General trigger-off replacement procedures for two-unit systems

A trigger-off replacement policy, suggested and analyzed in [1], for two-unit systems composed of identical units, is generalized and extended in this work in several ways. In the first part of the paper we obtain the appropriate integral equations for nonidentical units and then use them for a complete solution of the case of two units, whose lifetimes are distributed according to general Erlang distributions. In the second part of the paper we extend the trigger-off policy itself by allowing preventive replacements of units which reach a certain critical age. The system stops working if either one of the two units fails or reaches its critical age. Both cases present natural replacement possibilities for the remaining unit, provided that its age exceeds a predetermined critical age. Finally, we consider the question of which policy parameters, i.e. control limits and critical ages, to choose when facing a real-life situation. Using the criterion of expected costs per unit time in the long run, we show how to find the optimal parameters which minimize this objective function. The fact that a restricted optimization, within the class of trigger-off replacement policies, leads to the global optimal policy has been proved in [2] by a different approach.     

Optimal replacement for fault-tolerant systems

We consider the optimal replacement problem for a fault tolerant system comprised of N components. The components are distingushable, and the state of the system is given by knowing exactly which components are operationl and which have failed. The individual component failure rates depend on the state of the entire system. We assume that the rate at which the system produces income decreases as the system deteriorates and the system replacement cost rises. Individual components cannot be replaced. We give a greedy-type algorithm that produces the replacement policy that maximizes the long-run net system income per unit time.     

Maintaining systems with heterogeneous spare parts

We consider the problem of optimally maintaining a stochastically degrading, single‐unit system using heterogeneous spares of varying quality. The system's failures are unannounced; therefore, it is inspected periodically to determine its status (functioning or failed). The system continues in operation until it is either preventively or correctively maintained. The available maintenance options include perfect repair, which restores the system to an as‐good‐as‐new condition, and replacement with a randomly selected unit from the supply of heterogeneous spares. The objective is to minimize the total expected discounted maintenance costs over an infinite time horizon. We formulate the problem using a mixed observability Markov decision process (MOMDP) model in which the system's age is observable but its quality must be inferred. We show, under suitable conditions, the monotonicity of the optimal value function in the belief about the system quality and establish conditions under which finite preventive maintenance thresholds exist. A detailed computational study reveals that the optimal policy encourages exploration when the system's quality is uncertain; the policy is more exploitive when the quality is highly certain. The study also demonstrates that substantial cost savings are achieved by utilizing our MOMDP‐based method as compared to more naïve methods of accounting for heterogeneous spares.     

基于年龄更换策略的威布尔分布备件需求计算方法

备件需求量预测在装备保障工作中发挥着重要作用。预防性维修可以消除装备潜在的故障或避免装备发生故障后所导致的严重后果,从而使装备始终处于期望状态。预防性维修策略下备件需求量建模能够以可预见的可靠性数据为基础进行,其关键问题是最佳更换间隔时间的确定。针对威布尔分布装备备件,采用年龄更换策略,确定最佳更换间隔时间,并在此基础上导出威布尔型备件需求计算模型,为备件需求量的预测提供了一种简单有效的方法。     

A note on a modified block replacement policy for units with increasing marginal running costs

The model for a modified block replacement policy (MBRP) is extended to include running costs. An illustrative example is worked out for the case when item life is exponentially distributed and marginal running cost per unit time increases linearly with the age of the item.     

Replacement models under additive damage

A production system which generates income is subject to random failure. Upon failure, the system is replaced by a new identical one and the replacement cycles are repeated indefinitely. In our breakdown model, shocks occur to the system in a Poisson stream. Each shock causes a random amount of damage, and these damages accumulate additively. The failure time depends on the accumulated damage in the system. The income from the system and the cost associated with a planned replacement depend on the accumulated damage in the system. An additional cost is incurred at each failure in service. We allow a controller to replace the system at any stopping time T before failure time. We will consider the problem of specifying a replacement rule that is optimal under the following criteria: maximum total long-run average net income per unit time, and maximum total long-run expected discounted net income. Our primary goal is to introduce conditions under which an optimal policy is a control limit policy and to investigate how the optimal policy can be obtained. Examples will be presented to illustrate computational procedures.     

Optimal investment,pricing and replacement of computer resources

The problem of multiple-resource capacity planning under an infinite time horizon is analyzed using a nonlinear programming model. The analysis generalizes to the long term the short-run pricing model for computer networks developed in Kriebel and Mikhail [5]. The environment assumes heterogeneous resource capacities by age (vingate), which service a heterogeneous and relatively captive market of users with known demand functions in each time period. Total variable operating costs are given by a continuous psuedoconcave function of system load, capacity, and resource age. Optimal investment, pricing, and replacement decision rules are derived in the presence of economies of scale and exogenous technological progress. Myopic properties of the decision rules which define natural (finite) planning subhorizons are discussed.     

