Demand Censoring Research Articles

Optimal Policies for Dynamic Pricing and Inventory Control with Nonparametric Censored Demands

We study the classic model of joint pricing and inventory control with lost sales over T consecutive review periods. The firm does not know the demand distribution a priori and needs to learn it from historical censored demand data. We develop nonparametric online learning algorithms that converge to the clairvoyant optimal policy at the fastest possible speed. The fundamental challenges rely on that neither zeroth-order nor first-order feedbacks are accessible to the firm and reward at any single price is not observable due to demand censoring. We propose a novel inversion method based on empirical measures to consistently estimate the difference of the instantaneous reward functions at two prices, directly tackling the fundamental challenge brought by censored demands. Based on this technical innovation, we design bisection and trisection search methods that attain an [Formula: see text] regret for the case with concave reward functions, and we design an active tournament elimination method that attains [Formula: see text] regret when the reward functions are nonconcave. We complement the [Formula: see text] regret upper bound with a matching [Formula: see text] regret lower bound. The lower bound is established by a novel information-theoretical argument based on generalized squared Hellinger distance, which is significantly different from conventional arguments that are based on Kullback-Leibler divergence. Both the upper bound technique based on the “difference estimator” and the lower bound technique based on generalized Hellinger distance are new in the literature, and can be potentially applied to solve other inventory or censored demand type problems that involve learning. This paper was accepted by Jeannette Song, operations management. Supplemental Material: The data files and online appendix are available at https://doi.org/10.1287/mnsc.2023.4859 .

Offline Pricing and Demand Learning with Censored Data

We study a single product pricing problem with demand censoring in an offline data-driven setting. In this problem, a retailer has a finite amount of inventory and faces a random demand that is price sensitive in a linear fashion with unknown price sensitivity and base demand distribution. Any unsatisfied demand that exceeds the inventory level is lost and unobservable. We assume that the retailer has access to an offline data set consisting of triples of historical price, inventory level, and potentially censored sales quantity. The retailer’s objective is to use the offline data set to find an optimal price, maximizing his or her expected revenue with finite inventories. Because of demand censoring in the offline data, we show that the existence of near-optimal algorithms in a data-driven problem—which we call problem identifiability—is not always guaranteed. We develop a necessary and sufficient condition for problem identifiability by comparing the solutions to two distributionally robust optimization problems. We propose a novel data-driven algorithm that hedges against the distributional uncertainty arising from censored data, with provable finite-sample performance guarantees regardless of problem identifiability and offline data quality. Specifically, we prove that, for identifiable problems, the proposed algorithm is near-optimal and, for unidentifiable problems, its worst-case revenue loss approaches the best-achievable minimax revenue loss that any data-driven algorithm must incur. Numerical experiments demonstrate that our proposed algorithm is highly effective and significantly improves both the expected and worst-case revenues compared with three regression-based algorithms. This paper was accepted by J. George Shanthikumar, big data analytics. Funding: This work was supported by the MIT Data Science Laboratory. J. Bu was partially supported by a Hong Kong Polytechnic University Start-up Fund for New Recruits [Project ID P0039585]. Supplemental Material: Data and the online appendices are available at https://doi.org/10.1287/mnsc.2022.4382 .

Data-Driven Dynamic Pricing and Ordering with Perishable Inventory in a Changing Environment

We consider a retailer that sells a perishable product, making joint pricing and inventory ordering decisions over a finite time horizon of T periods with lost sales. Exploring a real-life data set from a leading supermarket chain, we identify several distinctive challenges faced by such a retailer that have not been jointly studied in the literature: the retailer does not have perfect information on (1) the demand-price relationship, (2) the demand noise distribution, (3) the inventory perishability rate, and (4) how the demand-price relationship changes over time. Furthermore, the demand noise distribution is nonparametric for some products but parametric for others. To tackle these challenges, we design two types of data-driven pricing and ordering (DDPO) policies for the cases of nonparametric and parametric noise distributions. Measuring performance by regret, that is, the profit loss caused by not knowing (1)–(4), we prove that the T-period regret of our DDPO policies are in the order of [Formula: see text] and [Formula: see text] in the cases of nonparametric and parametric noise distributions, respectively. These are the best achievable growth rates of regret in these settings (up to logarithmic terms). Implementing our policies in the context of the aforementioned real-life data set, we show that our approach significantly outperforms the historical decisions made by the supermarket chain. Moreover, we characterize parameter regimes that quantify the relative significance of the changing environment and product perishability. Finally, we extend our model to allow for age-dependent perishability and demand censoring and modify our policies to address these issues. This paper was accepted by David Simchi-Levi, Management Science Special Section on Data-Driven Prescriptive Analytics.

