Models and Customer-Centric System Performance Measures Using Fuzzy Reliability ZhaojunLi
Kailash C. Kapur
Industrial and Systems Engineering University of Washington Seattle, WA, 98195, USA
[email protected]
Abstract-This
paper
proposes
a
new
Industrial and Systems Engineering University of Washington Seattle, WA, 98195, USA
[email protected]
perspective
of fuzzy sets. For a continuously degrading system, it is difficult to precisely decide whether the system is in failure or success state. The definition of success and failure are vague and the success state can be considered as a fuzzy state based on using the substitute characteristic as a variable for the degree of membership for success. Higher values ofy result in better system performance and should have higher membership values for success using the theory of fuzzy sets. Thus the membership function defined overy is another way to model the multi-state or continuous state behavior of the system.
and
methodology to model the behavior of the system/component using the theory and methods for fuzzy sets. We use the indicator or performance or substitute variable which is well understood by the customer to fuzzity the states of the system. Thus, the success/failure events are treated as fuzzy sets. Fuzzy reliability is defined and compared to the traditional binary states and multi state reliability modeling. The concept of fuzzy random variable is introduced to model the dynamic behavior of time to fuzzy failure.
Customer-centric
developed examples
based show
system
performance
measures
on
fuzzy
reliability
modeling.
how
fuzzy
reliability
models
are
Numerical
and
Additionally, the fuzzy success membership function needs to be developed through interaction and communication with the customers and system designers. The membership function is a representation of customers' perception of performance levels or degrees of success for the system. Thus fuzzy reliability based on the success membership function is a customer-centric metric for reliability performance measure.
relevant
measures can be applied to evaluate complex system and the advantages over the classic reliability analysis in decision making. K�ords-Fuzzy reliability, substitute characteristics, fuzzy sets, multi-state reliability.
I.
INTRODUCTION
In the literature, the theory of fuzzy sets has been extensively investigated and applied in reliability analysis by treating some elements of the traditional reliability models such as component reliability/probability and failure rate as fuzzy numbers. Through fuzzy arithmetic operations, the resulting system reliability is also a fuzzy number [10, 11 12]. Cai et al. [13] introduced fuzzy reliability by considering system state as fuzzy state and derived fuzzy reliability under the assumption that the fuzzy success event is the standard complement of fuzzy failure event. Shiraishi and Furuta [14] investigated reliability of structures by defming the structure failure in a fuzzy way where the membership function for the failure event is based on the difference between the resistance and the load effect. This is also related to the traditional concept of safety margin. Compared with the existing fuzzy reliability models, the fuzzy reliability definition and model in this paper based on the fuzzy success event which is defmed over the substitute characteristic variable is more general and also does not assume the standard complement relationship between fuzzy success and fuzzy failure events.
It is well known that binary state modeling for the reliability of components as well as of the system is too simplistic and does not capture the reality for most systems which can have many levels of performance. This has been the motivation to use reliability models which consider multi-state systems with multi-state components [1, 2, 3, 4]. It has been well documented how these models with multi-state behavior are superior in terms of capturing the performance of the system and also making sure that the performance measures capture the experience of the customer over time with the system [5, 6, 7]. In this paper, a new perspective and methodology is proposed to model the behavior of the system/component using the theory and methods for fuzzy sets. The acceptable level of functionality for a system is related to some performance or indicator variabley, also called the substitute characteristic [8, 9]. We use the word substitute because this variable y is a means to measure the performance of the system. The functionality of the system is directly related to the values of this technicaVengineering substitute characteristic. To consider the degradable performance of the system, multi-state reliability model generalizes the binary state assumption and partitions the range of the substitute variable y into several groups or intervals with higher values representing higher levels of system performance. Another approach of modeling the continuous degradable performance is to apply the theory
The concept of using the substitute characteristic to develop the membership function for the fuzzy set has not been explored properly in the literature. We propose to use the substitute variable which is very well understood by the customer to fuzzify the states of the system/component. As generalizations of the traditional binary state and multi-state
978-1-4244-6588-0/10/$25.00 ©2010 IEEE 819
reliability models, both component and system fuzzy reliability defmitions and performance evaluation metrics are developed under the fuzzy state consideration. Numerical examples show the benefits of fuzzy reliability models and measures in supporting decision making where traditional binary and multi state models may not work. II. A.
