Extended Local Clustering Organization

with Rule-Based Neighborhood Searchfor Job-shop Scheduling Problem

Yasumasa Tamura, Hiroyuki Iizuka, and Masahito Yamamoto

Graduate School of Information Science and Technology, Hokkaido University, Japan{tamura,iizuka,masahito}@complex.ist.hokudai.ac.jp

Abstract. Job-shop scheduling problem (JSP) is one of the hardest com-binatorial optimization problems. Local clustering organization (LCO) isproposed by Furukawa et al. to solve such a combinatorial optimizationproblems as the metaheuristic algorithm. Its effectiveness for the JSP isverified by the comparison with genetic algorithm. However, since LCO isbased on the greedy search, the solution is often trapped in local minima.To improve the problem, this study proposes a novel neighborhood searchmethod using priority rules. This paper also shows the extended LCO in-tegrated with the search method.

Keywords: job-shop scheduling problem, local clustering organization,dispatching rules, kicking techniques.

1 Introduction

Job-shop scheduling problem (JSP) is a combinatorial optimization problemto determine a feasible and efficient schedule to process multiple jobs on mul-tiple machines [1]. The JSP is commonly described as follows. A set of jobsJ = {J1, J2, · · · , Jn} and a set of machines M = {M1,M2, · · · ,Mm} are given,where the notation n and m correspond to the number of jobs and the number ofmachines, respectively. Each job Ji consists of the sequence of consecutive oper-ations (oi1, oi2, · · · , oim) and an operation corresponds to each process of the jobon each machine. The sequential order of the processes is generally called tech-nological sequence and independently given to each job. The feasible solutions inthe JSP are restricted by the following constraints.

– All of operations are processed in accordance with the technological se-quence.

– A machine cannot process multiple operations simultaneously.

– Multiple operations belong to the same job cannot be processed on differentmachines simultaneously.

– Each operation cannot be interrupted and resumed its process while beingprocessed.

c© Springer International Publishing Switzerland 2015 465H. Handa et al. (eds.), Proc. of the 18th Asia Pacific Symp. on Intell. & Evol. Systems – Vol. 2,Proceedings in Adaptation, Learning and Optimization 2, DOI: 10.1007/978-3-319-13356-0_37

466 Y. Tamura, H. Iizuka, and M. Yamamoto

The efficiency of a schedule is generally evaluated by the makespan, which meansthe maximum completion time of all jobs.

The JSP has been studied as an important subject in the field of the op-erations research, the computer science and so on [2,3,4]. Garey et al. showedthat the computational complexity theory classifies JSP into NP-hard class assame as many other combinatorial optimization problems [5,6]. In addition, itis shown that the JSP is more difficult problem than the travelling salesmanproblem (TSP), a typical combinatorial optimization problem, by the compari-son with the flow-shop scheduling problem (FSP), a kind of the restricted JSP.Because of these characteristics, many recent studies to solve the JSP focus onthe approximate algorithms such as heuristic or metaheuristic methods insteadof the optimization algorithms such as branch and bound.

There are two typical heuristic methods for the JSP, a shifting bottleneck pro-cedure proposed by Adams et al. [7] and the dispatching priority rules [1]. Theyare known as the practical solution methods since they can find near-optimal so-lutions in the reasonable computational time. Metaheuristics can also find highquality solutions in the practical computation time. In particular, neighborhoodsearch (also called local search) with the critical paths or the critical blocks areknown as the effective solution methods for the JSP. The critical paths (blocks)are defined as sets of consecutive operations whose order directly affects themakespan. Van Laarhoven et al. proposed a solution method based on simulatedannealing (SA) [8]. They also showed that the neighborhood structure using thecritical paths can be used to search for the near-optimal solutions efficiently. Fu-rukawa et al. proposed local clustering organization as a metaheuristic methodwithout using critical paths [9]. The effectiveness of LCO for the JSP is verifiedin comparison to genetic algorithm (GA). The authors also proposed the hybridalgorithm based on LCO and SA without critical paths [10]. The study showedthat LCO is difficult to escape from local optima because of its searching mech-anism based on the greedy search. The hybrid algorithm improves the problemand can search for better solutions than the original LCO. On the other hand,the hybrid algorithm also requires relatively longer computational time than theoriginal LCO because of the annealing processes.

