Unobserved Lost Sales Research Articles

Demand Modeling in the Presence of Unobserved Lost Sales

We present an integrated optimization approach to parameter estimation for discrete choice demand models where data for one or more choice alternatives are censored. We employ a mixed-integer program (MIP) to jointly determine the prediction parameters associated with the customer arrival rate and their substitutive choices. This integrated approach enables us to recover proven, (near-) optimal parameter values with respect to the chosen loss-minimization (LM) objective function, thereby overcoming a limitation of prior multistart heuristic approaches that terminate without providing precise information on the solution quality. The imputations are done endogenously in the MIP by estimating optimal values for the probabilities of the unobserved choices being selected. Under mild assumptions, we prove that the approach is asymptotically consistent. For large LM instances, we derive a nonconvex-contvex alternating heuristic that can be used to obtain quick, near-optimal solutions. Partial information, user acceptance criteria, model selection, and regularization techniques can be incorporated to enhance practical efficacy. We test the LM model on simulated and real data and present results for a variety of demand-prediction scenarios: single-item, multi-item, time-varying arrival rate, large-scale instances, and a dual-layer estimation model extension that learns the unobserved market shares of competitors.This paper was accepted by Yinyu Ye, optimization.

Dynamic pricing and inventory management with demand learning: A bayesian approach

We consider a retail firm selling a durable product in a volatile market where the demand is price-sensitive and random but its distribution is unknown. The firm dynamically replenishes inventory and adjusts prices over time and learns about the demand distribution. Assuming that the demand model is of the multiplicative form and unmet demand is partially backlogged, we take the empirical Bayesian approach to formulate the problem as a stochastic dynamic program. We first identify a set of regularity conditions on demand models and show that the state-dependent base-stock list-price policy is optimal. We next employ the dimensionality reduction approach to separate the scale factor that captures observed demand information from the optimal profit function, which yields a normalized dynamic program that is more tractable. We also analyze the effect of demand learning on the optimal policy using the system without Bayesian update as a benchmark. We further extend our analysis to the case with unobserved lost sales and the case with additive demand.

Estimating the Demand Parameters for Single Period Problem, Markov-modulated Poisson Demand, Large Lot Size, and Unobserved Lost Sales

We consider a single-period single-item problem when the demand is a Markov-modulated Poisson process with hidden states, unknown intensities and continuous batch size distribution. The number of customers and lot size are assumed to be large enough. The estimators of demand mean and standard deviation for unobservable lost sales in the steady state are considered. The procedures are based on two censored samples: observed selling durations and the demands over the period. Numerical results are given.

Demand Estimation for Fast Moving Items and Unobservable Lost Sales

The paper is concerned with a single period problem when the demand is a stationary process, and the number of customers and order quantity are sufficiently large. Two estimators of the mean and standard deviation of the demand for unobserved lost sales are considered. The procedures are based on two censored samples: the observed selling durations, i.e. the moments of the last client’s arrivals during the stockout periods, and the demands. Marginal asymptotic distributions are used. The results of simulation for compound Poisson demand are given.

Estimating the Compound Poisson Demand’s Parameters for Single Period Problem for Large Lot Size

We consider a single period problem when the demand is a compound Poisson process with unknown intensity and continuous batch size distribution; and the lot size and the number of customers are large enough. The estimators of the mean and standard deviation of demand for unobserved lost sales are proposed. The procedures are based on the observed selling period’s durations and the demands, and use their marginal asymptotic distributions. Numerical results are given.

Confidence-based optimisation for the newsvendor problem under binomial, Poisson and exponential demand

We introduce a novel strategy to address the issue of demand estimation in single-item single-period stochastic inventory optimisation problems. Our strategy analytically combines confidence interval analysis and inventory optimisation. We assume that the decision maker is given a set of past demand samples and we employ confidence interval analysis in order to identify a range of candidate order quantities that, with prescribed confidence probability, includes the real optimal order quantity for the underlying stochastic demand process with unknown stationary parameter(s). In addition, for each candidate order quantity that is identified, our approach produces an upper and a lower bound for the associated cost. We apply this approach to three demand distributions in the exponential family: binomial, Poisson, and exponential. For two of these distributions we also discuss the extension to the case of unobserved lost sales. Numerical examples are presented in which we show how our approach complements existing frequentist—e.g. based on maximum likelihood estimators—or Bayesian strategies.

