Capacitated assortment optimization under the mixed multinomial logit model

Vetter, Oliver ; Sadeghi, Niloufar ; Schön, Cornelia

Document Type: Conference presentation
Year of publication: 2023
Conference title: 5th Meeting of AG MARKETING der Gesellschaft für Klassifikation e.V. (GfKl)
Location of the conference venue: Online
Date of the conference: 23.-24.11.2023
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Publication language: English
Institution: Business School > Service Operations Management (Schön 2014-)
Subject: 650 Management
Abstract: A key strategic decision for any company is the positioning and design of its product line. It is concerned with questions like "How many products to offer?", "Which products to offer?", and "How to differentiate and price the products in our line?". On the one hand, the implementation of product line decisions, like introducing new or redesigning existing products, is typically very costly. On the other hand, product line decisions determine the market success of the company. Only with a sufficiently differentiated product portfolio, diverse customer needs can be satisfied, and additional customers may be attracted. The problem of finding the most profitable product line under customer choice behavior has been separately addressed in the product line design (PLD) as well as in the assortment optimization (AO) literature. The AO problem under the mixed multinomial logit (MMNL) demand model is NP-hard. Existing exact methods for AO problems are suitable for small instances but computationally inefficient when applied to larger real-world instances. These instances can only be tackled with heuristic methods. We present novel exact solution procedures, an approximation scheme, and heuristic methods to solve the capacitated AO problem with discrete pricing under MMNL demand. We contribute to the existing literature in the following ways: - We improve the state-of-the-art approach of Şen et al. (2018) by adopting valid constraints from Méndez-Díaz et al. (2014) and applying a B&C technique. Computational tests show that this enhancement solves our problem instances on average between 32 % - 56 % faster. - We show that an FTPAS algorithm exists for the AO problem even if the discrete pricing decision is included. The algorithm is based on the work of Désir et al.(2022) with three changes. First, we include the pricing constraints. Second, we reduce the number of grid points by decomposing each fraction with a single variable (like in Shen, Zhang, & Wang, 2017). Third, we allow that the weights in the capacity constraints can have negative values which is a more general version of Désir et al. (2022). This allows the model to reflect situations where a lower bound on the capacity exists. - We review the current PLD and AO literature and show that under two conditions, the PLD and the AO problem formulations are equivalent, such that their solution results in the same product line. In addition, we propose three heuristics. As GAs are prominently used across both literature streams, we apply two different versions of GAs. The first uses the AO model and the second uses the PLD model formulation. The third heuristic is a hybrid approach that takes advantage of both the PLD and AO formulations. We name this approach two-step, and the others GA-AO and GA-PLD, respectively. Then, we test in a numerical performance analysis of solution quality and time whether solving the AO problem formulation is advantageous over solving the equivalent PLD problem formulation with state-of-the-art methods, or vice versa. The heuristics and exact methods are tested on synthetical and seven real-world conjoint studies.

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