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.     

The proper balancing of repair and preventive maintenance activities in the determination of coordinated shipboard allowance lists

The determination as to the cost-effective number of spares for given types of items or equipment to be carried on board various types of ships is studied. This spare pool, known as the ship's COSAL, must provide prespecified levels of protection against stockouts for all uses of that item on board the given ship. This article derives and illustrates the methodology for optimally trading off the reduction in the ship's COSAL that can be gained by improving repair/resupply capabilities or by lowering the failure rate of the equipment through more or different types of preventive maintenance. A flexible class of preventive maintenance/repair response functions to cost is studied which are nonlinear and exhibit realistic diminishing returns. Two different types of assumptions are possible regarding the interdependencies of the resupply times across different ship types. A tractable budget allocation method is presented which can be used in a multiitem, multiship, multiechelon repair environment where there is one budget to cover all spares, all repair/resupply, and preventive maintenance activities. The technique incorporates different criticalities of shortages by type of ship and item. It can be used either in a budget building mode or a budget execution mode.     

Queueing models for spares provisioning

A population of items which break down at random times and require repair is studied (the classic “machine repair problem with spares”). It is desired to determine the number of repair channels and spares required over a multiyear planning horizon in which population size and component reliability varies, and a service level constraint is imposed. When an item fails, a spare (if available) is immediately dispatched to replace the failed item. The failed item is removed, transported to the repair depot, repaired, and then placed in the spares pool (which is constrained to be empty not more than 10% of the time) unless there is a backlog of requests for spares, in which case it is dispatched immediately. The first model considered treats removal, transportation, and repair as one service operation. The second model is a series queue which allows for the separate treatment of removal, transportation, and repair. Breakdowns are assumed Poisson and repair times exponential.     

A deterministic inventory model for reparable items

A reparable inventory system has two distinct inventories within it—the inventory of items ready-for-issue and the inventory of carcasses available for repair. A reparable item is usually rebuilt upon failure, but the scrap rate in the repair process is generally positive. Consequently, new items must be procured from time to time to replace those item: which were scrapped. The ready-for-issue inventory has two input sources—procurement and repair, This paper develops a deterministic inbentory model for the reparable inventory system, and determines the optimal procurement and repair quantities.     

批量独立可修件的备件需求预测仿真算法

利用Monte Carlo随机仿真技术,采用事件调度法的仿真策略对可修件的使用与维修过程进行仿真,给出了在给定备件和维修分队数量下批量独立可修件的备件保障概率仿真算法,并用一系列计算实例证明了算法的可行性和正确性,该算法能有效解决安装在不同装备上,具有不同已工作时间、不同故障间隔时间分布、不同修复时间分布的可修件的备件保障概率计算问题,为备件的科学储备提供了有力的决策依据。     

Spares stocking policies for repairable items with dependent repair times

This paper analyzes the problem of determining desirable spares inventory levels for repairable items with dependent repair times. The problem is important for repairable products such as aircraft engines which can have very large investment in spares inventory levels. While existing models can be used to determine optimal inventory spares levels when repair times are independent, the practical considerations of limited repair shop capacity and prioritized shop dispatching rules combine to make repair times not independent of one another. In this research a simulation model of a limited capacity repair facility with prioritized scheduling is used to explore a variety of heuristic approaches to the spares stocking decision. The heuristics are also compared with use of a model requiring independent repair times (even though that assumption is not valid here). The results show that even when repair time dependencies are present, the performance of a model which assumes independent repair times is quite good.     

Customer delays in M/M/c repair systems with spares

A service center to which customers bring failed items for repair is considered. The items are exchangeable in the sense that a customer is ready to take in return for the failed item he brought to the center any good item of the same kind. This exchangeability feature makes it possible for the service center to possess spares. The focus of the article is on customer delay in the system—the time that elapses since the arrival of a customer with a failed item and his departure with a good one—when repaired items are given to waiting customers on a FIFO basis. An algorithm is developed for the computation of the delay distribution when the item repair system operates as an M/M/c queue.     

