A Unified Algorithmic Framework for Dynamic Assortment Optimization under MNL Choice

TL;DR

This work develops a unified relax-and-transform framework for dynamic assortment optimization under the MNL choice model, covering both DA (no personalization) and DAP (personalization) with dynamic stockouts and no replenishment. By leveraging a fluid CDLP relaxation and the DR-SO property, the authors design fast, provably good algorithms that achieve improved approximation guarantees and runtime across deterministic and stochastic horizons, including uncertain horizon length $T$ and various constraint types. A key technical advance is the generalized revenue-ordered structure and the Separability Lemma, enabling precise analysis of cannibalization and a clean reduction to tractable subproblems; these ideas underpin both the transformation from relaxation to practical policies and the multi-type extension for DAP. The framework yields strong constant-factor guarantees (e.g., up to 0.320–0.474 for DA under deterministic horizon, 0.5(1-1/e) for DAP with deterministic horizon, and 0.25–0.195 under stochastic horizons), scalable runtimes, and broad applicability to high-variance horizon distributions, marking a substantial step forward in dynamic, revenue-driven assortment planning.

Abstract

We consider assortment and inventory planning problems with dynamic stockout-based substitution effects, and without replenishment, in two different settings: (1) Customers can see all available products when they arrive, a typical scenario in physical stores. (2) The seller can choose to offer a subset of available products to each customer, which is more common on online platforms. Both settings are known to be computationally challenging, and the current approximation algorithms for the two settings are quite different. We develop a unified algorithm framework under the MNL choice model for both settings. Our algorithms improve on the state-of-the-art algorithms in terms of approximation guarantee and runtime, and the ability to manage uncertainty in the total number of customers and handle more complex constraints. In the process, we establish various novel properties of dynamic assortment planning (for the MNL choice model) that may be useful more broadly.

Figures (6)

• Figure 1: Example: FP$(\mathbf{x}+\delta_i \mathbf{e}_i,T)$ and its equivalent process.
• Figure 2: Left: $\textsc{fp} (\mathbf{x}^{\mathbf{c}^{+ij}})-\textsc{fp}(\mathbf{x}^{\mathbf{c}^{+j}})$. Right: $\textsc{fp} (\mathbf{x}^{\mathbf{c}^{+i}})-\textsc{fp}(\mathbf{x}^{\mathbf{c}})$
• Figure 3: Examples of Distributions with Different Number of Steps.
• Figure 4: Step Removal.
• Figure 5: Total Revenue in $\textsc{fp}(\underline{\mathbf{x}}^{\mathbf{c}},T)$, $\textsc{fp}(\mathbf{x}^{\mathbf{c}},T)$, $\textsc{fp}(\mathbf{x}^{\mathbf{c}},T)$.

Theorems & Definitions (67)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Remark 1
• Theorem 5
• Theorem 6
• Lemma 1: Generalized revenue-ordered optimal inventory
• Lemma 2: Separability
• Remark 2