Microelectronics Reliability

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1 Microelectronics Reliability 51 (2011) Contents lists available at ScienceDirect Microelectronics Reliability journal homepage: www. elsevier. com/ locate/ microrel Complex system maintainability verification with limited samples Qiang Miao a Liu Liu a Yuan Feng b Michael Pecht cd a School of Mechanical Electronic and Industrial Engineering University of Electronic Science and Technology of China Chengdu Sichuan China b Institute of Aero-equipment Beijing China c Center for Advanced Life Cycle Engineering (CALCE) University of Maryland College Park MD USA d Center for Prognostics and System Health Management City University of Hong Kong Hong Kong China a r t i c l e i n f o a b s t r a c t Article history: Received 28 March 2010 Accepted 14 September 2010 Available online 6 October 2010 Complex system maintainability verification is always a challenging problem due to limited sample sizes. Consequently conducting maintenance experiments in a laboratory environment is an appropriate way to obtain data for maintainability verification. In maintenance experiments faults are seeded in the equipment and maintenance activities are implemented to record repair time. In this process two problems arise when laboratory experimental data (in-lab data) are used together with field data during the operational test and evaluation stage. The first problem is the verification of segmental maintenance data and the second one is the combination of in-lab data and field data for integrative maintainability verification. Regarding the problems mentioned above this paper proposes a suitable methodology to solve them. Firstly the idea of segmentally weighted verification is adopted and the segmentally weighted verification (SWV) method is proposed to realize in-lab data verification. Secondly the Dempster Shafer (D S) evidence theory based integrative verification method is presented to solve the problem of in-lab and field data combination. A case study concerning radar system maintainability verification is presented as an example of the implementation of complex system maintainability verification in industry. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction Modern complex systems are usually defined as being systems with very complicated structures and a great number of entities. Aero-equipments are examples where maintainability and reliability play crucial roles in the system operation process. A perfect system would not only possess very high reliability to guarantee a long period of operation without failure but also it should be able to be promptly restored from a failure state through maintenance activities. Therefore maintainability is an important property that demonstrates the performance of aero-equipment maintenance through product design manufacturing and operation. The Mean Time to Repair (MTTR) is such a key metric to describe system maintainability. System maintainability is usually affected by various personnel and logistic factors. The degree of maintainability achieved depends upon the requirements imposed and management s emphasis on maintenance. The goal of maintainability verification is to validate whether the maintainability metric (e.g. MTTR) satisfies consumers requirements. Specifically it is necessary to follow certain standards to conduct maintenance design and verify system Corresponding author. Address: Department of Industrial Engineering University of Electronic Science and Technology of China 2006 iyuan Road Gaoxin Western District Chengdu Sichuan China. Tel.: address: mqiang@uestc.edu.cn (Q. Miao). maintainability before mass production so as to avoid critical design flaws at the beginning. MIL-STD-471A is a standard that provides uniform procedures test methods and requirements for verification demonstration and evaluation of the achievement of specified maintainability requirements for the assessment of the impact of planned logistic support [1]. According to MIL-STD- 471A for aero-equipment maintenance verification [1] the number of samples should be no less than 30. However it is almost impossible to obtain enough samples for maintainability verification at the equipment level from field tests during the operational test and evaluation stage due to the high cost of the tests. Generally we can deal with the problem of insufficient samples by utilizing probabilistic and statistic methods such as Bayesian techniques [23] Bootstrap methods [45] etc. The Bayesian method allows for the utilization of relevant historical