Optimal replacement policies for two-unit machines with running costs: II

In a previous paper [1] we have proved the optimality of a control limit policy for certain two-unit systems with running costs. Under such a policy when either of theunits fails we also replace the other unit if its age exceeds a predetermined control limit.In this work we develop a procedure for computing various important characteristics for any control limit policy including, of course, the optimal one. Some of the resulting characteristics can then be used for a restricted optimization within the class of control limit policies. The fact that this restricted optimization leads to the global optimal is implied by the previous paper. In this sense the two approaches are complementary.     

Maintenance optimization on parallel production units

The optimal preventive-maintenance schedule for a productionsystem consisting of N identical parallel production units isinvestigated. The lifetimes of the units are IFR-distributed,i.e. with an increasing failure rate, and are supposed to bestatistically independent. The relevant costs are due to productionlosses, which are increasing and convex in the number of unitsthat are out of operation simultaneously. Actual maintenancecosts (either preventive or corrective) are supposed to be negligibleas compared to the costs due to these production losses. First we consider the apparently trivial case of geometric (discrete-time)or exponential (continuous-time) lifetime distributions forthe units. In this situation, preventive maintenance cannotimprove the condition of a unit. Hence, apparently the onlyrelevant policy is to do corrective maintenance on failed units.However, the analysis reveals that this conclusion is not correct.It turns out that taking non-failed units out of operation deliberatelycan be better than restricting to corrective maintenance only. We first show that, in the case of geometrically distributedlifetimes and unit repair times, the optimal preventive–maintenancepolicy is characterized by a single control limit K. Wheneverthe number of working units is less than or equal to K, no unitsare taken out of operation, while i – K units are setapart whenever i ( > K) units are operational. Next we consider the case with exponentially distributed lifetimesand repair times. Moreover, we assume that the repair capacityis limited, in the sense that only s ( N) units can be underrepair simultaneously. We show that, also in this case, it canbe optimal to take a working unit out of operation until thenext decision epoch (which is either a failure epoch or a repaircompletion epoch). It is shown that the optimal policy has aweak monotonicity property: the number of units which remainin operation increases with the number of available units. However,it is not necessarily true that, under the optimal policy, thenumber of units in standby position increases with the numberof available units. Numerical examples are presented which illustrate that, fora wide range of parameter values, the easiest policy (only performcorrective maintenance on failed units) performs rather wellas compared to the overall optimal policy. Finally we consider the possible extension to the practicallymore interesting case of non-exponential lifetime distributions.In particular, we assume that the lifetimes are composed oftwo non-identical exponential phases. A unit in its first lifephaseis called ‘good’, while a unit in its second phaseis called "doubtful". In this situation, one has the optionto put a good or doubtful unit in standby position until thenext decision epoch or to perform preventive maintenance ona doubtful unit. The latter brings a unit back from the doubtfulinto the good state. An indication is given of the problemsthat arise in generalizing the results obtained for the exponentialcase.     

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.     

Preventive maintenance in a 1 out of n system: The uptime,downtime and costs

In repairable systems with redundancy, failed units can be replaced by spare units in order to reduce the system downtime. The failed units are sent to a repair shop or manufacturer for corrective maintenance and subsequently are returned for re-use. In this paper we consider a 1 out of n system with cold standby and we assume that repaired units are “as good as new”.When a unit has an increasing failure rate it can be advantageous to perform preventive maintenance in order to return it to its “as good as new” state, because preventive maintenance will take less time and tends to be cheaper. In the model we present we use age-replacement; a machine is taken out for preventive maintenance and replaced by a standby one if its age has reached a certain value, T_pm. In this paper we derive an approximation scheme to compute the expected uptime, the expected downtime and the expected costs per time unit of the system, given the total number of units and the age-replacement value, T_pm. Consequently the number of units and the value T_pm can be determined for maximum long-term economy.     

Optimizing costs of age replacement policies

This paper continues earlier work on the best implementation procedure for an age replacement policy. Under an age replacement policy, a stochastically failing unit is replaced at failure or after being in service for x units of time, whichever comes first. Sequentially estimating φ, the optimal replacement time, produces substantial cost savings. In this paper the rate of convergence of the actual costs to the theoretical optimal cost is studied. For any sequential procedure satisfying some mild measurability conditions, it is shown that with probability one the rate of convergence of the cost can be described based on the rate of convergence of the estimator of φ. Further, a sequential procedure is described whose cost converges to the optimal cost more rapidly than known competing procedures. For this procedure, the rate of convergence of the costs is further described by a result which states that an average actual cost per unit, when suitably standardized, converges in distribution to a normal random variable.     

