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Assortment Optimization And Pricing Under The Threshold Based Choice Models
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Book Synopsis Assortment Optimization and Pricing Under the Threshold-Based Choice Models by : Xu Tian
Download or read book Assortment Optimization and Pricing Under the Threshold-Based Choice Models written by Xu Tian and published by . This book was released on 2020 with total page 42 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we study revenue maximization assortment and pricing problems under threshold-based choice models, where a product is placed in a consumer's consideration set if its utility to the consumer exceeds the utility of an outside option by a specified threshold. We discuss two such models: the relative utility and absolute utility threshold-based choice models. For both models, the best revenue-ordered assortment and same-price policy can not achieve the optimal profit for the assortment problem or the pricing problem. Further, the revenue-maximizing assortment problem is NP-complete or NP-hard. However, we show that a performance guarantee relative to the optimal policy can be found for each model: for the relative utility model, by employing the best revenue-ordered assortment and same-price policy; for the absolute utility model, via a dynamic-program-based algorithm and a same-price policy. Finally, we show that our algorithms can be asymptotically optimal if the search cost of consumers is sufficiently small.
Book Synopsis Modeling Consumer Choice and Optimizing Assortment Under the Threshold Multinomial Logit Model by : Ruxian Wang
Download or read book Modeling Consumer Choice and Optimizing Assortment Under the Threshold Multinomial Logit Model written by Ruxian Wang and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper incorporates heterogeneous threshold effects into the classical multinomial logit (MNL) model, and studies the associated operations problems such as estimation and assortment optimization. The derived model is referred to as the threshold multinomial logit (TMNL) model and incorporates the recently proposed threshold Luce (T-Luce) model as a limiting case. Under the TMNL model, consumers first form their (heterogeneous) consideration set: If an alternative with significantly low utility is dominated by another one, it will not be included in the consideration set. The TMNL model can alleviate the restricted substitution patterns of MNL due to the independence of irrelevant alternatives (IIA) property, and therefore can model more flexible choice behavior. We develop a maximum likelihood based estimation to calibrate the proposed threshold model and further establish its statistical properties such as consistency and asymptotic normality under mild conditions. An efficient EM algorithm is also developed to handle the scenario with incomplete sales data. Our extensive numerical studies on synthetic and real datasets show that the new model can improve the goodness of fit and prediction accuracy of consumer choice behavior. In addition, we characterize the optimal strategies and provide efficient solutions for the associated assortment optimization problems under the TMNL model. Our theoretical and empirical results suggest that the threshold effects should be taken into account in firms' decision making such as demand estimation and operations management, and ignoring these effects could lead to sub-optimal solutions or even substantial losses for firms.
Book Synopsis Threshold Utility Model with Applications to Retailing and Discrete Choice Models by : Guillermo Gallego
Download or read book Threshold Utility Model with Applications to Retailing and Discrete Choice Models written by Guillermo Gallego and published by . This book was released on 2020 with total page 49 pages. Available in PDF, EPUB and Kindle. Book excerpt: We propose and study a threshold utility model (TUM) where consumers buy any product whose net utility exceeds a non-negative, product-specific threshold. The thresholds are selected to maximize the expected surplus of the representative consumer subject to a bound on the expected number of selected products. We show that at optimality the thresholds are product-invariant and that the generalized extreme value (GEV) model is a special case of the TUM. The TUM is shown to yield higher consumer surplus than observing all the products' utilities and selecting the best when the bound is an integer. The model can also be applied with proxy utilities as in on-line shopping, and portfolio management. Comparative statics are applied to the threshold, the purchase probabilities and the expected surplus. Extensions to multi-unit TUM, weighted TUM, multiplicative TUM, discontinuous utility and bound-induced copulas are also considered. We also provide solutions to pricing and assortment optimization problems under the TUM.
