Multi-period Newsvendor Problem Research Articles

Solving a Distribution-Free Multi-Period Newsvendor Problem With Advance Purchase Discount via an Online Ordering Solution

In this paper, we study a distribution-free multi-period newsvendor problem with advance purchase discount (APD). In addition to the regular-order placed at the beginning of each period, a decision-maker (DM) can also commit to an advance-order from the upstream supplier and receive discounts. The goal of the DM is to maximize total profits, and in this problem, the DM only has access to past demand data. To solve this problem, we apply an online method based on the theory of prediction and learning with expert advice to propose an explicit online ordering solution by using the fixed-stock policy as expert advice. With the properties of the gain function, we derive a theoretical result that guarantees, for any given advance-order quantity, the newsvendor’s cumulative gains achieved by the proposed online ordering solution converge to those from the best expert advice in hindsight for a sufficient large horizon. In addition, we extend the problem to the discrete case and obtain the corresponding explicit strategy and performance guarantee. Finally, numerical studies illustrate the effectiveness of the proposed solution, and the newsvendor’s total profits are comparable to the best expert advice. Sensitivity analysis also shows the robustness of the proposed solution.

The Exploration-Exploitation Trade-off 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 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? To what extent should one deviate from a myopic policy that takes the optimal decision for the current period without consideration for future periods? We analyze these questions in the context of a well-studied stationary multiperiod newsvendor problem in which the decision maker starts with a prior on parameters characterizing the demand distribution. We show that, under very general conditions in both perishable and nonperishable settings, the myopic policy will almost surely learn the optimal decision one would have taken with knowledge of the unknown parameters. Furthermore, in the perishable setting, we analyze finite time performance for a broad family of tractable cases. Through a combination of analytical parametric bounds and exhaustive exact analysis, we show that the myopic optimality gap is negligible for many practical instances. Funding: The third author was partially supported by the National Science Foundation [Grant CMMI-1235023]. Supplemental Material: The online supplement is available at https://doi.org/10.1287/stsy.2022.0093 .

Newsvendor Inventory and Pricing Decisions in the Presence of Stochastic Purchase Return

We consider a multi-period Newsvendor problem where the seller decides the initial inventory level and dynamically sets the prices in every periods. A purchase made in a period can be returned at a (random) future period following a return time distribution. Unlike existing works in the literature, which usually assume an exponential return time distribution with memoryless property for tractability, we consider a sufficiently general class of return time distributions that is not Markovian. The stochastic control formulation of this problem is intractable since the inventory dynamics depend on the timing of all past sales that have not been returned. We overcome this challenge by proposing two computationally tractable policies that are designed based on the solution of the deterministic counterpart of the intractable stochastic model. Our first policy (FP) uses a sequence of fixed prices throughout the horizon; our second policy (DBP) dynamically adjusts the prices based on the realized sales and returns. We show that our proposed policies are asymptotically optimal in the setting where the total market size for the whole selling season is large, with the second policy guaranteeing a stronger performance bound than the first one. Our analysis characterizes the dependency of price adjustments across periods. To test the performance of our policies on non-asymptotic setting, we run numerical experiments using relatively small per-period market size. We demonstrate that, in general, ignoring product returns in inventory planning and pricing decisions results in a 20-40% loss and that naive re-optimizations of a formulation that ignores (future) product returns do not help much; in contrast, our FP policy curbs the loss to 4-20% and our DBP policy further shrinks it to 1.5-13%. This highlights both the value of properly taking into account customers' return behavior in inventory and pricing decisions, and the power of properly devised simple policies.

Online ordering rules for the multi-period newsvendor problem with quantity discounts

In this paper, we study the multi-period newsvendor problem with quantity discounts and stationary demand, where order quantities need to be decided sequentially over a finite or infinite horizon without making statistical assumption on demands. The Weak Aggregating Algorithm (WAA), which is an online learning method of prediction with expert advice, is applied to the problem. We first consider the all-unit discount case, in which the reduced unit cost is applied to all units in an order. We present explicit online ordering rules by using the fixed-stock strategy as expert advice during the application of WAA. A modified gain function implying convoluted contracts between the newsvendor and his supplier is defined to obtain some theoretical guarantees, which ensure that the newsvendor’s average gains are almost as large as those from the best expert advice for a sufficiently large horizon. Further, we generalize the results to the multi-level incremental discount case, in which there are different unit costs for different quantity ranges. We also extend the results to cases where order quantities are integer-valued, and provide correspondingly online ordering rules and theoretical guarantees. Finally, numerical experiments are performed and show that the average gains obtained by the proposed online ordering rules are comparable to those offered by the best experts in hindsight. The results also indicate that the cumulative gains achieved by the ordering rules for the problem with all-unit discount are larger than those achieved by the ordering rules for the problem with incremental discount; and order quantities increase with the discount level for the all-unit discount while decrease for the incremental discount. The results obtained in this paper can provide competitive online ordering rules for industry managers who need to place long-term continuous orders for perishable products when the demand distribution is unknown.

