Comparing decoding algorithms in a weight-coded GA for TSP

A novel coding of candidate solutions in genetic algorithms for combinatorial problems associates numerical weights with elements of the target problem instance. The solution a chromosome of weights represents is made explicit by a heuristic algorithm for the problem whose actions the weights influence. The choice of the heuristic--called the decoding algorithm--is a crucial one in a genetic algorithm that employs such a coding. This paper describes a weighted coding of tours in a genetic algorithm for the traveling salesman problem and an investigation of nine heuristics for TSP as decoding algorithms in that GA. Two greedy heuristics performed poorly, but heuristics that build tours by iusertion-adding each new city so as to increase the tour length the least--did better. Several showed excellent performance on TSP instances of moderate size. The results indicate both the importance of the decoding algorithm in a GA that uses a weighted coding and the potential of such codings in GAs for combinatorial problems. 1. I N T R O D U C T I O N Combinatorial problems seek a selection, arrangement, or ordering of data elements so as to maximize or minimize an objective function on those elements. A classic example is the familiar traveling salesman problem (TSP), which seeks a tour that visits each of a collection of cities exactly once and returns to the start city in the shortest possible total distance. Combinatorial problems often have large and complex search spaces. In genetic algorithms for such problems, it can be difficult to design codings of candidate solutions that allow a GA, using the usual crossover and mutation ermission to make digital/hard copy of all or part of this work lbr personal or lassroom use is granted without lee provided that copies are not made or istributed tbr profit or commercial advantage, the copyright notice, the title of the tlblication and its date appear, and notice is given that copying is by permission of ,CM, Inc. To copy otherwise, to republish, to post on servers or to redistribute to sis, requires prior specific permission and/or a fee. © 1998 ACM 0-89791-969-6/98/0002 350 operators, to search the space effectively. GAs for combinatorial problems often use codings and operators tailored to the problems. This is true of GAs for the traveling salesman problem, for which investigators have described a wide variety of specialized codings and operators [8]. Some of these have addressed the blind TSP, in which the inter-city distances are used only to find tours' lengths. Others have used the distances during each tour's construction and thus have addressed the non-blind TSP. The GA described here is of this second kind. A novel coding of candidate solutions in GAs for combinatorial problems associates numerical weights with elements of the target problem instance. The solution a chromosome of weights represents is identified by a (non-genetic) heuristic for the problem whose actions the weights influence. The heuristic--called the decoding algorithm--makes explicit the structure a chromosome represents, and the fitness of that structure is the chromosome's fitness. Such codings have been used in GAs for several combinatorial problems [5, 9, 12]. In genetic algorithms whose chromosomes are strings of weights, the decoding algorithm is a critical design choice. This paper investigates that choice in a GA for the travehng salesman problem; this GA encodes tours as sequences of weights associated with the TSP instance's cities [5]. Nine heuristics are examined as decoding algorithms. Two are greedy, and the remaining seven apply i n s e r t i o n : a tour on some of the cities is extended by inserting a new city where it increases the tour's length the least. The resulting versions of the GA are tested on three well-known TSP instances, of 100, 105, and 150 cities. The choice of decoding algorithm strongly affects the GA's performance, and with some decoding algorithms the GA does very well on the test problems, The following sections describe the weighted coding of TSP tours, the TSP heuristics investigated as decoding algorithms, the weight-coded GA that serves as a test bed and the test problems, the performance of the GA with each heuristic as its decoding algorithm on the test problems, the performance of the GA with one of the heuristics on four larger TSP instances, and conclusions and implications for further work.