A PTAS for capacitated sum-of-ratios optimization

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Abstract

Motivated by an application in assortment planning under the nested logit choice model, we develop a polynomial-time approximation scheme for the sum-of-ratios optimization problem with a capacity constraint and a fixed number of product groups.

Section snippets

Motivation and introduction

Assortment planning is an important problem facing many retailers and has been studied extensively in the supply chain and operation management literature. Given a limited shelf capacity or inventory investment constraint, the retailer must determine the subset of products to offer that maximizes the total profit. The literature on assortment planning is broad and covers a range of operational issues. A comprehensive review of the literature in this area is given in [1]. The stream of research

Problem formulation

We are given a set of N products indexed by 1,N. For each product , let π and c denote its marginal profit (per-unit revenue minus marginal cost) and fixed cost, respectively. The fixed cost of each product might correspond to the cost of introducing the product into the store. We denote the option of no purchase by 0 and set π0=c0=0. Given an inventory investment or capacity constraint C, we wish to find the subset of products that gives the maximum expected profit.

We will model the

Connection to sum-of-ratios optimization

We will now describe how an approximation algorithm for a sum-of-ratios optimization problem can be used to obtained an approximation for the Assortment problem. For each g=1,,G, let αg=1τg and for each Hg, let v=eη/τg. Define a function J:R+R+ by: for each λR+, J(λ)=max{g=1GSg(πλ)v(Sgv)αgSgHgg and g=1GSgcC}. Using a technique pioneered by Megiddo [17], we give an alternative characterization of the optimal profit Y associated with the Assortment problem.

Lemma 3.1 Parametric Representation

J() is

A PTAS for SOR when G is fixed

In this section, we describe a PTAS for the SOR problem when the number of groups G is fixed. Let (A1,A2,,AG) denote the optimal solution associated with Z with AgHg for each g{1,2,,G} and g=1GAgcC. The main result of this paper is stated in the following theorem.

Theorem 4.1 PTAS

For each δ>0 and kZ+ with k2 , there exists an approximation algorithm Approx (k,δ) for the SOR problem that will generate an assortment (A1(k,δ),A2(k,δ),,AG(k,δ)) such that Ag(k,δ)Hg for each g{1,2,,G} , g=1GA

Acknowledgments

The first author would like to thank Mike Todd for insightful and stimulating discussions on the structure of the linear programs associated with this problem. This research is supported in part by the National Science Foundation through grants DMS-0732196, CMMI-0746844, CMMI-0621433, CCR-0635121, and CMMI-0727640.

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