An Approach with Fuzzy Arithmetic for Modeling Experts Prior Knowledge in Aging Repairable

AN APPROACH WITH FUZZY ARITHMETIC FOR MODELING EXPERTS’ PRIOR KNOWLEDGE IN AGING REPAIRABLE SYSTEMS

Yeu-Shiang Huang1), Chi-Chung Chang2)

1) National Cheng Kung University 2) Ming Chuan University

Abstract

This paper presents some ideas on the applications of fuzzy sets to decision making for aging repairable systems. A non-homogeneous Poisson process (NHPP) with a power-law intensity function is used in this study. In general, classical Bayesian decision methods presume that future states of nature can be characterized as probability events. However, we do not know what the future will entail so we devises a method to consider experts’ opinions, which are usually the absence of sharply defined criteria, and to develop a fuzzy Bayesian decision process for dealing with such situations. Two cases of the discrimination problem with aging repairable system are studied: (1) fuzzy states and exact information and (2) Fuzzy states and fuzzy information. The fuzzy decomposition and arithmetic derivation of the experts’ opinion are presented to facilitate the development of the Bayesian decision process for aging repairable systems

Keywords: Fuzzy Set; Fuzzy Bayesian Decision; Repairable System

1. Introduction

Human beings are constantly making decisions in the real world. In many situations, decision-making depends on numerous factors which limits human ability and increase difficulties to deal with (Asai and Okuda,1975). In such case, the use of fuzzy arithmetic method may be very helpful in solving the decision making problems of aging repairable systems (Bellman and Zadeh, 1970). In general, most complex systems, which are constructed by collecting more than one part to perform either single or multiple functions, are usually repaired rather than replace after failures, since such systems can be restored to fully implement the required functions by methods other than replacing the entire system

(Ascher and Feingold, 1984). However, the successive times between failures are not necessarily identically distributed, as in renewal processes. More generally, they can become smaller (an indication of deterioration), or conversely larger and larger (an indication of reliability growth) (Barlow and Proschan, 1965). If aging is detected, then the decision of when to overhaul or discard the system, given the costs of repairs and failures, is of fundamental importance. At the time of the decision, the degree of future deterioration, which is likely to be uncertain, is of primary interest for the decision maker. Decision analysis seems to be able to provide methods to deal with such uncertain situation (Freeling, 1984).

However, the decision structure is usually formulated in such a way as to imposingly quantify particular qualitative characteristics on human being as decision rules. The most important of these characteristics is that of human ability to precisely specify numerical values of ends and means in a decision process. These non-fuzzy decision rules are useful; however, they are limited in their applicability to real world situations where nearly all real human decision problems are imprecise, ill-definedness and vagueness (Dompere, 1982). In this paper, a fuzzy method is presented as a natural development from the Bayesian decision approach for aging repairable systems (Huang and Bier, 1998). The use of fuzzy sets in

2 modeling human’s perception of a system is not based on precise mathematical models. In particular, fuzzy set theorem provides another viewpoint to deal with uncertainties, especially when the uncertainty is the absence of sharply defined criteria rather than the presence of random variables.

2. Aging Repairable Systems

In order to model aging in repairable systems, the non-homogeneous Poisson process

(NHPP) was introduced since it seems more plausible for systems consisting of many components (Hartler, 1989). The system failure process is time-dependent and its intensity

function of the failure process is usually assumed to be of the form x   0 h; x, where 0 is the scale factor,  is the aging rate, x is the elapsed time, and h(.) can be any function that reflects the aging process. Suppose we have a system whose failure process is given by a non-homogeneous Poisson process and with a power law intensity function of the form

(Ascher and Feingold, 1984):

1 (x)   0x , 0> 0,  > 0, (2.1) The likelihood function of the first N = n failure times for the case of time-truncated data is given by

n n n -1  L(x1 , x2 ,⋯, xn |  0 ,)   0  (i=1 xi ) exp( 0 xn ) . (2.2) Huang and Bier (1998) proposed a natural conjugate prior distribution for the power law failure model, which is given by

m1 m1 m -1  f ( 0 ,)  K 0  {exp()ym } exp( 0 cym ) . (2.3) Comparing with other approaches, this natural conjugate prior distribution has many desirable properties, which are summarized as follows:

