Base-stock Policy Research Articles

Reliability-Driven Multiechelon Inventory Optimization With Applications to Service Spare Parts for Wind Turbines

There are multiple warehouses in a multiechelon inventory system, and the size of the state space increases exponentially with the number of warehouses. Therefore, the curse of dimensionality becomes unavoidable when performing steady-state analysis. Most existing studies calculate the inventory cost or supply chain reliability based on specific assumptions. For example, it often assumes that the lead time is either zero or an integral multiple of the review period, and that each warehouse adopts a base-stock policy. This article considers a more practical and prevalent situation where the lead time is less than a review period, and a more general (s, S) strategy is adopted. The curse of dimensionality during steady-state analysis is alleviated by decomposing transition probabilities. Then, the cost and supply chain reliability are derived from steady-state distributions. Finally, a case study involving spare part inventory of wind turbines is considered. Nondominated inventory strategies are obtained using the particle swarm optimization method to strike a balance between costs for the wind turbine manufacturer and wind farm owners.

Simulation-based inventory management of perishable products via linear discrete choice models

Retail inventory management of perishable items, like fresh food, is a relevant and complex problem. It is relevant in the light of trends towards the reduction of food waste, and because of potential cross-sales interaction with other item categories. It is complex, because of multiple sources of uncertainty in supply, demand, and quality, and other complicating factors like seasonality within the week, FIFO/LIFO consumer behavior, and potential substitutions between items, possibly because of a stockout. Similar items may be vertically differentiated due to intrinsic quality, which is also related with item age, or brand image, as it could be the case when a retail chain stocks both a brand item and a private label one. In the paper, we adapt a simple discrete choice model to represent consumers’ heterogeneity and different tradeoffs between price and quality, and apply simulation-based optimization to learn simple ordering rules for two vertically differentiated items, adapted to a seasonal case, in order to maximize long-term average profit under a lost sales assumption. While well-known constant and base-stock policies need not be optimal, they are simple to communicate and apply. We explore combinations of such rules for the two items, obtaining some useful managerial insights.

Fixed-interval order-up-to policies and myopic optimal warehouse stock allocation for one-warehouse multiple-retailer systems

We consider a centralized two-level distribution system that consists of one warehouse and multiple retailers. The retailers are non-identical and face independent Poisson demand. The system adopts a fixed-interval order-up-to policy, in which the warehouse and retailers each order in a fixed interval to raise the echelon inventory position to a fixed base stock level. Shortages in the system are fully backlogged but transshipments between retailers are not allowed. The fixed-interval order-up-to policy has been studied in the literature under virtual allocation that reserves a unit of warehouse stock immediately after a unit of demand occurs. We consider an inventory control system that is different in two aspects. First, the warehouse implements myopic optimal allocation that allocates stock to retailers only at points of delivery to minimize system cost. Second, the warehouse adopts a non-stationary base stock policy to order stock for retailers according to their replenishment schedules. We develop methods to exactly evaluate and fully optimize this inventory control system. In a numerical study, we demonstrate that myopic optimal allocation reduces system cost by an average of 5.89% with a range from 1.45% to 10.13% as compared to virtual allocation. In addition, we develop an iterative procedure to alleviate the problem of dimensionality for myopic optimal warehouse stock allocation and a close-to-optimal heuristic solution that can significantly reduce computation.

Asymptotic Scaling of Optimal Cost and Asymptotic Optimality of Base-Stock Policy in Several Multi-Dimensional Inventory Systems

Asymptotic Scaling of Optimal Cost and Asymptotic Optimality of Base-Stock Policy in Several Multi-Dimensional Inventory Systems

Stability of queueing-inventory systems with customers of different priorities

We study a production-inventory system with two customer classes with different priorities which are admitted to the system following a flexible admission control scheme. The inventory management is according to a base stock policy and arriving demand which finds the inventory depleted is lost (lost sales). We analyse the global balance equations of the associated Markov process and derive structural properties of the steady state distribution which provide insights into the equilibrium behaviour of the system. We derive a sufficient condition for ergodicity using the Foster-Lyapunov stability criterion. For a special case we show that the condition is necessary as well.