information available for prior knowledge modeling and it is a popular approach to dealing with insufficient samples. Some possible available sources of historical information are engineering design and test data operating data in different environments and operating experiences with similar equipments. For example Simkins and Bukowski [2] developed procedures for system reliability demonstration based on Bayesian approaches. Erto and Giorgio [3] presented a Bayesian solution to the problem of the estimation of the parameters of the Weibull-inverse power law model on the basis of a limited number of life tests. The Bootstrap method is another solution to the problem of insufficient sample data when no /$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi: /j.microrel

2 Q. Miao et al. / Microelectronics Reliability 51 (2011) prior information is available. The benefit of bootstrapping lies in the fact that it does not require an a priori assumption about the distribution and it can fully mine useful information from the limited samples through resampling. Laggoune et al. [4] proposed an opportunistic replacement policy for a multi-component series system in the context of data uncertainty and they applied the Bootstrap technique to deal with the problem of the small size of failure data samples. Zio and Pedroni [5] proposed a Bootstrapbased method to assess the reliability of thermal-hydraulic passive systems. It should be noted that the Bayesian and Bootstrap methods are trying to fully utilize information from historical data and limited operating test samples. However in case there is still not sufficient information for maintenance evaluation it is necessary to conduct maintenance experiments in a laboratory environment to obtain enough maintenance data. In these experiments faults are seeded in the system and maintenance activities are implemented to record repair time. Two problems exist when laboratory experimental data (in-lab data) are used together with field data during the operational test and evaluation stage for maintainability verification: (1) In experiments in a laboratory environment maintenance activities usually include three steps fault detection fault identification and maintenance disassembly/reassembly and their time values are collected in segments. The maintenance time can be defined as the summation of these three time values. In this situation the probability distribution of the maintenance time cannot be defined as a certain known distribution and how to utilize segmental information for maintainability verification is still a challenging problem. (2) The faults in laboratory experiments are manually seeded and the faults identified in the verification stage are naturally produced during operation. Therefore the two sets of maintenance data (i.e. the in-lab data and the field data) do not have a direct relation due to the differences in the experimental environment and the failure mechanism and it is not possible to simply put them together for maintainability verification. Data fusion is a necessary step before conducting integrative verification. The motivation of this research is a project supported by industry. The main objective of our research is to investigate the integrative methods for aero-equipment maintainability verification in the above context. Since the aforementioned issues are coming from engineering practices there are few similar studies. In view of the in-lab data verification we propose an SWV method to deal with the evaluation of the MTTR index using in-lab data. Then the D S evidence theory is applied to realize data fusion between inlab data and field data. The rest of the paper is organized as follows. In Section 2 an SWV method is proposed to evaluate maintainability using the in-lab maintenance data. In Section 3 a data fusion method based on the D S evidence theory is presented to solve the problem of integrative verification of maintainability with in-lab data and field data. Section 4 gives a case study using maintenance records of aero-equipment to validate the proposed methods. Finally conclusions are summarized in Section The SWV method for in-lab data 2.1. Problem description A maintenance activity usually includes three tasks namely fault detection (FD) fault identification (FI) and maintenance disassembly/reassembly (DR). In maintenance experiments under a laboratory environment the time spent on these tasks is measured separately according to the testing programs currently adopted in industry and the total time of a maintenance activity is the summation of all