Multinomial Logit

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Philip Kaminsky - One of the best experts on this subject based on the ideXlab platform.

Huseyin Topaloglu - One of the best experts on this subject based on the ideXlab platform.

  • assortment optimization under the multi purchase Multinomial Logit choice model
    Social Science Research Network, 2021
    Co-Authors: Jacob Feldman, Danny Segev, Huseyin Topaloglu, Laura Wagner, Yicheng Bai
    Abstract:

    In this paper, we introduce the Multi-Purchase Multinomial Logit choice model, which extends the random utility maximization framework of the classical Multinomial Logit model to a multiple-purchase setting. In this model, customers sample random utilities for each offered product as in the Multinomial Logit model. However, rather than focusing on a single product, they concurrently sample a ``budget'' parameter M, which indicates the maximum number of products that the customer is willing to purchase. Subsequently, the M highest utility products are purchased, out of those whose utilities exceed that of the no-purchase option. When fewer than M products satisfy the latter condition, only these products will be purchased. First and foremost, we propose the first multi-purchase choice model that can be fully operationalized. Specifically, we first provide a recursive procedure to compute the choice probabilities in this model, which in turn provides a framework to study its resulting assortment problem, where the goal is to select a subset of products to make available for purchase so as to maximize expected revenue. Our main algorithmic results consist of two distinct polynomial time approximations schemes (PTAS); the first, and simpler of the two, caters to a setting where each customer may buy only a constant number of products, whereas the second more nuanced algorithm applies to our multi-purchase model in its general form. Additionally, we study the revenue-potential of making assortment decisions that account for multi-purchase behavior in comparison to those that overlook this phenomenon. In particular, we relate both the structure and revenue performance of the optimal assortment under a traditional single-purchase model to that of the optimal assortment in the multi-purchase setting. Finally, we complement our theoretical work with an extensive set of computational experiments, where the efficacy of our proposed PTAS is tested against natural heuristics. Ultimately, we find that our approximation scheme outperforms these approaches by 1-5% on average.

  • joint assortment optimization and customization under a mixture of Multinomial Logit models on the value of personalized assortments
    Social Science Research Network, 2021
    Co-Authors: Omar El Housni, Huseyin Topaloglu
    Abstract:

    We consider a joint assortment optimization and customization problem under a mixture of Multinomial Logit models. In this problem, a firm faces customers of different types, each making a choice within an offered assortment according to the Multinomial Logit model with different parameters. The problem takes place in two stages. In the first stage, the firm picks an assortment of products to carry subject to a cardinality constraint. In the second stage, a customer of a certain type arrives into the system. Observing the type of the customer, the firm customizes the assortment that it carries by, possibly, dropping products from the assortment. The goal of the firm is to find an assortment to carry and a customized assortment for each customer type that can arrive in the second stage to maximize the expected revenue from a customer visit. The problem arises, for example, in online platforms, where retailers commit to a selection of products before the start of the selling season, but they could potentially customize the displayed assortments for each customer type. We refer to this problem as the Customized Assortment Problem (CAP). Letting m be the number of customer types, we show that the expected revenue of CAP can be $\Omega(m)$ times greater than the optimal expected revenue of the corresponding model without customization and this bound is tight. We establish that CAP is NP-hard to approximate within a factor better than (1-1/e), so we focus on providing an approximation framework for CAP. As our main technical contribution, we design a novel algorithm, which we refer to as Augmented Greedy, and building on it, we give a $\Omega(1/ \log m)$-approximation algorithm to CAP. Lastly, we present a fully polynomial-time approximation scheme for CAP when the number of customer types is constant. In our computational experiments, we demonstrate the value of customization by using a dataset from Expedia and check the practical performance of our approximation algorithm.

  • assortment optimization under the Multinomial Logit model with random choice parameters
    Production and Operations Management, 2014
    Co-Authors: Paat Rusmevichientong, David B Shmoys, Chaoxu Tong, Huseyin Topaloglu
    Abstract:

