Multinomial Logit Choice Model Research Articles

No Panic in Pandemic: The Impact of Individual Choice on Public Health Policy

Problem definition: Public health interventions, such as social distancing and lockdown, play an important role in containing infectious disease outbreaks, such as coronavirus disease 2019 (COVID-19). Yet, these interventions could cause significant financial losses because of the disruption to regular socioeconomic activities. Moreover, an individual’s activity level is influenced not only by public health policies but also by one’s perception of the disease burden of infection. Strategic planning is required to optimize the timing and intensity of these public health interventions by considering individual responses. Methodology/results: We use the multinomial logit choice model to characterize individual reactions to the risk of infection and public health interventions and integrate it into a repeated Stackelberg game with the susceptible-infected-recovered disease transmission dynamics. We find that the individual equilibrium activity level is higher than the socially optimal activity level because of an individual’s ignorance of the negative externality imposed on others. As a result, implementing lockdown and social distancing policies at moderate disease prevalence may be equivalently critical, if not more, compared with their implementations when the disease prevalence is at its peak level. To verify these findings, we conduct numerical studies based on representative COVID-19 data in Minnesota. Managerial implications: Our results call for policymakers’ attention to consider the impact of individuals’ responses in the planning for different pandemic containment measures. Individuals’ responses in the pandemic may significantly affect the optimal implementation of lockdown and social distancing policies. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0514 .

Does Crowdshipping of Parcels Generate New Passenger Trips? Evidence from the Netherlands

Crowdsourced shipping or crowdshipping is a promising solution to sustainable parcel delivery, owing to the potential to consolidate freight trips with preexisting passenger trips. Previous literature focuses on these consolidation benefits but does not address the possibility of new activity generation in crowdshipping. In this study, we investigate the willingness of private persons to accept shipments based on a newly generated home-based trip. We compare this to the choices of occasional carriers who build on the daily home–work commute to deliver parcels. Two stated preference experiments are conducted and a multinomial logit choice model and a latent class choice model are employed. These allow us to provide values of time of the occasional carriers, as an original contribution to the literature. The results show that commute-based carrier values of time are higher than those of home-based carriers. Concerning the trip generating power of crowdshipping, we find that low-income groups have a relatively high propensity to generate a home-based pickup and delivery trip. Finally, parcel lockers as delivery points positively influence acceptance of crowdshipping requests, as they allow for more flexibility in delivery times. Together, these results support the notion that crowdshipping can act as a potential trip generator in households.

Correction to: Optimal pricing for a multinomial logit choice model with network effects

Correction to: Optimal pricing for a multinomial logit choice model with network effects

Multiple-purchase choice model: estimation and optimization

Although multiple-purchase behavior is typical in retail practice, the choice model to portray such behavior is limited 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. 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.

Optimizing the economic and environmental benefits of ride‐hailing and pooling

Ride‐hailing platforms such as Uber and Lyft promise to reduce the negative externalities of driving and improve access to transportation. However, recent empirical evidence has been mixed about the impact of ride‐hailing on US cities, often resulting in a net increase in traffic congestion and greenhouse gas (GHG) emissions, largely due to increased travel demand and competition with public transit. Pooled rides, in which multiple passengers share a single vehicle, are an effective solution to improve the sustainability of ride‐hailing, reducing GHG emissions and traffic congestion and appealing to price‐sensitive population segments by offering relatively cheaper rides. Yet, most ride‐hailing trips are unprofitable currently, resulting from ride‐hailing rides being subsidized (especially pooled) to compete with cheaper transportation alternatives such as public transit. In this paper, we consider whether price optimization can be used to improve ride‐hailing revenues while also reducing the environmental impacts of ride‐hailing, particularly as the cost of ride‐hailing is expected to fall into the future with the introduction of automated vehicles. Using a discrete choice experiment and multinomial logit choice model with a representative sample of the US population, we estimate consumer preferences for the attributes of ride‐hailing services and use them to explore how ride prices affect the revenue of ride‐hailing platforms and the total vehicle miles traveled (VMT) by the ride‐hailing fleet. We show that as the costs of driving fall, continuously increasing the difference between the prices of individual and pooled rides is financially optimal for ride‐hailing platforms. Importantly, this pricing strategy also significantly reduces total VMT, resulting in a win–win for ride‐hailing platforms and cities. We perform extensive sensitivity analyses and show that our results are qualitatively robust under a wide range of consumer preferences and market conditions but that the optimal trajectory of prices and realized gains vary, highlighting opportunities for ride‐hailing services to influence the future of urban transportation.

