FU Yuqiang and MA Xiaoyang

1.School of Mathematics and Physics,University of Science and Technology Beijing,Beijing 100083,China;2.School of Information Management,Beijing Information Science &Technology University,Beijing 100192,China

Abstract:Component reallocation (CR) is receiving increasing attention in many engineering systems with functionally interchangeable and unbalanced degradation components.This paper studies a CR and system replacement maintenance policy of series repairable systems,which undergoes minimal repairs for each emergency failure of components,and considers constant downtime and cost of minimal repair,CR and system replacement.Two binary mixed integer nonlinear programming models are respectively established to determine the assignment of CR,and the uptime right before CR and system replacement with the objective of minimizing the system average maintenance cost and maximizing the system availability.Further,we derive the optimal uptime right before system replacement with maximization of the system availability,and then give the relationship between the system availability and the component failure rate.Finally,numerical examples show that the CR and system replacement maintenance policy can effectively reduce the system average maintenance cost and improve the system availability,and further give the sensitivity analysis and insights of the CR and system replacement maintenance policy.

Keywords:component reallocation (CR),system replacement,maintenance cost,availability,binary mixed integer nonlinear programming,minimal repair.

## 1.Introduction

For many multi-component systems,components can undergo unbalance degradation due to different workloads and environmental conditions,perform the same function,and can be exchanged with each other in the system.For example,the gravity and roll moment of a car leads to that the front tires wear more than the rear tires[1,2],and then swapping the positions of tires can even the degradation of tires and extend the usage of tires[3-11];the workloads make the rapid gravity filters in the water treatment system deteriorate at distinct rates[12],and changing the workloads can balance the health status of rapid gravity filters [10,11].Based on those systems,Fu et al.[7] comprehensively reported component reallocation (CR),a new maintenance policy arousing great interest in recent years.The core idea of CR is to balance the degradation of components by changing the positions of components during the operation of the system in order to optimize system performance.In the existing researches on a CR related problem,there are two key issues to be considered,the system maintenance strategy and the objective function.

From the perspective of the system maintenance strategy,reported studies mainly focused on the joint of classical maintenance strategy and CR.Fu et al.[3],Zhu et al.[4],and Ma et al.[5] proposed CR based preventive maintenance policies of components with unbalanced and determined degradation processes in a non-repairable system,to optimize the assignment and time for CR.Zhu and Hao [6] extended the research to components with stochastic degradation processes.Fu et al.[7,8] integrated the CR into the periodic maintenance policy for repairable systems to determine the optimum component reassignment time and assignment and the optimal time of system replacement.Fu and Wang [9] studied the joint optimization of CRs and system overhauls.Wang et al.[10] investigated the CR problem of a balanced system with multi-state components working in a shock environment,for which an optimization model is constructed to determine the optimum component reassignment time and strategy.Sun et al.[11] quantified the benefit of incorporating reallocation into the condition-based maintenance framework for series systems and then investigated the optimal control limits for reallocation and preventive replacement.Almuhtady et al.[13,14] introduced a degradation-based swapping optimization policy to optimally utilize batteries on non-repairable fleet level.However,those existing researches on CR related problem do not take the downtime of the minimal repair for failure component,CR and system replacement into consideration.

From the perspective of the objective function,the maximization of the system lifetime [3,4,6],maximization of the system reliability [5,10] and minimization of the system maintenance related cost [7-9,11,13,14] have been studied.Besides,traditional assignment problems generally adjust positions of components in the design phase of the system in order to improve system reliability [15-19] or reduce maintenance cost [20-22].However,whether CR problems or traditional assignment problems,the objective function of the optimal model is single and does not cover the system availability to the best of our knowledge.

