Extended optimal replacement policy for a two-unit system with failure rate interaction and external shocks

This study presents an extended replacement policy for a two-unit system which is subjected to shocks and exhibits failure rate from interaction. The external shocks that affect the system are of two types. A type I shock causes a minor failure of unit-A and the damage that is caused by such a failure affects unit-B, whereas a type II shock causes a total failure of the system (catastrophic failure). All unit-A failures can be recovered by making minimal repairs. The system also exhibits the interaction between the failure rates of units: a failure of any unit-A causes an internal shock that increases the failure rate of unit-B, whereas a failure of a unit-B causes instantaneous failure of unit-A. The goal of this study is to derive the long-run cost per unit time of replacement by introducing relative costs as a factor in determining optimality; then, the optimal replacement period, T*, and the optimal number of unit-A failures, n*, which minimise that cost can be determined. A numerical example illustrates the method.     

Optimal age and block replacement policies for a multi-component system with failure interaction

In this paper we consider a generalized age and block replacement policy for a multicomponent system with failure interaction. The i-th component (1 i N) has two types of failures. Type I and type II failures are age dependent. Type I failure (minor failure) is removed by a minimal repair, whereas type II failure (catastrophic failure) induces a total failure of the system (i.e. failure of all other components in the system) and is removed by an unplanned (or unscheduled) replacement of the system. For an age replacement maintenance policy, planned (or scheduled) replacements occur whenever an operating system reaches age T , whereas in the block replacement case, planned replacements occur every T units of time. The aim of this paper is to derive the expected long-run cost per unit time for each policy. The optimal T * which would minimize the cost rate is discussed. Various special cases are detailed. A numerical example is given to illustrate the method.     

The optimal used period of repairable product with leadtime after the warranty expiry

This paper considers the two-phase warranty models for repairable products. It defines the time-interval [0,?W] as the first phase (warranty period) and the time interval (W,?T?+?W) as the second phase (buyer survival period). The products have two types of failures: type I failures (minor failures) and type II failures (catastrophic failures). In the model, type I failures are also removed by minimal repairs in the first and the second phases, and type II failures are removed by replacements in the first phase. If type II failures take place in the second phase, then it is supposed the life of products will be ended. To buy a new product is conducted at time T+W or upon the type II failure. Whenever each replacement takes place, the spare unit is ordered and then delivered. Therefore, the lead-time is considered. This thesis considers three warranty and maintenance models for seller, buyer and the society. The objective is to obtain the optimal T?*. Finally, a numerical example is provided.     

A generalised maintenance policy with age-dependent minimal repair cost for a system subject to shocks under periodic overhaul

A system is subject to shocks that arrive according to a non-homogeneous Poisson process. As shocks occur, the system has two types of failures: type 1 failure (minor failure) is removed by a minimal repair, whereas type 2 failure (catastrophic failure) is removed by overhaul or replacement. The cost of minimal repair depends on age. A system is overhauled when the occurrence of a type 2 failure or at age T, whichever occurs first. At the N-th overhaul, the system is replaced rather than overhauled. A maintenance policy for determining optimal number of overhauls and optimal interval between overhauls which incorporate minimal repairs, overhauls and replacement is proposed. Under such a policy, an approach which using the concept of virtual age is adopted. It is shown that there exists a unique optimal policy which minimises the expected cost rate under certain conditions. Various cases are considered.     

Optimal number of repairs before replacement for a two-unit system subject to non-homogeneous pure birth process

We consider a discrete replacement model for a two-unit system subject to failure rate interaction and shocks. Two types of shocks occur according to a non-homogeneous pure birth process and can affect the two-unit system. Type I shock causes unit A to fail and can be rectified by a general repair, while type II shock results in a non-repairable failure and must be fixed by a replacement. Two-unit systems also exhibit failure rate interactions between the units: each failure of unit A causes some damage to unit B, while each failure of unit B causes unit A into an instantaneous failure. The occurrence of a particular type of shock is dependent on the number of shocks occurred since the last replacement. The objective of this paper is to determine the optimal number of minor failures before replacement that minimizes the expected cost rate. A numerical example is presented to illustrate application of the model.     

