A Model and a Method for Vehicle Routing Problem with Time Windows under Uncertainty(<Special Issue>Manufacturing Systems Division Conference 2010)

Abstract

This paper discusses the vehicle routing problem with time windows and stochastic demands. All customers have the penalty functions of the time to arrive at them, respectively (the penalty for time windows). The customers' demands are characterized by a known probability distribution and a customer's actual demand is only revealed when the vehicle reaches the customer location. When the total demands of customers on a route exceed the capacity of the vehicle, extra path to depot for replenishment must occur. This problem can be described as the problem of designing least travel cost routes from one depot to a set of customers. We express the travel cost by the sum of the planned distance, the expected distance of extra path to the depot and the expected penalty for time windows. The planned distance and the expected distance of extra path to the depot can be obtained immediately when a route is given. However the expected penalty for time windows cannot be obtained even if a route is given. Because we must determine the time to arrive at each customer in order to obtain the expected penalty for time windows, that is, we must consider the problem of determining the time to arrive at each customer in a given route so that the expected penalty for time windows is minimized. Therefore we propose the method for solving the problem of determining the time to arrive at each customer. Then we formulate the vehicle routing problem as a set covering problem and we propose the solution method using column generation approach.