An Evaluation of the Reliability of Complex Systems Using Shadowed Sets and Fuzzy Lifetime Data

Transcription

1 International Journal of Automation and Computing 2 (2006) An Evaluation of the Reliability of Complex Systems Using Shadowed Sets and Fuzzy Lifetime Data Olgierd Hryniewicz Systems Research Institute of the Polish Academy of Sciences, Warsaw , Poland Abstract: In this paper, we consider the problem of the evaluation of system reliability using statistical data obtained from reliability tests of its elements, in which the lifetimes of elements are described using an exponential distribution. We assume that this lifetime data may be reported imprecisely and that this lack of precision may be described using fuzzy sets. As the direct application of the fuzzy sets methodology leads in this case to very complicated and time consuming calculations, we propose simple approximations of fuzzy numbers using shadowed sets introduced by Pedrycz (1998). The proposed methodology may be simply extended to the case of general lifetime probability distributions. Keywords: Estimation of reliability, fuzzy reliability data, shadowed sets. 1 Introduction Statistical methods for the analysis of lifetime data have been intensively developed for the last fifty years. Numerous papers and textbooks dealing with different statistical aspects of the analysis of reliability data have been published. In [1 3] one can find practically all methods that could be used for the analysis of such data. However, practically all well known statistical procedures have been developed for the analysis of precisely measured lifetime data. Only in a few books and papers (e.g. in [3]), one can find information about the statistical analysis of interval data with precisely defined time intervals covering unknown lifetimes. Precisely reported lifetimes are common when data comes from specially designed life tests. In such a case a failure should be precisely defined, and all tested items should be continuously monitored. However, in real situations these test requirements might not be fulfilled. In an extreme case, reliability data comes from users whose reports are expressed in a vague way. The vagueness of reliability data coming from users has many different sources. In [4] these sources have been divided into three groups: vagueness caused by subjective and imprecise perception of failures by a user vagueness caused by imprecise recording of reliability data vagueness caused by imprecise recording of rate of usage Manuscript received September 23, 2005; revised January 9, address: hryniewi@ibspan.waw.pl The first source of vagueness is typically for noncatastrophic failures. A tested item may be considered as failed, or strictly speaking as nonconforming, when at least one value of its parameters falls beyond specification limits. In practice, however, a user does not have the possibility to measure all parameters and is not able to define precisely the moment of a failure. For example, if there exists a requirement for an admissible level of noise, it may usually be verified by a user only subjectively. A user can usually indicate only a moment when he noticed that a level of noise increased, and the moment when he (or she) considered it as obviously excessive. Therefore, it might be assumed that the first moment describes the time when the tested item (e.g. a car) may be considered as failed, and the second moment indicates the time of a sure failure. As a result, we obtain imprecise information about a real lifetime. Moreover, this type of vagueness causes situations in which even at the end of a test (i.e. at a censoring time), a user is not sure whether the tested item has failed or not. In such a case, we have not only imprecise values for lifetimes, but have imprecise information about the number of observed failures as well. This type of imprecise reliability data was considered in [4,5]. A second source of vagueness is typically retrospective data. Users do not record precisely the moments of failures, especially when they are not sure if they observed a real failure (see above). So when they are asked about failures which were observed some time ago, they sometimes provide imprecise information. A lifetime of an individual is the actual length of life of that individual, measured from some particular starting point. However, it may happen that the user cannot

