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The Fuzzy Arithmetic Operations of Transmission Average on Pseudo-Hexagonal Fuzzy Numbers and Its Application in Fuzzy System Reliability Analysis

The Fuzzy Arithmetic Operations of Transmission Average on Pseudo-Hexagonal Fuzzy Numbers and Its... FUZZY INFORMATION AND ENGINEERING 2021, VOL. 13, NO. 1, 58–78 https://doi.org/10.1080/16168658.2021.1915449 The Fuzzy Arithmetic Operations of Transmission Average on Pseudo-Hexagonal Fuzzy Numbers and Its Application in Fuzzy System Reliability Analysis a b F. Abbasi and T. Allahviranloo a b Department of Mathematics, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran; Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey ABSTRACT KEYWORDS Reliability analysis; fault tree In most natural science fields, triangular and trapezoidal fuzzy num- analysis; pseudo- hexagonal bers are commonly used while in engineering and social science fuzzy numbers; extension fields such as sociology and psychology while treating the uncer- principle (EP); transmission tainties these numbers are not applicable and fuzzy numbers with average (TA) more parameters and clear definitions of their arithmetic opera- tions are needed. In order to fill this gap in the literature, we pro- pose the new fuzzy arithmetic operations based on transmission average on pseudo-hexagonal fuzzy numbers, which was already implied in [1] in its rudimentary form and was finally presented in its fully fledged form in [2]. Several illustrative examples were given to show the accomplishment and ability of the proposed method. We present a new method for fuzzy system reliability analy- sis based on the fuzzy arithmetic operations of transmission average, where the reliabilities of the components of a system are repre- sented by pseudo- hexagonal fuzzy numbers defined in the uni- verse of discourse [0, 1]. The proposed method has the advantages of modeling and analyzing fuzzy system reliability in a more flex- ible and more intelligent manner. Finally, a marine power plant is considered in fuzzy environment. The reliability of components of the proposed model is considered as pseudo-hexagonal fuzzy numbers. 1. Introduction In most of the cases in social sciences triangular and trapezoidal fuzzy numbers [3,4]may not be enough to measure the attributes usually associated with opinions leading to an ordinal information which can be represented by more than four different points on the real line. Therefore, even the trapezoidal fuzzy numbers can not be enough to represent such cases arising from social science measurements. In this study, we propose the notion of pseudo-octagonal fuzzy numbers in order to fill this gap in literature. As regards fuzzy arithmetic operations using the extension principle (in the domain of the membership function) or the interval arithmetics (in the domain of the α- cuts), we have some problem in subtraction operator, division operator and obtaining the membership CONTACT F. Abbasia k.9121946081@gmail.com © 2021 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society of China & Operations Research Society of Guangdong Province. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons. org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 59 functions of operators. Although with the revised definitions on subtraction and division, usage of interval arithmetic for fuzzy operators have been permitted, because it always exists, but it’s not efficient, it means that result’s support is a major agent (dependence effect) and also complex calculations of interval arithmetic in determining the membership function of operators based on the extension principle, are not yet resolved. Therefore, we eliminated such deficiency with the new fuzzy arithmetic operations based on transitional spaces as TA [1,2]. In this paper, we propose the new fuzzy arithmetic operations based on TA on pseudo- hexagonal fuzzy numbers. The aim of this study is twofold: At first, providing the fuzzy arithmetic operations of transmission average on pseudo- hexagonal fuzzy numbers and then use such fuzzy num- bers with the proposed operations in fuzzy system reliability analysis. Reliability analysis is an important topic in engineering. Reliability is the probability that a component will not fail to perform within specified limits in a given time. In order to handle the insufficient infor- mation, the fuzzy approach is used to evaluate. Therefore fuzzy theory opened the way for facilitating the modeling and fuzziness aspect of system reliability. Several investigators pay attention to applying the fuzzy sets theory to reliability analysis [5–10]. In this paper, we have considered the failure distribution of the components with the pseudo- hexagonal fuzzy numbers based on the fuzzy arithmetic operations of TA. This is more flexible and more general than above mentioned methods including the interval arithmetic and α -cuts are used to evaluate fuzzy system reliability. The paper is organized as follows. In Section 2, we present the fuzzy arithmetic opera- tions of transmission average on pseudo- hexagonal fuzzy numbers. The analysis of fuzzy system reliability is investigated in Section 3. A technical example to illustrate applying the method and comparing the results of the new method with the previous methods are given in Section 4. Finally, conclusions and future research are drawn in Section 5. 2. The Fuzzy Arithmetic Operations of TA on Pseudo- Hexagonal Fuzzy Numbers At first, we propose the new fuzzy arithmetic operations based on transmission average on pseudo- hexagonal fuzzy numbers, which was already implied in [1] in its rudimentary form and was finally presented in its fully-fledged form in [2]. The properties of these proposed operations and their fundamental qualities are discussed. Several illustrative examples were given to show the accomplishment and ability of the proposed method. 