fCy)
o --
In classic reliability modeling, the states of component/system are assumed to be binary. The binary state assumption implies that the success and failure of a component can be precisely determined with respect to a threshold value of the substitute characteristic random variable Y. For "larger the better" substitute characteristic, the component is defmed as failure, if Y < Yo ; the component is defmed as success, if Y � Yo , where Yo is an exact threshold value which differentiates between failure and success (Figure 1). Under this binary state assumption, the precise defmition of success and failure can be expressed as, X=
{O, 1,
failure success
Y < Yo Y � Yo .
Pr[X = 1]
Let U be the universal set (any traditional crisp set whose element may not be real numbers). A fuzzy set can be defmed as,
A = { (y, IlA ( Y)); Y E U},
C. Fuzzy s tate representation by member ship functions
(2)
The success membership function Ils ( Y) in Figure 3 shows how success is defmed in fuzzy reliability modeling under the theory of fuzzy sets. This membership function for the fuzzy event of success is defined over the universal set IR = [O, YT], l where, Ils(y) = O,ifO:S; Y < Yl; 11 s(y) = __(y - Yl)' if
= E[lls ( Y)]· The umeliability of the component is R P[ Y < Yo ] , and R + R = 1. success �
;--____
=
P[X = 0]
=
(Y2-Yl)
/lsCy)
---1 1
----
o
I Yo
Y
(3)
where IlA (y) is called the membership function, which indicates the degree of membership of an element Y belonging to the set of A. Mathematically, IlA (y) is a function that maps the elements of A to the interval [0, 1], i.e., IlA: U -+ [0,1]. Fuzzy numbers are a special family of fuzzy sets which are defmed over the universal set of real number IR . The membership function of a fuzzy number can be expressed as, IlA: IR -+ [0,1]. It is noted that the membership function of a fuzzy set is the generalization of the characteristic function of a crisp set.
= E[X]
failure
Yo .... �
--
B. Basics of fuzzy set theory
(1)
= Pr[ Y � Yo ] = f.Y;;:Yo 1 dF (y)
...
y
Figure 2. Fuzzy state reliability modeling
The above expression can also be regarded as the characteristic function of the crisp set of success, i.e., Ils (y) = 0, if Y < Yo ; and Ils (y) = 1, if Y � Yo . The reliability of a component can be mathematically evaluated as,
fCy)
/ls(y) -:;...:: -�':::--""""-----1 1
STATIC FUZZY RELIABILITY MODELING
Motivation for fuzzy reliability model
R=
membership of success
�
Yl:s; Y < Yz; and Ils( Y) = 1, ifyz:s; y:s; YT. The membership function is a reflection of the customer/designer's cognition to the fuzzy success event. More specifically, ify E [O' Yl ], the component is considered as failure; if Y E (yvyz] , the component exhibits certain degrees of success; and if Y E ( Yz, YT] , the component is considered as success. Such continuous transition from success to failure state quantified using the fuzzy set concept is more realistic for degradable systems and components.
Figure 1. Binary state reliability modeling -------------------
From the fuzzy set theory point of view, the component's success and failure are treated as fuzzy events. For any given value of the substitute characteristicy, the component exhibits a certain degree of success Ils ( Y) as shown in Figure 2. Since it is difficult or umealistic to use a single threshold value to divide success from failure, fuzzy set theory is applied to deal with the fuzzy nature of the success and failure defmition. In other words, the events of success and failure are represented as fuzzy sets in fuzzy reliability modeling.
o
Yl
Y
/lsCy) -"A"'""--� 1
Y2
Figure 3. Fuzzy state representation
820
YT
Fuzzy reliability modeling is customer-centric and has advantages over the traditional reliability modeling. For example, given the probability distribution of the substitute characteristic, for the traditional binary and multi-state reliability modeling, the probability of a system being in success state or one of the multi-states heavily depends on a single threshold value and is very sensitive to this specified value. Such issues also exist for the reliability performance measures which utilize these probabilities. On the other hand, the fuzzy reliability utilizes the membership function of success, and the membership function incorporates customers' understanding and recognition to system/component's reliability performance. Once the membership function is appropriately estimated through interaction with customers and the system designers, the fuzzy reliability evaluation in (4) is more meaningful and realistic.
