extended precedence preservative crossover for job shop scheduling problems

Extended precedence preservative crossover for job shop schedulingproblemsChung Sin Ong, Noor Hasnah Moin, and Mohd Omar Citation: AIP Conf. Proc. 1522, 1346 (2013); doi: 10.1063/1.4801285 View online: http://dx.doi.org/10.1063/1.4801285 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1522&Issue=1 Published by the American Institute of Physics. Additional information on AIP Conf. Proc.Journal Homepage: http://proceedings.aip.org/ Journal Information: http://proceedings.aip.org/about/about_the_proceedings Top downloads: http://proceedings.aip.org/dbt/most_downloaded.jsp?KEY=APCPCS Information for Authors: http://proceedings.aip.org/authors/information_for_authors

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Extended Precedence Preservative Crossover for Job Shop Scheduling Problems

Ong Chung Sin, Noor Hasnah Moin and Mohd Omar

Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur

Abstract. Job shop scheduling problems (JSSP) is one of difficult combinatorial scheduling problems. A wide range of genetic algorithms based on the two parents crossover have been applied to solve the problem but multi parents (more than two parents) crossover in solving the JSSP is still lacking. This paper proposes the extended precedence preservative crossover (EPPX) which uses multi parents for recombination in the genetic algorithms. EPPX is a variation of the precedence preservative crossover (PPX) which is one of the crossovers that perform well to find the solutions for the JSSP. EPPX is based on a vector to determine the gene selected in recombination for the next generation. Legalization of children (offspring) can be eliminated due to the JSSP representation encoded by using permutation with repetition that guarantees the feasibility of chromosomes. The simulations are performed on a set of benchmarks from the literatures and the results are compared to ensure the sustainability of multi parents recombination in solving the JSSP.

Keywords: Genetic algorithms, Multi-parent crossover, Precedence preservative crossover (PPX), Job shop scheduling problems (JSSP). PACS: 02.70.-c, 02.10.Ab, 02.10.Ox

INTRODUCTION

Job shop scheduling problem (JSSP) is one of difficult combinatorial scheduling problems. The JSSP is more complicated to solve when the size of the problems increases. Since JSSP is a practical problem related to production scheduling, it received a lot of attentions from researchers. There are a lot of different strategies ranging from mathematical programming (exact algorithms) to metaheuristics (especially genetic algorithm (GA)) to solve the problems [1]. Kaschel [2] compares the different methods and concludes that the performance of GA is only average on many test cases but GA is still a powerful instrument because of their ability to adapt to new problem types. Therefore, there are a lot of studies and researches have been done on GA to be applied on the JSSP [3-4].

In recent years, since the first used of GA based algorithm to solve the JSSP proposed by Davis as early as 1985[1], GA has attracted many researchers to improve efficiency of the scheduling method and frequently used to solve scheduling problem [5-11]. Various GA strategies are introduced to increase the efficiency of GA to find the optimal or near optimal solutions for JSSP [4]. In the GA strategies, hybridization of GA with other methods or local search methods provided good results in solving the problems where GA capitalize on the strength of the local search in locating the optimal or near optimal solutions. Specifically, the local search procedure of Nowicki and Smutnicki [12] is embedded into GA by Goncalves et al. [8] due to the effectiveness of the local search that increases the performance of GA.

In addition to hybridization, the structure of the GA can be modified and enhanced to reduce the problems often encountered in GA such as premature convergence. Park et al. [7] retard the premature convergence in GA by using parallelization of GA (PGA) to find the near optimal solutions. Watanabe et al. [15] proposed the use of crossover search phase into the GA with search area adaption. This modified GA has capacity for adapting to the structure of the solutions space. Tail redundancy of chromosome is a part of the problems in GA. Thus, heuristic method is embedded into crossover by Ripon et al. [14] to reduce the tail redundancy of chromosome when implementing crossover operation.

All these researches show that GA is not restricted to a single procedure and perform well by modifying the GA structure or hybridization with other methods to increase the accuracy of searching solutions. Hence, the GA can be modified accordingly to suit the problem at hand including selecting several numbers of parents for the crossover operation.

In the current literature, most of the methods using GA focus on using two parents for crossover operation and the use of multi parents (more than two parents) crossover or recombination is still, to the best of our knowledge

Proceedings of the 20th National Symposium on Mathematical SciencesAIP Conf. Proc. 1522, 1346-1354 (2013); doi: 10.1063/1.4801285

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lacking for the JSSP. Although multi parent crossover has been used in other fields [16-19], only limited numbers are applied to combinatorial scheduling problem. In particular Eiben et al. [20] proposed multi parent for the adjacency based crossover and Ting et al. [21] developed multi parent extension of partially mapped crossover (MPPMX) for the travelling salesman problems (TSP). The Eiben et al. proposed two types of adjacency based crossover (ABC): occurrence based and fitness based. In both cases, multi parents are recombined to produce a single child. ABC is specifically designed for order based representation where the positions of the gene in the TSP are taken into consideration in the scanning process which is relatively important for rearrangement of the gene. However the tests carried out do not produce any significant improvement in the TSP.

