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Assortment Optimization Under The Multivariate Mnl Model
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Book Synopsis Assortment Optimization Under the Multivariate MNL Model by : Xin Chen
Download or read book Assortment Optimization Under the Multivariate MNL Model written by Xin Chen and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study an assortment optimization problem under a multi-purchase choice model in which customers choose a bundle of up to one product from each of two product categories. Different bundles have different utilities and the bundle price is the summation of the prices of products in it. For the uncapacitated setting where any set of products can be offered, we prove that this problem is strongly NP-hard. We show that an adjusted-revenue-ordered assortment provides a 1/2-approximation. Furthermore, we develop an approximation framework based on a linear programming relaxation of the problem and obtain a 0.74-approximation algorithm. This approximation ratio almost matches the integrality gap of the linear program, which is proven to be at most 0.75. For the capacitated setting, we prove that there does not exist a constant-factor approximation algorithm assuming the Exponential Time Hypothesis. The same hardness result holds for settings with general bundle prices or more than two categories. Finally, we conduct numerical experiments on randomly generated problem instances. The average approximation ratios of our algorithms are over 99%.
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 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 A Unified Analysis for Assortment Planning with Marginal Distributions by : Selin Ahipasaoglu
Download or read book A Unified Analysis for Assortment Planning with Marginal Distributions written by Selin Ahipasaoglu 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 problems under the marginal distribution model (MDM), a semiparametric choice model that only requires marginal error information without assuming independence. It is known that the multinomial logit (MNL) model belongs to MDM. In this paper, we further show that some multi-purchase choice models, such as the multiple-discrete-choice (MDC) model, and threshold utility model (TUM), also fall into the framework of MDM, although MDM does not explicitly model multi-purchase behavior. For the assortment problem under MDM, we characterize a general condition for the marginal distributions under which a strictly profit-nested assortment is optimal. Moreover, though the problem is shown to be NP-hard, we prove that the best strictly profit-nested assortment is a 1/2-approximate solution for all MDMs. We further construct a simple case of MDM such that the 1/2-approximate bound is tight. These results either generalize or improve existing results on assortment optimization under MNL, MDC, and TUM.
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 New Bounds for Assortment Optimization Under the Nested Logit Model by : Sumit Kunnumkal
Download or read book New Bounds for Assortment Optimization Under the Nested Logit Model written by Sumit Kunnumkal and published by . This book was released on 2020 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider the assortment optimization problem under the nested logit model and obtain new bounds on the gap between the optimal expected revenue and an upper bound based on a certain continuous relaxation of the assortment problem. Our bounds can be tighter than the existing bounds in the literature and provide more insight into the key drivers of tractability for the assortment optimization problem under the nested logit model. Moreover, our bounds scale with the nest dissimilarity parameters and we recover the well-known tractability results for the assortment optimization problem under the multinomial logit model when all the nest dissimilarity parameters are equal to one. We extend our results to the cardinality constrained assortment problem where there are constraints that limit the number of products that can be offered within each nest.
Book Synopsis Multiple-Purchase Choice Model by : Mengmeng Wang
Download or read book Multiple-Purchase Choice Model written by Mengmeng Wang and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Although multiple-purchase behavior is typical in retail practice, the choice model to portray such behavior is rare in existing research. This paper presents a new multiple-purchase (MP) choice model based on the multinomial logit (MNL) choice model, which allows customers to purchase more than one item in a single visit. We first prove that the log-likelihood function based on our MP choice model has a nice concave property such that we can efficiently estimate the parameters in the model with data. Then, we identify that a revenue-ordered assortment may not be optimal for an assortment optimization problem under the MP choice model. Moreover, there may be no inclusion between the optimal solutions obtained from the assortment problem based on our model and the MNL model. Next, we present an equivalent mixed-integer program for the multiple-purchase assortment optimization, which can be solved by state-of-the-art commercial solvers. Finally, we conduct extensive numerical experiments to evaluate the benefits from the MP choice model in both estimation and optimization problems. We first conduct a case study on a real-world dataset. The numerical results show that our MP choice model performs better in three estimation metrics and one revenue metric than the MNL choice model. Then, we demonstrate the advantage of the MP choice model on simulated data. Our model can provide significant realized revenue improvement compared with that obtained by the single-purchase MNL choice model in numerical results.
Book Synopsis Combinatorial Inference on the Optimal Assortment in Multinomial Logit Models by : Shuting Shen
Download or read book Combinatorial Inference on the Optimal Assortment in Multinomial Logit Models written by Shuting Shen and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Assortment optimization has received active explorations in the past few decades due to its practical importance. Despite the extensive literature dealing with optimization algorithms and latent score estimation, uncertainty quantification for the optimal assortment still needs to be explored and is of great practical significance. Instead of estimating and recovering the complete optimal offer set, decision-makers may only be interested in testing whether a given property holds true for the optimal assortment, such as whether they should include several products of interest in the optimal set, or how many categories of products the optimal set should include. This paper proposes a novel inferential framework for testing such properties. We consider the widely adopted multinomial logit (MNL) model, where we assume that each customer will purchase an item within the offered products with a probability proportional to the underlying preference score associated with the product. We reduce inferring a general optimal assortment property to quantifying the uncertainty associated with the sign change point detection of the marginal revenue gaps. We show the asymptotic normality of the marginal revenue gap estimator, and construct a maximum statistic via the gap estimators to detect the sign change point. By approximating the distribution of the maximum statistic with multiplier bootstrap techniques, we propose a valid testing procedure. We also conduct numerical experiments to assess the performance of our method.