Extended optimal replacement policy for a system subject to non-homogeneous pure birth shocks

This article studies the optimal replacement policy with general repairs for an operating system subject to shocks occurring to a non-homogeneous pure birth process (NHPBP). A shock causes that the system experiences one of two types of failures: type-I failure (minor failure) is rectified by a general repair, or type-II failure (catastrophic failure) is removed by an unplanned replacement. The probabilities of these two types of failures depend on the number of shocks since the last replacement. We consider a bivariate replacement policy (n, T) under which the system is replaced at planned life age T, or at the nth type-I failure, or at any type-II failure, whichever occurs first. The optimal replacement schedule which minimizes the expected cost rate model is derived analytically and discussed numerically.     

Optimal number of minimal repairs with cumulative repair cost limit for a two-unit system with failure rate interactions

A discrete replacement model is presented that includes a cumulative repair cost limit for a two-unit system with failure rate interactions between the units. We assume a failure in unit 1 causes the failure rate in unit 2 to increase, whereas a failure in unit 2 causes a failure in unit 1, resulting in a total system failure. If unit 1 fails and the cumulative repair cost till to this failure is less than a limit L, then unit 1 is repaired. If there is a failure in unit 1 and the cumulative repair cost exceeds L or the number of failures equals n, the entire system is preventively replaced. The system is also replaced at a total failure, and such replacement cost is higher than the preventive replacement cost. The long-term expected cost per unit time is derived using the expected costs as the optimality criterion. The minimum-cost policy is derived, and existence and uniqueness are proved.     

Optimal number of minimal repairs before replacement of a deteriorating system with inspections

In this study, we propose a generalised replacement model for a deteriorating system with failures that could only be detected through inspection work. The system is assumed to have two types of failures and is replaced at the Nth type I failure (minor failure) or first type II failure (catastrophic failure), depending on whichever occurs first. The probability of type I and II failures depends on the number of failures since the last replacement. Such systems can be repaired upon type I failure, but are stochastically deteriorating, that is, the lengths of the operating intervals are stochastically decreasing, whereas the durations of the repairs are stochastically increasing. Then, the expected net cost rate is obtained. Some special cases are considered. Finally, a numerical example is provided.     

Some replacement policies with minimal repairs and repair cost limit

Some extended replacement policies based on the number of failures, incorporating the concept of repair cost limit are discussed. Three models are considered as follows: (a) a unit is replaced at the nth failure, or when the estimated minimal repair cost exceeds a particular limit c; (b) a unit has two types of failures and is replaced at the nth type 1 failure, or type 2 failure, or when the estimated repair cost of type 1 failures exceeds a predetermined limit c—type 1 failures are minimal; failures, type 2 failures are catastrophic failures and both occur with constant probability; (c) a unit has two types of failures and the type 1 and type 2 failures are age dependent—the unit is replaced at the nth type 1 failure, type 2 failure, or when the estimated repair cost due to type 1 failures exceeds a predetermined limit c. Introducing costs due to replacements, inspections, and minimal repairs, an optimal number of minimal repairs before replacement is obtained, which minimizes the expected cost rate. Some particular cases are also derived. Finally, the application of these models to computer science is discussed.     

A shock model with two-type failures and optimal replacement policy

In this paper, a shock model for a repairable system with two-type failures is studied. Assume that two kinds of shock in a sequence of random shocks will make the system failed, one based on the inter-arrival time between two consecutive shocks less than a given positive value δ and the other based on the shock magnitude of single shock more than a given positive value γ. Under this assumption, we obtain some reliability indices of the shock model such as the system reliability and the mean working time before system failure. Assume further that the system after repair is ‘as good as new’, but the consecutive repair times of the system form a stochastic increasing geometric process. On the basis of the above assumptions, we consider a replacement policy N based on the number of failure of the system. Our problem is to determine an optimal replacement policy N* such that the long-run average cost per unit time is minimised. The explicit expression of long-run average cost per unit time is derived, and the corresponding optimal replacement policy can be determined analytically or numerically. Finally, a numerical example is given.     

