Fuzzy reliability using Beta distribution as dynamical membership function

Fuzzy reliability using Beta distribution as dynamical membership function S. Sardar Donighi 1, S. Khan Mohammadi 2 1 Azad University of Tehran, Iran Business Management Department (e-mail: soheila_sardar@ yahoo.com) 2 I.A.U., Science & Research campus, Iran (e-mail: khan@iaucss.org.com) Abstract. There are limitations for using conventional reliability in real world problems. Hence there is no global theory that can model all types of uncertainties and includes all kinds of causes of uncertainties. Fuzzy set based methods have been proved to be effective in handling multiple types of uncertainties in different fields, including reliability engineering. This paper presents a new approach for the fuzzy reliability based on the use of Beta type distribution as dynamical membership function and considers the effects of different factors on reliability as fuzzy sets. Based on experts' ideas a rule base is designed that can determine the level of reliability of each component by asking linguistic variables from operators. The outputs of the model are fuzzy sets. In order to determine the level of reliability, we calculate sum of squared errors between the calculated memberships and the memberships of any of verbal values for reliability, and we choose the verbal value with the minimum sum of squared error as current verbal reliability. Keywords: fuzzy reliability, Beta distribution, dynamical, membership function. 1 Introduction The concept of fuzzy reliability has been proposed and developed by several authors [Onisawa and Kacprzy, 1995], [Cai et al., 1995]. The conventional reliability is considered under the probability and binary-state assumptions. Cai et al., have given a different insight by introducing the possibility and the fuzzy-state assumptions to replace the probability and binary-state assumptions. According to Cai et al., various forms of fuzzy reliability theories, including profust, posbist, and posfust reliability models, can be considered by taking new assumptions, such as possibility or fuzzy-state, instead of probability or the binary-state assumptions. In the conventional systems, we always give an exactly failed or functioning probability for each component. However, in practice, when the stress or the strength or both of them are fuzzy variables, it is very difficult to compute the exact value for each component. So, currently, by investigating the fuzzy reliability of a system, the researchers always assume that the reliability of each component is a fuzzy variable [Mon and Cheng, 1994], [Utkin and Gurav, 1996], [Utkin et al., 1995]. The design of a fuzzy logic system (FLS) includes the design of a rule base, input scaled factors, output scaled factors, and membership functions. Some studies

have shown that FLS performance is more dependent on membership function design than rule base design [Cordon et al., 2000]. Other studies have discussed rule base design [Cordon et al., 2001], [Procyk and Mamdani, 1976], [Xiax-Tu, 1990]. In this paper, a new approach is introduced for designing dynamical membership functions with Beta type distribution and designing the rule base. Finally we can determine the reliability of each component based on experimental data. Although the probability approach has been applied successfully for many real world engineering reliability problems, there are some limitations to the probabilistic methods especially when the real world data have some uncertainties [Haofu, 2004]. 2 Uncertainty The most important aspects of uncertainty are: type of uncertainty, causes of uncertainty and theory used to model the uncertainty. Different types of uncertainties in reliability engineering are enumerated in table 1 [Haofu, 2004]. Types of Uncertainty Imprecision Incomplete data Engineering Example Failure time, Load simulation in the lab, Measurement accuracy, Modelling in simplification due to distribution proposed, Maintenance, Operational profile, External environment Censored testing data, Lack of data, Suspended test Vagueness Randomness Subjectivity Complexity Material property, Soft failure criteria, Software failure, Human error Operator (customer) description for the malfunction phenomenon, Linguistic description of characteristics of performance such as good, unacceptable, severe, Maintenance Component geometry variation, Material property variation, Loading and variation, Input signal and variation, External environment, Operating environment, Frequency of usage, Measurement error, Component failure Lack of knowledge, Expert judgment, Engineering experience Relationship between system and, components, Interaction between subsystems, Heuristic algorithms Table 1. Uncertainties in Reliability Engineering There is no single theory that can model all types of uncertainties and include all kinds of causes of uncertainties. In recent years the fuzzy set concept [Zadeh, 1978] was introduced to model linguistic-like variables. 3 Developing the membership function An important aspect about fuzzy modelling is designing the membership functions. In this paper, a new approach is introduced to assign more flexible

