Solving the clustered traveling salesman problem with ‐relaxed priority rule (original) (raw)

2020, International Transactions in Operational Research

The Clustered Traveling Salesman Problem with a Prespecified Order on the Clusters, a variant of the well-known traveling salesman problem is studied in literature. In this problem, delivery locations are divided into clusters with different urgency levels and more urgent locations must be visited before less urgent ones. However, this could lead to an inefficient route in terms of traveling cost. This priority-oriented constraint can be relaxed by a rule called d-relaxed priority that provides a trade-off between transportation cost and emergency level. Our research proposes two approaches to solve the problem with d-relaxed priority rule. We improve the mathematical formulation proposed in the literature to construct an exact solution method. A meta-heuristic method based on the framework of Iterated Local Search with problem-tailored operators is also introduced to find approximate solutions. Experimental results show the effectiveness of our methods. Keywords. Clustered traveling salesman problem, d-relaxed priority rule, mixed integer programming, iterated local search. the locations are supposed to have the same degrees of urgencies, i.e., they can be visited in any order. However, in a number of real-world routing applications, different levels of priorities at the delivery locations need to be taken into account in routing plans. For example, as a result of a natural disaster such as a storm, earthquake, tsunami, or hurricane, there are demands at many locations for relief supplies such as food, bottled water, blankets, or medical packs. Some locations are in more urgent need of supplies than other locations due to the relative position of the source of disasters, the damage status, or its importance (schools, hospitals, and government institutions should be considered as more important). Locations requiring the same level of urgency can be clusterized into groups. And the priority of a group during the relief process has to be considered, e.g., higher priority groups should be visited before others. In the example above, the priorities indicate the importance (or urgency) of the demand at each location. Typically, priority 1 nodes must be served before priority 2 nodes, priority 2 nodes must be served before priority 3 nodes, and so on. Such a problem is called the Clustered Traveling Salesman Problem with a Prespecified Order on the Clusters (CTSP-PO) and has been studied in [22, 17]. However, this rule is strict with respect to the priority and can lead to an inefficient route in terms of traveling cost. It may be relevant to visit some lower priority nodes while serving higher priority nodes. In [4, 5], the authors proposed a simple, but elegant rule called d-relaxed priority that provides flexibility to the decision maker in terms of capturing trade-offs between total distance and node priorities. In [5] and Chapter 14 of [6], the d-relaxed priority rule is defined as follows. Given a positive number d, at any point of the route, if p is the highest priority class among all unvisited locations, the relaxed rule allows the vehicle to visit locations with priority p, p + 1, ..., p + d before visiting all locations in class p. By changing the value of d, we can flexibly control to focus more on economic aspect or urgency level. Indeed, if we consider the 0-relaxed priority rule (i.e., d = 0), all the higher priority nodes must be visited before lower priority nodes. The problem is a CTSP-PO, the strictest version w.r.t priority. On the other hand, if d is set to g − 1, where g is the number of priorities, the problem becomes a typical TSP, all the node priorities being ignored.