Book Synopsis Assortment Optimization Under the Multinomial Logit Model with Utility-Based Rank Cutoffs by : Jacob Feldman
Download or read book Assortment Optimization Under the Multinomial Logit Model with Utility-Based Rank Cutoffs written by Jacob Feldman and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study assortment optimization problems under a natural variant of the multinomial logit model where the customers are willing to focus only on a certain number of products that provide the largest utilities. In particular, each customer has a rank cutoff, characterizing the number of products that she will focus on during the course of her choice process. Given that we offer a certain assortment of products, the choice process of a customer with rank cutoff k proceeds as follows. The customer associates random utilities with all of the products as well as the no-purchase option. She ignores all alternatives whose utilities are not within the k largest utilities. Among the remaining alternatives, the customer chooses the available alternative that provides the largest utility. Under the assumption that the~utilities follow Gumbel distributions with the same scale parameter, we provide a recursion to compute the choice probabilities. Considering the assortment optimization problem to find the revenue-maximizing assortment of products to offer, we show that the problem is NP-hard and give a polynomial-time approximation scheme. Since the customers ignore the products below their rank cutoffs in our variant of the multinomial logit model, intuitively speaking, our variant captures choosier choice behavior than the standard multinomial logit model. Accordingly, we show that the revenue-maximizing assortment under our variant includes the revenue-maximizing assortment under the standard multinomial logit model, so choosier behavior leads to larger assortments offered to maximize the expected revenue. We conduct computational experiments on both synthetic and real datasets to demonstrate that incorporating rank cutoffs can yield better predictions of customer choices and yield more profitable assortment recommendations.
Book Synopsis The Focal Multinomial Logit Model by : Lei Guan
Download or read book The Focal Multinomial Logit Model written by Lei Guan and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: {Problem Definition:} This paper considers the operational management problems under a newly proposed choice model that captures the effect of focality. The offered assortment is separated into the focal set and the non-focal set under this new model due to the bias of focality, which is identified by the focal sets and an assortment-dependent focal parameter. A prospective consumer is more likely to choose a product from the focal set, while she may still choose one from the non-focal set for a variety of reasons such as previous purchase experience or brand loyalty. This focal multinomial logit model generalizes the famous multinomial logit model and several well-studied consideration-set choice models. In addition, it has the capability to describe and explain a variety of irrational choice behaviors often observed in practice, such as the context effect, halo effect, and choice overload. {Methodology/results:} In this paper, we primarily focus on the threshold focal set and various focal parameter settings, including the constant, cardinality, and linear focal multinomial logit models, as well as a broader model that satisfies certain regularity conditions and subsumes the above models. We analyze the computational complexity and propose polynomial-time exact or approximation algorithms to solve the assortment optimization problems under different focal parameters. We then characterize the optimal strategy for the joint price and assortment optimization problem. Additionally, we develop a mixed integer conic programming reformulation method that converges to a global optimal estimator for the model calibration problem. {Managerial Implications:} We use these methods to conduct numerical experiments on both synthetic and real data sets. The results demonstrate the efficiency of our proposed algorithms, the predictive power, and the increase in revenue for the focal multinomial logit model. Our extensive analysis implies that in practice retailers may take into account the effect of focality in consumer purchase behavior because it could increase the accuracy of demand estimation and therefore improve operational performance.
Book Synopsis Assortment and Inventory Optimization by : Mohammed Ali Aouad
Download or read book Assortment and Inventory Optimization written by Mohammed Ali Aouad and published by . This book was released on 2017 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: Finding optimal product offerings is a fundamental operational issue in modern retailing, exemplified by the development of recommendation systems and decision support tools. The challenge is that designing an accurate predictive choice model generally comes at the detriment of efficient algorithms, which can prescribe near-optimal decisions. This thesis attempts to resolve this disconnect in the context of assortment and inventory optimization, through theoretical and empirical investigation. First, we tightly characterize the complexity of general nonparametric assortment optimization problems. We reveal connections to maximum independent set and combinatorial pricing problems, allowing to derive strong inapproximability bounds. We devise simple algorithms that achieve essentially best-possible factors with respect to the price ratio, size of customers' consideration sets, etc. Second, we develop a novel tractable approach to choice modeling, in the vein of nonparametric models, by leveraging documented assumptions on the customers' consider-then-choose behavior. We show that the assortment optimization problem can be cast as a dynamic program, that exploits the properties of a bi-partite graph representation to perform a state space collapse. Surprisingly, this exact algorithm is provably and practically efficient under common consider-then-choose assumptions. On the estimation front, we show that a critical step of standard nonparametric estimation methods (rank aggregation) can be solved in polynomial time in settings of interest, contrary to general nonparametric models. Predictive experiments on a large purchase panel dataset show significant improvements against common benchmarks. Third, we turn our attention to joint assortment optimization and inventory management problems under dynamic customer choice substitution. Prior to our work, little was known about these optimization models, which are intractable using modern discrete optimization solvers. Using probabilistic analysis, we unravel hidden structural properties, such as weak notions of submodularity. Building on these findings, we develop efficient and yet conceptually-simple approximation algorithms for common parametric and nonparametric choice models. Among notable results, we provide best-possible approximations under general nonparametric choice models (up to lower-order terms), and develop the first constant-factor approximation under the popular Multinomial Logit model. In synthetic experiments vis-a-vis existing heuristics, our approach is an order of magnitude faster in several cases and increases revenue by 6% to 16%.
