Considering that traditional failure trees have less historical failure data for basic events in practical applications, fuzzy logic is introduced. Using fuzzy numbers to represent the failure probability of each basic event, the establishment of a fault tree is no longer dependent on a large number of failure data. In the T-S fuzzy fault tree, the degree of each fault is usually represented by the fuzzy number in the interval [0, 1].

The trapezoidal membership function μ ( x ) , shown in Figure 2, is used as the membership function of fuzzy numbers.

μ ( x ) = ( m 0 , a l , b l , a r , b r ) (1)

The membership function expression of Figure 2 as follows:

μ ( x ) = { 0   0 ≤ x ≤ m 0 − a l − b l x − ( m 0 − a l − b l ) b l   m 0 − a l − b l < x ≤ m 0 − a l 1     m 0 − a l < x ≤ m 0 + a r m + a r + b r − x b r m 0 + a r < x ≤ m 0 + a r + b r 0       m 0 + a r + b r < x (2)

where m 0 is the center of a fuzzy number support set, a l and a r is supporting radius, b l and b r is area of fuzzy.

According to Figure 2, we can see, when a l = a r = 0 , trapezoid membership function is a triangle membership function; when b l = b r = 0 fuzzy number as a definite number.

The T-S fuzzy model is composed of a series of IF-THEN fuzzy rules. It is a nonlinear model, which is used to describe the association between events and form a T-S fuzzy gate. The rules of the model are expressed as follows [7] :

If x i ( i = 1 , 2 , ⋯ , n ) is basic event variable, y is superior variable, F l j ( j = 1 , 2 , ⋯ , n ) is fuzzy set, and the rules is l ( l = 1 , 2 , ⋯ , m ) ; if x 1 is F l 1 , x 2 is F l 2 , ⋯ , x n is F l n , then y is y l . If the membership function of the fuzzy set is μ F l j ( x j ) , then the output of T-S model as follow:

y = ∏ j = 1 n μ F l j ( x j ) / ∑ l = 1 m ∏ j = 1 n μ F l j ( x j ) y l (3)

So basic events and superior events T-S fuzzy gates are shown in Figure 3.

If the fuzzy number: { x 1 1 , x 1 2 , ⋯ , x 1 k 1 } , { x 2 1 , x 2 2 , ⋯ , x 2 k 2 } , ⋯ , { x n 1 , x n 2 , ⋯ , x n k n } and { y 1 , y 2 , ⋯ , y k y } indicates the degree of failure of the bottom event and the superior event respectively. Their range of values as follow:

{ 0 ≤ x 1 2 < x 1 2 < ⋯ < x 1 k 1 ≤ 1 0 ≤ x 2 1 < x 2 2 < ⋯ < x 2 k 2 ≤ 1                               ⋮ 0 ≤ x n 1 < x n 2 < ⋯ < x n k n ≤ 1 0 ≤ y 1 < y 2 < ⋯ < y k y ≤ 1 (4)

The T-S fuzzy gate algorithm is described as follows:

The rules is l ( l = 1 , 2 , ⋯ , m ) ; if x 1 is x 1 i 1 , x 2 is x 2 i 2 , ⋯ , x n is x n i n , then the probability, y is y 1 , is P l ( y 1 ) , the probability, y is y 2 , is P l ( y 2 ) , ⋯ , the probability, y is y k y , is P l ( y k y ) . Where i 1 = 1 , 2 , ⋯ , k 1 , i 2 = 1 , 2 , ⋯ , k 2 , ⋯ , i n = 1 , 2 , ⋯ , k n . The total number of rules is m, we can get the equation of m:

m = ∏ i = 1 n k i (5)

Assuming that the degree of failure of a basic event is P ( x 1 i 1 ) , P ( x 2 i 2 ) , ⋯ , P ( x n i n ) . Then the possibility of the rules l ( l = 1 , 2 , ⋯ , m ) execution as follows:

