# Optimal Selective Maintenance Decision-Making for Consecutive-Mission Systems with Variable Durations and Limited Maintenance Time

Authors: Huiying Gao; Xiaoqiang Zhang; Xiaoqiang Yang; Bo Zheng
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Maintenance is inevitable for repairable components or systems in modern industries. Since the maintenance effectiveness has a great influence on the subsequent operations and in addition, different maintenance options are possible for the components of the system during the break between any two successive missions, the optimal selective maintenance strategy needs to be determined for a system performing successive missions. A number of selective maintenance models were set up on the basis that the durations of each mission are predetermined, the maintenance time is negligible, and the states of the components at the end of the previous mission can be accurately obtained. However, in the actual industrial and military missions, these premises may not always hold. In this paper, a novel selective maintenance model under uncertainties and limited maintenance time is proposed to improve these deficiencies. The genetic algorithm is selected to solve the optimization problem, and an illustrative example is presented to demonstrate the proposed method. The optimal selective maintenance decision without the constraint of maintenance time is used for comparison.


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## Body

## 1. Introduction

Equipment maintenance refers to the technical activities to restore the function of the equipment after deterioration or failures, and it is a key factor affecting the development of industrial, military, aviation, and so on. In general, the maintenance of a component or system can be divided into planned maintenance and unplanned maintenance. The former, also known as preventive maintenance (PM), is usually a maintenance policy formulated in advance. In case of a PM, maintenance activities will be conducted at a preset time or under a predetermined system state. For unplanned maintenance, it is a maintenance policy taken after the system failure occurs and also called corrective maintenance (CM). What the decision-makers usually concern most contains two aspects: one is the reliability of the system after a maintenance activity and the other is the optimal maintenance strategy under limited maintenance time to minimize the maintenance cost. From the early 1960s, academia and industrial community have shown a strong interest in the optimization of maintenance strategies. In the past few decades, a large amount of research works with respect to maintenance have been reported. Scholars have built a number of maintenance models based on the assumption that the components are updated after repairs [1–3]. However, the assumption that the components become brand new after repairs is usually far from the truth. To describe the maintenance situation in the real world, the concept of imperfect maintenance is published. An imperfect maintenance (IM) refers to a maintenance activity after which the system returns to a state between minimal repair and replacement. Alaswad and Xiang [4], Wu and Zuo [5], and Nakagawa [6] have made several comprehensive reviews on imperfect maintenance. In these reviews, some IM models including p,q model [7], pt,qt model [8], Kijima I and Kijima II models [9, 10], age reduction models [11], intensity reduction model [12], quasirenewal process model [13], and so on are remarkable. How to choose a proper model is an application-dependent and thorny problem and has been recently discussed by Liu et al. [14].For the moment, the imperfect maintenance has been well studied but how to identify the optimal maintenance strategy under limited maintenance budget and time is still a hot potato for the decision-makers, especially, with the addition of uncertainties. The first reported work with respect to selective maintenance should date back to Rice et al. [15]. Immediately after that, Cassady et al. [16, 17] proposed some selective maintenance problems and a general selective maintenance model was constructed by considering different maintenance actions. In management science, selective maintenance is a class of optimization problems in nature. Rice et al. [15] and Rajagopalan and Cassady [18] proposed several enumerating methods to resolve the selective maintenance models, but they become computationally inefficient as the crescent complexity of selective maintenance models. Therefore, some heuristic algorithms such as simulated annealing algorithm [19], genetic algorithm [20, 21], ant colony optimization [22], and artificial neural networks [23] have been applied to identify an optimal maintenance strategy under acceptable computational efficiency.In practice, components or systems may be connected to each other in an arbitrary form and the components may be multistate. Therefore, Liu and Huang [24], Pandey et al. [25], and Liu et al. [26] studied the optimal selective maintenance strategy for a multistate system consisting of binary-state components or multistate components, respectively. Dao and Zuo [27] proposed a selective maintenance model of multistate systems based on the load-dependent degradation model. Besides, some selective maintenance optimizations based on various IM models such as the Kijima I model [28], the Kijima II model [24], and the hybrid model [29] have also been studied.Although a lot of research with respect to selective maintenance models have be done, but most of the models were set up on the basis that the durations of each mission were predetermined and the states of the components at the end of the previous mission can be accurately obtained. In the actual industrial tasks and military missions, these premises may not always hold. For example, an airplane may change the flight course according to the weather or the commands from air traffic controllers. In these situations, the mission duration will be variable. Jiang and Liu [30] and Khatab et al. [31] set up some selective maintenance models considering the uncertainties associated with the durations of missions but they do not concern the constraint of maintenance time. In this paper, a novel selective maintenance model is proposed under variable mission durations and limited maintenance time.The remainder of this paper is organized as follows. Section2 lists some hypotheses on which the selective maintenance model is set up, and imperfect maintenance based on the Kijima II model is introduced. Section 3 discusses the selective maintenance model under uncertainties. Section 4 presents an illustrative example. Conclusions are finally summarized in Section 5.