Bi-objective capacitated assortment planning under the MNL model: Trade-offs between revenue and market share

Assortment planning is a challenge for retailers because most current solutions only consider revenue maximization and ignore the trade-offs between revenue and market share. To address this challenge, we develop a bi-objective capacitated assortment planning (B-CAP) problem based on the multinomial logit (MNL) choice model and reveal managerial insights into the bi-objective function of the B-CAP problem. To solve this problem, we first examine the performance guarantees of the revenue-ordered assortment. Then, we provide parametric linear programming to generate candidate assortments and analyze the unimodality of the bi-objective function to simplify the computational complexity. Hence, the two-stage approach consisting of the geometric algorithm and Fibonacci search is designed to obtain the optimal solution. Finally, we present numerical experiments on both simulated and real data. The results indicate that the two-stage approach performs well in solving the B-CAP problem. Besides, an interesting finding is that market share may be significantly increased without a huge loss in total revenue when the trade-off parameter is small.

The Stability of MNL-Based Demand Under Dynamic Customer Substitution and Its Algorithmic Implications

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, that is, 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. Although this dynamic setting is extensively studied, the best known approximation algorithm guarantees an expected revenue of at least 0.139 times the optimum, assuming that the demand distribution has an increasing failure rate. In this paper, we establish novel stochastic inequalities showing that, for any given inventory level, the expected demand of each offered product is “stable” under basic algorithmic operations, such as scaling the MNL preference weights and shifting inventory across comparable products. We exploit this sensitivity analysis to devise the first approximation scheme for dynamic assortment planning under the MNL model.

Fulfillment and pricing optimization for omni-channel retailers considering shipment of in-store demand

Omni-channel retailers meet the demand through the e-commerce channel and the stores in different locations. They have specific centers to fulfill their online orders (e-fulfillment centers); however, they may fulfill their online orders using their in-store inventory. On the other hand, retailers can use pricing decisions to control the demand in each channel. We first model the customers’ demand by the multinomial logit (MNL) choice model. We then propose two mixed-integer nonlinear programming (MINLP) models to maximize the total profit by considering both fulfillment and pricing decisions. We assume e-fulfillment inventory can also be used as a backup inventory for in-store orders if stores are out of inventory, but only for a store in the same zone. We also assume limited initial inventory and no replenishment in a finite selling horizon. In the second MINLP model, we also allow for multiple shipping options and inventory decisions. By numerical experiments, we show that omni-channel retailers can increase their profit by having control over initial inventory allocation and fulfilling the in-store demand by e-fulfillment inventory.

Queues with Delayed Information: Analyzing the Impact of the Choice Model Function

In this paper, we study queueing systems with delayed information that use a generalization of the multinomial logit choice model as its arrival process. Previous literature assumes that the functional form of the multinomial logit model is exponential. However, in this work we generalize this to different functional forms. In particular, we compute the critical delay and analyze how it depends on the choice of the functional form. We highlight how the functional form of the model can be interpreted as an exponential model where the exponential rate parameter is uncertain. Furthermore, the rate parameter distribution is given by the inverse Laplace–Stieltjes transform of the functional form when it exists. We perform numerous numerical experiments to confirm our theoretical insights.

The impact of retailers’ loyalty programs in a supply chain context

This paper studies the impact of retailers’ loyalty programs (LPs) in a supply chain context. We consider a supply chain formed by one manufacturer selling its product through two competing retailers and focus on frequency loyalty programs that reward consumers with a free product after several purchases. Two different games are modeled: (1) a Stackelberg game, where the manufacturer is the leader and decides about the wholesale price, and the retailers are followers, and (2) a Nash game between the retailers, which decide about their price, whether to offer an LP and reward timing if they offer an LP. To do so, we formulate all players’ demand/profit using a multinomial logit choice model where consumers’ purchasing decisions depend on the utility they get from the different retailers’ products and rewards. To include different consumer mentalities, we consider two parameters: sensitivity to reward value and sensitivity to time distance. Using an iterative algorithm, we numerically solve the games for the market’s equilibrium prices and LP policies under serving consumers with different mentalities. We also study the market conditions under which retailers follow pre-set LP policies. Our main findings indicate that LPs can inflate prices at both levels of the supply chain, which can aggravate vertical externalities related to double marginalization. We show that supply chain managers should consider a profit-sharing agreement when retailers do not benefit from LPs. Further, we show that retailers should only offer LPs when consumers either have a low discount rate or/and they value rewards highly.

The Exponomial Choice Model for Assortment Optimization: An Alternative to the MNL Model?