System availability is an important objective function used to optimize the preventive maintenance policy and often has made a comparative study with the system maintenance related cost [23,24].Adhikary et al.[25]presented a preventive maintenance scheduling model for a continuous operating series system,and a multi-objective genetic algorithm is used to optimize the maximization of availability and minimization of maintenance cost.Qiu et al.[26] modeled the availability and optimal maintenance policies of a competing-risk system subject to multiple failure modes to obtain the optimal inspection interval that maximizes the system steady-state availability or minimizes the average long-run cost rate.Safaei et al.[27] dealt with a repairable system subject to three types of failures to find an optimal planned replacement time by minimizing the total expected discounted cost and maximizing the availability.Chen et al.[28] formulated a new condition-based maintenance optimization problem for continuously monitored degrading systems considering imperfect maintenance actions,which can balance the maintenance cost and the availability by searching the optimal preventive maintenance threshold.Urbani et al.[29] compared three maintenance policies for complex systems with non-identical components and economic dependencies,and then considered low total average maintenance cost and high availability of the system as desirable objectives.Hamdan et al.[30] introduced the optimal preventive maintenance models for the weightedk-out-of-nsystems based on average cost and availability criteria.

Based on above reviews,the system maintenance cost and system availability are important objectives for evaluating system performance of single-component or multicomponent systems.This article contributes by proposing and investigating the CR and system replacement maintenance policy based on availability and cost functions in series systems,which implements CR and system replacement designedly,and determines the optimal assignment of CR,and the optimal uptime right before CR and system replacement by means of establishing and optimizing the system average maintenance cost minimization model and the system availability maximization model.As an extension of the component assignment problem,the optimization of the CR and the system replacement maintenance policy is non-deterministic polynomial (NP)-hard for components with complex reliability functions,because the component assignment problem is NP-hard [31].The assignment of CR is coded by a series of binary variables.Thus,the difficulty lies on the modeling and analysis of CR and system replacement maintenance based on different objective functions.We first establish binary mixed integer nonlinear models according to the system average maintenance cost and the system availability separately.Then,some related analytical results are derived.Finally,numerical examples are designed to demonstrate efficiency and insights of the CR and system replacement maintenance policy.

The remainder of the paper is structured as follows.Section 2 provides the assumptions and the expected number of component failures.Section 3 establishes the optimization models to determine the decisions for CR and system replacement with the minimization of the system average maintenance cost and the maximization of the system availability,respectively,and then gives some analysis properties.Section 4 conducts the numerical examples to show the efficiency of the CR and system replacement maintenance policy.Finally,Section 5 gives conclusions.

## 2.Model assumptions and formulas

### 2.1 Model description and assumptions

For ann-component system,we consider the effect of CR on the maintenance policy with different rate functions subject to the system average maintenance cost and system availability.The main assumptions are summarized as follows：

(i) Components in system are brand new at initial,and undergo the unbalance degradation with the operation of the system because of the workload and environmental conditions.All components are mutually independent that the failure rate functions of components have the same mathematical formula in the same position and are different for different positions.Further,the reliability functions of components are differentiable and strictly monotonically decreasing.

(ii) All failures are repaired by the minimal repair,which restores a failed component to its state just before failure without changing its failure rate.Furthermore,the repair is detected and carried out instantly.

(iii) All components are functionally interchangeable,and the components and positions are one to one correspondence.Before system replacement,one CR is implemented.The CR changes positions of at least two components and does not affect the reliability of components at the uptime right before CR.

(iv) The minimal repair,CR,and system replacement are carried out with constant downtime and cost.

It should be noted that assumptions (i)-(iii) are general and widely used for the CR in repairable systems.The authors in [7-9] posed another assumption that the times for minimal repair,CR,and system replacement are negligible.It is clear that we relax this assumption as in(iv) to facilitate a more general research on the CR based problem.For example,repairing damaged tires,replacing tires and tire rotation all can stop the car for a period of time.In addition,the failures arrive according to a nonhomogeneous Poisson process,when all failures are repaired by the minimal repair.

