MA Ning, YE Jimin, and WANG Junyuan

1. College of Mathematics and Statistics, Xidian University, Xi’an 710071, China;

2. College of Sciences, China Jiliang University, Hangzhou 310018, China

Abstract: The maintenance model of simple repairable system is studied. We assume that there are two types of failure, namely type I failure (repairable failure) and type II failure (irrepairable failure). As long as the type I failure occurs, the system will be repaired immediately, which is failure repair (FR). Between the(n-1) t h and the nth FR, the system is supposed to be preventively repaired (PR) as the consecutive working time of the system reaches λn-1T , where λ and T are specified values. Further,we assume that the system will go on working when the repair is finished and will be replaced at the occurrence of the Nth type I failure or the occurrence of the first type II failure, whichever occurs first. In practice, the system will degrade with the increasing number of repairs. That is, the consecutive working time of the system forms a decreasing generalized geometric process(GGP) whereas the successive repair time forms an increasing GGP. A simple bivariate policy (T,N) repairable model is introduced based on GGP. The alternative searching method is used to minimize the cost rate function C(N,T), and the optimal(T,N)*is obtained. Finally, numerical cases are applied to demonstrate the reasonability of this model.

Keywords: renewal reward theorem, generalized geometric pro

1. Introduction

Power systems and network systems are closely related to the modern life. Once these systems fail, people’s lives and work will be paralyzed. In order to improve the system stability, reduce the probability of system failure and upgrade the operation efficiency, we try to find an optimal maintenance model and the corresponding optimal policy.Take the production line of a food processing plant for example. Once the system fails, the entire plant will stop running. The downtime loss is very huge. Therefore, how to minimize the sudden failure rate of the production line is an urgent problem we have to consider. If an optimal model of the production line and its optimal strategy result in the lowest operation cost or a higher rate of return of a production circle which starts from the work to the replacement of the production line, this will have a great impact on the factory, reducing the production cost. In the increasingly fierce food processing market, cost reduction is an effective strategy to gain competitive advantages and compete against others. Therefore, reducing the cost while maintaining quality and quantity of the products are the important goals pursued by the factory.

Generally, the perfect maintenance model is frequently studied, in which a failed system after repair will be as good as new. However, this does not suit all the systems. Another minimum maintenance model is proposed as well, which means that the system can continue to work after being repaired but the system performance will degrade[1,2]. Moreover, for some systems, it is unrealistic to have a person observe their status for 24 hours. Then, Barlow et al. first came up with a model to detect the status of the system at some specific time points [3]. This model can effectively reduce the system’s sudden failure rate, thus reducing the unit cost of the system operation. Nakagawa introduced a model in which the system is periodically checked to decide whether it needs to be replaced, and the optimal detection number that minimizes the system operating cost also is given [4]. Later on, Vaurio put forward a periodic inspection model with preventively repaired (PR) actions for normal operating systems and safety standby systems [5].Cheng and Li also presented a periodic inspection model based on the geometric process (GP) for simple repairable systems [6]. In their model, the system is supposed to be repaired when its working time reachesTor encounters a failure, and it should be replaced when the number of failures reaches the fixed timesN.

In practice, the accumulation of operation times,coupled with the increasing number of failure repair (FR)and the effect of environment, will cause loss to many systems. In other words, the consecutive working time of the system randomly decreases after FR while the repair time interval increases unexpectedly. The GP was proposed in [7,8] to model these monotonic processes. In Lam’s study, the system will be replaced when the working time of the system reachesTor the failure number reachesN(only considering repairable failures). The GP is theoretically reasonable and it has also been verified in numerical examples. Lam et al. applied the GP to simulate a real data set. By comparing the GP with the homogeneous Poisson process and two non-homogeneous Poisson processes, it is shown that the result based on the GP is better than the others in the simulation of the data set [9]. The monotonic process modelled by the GP is uniformly decreasing, that is, the expectation of the consecutive working time of the system is uniformly reducing with the increasing number of failures. However,that is unreasonable, because there are many factors that cause the loss of the system to be different each time,such as the influence of the cumulative effect of the failures, the degree of each failure and the difference of each maintenance team, as well as the effect after the repair.Many researchers have been devoted to improving GP.The generalized GP (GGP) was proposed in [10,11]. In GGP, the geometric ratio of every repair is different,which is more reasonable.

