A location‐inventory model with lateral transshipments

We study an infinite horizon periodic stochastic inventory system consisting of retail outlets and customers located on a homogenous line segment. In each period, the total demand, generated by the customers on the line, is normally distributed. To better match supply and demand, we incorporate lateral transshipments. We propose a compact model in which the strategic decisions—the number and locations of retail outlets—are determined simultaneously with the operational decisions—the inventory replenishment and transshipment quantities. We find the optimal balance between the risk‐pooling considerations, which drive down the optimal number of retail outlets, and lateral transshipments, which drive up the optimal number of retail outlets. We also explore the sensitivity of the optimal number of retail outlets to various problem parameters. This article presents a novel way of integrating lateral transshipments in the context of an inventory‐location model. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Lateral stock transshipments in a two‐location inventory system with fixed and joint replenishment costs

We address the problem of inventory management in a two‐location inventory system, in which the transshipments are carried out as means of emergency or alternative supply after demand has been realized. This model differs from previous ones as regards its replenishment costs structure, in which nonnegligible fixed replenishment costs and a joint replenishment cost are considered. The single period planning horizon is analyzed, with the form and several properties of the optimal replenishment and transshipment policies developed, discussed and illustrated. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 525–547, 1999     

Optimal pricing and inventory control policy in periodic‐review systems with fixed ordering cost and lost sales

This paper studies a periodic‐review pricing and inventory control problem for a retailer, which faces stochastic price‐sensitive demand, under quite general modeling assumptions. Any unsatisfied demand is lost, and any leftover inventory at the end of the finite selling horizon has a salvage value. The cost component for the retailer includes holding, shortage, and both variable and fixed ordering costs. The retailer's objective is to maximize its discounted expected profit over the selling horizon by dynamically deciding on the optimal pricing and replenishment policy for each period. We show that, under a mild assumption on the additive demand function, at the beginning of each period an (s,S) policy is optimal for replenishment, and the value of the optimal price depends on the inventory level after the replenishment decision has been done. Our numerical study also suggests that for a sufficiently long selling horizon, the optimal policy is almost stationary. Furthermore, the fixed ordering cost (K) plays a significant role in our modeling framework. Specifically, any increase in K results in lower s and higher S. On the other hand, the profit impact of dynamically changing the retail price, contrasted with a single fixed price throughout the selling horizon, also increases with K. We demonstrate that using the optimal policy values from a model with backordering of unmet demands as approximations in our model might result in significant profit penalty. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

The one‐warehouse multiretailer problem with an order‐up‐to level inventory policy

We consider a two‐level system in which a warehouse manages the inventories of multiple retailers. Each retailer employs an order‐up‐to level inventory policy over T periods and faces an external demand which is dynamic and known. A retailer's inventory should be raised to its maximum limit when replenished. The problem is to jointly decide on replenishment times and quantities of warehouse and retailers so as to minimize the total costs in the system. Unlike the case in the single level lot‐sizing problem, we cannot assume that the initial inventory will be zero without loss of generality. We propose a strong mixed integer program formulation for the problem with zero and nonzero initial inventories at the warehouse. The strong formulation for the zero initial inventory case has only T binary variables and represents the convex hull of the feasible region of the problem when there is only one retailer. Computational results with a state‐of‐the art solver reveal that our formulations are very effective in solving large‐size instances to optimality. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Joint inventory and pricing decisions for perishable products with two‐period lifetime

We consider a periodic review model over a finite horizon for a perishable product with fixed lifetime equal to two review periods. The excess demand in a period is backlogged. The optimal replenishment and demand management (using price) decisions for such a product depend on the relative order of consumption of fresh and old units. We obtain insights on the structure of these decisions when the order of consumption is first‐in, first‐out and last‐in, first‐out. For the FIFO system, we also obtain bounds on both the optimal replenishment quantity as well as expected demand. We compare the FIFO system to two widely analyzed inventory systems that correspond to nonperishable and one‐period lifetime products to understand if demand management would modify our understanding of the relationship among the three systems. In a counterintuitive result, we find that it is more likely that bigger orders are placed in the FIFO system than for a nonperishable product when demand is managed. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