2 146 International Journal of Automation and Computing 2 (2006) specify this starting point precisely, but only in a vague way. In such a situation, the lifetime of an item under study is also vague. A third source of vagueness is related to the fact that users who report their data in days (weeks, or months) use test items with different intensity. Two reports about items that failed after 20 weeks of work may have completely different meaning for the measurement of MTTF expressed in hours of continuous work, when the intensity of usage significantly differs in both of these cases. In practice, users are asked about the intensity of usage (for example, in hours per day), and their responses are, for obvious reasons, very often imprecise. The lack of precision of reliability field data comes from all of these sources, and in many cases cannot even be identified. Precise probability models can seldom be applied, only for clearly identified sources of vagueness, and are very often impractical, because many parameters are either unknown or difficult to estimate. Therefore, we have to admit that we often deal with extremely vague data expressed with imprecise words, and that this is the only source of information which can be used for the verification of hypotheses about the mean lifetime of test items. In the second section of this paper, we introduce the concept of the fuzzy modeling of imprecise lifetime data. The obtained results however may be too difficult for practitioners to implement. Therefore, there is a need to propose a much simpler approximate formulae. In the third section of this paper we present such results, using the concept of shadowed sets introduced by Pedrycz [6]. 2 Fuzzy modeling of imprecise life-data A frequently used reliability characteristic is mean time to failure MTTF. This can be easily estimated from a sample (W 1, W 2,, W n ) of the times to failure (lifetimes) when all lifetimes are known. However, in a majority of practical cases reliability tests are terminated before the failure of all items. Therefore, lifetime data are usually censored using fixed censoring times Z i > 0, i = 1,,n, n is the sample size. In such a case, we observe lifetimes W i, only if W i Z i. Therefore, lifetime data consists of pairs (T 1, Y 1 ),, (T n, Y n ), T i = min(w i, Z i ) (1) { 1, if Ti = W i Y i =. (2) 0, if T i = Z i Further analysis of censored lifetime data depends upon an assumed probability distribution. The simplest and most frequently used method in many areas of lifetime analysis is the exponential model. In this model, lifetime W is described by the probability density function: { 1 exp( w/θ), if w > 0 f(w) = θ (3) 0, if w 0 θ > 0 is mean lifetime. The exponential model is rather restrictive (constant hazard rate), but due to its simplicity is frequently used by practitioners. Therefore, in this paper we assume that an exponential distribution is used for the mathematical description of lifetimes. Let T = Z i (4) T i = i O W i + i C be total survival time (total test time), O and C denote the sets of items whose exact lifetimes are observed and censored respectively. Moreover, let r = Y i (5) be the number of observed failures. In the considered exponential model (r, T) is a minimally sufficient statistics for θ, and the maximum likelihood estimator of mean life time is given by: ˆθ = T r. (6) If we now suppose that the lifetimes (time to failure) and censoring times are not necessarily precisely reported, then it is possible to model this kind of vagueness using fuzzy numbers. This approach has recently been used by several authors, such as Viertl & Gurker [7], Grzegorzewski & Hryniewicz [5,8], and Hryniewicz [4]. Reference [6] proposed the generalization of an exponential model using the concept of fuzzy numbers and fuzzy random variables. Fuzzy subset A of the real line R with a membership function µ A : R [0, 1] is a fuzzy number iff: a) A is normal, i.e. there exists an element x 0 such that µ(x 0 ) = 1 b) A is fuzzy convex, i.e. µ A (λx 1 + (1 λ)x 2 ) µ A (x 1 ) µ A (x 2 ) x 1, x 2 R, λ [0, 1] c) µ A is upper semi-continuous, and d) supp A is bounded. From the above definition, one can easily find that for any fuzzy number A, there exist four real numbers a 1, a 2, a 3, and a 4, and two functions: non-decreasing function η A : R [0, 1], and non-increasing function