Inventory control with modulated demand and a partially observed modulation process

We consider a periodic review inventory control problem having an underlying modulation process that affects demand and that is partially observed by the uncensored demand process and a novel additional observation data (AOD) process. We present an attainability condition, AC, that guarantees the existence of an optimal myopic base stock policy if the reorder cost $$K=0$$ and the existence of an optimal (s, S) policy if $$K>0$$ , where both policies depend on the belief function of the modulation process. Assuming AC holds, we show that (i) when $$K=0$$ , the value of the optimal base stock level is constant within regions of the belief space and that each region can be described by two linear inequalities and (ii) when $$K>0$$ , the values of s and S and upper and lower bounds on these values are constant within regions of the belief space and that these regions can be described by a finite set of linear inequalities. A heuristic and bounds for the $$K=0$$ case are presented when AC does not hold. Special cases of this inventory control problem include problems considered in the Markov-modulated demand and Bayesian updating literatures.

Analysis of second order properties of production–inventory systems with lost sales

We consider a single-item production–inventory system under a base stock policy for inventory control. We model the system as a closed Gordon–Newell network. The population size of the network is equal to the base stock level, which is the sum of the finished goods and work-in-process inventory. Each exogenous demand, which follows a Poisson process, releases a production order for a new unit and increases the amount of the work-in-process inventory. When there are no items in the finished goods inventory available, arriving demand is lost. The replenishment network operates with state dependent service rates, which we assume to be increasing and concave. First, we analyze the queue length behavior of a two node system and provide conditions under which the mean queue length at the production server is convex in the number of customers in the system. We prove that this leads to convexity of a standard cost function. Using Norton’s theorem, we are able to generalize our results for arbitrarily large production–inventory systems.

Dynamic supplier competition and cooperation for buyer loyalty on service

A firm’s service-driven always-a-share behavior may have a significant effect on the competitive and cooperative inventory policies of its suppliers. To explore this effect, we consider a model of a repeat buyer (she) sharing her patronage among two heterogeneous newsvendor-type suppliers over an infinite horizon. To enjoy the best service advantage, the buyer plays one supplier (him) against the other by rewarding product availability with repurchase (loyalty) and punishing stockouts with switching (disloyalty) in the next period. Faced with this behavior, the optimal ordering policy of each supplier is a basestock policy with a non-negative “active” basestock level when the supplier has the buyer’s loyalty and a zero basestock level when he does not. Under competition, the optimal active basestock level of each supplier is greater than his myopic basestock level and increases in the other supplier’s active basestock level. Under a mild condition, the active basestock levels of both suppliers have at least one pure-strategy Nash equilibrium solution. If the suppliers cooperate, the optimal active basestock level of the supplier with the highest/lowest myopic profit is greater/smaller than his myopic basestock level. To better comprehend these results, we apply them to the case where the buyer’s demand is exponentially distributed. This allows us to obtain exact expressions for the optimal active basestock levels and payoff functions, which we then use in a numerical sensitivity analysis. We conclude with a discussion of the extension of the results to more than two suppliers.

Asymptotically Optimal Inventory Control for Assemble-to-Order Systems

We consider assemble-to-order (ATO) inventory systems with a general bill of materials and general deterministic lead times. Unsatisfied demands are always backlogged. We apply a four-step asymptotic framework to develop inventory policies for minimizing the long-run average expected total inventory cost. Our approach features a multistage stochastic program (SP) to establish a lower bound on the inventory cost and determine parameter values for inventory control. Our replenishment policy deviates from the conventional constant base stock policies to accommodate nonidentical lead times. Our component allocation policy differentiates demands based on backlog costs, bill of materials, and component availabilities. We prove that our policy is asymptotically optimal on the diffusion scale, that is, as the longest lead time grows, the percentage difference between the average cost under our policy and its lower bound converges to zero. In developing these results, we formulate a broad stochastic tracking model and prove general convergence results from which the asymptotic optimality of our policy follows as specialized corollaries. Funding: This study is based on work supported by the National Science Foundation [Grant CMMI-1363314].

