Assortment Optimization Problem Research Articles

The Exponomial Choice Model: Assortment Optimization and Application to Public Transit Choice Prediction in San Francisco

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.

Mallows-Smoothed Distribution Over Rankings Approach for Modeling Choice

Assortment optimization is an important problem that arises in many applications including retailing and online advertising. The goal in such problems is to determine a revenue/profit maximizing subset of products to offer from a large universe of products when customers exhibit stochastic substitution behavior. We consider a mixture of Mallows model for demand, which can be viewed as a “smoothed” generalization of the class of sparse rank-based choice models, designed to overcome some of its key limitations. In spite of these advantages, the Mallows distribution has an exponential support size and does not admit a closed-form expression for choice probabilities. We first conduct a case study using a publicly available data set involving real-world preferences on sushi types to show the benefits of Mallows-based smoothing. We show that smoothing significantly improves both the prediction and the decision accuracy on this data set. We then present an efficient procedure to compute the choice probabilities for any assortment under the mixture of Mallows model. Surprisingly, this finding allows to formulate a compact mixed integer program (MIP) that leads to a practical approach for solving the constrained assortment optimization problem under a general mixture of Mallows model. To complement this MIP formulation, we also exploit additional structural properties of the underlying distribution to propose several polynomial-time approximation schemes. These are the first efficient algorithms with provably near-optimal performance guarantees for the assortment optimization problem under the Mallows or the mixture of Mallows model in such generality.

A tractable discrete fractional programming: application to constrained assortment optimization

In this note, we consider a discrete fractional programming in light of a decision problem where limited number of indivisible resources are allocated to several heterogeneous projects to maximize the ratio of total profit to total cost. For each project, both profit and cost are solely determined by the amount of resources allocated to it. Although the problem can be reformulated as a linear program with $$O(m^2 n)$$O(m2n) variables and $$O(m^2 n^2)$$O(m2n2) constraints, we further show that it can be efficiently solved by induction in $$O(m^3 n^2 \log mn)$$O(m3n2logmn) time. In application, this method leads to an $$O(m^3 n^2 \log mn)$$O(m3n2logmn) algorithm for assortment optimization problem under nested logit model with cardinality constraints (Feldman and Topaloglu, Oper Res 63: 812---822, 2015).

New Bounds for Assortment Optimization Under the Nested Logit Model

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.

Consumer Choice Models with Endogenous Network Effects

Network externality arises when the utility of a product depends not only on its attributes but also on the number of consumers who purchase the same product. In this paper, we study consumer choice models that endogenize such network externality. We first characterize the choice probabilities under such models and conduct studies on comparative statics. Then we investigate the assortment optimization problem under such choice models. Although the problem is generally NP-hard, we show that a new class of assortments, called quasi-revenue-ordered assortments, which consist of a revenue-ordered assortment plus at most one additional item, are optimal under mild conditions. We also propose an iterative estimation method to calibrate such choice models, for both uncensored and censored data cases. An empirical study on a mobile game data set shows that our proposed model can provide better fits for the data, increase the prediction accuracy for consumer choices, and potentially increase revenue.This paper was accepted by Noah Gans, stochastic models and simulation.

The Multiproduct Newsvendor Problem with Customer Choice

We address the multiproduct newsvendor problem under a general specification of customer choice behavior. We develop a methodology that yields upper bounds on the optimal value as well as feasible inventory solutions. The methodology is based on an approximate Jordan decomposition of the state transition matrix. Two specializations of the methodology are presented: one leads to a decomposition by customer into a sequence of assortment optimization problems and the second leads to a decomposition by product into a collection of independent newsvendor problems. We conduct computational experiments and find that the proposed methodology outperforms existing bounds. The electronic companion is available at https://doi.org/10.1287/opre.2017.1654 .

Technical Note: Space Constrained Assortment Optimization under the Paired Combinatorial Logit Model

We study the space constrained assortment optimization problem under the paired combinatorial logit choice model. The goal in this problem is to choose a set of products to make available for purchase with the intention of maximizing the expected revenue from each arriving customer. Each offered product occupies a specific amount of space and there is a limit on the space consumed by all of the offered products. The purchasing decision of each customer is governed by the paired combinatorial logit choice model. We provide the first efficient constant factor approximation for this problem.

Technical Note—Assortment Optimization with Small Consideration Sets

In the paper “Assortment Optimization with Consideration Sets,” we study a customer choice model that captures purchasing behavior when there is a limit on the number of times that a customer will consider purchasing. Under this model, we assume each customer is characterized by a ranked preference list of products and, upon arrival, will purchase the highest ranking offered product. Because we restrict ourselves to settings in which customers consider a limited number of products, we assume that these rankings contain at most k products. We call this model the k-product nonparametric choice model. We focus on the assortment-optimization problem under this choice model. In this problem, the retailer wants to find the revenue-maximizing set of products to offer when the buying process of each customer is governed by the k-product nonparametric choice model. We develop a linear programming–based randomized rounding algorithm that gives the best known approximation guarantee.