This paper shows another approach to improve the problem in LCO. Thispaper proposes an extended LCO integrated with a novel neighborhood searchmechanism using priority rules. The proposed neighborhood search mechanismsearches effective solutions by applying a large-scale changes to the current so-lution without the annealing processes. In addition, to search good solutions ef-ficiently, this paper also proposes a novel mechanism to apply the priority rulesto the neighborhood search, in which the rules are adaptively optimized alongwith the search of the solution. By the collaboration between the large-scaleneighborhood search performed by the proposed method and the small-scaleneighborhood search performed by LCO, it is expected that the extended LCOwill search for good solutions efficiently. The effectiveness of the extended LCOand the proposed neighborhood search method is verified by some numericalexperiments.

Extended LCO with Rule-Based Neighborhood Search for JSP 467

Table 1. An instance of the JSP

—– 1 2 3

J1 (M1, 11) (M2, 9) (M3, 16)J2 (M2, 25) (M1, 10) (M3, 13)J3 (M2, 9) (M3, 11) (M1, 15)J4 (M3, 12) (M2, 14) (M1, 11)

10 20 30 40 50

Machine

Time

Fig. 1. A schedule for the instance shown in Table 1

2 Local Clustering Organization for the JSP

Local Clustering Organization is a probabilistic metaheuristic algorithm inspiredby the mechanism of the Self-Organizing Map (SOM), a kind of neural networksproposed by Kohonen [11], and it is originally proposed to solve the TSP [12].The searching mechanism of LCO is fundamentally based on the selection andthe modification of the local which means a part of the solution. In LCO, themodification processe is also named clustering.

This section briefly explains the overview of LCO for the JSP. The originalapplication of LCO to the JSP is studied in [9]. The work uses the permuta-tion with repetition as a solution representation. In addition, it provides someclustering methods to solve the JSP efficiently.

2.1 Solution Representation : Permutation with Repetition

The solution representation for the JSP based on the permutation with repeti-tion was originally proposed by Bierwirth[13]. The permutation consists of thesequence of jobs, in which each job occurs m times and xth occurrence of thejob Ji corresponds to the operation jxi . Decoding the permutation to a scheduleis performed by scanning it from left end to right sequentially. For example,the permutation with repetition S in eq. (1) represents one of the solutions forthe example problem given by Table 1, where (J1, 1) = (M1, 11) means the firstoperation of J1 has to be processed on M1 and it takes 11 units to process.

S = (J1, J3, J2, J4, J2, J3, J2, J4, J1, J3, J4, J1) (1)

468 Y. Tamura, H. Iizuka, and M. Yamamoto

localclustering

Fig. 2. An example of the clustering in the local

In this case, S is decoded into the sequence of consecutive operations Lo and italso represents the schedule shown in Fig. 1.

Lo = (o11, o31, o21, o41, o22, o32, o23, o42, o12, o33, o43, o13) (2)

The sequence Lo can also be used for solution representations. However, whensome entities in Lo are swapped recklessly, it can represent an unfeasible sched-ule. This problem makes the application of the solution algorithm difficult andcomplex. On the other hand, the solution representation using the permutationwith repetition is always decoded to a feasible schedule.

2.2 Overview of the Algorithm

In LCO for the JSP, a subsequence of the consecutive operations on the solu-tion representation is randomly selected as the local in each searching step. Thefollowing notations are used to determine the subsequence.

dc : the index of the center entity in selected localr(t) : the clustering radius in tth step

In the above notations, dc is determined at random in each step. The localcorresponds to a subsequence of the consecutive operations within a radius r(t)from the selected entity dc.

