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

AbstractWe consider the Inventory-Routing Problem where n geographically dispersed retailers must be supplied by a central facility. The retailers experience demand for a product at a deterministic rate and incur holding costs for keeping inventory. Distribution is performed by a fleet of capacitated vehicles. The objective is to minimize the average transportation and inventory costs per unit time over the infinite horizon. In this paper, we focus on the set of fixed partition policies. In a fixed partition policy, the retailers are partitioned into disjoint and collectively exhaustive sets. Each set of retailers is 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 fixed partition policy. A probabilistic analysis of the performance of this bound demonstrates that it is asymptotically 98.5%-effective. That is, as the number of retailers increases, the lower bound is very close to the cost of the optimal fixed partition policy

Multi-item lot-sizing with a joint set-up cost

We consider a multi-item lot-sizing problem in which there are demands, and unit production and storage costs. In addition production of any mix of items is measured in batches of fixed size, and there is a fixed set-up cost per batch in each period. Suppose that the unit production costs are constant over time, the storage costs are nonnegative, and for any two items the one that has a higher storage cost in one period has a higher storage cost in every period. Then we show that there is a linear program with O(mTexp.2) constraints and variables that solves the multi-item lot-sizing problem, thereby establishing that it is polynomially solvable, wheremis the number of items and T the number of time periods. This generalizes an earlier result of Anily and Tzur who presented a O(mTexp.m+5) dynamic programming algorithm for essentially the same problem. Under additional conditions, a similar linear programming result is shown to hold in the presence of backlogging when the batch size is arbitrarily large. Brief computational results on two instances with varying batch sizes are presented and discussed

The scheduling of maintenance service

We study a discrete problem of scheduling activities of several types under the constraint that at most a single activity can be scheduled to any one period. Applications of such a model are the scheduling of maintenance service to machines and multi-item replenishment of stock. In this paper we assume that the cost associated with any given type of activity increases linearly with the number of periods since the last execution of this type. The problem is to find an optimal schedule specifying at which periods to execute each of the activity types in order to minimize the long-run average cost per period. We investigate properties of an optimal solution and show that there is always a cyclic optimal policy. We propose a greedy algorithm and report on computational comparison with the optimal. We also provide a heuristic, based on regular cycles for all but one activity type, with a guaranteed worse case bound