacity of the machine is assumed to be 119862 andeach job 119895 has 119904119895 ⩽ 119862 Job list 119869 will be scheduledinto batches 119887 isin 119861 before they are processed where119861 denotes a batch list that is a feasible solution and|119861|means the number of batches in119861 Processing timeof each batch 119887 equals 119875119887 = max119901119895 | 119895 isin 119887

(3) The objective is to minimize makespan (119862max) whichis equal to the total batch processing time in a solution119861

Base on the assumptions and notations given above wecan get the following mathematic model of the problem

min 119862max = sum119887isin119861

119875119887 (1)

st sum119887isin119861

119909119895119887 = 1 forall119895 isin 119869 (2)

119899sum119895=1

119904119895 sdot 119909119895119887 le 119862 forall119887 isin 119861 (3)

119875119887 ge 119901119895119909119895119887 forall119895 isin 119869 119887 isin 119861 (4)

119909119895119887 isin 0 1 forall119895 isin 119869 119887 isin 119861 (5)

lceilsum119899119895=1 119904119895119862 rceil le |119861| le 119899 |119861| isin 119885+ (6)

Objective (1) is to minimize the makespan As only oneprocessing machine is considered the makespan is equal tothe total completion time of all batches formed Constraint(2) ensures that each job 119895 can be assigned exactly to onebatch Constraint (3) guarantees that total size of jobs in abatch does not exceed the machine capacity 119862 Constraint(4) explains that the batch processing time is determined bythe job with longest processing time in that batch Constraint(5) denotes the binary restriction of variable 119909119895119887 which isequal to 1 if job 119895 is assigned to batch 119861 and 119909119895119887 = 0otherwise Constraint (6) gives the upper and lower boundof the number of batches in a feasible solution 119861 The lowerbound is calculated when assuming jobs can be processedpartially across the batches [4] And the upper bound isgenerated when each batch accommodates only one job

3 MaxndashMin Ant System

MMAS [14] is one of the most successful variants in theframework of ant colony optimization (ACO) [15 16] whichhave been applied to many combinatorial optimization prob-lems such as scheduling problems [17] traffic assignmentproblems [18] and travelling salesman problems [15] InMMAS pheromone trail limits interval [120591min 120591max] is usedto prevent premature convergence and to exploit the bestsolutions found during the solution searchThe performance

Mathematical Problems in Engineering 3

of the algorithm is significantly affected by the values ofthe parameters thus parameters tuning is performed tooptimize algorithm performance A local search method isalso developed to enhance the search ability of MMAS

MMAS is a constructive metaheuristic algorithm whichis able to build a solution step by step It can be adaptedto various combinatory optimization problems with a fewmodifications Given a list of jobs MMAS will group thejobs into batches by adding jobs to the existing or newbatches one at a time The sequence of selecting jobs tobe constructed into batches depends on the state transitionprobability calculated according to density of pheromonetrail and heuristic information of each solution elementA solution is generated once all jobs are arranged intoa batch

31 Pheromone Trails When solving the problem of TSP(travelling salesman problem) by using ant colony optimiza-tion algorithms [15] the pheromone trails 120591119894119895 are definedby the expectation of choosing city 119895 from city 119894 that isthe amount of pheromone of the arc(119894 119895) The city withhigher density of pheromone trail will be selected with ahigher possibility However in the problem of schedulingbatch processing machine each solution is a set of batchesand the sequence of jobs in a batch does not affect thebatch processing time thus the pheromone trails imply theexpectation of arranging a job into a batch In this study wemeasure the expectation of adding a job to a batch by usingthe average pheromone trails between the job 119895 and each jobin batch 119887 as follows

120599119895119887 = 1|119887|sum119896isin119887

120591119895119896 (7)

where 120591119895119896 means the pheromone trail between job 119895 and theexisting job 119896 in batch 119887 120599119895119887 denotes the expectation of addingthe job 119895 to the current batch 119887 |119887| denotes the number of jobsin a batch 119887 And this expectation will be used as pheromonetrails in the calculation of state transition probability

32 Heuristic Information As the objective 119862max equals thetotal processing time of all batches formed the quality of asolution is affected by both the number of batches and theprocessing times of each batch in the solution Thus twokinds of heuristic information are considered in this studythat is the utilization of machine capacity and efficiency ofbatch processing time

Usually solutions with smaller number of batches gener-ate better results To reduce the unoccupied capacity of eachbatch we add job 119895 to batch 119887 with the most feasible capacityin priority according to FFD (first fit decreasing) algorithm inbin packing problem [19] The heuristic to add a feasible job119895 to batch 119887 is defined as follows

120583119895119887 = 119904119895 (8)

The processing time of a batch is determined by the jobwith the longest processing time in the batch Obviously jobswith similar processing times should be batched together toincrease the efficiency of batch processing timeThus we give

another heuristic information for adding job 119895 to batch 119887 asfollows

120578119895119887 = 11 + 10038161003816100381610038161003816119875119887 minus 11990111989510038161003816100381610038161003816 (9)

33 Solution Construction For each ant 119886 a solution isconstructed by selecting anunscheduled job 119895 and adding it toan existing batch 119887 according to a state transition probability119875119886119887119895 If no existing batches can accommodate the job 119895 anew empty batch will be created and accommodate it Asolution will be generated when all jobs are scheduled into abatch Since a solution construction depends on the sequenceof jobs chosen solution quality is significantly affected bystate transition probability The probability is determined bythe pheromone trails and heuristic information between thecurrent batch and job to be scheduled The state transitionprobability is defined as follows

119875119886119887119895 =

120599120572119887119895 sdot 120578120573119887119895 sdot 120583120574119887119895sum119895isin119881 120599120572119887119895 sdot 120578120573119887119895 sdot 120583120574119887119895 119895 isin 1198810 Else

(10)

where 119881 denotes the feasible jobs that can be added to thecurrent batch 119887 such that themachine capacity is not violated120572 120573 and 120574 show the relative importance of pheromonetrails and two kinds of heuristic information For each ant119886 a feasible job 119895 in 119881 will be selected with probability andadded to the current batch or a new batch until all jobs arescheduled

34 Update of Pheromone Trails The density of pheromonetrails is an important factor in the process of solution con-struction as it indicates the quality of solution componentsWhen all ants build its feasible solution the pheromonetrails on every solution component will be updated throughpheromone depositing and evaporating After each itera-tion the pheromone trails of each solution component willbe decreased with the evaporation rate 120588 while solutioncomponent of the iterative best solution or the global bestsolution will be increased by a quantity Δ120591119895119896 The pheromoneupdate for each ant 119886 at 119905th iteration between the solutioncomponents of job 119895 and job 119896 is performed according to (11)as follows120591119895119896 (119905 + 1) = 120588120591119895119896 (119905) + Δ120591119895119896 (11)

Δ120591119895119896=

1119862119886max if job 119895 and 119896 are of the same batch

0 otherwise(12)

where 120588 (0 le 120588 lt 1) controls the pheromone evaporationrate 119862119886max denotes the solution makespan get by ant 119886

The pheromone trails are limited in the interval of[120591min 120591max] in MMAS algorithm We set 120591max = 1[120588 times 119862lowastmax]and 120591min = 120591max2119899 in this study according to Stutzle andHoos [14] 119862lowastmax is the global best makespan that ants havefound

4 Mathematical Problems in Engineering

35 Local Search Algorithm Solution qualities can be effec-tively improved when local search methods are applied inmetaheuristics [20] That is because greedy strategies areusually adopted in local search methods and local optimalcan be easily found in the neighborhood of a given solutionMetaheuristicsrsquo ability in global optimal searching can beenhanced by combining their advantages in exploring solu-tion space with local search methods

As batch processing time is determined by the job withthe longest processing time in the batch jobs with smallerprocessing time have no effect on batch processing timeLocal search method can be applied to decrease batchprocessing time by batching jobs with similar processing timetogether

Definition 1 The job 119894 with longest processing time in batch119887 is called dominant job 119889119887 The total processing time of jobswithout dominant job is denoted as dominated job processingtime noted as119863119879Proposition 2 Themakespan will be minimized by maximiz-ing119863119879Proof We have 119862max = sum119887isin119861max119901119895 | 119895 isin 119887 for a givensolution 119861 According to Definition 1 119863119879 = sum119887isin119861(sum119895isin119887 119901119895 minusmax119901119895 | 119895 isin 119887) = sum119887isin119861sum119895isin119887 119901119895 minus sum119887isin119861max119901119895 | 119895 isin 119887thus we have 119862max = sum119887isin119861sum119895isin119887 119901119895 minus 119863119879 As the total jobprocessing timesum119887isin119861sum119895isin119887 119901119895 is constant for a given instancethemakespanwill beminimizedwhenDT ismaximized

Definition 3 For each batch 119887 the sum of its remain-ing capacity and the size of dominant job 119889119887 is calledexchangeable capacity EC119887 we denote EC119887 = 119862 minus sum119894isin119887 119904119894 +119904119895=argmax119901119895|119895isin119887

Proposition 4 Given 119901119887 ⩾ 119901119902 larger DT will be obtained byexchange 119889119887 of batch 119887 with 119898 jobs in batch 119902 where sum119898119894=1 119904119894 le119864119862119902 and max119901119894 | 119894 = 1 119898 le max119901119895 | 119895 isin 119902 while the119862max does not increase

Proof Given two batches 119887 and 119902 119901119887 ⩾ 119901119902 jobs in batch119887 can be divided into three sets 119889119887 119898 jobs 119872 and otherjobs 119874 where sum119898119894=1 119904119894 le EC119902 and max119901119894 | 119894 = 1 119898 lemax119901119895 | 119895 isin 119902 Jobs in batch 119902 can be divided into sets 119889119902and other jobs1198741015840 Therefore we have 119862max = 119901119889119887 +119901119889119902 Afterthe exchange is applied to batches we have 1198621015840max = 119901119889119887 + 1199011015840119889119902 where 1199011015840119889119902 = max119901119894 | 119894 isin 119872 cup 1198741015840 As 119901119889119902 ge 119901119894 | 119894 isin119872 cup 1198741015840 the relation 1198621015840max le 119862max is satisfied According toProposition 2 there is 119862max = sum119895isin119887cup119902 119901119895 minus 119863119879 a minor 119862maxindicates a larger DT Proposition 4 holds

According to Proposition 4 multiple jobs can beexchanged iteratively between batches to decrease batchprocessing time For a given solution 119878 the number of batchesis limited by the total number of jobs where each job isgrouped as one batchThe detailed procedure of the proposedlocal search algorithmMJE (Multiple Jobs Exchange) is listedas follows

M

Batch k

Size

Time

p q

Batch k minus 1

Δkminus1 Δ k

Figure 1 Notations description of AlgorithmMJE

Order batches and jobs for a given solution S

according to Proposition 4

No

End

Yes

Start

K = n

Exchange dominant job of k with m jobs in k minus 1

Initialize parameter k = 0 n = |S|

Figure 2 Flow chart of algorithm MJE

Algorithm MJE (Multiple Jobs Exchange)

Step 1 For the iterative best solution 119878 arrange the batchesin decreasing order of their processing time and order jobs ofeach batch in decreasing order of job size

Step 2 Initialize parameter 119896 = 0 set 119899 = |119878| Δ 119896 isthe remaining capacity of batch 119896 and 119901 and 119902 denote thedominated job of batch 119896 minus 1 and batch 119896 respectivelyStep 3 If 119896 = 119899 exit else 119896 = 119896 + 1Step 4 Exchange the job 119902 of batch 119896 with119898 (119898 gt 0) jobs119872in batch 119896 minus 1 if max119901119894 | 119894 isin 119872 le 119901119902sum119894isin119872 119904119894 le 119904119902 + Δ 119896 and119904119902 le sum119894isin119872 119904119894 + Δ 119896minus1 go to Step 3

Corresponding notations are illustrated in Figure 1Theflowchart of algorithmMJE is provided as in Figure 2

36 Algorithm Framework of MMAS with Local SearchAccording to basic framework of ant algorithms [16] theframework of MMAS to solve the problem under study isgiven as follows

Algorithm MMAS

Step 1 Initialize parameters in MMAS including 120572 120573 120574 and120591(0)

Mathematical Problems in Engineering 5

Start

Initialize ant population A job set J

For each Ant a in A

Existing jobs can be added to current batch

For each job j in J

Add job j to the current batch according to state transition probability

Yes

Create a new batch for ant a

Yes

No

All jobs have been assigned

Yes

Solutions have beenconstructed for all ants

NoNo

Yes

Apply local search method MJE

Update pheromone trails

Nc =

Yes

No

Output the global optimal solution

End

Initialize parameters (0) NcNcＧＲ

NcＧＲ

Figure 3 Flow chart of algorithmMMAS

Step 2 Initialize ant population

Step 3 Select a unscheduled job for each ant 119886 and add thejob to a new batch

Step 4 Arrange the next unscheduled feasible job with themaximum state transition probability calculated by (10) intothe current batch

Step 5 If there are existing jobs that can be added to thecurrent batch then go to Step 4Step 6 If there are jobs unscheduled then go to Step 3Step 7 Apply local search methodMJE to iteration solutions

Step 8 Update pheromone trails according to formula (11)

Step 9 If the termination condition is satisfied then outputthe solution Else go to Step 2

To illustrate the procedure of MMAS more logically theflow chart of algorithmMMAS is provided as in Figure 3

4 Experimentation

41 Experimental Design Random instances were generatedto verify the effectiveness of the proposed algorithm Threefactors were considered in the numeric experiment includingnumber of jobs 119869 job processing time 119875 and job size 11987824 (4 lowast 2 lowast 3) problem categories were generated anddenoted in the form of 119869119894119875119895119878119896 (119894 = 1 2 3 4 119895 = 1 2119896 = 1 2 3) Factors and levels are shown in Table 1The machine capacity is assumed to be 10 for all probleminstances

For example 119869111987521198783 represented the category ofinstances with 10 jobs jobsrsquo processing time were randomlygenerated from a discrete uniform [1 20] distribution andjobsrsquo sizes were randomly generated from a discrete uniform[4 8] distribution

6 Mathematical Problems in Engineering

gt660

lt645645ndash650650ndash655655ndash660

Cmax

S1 categories S2 categories S3 categories

10987654321

2 3 4 5 6 7 8 9 101

123456789

10

2 3 4 5 6 7 8 9 101

10987654321

2 3 4 5 6 7 8 9 101

gt360

lt345345ndash350350ndash355355ndash360

Cmax

gt1560

lt15421542ndash15501550ndash15551555ndash1560

Cmax

Figure 4 Contour lines of 119862max influenced by parameters 120572 120573 and 120574 for different problem categories

Table 1 Factors and levels

Factors Levels119869 1198691 = 10 1198692 = 20 1198693 = 50 1198694 = 100119875 1198751 119880[1 10] 1198752 119880[1 20]119878 1198781 119880[1 10] 1198782 119880[2 4] 1198783 119880[4 8]119880means data are generated from discrete uniform distribution

42 Parameter Tuning As each ant will build feasible solu-tions with probability a large population of ants will enhancethe algorithmrsquos ability in exploring solution space while moretimes of iteration help to make better use of pheromonetrails and heuristic information However the algorithm ismuch more time consuming with larger number of ants anditerations Preliminary tests were done and the results showedthat to increase the number of iteration yields better resultsin a given run time That is easy to be understood as thealgorithm inclines to perform randomly search in the initialstages and more empirical results are considered along withthe increasing of iterations To obtain a tradeoff betweensolution quality and time cost ant population was set to 30and iterations were set to 80 in this study

It can be seen from formula (10) that there is exponentialrelationship between state transition probability pheromonetrails and heuristic informationThus the probability ismoresensitive to these factors with larger parameters 120572 120573 and 120574To verify the influence of these parameters preliminary testswere done to choose the parameters levels as well 120572 = 1 wasused as a reference level to study the effect of parameters 120573and 120574 on makespan in the interval of [1 10] The results areshown in Figure 4

It can be observed from Figure 4 that medium 120573 andsmaller 120574 should be used for instances of 1198782 categories withsmall jobs to obtain bettermakespanwhile the larger 120574 shouldbe adopted for instances of 1198783 categorieswith large jobs on thecontrary For instances of 1198781 categories with mixed job sizessmaller values for 120573 and medium 120574 seem better compared

Table 2 Values for parameters

Run code 120573 1205741198781 3 51198782 6 31198783 6 8

with the former two categories All instances generate badresults when 120573 is too close to 1 Large size jobs usually havelower flexibility when arranged in batches compared withsmall size jobs So high level of 120574 is used to arrange large sizejobs first with high probability for instances with large jobsSimilarly big 120573 is used to batch jobs with similar processingtime together to increase the efficiency of batch processingtime According to the analysis above three levels for eachparameter are selected in this study as shown in Table 2

Pheromone trails evaporate at each iteration and thespeed is controlled by the parameter 120588 (1 minus 120588) meansevaporation rate A high evaporation rate leads to a constantchange of pheromone trails on each solution element while alow rate results in the pheromone trails on solution elementsthat cannot evaporate in time

Pheromone trails of each solution element is limitedin the interval of [120591min 120591max] in MMAS algorithm Thepheromone trails are usually set to a high value of 120591max atthe beginning of running in order to improve the searchingability in solution space If a solution element is always in thestate of evaporation and its pheromone trail decreases to 120591minjust after 119873119888 iterations then we have 120591min = 120591max sdot 120588119873119888 Since120591min = 120591max2119899 as mentioned above it can be derived that120588119873119888 = 12119899 that is 119873119888 = log12119899120588 0 lt 120588 lt 1 where 119899 is thenumber of jobs The relationship between 119873119888 and 120588 for 100jobs is described in Figure 5

It can be seen from Figure 5 that 119873119888 increases dramati-cally when 120588 gt 06 thus 120588 is set to 06 in this study in orderto ensure the pheromone trails evaporate not too quickly or

Mathematical Problems in Engineering 7

Table 3 Results of MMAS compared with CPLEX

Problem category MMAS CPLEXMin Max Avg Optimal () AvgTime (s) AvgTime (s)

