TY - JOUR

T1 - Assortment optimization under the paired combinatorial logit model

AU - Zhang, Heng

AU - Rusmevichientong, Paat

AU - Topaloglu, Huseyin

N1 - Funding Information:
Funding: This work was supported by Schmidt Sciences [City Logistics: Challenges and Opportunities in the Information Age Grant] and the National Science Foundation [Grants CMMI-1824860 and CMMI-1825406]. Supplemental Material: Computer code and data used in the computational experiments and the online appendices are available at https://doi.org/10.1287/opre.2019.1930.
Publisher Copyright:
© 2020 INFORMS.

PY - 2020/5

Y1 - 2020/5

N2 - 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.

AB - 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.

KW - Assortment optimization

KW - Customer choice modeling

KW - Paired combinatorial logit model

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U2 - 10.1287/opre.2019.1930

DO - 10.1287/opre.2019.1930

M3 - Article

AN - SCOPUS:85091626720

VL - 68

SP - 741

EP - 761

JO - Operations Research

JF - Operations Research

SN - 0030-364X

IS - 3

ER -