Book Synopsis Assortment Optimization Under Consider-Then-Choose Choice Models by : Ali Aouad
Download or read book Assortment Optimization Under Consider-Then-Choose Choice Models written by Ali Aouad and published by . This book was released on 2020 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Consider-then-choose models, borne out by empirical literature in marketing and psychology, explain that customers choose among alternatives in two phases, by first screening products to decide which alternatives to consider, before then ranking them. In this paper, we develop a dynamic programming framework to study the computational aspects of assortment optimization models posited on consider-then-choose premises. Although ranking-based choice models generally lead to computationally intractable assortment optimization problems, we are able to show that for many practical and empirically vetted assumptions on how customers consider and choose, the resulting dynamic program is efficient. Our approach unifies and subsumes several specialized settings analyzed in previous literature. Empirically, we demonstrate the versatility and predictive power of our modeling approach on a combination of synthetic and real industry datasets, where prediction errors are significantly reduced against common parametric choice models. In synthetic experiments, our algorithms lead to practical computation schemes that outperform a state-of-the-art integer programming solver in terms of running time, in several parameter regimes of interest.
Book Synopsis An Algorithm for Assortment Optimization Under Parametric Discrete Choice Models by : Tien Mai
Download or read book An Algorithm for Assortment Optimization Under Parametric Discrete Choice Models written by Tien Mai and published by . This book was released on 2019 with total page 37 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work concerns the assortment optimization problem that refers to selecting a subset of items that maximizes the expected revenue in the presence of the substitution behavior of consumers specified by a parametric choice model. The key challenge lies in the computational difficulty of finding the best subset solution, which often requires exhaustive search. The literature on constrained assortment optimization lacks a practically efficient method which that is general to deal with different types of parametric choice models (e.g., the multinomial logit, mixed logit or general multivariate extreme value models). In this paper, we propose a new approach that allows to address this issue. The idea is that, under a general parametric choice model, we formulate the problem into a binary nonlinear programming model, and use an iterative algorithm to find a binary solution. At each iteration, we propose a way to approximate the objective (expected revenue) by a linear function, and a polynomial-time algorithm to find a candidate solution using this approximate function. We also develop a greedy local search algorithm to further improve the solutions. We test our algorithm on instances of different sizes under various parametric choice model structures and show that our algorithm dominates existing exact and heuristic approaches in the literature, in terms of solution quality and computing cost.
Book Synopsis Customer Choice Models and Assortment Optimization by : James Mario Davis
Download or read book Customer Choice Models and Assortment Optimization written by James Mario Davis and published by . This book was released on 2015 with total page 424 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis handles a fundamental problem in retail: given an enormous variety of products which does the retailer display to its customers? This is the assortment planning problem. We solve this problem by developing algorithms that, given input parameters for products, can efficiently return the set of products that should be displayed. To develop these algorithms we use a mathematical model of how customers react to displayed items, a customer choice model. Below we consider two classic customer choice models, the Multinomial Logit model and Nested Logit model. Under each of these customer choice models we develop algorithms that solve the assortment planning problem. Additionally, we consider the constrained assortment planning problem where the retailer must display products to customers but must also satisfy operational constraints.
Book Synopsis When Prospect Theory Meets Consumer Choice Models by : Ruxian Wang
Download or read book When Prospect Theory Meets Consumer Choice Models written by Ruxian Wang and published by . This book was released on 2018 with total page 41 pages. Available in PDF, EPUB and Kindle. Book excerpt: Problem Definition: Reference prices arise as price expectations against which consumers evaluate products in their purchase scenarios. We investigate what will happen when prospect theory (e.g., reference prices) meets consumer choice models from the perspectives of both the consumers and the firm.Academic/Practical Relevance: Consumers see multiple relevant products on a particular purchase occasion, and often compare their prices to form the willingness to pay when considering whether to purchase a particular product. Reference prices, which are not included in many choice models, may impact consumer choice behavior, so we incorporate reference prices into consumer choice models and investigate the operations management problems.Methodology: We take the widely used multi-nomial logit model as a showcase to examine the effects of reference prices through analytical and empirical study. We consider the optimization problems on assortment planning and pricing under consumer choice models with a variety of reference prices, including the lowest price and the assortment variety.Results: Our empirical study on a real data set demonstrates that incorporating reference prices into choice models can significantly improve goodness-of-fit and prediction accuracy of consumer choice behavior. Furthermore, we characterize the optimal policies for the assortment planning and pricing problems under the consumer choice models with various reference prices. In particular for the pricing problems under the reference prices defined by either the lowest price or assortment variety, we show that the optimal pricing policy has the following structure: products can be grouped into several categories based on their costs; the products in the same category charge either the same profit markup or the same price.Managerial Implications: In practice, reference prices should be taken into account in model estimation and operations management. Ignoring reference prices may lead to substantial losses.
