Joint Assortment Optimization and Discrete Marketing Mix Allocation

Problem definition: Assortment selection and marketing mix allocation are critical decisions for retailers, directly influencing consumer choices. In this paper, we propose a multinomial logit (MNL) choice model in which consumer utility is influenced by marketing decisions such as advertising and promotions: a model widely utilized in empirical marketing literature. We then study the joint assortment and marketing mix allocation problem subject to either cardinality constraints or knapsack constraints on marketing mix allocations. Methodologies/results: We prove that the problem under cardinality constraints is already strongly NP-hard and does not admit constant ratio approximation. For the model with cardinality constraints, we provide an optimal ratio approximation algorithm and polynomial-time algorithms for special cases. With a constant number of marketing mix decisions, the problem can be solved using a linear program of polynomial size. Under knapsack constraints, we also provide an optimal ratio approximation algorithm and a fully polynomial-time approximation scheme (FPTAS) for special cases. With a constant number of marketing mixes, the problem admits an optimal polynomial-time approximation scheme (PTAS). Computational experiments with real-world NielsenIQ retail data show significant 2.05% revenue increases using our method over a two-stage “assortment-then-marketing mix allocation” heuristic approach. A complementary experiment using online transaction data from JD.com is also conducted to demonstrate the applicability of our method in online settings. Managerial implications: Our comprehensive numerical experiments across various scenarios demonstrate that neglecting the impact of marketing decisions in assortment selection can lead to a significant decline in profitability. This finding demonstrates the importance of jointly optimizing assortment and marketing mix allocation, particularly when the number of marketing mix decisions significantly exceeds the number of products and when resources for marketing mix allocation are limited. Funding: The first and fourth authors have been supported in part by National Science Foundation Grant 2246414 and the Office of Naval Research Grant N00014-24-1-2066. Supplemental Material: The online appendices are available at https://doi.org/10.1287/msom.2024.1406 .