Sequential variable neighborhood descent variants: an empirical study on the traveling salesman problem

Usually several neighborhood structures may be explored within a single local search algorithm. The simplest way is to define a single neighborhood as a union of all predefined neighborhood structures. The another possibility is to make an order (or sequence) of those neighborhoods and use them in the first improvement or the best improvement fashion, following that order. In this work, we firstly classify possible variants of sequential use of neighborhoods and then empirically analyze them in solving the classical travelling salesman problem. Most commonly used TSP neighborhood structures of the same size, such as 2-opt and insertion neighborhoods are explored in our empirical study. We in fact tested 76 different such heuristics on 15,200 random test instances. Several interesting observations are derived. In addition, the two best out of 76 heuristics (used as a local searches within variable neighborhood search (VNS)) are tested on 23 TSPLIB test instances. It appears that union of neighborhoods does not perform well.