Optimization methods for large-scale multiechelon stockage problems

I examine the problem of determining inventory stockage levels and locations of different parts in a multiechelon system. This stockage problem is complicated by parts commonality—each part may be used by several different end items. Stockage levels and locations of each part affect the availability of end items that use the part, since an end item will be out of service if it requires a part that is not available. Of course, if the part is available at another nearby location, then the end item will be out of service for a shorter period of time. An essential feature of any model for this problem is constraints on operational availability of the end items. Because these constraints would involve nonconvex functions if the stockage levels were allowed to vary continuously, I formulate a 0–1 linear optimization model of the stockage problem. In this model, each part can be stocked at any of a number of prespecified levels at each echelon. The model is to minimize stockage cost of the selected items subject to the end-item availability constraints and limits on the total weight, volume, and number of different parts stocked at each echelon. Advantages and disadvantages of different Lagrangian relaxations and the simplex method with generalized upper-bounding capability are discussed for solving this stockage model.     

An analysis of single item inventory systems with returns

Inventory systems with returns are systems in which there are units returned in a repairable state, as well as demands for units in a serviceable state, where the return and demand processes are independent. We begin by examining the control of a single item at a single location in which the stationary return rate is less than the stationary demand rate. This necessitates an occasional procurement of units from an outside source. We present a cost model of this system, which we assume is managed under a continuous review procurement policy, and develop a solution method for finding the policy parameter values. The key to the analysis is the use of a normally distributed random variable to approximate the steady-state distribution of net inventory. Next, we study a single item, two echelon system in which a warehouse (the upper echelon) supports N(N ? 1) retailers (the lower echelon). In this case, customers return units in a repairable state as well as demand units in a serviceable state at the retailer level only. We assume the constant system return rate is less than the constant system demand rate so that a procurement is required at certain times from an outside supplier. We develop a cost model of this two echelon system assuming that each location follows a continuous review procurement policy. We also present an algorithm for finding the policy parameter values at each location that is based on the method used to solve the single location problem.     

Optimal maintenance-repair policies for the machine repair problem

We consider a model with M + N identical machines. As many as N of these can be working at any given time and the others act as standby spares. Working machines fail at exponential rate λ, spares fail at exponential rale γ, and failed machines are repaired at exponential rate μ. The control variables are λ. μ, and the number of removable repairman, S, to be operated at any given time. Using the criterion of total expected discounted cost, we show that λ, S, and μ are monotonic functions of the number of failed machines M, N, the discount factor, and for the finite time horizon model, the amount of time remaining.     

Inventory rotation policies for slow moving items

In this article we present a stochastic model for determining inventory rotation policies for a retail firm which must stock many hundreds of distinctive items having uncertain heterogeneous sales patterns. The model develops explicit decision rules for determining (1) the length of time that an item should remain in inventory before the decision is made on whether or not to rotate the item out of inventory and (2) the minimum sales level necessary for retaining the item in inventory. Two inventory rotation policies are developed, the first of which maximizes cumulative expected sales over a finite planning horizon and the second of which maximizes cumulative expected profit. We also consider the statistical behavior of items having uncertain, discrete, and heterogeneous sales patterns using a two-period prediction methodology where period 1 is used to accumulate information on individual sales rates and this knowledge is then used, in a Bayesian context, to make sales predictions for period 2. This methodology assumes that over an arbitrary time interval sales for each item are Poisson with unknown but stationary mean sales rates and the mean sales rates are distributed gamma across all items. We also report the application of the model to a retail firm which stocks many hundreds of distinctive unframed poster art titles. The application provides some useful insights into the behavior of the model as well as some interesting aspects pertaining to the implementation of the results in a “real-world” situation.     

