Development of a dummy guided formulation and exact solution method for TSP

Abstract

A traveling salesman problem (TSP) is a problem whereby the salesman starts from an origin node and returns to it in such a way that every node in the network of nodes is visited once and that the total distance travelled is minimized. An efficient algorithm for the TSP is believed not to exist. The TSP is classified as NP-hard and coming up with an efficient solution for it will imply NP = P . The paper presents a dummy guided formulation for the traveling salesman problem. To do this, all sub-tours in a traveling salesman problem (TSP) network are eliminated using the minimum number of constraints possible. Since a minimum of three nodes are required to form a sub-tour, the TSP network is partitioned by means of vertical and horizontal lines in such a way that there are no more than three nodes between either the vertical lines or horizontal lines. In this paper, a set of all nodes between any pair of vertical lines or horizontal lines is called a block. Dummy nodes are used to connect one block to the next one. The reconstructed TSP is then used to formulate the TSP as an integer linear programming problem (ILP). With branching related algorithms, there is no guarantee that numbers of sub-problems will not explode to unmanageable levels. Heuristics or approximating algorithms that are sometimes used to make quick decisions for practical TSP models have serious economic challenges. The difference between the exact solution and the approximated one in terms of money is very big for practical problems such as delivering household letters using a single vehicle in Beijing, Tokyo, Washington, etc. The TSP model has many industrial applications such as drilling of printed circuit boards (PCBs), overhauling of gas turbine engines, X-Ray crystallography, computer wiring, order-picking problem in warehouses, vehicle routing, mask plotting in PCB production, etc.