Superadditivity-based valid inequalities and asymptotic bounds for the vehicle routing problem with stochastic demands

Abstract

Over the past thirty years, the vehicle routing problem with stochastic demands (VRPSD) has emerged as a canonical application of the integer L-shaped method. Recently, the disaggregated integer L-shaped (DL-shaped) method, which decomposes the recourse function by customer rather than treating it as an aggregate cost, has been proposed for the VRPSD under the classical detour-to-depot policy. However, its generalizability to other recourse policies has not been investigated. In this work, we identify the property that characterizes the validity of the DL-shaped reformulation: the superadditivity of the recourse function under path concatenation. We show that superadditivity holds under the optimal restocking policy, and rectify an incorrect argument from the original paper on the DL-shaped method, rigorously establishing its validity under the detour-to-depot policy. We then introduce a new family of valid inequalities, the edge-set cuts, which generalize the original DL-shaped cuts and are analytically shown to provide structural advantages over existing inequalities. Building on these results, we develop a DL-shaped algorithm for the VRPSD with optimal restocking. Our algorithm outperforms existing methods in the high customer-to-vehicle ratio regime and solves 14 open single-route instances. We further derive asymptotic bounds on the optimal value of the VRPSD in a Euclidean setting with i.i.d. customers. This analysis reveals an asymptotic equivalence between the VRPSD and the split-delivery vehicle routing problem. It also yields tight bounds on the cost of requiring the total expected demand on each route not to exceed the vehicle capacity, resolving an open question in our asymptotic setting.