Configuring information updating schemes of partial imperfect advance demand information in production-inventory systems

We consider a make-to-stock system, where a supplier produces a single product to meet a single customer’s order. The order quantity is fixed, but the order time is random. Before the order becomes due, it goes through multiple stages, and the transition time from one stage to the next is random. The customer provides advance demand information (ADI) by updating the order status information to the supplier when the order progresses to a key stage. We study the supplier’s optimal production policy. By formulating this problem as a Markov decision process, we show that the optimal production policy is a state-dependent base-stock policy, and the optimal base-stock level after receiving the update is higher than before. We extend the model to the case with multiple updates. We numerically examine the configuration of information updating schemes, i.e. the timing and number of updates. We find that partial ADI with one update can capture most of the benefits of full ADI in most cases. However, the timing of update is crucial for the benefit of ADI. For a few cases where only one update is less effective, increasing the number of updates slightly can capture most of the benefits of full ADI.

Load balancing in a network of queueing-inventory systems

We study a supply chain consisting of production-inventory systems at several locations which are coupled by a common supplier. Demand of customers arrives at each production system according to a Poisson process and is lost if the local inventory is depleted (“lost sales”). To satisfy a customer’s demand a server at the production system needs raw material from the associated local inventory. The supplier manufactures raw material to replenish the local inventories, which are controlled by a continuous review base stock policy. The routing of items depends on the on-hand inventory at the locations with the aim to obtain “load balancing”. We show that the stationary distribution has a product form of the marginal distributions of the production subsystem and the inventory-replenishment subsystem. For the marginal distribution of the production subsystem we derive an explicit solution and for the marginal distribution of the inventory-replenishment subsystem we deduce an explicit solution or a recursive algorithm for some special cases.

Data‐driven platelet inventory management under uncertainty in the remaining shelf life of units

Platelets are perishable (5–7 day shelf life) blood products required for a variety of clinical treatments. In North America, hospitals typically procure platelet units from a central supplier. As such, the remaining shelf life of the delivered units could be subject to high uncertainty. Our work focuses on developing new models that leverage the increasingly available data from hospital information systems to prescribe ordering decisions in the presence of this uncertainty. Specifically, we consider a periodic review, perishable inventory system with zero lead time and uncertainty in demand and remaining shelf life of orders, operating under an oldest‐unit, first‐out allocation policy. We consider a family of base stock policies and adopt an empirical risk minimization approach to estimate the required inventory at the beginning of each period. The required inventory level for each period is assumed to be a linear function of a set of observed features in that period and the coefficients of the linear model are obtained by minimizing an approximate measure of the in‐sample empirical cost, comprised of a weighted sum of shortage and expiry costs. Our fixed initial age model assumes a constant remaining shelf life for all units. Our robust model assumes that an adversary selects the remaining shelf life of units subject to an uncertainty budget determined through an endogenous uncertainty set. We investigate the out‐of‐sample performance of the proposed models in a case study using data from two Canadian hospitals and in comparison to the hospitals' historical performances as well as other benchmarks. Both models achieve significant improvements over the historical decisions. For instance, the fixed initial age model achieves a 53% and 93% reduction in the expiry rate and an 82% and 99% reduction in the shortage rate for the two hospitals, respectively. Further, it either outperforms or performs as well as the other benchmarks. The robust model achieves better out‐of‐sample generalizability and demonstrates a more “robust” performance under counterfactual remaining age distributions.

Inventory control under corporate income tax and accrual accounting

Corporate income tax constitutes a significant portion of financial costs in business operations. A typical tax system is characterized by the tax structure, accounting method, and loss carryover rule. In this article, we study the optimal inventory policies under taxation and accrual accounting. We consider a lost-sales periodic-review inventory system under a proportional tax with loss carryforward. We formulate the problem as a cyclic stochastic dynamic program with multiple accounting periods, each of which consists of multiple review periods. We show that an income-dependent basestock policy is optimal. We find that tax function convexity and loss carryforward may introduce conflicting incentives into inventory decisions: the former drives risk-averse decisions whereas the latter induces risk-seeking behavior. We identify two intertemporal effects of taxation, the intra-accounting-period effect and the inter-accounting-period effect, both inducing smaller basestock levels. We further extend the analysis to the progressive tax, backordering system, and cash accounting method. Our numerical study examines the effects of various characteristics of taxation on the optimal policies. Our results demonstrate that ignoring taxation in inventory decisions may lead to significant losses especially when the tax rate and demand uncertainty are sufficiently high, which reveals the value of tax considerations in inventory management.