Multi-Period Newsvendor Problem for the Management of Battery Energy Storage Systems in Support of Distributed Generation

Stakeholders’ interests on renewable and clean energy sources experienced a huge increase in the last decades, thanks to the remarkable benefits on climate, economic, and social issues. The integration of flexible energy storage systems represents a great chance for a further increased penetration as they support the mitigation of renewables’ main drawbacks (i.e., stochastic behavior) and guarantee the balance between energy supply and demand enabling non-simultaneous production and consumption. The increased focus on distributed generation and storage was also of interest to the research community which investigated both the economic and technical performances of the integrated systems. The operations management branch addressed this topic, since storage devices present many similarities with traditional inventory management applied to regular commodities. At a user level, the relation between energy production and storage can be studied by analogy with inventory models. Specifically, this study presents a multi-period newsvendor model for the management and optimal sizing of a battery energy storage system installed to increase the self-consumption rate by allowing loads shifting. This work aims to extend the traditional inventory management applying its concepts to energy systems operations in order to minimize the total energy cost. A numerical study is provided to show the behavior of the model.

Risk averse sourcing in a stochastic supply chain: A simulation-optimization approach

A growing need for global sourcing has forced firms to manage more complex supply chains with increasing risks of supply disruptions. Multi sourcing is a common method to hedge against these risks. In the presence of demand uncertainties and supply disruptions, minimizing the downside risk is necessary. Hence, in this paper a new multi-period and scenario based supply chain model consists of a number of unreliable suppliers and a number of retailers is developed in the form of a multi-period newsvendor problem with a risk averse objective function. In the model, there are two types of retailers both faced uncertain demands: risk sensitive and risk neutral. Retailers have three choices to respond to the customer demand: a forward contract, and two option contracts include reserving a certain capacity in the secondary supplier and buying from the spot market. The problem has also developed as an agent-based system. As a solution approach in the large scale problem instances, a simulation-optimization algorithm is developed. Two kinds of heuristics are compared in order to optimize the simulation procedure: genetic algorithm and q-learning. Results showed the efficiency of the q-learning algorithm.

Online ordering strategy for the discrete newsvendor problem with order value-based free-shipping

Suppliers always provide free-shipping for retailers whose total order value exceeds or equals an explicit promotion threshold. This paper incorporates a shipping fee in the discrete multi-period newsvendor problem and applies Weak Aggregating Algorithm (WAA) to offer explicit online ordering strategy. It further considers an extended case with salvage value and shortage cost. In particular, online ordering strategies are derived based on return loss function. Numerical examples serve to illustrate the competitive performance of the proposed ordering strategies. Results show that newsvendors' cumulative return losses are affected by the threshold of the order value-based free-shipping. Moreover, the introduction of salvage value and shortage cost greatly improves the competitive performance of online ordering strategies.

A simulation–optimization approach for a multi-period, multi-objective supply chain with demand uncertainty and an option contract

One of the main challenges in global procurement problems is the uncertainty in the demand and supply sides of supply chains. Besides, decision making in the stochastic supply chains is a complex problem. A powerful technique for decision analysis in complex stochastic problems is simulation. In this paper we propose a simulation-based optimization approach to solve a bi-objective (profit and service level) supply chain with uncertain customer demands and disruption events in the suppliers. The basic assumptions used in this paper are adopted from the multi-period newsvendor problem. In addition, based on the risk attitude of the buyers (retailers), to cope with the uncertainties, they can sign an option contract, reserving additional capacity in the secondary suppliers. Hence, a simulation approach is used to model the behavior (risk attitude) of the buyers. Indeed, because of the demand uncertainty, at the beginning of each contract period, buyers should decide on the amount of ordering from the primary suppliers. The risk attitude of the retailer (as a spectrum) is defined based on the amount of ordering from the primary supplier. Also, we use the Non-dominated Sorting Genetic Algorithm to optimize the bi-objective model. Finally, a numerical example has been solved with the proposed algorithm and the results are reported. The results showed that if the profit is more important than service level, the risk sensitive retailer prefers to show more risk averse behavior.

Learning from Clickstream Data in Online Retail

E-tailers now have the resources to record the sequence of clicks their online customers make, also known as their clickstream. Such detailed browsing information reveals customer preferences and, as such, can be used to improve product assortment and display decisions over time. In this paper, we consider a Bayesian framework to study how an e-tailer can draw inferences about customer preferences using clickstream and sales data, for utilitarian and hedonic product categories. For a given product category, the e-tailer dynamically determines the set of products to display on his website, via clickable pictures, and which products among the displayed ones to make available for purchase. We show that it may be optimal for the e-tailer to display phantom products, which are clickable, but later shown as unavailable for purchase. Under some conditions on the customers' clicking behavior on the website, we prove the optimality of an offer less strategy which is reminiscent of the well-known stock more result in multi-period newsvendor problem with demand learning. We also discuss the implications of our results on when the e-tailer should communicate product availability information with customers. Finally, we quantify the benefits which accrue from learning from clickstream data for the e-tailer. These benefits are substantial, especially for product categories for which customers do not have well-established preferences and go through an exploratory browsing process before making a purchase.