P 0 l = P ( x 1 i 1 ) P ( x 2 i 2 ) ⋯ P ( x n i n ) (6)

Therefore, the ambiguity of the superior incident is:

{ P ( y 1 ) = ∑ l = 1 m P 0 l P l ( y 1 ) P ( y 2 ) = ∑ l = 1 m P 0 l P l ( y 2 )                     ⋮ P ( y k y ) = ∑ l = 1 m P 0 l P l ( y k y ) (7)

If the x i ( i = 1 , 2 , ⋯ , n ) fault level is x ′ = { x ′ 1 , x ′ 2 , ⋯ , x ′ n } . According to the T-S fuzzy model, the fuzzy possibility of the fault level of the upper event can be calculated as follows:

{ P ( y 1 ) = ∑ l = 1 m β l ∗ ( x ′ ) P l ( y 1 ) P ( y 2 ) = ∑ l = 1 m β l ∗ ( x ′ ) P l ( y 2 )                       ⋮ P ( y k y ) = ∑ l = 1 m β l ∗ ( x ′ ) P l ( y k y ) (8)

where

β l ∗ ( x ′ ) = ∏ j = 1 n μ x j i j ( x ′ j ) / ∑ i = 1 m ∏ j = 1 n μ x j i j ( x ′ j ) (9)

Additionally, μ x j i j ( x ′ j ) is the degree of membership of the fuzzy set corresponding to the fault status of the j component in the l rule.

Therefore, taking Figure 1 as an example, if the fuzzy possibility of the degree of failure of the basic event is known, according to the T-S gate rule, the fuzzy possibility of the failure degree of the upper event can be estimated using formulas and formulas, respectively.

The T-S fuzzy importance degree 1 is the contribution of the failure of the minimum cut set of a component or system to the occurrence probability of the top event. It is a function of the time. The reliability parameters of the component and the system structure are widely used in practice [8] . The fuzzy subset of the failure probability of the known component x j fault status x j i j ( j = 1 , 2 , ⋯ , n ) is P ˜ x j i j , It’s membership function is μ P ˜ x j i j ; The fuzzy subset of the failure probability of the top event T fault status T q is P ˜ ( P ˜ x 1 i 1 , P ˜ x 2 i 2 , ⋯ , P ˜ x n i n ) , where ( i 1 = 1 , 2 , ⋯ , k 1 , i 2 = 1 , 2 , ⋯ , k 2 , ⋯ , i n = 1 , 2 , ⋯ , k n ), It’s membership function is μ P ˜ x j i j .

Definition: The fuzzy subset of the failure probability of component x j with failure status x j i j ( j = 1 , 2 , ⋯ , n ) is P ˜ x j i j . It’s T-S fuzzy importance for the system top event T is obtained by following equation:

I T q F u ( x j i j ) = E [ P ( T q , P ˜ x j i j = 1 ) − P ( T q , P ˜ x j i j = 0 ) ] = ∫ 0 1 x μ P ˜ x j i j , T q d x ∫ 0 1 μ P ˜ x j i j , T q d x − ∫ 0 1 x μ P ˜ x j i j , 0 d x ∫ 0 1 μ P ˜ x j i j , 0 d x (10)

where P ( T q , P ˜ x j i j = 1 ) indicate fuzzy subset, when P ˜ x j i j is 1, The probability of failure of its top event T. P ( T q , P ˜ x j i j = 0 ) indicate fuzzy subset, when P ˜ x j i j is 0, The probability of failure of its top event T. Combining Equations (6) and (7) with 0 and 1 instead of P ˜ x j i j , the two integral terms represent the center-of-gravity values of the fuzzy subset of the probability of failure of the system top event T failure degree T q when P ˜ x j i j is 0 and 1, respectively.

Definition: The following formula is expressed T-S fuzzy importance of basic events:

I T q F u ( x j ) = ∑ i j = 1 k ′ j I T q F u ( x j i j ) / k ′ j (11)

where, k ′ j indicate the number of non-zero fault conditions for the j part. If the fault status fuzzy number is represented by (0, 0.5, 1), then k ′ j is 2. I T q F u ( x j ) is the influence of each fault state on the average fault state of the system during the change of component fault status from 0 to 1.