A fixed interval ordering policy for joint optimization of age replacement and spare part provisioning

A new policy is presented for the joint optimization of age replacement and spare provisioning. The policy, referred to as a fixed interval ordering policy, is formulated by combining an age replacement policy with a periodic review ( t₀,q) type inventory policy, where t0 is the order interval and q is the order quantity. It is generally applicable to any operating system with either a single item or a number of identical items. A SLAM based simulation model has been developed to determine the optimal values of the decision variables by minimizing the total cost of replacement and inventory. The behaviour of the policy has been studied for a number of case problems specifically constructed by five-factor second-order rotatory design and the effects of different cost elements and item failure characterisics have been highlighted. The performance of the proposed policy has also been compared with that of the stocking policy which incorporates a continuous review ( s, S) type of inventory policy, where s is the stock reorder level and S is the maximum stock level. Simulation results clearly indicate that the optimal fixed interval ordering policy is less expensive than the optimal stocking policy when the system consists of a large number of operating units.     

A geometric process repair model for a two-component system with shock damage interaction

In this article, the repair-replacement problem for a two-component system with shock damage interaction and one repairman is studied. Assume that component 1 will be replaced as soon as it fails, and each failure of component 1 will induce a random shock to component 2. The shock damages may be accumulative, and whenever the total shock damage equals or exceeds a given threshold Δ, component 2 fails and the system breaks down. Component 2 is repairable, and it follows a geometric process repair. Under these assumptions, we consider a replacement policy N based on the number of failures of component 2. 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 minimised. The explicit expression of the average cost rate is derived by the renewal reward theorem, and the optimal replacement policy can be determined analytically or numerically. The existence and uniqueness of the optimal replacement policy N* is also proved under some mild conditions. Finally, two appropriate numerical examples are provided to show the effectiveness and applicability of the theoretic results in this article.     

An optimal replacement policy for repairable cold standby system with priority in use

In this article, a cold standby repairable system consisting of two nonidentical components and one repairman is studied. It is assumed that component 2 after a repair is “as good as new” while component 1 after a repair is not, but component 1 is given priority in use. Under these assumptions, by using the geometric process repair model, we consider a replacement policy N based on the number of failures of component 1 under which the system is replaced when the number of failures of component 1 reaches N. Our problem is to determine an optimal policy N* such that the long-run average cost per unit time (i.e. the average cost rate) of the system is minimized. The explicit expression of the average cost rate of the system is derived and the corresponding optimal replacement policy N* can be determined numerically. Finally, a special system with Weibull-distributed working time and repair time of component 1 is given to illustrate the theoretical results in this article.     

Optimal number of minimal repairs before replacement based on a cumulative repair-cost limit policy

In this paper, we consider a replacement model with minimal repair based on a cumulative repair-cost limit policy, where the information of all repair costs is used to decide whether the system is repaired or replaced. As a failure occurs, the system experiences one of the two types of failures: a type-I failure (repairable) with probability q, rectified by a minimal repair; or a type-II failure (non-repairable) with probability p (=1 − q) that calls for a replacement. Under such a policy, the system is replaced anticipatively at the nth type-I failure, or at the kth type-I failure (k < n) at which the accumulated repair cost exceeds the pre-determined threshold, or any type-II failure, whichever occurs first. The object of this paper is to find the optimal number of minimal repairs before replacement that minimizes the long-run expected cost per unit time of this polish. Our model is a generalization of several classical models in maintenance literature, and a numerical example is presented for illustration.     