The clustering method rearranges the sequence of the operations in the localwith the hill climbing search. Figure 2 partly shows the rearrangement performedby the clustering method. In general, LCO uses multiple clustering methods incombination and one of them is stochastically selected in each searching step(mixed clustering). According to some preliminary experiments, this paper usesthree clustering methods, SEM, SIM and IEM. SEM tries to rearrange the se-quence of operations in the local by swapping two entities iteratively and SIMinserts an entity into another place iteratively in the local. In addition, IEMtries to rearrange the sequence of operations in the local by inverting some sub-sequence iteratively in the local. It is important that these clustering methodsrearrange the sequence of operations in the local along with the greedy searchmechanism. Therefore, the current solution cannot become worse by each clus-tering method. The detail of them is described in our previous work [10] or theoriginal paper of LCO for the JSP [9].

Extended LCO with Rule-Based Neighborhood Search for JSP 469

The following procedure briefly shows the algorithm of LCO, where Sc meansthe current solution. The notation Sc is basically used to describe the currentsolution in this paper.

1. Set step t ← 1.2. Initialize Sc at random.3. Select a local on Sc.4. Select a clustering method stochastically.5. Apply the selected clustering method to the selected local on Sc.6. Replace t with t+ 1.7. If termination conditions of the algorithm are satisfied, stop the procedure.

Otherwise, go back to step 3.

In LCO, the clustering radius r(t) has a great effect on accuracy and computa-tional time of optimization. The smaller radius sometimes generates inaccuratesolutions because the widely covered optimization in the whole solution is notperformed. On the other hand, the bigger radius causes the delay of computa-tional time. In addition, the probabilities to select the clustering methods arealso important parameters of LCO to acquire high accurate solutions.

3 Extended LCO with Rule-Based Neighborhood Search

This paper integrates a novel large-scale neighborhood search method into LCO.In this section, firstly, the neighborhood structure and the searching mechanismusing priority rules are described. By combining the proposed neighborhoodsearch method with LCO, this section finally proposes the extended LCO.

3.1 Neighborhood Structure on the Permutations with Repetition

To define the large-scale neighborhood structure, the rearrangeable blocks areintroduced in this study. The rearrangeable blocks mean the subsequences ofthe solution representation which consist of the consecutive jobs processed onthe same machines. By the evaluation of the solution representation, each job inthe permutation with repetition is associated with the machine on which the jobis processed. The association between jobs and machines defines the sequence ofconsecutive jobs which are processed on the same machine as the rearrangeableblocks. For example, Fig. 3.1 shows the rearrangeable blocks on the solution Sshown in eq. (1). There are four rearrangeable blocks marked with the lines.

The neighborhood in our proposed method is defined as a set of the solutionsgenerated from a solution by only applying rearrangement to the sequences ofconsecutive jobs in the rearrangeable blocks on the permutation with repetition.In addition, as shown in Fig. 3.1, the rearrangement of the jobs in the rearrange-able blocks causes the change of the schedule (note that the rearrangements inthe solution representation does not necessarily changes the schedule due to theredundancy of the representations).

470 Y. Tamura, H. Iizuka, and M. Yamamoto

10 20 30 40 50

Machine

Time

Permutation with repetition

Schedule

Fig. 3. The rearrangeable blocks on the solution representation and the schedule

10 20

Machine

Time 10 20 Time

Job

Fig. 4. An example of rule-based schedule construction

3.2 Rearranging the Sequence of Operations Using Priority Rules

Since there are generally multiple rearrangeable blocks in a solution, the num-ber of neighborhood solutions generated from a solution sometimes increasesexponentially. The proposed method uses the priority rules to limit the numberof neighborhood solutions and to acquire the effective neighborhood solutionsefficiently.