3 (1) the marginal distribution of  is a gamma distribution with parameters m and  , m 1 expectation   , and coefficient of variation CV  .    m

(2) the conditional distribution of 0 given  is a gamma distribution with parameters m  and cym . m c    (3) the expectation of 0 is given by 0    , where zm=ln(ym). m    z m  m (m  1) (4) the coefficient of variation of 0 is given by CV0  1 , where m 2 z m  1  2 and zm=ln(ym).   2z m

These properties provide guidance on how to choose the parameters  , m, ym and c to achieve joint distributions with the desired prior moments. For example, m can be chosen to give the desired value of CV,  can be selected to give the desired value for , ym can be selected to give the desired value for CV0 and c can be selected to give the desired value for

0.

3. Fuzzy Bayesian Decision Process

In real Bayesian decision problems there are two different types of vagueness: states and information (Sylvia,1993). In this section we will discuss decision models for the case in which states and posterior additional information are fuzzified (Roubens,1997)( Stephen and

Donnell,1979). Finally, we will also present the decision flowchart and decision analysis process.

As mentioned in the previous section, Huang and Bier (1999) developed a joint natural conjugate prior distributions for 0 and  . This proposed conjugate prior distribution provide

4 guidance on how to choose the parameters  , m, ym and c by collecting the experts’ knowledge about the prior moments (i.e., , CV, 0, CV0). In other words, in order to apply the joint natural conjugate prior distribution into the decision process, the experts need to specify a specific value for each of these four moments, respectively. However, this is not necessarily realistic for real world cases, since it is usually a tough task for the experts to specify a value to a prior moment with enough confidence. Instead, the experts often think of a prior moment as a fuzzy number, that is, a range of numbers and each number within the range has a different membership function. Furthermore, since the prior moments are not exactly provided by the experts, the parameters (i.e.,  , m, c, and ym) are therefore formed as the functions of these fuzzy numbers. From the properties of the joint natural conjugate prior in the previous section, we can have

1   (  ,CV )  2 , (3.1)  CV 1 m  m(CV )  2 , (3.2) CV y  y ( ,CV ,CV ) m m   0

1 1 1 (3.3)  exp{[ m (CV ,CV )  1  [ m (CV ,CV )  1] m (CV ,CV )]},  0  0  0 and m c  c( ,CV , ,CV )  0   0 0 m    , (3.4)      ln(ym )  CV 2 1 (CV ,CV )  0 where  0 2 . CV  1 Based on the above discussion and equations (3.1) to (3.4), we start the discrimination problem with fuzzy states space F=(, CV, 0, CV0) and exact observation space X = {x}

(Dompere, 1982). Therefore, the fuzzy prior distribution of the fuzzy state Fj for the aging

5 repairable system is given by

f (F )  K  m1 m1{exp()y m } -1 exp( cy  ) j     0 m 0 m  CV  CV   0 0 (3.5)  ( ) (CV ) ( ) (CV )d dCV d dCV Fj  Fj  Fj 0 Fj 0   0 0 and the fuzzy posterior distribution of the fuzzy state Fj and additional exact data for the aging repairable system is given by

f (F j | x1 , x2 ,⋯, xn ) n  K'  mn1 mn1{ exp(  )y m x } -1 exp[  (cy   x  )]     0 m  i 0 m n . (3.6)  CV  CV i1   0 0  ( ) (CV ) ( ) (CV )d dCV d dCV Fj  Fj  Fj 0 Fj 0   0 0 where x1 , x2 ,⋯, xn means we observe additional exact data until the nth failures.

