Dynamic Programming Strategies for the Traveling Salesman Problem with Time Window and Precedence Constraints

Abstract

The Traveling Salesman Problem with Time Window and Precedence Constraints (TSP-TWPC) is to find an Hamiltonian tour of minimum cost in a graph G = (X, A) of n vertices, starting at vertex 1, visiting each vertex i ∈ X during its time window and after having visited every vertex that must precede i, and returning to vertex 1. The TSP-TWPC is known to be NP-hard and has applications in many sequencing and distribution problems. In this paper we describe an exact algorithm to solve the problem that is based on dynamic programming and makes use of bounding functions to reduce the state space graph. These functions are obtained by means of a new technique that is a generalization of the “State Space Relaxation” for dynamic programming introduced by Christofides et al. (Christofides, N., A. Mingozzi, P. Toth. 1981b. State space relaxation for the computation of bounds to routing problems. Networks 11 145–164.). Computational results are given for randomly generated test problems.

Exact algorithms for different classes of vehicle routing problems

“…We tested both BMR.ng and BMR.t on the seven classes of instances available at http: //iridia.ulb.ac.be/~manuel/tsptw-instances and the class proposed by Mingozzi et al [1997]. The classes proposed by Langevin et al [1993], Dumas et al [1995], and Mingozzi et al [1997] were easily solved (the most difficult instance was solved in 12 seconds by BMR.ng), so relative results are omitted.…”

“…The classes proposed by Langevin et al [1993], Dumas et al [1995], and Mingozzi et al [1997] were easily solved (the most difficult instance was solved in 12 seconds by BMR.ng), so relative results are omitted. The results on the other five classes are reported in the following and compared with the following state-of-the-art exact methods:…”

“…Algorithms based on DP were proposed by Dumas et al [1995], Mingozzi et al [1997], Balas and Simonetti [2001], and Li [2009]. Dumas et al [1995] described a DP algorithm that applies sophisticated elimination tests to reduce the state-space and the number of state transitions; they reported the solutions of instances involving up to 200 vertices.…”

“…Dumas et al [1995] described a DP algorithm that applies sophisticated elimination tests to reduce the state-space and the number of state transitions; they reported the solutions of instances involving up to 200 vertices. Mingozzi et al [1997] presented a DP algorithm for the TSP with time windows and precedence constraints based on the SSR technique and presented computational results for instances with up to 120 vertices. Balas and Simonetti [2001] considered a special case of the TSPTW where, for some initial ordering of the vertices, vertex i precedes vertex j if j ≥ i + k (for some value k > 0), and described a DP algorithm that is linear in the number of vertices and exponential in k. Li [2009] presented a DP algorithm based on a bi-directional resource-bounded label correcting algorithm (see Righini and Salani [2006]); Li reported the solutions of instances involving up to 233 vertices and solved a number of open instances to optimality.…”

Enhanced mixed integer programming techniques and routing problems

“…Dumas et al [74] proposed a dynamic-programming approach with sophisticated elimination tests to reduce the state space, while. Mingozzi, Bianco and Ricciardelli [126] presented a dynamic-programming algorithm with a generalization of the state-space-relaxation technique which can also be applied to TSPTW problems with precedence constraints. Also in the nineties, the problem started to receive quite a lot of attention by the Constraint Programming (CP) community.…”