An optimal replacement policy for a multistate degenerative simple system

In this paper, a degenerative simple system (i.e. a degenerative one-component system with one repairman) with k + 1 states, including k failure states and one working state, is studied. Assume that the system after repair is not “as good as new”, and the degeneration of the system is stochastic. Under these assumptions, we consider a new replacement policy T based on the system age. Our problem is to determine an optimal replacement policy T^∗ such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit expression of the average cost rate is derived, the corresponding optimal replacement policy can be determined, the explicit expression of the minimum of the average cost rate can be found and under some mild conditions the existence and uniqueness of the optimal policy T^∗ can be proved, too. Further, we can show that the repair model for the multistate system in this paper forms a general monotone process repair model which includes the geometric process repair model as a special case. We can also show that the repair model in the paper is equivalent to a geometric process repair model for a two-state degenerative simple system in the sense that they have the same average cost rate and the same optimal policy. Finally, a numerical example is given to illustrate the theoretical results of this model.     

Optimizing System Availability Under Minimal Repair with Non-Negligible Repair and Replacement Times

Availability measures are given for a repairable system under minimal repair with constant repair times. A new policy and an existing replacement policy for this type of system are discussed. Each involves replacement at the first failure after time T, with T representing total operating time in the existing model and total elapsed time (i.e. operating time + repair time) in the new model. Optimal values of T are found for both policies over a wide range of parameter values. These results indicate that the new and administratively easier policy produces only marginally smaller optimal availability values than the existing policy.     

Optimally maintaining a Markovian deteriorating system with limited imperfect repairs

We consider the problem of optimally maintaining a periodically inspected system that deteriorates according to a discrete-time Markov process and has a limit on the number of repairs that can be performed before it must be replaced. After each inspection, a decision maker must decide whether to repair the system, replace it with a new one, or leave it operating until the next inspection, where each repair makes the system more susceptible to future deterioration. If the system is found to be failed at an inspection, then it must be either repaired or replaced with a new one at an additional penalty cost. The objective is to minimize the total expected discounted cost due to operation, inspection, maintenance, replacement and failure. We formulate an infinite-horizon Markov decision process model and derive key structural properties of the resulting optimal cost function that are sufficient to establish the existence of an optimal threshold-type policy with respect to the system’s deterioration level and cumulative number of repairs. We also explore the sensitivity of the optimal policy to inspection, repair and replacement costs. Numerical examples are presented to illustrate the structure and the sensitivity of the optimal policy.     

Optimal age-replacement model with age-dependent type of failure and random lead time based on a cumulative repair-cost limit policy

In this paper, we consider an age-replacement model with minimal repair based on a cumulative repair cost limit and random lead time for replacement delivery. A cumulative repair cost limit policy uses information about a system’s entire repair cost history to decide whether the system is repaired or replaced; a random lead time models delay in delivery of a replacement once it is ordered. A general cost model is developed for the average cost per unit time based on the stochastic behavior of the assumed system, reflecting the costs of both storing a spare and of system downtime. The optimal age for preventive replacement minimizing that cost rate is derived, its existence and uniqueness is shown, and structural properties are presented. Various special cases are included, and a numerical example is given for illustration. Because the framework and analysis are general, the proposed model extends several existing results.     

Optimal maintenance scheduling for a complex manufacturing system subject to deterioration

We address the problem of determining inspection strategy and replacement policy for a deteriorating complex multi-component manufacturing system whose state is partially observable. We develop inspection and replacement scheduling models and other simple maintenance scheduling models via employing an imperfect repair model coupled with a damage process induced by operational conditions. The system state in performance of the imperfectly repaired system is modelled using a proportional intensity model incorporating a damage process and a virtual age process caused by repair. The system is monitored at periodic times and maintenance actions are carried out in response to the observed system state. Decisions to perform imperfect repair and replacement are based on the system state and crossing of a replacement threshold. The model proposed here aims at joint determination of a cost-optimal inspection and replacement policy along with an optimal level of maintenance which result in low maintenance cost and high operational performance and reliability of the system. To demonstrate the use of the model in practical applications a numerical example is provided. Solutions to optimal system parameters are obtained and the response of the model to these parameters is examined. Finally some features of the model are demonstrated. The approach presented provides a framework so that different scenario can be explored.     

Optimal replacement policies with minimal repair and age-dependent costs

In this paper, we study a modified minimal repair/replacement problem that is formulated as a Markov decision process. The operating cost is assumed to be a nondecreasing function of the system's age. The specific maintenance actions for a manufacturing system to be considered are whether to have replacement, minimal repair or keep it operating. It is shown that a control limit policy, or in particular a (t, T) policy, is optimal over the space of all possible policies under the discounted cost criterion. A computational algorithm for the optimal (t, T) policy is suggested based on the total expected discounted cost.     

