A Genetic Algorithm with a Penalty Function in the Selective Travelling Salesman Problem on a Road Network

The Selective Travelling Salesman Problem (STSP) is a version of the Travelling Salesman Problem (TSP) where it is not necessary to visit all vertices. Instead of it, with each vertex a number meaning a profit is associated. The problem is to find a cycle which maximizes collected profit but does not exceed a given cost constraint. A direct application of the STSP, e.g. in Intelligent Transportation Systems, is finding an optimal tour in road networks. However, while the classic STSP is defined on a complete graph, a road network is in general not complete and often has a rather sparse edge set. This paper presents the STSP defined on a road network (R-STSP). Since R-STSP is NP-hard and stands the problem with a constraint, the genetic algorithm (GA) with a penalty function is proposed. Computer experiments performed on the real road network in Poland have shown that this GA outperforms the GA searching only the feasible solution space.