In this paper, we consider the yet-uncharted assortment optimization problem under the exponomial choice model, in which 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, showing that the optimal expected revenue can be efficiently approached within any degree of accuracy. We synthesize several ideas related to approximate dynamic programming, intended to construct a compact discretization of the continuous state space by keeping track of “key statistics” in rounded form and by operating with a suitable bit precision complexity. We complement this result by a number of NP-hardness reductions to natural extensions of this problem. Moreover, we conduct empirical and computational evaluations of the exponomial choice model and our solution method. Focusing on choice models with a simple parametric structure, we provide new empirical evidence that the exponomial choice model can achieve higher predictive accuracy than the multinomial logit (MNL) choice model on several real-world data sets. We uncover that this predictive performance correlates with certain characteristics of the choice instance—namely, the entropy and magnitude of choice probabilities. Finally, we leverage fully ranked preference data to simulate the expected revenue of optimal assortments prescribed using the fitted exponomial and MNL models. On semisynthetic data, the exponomial-based approach can lift revenues by 3%–4% on average against the corresponding MNL benchmark. This paper was accepted by Chung Piaw Teo, optimization. Funding: The research of Danny Segev on this project was supported by the Israel Science Foundation [Grants 148/10 and 1407/20]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2022.4492 .

Online Personalized Assortment Optimization with High-Dimensional Customer Contextual Data

Problem definition: Consider an online personalized assortment optimization problem in which customers arrive sequentially and make their decisions (e.g., click an ad, purchase a product) following the multinomial logit choice model with unknown parameters. Utilizing a customer’s personal information that is high-dimensional, the firm selects an assortment tailored for each individual customer’s preference. Academic/practical relevance: High dimensionality of a customer’s contextual information is prevalent in real applications, and it creates tremendous computational challenge in online personalized optimization. Methodology: In this paper, an efficient learning algorithm is developed to tackle the computational complexity issue while maintaining satisfactory performance. The algorithm first applies a random projection for dimension reduction and incorporates an online convex optimization procedure for parameter estimation, thus overcoming the issue of linearly increasing computational requirement as data accumulates. Then, it integrates the upper confidence bound method to balance the exploration and revenue exploitation. Results: The theoretical performance of the algorithm in terms of regret is derived under some plausible sparsity assumption on personal information that is observed in real data, and numerical experiments using both synthetic data and a real data set from Yahoo! show that the algorithm performs very well, having scalability and significant advantage in computational time compared with benchmark methods. Managerial implications: Our findings suggest that practitioners should process high-dimensional sparse customer data with an appropriate feature engineering technique, such as random projection (instead of abandoning the sparse portion) to maximize the effectiveness of online optimization algorithms.

Robust product line pricing under the multinomial logit choice model

Incorporating consumer choice behavior into a product line design optimization model enhances the understanding of consumer choices and improves the opportunities to increase profit. Most product line optimization problems assume that parameters are precisely known in consumer choice model. However, the decision maker does not precisely know the model parameters because of insufficient sample data, measurement problems, and other factors. We investigate the problem of establishing robust product line pricing under a multinomial logit model to account for the uncertainty of the valuation parameter. First, we present a nominal product line model to maximize profit. We then establish a robust product line model to maximize the worst-case expected profit, where the valuation parameter lies in an uncertainty set. We consider both single and multiple products development and derive the optimal prices’ closed-form expressions. Through numerical experiments, we illustrate the benefit of robust product line pricing to address parameter uncertainty. We demonstrate that the difference between the expected nominal profit and the worst-case profit increases with the increase of the interval of the uncertainty set, and the robust profit relative to the worst-case nominal profit improves. The robust product line design can ensure steadier, even higher profit.

Stochastic approximation for uncapacitated assortment optimization under the multinomial logit model

AbstractWe consider dynamic assortment optimization with incomplete information under the uncapacitated multinomial logit choice model. We propose an anytime stochastic approximation policy and prove that the regret—the cumulative expected revenue loss caused by offering suboptimal assortments—after time periods is bounded by times a constant that is independent of the number of products. In addition, we prove a matching lower bound on the regret for any policy that is valid for arbitrary model parameters—slightly generalizing a recent regret lower bound derived for specific revenue parameters. Numerical illustrations suggest that our policy outperforms alternatives by a significant margin when and the number of products are not too small.

Network Revenue Management Under a Spiked Multinomial Logit Choice Model

Customers are sensitive to price when they shop from a group of substitutable products. Specifically, they are often attracted to the cheapest product in the group, even when the price difference to the next cheapest product is small. Such a phenomenon has been long noticed by the airline industry when customers purchase airline tickets. In “Network Revenue Management under a Spiked Multinomial Logit Choice Model,” Cao, Kleywegt, and Wang consider a new choice model to incorporate customers’ preference toward the cheapest product, which extends the popular multinomial logit choice model. They find that using this new choice model may improve airlines’ revenue. Moreover, (near) optimal revenue management policies under this choice model can be computed in an efficient manner.

