The traveling salesman problem (TSP) is a well-known optimization problem in graph theory, as well as in operations research that has nowadays received much attention because of its practical applications in industrial and service problems. In this problem, a salesman starts to move from an arbitrary place called depot and after visits all of the nodes, finally comes back to the depot. The obje...

The Traveling Salesman Problem (TSP) is perhaps the most studied discrete optimization problem. Its popularity is due to the facts that TSP is easy to formulate, difficult to solve, and has a large number of applications. It appears that K. Menger [31] was the first researcher to consider the Traveling Salesman Problem (TSP). He observed that the problem can be solved by examining all permutati...

The asymmetric traveling salesman problem (ATSP) is one of the most fundamental problems in combinatorial optimization. An instance of ATSP is a directed complete graph G = (V,E) with edge weights w : E → N that satisfy the triangle inequality, i. e., w(a, c) ≤ w(a, b) +w(b, c) for all distinct vertices a, b, c ∈ V . The goal is to find a Hamiltonian cycle of minimum weight. The weight of a Ham...

This paper analyzes the elastic net approach (Durbin and Willshaw 1987) to the traveling salesman problem of finding the shortest path through a set of cities. The elastic net approach jointly minimizes the length of an arbitrary path in the plane and the distance between the path points and the cities. The tradeoff between these two require­ ments is controlled by a scale parameter K. A global...

We consider the Railway Traveling Salesman Problem. We show that this problem can be reduced to a variant of the generalized traveling salesman problem, defined on an undirected graph G = (V, E) with the nodes partitioned into clusters, which consists in finding a minimum cost cycle spanning a subset of nodes with the property that exactly two nodes are chosen from each cluster. We describe an ...

This paper mainly investigates the traveling salesman problem under uncertain environment, in which not all the edge weights are deterministically present, but are present with uncertain variables. As a result, five types optimal route models for the uncertain traveling salesman problem, i.e., expected optimal route, α-optimal route, distribution optimal route, expected approximation optimal ro...

In order to solve traveling salesman problems that employed completiontimeshortest as the evaluating rule, an encoding method and improved differential evolution algorithm were proposed. In these methods, real number encoding and roulette wheel selection were adopted for improved differential evolution and neighborhood search operator was devised. It was fit for solving symmetric and asymmetric...

Since the computation of an exact solution is very expensive (under the assumption that P 6= NP ), we are interested in approximate solutions – at least for special cases. Particularly, we look for an Polynomial Time Approximation Scheme (PTAS), i.e., for any fixed error parameter ε > 0, the running time is bounded by a polynomial in n and the costs of the computed tour does not exceed (1+ε) ·O...

We propose a heuristic for a class of vehicle routing problems (VRPs) with minmax objective. These include the Capacitated VRP, the Capacitated VRP with multiple use of vehicles, and the Traveling Salesman Problem with multiple use of vehicles. A tabu search based adaptive memory procedure of instances indicate that the method produces very good solutions within reasonable computing times.