Particle Swarm Optimization Algorithm for Single Machine Total Weighted Tardiness Problem M. Fatih Tasgetiren Dept. of Management, Fatih University 34500 Buyukcekmece, Istanbul, Turkey Email: ftasgetiren@fatih.edu.tr Mehmet Sevkli Dept. of Industrial Engineering, Fatih University 34500 Buyukcekmece, Istanbul, Turkey Email: msevkli@fatih.edu.tr Yun-Chia Liang Dept. of Industrial Engineering and Management Yuan Ze University No 135 Yuan-Tung Road, Chung-Li, Taoyuan County, Taiwan 320, R.O.C. Email: ycliang@saturn.yzu.edu.tw Gunes Gencyilmaz Dept. of Management, Istanbul Kultur University E5 Karayolu Uzeri, Sirinevler, Istanbul, Turkey Email: g.gencyilmaz@iku.edu.tr Abstract: In this paper we present a particle swarm optimization algorithm to solve the single machine total weighted tardiness problem. A heuristic rule, the Smallest Position Value (SPV) rule, is developed to enable the continuous particle swarm optimization algorithm to be applied to all classes of sequencing problems, which are NP- hard in the literature. A simple but very efficient local search method is embedded in the particle swarm optimization algorithm. The computational results show that the particle swarm algorithm is able to find the optimal and best-known solutions on all instances of widely used benchmarks from the OR libary. I. INTRODUCTION Particle Swarm Optimization (PSO) is one of the latest population-based optimization methods, which does not use the filtering operation (such as crossover and/or mutation) and the members of the entire population are maintained through the search procedure. In a PSO algorithm, each member is called “particle”, and each particle flies around in the multi-dimensional search space with a velocity, which is constantly updated by the particle’s own experience and the experience of the particle’s neighbors or the experience of the whole swarm. Two variants of the PSO algorithm are developed, namely PSO with a local neighborhood, and PSO with a global neighborhood. According to the global neighborhood, each particle moves towards its best previous position and towards the best particle in the whole swarm, called gbest model. On the other hand, according to the local variant so called lbest, each particle moves towards its best previous position and towards the best particle in its restricted neighborhood [1]. Since PSO was first introduced by Kennedy and Eberhart [2, 3], it has been successfully applied to optimize various continuous nonlinear functions. Although the applications of PSO on combinatorial optimization problems are still limited, PSO has certain advantages such as easy to implement and computationally efficient. Therefore, this paper is the first to employ PSO on solving single machine total weighted tardiness (SMTWT) problem which is a typical combinatorial optimization problem. McNaughton [4] first presented a scheduling problem that the objective is to minimize total penalty cost. He proved that an optimal solution exists in which no job is split, so that only permutation schedules of the n jobs need to be considered. Therefore, the SMTWT problem can be stated as follows. Each of n jobs ( n j ,..., 1 = ) is to be processed without preemption on a single machine that can handle no more than one job at a time. The processing and set-up requirements of any job are independent of its position in the sequence. The release time of all jobs is zero. Thus, job j ( n j ,..., 1 = ) becomes available for processing at time zero, requires an uninterrupted positive processing time j p , which includes set-up and knock-down times on the machine, has a positive weight j w , and has a due date j d by which job j should ideally be finished. For a given processing 0-7803-8515-2/04/$20.00 ©2004 IEEE 1412