Models and Algorithms for Stochastic and Robust Vehicle Routing with Deadlines

Abstract

We consider the vehicle routing problem with deadlines under travel time uncertainty in the contexts of stochastic and robust optimization. The problem is defined on a directed graph where a fleet of vehicles is required to visit a given set of nodes and deadlines are imposed at a subset of nodes. In the stochastic vehicle routing problem with deadlines (SVRP-D), the probability distribution of the travel times is assumed to be known and the problem is solved to minimize the sum of probability of deadline violations. In the robust vehicle routing problem with deadlines (RVRP-D), however, the exact probability distribution is unknown but it belongs to a certain family of distributions. The objective of the problem is to optimize a performance measure, called lateness index, which represents the risk of violating the deadlines. Although novel mathematical frameworks have been proposed to solve these problems, the size of problem that those approaches can handle is relatively small. Our focus in this paper is the computational aspects of the two solution schemes. We introduce formulations that can be applied for the problems with multiple capacitated vehicles and discuss the extensions to the cases of incorporating service times and soft time windows. Furthermore, we develop an algorithm based on a branch-and-cut framework to solve the problems. The experiments show that these approaches provide substantial improvements in computational efficiency compared to the approaches in the literature.

The Robust Vehicle Routing Problem with Time Windows: Compact Formulation and Branch-Price-and-Cut Method

“…The CVRP with deadlines under travel time uncertainty was studied by Adulyasak and Jaillet (2016), who developed branch-and-cut and Benders decomposition algorithms for stochastic programming and RO versions of the problem. The SP model was based on a known probability distribution and its main focus was to minimize the expected number of deadline violations.…”

Robust Optimization for a Maritime Inventory Routing Problem

“…Adulyasak and Jaillet [2] consider the vehicle routing problem with deadlines under travel time uncertainty, discussing both stochastic and robust problem variants.…”

“…We consider an uncertainty set with Γ = 2 and T ijv = 1 for all (i, j, v). The plus 1 next to the values assigned to arcs (1, 1, 2, 1) and (2,1,3,2) represent the delay of 1 unit. The values α((i, n), γ) are given for the critical path leading to node (3,2).…”

“…The plus 1 next to the values assigned to arcs (1, 1, 2, 1) and (2,1,3,2) represent the delay of 1 unit. The values α((i, n), γ) are given for the critical path leading to node (3,2). The value α((1, 1), 0) is given by A 11 which is computed from the initial stock (S 0 1 = 22), the production rate R 1 = 5, and the load quantity (q 111 = 37).…”

“…In Example 1 we can see that for the same routing decisions there are alternative values of load and unload quantities. For instance, by decreasing q 111 to 32 and increasing q 122 to 50, the earliest time for visit (1,1) is now set to 2, (α(1, 1), 0) = 2, which leads to a situation where no shortage occurs at node (3,2). A lower bound for the worst case delay can be computed as the sum of the min{k, Γ} highest values ofT ijv in the route to node (i, m), where k is the number of arcs in that route.…”