ORIGINAL ARTICLE

A clustering-based modified variable neighborhood searchalgorithm for a dynamic job shop scheduling problem

Mohammad Amin Adibi & Jamal Shahrabi

Received: 8 October 2011 /Accepted: 26 September 2013# Springer-Verlag London 2013

Abstract The dynamic job shop scheduling (DJSS) problemoccurs when some real-time events are taken into account inthe ordinary job shop scheduling problem. Most researchesabout the DJSS problem have focused on methods in whichthe problem’s input data structure and their probable relation-ship are not considered in the optimization process while someuseful information can be extracted from such data. In thispaper, the variable neighborhood search (VNS) combinedwith the k-means algorithm as a modified VNS (MVNS)algorithm is proposed to address the DJSS problem. The k-means algorithm as a cluster analysis algorithm is used toplace similar jobs according to their processing time into thesame clusters. Jobs from different clusters are considered tohave greater probability to be selected when an adjacent for asolution is made in an optimization process using the MVNSalgorithm. To deal with the dynamic nature of the problem, anevent-driven policy is also selected. Computational resultsobtained using the proposed method in comparison with VNSand other common algorithms illustrate better performance in avariety of shop floor conditions.

Keywords Dynamic job shop scheduling . Variableneighborhood search . Cluster analysis . K-means algorithm

1 Introduction

The job shop scheduling (JSS) problem is an optimizationproblem in which a schedule of jobs that have known opera-tional sequences and time, in a multi-machine environment, isdetermined. This is one of the most interesting realms for

researchers and practitioners, having attracted extensive in-vestigations both in academic and applied fields. The JSSproblem is also a combinatorial optimization problem andwell-known to be NP-hard. The dynamic job shop scheduling(DJSS) problem emerges when real-time events such as ran-dom job arrivals and machine breakdowns are taken intoaccount in ordinary JSS problems. Most researches in theDJSS problem are about introducing new solving methodsor enhance existing methods to improve solution quality onlyby focusing on the solving method capabilities ignoring inputdata structure and their probable relationship. Qi et al. [1]proposed a genetic algorithm (GA) approach to address theDJSS problem. Chryssolouris and Subramanian [2] used ascheduling method based on GA for a dynamic job shop withunreliable machines, multiple routes, and multiple schedulingcriteria. Dewan and Joshi [3] presented an auction-baseddistributed scheduling mechanism to address the DJSS prob-lem. Dominic et al. [4] proposed combined dispatching rulesfor scheduling in a dynamic job shop environment. Yang et al.[5] proposed a layered hybrid algorithm for the DJSS problemthat was composed of an improved ant colony optimization(ACO) algorithm to select machines in the outer layer and aGAwith neighborhood search to sort jobs in the inner layer. Liand Chen [6] presented a hybrid approach combining the backpropagation neural network (BPNN) with GA for solving theproblem when machine breakdown and new job arrivals oc-cur. In their research, the BPNN was used to obtain feasiblesolution during the iterations, and GA is adapted to the globaloptimal searching. Cunli [7] proposed an assorted dynamicjob shop scheduling algorithm based on impact degree. Vinodand Sridharan [8] developed a discrete-event simulation modelfor a DJSS problemwith sequence-dependent setup time. Zhouet al. [9] explored the applicability of the ACO algorithm inDJSS problems comparing the most adopted approaches in thereal world. Wang et al. [10] developed an improved hybriddiscrete genetic algorithm particle swarm optimization to solve

M. A. Adibi : J. Shahrabi (*)Industrial Engineering and Management Systems Department,Amirkabir University of Technology, Tehran, Irane-mail: jamalshahrabi@aut.ac.ir

Int J Adv Manuf TechnolDOI 10.1007/s00170-013-5354-6

the JSS and DJSS problem. Fatemi Ghomi and Iranpoor [11]proposed a mixed integer programming mathematical formu-lation of the DJSS problem and developed a metaheuristicalgorithm which is composed of GA and simulated annealingcalled GA-SA algorithm to solve it. Vinod and Sridharan [12]studied the effects of due-date assignment methods and sched-uling rules on the performance of a job shop production systemin a dynamic environment.