2.1. Preliminaries and Notations In this subsection, some notations and background about the concept are brought. Definition 2.1.1: [11]Let X = X ×··· × X , be the Cartesian product of universes, and 1 n A = A ×··· × A be fuzzy sets in each universe respectively. Also let Y be another universe and 1 n B∈Y be a fuzzy set such that B = f(A × ··· × A ), where f : X → Y is a monotonic mapping. 1 n Then Zadeh’s EP is defined as follows: μ (y) = sup min (μ (x ), ... , μ (x )) B A 1 A 1 1 n (1) −1 (x × ··· × x ) ∈ f (y) 1 n −1 where f (y) is the inverse function of y = f (x × ··· × x ). 1 n 60 F. ABBASI AND T. ALLAHVIRANLOO Definition 2.1.2: [11](α -cut Representation) A fuzzy set, A can be represented (decom- posed) as, A = α.A , A = [A , A ]. (2) α α α α∈(0,1] α where A ={x|μ (x) ≥ α}, α A (3) α.A ={(x, α)|x ∈ A }. α α It is easy to check that the following holds: μ (x) = sup α. (4) x∈A Definition 2.1.3: [12] (Fuzzy number) A fuzzy set A in R is called a fuzzy number if it satisfies the following conditions: (i) A is normal, (ii) A is a closed interval for every α ∈ (0, 1], (iii) the support of A is bounded. According to the definition of fuzzy number mentioned above and our emphasis on pseudo-geometric fuzzy numbers, we define a pseudo- hexagonal fuzzy number as follows: Definition 2.1.4: (Pseudo- hexagonal fuzzy numbers) A fuzzy number à is called a Pseudo- hexagonal fuzzy number if its membership function μ (x) is given by l (x), a ≤ x ≤ a , 1 2 A1 l (x), a ≤ x ≤ a, ⎪ A2 1, a ≤ x ≤ a ¯, μ (x) = ⎪ (5) ⎪ ¯ r (x), a ≤ x ≤ a , ⎪ A2 r (x), a ≤ x ≤ a , 3 4 A1 0, otherwise. Where the pair of functions (l (x), l (x))and (r (x), r (x)) are nondecreasing and A1 A2 A2 A1 nonincreasing functions, respectively. Also, l (a ) = l (a ) = , 2 2 A1 A2 (6) r (a ) = r (a ) = . 3  3 A2 A1 The pseudo- hexagonal fuzzy number A is denoted by (a , a , a, a ¯, a , a , (l (x), l (x)), (r (x), r (x)), 1 2 3 4 A1 A2 A2 A1 FUZZY INFORMATION AND ENGINEERING 61 and the hexagonal fuzzy number by (a , a , a, a, a , a , (−, −), (−, −)), 1 2 3 4 that, (−, −), (−, −) means l (x), l (x), r (x) and r (x) are linear. A1 A2 A2 A1 Definition 2.1.5: [13] (Fuzzy arithmetic operations based on interval arithmetic(α-cut)) A popular way to carry out fuzzy arithmetic operations is by way of interval arithmetic. This is possible because any α- cut of a fuzzy number is always an interval. Therefore, any fuzzy number may be represented as a series of intervals. Let us consider two interval numbers [a,b] and [c,d] where a ≤ b and c ≤ d. Then the following arithmetic operations proceed as shown below: (i) addition: [a, b ] +[c,d] = [a + c, b +d], (ii) subtraction: [ a, b ]−[c,d] = [ a–d, b–c ], (iii) multiplication: [ a, b ] . [ c, d ] = [min { ac,ad, bc,bd },max { ac,ad, bc,bd } ], (iv) division: [ a, b ]/[ c, d ] = [a,b].[1/d,1/c]. 