Fuzzy system reliability analysis based on level (k, 1) interval-valued fuzzy numbers

- Fuzzy system reliability analysis based on level (k, 1) interval-valued fuzzy numbers.
- Reliability Fuzzy number.
- Level (k, 1) interval-valued fuzzy numbers Fuzzy reliability.
- This study uses Level (k, 1) interval-valued fuzzy numbers to examine the fuzzy reliability of a serial system and a parallel system and obtain the estimated reliability of both systems in the fuzzy sense..
- The probability assumption requires full characterization of the system behavior in the context of the probabil- ity evaluation, whereas the binary-state assumption assumes that the system has only two states at any time.
- This modified assumption asserts that at any given time, the system has only two states, namely, the fuzzy success state, and the fuzzy failure state.
- Cheng and Mon [6] used the a -cut of Level-1 fuzzy numbers to obtain the intervals and determine the fuzzy reliability of the serial system.
- In addition, they successfully iden- tified the fuzzy reliability of the parallel system.
- Chen [7] used fuzzy numbers to determine the fuzzy reliability of the two systems, whereas Singer [12] used LR-type fuzzy numbers to consider the fuzzy reliability problem.
- As shown in Section 3, the reliability R j of the subsystem was fuzzified by designating T j as the operating time for subsystem P j of certain systems.
- This study used Level (k, q ) interval-valued fuzzy numbers to determine the fuzzy reliability of both systems and obtain the estimated reli- ability of both systems in the fuzzy sense, using the signed distance method.
- Furthermore, to ensure easy defuzzification, the signed distance proposed by Abbasbandy and Asady [1] must be considered and modified into the signed distance of an interval-valued fuzzy number.
- Level (k, q ) interval-valued fuzzy numbers.
- The following definitions are proposed to use fuzzy numbers and Level (k, q ) interval-valued fuzzy sets in the fuzzy reliability of serial and parallel systems..
- A e is called a Level k triangular fuzzy number 0 <.
- When k = 1, it is called a triangular fuzzy number.
- When a = b = c, Level 1 fuzzy number (a,b,c.
- The i v fuzzy set A e indicates that, when the membership grade of x belongs to the interval l e.
- q ỡ, which is called the Level (k, q ) i v fuzzy number.
- 1 shows that, when a = p, c = r, k = 0, the Level (k, q ) i v fuzzy number [(a,b,c.
- reduces to the Level q triangular fuzzy number (p,b,r.
- Using the decomposition theory, we obtained the following result:.
- First, we considered the definition of the signed distance on R..
- 0 indicates that a lies to the right of the origin 0, and the distance between a and 0 is denoted by d ⁄ (a, 0.
- Level (k, q ) i v fuzzy numbers e A;0 <.
- a -cut of level (k, q ) i v fuzzy number e A..
- Therefore, the signed distance of the interval h A U l đ a ỡ.
- Therefore, we obtained the following definition:.
- e Similar to Definition 6, we obtained the following definition:.
- 1 ~ 2 F P : For 0 6 k 6 q 6 1, let the Level (k, q ) i v fuzzy numbers e A and B e be.
- r j , j = 1, 2, the extension principle can be used to find the graph of the membership function of e A L e B L , as shown in Fig.
- Therefore, we obtained the following:.
- Using (13), we obtained e A e B ơđa 1 p 1 .
- Fuzzy system reliability based on Level (k,1) fuzzy numbers.
- This section presents the fuzzy reliability of a serial system and a parallel system..
- An uncertainty evaluation error occurred when evaluating the reliability R j of the subsystemP j (j = 1,2.
- fuzzy number.
- n, where e R j is a Level (k,1)i v fuzzy number (Fig.
- Let the Level (k, 1)i v fuzzy number R e j be e R j Ử h e R L j .
- Based on Note 3, which corresponds to interval [R j d 2j , R j + d 3j ] D k , we can obtain the Level k triangular fuzzy number (R j d 2j , R j , R j + d 3j .
- k) and the interval [R j d 1j , R j + d 4j ] D 1 maps to a triangular fuzzy number (R j d 1j , R j , R j + d 4j .
- 7, the Level (k, 1)i v fuzzy number [(R j d 2j , R j , R j + d 3j .
- Hence, the following Level k triangular fuzzy number (Fig.
- Level k triangular fuzzy number (a 1 , P 1 , b 1 q 1 ,c 1 r 1 ,k)..
- grade and confidence level, it is possible to set up a Level k triangular fuzzy number (R j d 2j , R j , R j + d 3j .
- Similarly, we can set up a triangular fuzzy number (R j d 1j , R j , R j + d 4j .
- In conventional reliability theory, the reliability of a serial system is Y n.
- and the reliability of a parallel system is 1 Y n.
- n as (14) of a serial system, (a) The fuzzy reliability of the serial system is.