    We consider assortment optimization problems under the Multinomial Logit model, where the parameters of the choice model are random. The randomness in the choice model parameters is motivated by the fact that there are multiple customer segments, each with different preferences for the products, and the segment of each customer is unknown to the firm when the customer makes a purchase. This choice model is also called the mixture-of-Logits model. The goal of the firm is to choose an assortment of products to offer that maximizes the expected revenue per customer, across all customer segments. We establish that the problem is NP complete even when there are just two customer segments. Motivated by this complexity result, we focus on assortments consisting of products with the highest revenues, which we refer to as revenue-ordered assortments. We identify specially structured cases of the problem where revenue-ordered assortments are optimal. When the randomness in the choice model parameters does not follow a special structure, we derive tight approximation guarantees for revenue-ordered assortments. We extend our model to the multi-period capacity allocation problem, and prove that, when restricted to the revenue-ordered assortments, the mixture-of-Logits model possesses the nesting-by-fare-order property. This result implies that revenue-ordered assortments can be incorporated into existing revenue management systems through nested protection levels. Numerical experiments show that revenue-ordered assortments perform remarkably well, generally yielding profits that are within a fraction of a percent of the optimal.

  • joint stocking and product offer decisions under the Multinomial Logit model
    Production and Operations Management, 2013
    Co-Authors: Huseyin Topaloglu
    Abstract:

    This article studies a joint stocking and product offer problem. We have access to a number of products to satisfy the demand over a finite selling horizon. Given that customers choose among the set of offered products according to the Multinomial Logit model, we need to decide which sets of products to offer over the selling horizon and how many units of each product to stock so as to maximize the expected profit. We formulate the problem as a nonlinear program, where the decision variables correspond to the stocking quantity for each product and the duration of time that each set of products is offered. This nonlinear program is intractable due to its large number of decision variables and its nonseparable and nonconcave objective function. We use the structure of the Multinomial Logit model to formulate an equivalent nonlinear program, where the number of decision variables is manageable and the objective function is separable. Exploiting separability, we solve the equivalent nonlinear program through a dynamic program with a two dimensional and continuous state variable. As the solution of the dynamic program requires discretizing the state variable, we study other approximate solution methods. Our equivalent nonlinear program and approximate solution methods yield insights for good offer sets.

  • robust assortment optimization in revenue management under the Multinomial Logit choice model
    Operations Research, 2012
    Co-Authors: Paat Rusmevichientong, Huseyin Topaloglu
    Abstract:

    We study robust formulations of assortment optimization problems under the Multinomial Logit choice model. The novel aspect of our formulations is that the true parameters of the Logit model are assumed to be unknown, and we represent the set of likely parameter values by a compact uncertainty set. The objective is to find an assortment that maximizes the worst-case expected revenue over all parameter values in the uncertainty set. We consider both static and dynamic settings. The static setting ignores inventory consideration, whereas in the dynamic setting, there is a limited initial inventory that must be allocated over time. We give a complete characterization of the optimal policy in both settings, show that it can be computed efficiently, and derive operational insights. We also propose a family of uncertainty sets that enables the decision maker to control the trade-off between increasing the average revenue and protecting against the worst-case scenario. Numerical experiments show that our robust approach, combined with our proposed family of uncertainty sets, is especially beneficial when there is significant uncertainty in the parameter values. When compared to other methods, our robust approach yields over 10% improvement in the worst-case performance, but it can also maintain comparable average revenue if average revenue is the performance measure of interest.

Fred L Mannering - One of the best experts on this subject based on the ideXlab platform.

  • markov switching Multinomial Logit model an application to accident injury severities
    Accident Analysis & Prevention, 2009
    Co-Authors: Nataliya V Malyshkina, Fred L Mannering
    Abstract:

    In this study, two-state Markov switching Multinomial Logit models are proposed for statistical modeling of accident-injury severities. These models assume Markov switching over time between two unobserved states of roadway safety as a means of accounting for potential unobserved heterogeneity. The states are distinct in the sense that in different states accident-severity outcomes are generated by separate Multinomial Logit processes. To demonstrate the applicability of the approach, two-state Markov switching Multinomial Logit models are estimated for severity outcomes of accidents occurring on Indiana roads over a four-year time period. Bayesian inference methods and Markov Chain Monte Carlo (MCMC) simulations are used for model estimation. The estimated Markov switching models result in a superior statistical fit relative to the standard (single-state) Multinomial Logit models for a number of roadway classes and accident types. It is found that the more frequent state of roadway safety is correlated with better weather conditions and that the less frequent state is correlated with adverse weather conditions.

