Capacitated Assortment Optimization: Hardness and Approximation

2021 ◽  
Author(s):  
Antoine Désir ◽  
Vineet Goyal ◽  
Jiawei Zhang
Keyword(s):  
Large Class ◽  
Logit Model ◽  
Multinomial Logit Model ◽  
Multinomial Logit ◽  
Choice Models ◽  
Capacity Constraint ◽  
Linear Functions ◽  
Nested Logit ◽  
Assortment Optimization ◽  
Unconstrained Problem

Assortment optimization is an important problem arising in various applications. In many practical settings, the assortment is subject to a capacity constraint. In “Capacitated Assortment Optimization: Hardness and Approximation,” Désir, Goyal, and Zhang study the capacitated assortment optimization problem. The authors first show that adding a general capacity constraint makes the problem NP-hard even for the simple multinomial logit model. They also show that under the mixture of multinomial logit model, even the unconstrained problem is hard to approximate within any reasonable factor when the number of mixtures is not constant. In view of these hardness results, the authors present near-optimal algorithms for a large class of parametric choice models including the mixture of multinomial logit, Markov chain, nested logit, and d-level nested logit choice models. In fact, their approach extends to a large class of objective functions that depend only on a small number of linear functions.

Assortment Optimization under the Multinomial Logit Model with Random Choice Parameters

2014 ◽  
Vol 23 (11) ◽  
pp. 2023-2039 ◽  
Author(s):  
Paat Rusmevichientong ◽  
David Shmoys ◽  
Chaoxu Tong ◽  
Huseyin Topaloglu
Keyword(s):  
Logit Model ◽  
Multinomial Logit Model ◽  
Multinomial Logit ◽  
Random Choice ◽  
Assortment Optimization

Assortment Optimization Under the Multinomial Logit Model with Sequential Offerings

2020 ◽  
Vol 32 (3) ◽  
pp. 835-853 ◽  
Author(s):  
Nan Liu ◽  
Yuhang Ma ◽  
Huseyin Topaloglu
Keyword(s):  
Logit Model ◽  
Optimization Problem ◽  
Approximation Scheme ◽  
Optimization Problems ◽  
Multinomial Logit Model ◽  
Multinomial Logit ◽  
Polynomial Time Approximation Scheme ◽  
Assortment Optimization ◽  
Last Stage ◽  
Expected Revenue

We consider assortment optimization problems, where the choice process of a customer takes place in multiple stages. There is a finite number of stages. In each stage, we offer an assortment of products that does not overlap with the assortments offered in the earlier stages. If the customer makes a purchase within the offered assortment, then the customer leaves the system with the purchase. Otherwise, the customer proceeds to the next stage, where we offer another assortment. If the customer reaches the end of the last stage without a purchase, then the customer leaves the system without a purchase. The choice of the customer in each stage is governed by a multinomial logit model. The goal is to find an assortment to offer in each stage to maximize the expected revenue obtained from a customer. For this assortment optimization problem, it turns out that the union of the optimal assortments to offer in each stage is nested by revenue in the sense that this union includes a certain number of products with the largest revenues. However, it is still difficult to figure out the stage in which a certain product should be offered. In particular, the problem of finding an assortment to offer in each stage to maximize the expected revenue obtained from a customer is NP hard. We give a fully polynomial time approximation scheme for the problem when the number of stages is fixed.

Pricing Competition Under Specific Discrete Choice Models

2020 ◽  
Vol 37 (02) ◽  
pp. 2050008
Author(s):  
Farhad Etebari
Keyword(s):  
Discrete Choice ◽  
Logit Model ◽  
Choice Model ◽  
Public Information ◽  
Multinomial Logit Model ◽  
Multinomial Logit ◽  
Discrete Choice Models ◽  
Nested Logit ◽  
Nested Logit Model ◽  
Substitution Pattern

Recent developments of information technology have increased market’s competitive pressure and products’ prices turned to be paramount factor for customers’ choices. These challenges influence traditional revenue management models and force them to shift from quantity-based to price-based techniques and incorporate individuals’ decisions within optimization models during pricing process. Multinomial logit model is the simplest and most popular discrete choice model, which suffers from an independence of irrelevant alternatives limitation. Empirical results demonstrate inadequacy of this model for capturing choice probability in the itinerary share models. The nested logit model, which appeared a few years after the multinomial logit, incorporates more realistic substitution pattern by relaxing this limitation. In this paper, a model of game theory is developed for two firms which customers choose according to the nested logit model. It is assumed that the real-time inventory levels of all firms are public information and the existence of Nash equilibrium is demonstrated. The firms adapt their prices by market conditions in this competition. The numerical experiments indicate decreasing firm’s price level simultaneously with increasing correlation among alternatives’ utilities error terms in the nests.