Incentive Design for Operations-Marketing Multitasking

A firm hires an agent (e.g., a store manager) to undertake both operational and marketing tasks. Marketing tasks boost demand, but for demand to translate into sales, operational effort is required to maintain adequate inventory. The firm designs a compensation plan to induce the agent to put effort into both marketing and operations while facing demand censoring (i.e., demand in excess of available inventory is unobservable). We formulate this incentive-design problem in a principal-agent framework with a multitasking agent subject to a censored signal. We develop a bang-bang optimal control approach, with a general optimality structure applicable to a broad class of incentive-design problems. Using this approach, we characterize the optimal compensation plan, with a bonus region resembling a “mast” and “sail” such that a bonus is paid when either all inventory above a threshold is sold or the sales quantity meets an inventory-dependent target. The optimal mast-and-sail compensation plan implies nonmonotonicity, where the agent can be less likely to receive a bonus for achieving a better outcome. This gives rise to an ex post moral hazard issue where the agent may “hide” inventory to earn a bonus. We show that this ex post moral hazard issue is a result of demand censoring. If available information includes a waiting list (or other noisy signals) to gauge unsatisfied demand, no ex post moral hazard issues remain. This paper was accepted by Vishal Gaur, operations management.

Exploring the nonlinear dynamics of the lost-sales order-up-to policy

With most inventory theory investigating linear models, the dynamics of nonlinear inventory systems is not well understood. We explore the dynamics of the order-up-to policy under lost-sales for the case of i.i.d. normally distributed demand and unit lead times. We consider the ideal minimum mean-squared error forecast and two alternative scenarios: partial demand observation and dynamic demand forecasting, providing a broad understanding of the operational performance of lost-sales systems. In each scenario, we obtain analytical expressions for the order, inventory, and satisfied demand distributions. This allows us to quantify the Bullwhip and inventory variance amplification ratios as well as the fill rate and the inventory cover. We show the lost sales nonlinearity induces complex behaviours in inventory systems. Interestingly, lost sales smooth supply chain dynamics, significantly affecting the trade-off between service level and average inventory holding. We also reveal the inventory downsides of demand censoring and the production damages induced by dynamic forecasts. We identify a key parameter, the relative safety margin, that characterises the performance of lost-sales systems. We finish by offering some prescriptive results for the optimal safety stock and capacity level in both a retail and a manufacturing lost-sales setting.

Offline Pricing and Demand Learning with Censored Data

We study a single product pricing problem with demand censoring in an offline data-driven setting. In this problem, a retailer is given a finite amount of inventory, and faces a random demand that is price sensitive in a linear fashion with unknown price sensitivity and base demand distribution. Any unsatisfied demand that exceeds the inventory level is lost and unobservable. We assume that the retailer has access to an offline data set consisting of triples of historical price, inventory level and potentially censored sales quantity. The retailer's objective is to use the offline data set to find an optimal price, maximizing her expected revenue with finite inventories. Due to demand censoring in the offline data, we show that the existence of near-optimal algorithms in a data-driven problem -- which we call problem identifiability -- is not always guaranteed. We develop a necessary and sufficient condition for problem identifiability, and show that it is directly related to some quality metrics of the censored data. We propose a novel data-driven algorithm that hedges against the distribution uncertainty arising from censored data, and admits provable finite-sample performance guarantees regardless of problem identifiability and offline data quality. Specifically, we prove that, for identifiable problems, the proposed algorithm is near-optimal, and, for unidentifiable problems, its worst-case revenue loss approaches the best-achievable performance guarantee among all data-driven algorithms. The numerical experiments demonstrate that our algorithm is highly effective and significantly improves both the expected revenue and worst-case revenue compared with three baseline algorithms.