Establish T-S fuzzy fault tree [9] [10] as shown in Figure 4, x 1 , x 2 , ⋯ , x 16 is basic events, T is top event, y 1 , y 2 , y 3 is intermediate events. According to the actual fault condition, suppose that there are three kinds of states in the degree of failure of the basic fault and intermediate event of the fuzzy fault tree of the automatic transmission T-S (Table 1): no fault, minor fault, and complete fault, which are (0, 0.5, 1), respectively. The parameter is a l = a r = 0.2 , b l = b r = 0.3 .

T-S fuzzy gate rule algorithm can be obtained from expert experience and historical data in Tables 2-4. In Tables 2-4, each line represents a fuzzy rule, for example, the second line in Table 2 represents the rule. The degree of failure x₁ is 0, x₂ is 0, x₃ is 0, x₄ is 0, x₅ is 0.5, then the possibility of y₁, when its degree of fault is (0, 0.5, 1), is (0.3, 0.4, 0.3).

According to Tables 2-4, the fuzzy possibility of a superior event fault can be calculated by the fuzzy possibility or the degree of failure of the basic events, and then the fuzzy possibility of the top event can be estimated.

In practical engineering appraisal applications, weighted average expert surveys are often used to obtain inaccurate or incomplete failure data [11] [12] . To study the application of T-S fuzzy fault tree in transmission slip diagnosis. This article intends to obtain the fuzzy failure probability of the basic event through the expert survey method based on the revision of expert confidence index. The data of the experts’ selection, rating and experts’ judgments are all taken from the literature [11] . Experts are selected according to academic qualifications, length of service and other conditions. The experts involved in the investigation come from first-line management, maintenance and design personnel. The composition of the experts is shown in Table 5. Calculation formula such as formula (11)

ω r = ν r / ∑ r = 1 4 ( m r v r ) (11)

Ordinary expert survey methods generally assume that all experts have full confidence in their own judgments, but they are not practical when applied. In order to improve the accuracy of the fuzzy failure probability, this article introduces the “confidence index method” of the literature [11] to further correct the data obtained by the common expert survey method. Table 6 is the survey table of failure rates of basic events in the fault tree of the automatic transmission shift failure. The probability range is based on incomplete statistical data and is combined with personal work experience. Fill in the confidence index and select the confidence index for yourself. It is 0.1 ~ 1.

Based on the expert survey method modified by the confidence index and using the weighted average idea, the overall evaluation process for the probability of a certain basic event by all experts is as follows.

1) If the number of experts participating in the survey is m. There are N basic events. According to Table 5, the calculation weight of the ith expert is w i , and he judges the probability of occurrence of the jth basic event as [ L i j , R i j ] , and the confidence index of judging himself as K i j ( 0 < K i j ≤ 1 ) . The confidence index is 1, indicating that the expert has full confidence in his judgment and high credibility; and the confidence index is 0.1, indicating that the credibility of the expert’s judgment result is very low.

2) The interval of the probability interval of the expert judgment is obtained by Δ i j = R i j − L i j .

3) If m i j = Δ i j / 2 , then the contribution of the ith expert to the final cumulative result of the jth basic event occurrence probability is:

P i j = w i [ m i j − ( 1 − k i j ) Δ i j , m i j , m i j + ( 1 − k i j ) Δ i j ] (12)

4) The probability of the occurrence of the jth basic event, the final result of the confidence index correction and the m expert weight accumulation, such as the follow Equation (13)

p j = ∑ i = 1 m p i j (13)

p j is an fuzzy number of the isosceles triangular.

According to the statistical analysis of the automatic transmission slippage fault data and the expert survey with the confidence index correction, the reliability data of the components are shown in Table 6.