Optimal replacement under additive damage and other failure models

A machine or production system is subject to random failure. Upon failure the system is replaced by a new one, and the process repeats. A cost is associated with each replacement, and an additional cost is incurred at each failure in service. Thus, there is an incentive for a controller to attempt to replace before failure occurs. The problem is to find an optimal control strategy that balances the cost of replacement with the cost of failure and results in a minimum total long-run average cost per unit time. We attack this problem under the cumulative damage model for system failure. In this failure model, shocks occur to the system in accordance with a Poisson process. Each shock causes a random amount of damage or wear and these damages accumulate additively. At any given shock, the system fails with a known probability that depends on the total damage accumulated to date. We assume that the cumulative damage is observable by the controller and that his decisions may be based on its current value. Supposing that the shock failure probability is an increasing function of the cumulative damage, we show that an optimal policy is to replace either upon failure or when this damage first exceeds a critical control level, and we give an equation which implicitly defines the optimal control level in terms of the cost and other system parameters. Also treated are some more general models that allow for income lost during repair time and other extensions.     

Optimal burn‐in procedure for periodically inspected systems

Burn‐in is a widely used method to improve the quality of products or systems after they have been produced. In this paper, we study burn‐in procedure for a system that is maintained under periodic inspection and perfect repair policy. Assuming that the underlying lifetime distribution of a system has an initially decreasing and/or eventually increasing failure rate function, we derive upper and lower bounds for the optimal burn‐in time, which maximizes the system availability. Furthermore, adopting an age replacement policy, we derive upper and lower bounds for the optimal age parameter of the replacement policy for each fixed burn‐in time and a uniform upper bound for the optimal burn‐in time given the age replacement policy. These results can be used to reduce the numerical work for determining both optimal burn‐in time and optimal replacement policy. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

A dynamic nonlinear constrained optimal replacement model

A stochastically constrained optimal replacement model for capital equipment is constructed. Each piece of capital equipment, or machine, is characterized by its age and “utility” or “readiness” class. The readiness of a machine at any age is a stochastic function of its initial utility class and its age. The total discounted replacement cost of several replacement streams, each commencing with an initial machine, is minimized with respect to the replacement age and initial utility class of each machine, subject to a readiness constraint stating the lower bound on the expected number of machines in each utility class at any time. A general solution procedure is outlined and a specific case is solved in detail.     

Periodic replacement with minimal repair at failure and adjustment costs

We investigate periodic replacement policies with minimal repair at failure, thereby, minimizing the average expected cost per unit time over an infinite time span. The standard cost structure is modified by the introduction of a term which takes adjustment costs into account.     

复杂系统复合维修间隔期优化模型

采用组合策略对复杂系统辅以功能检测的定期更换维修工作进行综合优化.在分析复杂系统维修费用结构和组成的基础上,建立了其无限使用期条件下单位时间期望费用的数学模型,从而获得系统最佳的功能检测间隔期、定期更换周期内的检测次数和最优总费用.最后通过一个算例验证了该策略和模型的有效性.     

Optimal policy of continuous and discrete replacement with minimal repair at failure

This article considers combined continuous and discrete replacement with minimal repair at failure, in which a unit is replaced at time T or at number N of uses. Both optimal time T* and number N* to minimize the expected cost rate are discussed. They are found by unique solutions of equations when the hazard rates are monotonously increasing. A numerical example is given.     

Planning horizon procedures for machine replacement models with several possible replacement alternatives

This paper develops a forward algorithm and planning horizon procedures for an important machine replacement model where it is assumed that the technological environment is improving over time and that the machine-in-use can be replaced by any of the several different kinds of machines available at that time. The set of replacement alternatives may include (i) new machines with different types of technologies such as labor- and capital- intensive, (ii) used machines, (iii) repairs and/or improvements which affect the performance characteristics of the existing machine, and so forth. The forward dynamic programming algorithm in the paper can be used to solve a finite horizon problem. The planning horizon results give a procedure to identify the forecast horizon T such that the optimal replacement decision for the first machine based on the forecast of machine technology until period T remains optimal for any problem with horizon longer than T and, for that matter, for the infinite horizon problem. A flow chart and a numerical example have been included to illustrate the algorithm.     

