Combining Traveling Salesman and Traveling Repairman Problems: A multi-objective approach based on multiple scenarios

This paper analyzes a multi-objective variant of the well-known Traveling Salesman Problem (TSP) and the Traveling Repairman Problem (TRP) in order to address the classical conflict between cost minimization (represented by the TSP) and customer waiting time minimization (represented by the TRP). By simultaneously considering different scenarios with individual travel times, uncertainty in travel data is handled. We interpret each travel time scenario as an individual objective function and introduce deterministic multi-objective counterpart models denoted as the Multi-Objective TSP (MOTSP), Multi-Objective TRP (MOTRP), and the combined Multi-Objective TSP and TRP models (MOTSRP), respectively. Problems with and without additional deadline restrictions are considered, and the complexity status of computing the Pareto fronts of various problem variants for different underlying networks is resolved. As a particularly interesting case, we consider the MOTSRP with deadlines on a line and show that the problem is intractable even in this simple setting. Nevertheless, we propose a Dynamic Programming approach that solves random instances to optimality in reasonable time. Moreover, the computational study additionally evaluates the average complexity of the Line-MOTSRP with deadlines for different numbers of scenarios. The computational study also analyzes the Pareto fronts that are generated for specifically designed extremal instances.