In addition to solvingmethod capabilities to improve, someinformation in the problem’s input data such as necessaryoperations for each job, operation processing time, operationsequence, and etc. can be used in the optimization process toenhance solution quality and reduce required processing time.Chang [13] proposed an approach that can be used to providereal-time estimates of the queuing times for the remainingoperations of the jobs in a dynamic job shop and to incorpo-rate this estimated queuing time information into existingscheduling heuristics to improve their performance. Aydinand Oztmel [14] used a reinforcement learning agent to selectappropriate dispatching rules for scheduling according to theshop floor condition in the real-time job shop. Wei et al. [15]presented an iterative optimization framework for the dynamicscheduling system using reinforcement learning. Wang et al.[16] used the ACO algorithm to minimize make span in jobshop scheduling in which the ACO algorithm parameters areupdated according to shop floor conditions. Zandieh andAdibi [17] used a trained artificial neural network to updateparameters of a metaheuristic method at any reschedulingpoint in a DJSS problem according to the problem conditions.Chen et al. [18] proposed a rule driven dispatching methodbased on data envelopment analysis and reinforcement learn-ing for a multi-objective dynamic scheduling problem.Kapanoglo and Alikalfa [19] introduced an unsupervisedlearning scheduling system that builds “state” and “priorityrule” pairs depending on intervals of queue by using a rule-basedGA for a DJSS problem.

Among all approaches to knowledge extraction, clusteringis a data analysis task with numerous applications in a varietyof areas including bioinformatics, data mining, informationretrieval, information theory, machine learning, object, char-acter and pattern recognition, etc. [20]. Generally, clusteringtechniques are used to discover natural groups in datasets andto identify abstract structures that might exist without havingany background knowledge of the characteristics of the data.

In this study, to use both capable solving methods andextracted information from the problem’s input data, variableneighborhood search (VNS) [21] combined by cluster analysisalong with an event-driven policy is selected as a schedulingmethod to obtain an optimum solution (or a near optimum) atany rescheduling point for the DJSS problem.

VNS brings together a lot of desirable properties for ametaheuristic such as simplicity, efficiency, effectiveness,generality, etc. that has been widely used for combinatorial

optimization problems in recent years [22]. To enhance theefficiency and effectiveness of VNS, cluster analysis on inputdata at any rescheduling point is performed. As explained indetails in Sect. 4, by placing similar jobs according to theiroperational processing time into the same classes, uselessmovement in the VNS optimization process can be reducedby preventing similar sequence selections in replacementmechanism. K-means as a popular algorithm in clustering isalso used in this paper to identify similar sequences because ofits simplicity and low CPU time needs.

The rest of the paper is organized as follows: in Sect. 2, thedynamic job shop scheduling problem is defined in details.The k-means clustering analysis technique is described inSect. 3. The modified variable neighborhood search algorithmis argued in Sect. 4. The simulation study is presented inSect. 5, and the conclusion is put forth in Sect. 6.

2 Dynamic job shop scheduling problem

In the general job shop scheduling problem, n jobs should beprocessed on m machines, while minimizing a function ofcompletion time of jobs is considered alongwith the followingtechnological constraints and assumptions [17]:

& Each machine can perform only one operation at a time onany job.

& An operation of a job can be performed by only onemachine at a time.

& Once an operation has begun on a machine, it must not beinterrupted.

& An operation of a job cannot be performed until its precedingoperations are completed.

& There are no alternate routings, i.e., an operation of a jobcan be performed by only one type of machine.

& Operation processing time and the number of operablemachines are known in advance.

In this paper, mean flow timeminimization is selected as anoptimization objective. The flow time for the j th job can bedefined as f j=c j−r j;j =1,2,…n ., where r j and c j are j th jobready time (time at which a job arrives in the shop floor and is

Fig. 1 Gantt diagram of the solution represented by [2 1 3 2 3 3 1 1 2]

Int J Adv Manuf Technol

equal to 0 in case of ordinary or static JSS problems) andcompletion time, respectively. Thus, mean flow time is equalto∑ j =1

n (c j−r j)/n over the manufacturing horizon. The aim ofsolving a JSS problem is to find a sequence of jobs beingprocessed on each machine so that the objective function isminimized.

The operation-based representation method [23] whichencodes a schedule (equal to a JSS problem solution) as asequence of numbers is also selected to present a JSS problemsolution. In this method, each number stands for one opera-tion. The specific operation represented by a number isinterpreted according to the order of the numbers in the string.Each of the n numbers, representing n jobs, will appear mtimes spread over the entire string. For instance, consider onetime unit for each operation and [1 2 3; 2 3 1; 3 2 1] as a pre-defined required operation sequence for jobs 1, 2, and 3. Weare given a solution like 2 1 3 2 3 3 1 1 2½ � ,where {1, 2, 3} represents {job1, job2, job3}, respectively.Obviously, there are a total of nine operations, but threedifferent integers are repeated three times. The first integer,2, represents the first operation of the second job, O21, to beprocessed first on the corresponding machine. Likewise, thesecond integer, 1, represents the first operation of the first job,O11. Thus, the set of 2 1 3 2 3 3 1 1 2½ � isunderstood as [O21,O11,O31,O22,O32,O33,O12,O13,O23],where Oij stands for the j th operation of the i th job. Figure 1illustrates the Gantt diagram of the solution.

