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We propose a method for estimating substitute and lost demand when only sales and product availability data are observable, not all products are displayed in all periods (e.g., due to stockouts or availability controls), and the seller knows its aggregate market share. The model combines a multinomial logit (MNL) choice model with a nonhomogeneous Poisson model of arrivals over multiple periods. Our key idea is to view the problem in terms of primary (or first-choice) demand; that is, the demand that would have been observed if all products had been available in all periods. We then apply the expectation-maximization (EM) method to this model, and we treat the observed demand as an incomplete observation of primary demand. This leads to an efficient, iterative procedure for estimating the parameters of the model. All limit points of the procedure are provably stationary points of the incomplete data log-likelihood function. Every iteration of the algorithm consists of simple, closed-form calculations. We illustrate the effectiveness of the procedure on simulated data and two industry data sets.

We analyze a revenue management problem in which a seller facing a Poisson arrival stream of consumers operates an online multiunit auction. Consumers can get the product from an alternative list price channel. We consider two variants of this problem: In the first variant, the list price is an external channel run by another firm. In the second one, the seller manages both the auction and the list price channels. Each consumer, trying to maximize his own surplus, must decide either to buy at the posted price and get the item at no risk, or to join the auction and wait until its end, when the winners are revealed and the auction price is disclosed. Our approach consists of two parts. First, we study structural properties of the problem, and show that the equilibrium strategy for both versions of this game is of the threshold type, meaning that a consumer will join the auction only if his arrival time is above a function of his own valuation. This consumer’s strategy can be computed using an iterative algorithm in a function space, provably convergent under some conditions. Unfortunately, this procedure is computationally intensive. Second, and to overcome this limitation, we formulate an asymptotic version of the problem, in which the demand rate and the initial number of units grow proportionally large. We obtain a simple closed-form expression for the equilibrium strategy in this regime, which is then used as an approximate solution to the original problem. Numerical computations show that this heuristic is very accurate. The asymptotic solution culminates in simple and precise recipes of how bidders should behave, as well as how the seller should structure the auction, and price the product in the dual-channel case.

During the past few years, there has been a trend to enrich traditional revenue management models built upon the independent demand paradigm by accounting for customer choice behavior. This extension involves both modeling and computational challenges. One way to describe choice behavior is to assume that each customer belongs to a segment , which is characterized by a consideration set , i.e., a subset of the products provided by the firm that a customer views as options. Customers choose a particular product according to a multinomial-logit criterion, a model widely used in the marketing literature. In this paper, we consider the choice-based, deterministic, linear programming model (CDLP) of Gallego et al. (2004) [Gallego, G., G. Iyengar, R. Phillips, A. Dubey. 2004. Managing flexible products on a network. Technical Report CORC TR-2004-01, Department of Industrial Engineering and Operations Research, Columbia University, New York], and the follow-up dynamic programming decomposition heuristic of van Ryzin and Liu (2008) [van Ryzin, G. J., Q. Liu. 2008. On the choice-based linear programming model for network revenue management. Manufacturing Service Oper. Management 10 (2) 288–310]. We focus on the more general version of these models, where customers belong to overlapping segments. To solve the CDLP for real-size networks, we need to develop a column generation algorithm. We prove that the associated column generation subproblem is indeed NP-hard and propose a simple, greedy heuristic to overcome the complexity of an exact algorithm. Our computational results show that the heuristic is quite effective and that the overall approach leads to high-quality, practical solutions.

In retail operations, customer choices may be affected by stockout and promotion events. Given panel data with the transaction history of customers, and product availability and promotion data, our goal is to predict future individual purchases. We use a general nonparametric framework in which we represent customers by partial orders of preferences. In each store visit, each customer samples a full preference list of the products consistent with her partial order, forms a consideration set, and then chooses to purchase the most preferred product among the considered ones. Our approach involves: (a) defining behavioral models to build consideration sets as subsets of the products on offer, (b) proposing a clustering algorithm for determining customer segments, and (c) deriving marginal distributions for partial preferences under the multinomial logit model. Numerical experiments on real-world panel data show that our approach allows more accurate, fine-grained predictions for individual purchase behavior compared to state-of-the-art alternative methods. The online appendix is available at https://doi.org/10.1287/mnsc.2016.2683 . This paper was accepted by Vishal Gaur, operations management.

We propose an expectation-maximization (EM) method to estimate customer preferences for a category of products using only sales transaction and product availability data. The demand model combines a general, rank-based discrete choice model of preferences with a Bernoulli process of customer arrivals over time. The discrete choice model is defined by a probability mass function (pmf) on a given set of preference rankings of alternatives, including the no-purchase alternative. Each customer is represented by a preference list, and when faced with a given choice set is assumed to either purchase the available option that ranks highest in her preference list, or not purchase at all if no available product ranks higher than the no-purchase alternative. We apply the EM method to jointly estimate the arrival rate of customers and the pmf of the rank-based choice model, and show that it leads to a remarkably simple and highly efficient estimation procedure. All limit points of the procedure are provably stationary points of the associated incomplete data log-likelihood function, and the output produced are maximum likelihood estimates (MLEs). Our numerical experiments confirm the practical potential of the proposal. The online appendix is available at https://doi.org/10.1287/opre.2016.1559 .

