Exact analysis of a two-echelon inventory system for recoverable items under batch inspection policy

We investigate a two-echelon (base-depot) inventory system of recoverable (repairable) items. The arrivals of demand at the bases are in a Poisson manner and the order sizes are random. The failed units can be repaired either at the base or at the depot, and the units beyond economic repair are condemned. Inspection of the failed units is carried out in the batches they arrive, that is, arrival batches are not broken up. The exact expressions for stationary distribution of depot inventory position, and of the number of backorders, onhand inventory, in-repair inventory at all locations are derived under the assumptions of constant repair and lead times. Special cases of complete recoverability, nonrecoverability, and of the unit order size are also discussed.     

An approximation to the stationary distribution of a multiechelon repairable-item inventory system with finite sources and repair channels

We consider a multiechelon repairable-item inventory system where several bases are supported by a central depot. Unlike METRIC-based models, there are only a finite number of repairmen at each base and the depot, and the failure rates at the bases depend on the current number of items online. The principal objective of this article is to develop a quick and accurate approximation to the steady-state distribution of this system. A secondary objective is to compare the solution of this system with a comparable METRIC solution.     

A three-echelon,multi-item model for recoverable items

The main objective of this paper is to develop a mathematical model for a particular type of three-echelon inventory system. The proposed model is being used by the Air Force to evaluate inventory investment requirements for alternative logistic structures. The system we will model consists of a group of locations, called bases, and a central depot. The items of concern in our analysis are called recoverable items, that is, items that can be repaired when they fail. Furthermore, each item has a modular or hierarchical design. Briefly, the model is used to determine the stock levels at each location for each item so as to achieve optimum inventory-system performance for a given level of investment. An algorithm for the computation of stock levels for each item and location is developed and illustrated. Some of the ways the model can be used are illustrated with Air Force data.     

Risk pooling in a two-period,two-echelon inventory stocking and allocation problem

The object of this article is to investigate the risk-pooling effect of depot stock in two-echelon distribution system in which the depot serves n retailers in parallel, and to develop computationally tractable optimization procedures for such systems. The depot manager has complete information about stock levels and there are two opportunities to allocate stock to the retailers within each order cycle. We identify first- and second-order aspects to the risk-pooling effect. In particular, the second-order effect is the property that the minimum stock available to any retailer after the second allocation converges in probability to a constant as the number of retailers in the system increases, assuming independence of the demands. This property is exploited in the development of efficient procedures to determine near-optimal values of the policy parameters.     

Inventory management under random supply disruptions and partial backorders

We explore the management of inventory for stochastic-demand systems, where the product's supply is randomly disrupted for periods of random duration, and demands that arrive when the inventory system is temporarily out of stock become a mix of backorders and lost sales. The stock is managed according to the following modified (s, S) policy: If the inventory level is at or below s and the supply is available, place an order to bring the inventory level up to S. Our analysis yields the optimal values of the policy parameters, and provides insight into the optimal inventory strategy when there are changes in the severity of supply disruptions or in the behavior of unfilled demands. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 687–703, 1998     

Optimal division of inventory between depot and bases

We consider a problem of optimal division of stock between a logistic depot and several geographically dispersed bases, in a two‐echelon supply chain. The objective is to minimize the total cost of inventory shipment, taking into account direct shipments between the depot and the bases, and lateral transshipments between bases. We prove the convexity of the objective function and suggest a procedure for identifying the optimal solution. Small‐dimensional cases, as well as a limit case in which the number of bases tends to infinity, are solved analytically for arbitrary distributions of demand. For a general case, an approximation is suggested. We show that, in many practical cases, partial pooling is the best strategy, and large proportions of the inventory should be kept at the bases rather than at the depot. The analytical and numerical examples show that complete pooling is obtained only as a limit case in which the transshipment cost tends to infinity. © 2017 Wiley Periodicals, Inc. Naval Research Logistics, 64: 3–18, 2017     

The distribution of recoverable inventory items from a repair center when the number of consumption centers is large

This paper considers real-time decision rules for an inventory system where items are repaired than “used up.” The problem is to decide which user in the system has the greatest need for the newly available inventory items coming out of repair. The main result shows that two published approahes, the Transportation Time Look Ahead policy and METRIC, are optimal when the number of users gets large. A useful byproduct of the proof is a lower bound on the average backorder rate for a repair-inventory system of any size.     

