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Multiproduct Price Optimization Under The Multilevel Nested Logit Model
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Book Synopsis Multiproduct Price Optimization Under the Multilevel Nested Logit Model by : Hai Jiang
Download or read book Multiproduct Price Optimization Under the Multilevel Nested Logit Model written by Hai Jiang and published by . This book was released on 2014 with total page 34 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study the multiproduct price optimization problem under the multilevel nested logit model, which includes the multinomial logit and the two-level nested logit models as special cases. When the price sensitivities are identical within each primary nest, that is, within each nest at level 1, we prove that the profit function is concave with respect to the market share variables. We proceed to show that the markup, defined as price minus cost, is constant across products within each primary nest, and that the adjusted markup, defined as price minus cost minus the reciprocal of the product between the scale parameter of the root nest and the price-sensitivity parameter of the primary nest, is constant across primary nests at optimality. This allows us to reduce this multidimensional pricing problem to an equivalent single-variable maximization problem involving a unimodal function. Based on these findings, we investigate the oligopolistic game and characterize the Nash equilibrium. We also develop a dimension reduction technique which can simplify price optimization problems with flexible price-sensitivity structures.
Book Synopsis Multi-Product Price Optimization and Competition Under the Nested Logit Model with Product-Differentiated Price Sensitivities by : Guillermo Gallego
Download or read book Multi-Product Price Optimization and Competition Under the Nested Logit Model with Product-Differentiated Price Sensitivities written by Guillermo Gallego and published by . This book was released on 2014 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study firms that sell multiple substitutable products and customers whose purchase behavior follows a Nested Logit model, of which the Multinomial Logit model is a special case. Customers make purchasing decision sequentially under the Nested Logit model: they first select a nest of products and subsequently purchase one within the selected nest. We consider the multi-product pricing problem under the general Nested Logit model with product-differentiated price sensitivities and arbitrary nest coefficients. We show that the adjusted markup, defined as price minus cost minus the reciprocal of price sensitivity, is constant for all products within a nest at optimality. This reduces the problem's dimension to a single variable per nest. We also show that the adjusted nest-level markup is nest-invariant for all the nests, which further reduces the problem to maximizing a single-variable unimodal function under mild conditions. We also use this result to simplify the oligopolistic multi-product price competition and characterize the Nash equilibrium. We also consider more general attraction functions that include the linear utility and the multiplicative competitive interaction models as special cases, and show that similar techniques can be used to significantly simplify the corresponding pricing problems.
Book Synopsis Optimal Pricing of Correlated Product Options Under the Paired Combinatorial Logit Model by : Hongmin Li
Download or read book Optimal Pricing of Correlated Product Options Under the Paired Combinatorial Logit Model written by Hongmin Li and published by . This book was released on 2016 with total page 32 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we study price optimization with price-demand relationships captured by the paired combinatorial logit (PCL) model which overcomes restrictions of the well-studied multinomial logit (MNL) and nested logit (NL) models. The PCL model allows for choice-correlation and, like the NL model, includes the MNL model as a special case. Compared to the NL models, the PCL model does not restrict the sequence of the choice structure and allows for different covariances among all pairs of choices. This additional flexibility in structure enables a more accurate representation of some choice settings and broadens its empirical applications. Hence, it is of both theoretical and practical interests to extend the normative studies on the MNL and NL models to the PCL model and examine the pricing problem under this model. Due to cross-nesting of choice alternatives, the pricing problem under the PCL model poses a greater challenge than the MNL and NL models. However, using the concept of P-matrix, we are able to identify conditions for a unique optimal price solution and develop an efficient and theoretically sound approach for finding the optimal prices. We show that the analysis and solution approach are generalizable to other GEV family models with cross-nested alternatives.