## 2. Model Hypotheses and Imperfect Maintenance Based on Kijima II Model

In military operations, a weapon may change mission mode according to the actual situation and the mission durations are correspondingly variable. To maintain the warfare capability, the maintenance activities must maximize the reliability of the system within limited maintenance time. In this paper, to set up the novel selective maintenance model, some basic hypotheses are made as follows:(1)
The system for selective maintenance is composed ofn repairable components in any combination ways. The states of all the components as well as the system can be identified at any time and under a binary-state (functioning or failed).(2)
As shown in Figure1, the maintenance activities can be only conducted in the break between any two successive missions and the duration of each break is a limited constant, denoting as TM.(3)
Several maintenance options are available for components during the breaks, including do nothing (DN), minimal repair (MR), replacement maintenance (RM), and imperfect preventive/corrective maintenance (IPM/ICM).(4)
The state of elementi at the beginning and the end of the kth mission can be described by binary variables Si,kB and Si,kE, respectively. Moreover, it has(1)Si,kB=1,if component is functioning,0,otherwise,Si,kE=1,if component is functioning,0,otherwise.(5)
All the components are brand new before the first mission and the duration of the first mission is a given constantt1, the state of element ii∈1,2,⋯,n at the end of the first mission is known as Si,1E, and the virtual age of each element at the end of the first mission is also predetermined, whereas it is uncertain for the subsequent missions.(6)
The duration of each future mission (except the first mission) can be denoted as a discrete variableTkk∈2,⋯,N in this paper, which has wk possible values and the probability for the jthj∈1,2,⋯,wk possible value tkj is pkj.Figure 1
The successive missions and breaks.For componentii∈1,2,⋯n, it is assumed that the virtual ages at the beginning and the end of the kth mission are Vi,kB and Vi,kE, respectively; the virtual age after the kthk∈1,2,⋯,N−1 maintenance is Vi,kM (obviously, Vi,k+1B=Vi,kM and the minimum is zero); the failure time of component i from the beginning of the kth mission is represented by Ti,kFTi,kF∈0,+∞; the actual service time of component i in the kth mission can be denoted by a random variable Ti,kΘTi,kΘ∈0,tk, where tk is the duration of the kth mission. Then, it is obvious that Ti,kΘ∈0,tk in the case that the component fails in the kth mission; otherwise, the component works properly all through the kth mission and it has Ti,kΘ=tk. In addition, since the maintenance activity can only be conducted during the break between the two successive missions, it is easy to gain the following:(2)Vi,kE=Vi,kB+Ti,kΘ.According to the Kijima II model and equation (2), the virtual age of component i after the kth maintenance can be defined as follows:(3)Vi,k+1B=βi,kVi,kE=βi,kVi,kB+Ti,kΘ,where βi,kβi,k∈0,1 is the age reduction factor, and it is closely related to the maintenance effectiveness.In maintenance activities, the maintenance cost can be roughly divided into the fixed maintenance cost and the reparative maintenance cost. As a result, the total maintenance cost of componenti allocated in the kth break can be expressed as(4)Ci,k=Ci,k0+Ci,kM,where Ci,k0 represents the inevitable fixed cost such as the payment for the serviceman and Ci,kM represents the reparative maintenance cost used to reduce the virtual age of component i and it is controllable.Generally, the more reparative maintenance cost is, the better or more