119878111986911198751 2000 6000 3664 10000 011 06411986911198752 3300 12700 7198 10000 010 06311986921198751 4300 10500 6913 10000 033 11811986921198752 8100 19800 13170 9600 033 142119878211986911198751 1300 2900 2090 10000 022 04411986911198752 2300 5800 4024 10000 022 04511986921198751 2800 4800 3797 10000 081 280411986921198752 4600 9400 7194 9900 082 2690119878311986911198751 2700 6400 4412 10000 008 05911986911198752 5000 12400 8553 10000 008 05811986921198751 5600 12200 8784 10000 023 10711986921198752 10600 21700 16718 10000 023 112

n = 50n = 100n = 200

0010203040506070809

1

Evap

orat

ion

para

met

er (

)

10 20 30 40 50 600Maximum number of iterations (Nc)

Figure 5 Relationship between119873119888 and 120588

slowly and the solution space can be explored in a reasonablerun time

43 Performance Comparison for Small Scale Instances Asshown in Table 3 the results obtained from MMAS iscompared with that from CPLEX (a commercial solverfor linear and mixed-integer problems) Due to the NP-hardness of the problem only small scale instances of 1198691and 1198692 problem categories are solved by using CPLEX Theminimum maximum and average value of 119862max reportedfrom MMAS are listed in columns 3 to 5 The average runtime of MMAS and CPLEXT are given in columns 7 and8 Column 6 indicates the percent of the optimal valuesobtained by using MMAS compared with the results fromCPLEX

Nearly all small scale instances can be solved optimallyby using MMAS except several instances from 119869211987521198781 and119869211987521198782 with relative larger solution space MMAS is com-petitive in computational time compared with CPLEX for allinstances Computational time of all instances fromMMAS isless than 1 second The computational time of CPLEX variesgreatly depending on the solution space of an instance

44 Algorithm Performance Evaluation To evaluate the per-formance of the algorithm for all problem categories MMASdesigned in this paper was comparedwithGA [9] A basic antalgorithm ant cycle (AC) was coded to solve the problem aswell and the parameters used were as prescribed in Chenget al [21] Two well-known heuristics FFLPT and BFLPTwere also compared withMMAS in the experiment study 100instances were generated for each problem category and thebest result of 10 runs for each instance was eventually used

Comparison of various algorithms is summarized inTable 4 Column 1 in Table 4 presents the run code Column 2shows the results of MMAS compared with other algorithmswhere B E and I mean themakespan obtained byMMAS arebetter than equal to or inferior to that of other algorithmsrespectively Columns 3ndash6 report the proportion that themakespan obtained by using MMAS compared with that ofother algorithms in 100 instances of 1198781 problem categoriesColumns 7ndash10 and columns 11ndash14 are similar to columns 3ndash6For example the first number of 001 indicates that there isa proportion of 001 results reported by MMAS better thanthat generated by AC As the problem under study is stronglyNP-hard the solution space is to explode along with thepopulation size increasing form 1198691 to 1198694 On the other handsmaller job sizes give more combination of batches thus thesolution space becomes smaller for 1198783 problem categoriescompared with 1198781 for a given number of jobs

It can be seen from Table 4 that MMAS outperformsother algorithms on the whole Several results reported byMMAS inferior to those generated by other algorithms have

8 Mathematical Problems in Engineering

Table 4 Results of MMAS compared with other algorithms

Run code Cp 1198781 1198782 1198783AC GA BFLPT FFLPT AC GA BFLPT FFLPT AC GA BFLPT FFLPT

11986911198751 B 001 002 016 017 000 010 016 016 000 001 007 007E 099 098 084 083 100 090 084 084 100 099 093 093I 000 000 000 000 000 000 000 000 000 000 000 000

11986911198752 B 002 002 010 013 001 014 019 019 000 003 008 011E 098 098 090 087 099 086 081 081 100 097 092 089I 000 000 000 000 000 000 000 000 000 000 000 000

11986921198751 B 004 045 047 059 001 031 032 032 000 007 014 020E 096 055 053 041 099 069 068 068 100 093 086 080I 000 000 000 000 000 000 000 000 000 000 000 000

11986921198752 B 013 042 041 057 006 032 033 034 000 014 019 029E 087 058 059 043 094 068 067 066 100 086 081 071I 000 000 000 000 000 000 000 000 000 000 000 000

11986931198751 B 043 084 083 093 023 089 091 091 000 030 030 043E 057 016 017 007 075 011 009 009 100 070 070 057I 000 000 000 000 002 000 000 000 000 000 000 000

11986931198752 B 060 090 085 099 039 087 087 087 012 027 027 039E 040 010 015 001 060 013 013 013 088 073 073 061I 000 000 000 000 001 000 000 000 000 000 000 000

11986941198751 B 078 093 090 100 049 099 099 099 005 018 018 030E 021 006 009 000 049 001 001 001 095 082 082 070I 001 001 001 000 002 000 000 000 000 000 000 000

11986941198752 B 092 095 092 098 051 099 099 099 035 022 022 030E 005 004 006 002 043 000 000 000 065 078 078 070I 003 001 002 000 006 001 001 001 000 000 000 000

been earmarked by rectangle For 1198691 problem categoriesresults obtained from MMAS are mostly equal to that fromother algorithms A percentage of about 16 results fromMMAS is better than that fromalgorithmsBFLPT andFFLPTfor 1198781 and 1198782 problem categories The percentage is 7 for1198783 problem categories which is because more results arethe same for all algorithms in a small solution space Upto 14 reported by MMAS are better than GA for 119869111987521198783problem category The advantage of MMAS to algorithmslike GA BFLPT and FFLPT is much more remarkable for1198692 problem categories More better results can be found byMMAS than AC and the percentage is up to 6 to 13 for11986921198752 problem categories More than 80 results obtainedfrom MMAS are better than that from GA BFLPT andFFLPT for 11986931198781 and 11986931198782 problem categories About 23 to43 percent of results fromMMAS are better than AC whilethe latter reports few better results For 1198694 problem categoriesmore than 90 results generated by MMAS are better thanthat from algorithms AC GA BFLPT and FFLPT for largejob population instances Almost all other algorithms reportseveral better results than MMAS for 1198694 problem categories

Based on the analysis given above MMAS possesseshigher ability in solution space exploring It outperforms allother algorithms for almost all instances especially for largejob population As MMAS is exploring solution space inprobability the optimal solution cannot be guaranteed witha limited population and iterations Therefore several results

obtained from MMAS are inferior to other algorithms evento effective heuristics like BFLPT and FFLPT

The makespan increases for 1198752 problem categories com-pared with 1198751 problem categories but there is no significantperformance change for various algorithms

To illustrate the gap between MMAS and other algo-rithms for different population size the average percentage ofresults with better makespan obtained by MMAS comparedwith other algorithms is given in Figure 6

Averagely a percentage of 10 results obtained fromMMAS are better than that from other algorithms for 1198691problem categories while the number is 70 for 1198694 Algo-rithm AC is the second best algorithm compared to GA andthe other two heuristics BFLPT and FFLPT are all effectivein solving the problem especially for small population sizeBFLPT is generally better than FFLPT for almost all problemcategories and BFLPT is also used in GA to generate initialsolutions

5 Conclusions and Future Work

The problem of scheduling single batch processing machinewas studied in this paper An improved ant algorithmMMAS(MaxndashMin Ant System) was designed and applied to solvethe problem Parameters of MMAS were determined bypreliminary experiment and a local search method MJEwas also presented to improve the performance of MMAS

Mathematical Problems in Engineering 9

J1 J2 J3 J4

Number of jobs

ACGA

BFLPTFFLPT

01020304050607080

Perc

enta

ge (

)

Figure 6 Average percentage of results obtained from MMAScompared with that from other algorithms

Optimal objectives can be obtained by using MMAS in lesscomputational time compared to CPLEX for small scaleproblems To evaluate the performance of MMAS it wascompared with basic ant algorithm AC (Ant Cycle) GA(Genetic Algorithm) and the other two well-known heuris-tics of scheduling batch processing machine that is FFLPT(First Fit Longest Processing Time) and BFLPT (Best FitLongest Processing Time) The numeric experiment showedthat MMAS outperformed other algorithms for almost allinstances in various problem categories especially for largerpopulation size

Considering the good performance of MMAS betterheuristics mechanism can be designed and applied to otherproblems of scheduling batch processing machine Theproblem under study with other objectives and problemconstraints including release times and due dates can bestudied as well

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (no 2014QNA48) and NationalNatural Science Foundation of China (no 71401164)

References

[1] Y Ikura and M Gimple ldquoEfficient scheduling algorithms for asingle batch processing machinerdquo Operations Research Lettersvol 5 no 2 pp 61ndash65 1986

[2] C-Y Lee R Uzsoy and L AMartin-Vega ldquoEfficient algorithmsfor scheduling semiconductor burn-in operationsrdquo OperationsResearch vol 40 no 4 pp 764ndash775 1992

[3] C S Sung and Y I Choung ldquoMinimizing makespan on a singleburn-in oven in semiconductormanufacturingrdquoEuropean Jour-nal of Operational Research vol 120 no 3 pp 559ndash574 2000

[4] R Uzsoy ldquoScheduling a single batch processing machine withnon-identical job sizesrdquo International Journal of ProductionResearch vol 32 no 7 pp 1615ndash1635 1994

[5] L Dupont and F Jolai Ghazvini ldquoMinimizing makespan on asingle batch processing machine with non-identical job sizesrdquoJournal Europeen des Systemes Automatises vol 32 no 4 pp431ndash440 1998

[6] L Dupont and C Dhaenens-Flipo ldquoMinimizing the makespanon a batch machine with non-identical job sizes An exactprocedurerdquo Computers amp Operations Research vol 29 no 7 pp807ndash819 2002

[7] X Li Y Li andYWang ldquoMinimisingmakespan on a batch pro-cessing machine using heuristics improved by an enumerationschemerdquo International Journal of Production Research vol 55no 1 pp 176ndash186 2017

[8] S Melouk P Damodaran and P-Y Chang ldquoMinimizingmakespan for single machine batch processing with non-identical job sizes using simulated annealingrdquo InternationalJournal of Production Economics vol 87 no 2 pp 141ndash147 2004

[9] P Damodaran P Kumar Manjeshwar and K Srihari ldquoMin-imizing makespan on a batch-processing machine with non-identical job sizes using genetic algorithmsrdquo InternationalJournal of Production Economics vol 103 no 2 pp 882ndash8912006

[10] A Husseinzadeh Kashan M Husseinzadeh Kashan and SKarimiyan ldquoA particle swarm optimizer for grouping prob-lemsrdquo Information Sciences vol 252 pp 81ndash95 2013

[11] P Damodaran O Ghrayeb and M C Guttikonda ldquoGRASPto minimize makespan for a capacitated batch-processingmachinerdquoThe International Journal of AdvancedManufacturingTechnology vol 68 no 1-4 pp 407ndash414 2013

[12] S Zhou M Liu H Chen and X Li ldquoAn effective discretedifferential evolution algorithm for scheduling uniform parallelbatch processing machines with non-identical capacities andarbitrary job sizesrdquo International Journal of Production Eco-nomics vol 179 pp 1ndash11 2016

[13] L Jiang J Pei X Liu P M Pardalos Y Yang and XQian ldquoUniformparallel batchmachines scheduling consideringtransportation using a hybrid DPSO-GA algorithmrdquoThe Inter-national Journal of AdvancedManufacturing Technology vol 89no 5-8 pp 1887ndash1900 2017

[14] T Stutzle and H H Hoos ldquoMax-min ant systemrdquo FutureGeneration Computer Systems vol 16 no 8 pp 889ndash914 2000

[15] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[16] M Dorigo and C Blum ldquoAnt colony optimization theory asurveyrdquoTheoretical Computer Science vol 344 no 2-3 pp 243ndash278 2005

[17] R Xu H Chen and X Li ldquoMakespan minimization onsingle batch-processing machine via ant colony optimizationrdquoComputers amp Operations Research vol 39 no 3 pp 582ndash5932012

[18] L DrsquoAcierno M Gallo and B Montella ldquoAn ant colony optimi-sation algorithm for solving the asymmetric traffic assignmentproblemrdquo European Journal of Operational Research vol 217 no2 pp 459ndash469 2012

[19] D S Johnson A Demers J D Ullman M R Garey and RL Graham ldquoWorst-case performance bounds for simple one-dimensional packing algorithmsrdquo SIAM Journal on Computingvol 3 pp 299ndash325 1974

[20] M Pirlot ldquoGeneral local search methodsrdquo European Journal ofOperational Research vol 92 no 3 pp 493ndash511 1996

10 Mathematical Problems in Engineering

[21] B-Y Cheng H-P Chen and S-S Wang ldquoImproved ant colonyoptimization method for single batch-processing machine withnon-identical job sizesrdquo Journal of System Simulation vol 21 no9 pp 2687ndash2695 2009

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Mathematical Problems in Engineering 3

of the algorithm is significantly affected by the values ofthe parameters thus parameters tuning is performed tooptimize algorithm performance A local search method isalso developed to enhance the search ability of MMAS

MMAS is a constructive metaheuristic algorithm whichis able to build a solution step by step It can be adaptedto various combinatory optimization problems with a fewmodifications Given a list of jobs MMAS will group thejobs into batches by adding jobs to the existing or newbatches one at a time The sequence of selecting jobs tobe constructed into batches depends on the state transitionprobability calculated according to density of pheromonetrail and heuristic information of each solution elementA solution is generated once all jobs are arranged intoa batch

31 Pheromone Trails When solving the problem of TSP(travelling salesman problem) by using ant colony optimiza-tion algorithms [15] the pheromone trails 120591119894119895 are definedby the expectation of choosing city 119895 from city 119894 that isthe amount of pheromone of the arc(119894 119895) The city withhigher density of pheromone trail will be selected with ahigher possibility However in the problem of schedulingbatch processing machine each solution is a set of batchesand the sequence of jobs in a batch does not affect thebatch processing time thus the pheromone trails imply theexpectation of arranging a job into a batch In this study wemeasure the expectation of adding a job to a batch by usingthe average pheromone trails between the job 119895 and each jobin batch 119887 as follows

120599119895119887 = 1|119887|sum119896isin119887

120591119895119896 (7)

where 120591119895119896 means the pheromone trail between job 119895 and theexisting job 119896 in batch 119887 120599119895119887 denotes the expectation of addingthe job 119895 to the current batch 119887 |119887| denotes the number of jobsin a batch 119887 And this expectation will be used as pheromonetrails in the calculation of state transition probability

32 Heuristic Information As the objective 119862max equals thetotal processing time of all batches formed the quality of asolution is affected by both the number of batches and theprocessing times of each batch in the solution Thus twokinds of heuristic information are considered in this studythat is the utilization of machine capacity and efficiency ofbatch processing time

Usually solutions with smaller number of batches gener-ate better results To reduce the unoccupied capacity of eachbatch we add job 119895 to batch 119887 with the most feasible capacityin priority according to FFD (first fit decreasing) algorithm inbin packing problem [19] The heuristic to add a feasible job119895 to batch 119887 is defined as follows

120583119895119887 = 119904119895 (8)

The processing time of a batch is determined by the jobwith the longest processing time in the batch Obviously jobswith similar processing times should be batched together toincrease the efficiency of batch processing timeThus we give

another heuristic information for adding job 119895 to batch 119887 asfollows

120578119895119887 = 11 + 10038161003816100381610038161003816119875119887 minus 11990111989510038161003816100381610038161003816 (9)

33 Solution Construction For each ant 119886 a solution isconstructed by selecting anunscheduled job 119895 and adding it toan existing batch 119887 according to a state transition probability119875119886119887119895 If no existing batches can accommodate the job 119895 anew empty batch will be created and accommodate it Asolution will be generated when all jobs are scheduled into abatch Since a solution construction depends on the sequenceof jobs chosen solution quality is significantly affected bystate transition probability The probability is determined bythe pheromone trails and heuristic information between thecurrent batch and job to be scheduled The state transitionprobability is defined as follows

119875119886119887119895 =

120599120572119887119895 sdot 120578120573119887119895 sdot 120583120574119887119895sum119895isin119881 120599120572119887119895 sdot 120578120573119887119895 sdot 120583120574119887119895 119895 isin 1198810 Else

(10)

where 119881 denotes the feasible jobs that can be added to thecurrent batch 119887 such that themachine capacity is not violated120572 120573 and 120574 show the relative importance of pheromonetrails and two kinds of heuristic information For each ant119886 a feasible job 119895 in 119881 will be selected with probability andadded to the current batch or a new batch until all jobs arescheduled

34 Update of Pheromone Trails The density of pheromonetrails is an important factor in the process of solution con-struction as it indicates the quality of solution componentsWhen all ants build its feasible solution the pheromonetrails on every solution component will be updated throughpheromone depositing and evaporating After each itera-tion the pheromone trails of each solution component willbe decreased with the evaporation rate 120588 while solutioncomponent of the iterative best solution or the global bestsolution will be increased by a quantity Δ120591119895119896 The pheromoneupdate for each ant 119886 at 119905th iteration between the solutioncomponents of job 119895 and job 119896 is performed according to (11)as follows120591119895119896 (119905 + 1) = 120588120591119895119896 (119905) + Δ120591119895119896 (11)

Δ120591119895119896=

1119862119886max if job 119895 and 119896 are of the same batch

0 otherwise(12)

where 120588 (0 le 120588 lt 1) controls the pheromone evaporationrate 119862119886max denotes the solution makespan get by ant 119886

The pheromone trails are limited in the interval of[120591min 120591max] in MMAS algorithm We set 120591max = 1[120588 times 119862lowastmax]and 120591min = 120591max2119899 in this study according to Stutzle andHoos [14] 119862lowastmax is the global best makespan that ants havefound

4 Mathematical Problems in Engineering

35 Local Search Algorithm Solution qualities can be effec-tively improved when local search methods are applied inmetaheuristics [20] That is because greedy strategies areusually adopted in local search methods and local optimalcan be easily found in the neighborhood of a given solutionMetaheuristicsrsquo ability in global optimal searching can beenhanced by combining their advantages in exploring solu-tion space with local search methods

As batch processing time is determined by the job withthe longest processing time in the batch jobs with smallerprocessing time have no effect on batch processing timeLocal search method can be applied to decrease batchprocessing time by batching jobs with similar processing timetogether