Book Synopsis Assortment and Price Optimization Under an Endogenous Context-Dependent Multinomial Logit Model by : Yicheng Bai
Download or read book Assortment and Price Optimization Under an Endogenous Context-Dependent Multinomial Logit Model written by Yicheng Bai and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Motivated by empirical evidence that the utility of each product depends on the assortment of products offered along with it, we propose an endogenous context-dependent multinomial logit model (Context-MNL) under which the utility of each product depends on both the product's intrinsic value and the deviation of the intrinsic value from the expected maximum utility among all the products in the offered assortment. Under the Context-MNL model, an assortment provides a context in which customers evaluate the utility of each product. Our model generalizes the standard multinomial logit model and allows the utility of each product to depend on the offered assortment. The model is parsimonious, requires only one parameter more than the standard multinomial logit model, captures the assortment-dependent effect endogenously, and does~not require the decision-maker to determine in advance the relevant attributes of the assortment that might affect the product utility. The Context-MNL model also admits tractable maximum likelihood estimation and is operationally tractable, with efficient solution methods for solving assortment and price optimization problems. Our numerical study, which is based on data from Expedia, shows that compared to the standard multinomial logit model, the Context-MNL model substantially improves out-of-sample goodness of fit and prediction accuracy.
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 Price Optimization Under the Finite-Mixture Logit Model by : Ruben van de Geer
Download or read book Price Optimization Under the Finite-Mixture Logit Model written by Ruben van de Geer and published by . This book was released on 2019 with total page 51 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider price optimization under the finite-mixture logit model. This model assumes that customers belong to one of a number of customer segments, where each customer segment chooses according to a multinomial logit model with segment-specific parameters. We reformulate the corresponding price optimization problem and develop a novel characterization. Leveraging this new characterization, we construct an algorithm that obtains prices at which the revenue is guaranteed to be at least (1-epsilon) times the maximum attainable revenue for any prespecified epsilon>0. Existing global optimization methods require exponential time in the number of products to obtain such a result, which practically means that the prices of only a handful of products can be optimized. The running time of our algorithm, however, is exponential in the number of customer segments and only polynomial in the number of products. This is of great practical value, since in applications the number of products can be very large, while it is has been found in various contexts that a low number of segments is sufficient to capture customer heterogeneity appropriately. The results of our numerical study show that our algorithm runs fast on a broad range of problem instances.
Book Synopsis Assortment of Optimization and Pricing Problems Under Multi-stage Multinomial Logit Models by : Yuhang Ma
Download or read book Assortment of Optimization and Pricing Problems Under Multi-stage Multinomial Logit Models written by Yuhang Ma and published by . This book was released on 2019 with total page 163 pages. Available in PDF, EPUB and Kindle. Book excerpt: In most E-commerce scenarios such as hotel booking and online shopping, products are not offered to customers simultaneously. Instead, they are divided into different webpages and presented to customers sequentially. In this thesis, we focus on solving a common problem faced by online retailers: when products are revealed to customers sequentially, which products should the retailers display at each stage and what prices should the retailers charge for each product so that the expected revenue can be maximized? To solve those problems, we generalize the classical multinomial logit model to capture the customer's choice behavior under the sequential setting and present efficient algorithms for different generalized choice models and different operational constraints.
Book Synopsis Multi-Product Pricing Under the Multinomial Logit Model with Local Network Effects by : Mohan Gopalakrishnan
Download or read book Multi-Product Pricing Under the Multinomial Logit Model with Local Network Effects written by Mohan Gopalakrishnan and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Motivated by direct interactions with practitioners and real-world data, we study a monopoly firm selling multiple substitute products to customers characterized by their different social network degrees. Under the multinomial logit model framework, we assume that the utility a customer with a larger network degree derives from the seller's products is subject to more impact from her neighbors and describe the customers' choice behavior by a Bayesian Nash game. We show that a unique equilibrium exists as long as these network effects are not too large. Furthermore, we study how the seller should optimally set the prices of the products in this setting. Under the homogeneous product-related parameter assumption, we show that if the seller optimally price-discriminates all customers based on their network degrees, the products' markups are the same for each customer type. Building on this, we characterize the sufficient and necessary condition for the concavity of the pricing problem, and show that when the problem is not concave, we can convert it to a single-dimensional search and solve it efficiently. We provide several further insights about the structure of optimal prices, both theoretically and numerically. Furthermore, we show that we can simultaneously relax the multinomial logit model and homogeneous product-related parameter assumptions and allow customer in- and out-degrees to be arbitrarily distributed whilemaintaining most of our conclusions robust.
