Heuristics for the black and white traveling salesman problem

Abstract

The black and white traveling salesman problem (BWTSP) is defined on a graph G whose vertex set is partitioned into black and white vertices. The aim is to design a shortest Hamiltonian tour on G subject to two constraints: both the number of white vertices as well as the length of the tour between two consecutive black vertices are bounded above. The BWTSP has applications in airline scheduling and in telecommunications. This article proposes and compares several heuristics for the BWTSP. Computational results are reported for instances involving up to 200 vertices. Scope and purpose This article describes heuristics for a hard combinatorial problem called the black and white traveling salesman problem consisting of designing a tour containing black and white vertices subject to length and cardinality constraints. The main applications of this problem are found in the design of telecommunication fiber networks using the SONET technology (IEEE Trans. Reliab. 40 (1991) 428; Interfaces 25(1) (1995) 20; in: Sansò, Soriano (Eds.), Telecommunications and Planning, Kluwer, Boston, 1998, p. 147) and the scheduling of airline operations that incorporate maintenance connections (Talluri, Transp. Sci. 32 (1998) 43; Mak and Boland, Int. Trans. Oper. Res. 7 (2000) 431). In the first case, the aim is to determine a circular sequence of hubs (white vertices) and ring offices (black vertices) in such a way that the number of hubs and the distance between two consecutive ring offices do not exceed preset values. In the second case, a circular sequence of flight legs (white vertices) and maintenance stops (black vertices) must be computed. Again, both the distance and the number of flight legs between two consecutive maintenance stops are limited.