2.2. The new Fuzzy Arithmetic Operations based on TA As regards fuzzy arithmetic operations using of the extension principle (in the domain of the membership function) or the interval arithmetics (in the domain of the α- cuts), we have some problem in subtraction operator, division operator and obtaining the mem- bership functions of operators. Although with the revised definitions on subtraction and division, usage of an interval arithmetic for fuzzy operators have been permitted, because it always exists, but it is not efficient, it means that results support is major agent (dependence effect) and also complex calculations of interval arithmetic in determining the membership function of operators based on the extension principle, are not yet resolved. Therefore, we eliminated such deficiency with the fuzzy arithmetic operations based on TA [1]. We define fuzzy arithmetic operations based on TA for addition, subtraction, multiplica- tion and division on pseudo-hexagonal fuzzy numbers as follows: Consider two pseudo- hexagonal fuzzy numbers, a + a ¯ A = (a , a , a, a ¯, a , a , (l (x), l (x)), (r (x), r (x)), a = , 1 2 3 4 A1 A2 A2 A1 b + b ˜ ¯ B = (b , b , b, b, b , b , (l (x), l (x)), (r (x), r (x)), b = , 1 2 3 4 B1 B2 B2 B1 with the following α-cut forms: A = α.A α .A , 11α 22α 1 1 α∈ 0, α∈ ,1 2 2 −1 −1 −1 −1 A = [l (α), r (α)], A = [l (α), r (α)], 11α 22α A1 A1 A2 A2 −1 −1 B = α.B α B , B = [l (α), r (α)], B 11α 22α 11α 22α 1 1 B1 B1 α∈ 0, α∈ ,1 2 2 −1 −1 = [l (α), r (α)]. B2 B2 62 F. ABBASI AND T. ALLAHVIRANLOO Then, (i) addition, A + B = α.(A + B) α.(A + B) , (7) 1 11α 1 22α α∈ 0, α∈ ,1 2 2 where −1 −1 −1 −1 l (α) + l (α) r (α) + r (α) a + b a + b A1 A1 B1 B1 (8) (A + B) = + , + , 11α 2 2 2 2 −1 −1 −1 −1 l (α) + l (α) r (α) + r (α) a + b a + b A2 B2 A2 B2 (9) (A + B) = + , + , 22α 2 2 2 2 (ii) subtraction, Firstly, −B = α.(−B) α.(−B) , (10) 11α 22α 1 1 α∈ 0, α∈ ,1 2 2 where −1 −1 (−B) = [−2b + l (α), −2b + r (α)], (11) 11α B1 B1 −1 −1 (−B) = [−2b + l (α), −2b + r (α)], (12) 22α B2 B2 finally, A − B = A + (−B), A − B = α. (A − B) α.(A − B) , (13) 1 11α 1 22α α∈ 0, α∈ ,1 2 2 −1 −1 −1 −1 l (α) + l (α) r (α) + r (α) a − 3b a − 3b A1 B1 A1 B1 (14) (A − B) = + , + , 11α 2 2 2 2 −1 −1 −1 −1 l (α) + l (α) r (α) + r (α) a − 3b a − 3b A2 B2 A2 B2 (A − B) = + , + , (15) 22α 2 2 2 2 (iii) multiplication, A.B = α.(A − B) α.(A − B) , (16) 1 11α 1 22α α∈ 0, α∈ ,1 2 2 FUZZY INFORMATION AND ENGINEERING 63 where b a b a −1 −1 −1 −1 l (α) + l (α), r (α) + r (α) , a ≥ 0, b ≥ 0 A1 B1 A1 B1 ⎪ 2 2 2 2 b a b a −1 −1 −1 −1 ⎪ r (α) + l (α), l (α) + r (α) , a ≥ 0, b ≤ 0, A1 B1 A1 B1 2 2 2 2 (A.B) = 11α ⎪ b a b a ⎪ −1 −1 −1 −1 r (α) + r (α), l (α) + l (α) , a ≤ 0, b ≤ 0, ⎪ A1 B1 A1 B1 2 2 2 2 b a b a −1 −1 −1 −1 l (α) + r (α), r (α) + l (α) , a ≤ 0, b ≥ 0, A1 B1 A1 B1 2 2 2 2 (17) b a b a −1 −1 −1 −1 ⎪ l (α) + l (α), r (α) + r (α) , a ≥ 0, b ≥ 0 A2 B2 A2 B2 2 2 2 2 ⎪ b a b a ⎪ −1 −1 −1 −1 r (α) + l (α), l (α) + r (α) , a ≥ 0, b ≤ 0, A2 B2 A2 B2 2 2 2 2 (A.B) = 22α ⎪ b a b a ⎪ −1 −1 −1 −1 r (α) + r (α), l (α) + l (α) , a ≤ 0, b ≤ 0, ⎪ A2 B2 A2 B2 2 2 2 2 b a b a −1 −1 −1 −1 l (α) + r (α), r (α) + l (α) , a ≤ 0, b ≥ 0, A2 B2 A2 B2 2 2 2 2 (18) (iv) division, Firstly, −1 −1  −1 B = α.(B ) α.(B ) , (19) 1 1 11α 22α α∈ 0, α∈ ,1 2 2 where 1 1 −1 −1 −1 (B ) = l (α), r (α) , (20) 11α 2 B1 2 B1 b b 1 1 −1 −1 −1 (B ) = l (α), r (α) , (21) 22α 2 2 B2 B2 b b finally, −1 −1 −1 A.B = α.