3 O. Hryniewicz/An Evaluation of the Reliability of Complex Systems Using Shadowed 147 ξ A : R [0, 1], such that membership function µ A is given by 0, if x < a 1 η A (x), if a 1 x < a 2 µ A (x) = 1, if a 2 x < a 3. (7) ξ A (x), if a 3 x < a 4 0, if a 4 < x Functions η A and ξ A are called the left and right side of a fuzzy number A, respectively. Fuzzy numbers may also be described using a set of α-cuts. The α-cut of fuzzy number A is a non-fuzzy set defined as A α = {x R : µ A (x) α}. (8) A family {A α : α (0, 1]} is a set representation of fuzzy number A. From the resolution identity we obtain µ A (x) = sup{αi Aα (x) : α (0, 1]} (9) I Aα (x) denotes the characteristic function of A α. Every α-cut of a fuzzy number is a closed interval. Hence we have A α = A L α, AU α, A L α = inf{x R : µ A(x) α} A U α = sup{x R : µ A (x) α}. (10) Membership functions of fuzzy numbers that are defined as functions of other fuzzy numbers may be calculated using the following extension principle, introduced in [9] and described in [10] as follows. Let X be the Cartesian product of universe X = X 1 X 2 X r, and A 1,, A r be r fuzzy sets in X 1,, X r, respectively. Let f be a mapping from X = X 1 X 2 X r to a universe Y such that y = f(x 1,, x r ). The extension principle allows us to induce from r fuzzy sets A i a fuzzy set B on Y through f, such that µ B (y) = sup min µ A1 (x 1 ),, µ Ar (x r ) x 1,,x r;y=f(x 1,,x r) (11) µ B (y) = 0 if f 1 (y) =. (12) Fuzzy numbers may be treated like a generalization of real numbers. Similarly, real-valued random variables may be generalized using fuzzy random variables. There exist many definitions for a fuzzy random variable. The following definition is similar to that given in [11]. Suppose a random experiment is described using probability space (Ω, A, P), Ω is the set of all possible outcomes of the experiment, A is an σ-algebra of the subsets of Ω (the set of all possible events), and P is a probability measure. A mapping X : Ω FN FN is a space of all fuzzy numbers is called a fuzzy random variable if it satisfies the following properties a) {X α (ω) : α [0, 1]} is a set representation of X(ω) for all Ω, and b) for each α [0, 1] both and X L α = X L α(ω) = inf X α (ω) X U α = XU α (ω) = supx α(ω) are usual real-valued random variables in (Ω, A, P). Therefore, a fuzzy random variable X is considered a perception of an unknown (unobservable) usual random variable V, called the original of X. If only vague data is available, we can calculate the grade of acceptability that fixed random variable V is the original of fuzzy random variable X [11]. Fuzzy random variables are described using probability distributions that are similar to usual probability distributions, except for their parameters which are described using fuzzy numbers. The fuzzy parameters of a fuzzy random variable X, may be considered as fuzzy perceptions of unknown parameters of its original V. Therefore, in the case of imprecise fuzzy lifetime data, important reliability characteristics, like the mean time to failure, are also fuzzy. Let us consider the case of imprecise lifetime data. In the exponential model, time to failure and censoring times equally contribute, see (4), to the total test time T. Therefore, each of these times may be regarded as one lifetime datum, denoted T i, i = 1,,n. If we suppose that times T i may be described by fuzzy numbers, in order to simplify calculations, we assume that for each i = 1,,n it is described using trapezoidal fuzzy numbers defined by the following membership function: 0, if t < t 1,i (t t 1,i )/(t 2,i t 1,i ), if t 1,i t < t 2,i µ Ti (t) = 1, if t 2,i t < t 3,i. (t 4,i t)/(t 4,i t 3,i ), if t 3,i t < t 4,i 0, if t 4,i t (13) It is easy to note that precise lifetime data can be considered as a special case of (13), when the following equality holds: t 1,i = t 2,i = t 3,i = t 4,i = t i. If the following equalities hold: t 1,i = t 2,i = t l,i and t 3,i = t 4,i = t u,i, then expression (13) represents well known interval data. On the other hand, if we have t 2,i = t 3,i = t p,i, then (13) represents typical imprecise information t i is about t i,p. Now let us consider the fuzzy equivalent of the total test time, when individual fuzzy data are described