Book Synopsis When Advertising Meets Assortment Planning by : Chenhao Wang
Download or read book When Advertising Meets Assortment Planning written by Chenhao Wang and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Although assortment optimization has been extensively studied, not much is known about how it is affected by advertising. In this paper, we address this gap by considering a novel joint advertising and assortment optimization problem. To capture the effect of advertising in the context of assortment planning, we assume that one can increase the preference weight of a product by advertising it, and the degree of improvement is decided by the effectiveness of advertising, which could be product-specific, and the amount of advertising efforts allocated to that product. Given budget constraints on advertising, our objective is to find a solution, which is composed of an advertising strategy and an assortment of products, that maximizes the expected revenue. We analyze the structural properties of this problem and derive effective solutions under different settings. If there is no capacity constraint on the number of products displayed to consumers, we show that revenue-ordered assortments still maintain optimality, and we leverage this result to derive an optimal solution. For the cardinality constrained case, it is difficult to solve the optimization problem directly; therefore, we show by relaxation that a near-optimal solution can be found efficiently.
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 Assortment Optimization Under Multinomial Logit Choice Model with Tree Structured Consideration Sets by : Qingwei Jin
Download or read book Assortment Optimization Under Multinomial Logit Choice Model with Tree Structured Consideration Sets written by Qingwei Jin 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 multinomial logit choice model with two tree structured consideration set models, i.e., the subtree model and the induced paths model. In each model, there are multiple customer types and each customer type has a different consideration set. A customer of a particular type only purchases product within his consideration set. The tree structure means all products form a tree with each node representing one product and all consideration sets are induced from this tree. In the subtree model, each consideration set consists of products in a subtree and in the induced paths model, each consideration set consists of products on the path from one node to the root. All customers make purchase decisions following the same multinomial logit choice model except that different customer types have different consideration sets. The goal of the assortment optimization is to determine a set of products offered to customers such that the expected revenue is maximized. We consider both unconstrained problem and capacitated problem. We show that these problems are all NP-hard problems and propose a unified framework, which captures the tree structure in both models, to design fully polynomial time approximation schemes (FPTAS) for all these problems. Besides, we identify a special case under the induced paths model, showing that it can be solved in $O(n)$ operations.
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 Assortment Optimization Under General Choice by : Srikanth Jagabathula
Download or read book Assortment Optimization Under General Choice written by Srikanth Jagabathula and published by . This book was released on 2016 with total page 51 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider the key operational problem of optimizing the mix of offered products to maximize revenues when product prices are exogenously set and product demand follows a general discrete choice model. The key challenge in making this decision is the computational difficulty of finding the best subset, which often requires exhaustive search. Existing approaches address the challenge by either deriving efficient algorithms for specific parametric structures or studying the performance of general-purpose heuristics. The former approach results in algorithms that lack portability to other structures; whereas the latter approach has resulted in algorithms with poor performance. We study a portable and easy-to-implement local search heuristic. We show that it efficiently finds the global optimum for the multinomial logit (MNL) model and derive performance guarantees for general choice structures. Empirically, it is better than prevailing heuristics when no efficient algorithms exist, and it is within 0.02% of optimality otherwise.
Book Synopsis Assortment Optimization for the Multinomial Logit Model with Repeated Customer Interactions by : Ningyuan Chen
Download or read book Assortment Optimization for the Multinomial Logit Model with Repeated Customer Interactions written by Ningyuan Chen and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper presents the multinomial logit model with repeated customer interactions. In each period, the same customer selects a product from the assortment recommended in that period or opts out. It captures the essence of an increasingly popular business model called the subscription box, exemplified by Stitch Fix and Wantable. From the seller's perspective, the choice probability is updated based on the purchase history. We study the adaptive assortment recommendation strategy for all the periods. Although the problem is generally NP-hard as we show, when the customer interacts with the seller for two periods, we discover the structures of the optimal assortment when the available products in the two periods are identical and develop approximation algorithms in other cases. For more than two periods, although the optimal assortments are intractable, we find that the optimal fixed assortments that are not adapted to the purchase history can achieve 68.47% or 50% of the optimal expected revenue, respectively, when the available products across periods are disjoint or not. Using two public datasets, we demonstrate that the model with repeated customer interactions can better predict the purchase behavior and generates higher revenues.
