Joint optimization of LORA and spares stocks considering corrective maintenance time

Linhan Guo,Jiujiu Fan,Meilin Wen,and Rui Kang

School of Reliability and Systems Engineering,Beihang University,Beijing 100191,China

Joint optimization of LORA and spares stocks considering corrective maintenance time

Linhan Guo,Jiujiu Fan*,Meilin Wen,and Rui Kang

School of Reliability and Systems Engineering,Beihang University,Beijing 100191,China

Level of repair analysis(LORA)is an important method of maintenance decision for establishing systems of operation and maintenance in the equipment development period.Currently,the research on equipment of repair level focuses on economic analysis models which are used to optimize costs and rarely considers the maintenance time required by the implementation of the maintenance program.In fact,as to the system requiring high mission complete success,the maintenance time is an important factor which has a great infuence on the availability of equipment systems.Considering the relationship between the maintenance time and the spares stocks level,it is obvious that there are contradictions between the maintenance time and the cost.In order to balance these two factors,it is necessary to build an optimization LORA model.To this end,the maintenance time representing performance characteristic is introduced,and on the basis of spares stocks which is traditionally regarded as a decision variable,a decision variable of repair level is added,and a multi-echelon multiindenture(MEMI)optimization LORA model is built which takes the best cost-effectiveness ratio as the criterion,the expected number of backorder(EBO)as the objective function and the cost as the constraint.Besides,the paper designs a convex programming algorithm of multi-variable for the optimization model,provides solutions to the non-convex objective function and methods for improving the effciency of the algorithm.The method provided in this paper is proved to be credible and effective according to the numerical example and the simulation result.

level of repair,spares,convex optimization,multiechelon multi-indenture(MEMI).

1.Introduction

As we all know,equipment such as aircraft is expensive and technically complex,with a high downtime cost.Before the equipment is deployed,several tactical level decisions concerning its corrective maintenance need to be made:(i)which components to repair upon failure and which to discard,(ii)where to perform the repair,(iii)the amount of spares to stock at each site in the repair network. These decisions should be solved so that a target availability of the installed base is achieved against the lowest cost.Generally,the frst two decisions are taken explicitly through level of repair analysis(LORA)and the third decision can be solved by spares stocks through general multiechelon technique for recoverable item control(METRIC) type methods.

LORA is to decide the repair level of the product of downtime as soon as possible caused by maintenance with certain cost constraints in order to improve system availability and reduce maintenance cost as much as possible. Generally,LORA is carried out in the sequence of noneconomic analysis and then economic analysis to identify all the objects’repair level.There are many qualitative constraints in the non-economic analysis,such as motility requirements of deployment,restricts of current support systems,security requirements,special transportation requirements,practicability of repair technique,confdentiality constraints,and personnel and technical levels. However,the repair level of some products cannot be identifed only by these qualitative constraint conditions in the product detailed design phase,that is to say,there can be either lack of these products’constraints or constraints existing in several repair levels.For reasons outlined above, additional quantitative methods are needed,like the economic analysis method proposed early[1,2],which decide the repair level of products mainly by comparing maintenance costs of spares,personnel,materials,support equipment and facilities,and training in different repair levels. LORA needs to consider many infuence factors,but a majority of the infuence factors are used in the non-economic analysis method,so it is not necessary to consider all factors in the optimization modeling.At the beginning of the product design phase,too much data really cannot be obtained.However,the engineering background of this paper is in the product detailed design phase,and the inputdata involved in the proposed model,including component failure rate,repair time,transport time,spare unit cost and maintenance resource cost except spares,can totally be obtained by design and analysis methods of reliability and maintainability in the detailed design phase.For example, the component failure rate can be output from reliability prediction and allocation,the repair time can be output from maintainability prediction and allocation,the spare unit cost and maintenance resource cost except spares can be acquired from support resources planning and cost analysis,and the transport time can be output from early felding analysis.This paper aims at the products whose repair level cannot be decided by qualitative analysis and takes the quantitative analysis and optimization to obtain the optimal repair level.