4 148 International Journal of Automation and Computing 2 (2006) by (13). From the extension principle, one can easily find that the total fuzzy test time T is defined by the following membership function: 0, if t < T 1 (t T 1 )/(T 2 T 1 ), if T 1 t < T 2 µ T (t) = 1, if T 2 t < T 3 (14) (T 4 t)/(t 4 T 3 ), if T 3 t < T 4 0, if T 4 t T 1 = t 1,i (15) T 2 = T 3 = T 4 = t 2,i (16) t 3,i (17) t 4,i. (18) Now let us suppose that the number of failures r is precisely defined. This means that all failures may occur at imprecisely reported times, but their existence is certain. (The case of imprecise (fuzzy) failure counting is considered in [5]). In such a case, estimated mean time to failure θ is described by a membership function similar to that given in (14) (18). The only difference is in the scale of the membership function, as everything between (14) (18) must be divided by r. The situation becomes more complicated if we are interested in the evaluation of other reliability characteristics, e.g. the probability of failure: P(t) = 1 e t/θ, t > 0. (19) When the total test time is imprecise, and described by membership function (14), we may apply the extension principle to find a fuzzy version of the estimated probability of failure. By straightforward calculation, we can find that in this case, the membership function is given by the following expression 0, if p < p 1 T 4 + rt ln(1 p), if p 1 p < p 2 T 4 T 3 µ P (p) = 1, if p 2 p < p 3 (20) T 1 rt ln(1 p), if p 3 p < p 4 T 2 T 1 0, if p 4 p p 1 = 1 e rt/t4 (21) p 2 = 1 e rt/t3 (22) p 3 = 1 e rt/t2 (23) p 4 = 1 e rt/t1. (24) The situation is even more complicated if we want to evaluate the probability of failure of a system when the probabilities of failures of its elements are estimated from imprecise fuzzy lifetime data. Consider a coherent system whose elements have lifetimes described by exponential probability distributions. In the most general case we can write P S (t) = h(p 1 (t), P 2 (t),, P m (t)) (25) probabilities P 1 (t), P 2 (t),, P m (t) have a form given by (19). It is a well known fact that for a coherent system the probability of failure P S (t) is a non-decreasing function of its arguments. We can use this property for the calculation of the membership function of PS (t) when the membership functions of P 1 (t), P 2 (t),, P m (t) have a form given by (20). To calculate this membership function, let us consider the α-cut representation of µ P (p). For a given value of α [0, 1], we can define the following α-cuts P α,j (t) = [P L α,j, P U α,j], j = 1,,m (26) ( ) Pα,j L r j t = 1 exp, j = 1,,m T 4,j α(t 4,j T 3,j ) (27) ( Pα,j U = 1 exp r j t T 1,j α(t 2,j T 1,j ) ), j = 1,,m. (28) Now we can find the α-cut representation of the membership function of P S (t) using the following α-cuts P α,s (t) = [P L α,s, P U α,s] (29) P L α,s = h(p L α,1, P L α,2,,p L α,m) (30) P U α,s = h(p U α,1, P U α,2,,p U α,m). (31) However, all of these calculations may be prohibitively complicated for systems with a complicated function structure h(p 1 (t), P 2 (t),, P m (t)). Therefore, there is a need to simplify necessary calculations. In the next section we propose a method for such a simplification that is based on the concept of shadowed sets. 3 Description of vague lifedata using shadowed sets The theory of fuzzy sets provides a precise methodology for dealing with vagueness. However, in many

