Solving The Clustered Traveling Salesman Problem Withd-Relaxed Priority Rule

The clustered traveling salesman problem with a prespecified order on the clusters, a variant of the well-known traveling salesman problem, is studied in the 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 calledd-relaxed priority that provides a trade-off between transportation cost and emergency level. Given a positive integerd, at any point along the route, thed-relaxed rule allows the vehicle to visit locations with priorityp,p+1, horizontal ellipsis ,p+dbefore visiting all locations in classp, wherepis the highest priority class among all unvisited locations. Our research proposes two approaches to solve the problem withd-relaxed priority rule. We improve the mathematical formulation proposed in the literature to construct an exact solution method. A metaheuristic 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.