Dynamic inventory management with cash flow constraints

In this article, we consider a classic dynamic inventory control problem of a self‐financing retailer who periodically replenishes its stock from a supplier and sells it to the market. The replenishment decisions of the retailer are constrained by cash flow, which is updated periodically following purchasing and sales in each period. Excess demand in each period is lost when insufficient inventory is in stock. The retailer's objective is to maximize its expected terminal wealth at the end of the planning horizon. We characterize the optimal inventory control policy and present a simple algorithm for computing the optimal policies for each period. Conditions are identified under which the optimal control policies are identical across periods. We also present comparative statics results on the optimal control policy. © 2008 Wiley Periodicals, Inc. Naval Research Logistics 2008     

An exact analysis of a joint production‐inventory problem in two‐echelon inventory systems

We consider a two‐echelon inventory system with a manufacturer operating from a warehouse supplying multiple distribution centers (DCs) that satisfy the demand originating from multiple sources. The manufacturer has a finite production capacity and production times are stochastic. Demand from each source follows an independent Poisson process. We assume that the transportation times between the warehouse and DCs may be positive which may require keeping inventory at both the warehouse and DCs. Inventory in both echelons is managed using the base‐stock policy. Each demand source can procure the product from one or more DCs, each incurring a different fulfilment cost. The objective is to determine the optimal base‐stock levels at the warehouse and DCs as well as the assignment of the demand sources to the DCs so that the sum of inventory holding, backlog, and transportation costs is minimized. We obtain a simple equation for finding the optimal base‐stock level at each DC and an upper bound for the optimal base‐stock level at the warehouse. We demonstrate several managerial insights including that the demand from each source is optimally fulfilled entirely from a single distribution center, and as the system's utilization approaches 1, the optimal base‐stock level increases in the transportation time at a rate equal to the demand rate arriving at the DC. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Performance evaluation of a production/inventory system with periodic review and endogenous lead times

This paper considers a discrete time, single item production/inventory system with random period demands. Inventory levels are reviewed periodically and managed using a base‐stock policy. Replenishment orders are placed with the production system which is capacitated in the sense that there is a single server that sequentially processes the items one at a time with stochastic unit processing times. In this setting the variability in demand determines the arrival pattern of production orders at the queue, influencing supply lead times. In addition, the inventory behavior is impacted by the correlation between demand and lead times: a large demand size corresponds to a long lead time, depleting the inventory longer. The contribution of this paper is threefold. First, we present an exact procedure based on matrix‐analytic techniques for computing the replenishment lead time distribution given an arbitrary discrete demand distribution. Second, we numerically characterize the distribution of inventory levels, and various other performance measures such as fill rate, base‐stock levels and optimal safety stocks, taking the correlation between demand and lead times into account. Third, we develop an algorithm to fit the first two moments of the demand and service time distribution to a discrete phase‐type distribution with a minimal number of phases. This provides a practical tool to analyze the effect of demand variability, as measured by its coefficient of variation, on system performance. We also show that our model is more appropriate than some existing models of capacitated systems in discrete time. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Exact analysis of (R,s, S) inventory control systems with lost sales and zero lead time

We study an (R, s, S) inventory control policy with stochastic demand, lost sales, zero lead‐time and a target service level to be satisfied. The system is modeled as a discrete time Markov chain for which we present a novel approach to derive exact closed‐form solutions for the limiting distribution of the on‐hand inventory level at the end of a review period, given the reorder level (s) and order‐up‐to level (S). We then establish a relationship between the limiting distributions for adjacent values of the reorder point that is used in an efficient recursive algorithm to determine the optimal parameter values of the (R, s, S) replenishment policy. The algorithm is easy to implement and entails less effort than solving the steady‐state equations for the corresponding Markov model. Point‐of‐use hospital inventory systems share the essential characteristics of the inventory system we model, and a case study using real data from such a system shows that with our approach, optimal policies with significant savings in inventory management effort are easily obtained for a large family of items.     

Coordinated pricing and inventory control problems with capacity constraints and fixed ordering cost

This article addresses a single‐item, finite‐horizon, periodic‐review coordinated decision model on pricing and inventory control with capacity constraints and fixed ordering cost. Demands in different periods are random and independent of each other, and their distributions depend on the price in the current period. Each period's stochastic demand function is the additive demand model. Pricing and ordering decisions are made at the beginning of each period, and all shortages are backlogged. The objective is to find an optimal policy that maximizes the total expected discounted profit. We show that the profit‐to‐go function is strongly CK‐concave, and the optimal policy has an (s,S,P) ‐like structure. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

Deterministic inventory lot‐size models under inflation with shortages and deterioration for fluctuating demand