Generally,the quantitative analysis,namely economic analysis mentioned above,is only aimed at repair levels without considering several stations in the same level. In the support organization of multi-echelon multi-station, spare stock analysis is also to reduce maintenance downtime when failures occur and a certain amount of spare stock allocation is needed in each station.However, according to the overall target,there is a restrictive relation between the repair level and the spare stock.On the premise of the same amount of spares in different repair levels,repairing failure parts in lower stations can cause shorter maintenance downtime but higher maintenance cost,since for maintenance of a certain product, the amount of resources needed to be allocated in upper stations is much smaller than the total of sub-stations under the condition that an upper station supports many substations.However,under certain cost constraints,the spare allocation will be reduced thus the maintenance waiting time will be increased.It shows that respective optimizations for the repair level and spares will reduce the optimal effectiveness respectively.Only by establishing a joint optimal model with considerations of transferring and coupling relations of spare demands in the support organization of multi-echelon multi-station to synthetically balance the restrictive degree of the repair level and the spare allocation,can the global optimal maintenance plan be realized.

Current researches on optimal models of LORA are focused on repair levels of products that cannot be identifed by qualitative factors,and take the maintenance cost as the optimal object to identify the repair levels of these products.And prior researches mainly optimize the repair level and stock allocation respectively without considering the interaction between these two optimal issues.

Barros[3]proposed the frst multi-echelon multiindenture(MEMI)LORA model.Barros assumes that the same decisions are taken at all locations at one echelon level and all components at a certain indenture level require the same resource and that the resources are infnite and she models the problem as an integer linear programming model which she solves by using Cplex.Barros and Riley[4]used the same model as Barros did and solved it by using a branch-and-branch approach.Saranga and Dinesh Kumar[5]made the same assumptions as Barros [3],except that each component requires its own unique resources.They solved the model by using a genetic algorithm.Basten et al.[6]generalized the two aforementioned models by allowing for components requiring multiple resources and multiple components requiring the same resource.Finally,Basten et al.[7]generalized the model of [6]by allowing for different decisions at various locations at one echelon level.They show that the LORA problem can be modeled effciently as a generalized minimum cost fow model.Basten et al.[8]proposed a number of extensions to the model of[7].Brick and Uchoa[9]also used similar assumptions as those in[7],but considered resources have a maximum capacity and assumed two indenture levels.

In the literature about spares,the METRIC type models are the most clastic.Sherbrook[10]developed the METRIC model,which is the basis for a huge stream of METRIC type models.The goal in these models is to fnd the most cost effective allocation of spares in a network.Muckstadt[11]extended the work of Sherbrooke(two-echelon,single-indenture)by allowing for two indenture levels,leading to the so-called MODMETRIC model.Simon[12]considered the two-echelon, single-indenture,single-item problem,which is later extended to the general multi-echelon problem by Kruse [13].Graves[14]proposed a more accurate approximation for the two-echelon,single-indenture problem,the VARIMETRIC model,which Sherbrooke[15]extended to the two-indenture level.Axsater[16]provided an exact evaluation and enumeration but with penalty costs instead of a service level constraint.Rustenburg et al.[17]gave an exact evaluation for the general MEMI problem.Kim[18] proposed an algorithm for a multi-echelon repairable item stocks system with depot spares and general repair time distribution.

Some researchers solve the problem of LORA and spares stocks jointly.Alfredsson[19]frstly proposed a two-echelon,single-indenture model joint LORA and spares stocks.He assumes it is required at the same location that each component requires one resource and all components require the same resource.Basten et al.[20] proposed the same model as that in[19],but they allowed for more general component-resource and components may share resources.

However,in the existing literature about spares inven-tory and LORA,a majority of models mainly consider how to minimize the costs,while disregard the maintenance time.Although Alfredsson[19]considered the system downtime,he omitted the spares waiting time which is the major part of the downtime.In this paper,we take a research on the joint optimization of LORA and spares stocks by considering the maintenance time resulting from maintenance action.In the spares inventory,higher maintenance time will lead a larger number of spares to be stocked to achieve the same availability.Availability is defned by the up time(mean time between failure,MTBF) and the down time(mean time due to spares,maintenance and other delays resulting from maintenance action,MDT) [21].As is known,availability is an important factor in equipment systems,which is signifcantly infuenced by the maintenance time.Since MTBF is a reliability parameter related to the equipment’s own design away from the infuence of the optimization,MDT is the main factor to affect the availability of the equipment.Therefore,MDT can represent the system effectiveness instead of the availability of the equipment in our optimization model.Disregarding human factors or management delay in maintenance,MDT mainly means the turn-around time(TAT) which includes pure repair time,transportation time and spares waiting time.The pure repair time is the execution time depending on the specifc steps to fnish the job when all resources are ready,and it is generally a constant value plus the transportation time.Shortening the pure repair time or other resources waiting time generally has no obvious effect on the maintenance time,therefore,in the optimization,by shortening the spares waiting time,we can achieve a target availability of equipment systems,in other words,the optimization goal is to shorten the spares waiting time.Accordingto the Little Theory,we can transform the spares waiting time into the expected number of backorder(EBO),and further we can transform the maintenance time-cost balance into the EBO-cost balance.In the model,we will take EBO as the objective function,maintenance cost as the constraint condition.

