Abstract—A job shop scheduling problem with total tardiness and the maximum tardiness as objectives is addressed. We solve it by a rule-coded genetic algorithm. Characteristics of three existing fitness assignment mechanisms are identified and then combined through the proposed cyclic fitness assignment mechanism. Experiments are conducted on a public benchmark problem set, and the results show that the proposed algorithm outperforms the existing ones. I. INTRODUCTION Job shop scheduling is an attractive research topic for both researchers in the academia and engineers in the industry for last several decades. Theoretically, this problem has NP-hard complexity, which means that it is almost impossible to develop an algorithm to solve it optimally in polynomial time. This challenge makes it one representative problem in the category of hard combinatorial optimization problems. Practically, production control engineers in the industry face scheduling problems everyday. They need to determine how to arrange the resources to produce products to satisfy the concerned criteria such as customers’ due dates. The job shop scheduling problem shows essential similarities with practical scheduling problems, and solutions to this problem can be easily applied to the real-world problem. Scheduling refers to a task of allocation of resources over time to requests under certain constraints so as to satisfy or optimize the concerned criteria. In a typical job shop scheduling problem, we have M machines as resources and J jobs as requests. The common constraints include: (1) Each job needs to be processed by each machine exactly once in a predefined order. (2) Processing of a job on a machine is called an operation. The processing time of an operation is a constant and is known in advance. (3) There exists a precedence constraint between operations belonging to the same job, which means that the succeeding operation can start processing only after the preceding operation was finished. (4) Each machine can process only one job at a time. (5) The processing of an operation is uninterruptible. (6) There is no machine breakdown, transportation time, and setup time. (7) All jobs are ready at time zero. For each job j, denote its completion time by C j and due date by D j , and then its tardiness T j is calculated by max{0, C j – D j }. On one hand, we want to minimize the total tardiness so that customers’ jobs can be finished within due dates as much as possible. On the other hand, we also want to minimize the maximum tardiness to prevent some customer’s job from delaying too long. Thus, we define two objectives in this work: ∑ = = J j j T f ... 1 1 and } { max ... 1 2 j J j T f = = . Given more than one objective, solving job shop scheduling problems becomes more difficult since sometimes we cannot improve multiple objectives simultaneously. Let us use the problem listed in Table I as an example. Given the problem, two possible schedules A and B are depicted in Fig. 1. In schedule A, we have T 1 = 0 and T 2 = 7, and consequently f 1 = 7 and f 2 = 7. In schedule B, we have T 1 = 6 and T 2 = 2, and consequently f 1 = 8 and f 2 = 6. As we can see, there is not a single schedule which is optimal with respect to both f 1 and f 2 . TABLE I. A 2×2 JOB SHOP SCHEDULING PROBLEM Op1(Machine/Time) Op2(Machine/Time) Due date Job 1 M1/5 M2/5 12 Job 2 M1/8 M2/4 10 Fig. 1 Trade-off between multiple objectives In the literature, multiobjective optimization problems were generally treated in two ways. The first category of approach converted multiple objectives into a single objective through some aggregation function (for example, the linear weighted summation function), and then applied the single-objective optimization method to find the (single) optimal solution. Hence, the decision makers are usually burdened with the requirement of defining the suitable aggregation function (for example, defining the weight of each objective). Instead of finding a single optimal solution, Multiobjective Job Shop Scheduling using Genetic Algorithm with Cyclic Fitness Assignment Tsung-Che Chiang Li-Chen Fu National Taiwan University, Taiwan, R.O.C. National Taiwan University, Taiwan, R.O.C. tcchiang@ieee.org lichen@ntu.edu.tw D 2 = 10 D 1 = 12 10 17 18 12 Job 1 Job 2 Job 1 Job 2 Schedule B Schedule A f 1 f 2 7 8 7 6 A B 0-7803-9487-9/06/$20.00/©2006 IEEE 2006 IEEE Congress on Evolutionary Computation Sheraton Vancouver Wall Centre Hotel, Vancouver, BC, Canada July 16-21, 2006 3266