The parallel replacement problem with demand and capital budgeting constraints

A generalized parallel replacement problem is considered with both fixed and variable replacement costs, capital budgeting, and demand constraints. The demand constraints specify that a number of assets, which may vary over time, are required each period over a finite horizon. A deterministic, integer programming formulation is presented as replacement decisions must be integer. However, the linear programming relaxation is shown to have integer extreme points if the economies of scale binary variables are fixed. This allows for the efficient computation of large parallel replacement problems as only a limited number of 0–1 variables are required. Examples are presented to provide insight into replacement rules, such as the “no‐splitting‐rule” from previous research, under various demand scenarios. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 40–56, 2000     

A bivariate optimal replacement policy for a cold standby repairable system with repair priority

In this article, an optimal replacement policy for a cold standby repairable system consisting of two dissimilar components with repair priority is studied. Assume that both Components 1 and 2, after repair, are not as good as new, and the main component (Component 1) has repair priority. Both the sequence of working times and that of the components'repair times are generated by geometric processes. We consider a bivariate replacement policy (T,N) in which the system is replaced when either cumulative working time of Component 1 reaches T, or the number of failures of Component 1 reaches N, whichever occurs first. The problem is to determine the optimal replacement policy (T,N)* such that the long run average loss per unit time (or simply the average loss rate) of the system is minimized. An explicit expression of this rate is derived, and then optimal policy (T,N)* can be numerically determined through a two‐dimensional‐search procedure. A numerical example is given to illustrate the model's applicability and procedure, and to illustrate some properties of the optimal solution. We also show that if replacements are made solely on the basis of the number of failures N, or solely on the basis of the cumulative working time T, the former class of policies performs better than the latter, albeit only under some mild conditions. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

工龄更换策略下的备件库存决策研究

针对单部件系统工龄更换策略下备件需求的特点,建立了工龄更换策略与备件库存控制的联合优化模型。该模型通过分析一个订购期内工龄更换间隔期T及备件最大库存水平S对系统寿命分布的影响,建立了工龄更换间隔期、订购间隔期及最大库存水平与单位时间总费用（包括维修费用和库存费用）的关系,然后以单位时间总费用最小为目标,优化工龄更换间隔期T、订购间隔期t0及最大库存水平S。最后,基于案例,运用Matlab对模型进行数值计算,结果表明模型能有效地降低单位时间的总费用。     

On parallel machine replacement problems with general replacement cost functions and stochastic deterioration

The parallel machine replacement problem consists of finding a minimum cost replacement policy for a finite population of economically interdependent machines. In this paper, we formulate a stochastic version of the problem and analyze the structure of optimal policies under general classes of replacement cost functions. We prove that for problems with arbitrary cost functions, there can be optimal policies where a machine is replaced only if all machines in worse states are replaced (Worse Cluster Replacement Rule). We then show that, for problems with replacement cost functions exhibiting nonincreasing marginal costs, there are optimal policies such that, in any stage, machines in the same state are either all kept or all replaced (No‐Splitting Rule). We also present an example that shows that economies of scale in replacement costs do not guarantee optimal policies that satisfy the No‐Splitting Rule. These results lead to the fundamental insight that replacement decisions are driven by marginal costs, and not by economies of scale as suggested in the literature. Finally, we describe how the optimal policy structure, i.e., the No‐Splitting and Worse Cluster Replacement Rules, can be used to reduce the computational effort required to obtain optimal replacement policies. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

Multivariate age-dependent imperfect repair

In this article we consider models of systems whose components have dependent life lengths with specific multivariate distributions. Upon failure, components are repaired. Two types of repair are distinguished. After perfect repair, a unit has the same life distribution as a new item. After imperfect repair, a unit has the life distribution of an item which is of the same age but has never failed. We study a model in which the mechanism for determining the nature of the repair is age dependent.     

Heuristics for multicomponent joint replacement: Applications to aircraft engine maintenance

This paper considers the maintenance of aircraft engine components where economies exist for joint replacement because (a) the aircraft must be pulled from service for maintenance and (b) repair of some components requires removal and disassembly of the engine. It is well known that the joint replacement problem is difficult to solve exactly, because the optimal solution does not have a simple structured form. Therefore, we formulate three easy-to-implement heuristics and test their performance against a lower bound for various numerical examples. One of our heuristics, the base interval approach, in which replacement cycles for all components are restricted to be multiples of a specified interval, is shown to be robustly accurate. Moreover, this heuristic is consistent with maintenance policies used by commercial airlines in which periodic maintenance checks are made at regular intervals. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 435–458, 1998