Assortment Optimization Under the Paired Combinatorial Logit Model

When retailers decide which assortment of products to offer, they can make use of a choice model that describes how customers choose and substitute among the products. The key is to use a choice model that faithfully captures the choice process of customers, while making sure that the corresponding problem of finding the revenue-maximizing assortment remains tractable. In “Assortment Optimization Under the Paired Combinatorial Logit Model,” Zhang, Rusmevichientong, and Topaloglu consider the paired combinatorial logit model to capture the choice process of customers. This choice model uses a utility maximization framework to capture the customer choices, and the utilities of the products can have a rather general correlation structure. The authors demonstrate that one can construct algorithms with performance guarantees to solve the assortment optimization problem under this choice model.

Technical Note: Assortment Optimization with Small Consideration Sets

We study a customer choice model that captures purchasing behavior when there is a limit on the number of times that a customer will substitute among the offered products. Under this model, we assume each customer is characterized by a ranked preference list of products and, upon arrival, will purchase the highest ranking offered product. Since we restrict ourselves to settings in which customers consider a limited number of products, we assume that these rankings contain at most k products. We call this model the k-product nonparametric choice model. We focus on the assortment optimization problem under this choice model. In this problem, the retailer wants to find the revenue maximizing set of products to offer when the buying process of each customer is governed by the k-product nonparametric choice model. First, we show that the assortment problem is strongly NP-hard even for k = 2. Motivated by this result, we develop a linear programming-based randomized rounding algorithm that gives the best known approximation guarantee. We tighten the approximation guarantee further when each preference list contains at most two products and consider the case where there is a limit on the number of products that can be offered to the customers.

SAA-regularized methods for multiproduct price optimization under the pure characteristics demand model

Utility-based choice models are often used to determine a consumer’s purchase decision among a list of available products; to provide an estimate of product demands; and, when data on purchase decisions or market shares are available, to infer consumers’ preferences over observed product characteristics. These models also serve as a building block in modeling firms’ pricing and assortment optimization problems. We consider a firm’s multiproduct pricing problem, in which product demands are determined by a pure characteristics model. A sample average approximation (SAA) method is used to approximate the expected market share of products and the firm profit. We propose an SAA-regularized method for the multiproduct price optimization problem. We present convergence analysis and numerical examples to show the efficiency and the effectiveness of the proposed method.

Assortment Optimization Under a Single Transition Model

In this paper, we consider a Markov chain choice model with single transition. In this model, 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 generally NP-Hard even if each product could only transit to at most two products. Despite the complexity of the problem, we provide polynomial time algorithms for several special cases, such as when the transition probabilities are homogeneous with respect to the starting point, or when each product can only transit to one other product. We also provide a tight performance bound for revenue-ordered assortments. In addition, we propose a compact mixed integer program formulation that can solve this problem of large size. Through extensive numerical experiments, we show that the proposed algorithms can solve the problem efficiently and the obtained assortments could significantly improve the revenue of the seller than under the Markov chain choice model.

Thompson Sampling for Online Personalized Assortment Optimization Problems with Multinomial Logit Choice Models

Motivated by online retail applications, we study the online personalized assortment optimization problem. A seller conducts sales by offering assortments of products to a stream of arriving customers. The customers' purchase behavior follows their respective personalized Multinomial Logit choice models, which vary according to their individual attributes. The seller aims to maximize his revenue by offering personalized assortments to the customers, notwithstanding his uncertainty about the customers' choice models. We propose a Thompson Sampling based policy, policy Pao-Ts, where surrogate models for the latent choice models are constructed using samples from a progressively updated posterior distribution. We derive bounds on the revenue loss, namely Bayesian regret, incurred by policy Pao-Ts, in comparison to the optimal policy which is provided with the latent models. The regret bounds hold even when the customers' attributes vary arbitrarily, but not independently and identically distributed.

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

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.

A Markov Chain Approximation to Choice Modeling

Assortment planning is an important problem that arises in many industries such as retailing and airlines. One of the key challenges in an assortment planning problem is to identify the “right” model for the substitution behavior of customers from the data. Error in model selection can lead to highly suboptimal decisions. In this paper, we consider a Markov chain based choice model and show that it provides a simultaneous approximation for all random utility based discrete choice models including the multinomial logit (MNL), the probit, the nested logit and mixtures of multinomial logit models. In the Markov chain model, substitution from one product to another is modeled as a state transition in the Markov chain. We show that the choice probabilities computed by the Markov chain based model are a good approximation to the true choice probabilities for any random utility based choice model under mild conditions. Moreover, they are exact if the underlying model is a generalized attraction model (GAM) of which the MNL model is a special case. We also show that the assortment optimization problem for our choice model can be solved efficiently in polynomial time. In addition to the theoretical bounds, we also conduct numerical experiments and observe that the average maximum relative error of the choice probabilities of our model with respect to the true probabilities for any offer set is less than 3% where the average is taken over different offer sets. Therefore, our model provides a tractable approach to choice modeling and assortment optimization that is robust to model selection errors. Moreover, the state transition primitive for substitution provides interesting insights to model the substitution behavior in many real-world applications.