effective the maintenance will be. A great number of studies with respect to the age reduction factor can be found. Diallo et al. [32] indicated that the reparative maintenance cost and the current virtual age of the component were the two principal factors affecting the degree of system repair. For convenience, the maintenance cost for performing an minimal repair (MR), an maximum imperfect maintenance (MIM), and a replacement maintenance (RM) are denoted by Ci,kMR, Ci,kMIM, and Ci,kRM hereinafter, respectively. An ICM procedure for a failed component can be split up into an MR and a further maintenance to reduce the virtual age. Therefore, performing an MIM to a failed component is more expensive compared with a functioning component. When Ci,kM=Ci,kMR, MR is performed, the degree of repair of the component is zero, and the repaired component is restored to “as bad as old (ABAO)”; if Ci,k=0, then it corresponds to DN. Particularly, if Ci,kM=Ci,kRM, the repaired component is restored to “as good as new (AGAN).” It is easy to understand that when the virtual age of the component is small, the reparative cost of maintenance is relatively low and it can achieve a good maintenance effect, whereas, as the virtual age of the component continues to increase, the cost of reparative maintenance will increase and the corresponding maintenance effect will gradually decrease meanwhile. Nguyen et al. [33] described the relationships among preventive maintenance cost, replacement maintenance cost, and the degree of rejuvenation after system repair. Pandey et al. [29] defined the age reduction factor as a function related to maintenance cost as follows:(5)βi,k=1−Ci,kM−Ci,kMRCi,kMIM1/mi,k,forSi,kE=0andCi,kM∈Ci,kMR,Ci,kMR+Ci,kMIM,1−Ci,kMCi,kMIM1/mi,k,forSi,kE=10andCi,kM∈0,Ci,kMIM,where mi,k is the characteristic constant and it determines the exact relationship between the reparative maintenance cost Ci,kM and the age reduction factor βi,k. As shown in Figure 2, if the component is in functioning state, when the ratio between Ci,kM and Ci,kMIM is a constant, the variable mi,k will lead to different age reduction factors.Figure 2
The relationship betweenβi,k and Ci,kM/Ci,kMIM.From Figure2, it can be found that the larger the value of mi,k is, the more obvious the maintenance effect is. As discussed in [26, 34], this scenario corresponds to the component with smaller virtual age; whereas, the value of mi,k decreases as the component aging and the effect would be decreased for even a large amount of maintenance cost. Pandey et al. [29] defined the characteristic constant mi,k, the mean residual life (MRL) Ti,kMRL, and the virtual age Vi,kE and discussed the relationship of the three, and it has the following:(6)mi,k=Ti,kMRLVi,kE.Assuming that the virtual ageVi,kB at the beginning of the kth mission is a given constant vi,kb, the actual service time in the kth mission is tΘ, then the mean residual life Ti,kMRL can be calculated as follows:(7)Ti,kMRL=∫tΘ+∞Ri,kx|vi,kbdxRi,ktΘ|vi,kb,where Ri,ktΘ|vi,kb is a conditional probability, and it represents the survival probability of component i at the end of the kth mission under the condition that Vi,kB=vi,kb.Substituting equation (7) into equation (6),(8)mi,k=Ti,kMRLVi,kE=∫tΘ+∞Ri,kx|vi,kbdxvi,kb+tΘRi,ktΘ|vi,kb.In equation (8), if Ti,kMRL≥Vi,kE, i.e., mi,k≥1, the virtual age at the end of the kth mission is less than half of the mean residual life under the condition that Vi,kB=vi,kb, then component i is young; whereas, if Ti,kMRL<Vi,kE, i.e., mi,k<1, then component i is in aging state and the performance of the component will not be significantly improved even more maintenance cost is allocated.