We proposed the composite priority rule in the previous work [14]. The com-posite priority rule consists of some simple priority rules, such as the shortestprocessing time (SPT) and least work remains (LWKR) [1], and it gives the de-gree of relative priority P (oix) to each operation oix. This study uses followingthree notations to describe the composite priority rules.

pix : the processing time of the operation oixqix : the total processing time of the following operations of the operation oixwix : the waiting time of the operation oix for the machines

The example of those notations for the operation o12 is shown in Fig. 4. Itis commonly known that pix is used to describe SPT, qix is used to describeLWKR and wix is usd to describe a simple rule called first come first serve(FCFS).

In this study, the proposed method uses the linear combination rules as thepriority rules. The linear combination rule gives a degree of relative priority

Extended LCO with Rule-Based Neighborhood Search for JSP 471

P (oix) to a operation oix using the linear combination shown in eq. (3), wherethe notations α1, α2 and α3 correspond to the weights for the linear combination.

P (oix) = α1pix

max pix+ α2

qixmax qix

+ α3wix

maxwix(3)

Before applying the rules to generate the neighborhood solutions, the prioritiesof operations are originally defined by the current solution. To consider the effectsof the original priorities, this paper introduces a threshold model into rearrangingmechanism. When two operations oix and oi′x′ in the same rearrangeable blockhas the original precedence relation oix ≺ oi′x′ which means oix is processedpreferentially over oi′x′ , the precedence relation is inverted if eq. (4) is satisfied.Otherwise, the original precedence relation is kept as it is.

P (oix)− P (oi′x′) ≤ θ (4)

The notation θ in eq. (4) means the threshold value and this study regards θ asa part of the priority rule. In other words, the priority rule used in this study isdynamically composed of the real coded vector (α1, α2, α3, θ).

A set of priority rules is created by generating individual vectors (α1, α2, α3, θ).The number of priority rules is a parameter of the proposed method and it corre-sponds to the number of neighborhood solutions searched per step. In addition,because the effective rules can be different in each instance and the best ruleis generally unpredictable, the rules are optimized along with the searches ofthe solutions (schedules). In this study, the optimization of the rules is per-formed by genetic algorithm (GA). The fitness of each rule is evaluated by themakespan of the solution generated by the rule. From this mechanism, someeffective rules which generate effective neighborhood solutions are adaptivelyobtained in searches, and the optimization of the schedule is progressed usingthe effective rules.

3.3 Integration into LCO

The extended LCO alternately applies the local clustering and the large-scaleneighborhood search using rules to the current solution. The following procedurebriefly shows the algorithm.

1. Generate an initial solution Sc at random.2. Generate initial rules at random and evaluate each rule.3. Select a local on Sc and a clustering method stochastically.4. Apply the selected clustering method to the selected local on Sc.5. Apply genetic operations, selection, crossover and mutation, to the rules.6. Generate neighborhood solutions from Sc using each rule and replace the

best neighborhood solution with Sc.7. Evaluate each rule.8. If termination conditions of the algorithm are satisfied, stop the procedure.

Otherwise, go back to step 3.

472 Y. Tamura, H. Iizuka, and M. Yamamoto

In the procedure, the steps 3-4 correspond to the searches performed by theconventional LCO and the step 6 corresponds to the proposed neighborhoodsearch. The step 5 and step 7 correspond to the optimization processes for therules in the procedure. In addition, because the current solution can becomeworse in the step 6, the best solution found in searches should be memorized.

4 Numerical Experiments

To examine the effectiveness of the proposed method, this section provides theexperimental results and discussion. In the experiments, this paper uses 25 well-known benchmark problems introduced by Lawrence [15]. To evaluate the effec-tiveness of the extended LCO in different problem sizes, the benchmarks prob-lems are classified into 5 groups in terms of the problem size. The effectivenessis evaluated using the relative error rate RE shown in eq. (5), where C meansthe obtained makespan by the searching algorithm and Co means the optimalmakespan show in [16].

RE =C − Co

Co× 100 [%] (5)

In this paper, the effectiveness of the extended LCO is compared with theconventional LCO and SA proposed in [8]. All of methods are tested 100 timesin each benchmark problems. The conventional LCO and the extended LCO areterminated when 10 seconds have passed from the algorithm is started.