Book Synopsis An exact method for assortment optimization under the nested logit model by : Laurent Alfandari
Download or read book An exact method for assortment optimization under the nested logit model written by Laurent Alfandari and published by . This book was released on 2020 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Tractable Multi-product Pricing Under Discrete Choice Models by : Philipp Wilhelm Keller
Download or read book Tractable Multi-product Pricing Under Discrete Choice Models written by Philipp Wilhelm Keller and published by . This book was released on 2013 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider a retailer offering an assortment of differentiated substitutable products to price-sensitive customers. Prices are chosen to maximize profit, subject to inventory/ capacity constraints, as well as more general constraints. The profit is not even a quasi-concave function of the prices under the basic multinomial logit (MNL) demand model. Linear constraints can induce a non-convex feasible region. Nevertheless, we show how to efficiently solve the pricing problem under three important, more general families of demand models. Generalized attraction (GA) models broaden the range of nonlinear responses to changes in price. We propose a reformulation of the pricing problem over demands (instead of prices) which is convex. We show that the constrained problem under MNL models can be solved in a polynomial number of Newton iterations. In experiments, our reformulation is solved in seconds rather than days by commercial software. For nested-logit (NL) demand models, we show that the profit is concave in the demands (market shares) when all the price-sensitivity parameters are sufficiently close. The closed-form expressions for the Hessian of the profit that we derive can be used with general-purpose nonlinear solvers. For the special (unconstrained) case already considered in the literature, we devise an algorithm that requires no assumptions on the problem parameters. The class of generalized extreme value (GEV) models includes the NL as well as the cross-nested logit (CNL) model. There is generally no closed form expression for the profit in terms of the demands. We nevertheless how the gradient and Hessian can be computed for use with general-purpose solvers. We show that the objective of a transformed problem is nearly concave when all the price sensitivities are close. For the unconstrained case, we develop a simple and surprisingly efficient first-order method. Our experiments suggest that it always finds a global optimum, for any model parameters. We apply the method to mixed logit (MMNL) models, by showing that they can be approximated with CNL models. With an appropriate sequence of parameter scalings, we conjecture that the solution found is also globally optimal.
Book Synopsis An Exact Method for Assortment Optimization Under the Nested Logit Model by : Laurent Alfandari
Download or read book An Exact Method for Assortment Optimization Under the Nested Logit Model written by Laurent Alfandari and published by . This book was released on 2020 with total page 39 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study the problem of finding an optimal assortment of products maximizing the expected revenue, in which customer preferences are modeled using a Nested Logit choice model. This problem is known to be polynomially solvable in a specific case and NP-hard otherwise, with only approximation algorithms existing in the literature. For the NP-hard cases, we provide a general exact method that embeds a tailored Branch-and-Bound algorithm into a fractional programming framework. Contrary to the existing literature, in which assumptions are imposed on either the structure of nests or the combination and characteristics of products, no assumptions on the input data are imposed, and hence our approach can solve the most general problem setting. We show that the parameterized subproblem of the fractional programming scheme, which is a binary highly non-linear optimization problem, is decomposable by nests, which is a main advantage of the approach. To solve the subproblem for each nest, we propose a two-stage approach. In the first stage, we identify those products that are undoubtedly beneficial to offer, or not, which can significantly reduce the problem size. In the second stage, we design a tailored Branch-and-Bound algorithm with problem-specific upper bounds. Numerical results show that the approach is able to solve assortment instances with up to 5,000 products per nest. The most challenging instances for our approach are those in which the dissimilarity parameters of nests can be either less or greater than one.
Book Synopsis Assortment and Price Optimization Under MNL Model with Price Range Effect by : Stefanus Jasin
Download or read book Assortment and Price Optimization Under MNL Model with Price Range Effect written by Stefanus Jasin and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we study the assortment and price optimization problems under Multinomial Logit (MNL) model with price range effect, where the utility of a product is affected by the relative position of its price with respect to the highest and the lowest prices in the offer set. This model is motivated by the so-called Range Theory popularized in the behavioral economics and psychology literature. It addresses the limitation of a single-point interpretation of reference price, which ignores the impact of all other distributional information. We investigate the pure assortment problem, the pure pricing problem, and the joint assortment and pricing problem under the MNL model with price range effect. For each model, we first identify the structure of the optimal policy, and then we propose tractable algorithms that either output the optimal solution in polynomial time or admit an Fully Polynomial-Time Approximation Scheme (FPTAS).
