Stability of Solutions to Classes of Traveling Salesman Problems

Abstract

By performing stability analysis on an optimal tour for problems belonging to classes of the traveling salesman problem (TSP), this paper derives margins of optimality for a solution with respect to disturbances in the problem data. Specifically, we consider the asymmetric sequence-dependent TSP, where the sequence dependence is driven by the dynamics of a stack. This is a generalization of the symmetric non sequence-dependent version of the TSP. Furthermore, we also consider the symmetric sequence-dependent variant and the asymmetric non sequence-dependent variant. Amongst others these problems have applications in logistics and unmanned aircraft mission planning. Changing external conditions such as traffic or weather may alter task costs, which can render an initially optimal itinerary suboptimal. Instead of optimizing the itinerary every time task costs change, stability criteria allow for fast evaluation of whether itineraries remain optimal. This paper develops a method to compute stability regions for the best tour in a set of tours for the symmetric TSP and extends the results to the asymmetric problem as well as their sequence-dependent counterparts. As the TSP is NP-hard, heuristic methods are frequently used to solve it. The presented approach is also applicable to analyze stability regions for a tour obtained through application of the k -opt heuristic with respect to the k -neighborhood. A dimensionless criticality metric for edges is proposed, such that a high criticality of an edge indicates that the optimal tour is more susceptible to cost changes in that edge. Multiple examples demonstrate the application of the developed stability computation method as well as the edge criticality measure that facilitates an intuitive assessment of instances of the TSP.

Optimization approaches for civil applications of unmanned aerial vehicles (UAVs) or aerial drones: A survey

“…To address the problem of unreliable data, Niendorf et al compute stability regions for problem parameters, that is, parameter intervals, for which the computed solution will remain optimal. Motivated by surveillance for maintenance of bridges and other infrastructure subject to wind gusts between pylons, Guerrero and Bestaoui elaborate on the shortest path calculation in a windy environment based on Zermelo's navigation equations.…”

Stability analysis of multi-objective planning problems for unmanned aircraft

“…Stability regions for single objective TSPs and tolerances for a best tour in a set of tours in the context of mission planning for unmanned aircraft have been studied in previous work [20], [21]. Using similar techniques stability regions for combinatorial optimization problems with Markovian problem data have been described [22].…”

“…This section presents the computation of stability regions for solutions to single-objective traveling salesman problems based on the work in [21].…”

“…Using the method in [21] the stability region of the optimal solution x * to optimization problem (2) can be computed as follows. The reduced cost vector, q, is given by:…”

“…In order to maintain x * as optimal solution, it is necessary and sufficient [21] for the vector parameter ∆c to satisfy inequality: H∆c ≤ −q.…”