Discrete-item Inventory Control involving Fixed Setup Costs, Demand Censoring, and Unknown Distribution

We investigate a dynamic inventory control problem in which orderings involve fixed setup costs and unsatisfied demands are lost. Stepping off from the traditional literature which stresses the randomness of demand realizations, we move on to tolerate ambiguity. Now even the demand distribution $f\equiv (f(d))_{d=0,1,...}$ itself, out of which random realizations are sampled, can come from a vast and definitely non-singleton set. Lost sales and demand ambiguity would together complicate the problem through censoring, namely, the inability of the firm to observe the lost portion of the demand. Our main policy idea advocates periodically ordering up to seemingly wastefully high levels just to learn and in intervening periods, cleverly exploiting the information gained in these learning periods. By regret, we mean the price paid for ambiguity in long-run average performances. When demand support is unlimited, regret bounds in the orders of $\mathcal{O}(T^{8/9})$ and $\mathcal{O}(T^{(2+\sqrt{2})/4})\simeq \mathcal{O}(T^{0.854})$ can be established for, respectively, the case where $f(0)$ is bounded away from one and the other where the restriction is relaxed with the firm allowed to remove items from the inventory. When demand has a finite support, both bounds could be improved to the order of $\mathcal{O}(T^{2/3}\cdot (\ln T)^{1/2})$. We also propose other policies. % based on the general learning-while-doing idea but the different Kaplan-Meier (KM) estimation. Our simulation demonstrates the merits of the various policy ideas and the hurdles posed by the prospect of $f(0)\longrightarrow 1^-$.

Data-driven Dynamic Pricing and Ordering with Perishable Inventory in a Changing Environment

We consider a retailer that sells a perishable product, making joint pricing and inventory ordering decisions over a finite time horizon of T periods with lost sales. Exploring a real-life data set from a leading supermarket chain, we identify several distinctive challenges faced by such a retailer that have not been jointly studied in the literature: the retailer does not have perfect information on (1) the demand-price relationship, (2) the demand noise distribution, (3) the inventory perishability rate, and (4) how the demand-price relationship changes over time. Furthermore, the demand noise distribution is nonparametric for some products but parametric for others. To tackle these challenges, we design two types of data-driven pricing and ordering (DDPO) policies for the cases of nonparametric and parametric noise distributions. Measuring performance by regret, i.e., the profit loss caused by not knowing (1)-(4), we prove that the T-period regret of our DDPO policies are in the order of T^{2/3}(logT)^{1/2} and T^{1/2}logT in the cases of nonparametric and parametric noise distributions, respectively. These are the best achievable growth rates of regret in these settings (up to logarithmic terms). Implementing our policies in the context of the aforementioned real-life data set, we show that our approach significantly outperforms the historical decisions made by the supermarket chain. Moreover, we characterize parameter regimes that quantify the relative significance of the changing environment and product perishability. Finally, we extend our model to allow for age-dependent perishability and demand censoring, and modify our policies to address these issues.

Incentive Design for Operations-Marketing Multitasking

A firm hires an agent (e.g., store manager) to undertake both operational and marketing tasks. Marketing tasks boost demand, but for demand to translate into sales, operational effort is required to maintain adequate inventory. The firm designs a compensation plan to induce the agent to put effort into both marketing and operations while facing “demand censoring” (i.e., demand in excess of available inventory is unobservable). We formulate this incentive-design problem in a principal-agent framework with a multitasking agent subject to a censored signal. We develop a bang-bang optimal control approach, with a general optimality structure applicable to a broad class of incentive-design problems. Using this approach, we characterize the optimal compensation plan, with a bonus region resembling a “mast” and “sail,” such that a bonus is paid when either all inventory above a threshold is sold or the sales quantity meets an inventory-dependent target. The optimal “mast and sail” compensation plan implies non-monotonicity, where the agent can be less likely to receive a bonus for achieving a better outcome. This gives rise to an ex post moral hazard issue where the agent may “hide” inventory to earn a bonus. We show this ex post moral hazard issue is a result of demand censoring. If available information includes a waitlist (or other noisy signals) to gauge unsatisfied demand, no ex post moral hazard issues remain.