On (Re-Scaled) Multi-Attempt Approximation of Customer Choice Model and its Application to Assortment Optimization

Motivated by the classic exogenous demand model and the recently developed Markov chain model, we propose a new approximation to the general customer choice model based on random utility called multi-attempt model, in which a customer may consider several substitutes before finally deciding to not purchase anything. We show that the approximation error of multi-attempt model decreases exponentially in the number of attempts. However, despite its strong theoretical performance, the empirical performance of multi-attempt model is not satisfactory. This motivates us to construct a modification of multi-attempt model called re-scaled multi-attempt model. We show that re-scaled 2-attempt model is exact when the underlying true choice model is Multinomial Logit (MNL); if, however, the underlying true choice model is not MNL, we show numerically that the approximation quality of re-scaled 2-attempt model is very close to that of Markov chain model. The key feature of our proposed approach is that the resulting approximate choice probability can be explicitly written. From a practical perspective, this allows the decision maker to use off-the-shelf solvers, or borrow existing algorithms from literature, to solve a general assortment optimization problem with a variety of real-world constraints.

Robust optimization of the 0–1 knapsack problem: Balancing risk and return in assortment optimization

Retailers face the important but challenging task of optimizing their product assortments. The challenge is to find, for every category in every store, the assortment that maximizes (expected) category profit. Adding to the complexity of this 0–1 knapsack problem, retailers should also consider the risk associated with every assortment. While every product in the assortment offers an expected return, there is also uncertainty around its expected demand and profit contribution. Therefore, retailers face the difficult task of designing a portfolio of products that balances risk and return. In this paper, we develop a robust approach to optimize retail assortments that offers this balance. Since the dimensionality of this robust 0–1 knapsack problem in practice often precludes full enumeration, we propose a novel, efficient and real-time heuristic that solves this problem. The heuristic constructs an approximation of the risk-return Efficient Frontier of assortments. We find that the robust solutions offer the retailer a considerable reduction in risk (variance), yet only imply a small reduction in expected return. The constructed approximations contain assortments that are optimal solutions to the robust assortment optimization problem. Moreover, they represent insightful visualizations of the solution space, allowing for interactivity (“what risk premium should the retailer pay?”) in real-time (matter of seconds).

Assortment optimization over time

In this note we introduce the problem of assortment optimization over time. We have a sequence of time steps and can introduce one new product per time step. Once introduced a product cannot be removed. The goal is to determine which products to introduce so as to maximize revenue over all time steps under some choice model. Given a 1/α-approximation algorithm for the capacitated assortment optimization problem we give a 1/2α-approximation algorithm for this problem.

Capacity Constraints Across Nests in Assortment Optimization Under the Nested Logit Model

We consider assortment optimization problems when customers choose according to the nested logit model and there is a capacity constraint limiting the total capacity consumption of all products offered in all nests. When each product consumes one unit of capacity, our capacity constraint limits the cardinality of the offered assortment. For the cardinality constrained case, we develop an efficient algorithm to compute the optimal assortment. When the capacity consumption of each product is arbitrary, we give an algorithm to obtain a 4-approximate solution. We show that we can compute an upper bound on the optimal expected revenue for an individual problem instance by solving a linear program. In our numerical experiments, we consider problem instances involving products with arbitrary capacity consumptions. Comparing the expected revenues from the assortments obtained by our 4-approximation algorithm with the upper bounds on the optimal expected revenues, our numerical results indicate that the 4-approximation algorithm performs quite well, yielding less than 2% optimality gap on average.

Capacity Constrained Assortment Optimization Under the Markov Chain Based Choice Model

Assortment optimization is an important problem that arises in many practical applications such as retailing and online advertising. In such settings, the goal is to select a subset of items to offer from a universe of substitutable items in order to maximize expected revenue when consumers exhibit a random substitution behavior. We consider a capacity constrained assortment optimization problem under the Markov Chain based choice model, recently considered by Blanchet et al. (2013). In this model, the substitution behavior of customers is modeled through transitions in a Markov chain. Capacity constraints arise naturally in many applications to model real-life constraints such as shelf space or budget limitations. We show that the capacity constrained problem is APX-hard even for the special case when all items have unit weights and uniform prices, i.e., it is NP-hard to obtain an approximation ratio better than some given constant. We present constant factor approximations for both the cardinality and capacity constrained assortment optimization problem for the general Markov chain model. Our algorithm is based on a paradigm that allows us to transform a non-linear revenue function into a linear function. The local-ratio based algorithmic paradigm also provides interesting insights towards the optimal stopping problem as well as other assortment optimization problems.