membership functions under different situations with optimistic, most likely and pessimistic conditions. In fact, it is a dynamical membership function with a Beta type distribution. The Beta distribution shows the optimistic, most likely and pessimistic states, which can be calculated by: Γ ( a ). Γ ( b ) a b β ( x, a, b ) = x (1 x ) (1) Γ ( a + b ) Where "a" and "b" are parameters of Beta distribution function. To be able to use the Beta distribution function as the membership function of fuzzy reliability, equation (1) is normalized. That is, the probabilistic distribution, Eq. (1), is changed to membership function as [Khanmohammadi et al., 2000]: a b x (1 x ) µ ( x ) = β ( x, a, b ) = (2) a a a b ( ) (1 ) a + b a + b The parameters "a" and "b" may be multiplied by a suitable factor α to have appropriate shapes. Fig.1 shows some typical Beta shaped membership functions. Fig. 1. Typical Beta type membership functions (a 1 ) x =.1, pessimistic, (a 2 ) x=.5, most likely, (a 3 ) x=.9, optimistic 4 Dynamical memberships Temporal dynamics can be taken into account in fuzzy models by using timedependent membership functions. In [Virant and Zimic,1996] and [Khanmohammadi and Jassbi, 2004], the first ideas on time-dependent membership functions are introduced by extending the basic definitions about a fuzzy set A with membership µ A (x) towards a time-dependent fuzzy set A(t) with membership µ A (x, t) [Cerrada et al.,2005].

The projection on the plane (µ, t) for a given x is called a dynamical membership function. In this sense, the membership grade of x may be time dependent. Fig.2 depicts the idea behind the definition of dynamic membership functions. Fig.2. dynamic Beta shaped membership functions. a 1 (t=5), a 2 (t=10), a 3 (t=15) In this work, a fuzzy model with dynamical membership functions is proposed by extending eq (2) for determining dynamical levels of failures. u c( t ) s ( t ). c ( t ) (1 u ) (1 c( t )) s ( t ).( 1 c ( t )) µ A (u,t)=β(u,c(t),s(t))= s ( t ). c ( t ) s ( t ).( 1 c ( t )) Where A is any fuzzy set, µ A (u,t) is the array of memberships of elements of u in fuzzy set A, u is the array of elements of universe, c(t) and s(t) are pivot element and shape factor of fuzzy set A at time t respectively. In the introduced dynamic model it is considered that c(t) {.1,.3,.5,.7,.9}. By this way we can generate verbal values for levels of failures as table2. Cf =β (u,.1,.5) F = β (u,.3,.5) Sf = β (u,.5,.5) completely failure failure semi-failure H= β (u,.7,.5) healthy Ch= β (u,.9,.5) completely healthy Table 2. Levels of failures 5 Case study As we know, the reliabilities of components depend on different factors in various conditions. The reliability R i of each component i can be presented as a function of different factors as follow: (3) R i =f(m, E i, E, L) (4) Where R i is reliability of ith component, M is the material of each component, E i is the expert's idea, on the level of failure (It contains linguistic variables such as completely failure, failure, semi-failure, healthy and completely healthy) that can be determined by dynamical membership function with Beta type distribution as

presented by Eq.(3), and E is environment variable such as sound, frequency of vibrations, smell. Sounds of components are used to determine the reliability of components. They can take linguistic variables such as very much, much, medium, a little and feeble, calculated by bell shape membership function: 1 µ A (u,t)=bell(u,c(t),d)= (5) 2 1+ d( u c( t)) Where µ A (u,t) is membership function of each member, d is a parameter that determines shape of function (It is selected 0.0625 by trial and error in this work), u is the array of universe, c is the pivot for fuzzy value A at time t. Vibration, Smell and Lifetime of components can be defined by linguistic variables calculated by bell shape membership function Eq(5). The procedure of application of fuzzy rule base for determining the reliability of component is denoted by the following stages at each time t. Stage1 verbal values Step0. Determine the universes of discourses for failure, sound, smell, vibration, life time and reliability. Step1. Determine linguistic values for different parameters affecting the reliability for different components: levels of failures: sound: smell: vibration: lifetime: reliability: completely failure, failure, semi-failure, healthy and completely healthy. very much, much, medium, little and feeble. very much, much, medium, little and feeble. always, usually, often, seldom and never. very old, old, medium, new, very new. Completely reliable, reliable, rarely reliable, unreliable and completely unreliable. Table 3. Linguistic variables Step2. Determine the centre of gravities and shape factors for different linguistic values, determined on step 1. Step3. Calculate the membership functions of linguistic values at time t, using Eq. (3) for failure, and Eq. (5) for sound, smell, vibration, lifetime, and reliability. Stage2 Rule base If level of failure is a i and level of sound is b i and level of smell is c i and level of vibration d i and level of lifetime e i then the level of reliability r i for different components. Or simply: If S i then r i ==> R i Where S i is the composition of the fuzzy sets of different factors at any particular time with special conditions, r i is level of the reliability of component at the particular time with ith condition and R i is the ith rule at the particular time for ith condition. In this work 50 conditions are considered.