Salesforce Contracting Under Supply Uncertainty

We consider the compensation design problem of a firm that hires a salesperson to exert effort to increase demand. We assume both demand and supply to be uncertain, with sales being the smaller of demand and supply, and assume that if demand exceeds supply then unmet demand is unobservable (demand censoring). Under single moral hazard (i.e., when the salesperson’s effort is unobservable to the firm), we show that the optimal contract has an extreme convex form in which a bonus is provided only for achieving the highest sales outcome, even if low realized sales are due to low realized supply (on which the salesperson has no influence). Under double moral hazard (i.e., when the firm can also take supply-related actions that are unobservable to the salesperson), we show that the optimal contract is smoother as it involves positive compensation for intermediate sales outcomes to assure the salesperson that the firm does not have an incentive to deviate to an action that will hurt the agent; in fact, under certain conditions, the contract is concave in sales. We also determine conditions under which, if possible, the firm should postpone contracting until after supply is realized.

Optimal Merchandise Testing with Limited Inventory

We consider the allocation of inventory to stores in a “merchandise test,” whereby a fashion retailer deploys a new product to stores in limited quantities in order to learn about demand prior to the main selling season. Our problem formulation includes practical considerations like fixed costs and multiperiod inventory considerations but is challenging to analyze directly. Instead, we take a bounding approach that isolates the novel aspect of our problem: the impact of test inventory allocation on demand learning. We specifically examine a trade-off between the quantity of stores tested and the quality of observations, which can be impacted by demand censoring due to inventory stockouts. We find that the visibility into the timing of each sales transaction has a pivotal impact on the allocation decisions that optimize learning. When such timing information is unobservable, the retailer may need to consolidate inventory in few stores to increase service levels during the test and thereby to minimize the negative impacts of demand censoring. When sales timing information is observable, the retailer optimizes learning by maximizing the number of sales during the test period without regard to stockouts in individual stores. In a numerical study, we show that heuristics motivated by our analyses are near optimal and that inefficient allocations can considerably reduce the learning value of the merchandise test. We also examine the interplay between the retailer’s learning incentive and the natural incentives brought by fixed costs and profits earned during the test itself. We find that it is sufficient for the retailer to consider only the learning incentive when the main selling period is relatively long compared with the testing period. The online appendix is available at https://doi.org/10.1287/opre.2017.1596 .

Optimal Merchandise Testing with Limited Inventory

We consider the allocation of inventory to stores in a \merchandise test, whereby a fashion retailer deploys a new product to stores in limited quantities in order to learn about demand prior to the main selling season. Our problem formulation includes practical considerations like fixed costs and multiperiod inventory considerations but is challenging to analyze directly. Instead, we take a bounding approach that isolates the novel aspect of our problem: the impact of test inventory allocation on demand learning. We specifically examine a tradeoff between the quantity of stores tested and the quality of observations, which can be impacted by demand censoring due to inventory stockouts. We find that the visibility into the timing of each sales transaction has a pivotal impact on the allocation decisions that optimize learning. When such timing information is unobservable, the retailer may need to consolidate inventory in few stores to increase service levels during the test and thereby to minimize the negative impacts of demand censoring. When sales timing information is observable, the retailer optimizes learning by maximizing the number of sales during the test period without regard to stockouts in individual stores. In a numerical study, we show that heuristics motivated by our analyses are near-optimal and that inefficient allocations can considerably reduce the learning value of the merchandise test. We also examine the interplay between the retailer's learning incentive and the natural incentives brought by fixed costs and profits earned during the test itself. We find that that it is sufficient for the retailer to consider only the learning incentive when the main selling period is relatively long compared with the testing period.

