Multiobjective Job-Shop Scheduling With Genetic Algorithms Using a New Representation and Standard Uniform Crossover J. Garen 1 1. Department of Economics, University of Osnabrück, Katharinenstraße 3, 49074 Osnabrück - Germany Joost.Garen@Bertelsmann.de Abstract: This paper focuses on multi-objective job-shop scheduling using genetic algorithms. Although dealing with multiple objectives has received more and more attention over the last few years, scheduling is still dominated by unrealistic single-objective approaches. Representation of solutions is a key issue for implementing efficient multi-objective genetic algorithms, especially for the heavily constrained job-shop. We introduce a new kind of coding which allows the use of standard recombination operators without losing solution feasibility. The crossover chosen is Syswerda s uniform-crossover. First experiments with two problem instances taken from [Bag99] show the usefulness of the proposed method. Keywords: Multiple Objectives - Scheduling - Job-shop - Genetic Algorithms - Representation 1 Introduction The classical job-shop scheduling problem (JSP) is one of most difficult combinatorial optimization problems. During the last decades a great deal of attention has been paid to solving these problems with genetic algorithms (GAs). In a single-objective context some of the recent approaches have shown quite promising results [Mat96, OYK96]. But real world scheduling problems naturally involve multiple objectives. There are only few attempts to tackle the multiobjective JSP [Bag99]. The remainder of the paper is organized as follows. Section 2 describes the general JSP. Section 3 introduces the new representation. Section 4 shows some experimental results. Finally, section 5 presents the conclusion of this work. 2 Problem Statement The classical JSP consists of a set J of n jobs that must be processed in a set M of m machines. Each job j J consists of a chain of m operations describing the processing order. Each operation o ij, representing the i-th operation of job j, is characterized by the processing time t ij and the machine required. There are several constraints on jobs and machines, such as 1. a job does not visit the same machine twice 2. each machine can process only one operation at a time 3. operations cannot be interrupted 4. there are no precedence constraints among operations of different jobs 5. setup times for the operations are sequence-independent and included in the processing times 6. there is only one of each type of machine 7. machines are available at any time 1
A schedule is a description of when to process each of the operations satisfying the constraints. The goal of single-objective job-shop scheduling is to find the optimal schedule, which minimizes the predefined performance measure. In a multi-objective context we hope to find as much different schedules as possible, which are non-dominated with regard to two or more objectives. Some frequently used performance measures are makespan, mean flow-time and mean tardiness. Makespan is defined as the maximum completion time of all jobs, mean flow-time is the average of the flow-times of all jobs. Mean tardiness is defined as the average of tardiness of all jobs. 3 Representation of Solutions GAs require the parameter set of the underlying optimization problem to be coded as a socalled chromosome. While the evolution works on the chromosomes (genotype), evaluation is done with the decoded solution (phenotype). The way of encoding solutions to chromosomes is a key issue for GAs [CGT96]. Several different representations have been proposed for the JSP, such as: 1. operation-based representation 2. job-based representation 3. preference list-based representation 4. job pair relation-based representation 5. priority rule-based representation 6. disjunctive graph-based representation 7. completion time-based representation 8. machine-based representation 9. random keys representation Apart from random keys representation all of the above codings need non-standard crossover operators, which guaranties solution feasibility. We agree with Jain and Meeran that crossover operators for job-shop scheduling generally lose their efficiency by generating feasible schedules [JM98]. In a single-objective context the use of local search neighborhoods is one possible way out. These approaches are well known as genetic local search, population based local search or mementic search. It has been shown that the relevant neighborhood for the objective makespan is very small [JRM]. However, using genetic local search in multi-objective optimization is not straightforward because we could not restrict the neighborhood of a solution to some operations lying on the critical path. For this reason representation of solutions has to provide for an effective search. 3.1 Formal Description of the Proposed Representation A chromosome, consisting of n m genes represents a solution. Each gene symbolizes one operation. For each solution the order of genes within the chromosome is exactly the same. The first m genes symbolize all operations of the first job, the second m genes symbolize all operations of the second job and so forth. Thus a chromosome C is formally defined as C : [o 11, o 21,..., o m1,..., o 1n, o 2n,..., o mn ] (1) Obviously, that the gene at locus 3 m for example always corresponds to the last operation of job three. 1 Each operation holds a number which is used as a sort key to decode the solution. 1 Except in the phase of decoding. See section 3.2. 2
