Pergamon 0969-6016(94)E00012-H Int. l)'ans. Opl Res. Vol. I, No. 3, pp.317 336, t994 Elsevier Science Ltd. Printed in Great Britain. 0969-6016:94 $7.0('1+ ('1.00 Capacitated Clustering Problems by Hybrid Simulated Annealing and Tabu Search IBRAHIM H. OSMAN* and NICOS CHRISTOFIDESt *University of Kent, U.K. and tUniversity of London, U.K. The capacitated clustering problem (CCP) is the problem in which a given set of weighted objects is to be partitioned into clusters so that the total weight of objects in each cluster is less than a given value (cluster "capacity'). The objective is to minimize the total scatter of objects from the 'centre" of the cluster to which they have been allocated. A simple constructive heuristic, a ).-interchange generation mechanism, a hybrid simulated annealing (SA) and tabu search (TS) algorithm which has computationally desirable features using a new non-monotonic cooling schedule, are developed. A classification of the existing SA cooling schedules is presented. The effects on the final solution quality of the initial solutions, the cooling schedule parameters and the neighbourhood search strategies are investigated. Computational results on randomly generated problems with size ranging from 50 to 100 customers indicate that the hybrid SA/TS algorithm out-performs previous simulated annealing algorithms, a simple tabu search and local descent algorithms. Key words: capacitated clustering problems, capacitated p-median, plant location, heuristic, simulated annealing, tabu search, k, cal search INTRODUCTION The capacitated clustering problem, CCP, is the problem in which a given set of objects (or customers) is to be partitioned into a set of clusters. Each object has an associated weight (or demand) and must be assigned to exactly one cluster. Each cluster has a given capacity which must not be exceeded by the total weight of objects in the cluster. The dissimilarity measure is a cost (or distance) between any two objects. For a given cluster, a centre is that object of the cluster from which the sum of the dissimilarities to all other objects in the cluster is minimized. This sum is called the scatter of the cluster. The objective is to find a set of centres which minimizes the total scatter of all clusters. A pictorial representation of the CCP is given in Fig. 1. Clustering of related objects is of practical importance and can be found in diverse fields such as biology, economics, engineering, marketing, operations research, pattern recognition and statistics. Furthermore, the CCP is a very special case of the capacitated plant location problem with single source constraints, and many other combinatorial problems. As a consequence, it can be shown to be NP-complete (Garey and Johnson, 1979). Optimization algorithms by general-purpose branch and bound codes, such as the CPLEX mixed integer optimizer are available though they are ineffective for large-sized instances of the CCP. It seems reasonable to devise approximate algorithms (or heuristics) which are effective in solving large-sized problems. There are two types of heuristics. The first type uses a tailored approach which constructs a solution from the data by looking to the characteristics of the problem to be solved. This type is known as constructive heuristic and is generally difficult to generalize to different applications. The second type uses an iterative improvement (or local search) approach which starts from an initial solution and iteratively attempts to improve upon it by a series of local changes, The main drawback of local search heuristics is that the search might terminate at a local optima which may be far from a global optima. The quality of the final solution depends on the starting solution and the rules used to generate neighbouring solutions. Recently, a great deal of attention has focussed on two local search heuristics when solving hard combinatorial optimization problems: simulated annealing, SA (Kirkpatrick et al., 1983) and tabu search, TS (Glover, 1986). Both methods help reduce the effects of local optimality and produce near optimal solutions using strategies based on ideas from statistical physics and strategic oscillation ideas from the tenet of intelligent problem solving to explore the search space, respectively. In this paper, we introduce a 2-interchange mechanism for the CCP which has similar properties to Correspondence: 1. tt. Osman. Institute ¢~]Mathematics and Statistics. Unirersity ~['Kent. Canterhtwy. Kern. CT2 7 N F, U.K. 317