Ornamental Plants in the United States: An Econometric Analysis of a Household‐Level Demand System

ABSTRACTThis study provides an empirical analysis of demand for a large group of ornamental plants using a theory‐based demand model. Specifically, consumer preferences are represented by the Almost Ideal Demand System where allowance is made for demand censoring. Given that revealed‐preference data are usually limited to certain ornamental plants and geographical locations, we exploit unique hypothetical purchase data collected via an online survey regarding 16 annual, perennial, and foliage plants from across the United States. The effect of various socio‐economic and demographic factors on demand for plants is quantified. Our findings indicate that ornamental plants are predominantly price‐elastic with foliage plants being more price‐responsive vis‐a‐vis other plant categories. Further, a majority of plants are expenditure‐elastic with the estimates of foliage plants manifesting the greatest variability. Finally, plants in the same category appear to be closer substitutes. Results benefit ornamental plant industry stakeholders as they determine the best pricing strategies in their specific markets.

Nonparametric Algorithms for Joint Pricing and Inventory Control with Lost-Sales and Censored Demand

We consider the classical joint pricing and inventory control problem with lost-sales and censored demand in which the customer's response to selling price and the demand distribution are not known a priori, and the only available information for decision-making is the past sales data. Conventional approaches, such as stochastic approximation, online convex optimization, and continuum-armed bandit algorithms, cannot be employed since neither the realized values of the pro t function nor its derivatives are known. A major difficulty of this problem lies in the fact that the estimated pro t function from observed sales data is multimodal even when the expected pro t function is concave. We develop a nonparametric data-driven algorithm that actively integrates exploration and exploitation through carefully designed cycles. The algorithm searches the decision space through a sparse discretization scheme to jointly learn and optimize a multimodal (sampled) pro t function, and corrects the estimation biases caused by demand censoring. We show that the algorithm converges to the optimal policy as the planning horizon increases, and obtain the convergence rate of regret. Numerical experiments show that the proposed algorithm performs very well.

Technical Note—Impact of Inventory on Quota-Bonus Contracts with Rent Sharing

We study the impact of limited inventory on optimal sales-force compensation contracts. We adopt a principal-agent framework, characterized by limited liability and rent sharing with the agent. A commonly invoked assumption in the inventory management literature is that the demand distribution satisfies the increasing failure rate (IFR) property. Under this assumption, however, past research has established that a quota-bonus contract—a widely adopted sales-force compensation contract in the practice—cannot sustain in equilibrium. We show that because of demand censoring in the presence of limited inventory (i.e., demand realizations higher than the inventory level are unobservable), a quota-bonus contract is the optimal equilibrium contract, and it exists, even for a demand distribution with the IFR property. Since most well-known distributions satisfy the IFR property, and inventory constraints are operative in many real-world situations, our results significantly extend the scope of the optimality of quota-bonus contracts and underscore the importance of considering the inventory aspect while making sales-force compensation decisions.

The Exploration-Exploitation Tradeoff in the Newsvendor Problem

When an inventory manager attempts to construct probabilistic models of demand based on past data, demand samples are almost never available: only sales data can be used. This limitation, referred to as demand censoring, introduces an exploration-exploitation trade-off as the ordering decisions impact the information collected. Much of the literature has sought to understand how operational decisions should be modified to incorporate this trade-off. We ask an even more basic question: when does the exploration-exploitation trade-off matter? Further, what is the cost imposed by having access only to sales data as opposed to underlying demand samples? We analyze these questions in the context of a well-studied stationary multi-period newsvendor problem in which the decision-maker starts with a prior on the demand distribution. Quite remarkably, we show that, for a broad family of tractable cases, the exploration-exploitation trade-off is negligible for most practical instances; we further articulate the conjunction of conditions that can lead to a non-trivial value. Moreover, we establish that losses stemming from demand censoring are limited, but these are of higher order than those due to ignoring the exploration-exploitation trade-off. In other words, effort to improve information collection about lost sales is more valuable than analytic or computational effort to pin down the optimal policy in the presence of censoring.