Definition 1 The job 119894 with longest processing time in batch119887 is called dominant job 119889119887 The total processing time of jobswithout dominant job is denoted as dominated job processingtime noted as119863119879Proposition 2 Themakespan will be minimized by maximiz-ing119863119879Proof We have 119862max = sum119887isin119861max119901119895 | 119895 isin 119887 for a givensolution 119861 According to Definition 1 119863119879 = sum119887isin119861(sum119895isin119887 119901119895 minusmax119901119895 | 119895 isin 119887) = sum119887isin119861sum119895isin119887 119901119895 minus sum119887isin119861max119901119895 | 119895 isin 119887thus we have 119862max = sum119887isin119861sum119895isin119887 119901119895 minus 119863119879 As the total jobprocessing timesum119887isin119861sum119895isin119887 119901119895 is constant for a given instancethemakespanwill beminimizedwhenDT ismaximized

Definition 3 For each batch 119887 the sum of its remain-ing capacity and the size of dominant job 119889119887 is calledexchangeable capacity EC119887 we denote EC119887 = 119862 minus sum119894isin119887 119904119894 +119904119895=argmax119901119895|119895isin119887

Proposition 4 Given 119901119887 ⩾ 119901119902 larger DT will be obtained byexchange 119889119887 of batch 119887 with 119898 jobs in batch 119902 where sum119898119894=1 119904119894 le119864119862119902 and max119901119894 | 119894 = 1 119898 le max119901119895 | 119895 isin 119902 while the119862max does not increase

Proof Given two batches 119887 and 119902 119901119887 ⩾ 119901119902 jobs in batch119887 can be divided into three sets 119889119887 119898 jobs 119872 and otherjobs 119874 where sum119898119894=1 119904119894 le EC119902 and max119901119894 | 119894 = 1 119898 lemax119901119895 | 119895 isin 119902 Jobs in batch 119902 can be divided into sets 119889119902and other jobs1198741015840 Therefore we have 119862max = 119901119889119887 +119901119889119902 Afterthe exchange is applied to batches we have 1198621015840max = 119901119889119887 + 1199011015840119889119902 where 1199011015840119889119902 = max119901119894 | 119894 isin 119872 cup 1198741015840 As 119901119889119902 ge 119901119894 | 119894 isin119872 cup 1198741015840 the relation 1198621015840max le 119862max is satisfied According toProposition 2 there is 119862max = sum119895isin119887cup119902 119901119895 minus 119863119879 a minor 119862maxindicates a larger DT Proposition 4 holds

According to Proposition 4 multiple jobs can beexchanged iteratively between batches to decrease batchprocessing time For a given solution 119878 the number of batchesis limited by the total number of jobs where each job isgrouped as one batchThe detailed procedure of the proposedlocal search algorithmMJE (Multiple Jobs Exchange) is listedas follows

M

Batch k

Size

Time

p q

Batch k minus 1

Δkminus1 Δ k

Figure 1 Notations description of AlgorithmMJE

Order batches and jobs for a given solution S

according to Proposition 4

No

End

Yes

Start

K = n

Exchange dominant job of k with m jobs in k minus 1

Initialize parameter k = 0 n = |S|

Figure 2 Flow chart of algorithm MJE

Algorithm MJE (Multiple Jobs Exchange)

Step 1 For the iterative best solution 119878 arrange the batchesin decreasing order of their processing time and order jobs ofeach batch in decreasing order of job size

Step 2 Initialize parameter 119896 = 0 set 119899 = |119878| Δ 119896 isthe remaining capacity of batch 119896 and 119901 and 119902 denote thedominated job of batch 119896 minus 1 and batch 119896 respectivelyStep 3 If 119896 = 119899 exit else 119896 = 119896 + 1Step 4 Exchange the job 119902 of batch 119896 with119898 (119898 gt 0) jobs119872in batch 119896 minus 1 if max119901119894 | 119894 isin 119872 le 119901119902sum119894isin119872 119904119894 le 119904119902 + Δ 119896 and119904119902 le sum119894isin119872 119904119894 + Δ 119896minus1 go to Step 3

Corresponding notations are illustrated in Figure 1Theflowchart of algorithmMJE is provided as in Figure 2

36 Algorithm Framework of MMAS with Local SearchAccording to basic framework of ant algorithms [16] theframework of MMAS to solve the problem under study isgiven as follows

Algorithm MMAS

Step 1 Initialize parameters in MMAS including 120572 120573 120574 and120591(0)

Mathematical Problems in Engineering 5

Start

Initialize ant population A job set J

For each Ant a in A

Existing jobs can be added to current batch

For each job j in J

Add job j to the current batch according to state transition probability

Yes

Create a new batch for ant a

Yes

No

All jobs have been assigned

Yes

Solutions have beenconstructed for all ants

NoNo

Yes

Apply local search method MJE

Update pheromone trails

Nc =

Yes

No

Output the global optimal solution

End

Initialize parameters (0) NcNcＧＲ

NcＧＲ

Figure 3 Flow chart of algorithmMMAS

Step 2 Initialize ant population

Step 3 Select a unscheduled job for each ant 119886 and add thejob to a new batch

Step 4 Arrange the next unscheduled feasible job with themaximum state transition probability calculated by (10) intothe current batch

Step 5 If there are existing jobs that can be added to thecurrent batch then go to Step 4Step 6 If there are jobs unscheduled then go to Step 3Step 7 Apply local search methodMJE to iteration solutions

Step 8 Update pheromone trails according to formula (11)

Step 9 If the termination condition is satisfied then outputthe solution Else go to Step 2

To illustrate the procedure of MMAS more logically theflow chart of algorithmMMAS is provided as in Figure 3

4 Experimentation

41 Experimental Design Random instances were generatedto verify the effectiveness of the proposed algorithm Threefactors were considered in the numeric experiment includingnumber of jobs 119869 job processing time 119875 and job size 11987824 (4 lowast 2 lowast 3) problem categories were generated anddenoted in the form of 119869119894119875119895119878119896 (119894 = 1 2 3 4 119895 = 1 2119896 = 1 2 3) Factors and levels are shown in Table 1The machine capacity is assumed to be 10 for all probleminstances

For example 119869111987521198783 represented the category ofinstances with 10 jobs jobsrsquo processing time were randomlygenerated from a discrete uniform [1 20] distribution andjobsrsquo sizes were randomly generated from a discrete uniform[4 8] distribution

6 Mathematical Problems in Engineering

gt660

lt645645ndash650650ndash655655ndash660

Cmax

S1 categories S2 categories S3 categories

10987654321

2 3 4 5 6 7 8 9 101

123456789

10

2 3 4 5 6 7 8 9 101

10987654321

2 3 4 5 6 7 8 9 101

gt360

lt345345ndash350350ndash355355ndash360

Cmax

gt1560

lt15421542ndash15501550ndash15551555ndash1560

Cmax

Figure 4 Contour lines of 119862max influenced by parameters 120572 120573 and 120574 for different problem categories

Table 1 Factors and levels

Factors Levels119869 1198691 = 10 1198692 = 20 1198693 = 50 1198694 = 100119875 1198751 119880[1 10] 1198752 119880[1 20]119878 1198781 119880[1 10] 1198782 119880[2 4] 1198783 119880[4 8]119880means data are generated from discrete uniform distribution

42 Parameter Tuning As each ant will build feasible solu-tions with probability a large population of ants will enhancethe algorithmrsquos ability in exploring solution space while moretimes of iteration help to make better use of pheromonetrails and heuristic information However the algorithm ismuch more time consuming with larger number of ants anditerations Preliminary tests were done and the results showedthat to increase the number of iteration yields better resultsin a given run time That is easy to be understood as thealgorithm inclines to perform randomly search in the initialstages and more empirical results are considered along withthe increasing of iterations To obtain a tradeoff betweensolution quality and time cost ant population was set to 30and iterations were set to 80 in this study

It can be seen from formula (10) that there is exponentialrelationship between state transition probability pheromonetrails and heuristic informationThus the probability ismoresensitive to these factors with larger parameters 120572 120573 and 120574To verify the influence of these parameters preliminary testswere done to choose the parameters levels as well 120572 = 1 wasused as a reference level to study the effect of parameters 120573and 120574 on makespan in the interval of [1 10] The results areshown in Figure 4

It can be observed from Figure 4 that medium 120573 andsmaller 120574 should be used for instances of 1198782 categories withsmall jobs to obtain bettermakespanwhile the larger 120574 shouldbe adopted for instances of 1198783 categorieswith large jobs on thecontrary For instances of 1198781 categories with mixed job sizessmaller values for 120573 and medium 120574 seem better compared

Table 2 Values for parameters

Run code 120573 1205741198781 3 51198782 6 31198783 6 8

with the former two categories All instances generate badresults when 120573 is too close to 1 Large size jobs usually havelower flexibility when arranged in batches compared withsmall size jobs So high level of 120574 is used to arrange large sizejobs first with high probability for instances with large jobsSimilarly big 120573 is used to batch jobs with similar processingtime together to increase the efficiency of batch processingtime According to the analysis above three levels for eachparameter are selected in this study as shown in Table 2

Pheromone trails evaporate at each iteration and thespeed is controlled by the parameter 120588 (1 minus 120588) meansevaporation rate A high evaporation rate leads to a constantchange of pheromone trails on each solution element while alow rate results in the pheromone trails on solution elementsthat cannot evaporate in time

Pheromone trails of each solution element is limitedin the interval of [120591min 120591max] in MMAS algorithm Thepheromone trails are usually set to a high value of 120591max atthe beginning of running in order to improve the searchingability in solution space If a solution element is always in thestate of evaporation and its pheromone trail decreases to 120591minjust after 119873119888 iterations then we have 120591min = 120591max sdot 120588119873119888 Since120591min = 120591max2119899 as mentioned above it can be derived that120588119873119888 = 12119899 that is 119873119888 = log12119899120588 0 lt 120588 lt 1 where 119899 is thenumber of jobs The relationship between 119873119888 and 120588 for 100jobs is described in Figure 5

It can be seen from Figure 5 that 119873119888 increases dramati-cally when 120588 gt 06 thus 120588 is set to 06 in this study in orderto ensure the pheromone trails evaporate not too quickly or

Mathematical Problems in Engineering 7

Table 3 Results of MMAS compared with CPLEX

Problem category MMAS CPLEXMin Max Avg Optimal () AvgTime (s) AvgTime (s)

119878111986911198751 2000 6000 3664 10000 011 06411986911198752 3300 12700 7198 10000 010 06311986921198751 4300 10500 6913 10000 033 11811986921198752 8100 19800 13170 9600 033 142119878211986911198751 1300 2900 2090 10000 022 04411986911198752 2300 5800 4024 10000 022 04511986921198751 2800 4800 3797 10000 081 280411986921198752 4600 9400 7194 9900 082 2690119878311986911198751 2700 6400 4412 10000 008 05911986911198752 5000 12400 8553 10000 008 05811986921198751 5600 12200 8784 10000 023 10711986921198752 10600 21700 16718 10000 023 112

n = 50n = 100n = 200

0010203040506070809

1

Evap

orat

ion

para

met

er (

)

10 20 30 40 50 600Maximum number of iterations (Nc)

Figure 5 Relationship between119873119888 and 120588

slowly and the solution space can be explored in a reasonablerun time

43 Performance Comparison for Small Scale Instances Asshown in Table 3 the results obtained from MMAS iscompared with that from CPLEX (a commercial solverfor linear and mixed-integer problems) Due to the NP-hardness of the problem only small scale instances of 1198691and 1198692 problem categories are solved by using CPLEX Theminimum maximum and average value of 119862max reportedfrom MMAS are listed in columns 3 to 5 The average runtime of MMAS and CPLEXT are given in columns 7 and8 Column 6 indicates the percent of the optimal valuesobtained by using MMAS compared with the results fromCPLEX

Nearly all small scale instances can be solved optimallyby using MMAS except several instances from 119869211987521198781 and119869211987521198782 with relative larger solution space MMAS is com-petitive in computational time compared with CPLEX for allinstances Computational time of all instances fromMMAS isless than 1 second The computational time of CPLEX variesgreatly depending on the solution space of an instance

44 Algorithm Performance Evaluation To evaluate the per-formance of the algorithm for all problem categories MMASdesigned in this paper was comparedwithGA [9] A basic antalgorithm ant cycle (AC) was coded to solve the problem aswell and the parameters used were as prescribed in Chenget al [21] Two well-known heuristics FFLPT and BFLPTwere also compared withMMAS in the experiment study 100instances were generated for each problem category and thebest result of 10 runs for each instance was eventually used

Comparison of various algorithms is summarized inTable 4 Column 1 in Table 4 presents the run code Column 2shows the results of MMAS compared with other algorithmswhere B E and I mean themakespan obtained byMMAS arebetter than equal to or inferior to that of other algorithmsrespectively Columns 3ndash6 report the proportion that themakespan obtained by using MMAS compared with that ofother algorithms in 100 instances of 1198781 problem categoriesColumns 7ndash10 and columns 11ndash14 are similar to columns 3ndash6For example the first number of 001 indicates that there isa proportion of 001 results reported by MMAS better thanthat generated by AC As the problem under study is stronglyNP-hard the solution space is to explode along with thepopulation size increasing form 1198691 to 1198694 On the other handsmaller job sizes give more combination of batches thus thesolution space becomes smaller for 1198783 problem categoriescompared with 1198781 for a given number of jobs

It can be seen from Table 4 that MMAS outperformsother algorithms on the whole Several results reported byMMAS inferior to those generated by other algorithms have

8 Mathematical Problems in Engineering

Table 4 Results of MMAS compared with other algorithms

Run code Cp 1198781 1198782 1198783AC GA BFLPT FFLPT AC GA BFLPT FFLPT AC GA BFLPT FFLPT

11986911198751 B 001 002 016 017 000 010 016 016 000 001 007 007E 099 098 084 083 100 090 084 084 100 099 093 093I 000 000 000 000 000 000 000 000 000 000 000 000

11986911198752 B 002 002 010 013 001 014 019 019 000 003 008 011E 098 098 090 087 099 086 081 081 100 097 092 089I 000 000 000 000 000 000 000 000 000 000 000 000

11986921198751 B 004 045 047 059 001 031 032 032 000 007 014 020E 096 055 053 041 099 069 068 068 100 093 086 080I 000 000 000 000 000 000 000 000 000 000 000 000

11986921198752 B 013 042 041 057 006 032 033 034 000 014 019 029E 087 058 059 043 094 068 067 066 100 086 081 071I 000 000 000 000 000 000 000 000 000 000 000 000

11986931198751 B 043 084 083 093 023 089 091 091 000 030 030 043E 057 016 017 007 075 011 009 009 100 070 070 057I 000 000 000 000 002 000 000 000 000 000 000 000

11986931198752 B 060 090 085 099 039 087 087 087 012 027 027 039E 040 010 015 001 060 013 013 013 088 073 073 061I 000 000 000 000 001 000 000 000 000 000 000 000

11986941198751 B 078 093 090 100 049 099 099 099 005 018 018 030E 021 006 009 000 049 001 001 001 095 082 082 070I 001 001 001 000 002 000 000 000 000 000 000 000

11986941198752 B 092 095 092 098 051 099 099 099 035 022 022 030E 005 004 006 002 043 000 000 000 065 078 078 070I 003 001 002 000 006 001 001 001 000 000 000 000

been earmarked by rectangle For 1198691 problem categoriesresults obtained from MMAS are mostly equal to that fromother algorithms A percentage of about 16 results fromMMAS is better than that fromalgorithmsBFLPT andFFLPTfor 1198781 and 1198782 problem categories The percentage is 7 for1198783 problem categories which is because more results arethe same for all algorithms in a small solution space Upto 14 reported by MMAS are better than GA for 119869111987521198783problem category The advantage of MMAS to algorithmslike GA BFLPT and FFLPT is much more remarkable for1198692 problem categories More better results can be found byMMAS than AC and the percentage is up to 6 to 13 for11986921198752 problem categories More than 80 results obtainedfrom MMAS are better than that from GA BFLPT andFFLPT for 11986931198781 and 11986931198782 problem categories About 23 to43 percent of results fromMMAS are better than AC whilethe latter reports few better results For 1198694 problem categoriesmore than 90 results generated by MMAS are better thanthat from algorithms AC GA BFLPT and FFLPT for largejob population instances Almost all other algorithms reportseveral better results than MMAS for 1198694 problem categories

Based on the analysis given above MMAS possesseshigher ability in solution space exploring It outperforms allother algorithms for almost all instances especially for largejob population As MMAS is exploring solution space inprobability the optimal solution cannot be guaranteed witha limited population and iterations Therefore several results

obtained from MMAS are inferior to other algorithms evento effective heuristics like BFLPT and FFLPT

The makespan increases for 1198752 problem categories com-pared with 1198751 problem categories but there is no significantperformance change for various algorithms

To illustrate the gap between MMAS and other algo-rithms for different population size the average percentage ofresults with better makespan obtained by MMAS comparedwith other algorithms is given in Figure 6

Averagely a percentage of 10 results obtained fromMMAS are better than that from other algorithms for 1198691problem categories while the number is 70 for 1198694 Algo-rithm AC is the second best algorithm compared to GA andthe other two heuristics BFLPT and FFLPT are all effectivein solving the problem especially for small population sizeBFLPT is generally better than FFLPT for almost all problemcategories and BFLPT is also used in GA to generate initialsolutions

5 Conclusions and Future Work

The problem of scheduling single batch processing machinewas studied in this paper An improved ant algorithmMMAS(MaxndashMin Ant System) was designed and applied to solvethe problem Parameters of MMAS were determined bypreliminary experiment and a local search method MJEwas also presented to improve the performance of MMAS

Mathematical Problems in Engineering 9

J1 J2 J3 J4

Number of jobs

ACGA

BFLPTFFLPT

01020304050607080

Perc

enta

ge (

)

Figure 6 Average percentage of results obtained from MMAScompared with that from other algorithms

Optimal objectives can be obtained by using MMAS in lesscomputational time compared to CPLEX for small scaleproblems To evaluate the performance of MMAS it wascompared with basic ant algorithm AC (Ant Cycle) GA(Genetic Algorithm) and the other two well-known heuris-tics of scheduling batch processing machine that is FFLPT(First Fit Longest Processing Time) and BFLPT (Best FitLongest Processing Time) The numeric experiment showedthat MMAS outperformed other algorithms for almost allinstances in various problem categories especially for largerpopulation size

Considering the good performance of MMAS betterheuristics mechanism can be designed and applied to otherproblems of scheduling batch processing machine Theproblem under study with other objectives and problemconstraints including release times and due dates can bestudied as well

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (no 2014QNA48) and NationalNatural Science Foundation of China (no 71401164)

