A geometric-process repair-model with good-as-new preventive repair

This paper studies a deteriorating simple repairable system. In order to improve the availability or economize the operating costs of the system, the preventive repair is adopted before the system fails. Assume that the preventive repair of the system is as good as new, while the failure repair of the system is not, so that the successive working times form a stochastic decreasing geometric process while the consecutive failure repair times form a stochastic increasing geometric process. Under this assumption and others, by using geometric process we consider a replacement policy N based on the failure number of the system. Our problem is to determine an optimal replacement policy N such that the average cost rate (i.e., the long-run average cost per unit time) is minimized. The explicit expression of the average cost rate is derived, and the corresponding optimal replacement policy can be determined analytically or numerically. And the fixed-length interval time of the preventive repair in the system is also discussed. Finally, an appropriate numerical example is given. It is seen from that both the optimal policies N** and N* are unique. However, the optimal policy N** with preventive repair is better than the optimal policy N* without preventive repair     

A Geometric Process $delta$-Shock Maintenance Model

A geometric process $delta$ -shock maintenance model for a repairable system is introduced. If there exists no shock, the successive operating time of the system after repair will form a geometric process. Assume that the shocks will arrive according to a Poisson process. When the interarrival time of two successive shocks is smaller than a specified threshold, the system fails, and the latter shock is called a deadly shock. The successive threshold values are monotone geometric. The system will fail at the end of its operating time, or the arrival of a deadly shock, whichever occurs first. The consecutive repair time after failure will constitute a geometric process. A replacement policy $N$ is adopted by which the system will be replaced by a new, identical one at the time following the $N$th failure. Then, for the deteriorating system, and the improving system, an optimal policy $N^{ast}$ for minimizing the long-run average cost per unit time is determined analytically.      

Optimal periodic preventive repair and replacement policy assuming geometric process repair

In this paper, a simple deteriorating system with repair is studied. When failure occurs, the system is replaced at high cost. To extend the operating life, the system can be repaired preventively. However, preventive repair does not return the system to a "good as new" condition. Rather, the successive operating times of the system after preventive repair form a stochastically decreasing geometric process, while the consecutive preventive repair times of the system form a stochastically increasing geometric process. We consider a bivariate preventive repair policy to solve the efficiency for a deteriorating & valuable system. Thus, the objective of this paper is to determine an optimal bivariate replacement policy such that the average cost rate (i.e., the long-run average cost per unit time) is minimized. The explicit expression of the average cost rate is derived, and the corresponding optimal replacement policy can be determined numerically. An example is given where the operating time of the system is given by a Weibull distribution.     

An optimal replacement policy for a three-state repairable system with a monotone process model

In this paper, a deteriorating simple repairable system with three states, including two failure states and one working state, is studied. Assume that the system after repair cannot be "as good as new", and the deterioration of the system is stochastic. Under these assumptions, we use a replacement policy N based on the failure number of the system. Then our aim is to determine an optimal replacement policy N/sup */ such that the average cost rate (i.e., the long-run average cost per unit time) is minimized. An explicit expression of the average cost rate is derived. Then, an optimal replacement policy is determined analytically or numerically. Furthermore, we can find that a repair model for the three-state repairable system in this paper forms a general monotone process model. Finally, we put forward a numerical example, and carry through some discussions and sensitivity analysis of the model in this paper.     

Extended block replacement policy of a system subject to shocks

A generalization of the block replacement (BR) policy is proposed and analyzed for a system subject to shocks. Under such a policy, an operating system is preventively replaced by new ones at times i·T (i=1,2,3,...) independently of its failure history. If the system fails in: (a) ((i-1)·T, (i-1)·T+T₀), it is either replaced by a new one or minimally repaired; or (b) ((i-1)·T+T₀, i·T), it is either minimally repaired or remains inactive until the next planned replacement. The choice of these two actions is based on some mechanism (modeled as random) which depends on the number of shocks since the latest replacement. The average cost rate is obtained using the results of renewal reward theory. The model with two variables is transformed into a model with one variable and the optimum policy is discussed. Various special cases are considered. The results extend many of the well-known results for BR policies     

Optimal replacement policy (N, n) for a parallel system with common cause failure and geometric repair time

The purpose of this article is to present an improved replacement model for a parallel system of N identical units, by bringing in common cause failure (CCF), maintenance cost and repair cost per unit time additionally, and to develop a procedure to obtain the optimal redundant units N^* and optimal number of repairs n^* with the conditions that the system is allowed to undergo at most a prefixed number of repairs before to be replaced and the successive reapir times after failures constitute a non-decreasing Geometric process. Several conditions for the existence of the optimal N^* and n^* is stated and the results are illustrated by a numerical example.     

A maintenance model for two-unit redundant system

In this paper, a maintenance model for two-unit redundant system with one repairman is studied. At the beginning, unit 1 is operating, unit 2 is the standby unit. The costs include the operating reward, repair cost and replacement cost, besides, a penalty cost is incurred if the system breaks down. Two kinds of replacement policy, based on the number of failures for two units and the working age, respectively are used. The long-run average cost per unit time for each kind of replacement policy is derived. Also, a particular model in which the system is deteriorative, two units are identical and the penalty cost rate is high, is thoroughly studied.     