Book Synopsis Multi-Objective Assortment Optimization by : Zhen Chen
Download or read book Multi-Objective Assortment Optimization written by Zhen Chen and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Assortment optimization is a fundamental problem in revenue management, in which the objective usually is to select a subset of products to offer to customers in order to maximize expected revenue or profit. However, business practices often involve multiple, and potentially conflicting goals. In this work, we propose a general framework and a novel reformulation method for solving multi-objective assortment optimization problems. Specifically, we consider problems with a separable sum of multiple convex objective functions on linear combinations of choice probabilities, and we present a reformulation that effectively "linearizes" the problem. We prove that the reformulated problem is equivalent to the original problem and that it leads to a unified solution approach to multi-objective assortment optimization problems in various contexts. We show that the approach encompasses a wide range of operational objectives, such as risk, customer utility, market share, costs with economies of scale, and dualized convex constraints. We first illustrate our approach with the multinomial logit model without any constraints or with allowance for totally unimodular constraints. We further show that our framework leads to tractable solutions under the nested logit model and the Markov chain choice model. Together with large-scale numerical experiments to demonstrate the efficiency and practicality of our methods, we highlight that our work provides a powerful and flexible tool for solving multi-objective assortment problems, which arise frequently in practical revenue management settings.
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 Price Optimization for a Multi-Stage Choice Model by : Jiaqi Shi
Download or read book Price Optimization for a Multi-Stage Choice Model written by Jiaqi Shi and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Considering the real-world situations where past purchases could influence future prices, this research examines the multi-product price optimization problem under a multi-stage choice model. Particularly, the seller commits to a multi-stage pricing policy and determines product prices based on the customer's purchase history, and the customer makes purchase decisions such that the expected utility is maximized. We show that the pricing problem has a unique optimal solution under some mild conditions and the optimal solution satisfies a modified equal adjusted markup property. Based on the property, the problem can be solved efficiently by reducing it to a single-dimensional search problem. Moreover, the optimal pricing policy has an important property, namely, the product with a higher adjusted markup in earlier stages should always lead to lower prices in subsequent stages. We also show that compared to customers that are myopic, the seller should offer higher first-stage prices and lower second-stage prices to forward-looking customers, which will lead to a higher profit. Numerical analyses are also conducted to demonstrate the above results.
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 Multi-Product Dynamic Pricing with Reference Effects Under Logit Demand by : Mengzi Amy Guo
Download or read book Multi-Product Dynamic Pricing with Reference Effects Under Logit Demand written by Mengzi Amy Guo and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider an infinite-horizon multi-product dynamic pricing problem with reference effects in a monopolistic setting, where the reference price is an exponentially weighted average of historical prices. In each period, the demand follows the multinomial logit (MNL) model, where the utility depends on both the current price and the reference price, and consumers can have product-differentiated sensitivities to the price and the reference price. We conduct thorough analyses of the myopic pricing policy, including its solution, long-run convergence behavior, and performance guarantee compared to the long-term discounted revenue of the optimal pricing policy. Furthermore, we establish the structural properties of the optimal pricing policy. When consumers are loss-neutral towards all products, we explicitly characterize the optimal pricing policy if it converges to a steady state, and based on our characterization we show that the steady state price can be computed efficiently by a binary search. Interestingly, we find that such a convergence behavior of the optimal pricing policy heavily relies on the upper bound of the admissible price range, and a low price upper bound facilitates the policy to converge. In contrast, when consumers are gain-seeking towards all products, we prove that the optimal pricing policy admits no steady state regardless of the price range. Nevertheless, if consumers are only gain-seeking towards certain but not all products, the optimal pricing policy can potentially be convergent. In addition, our numerical experiments show that loss-aversion over all products does not rule out price fluctuations. This finding is at odds with the classic belief on loss-averse consumers and hence, highlights the significance of accounting for cross-product effects through the MNL demand.