Capacitated warehouse location model with risk pooling

In this article, we introduce the capacitated warehouse location model with risk pooling (CLMRP), which captures the interdependence between capacity issues and the inventory management at the warehouses. The CLMRP models a logistics system in which a single plant ships one type of product to a set of retailers, each with an uncertain demand. Warehouses serve as the direct intermediary between the plant and the retailers for the shipment of the product and also retain safety stock to provide appropriate service levels to the retailers. The CLMRP minimizes the sum of the fixed facility location, transportation, and inventory carrying costs. The model simultaneously determines warehouse locations, shipment sizes from the plant to the warehouses, the working inventory, and safety stock levels at the warehouses and the assignment of retailers to the warehouses. The costs at each warehouse exhibit initially economies of scale and then an exponential increase due to the capacity limitations. We show that this problem can be formulated as a nonlinear integer program in which the objective function is neither concave nor convex. A Lagrangian relaxation solution algorithm is proposed. The Lagrangian subproblem is also a nonlinear integer program. An efficient algorithm is developed for the linear relaxation of this subproblem. The Lagrangian relaxation algorithm provides near‐optimal solutions with reasonable computational requirements for large problem instances. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Optimal design of a multi-item,multi-location,multi-repair type repair and supply system

The design of a system with many locations, each with many items which may fail while in use, is considered. When items fail, they require repair; the particular type of repair being governed by a probability distribution. As repairs may be lengthy, spares are kept on hand to replace failed items. System ineffectiveness is measured by expected weighted shortages over all items and locations, in steady state. This can be reduced by either having more spares or shorter expected repair times. Design consists of a provisioning of the number of spares for each item, by location; and specifying the expected repair times for each type of repair, by item and location. The optimal design minimizes expected shortages within a budget constraint, which covers both (i) procurement of spares and (ii) procurement of equipment and manning levels for the repair facilities. All costs are assumed to be separable so that a Lagrangian approach is fruitful, yielding an implementable algorithm with outputs useful for sensitivity analysis. A numerical example is presented.     

More ado about economic order quantities (EOQ)

This paper is concerned with the determination of explicit expressions for economic order quantities and reorder levels, such that the cost of ordering and holding inventory is minimized for specific backorder constraints. Holding costs are applied either to inventory position or on-hand inventory, and the backorder constraint is considered in terms of the total number of backorders per year or the average number of backorders at any point in time. Through the substitution of a new probability density function in place of the normal p.d.f., explicit expressions are determined for the economic order quantities and the reorder points. The resulting economic order quantities are independent of all backorder constraints. It is also concluded that under certain conditions, the minimization of ordering costs and inventory holding costs (applied to inventory position), subject to a backorder constraint, is equivalent in terms of reorder levels to minimization of the safety level dollar investment subject to the same backorder constraint.     

Production-allocation scheduling and capacity expansion using network flows under uncertainty

This paper extends Connors and Zangwill's work in network flows under uncertainty to the convex costs case. In this paper the extended network flow under uncertainty algorithm is applied to compute N-period production and delivery schedules of a single commodity in a two-echelon production-inventory system with convex costs and low demand items. Given an initial production capacity for N periods, the optimal production and delivery schedules for the entire N periods are characterized by the flows through paths of minimal expected discounted cost in the network As a by-product of this algorithm the multi-period stochastic version of the parametric budget problem for the two-echelon production-inventory system is solved.     

Multiechelon inventory systems with lateral supply

We develop approximations to estimate the expected backorders in a multiechelon system in which lateral supply actions between bases are allowed when a backorder occurs. These approximations are easy to compute, and the average absolute error over a wide range of parameter values is less than 4% when items are depot repairable, even when bases are dissimilar. With lateral supply, backorder reductions of 30-50% are not uncommon, and a 72% reduction was observed in two cases. Lateral supply becomes more important with low demand rates. A similar approach was unsuccessful for base-repairable items. However, lateral supply has a beneficial effect only when the lateral supply time is very short, 1/4 or less of the average base repair time. Even in such cases lateral supply is unlikely to be important in an actual application, because base management can expedite repair of critical items.     

Return policies for an inventory system with positive and negative demands

We consider a single item inventory system with positive and negative stock fluctuations. Items can be purchased from a central stock, n items can be returned for a cost R + rn, and a linear inventory carrying cost is charged. It is shown that for minimizing the asymptotic cost rate when returns are a significant fraction of stock usage, a two-critical-number policy (a,b) is optimal, where b is the trigger level for returns and b – a is the return quantity. The values for a and b are found, as well as the operating characteristics of the system. We also consider the optimal return decision to make at time zero and show that it is partially determined by a and b.     