(A.B ) α.(A.B ) , (22) 1 11α 1 22α α∈ 0, α∈ ,1 2 2 1 a 1 a −1 −1 −1 −1 l (α) + l (α), r (α) + r (α) , a ≥ 0, b > 0 ⎪  2   2 A1 B1 A1 B1 ⎪ 2b 2b 2b 2b 1 a 1 a −1 −1 −1 −1 ⎪ r (α) + l (α), l (α) + r (α) , a ≥ 0, b < 0, A1 2 A1 2 B1 B1 2b 2b 2b 2b −1 (A.B ) = 11α ⎪ 1 a 1 a −1 −1 −1 −1 r (α) + r (α), l (α) + l (α) , a ≤ 0, b < 0, ⎪  2   2 A1 B1 A1 B1 ⎪ 2b 2b 2b 2b 1 a 1 a −1 −1 −1 −1 ⎩ l (α) + r (α), r (α) + l (α) , a ≤ 0, b > 0, 2 2 A1 B1 A1 B1 2b 2b 2b 2b (23) 64 F. ABBASI AND T. ALLAHVIRANLOO 1 a 1 a −1 −1 −1 −1 l (α) + l (α), r (α) + r (α) , a ≥ 0, b > 0 A2 2 B2 A2 2 B2 ⎪ 2b 2b 2b 2b 1 a 1 a −1 −1 −1 −1 ⎪ r (α) + l (α), l (α) + r (α) , a ≥ 0, b < 0, 2 2 A2 B2 A2 B2 2b 2b 2b 2b −1 (A.B ) = 22α ⎪ 1 a 1 a ⎪ −1 −1 −1 −1 r (α) + r (α), l (α) + l (α) , a ≤ 0, b < 0, ⎪ A2 2 B2 A2 2 B2 ⎪ 2b 2b 2b 2b 1 a 1 a −1 −1 −1 −1 ⎩ l (α) + r (α), r (α) + l (α) , a ≤ 0, b > 0, 2 2 A2 B2 A2 B2 2b 2b 2b 2b (24) Remark 2.2.1: The division operation on the pseudo-hexagonal fuzzy number 0 = (a , a , − a, a, a , a , l (x), l (x), r (x), r (x)), a > 0, 1 2 3 4 ˜ ˜ ˜ ˜ 01 02 02 01 is not able to be defined. Remark 2.2.2: Since the pseudo-hexagonal fuzzy numbers are a special case of pseudo- geometric fuzzy numbers, we have the lemma and theorems from the ref. [1] for the pseudo-hexagonal fuzzy numbers. 2.3. Numerical Examples In this subsection, we provided several numerical samples to illustrate the application of the proposed method on pseudo-hexagonal and hexagonal fuzzy numbers. We also compared the results of the new method with the previous methods. Example 2.3.1: In this example, we compare the results TA method with EP (α-cut) method. Let A = (1, 2, 4, 6, 7, 9, (−, −), (−, −)), B = (4, 5, 8, 9, 10, 12, (−, −), (−, −)), with the following α-cut forms: (See Figure 1) A = α.A α A , A = [2α + 1, −4α + 9], 11α 22α 11α 1 1 α∈ 0, α∈ ,1 2 2 A = [4α, −2α + 8], 22α B = α.B α.B , B = [2α + 4, −4α + 12], 11α 22α 11α 1 1 α∈ 0, α∈ ,1 2 2 B = [6α + 2, −2α + 11]. 22α Then using the elementary fuzzy arithmetic operations based on the EP (α -cut) and TA, we get: FUZZY INFORMATION AND ENGINEERING 65 1. Based on the EP (α -cut): A + B = (5, 7, 12, 15, 17, 21, (−, −), (−, −)), − B = (−12, −10, −9, −8, −5, −4, (−, −), (−, −)), A − B = (−11, −8, −5, −2, 2, 5, (−, −), (−, −)), −1 A.B = (4, 10, 32, 54, 70, 108(−, −), (−, −)), B 1 1 1 1 1 1 = , , , , , , (−, −), (−, −) , 12 10 9 8 5 4 1 2 4 6 7 9 −1 A.B = , , , , , , (−, −), (−, −) . 12 10 9 8 5 4 2. Based on the TA: 37 41 51 57 61 69 A + B = , , , , , , (−, −), (−, −) , − 4 4 4 4 4 4 B = (−13, −12, −9, −8, −7, −5, (−, −), (−, −)), A − B −31 −27 −17 −11 −7 −1 = , , , , , , (−, −), (−, −) , A.B 4 4 4 4 4 4 57 219 273 −1 = , 21, 37, 38, , (−, −), (−, −) , B 4 4 4 16 20 32 36 40 48 −1 = , , , , , , (−, −), (−, −) , A.B 2 2 2 2 2 2 17 17 17 17 17 17 57 84 148 192 219 273 = , , , , , , (−, −), (−, −) . 