## 3. The Proposed Selective Maintenance Model under Uncertainties and Limited Maintenance Time

The survival probability and the virtual age of componenti at the end of the kth mission are closely related to its state at the beginning of the kth mission. The detailed evaluation method will be discussed in the following:(a)
Si,kB=0.In this case, componenti is failed at the beginning of the kth mission. Since the maintenance activities can only be conducted in the breaks between any two successive missions, as a result, component i does not work in reality during this mission, and the survival probability at the end of the kth mission as well as the incremental virtual age of component i throughout this mission are both zero.(b)
Si,kB=1.In this case, componenti is in functioning state at the beginning of the kth mission. According to the state of component i at the end of the kth mission, two cases can be subdivided further.(b1)
Si,kB=1,Si,kE=1.In this case, componenti is operating smoothly during the kth mission and it remains in functioning state at the end of the kth mission. Therefore, the survival probability of component i in the kth mission, under the condition that Vi,kB=vi,kb, can be denoted as a conditional probability Ri,kt|vi,kb, which satisfies the following:(9)Ri,kt|vi,kb=Rit+vi,kbRivi,kb,wheret is the time interval since the beginning of the kth mission and Rivi,kb is the survival probability of component i in the case that the virtual age is vi,kb. Since component i is in functioning state at the end of the kth mission, the actual service time for component i throughout the kth mission satisfies Ti,kΘ=Tk=tk. The virtual age and the survival probability at the end of the kth mission can be expressed as Vi,kE=vi,kb+tk and Ri,ktk|vi,kb, respectively.(b2)
Si,kB=1,Si,kE=0.On the contrary, componenti may fail within the kth mission. In this case, the value of failure time Ti,kΘ lies in the time interval 0,tk, i.e., Ti,kΘ∈0,tk. The probability density function (PDF) of Ti,kΘ under the condition that Vi,kB=vi,kb,Tk=tk can be calculated by the following:(10)fTi,kΘ|Vi,kB,Tkt|vi,kb,tk=fTi,kΘ|Vi,kBt|vi,kb1−Ri,ktk|vi,kb,0<t<tk,wherefTi,kΘ|Vi,kBt|vi,kb is the PDF of failure time Ti,kΘ. Then, the virtual age of component i at the end of the kth mission is Vi,kE=Vi,kB+Ti,kΘ; furthermore, the cumulative distribution function (CDF) and PDF of Vi,kE can be derived as follows:(11)FVi,kE|Vi,kB,Tkvi,ke|vi,kb,tk=FTi,kΘ|Vi,kB,Tkvi,ke−vi,kb|vi,kb,tk,(12)fVi,kE|Vi,kB,Tkvi,ke|vi,kb,tk=dFTi,kΘ|Vi,kB,Tkvi,ke−vi,kb|vi,kb,tkdvi,ke=fTi,kΘ|Vi,kB,Tkvi,ke−vi,kb|vi,kb,tk,whereFTi,kΘ|Vi,kB,Tkvi,ke−vi,kb|vi,kb,tk and fTi,kΘ|Vi,kB,Tkvi,ke−vi,kb|vi,kb,tk are the CDF and PDF of Ti,kΘ, respectively; FVi,kE|Vi,kB,Tkvi,ke|vi,kb,tk and fVi,kE|Vi,kB,Tkvi,ke|vi,kb,tk are the CDF and PDF of Vi,kE, respectively.If ICM is selected for a failed componenti, it will be restored to functioning state after the maintenance activity and it has Si,k+1B=1, Vi,k+1B=βi,kVi,kB+Ti,kΘ,βi,k∈0,1; in this scenario, the CDF and PDF of Vi,k+1B under the condition that Vi,kB=vi,kb,Tk=tk can be expressed as