Dual-Channel Retailer’s Multi-Period Inventory and Pricing Policy with the Quantity-Sales Mode

This paper investigates a dual-channel retailer’s inventory and pricing integrated decision-making problem over multiple periods. The quantity-sales mode is adopted in the online channel while the unit-sales mode is used offline. The customer demand of each sales channel is stochastic, and depends on visit costs, customer return costs, sales modes and pricing levels. To achieve the long-term profit maximization, the dual-channel retailer needs to optimize the pricing levels of both channels and the ordering quantity. For this problem, we first formulate a stochastic dynamic programming model, and then prove the joint concavity and the supermodularity of the objective function. With these properties, the optimality of the base-stock policy is proved, and the optimum price policy is characterized. Specifically, if the retailer’s optimum stock is neither too low nor too high, both sales channels need to be operated simultaneously, otherwise, only operating one channel is optimal. The online price is invariant with the retailer’s stock level while the offline price is decreasing in the retailer’s stock level. Finally, numerical results show that the quantity-sales mode, dual-channel sales model, and the multi-period dynamic decisions could benefit the dual-channel retailer. In addition, influences of cost parameters on the total profit are verified numerically.

Bandit-based inventory optimisation: Reinforcement learning in multi-echelon supply chains

Even though base-stock policies are per se straightforward, determining them in complex, stochastic multi-echelon supply chains is often cumbersome or even analytically impossible. Therefore, a wide range of heuristics has been proposed for this purpose. This is the first study considering the problem as a multi-armed bandit problem. In this context, we investigate two algorithms: first, we propose an approach that is based on upper confidence bounds and priority queues. This so-called PQ-UCB algorithm allows us to drastically reduce the runtime of upper confidence bound allocation strategies in problems with large action spaces. Subsequently, we apply the parameter-free sequential halving (SH) algorithm. We investigate various scenarios to compare the performance of both algorithms with the performance of a genetic algorithm and a simulated annealing algorithm taken from the literature. PQ-UCB as well as SH outperform both benchmark metaheuristics and require substantially less effort related to parameter tuning (or even no effort in the case of SH). As multi-armed bandits are not common in inventory optimisation so far, we aim to emphasise their strengths and hope to promote their dissemination also in other domains of supply chain management.

A model of a queuing system with a base stock policy and two supply sources

We propose a Markov model of a two-source queuing-inventory system with an (S-1,S) inventory control policy. Replenishments from different sources are carried out with different delays. If the inventory level falls to a certain threshold value, an emergency order to the fast source is instantly generated, with the fast source being expensive. Events occur in the system, as a result of which the inventory level instantly decreases. The ergodicity conditions of the system under study are found, its stationary distribution is calculated and formulas for finding and optimizing its characteristics are proposed.