Furthermore, in the part of observe additional fuzzy data, the fuzzy likelihood function of observing the nth failure times is given by

' ' ' Lik(x1 , x2 ,..., xn 0 ,  ) n  ⋯ n  n ( x ')  1 exp( x ') (x ')⋯ (x ')dx '⋯dx ' (3.7)   0  i 0 n 1 1 n n 1 n i1 X1 X n ' ' ' where x1 , x2 ,..., xn means we observe additional fuzzy data until the nth failures. If we

' ' ' observe fuzzy state Fj and additional fuzzy data ( x1 , x2 ,..., xn ), then the posterior distribution will satisfy

' ' ' f (F j | x1 , x2 ,..., xn ) n  K'' ⋯  mn1 mn1{ exp(  )y m x } -1 exp[  (cy   x  )]       0 m  i 0 m n  CV  CV X  X i1   0 0 1 n  (x ')⋯ (x ') ( ) (CV ) ( ) (CV ) 1 1 n n Fj  Fj  Fj 0 Fj 0 (3.8) dx '⋯dx 'd dCV d dCV 1 n   0 0 where K ,K’ and K'' in equations (3.6), (3.7) and (3.8) are normalizing factors.

Figure 3.1 shows a fuzzy Bayesian analysis in aging repairable system with experts’

6 prior knowledge.

Fit in with the assumption of NHPP? No Yes Fuzzy states space？

Expert provides exact parameters Expert provides fuzzy parameters ' ' ' ' E0 ,CV 0 , E,CV E 0 ,CV 0 , E ,CV 

Exact states of prior distribution Fuzzy states of prior distribution 

f (0 ,  ) (0 ,  ) No Yes satisfied？

Collecting additional No Making a decision by information? prior distribution Yes

Fuzzy posterior Yes information? No

Non-fuzzy posterior Fuzzy posterior   distribution f (0 ,  ) distribution (0 ,  )

Making a decision by End posterior distribution

Figure 3.1 Fuzzy Bayesian Decision Flowchart with Aging Repairable System

Now, the basic elements of a Bayesian decision analysis for an aging repairable system are as follows:

(a) Parameter space :{(0,b)| 0>0}.

(b) Action space A:{a1,a2}, where a1 is the status quo, and a2 is the risk reduction action.

7 (c) Loss function L: a real function defined on ´A. If we decide to keep the system

operating, then the loss we face is L(,a1); if we decide to take the risk reduction

action, then the loss we face is L(,a2).

(d) Sample space X: The additional information available to be collected (e.g.,

successive failure times). The cost of collecting this additional data or information

should also be reflected in the decision process.

We assume (i) that the system's status after a repair is essentially the same as it was immediately before failure occurred (as good as old); and (ii) that the repair times can be neglected. The following terminology will be used in the decision analysis process:

CF: the cost of a failure if it occurs.

CR: the cost of the proposed risk reduction action.

CI: the cost of collecting additional information.

: the reduction in failure rate that would result from the proposed risk reduction action

T: the time horizon under consideration.

t: the time at which the decision is being made.

: the expected number of failures during the time period [t,T] under the status quo.

Suppose that the repairable system has a planned lifetime (i.e., time horizon) T, and the decision of whether to maintain the status quo or perform some risk reduction action must be made at time t. The decision variable we are dealing with is then the expected number of failures during the time period [t,T]. Since the system failure times are assumed to be drawn from a non-homogeneous Poisson process with power law intensity function, the expected number of failures in [t,T] under the status quo is given by

8 T T 1   (T,t,0, )= t (s)ds  t λ 0β s ds =0(T -t ). (3.9)

Suppose that the risk reduction action will reduce the failure intensity by a fraction r, where 0 <  < 1. Then the expected number of failures in [t,T] if the risk reduction action is performed is given by

T t (s)(1 )ds =(1- )  . (3.10)

On the basis of the assumptions given above, we therefore have a two-action problem with a linear loss function, where the loss for taking action a1 (i.e., continuing with the status quo) is CF and the loss for taking action a2 (i.e., undertaking the risk reduction action) is

C F (1-) + C R . The expected loss for the status quo is simply CFE{ }, and the expected

loss for the risk reduction action is C F (1-ρ )E{} + C R . Since the fuzzy prior and posterior density functions for  are available by using fuzzy arithmetic transformation for equations

(3.5) to (3.8), the fuzzy prior and posterior mean values of  can be evaluated. Therefore,

Fuzzy Bayesian decision analyses can be performed by comparing the fuzzy prior and posterior mean values of  with the cutoff value C=CR/(CF). If the relevant mean is smaller than C, then we should keep the system operating as in the status quo; if not, then we should perform the risk reduction action.

