Journal of Statistical Physics, Vol. 50, Nos. 1/2, 1988 A Quantitative Analysis of the Simulated Annealing Algorithm: A Case Study for the Traveling Salesman Problem Emile H. L. Aarts, 1 Jan H. M. Korst, 1 and Peter J. M. van Laarhoven ~ Received May 22, 1987," revision received September 1, 1987 A quantitative study is presented of the typical behavior of the simulated annealing algorithm based on a cooling schedule presented previously by the authors. The study is based on the analysis of numerical results obtained by systematically applying the algorithm to a 100-city traveling salesman problem. The expectation and the variance of the cost are analyzed as a function of the control parameter of the cooling schedule. A semiempirical average-case perfor- mance analysis is presented from which estimates are obtained on the expec- tation of the average final result obtained by the simulated annealing algorithm as a function of the distance parameter, which determines the decrement of the control parameter. KEY WORDS: Combinatorial optimization; simulated annealing; traveling salesman problem; performance analysis. 1. INTRODUCTION Ever since Kirkpatrick etal. (8) and t~erny (4) introduced the concepts of annealing into the field of combinatorial optimization, much effort has been devoted to investigating the theory of the simulated annealing algorithm 2 and many applications to a wide variety of problems in various disciplines have been presented. For an extensive treatment of the theory and the applications the reader is referred to Ref. 10. The annealing algorithm is based on Monte Carlo techniques applying the Metropolis algorithm from statistical physics (13) and can be modeled mathematically i Philips Research Laboratories, 5600 JA Eindhoven, the Netherlands. 2 Other names used to denote the method are statistical cooling (1) and probabilistic hill climbing.115) 187 0022-4715/88/0100-0187506.00/0 9 1988PlenumPublishing Corporation