follows:(13)FVi,k+1B|Vi,kB,Tkvi,k+1b|vi,kb,tk=FTi,kΘ|Vi,kB,Tkvi,k+1b/βi,k−vi,kb|vi,kb,tk,(14)fVi,k+1B|Vi,kB,Tkvi,k+1b|vi,kb,tk=dFTi,kΘ|Vi,kB,Tkvi,k+1b/βi,k−vi,kb|vi,kb,tkdvi,k+1b=fTi,kΘ|Vi,kB,Tkvi,k+1b/βi,k−vi,kb|vi,kb,tkβi,k,where FVi,k+1B|Vi,kB,Tkvi,k+1b|vi,kb,tk and fVi,k+1B|Vi,kB,Tkvi,k+1b|vi,kb,tk are the CDF and PDF of Vi,k+1B, respectively.In a number of actual industrial and military environments, the durations are variables, together with the inherent randomness of the failure timeTi,kF; as a result, the actual service time Ti.kΘ is a random variable. In addition to these uncertainties, the virtual age at the beginning of the kth mission and the age reduction factor βi,k associated with the maintenance options are uncertain; thus, the virtual age at the beginning of the k+1st mission is stochastic. The propagation of a variety of uncertainties between the successive missions is illustrated in Figure 3.Figure 3
The propagation of uncertainty between the successive missions.The survival probability of componenti in the kth mission under the condition that Tk−1=tk−1=t1,t2,⋯,tk−1 can be expressed as follows:(15)Ri,kt|tk−1=PrTi,k>t|T1=t1,T2=t2,⋯,Tk−1=tk−1.Similarly, the reliability for componenti in successfully completing the kth mission under the condition that Tk=tk=t1,t2,⋯,tk can be denoted as follows:(16)Ri,ktk=PrSi,kE=1|T1=t1,T2=t2,⋯,Tk=tk=Ri,ktk|tk−1,where t is the elapsed time since the beginning of the kth mission.In case ofTk=tk=t1,t2,⋯,tk, the reliability of the system in successfully completing the kth mission satisfies the following:(17)Rs,ktk=Rs,kt1,t2,⋯,tk=PrSs,kE=1|T1=t1,T2=t2,⋯,Tk=tk,where Ss,kE=1 indicates that the system is in functioning state at the end of the kth mission.According to the sixth hypothesis inSection 2, the durations in this study are assumed to be discrete variables. Consequently, the reliability of the system in successfully completing the kth mission can be derived on the basis of equation (17) as follows:(18)Rs,kD=PrSs,kE=1=∑j1=1w1p1j1∑j2=1w2p2j2⋯∑jk=1wkpkjkRs,kt1j1,t2j2,⋯,tkjk=∑j1=1w1∑j2=1w2⋯∑jk=1wk∏v=1kpvjvRs,kt1j1,t2j2,⋯,tkjk,where wvv∈1,2,⋯,k and tvjvjv∈1,2,⋯,wv are the number of possible values and the jvth possible value for tv, respectively, and pvjv is the corresponding probability mass.The purpose of the selective maintenance is to maximize the probability for completing all the missions subject to the total maintenance budget. According to the Hypothesis 2, the first maintenance activity is conducted after mission 1 is over. In other words, the maintenance activities have nothing to do withRs,1D. Thus, the selective optimization model can be simplified as follows:(19)maxRs=∏k=2NRs,kD,s.t.∑j=1N−1∑i=1nCi,j≤Cδ;0≤Ci,j≤Ci,j0+Ci,jRM;∑i=1nti,jM≤TM;i∈1,2,⋯,n;j∈1,2,⋯,N−1.where Ci,k is the maintenance cost allocated to component i in the kthk∈1,2,⋯,N−1 break, and Cδ is the limited maintenance budget for all the maintenance activities.According to equation (19), the selective maintenance decision is a complex nonlinear programming problem with continuous decision variables. It is scarcely possible to solve the problem within the limited time by traditional reliability analysis methods or optimization algorithms [35–40]. However, some advanced intelligent optimization algorithms such