4. Fuzzy Arithmetic

In this section we will examine the fuzzy aggregation operators in two ways: through

 ,CV ,  ,CV fuzziness of prior parameters   0 0 and through fuzziness of failure data record. However, a fuzzy number is not a measurement. In other word, a fuzzy number is a

9 subjective valuation assigned by one or more human operators. In order to transfer the subjective valuation into real valuation. We have to use the concept of fuzzy of two dimensions and the fuzzy aggregation method , which are introduced as below:

If we consider a fuzzy graph G  R  R ,that is, a functional fuzzy relation in R 2 such

2 that its membership function G (x, y) ,(x, y)  R , G (x, y) 0,1,has the following properties:

1. x0  R, G (x0 , y) 0,1 (4.1)

2. y0  R, G (x, y0 ) 0,1 (4.2) both(4.1) and (4.2) are convex membership function.

3. 0,1 G (x, y)   (4.3)   cut  is a convex surface. 2 4. (x2 , y2 )  R ,  G (x, y)  1 (4.4) If conditions (4.1) ~ (4.4) are satisfied, the fuzzy subset G  R 2 is called a fuzzy number of dimension 2 and Figure 4.1 shows such a fuzzy number.

 1 X  0 

X (X 2 ,Y2 )  0

 Y (X 2 ,Y2 )

Figure 4.1 A Fuzzy number of dimension 2. Figure 4.2   cut of two-dimensional fuzzy number

According the pyramidal-shape in Figure 4.1, we now consider fuzzy pyramidal number in R 2 with edges having rectangular bases parallel to the axis x and y, as shown in Figure

4.2. Furthermore, the fuzzy arithmetic rule as follows:

10 1. (x),(y),(z)  R 2  (z)  ( (x)   (y)) (4.5) Dc  Da Db (z)(x)( y) 2. (x),(y),(z)  R 2  (z)  ( (x)   (y)) (4.6) Dc  Da Db (z)( x)( y) 3. (x),(y),(z)  R2  (z)  ( (x)   (y)) (4.7) Dc  Da Db (z)(x)( y)  we use   cut to express   cut to avoid reduplication. 4. (x),(y),(z)  R 2  (z)  ( (x)   (y)) (4.8) Dc  Da Db (z)(x):( y) 5. (x),(y),(z) R 2 where A( )  a( ) ,b( )  R  A1( )  1/b( ) ,1/ a( )  (4.9) ( ) ( ) 6. e x (A( ) )  [e x / b ,e x / a ] where ( ) ( ) ( )  x  [x1 , x2 ]  R ( ) x( ) x( ) (4.10) e x  [e 1 ,e 2 ] These equations (4.5)~(4.10) can be adapted to subtraction, multiplication, division, and maximum and minimum operations.

The development of the approach of fuzzy arithmetic has been presented as a natural extension of the theory of non-fuzzy arithmetic through canonically set-theoretic representations. First of all, we will start our investigations with fuzzy states and consider the

 ,CV ,  ,CV membership functions of the   0 0 where provided by experts with the fuzzy

 l, L i   ,CV ,  ,CV interval data i  ,   0 0 and i  R . Second, we use the operation of max- min convolution method transformed non-normalize membership into normalization membership. Therefore, we will get the normalized triangular membership function on the

11 axis fuzzy parameters. Subsequently, let us consider the importance weights of the prior parameters are assessed in linguistic terms represented by fuzzy numbers, such as“VL”(Very

Low)、“L”(Low)、“M”(Medium)、“H”(High)、“VH”(Very High), and the membership functions of the five linguistic terms are shows in Figure 4.3.

1.0 VL L M H VH

Linguistic Weight 0 1 3 5 7 9 10

Figure 4.3 Membership functions for linguistic weighting values

More generally, we can carry out these operators (4.5)~(4.10) on domain by use the fuzzy extend method and shown in Figure 4.4 (Kaufmann and Gupta, 1991). We now consider the () () () () 2 CV  CV  rectangular-shape A , B ,C , D in R with   level to indicate  ,  , 0 , 0 , respectively (Figure 4.5) and by applying the fuzzy arithmetic approach transfer the functions (3.1)~(3.4) into (4.15)~(4.18).