### 2.2 Total expected number of component failures

Define binary variablesxi,jfori,j=1,2,···,nthat binary variablexi,j=1 if the component at positioniis reassigned to positionj,and 0 otherwise.By assumption (iii),the CR satisfies the one-to-one correspondence between components and positions as

The CR swaps at least two components,that is,

Denoterj(t) as the initial reliability function of the component at positionjwithrj(0)=1 and0 ≤rj(t)≤1 forj=1,2,···,n.DenoteNanddMas the total expected number of component failures during the operation of the system,and the downtime of minimal repair for failure component,respectively.Suppose that components are changed at uptime τ,i.e.,the uptime right before CR,and the system is replaced with a new one at uptimeT,i.e.,the uptime right before system replacement.Based on assumption (ii),Fu et al.[7] gave the expression form ofNby CR as follows:

## 3.Optimization models

Subsections 3.1 and 3.2 establish binary mixed integer nonlinear programming models based on minimization of the system average maintenance cost and maximization of the system availability,respectively.Subsection 3.3 derives some properties for analyzing the maximization model of the system availability.

### 3.1 Minimization of the system average maintenance cost

DenotedRanddPas the downtime of CR and system replacement,respectively.A life cycle is from the initial time to the finish of system replacement.Then,the expected length of a life cycle,denoted as E (L),is

whereLrepresents the length of a life cycle,anddMN+dR+dPis the expected downtime of the system in a life cycle.

For the series system,the total expected number of system failures is equal to the total expected number of components failures in (3).Based on key renewal theorem[32],(3) and (5),the system average maintenance cost of the CR and system replacement maintenance policy in a life cycle,denoted asC(τ,X,T),is

whereZ,cM,cR,andcPare respectively the total maintenance cost in a life cycle,the cost rate of the failure component,the cost of CR and the cost of system replacement,and thencMNis the total expected cost of the minimal repair.

To determine the optimal assignment of CR (i.e.,X),and the optimal uptime right before CR and system replacement (i.e.,τ andT),a binary mixed integer nonlinear programming model of the CR and system replacement maintenance policy with minimizing the system average maintenance cost can be formulated as

Consistent with (4) and (6),the system average maintenance cost of the single system replacement maintenance policy without CR in a life cycle,denoted asCw(T),is

### 3.2 Maximization of the system availability

Similar to (6),the system availability of the CR and system replacement maintenance policy in a life cycle,denoted asA(τ,X,T),is

where E(T) is the expected uptime of system in a life cycle.

In terms of (1),(2) and (9),a binary mixed integer nonlinear programming model of the CR and system replacement maintenance policy with maximizing the system availability can be formulated

Consistent with (4) and (9),the system availability of the single system replacement maintenance policy without CR in a life cycle,denoted asAw(T),is

### 3.3 Analytical results for maximization of the system availability

ProofBy Theorem 1 and Corollary 1,T∗andare the optimal uptime right before system replacement with and without CR by maximizing the system availability.

By

Then,for given uptime of CR and assignment of CR

By (15) and (16),we can obtain the result.□

Theorem 2 shows that the CR and system replacement maintenance policy can improve the system availability by reducing the total of the failure rates of components.Further,balancing the degradation of components can improve system performance,i.e.,minimization of the system average maintenance cost and maximization of the system availability.

## 4.Numerical example

In this section,we compare the solutions of minimization of the system average maintenance cost by solving model(7) and maximization of the system availability by solving model (10).For a given assignment of CR,models (7)and (10) decrease to continuous nonlinear programming models,which can be solved by the interior point algorithm [33].Therefore,for a small size series system,we can list all possible assignments of CR using the enumeration method,and then solve models (7) and (10) for each given assignment by the interior point algorithm.Finally,the best solution of all assignments is the optimal solutions.We solve the continuous nonlinear programming models using solver “fmincon” in Matlab R2017b for all numerical examples.Subsection 4.1 specifies the models in Section 3,in which the reliability functions of components in positions follow the Weibull distributions in a series system,and presents the parameters setting.Subsection 4.2 gives the optimal solutions by solving models(7) and (10).Subsections 4.3 and 4.4 show the sensitivity of CR and system replacement maintenance policy from different perspectives.