The parameters for LCO, the clustering radius r(t) and the selection proba-bilities for each clustering method, are determined by some preliminary exper-iments. The clustering radius r(t) is determined by eq. (6) in each search step,where URAND(a, b) generates a uniform random number in the range [a, b), nmeans the number of jobs and m means the number of machines.

r(t) = URAND

(1

4,1

3

)× n×m (6)

The effectiveness of each individual clustering method, SEM, SIM and IEM isalso verified in the preliminary experiments. Considering the effectiveness ofLCO, the selection probabilities should generally be set to (SIM) ≥ (SEM) �(IEM), where (SIM), (SEM) and (IEM) respectively mean the selection proba-bilities of SIM, SEM and IEM. In this study, the selection probabilities are setas eq. (7).

(SIM) : (SEM) : (IEM) = 50% : 40% : 10% (7)

These parameters are also used in the extended LCO as well as the conventionalLCO. In addition, the extended LCO generates 10 neighborhood solutions using10 kinds of the priority rules in the searching processes to apply the proposedneighborhood search method.

Extended LCO with Rule-Based Neighborhood Search for JSP 473

Table 2. Comparison with the Conventional LCO

Instances Jobs × MachinesExtended LCO Conventional LCO

Best [%] Avg. [%] Best [%] Avg. [%]

la16 - la20 10 × 10 0.160 0.410 0.730 2.23la21 - la25 15 × 10 0.760 1.15 0.920 2.36la26 - la30 20 × 10 1.25 1.71 1.52 2.18la31 - la35 30 × 10 0.000 0.000 0.000 0.000la36 - la40 15 × 15 2.09 3.49 2.78 4.43

Avg. 0.850 1.35 1.19 2.24

Table 3. Comparison with SA shown in [8]

Instances Jobs × MachinesExtended LCO SA

Best [%] Avg. [%] Best [%] Avg. [%] T [sec]

la16 - la20 10 × 10 0.160 0.410 0.707 1.27 715.20la21 - la25 15 × 10 0.760 1.15 1.23 2.02 2095.6la26 - la30 20 × 10 1.25 1.71 1.83 2.41 4319.0la31 - la35 30 × 10 0.000 0.000 0.000 0.744 1740.6la36 - la40 15 × 15 2.09 3.49 1.67 2.50 5450.4

Avg. 0.850 1.35 1.09 1.79 2864.2

4.1 Comparison with the Conventional LCO and SA

Table 2 compares the results of the extended LCO and the conventional LCOand Table 3 compares the results of the extended LCO and SA. To compare themean performance of each method, this paper shows the mean of the best RE(the column Best) and the average RE (the column Avg.) for each instance over100 trials. In addition, the column T [sec] in the results of SA shows the averagecomputational time of SA. Because the architecture of computers used in [8] isfar different from the one used in this study, the column T [sec] cannot be usedto the statistical comparison. However, the values can be used to understandtrends of the search performed by SA.

From Table 2, the extended LCO can obviously search for better solutionsthan the conventional LCO, which means the extended LCO is averagely moreeffective than the conventional method in all problem sizes. The results suggestthat the proposed neighborhood search method can improve the searching mech-anism of LCO. On the other hand, in Table 3, while most results show the similartrends as Table 2, the effectiveness of the extended LCO is less than that of SAin the group la36-la40. Because SA requires much long computational time forthe instances la36-la40, it is considered that the extended LCO has not muchcomputational time for those instances in the experiments. The performance ofthe proposed method and the extended LCO is investigated in following discus-sion, including the cause of the ineffectiveness of the extended LCO shown inthe comparison with SA.