three time variables. Since these variables may follow different probability distributions the distribution of the total maintenance time may be very complicated [6]. Therefore we cannot directly utilize these in-lab data collected from the laboratory with the verification methods available in the maintenance program (such as MIL-STD-471A) to conduct maintainability verification. Currently the available literature about the integrative verification of segmental maintenance time is very limited. Goulden [7] proposed the idea of a weighted method at the equipment-level for system-level maintainability verification. In this study we propose the SWV method for in-lab MTTR verification. In this process the weights of maintenance activities such as FD FI and DR are obtained respectively. Through the reconstruction of segmental results the MTTR verification can be accomplished. If the number of samples is insufficient for the estimation and verification of the segmental time variables (which is often true since the experimental cost is very high) the Bootstrap method can be adopted The Bootstrap method The Bootstrap method is a technique for uncertainty analysis that does not require any a priori assumptions for the unknown distribution of experimentally observed data. It can utilize the given population s information and simulate the unknown distribution through resampling of the available data. Resampling methods create an ensemble of data sets where each set is replicated from the original samples. Compared with the jack-knife algorithm the Bootstrap method creates new data sets by sampling with replacement while the jack-knife may produce highly inconsistent estimates for the standard error and/or other measures [8]. For this reason the Bootstrap method has been used in this research for the estimation of distribution parameters of the segmental time variables with insufficient sample size. Let = ( n ) be a sample of size n from the population characterized by an unknown probability function F(). h = h() are the unknown parameters of the population. The empirical distribution function F n () can be obtained through the current available sample = ( n ) by assigning a probability of 1/n at each point n of the sample. The Bootstrap sample ¼ f 1 ; 2 ;... ; ng can be obtained through resampling with replacement from F n. The sample * can be considered a randomly resampled version of. Similarly F nðþ is the empirical distribution function derived from *. The estimate of h given the Bootstrap sample * can be defined as ^h ¼ ^hð Þ ð1þ Repeat the generation of a Bootstrap sample N times and this leads to the Bootstrap samples ðjþ ¼ f ðjþ 1 ; ðjþ 2 ;... ; ðjþ n g; j ¼ 1; 2;... ; N ð2þ where *(j) is the jth Bootstrap sample. With each Bootstrap sample *(j) j = 12...N a Bootstrap replication ^h ðjþ can be obtained as ^h ðjþ ¼ ^hð ðjþ Þ. The Bootstrap replications ^h ðjþ ; j ¼ 1; 2;... ; N yield an estimator for the expectation of the h(): hðþ ¼ 1 N N j¼1 ^hð ðjþ Þ ð3þ

3 296 Q. Miao et al. / Microelectronics Reliability 51 (2011) The standard error SEð^hÞ is then estimated by the standard deviation of the N replications: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N h i 2 SEð^hÞ ¼ ^h N 1 ðjþ hðþ ð4þ j¼ The proposed SWV method For simplification define T as the total maintenance time and the variables Y Z are the time spent on fault detection fault identification and maintenance disassembly/reassembly respectively. Therefore T ¼ þ Y þ Z The maintainability index MTTR is the expectation of T which can be expressed as: MTTR ¼ EðTÞ and the expectation of T can be expressed as the summation of the expectations of the variables Y Z. EðTÞ ¼ Eð þ Y þ ZÞ ¼ EðÞ þ EðYÞ þ EðZÞ The weights of Y Z in the verification of MTTR are defined as EðÞ k ¼ EðÞ þ EðYÞ þ EðZÞ EðYÞ k Y ¼ EðÞ þ EðYÞ þ EðZÞ EðZÞ k Z ¼ EðÞ þ EðYÞ þ EðZÞ : In most cases the costs of in-lab tests are very expensive and it is difficult to obtain enough samples for MTTR verification. Therefore to calculate the segmental weight values k k Y k Z it is necessary to obtain the probability distribution functions f() f(y) f(z) and the means l l Y l Z using the aforementioned Bootstrap method. Here l = E() l Y = E(Y) and l Z = E(Z). Given the segmental weight values k k Y k Z we can implement the maintainability verification now. Given that the quality of the collected data is accurate (since reasonably precise estimates of the sample variance can be obtained through the Bootstrap method [5]) the verification method with the distribution unknown and the variance known can be selected according