Online ordering policies for a two-product, multi-period stationary newsvendor problem

This paper formulates a two-product, multi-period newsvendor problem in which the two products' total demands are fixed and the newsvendor must decide his order quantity for each product in the subsequent period. This paper adopts the online learning method advanced in prediction with expert advice to study the formulated newsvendor problem. Following stationary expert advice that the order quantities for each product be kept at the same values throughout all periods, this study begins by providing a newsvendor online ordering policy that determines real-valued order quantities. Then, it obtains theoretical results that guarantee that the online ordering policy can achieve competitive cumulative gain compared with the best expert advice. An online ordering policy for deciding integer-valued order quantities and its theoretical guarantee are then proposed. Finally, computational experiments are presented to illustrate the effectiveness of the online ordering policies proposed herein.

Channel Coordination in a Multi-Period Newsvendor Model with Dynamic, Price-Dependent Stochastic Demand

In this study, we extend the single-period newsvendor problem with stochastic demand into a multi-period and time-dependent one and find a solution for it. We analyze the multi-period newsvendor problem with stochastic demand in a Stackelberg framework where the wholesaler is the leader and the retailer (newsvendor) is the follower. We use an additive-multiplicative structure for the demand so that both its mean and variance are considered as functions of the current retail price. Moreover, in our model, the demand mean and variance can be either time-dependent or autonomous with respect to time. A price-dependent memory function is also embedded in this representation of demand that carries the effect of demand level at present over to the demand in future. This leads to a strategic game in which the players must balance immediate high profits with reduced future earnings. We propose a complete solution to this stochastic multi-period Stackelberg game, covering cases with finite and infinite horizons. The theory is illustrated by using Cobb-Douglas demand functions as the deterministic part, while a random variable brings in stochasticity into the model. Because our theory is very flexible with respect to the assumptions made about the demand, market memory, and the behavior of the two agents, it is applicable to a wide variety of functional forms and capable to model many different economic contexts.

Optimal inventory control in a multi-period newsvendor problem with non-stationary demand

The optimal control of inventory in supply chains plays a key role in the competiveness of a corporation. The inventory cost can account for half of company’s logistics cost. The classical inventory models, e.g., newsvendor and EOQ models, assume either a single or infinite planning periods. However, these models may not be applied to perishable products which usually have a certain shelf life. To optimize the total logistic cost for perishable products, this paper presents a multi-period newsvendor model, and the problem is formulated as a multi-stage stochastic programming model with integer recourse decisions. We extend the progressive hedging method to solve the model efficiently. A numerical example and its sensitivity analysis are demonstrated.

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.

Probability-free solutions to the non-stationary newsvendor problem

This paper concerns the multi-period newsvendor problem. In this problem, the decision maker has to decide the order quantity of an item in the subsequent period in which the demand is usually unknown. No statistical assumptions are made about the unknown demand. We adopt an online learning method from the field of prediction with expert advice to study the non-stationary newsvendor problem. We propose newsvendor strategies for both real-valued and integer order quantities. Taking the non-stationary strategies that can switch between different order quantities as benchmark, we prove that our proposed strategies can guarantee that the newsvendor’s cumulative gains are almost as large as those of the best switching strategies with not too many switches. Simple computational experiments are further performed to illustrate the effectiveness of our strategies.

Optimal inventory decisions in a multiperiod newsvendor problem with partially observed Markovian supply capacities

Abstract This paper considers a multi-period news-vendor problem with partially observed supply-capacity information which evolves as a Markovian Process. The supply capacity is fully observed by the buyer when the capacity is smaller than the buyer’s ordering quantity. Otherwise, the buyer knows that the current-period supply capacity is greater than its ordering quantity. Based on these two observations, the buyer updates the future supply-capacity forecasting accordingly. With a dynamic programming formulation, we prove the existence of an optimal ordering policy. We also prove that the optimal order quantity is greater than the myopic order quantity.

A Multiperiod Newsvendor Problem with Partially Observed Demand

We consider a newsvendor problem with partially observed Markovian demand. Demand is observed if it is less than the inventory. Otherwise, only the event that it is larger than or equal to the inventory is observed. These observations are used to update the demand distribution from one period to the next. The state of the resulting dynamic programming equation is the current demand distribution, which is generally infinite dimensional. We use unnormalized probabilities to convert the nonlinear state transition equation to a linear one. This helps in proving the existence of an optimal feedback ordering policy. So as to learn more about the demand, the optimal order is set to exceed the myopic optimal order. The optimal cost decreases as the demand distribution decreases in the hazard rate order. In a special case with finitely many demand values, we characterize a near-optimal solution by establishing that the value function is piecewise linear.