Short CommunicationOn the product line selection problem under attraction choice models of consumer behavior
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.
Section snippets
Products and prices
Corresponding to the two-step product line design approach, let be a given set of candidate products to offer from which the seller selects the final product line. Each product (profile) is characterized by a unique combination of non-price related attribute levels as follows: given a set of non-price product attributes, and sets of levels for attributes , each product can be represented by a vector of dimension with elements if product has level
Structural properties and solution approach
The major difficulty associated with the seller’ product line selection problem (5), (6), (7), (8) is that under a probabilistic choice model (1), the profit objective function as well as certain constraints such as (7) are generally non-concave in price (see Hanson and Martin, 1996, Kraus and Yano, 2003). To overcome this, we do not follow the conventional approach to view price as the primary decision variable; rather, we show in the following that the existence of the inverse attraction
Conclusion
We have presented an exact approach for the profit-oriented product line selection problem with continuous prices when choice behavior is either modelled at the aggregate level or when it is disaggregated into customer segments that can be targeted with a customized pricing strategy increasingly encountered in the era of e-business due to its huge profit potential. Our optimization model can accommodate a variety of attraction models including the MNL, the BTL, the MCI, and approximately the
References (36)
Market-Share Models
- et al.
Profitability in product line pricing and composition with manufacturing commonalities
European Journal of Operational Research
(2006) - et al.
Research for product positioning and design decisions: An integrative review
International Journal of Research in Marketing
(1995) - et al.
Optimal product design using conjoint analysis: computational complexity and algorithms
European Journal of Operational Research
(1989) - et al.
Product line selection and pricing under a share-of-surplus choice model
European Journal of Operational Research
(2003) - et al.
Perceptual maps and the optimal location of new products: An integrative essay
Internat. J. Res. Marketing
(1988) Price discrimination
- et al.
Putting one-to-one marketing to work: personalization, customization and choice
Marketing Letters
(2008) - et al.
A market share theorem
Journal of Marketing Research
(1975) - et al.
Optimizing product line designs: Efficient methods and comparisons
Management Science
(2008)
An overview of pricing models for revenue management
Manufacturing & Service Operations Management
Conjoint optimization: An exact branch-and-bound algorithm for the share-of-choice problem
Management Science
Personalized pricing and quality differentiation
Management Science
Market Share Analysis
Positioning and pricing a product line
Marketing Science
Optimal product line design: genetic algorithm approach to mitigate cannibalization
Journal of Optimization Theory and Applications
Gewinnorientierte Produktliniengestaltung unter Beriicksichtigung des Kundennutzens
Zeitschrift fur Betriebswirtschaft
Models and heuristics for product line selection
Marketing Science
Cited by (44)
An integrated bi-objective optimization model accounting for the social acceptance of renewable fuel production networks
2024, European Journal of Operational ResearchOptimal pricing for dual-channel retailing with stochastic attraction demand model
2024, International Journal of Production EconomicsPrice matching and product differentiation strategies considering showrooming
2023, Journal of Retailing and Consumer ServicesThe rank pricing problem with ties
2021, European Journal of Operational ResearchCitation Excerpt :In the probabilistic choice behavior, each customer (or type of customer) probabilistically chooses from the available options. Some references are those by Chen and Hausman (2000), Schon (2010a,b) and Kraus and Yano (2003), among others. In the first-choice rule, customers deterministically select the product from the offered line that maximizes their utility.
Product line selection of fast-moving consumer goods
2021, Omega (United Kingdom)Citation Excerpt :The model supports multiple segments, allowing price discrimination, and is the first work in PLS using linear constraints to implement the attraction model. In a technical note, Schön [26] extends her work by considering price as a bounded continuous variable and associating fix costs with capacitated attribute levels. In both works, the author takes advantage of the problem structure to make the solution tractable.
Product line optimization in the presence of preferences for compromise alternatives
2021, European Journal of Operational ResearchCitation Excerpt :We assume that the prices for the products, from which the product line is built, are given. Concerning these characteristics, our model is closely related to, e.g., Aydin and Ryan (2000), Chen and Hausman (2000), or Schön (2010a, 2010b). Compromises are modelled according to the CVM presented in Chorus and Bierlaire (2013).