Random job arrivals and machine breakdowns that belongto job-related and resource-related real-time events, respec-tively, are considered in this paper in order to impose morereality in job shops. So contrary to ordinary JSS problem, alljobs are not ready at the beginning of the planning process butcome to the shop floor gradually. In addition, some machinerymay break down during the processing jobs. Finding jobsequences to be processed on machinery in such environmentis known as the DJSS problem [1].

In the job shop environment, the distribution of the jobarrival process closely follows the Poisson distribution.Hence, the time between arrivals of jobs is distributed expo-nentially [24–26]. The time between two breakdowns andrepair time are also assumed to follow an exponential distri-bution. Thus, the mean time between failure (MTBF) and themean time to repair (MTTR) are two parameters related tomachine breakdown.

In the dynamic job shop framework, considering an event-driven policy [27], in which rescheduling is triggered inresponse to a dynamic event that changes the current condition,a static job shop problem is generated whenever a new jobarrives or a machine breaks down. At that time, we consider anew job with the remaining operation of previous jobs or abroken machine that needs time to repair.

The new static problem is fed to the proposed schedulingmethod (MVNS) to produce an optimum or a nearly optimumschedule. The optimum solution will be used as productionschedule until the next event takes place.

3 K-means algorithm

Given n patterns, or data points, in d-dimensional space, theclustering problem refers to the process of partitioning the npatterns into k groups or clusters based on some similaritymetrics. An optimal clustering is a partitioning that minimizes

Step 1. Randomly choose k cluster center from all pattern in the data set D.

Step 2. Assign each pattern in D to the closest cluster center; then update the cluster centers.

Step 3. If the new cluster centers and the previous ones are the same, then terminate; otherwise, return to Step 2.

Fig. 2 Steps of the k-means algorithm

Start

Consider Jobsexist in shop floor

at beginning

Are you in planninghorizon?

Run optimizationalgorithm(MVNS) which uses

information extracted frominput data by clustering

Execute optimumschedule until next realtime event takes place

Finish

yes

No

Perform jobclustering

Fig. 3 Proposed strategyfor dynamic job shopscheduling problem

Int J Adv Manuf Technol

the intra-cluster distance and maximizes the inter-clusterdistance. In practice, the most popular metric is the sum ofsquared errors which is defined as Eq. (1).

SSE ¼Xk

i¼1

Xn

j¼1X ij−Ci

�� ��2 ð1Þ

Where

Ci ¼ 1

ni

Xn

j¼1X ij ð2Þ

denotes the mean of the i th cluster, k the number of clusters,Xij the j th pattern in the i th cluster, ni the number of patternsin the i th cluster, and

n ¼Xk

i¼1ni: ð3Þ

In this paper, a heuristic algorithm illustrated in Fig. 2[20,28] is used to do the clustering of n jobs based onsimilarity between m dimensional processing time vectorsfor each job in forms of the k-means clustering problem. It is

notable that the k-means clustering problem is computationallyhard (NP hard), but the selected heuristic algorithm is efficient interms of execution time and converges to a clustering structurequickly even though the obtained clustering structure is not aglobal optimum [29].

At any rescheduling point, the remaining jobs are clusteredby the k-means algorithm, and distances between clustercenters are calculated to be considered as a base to determinethe probability of selecting jobs to make neighborhoods asexplained in details in Sect. 4 (Fig. 3).

4 Modified VNS for the DJSS problem

The VNS algorithm as a metaheuristic method is widely usedto solve combinatorial problems. The algorithm searches theneighborhood of a solution until another solution better thanthe incumbent is found and then jumps to the new one. VNStries to escape from a local optimum by changing the neigh-borhood structure (NS) that is the manner in which the

Fig. 4 Steps of the modifiedVNS (MVNS) algorithm

Fig. 5 Steps of the local searchalgorithm

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neighborhood is defined. Neighborhoods are usually rankedin such a way that solutions increasingly far from the currentone are explored [30,31].

The VNS algorithm begins with an initial solution, x ∊S ,where S is the set of search space, and uses a two-nested loop.The inner loop alters and explores using two main functions,namely “shake” and “local search,” respectively. The outerloop works as a refresher reiterating the inner loop. The localsearch explores a better solution within the local neighbor-hood, while the shake diversifies the solution by switching toanother local neighborhood. In this paper, using the k-meansalgorithm, neighborhoods are made by a knowledge extractedfrom input data so that more diversification is assured byselecting jobs from different clusters. In the proposed selec-tion mechanism, the distance between cluster centers is con-sidered to calculate the probability of selecting a job numberto make a new adjacent. In other words, in this approach, a jobthat belongs to a farther cluster has a greater probability to beselected in making a new adjacent than one belonging to acloser cluster. The rate of similar jobs that have operationswith zero processing time can be high in dynamic environ-ment because when a real-time event takes place, with a highprobability, some operations of the current jobs are done.Considering an event-driven policy, it causes a reduction inVNS efficiency.

