An efficient two-phase metaheuristic algorithm for the time dependent traveling Salesman problem

The Time Dependent Traveling Salesman Problem (TDTSP) is a class of NP-hard combinatorial optimization problems which has many practical applications. To the best of our knowledge, developing metaheuristic algorithm for the problem has not been studied much before, even though it is a natural and general extension of the Minimum Latency Problem (MLP) or Traveling Salesman Problem (TSP). In this paper, we propose an effective two-phase metaheuristic which combines the Insertion Heuristic (IH), Variable Neighborhood Search (VNS) and the tabu search (TS) to solve the problem. In a construction phase, the IH is used to create an initial solution that is good enough. In an improvement phase, the VNS is employed to generate diverse and various neighborhoods, while the main attribute of tabu search is to prohibit our algorithm from getting trapped into cycles, and to guide the search to escape local optima. Moreover, we introduce a novel neighborhoods’ structure in VNS and present a O(1) operation for calculating the cost of each neighboring solution in a special case of TDTSP where the TDTSP becomes the MLP. Extensive computational experiments on 355 benchmark instances show that our algorithm can find the optimal solutions for small instances with up to 100 vertices in a reasonable amount of time. For larger instances, our algorithm obtains the new best solutions in comparison with the state-of-the-art algorithm solutions