A Column Generation Approach to the Vehicle Routing Problem

Abstract

We apply column generation with branch-and-price optimization to the multi-target, multi-task assignment problem, with precedence constraints. The cost of any feasible UAV trajectory is calculated using Dubin’s ∞yable paths. We present results for the multitarget, single-task assignment problem. Along with the multi-target, multi-task assignment problem with precedence constraints. olumn generation is a means of solving linear programs. 1 In this paper, we apply column generation to the multi-target, multi-task assignment problem, with precedence constraints. Currently task assignment algorithms are calculated at a central (base) location, with assignments being fed to each agent in the team. The motivation for this work to to investigate the possibility of using column generation as a distributed means of calculating optimal assignments. Column generation is based on the Dantzig-Wolfe decomposition 2 and permits distribution of certain linear programs depending on the constraints. Rather than optimizing a large linear program, column generation divides and conquers by repeatedly iterating between a so called master problem with coupling constraints and a series of sub-problems, each with its own respective local constraints. 1,3,4 Some interesting properties of column generation are that by relaxing the integrality constraints of the master problem, the resulting bound is tighter than if the original problem were relaxed. 5 Furthermore, by applying the Dantzig-Wolfe decomposition, an optimization problem with non-linear costs and constraints becomes a linear program if separability holds. 6,7 This last property permits us to solve the task assignment problem with non-linear costs and precedence constraints determined by Dubin’s ∞yable paths as a linear program. Similar works, and the references therein, include Ref. 8 which is a single-assignment problem without precedence constraints, Ref. 9 which solves the routing with time windows using column generation and a branch-and-price scheme, and Ref. 10 which uses column generation to solve large tra‐c scheduling problems. We begin this paper by expressing the vehicle routing problem with precedence constraints in Section II. Then we explain column generation using a simple ∞ow problem in Section III. This toy problem explains column generation in an intuitive fashion, since it is easy to visualize. Then, in Section IV, we apply the Dantzig-Wolfe decomposition to the vehicle routing problem with precedence constraints and set up the column generation problem. We conclude with some illustrative examples and remarks in Sections V and VI.