Considering the maintenance and replacement of the system, many people have done research on two inspection models, including periodic inspection and random inspection. In [12], Wang considered two types of checks in the delay time setting. Nakagawa et al. summarized the strategies for periodic inspection and random inspection[13]. Chen et al. applied periodic and random inspection strategies to computer systems [14]. Cheng and Li studied the GP maintenance model proposed by Lam. They assumed that when the system failed, it can be detected by inspection [6]. Chen et al. introduced two kinds of failure competition, which are degradation failure and sudden failure. The preventive maintenance is carried out when the system performance level is degraded within a certain range, the degradation failure repair maintenance is carried out when the components are completely degraded, and the sudden failure repair maintenance is performed when the system performance level is in a certain range [15]. In those researches, they dealt with only one failure type. However, it is not enough to consider only one failure type in real life. Sheu et al. introduced a generalized replacement model, which attempts to deal with two failure types. And the system is replaced when theNth type I failure or the first type II failure occurs, whichever occurs first. And the probability of occurrence of type II failure is related to the number of type I failure that has occurred since the last replacement [16,17].

Based on the above research, it can be found that regular inspection and preventive repair of the running system can effectively reduce the system’s sudden failure rate and extend its life. Therefore, it is a meaningful measure to consider preventive repair and FR together.For more information on the application of preventive maintenance, please refer to [18-21]. Considering the effects of FR and cumulative operation of the system, the system is gradually deteriorating. Therefore, the continuous working time of the system decreases randomly and the continuous repair time increases randomly. Compared with GP and extended GP (EGP), GGP are more extensive. More applications of GP and EGP can be found in [22-29]. The GP describes a monotonic process of uniform deterioration, which is obviously more limited. The EGP combines the “perfect repair model” and the deterioration model after repair. Among them, the monotonic process is described by the GP. This model considering two situations of repair, which are “repair as new” and uniform deterioration. While the GGP can describe the monotonic process that its degree of deterioration has gradually increased unevenly. The deterioration process depends on the environment, conditions and effects of each FR, that is, it is reasonable that the degree of deterioration is increasing. With the increasing number of failures, the system becomes worse, and the degree of deterioration becomes larger. Therefore, the GGP can be broader and more reasonable to describe the monotonic process with different degrees of deterioration. In addition, considering two replacement modes, the maintenance model of the system is more abundant.

This paper introduces an optimal policy to solve this problem. We consider the model with two kinds of repair,which are PR and FR. When the system breaks down before the due testing time, it is assumed that there are two types of failure, namely type I and type II. As long as the type I failure occurs, the system will be FR. Between the(n-1) t h and thenth FR, the system is supposed to be PR as theconsecutiveworking time of thesystemreaches t ime interval ofinspection is λn-1T,and λ is an indicator λn-1T,where λandTare specifiedvalues. That is, the meaning that the detection interval is shortened, andTis a parameter. The system will go on working as soon as the repair is finished and will be replaced at the occurrence of theNth type I failure or the first type II failure,whichever happens first. In practice, the system will degrade with the increasing number of repairs. That is, the consecutive working time of the system forms a decreasing process, while the time interval of repairs forms an increasing process. The GGP is used to illustrate these two processes, and a simple bivariate policy repairable model is introduced based on GGP. The alternative searching method is used to minimize the cost rate functionC(N,T),then the optimal (T,N)*is obtained. Finally, the numerical cases are applied to demonstrate the reasonability of this model.

Section 2 is devoted to the model establishment. Section 3 is devoted to the optimizing of the objective function,andanalgorithmispresentedtofind the optimal solution(T,N)*. Section4includes thenumericalexamples and the conclusions are given in Section 5.

2. Model analysis

2.1 Definitions

We give the definitions of stochastic order, GP and GGP.

Definition 1Given two random variablesXandY, if for allt, there is

thenXis said to be stochastically larger thanY, which is denoted asX＞stYorX＜stY[30].