M.F 1.0

c (x( ) , y ( ) ) c (x( ) , y ( ) ) 4 c4 c4 3 c3 c3 Fuzzy Parameters

 0 a (x ( ) , y ( ) ) a (x ( ) , y ( ) ) 4 a4 a4 3 a3 a3 c (x( ) , y ( ) ) c (x( ) , y ( ) ) 1 c1 c1 2 c2 c2  

( ) ( ) ( ) ( ) a (x , y ) a (x , y ) ( ) ( ) ( ) ( ) 1 a1 a1 2 a2 a2 d (x , y ) d (x , y ) 4 d4 d 4 3 d3 d3

CV 0 b (x( ) , y ( ) ) b (x ( ) , y ( ) ) 4 b4 b4 3 b3 b3 d (x ( ) , y ( ) ) d (x( ) , y( ) ) 1 d1 d1 2 d2 d2 CV Linguistic b (x( ) , y ( ) ) b (x ( ) , y ( ) ) Weight 1 b1 b1 2 b2 b2

Figure 4.4 Sum of two fuzzy pyramidal numbers

12 Linguistic Weight

C () A()

B()

D()

Fuzzy Parameters CV  CV  Figure 4.5 The rectangular coordinate of  ,  , 0 , 0 under the   cut

A()  [(x  , y  ),(x  , y  ),(x  , y  ),(x  , y  )] (4.11) a1 a1 a2 a2 a3 a3 a4 a4 B ()  [(x  , y  ),(x  , y  ),(x  , y  ),(x  , y  )] (4.12) b1 b1 b2 b2 b3 b3 b4 b4 C ()  [(x  , y  ),(x  , y  ),(x  , y  ),(x  , y  )] (4.13) c1 c1 c2 c2 c3 c3 c4 c4 D()  [(x , y  ),(x , y  ),(x , y  ),(x , y  )] (4.14) d1 d1 d2 d2 d3 d3 d4 d4

Then  , m, ym and c can be transferred shown in (4.15)~(4.18)

2() () 1 (4.15)   A  B  2   CV 1 1 1 1 1 1 1 1 (4.16) m  [( , ),( , ),( , ),( , )] x 2 y 2 x 2 y 2 x 2 y 2 x 2 y 2 a1 a1 a2 a2 a3 a3 a4 a4 2  1  1   1     x2 1 x2 1 y2    x ln( c2 ) x ln( c2 ) y ln( c2 )   a2 2  a2 2   a2 2   xa 1 xa 1 ya  e 2 e 2 1, e 2                          2  1  1   1     x2 1 x2 1 y2    c1  c1  c1    xa ln( )  xa ln( )   ya ln( )    1 x2 1 1 x2 1 1 y2   e a1 e a1 1, e a1      (4.17)              1        y    e{  ln } m  2  2  1  1   1    x2 1 x2 1 y2   c4  c4  c4   xa ln( )  xa ln( )   ya ln( )    4 x2 1 4 x2 1 4 y2  e a3 e a3 1, e a3                      2   1  1   1    x2 1 x2 1 y2   c3  c3  c3  xa ln( )  xa ln( )   ya ln( )   3 x2 1 3 x2 1 3 y2  e a4 e a4 1, e a4                    

13 m 0 (4.18) c  m        ln(ym ) 

In the part of fuzziness failure data, we also use the concept of fuzzy interval and consider the linguistic weight. From a practical point of view, we will calculate that hour to substitute for a date in collected failure data record and show it in Figure 4.6.

Linguistic Y Weight

X/hr. Failure Data Figure 4.6 The relationship of fuzzy data and linguistic weight

5. Application

Usually, in a quantitative setting, the information is expressed by means of numerical values(Crow, 1974). However, when we work in a qualitative setting, that is, with vague or imprecise knowledge, the information cannot be estimated with an exact numerical value

Herrera and Herrera,2000). In that case, a more realistic approach may be used to linguistic assessments instead of numerical values, that is, to suppose that the variables which participate in the problem are assessed by means of linguistic terms (Hisdal, 1984) (Tong and

Bonissone, 1984). This approach is appropriate for a lot of problems, since it allows a representation of the information in a more direct and adequate form if we are unable to express it with precision. In this section we will show the discrimination problems with both fuzzy states and exact information and exact states and fuzzy information, which are studied from the viewpoint of fuzzy arithmetic measures.