Optimal number of minimal repairs before replacement based on a cumulative repair-cost limit policy

In this paper, we consider a replacement model with minimal repair based on a cumulative repair-cost limit policy, where the information of all repair costs is used to decide whether the system is repaired or replaced. As a failure occurs, the system experiences one of the two types of failures: a type-I failure (repairable) with probability q, rectified by a minimal repair; or a type-II failure (non-repairable) with probability p (=1 − q) that calls for a replacement. Under such a policy, the system is replaced anticipatively at the nth type-I failure, or at the kth type-I failure (k < n) at which the accumulated repair cost exceeds the pre-determined threshold, or any type-II failure, whichever occurs first. The object of this paper is to find the optimal number of minimal repairs before replacement that minimizes the long-run expected cost per unit time of this polish. Our model is a generalization of several classical models in maintenance literature, and a numerical example is presented for illustration.     

Discrete-time spare ordering policy with randomized lead times and discounting

The paper considers a generalized discrete‐time order‐replacement model for a single unit system, which is subject to random failure when in operation. Two types of discrete randomized lead times are considered for a spare unit; one is for regular (preventive) order and another is for expedited (emergency) order. The model is formulated based on the discounted cost criterion. The underlying two‐dimensional optimization problem is reduced to a simple one‐dimensional one and then the optimal ordering policy for the spare unit is characterized under two extreme conditions: (i) unlimited inventory time and (ii) zero inventory time for the spare unit. A numerical example is used to determine the optimal spare‐ordering policy numerically and to examine the sensitivity of the model parameters.     

A multi-criteria optimal replacement policy for a system subject to shocks

A system is subject to shocks that arrive according to a non-homogeneous Poisson process. As these shocks occur, the system experiences one of two types of failures: a type-I failure (minor), rectified by a minimal repair; or a type-II failure (catastrophic) that calls for a replacement. In this study, we consider a multi-criteria replacement policy based on system age, nature of failure, and entire repair-cost history. Under such a policy, the system is replaced at planned life time T, or at the nth type-I failure, or at the kth type-I failure (k < n) at which the accumulated repair cost exceeds the pre-determined limit, or at the first type-II failure, whichever occurs first. An optimal policy over the control parameters is studied analytically by showing its existence, uniqueness, and structural properties. This model is a generalization of several existing models in the literature. Some numerical examples are presented to show several useful insights.     

A deteriorating repairable system with stochastic lead time and replaceable repair facility

In this paper, a deteriorating repairable system with stochastic lead time and replaceable repair facility is studied. We assume that the spare system for replacement is available only by an order and the lead time for delivering the spare follows exponential distribution. Moreover, we also suppose that the repair facility may be subject to failure during the repair period. Under these assumptions, by using the geometric process and the supplementary variable technique, some important reliability indices such as the system availability, rate of occurrence of failure (ROCOF) and the probability that the system is waiting for replacement are derived. An ordering policy N − 1 and a replacement policy N based on the number of failures of the system are also considered. Furthermore, employing several Lemmas, the explicit expression of the average cost rate is derived. Meanwhile, the optimum value N^∗ for minimizing the average cost rate could be determined numerically.     

Replacement problems for a cold standby repairable system

In this paper, a cold standby repairable system consisting of two identical components and one repairman is studied. Assume that each component after repair is not “as good as new”, by using a geometric process, we consider two kinds of repair replacement policy, one based on the working age T of component 1 under which the system is replaced when the working age of component 1 reaches T, and the other based on the failure number N of component 1 under which the system is replaced when the failure number of component 1 reaches N. Our problem is to choose optimal replacement policies T* and N* respectively such that the long-run average cost per unit time of the system is minimized. And we can prove under some mild conditions that the optimal policy N* is better than the optimal policy T*. Finally, a numerical example for policy N is given.     

Optimal replacement policy for a deteriorating production system with preventive maintenance

This paper presents a new policy for determining the optimal replacement time of a deteriorating production system. The optimal replacement time is expressed in terms of the accumulated number of failures that the system has experienced. The provision of preventive maintenance is incorporated in the system model and the objective function is cost efficiency (i.e. the long-run average cost per unit working time). A numerical example is given in the paper. The basic concept used in this paper parallels the geometric process replacement policy N introduced by Lam in 1988. The work in this paper generalizes and modifies Lam's 1988 work.