Information‐Sensitive Replenishment when Inventory Records Are Inaccurate

Inspired by recent empirical work on inventory record inaccuracy, we consider a periodic review inventory system with imperfect inventory records and unobserved lost sales. Record inaccuracies are assumed to arrive via an error process that perturbs physical inventory but is unobserved by the inventory manager. The inventory manager maintains a probability distribution around the physical inventory level that he updates based on sales observations using Bayes Theorem. The focus of this study is on understanding, approximating, and evaluating optimal forward‐looking replenishment in this environment. By analyzing one‐ and two‐period versions of the problem, we demonstrate several mechanisms by which the error process and associated record inaccuracy can impact optimal replenishment. Record inaccuracy generally brings an incentive for a myopic manager to increase stock to buffer the added uncertainty. On the other hand, a forward‐looking manager will stock less than a myopic manager, in part to improve information content for future decisions. Using an approximate partially observed dynamic programming policy and associated bound, we numerically corroborate our analytical findings and measure the effectiveness of an intelligent myopic heuristic. We find that the myopic heuristic is likely sufficiently good in practical settings targeting high service levels.

Inventory and sales effort management under unobservable lost sales

A retailer needs to make decisions regarding how much to order and how much sales effort to exert in an environment with uncertain demand. One intrinsic complexity in a typical retail environment is caused by the fact that the retailer can obtain information about demand only based on sales, as demand itself is unobservable. Taking a Bayesian approach, Lariviere and Porteus (1999) show that in such a setting a retailer should stock more to increase the probability of an exact demand observation. In this article, we extend their work by allowing the retailer to control both the stocking quantity and sales effort, which can be used to affect demand. We show that their insights with respect to information stalking carry over to this setting. In addition, our model allows gaining a better understanding of optimal sales effort strategies. We find that demand management has a dual role in supporting information gathering: while at the beginning of a product life cycle it is optimal to support learning effects by sharply reducing sales effort, at later stages of the product life cycle an aggressive strategy of increased promotional activities can be used to harvest the information gathered in earlier periods.

Bounds and Heuristics for Optimal Bayesian Inventory Control with Unobserved Lost Sales

In most retail environments, when inventory runs out, the unmet demand is lost and not observed. The sales data are effectively censored by the inventory level. Factoring this censored data effect into demand estimation and inventory control decision makes the problem difficult to solve. In this paper, we focus on developing bounds and heuristics for this problem. Specifically, we consider a finite-horizon inventory control problem for a nonperishable product with unobserved lost sales and a demand distribution having an unknown parameter. The parameter is estimated sequentially by the Bayesian updating method. We first derive a set of solution upper bounds that work for all prior and demand distributions. For a fairly general monotone likelihood-ratio distribution family, we derive relaxed but easily computable lower and upper bounds along an arbitrary sample path. We then propose two heuristics. The first heuristic is derived from the solution bound results. Computing this heuristic solution only requires the evaluation of the objective function in the observed lost-sales case. The second heuristic is based on the approximation of the first-order condition. We combine the first-order derivatives of the simpler observed lost-sales and perishable-inventory models to obtain the approximation. For the latter case, we obtain a recursive formula that simplifies the computation. Finally, we conduct an extensive numerical study to evaluate and compare the bounds and heuristics. The numerical results indicate that both heuristics perform very well. They outperform the myopic policies by a wide margin.

Analysis of Perishable-Inventory Systems with Censored Demand Data

We consider a multiperiod inventory system of a perishable product with unobservable lost sales. Demand distribution parameters are unknown and are updated periodically using the Bayesian approach based on the censored historical sales data. We develop an explicit expression of the first-order condition for optimality that demonstrates the key trade-off of the problem. The result generalizes partial characterizations of this trade-off in the literature. It shows that the myopic solution is a lower bound on the optimal inventory level. It also enables us to quantify the expected marginal value of information.

Dynamic Inventory Management with Learning About the Demand Distribution and Substitution Probability

Awell-known result in the Bayesian inventory management literature is: If lost sales are not observed, the Bayesian optimal inventory level is larger than the myopic inventory level (one should “stock more” to learn about the demand distribution). This result has been proven in other studies under the assumption that inventory is perishable, so the myopic inventory level is equal to the Bayesian optimal inventory level with observed lost sales. We break that equivalence by considering nonperishable inventory. We prove that with nonperishable inventory, the famous “stock more” result is often reversed to “stock less,” in that the Bayesian optimal inventory level with unobserved lost sales is lower than the myopic inventory level. We also prove that making lost sales unobservable increases the Bayesian optimal inventory level; in this specific sense, the famous “stock more” result of other studies generalizes to the case of nonperishable inventory. When the product is out of stock, a customer may accept a substitute or choose not to purchase. We incorporate learning about the probability of substitution. This reduces the Bayesian optimal inventory level in the case that lost sales are observed. Reducing the inventory level has two beneficial effects: to observe and learn more about customer substitution behavior and (for a nonperishable product) to reduce the probability of overstocking in subsequent periods. Finally, for a capacitated production-inventory system under continuous review, we derive maximum likelihood estimators (MLEs) of the demand rate and probability that customers will wait for the product. (Accepting a raincheck for delivery at some later time is a common type of substitution.) We investigate how the choice of base-stock level and production rate affect the convergence rate of these MLEs. The results reinforce those for the Bayesian, uncapacitated, periodic review system.