Book Synopsis Capacitated Assortment and Price Optimization Under the Multinomial Logit Model by : Ruxian Wang
Download or read book Capacitated Assortment and Price Optimization Under the Multinomial Logit Model written by Ruxian Wang and published by . This book was released on 2014 with total page 7 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider an assortment and price optimization problem where a retailer chooses an assortment of competing products and determines their prices to maximize the total expected profit subject to a capacity constraint. Customers' purchase behavior follows the multinomial logit choice model with general utility functions. This paper simplifies it to a problem of finding a unique fixed point of a single-dimensional function and visualizes the assortment optimization process. An efficient algorithm to find the optimal assortment and prices is provided.
Book Synopsis The Exponomial Choice Model by : Aydin Alptekinoglu
Download or read book The Exponomial Choice Model written by Aydin Alptekinoglu and published by . This book was released on 2015 with total page 50 pages. Available in PDF, EPUB and Kindle. Book excerpt: We investigate the use of a canonical version of a discrete choice model due to Daganzo (1979) in optimal pricing and assortment planning. In contrast to multinomial and nested logit (the prevailing choice models used for optimizing prices and assortments), this model assumes a negatively skewed distribution of consumer utilities, an assumption we motivate by conceptual arguments as well as published work. The choice probabilities in this model can be derived in closed-form as an exponomial (a linear function of exponential terms). The pricing and assortment planning insights we obtain from the Exponomial Choice (EC) model differ from the literature in two important ways. First, the EC model allows variable markups in optimal prices that increase with expected utilities. Second, when prices are exogenous, the optimal assortment may exhibit leapfrogging in prices, i.e., a product can be skipped in favor of a lower-priced one depending on the utility positions of neighboring products. These two plausible pricing and assortment patterns are ruled out by multinomial logit (and by nested logit within each nest). We provide structural results on optimal pricing for monopoly and oligopoly cases, and on the optimal assortments for both exogenous and endogenous prices. We also demonstrate how the EC model can be easily estimated--by establishing that the loglikelihood function is concave in model parameters and detailing an estimation example using real data.
Book Synopsis Branch-and-Bound Algorithms for Assortment Optimization Under Weakly Rational Choice by : Clark Pixton
Download or read book Branch-and-Bound Algorithms for Assortment Optimization Under Weakly Rational Choice written by Clark Pixton and published by . This book was released on 2016 with total page 29 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study the static assortment optimization problem under weakly rational choice models, i.e. models in which adding a product to an assortment does not increase the probability of purchasing a product already in that assortment. This setting applies to most choice models studied and used in practice, such as the multinomial logit and random parameters logit models. We give a mixed-integer linear optimization formulation with an exponential number of constraints, and present two branch-and-bound algorithms for solving this optimization problem. The formulation and algorithms require only black-box access to purchase probabilities, and thus provide exact solution methods for a general class of discrete choice models, in particular those models without closed-form choice probabilities. We show that one of our algorithms is a PTAS for assortment optimization under weakly rational choice when the no-purchase probability is small, and give an approximation guarantee for the other algorithm which depends only on the prices of the products. Finally, we test the performance of our algorithms with heuristic stopping criteria, motivated by the fact that they discover the optimal solution very quickly.
Book Synopsis Capacity, Pricing and Assortment Management Under Discrete Choice Model with Anticipated Wait by : Ruxian Wang
Download or read book Capacity, Pricing and Assortment Management Under Discrete Choice Model with Anticipated Wait written by Ruxian Wang and published by . This book was released on 2020 with total page 40 pages. Available in PDF, EPUB and Kindle. Book excerpt: Customers often face multiple choices when purchasing a product or service. After making a choice, they sometimes have to wait for a while before receiving their purchased item due to the firm's limited capacityto process orders. This paper incorporates the anticipated wait for receiving purchased products or services into customers' choice behavior. The resulting choice model shares the same spirit of the rational expectation equilibrium, and captures the effects of negative externality caused by the anticipated wait, because all orders may be processed by a common facility. Our analysis shows that the anticipated wait may change the substitution patterns dramatically. We further investigate the effects of the anticipated wait on the decisions of capacity investment, product pricing and assortment planning. We establish the one-to-one mapping between the price vector and the choice probability vector, and show that the equivalent profit function of the choice probabilities is explicitly defined and more tractable. We characterize the multi-product price optimization problem under the MNL model with waiting. In addition to price competition, we also study the Cournot competition, in which the decision is the choice probability for each firm, and show that there exists a Nash equilibrium. For the assortment optimization, we identify the conditions under which the optimality of the revenue-ordered assortment still holds. Because the assortment problem with waiting is generally NP-hard, we develop efficient approximations with performance guarantee and also provide an easy-to-compute tight upper bound. The new model has the potential to increase prediction accuracy for customers' choice behavior especially when customers faced with multiple choices are aware of the possible waiting for their purchased products. Failure to take into account the effects of the anticipated wait in customers' purchase behavior may result in substantial losses to firms.
