Multi-objective Solution of Traveling Salesman Problem with Time

Abstract

The traveling salesman problem (TSP) is a challenging problem in combinatorial optimization. No general method of solution is known, and the problem is NP-hard. In this paper, we consider the multi-objective TSP which encompasses the optimization of two conflicting and competing objectives: here the dual minimization of the total travel distance and total travel time at various traffic flow conditions. It is well known that travellers can experience extra travel time during peak hours (i.e., congestion conditions) compared to free flow conditions (i.e., un-congested conditions), therefore and under some conditions, minimizing traveled time could conflict and compete with travel distance and vice versa. This problem has been studied in the form of a single objective problem, where either the two objectives have been combined in a single objective function or one of the objectives has been treated as a constraint. The purpose of this paper is to find a set of non-dominated solutions (i.e., the sequence of cities) using the notion of Pareto optimality where none of the objective functions can be improved in value without degrading one or more of the other objective values. The traveller then has the chance to choose a solution that fits his/her needs at each congestion level. In this paper, a multi-objective genetic algorithm (MOGA) for searching for efficient solutions is investigated. Here, an initial population composed of an approximation to the extreme supported efficient solutions is generated. A Pareto local search is then applied to all solutions of the initial population. The method is applied to a simulated problem and to a real-world problem where distances and real estimates of the travel duration for multiple origins and destinations for specific transport modes are obtained from Google Maps Platform using a Google Distance Matrix API. Results show that solving a TSP as a multi-objective optimization problem can provide more realistic solutions. The proposed approach can be used for recommending routes based on variable duration matrix and cost.