14 Example 1. Fuzzy States and Exact Information

First, assume that the four fuzzy parameters states spaces and the linguistic importance weight of each parameters assigned by experts is shown in Table 5.1.

Fuzzy interval Linguistic The measure indexes of aging system value Weight

   2,5,7 The expected value 0 0 L1,3,5 Initial failure rate 0 CV  cv 1,2,3 Coefficient of variation 0 0 M 3,5,7   6,7,8 The expected value   M 3,5,7 The aging rate  CV  4,5,6 Coefficient of variation  cv H5,7,9

Table 5.1 The fuzzy prior parameters provided by experts

Furthermore, we use the fuzzy extend method with applying formula (4.5)~(4.10), then the fuzzy number can be defuzzified into a crisp value (Figure 5.1). Finally, equations (3.5) and (3.6) can be used to study the prior and posterior decision for the decision maker when dealing with the decision problem for aging repairable system.

Membership Function 1.0

m R Fuzzy parameter

1 1 m  1 1 ( 2(0.5) , 2(0.5) ) ( , ) x y 2(0.5) 2(0.5) a3 a3 x y a4 a4 B 1 1 1 1 ( , ) ( , ) x2(0.5) y 2(0.5) x2(0.5) y 2(0.5) Linguistic a1 a1 a2 a2 Weight Figure 5.1 The defuzzied point m of membership function m with   cut  0.5

15 Example 2. Fuzzy States and Fuzzy Information

If the addition information is fuzzy, then the functional value of the likelihood function

Lik(x1 ', x2 ',..., xn ' 0 ,  ) is fuzzy, whereas the functional value of the prior is also a fuzzy number. Therefore, beside the work for the fuzzy prior as described in the previous example, we have to take the failure data of the system and show it in Table5.2. Further, we also use fuzzifer method mentioned in section 4 and present it with Table5.3.

Table 5.2 Failure Data of system In-Service Data Observation Period Failure Dates 1973.5.1 1988.1.1 1989.7.6 Failure Dates Fuzzy intervalto values/Hour 1989.10.23Linguistic Weight 1990.12.31 1990.1.12 1989.07.06 ( D ) 8,10,12 L1,3,5 1 1990.9.8 1989.10.23 ( D2 ) 10,12,14 1990.11.14M 3,5,7 D 14,16,18 M 3,5,7 1990.01.12 ( 3 )     Table 5.3 1990.09.08 ( D ) 8,10,12 H5,7,9 The 4 fuzzy 1990.11.14 ( D5 ) 13,15,17 H5,7,9 posterior parameters provided by experts

Membership Function 1.0

c c 4 3 Fuzzy Parameters

a4 a3 c1 c2 D 1 a2 e4 e3 a1

e1 e2 b4 b3

d4 d 3 D2 D4 Linguistic b1 b2 Weight d1 d2

Figure 5.2 Sum of fuzzy failure data and weight pyramidal numbers

16 By way of the fuzzy extend formula (4.5)~(4.10) and show it in Figure 5.2, then the fuzzy number can be defuzzified into a crisp value (with   cut  0.5) and form the likelihood function. Finally, equations (3.7) and (3.8) can be used to study the prior and posterior decision for the decision maker when dealing with the decision problem for aging repairable system.

In this application, for each fuzzy state the fuzzy prior and posterior moments can be computed by fuzzy arithmetic approach to support decision makers for dealing with the decision problem in aging repairable systems. Further investigation is needed for the completion of this study.

5. Conclusion

In this paper, we have presented a new method to solve the decision problem of aging repairable systems and we also present an example to illustrate the fuzzy arithmetic approach for modeling experts’ prior knowledge in aging repairable systems. However, this decision process is useful in selecting the best alternative when the aging repairable system associated with alternatives are known in terms of linguistic variables (Zadeh, 1975a, 1975b, 1975c), in particular, when these linguistic variables can be modeled by fuzzy numbers. In real world situations, the aging phenomena are usually expressed as some degrees of severity. In such case, the proposed decision process can provide more realistic solutions.

