Discrete Differential Evolution with local search to solve the Traveling Salesman Problem: Fundamentals and case studies

Traveling salesman problem (TSP) is one of the most difficult problems in the combinatorial optimization area. The goal of TSP is to find one path that can travel between all the nodes (instances) of the graph just once (Hamiltonian tour) in the smallest tour, that is, smallest Euclidian distance. In this context, many other meta-heuristics techniques based on evolutionary algorithms have been propose in literature to solve TSP problems. Differential evolution (DE) is a relatively new simple evolutionary algorithm, which is an effective adaptive approach to global optimization over continuous search spaces. The design principles of DE are simplicity and efficiency. The most distinct feature of DE is that it mutates vectors by adding weighted, random vector differentials to them. Since its invention, DE has been applied with high success on many numerical optimization problems outperforming other more popular meta-heuristics such as the genetic algorithms. Recently, some researchers extended with success the application of DE to complex combinatorial optimization problems with discrete decision variables such as the traveling salesman problem, the machine layout problem, the flow-shop scheduling problem. In this paper, the following discrete DE approaches for the TSP are proposed and evaluated: i) DE approach without local search, ii) DE with local search based on Lin-Kernighan-Heulsgaun (LKH) method, and iii) DE with local search based on variable neighborhood search (VNS) and together with LKH method. Numerical study is carried out using the TSPLIB of test TSP problems. In this context, the computational results are compared with the other results in the recent TSP literature. The obtained results show that LKH method is the best method to reach optimal results for TSPLIB benchmarks, but for largest problems, the ED+VNS improve the quality of obtained results.