Asymptotic Optimality of Base-Stock Policies for Perishable Inventory Systems

We consider periodic review perishable inventory systems with a fixed product lifetime. Unsatisfied demand can be either lost or backlogged. The objective is to minimize the long-run average holding, penalty, and outdating cost. The optimal policy for these systems is notoriously complex and computationally intractable because of the curse of dimensionality. Hence, various heuristic replenishment policies are proposed in the literature, including the base-stock policy, which raises the total inventory level to a constant in each review period. Whereas various studies show near-optimal numerical performances of base-stock policies in the classic system with zero replenishment lead time and a first-in-first-out issuance policy, the results on their theoretical performances are very limited. In this paper, we first focus on this classic system and show that a simple base-stock policy is asymptotically optimal when any one of the product lifetime, demand population size, unit penalty cost, and unit outdating cost becomes large; moreover, its optimality gap converges to zero exponentially fast in the first two parameters. We then study two important extensions. For a system under a last-in-first-out or even an arbitrary issuance policy, we prove that a simple base-stock policy is asymptotically optimal with large product lifetime, large unit penalty costs, and large unit outdating costs, and for a backlogging system with positive lead times, we prove that our results continue to hold with large product lifetime, large demand population sizes, and large unit outdating costs. Finally, we provide a numerical study to demonstrate the performances of base-stock policies in these systems. This paper was accepted by Victor Martinez de Albéniz, operations management. Funding: J. Bu was partially supported by a Hong Kong Polytechnic University Start-up Fund for New Recruits [Grant P0039585]. X. Gong was partially supported by a Chinese University of Hong Kong (CUHK) Direct Grant [Grant 4057147] and the Hong Kong Research Grants Council (RGC) General Research Fund [Grant CUHK14500120]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2022.4400 .

Minimizing conditional value‐at‐risk under a modified basestock policy

Consider a single‐product, make‐to‐stock, risk‐averse company that manages its finished goods inventory by a basestock policy. Item demand is modeled as a Poisson process, and unit production time is modeled as an exponential random variable. Annual cost is equal to the sum over one year of the inventory‐holding and back‐ordering costs. The company wants to minimize the annual cost conditional value‐at‐risk by an appropriate basestock selection called the optimal basestock. After offering evidence that the optimal basestock is analytically intractable, we derive a closed‐form approximation of the optimal basestock. In two independent simulation experiments, we found that the average penalty for using the approximation is less than 1%; we believe this accuracy would be acceptable for practical applications. Leveraging the approximation's accuracy and simplicity, we establish several results that may serve as predictors of the company's appropriate basestock adjustments to changes in operational, economic, and risk aversion parameters. We illustrate how our research results were successfully applied in a Fortune 10 energy company. We extend the problem setting in two different directions to widen the applicability of our research results.

Learning in Structured MDPs with Convex Cost Functions: Improved Regret Bounds for Inventory Management

A fundamental yet notoriously difficult problem in operations management is the periodic inventory control problem under positive lead time and lost sales. More recently, there has been interest in the problem setting where the demand distribution is not known a priori and must be learned from the observations made during the decision-making process. In “Learning in Structured MDPs with Convex Cost Functions: Improved Regret Bounds for Inventory Management,” Agrawal and Jia present a reinforcement learning algorithm that uses the observed outcomes of past decisions to implicitly learn the underlying dynamics and adaptively improve the decision-making strategy over time. They show that, compared with the best base-stock policy, their algorithm achieves an optimal regret bound in terms of the time horizon and scales linearly with the lead time of the inventory ordering process. Furthermore, they demonstrate that their approach is not restricted to the inventory problem and can be applied in an almost black box manner to more general reinforcement learning problems with convex cost functions.

Optimal Policies for a Multi-Echelon Inventory Problem with Service Time Target and Expediting

Problem definition: We study the optimal inventory ordering, expediting, and allocation decisions in a multiechelon supply chain over a finite horizon, in which customer orders are quoted with a fixed fulfillment time window, termed the service time target (STT). Academic/practical relevance: Service time target is commonly used as a marketing strategy to increase customer satisfaction and strengthen firms’ competitive edge. However, how to efficiently manage a multistage supply chain to meet the target has received relatively scant attention in the literature. Our study fills this gap. Methodology: We use dynamic programming to characterize the optimal policy. Results: We show that an echelon base stock policy and a rationing policy are optimal for inventory ordering and inventory allocation/expediting, respectively. We also develop a polynomial-time algorithm to compute the optimal policy. To derive these results, we uncover a new functional property named the decomposable of degree 2 property, which is a nontrivial generalization of the celebrated Clark-Scarf decomposition. This property further allows us to derive induced penalty and compensation to coordinate a decentralized serial system with STT and expediting. Managerial implications: Our result provides an efficient decision tool for managing centralized and decentralized serial supply chains with STT and expediting. Our model can be used to quantify the tradeoff between marketing and operational decisions, such as the impact of a marginal reduction in STT on system cost and expedition frequency, as explored in our numerical studies.