On the Heterogeneous Vehicle Routing Problem

On the Heterogeneous Vehicle Routing Problem under Demand Uncertainty∗ Mahdi Noorizadegan†

Laura Galli‡

Bo Chen§

September 2012

Abstract In this paper we study the heterogeneous vehicle routing problem under demand uncertainty, on which there has been little research to our knowledge. The focus of the paper is to provide a strong formulation that also easily allows tractable robust and chance-constrained counterparts. To this end, we propose a basic Miller-Tucker-Zemlin (MTZ) formulation with the main advantage that uncertainty is restricted to the right-hand side of the constraints. This leads to compact and tractable counterparts of demand uncertainty. On the other hand, since the MTZ formulation is well known to provide a rather weak linear programming relaxation, we propose to strengthen the initial formulation with valid inequalities and lifting techniques and, furthermore, to dynamically add cutting planes that successively reduce the polyhedral region using a branch-and-cut algorithm. We complete our study with extensive computational analysis with different performance measures on different classes of instances taken from the literature. In addition, using simulation, we conduct a scenario-based risk level analysis for both cases where either unmet demand is allowed or not. Keywords: stochastic vehicle routing, robust optimization, chance constrained programming ∗

Presented at The 21st International Symposium on Mathematical Programming, 19– 24 August 2012, Berlin, Germany † Centre for Discrete Mathematics and Its Applications, Warwick Business School, University of Warwick, Coventry CV4 7AL, United Kingdom ‡ Department of Computer Science, University of Pisa, Largo B. Pontecorvo 3, 56124 Pisa, Italy § Corresponding author: Centre for Discrete Mathematics and Its Applications, Warwick Business School, University of Warwick, Coventry CV4 7AL, United Kingdom