Time and inventory dependent optimal maintenance policies for single machine workstations: An MDP approach

In this work the problem of obtaining an optimal maintenance policy for a single-machine, single-product workstation that deteriorates over time is addressed, using Markov Decision Process (MDP) models. Two models are proposed. The decision criteria for the first model is based on the cost of performing maintenance, the cost of repairing a failed machine and the cost of holding inventory while the machine is not available for production. For the second model the cost of holding inventory is replaced by the cost of not satisfying the demand. The processing time of jobs, inter-arrival times of jobs or units of demand, and the failure times are assumed to be random. The results show that in order to make better maintenance decisions the interaction between the inventory (whether in process or final), and the number of shifts that the machine has been working without restoration, has to be taken into account. If this interaction is considered, the long-run operational costs are reduced significantly. Moreover, structural properties of the optimal policies of the models are obtained after imposing conditions on the parameters of the models and on the distribution of the lifetime of a recently restored machine.     

Stochastic Comparison of Age-Dependent Block Replacement Policies

The age-dependent block replacement policy is a modified block replacement policy with an age limit for preventive replacements. Under this policy, any failed component is repaired, but only the components whose ages exceed a fixed age limit are replaced preventively at the scheduled maintenance times. Using the compensator method, we compare stochastically the failure counting processes of the age-dependent block replacement policies with different parameters, and show that the age-dependent block replacement policy, although cost effective, leads to more failures than the age and block replacement policies. AMS 2000 Subject Classification 60K10     

A replacement model for an additive damage model with restoration

A system existing in a random environment receives shocks at random points of time. Each shock causes a random amount of damage which accumulates over time. A breakdown can occur only upon the occurrence of a shock according to a known failure probability function. Upon failure the system is replaced by a new identical one with a given cost. When the system is replaced before failure, a smaller cost is incurred. Thus, there is an incentive to attempt to replace the system before failure. The damage process is controlled by means of a maintenance policy which causes the accumulated damage to decrease at a known restoration rate. We introduce sufficient conditions under which an optimal replacement policy which minimizes the total expected discounted cost is a control limit policy. The relationship between the undiscounted case and the discounted case is examined. Finally, an example is given illustrating computational procedures.     

Renewal analysis of a replacement process

We consider a unit with a random lifetime which is replaced at renewal times by a new identical one regardless of whether it has failed before or not. For this random periodic replacement policy, we derive exact formulas for the cycle length, defined as the time between the replacements of two successive failed units, the stationary probability of the current unit to have failed, and the stationary and the transient distributions of the residual lifetime of the current unit.     

两部件冷备系统的可靠性分析及其最优更换策略

本文研究了两个不同部件、一个修理工组成的冷贮备可修系统,假定它们的寿命分布和维修分布均匀为指数分布,但故障后均不能修复如新时,我们利用几何过程和补充变量法求得了一些可靠性指标,并以故障次数为策略,以长期运行单位时间内的期望效益为目标函数,确定了最优的故障次数,便得目标函数达到最大值,从而保证了系统的可用度。     

The average optimality of a repair-limit replacement policy

We consider a minimal-repair and replacement problem of a reliability system whose state at a failure is described by a pair of two attributes, i.e., the total number of its past failures and the current failure level. It is assumed that the system is bothered by more frequent and more costly failures as time passes. Our problem is to find and/or characterize a minimal-repair and replacement policy of minimizing the long-run average expected maintenance cost per unit time over the infinite time horizon. Formulating the problem as a semi-Markov decision process, we show that a repairlimit replacement policy is average optimal. That is, for each total number of past system failures, there exists a threshold, called a repair limit, such that it is optimal to repair minimally if the current failure level is lower than the repair limit, and to replace otherwise. Furthermore, the repair limit is decreasing in the total number of past system failures.     

Optimal design of unreliable production–inventory systems with variable production rate

In this article, we study an economic manufacturing quantity (EMQ) problem for an unreliable production facility where the production rate is treated as a decision variable. As the stress condition of the machine changes with the production rate, the failure rate of the machine is assumed to be dependent on the production rate. The unit production cost is also taken as a function of the production rate, as the machine can be operated at different production rates resulting in different unit production costs. The basic EMQ model is formulated under general failure and general repair time distributions and the optimal production policy is derived for specific failure and repair time distributions viz., exponential failure and exponential repair time distributions. Considering randomness of the time to machine failure and corrective repair time, the model is extended to the case where certain safety stocks in inventory may be useful to improve service level to customers. Optimal production policies of the proposed models are derived numerically and the sensitivity of the optimal results with respect to those parameters which directly influence the machine failure and repair rates is also examined.     