According to Table 6, the failure rate of these components is the failure rate when the failure state is 0.5, and the failure rate assuming that the failure level is 0.5 is equal to the failure rate when the failure degree is 1. Combining Equations (6) and (7) with MATLAB, the likelihood of an intermediate event blurring is as follows:

P ( y 2 = 0.5 ) = ∑ l = 1 81 p 0 l p l ( y 2 = 0.5 ) = p 0 1 p 1 ( y 2 = 0.5 ) + p 0 2 p 2 ( y 2 = 0.5 ) + ⋯ + p 0 81 p 81 ( y 2 = 0.5 ) = ( 4.21 , 5.66 , 6.32 ) × 10 − 7

P ( y 2 = 1 ) = ∑ l = 1 81 p 0 l p l ( y 2 = 1 ) = p 0 1 p 1 ( y 2 = 1 ) + p 0 2 p 2 ( y 2 = 1 ) + ⋯ + p 0 81 p 81 ( y 2 = 1 ) = ( 13.67 , 15.23 , 16.38 ) × 10 − 7

Similarly, the fuzzy possibility of different states of other intermediate events can be calculated. We can know in Table 7.

In the same way, the likelihood of the top event being calculated is as follows:

P ( T = 0.5 ) = ( 18.113 , 20.325 , 23.569 ) × 10 − 7

P ( T = 1 ) = ( 38.863 , 46.729 , 55.686 ) × 10 − 7

The calculation results show that the probability of failure of automatic transmission slipping is the same order of magnitude as the probability of failure of each component. The possibility of failure of the T-S fuzzy fault treetop event is far greater than the possibility of failure of each component. This result is in accordance with the actual situation, which verifies the accuracy and feasibility of the T-S fuzzy fault number in the automatic transmission fault analysis process.

Take the gravity value of each event as its failure probability, and according to formula (10), obtain the T-S fuzzy importance degree when y is 0.5 when component x is 0.5, as follows:

I 0.5 F u ( x 6 1 ) = E [ P ( 0.5 , P ˜ x 6 1 = 1 ) − P ( 0.5 , P ˜ x 6 1 = 0 ) ] = 0.166

Similarly, the T-S fuzzy significance of the fault states of the components of 0.5 and 1 can be obtained as shown in Table 8.

According to Equation (11), combined with the T-S fuzzy importance of the fault states 0.5 and 1, the T-S fuzzy significance at 0.5 is obtained.

I 0.5 F u ( x 6 ) = [ I 0.5 F u ( x 6 0.5 ) + I 0.5 F u ( x 6 1 ) ] / 2 = 0.4

Similarly, the T-S fuzzy importance of each component is obtained, as shown in Table 9.

Given the fuzzy subset of the failure probability of the component’s fault state, it can be known from Table 9 that when the system is in a half-fault state (0.5), the importance of x is greatest, which is the weak link in the process of transmission slip. Then we can get the troubleshooting sequence:

x 6 , x 1 ( x 2 , x 5 , x 14 ) , x 12 ( x 15 ) , x 9 , x 8 , x 13 , x 11 , x 3 , x 7 , x 4 ( x 10 , x 16 ) . When the automatic transmission is in a complete fault state (1), the importance of each component is from large to small: x 2 , x 14 , x 1 , x 6 , x 13 , x 4 , x 15 , x 3 ( x 10 ) , x 8 , x 16 , x 11 , x 9 , x 12 , x 7 , x 5 , It can be seen that when the system completely fails, the importance of the basic event x 2 , x 14 , x 1 , x 6 is large, the impact on the top event is also the greatest. Other components also have a certain influence, but the degree of influence is small.

In summary, the analysis result of the slip fault of the automatic transmission based on the T-S fuzzy fault tree is basically in accordance with the situation on

site. The level of oil level, clutch friction, filter plugging, etc. are all weak points. If there is slippage fault, it should be a key point for troubleshooting during troubleshooting, so as to reduce system failure rate and improve normal reliability operation.