Some applications of the clustered travelling salesman problem

The purpose of this article is to describe several applications of the clustered travelling salesman problem (CTSP), a variant of the well-known travelling salesman problem (TSP). The CTSP is defined on a complete graph G 1⁄4 ðV ;EÞ in the undirected case or (V, A) in the directed case, where V 1⁄4 fv1; . . . ; vng is a vertex set partitioned into m clusters V1; . . . ;Vm. The set E 1⁄4 fðvi; vjÞ : vi; vj 2 V ; i < jg is an edge set, and A 1⁄4 fðvi; vjÞ : vi; vj 2 V ; i 61⁄4 jg is an arc set, so that both the undirected and the directed case can be considered. A cost matrix C 1⁄4 ðcijÞ is defined on E or A. The CTSP consists of determining a least cost Hamiltonian cycle or circuit on G in which the vertices of any cluster Vk are contiguous. The CTSP coincides with the TSP whenever all clusters are singletons. In general, the CTSP can readily be transformed into a TSP by adding an arbitrarily large constant M to inter-cluster costs, or by subtracting M from intra-cluster costs. However, such transformations introduce a fair amount of degeneracy in the problem and it may be preferable to solve the CTSP directly by means of a specialized algorithm. The CTSP was introduced by Chisman and early algorithms were provided by Lokin, and Jongens and Volgenant. These authors have described simple applications in the areas of vehicle routing, warehouse order picking and manufacturing. Our aim is to provide a much wider range of well-known and new applications of the CTSP, and thereby to show the potential of this problem as a modelling tool. This article follows two similar contributions on the TSP and on the generalized travelling salesman problem (GTSP) to be described in the following section. Applications of the CTSP arise in areas as diverse as vehicle routing, manufacturing, computer operations, examination timetabling, cytological reading, integrated circuit testing, and computer program restructuring. Before describing these in the following sections, we outline a key relationship between the CTSP and the GTSP.