State-dependent Base-stock Policy Research Articles

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.

Joint inventory and pricing control with lagged price responses

This paper studies the single stage, multiple period joint inventory and pricing planning problem with lagged price response. We prove that the optimal joint inventory and pricing policy takes the form of a state dependent list price base stock policy for the additive linear demand function with a first order auto-regressive distributed lag. Both the base stock level and the list price decisions are determined as functions of the effective price, which captures the current and past pricing impacts. Numerical studies are conducted to examine the impact of the lagged price response on the optimal decisions and the total profit.

Reducing risks in spare parts service contracts with a long downtime constraint

This article investigates spare parts service contracts for capital goods. We consider a single-item, single-location inventory system that serves one customer with multiple machines. During the contract execution phase, the true demand rate is observed. It can differ from the estimated demand rate because of two factors: increased demand variation in finite horizon settings and a shift in the mean utilization of the machines by the user during the contract. When the true demand rate is higher than the estimated demand rate, the Original Equipment Manufacturer (OEM) is faced with higher-than-expected costs for the execution of the contract, and the asset user is generally faced with a higher number of extreme long downtime events. Therefore, we introduce the flexible-time contract, which ends after a predetermined number of demands. Using a Markov decision process, we prove that a state-dependent base stock policy is optimal under a flexible-time contract. Using simulation, we compare the flexible-time contract with the standard fixed-time contract. Our results show that the flexible-time contract reduces the costs for the OEM by up to 35% and prevents not meeting the agreed-on service level. We obtain similar results in a multi-item setting.

Markov-modulated analysis of a spare parts system with random lead times and disruption risks

Spare parts supply chains are highly dependent on the dynamics of their installed bases. A decreasing number of capital products in use increases the nonstationary supply-side risk especially towards the end-of-life of capital products. This supply-side risk appears to present itself through varying lead times coupled with supply disruptions. To model the nonstationary supply-side risk, we consider an exogenous Markov chain that modulates random lead times and disruption probabilities. Assuming that order crossovers do not occur, we prove the optimality of a state-dependent base stock policy. Later, we conduct an impact study to understand the value of considering stochastic lead times and supply disruption risk in spare parts inventory control. Our results indicate that the coupled effect of random lead times and disruptions can be larger than the summation of individual effects even for moderate lead time variances. Also, the effect of nonstationarity on total cost can be as large as the summation of all risk factors combined. In addition to this managerial insight we present a procedure for supply risk mitigation based on an empirical model and our mathematical model. Experiments on a real business case indicate that the procedure is capable of reducing costs while making the inventory system more prepared for disruptions.

Dynamic Inventory Management with Inventory-Based Financing

This paper considers a stochastic inventory management problem where a capital-constrained firm can obtain additional working capital through an inventory-based financing facility by pledging its inventory to obtain loans from a lender. We show that it is optimal to deploy a state-dependent base-stock policy through which the firm may order new inventory up to or salvage existing inventory down to certain levels in each period depending on whether it has an initial net capital or debt defined as the on-hand capital minus all outstanding loans and interests. Through analytical and numerical studies, we offer a number of new insights regarding the optimal inventory management for a capital-constrained firm with access to inventory-based financing. In particular, we find that when the firm anticipates a shortage of capital with which to meet high demands in a future period, it may strategically over-stock its inventory in earlier periods in order to secure the necessary capital. Through extensive numerical experiments, our paper also demonstrates how some of the key parameters may affect the optimal inventory and financing decisions and the associated profits. Finally, we discuss three model extensions that allow finite storage capacity, per-unit inventory holding cost, and demand backlogging.

Optimal Inventory Control with Advance Supply Information

It has been shown in numerous situations that sharing information between the companies leads to improved performance of the supply chain. We study a positive lead time periodic-review inventory system of a retailer facing stochastic demand from his customer and stochastic limited supply capacity of the manufacturer supplying the products to him. The consequence of stochastic supply capacity is that the orders might not be delivered in full, and the exact size of the replenishment might not be known to the retailer. The manufacturer is willing to share the so-called advance supply information (ASI) about the actual replenishment of the retailer's pipeline order with the retailer. ASI is provided at a certain time after the orders have been placed and the retailer can now use this information to decrease the uncertainty of the supply, and thus improve its inventory policy. For this model, we develop a dynamic programming formulation, and characterize the optimal ordering policy as a state-dependent base-stock policy. In addition, we show some properties of the base-stock level. While the optimal policy is highly complex, we obtain some additional insights by comparing it to the state-dependent myopic inventory policy. We conduct the numerical analysis to estimate the in uence of the system parameters on the value of ASI. While we show that the interaction between the parameters is relatively complex, the general insight is that due to increasing marginal returns, the majority of the benets are gained only in the case of full, or close to full, ASI visibility. DOI: 10.15458/85451.28