In VNS, the inner loop iterates as long as it keeps improv-ing the solutions, where an integer ,i , controls the length of the

loop. Once an inner loop is completed, the outer loop reiteratesuntil the termination condition is met. The steps of the modifiedVNS (MVNS) structure are illustrated in Fig. 4.

To reduce computational time and enhance efficiency andeffectiveness, two famous neighborhood structures, insertionand swap [32], are modified to be used in the proposedoptimization algorithm process for both shake and local searchfunction. Such modification consists of determining theprobability of job selection in NSs using job clusteringresults. The two neighborhood structures employed in theproposed algorithm are defined below.

1. Conducted insertion : Identifies two particular operationsbased on probability obtained from clustering results andplaces one operation in the position that directly precedesthe other operation.

2. Conducted swap : Identifies two particular operationsbased on probability obtained from clustering resultsand places each operation in the position previously oc-cupied by the other operation.

A threshold accepting method local search [33] based onthe two mentioned NSs is used in this paper. The thresholdaccepting method is an iterative procedure in which x ′←x ′′

when f(x ′′)−f(x ′)≤dr ; where dr is the acceptance level. Localsearch function steps are illustrated in Fig. 5.

5 Simulation

To demonstrate the performance of the proposed method, ajob shop environment consisting of ten machines is simulated.It is noteworthy that a job shop with more than six machinespresents the complexity involved in a large dynamic job shopscheduling problem [2]. The simulation starts consideringinitial jobs in the shop floor at the beginning and continuesuntil the number of new job arriving at the shop floor reaches1,200. The mean flow time of the latest 1,000 jobs that leavethe shop floor during the planning horizon is selected as theperformance measure. The proposed method is compared toVNS in its basic form and to make a fair comparison betweenVNS andMVNS; optimization parameters are selected so thatthe same CPU time is generated. It is also compared to three

Table 1 Comparison of the MVNS and other algorithms

Simulated shop floor condition Performancemeasure forSPT

Performancemeasure forLIFO

Performancemeasure forFIFO

Performancemeasure forVNS

Performancemeasure forMVNS

MTBJA=300, MTBF=5700, MTTR=300 2224.90 2223.80 2290.70 1921.40 1900.8

MTBJA=250, MTBF=5700, MTTR=300 2359.20 2395.90 2354.40 2011.4 1981.7

MTBJA=200, MTBF=5700, MTTR=300 2945.90 2799.60 2785.10 2428.2 2389.3

MTBJA=150, MTBF=5700, MTTR=300 5825.1 5532.4 6083.20 3132.4 3068.5

1.07

1.481.60

2.04

0.75

0.95

1.15

1.35

1.55

1.75

1.95

2.15

300 250 200 150

Imp

rove

men

t P

erce

nta

ge

MTBJA

Fig. 6 Improvement percentages over various MTBJA

Int J Adv Manuf Technol

common heuristics that are widely used in the DJSS literaturenamely SPT, LIFO, and FIFO.

Furthermore, it is assumed that all machines have the sameMTTR and MTBF equal to 300 and 5,700 time unit, respec-tively. Thus, on an average of 5,700 time unit, a machine isavailable and then breaks down with a mean time to repair of300 time units. Increasing mean time between job arrivals(MTBJA) and considering fixed values of MTBF and MTTRleads to an increase in the number of jobs that simultaneouslyexist in the shop floor at the rescheduling point. During thesimulation, the number of jobs that simultaneously exist in theshop floor may reach more than 200 jobs. Simulation isrepeated at four states of MTBJA, 300, 250, 200, and 150time unit. The number of clusters is also considered over thesimulation process as a tradeoff between exploration andexploitation in the MVNS algorithm. Results are presentedin Table 1.

As presented in Table 1, the proposed method improves theperformance measure in comparison with VNS.

In Fig. 6, improvement percentages over different MTBJAare illustrated. It is obvious that as the problem gets harder, theproposed method generates better schedules.

6 Conclusion

In this paper, a cluster analysis algorithm is used to extractknowledge from the problem’s input data in a dynamic jobshop scheduling problem in order to improve optimizationresults. In fact, the extracted knowledge is used in optimiza-tion process by variable neighborhood search algorithm. In theoptimization process that is triggered when a real-time eventoccurs to make a new adjacent solution, jobs in the fartherclusters have greater probability to be selected in the replacementmechanism. The proposed method (MVNS) was compared toVNS and some common heuristics using a simulated jobshop under a range of different conditions. Results indicatethat the performance of the proposed method is significantlybetter than VNS and other methods including three heuristicsnamely SPT, LIFO, and FIFO.

Conflict of interest The authors declare that they have no conflict ofinterest.

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