References

[1] Y Ikura and M Gimple ldquoEfficient scheduling algorithms for asingle batch processing machinerdquo Operations Research Lettersvol 5 no 2 pp 61ndash65 1986

[2] C-Y Lee R Uzsoy and L AMartin-Vega ldquoEfficient algorithmsfor scheduling semiconductor burn-in operationsrdquo OperationsResearch vol 40 no 4 pp 764ndash775 1992

[3] C S Sung and Y I Choung ldquoMinimizing makespan on a singleburn-in oven in semiconductormanufacturingrdquoEuropean Jour-nal of Operational Research vol 120 no 3 pp 559ndash574 2000

[4] R Uzsoy ldquoScheduling a single batch processing machine withnon-identical job sizesrdquo International Journal of ProductionResearch vol 32 no 7 pp 1615ndash1635 1994

[5] L Dupont and F Jolai Ghazvini ldquoMinimizing makespan on asingle batch processing machine with non-identical job sizesrdquoJournal Europeen des Systemes Automatises vol 32 no 4 pp431ndash440 1998

[6] L Dupont and C Dhaenens-Flipo ldquoMinimizing the makespanon a batch machine with non-identical job sizes An exactprocedurerdquo Computers amp Operations Research vol 29 no 7 pp807ndash819 2002

[7] X Li Y Li andYWang ldquoMinimisingmakespan on a batch pro-cessing machine using heuristics improved by an enumerationschemerdquo International Journal of Production Research vol 55no 1 pp 176ndash186 2017

[8] S Melouk P Damodaran and P-Y Chang ldquoMinimizingmakespan for single machine batch processing with non-identical job sizes using simulated annealingrdquo InternationalJournal of Production Economics vol 87 no 2 pp 141ndash147 2004

[9] P Damodaran P Kumar Manjeshwar and K Srihari ldquoMin-imizing makespan on a batch-processing machine with non-identical job sizes using genetic algorithmsrdquo InternationalJournal of Production Economics vol 103 no 2 pp 882ndash8912006

[10] A Husseinzadeh Kashan M Husseinzadeh Kashan and SKarimiyan ldquoA particle swarm optimizer for grouping prob-lemsrdquo Information Sciences vol 252 pp 81ndash95 2013

[11] P Damodaran O Ghrayeb and M C Guttikonda ldquoGRASPto minimize makespan for a capacitated batch-processingmachinerdquoThe International Journal of AdvancedManufacturingTechnology vol 68 no 1-4 pp 407ndash414 2013

[12] S Zhou M Liu H Chen and X Li ldquoAn effective discretedifferential evolution algorithm for scheduling uniform parallelbatch processing machines with non-identical capacities andarbitrary job sizesrdquo International Journal of Production Eco-nomics vol 179 pp 1ndash11 2016

[13] L Jiang J Pei X Liu P M Pardalos Y Yang and XQian ldquoUniformparallel batchmachines scheduling consideringtransportation using a hybrid DPSO-GA algorithmrdquoThe Inter-national Journal of AdvancedManufacturing Technology vol 89no 5-8 pp 1887ndash1900 2017

[14] T Stutzle and H H Hoos ldquoMax-min ant systemrdquo FutureGeneration Computer Systems vol 16 no 8 pp 889ndash914 2000

[15] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[16] M Dorigo and C Blum ldquoAnt colony optimization theory asurveyrdquoTheoretical Computer Science vol 344 no 2-3 pp 243ndash278 2005

[17] R Xu H Chen and X Li ldquoMakespan minimization onsingle batch-processing machine via ant colony optimizationrdquoComputers amp Operations Research vol 39 no 3 pp 582ndash5932012

[18] L DrsquoAcierno M Gallo and B Montella ldquoAn ant colony optimi-sation algorithm for solving the asymmetric traffic assignmentproblemrdquo European Journal of Operational Research vol 217 no2 pp 459ndash469 2012

[19] D S Johnson A Demers J D Ullman M R Garey and RL Graham ldquoWorst-case performance bounds for simple one-dimensional packing algorithmsrdquo SIAM Journal on Computingvol 3 pp 299ndash325 1974

[20] M Pirlot ldquoGeneral local search methodsrdquo European Journal ofOperational Research vol 92 no 3 pp 493ndash511 1996

10 Mathematical Problems in Engineering

[21] B-Y Cheng H-P Chen and S-S Wang ldquoImproved ant colonyoptimization method for single batch-processing machine withnon-identical job sizesrdquo Journal of System Simulation vol 21 no9 pp 2687ndash2695 2009

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4 Mathematical Problems in Engineering

35 Local Search Algorithm Solution qualities can be effec-tively improved when local search methods are applied inmetaheuristics [20] That is because greedy strategies areusually adopted in local search methods and local optimalcan be easily found in the neighborhood of a given solutionMetaheuristicsrsquo ability in global optimal searching can beenhanced by combining their advantages in exploring solu-tion space with local search methods

As batch processing time is determined by the job withthe longest processing time in the batch jobs with smallerprocessing time have no effect on batch processing timeLocal search method can be applied to decrease batchprocessing time by batching jobs with similar processing timetogether

Definition 1 The job 119894 with longest processing time in batch119887 is called dominant job 119889119887 The total processing time of jobswithout dominant job is denoted as dominated job processingtime noted as119863119879Proposition 2 Themakespan will be minimized by maximiz-ing119863119879Proof We have 119862max = sum119887isin119861max119901119895 | 119895 isin 119887 for a givensolution 119861 According to Definition 1 119863119879 = sum119887isin119861(sum119895isin119887 119901119895 minusmax119901119895 | 119895 isin 119887) = sum119887isin119861sum119895isin119887 119901119895 minus sum119887isin119861max119901119895 | 119895 isin 119887thus we have 119862max = sum119887isin119861sum119895isin119887 119901119895 minus 119863119879 As the total jobprocessing timesum119887isin119861sum119895isin119887 119901119895 is constant for a given instancethemakespanwill beminimizedwhenDT ismaximized

Definition 3 For each batch 119887 the sum of its remain-ing capacity and the size of dominant job 119889119887 is calledexchangeable capacity EC119887 we denote EC119887 = 119862 minus sum119894isin119887 119904119894 +119904119895=argmax119901119895|119895isin119887

Proposition 4 Given 119901119887 ⩾ 119901119902 larger DT will be obtained byexchange 119889119887 of batch 119887 with 119898 jobs in batch 119902 where sum119898119894=1 119904119894 le119864119862119902 and max119901119894 | 119894 = 1 119898 le max119901119895 | 119895 isin 119902 while the119862max does not increase

Proof Given two batches 119887 and 119902 119901119887 ⩾ 119901119902 jobs in batch119887 can be divided into three sets 119889119887 119898 jobs 119872 and otherjobs 119874 where sum119898119894=1 119904119894 le EC119902 and max119901119894 | 119894 = 1 119898 lemax119901119895 | 119895 isin 119902 Jobs in batch 119902 can be divided into sets 119889119902and other jobs1198741015840 Therefore we have 119862max = 119901119889119887 +119901119889119902 Afterthe exchange is applied to batches we have 1198621015840max = 119901119889119887 + 1199011015840119889119902 where 1199011015840119889119902 = max119901119894 | 119894 isin 119872 cup 1198741015840 As 119901119889119902 ge 119901119894 | 119894 isin119872 cup 1198741015840 the relation 1198621015840max le 119862max is satisfied According toProposition 2 there is 119862max = sum119895isin119887cup119902 119901119895 minus 119863119879 a minor 119862maxindicates a larger DT Proposition 4 holds

According to Proposition 4 multiple jobs can beexchanged iteratively between batches to decrease batchprocessing time For a given solution 119878 the number of batchesis limited by the total number of jobs where each job isgrouped as one batchThe detailed procedure of the proposedlocal search algorithmMJE (Multiple Jobs Exchange) is listedas follows

M

Batch k

Size

Time

p q

Batch k minus 1

Δkminus1 Δ k

Figure 1 Notations description of AlgorithmMJE

Order batches and jobs for a given solution S

according to Proposition 4

No

End

Yes

Start

K = n

Exchange dominant job of k with m jobs in k minus 1

Initialize parameter k = 0 n = |S|

Figure 2 Flow chart of algorithm MJE

Algorithm MJE (Multiple Jobs Exchange)

Step 1 For the iterative best solution 119878 arrange the batchesin decreasing order of their processing time and order jobs ofeach batch in decreasing order of job size

Step 2 Initialize parameter 119896 = 0 set 119899 = |119878| Δ 119896 isthe remaining capacity of batch 119896 and 119901 and 119902 denote thedominated job of batch 119896 minus 1 and batch 119896 respectivelyStep 3 If 119896 = 119899 exit else 119896 = 119896 + 1Step 4 Exchange the job 119902 of batch 119896 with119898 (119898 gt 0) jobs119872in batch 119896 minus 1 if max119901119894 | 119894 isin 119872 le 119901119902sum119894isin119872 119904119894 le 119904119902 + Δ 119896 and119904119902 le sum119894isin119872 119904119894 + Δ 119896minus1 go to Step 3

Corresponding notations are illustrated in Figure 1Theflowchart of algorithmMJE is provided as in Figure 2

36 Algorithm Framework of MMAS with Local SearchAccording to basic framework of ant algorithms [16] theframework of MMAS to solve the problem under study isgiven as follows

Algorithm MMAS

Step 1 Initialize parameters in MMAS including 120572 120573 120574 and120591(0)

Mathematical Problems in Engineering 5

Start

Initialize ant population A job set J

For each Ant a in A

Existing jobs can be added to current batch

For each job j in J

Add job j to the current batch according to state transition probability

Yes

Create a new batch for ant a

Yes

No

All jobs have been assigned

Yes

Solutions have beenconstructed for all ants

NoNo

Yes

Apply local search method MJE

Update pheromone trails

Nc =

Yes

No

Output the global optimal solution

End

Initialize parameters (0) NcNcＧＲ

NcＧＲ

Figure 3 Flow chart of algorithmMMAS

Step 2 Initialize ant population

Step 3 Select a unscheduled job for each ant 119886 and add thejob to a new batch

Step 4 Arrange the next unscheduled feasible job with themaximum state transition probability calculated by (10) intothe current batch

Step 5 If there are existing jobs that can be added to thecurrent batch then go to Step 4Step 6 If there are jobs unscheduled then go to Step 3Step 7 Apply local search methodMJE to iteration solutions

Step 8 Update pheromone trails according to formula (11)

Step 9 If the termination condition is satisfied then outputthe solution Else go to Step 2

To illustrate the procedure of MMAS more logically theflow chart of algorithmMMAS is provided as in Figure 3

4 Experimentation

41 Experimental Design Random instances were generatedto verify the effectiveness of the proposed algorithm Threefactors were considered in the numeric experiment includingnumber of jobs 119869 job processing time 119875 and job size 11987824 (4 lowast 2 lowast 3) problem categories were generated anddenoted in the form of 119869119894119875119895119878119896 (119894 = 1 2 3 4 119895 = 1 2119896 = 1 2 3) Factors and levels are shown in Table 1The machine capacity is assumed to be 10 for all probleminstances

For example 119869111987521198783 represented the category ofinstances with 10 jobs jobsrsquo processing time were randomlygenerated from a discrete uniform [1 20] distribution andjobsrsquo sizes were randomly generated from a discrete uniform[4 8] distribution

6 Mathematical Problems in Engineering

gt660

lt645645ndash650650ndash655655ndash660

Cmax

S1 categories S2 categories S3 categories

10987654321

2 3 4 5 6 7 8 9 101

123456789

10

2 3 4 5 6 7 8 9 101

10987654321

2 3 4 5 6 7 8 9 101

gt360

lt345345ndash350350ndash355355ndash360

Cmax

gt1560

lt15421542ndash15501550ndash15551555ndash1560

Cmax

Figure 4 Contour lines of 119862max influenced by parameters 120572 120573 and 120574 for different problem categories

Table 1 Factors and levels

Factors Levels119869 1198691 = 10 1198692 = 20 1198693 = 50 1198694 = 100119875 1198751 119880[1 10] 1198752 119880[1 20]119878 1198781 119880[1 10] 1198782 119880[2 4] 1198783 119880[4 8]119880means data are generated from discrete uniform distribution

42 Parameter Tuning As each ant will build feasible solu-tions with probability a large population of ants will enhancethe algorithmrsquos ability in exploring solution space while moretimes of iteration help to make better use of pheromonetrails and heuristic information However the algorithm ismuch more time consuming with larger number of ants anditerations Preliminary tests were done and the results showedthat to increase the number of iteration yields better resultsin a given run time That is easy to be understood as thealgorithm inclines to perform randomly search in the initialstages and more empirical results are considered along withthe increasing of iterations To obtain a tradeoff betweensolution quality and time cost ant population was set to 30and iterations were set to 80 in this study

It can be seen from formula (10) that there is exponentialrelationship between state transition probability pheromonetrails and heuristic informationThus the probability ismoresensitive to these factors with larger parameters 120572 120573 and 120574To verify the influence of these parameters preliminary testswere done to choose the parameters levels as well 120572 = 1 wasused as a reference level to study the effect of parameters 120573and 120574 on makespan in the interval of [1 10] The results areshown in Figure 4

It can be observed from Figure 4 that medium 120573 andsmaller 120574 should be used for instances of 1198782 categories withsmall jobs to obtain bettermakespanwhile the larger 120574 shouldbe adopted for instances of 1198783 categorieswith large jobs on thecontrary For instances of 1198781 categories with mixed job sizessmaller values for 120573 and medium 120574 seem better compared

Table 2 Values for parameters

Run code 120573 1205741198781 3 51198782 6 31198783 6 8

with the former two categories All instances generate badresults when 120573 is too close to 1 Large size jobs usually havelower flexibility when arranged in batches compared withsmall size jobs So high level of 120574 is used to arrange large sizejobs first with high probability for instances with large jobsSimilarly big 120573 is used to batch jobs with similar processingtime together to increase the efficiency of batch processingtime According to the analysis above three levels for eachparameter are selected in this study as shown in Table 2

Pheromone trails evaporate at each iteration and thespeed is controlled by the parameter 120588 (1 minus 120588) meansevaporation rate A high evaporation rate leads to a constantchange of pheromone trails on each solution element while alow rate results in the pheromone trails on solution elementsthat cannot evaporate in time

Pheromone trails of each solution element is limitedin the interval of [120591min 120591max] in MMAS algorithm Thepheromone trails are usually set to a high value of 120591max atthe beginning of running in order to improve the searchingability in solution space If a solution element is always in thestate of evaporation and its pheromone trail decreases to 120591minjust after 119873119888 iterations then we have 120591min = 120591max sdot 120588119873119888 Since120591min = 120591max2119899 as mentioned above it can be derived that120588119873119888 = 12119899 that is 119873119888 = log12119899120588 0 lt 120588 lt 1 where 119899 is thenumber of jobs The relationship between 119873119888 and 120588 for 100jobs is described in Figure 5

It can be seen from Figure 5 that 119873119888 increases dramati-cally when 120588 gt 06 thus 120588 is set to 06 in this study in orderto ensure the pheromone trails evaporate not too quickly or

Mathematical Problems in Engineering 7

Table 3 Results of MMAS compared with CPLEX

Problem category MMAS CPLEXMin Max Avg Optimal () AvgTime (s) AvgTime (s)

119878111986911198751 2000 6000 3664 10000 011 06411986911198752 3300 12700 7198 10000 010 06311986921198751 4300 10500 6913 10000 033 11811986921198752 8100 19800 13170 9600 033 142119878211986911198751 1300 2900 2090 10000 022 04411986911198752 2300 5800 4024 10000 022 04511986921198751 2800 4800 3797 10000 081 280411986921198752 4600 9400 7194 9900 082 2690119878311986911198751 2700 6400 4412 10000 008 05911986911198752 5000 12400 8553 10000 008 05811986921198751 5600 12200 8784 10000 023 10711986921198752 10600 21700 16718 10000 023 112

n = 50n = 100n = 200

0010203040506070809

1

Evap

orat

ion

para

met

er (

)

10 20 30 40 50 600Maximum number of iterations (Nc)

Figure 5 Relationship between119873119888 and 120588

slowly and the solution space can be explored in a reasonablerun time

43 Performance Comparison for Small Scale Instances Asshown in Table 3 the results obtained from MMAS iscompared with that from CPLEX (a commercial solverfor linear and mixed-integer problems) Due to the NP-hardness of the problem only small scale instances of 1198691and 1198692 problem categories are solved by using CPLEX Theminimum maximum and average value of 119862max reportedfrom MMAS are listed in columns 3 to 5 The average runtime of MMAS and CPLEXT are given in columns 7 and8 Column 6 indicates the percent of the optimal valuesobtained by using MMAS compared with the results fromCPLEX

Nearly all small scale instances can be solved optimallyby using MMAS except several instances from 119869211987521198781 and119869211987521198782 with relative larger solution space MMAS is com-petitive in computational time compared with CPLEX for allinstances Computational time of all instances fromMMAS isless than 1 second The computational time of CPLEX variesgreatly depending on the solution space of an instance

44 Algorithm Performance Evaluation To evaluate the per-formance of the algorithm for all problem categories MMASdesigned in this paper was comparedwithGA [9] A basic antalgorithm ant cycle (AC) was coded to solve the problem aswell and the parameters used were as prescribed in Chenget al [21] Two well-known heuristics FFLPT and BFLPTwere also compared withMMAS in the experiment study 100instances were generated for each problem category and thebest result of 10 runs for each instance was eventually used

Comparison of various algorithms is summarized inTable 4 Column 1 in Table 4 presents the run code Column 2shows the results of MMAS compared with other algorithmswhere B E and I mean themakespan obtained byMMAS arebetter than equal to or inferior to that of other algorithmsrespectively Columns 3ndash6 report the proportion that themakespan obtained by using MMAS compared with that ofother algorithms in 100 instances of 1198781 problem categoriesColumns 7ndash10 and columns 11ndash14 are similar to columns 3ndash6For example the first number of 001 indicates that there isa proportion of 001 results reported by MMAS better thanthat generated by AC As the problem under study is stronglyNP-hard the solution space is to explode along with thepopulation size increasing form 1198691 to 1198694 On the other handsmaller job sizes give more combination of batches thus thesolution space becomes smaller for 1198783 problem categoriescompared with 1198781 for a given number of jobs

It can be seen from Table 4 that MMAS outperformsother algorithms on the whole Several results reported byMMAS inferior to those generated by other algorithms have