as genetic algorithm, simulated annealing algorithm, Bayesian networks [41–43], and ant colony algorithm can find the global approximate optimal solution in a reasonable amount of time. Without loss of generality, the widely used genetic algorithm is selected in this study.The key point to solve optimization problems with the genetic algorithm lies in how to convert the feasible solutions into population individuals in the genetic algorithm. Theoretically, the maintenance cost allocated to each component can be any real number in the feasible region. To reduce the computational complexity, the decision variables are converted into integer variables and the feasible solutions corresponding to the population individuals are expressed as an integer vectorL composed of n×N−1 elements and yields the following:(20)L=l1,1,l2,1,⋯,ln,1︸the1stbreak,⋯,l1,k,l2,k,⋯,ln,k︸thekthbreak,⋯,l1,N−1,l2,N−1,⋯,ln,N−1︸theN−1thbreak,where li,k0≤li,k≤WL is the maintenance level for component i in the kth break and WL is the number of maintenance levels determined by decision-makers. It is easy to understand that the larger WL is, the more accurate the optimal solutions will be while more computation complexity. Therefore, the decision-makers need to make a compromise between the accuracy and the computation complexity. The different values of WL correspond to various maintenance costs and maintenance times. The detailed information with respect to maintenance activities is tabulated in Table 1.Table 1
The maintenance actions and relevant information corresponding to different maintenance levels.
Levelsli,kCostsCi,kTimeti,kMMaintenance activitiesVi,kE=0Vi,kE=1000DNDN1Ci,k0+Ci,kMRti,kM1MRIPM⋮⋮⋮⋮⋮li,kCi,k0+Ci,kMR+li,k−1Ci,kMIM−Ci,kMR/WL−3ti,kMli,kICMIPM⋮⋮⋮⋮⋮WL−2Ci,k0+Ci,kMIMti,kMWL−2ICMMIMWL−1Ci,k0+Ci,kMR+Ci,kMIMti,kMWL−1MIMMIMWLCi,k0+Ci,kRMti,kMWLRMRMThe process of the optimization for selective maintenance based on the genetic algorithm is illustrated with Figure4. It contains mainly six steps, and the pseudoprogram can described as follows.Step 1. Initialization: set k=1; determine the system structure as well as the initial state and the virtual age of each component at the end of the first mission;Step 2. Feasible population: generate initial feasible individuals according to the number of population individuals set by the genetic algorithm;Step 3. Virtual age and survival probability for component: convert the population individuals into the maintenance cost allocated to each component; calculate the virtual age Vi,kE and the survival probability Ri,ktk for each component;Step 4. Survival probability and maintenance cost for the system: evaluate the survival probability Rs,kΔ according to the system structure; calculate the actual total maintenance cost Cδ;Step 5. Stopping criterion: if the number of iteration optimizations is less than the specified value Nc, a new population individual will be generated according to the selection, crossover, and mutation operation rules of the genetic algorithm, and the process goes to step 3; otherwise, the process proceeds to step 6;Step 6. End: output the optimal solutions.Figure 4
The process of the genetic algorithm.