474 Y. Tamura, H. Iizuka, and M. Yamamoto

Fig. 5. Comparison with the random selection

4.2 The Effectiveness of the Rule-Based Neighborhood Search

To evaluate the effectiveness of the rule-based neighborhood search method, thispaper compares it with the neighborhood search not using the priority rules.The comparison search method generates neighborhood solutions based on thesame neighborhood structure as the rule-based one. On the other hand, thecomparison method selects the limited number of neighborhood solutions (inthis case 10 solutions) at random and the current solution is replaced with thebest one in the selected solutions. Figure 5 shows the results of the comparison foreach instance, where the red boxes correspond to the average RE of rule-basedselection and the blue ones correspond to the average RE of random selectionfor each instance. The average RE is calculated from the results obtained over100 trials as well as the previous experiments. Figure 5 explains that the rule-based neighborhood search leads better results than the random neighborhoodsearch in most cases. The results suggest that the proposed method can searchfor effective neighborhood solutions efficiently when the number of generatedneighborhood solutions is limited.

On the other hand, Fig. 5 also shows that there is hardly any differencebetween the effectiveness of the rule-based neighborhood search and that of therandom neighborhood search in the group la36-la40. It is inferred that this resultis related with the fact that the extended LCO is less effective than SA for theinstances in the group la36-la40. The rest of this section discusses the behaviorof the proposed method for the instances la36-la40.

Extended LCO with Rule-Based Neighborhood Search for JSP 475

1400

1450

1500

1550

1600

1650

1700

0 2000 4000 6000 8000 10000

mak

espa

n

elapsed time [msec]

BestTemporal

Fig. 6. Changes of makespan in a trial for la37

Figure 6 shows the changes of makespan which the best solution and thetemporal solution have, where the temporal solution corresponds to the currentsolution in searches. From Fig. 6, while the makespan of the temporal solutiondecreases in the early stage of the search (approximately from 0 to 1000 [msecs])as well as the best solution, it often increases in the middle and late part ofsearch. This result means the proposed method can assist LCO in obtainingeffective solution in the early part of search, which is considered as an effectiveadvantage of the proposed method. In addition, while the temporal solutionbecome worse, the best solution is improved in the middle stage of the search.It means the proposed method successfully help the current solution to escapefrom local minima as a kind of kicking methods. Although these results show theadvantages of the proposed method, the behavior of the method in the latterstage of the search (after 8000 [msecs]) shows its disadvantage. The makespanof the temporal solution is much worse than the best solution the latter stageof the search (after 8000 [msecs]), which makes the search ineffective. From Fig.6, such a problem is caused by the difficulties of obtaining effective rules orthe performance limit of the rules. If the problem is caused by the difficultiesof obtaining the effective rules, the problem can be improved by extending thecomputational time or devising the optimization algorithm for the rules. It isalso considered that if the problem caused by the performance limit of the rules,we should revise the searching mechanism. Since the consecutive application ofthe rule-based neighborhood search seems to make the current solution worse,the alternate application of LCO and the proposed method should particularlybe revised. Our future work is to verify those consideration and to improve theproposed method.

476 Y. Tamura, H. Iizuka, and M. Yamamoto

5 Conclusion

This paper proposes an improved LCO integrated with a novel large-scale neigh-borhood search method. The large-scale neighborhood search method has somesignificant characteristics as follows:

– The neighborhood solutions are generated by applying some large changesto the current solution.

– Some effective neighborhood solutions are generated by some dynamic pri-ority rules.

– The number of neighborhood solutions are limited by the number of rulesnot to deteriorate the efficiency of searches.

– The priority rules are also optimized along with searches of the schedule.

The effectiveness of the improved LCO is verified by some numerical experi-ments. The experimental results are summarized as follows:

– Improved LCO can averagely search for better solutions than the conven-tional LCO.

– Rule-based selection of the neighborhood solutions is more effective thanrandom selection, when the number of neighborhood solutions is limited.

– The results suggest the proposed neighborhood search method improves thecurrent solution in the early part of search, and it also performs the kickingmechanism in the middle part of search.

Acknowledgement. This study was supported by JSPS KAKENHI, Grant-in-Aid for JSPS Fellows, 26·1342. This study was also supported by the members ofthe Laboratory of Autonomous Systems Engineering, Hokkaido University. Wewould like thank them for their numerous suggestions.

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