In Section 2,we outline our model and design a convex programming algorithm to solve the model in Section 3.In Section 4,we design a numerical experiment and confrm the correctness of our algorithm by simulation.Finally,we give some conclusions and recommendations for further research in Section 5.

2.Model

2.1Assumptions and notations

We use the underlying assumptions in the MEMI LORA model:

(i)Line replaceable unit(LRU)failure time is exponential distribution;

(ii)For each component at each location,the(s?1,s) continuous review stocks control policy is used;

(iii)Except for spares,other maintenance resources are always adequate;

(iv)A failure of LRU is caused by a failure in at most one shop replaceable unit(SRU).

We index the depot-level sites by Dc(c=1,2,...,C). Each depot-level site supports multiple intermediate-level sites which are indexed by Ib(b=1,2,...,B).Each intermediate-level site supports multiple organization-level sites which are indexed by Oa(a=1,2,...,A).Let 0 denote the LRU and index the LRU by i(i=1,2,...,I). Each LRU contains multiple SRUs which are indexed by x(x=1,2,...,X).

mOa:the demand rate of the site Oa;

mLi0: the demand rate of the LRUiat the site L(L: Oa,Ib,Dc);

mLix: the demand rate of the SRUixat the site L(L: Oa,Ib,Dc);

pOi1a:probability of the failure LRUiat the site Oa repaired at organization-level;

pOi2a:probability of the failure LRUiat the site Oa delivered to intermediate-level to repair;

pOi3a:probability of the failure LRUiat the site Oa delivered to depot-level to repair;

pIi2b:probability of the failure LRUiat the site Ib repaired at intermediate-level;

pIi3b:probability of the failure LRUiat the site Ib delivered to depot-level to repair;

fiOxa1:probability of the failure SRUixat the site Oa repaired at organization-level;

fiOxa2:probability of the failure SRUixat the site Oa delivered to intermediate-level to repair;

fiOxa3:probability of the failure SRUixat the site Oa delivered to depot-level to repair;

fiIxb2:probability of the failure SRUixat the site Ib repaired at intermediate-level;

fiIxb3:probability of the failure SRUixat the site Ib delivered to depot-level to repair;

qiLx:probability of the failure LRUiat the site L caused by SRUix(L:Oa,Ib,Dc);

TiL0:repair time of the LRUiat the site L(L:Oa, Ib,Dc);

TiLx:repair time of the SRUixat the site L(L:Oa, Ib,Dc);

Tw:spares waiting time of organization-level;

TwLi0:spares waiting time of the LRUiat the site L(L:Oa,Ib,Dc);

TwLix:spares waiting time of the SRUixat the site L(L:Oa,Ib,Dc);

TiD0cIb:transportation time of the site Dc to the site Ib;

TiD0cOa:transportationtimeofthesite DctothesiteOa;

TiI0bOa:transportation time of the site Ib to the site Oa;

XiL0:the number of the LRUiin repairing at the site L(L:Oa,Ib,Dc);

XiLx:the number of the SRUixin repairing at the site L(L:Oa,Ib,Dc);

sLi0: the stock level of the LRUiat the site L(L:Oa, Ib,Dc);

sLi0: the stock level of the SRUixat the site L(L:Oa, Ib,Dc);

E[XiL0]:the expected pipeline for the LRUiat the site L(L:Oa,Ib,Dc);

E[XiLx]:the expected pipeline for the SRUixat the site L(L:Oa,Ib,Dc);

EBOOia:the expected LRUibackorder at the site Oa;

EBO(sLi0|E[XiL0]):the expected LRUibackorder at the site L when the stock is sLi0and the expected pipeline is E[XiL0](L:Oa,Ib,Dc);

EBO(sLix|E[XiLx]):the expected SRUixbackorder at the site L when the stock is sLixand the expected pipeline is E[XiLx](L:Oa,Ib,Dc).

2.2Mathematical model

In this section,we mainly consider the three-echelon,twoindenture support system,and based on the analysis,we build a MEMI LORA optimization model as shown in Fig.1.

