A hybrid metaheuristic to solve the vehicle routing problem with stochastic demand and probabilistic duration constraints

The vehicle routing problem with stochastic demands and probabilistic distance constraints (VRPSD-PDC) can be defined on a complete and undirected graph G = (V, E), where V = {0, . . . , n} is the vertex set and E = {(v, u) : v, u ∈ V, v ̸= u} is the edge set. Vertices v = 1, . . . , n represent the customers and vertex v = 0 represents the depot. A distance de is associated with edge e = (v, u) = (u, v) ∈ E , and it represents the travel cost between vertices v and u. Each customer v has a random demand ξv for a given product. Customer demands are met using an unlimited fleet of homogeneous vehicles located at the depot. Each vehicle has a maximum capacity Q and a maximum travel distance L. The exact quantity demanded by each customer is not known until the vehicle arrives at the customer location. It is assumed, however, that each customer’s demand follows an independent and known probability distribution and that all demand realizations (actual quantities) are nonnegative and less than the capacity of the vehicle. We formulate the VRPSD-PDC as an extension of the classical two-stage stochastic programming formulation for the VRPSD that includes chance constraints on the maximum travel distances of the routes. As the name suggests, the two-stage stochastic