In this paper, we extend the inventory lot‐size models to allow for inflation and fluctuating demand (which is more general than constant, increasing, decreasing, and log‐concave demand patterns). We prove that the optimal replenishment schedule not only exists but is also unique. Furthermore, we show that the total cost associated with the inventory system is a convex function of the number of replenishments. Hence, the search for the optimal number of replenishments is simplified to finding a local minimum. Finally, several numerical examples are provided to illustrate the results. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 144–158, 2001     

A computationally efficient approach for determining inventory levels in a capacitated multiechelon production‐distribution system

The system under study is a single item, two‐echelon production‐inventory system consisting of a capacitated production facility, a central warehouse, and M regional distribution centers that satisfy stochastic demand. Our objective is to determine a system base‐stock level which minimizes the long run average system cost per period. Central to the approach are (1) an inventory allocation model and associated convex cost function designed to allocate a given amount of system inventory across locations, and (2) a characterization of the amount of available system inventory using the inventory shortfall random variable. An exact model must consider the possibility that inventories may be imbalanced in a given period. By assuming inventory imbalances cannot occur, we develop an approximation model from which we obtain a lower bound on the per period expected cost. Through an extensive simulation study, we analyze the quality of our approximation, which on average performed within 0.50% of the lower bound. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 377–398, 2000     

On the (S − 1, S) lost sales inventory model with priority demand classes

In this paper an inventory model with several demand classes, prioritised according to importance, is analysed. We consider a lot‐for‐lot or (S ? 1, S) inventory model with lost sales. For each demand class there is a critical stock level at and below which demand from that class is not satisfied from stock on hand. In this way stock is retained to meet demand from higher priority demand classes. A set of such critical levels determines the stocking policy. For Poisson demand and a generally distributed lead time, we derive expressions for the service levels for each demand class and the average total cost per unit time. Efficient solution methods for obtaining optimal policies, with and without service level constraints, are presented. Numerical experiments in which the solution methods are tested demonstrate that significant cost reductions can be achieved by distinguishing between demand classes. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 593–610, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10032     

Dynamic pricing in a dual‐market environment

This article is concerned with the determination of pricing strategies for a firm that in each period of a finite horizon receives replenishment quantities of a single product which it sells in two markets, for example, a long‐distance market and an on‐site market. The key difference between the two markets is that the long‐distance market provides for a one period delay in demand fulfillment. In contrast, on‐site orders must be filled immediately as the customer is at the physical on‐site location. We model the demands in consecutive periods as independent random variables and their distributions depend on the item's price in accordance with two general stochastic demand functions: additive or multiplicative. The firm uses a single pool of inventory to fulfill demands from both markets. We investigate properties of the structure of the dynamic pricing strategy that maximizes the total expected discounted profit over the finite time horizon, under fixed or controlled replenishment conditions. Further, we provide conditions under which one market may be the preferred outlet to sale over the other. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 531–549, 2015     

Optimal control of a production‐inventory system with both backorders and lost sales

We consider the optimal control of a production inventory‐system with a single product and two customer classes where items are produced one unit at a time. Upon arrival, customer orders can be fulfilled from existing inventory, if there is any, backordered, or rejected. The two classes are differentiated by their backorder and lost sales costs. At each decision epoch, we must determine whether or not to produce an item and if so, whether to use this item to increase inventory or to reduce backlog. At each decision epoch, we must also determine whether or not to satisfy demand from a particular class (should one arise), backorder it, or reject it. In doing so, we must balance inventory holding costs against the costs of backordering and lost sales. We formulate the problem as a Markov decision process and use it to characterize the structure of the optimal policy. We show that the optimal policy can be described by three state‐dependent thresholds: a production base‐stock level and two order‐admission levels, one for each class. The production base‐stock level determines when production takes place and how to allocate items that are produced. This base‐stock level also determines when orders from the class with the lower shortage costs (Class 2) are backordered and not fulfilled from inventory. The order‐admission levels determine when orders should be rejected. We show that the threshold levels are monotonic (either nonincreasing or nondecreasing) in the backorder level of Class 2. We also characterize analytically the sensitivity of these thresholds to the various cost parameters. Using numerical results, we compare the performance of the optimal policy against several heuristics and show that those that do not allow for the possibility of both backordering and rejecting orders can perform poorly.© 2010 Wiley Periodicals, Inc. Naval Research Logistics 2010     