In this paper, we have assumed that the importance weights of different criteria are assessed in linguistic terms represented by triangular fuzzy numbers. However, there are still several limitations of this paper and further study may undergo by considering other kinds of

17 fuzzy membership function, since it still leaves lots of space for extension.

Reference

1) ASAI, K., TANAKA, H., AND OKUDA, T. (1975) “Decision Making And Its Goal In A Fuzzy Environment”, In Zadeh et al (eds) Fuzzy Sets and Their Applications to Cognitive and Decision Processes, New York, London.

2) ASCHER, H. E. AND FEINGOLD, H. (1984) Repairable Systems Reliability, New York: Marcel Deker.

3) BELLMAN, R. AND ZADEH, L. A. (1970)“Decision Making In A Fuzzy Environment”, Management Science, Vol. 17, pp. B-141-164.

4) BARLOW, R.E AND PROSCHAN, F. (1965) Mathematical T heory of R eliability, Wiley, New York

5) CROW, L. H. (1974) “Reliability Analysis For Complex Repairable Systems” in: Proschan, F. and Serfling, R. J. (eds), Reliability and biometry , Philadelphia, Society for Industrial and Applied Mathematics, pp. 379-410.

6) Dompere, K.K. (1982.) “The Theory Of Fuzzy Decisions”, Approximate reasoning in decision analysis M.M. Gupta and E. Sanchez (eds.), North-Holland Publishing company, pp. 365-379.

7) FREELING, A. N. S. (1984) “Possibilities Versus Fuzzy Probabilities – Two Alternative Decision Aids”. In Zimmermann et al (eds.) Fuzzy Sets and Decision Analysis, Amsterdam, New York, Oxford, pp. 67-82.

8) HARTLER, G. (1989) “The Non-Homogeneous Poisson Process-A Model For The Reliability Of Complex Repairable Systems”, Microelectronics and Reliability, Vol. 29, No. 3, pp.381-386.

9) HERRERA,F AND HERRERA,V (2000) “Linguistic Decision Analysis: Steps For Solving Decision Problems Under Linguistic Information”, Fuzzy Sets And System, Vol. 115, No. 3, pp. 62-82.

10) HISDAL,E. (1984) “Decision Based On Statements In Natural Language”, TIMS/Studies in the management Sciences, Vol.20, pp. 357-381

11) HUANG, Y.-S.(1996) “Non-Homogeneous Poisson Process With Linear, Power Law, Or

18 Exponential Intensity Functions For Modeling Aging In Repairable System”, a dissertation of doctor of philosophy ,University of Wisconsin-Madison.

12) HUANG, Y.-S. and Bier, V.M.(1998) “A Natural Conjugate Prior for The Non- Homogeneous Poisson Process With a Power Law Intensity Functions”, Communications in Statistic-Simulations and Computation. Vol.27, No.2, pp.525-551.

13) HUANG, Y.-S. and Bier, V.M.(1999) “A Natural Conjugate Prior for The Non- Homogeneous Poisson Process With a Exponential Intensity Functions”, Communications in Statistic-Simulations and Computation. Vol.28, No.6, pp.1479-1509.

14) KAUFMANN, A. and GUPTA,M.(1991) Introduction Fuzzy Arithmetic, Thomson.

15) ROUBENS, M.(1997) “Fuzzy And Decision Analysis”, Fuzzy Sets and System, Vol. 90, pp.199-206.

16) STEPHEN, R. W. AND DONNELL, M. L. (1979) “Fuzzy Decision Analysis”, IEEE Transactions on systems, man, and cybernetics, Vol. SMC-9, No.1, January.

17) SYLVIA FRUHWIRTH-SCHNATTER (1993) “On Fuzzy Bayesian Inference”, Fuzzy Sets and System, Vol. 60, pp. 41-58.

18) TONG, R.M. and BONISSONE,P.P.(1984) “Linguistic Solutions To Fuzzy Decision Problem”, T IMS/studies in the management Sciences, Vol. 20, pp. 323-334.

19) ZADEH, L.A. (1975) “The Concept of a Linguistic Variable and Its Application to Approximate Reasoning – I”, Inform ation Sci ence. Vol. 8, pp. 199-249.