Number-dependent replacement model with random lead-time for systems under maintenance continuously

This article considers a number-dependent replacement policy where a system has two types of failures and is replaced at the nth type I failure (minor failure) or first type II failure (catastrophic failure), whichever occurs first where type I and type II failures are age dependent. Type I failures can be removed by restoration without any cost, since the maintenance work is executed continuously during the operation of the system. However, type II failure can be removed only through a replacement at a replacement cost. A spare unit for replacement can be delivered upon order and is available only when the random lead-time is finished. A model is developed for the average cost per unit time and is based on the stochastic behaviour of the assumed system and reflects the cost of storing a spare as well as system downtime. The optimal number for a minimum-cost policy is described and discussed. It is shown that the optimal number n* which minimises the cost rate is given by a unique solution of the equation under certain conditions.     

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.     

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 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.     

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.     

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.     

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.     

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.     

Optimum block replacement policy over a random time horizon

In this paper we consider the block replacement policy (BRP) for a system operating over a random time horizon. Under such a policy, a system is replaced by a new one either at failure or at a given time interval. The optimality criterion is the expected total replacements cost. Conditions under which an optimal replacement period exits are given. It is shown that BRP over an infinite time horizon is obtained as a particular case of the present work. A numerical example is given to illustrate the proposed replacement model.     

Optimum preventive maintenance policies for systems subject to random working times,replacement, and minimal repair

This paper proposes, from the economical viewpoint of preventive maintenance in reliability theory, several preventive maintenance policies for an operating system that works for jobs at random times and is imperfectly maintained upon failure. As a failure occurs, the system suffers one of two types of failure based on a specific random mechanism: type-I (repairable) failure is rectified by a minimal repair, and type-II (non-repairable) failure is removed by a corrective replacement. First, a modified random and age replacement policy is considered in which the system is replaced at a planned time T, at a random working time, or at the first type-II failure, whichever occurs first. Next, as one extended model, the system may work continuously for N jobs with random working times. Finally, as another extended model, we might consider replacing an operating system at the first working time completion over a planned time T. For each policy, the optimal schedule of preventive replacement that minimizes the mean cost rate is presented analytically and discussed numerically. Because the framework and analysis are general, the proposed models extend several existing results.     

Warranty cost analysis for nonrepairable services products

Many service products are installed in a complex system that is operated only when the entire system is completed. The time from their installation to commissioning, called a dormant state in this article, may take several years for systems such as complete buildings or aircraft. Warranties for the products may cover the time starting from their installation to a certain time. Warranty cost on replacements for such products is different from normal products without any dormant state. This article analyses the replacement cost for nonrepairable services products from a manufacturer perspective. We consider four warranty policies, which include two types of warranty terms (i.e., nonrenewing or renewing) and two types of replacements (i.e., with preventive replacement or replacement only upon failures). Relationships between the failure patterns at the dormant state and at the operating state are also discussed. Numerical examples and sensitivity analysis are presented to demonstrate the applicability of the methodology described in this article.     

Analysis of maintenance policies for finite life-cycle multi-state systems

Maintenance policies for multi-state systems (MSS) are often analyzed under infinite horizon assumptions. In practice, it is important to consider maintenance policies under a finite horizon because the life cycles of most systems are finite. In this paper, we consider a finite life-cycle MSS that is subject to both degradation and Poisson failures. We study two classes of maintenance policies – preventive replacements and corrective replacements, and their effectiveness in controlling the customer’s expected discounted maintenance cost (EDMC). For both policies, replacement decisions are modelled via two control parameters – a threshold on the current system state and a threshold on the residual life cycle, which is measured as the time span from present to the end of life cycle. We derive close-to-explicit forms of the cost models under each of the policy. Methodologies for optimizing the maintenance thresholds are further proposed. Computational results verify that preventive replacements outperform corrective replacements typically when the downtime cost per failure is relatively high compared to the repair cost.     