退化可修系统最优更换数学模型研究

研究了单部件组成的退化可修系统,在假定故障部件“修复非新”的条件下,以系统中部件的故障次数N为更换策略进行了研究,我们推导出系统经长期运行后,单位时间内期望效益的明显表达式,而且在一定条件下证明了最优策略N*是所有更换策略中最优的.最后还通过几何过程对此进行了讨论.     

A general age-replacement model with minimal repair under renewing free-replacement warranty

A general age-replacement model in which incorporates minimal repair, planned and unplanned replacement, is considered in this paper for products under a renewing free-replacement warranty policy. For both warranted and non-warranted products, cost models from the user’s perspective are developed, and the corresponding optimal replacement ages are derived such that the long-run expected cost rate is minimized. The impacts of a product warranty on the optimal replacement model are investigated analytically. Furthermore, we show that the optimal replacement age for a warranted product is closer to the end of the warranty period than for a non-warranted product. Finally, numerical examples are given for illustration.     

Optimal replacement model with age-dependent failure type based on a cumulative repair-cost limit policy

This paper presents a replacement model with age-dependent failure type based on a cumulative repair-cost limit policy, whose concept uses the information of all repair costs to decide whether the system is repaired or replaced. As failures occur, the system experiences one of the 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. A critical type-I failure means a minor failure at which the accumulated repair cost exceeds the pre-determined limit for the first time. The system is replaced at the nth type-I failure, or at a critical type-I failure, or at first type-II failure, whichever occurs first. The optimal number of minimal repairs before replacement which minimizes the mean cost rate is derived and studied in terms of its existence and uniqueness. Several classical models in maintenance literature are special cases of our model.     

Costing a Finite Minimal Repair Replacement Policy

In this paper an integral equation technique is used to evaluate the expected cost for the period (0, t] of a policy involving minimal repair at failure with replacement after N failures. This cost function provides an appropriate criterion to determine the optimal replacement number N* for a system required for use over a finite time horizon. In an example, it is shown that significant cost savings can be achieved using N* from the new finite time horizon model rather than the value predicted by the usual asymptotic model.     

New failure and minimal repair processes for repairable systems in a random environment

Most often, minimal repair is defined as a replacement of a failed item by an operable item that has the same distribution of the remaining lifetime as the failed one just prior a failure. This is the so‐called statistical minimal repair extensively explored in the literature. Another well‐known type of minimal repair takes into account the state of a system prior to a failure (the information‐based minimal repair). In this paper, we suggest the new type of minimal repair to be called conditional statistical minimal repair. Our approach goes further and deals with the corresponding minimal repair processes for systems operating in a random environment. Moreover, we also consider heterogeneous populations of items, which makes the model more realistic. Both of these aspects that affect the failure mechanism of items are studied. Environment is modeled by the nonhomogeneous Poisson shock process. Two models for the failure mechanism defined by the extreme shock model and the cumulative shock model, respectively, are considered. Some examples illustrating our findings are presented.     

An optimal repair–replacement policy for a cold standby system with use priority

In this paper, the maintenance problem for a cold standby system consisting of two dissimilar components and one repairman is studied. Assume that both component 1 and component 2 after repair follow geometric process repair and component 1 is given priority in use when both components are workable. Under these assumptions, using geometric process repair model, we consider a replacement policy N under which the system is replaced when the number of failures of component 1 reaches N. Our purpose is to determine an optimal replacement policy N1 such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit expression for the average cost rate of the system is derived and the corresponding optimal replacement policy N1 can be determined analytically or numerically. Finally, a numerical example is given to illustrate some theoretical results and the model applicability.     

Optimal Replacement Policy for a General Model with Imperfect Repair

A general model is considered which incorporates imperfect repair and repair cost which depends on time and on the number of repairs in the cycle. This model is an extension of models examined previously in the literature. The objective of this paper is to find the optimal replacement policy and compare it with the replacement policies considered earlier for some variants of this model. The form of the optimal replacement policy is found in the general case and the expected average cost per unit time is derived in two special cases. Numerical examples show that the optimal policy is considerably better than the optimal periodic policy. This paper generalizes and unifies previous research in the area.     

