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 generalized age replacement policy with random delivery time and inspection

This paper is concerned with an age replacement policy that is more general because spare part random delivery time, age-dependent minimal repair and inspection are considered. Introducing the cost due to ordering, repairs, shortage, holding and inspection, we derive the expected cost per unit time in the long run as a criterion of optimality and seek the optimum age replacement policy by minimizing that cost. Various special cases are discussed and a numerical example is finally given to illustrate the method.     

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.     

Preventive replacement policies with time of operations,mission durations,minimal repairs and maintenance triggering approaches

When a mission arrives at a random time and lasts for a duration, it becomes an interesting problem to plan replacement policies according to the health condition and repair history of the operating unit, as the reliability is required at mission time and no replacement can be done preventively during the mission duration. From this viewpoint, this paper proposes that effective replacement policies should be collaborative ones gathering data from time of operations, mission durations, minimal repairs and maintenance triggering approaches. We firstly discuss replacement policies with time of operations and random arrival times of mission durations, model the policies and find optimum replacement times and mission durations to minimize the expected replacement cost rates analytically. Secondly, replacement policies with minimal repairs and mission durations are discussed in a similar analytical way. Furthermore, the maintenance triggering approaches, i.e., replacement first and last, are also considered into respective replacement policies. Numerical examples are illustrated when the arrival time of the mission has a gamma distribution and the failure time of the unit has a Weibull distribution. In addition, simple case illustrations of maintaining the production system in glass factories are given based on the assumed data.     

Optimum ordering policies with age-dependent minimal repair and two types of lead times

Two ordering policies for a complex system with age-dependent minimal repair and two types of lead times are considered. Introducing costs due to ordering, repairs, shortage and holding, the expected cost per unit time is derived in the long run as a criterion of optimality and the optimum ordering policies found by minimizing that cost. We show that, under certain conditions, there exists a finite and unique optimum policy. Various special cases are discussed.     

Determining optimal warranty periods from the seller's perspective and optimal out-of-warranty replacement age from the buyer's perspective

This paper considers a general repairable product sold under a failure-free renewing warranty agreement. In the case of a general repair model, there can be two types of failure: Type I failure (a minor failure), which can be rectified by minimal repairs; and type II failure (a catastrophic failure), which can be rectified only by replacement. After a minimal repair, the product is operational but the failure rate of the product remains unchanged. The aim of this paper is to determine the optimal warranty period and the optimal out-of-warranty replacement age, from the perspective of the seller (manufacturer) and the buyer (consumer), respectively, while minimizing the corresponding cost functions. We prove under mild conditions, that the optimal solution of minimizing the cost function exists and is finite. Further, a concise numerical example is demonstrated, and the sensitivity analysis of some of the parameters related to costs is carried out as well. Finally, some practical aspects of renewing the warranty policy for the practitioner and reader are addressed.     

Generalized spare ordering policies with allowable inventory time

In this paper, a generalized ordering policy of a one-unit system with age-dependent minimal repair and random lead time is considered. We treat the general order-replacement model with two decision variables: ordering time and allowable inventory time. The allowable inventory period is measured from the time instant that the ordered spare is delivered. By introducing costs due to ordering, repairs, shortage, and holding, we derive the expected cost per unit time in the long run as a criterion of optimality, and seek the optimum policy by minimizing that cost. And we can prove under some mild conditions that there exists a finite and unique optimum policy. Various special cases are discussed.     

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.     

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 replacement policy for a two-dissimilar-component cold standby system with different repair actions

In this paper, a cold standby repairable system consisting of two dissimilar components and one repairman is studied. When failures occur, the repair of both component 1 and component 2 are not ‘as good as new’. The consecutive operating times of component 1 after repair constitute a decreasing geometric process, while the repair times of component 1 are independent and identically distributed. For component 2, its failure is rectified by minimal repair, and the repair time is negligible. Component 1 has priority in use when both components are good. The replacement policy N is based on the failure number of component 1. Under policy N, we derive the explicit expression of the long-run average cost rate C(N) as well as the average number of repairs of component 2 before the system replaced. The optimal replacement policy N*, which minimises the long-run average cost rate C(N), is obtained theoretically. If the failure rate r(t) of component 2 is increasing, the existence and uniqueness of the optimal policy N* is also proved. Finally, a numerical example is given to validate the developed theoretical model. Some sensitivity analyses are provided to show the influence of some parameters, such as the costs for replacement and repair, and the parameters of the lifetime and repair time distributions of both components, to the optimal replacement policy N* and corresponding average cost rate C(N*).     

Optimal replacement period of a two-unit system with failure rate interaction and external shocks

In this article, a periodical replacement model for a two-unit system which is both subjected to failure rate interaction and external shocks will be presented. Without external shocks, each unit 1, whenever it fails, will act as an interior shock to affect the failure rate of unit 2 and increase the failure rate of unit 2 to a certain degree, while each unit 2 failure causes unit 1 into instantaneous failure. Besides failure rate interaction between units, the system is also subjected to external shocks which can be divided into two types. Type A shock causes unit 1 into failure and then converts the damage of such a failure to unit 2, while type B shock makes the system total breakdown. All unit 1 failures are corrected by minimal repairs. The aim of this article is to derive the expected cost rate per unit time by introducing relative costs as a criterion of optimality, and then the optimal replacement period which minimizes that cost will be determined. A numerical example is given to illustrate the method.     

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.     

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.     

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.     

Comparison of group replacement policies under minimal repair

Three replacement policies for a group of identical units are compared. ( 1 ) Units are periodically replaced all together. Between the periodic group replacements, minimal repairs are performed at failures. ( 2 ) The group replacement interval is divided into repair and waiting intervals. Minimal repairs are performed at failures during the repair interval, but no repair is made in the waiting interval, and the unit remains failed until the group replacement time comes. ( 3 ) Each unit undergoes minimal repair at failure during the repair interval. Beyond the interval, no repair is made until a number of failures. The expected cost rate expressions under each policy are derived. It is shown that the third policy is better economically than the other two policies. Numerical examples are given to demonstrate the results.     

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 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.     

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.     

Maintenance policy of aircraft according to multiple criteria

A model has been developed to derive scheduled replacement operating-ago maintenance policies for the critical parts of aircraft based on multiple criteria : the lowest possible operating cost per unit of time, minimal mission reliability, and minimal expected inflight failure cost. An example has been given, using the Weibull distribution.