An asymptotic 98.5%-effective lower bound on fixed partition policies for the inventory-routing problem

We consider the Inventory-Routing Problem where n geographically dispersed retailers must be supplied by a central 9 facility. The retailers experience demand for a product at a deterministic rate and incur holding costs for keeping inventory. Distribution is performed by a eet of capacitated vehicles. The objective is to minimize the average transportation and 11 inventory costs per unit time over the in nite horizon. In this paper, we focus on the set of xed partition policies. In a xed partition policy, the retailers are partitioned into disjoint and collectively exhaustive sets. Each set of retailers is 13 served independently of the others and at its optimal replenishment rate. We derive a deterministic (O(n)) lower bound on the cost of the optimal xed partition policy. A probabilistic analysis of the performance of this bound demonstrates 15 that it is asymptotically 98.5%-e ective. That is, as the number of retailers increases, the lower bound is very close to the cost of the optimal xed partition policy. 17 c © 2003 Published by Elsevier B.V.