to MIL-STD-471A. Assume that the required maintainability index for the equipment is MTTR = T 0. The maintainability index for the fault detection is k T 0. Then the hypothesis test can be defined as: H 0 : l ¼ k T o ; H 1 : l > k T o ð9þ And the decision criteria are: Accept H 0 ; Reject H 0 ; if ^l 6 k T 0 þ p ^r ffiffiffi Z 1 a n if ^l > k T 0 þ p ^r ffiffiffi Z 1 a n ð5þ ð6þ ð7þ ð8þ ð10þ Here n is the sample size ^l is the estimate of the mean of the variable ; ^r is the estimate of the standard deviation of the variable a is the producer s risk and Z 1 a is the (1 a) lower percentile point of the Standard Normal Distribution N(0 1). The verification result can be written as: H ¼ 1; ðaccept H 0Þ 0; ðreject H 0 Þ ð11þ Similarly we can get the verification results of the fault identification Y and the maintenance disassembly/reassembly Z as H Y H Z respectively. At this moment given the H H Y and H Z the integrative verification result can be calculated as: H T ¼ k H þ k Y H Y þ k Z H Z ð12þ Here H T can be seen as a fuzzy number and is in the range of [01]. Redefine the hypothesis test for the maintainability verification of MTTR as below: H 0 : l T ¼ T o ; H 1 : l T > T o ð13þ According to the idea of hypothesis testing of fuzzy logic [9] a final conclusion should be drawn based on the following criteria: (1) When H T P 1 h accept the H 0. (2) When h < H T < 1 h go on with the experiment. (3) When H T 6 h reject the H 0. Here h 2 [00.5]. If h < H T < 1 h occurs it means the maintainability index of the current batch of equipment is very close to the contract s requirements. 3. The integrative verification of maintainability using the D S evidence theory 3.1. Problem description In the verification of the maintainability index MTTR in-lab maintenance data collected from the laboratory experiments should be used to supplement the maintenance data from the field during the operational test and evaluation stage. However due to the differences in the experiment environment and the failure production mechanism between these two data sets it is necessary to implement data fusion before the integrative verification. Data fusion has received significant attention in many application areas [10 13]. The aim of data fusion is to integrate different sources of data in order to achieve refined/improved information than can be derived from each single source alone [10]. One of the simplest and most intuitive general methods of fusion is to take a weighted average of redundant information provided by multiple sources of data [11]. The Bayesian approach provides another solution to data fusion by combining the associated probability distributions of each source of data into a joint posterior distribution fusion [12]. The D S evidence theory which is an extension of the Bayesian approach makes explicit any lack of information concerning a proposition s probability by separating firm support for the proposition from its plausibility [1113]. It is suitable for taking into account the disparity between knowledge types because it can provide a federative framework and it combines cumulative evidence for changing prior opinions in the light of new evidence [14]. Suppose the in-lab data and the field data are two different characterizations from the same sample under two different conditions. Because of the randomness of the data these two different data sets can be considered as two evidences from different information sources in the same frame of discernment in the D S evidence theory [15]. Therefore the D S evidence theory is applied in this research to fuse the two different data sources in the process of integrative maintainability verification Fundamentals of the D S evidence theory The D S evidence theory is concerned with the question of belief in a hypothesis. It calculates the probability that a piece of evidence supports the hypothesis and offers an alternative approach for uncertainty reasoning from imprecise and uncertain information. In this section the fundamentals of the D S evidence theory will be given.