The performance of cannibalization policies in a maintenance system with spares,repair, and resource constraints

This article presents the results of comparing the performance of several cannibalization policies using a simulation model of a maintenance system with spares, repair, and resource constraints. Although the presence of cannibalization has been incorporated into a number of maintenance system models reported in the literature, the questions of whether cannibalization should be done and what factors affect canibalization have received little attention. Policies tested include both no cannibalization and unlimited cannibalization as well as other based on the number of maintenance personnel available, the short-term machine failure rate at the time of cannibalization, and the relationship between the mean cannibalization and repair rates. The best policies found are those that allow cannibalization only when it can be done quickly relative to repair or when it can be done without delaying part repair actions. The policy of complete cannibalization (always cannibalize when it is possible) is found to perform poorly except when either average maintenance personnel utilization is very low or when mean cannibalization times are very short relative to mean repair times. The latter result casts doubts on the appropriateness of the assumption of complete cannibalization in many models in the literature.     

The initial provisioning decision for insurance type items

Initial provisioning decisions (inventory stocking requirements) for low demand items often have to be made without much knowledge of what future demand rates will be. When the nature of an item is such that little demand for it is expected, the problem of whether to stock initially or risk not stocking the item is most critical. This report discusses this problem and presents decision procedures which can be used to handle this aspect of initial provisioning. The procedures relate an item's provisioning desirability to its provisioning characteristics, such as expected cost, expected resupply time, current information on its likely demand rate, and to an overall operating policy or criterion. The criterion function measures the total system degredation as a function of the events of having items out of stock when demand occurs. Several different policy functions are discussed and the provisioning decision rules which apply to each are presented. Demand rate information is handled through a Bayesian type approach. The decision rules presented in this report can be utilized to either determine stocking requirements within a budgetary constraint, or determine the relative stocking desirability on an item-by-item basis.     

Multi-item service constrained (s,S) policies for spare parts logistics systems

Many organizations providing service support for products or families of products must allocate inventory investment among the parts (or, identically, items) that make up those products or families. The allocation decision is crucial in today's competitive environment in which rapid response and low levels of inventory are both required for providing competitive levels of customer service in marketing a firm's products. This is particularly important in high-tech industries, such as computers, military equipment, and consumer appliances. Such rapid response typically implies regional and local distribution points for final products and for spare parts for repairs. In this article we fix attention on a given product or product family at a single location. This single-location problem is the basic building block of multi-echelon inventory systems based on level-by-level decomposition, and our modeling approach is developed with this application in mind. The product consists of field-replaceable units (i.e., parts), which are to be stocked as spares for field service repair. We assume that each part will be stocked at each location according to an (s, S) stocking policy. Moreover, we distinguish two classes of demand at each location: customer (or emergency) demand and normal replenishment demand from lower levels in the multiechelon system. The basic problem of interest is to determine the appropriate policies (s_i S_i) for each part i in the product under consideration. We formulate an approximate cost function and service level constraint, and we present a greedy heuristic algorithm for solving the resulting approximate constrained optimization problem. We present experimental results showing that the heuristics developed have good cost performance relative to optimal. We also discuss extensions to the multiproduct component commonality problem.     

Expected time delay in multi‐item inventory systems with correlated demands

This paper considers multi‐item inventory systems where a customer order may require several different items (i.e., demands are correlated across items) and customer satisfaction is measured by the time delays seen by the customers. Most inventory models on time delay in the literature assume each demand only requires one item (i.e., demands are not correlated across items or are independent). In this paper, we derive an exact expression for the expected total time delay. We show that when items are actually correlated, assuming items are independent leads to an overestimate of the total time delay. However, (1) it is extremely difficult in practice to obtain the demand information for all demand types (especially in a system with tens of thousands of part numbers), and (2) the problem becomes too complicated to be of practical interest when the correlation is considered. We then explore the possibility of including the demand information partially and develop bounds for the time delays. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 671–688, 1999     

