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Assortment Optimization Under the Paired Combinatorial Logit Model

Author

Listed:
  • Heng Zhang

    (Marshall School of Business, University of Southern California, Los Angeles, California 90089)

  • Paat Rusmevichientong

    (Marshall School of Business, University of Southern California, Los Angeles, California 90089)

  • Huseyin Topaloglu

    (School of Operations Research and Information Engineering, Cornell Tech, New York, New York 10044)

Abstract

We consider uncapacitated and capacitated assortment problems under the paired combinatorial logit model, where the goal is to find a set of products to offer to maximize the expected revenue obtained from a customer. In the uncapacitated setting, we can offer any set of products, whereas in the capacitated setting, there is an upper bound on the number of products that we can offer. We establish that even the uncapacitated assortment problem is strongly NP-hard. To develop an approximation framework for our assortment problems, we transform the assortment problem into an equivalent problem of finding the fixed point of a function, but computing the value of this function at any point requires solving a nonlinear integer program. Using a suitable linear programming relaxation of the nonlinear integer program and randomized rounding, we obtain a 0.6-approximation algorithm for the uncapacitated assortment problem. Using randomized rounding on a semidefinite programming relaxation, we obtain an improved 0.79-approximation algorithm, but the semidefinite programming relaxation can become difficult to solve in practice for large problem instances. Finally, using iterative rounding, we obtain a 0.25-approximation algorithm for the capacitated assortment problem. Our computational experiments on randomly generated problem instances demonstrate that our approximation algorithms, on average, yield expected revenues that are within 1.1% of an efficiently computable upper bound on the optimal expected revenue.

Suggested Citation

  • Heng Zhang & Paat Rusmevichientong & Huseyin Topaloglu, 2020. "Assortment Optimization Under the Paired Combinatorial Logit Model," Operations Research, INFORMS, vol. 68(3), pages 741-761, May.
  • Handle: RePEc:inm:oropre:v:68:y:2020:i:3:p:741-761
    DOI: 10.1287/opre.2019.1930
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    References listed on IDEAS

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    Cited by:

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    2. Kitthamkesorn, Songyot & Chen, Anthony & Ryu, Seungkyu & Opasanon, Sathaporn, 2024. "Maximum capture problem based on paired combinatorial weibit model to determine park-and-ride facility locations," Transportation Research Part B: Methodological, Elsevier, vol. 179(C).
    3. Xiangyu Gao & Stefanus Jasin & Sajjad Najafi & Huanan Zhang, 2022. "Joint Learning and Optimization for Multi-Product Pricing (and Ranking) Under a General Cascade Click Model," Management Science, INFORMS, vol. 68(10), pages 7362-7382, October.
    4. Ali Aouad & Jacob Feldman & Danny Segev, 2023. "The Exponomial Choice Model for Assortment Optimization: An Alternative to the MNL Model?," Management Science, INFORMS, vol. 69(5), pages 2814-2832, May.
    5. Markus Leitner & Andrea Lodi & Roberto Roberti & Claudio Sole, 2024. "An Exact Method for (Constrained) Assortment Optimization Problems with Product Costs," INFORMS Journal on Computing, INFORMS, vol. 36(2), pages 479-494, March.

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