Definition 2Given a random process{Mk,k=1,2,···} , for alln, there is

Then, {Mk,k=1,2,···} is called the randomly increasing (decreasing) process.

2.2 Model assumptions

The maintenance model of the system is based on the following assumptions.

Assumption 1A new system is installed att=0.The process for a system from being newly installed to being replaced is called a cycle. The process from the new installation to the completion of the first FR is called the first period of a cycle. The process from the completion of the (n-1)th FR to the completion of thenth FR is called thenth period in a cycle, and the process from the completion of thenth FR to the replacement is called theNth period in a cycle .

Assumption 2In the first period of a cycle, the inspection interval of the system after FR isT. Because the successive working time is a decrease process with the period index, we assume that the inspection interval of thenth period is λn-1T, where 0＜λ＜1. In a period, after PR, the system is “good as it is before PR”. In any period of a cycle, when the consecutive working time of the system reaches λn-1T, the worker will detect the system and perform PR. When the consecutive working time of the system is less than λn-1T, a failure occurs, the worker will carry out FR to the system, and the system enters the next period.

Assumption 3The system is replaced by an identical new one at the occurrence of theNth type I failure or the first type II failure, whichever occurs first. Type II failure may occur in any period of a cycle. As shown in Fig. 1 and Fig. 2.

Fig. 1 System replaced when the number of type I failure reachesN

whereaidenotes the geometric ratio of theith FR about consecutiveworkingtime;bidenotes the geometric ratio of theith FRaboutconsecutivePR time;cidenotesthe geometric ratio of theithFRabout the consecutiveFR time.ai＞1,0＜bi,ci＜1andbi＜ci,because theimpact ofsystemdegradationon PRislessthanthe impacton FR. E(X1)=λ1, E(Y1)=μp, E(Z1)=μf.

2.3 Model analysis

Therefore, when thekth failure occurs, it is classified into either a type I failure with probabilityor a type II failure with probability θk=1-ηk. AndP(L＞N)is the probability of the first replacement mode;P(L=k),k≤Nis the probability of the second replacement mode.

The objective function of the maintenance model is the long-run average cost per unit of operating timeC(N,T),according to the updated reward theorem, the long-run average cost per unit of operating time is the average cost per unit time of a cycle. Hence, only consider one cycle.Thus operatingtime in the firstreplacement mode, andWIIbe the operating time in the secondreplacement mode.LetR

LetWbe theoperatingtimein onecycle,WIbe the bethe cost of thesystem inone cycle,RIbethecost in the firstreplacementmode,andRIIbe thecostin the second replacement mode.From Fig. 1andFig.2, it is easy to obtain

According to the full probability formula, the total operating time of the system in a cycle is

The expectation ofWis

The cost of system operating is

According to the full probability formula, the total cost of the system in a cycle is

The expectation ofRis

Hence, the long-run average cost per unit of operating time is

wherepn=Fn(λn-1T) andqn=1-Fn(λn-1T), the expectation ofMnis

LetXnbe the total working time in thenth period of a cycle,

Then, the expectation ofXnis

According to the double expectation formula, the mean ofis

The mean ofZnandW0is

According to (3)-(8), we obtain

3. Optimization

3.1 Replacement policy

Lam researched two kinds of univariate replacement policies [7,8]. One is policyT, where the system is replaced when its operating time reachesT. The other is policyN, where the system is replaced when its failure number reachesN. Zhang introduced a bivariate policy(N,T), in which the system is replaced at the operating ageTor at the time of theNth type II failure occurs,whenever occurs first [30]. It is showed that the policy(N,T) is better than both policyTand policyN. This paper aims to find the optimal (T,N)*by minimizingC(N,T).

First, for any fixedT0＞0 , we can findN1that makesC(N,T0) the smallest,Nis a countable discrete variable,N1satisfies the following formula:

where

whereN1found in the above formula satisfies

Second, letN=N1, we can findT1that makesC(N1,T)the smallest.Tis a continuous random variable, andC(N1,T) is differentiated with respect toT, thenTsatisfies the following formula:

where

where ( ·)′denotes the operator ∂(·)′/∂T.

DifferentiatingC(N1,T) with respect toTtwice, we obtain

where

(·)′and (·)′′denote the operator ∂(·)/∂Tand∂2(·)/∂T2respectively.