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 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.     

A number-dependent replacement policy for a system with continuous preventive maintenance and random lead times

This paper considers a number-dependent replacement policy for a system with two failure types that is replaced at the nth type I (minor) failure or the first type II (catastrophic) failure, whichever occurs first. Repair or replacement times are instantaneous but spare/replacement unit delivery lead times are random. Type I failures are repaired at zero cost since preventive maintenance is performed continuously. Type II failures, however, require costly system replacement. A model is developed for the average cost per unit time based on the stochastic behavior of the system and replacement, storage, and downtime costs. The cost-minimizing policy is derived and discussed. We show that the optimal number of type I failures triggering replacement is unique under certain conditions. A numerical example is presented and a sensitivity analysis is performed.     

Performance modeling of state dependent system with mixed standbys and two modes of failure

The present investigation deals with a multicomponent repairable system with state dependent rates. For smooth functioning of the system, mixed standbys (warm and cold) are provided so that the failed units are immediately replaced by standbys if available. To prevent congestion in the system due to failure of units, permanent along with additional repairmen are provided to restore the failed units. It is assumed that the units may fail in two modes. The units have exponential life time and repair time distributions. The failed unit may balk in case of heavy load of failed units. The failed units may also wait in the queue and renege on finding the repairmen busy according to a pre-specified rule. The Chapman–Kolmogorov equations, governing the model in the form of matrix are constructed using transition flow rates of different states. The steady state solution of queue size distribution is derived using product formula. A cost function is suggested to determine the optimal number of warm and cold standbys units required for the desired level of quality of service. The numerical illustrations are carried out to explore the effect of different parameters on performance measures.     

Production and replacement policies for a deteriorating manufacturing system under random demand and quality

This work investigates the production planning of an unreliable deteriorating manufacturing system under uncertainties. The effect of the deterioration phenomenon on the machine is mainly observed in its availability and the quality of the parts produced, with the rates of failure and defectives increasing with the age of the machine. The option to replace the machine should be considered to mitigate the effect of deterioration in order to ensure long-term satisfaction of demand. The objective of this paper is to find the production rate and the replacement policy that minimize the total discounted cost, which includes inventory, backlog, production, repair and replacement costs, over an infinite planning horizon. We formulate the stochastic control problem in the framework of a semi-Markov decision process to consider the machine's history. The integration of random demand and quality behaviour led us to propose a new modeling approach by developing optimality conditions in terms of a second-order approximation of Hamilton–Jacobi–Bellman (HJB) equations. Numerical methods are used to obtain the optimal control policies. Finally, a numerical example and a sensitivity analysis are presented in order to illustrate and confirm the structure of the optimal solution obtained.     

Economic and economic-statistical design of a chi-square chart for CBM

In this paper, the economic and economic-statistical design of a χ² chart for a maintenance application is considered. The machine deterioration process is described by a three-state continuous time Markov chain. The machine state is unobservable, except for the failure state. To avoid costly failures, the system is monitored by a χ² chart. The observation process stochastically related to the machine condition is assumed to be multivariate, normally distributed. When the chart signals, full inspection is performed to determine the actual machine condition. The system can be preventively replaced at a sampling epoch and must be replaced upon failure; preventive replacement costs less than failure replacement. The objective is to find the optimal control chart parameters that minimize the long-run average maintenance cost per unit time. For the economic-statistical design, an additional constraint guaranteeing the occurrence of the true alarm signal on the chart before failure with given probability is considered. For both designs, the objective function is derived using renewal theory.     

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.     

Replacement Policies for Components that Deteriorate

For many industrial processes the cost of a component failing in service is sufficient to warrant replacement before failure, but intensive operation of the processes restricts replacement opportunities. A model is proposed where at each opportunity replacement is optional. Dynamic programming methods are used to show that for components that deteriorate as they are used, the best policy is to replace the component if its age exceeds a control limit otherwise to defer replacement.Numerical results are given when the time to failure of the component has a gamma distribution and replacement opportunities occur at random, or are entirely regular. A "rule of thumb" is given for calculating the control limit, and it is shown to be nearly optimal.