TY - JOUR
T1 - Multiproduct pricing under the generalized extreme value models with homogeneous price sensitivity parameters
AU - Zhang, Heng
AU - Rusmevichientong, Paat
AU - Topaloglu, Huseyin
N1 - Funding Information:
This work was supported in part by the National Science Foundation [Grants CMMI-1157569, CMMI-1433396, and CMMI-1433398].
Funding Information:
Funding: This work was supported in part by the National Science Foundation [Grants CMMI-1157569, CMMI-1433396, and CMMI-1433398]. SupplementalMaterial: The e-companion is available at https://doi.org/10.1287/opre.2018.1740.
Publisher Copyright:
© 2018 INFORMS.
PY - 2018/11
Y1 - 2018/11
N2 - We consider unconstrained and constrained multiproduct pricing problems when customers choose according to an arbitrary generalized extreme value (GEV) model and the products have the same price sensitivity parameter. In the unconstrained problem, there is a unit cost associated with the sale of each product. The goal is to choose the prices for the products to maximize the expected profit obtained from each customer. We show that the optimal prices of the different products have a constant markup over their unit costs. We provide an explicit formula for the optimal markup in terms of the Lambert-W function. In the constrained problem, motivated by the applications with inventory considerations, the expected sales of the products are constrained to lie in a convex set. The goal is to choose the prices for the products to maximize the expected revenue obtained from each customer, while making sure that the constraints for the expected sales are satisfied. If we formulate the constrained problem by using the prices of the products as the decision variables, then we end up with a nonconvex program. We give an equivalent market-share-based formulation, where the purchase probabilities of the products are the decision variables. We show that the market-share-based formulation is a convex program, the gradient of its objective function can be computed efficiently, and we can recover the optimal prices for the products by using the optimal purchase probabilities from the market-share-based formulation. Our results for both unconstrained and constrained problems hold for any arbitrary GEV model.
AB - We consider unconstrained and constrained multiproduct pricing problems when customers choose according to an arbitrary generalized extreme value (GEV) model and the products have the same price sensitivity parameter. In the unconstrained problem, there is a unit cost associated with the sale of each product. The goal is to choose the prices for the products to maximize the expected profit obtained from each customer. We show that the optimal prices of the different products have a constant markup over their unit costs. We provide an explicit formula for the optimal markup in terms of the Lambert-W function. In the constrained problem, motivated by the applications with inventory considerations, the expected sales of the products are constrained to lie in a convex set. The goal is to choose the prices for the products to maximize the expected revenue obtained from each customer, while making sure that the constraints for the expected sales are satisfied. If we formulate the constrained problem by using the prices of the products as the decision variables, then we end up with a nonconvex program. We give an equivalent market-share-based formulation, where the purchase probabilities of the products are the decision variables. We show that the market-share-based formulation is a convex program, the gradient of its objective function can be computed efficiently, and we can recover the optimal prices for the products by using the optimal purchase probabilities from the market-share-based formulation. Our results for both unconstrained and constrained problems hold for any arbitrary GEV model.
KW - Choice models
KW - Concave maximization
KW - Generalized extreme value models
KW - Homogenous price sensitivity
KW - Multiproduct pricing
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U2 - 10.1287/opre.2018.1740
DO - 10.1287/opre.2018.1740
M3 - Article
AN - SCOPUS:85061600542
SN - 0030-364X
VL - 66
SP - 1559
EP - 1570
JO - Operations Research
JF - Operations Research
IS - 6
ER -