Analysis of maintenance policies for finite life-cycle multi-state systems

Maintenance policies for multi-state systems (MSS) are often analyzed under infinite horizon assumptions. In practice, it is important to consider maintenance policies under a finite horizon because the life cycles of most systems are finite. In this paper, we consider a finite life-cycle MSS that is subject to both degradation and Poisson failures. We study two classes of maintenance policies – preventive replacements and corrective replacements, and their effectiveness in controlling the customer’s expected discounted maintenance cost (EDMC). For both policies, replacement decisions are modelled via two control parameters – a threshold on the current system state and a threshold on the residual life cycle, which is measured as the time span from present to the end of life cycle. We derive close-to-explicit forms of the cost models under each of the policy. Methodologies for optimizing the maintenance thresholds are further proposed. Computational results verify that preventive replacements outperform corrective replacements typically when the downtime cost per failure is relatively high compared to the repair cost.     

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.     

Extended periodic imperfect preventive maintenance model of a system subjected to shocks

This article deals with a periodic imperfect preventive maintenance (PM) model of a system subjected to random shocks. A system is subject to shocks that arrive according to a non-homogeneous Poisson process. As shocks occur, the system experiences one of the two types of failures: type-I failure (minor) and type-II failure (catastrophic). Type-I failures are rectified by minimal repair. The system is maintained following the occurrence of a type-II failure or at age T, whichever takes place first. At the N-th PM, the system is replaced. An approach that generalises the existing works on the periodic imperfect PM policy is proposed. The imperfect PM model adopted is hybrid in the sense that it not only reduces the effective age of the system but also alters the system hazard rate. Taking random minimal repair costs into consideration, the objective consists of finding the optimal PM and replacement schedules that minimise the expected cost per unit time over an infinite time-horizon.     

Tool maintenance optimization for multi-station machining systems with economic consideration of quality loss and obsolescence

Tools used in a machining process are vulnerable to frequent wear-outs and failures during their useful life. Maintenance is thus considered essential under such conditions. Additionally, it is widely recognized that the maintenance of manufacturing equipments and the quality of manufactured product are highly interrelated. However, few detailed study has been found in the literature dealing with the effects of maintenance policies on the operational performance of such a system, especially the long-term average cost. The need for a method to determine the optimal tool maintenance policy has become increasingly important. Since the multiple tools in a multi-station machining system generally have significant interactive impacts on the product quality loss, the optimal multi-component maintenance models for several policies are investigated to address the interdependence among these tools. Three distinctive multi-component maintenance policies, i.e., age replacement, block replacement, and block replacement with minimal repair, are identified and analyzed. The proposed approach focuses on these maintenance policies with consideration of both component catastrophic failures, and the interdependence of component degradations on the product quality loss as well as the obsolescence cost. The effects of various maintenance policies on the system performance are simulated, and they are used to determine the best policy for a given system. An illustrative example is used to demonstrate effectiveness and applicability of the proposed approach. The results presented a comparative analysis of specified maintenance policies with respect to the total maintenance cost with consideration of the product quality loss and the obsolescence cost.     

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.     

Optimum random and age replacement policies for customer-demand multi-state system reliability under imperfect maintenance

This paper proposes the generalised random and age replacement policies for a multi-state system composed of multi-state elements. The degradation of the multi-state element is assumed to follow the non-homogeneous continuous time Markov process which is a continuous time and discrete state process. A recursive approach is presented to efficiently compute the time-dependent state probability distribution of the multi-state element. The state and performance distribution of the entire multi-state system is evaluated via the combination of the stochastic process and the Lz-transform method. The concept of customer-centred reliability measure is developed based on the system performance and the customer demand. We develop the random and age replacement policies for an aging multi-state system subject to imperfect maintenance in a failure (or unacceptable) state. For each policy, the optimum replacement schedule which minimises the mean cost rate is derived analytically and discussed numerically.     

An optimal replacement period for a k-out-of-n: F system subject to shocks

A k-out-of-n: F system, which consists of n components and fails if at least k of the n components fail, is subject to shocks that arrive according to an inhomogeneous Poisson process. The k-out-of-n: F system is completely replaced (planned replacement) whenever it reaches age T (T > 0) at a fixed cost R₂. If the mth shock arrives at age S_m < T, it causes the simultaneous failure of j components at the same time with probability p_j