We propose an approach for estimating customer preferences for a set of substitutable products using only sales transactions and product availability data. The underlying demand framework combines a general, nonparametric discrete choice model with a Bernoulli process of arrivals over time. The choice model is defined by a discrete probability mass function (pmf) on a set of possible preference rankings of alternatives, and it is compatible with any random utility model. An arriving customer is assumed to purchase the available option that ranks highest in her preference list. The problem we address is how to jointly estimate the arrival rate and the pmf of the rank-based choice model under a maximum likelihood criterion. Since the potential number of customer types is factorial, we propose a market discovery algorithm that starts with a parsimonious set of types and enlarge it by automatically generating new types that increase the likelihood value. Numerical experiments confirm the potential of our proposal. For a realistic data set in the hospitality industry, our approach improves the root mean square errors between predicted and observed purchases computed under independent demand model estimates by 67% to 93%. This paper was accepted by Serguei Netessine, operations management.

A technology firm launches newer generations of a given product over time. At any moment, the firm decides whether to release a new version of the product that captures the current technology level at the expense of a fixed launch cost. Consumers are forward-looking and purchase newer models only when it maximizes their own future discounted surpluses. We start by assuming that consumers have a common valuation for the product and consider two product launch settings. In the first setting, the firm does not announce future release technologies and the equilibrium of the game is to release new versions cyclically with a constant level of technology improvement that is optimal for the firm. In the second setting, the firm is able to precommit to a schedule of technology releases and the optimal policy generally consists of alternating minor and major technology launch cycles. We verify that the difference in profits between the commitment and no-commitment scenarios can be significant, varying from 4% to 12%. Finally, we generalize our model to allow for multiple customer classes with different valuations for the product, demonstrating how to compute equilibria in this case and numerically deriving insights for different market compositions. This paper was accepted by Yossi Aviv, operations management.

In retail operations, customer choices may be affected by stockout and promotion events. Given panel data with the transaction history of customers, and product availability and promotion data, our goal is to predict future individual purchases. We use a general nonparametric framework in which we represent customers by partial orders of preferences. In each store visit, each customer samples a full preference list of the products consistent with her partial order, forms a consideration set, and then chooses to purchase the most preferred product among the considered ones. Our approach involves: (a) defining behavioral models to build consideration sets as subsets of the products on offer, (b) proposing a clustering algorithm for determining customer segments, and (c) deriving marginal distributions for partial preferences under the multinomial logit model. Numerical experiments on real-world panel data show that our approach allows more accurate, fine-grained predictions for individual purchase behavior compared to state-of-the-art alternative methods.

In retail operations, customer choices may be affected by stockout and promotion events. Given panel data with the transaction history of customers, and product availability and promotion data, our goal is to predict future individual purchases. We use a general nonparametric framework in which we represent customers by partial orders of preferences. In each store visit, each customer samples a full preference list of the products consistent with her partial order, forms a consideration set, and then chooses to purchase the most preferred product among the considered ones. Our approach involves: (a) defining behavioral models to build consideration sets as subsets of the products on offer, (b) proposing a clustering algorithm for determining customer segments, and (c) deriving marginal distributions for partial preferences under the multinomial logit model. Numerical experiments on real-world panel data show that our approach allows more accurate, fine-grained predictions for individual purchase behavior compared to state-of-the-art alternative methods.

We analyze a dynamic auction, in which a seller with C units to sell faces a sequence of buyers separated into T time periods. Each group of buyers has independent, private values for a single unit. Buyers compete directly against each other within a period, as in a traditional auction, and indirectly with buyers in other periods through the opportunity cost of capacity assessed by the seller. The number of buyers in each period, as well as the individual buyers' valuations, are random. The model is a variation of the traditional single leg, multiperiod revenue management problem, in which consumers act strategically and bid for units of a fixed capacity over time. For this setting, we prove that dynamic variants of the first–price and second–price auction mechanisms maximize the seller's expected revenue. We also show explicitly how to compute and implement these optimal auctions. The optimal auctions are then compared to a traditional revenue management mechanism—in which list prices are used in each period together with capacity controls—and to a simple auction heuristic that consists of allocating units to each period and running a sequence of standard, multiunit auctions with fixed reserve prices in each period. The traditional revenue management mechanism is proven to be optimal in the limiting cases when there is at most one buyer per period, when capacity is not constraining, and asymptotically when the number of buyers and the capacity increases. The optimal auction significantly outperforms both suboptimal mechanisms when there are a moderate number of periods, capacity is constrained, and the total volume of sales is not too large. The benefit also increases when variability in the dispersion in buyers' valuations or in the number of buyers per period increases.