Admission and inventory control of a single-component make-to-order production system with replenishment setup cost and lead time

We consider the admission control and inventory management problems of a single-component make-to-order production system. Components are purchased from suppliers in batches of fixed size subject to stochastic lead times and setup costs. A control policy specifies when a batch of components is purchased, and whether the demand for each MTO production is accepted upon arrival. We formulate the problem as a Markov decision process (MDP) model, and characterize the structure of optimal admission control and inventory replenishment policies. We show that a state dependent base-stock policy is optimal for the inventory replenishment, although the MDP value function is not necessarily convex. We also show that the optimal admission control can be identified as a lattice dependent policy. A sensitivity analysis is conducted to show how the optimal policy changes as a function of the system parameters. To effectively coordinate admission and inventory control decisions, we propose simple, implementable, and yet effective heuristic policies. Our extensive numerical results suggest that the proposed heuristics can greatly help firms to effectively coordinate their admission and inventory control activities.

Benefits of leadtime information and of its combination with demand forecast information

This paper studies both the benefits of leadtime information and the interrelationships among demand forecast information, leadtime uncertainty, and leadtime information, for improving inventory control performance. To this end, it explicitly incorporates both leadtime information and demand forecast information into an inventory model although they have conventionally been dealt with individually in the literature. Specifically, it considers a periodic-review inventory model with uncertain leadtimes, in which both leadtime information and demand forecast information are available. The leadtime information is modeled as the information of the arrival of outstanding orders and the demand forecast information as the Martingale Model of Forecast Evolution (MMFE).The paper analyzes the way to use both leadtime information and demand forecast information for inventory control operations. Specifically, under certain conditions, the optimality of the state-dependent base-stock policy is shown, and an effective heuristic policy is developed.It also derives some managerial insights for exploiting both leadtime information and demand forecast information in the case where leadtimes are uncertain. For instance, while leadtime information itself significantly improves inventory control performance, it also enhances the benefits of demand forecast information. Specifically, the benefits of demand forecast information may significantly decrease as leadtime uncertainty increases. However, the decrease of the benefits of demand forecast information may be alleviated with leadtime information.

Incremental quantity discounts in a periodic-review stochastic inventory model with a general demand distribution

We examine a periodic–review stochastic inventory model in which the buyer places orders to meet stochastic demands under an incremental quantity discount schedule with a single price–break point. With a general demand distribution, we prove the optimality of a state–dependent base–stock policy for the single–period problem and provide simple bounds on the optimal order–up–to levels for the multiple–period problem. We also solve the infinite horizon problem by using the policy iteration method. Our computational results suggest that a state–dependent base–stock policy is also optimal for the multiple–period and infinite–horizon problems.

Managing an assemble-to-order system with after sales market for components

Abstract In this paper, we consider an assemble-to-order manufacturing system producing a single end product, assembled from n components, and serving an after sales market for individual components. Components are produced in a make-to-stock fashion, one unit at a time, on independent production facilities. Production times are exponentially distributed with finite production rates. The components are stocked ahead of demand and therefore incur a holding cost rate per unit. Demand for the end product as well as for the individual components occurs continuously over time according to independent Poisson streams. In order to characterize the optimal production and inventory rationing policies, we formulate such a problem using a Markov decision process framework. In particular, we show that the optimal component production policy is a state-dependent base-stock policy. We also show that the optimal component inventory rationing policy is a rationing policy with state-dependent rationing levels. Recognizing that such a policy is generally not only difficult to obtain numerically but also is difficult to implement in practice, we propose three heuristic policies that are easier to implement in practice. We show that two of these heuristics are highly efficient compared to the optimal policy. In particular, we show that one of the two heuristics strikes a balance between high efficiency and computational effort and thus can be used as an effective substitute of the optimal policy.