Dynamic learning, pricing, and ordering by a censored newsvendor

AbstractWe address the problem of determining optimal ordering and pricing policies in a finite‐horizon newsvendor model with unobservable lost sales. The demand distribution is price‐dependent and involves unknown parameters. We consider both the cases of perishable and nonperishable inventory. A very general class of demand functions is studied in this paper. We derive the optimal ordering and pricing policies as unique functions of the stocking factor (which is a linear transformation of the safety factor). An important expression is obtained for the marginal expected value of information. As a consequence, we show when lost sales are unobservable, with perishable inventory the optimal stocking factor is always at least as large as the one given by the single‐period model; however, if inventory is nonperishable, this result holds only under a strong condition. This expression also helps to explain why the optimal stocking factor of a period may not increase with the length of the problem. We compare this behavior with that of a full information model. We further examine the implications of the results to the special cases when demand uncertainty is described by additive and multiplicative models. For the additive case, we show that if demand is censored, the optimal policy is to order more as well as charge higher retail prices when compared to the policies in the single‐period model and the full information model. We also compare the optimal and myopic policies for the additive and multiplicative models. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007

Can the desired service level be achieved when the demand and lost sales are unobserved?

In this paper we consider an inventory management problem in the retail industry with unobserved lost sales. The retailer does not know the demand for a particular product but she has access to Point-Of-Sale (POS) data. The retailer uses a fixed review period, order-up-to level system to control the inventory. The objective of the retailer is to achieve a pre-specified service level. We define the service level as the fraction of demand satisfied from inventory. However, due to the unobserved lost sales nature of the problem, the retailer cannot exactly measure the current service level. We propose a POS data-based mechanism for periodic updating of the order-up-to level. We show that the periodic updating approach yields the desired service level without using any inventory information. Once the periodic updating scheme converges, it also gives the actual demand distribution.

The Censored Newsvendor and the Optimal Acquisition of Information

This paper investigates the effect of demand censoring on the optimal policy in newsvendor inventory models with general parametric demand distributions and unknown parameter values. We show that the newsvendor problem with observable lost sales reduces to a sequence of single-period problems, while the newsvendor problem with unobservable lost sales requires a dynamic analysis. Using a Bayesian Markov decision process approach we show that the optimalin ventory level in the presence of censored demand is higher than would be determined using a Bayesian myopic policy. We explore the economic rationality for this observation and illustrate it with numerical examples.

Stalking Information: Bayesian Inventory Management with Unobserved Lost Sales

Retailers are frequently uncertain about the underlying demand distribution of a new product. When taking the empirical Bayesian approach of Scarf (1959), they simultaneously stock the product over time and learn about the distribution. Assuming that unmet demand is lost and unobserved, this learning must be based on observing sales rather than demand, which differs from sales in the event of a stockout. Using the framework and results of Braden and Freimer (1991), the cumulative learning about the underlying demand distribution is captured by two parameters, a scale parameter that reflects the predicted size of the underlying market, and a shape parameter that indicates both the size of the market and the precision with which the underlying distribution is known. An important simplification result of Scarf (1960) and Azoury (1985), which allows the scale parameter to be removed from the optimization, is shown to extend to this setting. We present examples that reveal two interesting phenomena: (1) A retailer may hope that, compared to stocking out, realized demand will be strictly less than the stock level, even though stocking out would signal a stochastically larger demand distribution, and (2) it can be optimal to drop a product after a period of successful sales. We also present specific conditions under which the following results hold: (1) Investment in excess stocks to enhance learning will occur in every dynamic problem, and (2) a product is never dropped after a period of poor sales. The model is extended to multiple independent markets whose distributions depend proportionately on a single unknown parameter. We argue that smaller markets should be given better service as an effective means of acquiring information.

Estimating negative binomial demand for retail inventory management with unobservable lost sales

The importance of effective inventory management has greatly increased for many major retailers because of more intense competition. Retail inventory management methods often use assumptions and demand distributions that were developed for application areas other than retailing. For example, it is often assumed that unmet demand is backordered and that demand is Poisson or normally distributed. In retailing, unmet demand is often lost and unobserved. Using sales data from a major retailing chain, our analysis found that the negative binomial fit significantly better than the Poisson or the normal distribution. A parameter estimation methodology that compensates for unobserved lost sales is developed for the negative binomial distribution. The method's effectiveness is demonstrated by comparing parameter estimates from the complete data set to estimates obtained by artificially truncating the data to simulate lost sales. © 1996 John Wiley & Sons, Inc.