On the other hand, MPPMX performs well in the TSP. MPPMX is derived from the partially mapped crossover (PMX) which is one of the two parents’ recombination operators. They increase the number of parents for the crossover operation implementation instead of using two parents only for the process. Substring in the chromosomes are selected and exchanged with other parents by mapping, and the offspring will go through the legalization process. Due to the notable improvement of this method, multi parent crossover is able to be modified and consequently increases the efficiency of the GAs.

In this paper, we proposed extended precedence preservative crossover (EPPX) as a multi parents crossover. EPPX is based on the precedence preservative crossover (PPX) proposed by Bierwirth et al. [22]. PPX is used as our crossover references because of its capability to preserve the phenotypical properties of the schedules. EPPX as crossover operator will retain this advantage in the GA. The aim of this research is to ensure the sustainability of multi-parents recombination using EPPX to solve JSSP in conjunction with simple local search.

JOB SHOP SCHEDULING PROBLEMS

The job shop scheduling problems (JSSP) is a combinatorial scheduling problem. In the JSSP, a set of n jobs need to be processed on a set of m machines. A job consists of a set of operations J, which is denoted as Oij, i (1� i � n) is the number of jobs and j (1 � j � J) denotes the index of the operation order in the job. The technological requirement for each operation processing time is denoted as pij and a set of machines denoted as Mk (1� k � m). Precedence constraint of the JSSP is defined as:

� Operation jth must finish before operation jth+1 in the job. � Only one operation can be processed in the machine at the time for one time. � The delay time for the job transfer machine will be neglected and operation allocation for machine will be

predefined. � No interruption of the operation (preemptive is not allowed) � Same priority level of all operation

The main objective of JSSP is to find the minimum makespan for the scheduling (Cmax). The finish time of job i can represent by FiJ. Let G(k) be the set of operations being processed in the machine k,

and let kOijX , =1 if Oij has been assign to machine k and kOij

X , = 0 otherwise.

The conceptual model of the JSSP can be described as below: Minimize � �� max iJF (1)

1 1 , 1, 2,... , for allij ij ijF F p j J i� �� (2)

,( )

1 , for allij

ij

O kO G k

X k�

�� (3)

The objective function represented by equation (1) minimizes the maximum finish time in the set of the jobs i, therefore minimizes the makespan. Equation (2) satisfies precedence relationships between operations and equation (3) imposes that an operation can only be assigned to a machine at a time. An example of 3 jobs, 3 machines and their sequences are given in Table 1.

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TABLE (1). Example for 3 jobs 3 machines problem. Job Operation routing

1 2 3 1 3 3 2 Processing time 2 1 5 3 3 3 2 3 1 M1 M2 M3 Machine sequence 2 M1 M3 M2 3 M2 M1 M3

GENETIC ALGORITHMS

GA is search heuristic that mimic the evolutionary processes in biological systems. Evolutionary processes such as reproduction, selection, crossover and mutation which inspired by natural evolution are used to generate solutions for optimization problems. Those techniques are translated into the form of computer simulations. GA begins with a population, which represents a set of potential solutions in the search space. They attempts to combine the good features in each individual in the population using random search information exchange in order to construct individuals which are better suited than previous individuals. Through the evolution, individuals that are poor tend to replaced by fitter individuals to generate a new population which estimated that the desired optimal solutions will be found.

Representation

Representation of JSSP in chromosome can be classified in two approaches: direct and indirect [3]. Direct approach is represented as binary string and it evolves to find the better schedule. Indirect approach encodes an integer representation of the jobs in the chromosome. Indirect method is applied in our test. The chromosome is represented as permutation integer with repetition [5]. In this representation, number i=1, 2, 3… which represent the number of jobs and i is repeated according the number of operations required. Figure 1 illustrates the representation of 3 jobs and 3 operations/machines. The chromosome is represented as [1 2 3 3 2 2 3 1 1], where number 1, 2, and 3 in the chromosome represents job 1, 2, and 3 respectively. Each job consists of three operations so it is repeated three times in the chromosome. The chromosome is scanned from left to right, and at the j-th occurrence of a job number refers to the j-th operation in the technological sequence of this job. The chromosome created is always feasible and legal.