Approximation algorithms for optimal purchase/inventory policy when purchase price and demand are stochastic

We consider a purchase/inventory control problem in which the purchase price and demand are stochastic, a common situation encountered by firms that replenish in a foreign currency or from commodity markets. More specifically, we assume that the demand follows a Poisson arrival process and that the log-price evolves according to a general Wiener process. Under these circumstances, the optimal policy is a state dependent base-stock policy that can be described as a series of threshold prices. An iterative procedure for determining the optimal thresholds has been derived earlier but, even for the simplest price process, the solution quickly becomes numerically intractable. To deal with this, we propose an approximation that allows us to derive simple heuristics for finding thresholds that are close to optimal. For certain price processes the heuristics are just a series of closed-form expressions. The computational complexity is reduced significantly, and the numerical study shows that the new heuristics perform considerably better than earlier suggested heuristics.

The pediatric vaccine stockpiling problem

The U.S. has experienced many major interruptions of its pediatric vaccine production in the past decade. The Centers for Disease Control and Prevention (CDC) copes with these shortages by building a national stockpile of pediatric vaccines, which it makes accessible to the public in the event of a shortage. The management of this stockpile is difficult due to limited production capacity and long and unpredictable production interruptions. In this paper, we address policies for managing the stockpile. We provide sufficient conditions for the optimal policy to be a modified state-dependent base-stock policy, with the base-stock level decreasing in the pipeline inventory. Since the optimal policy is in general difficult to evaluate, we derive bounds on the optimal decision in each period. We develop an efficient policy that performs on average within 1% of optimality in simulations. We show that stocking the same supply of vaccine of every type can be over-conservative in some cases, and inadequate in others by large factors. We also quantify the substantial reduction in inventory level that can be achieved when there are multiple suppliers in the market.

Optimal Pricing and Production Control in an Assembly System With a General Stockout Cost

We study the optimal control of an assembly system used to produce an assemble-to-order end product. We determine how to manage component production and end product price in order to maximize total discounted profit over an infinite horizon. We show the optimal production control is a state dependent base-stock policy and the optimal price adjustment is given by a threshold switching policy.

A Multiordering Newsvendor Model with Dynamic Forecast Evolution

We consider a newsvendor who dynamically updates her forecast of the market demand over a finite planning horizon. The forecast evolves according to the martingale model of forecast evolution (MMFE). The newsvendor can place multiple orders with increasing ordering cost over time to satisfy demand that realizes at the end of the planning horizon. In this context, we explore the trade-off between improving demand forecast and increasing ordering cost. We show that the optimal ordering policy is a state-dependent base-stock policy and analytically characterize that the base-stock level depends on the information state in a linear (log-linear) fashion for additive (multiplicative) MMFE. We also study a benchmark model where the newsvendor is restricted to order only once. By comparing the multiordering and single-ordering models, we quantify the impact of the multiordering strategy on the newsvendor's expected profit and risk exposure.

Inventory management with random supply and imperfect information: A hidden Markov model

In most of the papers on inventory models operating in a random environment, the state of the environment in each period is assumed to be fully observed with perfect information. However, this assumption is not realistic in most real-life situations and we provide a remedy in this paper by assuming that the environment is only partially observed with imperfect information. We accomplish this by analyzing two formulations of single-item models with periodic-review and random supply in a random environment. In the first one, supply is random due to random capacity of production and random availability of transportation. We show that state-dependent base-stock policy is optimal if the capacity and all costs are observed, while demand and availability are unobserved. In the second model, we consider a model with random availability only with fixed-ordering cost. We show that state-dependent (s,S) policy is optimal if the availability process is observable.

TECHNICAL NOTE—Optimal Control of an Assembly System with Multiple Stages and Multiple Demand Classes

We consider an assembly system with multiple stages, multiple items, and multiple customer classes. The system consists of m production facilities, each producing a different item. Items are produced in variable batch sizes, one batch at a time, with exponentially distributed batch production times. Demand from each class takes place continuously over time according to a compound Poisson process. At each decision epoch, we must determine whether or not to produce an item and, should demand from a particular class arise, whether or not to satisfy it from existing inventory, if any is available. We formulate the problem as a Markov decision process and use it to characterize the structure of the optimal policy. In contrast to systems with exogenous and deterministic production lead times, we show that the optimal production policy for each item is a state-dependent base-stock policy with the base-stock level nonincreasing in the inventory level of items that are downstream and nondecreasing in the inventory level of all other items. For inventory allocation, we show that the optimal policy is a multilevel state-dependent rationing policy with the rationing level for each demand class nonincreasing in the inventory level of all nonend items. We also show how the optimal control problem can be reformulated in terms of echelon inventory and how the essential features of the optimal policy can be reinterpreted in terms of echelon inventory.