Maintenance-policy cost-analysis for a series system withhighly-censored data

This paper considers the problem of estimating the optimal age-replacement time for a series arrangement of functional subsystems when data are subject to high levels of random censoring on the right. The system does not have redundant components. Simulation is used to compare the performance of the Kaplan-Meier Estimator (KME), the Piecewise Exponential Estimator (PEXE) and the Maximum Likelihood Estimator (MLE) in estimating the optimal replacement time for the system, as well as for each component, under high levels of random censorship. Monte Carlo analysis is used to estimate `average optimal age replacement times' determined using total time on test (TTT) transforms based on the KME, PEXE, and MLE methods. The optimal replacement time is used to calculate a value which is used to compare the relative long-run cost per unit-time for each method. The differences between using system-level data vs. component-level data to construct a maintenance policy are examined. With respect to cost effectiveness, the results identify the crucial factor in determining whether to perform system-level maintenance or component-level maintenance; that factor is the ratio of the `cost of performing preventive maintenance' and the `penalty cost of experiencing a system failure'. For the ratios used in this study (0.1 to 0.5) a `component-level maintenance policy' is more cost effective than a `system-level maintenance policy'. The results also show that for a correctly specified model and for large sample sizes, the age replacement times provided by the MLE are more accurate than those provided by the KME and PEXE, especially under high levels of censoring     

An optimal maintenance model using a number of different actions

In this paper, we study an optimal maintenance model. The state of a system is determined by the distribution of its operating time. Whenever the system fails, a number of actions can be chosen, including the repair actions, the replacement actions and the action of discarding the system. The objective of this paper is to determine an optimal policy which maximizes the expected total discounted reward. By using the semi-Markov decision process approach, the method of successive approximations is suggested for determining the optimal reward function and the corresponding optimal policy.     

Age Replacement In Simple Systems With Increasing Loss Functions

Some age replacement policies are investigated and conditions for the unique existence of an optimum policy are derived. The optimum policy is the one which minimizes the expected cost per unit time over an infinite time span or maximizes the proportion of time during which the system is in operation. Losses have been expressed through increasing operating cost, if the objective is to minimize the expected total cost per unit time and through increasing renewal times if the objective is to maximize the availability of the system.     

An inspection-maintenance model for systems with multiple competing processes

In some applications, the failure rate of the system depends not only on the time, but also upon the status of the system, such as vibration level, efficiency, number of random shocks on the system, etc., which causes degradation. In this paper, we develop a generalized condition-based maintenance model subject to multiple competing failure processes including two degradation processes, and random shocks. An average long-run maintenance cost rate function is derived based on the expressions for the degradation paths & cumulative shock damage, which are measurable. A geometric sequence is employed to develop the inter-inspection sequence. Upon inspection, one needs to decide whether to perform a maintenance, such as preventive or corrective, or to do nothing. The preventive maintenance thresholds for degradation processes & inspection sequences are the decision variables of the proposed model. We also present an algorithm based on the Nelder-Mead downhill simplex method to calculate the optimum policy that minimizes the average long-run maintenance cost rate. Numerical examples are given to illustrate the results using the optimization algorithm.     

An age replacement policy with minimal repair and general random repair cost

An age replacement policy is introduced which incorporates minimal repair, replacement, and general random repair costs. If an operating unit fails at age y<T, it is either replaced by a new unit with probability p(y) at a cost c₀, or it undergoes minimal repair with probability q(y) = 1−p(y). Otherwise, a unit is replaced when it fails for the first time after age T. The cost of the i-th minimal repair of an unit at age y depends on the random part C(y) and the deterministic part c_i(y). The aim of the paper is to find the optimal T which minimizes the long run expected cost per unit time of the policy. Various special cases are considered.     

Replacement vs. repair of failed components for a system with arandom lifetime

The computation of a variety of reliability measures for a system that is observed over a random time horizon is examined. This time interval, which can correspond to the system mission, the system lifetime, etc., is assumed to be distributed as a random variable of phase type. Models are developed that aid a decision-maker to choose among the actions of replace, repair, or do nothing, whenever a system component fails, under several assumptions concerning the amount of information available to the decision-maker. In addition, a policy of replacement after a fixed number of repairs is examined. An example computation of several reliability measures and the application of a cost model are included to illustrate the tractability of the results and their usefulness in design decisions     

Modeling and Analyzing a Joint Optimization Policy of Block-Replacement and Spare Inventory With Random-Leadtime

We consider a generalized joint optimization policy of block replacement & periodic review spare inventory with random lead time. According to the relationship between geometric area in the graph of inventory level over time, and holding or shortage costs, a model analyzing four mutually exclusive & exhaustive possibilities is developed for the expected average cost per unit time, and is based on the stochastic behavior of the assumed system. The model reflects the cost of inventory holding, spare shortage, replacement, and ordering. And for the first time known to the authors, we deliver the sufficient and necessary conditions of the existence and uniqueness of the minimum in the joint models of this type. Because the model and its analysis are general, one existing result is shown to be subsumed by this model with some modifications. Some numerical cases verify the deduction, and give a general searching solution procedure. Finally, we introduce some discussions related to the models. The models mentioned in the paper can be readily applied in many fields such as economical fields, financial engineering, armament administration, and even medical fields, with some modifications. And the mathematical deduction in the paper will be a guideline for analyzing related stochastic models.     

Optimum Ordering Policy under Random Lead-Time for Equipment with a Sensor

A replacement model is presented for a single equipment with a sensing device attached, where two kinds of random lead times (one for regular order and the other for expedited order) are considered. Both the equipment and device can fail, and the state of the equipment (good or failed) is monitored by the device and also inspected manually at each ordering time (fixed). Our model shows, under certain conditions, that there exists a finite and unique ordering policy maximizing the cost effectiveness which is used as the criterion for optimality.     

Age replacement of components during IFR delay time

This paper proposes two alternative policies for preventive replacement of a component, which shows sign of occurrence of a fault, and operates for some random time with degraded performance, before its final failure. The time between fault occurrence and component failure is termed as delay time. The first policy, namely age replacement during delay time policy (ARDTP), recommends replacement of a faulty component on failure or preventive replacement of the same after a fixed time during its delay time. It considers the performance degradation during delay time to develop an age replacement policy. It is also shown that the policy is a feasible proposition for a component that has positive (nonnegative) performance degradation during its CFR (IFR) delay time. The second policy, OARDTP, extends ARDTP to opportunistic age replacement policy where a faulty component is replaced at the first available randomly occurring maintenance opportunity, after a fixed time from occurrence of fault, or on failure. The time between opportunities (TBO) is considered to be exponentially distributed. This policy reduces the number of forced shutdowns, which is essential to ARDTP. It is shown that the second policy is superior to the first policy if the cost of a preventive replacement with forced shutdown is more than the preventive replacement cost during an opportunity. The policies are appropriate for complex process plants, where the tracking of the entire service life of each component is difficult. Their implementation requires tracking of components' delay time only, and estimation of mean time to occurrence of faults. The policies are relatively insensitive to estimation error in failure replacement cost. As their implementation requires immediate capturing of fault occurrence information, they are particularly attractive to organizations where operators are involved in the maintenance of machines.     

Optimal replacement policies for systems with multiple standbycomponents

The authors consider two new preventive replacement policies for a multiple-component cold-standby system. The failure rate of the component in operation is constant. The system is inspected at random points over time to determine whether it is to be replaced. The replacement decision is based on the number of failed components at the time of inspection. There are two replacement options if the complete system fails during operation: (i) replace the system if an inspection reveals that it has failed (system failure is not self-announcing), and (ii) replace the system the instant it fails (system failure is self-announcing). There is a threshold value on the number of failed components (at the time of inspection) which minimizes the mean total cost. The authors develop a simple efficient procedure to find the optimal threshold value. They compare the cost of operating a system that is inspected at random points in time, with the cost of operating a system that is monitored continuously through an attached monitoring device, and discuss cost tradeoffs     

Cost analysis of a two dissimilar-unit cold standby redundant system subject to inspection and two types of repair

This paper deals with the cost analysis of a two dissimilar-unit cold standby redundant system subject to inspection and two types of repair where each unit of the system has two modes, normal and failed. It is assumed that the failure, repair, replacement and inspection times are stochastically independent random variables each having an arbitrary distribution. The cold standby unit replaces the failed operative unit after a random amount of time. An inspection is required to decide whether it needs type I (minor repair) or type 2 (major repair). In this system the repairman is not always available with the system, but is called whenever the operative unit fails. The system is analysed by the semi-Markov process technique. Some reliability measures of interest to system designers as well as operations managers have been obtained. Pointwise availability, steady-state availability, busy period by a server and the expected cost per unit time of the system are obtained. Certain important results have been derived as particular cases.     

Optimal control of a removable and non-reliable server in a markovian queueing system with finite capacity

This paper studies the economic behavior of a removable and non-reliable server in an Markovian queueing system with finite capacity under steady-state conditions. The removable server applies the N policy which turns the server on when the queue length reaches the value N, and turns the server off when the system is empty. The server may break down only if operating and require repair at a repair facility. Interarrival and service times of the customers, and breakdown and repair times of the server, are assumed to follow a negative exponential distribution. A cost model is developed to determine the optimal operating N policy numerically in order to minimize the total expected cost per unit time.     

Cost limit replacement policy under minimal repair

This paper presents a policy for either repairing or replacing a system that has failed. When a system requires repair, it is first inspected and the repair cost is estimated. Repair is only then undertaken if the estimated cost is less than the “repair cost limit”. However, the repair cannot return the system to “as new” condition but instead returns it to the average condition for a working system of its age. Examples include complex systems where the repair or replacement of one component does not materially affect the condition of the whole system. A Weibull distribution of time to failure and a negative exponential distribution of estimated repair cost are assumed for analytic amenability. An optimal “repair cost limit” policy is developed that minimizes the average cost per unit time for repairs and replacement. It is shown that the optimal policy is finite and unique.