Fig.1 Three-echelon,two-indenture support structure

As shown in Fig.1(a),the support organization has three echelons,in which one depot-level site supports a number of repair sites called intermediate-level and one intermediate-level supports a number of repair sites called organization-level.The product structure can be abstracted as a two-indenture system,which is composed of multiple LRUs consisting of a number of SRUs in series,as shown in Fig.1(b).

In Section 1,we have analyzed that the objective function of model is the EBO which is transformed by the spares waiting time.Therefore,the real objective function is the spares waiting time.Now,we deduce the relationship between the spares waiting time and the EBO.

In the multi-indenture support organization,as we know, the mean waiting time is expressed as

And TOawcan be expressed as

where I(a)is the number of LRUiat the site Oa.We can get TOawi0from the Little formula:

Combining(1),(2)and(3),we can get

In Fig.2,we present how to compute the objective function EBO.According to the Palm’s Theorem[21,22],we can obtain the EBO formula.

Meantime,the demand rate of spares at each site can be computed as follows:

Fig.2 Computing fow for MEMI objective function

During the optimization process,it needs to consider two constrains conditions:cost and repair level decision variable.As mentioned in Section 1,the costs mainly include the spares cost and the maintenance resource cost except spares.If we defne CSias the LRU total spares cost,as the LRU total maintenance resource cost,csi0as the LRU cost,as the SRU cost,cLi0as the LRU maintenance resource cost at the site L(L:Oa,Ib,Dc),cLixas the SRU maintenance resource cost at the site L(L:Oa,Ib,Dc), and T as the support time(year),then we can get total spares and maintenance resource costs as

Here the constraint condition is

We defne according to the computing fow shown in Fig.2.Equations(18)–(20)as cost constraint conditions,(21)–(26)as repairleveldecision variable constraint conditions,then we can get a MEMI LORA optimization model,as shown in(27):

3.Optimization algorithm

3.1Analysis of objective function EBO(s,p)

Different from the objective function EBO(s)in the traditional METRIC type model,we add a repair level decision variable p in our model,so inventories are not the only variable in the objective function EBO(s,p).As we know, the function EBO(s)is convex,while whether the function EBO(s,p)is convex or not still needs further study.

Now,we use the three-echelon,single-spare support organization for the object to verify the characteristic of the function.

Assuming there is only one site in each echelon,and the probability that the LRU is repaired at organization-level, intermediate-level or depot-level is p={p1,p2,p3},then there are only three combinations for p:p1={1,0,0}, p2={0,1,0},p3={0,0,1}.Assuming the spares stocks are zero at each echelon,the EBO is equal to the quantity demanded at each echelon,and the EBO at organizationlevel is

We defne the maintenance time of organization-level, intermediate-level and depot-level as T1,T2,and T3respectively,and the delivery time of organization-level to intermediate-level,and intermediate-level to depotlevel as O2,and O3respectively.Assuming EBO(p3)＞EBO(p2)＞EBO(p1),we can get

The second difference formula EBO(p)is

By adjusting the distance between sites or designing the appropriate maintenance time,we can make Δ2EBO(p)＜0,so EBO(p)is non-convex.According to the convex optimization theory,EBO(s,p)needs to traverse the inventories(s)and repair level(p)respectively.However, EBO(p)is non-convex,and it will affect the application of the convex optimization algorithm which needs to be improved.

3.2Algorithm

In Section 3.1,we have analyzed the objective function, next we should improve the algorithm to make EBO(s,p) convex.In this section,we frstly construct a convex function for non-convex function EBO(p),and then present the whole algorithm fow.

When traversing the repair level combination of all kinds of spares,the circulation of the repair level combination can be inserted in the circulation of the spares type.Therefore,if we have structured the EBO convex curve of all kinds of spares according to the repair level decision variables before traversing the spares types,all kinds of spares can be turned into convex optimization according to the margin iteration of the unit cost effect.Thus,we can conclude that the key step of solving the non-convex function optimization algorithm is to structure a convex function respectively for non-convex function EBO(p)of all kinds of spares.

The method of structuring convex function EBO(p)-cost of all kinds of spares is the same as that of constructing the EBO(s)-cost optimization curve,which enhances the unit cost effect of the certain spare to make an optimization decision.To a certain spare,there may be several repair levels to choose.Here,we fx the stocks sof EBO(s,p),and decide which kind of choice can make EBO(p)minimum. Based on this,we choose the minimum EBO(p)of repair levels in each step of increasing spares stocks,and get the optimization EBO(p)-cost convex function which is iterated in gradient direction.

