On the product line selection problem under attraction choice models of consumer behavior

Abstract

Product line design (PLD) involves important decisions at the interface of operations and marketing that are very costly to implement and change, and, simultaneously, determinant for market success. To evaluate the financial performance of a product line, a number of mathematical programming approaches have been proposed. Problem formulations are typically mixed or pure integer non-linear optimization models that are intractable for exact solution - in particular when empirically supported consumer choice models are incorporated.

In this note, we present an exact approach for determining a profit-maximizing product line with continuous prices when consumers choose among available products according to a general and widely applied attraction choice model including the MNL, the BTL, the MCI, and approximately the first choice model. In particular, we show how to efficiently exploit the structural properties resulting from attraction models when consumer behavior is (a) modelled at the aggregate level or (b) disaggregated into customer segments in such a way that each segment can be offered a customized price - a strategy that firms more and more engage in, recognizing it to be not only very profitable but also implementable in the era of e-business. Under these assumptions, we can transform the standard MINLP formulation of the PLD problem into a more convenient convex MIP that can be solved globally with current solvers even for large instances with ten-thousands of products in reasonable time. Therefore our work contributes by accommodating a new trend increasingly encountered in practice and by providing an efficient exact approach to profit-driven PLD for real-world applications.

Introduction

In today’s increasingly competitive business environment, product line design and pricing are among the most critical decisions determinant for a firm’s market success and, simultaneously, very costly to implement and change. Recognizing heterogeneity in demand, firms across industries are incited to differentiate their products and prices in order to closely match the diverse needs of different customers and thereby increase the probability of making a sale (e.g. Kekre and Srinivasan (1990)). However, the seller must carefully balance the potential gains from increasing product variety against the costs and possible cannibalization effects on existing products in the line. The core problem in product line design involves to simultaneously determine how many products to offer, how to differentiate them along their key attributes, and how to price them - given that products are linked by cross effects in demand and by shared resources that jointly drive revenue and costs and thus have an interdependent impact on total profit.

To provide systematic support for making PLD decisions, a number of conjoint data-based mathematical programming approaches have been developed over the last three decades where a product is represented as a set of attributes each of which can take on one of several possible levels that are determinant for customer preference and choice (see the reviews by Belloni et al., 2008, Green and Srinivasan, 1990, Kaul and Rao, 1995, Krishnan and Ulrich, 2001, Yano and Dobson, 1998). Although many of these mathematical programming approaches have been successfully implemented and applied for decision support, an old technical challenge persists: PLD is an inherently complex problem entailing substantial computational difficulties, especially for larger problem sizes and when realistic models of customer preference are incorporated. There is a common agreement among researchers and practitioners alike that the development of efficient approaches to the PLD problem is a high-priority research task (Schmalensee and Thisse, 1988, Hanson and Martin, 1996) to which this technical note contributes.

PLD problem formulations are commonly distinguished by the objective function (e.g. buyers’ welfare, seller’s share-of-choice, profit/contribution), where we focus on profit-driven PLD, i.e. the seller’s objective is to configure the products and prices such that profit is maximized, given consumers’ willingness-to-pay (gross utility) and the products’ fixed and variable costs. Depending on how the decision variables are defined (at attribute or product level), an optimal product line is either determined by directly assigning attribute levels to the products’ attributes (one-step approach to PLD), or it is composed in two steps by first constructing a set of candidate products from which the final menu of products to be offered is then selected (two-step approach to PLD, or product line selection).

Furthermore, PLD problem formulations can be distinguished by the underlying customer choice model (first or probabilistic choice behavior) and by the assumed fixed cost structure (i.e. by the model’s ability to accommodate fixed costs for development, manufacturing, and marketing of a new product). Most PLD models incorporate the deterministic first choice (FC) model under which each consumer self-selects with certainty the product from the menu of offers that provides him or her the highest surplus (net utility) (e.g. Zufryden, 1977, Green and Krieger, 1985, McBride and Zufryden, 1988, Dobson and Kalish, 1988, Fruchter et al., 2006, Day and Venkataramanan, 2006)). The resulting problem formulations under the first choice rule and with discrete (continuous) prices are typically pure (mixed) 0-1 non-linear optimization problems (NLP). Kohli and Krishnamurti (1989) have shown that the product line design problem is NP-complete. Accordingly, various heuristic solution methods have been proposed in the literature cited above.