Some approximations in multi-item,multi-echelon inventory systems for recoverable items

The optimization problem as formulated in the METRIC model takes the form of minimizing the expected number of total system backorders in a two-echelon inventory system subject to a budget constraint. The system contains recoverable items – items subject to repair when they fail. To solve this problem, one needs to find the optimal Lagrangian multiplier associated with the given budget constraint. For any large-scale inventory system, this task is computationally not trivial. Fox and Landi proposed one method that was a significant improvement over the original METRIC algorithm. In this report we first develop a method for estimating the value of the optimal Lagrangian multiplier used in the Fox-Landi algorithm, present alternative ways for determining stock levels, and compare these proposed approaches with the Fox-Landi algorithm, using two hypothetical inventory systems – one having 3 bases and 75 items, the other 5 bases and 125 items. The comparison shows that the computational time can be reduced by nearly 50 percent. Another factor that contributes to the higher requirement for computational time in obtaining the solution to two-echelon inventory systems is that it has to allocate stock optimally to the depot as well as to bases for a given total-system stock level. This essentially requires the evaluation of every possible combination of depot and base stock levels – a time-consuming process for many practical inventory problems with a sizable system stock level. This report also suggests a simple approximation method for estimating the optimal depot stock level. When this method was applied to the same two hypotetical inventory systems indicated above, it was found that the estimate of optimal depot stock is quite close to the optimal value in all cases. Furthermore, the increase in expected system backorders using the estimated depot stock levels rather than the optimal levels is generally small.     

Approximations for the consolidation of multiechelon repair facilities

This paper develops and illustrates an approximate approach for analytically assessing the impacts on both costs and service of consolidation of repair facilities. The repair facilities are two echelon generalizations of the classical repairmen problem in which two types of failures, say major and minor, can occur, each type requiring repair at a different echelon: The questions addressed are the reductions possible in spares, repairmen, and service rates due to the consolidated system's increased efficiency, as well as the physical separation between the users and the consolidated repair facility that is economical. The method of analysis is based upon asymptotic approximations developed for the repairmen problem, valid when the number of operational equipments is large (greater than 50); it is helpful since it provides a tractable means for predicting the steady-state performance of the decentralized and consolidated installations as a function of the many parameters involved without having to resort to an exhaustive computation of all the exact steady-state probabilities.     

A modified block replacement policy

A well known preventive replacement policy is the block replacement policy (BRP). In such a policy the item undergoes a planned replacement at a sequence of equally spaced time points independent of failure history. The main advantage of a BRP is its simplicity, because under this policy it is unnecessary to keep detailed records about times of failures or ages of items. The main drawback of a BRP is that at planned replacement times we may be replacing practically new items. In this paper we study a modified BRP which is free of this drawback. We calculate the expected cost of following a modified BRP for lifetime distributions possessing a special structure and illustrate it for the case of an Erlang distribution. A numerical comparison is made between a modified BRP and a standard BRP for the special case of a two stage Erlang distribution.     

An economic sequential screening procedure for limited failure populations

An economic sequential screening procedure is considered for limited failure populations in which defective items fail soon after they are put in operation and nondefective ones never fail during the technical life of the items. A cost model is constructed that involves screening test cost and external failure cost. A sequential scheme that minimizes the expected cost is derived from the solution of a dynamic programming formulation and the optimal decision at each stage is obtained in a closed form. © 1994 John Wiley & Sons, Inc.     

A comparison of usage data among types of aircraft

It has long been an accepted proposition that base level usage df technical repair items for aircraft is related to program elements. The program element most often cited in this connection is flying hours. Evidence of this relationship on a line item basis, however, has been sparse. This study suggests that although the supposed relation is intuitively plausible, for most line items it cannot be effectively utilized in projecting base level requirements. The basis of this conclusion is an analysis of demand characteristics for technical repair items for a variety of naval aircraft.