Book Synopsis The Stability of MNL-Based Demand Under Dynamic Customer Substitution and Its Algorithmic Implications by : Ali Aouad
Download or read book The Stability of MNL-Based Demand Under Dynamic Customer Substitution and Its Algorithmic Implications written by Ali Aouad and published by . This book was released on 2019 with total page 53 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study the dynamic assortment planning problem under the widely-utilized Multinomial Logit choice model (MNL). In this single-period assortment optimization and inventory management problem, the retailer jointly decides on an assortment, i.e., a subset of products to be offered, as well as on the inventory levels of these products, aiming to maximize the expected revenue subject to a capacity constraint on the total number of units stocked. The demand process is formed by a stochastic stream of arriving customers, who dynamically substitute between products according to the MNL model. This modeling approach has motivated a growing line of research on joint assortment and inventory optimization, initiated by the seminal papers of Bassok et al. (1999) and Mahajan and van Ryzin (2001). The currently best-known provably-good approximation in the dynamic setting considered, recently devised by Aouad et al. (2018b), leads to an expected revenue of at least 0.139 times the optimum under increasing-failure rate demand distributions, far from being satisfactory in practical revenue management applications. In this paper, we establish novel stochastic inequalities showing that, for any given inventory levels, the expected demand of each offered product is "stable" under basic algorithmic operations, such as scaling the MNL preference weights and shifting inventory across certain products. By exploiting this newly-gained understanding, we devise the first approximation scheme for dynamic assortment planning under the MNL model, allowing one to efficiently compute inventory levels that approach the optimal expected revenue within any degree of accuracy. Our approximation scheme is employed in extensive computational experiments to concurrently measure the performance of various algorithmic practices proposed in earlier literature. These experiments provide further insights regarding the value of dynamic substitution models, in comparison to simple inventory models that overlook stock-out effects, and shed light on their real-life deployability.
Book Synopsis Online Assortment Optimization with High-Dimensional Data by : Xue Wang
Download or read book Online Assortment Optimization with High-Dimensional Data written by Xue Wang and published by . This book was released on 2020 with total page 62 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this research, we consider an online assortment optimization problem, where a decision-maker needs to sequentially offer assortments to users instantaneously upon their arrivals and users select products from offered assortments according to the contextual multinomial logit choice model. We propose a computationally efficient Lasso-RP-MNL algorithm for the online assortment optimization problem under the cardinality constraint in high-dimensional settings. The Lasso-RP-MNL algorithm combines the Lasso and random projection as dimension reduction techniques to alleviate the computational complexity and improve the learning and estimation accuracy under high-dimensional data with limited samples. For each arriving user, the Lasso-RP-MNL algorithm constructs an upper-confidence bound for each individual product's attraction parameter, based on which the optimistic assortment can be identified by solving a reformulated linear programming problem. We demonstrate that for the feature dimension $d$ and the sample size dimension $T$, the expected cumulative regret under the Lasso-RP-MNL algorithm is upper bounded by $ tilde{ mathcal{O}}( sqrt{T} log d)$ asymptotically, where $ tilde{ mathcal{O}}$ suppresses the logarithmic dependence on $T$. Furthermore, we show that even when available samples are extremely limited, the Lasso-RP-MNL algorithm continues to perform well with a regret upper bound of $ tilde{ mathcal{O}}( T^{ frac{2}{3}} log d)$. Finally, through synthetic-data-based experiments and a high-dimensional XianYu assortment recommendation experiment, we show that the Lasso-RP-MNL algorithm is computationally efficient and outperforms other benchmarks in terms of the expected cumulative regret.
Book Synopsis Capacitated Assortment Optimization by : Antoine Désir
Download or read book Capacitated Assortment Optimization written by Antoine Désir and published by . This book was released on 2020 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Assortment optimization is an important problem that arises in many practical applications such as retailing and online advertising. In this problem, the goal is to select a subset of items that maximizes the expected revenue in the presence of (1) the substitution behavior of consumers specified by a choice model, and (2) a potential capacity constraint bounding the total weight of items in the assortment. The latter is a natural constraint arising in many applications. We begin by showing how challenging these two aspects are from an optimization perspective. First, we show that adding a general capacity constraint makes the problem NP-hard even for the simplest choice model, namely the multinomial logit model. Second, we show that even the unconstrained assortment optimization for the mixture of multinomial logit model is hard to approximate within any reasonable factor when the number of mixtures is not constant.In view of these hardness results, we present near-optimal algorithms for the capacity constrained assort- ment optimization problem under 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, we develop near-optimal algorithms for a general class of capacity constrained optimization problems whose objective function depends on a small number of linear functions. For the mixture of multinomial logit model (resp. Markov chain model), the running time of our algorithm depends exponentially on the number of segments (resp. rank of the transition matrix). Therefore, we get efficient algorithms only for the case of constant number of segments (resp. constant rank). However, in light of our hardness result, any near-optimal algorithm will have a super polynomial dependence on the number of mixtures for the mixture of multinomial logit choice model.