  • markov switching Multinomial Logit model an application to accident injury severities
    arXiv: Applications, 2008
    Co-Authors: Nataliya V Malyshkina, Fred L Mannering
    Abstract:

    In this study, two-state Markov switching Multinomial Logit models are proposed for statistical modeling of accident injury severities. These models assume Markov switching in time between two unobserved states of roadway safety. The states are distinct, in the sense that in different states accident severity outcomes are generated by separate Multinomial Logit processes. To demonstrate the applicability of the approach presented herein, two-state Markov switching Multinomial Logit models are estimated for severity outcomes of accidents occurring on Indiana roads over a four-year time interval. Bayesian inference methods and Markov Chain Monte Carlo (MCMC) simulations are used for model estimation. The estimated Markov switching models result in a superior statistical fit relative to the standard (single-state) Multinomial Logit models. It is found that the more frequent state of roadway safety is correlated with better weather conditions. The less frequent state is found to be correlated with adverse weather conditions.

  • an exploratory Multinomial Logit analysis of single vehicle motorcycle accident severity
    Journal of Safety Research, 1996
    Co-Authors: Viswanathan Shankar, Fred L Mannering
    Abstract:

    Abstract Most previous research on motorcycle accident severity has focused on univariate relationships between severity and an explanatory variable of interest (e.g., helmet use). The potential ambiguity and bias that univariate analyses create in identifying the causality of severity has generated the need for multivariate analyses in which the effects of all factors that influence accident severity are considered. This paper attempts to address this need by presenting a Multinomial Logit formulation of motorcyclerider accident severity in single-vehicle collisions. Five levels of severity are considered: 1. (a) property damage only, 2. (b) possible injury, 3. (c) evident injury, 4. (d) disabling injury, and 5. (e) fatality. Using 5-year statewide data on single-vehicle motorcycle accidents from the state of Washington, we estimate a multivariate model of motorcycle-rider severity that considers environmental factors, roadway conditions, vehicle characteristics, and rider attributes. Our findings show that the Multinomial Logit formulation that we use is a promising approach to evaluate the determinants of motorcycle accident severity.

Alper şen - One of the best experts on this subject based on the ideXlab platform.

Garud Iyengar - One of the best experts on this subject based on the ideXlab platform.

  • Multinomial Logit contextual bandits provable optimality and practicality
    arXiv: Machine Learning, 2021
    Co-Authors: Garud Iyengar
    Abstract:

    We consider a sequential assortment selection problem where the user choice is given by a Multinomial Logit (MNL) choice model whose parameters are unknown. In each period, the learning agent observes a $d$-dimensional contextual information about the user and the $N$ available items, and offers an assortment of size $K$ to the user, and observes the bandit feedback of the item chosen from the assortment. We propose upper confidence bound based algorithms for this MNL contextual bandit. The first algorithm is a simple and practical method which achieves an $\tilde{\mathcal{O}}(d\sqrt{T})$ regret over $T$ rounds. Next, we propose a second algorithm which achieves a $\tilde{\mathcal{O}}(\sqrt{dT})$ regret. This matches the lower bound for the MNL bandit problem, up to logarithmic terms, and improves on the best known result by a $\sqrt{d}$ factor. To establish this sharper regret bound, we present a non-asymptotic confidence bound for the maximum likelihood estimator of the MNL model that may be of independent interest as its own theoretical contribution. We then revisit the simpler, significantly more practical, first algorithm and show that a simple variant of the algorithm achieves the optimal regret for a broad class of important applications.

  • thompson sampling for Multinomial Logit contextual bandits
    Neural Information Processing Systems, 2019
    Co-Authors: Garud Iyengar
    Abstract:

    We consider a dynamic assortment selection problem where the goal is to offer a sequence of assortments that maximizes the expected cumulative revenue, or alternatively, minimize the expected regret. The feedback here is the item that the user picks from the assortment. The distinguishing feature in this work is that this feedback has a Multinomial logistic distribution. The utility of each item is a dynamic function of contextual information of both the item and the user. We propose two Thompson sampling algorithms for this Multinomial Logit contextual bandit. Our first algorithm maintains a posterior distribution of the true parameter and establishes $\tilde{O}(d\sqrt{T})$ Bayesian regret over $T$ rounds with $d$ dimensional context vector. The worst-case computational complexity of this algorithm could be high when the prior distribution is not a conjugate. The second algorithm approximates the posterior by a Gaussian distribution, and uses a new optimistic sampling procedure to address the issues that arise in worst-case regret analysis. This algorithm achieves $\tilde{O}(d^{3/2}\sqrt{T})$ worst-case (frequentist) regret bound. The numerical experiments show that the practical performance of both methods is in line with the theoretical guarantees.

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