Stage3 Determining final reliability: In this stage, with having the rule base and fuzzy values of factors we determine the levels of reliabilities of components as fuzzy sets at various times. In order to determine exact levels of reliabilities, we calculate sum of squared errors between the calculated memberships and the memberships of any of verbal values for reliability, and we choose the verbal value with the minimum sum of squared error for the current reliability. For example suppose our calculated memberships are, r = 0.2373 0.2373 0.2373 0.3333 0.3333, then we are considering different levels of reliabilities as shown in table 4. The current reliability is "Reliable". Level of reliability (r ) 0.2373 0.2373 0.2373 0.3333 0.3333 sum(r-ri)^2 Completely unreliable Unreliable Rarely reliable Reliable Completely reliable 1.0000 0.3333 0.1111 0.0526 0.0303 0.3333 1.0000 0.3333 0.1111 0.0526 0.1111 0.3333 1.0000 0.3333 0.1111 0.0526 0.1111 0.3333 1.0000 0.3333 0.0303 0.0526 0.1111 0.3333 1.0000 0.7775 0.7283 0.6562 0.5037 * 0.5373 Table 4. Levels of reliability 6 Numerical example We would like to determine the reliability of a component for 4 weeks. At first; we must determine the factors affecting the reliability as inputs of the model. R i =f(m, E i, E, L) Figure 3 shows the membership functions for verbal values of Level of failure, Sound, vibration and smell in 4 weeks. Fig. 3. Membership functions for verbal values of level of failure, Sound, vibration and smell in 4 weeks.

Figure 4 shows the surface plots of final rule and reduced rule, calculated in stage 2 and stage 3. Fig. 4. Surface plots of final rule and reduced rule, calculated in stage 2 and stage 3. After determining the fuzzy values of factors, the composed characteristics of components can be obtained using stage 3 for 4 weeks: First week, µ s1 (x)= 0.0000 0.0642 0.3333 0.1111 0.0000 Second week, µ s2 (x)= 0.0000 0.1111 0.3333 0.0642 0.0000 Third week, µ s3 (x)= 0.0526 0.1111 0.0313 0.0010 0.0000 Forth week, µ s4 (x)= 0.0000 0.0010 0.0313 0.2373 0.3333 By applying these values to rule matrix R, we can determine the levels of reliabilities as fuzzy sets in each week: rw 1 = 0.1111 0.1111 0.1111 0.1111 0.1111 rw 2 = 0.3333 0.3333 0.3333 0.3333 0.3333 rw 3 = 0.0526 0.0526 0.0526 0.0526 0.0526 rw 4 = 0.2373 0.2373 0.2373 0.3333 0.3333 In order to determine the level of reliability of the component as outputs of the model, we calculate sum of squared errors between the calculated memberships for different weeks by verbal values of reliabilities and the memberships of any of verbal values for reliability, and we choose the verbal value with the minimum sum of squared error for the current reliability such as: sse1=sum((rw 1 -Cu).^2)= 0.1347* sse2=sum ((rw 1 -Ur).^2)= 0.9417 sse3=sum ((rw 1 -Rr).^2)= 1.2346 sse4=sum ((rw 1 -Rl).^2)= 1.4407 sse5=sum ((rw 1 -Cr).^2= 1.8586 Reliability of the component is completely unreliable for the first week. Using the same procedure the verbal levels of probabilities can be determined for 4 weeks. 7 Conclusions In this work, a new approach for the fuzzy reliability based on the use of Beta type distribution as dynamical membership function is presented. Based on experts' ideas on the fuzzy effects of factors on reliability a rule base is designed

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