Assortment Optimization under the Multinomial Logit Model with Covering Constraints

TL;DR

The paper studies assortment optimization under the multinomial logit (MNL) model with general covering constraints that require offering a minimum number of items from each category. It develops a spectrum of results across deterministic and randomized settings, and across single-segment and multi-segment demand: (i) a near-hardness-tight $1/(\\log K+2)$-approximation for deterministic single-segment; (ii) a polynomial-time linear-programming formulation for randomized single-segment; (iii) $$(1-\\epsilon)/(\\log K+2)$$-approximation for constant number of segments in the deterministic multi-segment case and a $1/(m(\\log K+2))$-approximation for general $m$; (iv) polynomial-time solvability of randomized multi-segment via LP; and (v) numerical experiments on real data showing limited revenue loss from covering constraints and that randomized solutions typically involve only a few assortments. The results provide both theoretical guarantees and practical guidance for enforcing coverage/diversity constraints in MNL-based assortment design, with LP-based methods enabling scalable solutions for randomized variants. Practical impact is demonstrated through real-data experiments indicating small revenue penalties and limited need for extensive randomization. All mathematical notation is kept explicit, with $K$ categories, $m$ segments, and covering thresholds $\\ell_k$ consistently represented in $LaTeX$-style formatting wrapped in $...$.

Abstract

We consider an assortment optimization problem under the multinomial logit choice model with general covering constraints. In this problem, the seller offers an assortment that should contain a minimum number of products from multiple categories. We refer to these constraints as covering constraints. Such constraints are common in practice due to service level agreements with suppliers or diversity considerations within the assortment. We consider both the deterministic version, where the seller decides on a single assortment, and the randomized version, where they choose a distribution over assortments. In the deterministic case, we provide a $1/(\log K+2)$-approximation algorithm, where $K$ is the number of product categories, matching the problem's hardness up to a constant factor. For the randomized setting, we show that the problem is solvable in polynomial time via an equivalent linear program. We also extend our analysis to multi-segment assortment optimization with covering constraints, where there are $m$ customer segments, and an assortment is offered to each. In the randomized setting, the problem remains polynomially solvable. In the deterministic setting, we design a $(1 - ε) / (\log K + 2)$-approximation algorithm for constant $m$ and a $1 / (m (\log K + 2))$-approximation for general $m$, which matches the hardness up to a logarithmic factor. Finally, we conduct a numerical experiment using real data from an online electronics store, categorizing products by price range and brand. Our findings demonstrate that, in practice, it is feasible to enforce a minimum number of representatives from each category while incurring a relatively small revenue loss. Moreover, we observe that the optimal expected revenue in both deterministic and randomized settings is often comparable, and the optimal solution in the randomized setting typically involves only a few assortments.

Figures (1)

• Figure 1: Revenue loss compared to unconstrained optimum in the randomized setting

Theorems & Definitions (39)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• proof : Proof of Theorem \ref{['thm:single_segment']}
• Remark 4: Extensions to Cardinality Constraint
• Theorem 5
• Lemma 6
• proof
• Lemma 7