Insurance type item provisioning guidelines

Policy decisions for insurance type items, where zero or one unit is maintained at the depot, are more difficult and more critical than decisions for common supply items. This report presents results of developing initial provisioning guidelines for insurance type items. The guidelines are based on examination of lifetime costs and benefits. Costs of stocking an item as compared with not stocking are developed through a sinking fund annual payment formulation. Benefits of stocking are developed as stationary reduction in time weighted backorders experienced. A resource allocation formulation yields an optimal policy for allocating a fixed budget. The guideline is presented with refinements based on a sample of items. A figure of merit is calculated for each item, and if it is large the item is stocked while if small it is not stocked. Empirical definitions for large and small are developed based on sample data. Estimation techniques are discussed for deriving all of an item's parameters needed to compute its figure of merit. A Bayes procedure is suggested based on family group Experienced Demand Replacement Factors. This and other techniques are discussed.     

A two-echelon inventory model with purchases,dispositions, shipments,returns and transshipments

This paper presents a one-period two-echelon inventory model with one warehouse in the first echelon and n warehouses in the second echelon. At the beginning of the period the stock levels at all facilities are adjusted by purchasing or disposing of items at the first echelon, returning or shipping items between the echelons and transshipping items within the second echelon. During the period, demands (which may be negative) are placed on all warehouses in the second echelon and an attempt is made to satisfy shortages either by an expedited shipment from the first echelon to the second echelon or an expedited transshipment within the second echelon. The decision problem is to choose an initial stock level at the first echelon (by a purchase or a disposition) and an initial allocation so as to minimize the initial stock movement costs during the period plus inventory carrying costs and system shortage costs at the end of the period. It is shown that the objective function takes on one of four forms, depending on the relative magnitudes of the various shipping costs. All four forms of the objective function are derived and proven to be convex. Several applications of this general model are considered. We also consider multi-period extensions of the general model and an important special case is solved explicitly.     

Markovian multiechelon repairable inventory system

In this article we model a two-echelon (two levels of repair, one level of supply) repairable-item inventory system using continuous-time Markov processes. We analyze two models. In the first model we assume a system with a single base. In the second model we expand this model to include n bases. The Markov approach gives rise to multidimensional state spaces that are large even for relatively small problems. Because of this, we utilize aggregate/disaggregate techniques to develop a solution algorithm for finding the steady-state distribution. This algorithm is exact for the single-base model and is an approximation for the n-base model, in which case it is found to be very accurate and computationally very efficient.     

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     

Optimal disposal policies for a single-item inventory system with returns

We consider a single-item inventory system in which the stock level can increase due to items being returned as well as decrease when demands occur. Returned items can be repaired and then used to satisfy future demand, or they can be disposed of. We identify those inventory levels where disposal is the best policy. It is shown that this problem is equivalent to a problem of controlling a single-server queue. When the return and demand processes are both Poisson, we find the optimal policy exactly. When the demand and return processes are more general, we use diffusion approximations to obtain an approximate model, which is then solved. The approximate model requires only mean and variance data. Besides the optimal policy, the output of the models includes such characteristics as the operating costs, the purchase rate for new items, the disposal rate for returned items and the average inventory level. Several numerical examples are given. An interesting by-product of our investigation is an approximation for the steady-state behavior of the bulk GI/G/1 queue with a queue limit.     

Distribution free bounds for service constrained (Q,r) inventory systems

A classical and important problem in stochastic inventory theory is to determine the order quantity (Q) and the reorder level (r) to minimize inventory holding and backorder costs subject to a service constraint that the fill rate, i.e., the fraction of demand satisfied by inventory in stock, is at least equal to a desired value. This problem is often hard to solve because the fill rate constraint is not convex in (Q, r) unless additional assumptions are made about the distribution of demand during the lead‐time. As a consequence, there are no known algorithms, other than exhaustive search, that are available for solving this problem in its full generality. Our paper derives the first known bounds to the fill‐rate constrained (Q, r) inventory problem. We derive upper and lower bounds for the optimal values of the order quantity and the reorder level for this problem that are independent of the distribution of demand during the lead time and its variance. We show that the classical economic order quantity is a lower bound on the optimal ordering quantity. We present an efficient solution procedure that exploits these bounds and has a guaranteed bound on the error. When the Lagrangian of the fill rate constraint is convex or when the fill rate constraint does not exist, our bounds can be used to enhance the efficiency of existing algorithms. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 635–656, 2000