5 O. Hryniewicz/An Evaluation of the Reliability of Complex Systems Using Shadowed 149 practical cases this precision seems excessive. See for example a description of imprecise life data and its consequences. A very simple and intuitively well understood fuzzy description of imprecise total test time (14) is transformed to a more complicated description of the imprecise probability of failure (20), and further to an incomprehensible (for practitioners) fuzzy description of the probability of failure of a coherent system, with a general reliability structure. Therefore, there is a practical need to simplify this description using some approximation of fuzzy sets. One such approximation has been proposed by Pedrycz [6] who introduced the notion of shadowed sets. Let us introduce the concept of a shadowed set in an informal way. We suppose that a vague lifetime is described by a fuzzy number. If the value of the membership function for a given value of lifetime is small, we can say that such a lifetime is not very plausible. On the other hand, if such a value is high (close to 1) we can regard that lifetime as very plausible. The concept of shadowed sets consists in elevating to 1 all values of membership functions that are high enough, and in reducing to 0 all values of membership functions that are small enough. The membership function for a region it is not possible either to elevate it to 1, or reduce it to 0, becomes undefined. Therefore, a fuzzy number A can be approximated using a set of intervals: a central interval with a membership grade equal to one, and two adjacent intervals with undefined (shadowed) grades of membership, and open side intervals the grade of membership is equal to 0. Consider a fuzzy number A with a membership function given by (7). The elevation of its membership function to 1 can be produced for any and {x [a 2, a 2 ], a 2 = η 1 A (α U)} {x [a 3, a 3 ], a 3 = ξ 1 A (α U)} α U < 1 is a given upper threshold. Similarly, the reduction of its membership function to 0 can be conducted for any and {x [a 1, a 1], a 1 = η 1 (α L )} {x [a 4, a 4], a 4 = ξ 1 (α L )} α L < α U is a given lower threshold. Therefore, each fuzzy number can be represented by a shadowed fuzzy number defined by the set of four real numbers {a 1, a 2, a 3, a 4 } that have been defined above. Further transformations of shadowed fuzzy numbers can be produced using interval arithmetic. Reference [6] proposed a general methodology for finding such thresholds that α L = α, α < 0, 5, and α U = 1 α. The value of α can be found by taking a 1 = η 1 A (α) a 2 = η 1 A (1 α) a 3 = η 1 A (1 α) a 4 = η 1 A (α) and solving the following equation 1 a 1 η A (x)dx+ 4 a 4 ξ A (x)dx+ 2 a (1 η A (x))dx η A (x)dx+ a 1 a 3 (1 ξ A (x))dx 4 a 3 ξ A (x)dx=0. (32) In the case of the fuzzy probability of failure defined in (20), solving (32) is extremely difficult, as it involves the numerical evaluation of special functions named integral logarithms. The situation may be simplified when we assume that the probability of failure is small. In such a case we can use the following approximation: 0, if p < p 1 T 4 rt p, if p 1 p < p 2 T 4 T 3 µ P (p) 1, if p 2 p < p 3 (33) T 1 + rt p, if p 3 p < p 4 T 2 T 1 0, if p 4 p p 1, p 2, p 3, and p 4 are given by (21) (24), respectively. The solution to (32), when the fuzzy number is described by (33), is much easier, but it can be produced only numerically. The amount of computation is comparable with what is necessary when we use the concept of α-cuts, and therefore there is a practical need to find a simpler approximate solution. A very simple solution can be found however when we apply the methodology of shadowed sets at the level of fuzzy total test time. Reference [6] has shown that in the case of linear functions η A (x) and ξ A (x), the value of α that fulfills (32) is always equal to 0,414. Simple calculations lead to the representation of fuzzy test time T described in (14) by a set of four numbers {t 1, t 2, t 3, t 4 }, defined in the following expressions t 1 = T 1 + 0, 414(T 2 T 1 ) (34) t 2 = T 1 + 0, 586(T 2 T 1 ) (35) t 3 = T 4 0, 586(T 4 T 3 ) (36) t 4 = T 4 0, 414(T 4 T 3 ). (37)

