Scalable Approximately Optimal Policies for Multi-Item Stochastic Inventory Problems

Abstract

We analyze an inventory system with an arbitrary number I of items sharing a limited storage capacity or inventory budget in each of T periods of the planning horizon. Demands for the different items follow a general multivariate Normal distribution allowing for general correlation structures. Inventories may be adjusted by placing orders which arrive after a given lead time, or by salvaging part of the inventory. The capacity constraints are modeled as chance constraints that impose an upper bound on the overflow probability in each period.

We design a heuristic which is asymptotically optimal in I when demands are correlated only among items within a common product line. The complexity grows quadratically in I and like O(T^{\frac{3}{2}}) in T.

Thereafter, we design a practical heuristic which we recommend for moderate values of I, and which is of similar worst case complexity as the asymptotically optimal heuristic.

An extensive numerical study involving more than 28,000 instances with up to 40 items, shows that the average gap between the upper bound and lower bound is 1.05\%, with 98.2\% exhibiting a gap smaller than 5\%. Empirically, we observe that the runtime of the inventory reduction algorithm grows linearly with I.

We provide a scalable methodology to identify near optimal procurement strategies for a problem that is central to most brick-and-mortar and online retailers and distributors. Additionally, this methodology can be used to guide capacity planning as well as assortment decisions.

Keywords: Dynamic Programming, Inventory Production, Optimal Control