Online Learning and Optimization for Revenue Management Problems with Add-on Discounts

We study in this paper a revenue-management problem with add-on discounts. The problem is motivated by the practice in the video game industry by which a retailer offers discounts on selected supportive products (e.g., video games) to customers who have also purchased the core products (e.g., video game consoles). We formulate this problem as an optimization problem to determine the prices of different products and the selection of products for add-on discounts. In the base model, we focus on an independent demand structure. To overcome the computational challenge of this optimization problem, we propose an efficient fully polynomial-time approximation scheme (FPTAS) algorithm that solves the problem approximately to any desired accuracy. Moreover, we consider the problem in the setting in which the retailer has no prior knowledge of the demand functions of different products. To solve this joint learning and optimization problem, we propose an upper confidence bound–based learning algorithm that uses the FPTAS optimization algorithm as a subroutine. We show that our learning algorithm can converge to the optimal algorithm that has access to the true demand functions, and the convergence rate is tight up to a certain logarithmic term. We further show that these results for the independent demand model can be extended to multinomial logit choice models. In addition, we conduct numerical experiments with the real-world transaction data we collect from a popular video gaming brand’s online store on Tmall.com. The experiment results illustrate our learning algorithm’s robust performance and fast convergence in various scenarios. We also compare our algorithm with the optimal policy that does not use any add-on discount. The comparison results show the advantages of using the add-on discount strategy in practice. This paper was accepted by J. George Shanthikumar, big data analytics.

Assortment Optimization under Multinomial Logit Choice Model with Tree Structured Consideration Sets

Assortment Optimization under Multinomial Logit Choice Model with Tree Structured Consideration Sets

Sparse assortment personalization in high dimensions

The data-driven conditional multinomial logit choice model with customer features performs well in the assortment personalization problem when the low-rank structure of the parameter matrix is considered. However, despite recent theoretical and algorithmic advances, parameter estimation in the choice model still poses a challenging task, especially when there are more predictors than observations. For this reason, we suggest a penalized likelihood approach based on a feature matrix to recover the sparse structure from populations and products toward the assortment. Our proposed method considers simultaneously low-rank and sparsity structures, which can further reduce model complexity and improve its estimation and prediction accuracy. A new algorithm, sparse factorial gradient descent (SFGD), was proposed to estimate the parameter matrix, which has high interpretability and efficient computing performance. As a first-order method, the SFGD works well in high-dimensional scenarios because of the absence of the Hessian matrix. Simulation studies show that the SFGD algorithm outperforms state-of-the-art methods in terms of estimation, sparsity recovery, and average regret. We also demonstrate the effectiveness of our proposed method using advertising behavior data analysis.

Dynamic pricing and cross-channel fulfillment for omnichannel retailing industry: An approximation policy and implications

Emerging to satisfy the needs of consumers in the e-commerce era, omni-channel retailing is the integration of the online channel and the offline store network with the key features of channel integration and inventory sharing. As such, the omni-channel retailer faces more challenging problems of pricing and order fulfillment compared with the pure e-commence retailer. With the omni-channel features, we study the joint dynamic pricing and (cross-channel) order fulfillment problem of an omni-channel retailer with a network of physical stores and fulfillment centers facing limited initial inventory and no replenishment opportunity in a finite selling season. The classical multinomial logit (MNL) choice model is used to characterize the customer demand with the above omni-channel features. Then, the omni-channel joint dynamic pricing and order fulfillment policy (OPFP) is identified based on the optimal solution of the deterministic relaxation to the stochastic control formulation of the original problem. Theoretically, we demonstrate that the proposed approximation policy is asymptotically optimal. We examine some variants of this OPFP to address the problem under various practical constraints and show that they maintain sufficiently good performance. In addition, we show that the proposed OPFP can yield a lower average order fulfillment cost than the heuristic order fulfillment policy that chooses the bricks-and-mortar store with the lowest fulfillment cost. We also find that the improved performance of the re-optimization policy can be offset by the loss of sales volume transfer between channels for large-scale problems.

A regret lower bound for assortment optimization under the capacitated MNL model with arbitrary revenue parameters

AbstractIn this note, we consider dynamic assortment optimization with incomplete information under the capacitated multinomial logit choice model. Recently, it has been shown that the regret (the cumulative expected revenue loss caused by offering suboptimal assortments) that any decision policy endures is bounded from below by a constant times $\sqrt {NT}$, where $N$ denotes the number of products and $T$ denotes the time horizon. This result is shown under the assumption that the product revenues are constant, and thus leaves the question open whether a lower regret rate can be achieved for nonconstant revenue parameters. In this note, we show that this is not the case: we show that, for any vector of product revenues there is a positive constant such that the regret of any policy is bounded from below by this constant times $\sqrt {N T}$. Our result implies that policies that achieve ${{\mathcal {O}}}(\sqrt {NT})$ regret are asymptotically optimal for all product revenue parameters.