Optimizing assortment and pricing of multiple retail categories with cross-selling

Abstract

This paper investigates the joint optimization of assortment and pricing decisions for complementary retail categories. Each category comprises substitutable items (e.g., different coffee brands) and the categories are related by cross-selling considerations that are empirically observed in marketing studies to be asymmetric in nature. That is, a subset of customers who purchase a product from a primary category (e.g., coffee) can opt to also buy from one or several complementary categories (e.g., sugar and/or coffee creamer). We propose a mixed-integer nonlinear program that maximizes the retailer’s profit by jointly optimizing assortment and pricing decisions for multiple categories under a classical deterministic maximum-surplus consumer choice model. A linear mixed-integer reformulation is developed which effectively enables an exact solution to relatively large problem instances using commercial optimization solvers. This is encouraging, because simpler product line optimization problems in the literature have posed significant computational challenges over the last decades and have been mostly tackled via heuristics. Moreover, our computational study indicates that overlooking cross-selling between retail categories can result in substantial profit losses, suboptimal (narrower) assortments, and inadequate prices.

Appendix: Data generation scheme

A set of simulated problem instances are generated using the following data generation scheme:

• Number of customers in each customer segment ($s^\ell _i$):

     Randomly set $s^\ell _i \leftarrow$ floor(Uniform(150, 400)).

• Fixed assortment cost ($f^\ell _j$):

     Randomly set $f^\ell _j \leftarrow$ floor(Uniform(500, 2000)).

• Unit ordering cost ($c^\ell _{jt}$):

     Randomly set $c^\ell _{j} \leftarrow$ round(Uniform(100, 140), 1). (Rounded to 1 digit past the decimal)

• Customer reservation prices:

     Randomly set $\alpha ^\ell _{ij}\,\leftarrow$ round($c^{\ell }_{j}$ * Uniform(0.99, 1.04), 2). (Rounded to 2 digits past the decimal)

     Randomly set $\beta ^\ell _{ij} \leftarrow$ round($c^{\ell }_{j}$ * Uniform(0.99, 1.04), 2) (for $\ell \in \mathcal {L} \setminus \{1\}$).

• Cross-selling matrix ($\gamma ^k_{i}$):

     Randomly set $\gamma ^k_{i} \leftarrow$ round(Uniform(0,0.6), 2).

Although the data generator is not based on a specific application, certain relationships were enforced between input parameters to yield meaningful instances. In additional, the reservation prices ($\alpha$- and $\beta$-parameters) were randomly generated by multiplying the unit ordering cost by a scalar randomly generated using a uniform distribution over the interval (0.99, 1.04). As such, reservations prices in our instances tend to be within 4 % above the ordering cost and occasionally slightly below the ordering cost in order to reflect exigent customer segment in a highly competitive market. Finally, the fraction of customers of a given segment of the primary category that considers cross-selling from a secondary category (the $\gamma$ parameter) is chosen between 0 and 60 %. This covers a wide range of customer behavior from not being interested in cross-selling to extensively exercising it.