We construct the EBO(p)-cost curve according to the theory of marginal analysis,and it is obvious that the curve is convex.Using the point on the EBO(p)-cost convex curve to make optimization analysis among all kinds of spares,we can get the optimization curve of EBO(s,p)-cost.The whole algorithm fow is shown in Fig.3.

Fig.3 Algorithm fow diagram

3.3Method of improving optimization algorithm effciency

In the optimization model joint LORA and spares stocks, the EBO is affected by stocks(s)and repair level(p). Since the repair level probability p and f are 0-1 variables, andcan be obtained at corresponding sites,thus the number of repair level probability combinations at all sites in organization-level can reach 2(3A+2B)(1+X).This means that when we compute Δ,the running speed of the algorithm will be affected seriously because of a large number of cycles of the EBO.However,in the algorithm iterative process,when we increase a certain type of spares, other types do not change with the increase of this type, which means there are many cases that can be disregarded when we compute Δ.Therefore,we list all possible combinations to improve the effciency of the algorithm in Table 1.

Table 1 Relationship of values of the repair level probability

Different combinations of repair level probability will lead to different EBO expressions,so it needs to compute the objective function value of all kinds of spares at each site when using the repair level.

Plus one LRU at each echelon site,we can get the number of combinations of each echlon site according to(5), (6)and(7),as shown in Table 2.

Table 2 Values of p and f when plus one LRU at each site

Plus one SRU at each echelon site,we can get the number of combinations of each echelon site according to(8), (9)and(10),as shown in Table 3.

Table 3 Values of p and f when plus one SRU at each site

4.Application case

Taking equipment maintenance planning as the background,we introduce how to apply the convex optimization algorithm based on the optimization theory proposed in this paper.

As shown in Fig.1(a),the support organization consists of three echelons:one depot-level site(D)supports two intermediate-level sites(I1,I2)which respectively support two organization-level sites(O1,O2,O3,O4).Eachorganization-level site supports ten systems,and these ten systems which contain eight types of LRUs are identical at the same site.Product structure is shown in Fig.1(b).

(i)Input data

The uncertainty factors in the model are mainly demand rate of spares,unit cost and maintenance time.This case will present the infuence of these factors on the repair level and the inventory with the simple variable method. As shown in Table 4,the maintenance time is the only difference between LRU1and LRU2,the unit cost is the only difference between LRU3and LRU4,the demand rate is the only difference between LRU5and LRU6,and the data of LRU7and LRU8are different with the frst six LRUs. The maintenance resource cost except spares in each repair level is shown in Table 5.

Table 4 Input date of multi-site multi-type LRUs

Table 5 Maintenance resource cost of LRUs in each repair level (thousand dollars)

(ii)Output

In the cost constrain of 450 thousand dollars,the total maintenance resource cost of all LRUs is 262.82 thousand dollars and the total spare cost is 187.18 thousand dollars, and the optimization value of the EBO is 5.064.

In the convex optimization algorithm,according to the principle of the highest unit cost-effectiveness,selecting sequence of spares based on marginal analysis is shown in Fig.4,the Y-axis represents the type of LRU,while the X-axis represents selecting sequence,the length of which represents cost.The length of each segment represents unitcost and the label of which respectively represents the stocking site and selecting sequence of each LRU.

Fig.4 Selecting sequence of spares

The optimal repair level of each LRU is shown in Table 6.Table 6(a)presents the optimal repair level of LRUs which are respectively deployed in site O1and site O2under the intermediate-level site I1.Table 6(b)presents the optimal repair level of LRUs which are respectively deployed in site O3and site O4under the intermediate-level site I2.

Table 6(a) Optimal repair level of each LRU

Table 6(b) Optimal repair level of each LRU

The optimal stocks allocation of each LRU is shown in Table 7.

Table 7 Stocks allocation of multi-site multi-type LRUs

Fig.5 shows the EBOi-cost optimal curve of each LRU and the total EBO-cost optimal curve,from which we can see the optimal EBO of eight LRUs in different total costs. In this case,the total EBO is 5.064 under spares cost of 187.18,and the red dot on the total EBO-cost curve is the optimal selecting dot by the marginal analysis which also can be seen in details in Fig.4.

In order to verify our algorithm,we simulate the model with the optimal inventory allocation and repair level by the SIMLOX,which is a simulation software for logistic support systems.We take the result of the optimal inventory allocation and repair level under the cost constraint of 450 provided by our algorithm as input data,and let these forty systems carry out an identical mission for one year. Fig.6 presents the simulation curve,from which it can be found that with the same input of Table 4,the mean EBO of the forty systems carrying out the identical mission for one year is 5.152.