8 Mathematical Problems in Engineering

Table 4 Results of MMAS compared with other algorithms

Run code Cp 1198781 1198782 1198783AC GA BFLPT FFLPT AC GA BFLPT FFLPT AC GA BFLPT FFLPT

11986911198751 B 001 002 016 017 000 010 016 016 000 001 007 007E 099 098 084 083 100 090 084 084 100 099 093 093I 000 000 000 000 000 000 000 000 000 000 000 000

11986911198752 B 002 002 010 013 001 014 019 019 000 003 008 011E 098 098 090 087 099 086 081 081 100 097 092 089I 000 000 000 000 000 000 000 000 000 000 000 000

11986921198751 B 004 045 047 059 001 031 032 032 000 007 014 020E 096 055 053 041 099 069 068 068 100 093 086 080I 000 000 000 000 000 000 000 000 000 000 000 000

11986921198752 B 013 042 041 057 006 032 033 034 000 014 019 029E 087 058 059 043 094 068 067 066 100 086 081 071I 000 000 000 000 000 000 000 000 000 000 000 000

11986931198751 B 043 084 083 093 023 089 091 091 000 030 030 043E 057 016 017 007 075 011 009 009 100 070 070 057I 000 000 000 000 002 000 000 000 000 000 000 000

11986931198752 B 060 090 085 099 039 087 087 087 012 027 027 039E 040 010 015 001 060 013 013 013 088 073 073 061I 000 000 000 000 001 000 000 000 000 000 000 000

11986941198751 B 078 093 090 100 049 099 099 099 005 018 018 030E 021 006 009 000 049 001 001 001 095 082 082 070I 001 001 001 000 002 000 000 000 000 000 000 000

11986941198752 B 092 095 092 098 051 099 099 099 035 022 022 030E 005 004 006 002 043 000 000 000 065 078 078 070I 003 001 002 000 006 001 001 001 000 000 000 000

been earmarked by rectangle For 1198691 problem categoriesresults obtained from MMAS are mostly equal to that fromother algorithms A percentage of about 16 results fromMMAS is better than that fromalgorithmsBFLPT andFFLPTfor 1198781 and 1198782 problem categories The percentage is 7 for1198783 problem categories which is because more results arethe same for all algorithms in a small solution space Upto 14 reported by MMAS are better than GA for 119869111987521198783problem category The advantage of MMAS to algorithmslike GA BFLPT and FFLPT is much more remarkable for1198692 problem categories More better results can be found byMMAS than AC and the percentage is up to 6 to 13 for11986921198752 problem categories More than 80 results obtainedfrom MMAS are better than that from GA BFLPT andFFLPT for 11986931198781 and 11986931198782 problem categories About 23 to43 percent of results fromMMAS are better than AC whilethe latter reports few better results For 1198694 problem categoriesmore than 90 results generated by MMAS are better thanthat from algorithms AC GA BFLPT and FFLPT for largejob population instances Almost all other algorithms reportseveral better results than MMAS for 1198694 problem categories

Based on the analysis given above MMAS possesseshigher ability in solution space exploring It outperforms allother algorithms for almost all instances especially for largejob population As MMAS is exploring solution space inprobability the optimal solution cannot be guaranteed witha limited population and iterations Therefore several results

obtained from MMAS are inferior to other algorithms evento effective heuristics like BFLPT and FFLPT

The makespan increases for 1198752 problem categories com-pared with 1198751 problem categories but there is no significantperformance change for various algorithms

To illustrate the gap between MMAS and other algo-rithms for different population size the average percentage ofresults with better makespan obtained by MMAS comparedwith other algorithms is given in Figure 6

Averagely a percentage of 10 results obtained fromMMAS are better than that from other algorithms for 1198691problem categories while the number is 70 for 1198694 Algo-rithm AC is the second best algorithm compared to GA andthe other two heuristics BFLPT and FFLPT are all effectivein solving the problem especially for small population sizeBFLPT is generally better than FFLPT for almost all problemcategories and BFLPT is also used in GA to generate initialsolutions

5 Conclusions and Future Work

The problem of scheduling single batch processing machinewas studied in this paper An improved ant algorithmMMAS(MaxndashMin Ant System) was designed and applied to solvethe problem Parameters of MMAS were determined bypreliminary experiment and a local search method MJEwas also presented to improve the performance of MMAS

Mathematical Problems in Engineering 9

J1 J2 J3 J4

Number of jobs

ACGA

BFLPTFFLPT

01020304050607080

Perc

enta

ge (

)

Figure 6 Average percentage of results obtained from MMAScompared with that from other algorithms

Optimal objectives can be obtained by using MMAS in lesscomputational time compared to CPLEX for small scaleproblems To evaluate the performance of MMAS it wascompared with basic ant algorithm AC (Ant Cycle) GA(Genetic Algorithm) and the other two well-known heuris-tics of scheduling batch processing machine that is FFLPT(First Fit Longest Processing Time) and BFLPT (Best FitLongest Processing Time) The numeric experiment showedthat MMAS outperformed other algorithms for almost allinstances in various problem categories especially for largerpopulation size

Considering the good performance of MMAS betterheuristics mechanism can be designed and applied to otherproblems of scheduling batch processing machine Theproblem under study with other objectives and problemconstraints including release times and due dates can bestudied as well

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (no 2014QNA48) and NationalNatural Science Foundation of China (no 71401164)

References

[1] Y Ikura and M Gimple ldquoEfficient scheduling algorithms for asingle batch processing machinerdquo Operations Research Lettersvol 5 no 2 pp 61ndash65 1986

[2] C-Y Lee R Uzsoy and L AMartin-Vega ldquoEfficient algorithmsfor scheduling semiconductor burn-in operationsrdquo OperationsResearch vol 40 no 4 pp 764ndash775 1992

[3] C S Sung and Y I Choung ldquoMinimizing makespan on a singleburn-in oven in semiconductormanufacturingrdquoEuropean Jour-nal of Operational Research vol 120 no 3 pp 559ndash574 2000

[4] R Uzsoy ldquoScheduling a single batch processing machine withnon-identical job sizesrdquo International Journal of ProductionResearch vol 32 no 7 pp 1615ndash1635 1994

[5] L Dupont and F Jolai Ghazvini ldquoMinimizing makespan on asingle batch processing machine with non-identical job sizesrdquoJournal Europeen des Systemes Automatises vol 32 no 4 pp431ndash440 1998

[6] L Dupont and C Dhaenens-Flipo ldquoMinimizing the makespanon a batch machine with non-identical job sizes An exactprocedurerdquo Computers amp Operations Research vol 29 no 7 pp807ndash819 2002

[7] X Li Y Li andYWang ldquoMinimisingmakespan on a batch pro-cessing machine using heuristics improved by an enumerationschemerdquo International Journal of Production Research vol 55no 1 pp 176ndash186 2017

[8] S Melouk P Damodaran and P-Y Chang ldquoMinimizingmakespan for single machine batch processing with non-identical job sizes using simulated annealingrdquo InternationalJournal of Production Economics vol 87 no 2 pp 141ndash147 2004

[9] P Damodaran P Kumar Manjeshwar and K Srihari ldquoMin-imizing makespan on a batch-processing machine with non-identical job sizes using genetic algorithmsrdquo InternationalJournal of Production Economics vol 103 no 2 pp 882ndash8912006

[10] A Husseinzadeh Kashan M Husseinzadeh Kashan and SKarimiyan ldquoA particle swarm optimizer for grouping prob-lemsrdquo Information Sciences vol 252 pp 81ndash95 2013

[11] P Damodaran O Ghrayeb and M C Guttikonda ldquoGRASPto minimize makespan for a capacitated batch-processingmachinerdquoThe International Journal of AdvancedManufacturingTechnology vol 68 no 1-4 pp 407ndash414 2013

[12] S Zhou M Liu H Chen and X Li ldquoAn effective discretedifferential evolution algorithm for scheduling uniform parallelbatch processing machines with non-identical capacities andarbitrary job sizesrdquo International Journal of Production Eco-nomics vol 179 pp 1ndash11 2016

[13] L Jiang J Pei X Liu P M Pardalos Y Yang and XQian ldquoUniformparallel batchmachines scheduling consideringtransportation using a hybrid DPSO-GA algorithmrdquoThe Inter-national Journal of AdvancedManufacturing Technology vol 89no 5-8 pp 1887ndash1900 2017

[14] T Stutzle and H H Hoos ldquoMax-min ant systemrdquo FutureGeneration Computer Systems vol 16 no 8 pp 889ndash914 2000

[15] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[16] M Dorigo and C Blum ldquoAnt colony optimization theory asurveyrdquoTheoretical Computer Science vol 344 no 2-3 pp 243ndash278 2005

[17] R Xu H Chen and X Li ldquoMakespan minimization onsingle batch-processing machine via ant colony optimizationrdquoComputers amp Operations Research vol 39 no 3 pp 582ndash5932012

[18] L DrsquoAcierno M Gallo and B Montella ldquoAn ant colony optimi-sation algorithm for solving the asymmetric traffic assignmentproblemrdquo European Journal of Operational Research vol 217 no2 pp 459ndash469 2012

[19] D S Johnson A Demers J D Ullman M R Garey and RL Graham ldquoWorst-case performance bounds for simple one-dimensional packing algorithmsrdquo SIAM Journal on Computingvol 3 pp 299ndash325 1974

[20] M Pirlot ldquoGeneral local search methodsrdquo European Journal ofOperational Research vol 92 no 3 pp 493ndash511 1996

10 Mathematical Problems in Engineering

[21] B-Y Cheng H-P Chen and S-S Wang ldquoImproved ant colonyoptimization method for single batch-processing machine withnon-identical job sizesrdquo Journal of System Simulation vol 21 no9 pp 2687ndash2695 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 5

Start

Initialize ant population A job set J

For each Ant a in A

Existing jobs can be added to current batch

For each job j in J

Add job j to the current batch according to state transition probability

Yes

Create a new batch for ant a

Yes

No

All jobs have been assigned

Yes

Solutions have beenconstructed for all ants

NoNo

Yes

Apply local search method MJE

Update pheromone trails

Nc =

Yes

No

Output the global optimal solution

End

Initialize parameters (0) NcNcＧＲ

NcＧＲ

Figure 3 Flow chart of algorithmMMAS

Step 2 Initialize ant population

Step 3 Select a unscheduled job for each ant 119886 and add thejob to a new batch

Step 4 Arrange the next unscheduled feasible job with themaximum state transition probability calculated by (10) intothe current batch

Step 5 If there are existing jobs that can be added to thecurrent batch then go to Step 4Step 6 If there are jobs unscheduled then go to Step 3Step 7 Apply local search methodMJE to iteration solutions

Step 8 Update pheromone trails according to formula (11)

Step 9 If the termination condition is satisfied then outputthe solution Else go to Step 2

To illustrate the procedure of MMAS more logically theflow chart of algorithmMMAS is provided as in Figure 3

4 Experimentation

41 Experimental Design Random instances were generatedto verify the effectiveness of the proposed algorithm Threefactors were considered in the numeric experiment includingnumber of jobs 119869 job processing time 119875 and job size 11987824 (4 lowast 2 lowast 3) problem categories were generated anddenoted in the form of 119869119894119875119895119878119896 (119894 = 1 2 3 4 119895 = 1 2119896 = 1 2 3) Factors and levels are shown in Table 1The machine capacity is assumed to be 10 for all probleminstances

For example 119869111987521198783 represented the category ofinstances with 10 jobs jobsrsquo processing time were randomlygenerated from a discrete uniform [1 20] distribution andjobsrsquo sizes were randomly generated from a discrete uniform[4 8] distribution

6 Mathematical Problems in Engineering

gt660

lt645645ndash650650ndash655655ndash660

Cmax

S1 categories S2 categories S3 categories

10987654321

2 3 4 5 6 7 8 9 101

123456789

10

2 3 4 5 6 7 8 9 101

10987654321

2 3 4 5 6 7 8 9 101

gt360

lt345345ndash350350ndash355355ndash360

Cmax

gt1560

lt15421542ndash15501550ndash15551555ndash1560

Cmax

Figure 4 Contour lines of 119862max influenced by parameters 120572 120573 and 120574 for different problem categories

Table 1 Factors and levels

Factors Levels119869 1198691 = 10 1198692 = 20 1198693 = 50 1198694 = 100119875 1198751 119880[1 10] 1198752 119880[1 20]119878 1198781 119880[1 10] 1198782 119880[2 4] 1198783 119880[4 8]119880means data are generated from discrete uniform distribution

42 Parameter Tuning As each ant will build feasible solu-tions with probability a large population of ants will enhancethe algorithmrsquos ability in exploring solution space while moretimes of iteration help to make better use of pheromonetrails and heuristic information However the algorithm ismuch more time consuming with larger number of ants anditerations Preliminary tests were done and the results showedthat to increase the number of iteration yields better resultsin a given run time That is easy to be understood as thealgorithm inclines to perform randomly search in the initialstages and more empirical results are considered along withthe increasing of iterations To obtain a tradeoff betweensolution quality and time cost ant population was set to 30and iterations were set to 80 in this study

It can be seen from formula (10) that there is exponentialrelationship between state transition probability pheromonetrails and heuristic informationThus the probability ismoresensitive to these factors with larger parameters 120572 120573 and 120574To verify the influence of these parameters preliminary testswere done to choose the parameters levels as well 120572 = 1 wasused as a reference level to study the effect of parameters 120573and 120574 on makespan in the interval of [1 10] The results areshown in Figure 4

It can be observed from Figure 4 that medium 120573 andsmaller 120574 should be used for instances of 1198782 categories withsmall jobs to obtain bettermakespanwhile the larger 120574 shouldbe adopted for instances of 1198783 categorieswith large jobs on thecontrary For instances of 1198781 categories with mixed job sizessmaller values for 120573 and medium 120574 seem better compared

Table 2 Values for parameters

Run code 120573 1205741198781 3 51198782 6 31198783 6 8

with the former two categories All instances generate badresults when 120573 is too close to 1 Large size jobs usually havelower flexibility when arranged in batches compared withsmall size jobs So high level of 120574 is used to arrange large sizejobs first with high probability for instances with large jobsSimilarly big 120573 is used to batch jobs with similar processingtime together to increase the efficiency of batch processingtime According to the analysis above three levels for eachparameter are selected in this study as shown in Table 2

Pheromone trails evaporate at each iteration and thespeed is controlled by the parameter 120588 (1 minus 120588) meansevaporation rate A high evaporation rate leads to a constantchange of pheromone trails on each solution element while alow rate results in the pheromone trails on solution elementsthat cannot evaporate in time

Pheromone trails of each solution element is limitedin the interval of [120591min 120591max] in MMAS algorithm Thepheromone trails are usually set to a high value of 120591max atthe beginning of running in order to improve the searchingability in solution space If a solution element is always in thestate of evaporation and its pheromone trail decreases to 120591minjust after 119873119888 iterations then we have 120591min = 120591max sdot 120588119873119888 Since120591min = 120591max2119899 as mentioned above it can be derived that120588119873119888 = 12119899 that is 119873119888 = log12119899120588 0 lt 120588 lt 1 where 119899 is thenumber of jobs The relationship between 119873119888 and 120588 for 100jobs is described in Figure 5

It can be seen from Figure 5 that 119873119888 increases dramati-cally when 120588 gt 06 thus 120588 is set to 06 in this study in orderto ensure the pheromone trails evaporate not too quickly or

Mathematical Problems in Engineering 7

Table 3 Results of MMAS compared with CPLEX

Problem category MMAS CPLEXMin Max Avg Optimal () AvgTime (s) AvgTime (s)

119878111986911198751 2000 6000 3664 10000 011 06411986911198752 3300 12700 7198 10000 010 06311986921198751 4300 10500 6913 10000 033 11811986921198752 8100 19800 13170 9600 033 142119878211986911198751 1300 2900 2090 10000 022 04411986911198752 2300 5800 4024 10000 022 04511986921198751 2800 4800 3797 10000 081 280411986921198752 4600 9400 7194 9900 082 2690119878311986911198751 2700 6400 4412 10000 008 05911986911198752 5000 12400 8553 10000 008 05811986921198751 5600 12200 8784 10000 023 10711986921198752 10600 21700 16718 10000 023 112

n = 50n = 100n = 200

0010203040506070809

1

Evap

orat

ion

para

met

er (

)

10 20 30 40 50 600Maximum number of iterations (Nc)

Figure 5 Relationship between119873119888 and 120588

slowly and the solution space can be explored in a reasonablerun time

43 Performance Comparison for Small Scale Instances Asshown in Table 3 the results obtained from MMAS iscompared with that from CPLEX (a commercial solverfor linear and mixed-integer problems) Due to the NP-hardness of the problem only small scale instances of 1198691and 1198692 problem categories are solved by using CPLEX Theminimum maximum and average value of 119862max reportedfrom MMAS are listed in columns 3 to 5 The average runtime of MMAS and CPLEXT are given in columns 7 and8 Column 6 indicates the percent of the optimal valuesobtained by using MMAS compared with the results fromCPLEX

Nearly all small scale instances can be solved optimallyby using MMAS except several instances from 119869211987521198781 and119869211987521198782 with relative larger solution space MMAS is com-petitive in computational time compared with CPLEX for allinstances Computational time of all instances fromMMAS isless than 1 second The computational time of CPLEX variesgreatly depending on the solution space of an instance

44 Algorithm Performance Evaluation To evaluate the per-formance of the algorithm for all problem categories MMASdesigned in this paper was comparedwithGA [9] A basic antalgorithm ant cycle (AC) was coded to solve the problem aswell and the parameters used were as prescribed in Chenget al [21] Two well-known heuristics FFLPT and BFLPTwere also compared withMMAS in the experiment study 100instances were generated for each problem category and thebest result of 10 runs for each instance was eventually used

Comparison of various algorithms is summarized inTable 4 Column 1 in Table 4 presents the run code Column 2shows the results of MMAS compared with other algorithmswhere B E and I mean themakespan obtained byMMAS arebetter than equal to or inferior to that of other algorithmsrespectively Columns 3ndash6 report the proportion that themakespan obtained by using MMAS compared with that ofother algorithms in 100 instances of 1198781 problem categoriesColumns 7ndash10 and columns 11ndash14 are similar to columns 3ndash6For example the first number of 001 indicates that there isa proportion of 001 results reported by MMAS better thanthat generated by AC As the problem under study is stronglyNP-hard the solution space is to explode along with thepopulation size increasing form 1198691 to 1198694 On the other handsmaller job sizes give more combination of batches thus thesolution space becomes smaller for 1198783 problem categoriescompared with 1198781 for a given number of jobs

