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Then the cofactor and determinant of A, matrix are given in (56) and (57): The next step is to find out the error matrix for A and Ap denoted by E, and E, , respectively. The expressions for these error matrices are given in (59) and (60):

Figure 4 Then the cofactor and determinant of A, matrix are given in (56) and (57): The next step is to find out the error matrix for A and Ap denoted by E, and E, , respectively. The expressions for these error matrices are given in (59) and (60):

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Fig. 1 Fuzzy variable with triangular membership function In general, mathematical definition of a Fuzzy set A is given in (1): In this section, fuzzy mathematics including membership func- tion, a-cut techniques on fuzzy variables and interval arithme- tic will be discussed. In the classical set theory, mathematical operation performs on crisp data. However, FST deals with imprecise data or uncertain data. The degree of impreciseness is defined by membership function on the scale of 0-1. Various types of membership functions such as triangular, trapezoidal, Gaussian, sigmoid, etc. are in general used in the FST. In this study, triangular membership function has been considered to take into account the variation of time-to-failure around a specified value. A graphical representation of a fuzzy variable (X ) with triangular membership function has been shown in Fig. 1. Base of the triangle (J — m — r) is known as support of the fuzzy variable X. In the interval notation fuzzy variable can be represented as [/, m, r]. It means that the value of the membership function at middle point (i.e., m) is unity; how- ever, zero at both left boundary (i.e., 7) and right boundary (.e., r). The value of membership function decides the degree of impreciseness. The value of membership function equal to unity represents the crisp (or precise) value of the fuzzy vari- able X. The degree of impreciseness increases when the value of membership function decreases from unity. It becomes maximum at the support of the fuzzy variable X. The meth- odology used for definition of imprecise variable with fuzzy membership function is known as fuzzification. In veneral. mathematical definition of a Fuzzy set A is given
Fig. 1 Fuzzy variable with triangular membership function In general, mathematical definition of a Fuzzy set A is given in (1): In this section, fuzzy mathematics including membership func- tion, a-cut techniques on fuzzy variables and interval arithme- tic will be discussed. In the classical set theory, mathematical operation performs on crisp data. However, FST deals with imprecise data or uncertain data. The degree of impreciseness is defined by membership function on the scale of 0-1. Various types of membership functions such as triangular, trapezoidal, Gaussian, sigmoid, etc. are in general used in the FST. In this study, triangular membership function has been considered to take into account the variation of time-to-failure around a specified value. A graphical representation of a fuzzy variable (X ) with triangular membership function has been shown in Fig. 1. Base of the triangle (J — m — r) is known as support of the fuzzy variable X. In the interval notation fuzzy variable can be represented as [/, m, r]. It means that the value of the membership function at middle point (i.e., m) is unity; how- ever, zero at both left boundary (i.e., 7) and right boundary (.e., r). The value of membership function decides the degree of impreciseness. The value of membership function equal to unity represents the crisp (or precise) value of the fuzzy vari- able X. The degree of impreciseness increases when the value of membership function decreases from unity. It becomes maximum at the support of the fuzzy variable X. The meth- odology used for definition of imprecise variable with fuzzy membership function is known as fuzzification. In veneral. mathematical definition of a Fuzzy set A is given
Let us assume the multiplication operation on these matrices results in a matrix in the form given in ( 20):
Let us assume the multiplication operation on these matrices results in a matrix in the form given in ( 20):
Then mathematical operations involved in each element are given in (21)-(28):
Then mathematical operations involved in each element are given in (21)-(28):
Other terms important in reliability analysis are reliability function, hazard function, mean time to failure (MTTF), etc. The Reliability hazard function and MTTF will be in the interval form due to a-cut interval of *7. These parameters are defined with a-cut notation as given in (32), (33) and (34), respectively.
Other terms important in reliability analysis are reliability function, hazard function, mean time to failure (MTTF), etc. The Reliability hazard function and MTTF will be in the interval form due to a-cut interval of *7. These parameters are defined with a-cut notation as given in (32), (33) and (34), respectively.
Table 1 Fuzzy failure time cycle data for ball bearing in ascending order Ball bearing failure time cycle data for 20 failure incidences in triangular membership function are considered for the demonstration on the methodology developed for reliability analysis. These data are given in Table |. In the table, the data given as a most likely value of fuzzy triangular variable represent the crisp value of failure time of the ball bear- ing. The crisp value also represents as the precise data. The degree of impreciseness increases when data deviate from
Table 1 Fuzzy failure time cycle data for ball bearing in ascending order Ball bearing failure time cycle data for 20 failure incidences in triangular membership function are considered for the demonstration on the methodology developed for reliability analysis. These data are given in Table |. In the table, the data given as a most likely value of fuzzy triangular variable represent the crisp value of failure time of the ball bear- ing. The crisp value also represents as the precise data. The degree of impreciseness increases when data deviate from
Fig. 3 Fuzzy representation of first failure time
Fig. 3 Fuzzy representation of first failure time
Table 5 Estimated slope and intercept Table 6 Slopes and intercepts for different a values
Table 5 Estimated slope and intercept Table 6 Slopes and intercepts for different a values
Fig.4 Fuzzy representation of slope
Fig.4 Fuzzy representation of slope
exercise has been carried out using the crisp value of failure time data and shown in Fig. 6:
exercise has been carried out using the crisp value of failure time data and shown in Fig. 6:
Fig.6 Linear fitting of crisp data of failure time
Fig.6 Linear fitting of crisp data of failure time
Fig. 7 Variation of B(10) life Fig.5 Fuzzy representation of intercept
Fig. 7 Variation of B(10) life Fig.5 Fuzzy representation of intercept
Table 7 Comparison of estimated model parameters between fuzzy and deterministic method
Table 7 Comparison of estimated model parameters between fuzzy and deterministic method
Table 8 Variation B(X) life ( hour ) for different a values
Table 8 Variation B(X) life ( hour ) for different a values
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MathematicsWeibull distribution
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