Selection in evolutionary algorithms for the traveling salesman problem

1. I N T R O D U C T I O N Selection takes place at two different phases of an evolut ionary algorithm (EA), leading the search into promising regions of the search space. The parent selection phase uses the fitness information to select parents for reproduction, while the replacement scheme determines how the next population is formed by the produced children and the old population. Algorithms like (#, A)-selection [1] and GENITOR [22] are usually considered as selection algorithms, but in fact they represent combined selection strategies consisting of a specific parent selection method and a specific replacement scheme. Some standard combinations are frequently used, although some other combination might be more successful for the application at hand. Many papers investigated measures for selection pressure and diversity [4; 5; 10; 11], but none of those measures reliably predicts an EA's success because of the strong interdependence of parameters and representation issues. Therefore, we compare several combined selection strategies, based on all well known parent selection algorithms and some representative replacement schemes, in Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advan~ge and that copies bear this notice and the ~ l l citation on the first page. To copy otherwise, to republish, to pest on servers or to redistribute to lists, requires prior specific permission and or fee. SAC'00 March 19-21 Como, Italy (c) 2000 ACM 1-58113-239-5/00/003>. >$5.00 b e g i n t e O initialize P(0) evaluate P(0) w h i l e n o t termination criterion d o b e g i n select parent population Pp(t) from P(t) produce child population Pc(t) from Pp(t) evaluate P¢(t) select P(t + 1) from P(t) and Pc(t) t+-t+l end end Figure 1: Evolutionary algorithm order to identify the interdependence between both phases. Our empirical investigation is based on EAs for the traveling salesman problem, examining the effects of selection in the context of a difficult problem and specific operator sets, being based on our new Doubleside~Edge crossover and the inver-over operator proposed by Tao and Michalewicz [21]. We employ the general EA structure shown in figure 1: The parent population Pp(t) of size u is selected from the current population P(t) of size/~. Then, operators like mutat ion and crossover are used to produce the child population Pc(t) of size A from Pv(t), followed by an application of the replacement scheme to obtain the new population P(t + 1) which might contain individuals from both, P(t) and Pc(t). In addition to the empirical analysis, we prove the expected fitness distributions of linear ranking and binary tournament selection (with replacement) being equivalent. This result enables us to interpret the empirical results concerning the relation between stochastic universal sampling and the implicit sampling of tournament selection. We provide relevant details concerning common parent selection methods and replacement schemes in sections 2 and 3, respectively. An introduction to the TSP is presented in section 4, followed by the empirical results in section 5. Conclusions are given in section 6. 2. P A R E N T S E L E C T I O N The selection of the parent population Pp(t) from P(t) can be identified by a mapping ps : P(t) ~ IN that satisfies ps(x~) = v . ziEP(t)