It can be seen from Table 4 that MMAS outperformsother algorithms on the whole Several results reported byMMAS inferior to those generated by other algorithms have

8 Mathematical Problems in Engineering

Table 4 Results of MMAS compared with other algorithms

Run code Cp 1198781 1198782 1198783AC GA BFLPT FFLPT AC GA BFLPT FFLPT AC GA BFLPT FFLPT

11986911198751 B 001 002 016 017 000 010 016 016 000 001 007 007E 099 098 084 083 100 090 084 084 100 099 093 093I 000 000 000 000 000 000 000 000 000 000 000 000

11986911198752 B 002 002 010 013 001 014 019 019 000 003 008 011E 098 098 090 087 099 086 081 081 100 097 092 089I 000 000 000 000 000 000 000 000 000 000 000 000

11986921198751 B 004 045 047 059 001 031 032 032 000 007 014 020E 096 055 053 041 099 069 068 068 100 093 086 080I 000 000 000 000 000 000 000 000 000 000 000 000

11986921198752 B 013 042 041 057 006 032 033 034 000 014 019 029E 087 058 059 043 094 068 067 066 100 086 081 071I 000 000 000 000 000 000 000 000 000 000 000 000

11986931198751 B 043 084 083 093 023 089 091 091 000 030 030 043E 057 016 017 007 075 011 009 009 100 070 070 057I 000 000 000 000 002 000 000 000 000 000 000 000

11986931198752 B 060 090 085 099 039 087 087 087 012 027 027 039E 040 010 015 001 060 013 013 013 088 073 073 061I 000 000 000 000 001 000 000 000 000 000 000 000

11986941198751 B 078 093 090 100 049 099 099 099 005 018 018 030E 021 006 009 000 049 001 001 001 095 082 082 070I 001 001 001 000 002 000 000 000 000 000 000 000

11986941198752 B 092 095 092 098 051 099 099 099 035 022 022 030E 005 004 006 002 043 000 000 000 065 078 078 070I 003 001 002 000 006 001 001 001 000 000 000 000

been earmarked by rectangle For 1198691 problem categoriesresults obtained from MMAS are mostly equal to that fromother algorithms A percentage of about 16 results fromMMAS is better than that fromalgorithmsBFLPT andFFLPTfor 1198781 and 1198782 problem categories The percentage is 7 for1198783 problem categories which is because more results arethe same for all algorithms in a small solution space Upto 14 reported by MMAS are better than GA for 119869111987521198783problem category The advantage of MMAS to algorithmslike GA BFLPT and FFLPT is much more remarkable for1198692 problem categories More better results can be found byMMAS than AC and the percentage is up to 6 to 13 for11986921198752 problem categories More than 80 results obtainedfrom MMAS are better than that from GA BFLPT andFFLPT for 11986931198781 and 11986931198782 problem categories About 23 to43 percent of results fromMMAS are better than AC whilethe latter reports few better results For 1198694 problem categoriesmore than 90 results generated by MMAS are better thanthat from algorithms AC GA BFLPT and FFLPT for largejob population instances Almost all other algorithms reportseveral better results than MMAS for 1198694 problem categories

Based on the analysis given above MMAS possesseshigher ability in solution space exploring It outperforms allother algorithms for almost all instances especially for largejob population As MMAS is exploring solution space inprobability the optimal solution cannot be guaranteed witha limited population and iterations Therefore several results

obtained from MMAS are inferior to other algorithms evento effective heuristics like BFLPT and FFLPT

The makespan increases for 1198752 problem categories com-pared with 1198751 problem categories but there is no significantperformance change for various algorithms

To illustrate the gap between MMAS and other algo-rithms for different population size the average percentage ofresults with better makespan obtained by MMAS comparedwith other algorithms is given in Figure 6

Averagely a percentage of 10 results obtained fromMMAS are better than that from other algorithms for 1198691problem categories while the number is 70 for 1198694 Algo-rithm AC is the second best algorithm compared to GA andthe other two heuristics BFLPT and FFLPT are all effectivein solving the problem especially for small population sizeBFLPT is generally better than FFLPT for almost all problemcategories and BFLPT is also used in GA to generate initialsolutions

5 Conclusions and Future Work

The problem of scheduling single batch processing machinewas studied in this paper An improved ant algorithmMMAS(MaxndashMin Ant System) was designed and applied to solvethe problem Parameters of MMAS were determined bypreliminary experiment and a local search method MJEwas also presented to improve the performance of MMAS

Mathematical Problems in Engineering 9

J1 J2 J3 J4

Number of jobs

ACGA

BFLPTFFLPT

01020304050607080

Perc

enta

ge (

)

Figure 6 Average percentage of results obtained from MMAScompared with that from other algorithms

Optimal objectives can be obtained by using MMAS in lesscomputational time compared to CPLEX for small scaleproblems To evaluate the performance of MMAS it wascompared with basic ant algorithm AC (Ant Cycle) GA(Genetic Algorithm) and the other two well-known heuris-tics of scheduling batch processing machine that is FFLPT(First Fit Longest Processing Time) and BFLPT (Best FitLongest Processing Time) The numeric experiment showedthat MMAS outperformed other algorithms for almost allinstances in various problem categories especially for largerpopulation size

Considering the good performance of MMAS betterheuristics mechanism can be designed and applied to otherproblems of scheduling batch processing machine Theproblem under study with other objectives and problemconstraints including release times and due dates can bestudied as well

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (no 2014QNA48) and NationalNatural Science Foundation of China (no 71401164)

References

[1] Y Ikura and M Gimple ldquoEfficient scheduling algorithms for asingle batch processing machinerdquo Operations Research Lettersvol 5 no 2 pp 61ndash65 1986

[2] C-Y Lee R Uzsoy and L AMartin-Vega ldquoEfficient algorithmsfor scheduling semiconductor burn-in operationsrdquo OperationsResearch vol 40 no 4 pp 764ndash775 1992

[3] C S Sung and Y I Choung ldquoMinimizing makespan on a singleburn-in oven in semiconductormanufacturingrdquoEuropean Jour-nal of Operational Research vol 120 no 3 pp 559ndash574 2000

[4] R Uzsoy ldquoScheduling a single batch processing machine withnon-identical job sizesrdquo International Journal of ProductionResearch vol 32 no 7 pp 1615ndash1635 1994

[5] L Dupont and F Jolai Ghazvini ldquoMinimizing makespan on asingle batch processing machine with non-identical job sizesrdquoJournal Europeen des Systemes Automatises vol 32 no 4 pp431ndash440 1998

[6] L Dupont and C Dhaenens-Flipo ldquoMinimizing the makespanon a batch machine with non-identical job sizes An exactprocedurerdquo Computers amp Operations Research vol 29 no 7 pp807ndash819 2002

[7] X Li Y Li andYWang ldquoMinimisingmakespan on a batch pro-cessing machine using heuristics improved by an enumerationschemerdquo International Journal of Production Research vol 55no 1 pp 176ndash186 2017

[8] S Melouk P Damodaran and P-Y Chang ldquoMinimizingmakespan for single machine batch processing with non-identical job sizes using simulated annealingrdquo InternationalJournal of Production Economics vol 87 no 2 pp 141ndash147 2004

[9] P Damodaran P Kumar Manjeshwar and K Srihari ldquoMin-imizing makespan on a batch-processing machine with non-identical job sizes using genetic algorithmsrdquo InternationalJournal of Production Economics vol 103 no 2 pp 882ndash8912006

[10] A Husseinzadeh Kashan M Husseinzadeh Kashan and SKarimiyan ldquoA particle swarm optimizer for grouping prob-lemsrdquo Information Sciences vol 252 pp 81ndash95 2013

[11] P Damodaran O Ghrayeb and M C Guttikonda ldquoGRASPto minimize makespan for a capacitated batch-processingmachinerdquoThe International Journal of AdvancedManufacturingTechnology vol 68 no 1-4 pp 407ndash414 2013

[12] S Zhou M Liu H Chen and X Li ldquoAn effective discretedifferential evolution algorithm for scheduling uniform parallelbatch processing machines with non-identical capacities andarbitrary job sizesrdquo International Journal of Production Eco-nomics vol 179 pp 1ndash11 2016

[13] L Jiang J Pei X Liu P M Pardalos Y Yang and XQian ldquoUniformparallel batchmachines scheduling consideringtransportation using a hybrid DPSO-GA algorithmrdquoThe Inter-national Journal of AdvancedManufacturing Technology vol 89no 5-8 pp 1887ndash1900 2017

[14] T Stutzle and H H Hoos ldquoMax-min ant systemrdquo FutureGeneration Computer Systems vol 16 no 8 pp 889ndash914 2000

[15] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[16] M Dorigo and C Blum ldquoAnt colony optimization theory asurveyrdquoTheoretical Computer Science vol 344 no 2-3 pp 243ndash278 2005

[17] R Xu H Chen and X Li ldquoMakespan minimization onsingle batch-processing machine via ant colony optimizationrdquoComputers amp Operations Research vol 39 no 3 pp 582ndash5932012

[18] L DrsquoAcierno M Gallo and B Montella ldquoAn ant colony optimi-sation algorithm for solving the asymmetric traffic assignmentproblemrdquo European Journal of Operational Research vol 217 no2 pp 459ndash469 2012

[19] D S Johnson A Demers J D Ullman M R Garey and RL Graham ldquoWorst-case performance bounds for simple one-dimensional packing algorithmsrdquo SIAM Journal on Computingvol 3 pp 299ndash325 1974

[20] M Pirlot ldquoGeneral local search methodsrdquo European Journal ofOperational Research vol 92 no 3 pp 493ndash511 1996

10 Mathematical Problems in Engineering

[21] B-Y Cheng H-P Chen and S-S Wang ldquoImproved ant colonyoptimization method for single batch-processing machine withnon-identical job sizesrdquo Journal of System Simulation vol 21 no9 pp 2687ndash2695 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Mathematical Problems in Engineering

gt660

lt645645ndash650650ndash655655ndash660

Cmax

S1 categories S2 categories S3 categories

10987654321

2 3 4 5 6 7 8 9 101

123456789

10

2 3 4 5 6 7 8 9 101

10987654321

2 3 4 5 6 7 8 9 101

gt360

lt345345ndash350350ndash355355ndash360

Cmax

gt1560

lt15421542ndash15501550ndash15551555ndash1560

Cmax

Figure 4 Contour lines of 119862max influenced by parameters 120572 120573 and 120574 for different problem categories

Table 1 Factors and levels

Factors Levels119869 1198691 = 10 1198692 = 20 1198693 = 50 1198694 = 100119875 1198751 119880[1 10] 1198752 119880[1 20]119878 1198781 119880[1 10] 1198782 119880[2 4] 1198783 119880[4 8]119880means data are generated from discrete uniform distribution

42 Parameter Tuning As each ant will build feasible solu-tions with probability a large population of ants will enhancethe algorithmrsquos ability in exploring solution space while moretimes of iteration help to make better use of pheromonetrails and heuristic information However the algorithm ismuch more time consuming with larger number of ants anditerations Preliminary tests were done and the results showedthat to increase the number of iteration yields better resultsin a given run time That is easy to be understood as thealgorithm inclines to perform randomly search in the initialstages and more empirical results are considered along withthe increasing of iterations To obtain a tradeoff betweensolution quality and time cost ant population was set to 30and iterations were set to 80 in this study

It can be seen from formula (10) that there is exponentialrelationship between state transition probability pheromonetrails and heuristic informationThus the probability ismoresensitive to these factors with larger parameters 120572 120573 and 120574To verify the influence of these parameters preliminary testswere done to choose the parameters levels as well 120572 = 1 wasused as a reference level to study the effect of parameters 120573and 120574 on makespan in the interval of [1 10] The results areshown in Figure 4

It can be observed from Figure 4 that medium 120573 andsmaller 120574 should be used for instances of 1198782 categories withsmall jobs to obtain bettermakespanwhile the larger 120574 shouldbe adopted for instances of 1198783 categorieswith large jobs on thecontrary For instances of 1198781 categories with mixed job sizessmaller values for 120573 and medium 120574 seem better compared

Table 2 Values for parameters

Run code 120573 1205741198781 3 51198782 6 31198783 6 8

with the former two categories All instances generate badresults when 120573 is too close to 1 Large size jobs usually havelower flexibility when arranged in batches compared withsmall size jobs So high level of 120574 is used to arrange large sizejobs first with high probability for instances with large jobsSimilarly big 120573 is used to batch jobs with similar processingtime together to increase the efficiency of batch processingtime According to the analysis above three levels for eachparameter are selected in this study as shown in Table 2

Pheromone trails evaporate at each iteration and thespeed is controlled by the parameter 120588 (1 minus 120588) meansevaporation rate A high evaporation rate leads to a constantchange of pheromone trails on each solution element while alow rate results in the pheromone trails on solution elementsthat cannot evaporate in time

Pheromone trails of each solution element is limitedin the interval of [120591min 120591max] in MMAS algorithm Thepheromone trails are usually set to a high value of 120591max atthe beginning of running in order to improve the searchingability in solution space If a solution element is always in thestate of evaporation and its pheromone trail decreases to 120591minjust after 119873119888 iterations then we have 120591min = 120591max sdot 120588119873119888 Since120591min = 120591max2119899 as mentioned above it can be derived that120588119873119888 = 12119899 that is 119873119888 = log12119899120588 0 lt 120588 lt 1 where 119899 is thenumber of jobs The relationship between 119873119888 and 120588 for 100jobs is described in Figure 5

It can be seen from Figure 5 that 119873119888 increases dramati-cally when 120588 gt 06 thus 120588 is set to 06 in this study in orderto ensure the pheromone trails evaporate not too quickly or

Mathematical Problems in Engineering 7

Table 3 Results of MMAS compared with CPLEX

Problem category MMAS CPLEXMin Max Avg Optimal () AvgTime (s) AvgTime (s)

119878111986911198751 2000 6000 3664 10000 011 06411986911198752 3300 12700 7198 10000 010 06311986921198751 4300 10500 6913 10000 033 11811986921198752 8100 19800 13170 9600 033 142119878211986911198751 1300 2900 2090 10000 022 04411986911198752 2300 5800 4024 10000 022 04511986921198751 2800 4800 3797 10000 081 280411986921198752 4600 9400 7194 9900 082 2690119878311986911198751 2700 6400 4412 10000 008 05911986911198752 5000 12400 8553 10000 008 05811986921198751 5600 12200 8784 10000 023 10711986921198752 10600 21700 16718 10000 023 112

n = 50n = 100n = 200

0010203040506070809

1

Evap

orat

ion

para

met

er (

)

10 20 30 40 50 600Maximum number of iterations (Nc)

Figure 5 Relationship between119873119888 and 120588

slowly and the solution space can be explored in a reasonablerun time

43 Performance Comparison for Small Scale Instances Asshown in Table 3 the results obtained from MMAS iscompared with that from CPLEX (a commercial solverfor linear and mixed-integer problems) Due to the NP-hardness of the problem only small scale instances of 1198691and 1198692 problem categories are solved by using CPLEX Theminimum maximum and average value of 119862max reportedfrom MMAS are listed in columns 3 to 5 The average runtime of MMAS and CPLEXT are given in columns 7 and8 Column 6 indicates the percent of the optimal valuesobtained by using MMAS compared with the results fromCPLEX

Nearly all small scale instances can be solved optimallyby using MMAS except several instances from 119869211987521198781 and119869211987521198782 with relative larger solution space MMAS is com-petitive in computational time compared with CPLEX for allinstances Computational time of all instances fromMMAS isless than 1 second The computational time of CPLEX variesgreatly depending on the solution space of an instance

44 Algorithm Performance Evaluation To evaluate the per-formance of the algorithm for all problem categories MMASdesigned in this paper was comparedwithGA [9] A basic antalgorithm ant cycle (AC) was coded to solve the problem aswell and the parameters used were as prescribed in Chenget al [21] Two well-known heuristics FFLPT and BFLPTwere also compared withMMAS in the experiment study 100instances were generated for each problem category and thebest result of 10 runs for each instance was eventually used

Comparison of various algorithms is summarized inTable 4 Column 1 in Table 4 presents the run code Column 2shows the results of MMAS compared with other algorithmswhere B E and I mean themakespan obtained byMMAS arebetter than equal to or inferior to that of other algorithmsrespectively Columns 3ndash6 report the proportion that themakespan obtained by using MMAS compared with that ofother algorithms in 100 instances of 1198781 problem categoriesColumns 7ndash10 and columns 11ndash14 are similar to columns 3ndash6For example the first number of 001 indicates that there isa proportion of 001 results reported by MMAS better thanthat generated by AC As the problem under study is stronglyNP-hard the solution space is to explode along with thepopulation size increasing form 1198691 to 1198694 On the other handsmaller job sizes give more combination of batches thus thesolution space becomes smaller for 1198783 problem categoriescompared with 1198781 for a given number of jobs

It can be seen from Table 4 that MMAS outperformsother algorithms on the whole Several results reported byMMAS inferior to those generated by other algorithms have

8 Mathematical Problems in Engineering

Table 4 Results of MMAS compared with other algorithms

Run code Cp 1198781 1198782 1198783AC GA BFLPT FFLPT AC GA BFLPT FFLPT AC GA BFLPT FFLPT

11986911198751 B 001 002 016 017 000 010 016 016 000 001 007 007E 099 098 084 083 100 090 084 084 100 099 093 093I 000 000 000 000 000 000 000 000 000 000 000 000

11986911198752 B 002 002 010 013 001 014 019 019 000 003 008 011E 098 098 090 087 099 086 081 081 100 097 092 089I 000 000 000 000 000 000 000 000 000 000 000 000

11986921198751 B 004 045 047 059 001 031 032 032 000 007 014 020E 096 055 053 041 099 069 068 068 100 093 086 080I 000 000 000 000 000 000 000 000 000 000 000 000

11986921198752 B 013 042 041 057 006 032 033 034 000 014 019 029E 087 058 059 043 094 068 067 066 100 086 081 071I 000 000 000 000 000 000 000 000 000 000 000 000

11986931198751 B 043 084 083 093 023 089 091 091 000 030 030 043E 057 016 017 007 075 011 009 009 100 070 070 057I 000 000 000 000 002 000 000 000 000 000 000 000

