A PTAS for Capacitated Sum-of-Ratios Optimization Paat Rusmevichientong * Zuo-Jun Max Shen † David B. Shmoys ‡ August 29, 2008 Abstract Motivated by an application in assortment planning under the nested logit choice model, we consider the sum-of-ratios optimization problem with a capacity constraint. When the number of product groups is fixed, we develop a polynomial-time approximation scheme and demonstrate how the scheme can be applied to the assortment planning problem. Keywords: Sum-of-Ratios, Polynomial-time Approximation Scheme, and Assortment Planning. 1. 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. Kok et al. (2006) provide a comprehensive review of the literature in this area. The stream of research most closely related to our work is on demand modeling and optimization algorithms. Earlier work in assortment planning assumes that the demand of each product is independent. Recent work focuses on more complex models of customer choice behavior, allowing for more intricate substitution patterns among products, and providing more realistic models of demand. One of the most commonly used choice models in economics, marketing, and operations management is the multinomial logit (MNL) model (see, for example, McFadden (1974), Ben-Akiva and Lerman (1985), Anderson et al. (1992), Mahajan and van Ryzin (1998), and the references therein). Pioneering work in assortment optimization with complex choice models include van Ryzin and Mahajan (1999), Chen and Hausman (2001), and Mahajan and van Ryzin (2001). Although the MNL model is analytically tractable, it exhibits the Independence of Irrelevant Alternative (IIA) property. Under this property, the likelihood of choosing between any two alternatives is independent of the assortment containing them. As discussed in McFadden (1974), McFadden (1981), Ben-Akiva and Lerman (1985), and many other work, this property is inconsistent with the actual customer choice behavior in many settings. Despite this shortcoming, the MNL model remains one of the most useful and popular choice * paatrus@cornell.edu, School of ORIE, Cornell University, Ithaca, NY 14853 † shen@ieor.berkeley.edu, Dept. of IEOR, University of California, Berkeley, CA 94720-1777 ‡ shmoys@cs.cornell.edu, School of ORIE and Dept. of Computer Science, Cornell University, Ithaca, NY 14853 1