An Algorithm For Assortment Optimization Under Parametric Discrete Choice Models

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An Algorithm for Assortment Optimization Under Parametric Discrete Choice Models

Author : Tien Mai
Publisher :
ISBN 13 :
Total Pages : 37 pages
Book Rating : 4.:/5 (13 download)

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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.

Branch-and-Bound Algorithms for Assortment Optimization Under Weakly Rational Choice

Author : Clark Pixton
Publisher :
ISBN 13 :
Total Pages : 29 pages
Book Rating : 4.:/5 (13 download)

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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.

Assortment Optimization Under Consider-Then-Choose Choice Models

Author : Ali Aouad
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (137 download)

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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.

Assortment Optimization Under Multiple-Discrete Customer Choices

Author : Heng Zhang
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (137 download)

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Book Synopsis Assortment Optimization Under Multiple-Discrete Customer Choices by : Heng Zhang

Download or read book Assortment Optimization Under Multiple-Discrete Customer Choices written by Heng Zhang and published by . This book was released on 2021 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider an assortment optimization problem where the customer may purchase multiple products and possibly more than one unit of each product purchased. We adopt the customer consumption model based on the Multiple-Discrete-Choice (MDC) model proposed by Huh and Li (2021). We identify conditions under which the profit-ordered sets are optimal. Without these conditions, we show that assortment optimization is NP-hard. Furthermore, we prove that a generalization of the profit-ordered sets achieves an approximation guarantee of 1/2. We also present an algorithm that computes an epsilon-optimal solution to the assortment problem in running time polynomial in 1/epsilon and the problem input size, once we impose the mild technical assumption that model parameters are bounded.

Assortment and Inventory Optimization

Author : Mohammed Ali Aouad
Publisher :
ISBN 13 :
Total Pages : 256 pages
Book Rating : 4.:/5 (12 download)

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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%.

Assortment Optimization Under General Choice

Author : Srikanth Jagabathula
Publisher :
ISBN 13 :
Total Pages : 51 pages
Book Rating : 4.:/5 (13 download)

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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.

The Exponomial Choice Model

Author : Ali Aouad
Publisher :
ISBN 13 :
Total Pages : 57 pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis The Exponomial Choice Model by : Ali Aouad

Download or read book The Exponomial Choice Model written by Ali Aouad and published by . This book was released on 2019 with total page 57 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we consider the yet-uncharted assortment optimization problem under the Exponomial choice model, where the objective is to determine the revenue maximizing set of products that should be offered to customers. Our main algorithmic contribution comes in the form of a fully polynomial-time approximation scheme (FPTAS), showing that the optimal expected revenue can be efficiently approached within any degree of accuracy. This result is obtained through a synthesis of ideas related to approximate dynamic programming, that enable us to derive a compact discretization of the continuous state space by keeping track of several key statistics in "rounded" form throughout the overall computation. Consequently, we obtain the first provably-good algorithm for assortment optimization under the Exponomial choice model, which is complemented by a number of hardness results for natural extensions. We show in computational experiments that our solution method admits an efficient implementation, based on additional pruning criteria.Furthermore, we conduct empirical evaluations of the Exponomial choice model. We present a number of case studies using real-world data sets, spanning retail, online platforms, and transportation. We focus on a comparison with the popular Multinomial Logit choice model (MNL), which is largely dominant in the choice modeling practice, as both models share a simple parametric structure with desirable statistical and computational properties. We identify several settings where the Exponomial choice model has better predictive accuracy than MNL and leads to more profitable assortment decisions. We provide implementation guidelines and insights about the performance of the Exponomial choice model relative to MNL.

Customer Choice Models and Assortment Optimization

Author : James Mario Davis
Publisher :
ISBN 13 :
Total Pages : 424 pages
Book Rating : 4.:/5 (932 download)

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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.

Capacitated Assortment Optimization

Author : Antoine Désir
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (137 download)

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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.