4 Q. Miao et al. / Microelectronics Reliability 51 (2011) Frame of discernment Define H as a finite and nonempty sample space with n elements which are mutually exclusive alternatives. The set consisting of all the subsets of H is called the power set of H and is denoted by 2 H. When H has n elements 2 H has 2 n subsets. H is also called the frame of discernment and it contains every possible hypothesis. An element in this space H can be a hypothesis an object or a fault Basic probability assignment Basic probability assignment (BPA) is a function m:2 H? [01] such that 8 < mð/þ ¼ 0 P : mðaþ ¼ 1 ð14þ A#H Here / is the empty set. Given a piece of evidence A a belief level between [01] denoted by m() is assigned to each subset of H. m(a) expresses the exact belief measure of the A. A ¼ H A; mðaþ þ mðaþ61. This shows that m(a) is not a probability Belief function For the hypothesis A the belief function Bel:2 H? [01] is defined as BelðAÞ ¼ B#A mðbþ; 8A#H ð15þ The belief function Bel(A) represents the belief level that a hypothesis lies in A or any subset of A. It measures the total amount of probability that must be distributed among the elements of A. According to (14) and (15) we have 8 < Belð/Þ ¼ Mð/Þ ¼ 0 BelðHÞ ¼ P : MðBÞ ¼ 1 ð16þ B#H Plausibility function For the hypothesis A the plausibility function Pl(A) represents the plausibility belief level and is defined as: PlðAÞ ¼ 1 BelðAÞ ¼ mðbþ; 8A#H; B#H ð17þ B\A / Pl(A) describes the degree that we fail to disbelieve the hypothesis A. The relationship between Bel(A) and Pl(A) can be summarized as: ( PlðAÞ ¼ 1 BelðAÞ ð18þ PlðAÞ P BelðAÞ Obviously the proposition that the hypothesis A is not false does not mean A is true which means the belief degree of A is not false (Pl(A)) should be greater than or equal to the belief degree of A is true (Bel(A)). That is Pl(A) P Bel(A). Therefore Pl(A) can be seen as an upper limit function on the probability of A. On the other hand the belief function Bel(A) behaves as a lower limit function on the probability of A. When Bel(A) = Pl(A) Pl(A) is fully certain for the belief measure of A. Both imprecision and uncertainty can be represented by the belief function Bel(A) and the plausibility function Pl(A). As shown in Fig. 1 [Bel(A) Pl(A)] is the confidence interval that describes the uncertainty of A. Pl(A) Bel(A) for A#H represents the ignorance level in hypothesis A. If the information for fusion is missing or unreliable the difference between Bel(A) and Pl(A) increases. This difference provides a measure of the imprecision and the uncertainty of the belief level in decision-making Dempster s combination rule Suppose m 1 m 2 are two different basic probability assignment functions based on information obtained from two different information sources A 1 A 2 in the same frame of discernment H. These different evidences can be fused using Dempster s combination rule which is also called the orthogonal sum of evidences. That is m = m 1 m 2 which represents the combination of m 1 and m 2. According to the Dempster s combination rule we have mð/þ ¼ 0 mðcþ ¼ K 1 where K ¼ 1 A 1 \A 2 ¼/ A 1 \A 2 ¼C m 1 ða 1 Þ m 2 ða 2 Þ m 1 ða 1 Þ m 2 ða 2 Þ ð19þ ð20þ In the combination process C = A 1 \ A 2 if C = / then the pieces of evidence conflict with each other and the belief level in hypothesis C is then null. K in (19) is called the uncertainty factor which is often interpreted as a measure of conflict between different sources. If K 0 then m is also a BPA in the same frame of discernment H. On the contrary if K = 0 m does not exist and m 1 and m 2 conflict with each other D S evidence theory based data fusion for integrative maintainability verification Suppose there are m 0 repair time records collected from field test t 1 ; t 2 ;... ; t m 0 and m 00 repair time records from the laboratory t 0 1 ; t0 2 ;... ; t0 m 00. Since the sample sizes (m0 and m 00 respectively) of these two data sets may be different and the combination rule requires equivalent information it is necessary to do some preprocessing steps before fusion. If m 0 < m 00 we can interpolate some points in the field data; if m 0 > m 00 we can interpolate some points in the in-lab data. Suppose the number of samples in each data set is n = max(m 0 m 00 ) after pre-processing then we can fuse them. In order to reduce the influence of uncertain factors caused by randomness in the data we rearrange the field data by descending order t 1 P t 2 P Pt n and rearrange the in-lab data by ascending order t t t0 n. Then we have: m 1 ðft 1 g; ft 2 g;... ; ft i g;... ; ft n gþ n n ¼ t 1 t i ; t 2 t i ;... ; t i n t i ;... ; t n n t i! ð21þ Plausibility Interval Belief Interval Uncertainty Interval Refusal Interval m 2 ðft 0 1 g; ft0 2 g;... ; ft0 i g;... ; ft0 n gþ ¼ t 0 1 t 0 i ; t0 2 n n n n t 0 i ;... ; t0 i t 0 i ;... ; t0 n t 0 i! ð22þ 0 Bel(A) Pl(A) 1 Fig. 1. The relationship between Bel(A) and Pl(A). Here i = 12...n. t i = P n t i in (21) and t 0 i =P n t0 i in (22) describe the uncertainties of the random repair times t i and t 0 i in the evidence theory respectively. They can be considered BPAs of hypotheses t i and t 0 i.