When ∂2C(N,T)/∂T≥0, it can prove that the existenceofT1makesC(N1,T1)=mTinC(N1,T).IfT0=T1,then(N1,T1)is theoptimalsolution tothemodel. Let(N,T)*=(N1,T1).

Otherwise, letT=T1and repeat the above steps to find anN2that satisfies (10). Then, letN=N2and find aT2that satisfies (11), ifT2=T1, then, (N2,T2) is the optimal solution to the model. Let (N,T)*=(N2,T2).

Otherwise, continue to repeat the above steps. Until the alternately updated sequence (Nk,Tk) stops or the amount of update reaches the set threshold. The alternative search algorithm for solving the optimal strategy is summarized as Algorithm 1. In practice, Algorithm 1 can always find the optimal solution to the model, (N,T)*=(Nk,Tk). In summary, (N,T)*is the optimal solution toC(N,T), that is, the optimal replacement strategy [29].

3.2 Special cases

Case 1Ifthat is, the probability of the occurrence of type II failure is zero, hence, the system will be replaced only at the time of theNth type I failure, then we obtain the following result:

Case 2If the time of PR, FR and replacement are negligible, the fixed cost separately areC0,C1,andd,then the model degenerates into

Case 3 IfN→∞ , thenthat is, the system is replaced only at the time of the occurrence of the first type II failure, then the model degenerates into

Case 4IfT→∞, then there is no inspection and PR.Only when the system fails, perform FR for the system.Then, the model is degenerated into

4. Numerical example

Assume that the CDF of operating timefor anyiis exponential, that is,

Then,

Then (9) is given by

where

We can see the influence of differentqandα onC(N,T). α=1 means that the probability of the occurrence of type I does not depend on the number of failures since the last replacement. Whenq=1 , thenthat is, Case 1 in the last section.we can findwhen(N,T)*=(4,11). Theoptimal valueC(N*,T*)is-9.073 9, whichmeansthat the system

Setq=0.8andα=0.5,shownas Fig. 3andTable2,should be replaced when the time interval of inspection in the first period of a cycle is 11 and the times of FR reaches4.Thentheoptimal expected netcost rateis-9.0739×104per unit time, that is the net profitis 9.0739×104perunittime.Fig. 4 and Fig. 5 are thecases thatTandNtake different values. And the situations in Figs. 3-5 where there is a sharp decrease in the value ofC(N,T)may be because of parameter values of the example or the properties of this model.

Table 1 Optimal value (× 1 04 ) of C (N,T) on different q andα

Fig. 3 Plot of C (N,T), N=[0,5,15],T=[0,10,50]

Table 2 Some results of C(N,T) with q=0.8,α=0.5

Fig. 4 Plot of C(N,T), N=[0,5,20],T=[0,10,50]

Fig. 5 Plot of C(N,T), N=[0,5,20],T=[0,20,100]

Whenq=1, this model is degenerated into the model find when (N,T)*=(4,10) , the optimal valueC(N*,T*) is of special case 1. Shown as Fig. 6 and Table 3, we can -7.859 6. Other special cases have similar results. Form Table 4 we can see that the mean ofXnis ran- means ofYnandZnare randomly increasing with the number of PR. And it is clear that E(Xn) is less than E(Zn), when the number of PR is greater than or equal to domly decreasing with the number of PR. While the 9. Hence, there is no need to repair the system when the number of PR is greater than 9.

Fig. 6 Plot of C (N,T) with q=1, N=[0,5,15],T=[0,10,50]

Table 3 Some results of C(N,T) withq=1

Table 4 Random change of the means of Xn, Y n andZn

5. Conclusions

This paper studies the maintenance model of a simple repairable system with two failure modes with PR, and models the consecutive working time and the consecutive repair time of the system based on the GGP, aiming at the minimum expected cost per unit time of the long-term operation of the system. We can find the optimal value by the alternative searching method. Then numerical examples are given. For the production line of a food processing plant we have mentioned, this model can effectively improve its stability, raise economic efficiency, as well as decrease the costs. It can be widely used in cold standby systems, power systems, network systems, etc.