### 4.1 Weibull distribution and parameters setting

Suppose that the reliability function of a component in positionjfollows Weibull distribution as

and the corresponding failure rate function is

where βjis the shape parameter,βj＞0,and θjis the scale parameter,θj＞0 forj=1,2,···,n.Note thatλj(t)is increasing for βj＞1,decreasing for 0 ＜βj＜1,and constant for βj=1.

By (3),the total expected number of component failures during the uptime of the system by CR is

Then,the system average maintenance cost in a life cycle based on CR by (6) is

And the system availability in a life cycle based on CR by (9) is

Corollary 2 shows that Theorem 1 also holds for this case,in which the failure rate functions of components in positions follow increasing Weibull distributions.

Corollary 2Suppose that λj(·) follows Weibull distribution in (18) with βj＞1 forj=1,2,···,n.Then,for given uptime right before CR and assignment of CR,the optimal uptime right before system replacement by solving model (10) isT∗=τ or satisfiesQ(T∗)=0.

ProofWe first showQ(∞)＜0.By (17) and (18),

and

Thus,the CR and system replacement maintenance policy performs better than the single system replacement maintenance policy without CR if and only if Imp.A,Red.C ＞0 .Note that the larger Imp.A andRed.C are,the better the effective of CR is.

In the following discussion,a series system of three positions and three components are considered.The reliability and failure rate of components follow the Weibull distributions in (17) and (18) with β1=2.27,β2=2.46,β3=1.23,θ1=4.18,θ2=1.01,and θ3=4.63 .SetcM=5,cR=1,cP=200,dM=0.1,dR=0.05,anddP=1.

### 4.2 Optimization solutions

By (1) and (2),there are five possible assignments of CR in total for a 3-component system.Table 1 lists the optimal solutions by solving model (7) when the assignment of CR is fixed from Case 1 to Case 5 in the second column.For the given assignment of CRX,columns “ τ”and “T” give the optimal uptime right before CR and system replacement,and columnsand “Red.C” show the corresponding total expected number of component failures,the summation of the failure rates of all components,the system average maintenance cost,and the reduction percentages on the system average maintenance cost at timeT,respectively.Table 1 is similar to Table 2 but obtained from model (10) by maximizing the system availability.

Table 1 Optimal solutions by minimizing system average maintenance cost for fixed assignment of CR

Table 2 Optimal solutions by maximizing system availability for fixed assignment of CR

Further,the reduction percentages of Case 1 in Table 1 is far smaller than that of the optimal solutions by solving model (7),i.e.,Case 5,and the improvement percentages of Case 1 in Table 2 is less than 0,which means that the assignment of CR is a key factor in order to save cost and improve availability.On the other hand,τ=Tin Case 1 and τ ＜Tin Cases 2-5 in Table 2,which is consistent with the results of Corollary 2.

### 4.3 Sensitivity of parameters of CR, system replacement, and minimal repair

For the instances in Subsection 4.1,Tables 3-5 present the optimal solutions by solving model (7) when the downtime and cost of CR (i.e.,dRandcR),system replacement (i.e.,dPandcP) and minimal repair (i.e.,dMandcM) are fixed at the values in the first and second columns,respectively.Table 6 presents the optimal solutions considering the different downtime of CR,system replacement minimal repair (i.e.,dR,dPanddM) by solving model (7) as shown in the second column.

Table 3 Optimal solutions under different downtime and cost of CR by minimizing system average maintenance cost

Table 4 Optimal solutions under different downtime and cost of system replacement by minimizing system average maintenance cost

Table 5 Optimal solutions under different downtime and cost of minimal repair by minimizing system average maintenance cost

Table 6 Optimal solutions under different downtime of CR, system replacement and minimal repair by maximizing system availability

Based on the results in Tables 3-6,we observe and illustrate the following points.