FIGURE 1. Permutation with repetition representation for 3 jobs 3 machines

Objective Function

In the JSSP, each job has different finish time due to the different schedule operation time. Cmax will be the maximum completion in the scheduling (please refer to equation (1)) .The objective is evaluated for each chromosome and it also determines the ranking of the chromosome which is used in the process of selection. Each chromosome competes with each other and the selected chromosome will survive to next generation based on the

Permutation with repetition

Machine 1 Machine 2 Machine 3

1 2 3 3 2 2 3 1 1

1 2 3 3 2 1 2 3 1

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objective function (fitness value). Chromosome with greater fitness indicates the greater probability to survive. The highest ranking chromosome in a population consider as the best solution. It is noted that the lower makespan is given the highest ranking.

Selection

In this method, we use the stochastic universal sampling (SUS). This fitness based proportionate selection will choose the chromosomes into recombination process with minimum spread and zero bias. After the recombination process, elitism strategy is applied and 10% of the best fitness new offspring to replace the 10% the bad individuals in the previous population to generate new population.

Proposed Extended Precedence Preservative Crossover (EPPX)

In GA, there are no limitations that the recombination process needs only two parents but multi parents consisting more than two parents is also acceptable. Therefore, some of the multi parents’ crossover operators are extended from the two parents’ crossover operators [16, 21] for recombination process. As mentioned before, EPPX is an extension of PPX. A crossover mask in the form of a vector is generated randomly to determine the genes in which parent, specified in the mask, to be selected for recombination. The multi parents will then recombine into a single offspring (Figure 2 (a)). Starting from the first element on the mask vector the first gene in that parent 1 is selected. The selected job (job 3) is eliminated in the other parents (Figure 2 (b, c)). The second element in the mask indicates that the first element (after deletion) is to be selected also from parent 1 (Figure 2 (c)). The third element in the mask shows that the first element in parent 3 is selected (Figure 2(d)). The process continues until all the elements in the mask have been examined. Pseudo code for EPPX presented as in Figure 3.

FIGURE 2. EPPX crossover

Parent 1 : 3 3 1 1 2 1 2 2 3

Parent 2 : 3 2 2 1 1 1 3 3 2 Parent 3 : 1 3 2 2 1 1 2 3 3

Vector : 1 1 3 2 3 3 1 1 2

Child : 3 Vector number 1= select first gene from Parent 1 Vector number 2= select first gene from Parent 2 Vector number 3= select first gene from Parent 3

Parent 1 : 3 3 1 1 2 1 2 2 3

Parent 2 : 3 2 2 1 1 1 3 3 2 Parent 3 : 1 3 2 2 1 1 2 3 3

Vector : 1 1 3 2 3 3 1 1 2

Child : 3 3

Parent 1 : 3 3 1 1 2 1 2 2 3 Parent 2 : 3 2 2 1 1 1 3 3 2

Parent 3 : 1 3 2 2 1 1 2 3 3

Vector : 1 1 3 2 3 3 1 1 2

Child : 3 3 1

(a) Example of EPPX

(c) Second step - Previous selected gene will be deleted, first gene (after deletion) in parent 1 selected and the job from the other parents are removed.

(d) Repeat - The process will continue until at the end of the vector

(b) First step - Vector number 1, the first gene in Parent 1 is selected and the same job from the other parents is removed.

Parent 1 : 3 3 1 1 2 1 2 2 3 Parent 2 : 3 2 2 1 1 1 3 3 2 Parent 3 : 1 3 2 2 1 1 2 3 3 Vector : 1 1 3 2 3 3 1 1 2 Child : 3 3 1 2 2 1 1 2 3

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FIGURE 3. Pseudo code for EPPX

Mutation

Mutation is a genetic operator, analogous to the biological mutation, which is used to maintain genetic diversity from one generation of a population of chromosomes to the next. The purpose of mutation in GA is to diversify, thus allowing the algorithm to avoid local minima by preventing the population of chromosomes from becoming too similar to each other, thus slowing or even stopping the evolution. This reasoning also explains the fact that most GA systems tend to avoid taking the fittest of the population only when generating the next chromosome but using rather a random selection (or pseudo-random with a weighting toward those that are fitter).