Book Synopsis Assortment Optimisation Under a General Discrete Choice Model by : Gerardo Berbeglia
Download or read book Assortment Optimisation Under a General Discrete Choice Model written by Gerardo Berbeglia and published by . This book was released on 2015 with total page 38 pages. Available in PDF, EPUB and Kindle. Book excerpt: A central problem in revenue management, known as the assortment problem, consists in deciding which subset of products to offer to consumers in order to maximise revenue. A simple and natural strategy is to select the best assortment out of all those that are constructed by fixing a threshold revenue r and then choosing all products with revenue at least r. This is known as the revenue-ordered assortments strategy. The main contribution of this paper is a precise analysis of how well revenue-ordered assortments approximate the optimum revenue when customers are rational in the following sense: the probability of selecting a specific product (including the no-choice option) from the set being offered cannot increase if the offer set is enlarged. The corresponding discrete choice models form a broad class of models which includes all discrete choice models based on random utility. Our analysis of revenue-ordered assortments match and unify known results for certain models, and improves the best known results for others, such as for the Mixed Multinomial Logit model recently studied by Rusmevichientong et al (2014). An appealing feature of our analysis is that it is simple and relies only on the above-mentioned rationality property, and yet it is best possible even for very specific models within the class. We also show that a large class of problems known as envy-free pricing problems can be seen as assortment problems for a specifically constructed discrete choice model that satisfies the rationality property. In this context, revenue-ordered assortments turn out to be equivalent to the well-studied uniform pricing strategy.
Book Synopsis Consumer Choice with Consideration Set by : Ruxian Wang
Download or read book Consumer Choice with Consideration Set written by Ruxian Wang and published by . This book was released on 2019 with total page 54 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper investigates the threshold Luce model, a recently proposed choice model with a threshold for the consideration-set formation. Under the threshold Luce model, consumers first form their consideration set: If an alternative with significantly low utility is dominated by another one, it will not be included in the consideration set. The threshold Luce model can alleviate the independence of irrelevant alternatives (IIA) property and allow more flexible substitution patterns. We characterize the optimal strategy and develop efficient solutions for price and assortment optimization problems. Under the threshold Luce model, the price competition may have zero, one, two, or infinite Nash equilibria, depending on the magnitude of the threshold effect. Moreover, we also develop an efficient estimation method to calibrate the threshold Luce model. Our numerical study on synthetic and real data sets shows that the new model can improve the goodness of fit and prediction accuracy of consumer choice behavior, which suggests the threshold effect should be taken into account in decision making.
Book Synopsis Thompson Sampling for Online Personalized Assortment Optimization Problems with Multinomial Logit Choice Models by : Wang Chi Cheung
Download or read book Thompson Sampling for Online Personalized Assortment Optimization Problems with Multinomial Logit Choice Models written by Wang Chi Cheung and published by . This book was released on 2017 with total page 37 pages. Available in PDF, EPUB and Kindle. Book excerpt: Motivated by online retail applications, we study the online personalized assortment optimization problem. A seller conducts sales by offering assortments of products to a stream of arriving customers. The customers' purchase behavior follows their respective personalized Multinomial Logit choice models, which vary according to their individual attributes. The seller aims to maximize his revenue by offering personalized assortments to the customers, notwithstanding his uncertainty about the customers' choice models. We propose a Thompson Sampling based policy, policy Pao-Ts, where surrogate models for the latent choice models are constructed using samples from a progressively updated posterior distribution. We derive bounds on the revenue loss, namely Bayesian regret, incurred by policy Pao-Ts, in comparison to the optimal policy which is provided with the latent models. The regret bounds hold even when the customers' attributes vary arbitrarily, but not independently and identically distributed.