The values of the numbers are restricted to the interval I = [1,..., m]. For a simple two-job two-operation problem a possible solution would be C : [o 11 = 2, o 21 = 1, o 12 = 1, o 22 = 2] (2) 3.2 Decoding of the Proposed Representation The way of decoding a solution is organized as follows. First the solution is sorted in ascending order using the numbers. If two operations have the same number, the operation with the shorter processing time (known as SPT priority rule) is preferred. Thus, the solution shown in formula 2 is modified to 2 C : [o 21 = 1, o 12 = 1, o 11 = 2, o 22 = 2] (3) Second, the dispatching sequence of operations is constructed in a simple step-by-step manner. The chromosome is scanned from left to right. If the predecessor of the currently viewed operation has already been considered in the dispatching sequence, the operation is scheduled. The procedure of scanning the chromosome is repeated until all operations are scheduled. Algorithm 1 clarifies the procedure. Algorithm 1: The simple schedule-builder Data : C = Array[1...n m] of sorted operations begin while not all operations scheduled do for (i = 0; i < n m; i + +) do if predecessor of C[i] is already scheduled then dispatch C[i] remove C[i] from the chromosome end The numbers consequently determine the relative position of the dispatching attempt(s). For the sorted solution shown in formula 3 the corresponding dispatching sequence would be C : [o 12, o 11, o 22, o 21 ] (4) Obviously the proposed representation which only decodes feasible solutions, is redundant. Thus one schedule (phenotype) can be represented in many different ways (genotype). But we expect the use of standard crossover operators independent of any constraints to over-compensate the disadvantage of redundancy. 4 Experiments 4.1 Single-objective Context To evaluate the proposed representation we perform some experiments in a single-objective context. The experiments are conducted using 100 chromosomes and run for 100 generations. The problem instances are Fisher & Thompson s ft10 and ft20 as well as Lawrence s la23, la27 and la32. Every solution of the current generation is selected for uniform-crossover together with another solution which is extract using roulette-wheel-selection. The probability of crossover is 0.9, while the rate of mutation is 0.04 3. Every offspring replaces the worst solution of the current generation. Table 1 shows the results obtained. 2 Under the assumption that t 21 t 12 and t 11 t 22. 3 The used mutation operator just swaps two numbers of two operations randomly. 3
Instance Optimum Best Average ft10 930 941 975.9 ft20 1165 1181 1215.9 la23 1032 1032 1032 la27 1235 1290 1307.3 la32 1850 1850 1850 Table 1: Results for single-objective JSP 4.2 Multi-objective Context Multi-objective optimization differs from single-objective optimization in many ways [Deb01]. For two or more conflicting objectives, each objective corresponds to a different optimal solution, but none of these trade-off solutions is optimal with respect to all objectives. Thus, multi-objective optimization does not try to find one optimal solution but all trade-off solutions. Apart from having multiple objectives, the fundamental difference is that multi-objective optimization deals with two goals. The first goal is to find a set of solutions as close as possible to the Pareto-optimal front. The second goal is to find a set of solutions as diverse as possible. For multi-objective scheduling the proposed genetic algorithm is modified in the following way. We introduce an archive which contains non-dominated solutions. If an offspring dominates a member C of the archive the offspring will become a new member of the archive, while C will be removed. The fitness-value of a chromosome C is calculated by dividing one by the number of solution which dominates C. Two JSP given by Bagchi [Bag99, p. 265, Tab. 12.1 and 12.2] are the basis of the following experiments. The first problem, called JSP1, is a ten job five machine instance. The second problem, called JSP2, is a ten job ten machine instance. Figure 1 and Figure 2 show all non-dominated solutions of a random initial population as gray circles and all non-dominated solutions of generation 200 as transparent circles for JSP1 respectively JSP2. The diameter of the circle symbolizes the mean tardiness of the corresponding solution. Apparently, the GA minimizes all objectives simultaneously. Mean flow-time Mean flow-time 150 200 140 190 130 180 120 170 160 170 180 190 Makespan 190 210 230 250 Makespan Figure 1: JSP1: Non-dominated solutions Figure 2: JSP2: Non-dominated solutions To maintain sufficient diversity in later generations and to evaluate significance of populationsize, the number of chromosomes is now increased to 400. The algorithm loops for 150 generations. Figure 3 and Figure 4 show the number of Pareto-optimal solutions for the process of evolution. Table 2 and Table 3 show the values of objectives for ten Pareto-optimal solutions. 4
Unique Pareto-solutions 40 32 24 16 8 50 100 150 Generations Figure 3: JSP1: Process in finding Paretooptimal solutions solution makespan flow-time tardiness 1 159 124.3 15.7 2 167 122.4 15.1 3 182 135.7 5.8 4 156 128.4 10.8 5 169 134.5 6.1 6 159 127.3 7.8 7 160 124.3 13.9 8 165 128.8 6.4 9 158 126.0 8.2 10 162 130.5 6.4 Table 2: JSP1: Ten Pareto-optimal solutions Unique Pareto-solutions 20 16 12 8 4 50 100 150 Generations Figure 4: JSP2: Process in finding Paretooptimal solutions solution makespan flow-time tardiness 1 228 189.1 30.7 2 254 186.7 29.2 3 238 188.1 28.2 4 204 174.8 31.3 5 230 179.4 29.3 6 196 174.7 32.2 7 203 173.4 32.2 8 201 176.1 31.8 9 199 174.6 33.0 10 212 174.5 31.6 Table 3: JSP2: Ten Pareto-optimal solutions 5 Conclusion In this report we have presented a multi-objective GA for job-shop scheduling with a new kind of representation that allows the use of simple recombination operators. The new representation has initially been tested in a single-objective context to evaluate its effectiveness with quite promising results. Afterwards the algorithm has been applied to two JSP presented by Bagchi [Bag99]. The simulation results clearly show that the proposed approach is able to find a set of solutions close to the Pareto-optimal front as well as find a set of diverse solutions. References [Bag99] T. P. Bagchi. Multiobjective Scheduling By Genetic Algorithms. Kluwer Academic Publishers, 1999. [CGT96] R. Cheng, M. Gen, and Y. Tsujimura. A tutorial of job-shop scheduling problems using genetic algorithms - i. representation. Computers industrial Engineering, 30:983 997, 1996. [Deb01] K. Deb. Multi-Objective Optimization Using Evolutionary Algorithms. John Wiley & Sons, 2001. 5
[JM98] [JRM] A. Jain and S. Meeran. A state-of-the-art review of job-shop scheduling techniques. Technical report, University of Dundee, Department of Applied Physics, Electronic and Mechanical Engineering, 1998. A. Jain, B. Rangaswamy, and S. Meeran. Job-shop neighbourhoods and move evaluation strategies. [Mat96] D.C. Mattfeld. Evolutionary Search and the Job Shop. Physica-Verlag, 1996. [OYK96] I. Ono, M. Yamamura, and S. Kobayashi. A genetic algorithm for job-shop scheduling problems using job-based order crossover. In Proceedings of ICEC 96, pages 547 552, 1996. 6