## 4. An Illustrative Example

To demonstrate the feasibility of the proposed method, an illustrative example with respect to selective maintenance will be studied in this section. The hydraulic system of an airplane is composed of three components, as shown in Figure5, and it is going to perform four different flights (four successive missions). Assuming that all the components are in brand new state at the beginning of the first mission, the lifetime of each component follows a Weibull distribution, the scale parameter ηii∈1,2,3, and the shape parameter γii∈1,2,3 as well as the maintenance cost, and the state and the virtual age at the end of the first mission are tabulated in Table 2. The maintenance times corresponding to different maintenance levels are tabulated in Table 3. The durations of the four missions are discrete variables, and the probability distributions are tabulated in Table 4. In this study, the maximum maintenance level WL is set as WL=5, the total budget for maintenance cost Cδ is 240,000 CNY, and the maximum allowable maintenance time is 6 hour. The optimal maintenance decision can be obtained, and it is tabulated in Table 5. Scenario 1 denotes that the limited maintenance time is considered in the optimal selective maintenance decision, whereas Scenario 2 does not. The optimal selective maintenance decisions in Scenario 2 are used for comparison.Figure 5
The reliability logic block diagram for the hydraulic system.Table 2
The parameters of each component (unit of maintenance cost: ¥1,000; unit of time: day).
iηiγiSi,1BSi,1EVi,1BTi,1ΘCi,k0Ci,kMRCi,kMIMCi,kRM1222.51106.50.281.628372161.81106.50.312.632403183.21003.50.422.22435Table 3
The maintenance time corresponding to different maintenance levels (unit of time: hour).
Component IDMaintenance levelTime1#0010.5023.2034.4045.1050.602#0010.2023.2034.8045.3150.423#0010.3521.2531.8043.4550.80Table 4
The probability distributions of the durations (unit of time: day).
T16.5——p11——T29.51010.5p20.20.60.2T38.89.6—p30.50.5—T49.310.210.8p40.30.40.3Table 5
The optimal selective maintenance decisions (unit of maintenance cost: ¥1,000; unit of time: hour).
iScenario 1Scenario 2Maintenance cost (levelli,k)Maintenance cost (levelli,k)k=1k=2k=3k=1k=2k=3114.08 (2)1.88 (1)37.28 (5)14.08 (2)14.08 (2)29.88 (4)240.31 (5)40.31 (5)17.61 (2)34.91 (4)32.31 (3)17.61 (2)335.42 (5)26.62 (4)24.42 (3)35.42 (5)13.52 (2)24.42 (3)Rs,k+1D0.98460.89230.96720.98320.93570.9656Cs,k89.8168.8179.3184.4159.9171.91∑i=1nti,kM4.224.375.69.319.2510.1C⋄—237.93——216.23—Rs—0.8497——0.8883—∗Cs,k represents the total maintenance budget allocated to the kth maintenance activity; C⋄ represents the actual maintenance cost for all the missions.It can be seen from Table5 that there are three selective maintenance decisions need to be made for the four missions. Empirically, the more maintenance cost is allocated to a component, the more reliable it is. The maximal reliability of the system in successfully completing the missions under two different scenarios is 0.8497 and 0.8883, respectively. It is easy to see that the maintenance cost in Scenario 1 is larger than the cost in Scenario 2, but the reliability is worse. The main reason for this result is the limited maintenance time.


## 5. Summary and Conclusions

In this paper, the conception of virtual age is used for the repairable components and the durations of the missions are regarded as discrete random variables, and a novel selective maintenance model under uncertainties and limited maintenance time is proposed. The imperfect maintenance based on the Kijima II model is introduced firstly, and then, the CDFs and PDFs of the virtual ages of components after a maintenance activity are derived. Subsequently, the survival probability of each component in different mission and the reliability of the system in successfully completing the missions are derived. Finally, the optimal selective maintenance decisions based on the genetic algorithm under limited maintenance budget and limited maintenance time are obtained. The proposed selective maintenance model takes the maintenance time into consideration, which is more consistent with the actual situation. In case of a limited maintenance time, the maintenance cost is larger and the reliability is worse, but they are sacrificed for saving time. The decision-maker must make a compromise between the three.In this paper, the states of each component are assumed to be binary variables and the durations of the missions are crisp values. It should be noted that the computation complexity will increase rapidly as the missions increase. Selective maintenance issues with respect to multistate systems as well as fuzzy durations will be studied in our future works.



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