Managing Demand Forecasting with Moral Hazard and Demand Censoring

Demand forecasting has recently become a prime candidate for outsourcing. This research investigates how to design an information quality incentive (IQI) mechanism to manage the quality of demand forecasting in a multi-stage model where the company uses a forecaster's demand forecast to manage the production activity. The posterior demand follows a normal distribution with a precision determined by the forecaster's effort. At the end of the planning horizon, the company conducts a review with the forecaster to determine the amount of a transfer payment. If the forecaster's ability is known and the company is allowed to charge a small penalty for any forecast that is out of an acceptable range, we propose an IQI contract that enables the company to achieve the first-best outcome by overcoming moral hazard and demand censoring. In a more difficult case where any negative transfer payment is banned, the company cannot avoid paying an information rent to the forecaster. We show that in terms of reducing information rents, a transfer payment function based on absolute forecast errors is more efficient than a counter part that is based on squared forecast errors. We also extend the analysis to the case where the forecaster's ability is private.

Impact of Inventory on Quota-Bonus Contracts with Rent Sharing

We study the impact of limited inventory on optimal salesforce compensation contracts. We use the framework of Oyer (2000), characterized by limited liability and rent sharing with the agent. A commonly invoked assumption in the inventory management literature is that the demand distribution satisfies the increasing failure rate (IFR) property. Under this assumption, however, Oyer (2000) shows that a quota-bonus contract -- a widely adopted salesforce compensation contract in the practice -- cannot sustain in equilibrium. We show that due to demand censoring in the presence of limited inventory (i.e., demand realizations higher than the inventory level are unobservable), a quota-bonus contract is the optimal equilibrium contract, and it exists, even for a demand distribution with the IFR property. Since many well-known distributions have the IFR property, and inventory constraints are operative in many real-world situations, our results significantly extend the scope of the optimality of quota-bonus contracts. Our results also underscore the importance of considering the inventory aspect while making salesforce compensation decisions.

On Implications of Demand Censoring in the Newsvendor Problem

We consider a repeated newsvendor problem in which the decision maker (DM) does not have access to the underlying demand distribution. The goal of this paper is to characterize the implications of demand censoring on performance. To that end, we compare the benchmark setting in which the DM has access to demand observations to a setting in which the DM may only rely on sales data. We measure performance in terms of regret: the difference between the cumulative costs of a policy and the optimal cumulative costs with knowledge of the demand distribution. Through upper and lower bounds, we characterize the optimal magnitude of the worst-case regret for the two settings, enabling one to isolate the implications of demand censoring. In particular, the results imply that the exploration–exploitation trade-off introduced by demand censoring is fundamentally different in the continuous and discrete demand cases, and that active exploration plays a much stronger role in the latter case. We further establish that in the discrete demand case, the need for active exploration almost disappears as soon as a lost sales indicator (that records whether demand was censored or not) becomes available, in addition to the censored demand samples. This paper was accepted by Gérard P. Cachon, stochastic models and simulation.

On Implications of Demand Censoring in the Newsvendor Problem

We consider a repeated newsvendor problem in which the decision-maker (DM) does not have access to the underlying demand distribution. The goal of the paper is to characterize the implications of demand censoring on performance. To that end, we compare the benchmark setting in which the DM has access to demand observations to a setting in which the DM may only rely on sales data. We measure performance in terms of regret: the difference between the cumulative costs of a policy and the optimal cumulative costs with knowledge of the demand distribution. Through upper and lower bounds, we characterize the optimal magnitude of the worst case regret for the two settings, enabling one to isolate the implications of demand censoring. In particular, the results imply that the exploration-exploitation trade-off introduced by demand censoring is fundamentally different in the continuous and discrete demand cases, and that active exploration plays a much stronger role in the latter case. We further establish that in the discrete demand case, the need for active exploration almost disappears as soon as a lost sales indicator (that records whether demand was censored or not) becomes available, in addition to the censored demand samples.