The multi-state approach (Figure 4) may classify the state of the component as 2 if y E (Yz,YT], state 1 if Y E (Yt,Yz], and state 0 if y E [0,Yt]. Such multi-state classification is an extension of the binary state reliability modeling, and it approximates the continuous performance levels by simplifying and discretizing them with fmite number of state values. Such approximation can also be understood as a special membership function where the range of the membership function values is discrete, i.e., 0, 1, and 2, instead of the continuous interval [0,1]. 2
fey
III.
o o
y
In the traditional dynamic reliability model, for a given failure threshold value of Yo , the time to failure is a random variable X(w) due to variation in the degradation process of individual units (Figure 5). When the system state is considered as a fuzzy event, it exhibits different degrees of failure from [0, 1] over the whole degradation range of substitute characteristic values of [YI ,YT] , where YI is the initial/ideal state of the system and YT is some critical value. The time to fuzzy failure is not only random but also fuzzy. For a given level of system failure degree a, the observed time to fuzzy failure is a random interval [X�(w),X�(w)] . The concept of fuzzy random variable X can be used to model the time to fuzzy failure [18]. Fuzzy random variable can be formally defmed over a probability space as follows: Let (fl,� ,P) be a probability space. A function X:fl -+ �o (R) is called a fuzzy random variable on(fl,� ,P), if for any a E (0,1] and w E fl,
yz
Figure 4. Multi-state classification
The above comparisons of state representation under fuzzy, binary, and multi-state assumptions show that the fuzzy state representation captures more information in terms of the system's continuous performance from the customer and system designer, while both binary and multi-state representations are simplification and discretization of the continuous performance levels. In addition, the fuzzy state representation using membership functions integrates the customer's preferences and judgments to system performance and can provide more meaningful and accurate information for decision making. D.
(5) {x:x E R,X (w)(x) � a} = [X�,X�] is a random interval, i.e., X� and X� are two random variables or measurable with respect to the a-algebra�, where�o (R) is a collection of all bounded and closed fuzzy numbers on R. The
Xu(w)
Fuzzy reliability model
Static fuzzy reliability model is developed in this section. The success and failure are treated as fuzzy events, which contains many elements or substitute characteristic values Y exhibiting different degrees of success or failure. As a natural extension of the traditional binary state reliability evaluation in (2), the fuzzy reliability of a component can be evaluated as the probability of the fuzzy event of success [15], which is uniquely determined by its membership function. Thus we defme fuzzy reliability as:
R
=
Pr[success] = f Ils(y)dF(y) = E [lls (Y)],
DYNAMIC FUZZY RELIABILITY MODEL
=
fuzzy random variable degenerates to a random variable if a = 1 since X� = X�.
(4)
where F(y) is the cumulative distribution function of the substitute characteristic variable y. This defmition was also used in [14] for the special case of reliability of structures. The above defmition based on the work of Zadeh [15] has been generalized to fuzzy probability space by Klement et al. [16]. The fuzzy reliability defmition in (4) degenerates to the classic binary reliability model when the membership functions of fuzzy sets are substituted with the characteristic functions of crisp sets. This fuzzy reliability defmition is also a counterpart to the system's performance measure of state expectation defmition E(X) = L�o i * Pr[X = i] in multi-state reliability modeling [17].
1 o
X;;Cw)
X;Cw)
Figure 5. Fuzzy random variable: time to fuzzy failure
The expectation of a fuzzy random variable which can be expressed by E[X] = (E[X�],E[X�]) is a special fuzzy set, i.e., a fuzzy number. The membership function of the expectation
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model, and the method used for deriving the structure function for binary state system cannot be directly applied in the multi state system. Likewise, the structure function method used in binary state system becomes inapplicable in fuzzy state system. So, we propose the evaluation of fuzzy system reliability as follows,
of fuzzy random variable, i.e., time to fuzzy failure, can be evaluated based on the Resolution Identity [19],
IlE[Xal(t ) = sUPO