(i) Parameters of CR,system replacement and minimal repair have great influences on the optimal uptime right before CR and system replacement (τ∗andT∗) in Tables 3-6.

(ii) The optimal assignments of CR in Table 3 and Table 6 remain the same.Thus,the optimal assignment of CR is rather robust for the downtime and cost of CR of minimizing the system average maintenance cost,and the downtime of CR,system replacement and minimal repair of maximizing the system availability.

(iii) The system average maintenance cost increases with the cost of CR,system replacement and minimal repair.However,the reduction percentages on the system average maintenance cost decreases with the cost of CR and system replacement and the downtime of minimal repair,and increases with the downtime of CR and the cost of minimal repair.WhencR=150,i.e.,the CR cost is relative high,it is no longer economical to implement the CR ( Red.C ＜0).Thus,the downtime of CR is acceptable,the cost of CR and system replacement the smaller the better,and the system requiring expensive repair costs would be better suited to implement the CR.

(iv) The increase of the downtime of CR,system replacement and minimal repair leads to the reduction of system availability.However,the improvement percentages on the system availability decreases with the downtime of CR,and increases with the downtime of system replacement and minimal repair.Thus,the CR becomes more effective,when the system replacement and minimal repair cause longer downtime and the downtime of CR is smaller.

### 4.4 Sensitivity of all parameters

To comprehensively investigate sensitivity of all the parameters in models (7) and (10) on the optimal solutions,Fig.1 presents the reduction percentages on the system average maintenance cost and the improvement percentages in (20) on the system availability of 3-component system,in which 100 instances are generated,each with a set of randomly generated values of all parameters.Referring to the setting of parameters in Subsection 4.1,the parametersdM,dR,dP,cM,cR,cP,andβjand θjforj=1,2,3 are randomly generated from uniform distributions on intervals [0.05,0.5],[0.01,0.1],[0.5,2],[1,20],[1.01,2.5],[0.1,5],[1.8,3.2],and [0.1,2.2],respectively.

Fig.1 Reduction percentages on the system average maintenance cost (Red.C) and the improvement percentages on the system availability (Imp.A) for 100 random instances

As shown in Fig.1,the reduction percentages on the system average maintenance cost ranges from -0.89% to 14.16%,and the improvement percentages on the system availability ranges from -1.28% to 28.88%.Over the same set of 100 instances,92 instances reduce the system average maintenance cost ( Red.C ＞0) and 82 instances improve the system availability ( Imp.A ＞0) by implementing the CR and system replacement maintenance policy.These results imply that the CR is an effective operation,and is not very sensitive to the estimation of all parameters within a wide range.In the 18 instances satisfying Imp.A ≤0,13 instances satisfies Red.C ＞0.That is,minimization of the system average maintenance cost and maximization of the system availability may not be achieved at the same time for some cases.

## 5.Conclusions

This article introduces the downtime of minimal repair,CR and system replacement into the CR and system replacement maintenance policy.Subject to the one to one correspondence between components and positions,two binary mixed integer nonlinear programming models are established to determine the assignment of CR,and the uptime right before CR and system replacement by optimizing the system average maintenance cost and the system availability.For maximization of the system availability,the expression of the optimal uptime right before system replacement is derived,and that the CR is effective if and only if the degradation of components is balanced by CR is proofed when the failure rate of components is increasing.

The numerical examples show that the CR and system replacement maintenance policy is effective and can reduce the system average maintenance cost and improve the system availability,especially when the downtime and cost of CR are not very high.In addition,the downtime of minimal repair and CR have opposite effects on reducing the system average maintenance cost and improving the system availability.Specially,the decision scheme of CR has a great impact on the performance of the system.In future research,there are a series of interesting extensions,such as the imperfect repair,stochastic downtime for failure components,and new method for large-size computational problems.