The Robust Traveling Salesman Problem with Interval Data

Abstract

The traveling salesman problem is one of the most famous combinatorial optimization problems and has been intensively studied. Many extensions to the basic problem have also been proposed, with the aim of making the resulting mathematical models as realistic as possible. We present a new extension to the basic problem, where travel times are specified as a range of possible values. This model reflects the intrinsic difficulties of estimating travel times in reality. We apply the robust deviation criterion to drive optimization over the interval data problem so obtained. Some interesting theoretical properties of the new optimization problems are identified and discussed, together with a new mathematical formulation and some exact and heuristic algorithms. Computational experiments are finally presented.

Models and Algorithms for Stochastic and Robust Vehicle Routing with Deadlines

“…This solution framework also offers the capability to solve the SVRP-D exactly if the probability for the tail of the convolution of the travel times can be computed. Third, we perform extensive computational experiments to evaluate the proposed Table 1 Summary of the related literature Author(s) Problem Uncertainty Recourse Deadline Network Approach description function restriction Laporte et al (1992) stochastic scenarios cost one deadline m-TSP branch-and-cut Lambert et al (1993) stochastic scenarios (2) cost one deadline m-TSP heuristic Kenyon and Morton (2003) stochastic scenarios (30) cost/prob one deadline m-TSP SAA/branch-and-cut Verweij et al (2003) stochastic scenarios (1000) cost one deadline SPP/TSP SAA/branch-and-cut Thomas (2008, 2009) stochastic scenarios (2) cost multiple deadlines PTSP heuristic Montemanni et al (2007) robust interval regret N/A TSP branch-and-cut/Benders Sungur et al (2010) stochastic scenarios (n/a) cost time windows VRP heuristic Lee et al (2012) robust budget of uncertainty n/a time windows VRP column generation Agra et al (2013) robust budget of uncertainty n/a time windows VRP column generation Jaillet et al (2014) robust unsatisfactory index n/a time windows SPP/TSP iterative procedure and their reformulation schemes. The algorithm for these problems are presented in Section 4 and the computational experiments are shown in Section 5.…”

Minmax regret combinatorial optimization problems: an Algorithmic Perspective

“…Montemanni instances. A similar procedure to proposed for TSP in [59] for generate interval costs is used. 80 instances were generated in total, ten for each problem size.…”

“…The most important effort devoted to exactly solve the robust counterpart of the TSP was done by Montemmani et al in [59]. Based on structural properties, an appropiate mathematical programming formulation is presented which allows the development of three exact approaches: B&B, B&C and Benders decomposition.…”

“…Montemanni et al [58] studied heuristics and preprocessing techniques for interval data MMR-TSP. In particular, they presented efficient tools for estimating the robustness of a given tour, producing heuristic solutions to the problem and preprocessing the problem in order to identify edges that will never be on an optimal robust tour.…”

A Memetic Algorithm for the Car Renter Salesman Problem

“…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).…”