In this study, the mutation is applied by selecting two genes in different positions and different job inside the same chromosome to be swapped. The process will be repeated if two genes selected are at same position or same job. FIGURE 4 illustrates the swapping of the two genes in the chromosome.

x y

y x

FIGURE 4. Mutation by swapping two genes in the chromosome

Algorithm 1: Extended Precedence Preservative Crossover (EPPX) Crossover vector generated randomly Three parents selected ->S1, S2, and S3 `for k=1 to length of the chromosome do Select vector number by position k-th starting from the left in vector case vector number of Vector number 1: Choose first gene at left most S1 Search same job number at left most in S2 and S3 Remove the first gene in S1 Remove the gene searched in S2 and S3 Vector number 2: Choose first gene at left most S2 Search same job number at left most in S1 and S3 Remove the first gene in S2 Remove the gene searched in S1 and S3 Vector number 3: Choose first gene at left most S3 Search same job number at left most in S1 and S2 Remove the first gene selected in S3 Remove the gene searched in S1 and S2 end case Selected gene insert to new chromosome by sequence from left to right end for

Before mutation

After mutation

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SCHEDULE BUILDER

Feasible schedule consists of three types: semi-active, active and non delay schedule. Optimal solution of the scheduling always lies in the active schedule with no permissible left shift in the schedule is possible. Recombination of such active schedule chromosome will produce good solutions [9]. In order to build the active schedule chromosome, the chromosome needs to be decoded into feasible schedule. This is achieved by applying a schedule builder that performs by a simple local search. Scheduling can be build by selecting the gene of the chromosome from left to right and inserting into schedule with a simple schedule builder and then deleted from the chromosome [23]. While placing the job in the schedule, it must meet the technological requirement and precedence constraints. At the mean time, the job will always find the earliest completion time to be inserted by using a simple local search. Figure 5(a) illustrates the scheduling without local search and the job will be placed by following the sequences encoded in the chromosome [1 1 3 …]. Applying the local search will enable the job to find the possible blank time interval before appending an operation at the last position (Figure 5(b)) and the chromosome encoded will become [1 3 1…].

FIGURE 5. Local search procedure

COMPUTATIONAL RESULTS

Experiment Setup

The benchmarks that we use are selected instances from OR library [24]. The ft10 and ft20 considered as difficult computational problem especially ft10 problem referred as “notorious” problem because it remains unsolved for 20 years. The ten tough problems proposed by Applegate and Cook [25] are more difficult than the ft10 problem. The table shows the selected instances from the library with their best known solutions (BKS) [8]. Note that instances with asterisks are a part of ten tough problems.

TABLE (2). Instances of JSSP

Instances No. of Jobs No. of Machines BKS ft 06 6 6 55 ft 10 10 10 930 ft 20 20 5 1165 la 01 10 5 666 la 10 15 5 958 la 21* 15 10 1046 la 27* 20 10 1235 la 35 30 10 1888 la 38* 15 15 1196

M1 J1

M2 J1 J3

M3

0 2 4 6 8 10

(a) Semi active schedule without simple local (b) Active schedule after simple local search

M1 J1

M2 J3 J1

M3

0 2 4 6 8 10

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Due to the different structure of the JSSP, the number of individuals in the population is calculated based on the number of operations. Some problems have very large solution space which contains many variables and large range of permissible values for the solutions. Therefore a fixed population is probably not enough because it simply doesn’t represent large enough space sample of the solution space, so the population will vary on different size of instances. The other parameters are fixed for all problems (Table 3).

TABLE (3). Parameters for GA Parameter Value Number of individuals Selection

Twice of the total operations Stochastic Universal Sampling (SUS)

Crossover rate Mutation rate Elitism Maximum number of generations

0.7 0.4

Replace 10% best offspring to the previous population

500

Discussion of the Results

Table (4) summarizes all the results of the selected instances running by our GA. The relatively less difficult problems (ft06, la01, la10, and la35) can be solved easily with the small average deviation. This shows that GA is capable to solve the JSSP. When the problems structure become more complicated (ft10, ft20, la21, la 27, and la38), they are hard to solve. The best solutions with EPPX are located on the different numbers of parents for different instances. Each instance differs in the different range of deviations. We observe that the nearest optimal best solution for the difficult problems, the deviations obtained ranging between 2%-12%. The acceptable ranges of comparable GAs are until 7% [5]. Deviation ranges more than these values are considered not effective. Results of the best and average solutions become deteriorate and worse when the number of parents for recombination increases and it is not advisable to increase the number of parents further. Note that computational times for larger number of parents reducing because on the less crossover operations in a population (multi parents are recombined to produce a single child).