11986931198752 B 060 090 085 099 039 087 087 087 012 027 027 039E 040 010 015 001 060 013 013 013 088 073 073 061I 000 000 000 000 001 000 000 000 000 000 000 000

11986941198751 B 078 093 090 100 049 099 099 099 005 018 018 030E 021 006 009 000 049 001 001 001 095 082 082 070I 001 001 001 000 002 000 000 000 000 000 000 000

11986941198752 B 092 095 092 098 051 099 099 099 035 022 022 030E 005 004 006 002 043 000 000 000 065 078 078 070I 003 001 002 000 006 001 001 001 000 000 000 000

been earmarked by rectangle For 1198691 problem categoriesresults obtained from MMAS are mostly equal to that fromother algorithms A percentage of about 16 results fromMMAS is better than that fromalgorithmsBFLPT andFFLPTfor 1198781 and 1198782 problem categories The percentage is 7 for1198783 problem categories which is because more results arethe same for all algorithms in a small solution space Upto 14 reported by MMAS are better than GA for 119869111987521198783problem category The advantage of MMAS to algorithmslike GA BFLPT and FFLPT is much more remarkable for1198692 problem categories More better results can be found byMMAS than AC and the percentage is up to 6 to 13 for11986921198752 problem categories More than 80 results obtainedfrom MMAS are better than that from GA BFLPT andFFLPT for 11986931198781 and 11986931198782 problem categories About 23 to43 percent of results fromMMAS are better than AC whilethe latter reports few better results For 1198694 problem categoriesmore than 90 results generated by MMAS are better thanthat from algorithms AC GA BFLPT and FFLPT for largejob population instances Almost all other algorithms reportseveral better results than MMAS for 1198694 problem categories

Based on the analysis given above MMAS possesseshigher ability in solution space exploring It outperforms allother algorithms for almost all instances especially for largejob population As MMAS is exploring solution space inprobability the optimal solution cannot be guaranteed witha limited population and iterations Therefore several results

obtained from MMAS are inferior to other algorithms evento effective heuristics like BFLPT and FFLPT

The makespan increases for 1198752 problem categories com-pared with 1198751 problem categories but there is no significantperformance change for various algorithms

To illustrate the gap between MMAS and other algo-rithms for different population size the average percentage ofresults with better makespan obtained by MMAS comparedwith other algorithms is given in Figure 6

Averagely a percentage of 10 results obtained fromMMAS are better than that from other algorithms for 1198691problem categories while the number is 70 for 1198694 Algo-rithm AC is the second best algorithm compared to GA andthe other two heuristics BFLPT and FFLPT are all effectivein solving the problem especially for small population sizeBFLPT is generally better than FFLPT for almost all problemcategories and BFLPT is also used in GA to generate initialsolutions

5 Conclusions and Future Work

The problem of scheduling single batch processing machinewas studied in this paper An improved ant algorithmMMAS(MaxndashMin Ant System) was designed and applied to solvethe problem Parameters of MMAS were determined bypreliminary experiment and a local search method MJEwas also presented to improve the performance of MMAS

Mathematical Problems in Engineering 9

J1 J2 J3 J4

Number of jobs

ACGA

BFLPTFFLPT

01020304050607080

Perc

enta

ge (

)

Figure 6 Average percentage of results obtained from MMAScompared with that from other algorithms

Optimal objectives can be obtained by using MMAS in lesscomputational time compared to CPLEX for small scaleproblems To evaluate the performance of MMAS it wascompared with basic ant algorithm AC (Ant Cycle) GA(Genetic Algorithm) and the other two well-known heuris-tics of scheduling batch processing machine that is FFLPT(First Fit Longest Processing Time) and BFLPT (Best FitLongest Processing Time) The numeric experiment showedthat MMAS outperformed other algorithms for almost allinstances in various problem categories especially for largerpopulation size

Considering the good performance of MMAS betterheuristics mechanism can be designed and applied to otherproblems of scheduling batch processing machine Theproblem under study with other objectives and problemconstraints including release times and due dates can bestudied as well

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (no 2014QNA48) and NationalNatural Science Foundation of China (no 71401164)

References

[1] Y Ikura and M Gimple ldquoEfficient scheduling algorithms for asingle batch processing machinerdquo Operations Research Lettersvol 5 no 2 pp 61ndash65 1986

[2] C-Y Lee R Uzsoy and L AMartin-Vega ldquoEfficient algorithmsfor scheduling semiconductor burn-in operationsrdquo OperationsResearch vol 40 no 4 pp 764ndash775 1992

[3] C S Sung and Y I Choung ldquoMinimizing makespan on a singleburn-in oven in semiconductormanufacturingrdquoEuropean Jour-nal of Operational Research vol 120 no 3 pp 559ndash574 2000

[4] R Uzsoy ldquoScheduling a single batch processing machine withnon-identical job sizesrdquo International Journal of ProductionResearch vol 32 no 7 pp 1615ndash1635 1994

[5] L Dupont and F Jolai Ghazvini ldquoMinimizing makespan on asingle batch processing machine with non-identical job sizesrdquoJournal Europeen des Systemes Automatises vol 32 no 4 pp431ndash440 1998

[6] L Dupont and C Dhaenens-Flipo ldquoMinimizing the makespanon a batch machine with non-identical job sizes An exactprocedurerdquo Computers amp Operations Research vol 29 no 7 pp807ndash819 2002

[7] X Li Y Li andYWang ldquoMinimisingmakespan on a batch pro-cessing machine using heuristics improved by an enumerationschemerdquo International Journal of Production Research vol 55no 1 pp 176ndash186 2017

[8] S Melouk P Damodaran and P-Y Chang ldquoMinimizingmakespan for single machine batch processing with non-identical job sizes using simulated annealingrdquo InternationalJournal of Production Economics vol 87 no 2 pp 141ndash147 2004

[9] P Damodaran P Kumar Manjeshwar and K Srihari ldquoMin-imizing makespan on a batch-processing machine with non-identical job sizes using genetic algorithmsrdquo InternationalJournal of Production Economics vol 103 no 2 pp 882ndash8912006

[10] A Husseinzadeh Kashan M Husseinzadeh Kashan and SKarimiyan ldquoA particle swarm optimizer for grouping prob-lemsrdquo Information Sciences vol 252 pp 81ndash95 2013

[11] P Damodaran O Ghrayeb and M C Guttikonda ldquoGRASPto minimize makespan for a capacitated batch-processingmachinerdquoThe International Journal of AdvancedManufacturingTechnology vol 68 no 1-4 pp 407ndash414 2013

[12] S Zhou M Liu H Chen and X Li ldquoAn effective discretedifferential evolution algorithm for scheduling uniform parallelbatch processing machines with non-identical capacities andarbitrary job sizesrdquo International Journal of Production Eco-nomics vol 179 pp 1ndash11 2016

[13] L Jiang J Pei X Liu P M Pardalos Y Yang and XQian ldquoUniformparallel batchmachines scheduling consideringtransportation using a hybrid DPSO-GA algorithmrdquoThe Inter-national Journal of AdvancedManufacturing Technology vol 89no 5-8 pp 1887ndash1900 2017

[14] T Stutzle and H H Hoos ldquoMax-min ant systemrdquo FutureGeneration Computer Systems vol 16 no 8 pp 889ndash914 2000

[15] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[16] M Dorigo and C Blum ldquoAnt colony optimization theory asurveyrdquoTheoretical Computer Science vol 344 no 2-3 pp 243ndash278 2005

[17] R Xu H Chen and X Li ldquoMakespan minimization onsingle batch-processing machine via ant colony optimizationrdquoComputers amp Operations Research vol 39 no 3 pp 582ndash5932012

[18] L DrsquoAcierno M Gallo and B Montella ldquoAn ant colony optimi-sation algorithm for solving the asymmetric traffic assignmentproblemrdquo European Journal of Operational Research vol 217 no2 pp 459ndash469 2012

[19] D S Johnson A Demers J D Ullman M R Garey and RL Graham ldquoWorst-case performance bounds for simple one-dimensional packing algorithmsrdquo SIAM Journal on Computingvol 3 pp 299ndash325 1974

[20] M Pirlot ldquoGeneral local search methodsrdquo European Journal ofOperational Research vol 92 no 3 pp 493ndash511 1996

10 Mathematical Problems in Engineering

[21] B-Y Cheng H-P Chen and S-S Wang ldquoImproved ant colonyoptimization method for single batch-processing machine withnon-identical job sizesrdquo Journal of System Simulation vol 21 no9 pp 2687ndash2695 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

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Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 7

Table 3 Results of MMAS compared with CPLEX

Problem category MMAS CPLEXMin Max Avg Optimal () AvgTime (s) AvgTime (s)

119878111986911198751 2000 6000 3664 10000 011 06411986911198752 3300 12700 7198 10000 010 06311986921198751 4300 10500 6913 10000 033 11811986921198752 8100 19800 13170 9600 033 142119878211986911198751 1300 2900 2090 10000 022 04411986911198752 2300 5800 4024 10000 022 04511986921198751 2800 4800 3797 10000 081 280411986921198752 4600 9400 7194 9900 082 2690119878311986911198751 2700 6400 4412 10000 008 05911986911198752 5000 12400 8553 10000 008 05811986921198751 5600 12200 8784 10000 023 10711986921198752 10600 21700 16718 10000 023 112

n = 50n = 100n = 200

0010203040506070809

1

Evap

orat

ion

para

met

er (

)

10 20 30 40 50 600Maximum number of iterations (Nc)

Figure 5 Relationship between119873119888 and 120588

slowly and the solution space can be explored in a reasonablerun time

43 Performance Comparison for Small Scale Instances Asshown in Table 3 the results obtained from MMAS iscompared with that from CPLEX (a commercial solverfor linear and mixed-integer problems) Due to the NP-hardness of the problem only small scale instances of 1198691and 1198692 problem categories are solved by using CPLEX Theminimum maximum and average value of 119862max reportedfrom MMAS are listed in columns 3 to 5 The average runtime of MMAS and CPLEXT are given in columns 7 and8 Column 6 indicates the percent of the optimal valuesobtained by using MMAS compared with the results fromCPLEX

Nearly all small scale instances can be solved optimallyby using MMAS except several instances from 119869211987521198781 and119869211987521198782 with relative larger solution space MMAS is com-petitive in computational time compared with CPLEX for allinstances Computational time of all instances fromMMAS isless than 1 second The computational time of CPLEX variesgreatly depending on the solution space of an instance

44 Algorithm Performance Evaluation To evaluate the per-formance of the algorithm for all problem categories MMASdesigned in this paper was comparedwithGA [9] A basic antalgorithm ant cycle (AC) was coded to solve the problem aswell and the parameters used were as prescribed in Chenget al [21] Two well-known heuristics FFLPT and BFLPTwere also compared withMMAS in the experiment study 100instances were generated for each problem category and thebest result of 10 runs for each instance was eventually used

Comparison of various algorithms is summarized inTable 4 Column 1 in Table 4 presents the run code Column 2shows the results of MMAS compared with other algorithmswhere B E and I mean themakespan obtained byMMAS arebetter than equal to or inferior to that of other algorithmsrespectively Columns 3ndash6 report the proportion that themakespan obtained by using MMAS compared with that ofother algorithms in 100 instances of 1198781 problem categoriesColumns 7ndash10 and columns 11ndash14 are similar to columns 3ndash6For example the first number of 001 indicates that there isa proportion of 001 results reported by MMAS better thanthat generated by AC As the problem under study is stronglyNP-hard the solution space is to explode along with thepopulation size increasing form 1198691 to 1198694 On the other handsmaller job sizes give more combination of batches thus thesolution space becomes smaller for 1198783 problem categoriescompared with 1198781 for a given number of jobs

It can be seen from Table 4 that MMAS outperformsother algorithms on the whole Several results reported byMMAS inferior to those generated by other algorithms have

8 Mathematical Problems in Engineering

Table 4 Results of MMAS compared with other algorithms

Run code Cp 1198781 1198782 1198783AC GA BFLPT FFLPT AC GA BFLPT FFLPT AC GA BFLPT FFLPT

11986911198751 B 001 002 016 017 000 010 016 016 000 001 007 007E 099 098 084 083 100 090 084 084 100 099 093 093I 000 000 000 000 000 000 000 000 000 000 000 000

11986911198752 B 002 002 010 013 001 014 019 019 000 003 008 011E 098 098 090 087 099 086 081 081 100 097 092 089I 000 000 000 000 000 000 000 000 000 000 000 000

11986921198751 B 004 045 047 059 001 031 032 032 000 007 014 020E 096 055 053 041 099 069 068 068 100 093 086 080I 000 000 000 000 000 000 000 000 000 000 000 000

11986921198752 B 013 042 041 057 006 032 033 034 000 014 019 029E 087 058 059 043 094 068 067 066 100 086 081 071I 000 000 000 000 000 000 000 000 000 000 000 000

11986931198751 B 043 084 083 093 023 089 091 091 000 030 030 043E 057 016 017 007 075 011 009 009 100 070 070 057I 000 000 000 000 002 000 000 000 000 000 000 000

11986931198752 B 060 090 085 099 039 087 087 087 012 027 027 039E 040 010 015 001 060 013 013 013 088 073 073 061I 000 000 000 000 001 000 000 000 000 000 000 000

11986941198751 B 078 093 090 100 049 099 099 099 005 018 018 030E 021 006 009 000 049 001 001 001 095 082 082 070I 001 001 001 000 002 000 000 000 000 000 000 000

11986941198752 B 092 095 092 098 051 099 099 099 035 022 022 030E 005 004 006 002 043 000 000 000 065 078 078 070I 003 001 002 000 006 001 001 001 000 000 000 000

been earmarked by rectangle For 1198691 problem categoriesresults obtained from MMAS are mostly equal to that fromother algorithms A percentage of about 16 results fromMMAS is better than that fromalgorithmsBFLPT andFFLPTfor 1198781 and 1198782 problem categories The percentage is 7 for1198783 problem categories which is because more results arethe same for all algorithms in a small solution space Upto 14 reported by MMAS are better than GA for 119869111987521198783problem category The advantage of MMAS to algorithmslike GA BFLPT and FFLPT is much more remarkable for1198692 problem categories More better results can be found byMMAS than AC and the percentage is up to 6 to 13 for11986921198752 problem categories More than 80 results obtainedfrom MMAS are better than that from GA BFLPT andFFLPT for 11986931198781 and 11986931198782 problem categories About 23 to43 percent of results fromMMAS are better than AC whilethe latter reports few better results For 1198694 problem categoriesmore than 90 results generated by MMAS are better thanthat from algorithms AC GA BFLPT and FFLPT for largejob population instances Almost all other algorithms reportseveral better results than MMAS for 1198694 problem categories

Based on the analysis given above MMAS possesseshigher ability in solution space exploring It outperforms allother algorithms for almost all instances especially for largejob population As MMAS is exploring solution space inprobability the optimal solution cannot be guaranteed witha limited population and iterations Therefore several results

obtained from MMAS are inferior to other algorithms evento effective heuristics like BFLPT and FFLPT

The makespan increases for 1198752 problem categories com-pared with 1198751 problem categories but there is no significantperformance change for various algorithms

To illustrate the gap between MMAS and other algo-rithms for different population size the average percentage ofresults with better makespan obtained by MMAS comparedwith other algorithms is given in Figure 6

Averagely a percentage of 10 results obtained fromMMAS are better than that from other algorithms for 1198691problem categories while the number is 70 for 1198694 Algo-rithm AC is the second best algorithm compared to GA andthe other two heuristics BFLPT and FFLPT are all effectivein solving the problem especially for small population sizeBFLPT is generally better than FFLPT for almost all problemcategories and BFLPT is also used in GA to generate initialsolutions

5 Conclusions and Future Work

The problem of scheduling single batch processing machinewas studied in this paper An improved ant algorithmMMAS(MaxndashMin Ant System) was designed and applied to solvethe problem Parameters of MMAS were determined bypreliminary experiment and a local search method MJEwas also presented to improve the performance of MMAS

Mathematical Problems in Engineering 9

J1 J2 J3 J4

Number of jobs

ACGA

BFLPTFFLPT

01020304050607080

Perc

enta

ge (

)

Figure 6 Average percentage of results obtained from MMAScompared with that from other algorithms

Optimal objectives can be obtained by using MMAS in lesscomputational time compared to CPLEX for small scaleproblems To evaluate the performance of MMAS it wascompared with basic ant algorithm AC (Ant Cycle) GA(Genetic Algorithm) and the other two well-known heuris-tics of scheduling batch processing machine that is FFLPT(First Fit Longest Processing Time) and BFLPT (Best FitLongest Processing Time) The numeric experiment showedthat MMAS outperformed other algorithms for almost allinstances in various problem categories especially for largerpopulation size

Considering the good performance of MMAS betterheuristics mechanism can be designed and applied to otherproblems of scheduling batch processing machine Theproblem under study with other objectives and problemconstraints including release times and due dates can bestudied as well

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (no 2014QNA48) and NationalNatural Science Foundation of China (no 71401164)

References

[1] Y Ikura and M Gimple ldquoEfficient scheduling algorithms for asingle batch processing machinerdquo Operations Research Lettersvol 5 no 2 pp 61ndash65 1986

[2] C-Y Lee R Uzsoy and L AMartin-Vega ldquoEfficient algorithmsfor scheduling semiconductor burn-in operationsrdquo OperationsResearch vol 40 no 4 pp 764ndash775 1992

[3] C S Sung and Y I Choung ldquoMinimizing makespan on a singleburn-in oven in semiconductormanufacturingrdquoEuropean Jour-nal of Operational Research vol 120 no 3 pp 559ndash574 2000

[4] R Uzsoy ldquoScheduling a single batch processing machine withnon-identical job sizesrdquo International Journal of ProductionResearch vol 32 no 7 pp 1615ndash1635 1994

[5] L Dupont and F Jolai Ghazvini ldquoMinimizing makespan on asingle batch processing machine with non-identical job sizesrdquoJournal Europeen des Systemes Automatises vol 32 no 4 pp431ndash440 1998

[6] L Dupont and C Dhaenens-Flipo ldquoMinimizing the makespanon a batch machine with non-identical job sizes An exactprocedurerdquo Computers amp Operations Research vol 29 no 7 pp807ndash819 2002