6 150 International Journal of Automation and Computing 2 (2006) Hence, we can represent the fuzzy probability of failure P(t) defined by (20) as its shadowed counterpart {p s 1, p s 2, p s 3, p s 4}, p s 1 = 1 e rt/t4 (38) p s 2 = 1 e rt/t3 (39) p s 3 = 1 e rt/t2 (40) p s 4 = 1 e rt/t1. (41) Now the calculation of the fuzzy probability of a system s failure P S (t) becomes much simpler, and this characteristic can be given as a shadowed fuzzy number {PS,1 s, P S,2 s, P S,3 s, P S,4 s }, P s S,1(t) = h(p s 1,1, p s 2,1,,p s m,1) (42) P s S,2 (t) = h(ps 1,2, ps 2,2,,ps m,2 ) (43) P s S,3(t) = h(p s 1,3, p s 2,3,,p s m,3) (44) P s S,4 (t) = h(ps 1,4, ps 2,4,,ps m,4 ) (45) and p s j,1, ps j,2, ps j,3, ps j,4, j = 1,,m are given by (38) (41) for each element of the system. 4 Conclusions In this paper, we have proposed a new methodology for the evaluation of reliability in a coherent system when the reliability of all its elements are described by an exponential model. The novelty of our approach consists of the assumption that the observed time to failure of a system s elements may be reported imprecisely. We have assumed that this imprecision can be described using fuzzy numbers. Unfortunately, the direct application of the fuzzy sets methodology to the considered problem requires significant computational effort. To alleviate these computational problems we propose simple approximations of fuzzy numbers using shadowed fuzzy sets introduced in [6]. The proposed methodology may be used for a more general problem when times to failure are distributed according to other probability distributions. In such a case we have to approximate individual observed fuzzy times to failure by their respective shadowed fuzzy sets. Further calculations should be performed using interval arithmetic rules and their respective formulae known from mathematical statistics. References [1] L. J. Bain, M. Englehardt. Statistical Analysis of Reliability and Life-Testing Models: Theory and Methods, 2nd ed. Marcel Dekker, New York, [2] W. Q. Meeker, L. A. Escobar. Statistical Methods for Reliability Data. John Wiley & Sons, Inc., New York, [3] J. F. Lawless. Statistical Models and Methods For Lifetime Data, 2nd ed. John Wiley & Sons, Inc., New York, [4] O. Hryniewicz. Lifetime Tests for Imprecise Data and Fuzzy Reliability Requirements, Reliability and Safety Analyses under Fuzziness, T.Onisawa, J.Kacprzyk (eds.), Physica Verlag, Heidelberg, pp , [5] P. Grzegorzewski, O. Hryniewicz. Computing with words and life data. International Journal of Applied Mathematics and Computer Scence, vol. 12, no. 2, pp , [6] W. Pedrycz. Shadowed sets: Representing and processing fuzzy sets. IEEE Transactions on Systems, Man and Cybernetics Part B: Cyberetics, vol. 28, no. 1, pp , [7] R. Viertl, W. Gurker. Reliability Estimation Based on Fuzzy Life Time Data, Reliability and Safety Analyses under Fuzziness, T.Onisawa, J.Kacprzyk.(eds.), Physica Verlag, Heidelberg, pp , [8] P. Grzegorzewski, O. Hryniewicz. Lifetime Tests for Vague Data, Computing with Words in Information/Intelligent Systems, J.Kacprzyk, L.Zadeh (eds.), Part 2, Physica Verlag, Heidelberg, pp , [9] L. Zadeh. The Concept of a Linguistic Variable and Its Application to Approximate Reasoning, Parts 1, 2, and 3. Information Sciences, vol. 8, pp , pp ; vol. 9: pp , [10] D. Dubois, H. Prade. Fuzzy Sets and Systems, Theory and Applications. Academic Press, New York, [11] R. Kruse, K. D. Meyer. Statistics with Vague Data. D. Riedel, Dordrecht, Olgierd Hryniewicz graduated from Warsaw University of Technology. In 1970 he received his M.Sc. degree in electronics. In 1976 he received his Ph.D. degree in cybernetics from the Institute of Management and Automatic Control of the Polish Academy of Sciences. In 1985 he received his D.Sc. (habilitation) degree in statistics from the Academy of Economy in Cracow. In 1996 he received the state title of Professor. He is currently the Director, and a Professor, at the Systems Research Institute of the Polish Academy of Sciences. His research interest is the application of statistical methods in reliability and quality control. He is also interested in the application of fuzzy sets in decision support systems. He is the author of more than 150 books, journal papers, contributions to books, and conference proceedings.