Inventory management with advance capacity information

An important aspect of supply chain management is dealing with demand and supply uncertainty. The uncertainty of future supply can be reduced if a company is able to obtain advance capacity information (ACI) about future supply/production capacity availability from its supplier. We address a periodic-review inventory system under stochastic demand and stochastic limited supply, for which ACI is available. We show that the optimal ordering policy is a state-dependent base-stock policy characterized by a base-stock level that is a function of ACI. We establish a link with inventory models that use advance demand information (ADI) by developing a capacitated inventory system with ADI, and we show that equivalence can only be set under a very specific and restrictive assumption, implying that ADI insights will not necessarily hold in the ACI environment. Our numerical results reveal several managerial insights. In particular, we show that ACI is most beneficial when there is sufficient flexibility to react to anticipated demand and supply capacity mismatches. Further, most of the benefits can be achieved with only limited future visibility. We also show that the system parameters affecting the value of ACI interact in a complex way and therefore need to be considered in an integrated manner. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011

Production control and stock rationing for a make-to-stock system with parallel production channels

This article considers the problem of production control and stock rationing in a make-to-stock production system with lost sales, multiple servers in parallel production channels, and several customer classes. It is assumed that independent stationary Poisson demand streams and exponential service times are in operation. At decision epochs, the control specifies whether or not to increase the number of active servers in conjunction with the stock allocation decision. Previously placed production orders cannot be cancelled. The system is modeled as an M/M/s make-to-stock queue, and properties of the optimal cost function and of the optimal production and rationing policies are characterized. It is shown that the optimal production policy is a state-dependent base-stock policy, and the optimal rationing policy is of threshold type. Furthermore, it is shown that the rationing levels are non-increasing in the number of active channels. It is also shown that the optimal ordering policy transforms into a bang-bang type policy when the model is relaxed by allowing order cancellations. Another model with partial order-cancellation flexibility is provided to fill the gap between the no-flexibility and the full-flexibility models. The additional gain that the optimal policy provides over the suboptimal base-stock policy proposed in the literature is qualified along with the value of the flexibility to cancel production orders.

Production-inventory systems with imperfect advance demand information and updating

We consider a supplier with finite production capacity and stochastic production times. Customers provide advance demand information (ADI) to the supplier by announcing orders ahead of their due dates. However, this information is not perfect, and customers may request an order be fulfilled prior to or later than the expected due date. Customers update the status of their orders, but the time between consecutive updates is random. We formulate the production-control problem as a continuous-time Markov decision process and prove there is an optimal state-dependent base-stock policy, where the base-stock levels depend upon the numbers of orders at various stages of update. In addition, we derive results on the sensitivity of the state-dependent base-stock levels to the number of orders in each stage of update. In a numerical study, we examine the benefit of ADI, and find that it is most valuable to the supplier when the time between updates is moderate. We also consider the impact of holding and backorder costs, numbers of updates, and the fraction of customers that provide ADI. In addition, we find that while ADI is always beneficial to the supplier, this may not be the case for the customers who provide the ADI. © 2011 Wiley Periodicals, Inc. Naval Research Logistics 58: 88-106, 2011

Optimal Control of an Assembly System with Multiple Stages and Multiple Demand Classes

We consider an assembly system with multiple stages, multiple items, and multiple customer classes. The system consists of m production facilities, each producing a different item. Items are produced in variable batch sizes, one batch at a time, with exponentially distributed batch production times. Demand from each class takes place continuously over time according to a compound Poisson process. At each decision epoch, we must determine whether or not to produce an item and should demand from a particular class arise whether or not to satisfy it from existing inventory, if any is available. We formulate the problem as a Markov decision process and use it to characterize the structure of the optimal policy. In contrast to systems with exogenous and deterministic production leadtimes, we show that the optimal production policy for each item is a state-dependent base-stock policy with the base-stock level non-increasing in the inventory level of items that are downstream and non-decreasing in the inventory level of all other items. For inventory allocation, we show that the optimal policy is a multi-level state-dependent rationing policy with the rationing level for each demand class non-increasing in the inventory level of all non-end items. We also show how the optimal control problem can be reformulated in terms of echelon inventory and how the essential features of the optimal policy can be reinterpreted in terms of echelon inventory.