[7] X Li Y Li andYWang ldquoMinimisingmakespan on a batch pro-cessing machine using heuristics improved by an enumerationschemerdquo International Journal of Production Research vol 55no 1 pp 176ndash186 2017

[8] S Melouk P Damodaran and P-Y Chang ldquoMinimizingmakespan for single machine batch processing with non-identical job sizes using simulated annealingrdquo InternationalJournal of Production Economics vol 87 no 2 pp 141ndash147 2004

[9] P Damodaran P Kumar Manjeshwar and K Srihari ldquoMin-imizing makespan on a batch-processing machine with non-identical job sizes using genetic algorithmsrdquo InternationalJournal of Production Economics vol 103 no 2 pp 882ndash8912006

[10] A Husseinzadeh Kashan M Husseinzadeh Kashan and SKarimiyan ldquoA particle swarm optimizer for grouping prob-lemsrdquo Information Sciences vol 252 pp 81ndash95 2013

[11] P Damodaran O Ghrayeb and M C Guttikonda ldquoGRASPto minimize makespan for a capacitated batch-processingmachinerdquoThe International Journal of AdvancedManufacturingTechnology vol 68 no 1-4 pp 407ndash414 2013

[12] S Zhou M Liu H Chen and X Li ldquoAn effective discretedifferential evolution algorithm for scheduling uniform parallelbatch processing machines with non-identical capacities andarbitrary job sizesrdquo International Journal of Production Eco-nomics vol 179 pp 1ndash11 2016

[13] L Jiang J Pei X Liu P M Pardalos Y Yang and XQian ldquoUniformparallel batchmachines scheduling consideringtransportation using a hybrid DPSO-GA algorithmrdquoThe Inter-national Journal of AdvancedManufacturing Technology vol 89no 5-8 pp 1887ndash1900 2017

[14] T Stutzle and H H Hoos ldquoMax-min ant systemrdquo FutureGeneration Computer Systems vol 16 no 8 pp 889ndash914 2000

[15] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[16] M Dorigo and C Blum ldquoAnt colony optimization theory asurveyrdquoTheoretical Computer Science vol 344 no 2-3 pp 243ndash278 2005

[17] R Xu H Chen and X Li ldquoMakespan minimization onsingle batch-processing machine via ant colony optimizationrdquoComputers amp Operations Research vol 39 no 3 pp 582ndash5932012

[18] L DrsquoAcierno M Gallo and B Montella ldquoAn ant colony optimi-sation algorithm for solving the asymmetric traffic assignmentproblemrdquo European Journal of Operational Research vol 217 no2 pp 459ndash469 2012

[19] D S Johnson A Demers J D Ullman M R Garey and RL Graham ldquoWorst-case performance bounds for simple one-dimensional packing algorithmsrdquo SIAM Journal on Computingvol 3 pp 299ndash325 1974

[20] M Pirlot ldquoGeneral local search methodsrdquo European Journal ofOperational Research vol 92 no 3 pp 493ndash511 1996

10 Mathematical Problems in Engineering

[21] B-Y Cheng H-P Chen and S-S Wang ldquoImproved ant colonyoptimization method for single batch-processing machine withnon-identical job sizesrdquo Journal of System Simulation vol 21 no9 pp 2687ndash2695 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Mathematical Problems in Engineering

Table 4 Results of MMAS compared with other algorithms

Run code Cp 1198781 1198782 1198783AC GA BFLPT FFLPT AC GA BFLPT FFLPT AC GA BFLPT FFLPT

11986911198751 B 001 002 016 017 000 010 016 016 000 001 007 007E 099 098 084 083 100 090 084 084 100 099 093 093I 000 000 000 000 000 000 000 000 000 000 000 000

11986911198752 B 002 002 010 013 001 014 019 019 000 003 008 011E 098 098 090 087 099 086 081 081 100 097 092 089I 000 000 000 000 000 000 000 000 000 000 000 000

11986921198751 B 004 045 047 059 001 031 032 032 000 007 014 020E 096 055 053 041 099 069 068 068 100 093 086 080I 000 000 000 000 000 000 000 000 000 000 000 000

11986921198752 B 013 042 041 057 006 032 033 034 000 014 019 029E 087 058 059 043 094 068 067 066 100 086 081 071I 000 000 000 000 000 000 000 000 000 000 000 000

11986931198751 B 043 084 083 093 023 089 091 091 000 030 030 043E 057 016 017 007 075 011 009 009 100 070 070 057I 000 000 000 000 002 000 000 000 000 000 000 000

11986931198752 B 060 090 085 099 039 087 087 087 012 027 027 039E 040 010 015 001 060 013 013 013 088 073 073 061I 000 000 000 000 001 000 000 000 000 000 000 000

11986941198751 B 078 093 090 100 049 099 099 099 005 018 018 030E 021 006 009 000 049 001 001 001 095 082 082 070I 001 001 001 000 002 000 000 000 000 000 000 000

11986941198752 B 092 095 092 098 051 099 099 099 035 022 022 030E 005 004 006 002 043 000 000 000 065 078 078 070I 003 001 002 000 006 001 001 001 000 000 000 000

been earmarked by rectangle For 1198691 problem categoriesresults obtained from MMAS are mostly equal to that fromother algorithms A percentage of about 16 results fromMMAS is better than that fromalgorithmsBFLPT andFFLPTfor 1198781 and 1198782 problem categories The percentage is 7 for1198783 problem categories which is because more results arethe same for all algorithms in a small solution space Upto 14 reported by MMAS are better than GA for 119869111987521198783problem category The advantage of MMAS to algorithmslike GA BFLPT and FFLPT is much more remarkable for1198692 problem categories More better results can be found byMMAS than AC and the percentage is up to 6 to 13 for11986921198752 problem categories More than 80 results obtainedfrom MMAS are better than that from GA BFLPT andFFLPT for 11986931198781 and 11986931198782 problem categories About 23 to43 percent of results fromMMAS are better than AC whilethe latter reports few better results For 1198694 problem categoriesmore than 90 results generated by MMAS are better thanthat from algorithms AC GA BFLPT and FFLPT for largejob population instances Almost all other algorithms reportseveral better results than MMAS for 1198694 problem categories

Based on the analysis given above MMAS possesseshigher ability in solution space exploring It outperforms allother algorithms for almost all instances especially for largejob population As MMAS is exploring solution space inprobability the optimal solution cannot be guaranteed witha limited population and iterations Therefore several results

obtained from MMAS are inferior to other algorithms evento effective heuristics like BFLPT and FFLPT

The makespan increases for 1198752 problem categories com-pared with 1198751 problem categories but there is no significantperformance change for various algorithms

To illustrate the gap between MMAS and other algo-rithms for different population size the average percentage ofresults with better makespan obtained by MMAS comparedwith other algorithms is given in Figure 6

Averagely a percentage of 10 results obtained fromMMAS are better than that from other algorithms for 1198691problem categories while the number is 70 for 1198694 Algo-rithm AC is the second best algorithm compared to GA andthe other two heuristics BFLPT and FFLPT are all effectivein solving the problem especially for small population sizeBFLPT is generally better than FFLPT for almost all problemcategories and BFLPT is also used in GA to generate initialsolutions

5 Conclusions and Future Work

The problem of scheduling single batch processing machinewas studied in this paper An improved ant algorithmMMAS(MaxndashMin Ant System) was designed and applied to solvethe problem Parameters of MMAS were determined bypreliminary experiment and a local search method MJEwas also presented to improve the performance of MMAS

Mathematical Problems in Engineering 9

J1 J2 J3 J4

Number of jobs

ACGA

BFLPTFFLPT

01020304050607080

Perc

enta

ge (

)

Figure 6 Average percentage of results obtained from MMAScompared with that from other algorithms

Optimal objectives can be obtained by using MMAS in lesscomputational time compared to CPLEX for small scaleproblems To evaluate the performance of MMAS it wascompared with basic ant algorithm AC (Ant Cycle) GA(Genetic Algorithm) and the other two well-known heuris-tics of scheduling batch processing machine that is FFLPT(First Fit Longest Processing Time) and BFLPT (Best FitLongest Processing Time) The numeric experiment showedthat MMAS outperformed other algorithms for almost allinstances in various problem categories especially for largerpopulation size

Considering the good performance of MMAS betterheuristics mechanism can be designed and applied to otherproblems of scheduling batch processing machine Theproblem under study with other objectives and problemconstraints including release times and due dates can bestudied as well

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (no 2014QNA48) and NationalNatural Science Foundation of China (no 71401164)

References

[1] Y Ikura and M Gimple ldquoEfficient scheduling algorithms for asingle batch processing machinerdquo Operations Research Lettersvol 5 no 2 pp 61ndash65 1986

[2] C-Y Lee R Uzsoy and L AMartin-Vega ldquoEfficient algorithmsfor scheduling semiconductor burn-in operationsrdquo OperationsResearch vol 40 no 4 pp 764ndash775 1992

[3] C S Sung and Y I Choung ldquoMinimizing makespan on a singleburn-in oven in semiconductormanufacturingrdquoEuropean Jour-nal of Operational Research vol 120 no 3 pp 559ndash574 2000

[4] R Uzsoy ldquoScheduling a single batch processing machine withnon-identical job sizesrdquo International Journal of ProductionResearch vol 32 no 7 pp 1615ndash1635 1994

[5] L Dupont and F Jolai Ghazvini ldquoMinimizing makespan on asingle batch processing machine with non-identical job sizesrdquoJournal Europeen des Systemes Automatises vol 32 no 4 pp431ndash440 1998

[6] L Dupont and C Dhaenens-Flipo ldquoMinimizing the makespanon a batch machine with non-identical job sizes An exactprocedurerdquo Computers amp Operations Research vol 29 no 7 pp807ndash819 2002

[7] X Li Y Li andYWang ldquoMinimisingmakespan on a batch pro-cessing machine using heuristics improved by an enumerationschemerdquo International Journal of Production Research vol 55no 1 pp 176ndash186 2017

[8] S Melouk P Damodaran and P-Y Chang ldquoMinimizingmakespan for single machine batch processing with non-identical job sizes using simulated annealingrdquo InternationalJournal of Production Economics vol 87 no 2 pp 141ndash147 2004

[9] P Damodaran P Kumar Manjeshwar and K Srihari ldquoMin-imizing makespan on a batch-processing machine with non-identical job sizes using genetic algorithmsrdquo InternationalJournal of Production Economics vol 103 no 2 pp 882ndash8912006

[10] A Husseinzadeh Kashan M Husseinzadeh Kashan and SKarimiyan ldquoA particle swarm optimizer for grouping prob-lemsrdquo Information Sciences vol 252 pp 81ndash95 2013

[11] P Damodaran O Ghrayeb and M C Guttikonda ldquoGRASPto minimize makespan for a capacitated batch-processingmachinerdquoThe International Journal of AdvancedManufacturingTechnology vol 68 no 1-4 pp 407ndash414 2013

[12] S Zhou M Liu H Chen and X Li ldquoAn effective discretedifferential evolution algorithm for scheduling uniform parallelbatch processing machines with non-identical capacities andarbitrary job sizesrdquo International Journal of Production Eco-nomics vol 179 pp 1ndash11 2016

[13] L Jiang J Pei X Liu P M Pardalos Y Yang and XQian ldquoUniformparallel batchmachines scheduling consideringtransportation using a hybrid DPSO-GA algorithmrdquoThe Inter-national Journal of AdvancedManufacturing Technology vol 89no 5-8 pp 1887ndash1900 2017

[14] T Stutzle and H H Hoos ldquoMax-min ant systemrdquo FutureGeneration Computer Systems vol 16 no 8 pp 889ndash914 2000

[15] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[16] M Dorigo and C Blum ldquoAnt colony optimization theory asurveyrdquoTheoretical Computer Science vol 344 no 2-3 pp 243ndash278 2005

[17] R Xu H Chen and X Li ldquoMakespan minimization onsingle batch-processing machine via ant colony optimizationrdquoComputers amp Operations Research vol 39 no 3 pp 582ndash5932012

[18] L DrsquoAcierno M Gallo and B Montella ldquoAn ant colony optimi-sation algorithm for solving the asymmetric traffic assignmentproblemrdquo European Journal of Operational Research vol 217 no2 pp 459ndash469 2012

[19] D S Johnson A Demers J D Ullman M R Garey and RL Graham ldquoWorst-case performance bounds for simple one-dimensional packing algorithmsrdquo SIAM Journal on Computingvol 3 pp 299ndash325 1974

[20] M Pirlot ldquoGeneral local search methodsrdquo European Journal ofOperational Research vol 92 no 3 pp 493ndash511 1996

10 Mathematical Problems in Engineering

[21] B-Y Cheng H-P Chen and S-S Wang ldquoImproved ant colonyoptimization method for single batch-processing machine withnon-identical job sizesrdquo Journal of System Simulation vol 21 no9 pp 2687ndash2695 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 9

J1 J2 J3 J4

Number of jobs

ACGA

BFLPTFFLPT

01020304050607080

Perc

enta

ge (

)

Figure 6 Average percentage of results obtained from MMAScompared with that from other algorithms

Optimal objectives can be obtained by using MMAS in lesscomputational time compared to CPLEX for small scaleproblems To evaluate the performance of MMAS it wascompared with basic ant algorithm AC (Ant Cycle) GA(Genetic Algorithm) and the other two well-known heuris-tics of scheduling batch processing machine that is FFLPT(First Fit Longest Processing Time) and BFLPT (Best FitLongest Processing Time) The numeric experiment showedthat MMAS outperformed other algorithms for almost allinstances in various problem categories especially for largerpopulation size

Considering the good performance of MMAS betterheuristics mechanism can be designed and applied to otherproblems of scheduling batch processing machine Theproblem under study with other objectives and problemconstraints including release times and due dates can bestudied as well

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (no 2014QNA48) and NationalNatural Science Foundation of China (no 71401164)

References

[1] Y Ikura and M Gimple ldquoEfficient scheduling algorithms for asingle batch processing machinerdquo Operations Research Lettersvol 5 no 2 pp 61ndash65 1986

[2] C-Y Lee R Uzsoy and L AMartin-Vega ldquoEfficient algorithmsfor scheduling semiconductor burn-in operationsrdquo OperationsResearch vol 40 no 4 pp 764ndash775 1992

[3] C S Sung and Y I Choung ldquoMinimizing makespan on a singleburn-in oven in semiconductormanufacturingrdquoEuropean Jour-nal of Operational Research vol 120 no 3 pp 559ndash574 2000

[4] R Uzsoy ldquoScheduling a single batch processing machine withnon-identical job sizesrdquo International Journal of ProductionResearch vol 32 no 7 pp 1615ndash1635 1994

[5] L Dupont and F Jolai Ghazvini ldquoMinimizing makespan on asingle batch processing machine with non-identical job sizesrdquoJournal Europeen des Systemes Automatises vol 32 no 4 pp431ndash440 1998

[6] L Dupont and C Dhaenens-Flipo ldquoMinimizing the makespanon a batch machine with non-identical job sizes An exactprocedurerdquo Computers amp Operations Research vol 29 no 7 pp807ndash819 2002

[7] X Li Y Li andYWang ldquoMinimisingmakespan on a batch pro-cessing machine using heuristics improved by an enumerationschemerdquo International Journal of Production Research vol 55no 1 pp 176ndash186 2017

[8] S Melouk P Damodaran and P-Y Chang ldquoMinimizingmakespan for single machine batch processing with non-identical job sizes using simulated annealingrdquo InternationalJournal of Production Economics vol 87 no 2 pp 141ndash147 2004

[9] P Damodaran P Kumar Manjeshwar and K Srihari ldquoMin-imizing makespan on a batch-processing machine with non-identical job sizes using genetic algorithmsrdquo InternationalJournal of Production Economics vol 103 no 2 pp 882ndash8912006

[10] A Husseinzadeh Kashan M Husseinzadeh Kashan and SKarimiyan ldquoA particle swarm optimizer for grouping prob-lemsrdquo Information Sciences vol 252 pp 81ndash95 2013

[11] P Damodaran O Ghrayeb and M C Guttikonda ldquoGRASPto minimize makespan for a capacitated batch-processingmachinerdquoThe International Journal of AdvancedManufacturingTechnology vol 68 no 1-4 pp 407ndash414 2013

[12] S Zhou M Liu H Chen and X Li ldquoAn effective discretedifferential evolution algorithm for scheduling uniform parallelbatch processing machines with non-identical capacities andarbitrary job sizesrdquo International Journal of Production Eco-nomics vol 179 pp 1ndash11 2016

[13] L Jiang J Pei X Liu P M Pardalos Y Yang and XQian ldquoUniformparallel batchmachines scheduling consideringtransportation using a hybrid DPSO-GA algorithmrdquoThe Inter-national Journal of AdvancedManufacturing Technology vol 89no 5-8 pp 1887ndash1900 2017

[14] T Stutzle and H H Hoos ldquoMax-min ant systemrdquo FutureGeneration Computer Systems vol 16 no 8 pp 889ndash914 2000

[15] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[16] M Dorigo and C Blum ldquoAnt colony optimization theory asurveyrdquoTheoretical Computer Science vol 344 no 2-3 pp 243ndash278 2005

[17] R Xu H Chen and X Li ldquoMakespan minimization onsingle batch-processing machine via ant colony optimizationrdquoComputers amp Operations Research vol 39 no 3 pp 582ndash5932012

[18] L DrsquoAcierno M Gallo and B Montella ldquoAn ant colony optimi-sation algorithm for solving the asymmetric traffic assignmentproblemrdquo European Journal of Operational Research vol 217 no2 pp 459ndash469 2012

[19] D S Johnson A Demers J D Ullman M R Garey and RL Graham ldquoWorst-case performance bounds for simple one-dimensional packing algorithmsrdquo SIAM Journal on Computingvol 3 pp 299ndash325 1974

[20] M Pirlot ldquoGeneral local search methodsrdquo European Journal ofOperational Research vol 92 no 3 pp 493ndash511 1996

10 Mathematical Problems in Engineering

[21] B-Y Cheng H-P Chen and S-S Wang ldquoImproved ant colonyoptimization method for single batch-processing machine withnon-identical job sizesrdquo Journal of System Simulation vol 21 no9 pp 2687ndash2695 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Mathematical Problems in Engineering

[21] B-Y Cheng H-P Chen and S-S Wang ldquoImproved ant colonyoptimization method for single batch-processing machine withnon-identical job sizesrdquo Journal of System Simulation vol 21 no9 pp 2687ndash2695 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom