Periodic replacement when minimal repair costs depend on the age and the number of minimal repairs for a multi-unit system

A policy of periodic replacement with minimal repair at failure is considered for a multi-unit system which has a specific multivariate distribution. Under such a policy the system is replaced at multiples of some period T while minimal repair is performed for any intervening component failure. The cost of a minimal repair to the component is assumed to be a function of its age and the number of minimal repairs. A simple expression is derived for the expected minimal repair cost in an interval in terms of the cost function and the failure rate of the component. The necessary and sufficient conditions for the existence of an optimal replacement interval are found.     

Periodic replacement when minimal repair costs depend on the age and the number of minimal repair for a multi-unit system

A policy of periodic replacement with minimal repair at failure is considered for the multi-unit system which have the specific multivariate distribution. Under such a policy the system is replaced at multiples of some period T while minimal repair is performed at any intervening component failures. The cost of a minimal repair to the component is assumed to be a function of its age and the number of minimal repair. A simple expression is derived for the expected minimal repair cost in an interval in terms of the cost function and the failure rate of the component. Necessary and sufficient conditions for the existence of an optimal replacement interval are exhibited.     

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.     

A multicomponent two-unit cold standby system with three modes

This paper investigates a mathematical model of a two-unit cold standby redundant system with three possible states of each unit—normal, partially failed and failed. Each unit has n components, each having a constant failure rate and a repair rate, an arbitrary function of the time spent. These vary from component to component. Steady-state probabilities, steady-state pointwise availability, mean time to system failure and Laplace transforms of various transient probabilities have been obtained. Several earlier results are verified as special cases.     

Periodic replacement with minimal repair at failure and general random repair cost for a multi-unit system

A policy of periodic replacement with minimal repair at failure is considered for the multi-unit system which have the specific multivariate distribution. Under such a policy an operating system is completely replaced whenever it reaches age T (T > 0) at a cost c₀ while minimal repair is performed at any intervening component failures. The cost of the j-th minimal repair to the component which fails at age y is g(C(y),c_j(y)), where C(y) is the age-dependent random part, c_j(y) is the deterministic part which depends on the age and the number of the minimal repair to the component, and g is an positive nondecreasing continuous function. A simple expression is derived for the expected minimal repair cost in an interval in terms of the cost function and the failure rate of the component. Necessary and sufficient conditions for the existence of an optimal replacement interval are exhibited.     

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     

On optimal burn-in procedures - a generalized model

Burn-in is a manufacturing technique that is intended to eliminate early failures. In this paper, burn-in procedures for a general failure model are considered. There are two types of failure in the general failure model. One is Type I failure (minor failure), which can be removed by a minimal repair or a complete repair; and the other is Type II failure (catastrophic failure), which can be removed only by a complete repair. During the burn-in process, two types of burn-in procedures are considered. In Burn-In Procedure I, the failed component is repaired completely regardless of the type of failure; whereas, in Burn-In Procedure II, only minimal repair is done for the Type I failure, and a complete repair is performed for the Type II failure. Under the model, various additive cost functions are considered. It is assumed that the component before undergoing the burn-in process has a bathtub-shaped failure rate function with the first change point t/sub 1/, and the second change point t/sub 2/. The two burn-in procedures are compared in cases when both the procedures are applicable. It is shown that the optimal burn-in time b/sup */ minimizing the cost function is always before t/sub 1/. It is also shown that a large initial failure rate justifies burn-in, i.e., b/sup */>0. The obtained results are applied to some examples.     

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 economic discrete replacement policy for a shock damage model with minimal repairs

A discrete replacement model for a repairable system which is subject to shocks and minimal repairs is discussed. Such shocks can be classified, depending on its effect to the system, into two types: Type I and Type II shocks. Whenever a type II shock occurs causes the system to go into failure, such a failure is called type II failure and can be corrected by a minimal repair. A type I shock does damage to the system in the sense that it increases the failure rate by a certain amount and the failure rate also increases with age due to aging process without external shocks; furthermore, the failure occurred in this condition is called type I failure. The system is replaced at the time of the first type I failure or the n-th type Il failure, whichever occurs first. Introducing costs due to replacement and mininal repairs, the long-run expected cost per unit time is derived as a criterion of optimality and the optimal number n∗ found by minimizing that cost. It is shown that, under certain conditions, there exists a finite and unique optimal number n∗.     

Preventive replacement in systems with dependent components

A multicomponent series system includes a component which deteriorates over time, changing its operating characteristics and, consequently, increasing the failure rates of neighboring components. Preventive replacement of the deteriorating component can be beneficial. Replacement policies that include inspecting the deteriorating component at system failure instances and replacing it if the deterioration exceeds a critical level, or continuously monitoring the deteriorating component are considered. The system is modeled as a Markov chain solved by an efficient algorithm that exploits the system structure. For a two-component system, a closed-form equation gives the critical level for the minimum-average-cost failure-replacement policy. For the general case, replacement policies are evaluated by mean cost rate and by the ratio of the reduction in the number of failures to the number of preventive replacements     

Cost analysis of a two-unit cold standby system with two types of operation and repair

This paper deals with the cost analysis of a single-server two-identical unit cold standby system and two types of repair—minor and major. The unit requires minor repair if it fails for the first time. The major repair is required only when the unit fails after the minor repair. Upon minor repair the unit does not work as a normal unit but as a quasi-normal unit which has a different (increased) failure rate from that of a new one. Upon major repair the unit works as good as new (normal unit). Failure time distributions are negative exponential whereas repair time distributions are general. Using regeneration point technique the system characteristics of interest to system designers and operations managers have been obtained.     

Repairable consecutive-2-out-of-n:F system

In this paper, a linear consecutive-2-out-of-n:F repairable system is studied. Assume that the working time and the repair time of each component are both exponentially distributed, and each component after repair is as good as new. By using the definition of generalized transition probability, we derive the state transition probability of the system. When n is given, we obtain the exact formulas of the system reliability (or its Laplace transform) and the system mean time to first failure.     

Direct computation of expected numbers of failures and repairs via integral equation approach

This paper develops a method for computing the expected number of failures and the expected number of repairs of a component in a prescribed time interval. Our method computes directly the above mentioned quantities without passing through a conventional step of calculating the unconditional failure and repair intensities over the corresponding time interval. Our method is constructed via integral equation formulation with its operator equation representation. It is shown that the expected number of failures and the expected number of repairs can be computed with the same precision of accuracy as that of the unconditional failure and repair intensities, which cannot be possible by a conventional approach.     

Optimal Selection Methods for Failure & Repair Rates Using Nomographs

This paper explains the optimal selection methods using nomographs to solve two essentially different problems. The one is the problem of unit level, and the other is the one of system level. The unit level assumes that the cost information as a function of failure rate ? and repair rate ? are empirically known. The paper presents a method, by which a nomograph is used to select easily the optimum pair from the infinitely many pairs (?, ?) of feasible solutions, to gain the required unit availability at minimum cost of this assumption. At the system level, the system is composed of n serial i-units which are selectable from a group provided for each i-unit (i = 1, 2, ..., n), several different repair plans are available for the unit. Each unit has a specific failure rate and associated cost, and each repair plan has a specific repair rate and associated cost. There are service personnel for each unit. The failure and repair rates are constant. The paper presents: 1) A method using nomographs to select the optimum pairs from the many pairs (?i, ?i) (i = 1, 2, ..., n) of feasible solutions, to gain the required system availability at the minimum system cost, 2) A method to select the optimum pairs, from the many pairs (?i, ?i (i = 1, 2, .     

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.     

Pivotal Decomposition to Find Availability and Failure-Frequency of Systems with Common-Cause Failures

The purpose of this article is to extend the pivotal decomposition method for system availability and failure frequency from the case where components are statistically independent to that where components are also subject to common-cause failures. This method requires as input data the failure rate and the mean repair time of each component, the occurrence rate of each common-cause failure, and the mean-time for repairing it.     

System Availability and Optimum Spare Units

The steady-state availability of a repairable system with cold standbys and nonzero replacement time is maximized under constraints of total cost and total weight. Likewise the cost can be minimized under constraints of steady-state availability and total weight. A new, more efficient algorithm is used for the constrained optimization. The problem is formulated as a nonlinear integer programming problem. Since the objective functions are monotone, it is easy to obtain optimal solutions. These new algorithms are natural extensions of the Lawler-Bell algorithm. Availability is adjusted by the number of spares allowed. Other measures of system goodness are considered, viz, failure rate, weight, price, mean repair time, mean repair cost, mean replacement time, and mean replacement cost of a unit.     

Assignment of Priority in Improving System Reliability

In many maintenance situations for certain weapon systems, such as anti-aircraft systems, the problems confronted are: 1) which priority for repair is to be assigned, and 2) which type of component should be assigned priority for repair. This can be done on the basis of mean time to system failure. This paper discusses the reliability characteristics of a system of two paralleled radars working in conjunction with two paralleled computers. The system is in up stage even if one computer and/or one radar fails. The system failure takes place only when both the computers or both the radars are in failed condition. The distribution of time to system failure and its expectation have been derived assuming that the failures occur following Poisson distribution and the repair times follow the negative exponential distribution for these two types of subsystems imposing head-of-the-line priority and preemptive resume priority for the repair process. The results are discussed with reference to numerical examples. It has been observed that the mean time to system failure is higher when the head-of-the-line priority discipline is adopted for repair of components, especially when the repair times are shorter.     

Approximate Markov Modeling of High-Reliability Telecommunications Systems

The reliability model of a system with redundancy but only a single repairman is Markov only if the component failure rates and the repair rate are constants. This paper introduces a method with which a reliability analyst can formulate an approximate (time-homogeneous) Markov model for a system with 1-out-of-2 redundancy when repair is nonexponential (repair rate is time-dependent). This approximate model yields accurate steady-state predictions of system reliability when time-to-repair is orders of magnitude smaller than timeto-component-failure, as is typical in high-reliability telecommunications systems. Transition rates and error bounds for the approximate model are given based on the first three moments of the repair time distribution. An application of the method is shown for the system in which repair time is composed of a "next day" parts delivery phase followed by an on-site repair phase.     

Cost analysis of a system with common cause failure and two types of repair facilities

A mathematical model to predict the cost involved to run an n-component single unit system which can fail in n-mutually exclusive ways of total failure or due to common cause, has been developed. Each component has two modes (normal and failure) with two types of repair facilities. Repair rates are arbitrary functions of the time spent. All other transition rates are constant. Laplace transform of the state probabilities are developed along with steady-state behaviour of the system. Inversions are computed to determine the expected profit and availability of the system at any time.