The first choice rule is known to predict more extreme outcomes than empirically observed, i.e. it overestimates the share for the most attractive product and underestimates it for other products (Huber et al. (2007)). As an alternative to the first choice rule, probabilistic choice models, in particular attraction models (Bell et al., 1975, Cooper and Nakanishi, 1988) have been incorporated into conjoint simulators (Green and Krieger (1988)), and more sporadically into optimization problems for normative product line design. Attraction models, introduced formally in Section 2.1, include the classical multinomial logit (MNL), the (extended) Bratley-Terry-Luce (BTL), the multiplicative competitive interaction (MCI), and asymptotically the first choice (FC) model. Attraction models (in particular the MNL model) are among the most commonly applied choice models in empirical studies across industries since they capture demand interdependencies when customers choose among a set of alternative products (see e.g. Train, 2003, Hopp and Xu, 2005). Furthermore, attraction models are analytically tractable and standard software for estimation is widely available.

Despite the popularity of attraction models, only a few papers in the normative literature address the PLD problem under probabilistic choice behavior (Gaul et al., 1995, Hanson and Martin, 1996, Hausman and Chen, 2000, Steiner and Hruschka, 2002, Kraus and Yano, 2003). A major difficulty is that when choice probabilities are multiplied with product contribution margins, the resulting profit function is generally non-concave in price with potentially many local optima far from the global one. Accordingly, most existing PLD approaches under probabilistic choice behavior involve non-linear integer problem formulations and heavily rely on heuristic solution procedures. As an exception, Hausman and Chen (2000) proposed an exact approach for the product line selection problem with discrete prices in order to maximize contribution (no fixed cost) when there is just a single consumer segment whose choice behavior can be described by an aggregate MNL model. The authors have shown that the problem in this case can be formulated as a binary linear fractional programming problem with totally unimodular constraint coefficient matrix. It can thus be solved efficiently by using standard methods of fractional or convex programming.

In this note, we present an exact two-step approach for determining a profit-maximizing product line with continuous prices when consumers choose among available products according to a general attraction model. In particular, we show how to efficiently exploit the structural properties resulting from attraction models when consumer choice behavior is (a) modelled at the aggregate level or (b) disaggregated into customer segments in such a way that each segment can be offered a customized price. The first case is commonly encountered in practice when choice-based conjoint data is used to model consumer preferences (see e.g. Hausman and Chen (2000)). The case of price customization refers to the spreading practice of selling a product or service at a tailored price to an individual customer or a customer segment (e.g. based on an observable group identity) in order to capitalize on heterogeneous willingness-to-pay (Varian (1989)).¹ Firms increasingly engage in customizing prices, recognizing this strategy to be not only very profitable but also implementable in the era of e-business (Shapiro and Varian (1999)). Famous examples include Online retailers such as Amazon or Dell, car manufacturers and dealers, professional and financial service providers, and the travel, hospitality and entertainment industries (Arora et al. (2008), Choudhary et al. (2005)). Despite its growing diffusion, the marketing strategy to customize prices has to the author’s knowledge heretofore not been considered in normative product line design models - with the recent exception of Schön (2010) analyzing a product line selection problem with price discrimination when prices are constrained to discrete values as a matter of policy. On the operations side, our problem formulation can easily accommodate various fixed cost structures and capacity constraints for resources related to product features, products and/or subsets of products.

Given these assumptions, we show that the initial intuitive MINLP problem formulation for PLD can be transformed to a mixed 0-1 maximization problem with concave objective function and linear constraints where the number of binary variables corresponds to the number of fixed cost categories considered. This convenient problem structure is achieved by viewing demand rather than price as the decision variable and applying a generalized Charnes-Cooper transformation. By controlling demand for each product, we do not only overcome the severe non-concavities of the profit objective function in price under probabilistic choice behavior but also avoid additional binary variables for controlling a product’s impact on contribution from the product line depending on whether the product is offered or not. In contrast to previous product line selection approaches, even problem instances with ten-thousands of product candidates can be solved exactly in reasonable time with standard mixed-integer convex programming techniques due to this structure.

The paper is structured as follows. Our basic model is presented in Section 2 while the structural properties, transformation steps and potential solution procedures are discussed in Section 3. Section 4 provides a conclusion and future research directions. A thorough numerical study that demonstrates the performance of the approach can be obtained from the author.