Assortment Optimization and Pricing Under the Threshold-Based Choice Models

Author : Xu Tian
Publisher :
ISBN 13 :
Total Pages : 42 pages
Book Rating : 4.:/5 (129 download)

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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.

A Unified Analysis for Assortment Planning with Marginal Distributions

Author : Selin Ahipasaoglu
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (137 download)

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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.

An Exact Method for Assortment Optimization Under the Nested Logit Model

Author : Laurent Alfandari
Publisher :
ISBN 13 :
Total Pages : 39 pages
Book Rating : 4.:/5 (13 download)

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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.

Context Dependent Discrete Choice Models and Assortment Optimization for Online Retail

Author : Uzma Mushtaque
Publisher :
ISBN 13 :
Total Pages : 420 pages
Book Rating : 4.:/5 (126 download)

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Book Synopsis Context Dependent Discrete Choice Models and Assortment Optimization for Online Retail by : Uzma Mushtaque

Download or read book Context Dependent Discrete Choice Models and Assortment Optimization for Online Retail written by Uzma Mushtaque and published by . This book was released on 2017 with total page 420 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Assortment Optimization Under the Multinomial Logit Model with Utility-Based Rank Cutoffs

Author : Jacob Feldman
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (137 download)

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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.

Assortment Optimization Under a Single Transition Model

Author : Kameng Nip
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (137 download)

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Book Synopsis Assortment Optimization Under a Single Transition Model by : Kameng Nip

Download or read book Assortment Optimization Under a Single Transition Model written by Kameng Nip and published by . This book was released on 2020 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we consider a new customer choice model which we call the single transition choice model. In this model, there is a universe of products and customers arrive at each product with a certain probability. If the arrived product is unavailable, then the seller can recommend a subset of available products to the customer and the customer will purchase one of the recommended products or choose not to purchase with certain transition probabilities. The distinguishing features of the model are that the seller can control which products to recommend depending on the arrived product and that each customer either purchases a product or leaves the market after one transition.We study the assortment optimization problem under this model. Particularly, we show that this problem is NP-Hard even if the customer can transition from each product to at most two products. Despite the complexity of the problem, we provide polynomial time algorithms or approximation algorithms for several special cases, such as when the customer can only transition from each product to at most a given number of products and the size of each recommended set is at most a given number. We also provide a tight worst-case performance bound for revenue-ordered assortments. In addition, we propose a compact mixed integer program formulation for this problem, which is efficient for problems of moderate size. Finally, we conduct numerical experiments to demonstrate the effectiveness of the proposed algorithms.

Capacity, Pricing and Assortment Management Under Discrete Choice Model with Anticipated Wait

Author : Ruxian Wang
Publisher :
ISBN 13 :
Total Pages : 40 pages
Book Rating : 4.:/5 (129 download)

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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.

An Optimal Greedy Heuristic with Minimal Learning Regret for the Markov Chain Choice Model

Author : Guillermo Gallego
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (137 download)

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Book Synopsis An Optimal Greedy Heuristic with Minimal Learning Regret for the Markov Chain Choice Model by : Guillermo Gallego

Download or read book An Optimal Greedy Heuristic with Minimal Learning Regret for the Markov Chain Choice Model written by Guillermo Gallego and published by . This book was released on 2021 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study the assortment optimization problem and show that local optima are global optima for all discrete choice models that can be represented by the Markov Chain model. We develop a forward greedy heuristic that finds an optimal assortment for the Markov Chain model and runs in $O(n^2)$ iterations. The heuristic has performance bound $1/n$ for any regular choice model which is best possible among polynomial heuristics. We also propose a backward greedy heuristic that is optimal for Markov chain model and requires fewer iterations. Numerical results show that our heuristics performs significantly better than the estimate then optimize method and the revenue-ordered assortment heuristic when the ground truth is a latent class multinomial logit choice model. Based on the greedy heuristics, we develop a learning algorithm that enjoys asymptotic optimal regret for the Markov chain choice model and avoids parameter estimations, focusing instead on binary comparisons of revenues.