5 298 Q. Miao et al. / Microelectronics Reliability 51 (2011) Then the two evidences in (21) and (22) can be combined by the Dempster s combination rule as shown in (19) and we have: mðfs i gþ ¼ m 1 m 2 ¼ K 1 m 1 ðft i gþ m 2 ðft 0 j gþ ð23þ B i \C j ¼fs i g Here B i means the ith element of m 1 in (21) and C j means the jth element of m 2 in (22). According to (20) K is K ¼ 1 m 1 ðt i Þm 2 ðt 0 j Þ ð24þ B i \C j ¼/ Suppose the combined evidence is: mðft 1 g; ft 2 g;... ; ft i g;... ; ft n gþ ¼ ðs 1 ; s 2 ;... ; s i ;... ; s n Þ ð25þ Here T i is the repair time after data fusion; m({t i }) in (25) is a BPA of hypothesis T i which is combined by t i and t 0 i. Like the definitions of m 1 ({t i }) and m 2 ðft 0 i gþ here we denote m({t i}) = s i by the same definition as in (21) and (22). This means that s i can be considered the ith combined repair time divided by the summation of combined repair time. That is: s n i ¼ T i T i ð26þ Suppose P n T i is weighted by P n t i and P n t0 i we have T i as " # T i ¼ k n t i þ ð1 kþ n s i ; k 2 ½0; 1Š ð27þ t 0 i where k is a constant and its range is defined as k 2 [01]. The weighted coefficient k can be obtained from a priori knowledge or by expert experience. With the fused data set T = {T 1 T 2...T n } we can implement the integrative maintainability verification according to the hypothesis testing procedures in [1]. 4. Case study 4.1. Validation of the SWV method using in-lab data In this section maintainability verification using in-lab data obtained from a certain piece of aero-equipment is implemented. Table 1 shows the maintenance time records of this equipment collected from the laboratory in which the segmental time information includes fault detection time fault identification time Y and maintenance disassembly/reassembly time Z. Assume that the required maintainability index for this equipment is MTTR = 4 and the producer s and consumer s risks are a = b = Table 1 Maintenance records of a certain piece of equipment. No. Segmental maintenance time record (h) FD FI Y DR Z Since the sample size is less than 30 it is necessary to utilize the Bootstrap method described in Section 2.2 to estimate the distribution parameters (e.g. the mean) of Y Z. For example after times resampling on the fault detection time we can get the estimate of the mean for as E() = E( * ) = in which * is the sample after resampling. Similarly the estimate of the mean for Y and Z can be obtained as E(Y) = E(Y * ) = and E(Z) = E(Z * ) = respectively. According to Eq. (8) the segmental weight values of Y Z can be calculated as: EðÞ k ¼ EðÞ þ EðYÞ þ EðZÞ ¼ 1:040 1:040 þ 1:266 þ 1:912 ¼ 0:247 EðYÞ k Y ¼ EðÞ þ EðYÞ þ EðZÞ ¼ 1:266 1:040 þ 1:266 þ 1:912 ¼ 0:300 EðYÞ k Z ¼ EðÞ þ EðYÞ þ EðZÞ ¼ 1:912 1:040 þ 1:266 þ 1:912 ¼ 0453 With these weight values the required MTTR index can be allocated to each time variable. The maintainability indexes for the three time variables Y Z can be expressed as: T ¼ k MTTR ¼ 0:988 T Y ¼ k Y MTTR ¼ 1:200 T Z ¼ k Z MTTR ¼ 1:812 The verification of segmental time variables can follow the methods provided in [1]. Assume that the verification results are H() = 0 H(Y) = 1 and H(Z) = 1. Then according to Eq. (12) H T = With h = 0.3 a conclusion can be drawn to reject the original hypothesis H 0. That is the maintainability of this equipment fails to satisfy the requirement Validation of the D S evidence theory based integrative maintainability verification method In this section the example of a radar system is used to investigate the proposed D S evidence theory based integrative maintainability verification method. Table 2 shows the maintenance records collected from the field test and the laboratory experiments. The unit is hours and the required maintainability index MTTR is 2 h Data fusion using D S evidence theory Firstly the samples from the field data were sorted in descending order as: t ¼ ð2:5; 2:5; 2:5; 2; 2; 2; 1:5; 1:5; 1; 0:5Þ and the samples from the in-lab data were sorted in ascending order as: t 0 ¼ ð0:5; 1; 1; 1:5; 1:5; 1:8; 2; 2; 2:5; 2:5Þ Transform the data as in the form of (21) and (22): m 1 ðft 1 g; ft 2 g; ; ft i g; ; ft n gþ ¼ m 1 ðf2:5g; f2:5g; f2:5g; f2g; f2g; f2g; f1:5g; f1:5g; f1g; f0:5gþ ¼ 2:5 18 ; 2:5 18 ; 2:5 18 ; 2 18 ; 2 18 ; 2 18 ; 1:5 18 ; 1:5 18 ; 1 18 ; 0:5 18 ; Table 2 Maintenance records of a certain type of radar. System Radar Required MTTR 2 Number of samples 10 Maintenance records from field test Maintenance records from laboratory experiment

