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

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In this paper, we present two general variable neighborhood search (GVNS) based variants for solving the traveling salesman problem with draft limits (TSPDL), a recent extension of the traveling salesman problem. TSPDL arises in the context of maritime transportation. It consists of finding optimal Hamiltonian tour for a given ship which has to visit and deliver products to a set of ports while respecting the draft limit constraints. The proposed methods combine ideas in sequential variable neighborhood descent within GVNS. They are tested on a set of benchmarks from the literature as well as on a new one generated by us. Computational experiments show remarkable efficiency and effectiveness of our new approach. Moreover, new set of benchmarks instances is generated.

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