Investigation of Stochastic Network Loading Procedures

1998 ◽  
Vol 1645 (1) ◽  
pp. 94-102 ◽  
Author(s):  
J. N. Prashker ◽  
S. Bekhor
Keyword(s):  
Logit Model ◽  
Route Choice ◽  
Multinomial Logit Model ◽  
Multinomial Logit ◽  
Nested Logit ◽  
Stochastic Network ◽  
Nested Logit Model ◽  
Network Loading ◽  
Stochastic Loading ◽  
The Cross

The network loading process of stochastic traffic assignment is investigated. A central issue in the assignment problem is the behavioral assumption governing route choice, which concerns the definition of available routes and the choice model. These two problems are addressed and reviewed. Although the multinomial logit model can be implemented efficiently in stochastic network loading algorithms, the model suffers from theoretical drawbacks, some of them arising from the independence of irrelevant alternatives property. As a result, the stochastic loading on routes that share common links is overloaded at the overlapping parts of the routes. Other logit-family models recently have been proposed to overcome some of the theoretical problems while maintaining the convenient analytical structure. Three such models are investigated: the C-logit model, which was specifically defined for route choice; and two general discrete-choice models, the cross-nested logit model and the paired combinatorial logit model. The two latter models are adapted to route choice, and simple network examples are presented to illustrate the performance of the models with respect to the overlapping problem. The results indicate that all three models perform better than does the multinomial logit model. The cross-nested logit model has an advantage over the two other generalized models because it enables performing stochastic loading without route enumeration. The integration of this model with the stochastic equilibrium problem is discussed, and a specific algorithm using the cross-nest logit model is presented for the stochastic loading phase.

scholarly journals Assortment optimization under a multinomial logit model with position bias and social influence

4OR ◽  
2015 ◽  
Vol 14 (1) ◽  
pp. 57-75 ◽  
Author(s):  
Andrés Abeliuk ◽  
Gerardo Berbeglia ◽  
Manuel Cebrian ◽  
Pascal Van Hentenryck
Keyword(s):  
Social Influence ◽  
Logit Model ◽  
Multinomial Logit Model ◽  
Multinomial Logit ◽  
Position Bias ◽  
Assortment Optimization

Assortment Optimization and Pricing Under the Multinomial Logit Model with Impatient Customers: Sequential Recommendation and Selection

2021 ◽  
Author(s):  
Pin Gao ◽  
Yuhang Ma ◽  
Ningyuan Chen ◽  
Guillermo Gallego ◽  
Anran Li ◽  
...  
Keyword(s):  
Logit Model ◽  
Choice Model ◽  
Multinomial Logit Model ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Multinomial Logit ◽  
Impatient Customers ◽  
Purchasing Decisions ◽  
Assortment Optimization ◽  
Decision Variables

Sequential Recommendation Under the Multinomial Logit Model with Impatient Customers In many applications, customers incrementally view a subset of offered products and make purchasing decisions before observing all the offered products. In this case, the decision faced by a firm is not only what assortment of products to offer, but also in what sequence to offer the products. In “Assortment Optimization and Pricing Under the Multinomial Logit Model with Impatient Customers: Sequential Recommendation and Selection”, Gao, Ma, Chen, Gallego, Li, Rusmevichientong, and Topaloglu propose a choice model where each customer incrementally view the assortment of products in multiple stages, and their patience level determines the maximum number of stages. Under this choice model, the authors develop a polynomial-time algorithm that finds a revenue-maximizing sequence of assortments. If the sequence of assortments is fixed, the problem of finding revenue-maximizing prices can be transformed to a convex program. They combine these results to develop an effective approximation algorithm when both the sequence of assortments and prices are decision variables.

Revenue-Utility Tradeoff in Assortment Optimization Under the Multinomial Logit Model with Totally Unimodular Constraints

2020 ◽  
Author(s):  
Mika Sumida ◽  
Guillermo Gallego ◽  
Paat Rusmevichientong ◽  
Huseyin Topaloglu ◽  
James Davis
Keyword(s):  
Expected Utility ◽  
Logit Model ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Multinomial Logit Model ◽  
Multinomial Logit ◽  
Solution Quality ◽  
Assortment Optimization ◽  
Unimodular Matrix ◽  
Expected Revenue

We examine the revenue–utility assortment optimization problem with the goal of finding an assortment that maximizes a linear combination of the expected revenue of the firm and the expected utility of the customer. This criterion captures the trade-off between the firm-centric objective of maximizing the expected revenue and the customer-centric objective of maximizing the expected utility. The customers choose according to the multinomial logit model, and there is a constraint on the offered assortments characterized by a totally unimodular matrix. We show that we can solve the revenue–utility assortment optimization problem by finding the assortment that maximizes only the expected revenue after adjusting the revenue of each product by the same constant. Finding the appropriate revenue adjustment requires solving a nonconvex optimization problem. We give a parametric linear program to generate a collection of candidate assortments that is guaranteed to include an optimal solution to the revenue–utility assortment optimization problem. This collection of candidate assortments also allows us to construct an efficient frontier that shows the optimal expected revenue–utility pairs as we vary the weights in the objective function. Moreover, we develop an approximation scheme that limits the number of candidate assortments while ensuring a prespecified solution quality. Finally, we discuss practical assortment optimization problems that involve totally unimodular constraints. In our computational experiments, we demonstrate that we can obtain significant improvements in the expected utility without incurring a significant loss in the expected revenue. This paper was accepted by Omar Besbes, revenue management and market analytics.

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