An efficient probability-based VNS algorithm for delivery territory design

This paper deals with the Delivery Territory Design Problem (DTDP), a districting problem that often occurs in delivery operations. The goal of the problem is to construct clusters of nodes (territories) such that the maximum diameter of a territory is minimized, while the territories designed are balanced w.r.t. some performance measures. We propose to solve the DTDP using a Probabilistic Variable Neighborhood Search (ProbVNS) algorithm based on two local search procedures: a tailored randomized shake procedure that targets both a reduction of infeasibility and diversification, and a deterministic local search based on a linear combination of objective and constraint violation. In addition to searching in different neighborhoods, the ProbVNS also changes the search direction by exploring different penalties for violating constraints. Numerical experiments show that ProbVNS outperforms a recent GRASP with the Path-Relinking (PR) algorithm proposed in the literature in terms of feasibility and objective value. In the tested instances, ProbVNS obtained a lower infeasibility measure in 90% of the instances. For these instances, the average decrease in the objective value was 8.3%, with a maximum decrease of 51%. Finally, the running times of ProbVNS are, on average, 2.7 times lower than those of PR.