6 Q. Miao et al. / Microelectronics Reliability 51 (2011) and m 2 ðft 0 1 g; ft0 2 g; ; ft0 i g; ; ft0 n gþ ¼ m 2 ðf0:5g; f1g; f1g; f1:5g; f1:5g; f1:8g; f2g; f2g; f2:5g; f2:5gþ ¼ 0:5 16:3 ; 1 16:3 ; 1 16:3 ; 1:5 16:3 ; 1:5 16:3 ; 1:8 16:3 ; 2 16:3 ; 2 16:3 ; 2:5 16:3 ; 2:5 16:3 According to (23) and (24) we have: mðft 1 gþ ¼ m 1 m 2 ¼ K 1 m 1 ðft i gþ m 2 ðft 0 j gþ B i \C j ¼fs 1 g ¼ 293:4 25:6 1:25 293:4 ¼ 1:25 25:6 In the same way we can get s 2 s 3...s 10 as below: mðft 1 g; ft 2 g; ; ft i g; ; ft n gþ k n ¼ ðs 1 ; s 2 ; ; s i ; ; s n Þ ¼ 1:25 25:6 ; 2:5 25:6 ; 2:5 :6 25:6 ; 2:5 25:6 ; 1:25 : 25:6 Suppose k = 0.8 in (27) then: t i þ ð1 kþ n t 0 i ¼ 17:66 Then we can get T i according to (27): T ¼ ð0:862; 1:725; 1:725; 2:070; 2:070; 2:483; 2:070; 2:070; 1:725; 0:862Þ Verification of MTTR Here we use the field data the in-lab data and the combined data obtained by D S evidence theory based data fusion to evaluate the MTTR. The method used in this research to evaluate the MTTR is from Ref. [1]. Consider the samples after data fusion: 2:5;1:5;2;0:5;1;2;2:5;1:5;2;2:5;1:8;1;2;2:5;1:5;1:5;2:5;0:5;2;1; 0:862; 1:725; 1:725; 2:070; 2:070; 2:483; 2:070; 2:070; 1:725; 0:862 The mean of the sample is ct ¼ 1 n n i ¼ 1:732 The variance of the sample is ^d 2 ct ¼ 1 n ð i ct Þ 2 ¼ 0:369 n 1 And the standard deviation qffiffiffiffiffiffi ^d ct ¼ ^d 2 ct ¼ 0:608 Assume the desired value l 0 = 2 and the maximum tolerable value l 1 = 2.4. Assume the producer s risk is a = 0.05 and the consumer s risk is b = Then we have Z 1 a = Z 1 b = 1.65 from the Standard Normal Distribution Table. The quantity of sample needed in this evaluation is n ¼ ^r 2 2 ¼ 1:65 þ 1:65 0:608 ¼ 25:16 < 30 2:4 2 Z 1 a þ Z 1 b l 1 l 0 So here n should be 30 and the critical value is K ¼ l 1 z 1 a þ l 0 z 1 b z 1 a þ z 1 b ¼ 2:4 1:65 þ 2 1:65 ¼ 2:2 1:65 þ 1:65 Decision procedure: accept H 0 if ct 6 K and reject H 0 if ct P K. In this example since ct < K H 0 should be accepted. 5. Conclusions In the process of complex system (such as aero-equipment) maintainability verification insufficient sample size is always a challenging problem due to the high cost during the operational test and evaluation stage. Therefore conducting maintenance experiments in a laboratory environment is an appropriate practice that has been accepted by industry. The in-lab data can be used as a supplement to the field test data to implement maintainability verification. However this results in two issues: one is how to utilize in-lab data for maintainability verification and the other is the data fusion of in-lab data and field data for integrative maintainability verification. Regarding the issues mentioned above this paper proposed a suitable methodology to solve them. Firstly the idea of segmentally weighted verification was adopted and the SWV based method was proposed to realize the in-lab data verification. Secondly a D S evidence theory based integrative verification method was presented to solve the second issue. A case study concerning a radar system maintainability verification was presented as an example for the implementation of complex system maintainability verification in industry. Acknowledgments This research was partially supported by the National Natural Science Foundation of China (Grant No ) Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No ) and the Fundamental Research Funds for the Central Universities (Grant No. ZYG ). The work described in this paper was also partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region China (CityU8/CRF/09). References [1] MIL-STD-471A. Maintainability verification/demonstration/evaluation; [2] Simkins DJ Bukowski JV. Bayesian reliability evaluation of computer systems. Comp Electr Eng 1984;11(2-3): [3] Erto P Giorgio M. 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