Robust assortment optimization using worst-case CVaR under the multinomial logit model

We consider robust assortment optimization problems with partial distributional information of parameters in the multinomial logit choice model. The objective is to find an assortment that maximizes a revenue target using a distributionally robust chance constraint, which can be approximated by the worst-case Conditional Value-at-Risk. We show that our problems are equivalent to robust assortment optimization problems over special uncertainty sets of parameters, implying the optimality of revenue-ordered assortments under certain conditions.     

Price optimization under the extended nested logit model

We examine multi-product price optimization of the extended nested logit (ENL) model proposed by Kovach and Tserenjigmid, including the nested logit, increasing linear nested stochastic, and two-stage nested attraction models as special cases. The well-known constant adjusted markup and adjusted nest-level markup properties are extended to ENL. In addition, we present sufficient conditions under which the objective function is unimodal. We also provide the upper and lower bounds of the optimal adjusted nest-level markup when preconditions are not met.     

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

Motivated by applications in retail, online advertising, and cultural markets, this paper studies the problem of finding an optimal assortment and positioning of products subject to a capacity constraint in a setting where consumers preferences can be modeled as a discrete choice under a multinomial logit model that captures the intrinsic product appeal, position biases, and social influence. For the static problem, we prove that the optimal assortment and positioning can be found in polynomial time. This is despite the fact that adding a product to the assortment may increase the probability of selecting the no-choice option, a phenomenon not observed in almost all models studied in the literature. We then consider the dynamics of such a market, where consumers are influenced by the aggregate past purchases. In this dynamic setting, we provide a small example to show that the natural and often used policy known as popularity ranking, that ranks products in decreasing order of the number of purchases, can reduce the expected profit as times goes by. We then prove that a greedy policy that applies the static optimal assortment and positioning at each period, always benefits from the popularity signal and outperforms any policy where consumers cannot observe the number of past purchases (in expectation).     

A comparison of linear reformulations for multinomial logit choice probabilities in facility location models

In the last decade several papers appeared on facility location problems that incorporate customer demand by the multinomial logit model. Three linear reformulations of the original non-linear model have been proposed so far. In this paper, we discuss these models in terms of solvability. We present empirical findings based on synthetic data.     

Assortment optimization with repeated exposures and product-dependent patience cost

In this paper, we study the assortment optimization problem faced by many online retailers such as Amazon. We develop a cascade multinomial logit model, based on the classic multinomial logit model, to capture the consumers' purchasing behavior across multiple stages. Unlike most of existing studies, our model allows for repeated exposures of a product. In addition, each consumer has a patience budget that is sampled from a known distribution and each product is associated with a patience cost, which is the required amount of the cognitive efforts on browsing that product. We propose an approximation solution to the assortment optimization problem under cascade multinomial logit model.     

A greedy algorithm for the two-level nested logit model

We consider the assortment optimization problem under the classical two-level nested logit model. We establish a necessary and sufficient condition for the optimal assortment and develop a simple and fast greedy algorithm that iteratively removes at most one product from each nest to compute an optimal solution.     

A comparison of generalized multinomial logit and latent class approaches to studying consumer heterogeneity with some extensions of the generalized multinomial logit model

We calibrate and contrast the recent generalized multinomial logit model and the widely used latent class logit model approaches for studying heterogeneity in consumer purchases. We estimate the parameters of the models on panel data of household ketchup purchases, and find that the generalized multinomial logit model outperforms the best‐fitting latent class logit model in terms of the Bayesian information criterion. We compare the posterior estimates of coefficients for individual customers based on the two different models and discuss how the differences could affect marketing strategies (such as pricing), which could be affected by applying each of the models. We also describe extensions to the scale heterogeneity model that includes the effects of state dependence and purchase history. Copyright © 2011 John Wiley & Sons, Ltd.     

Product sequencing and pricing under cascade browse model

We study the assortment optimization problem under the newly proposed cascade browse model. We propose a constant approximate solution to this problem. As a byproduct, we propose the first fully polynomial-time approximation scheme (FPTAS) for the classic assortment optimization problem subject to one capacity constraint and one cardinality constraint. We also studied a joint pricing and sequencing problem under the above model and develop a constant approximate solution to this problem.     

Asymptotic laws for disparity statistics in product multinomial models

The paper presents asymptotic distributions of φ-disparity goodness-of-fit statistics in product multinomial models, under hypotheses and alternatives assuming sparse and nonsparse cell frequencies. The φ-disparity statistics include the power divergences of Read and Cressie (Goodness-of-fit Statistics for Discrete Multivariate Data, Springer, New York, 1988), the φ-divergences of Ciszár (Studia Sci. Math. Hungar. 2 (1967) 299) and the robust goodness of fit statistics of Lindsay (Ann. Statist. 22 (1994) 1081).     