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TABLE (4). Results of the selected JSSP instances JSSP BKS Number of Parents

3 4 5 6 7 8 9 10

ft06 55 a 55 55 55 55 55 55 55 55 b 55.9 (1.64%) 56.2 (2.24%) 56.4 (2.60%) 56.8 (3.21%) 56.7 (3.15%) 56.7 (3.15%) 57.2 (4.06%) 56.63 (2.97%)

c 10.15 sec 9.75 sec 9.27 sec 9.45 sec 7.98 sec 7.59 sec 7.99 sec 6.20 sec

ft10 930 953 (2.47%) 955 (2.69%) 955 (2.69%) 950 (2.15%) 955 (2.69%) 990 (6.45%) 976 (4.95%) 960 (3.23%) 1008.4 (8.43%) 1014.6 (9.09%) 1020.2 (9.70%) 1030.97 (10.86%) 1032.2 (10.99%) 1039.3 (11.75%) 1041.4 (11.98%) 1035.2 (11.31%)

166.68 sec 132.55 sec 110.84 sec 95.46 sec 84.42 78.54 71.79 sec 67.81 sec

ft20 1165 1204 (3.35%) 1208 (3.69%) 1211 (3.95%) 1206 (3.52%) 1228 (5.41%) 1236 (6.09%) 1254 (7.64%) 1250 (7.30%) 1240.4 (6.47%) 1245.7 (6.93%) 1265.2 (8.60%) 1267.1 (8.76%) 1271.3 (9.12%) 1277.1 (9.62%) 1299.2 (11.52%) 1298.8 (11.49%)

168.88 sec 132.24 sec 112.95 sec 96.77 sec 85.38 sec 79.34 sec 72.78 sec 68.14 sec

la01 666 666 666 666 666 666 666 666 666 667.6 (0.235%) 669.6 (0.546%) 669.9 (0.591%) 670.8 (0.726%) 670.8 (0.726%) 674.5 (1.28%) 672.6 (1.00%) 675.3 (1.40%)


la10 958 958 958 958 958 958 958 958 958 958 (0%) 958 (0%) 958 (0%) 958 (0%) 958 (0%) 958 (0%) 958 (0%) 958 (0%)


la21 1046 1082 (3.44%) 1084 (3.63%) 1108 (5.93%) 1092 (4.40%) 1103 (5.45%) 1118 (6.88%) 1115 (6.60%) 1111 (6.21%) 1110.7 (6.19%) 1121.5 (7.21%) 1125.3 (7.58%) 1129.8 (8.01%) 1133.9 (8.40%) 1138.4 (8.84%) 1146.4 (9.60%) 1148.2 (9.77%)


la27 1235 1287 (4.21%) 1299 (5.18%) 1290 (4.45%) 1294 (4.78%) 1310 (6.07%) 1313 (6.32%) 1312 (6.23%) 1323 (7.13%) 1312.4 (6.30%) 1328.5 (7.57%) 1327.4 (7.48%) 1335.1 (8.10%) 1340.6 (8.55%) 1344.6 (8.88%) 1346.4 (9.02%) 1355.4 (9.75%)


la35 1888 1888 1888 1888 1888 1888 1888 1888 1888 1889.7 (0.0882%) 1890.1 (0.109%) 1889.1 (0.0565%) 1890.6 (0.139%) 1891.5 (0.187%) 1892.0 (0.210%) 1892.0 (0.214%) 1894.4 (0.341%)


la38 1196 1268 (6.02%) 1280 (7.02%) 1286 (7.53%) 1291 (7.94%) 1309 (9.45%) 1290 (7.86%) 1313 (9.78%) 1307 (9.28%) 1303.7 (9.00%) 1315.3 (9.97%) 1327.7 (11.01%) 1325.9 (10.86%) 1339.1 (11.97%) 1336.6 (11.75%) 1341.3 (1215%) 1341.1 (12.13%) 937.59 sec 735.10 sec 619.27 sec 541.91 sec 486.71 sec 441.93 sec 410.43 sec 386.22 sec

a = best solutions (deviation) b = average solutions (deviation) c = computational time Each instances’ result will appear same as ft06

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CONCLUSIONS

EPPX using multi parents are able to get the solutions within the range of the acceptable value of ranges of GA. Results show that certain range of number of parents for crossover will provide good solutions. In the future works, suitable number of parents for crossover needs to determined to solve JSSP and we anticipate that the efficiency of GA can be increased by embedding other local search into GA.

ACKNOWLEDGMENTS

The authors would like to thank Assoc. Prof. Dr. Noor Hasnah Moin and Prof. Dr. Mohd Omar, who provided valuable comments and ideas to the writing of the research summarized here. With their guidance and input, this paper can be finished successfully.

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extended precedence preservative crossover for job shop scheduling problems

Education

parents crossover

jssp representation

production scheduling

genetic algorithm ga

multiparent crossover

performance of ga

journal information

practical problem