Hierarchical decision making in production and repair/replacement planning with imperfect repairs under uncertainties

In this paper, we analyse an optimal production, repair and replacement problem for a manufacturing system subject to random machine breakdowns. The system produces parts, and upon machine breakdown, either an imperfect repair is undertaken or the machine is replaced with a new identical one. The decision variables of the system are the production rate and the repair/replacement policy. The objective of the control problem is to find decision variables that minimize total incurred costs over an infinite planning horizon. Firstly, a hierarchical decision making approach, based on a semi-Markov decision model (SMDM), is used to determine the optimal repair and replacement policy. Secondly, the production rate is determined, given the obtained repair and replacement policy. Optimality conditions are given and numerical methods are used to solve them and to determine the control policy. We show that the number of parts to hold in inventory in order to hedge against breakdowns must be readjusted to a higher level as the number of breakdowns increases or as the machine ages. We go from the traditional policy with only one high threshold level to a policy with several threshold levels, which depend on the number of breakdowns. Numerical examples and sensitivity analyses are presented to illustrate the usefulness of the proposed approach.     

Allocation of a relevation in redundancy problems

The relevation can be considered as a replacement or repair policy in reliability, in which, when a unit fails, the unit is restored to a working condition just previous to the failure, in the sense that the age of the unit is not changed but the failure rate changes. It can be also considered as a generalization of the minimal repair policy and the load‐sharing model. In this paper, we consider the problem of where to allocate a relevation in a system to increase the reliability of the system and the particular cases of load‐sharing and minimal repair policies.     

Replacement Policies under Minimal Repair

In many situations where system failures occur the concept of ‘minimal repair’ is important. A minimal repair occurs when the failed system is not treated so as to return it to ‘as new’ condition but is instead returned 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.For a system with decreasing reliability it will become increasingly expensive to maintain operation by minimal repairs, and the question then arises as to when the entire system should be replaced. We consider cases where the failure distribution can be modelled by the Weibull distribution. Two policies have been suggested for this case. One is to replace at a fixed time and the other is to replace at a fixed number of failures. We consider a third policy, to replace at the next failure after a fixed time, and show that it is optimal.Expressions to decide the replacement point and the cost of this policy are derived. Unfortunately these do not give rise to explicit representations, and so they are used to provide extensive numerical comparisons of the policies in a search for effective explicit approximations. Conclusions are drawn from these comparisons regarding the relative effectiveness of the policies and approximations.     

Optimal Continuous Policies for Repair and Replacement

This paper investigates the problem of finding optimal replacement policies for equipment subject to failures with randomly distributed repair costs, the degree of reliability of the equipment being considered as a state of a Markov process. Algorithms have been devised to find optimal combined policies both for preventive replacement and for replacement in case of failure by using repair-limit strategies.First a simple procedure to obtain an optimal discrete policy is described. Then an algorithm is formulated in order to calculate an optimal continuous policy: it is shown how the optimal repair limit is the solution to an ordinary differential equation, and how the value of the repair limit determines the optimal preventive replacement policy.     

Optimal replacement policy for a multistate repairable system

In this paper, a deteriorating simple repairable system with k + 1 states, including k failure states and one working state, is studied. The system after repair is not ‘as good as new’ and the deterioration of the system is stochastic. Under these assumptions, we study a replacement policy, called policy N, based on the failure number of the system. The objective is to maximize the long-run expected profit per unit time. The explicit expression of the long-run expected profit per unit time is derived and the corresponding optimal solution may be determined analytically or numerically. Furthermore, we prove that the model for the multistate system in this paper forms a general monotone process model which includes the geometric process repair model as a special case. A numerical example is given to illustrate the theoretical results.     

修理工可单重休假的预防维修更换策略

本文研究了一个修理工带有单重休假的单部件可修系统.为了延长系统的使用寿命,在系统故障前考虑了预防维修,且假定预防维修能够“修复如新”,而故障维修为“修复非新”时,以系统的故障次数N为更换策略.通过更新过程和几何过程理论,得出系统经长期运行单位时间内期望费用的明显表达式,并对预防维修的定长间隔时间T及更换策略N进行了讨论,最后,通过实例分析,求出最优策略N’,使得目标函数取得最优值.     

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本文考虑了一个其产品保修期内免费小修的退化 生产系统的定期检修策略. 系统的退化过程包括三个状态: 可控制状态, 不可控制状态, 故障状态. 过程呆在可控制状态和不可控制状态的时间假设都服从指数分布. 生产系统在固定的时刻t或发生故障时进行检修, 两者以先发生为准. 本文讨论了使单位产品每周期期望成本最小的最优定期检修时间本文考虑了一个其产品保修期内免费小修的退化生产系统的定期检修策略.系统的退化过程包括三个状态:可控制状态,不可控制状态,故障状态.过程呆在可控制状态和不可控制状态的时间假设都服从指数分布.生产系统在固定的时刻t﹡或发生故障时进行检修,两者以先发生为准.本文讨论了使单位产品每周期期望成本最小的最优定期检修时间t﹡,三种特殊情况显示了最优值t的性质.此外,灵敏性分析和数字实例说明了模型中的参数对最优定期检修策略的影响.