Assortment planning for vertically differentiated products under a consider-then-choose model

Consider-then-choose models, borne out by empirical literature in marketing and psychology, note that customers follow a two-stage procedure to choose among alternatives. In this paper, we consider the assortment optimization problem of a retailer who manages a category of vertically differentiated products under customers’ consider-then-choose behavior. We characterize some structural results of the optimal assortment and find that the problem can be solved as the shortest path problem. Also, we develop an efficient algorithm to identify an optimal assortment.     

Estimation of failure probability using semi-definite logit model

We will propose a new and practical method for estimating the failure probability of a large number of small to medium scale companies using their balance sheet data. We will use the maximum likelihood method to estimate the best parameters of the logit function, where the failure intensity function in its exponent is represented as a convex quadratic function instead of a commonly used linear function. The reasons for using this type of function are : (i) it can better represent the observed nonlinear dependence of failure probability on financial attributes, (ii) the resulting likelihood function can be maximized using a cutting plane algorithm developed for nonlinear semi-definite programming problems.We will show that we can achieve better prediction performance than the standard logit model, using thousands of sample companies.Revised: December 2002,     

On the assortment optimization with endogenized market size

In this paper, we consider the joint effects of product substitution and market size endogenization. Under the substitution effects, a product’s demand may be cannibalized by other substitutable products; while the market size, measured by the number of customers who are interested in the products from the same category, may be largely influenced by the product offer set. We establish the computational complexity for the assortment problem under the joint effects, and develop a fully polynomial-time approximation scheme (FPTAS).     

基于Logit离散选择模型的品类优化问题综述

品类优化问题(Assortment Optimization Problem)是收益管理的经典问题.它研究零售商在满足运营约束的前提下,应如何从给定产品集合中选择一个子集提供给消费者,以最大化预期收益.该问题的核心在于如何准确地刻画消费者在面对细分产品时的选择行为、建立相应的优化模型并设计高效率的求解算法.基于Logit离散选择模型的品类优化问题:首先,介绍了基于Multinomial Logit模型的品类优化问题.然后介绍了两个更复杂的变种:第一个是基于两层以及多层Nested Logit模型的品类优化问题,这类问题可合理刻画细分产品之间的"替代效应";第二个是基于Mixtures of Multinomial Logits模型的品类优化问题,这类问题可充分考虑消费者群体的异质性.随后,介绍了数据驱动的品类优化问题的相关进展.最后,指出该问题未来可能的若干研究方向.     

A two step algorithm for solving a large scale semi-definite logit model

This paper proposes a two step algorithm for solving a large scale semi-definite logit model, which is appreciated as a powerful model in failure discriminant analysis. This problem has been successfully solved by a cutting plane (outer approximation) algorithm. However, it requires much more computation time than the corresponding linear logit model. A two step algorithm to be proposed in this paper is intended to reduce the amount of computation time by eliminating a certain portion of the data based on the information obtained by solving an associated linear logit model. It will be shown that this algorithm can generate a solution with almost the same quality as the solution obtained by solving the original large scale semi-definite model within a fraction of computation time.     

美国经济冲击对东亚经济波动的传导:基于面板logit模型的实证分析

美国经济冲击会通过具体渠道进行传导,影响东亚地区的经济波动.具体来看,与美国的双边进口贸易渠道和FDI渠道的传导会增加东亚地区经济波动概率,并且在控制国内经济规模后,边际效应进一步提高;与美国的双边出口贸易渠道和汇率渠道的传导反而会降低东亚地区经济波动的概率;并且传导效应在不同经济发展程度的国家呈现出明显的差异,发展中经济体受美国经济冲击的影响较大.     

Parameterized approximations for the two-sided assortment optimization

We consider the problem faced by an online service platform that matches suppliers with consumers. Unlike traditional matching models, which treat them as passive participants, we allow both sides of the market to exercise their choices. To model this setting, we introduce a two-sided assortment optimization model wherein each participant's choice is modeled using a multinomial logit choice function, and the platform's objective is to maximize its expected revenue. We first show that the problem is NP-hard even when the number of suppliers is limited to two and provide a mixed-integer linear programming formulation. Next, we discuss two simple greedy heuristics and argue that these can lead to arbitrarily bad solutions. We then develop relaxations that provide upper and lower bounds and investigate the tightness of these relaxations by obtaining parametric approximation guarantees. Finally, we present numerical results on synthetic data demonstrating the practical utility of these relaxations.     

Analysing head coach dismissals in the German “Bundesliga” with a mixed logit approach

This article analyzes the dismissal of head coaches in the top tier of German professional football, the “Bundesliga”. A random parameter logit model is used to analyse the characteristics that are associated with the probability of a coach to be dismissed, taking into account uncontrolled heterogeneity of the data. Policy implications are presented.