A vehicle routing problem with time windows and stochastic demands

Abstract This study presents an approach for considering a vehicle routing problem where customers’ pickup demands are uncertain and require serving within some settled time windows. Customers’ demands are assumed to follow given discrete probability distributions. This study proposes a nonlinear stochastic integer program with recourse to formulate the vehicle routing problem with stochastic demands and time windows (VRPTW‐SD, for short). The objective of the VRPTW‐SD is to minimize the total cost of the first‐stage solution and expected recourse cost of the second‐stage solution. The total cost of the first‐stage problem includes the total travel cost for all links and the total waiting cost at all nodes. When a vehicle capacity is attained or exceeded, recourse actions are needed and recourse costs incurred in order to finish the planned route schedules. Two categories of schedule failure are introduced in this work; the recourse costs derive from the variations in travel time travel time, waiting time, and penalties of late arrival for time windows. In addition, an optimization algorithm is developed for solving the VRPTW‐SD, according to the framework of the L‐shaped method. Numerical results are given to demonstrate its validity.