- (b) Using Definition 6, we obtained the following estimate of the reliability of the serial system in the fuzzy sense..
- Level (k, 1) i v fuzzy numbers e R j.
- Level k triangular fuzzy number..
- 0ỡ ~ as the fuzzy estimate of the reliability of the serial system..
- 1, we obtained 1.
- n as (14) of a parallel system, (a) The fuzzy reliability of the parallel system is.
- (b) Using Definition 6, we obtained the following estimate of the reliability of the parallel system in the fuzzy sense 1.
- Using Definition 9, we obtained (a)..
- 9), Properties 3 and 4 can be used to find the fuzzy reliability of the sys- tem and the estimated reliability of the system in the fuzzy sense..
- What is the possibility that people coming into the vicinity of the machines are injured by getting a chip in the eye? The people at highest risk are the operators, who are obliged to wear safety glasses, yet often fail to do so.
- Other people at risk include people in the vicinity of the machines, people bringing and remov- ing items to the area, and people entering the area for other reasons.
- Case 1: Symmetrical Level (0.9, 1)i v fuzzy numbers..
- 9, the truth function of the main system X can be written as.
- Using Definition 6, we defuzzified R e X and obtained the estimated reliability of the system in the fuzzy sense as.
- Chen [7] used a triangular fuzzy number.
- Subsequently, we obtained the triangular fuzzy number e R j Ử đR j d 1j .
- Therefore, the triangular fuzzy numbers used by Chen [7] are.
- 1ỡ We obtained.
- đ21ỡ Using Definition 8, we defuzzified e R đ1ỡ X and obtained the estimated reliability of the system in the fuzzy sense as d R e đ1ỡ X.
- For (19) and R j in Table 1, we obtained the reliability of the system in the crisp case, which was 0.102264.
- We obtained the difference 1 2 dđ e R X.
- Subsequently, we obtained the following:.
- 1ỡ Using Properties 3 and 4, we obtained the following results:.
- Using Definition 6, we defuzzified e R đ2ỡ X and obtained the estimated reliability of the system in the fuzzy sense as.
- đ23ỡ Using Definition 8, we defuzzified e R đ3ỡ X and obtained the estimated reliability of the system in the fuzzy sense as dđe R đ3ỡ X .
- đ24ỡ Using Definition 6, we defuzzified e R đ4ỡ X and obtained the estimated reliability of the system in the fuzzy sense as.
- Using Definition 8, we defuzzified e R đ5ỡ X and obtained the estimated reliability of the system in the fuzzy sense as d e R đ5ỡ X.
- Case 4: Asymmetrical Level (0.9, 1) i v fuzzy numbers..
- Subsequently, we obtained.
- 1ỡ Using Properties 3 and 4, we obtained the following i v fuzzy numbers..
- Using Definition 6, we defuzzified e R đ6ỡ X and obtained the estimated reliability of the system in the fuzzy sense as.
- Using Definition 8, we defuzzified e R đ7ỡ X and obtained the estimated reliability of the system in the fuzzy sense as d e R đ7ỡ X.
- [A] Triangular fuzzy numbers fall within a special case of Level (k,1)i v fuzzy numbers..
- 7, Level (k,1)i v fuzzy number e R j Ử ơđR j d 2j .
- As shown in the figure, the triangular fuzzy number (R j d 1j , R j , R j + d 4j .
- As stated in Remark 1, it is preferable to use a Level (k, 1)i v fuzzy number to defuzzify e R j than using a triangular fuzzy number in a real number..
- Chen [7] used a fuzzy number that was used as a triangular fuzzy number in this study.
- As stated in [A], the triangular fuzzy number is a special case of the Level (k, 1)i v fuzzy number.
- Chen did not consider the estimated reliability of the system in the fuzzy sense.
- The fuzzy reliability of the serial system is e P Ử e P 1 đỡ e P 2 đỡ đỡ e P n Ử a a 11 .
- The fuzzy reliability of the parallel system is e P Ử 1 Y n.
- This interval is obtained by a -cut of the triangular fuzzy numbers.
- Using the decomposition theorem, we obtained e P i Ử S.
- The reliability of production process experiments must be considered in relation to factory production processes.
- Therefore, this study used the fuzzy concept to address system reliability issues..
- This study used Level (k, 1) interval-valued fuzzy numbers to examine the fuzzy reliability of a serial system and a parallel system.
- The membership grade of reliability was transformed into a Level (k, 1)i v fuzzy number using the confidence level concept.
- The reliability of both the serial and parallel systems was fuzzified using the signed distance method to defuzzify the fuzzy reliability.
- This method was designed to obtain the estimated reliability of both systems in the fuzzy sense.
- The authors gratefully acknowledge the helpful comments and suggestions of the reviewers.
- Chen, Fuzzy system reliability analysis using fuzzy number arithmetic operations, Fuzzy Sets Syst

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