scholarly journals A regret lower bound for assortment optimization under the capacitated MNL model with arbitrary revenue parameters

2021 ◽  
pp. 1-9
Author(s):  
Yannik Peeters ◽  
Arnoud V. den Boer
Keyword(s):  
Lower Bound ◽  
Incomplete Information ◽  
Positive Constant ◽  
Time Horizon ◽  
Choice Model ◽  
Multinomial Logit ◽  
Asymptotically Optimal ◽  
Assortment Optimization ◽  
Mnl Model ◽  
Expected Revenue

Abstract In this note, we consider dynamic assortment optimization with incomplete information under the capacitated multinomial logit choice model. Recently, it has been shown that the regret (the cumulative expected revenue loss caused by offering suboptimal assortments) that any decision policy endures is bounded from below by a constant times $\sqrt {NT}$ , where $N$ denotes the number of products and $T$ denotes the time horizon. This result is shown under the assumption that the product revenues are constant, and thus leaves the question open whether a lower regret rate can be achieved for nonconstant revenue parameters. In this note, we show that this is not the case: we show that, for any vector of product revenues there is a positive constant such that the regret of any policy is bounded from below by this constant times $\sqrt {N T}$ . Our result implies that policies that achieve ${{\mathcal {O}}}(\sqrt {NT})$ regret are asymptotically optimal for all product revenue parameters.

Assortment Optimization under the Multi-Purchase Multinomial Logit Choice Model

2021 ◽  
Author(s):  
Jacob Feldman ◽  
Danny Segev ◽  
Huseyin Topaloglu ◽  
Laura Wagner ◽  
Yicheng Bai
Keyword(s):  
Choice Model ◽  
Multinomial Logit ◽  
Assortment Optimization

Optimal Policy for Dynamic Assortment Planning Under Multinomial Logit Models

2021 ◽  
Author(s):  
Xi Chen ◽  
Yining Wang ◽  
Yuan Zhou
Keyword(s):  
Choice Model ◽  
Current Knowledge ◽  
Search Algorithm ◽  
Multinomial Logit ◽  
Planning Problem ◽  
Assortment Planning ◽  
Substitutable Products ◽  
Dynamic Decisions ◽  
Mnl Model ◽  
Regret Bound

We study the dynamic assortment planning problem, where for each arriving customer, the seller offers an assortment of substitutable products and the customer makes the purchase among offered products according to an uncapacitated multinomial logit (MNL) model. Because all the utility parameters of the MNL model are unknown, the seller needs to simultaneously learn customers’ choice behavior and make dynamic decisions on assortments based on the current knowledge. The goal of the seller is to maximize the expected revenue, or, equivalently, to minimize the expected regret. Although dynamic assortment planning problem has received an increasing attention in revenue management, most existing policies require the estimation of mean utility for each product and the final regret usually involves the number of products [Formula: see text]. The optimal regret of the dynamic assortment planning problem under the most basic and popular choice model—the MNL model—is still open. By carefully analyzing a revenue potential function, we develop a trisection-based policy combined with adaptive confidence bound construction, which achieves an item-independent regret bound of [Formula: see text], where [Formula: see text] is the length of selling horizon. We further establish the matching lower bound result to show the optimality of our policy. There are two major advantages of the proposed policy. First, the regret of all our policies has no dependence on [Formula: see text]. Second, our policies are almost assumption-free: there is no assumption on mean utility nor any “separability” condition on the expected revenues for different assortments. We also extend our trisection search algorithm to capacitated MNL models and obtain the optimal regret [Formula: see text] (up to logrithmic factors) without any assumption on the mean utility parameters of items.

scholarly journals Multi-route choice modelling in a metropolitan context: A comparative analysis using Multinomial Logit and Fuzzy Logic based approaches

2020 ◽  
Vol 79 (ET.2020) ◽  
pp. 1-17
Author(s):  
Sowjanya Dhulipala
Keyword(s):  
Decision Making ◽  
Fuzzy Logic ◽  
Choice Model ◽  
Route Choice ◽  
Fuzzy Rule ◽  
Multinomial Logit ◽  
Choice Modelling ◽  
Choice Behaviour ◽  
Linguistic Rules ◽  
Mnl Model

Route choice plays a vital role in the traffic assignment and network building, as it involves decision making on part of riders. The vagueness in travellers’ perceptions of attributes of the available routes between any two locations adds to the complexities in modelling the route choice behaviour. Conventional Logit models fail to address the uncertainty in travellers’ perceptions of route characteristics (especially qualitative attributes, such as environmental effects), which can be better addressed through the theory of fuzzy sets and linguistic variables. This study thus attempts to model travellers’ route choice behaviour, using a fuzzy logic approach that is based on simple and logical ‘if-then’ linguistic rules. This approach takes into consideration the uncertainty in travellers’ perceptions of route characteristics, resembling humans’ decision-making process. Three attributes – travel time, traffic congestion, and road-side environment are adopted as factors driving people’s choice of routes, and three alternative routes between two typical locations in an Indian metropolitan city, Surat, are considered in the study. The approach to deal with multiple routes is shown by analyzing two-wheeler riders’ (e.g. motorcyclists’ and scooter drivers’) route choice behaviour during the peak-traffic time. Further, a Multinomial Logit (MNL) model is estimated, to enable a comparison of the two modelling approaches. The estimated Fuzzy Rule-Based Route Choice Model outperformed the conventional MNL model, accounting for the uncertain behaviour of travellers.

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.

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.

Dynamic Assortment Optimization with a Multinomial Logit Choice Model and Capacity Constraint

2010 ◽  
Vol 58 (6) ◽  
pp. 1666-1680 ◽  
Author(s):  
Paat Rusmevichientong ◽  
Zuo-Jun Max Shen ◽  
David B. Shmoys
Keyword(s):  
Choice Model ◽  
Multinomial Logit ◽  
Capacity Constraint ◽  
Assortment Optimization

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.

scholarly journals Multinomial Logit Bandit with Linear Utility Functions

2018 ◽  
Author(s):  
Mingdong Ou ◽  
Nan Li ◽  
Shenghuo Zhu ◽  
Rong Jin
Keyword(s):  
Choice Model ◽  
Multinomial Logit ◽  
Linear Functions ◽  
Underlying Structure ◽  
Set Size ◽  
Linear Utility ◽  
Mnl Model ◽  
Regret Bound ◽  
Exploration Exploitation ◽  
Candidate Set

Multinomial logit bandit is a sequential subset selection problem which arises in many applications. In each round, the player selects a K-cardinality subset from N candidate items, and receives a reward which is governed by a multinomial logit (MNL) choice model considering both item utility and substitution property among items. The player's objective is to dynamically learn the parameters of MNL model and maximize cumulative reward over a finite horizon T. This problem faces the exploration-exploitation dilemma, and the involved combinatorial nature makes it non-trivial. In recent years, there have developed some algorithms by exploiting specific characteristics of the MNL model, but all of them estimate the parameters of MNL model separately and incur a regret bound which is not preferred for large candidate set size N. In this paper, we consider the linear utility MNL choice model whose item utilities are represented as linear functions of d-dimension item features, and propose an algorithm, titled LUMB, to exploit the underlying structure. It is proven that the proposed algorithm achieves regret which is free of candidate set size. Experiments show the superiority of the proposed algorithm.

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