# Improved Approximation Algorithm for the One-Warehouse Multi-Retailer Problem

## Abstract

In this paper, we will consider a well-studied inventory model, called the *one-warehouse multi-retailer problem* (OWMR) and its special case the *joint replenishment problem* (JRP). As the name suggests, in this model there is one warehouse that orders a particular commodity from a supplier, in order to serve demand at *N* distinct retailers. We consider a discrete finite planning horizon of *T* periods, and are given the demand *d* _{ it } required for each retailer *i*=1,...,*N* in each time period *t*=1,...,*T*. There are two types of costs incurred: *ordering costs* (to model that there are fixed costs incurred each time the warehouse replenishes its supply on hand from the supplier, as well as the analogous cost for each retailer to be stocked from the warehouse) and *holding costs* (to model the fact that maintaining inventory, at both the warehouse and the retail store, incurs a cost). The aim of the model is to provide an optimization framework to balance the fact that ordering too frequently is inefficient for ordering costs, whereas ordering too rarely incurs excessive holding costs.

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