A Memetic Algorithm for the Car Renter Salesman Problem

Abstract

The Traveling Salesman Problem (TSP) is a classic Combinatorial Optimization problem. Given a graph G=(N,M), where N={1,...,n} is the set of nodes and M={1,...,m} is the set of edges, and costs, cij, associated with each edge connecting vertices i and j, the problem consists in finding the minimum length Hamiltonian cycle. The TSP is NP-hard (Garey & Johnson, 1979) and one of the combinatorial optimization problems more intensively investigated. The size of the larger non trivial TSP instance solved by an exact method evolved from 318 cities in the 80’s (Crowder & Padberg, 1980), to 7397 cities in the 90’s (Applegate et al., 1994) and 24978 cities in 2004. The best mark was reached in 2006 with the solution of an instance with 85900 cities (Applegate et al., 2006). The TSP has several important practical applications and a number of variants (Gutin & Punnen, 2002). Some of these variants are classic such as the Peripatetic Salesman (Krarup, 1975) and the M-tour TSP (Russel, 1977), other variants are more recent such as the Colorful TSP (Xiong et al., 2007) and the Robust TSP (Montemanni et al., 2007), among others. A new TSP variant is introduced in this chapter named The Car Renter Salesman Problem (CaRS). It models important applications in tourism and transportation areas and represents a complex variant that challenges the state of the art. In this paper the new problem and some variations are presented, its complexity is analyzed and some related problems are briefly overviewed. A memetic algorithm is proposed for the problem and it is compared to a hybrid GRASP/VND algorithm. CaRS Problem is introduced in Section 2, where several conditions under which this variant can be presented are introduced. Section 3 presents two metaheuristic methods for the investigated problem. In order to compare the performance of the proposed approaches, a set of instances introduced for the new problem, named CaRSLib. This set contains Euclidean and non-Euclidean symmetric instances with number of cities ranging from 14 to 300 and number of cars between 2 and 5. A set of 40 instances is used in the computational experiments. The heuristics proposed in Section 3 establish the first upper limits for solutions of CaRSLib of instances. The results of computational experiments comparing the performance of the proposed approaches are presented in Section 4. Statistical tests are applied to support conclusions on the behavior of the proposed algorithms. According to