(iii)Result analysis

According to the output,comparing with the results of LRU1and LRU2,the maintenance time affects the combination of the repair level.

Comparing with the results of LRU3and LRU4,the unit cost affects the stocks allocation at each site.Comparing with the results of LRU5and LRU6,the demand rate of spares affects the stocks allocation at each site.Drawing a conclusion,from the results,we can get that the maintenance time of LRU at each site can affect the combination of the repair level,demand rate and unit cost of LRU,and affect stocks allocation at each site.In addition,according to the simulation result,comparing the EBO of theoretical calculation with that of the simulation,we can see that there is only little difference.Thus,the correctness of our algorithm can be confrmed.

Fig.5 Optimal curve of multi-type LRUs

Fig.6 EBO simulation

5.Conclusions and further research

In this paper,we propose a method of LORA considering the maintenance time in equipment support systems, analyze the problem of level of repair systematically,deduce the LORA objective function and give the modeling condition.Then,we provide the optimal objective function formula and constraint condition at the MEMI support organization,and build the MEMI optimization model joint LORA and spares stocks.In addition,we analyze the characteristics of the optimization problem,design a multivariable convex optimization algorithm,explain the theory and fow of the algorithm,and provide solutions to then onconvex objective function and methods for improving the effciency of the algorithm.Finally,we confrm the correctness of the proposed algorithm through a numerical example by virtue of simulation.

The contribution of the paper is that it analyzes the restrictive relation of the repair level and spare stock,and the transferring and coupling relations of the number of failures sent to repair,spares demand and spare backorders random variables in multi-indenture product systems.This paper introduces the maintenance time into the traditional quantitative repair level analysis and establishes the joint optimal model of the repair level and spare stock.Meanwhile,the support organization of multi-echelon multistation in the model is asymmetric which means it is more truthful since the operation profle and failure of systems in all stations and the maintenance capability can be different.In addition,the paper designes the multi-variable convex optimization algorithm for the model,which can be highly effective when applied to complicated support systems,and verifes the correctness of the algorithm and the model by simulation.

In further research,we will consider all maintenance resources in our model,build optimization models of limited maintenance capability,and take the correlation of interdepot maintenance resources into account.

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Biographies

Linhan Guo was born in 1979.He received his Ph.D.degree in system engineering from Beihang University in 2006.From 2006 to 2010,he was a lecturer of the Reliability and System Engineering Department in Beihang University.Since 2010,he has been an associate professor with the Reliability and System Engineering Department in Beihang University.He is the author of four books,more than 30 papers,and two patents.He is the reviewer of European Journal of Operational Research,Journal of System Engineering and Electronics,and Chinese Journal of Aeronautics.His research interests include reliability and maintainability modeling and simulation.Besides,he is also the expert in complex supply chain and network reliability theory and methods.

E-mail:linhanguo@buaa.edu.cn

Jiujiu Fan was born in 1989.He received his B.E. degree in automation from Wuhan University of Technology in 2012.Since 2012,he is a master graduate student with the School of Reliability and System Engineering,Beihang University.His research interests include the optimization of spares and maintenance decision.

E-mail:mrfan99@163.com

Meilin Wen was born in 1980.She received her B.S.degree in mathematical sciences from Tianjin University in 2003 and her M.S.and Ph.D.degrees in mathematics from Tsinghua University in 2008. Since 2009,she is a lecturer of the Reliability and System Engineering Department in Beihang University.She is the author of 10 SCI papers.Her research interests are uncertainty theory and uncertainty reliability theory.

E-mail:wenmeilin@buaa.edu.cn

Rui Kang was born in 1966.He received his B.S degree in electric engineering from Beihang University in 1987 and M.S.degree in 1990.Nowadays, He is the managing director in the Reliability Engineering Center and the chief engineer.Besides that, he has been the director in the Reliability Engineering Institute of Beihang University.He was awarded as“Changjiang Scholar”by Ministry of Education in 2013.He is the author of seven books,more than 240 papers,and 18 patents.He is the member of IEEE.He was a recipient of the Aviation Corporation Reliability of Advanced Individual in 1993,and the Beijing Youth Science Leader in 1994.His research interests include reliability engineering theory,the complex system reliability maintainability and affordable CAD technology.

E-mail:kangrui@buaa.edu.cn

10.1109/JSEE.2015.00012

Manuscript received January 06,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(61104132;61304148).