Just-in-time scheduling problem is the problem of finding an optimal schedule such that each job finishes exactly at its due date. We study the problem under a realistic assumption called periodic time slots. In this paper, we prove that this problem cannot be approximated, assuming P≠NP. Next, we present a heuristic algorithm, assuming that the number of machines is one. The key idea is a reduction of the problem to a network flow problem. The heuristic algorithm is fast because its main part consists of computation of the minimum cost flow that dominates the total time. Our algorithm is O(n3) in the worst case, where n is the number of jobs. Next, we show some simulation results. Finally, we show cases in which our algorithm returns an optimal schedule and is a factor 1.5 approximation algorithm, respectively, and also give an approximation ratio depending on the upper bound of set-up times.

A job shop system typically seen in flexible manufacturing systems (FMS) is a system composed of a set of machines and a various kind of jobs processed with the machines. A production system of semiconductor fabrication is an example of job shop systems, which has main features of repetitive processes of one part and set-up times required for machines processing different types of parts. On the other hand, timed Petri nets are used for modelling and analyzing a wide variety of discrete event systems. There are many applications of timed Petri nets to the scheduling problems of job shop systems. The performance evaluation and steady state behaviors are studied by using the maximum cycle time of timed marked graphs. The aim of this paper is to propose a new model for production systems including repetitive processes and set-up time requirements which enables the quantitative analysis of real time system performance. In job shop systems such as a semiconductor fabrication system, it takes considerable amount of set-up time to prepare different types of chemical reactions and the model should take account of a set-up time for each machine. We focus upon the relationship between facility utilization factor and production cycle time in the steady state. In the proposed model, the minimum total set-up time can be attained. Quantitative relationship between utilization factor and production cycle time is derived by using the proposed model. A utilization factor of a system satisfying a given limit of the cycle time is evaluated, and the improvement of the utilization factor is considered. Conversely, we consider the improvement of the cycle time of a system satisfying a given limit of utilization factor.