Inventory control and periodic price discounting campaigns

This paper develops an inventory model that determines replenishment strategies for buyers facing situations in which sellers offer price‐discounting campaigns at random times as a way to drive sales or clear excess inventory. Specifically, the model deals with the inventory of a single item that is maintained to meet a constant demand over time. The item can be purchased at two different prices denoted high and low. We assume that the low price goes into effect at random points in time following an exponential distribution and lasts for a random length of time following another exponential distribution. We highlight a replenishment strategy that will lead to the lowest inventory holding and ordering costs possible. This strategy is to replenish inventory only when current levels are below a certain threshold when the low price is offered and the replenishment is to a higher order‐up‐to level than the one currently in use when inventory depletes to zero and the price is high. Our analysis provides new insight into the behavior of the optimal replenishment strategy in response to changes in the ratio of purchase prices together with changes in the ratio of the duration of a low‐price period to that of a high‐price period. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007.     

Centralized inventory control in a two‐level distribution system with Poisson demand

This paper introduces a new replenishment policy for inventory control in a two‐level distribution system consisting of one central warehouse and an arbitrary number of nonidentical retailers. The new policy is designed to control the replenishment process at the central warehouse, using centralized information regarding the inventory positions and demand processes of all installations in the system. The retailers on the other hand are assumed to use continuous review (R, Q) policies. A technique for exact evaluation of the expected inventory holding and backorder costs for the system is presented. Numerical results indicate that there are cases when considerable savings can be made by using the new (α₀, Q₀) policy instead of a traditional echelon‐ or installation‐stock (R, Q) policy. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 798–822, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10040     

Production/distribution system design with inventory considerations

This study addresses the design of a three‐stage production/distribution system where the first stage includes the set of established retailers and the second and third stages include the sets of potential distribution centers (DCs) and potential capacitated suppliers, respectively. In this problem, in addition to the fixed location/operating costs associated with locating DCs and suppliers, we consider the coordinated inventory replenishment decisions at the located DCs and retailers along with the appropriate inventory costs explicitly. In particular, we account for the replenishment and holding costs at the retailers and selected DCs, and the fixed plus distance‐based transportation costs between the selected plants and their assigned DCs, and between the selected DCs and their respective retailers, explicitly. The resulting formulation is a challenging mixed‐integer nonlinear programming model for which we propose efficient heuristic solution approaches. Our computational results demonstrate the performance of the heuristic approaches as well as the value of integrated decision‐making by verifying that significant cost savings are realizable when the inventory decisions and costs are incorporated in the production distribution system design. © 2012 Wiley Periodicals, Inc. Naval Research Logistics 59: 172–195, 2012     

Fixed‐interval joint‐replenishment policies for distribution systems with multiple retailers and stochastic demand

We consider a distribution system consisting of a central warehouse and a group of retailers facing independent stochastic demand. The retailers replenish from the warehouse, and the warehouse from an outside supplier with ample supply. Time is continuous. Most previous studies on inventory control policies for this system have considered stock‐based batch‐ordering policies. We develop a time‐based joint‐replenishment policy in this study. Let the warehouse set up a basic replenishment interval. The retailers are replenished through the warehouse in intervals that are integer multiples of the basic replenishment interval. No inventory is carried at the warehouse. We provide an exact evaluation of the long‐term average system costs under the assumption that stock can be balanced among the retailers. The structural properties of the inventory system are characterized. We show that, although it is well known that stock‐based inventory control policies dominate time‐based inventory control policies at a single facility, this dominance does not hold for distribution systems with multiple retailers and stochastic demand. This is because the latter can provide a more efficient mechanism to streamline inventory flow and pool retailer demand, even though the former may be able to use more updated stock information to optimize system performance. The findings of the study provide insights about the key factors that drive the performance of a multiechelon inventory control system. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 637–651, 2013     

A duality‐based relaxation and decomposition approach for inventory distribution systems

We propose a new method for making the inventory replenishment decisions in distribution systems. In particular, we consider distribution systems consisting of multiple retailers that face random demand and a warehouse that supplies the retailers. The method that we propose is based on formulating the distribution problem as a dynamic program, and relaxing the constraints that ensure the nonnegativity of the shipments to the retailers, by associating Lagrange multipliers with them. We show that our method provides lower bounds on the value functions, and a good set of values for the Lagrange multipliers can be obtained by maximizing a concave function in a relatively straightforward manner. Computational experiments indicate that our method can provide significant improvements over the traditional approaches for making the inventory replenishment decisions, in terms of both the tightness of the lower bounds on the value functions and the performance of the policies. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008