2 2 2 2 2 2 17 17 17 17 17 17 The graphical comparison is shown in Figures 2–5. Example 2.3.2: Let A= α.A α.A , 11α 22α 1 1 α∈ 0, α∈ ,1 2 2 √ √ √ √ A = 2 − 4 4 − 4α,4 + 16 − 16α , A = 2 − 8 2(1 − α),4 − 4 2(α − 1) , 11α 22α B= α.B α.B , 11α 22α 1 1 α∈ 0, α∈ ,1 2 2 5 33 7 1 2 2 B = α − 1, − α − , B = [(α + 1) − 2, (3 − α) + 2]. 11α 22α 2 4 2 2 Then using the elementary fuzzy arithmetic operations based on the TA, we get: A + B = α.(A + B) α.(A + B) , (25) 11α 22α 1 1 α∈ 0, α∈ ,1 2 2 66 F. ABBASI AND T. ALLAHVIRANLOO Figure 1. The hexagonal fuzzy numbers of example 2.3.1. Figure 2. The red graph based on the extension principle (α-cut). The black graph is based on the transmission average. where 2−4 4−4α+ α−1 3 + 4 3 + 4 (A + B) = + , 11α 2 2 2 (26) 33 7 1 4 + 16 − 16α + − α − 4 2 2 + , 2 FUZZY INFORMATION AND ENGINEERING 67 Figure 3. The red graph based on the extension principle (α-cut). The black graph is based on the transmission average. Figure 4. The red graph based on the extension principle (α-cut). The black graph is based on the transmission average. 3 + 4 2−8 2(1−α)+(α+1) −2 3 + 4 (A + B) = + , 22α 2 2 2 (27) 4 − 4 2(α − 1) + (3 − α) + 2 + . −B = α.(−B) α.(−B) , (28) 1 11α 1 22α α∈ 0, α∈ ,1 2 2 where 5 33 7 1 (−B) = −2 × 4 + α − 1, −2 × 4 + − α − , (29) 11α 2 4 2 2 68 F. ABBASI AND T. ALLAHVIRANLOO Figure 5. The red graph based on the extension principle (α-cut). The black graph is based on the transmission average. 2 2 (−B) = [−2 × 4 + (α + 1) − 2, −2 × 4 + (3 − α) + 2]. (30) 22α A − B = A + (−B), A − B = α.(A − B) α.(A − B) , (31) 11α 22α 1 1 α∈ 0, α∈ ,1 2 2 where 2 − 4 4 − 4α + α − 1 3 − 3 × 4 3 − 3 × 4 (A − B) = + , 11α 2 2 2 33 7 1 4 + 16 − 16α + − α − 4 2 2 + , (32) 3 − 3 × 4 2 − 8 2(1 − α) + (α + 1) − 2 3 − 3 × 4 (A − B) = + , 22α 2 2 2 √ (33) 4 − 4 2(α − 1) + (3 − α) + 2 + . A.B = α.(A.B) α.(A.B) , (34) 1 11α 1 22α α∈ 0, α∈ ,1 2 2 where √ √ 4 3 5 4 (A.B) = 2 − 4 4 − 4α + α − 1 , 4 + 16 − 16α 11α 2 2 2 2 3 33 7 1 + − α − (35) 2 4 2 2 FUZZY INFORMATION AND ENGINEERING 69 √ √ 4 3 4 (A.B) = 2 − 8 2(1 − α) + ((α + 1) − 2), 4 − 4 2(α − 1) 22α 2 2 2 + ((3 − α) + 2) (36) −1 −1 −1 B = α..(B ) α..(B ) , (37) 1 11α 1 22α α∈ 0, α∈ ,1 2 2 where 1 5 1 33 7 1 −1 (B ) = α − 1 , − α − , (38) 11α 2 2 4 2 4 4 2 2 1 1 −1 2 2 (B ) = ((α + 1) − 2), ((3 − α) + 2) , (39) 22α 2 2 4 4 −1 −1 −1 A.B = α.(A.B ) α.(A.B ) , (40) 11α 22α 1 1 α∈ 0, α∈ ,1 2 2 −1 (A.B ) 11α √ √ 1 3 5 1 = 2 − 4 4 − 4α + α − 1 , 4 + 16 − 16α 2 × 4 2 × 4 2 2 × 4 3 33 7 1 + − α − (41) 2 × 4 4 2 2 −1 (A.B ) 22α 1 3 1 = 2 − 8 2(1 − α) + ((α + 1) − 2), 2 × 4 2 × 4 2 × 4 × 4 − 4 2(α − 1) + ((3 − α) + 2) (42) 2 × 4 3. Reliability Analysis of Fuzzy System Using TA-based Arithmetic Operations Using TA-based fuzzy number arithmetic operations, a new procedure for analyzing fuzzy system reliability is shown in this section; the reliability of each system component is denoted by a pseudo- hexagonal fuzzy number. This is a more flexible and more generic method than all the aforementioned methods (including the interval arithmetic), and α-cuts are utilized for assessing fuzzy system reliability. 3.1. Fault Tree Analysis A fault tree usually includes the top event, the basic events and the logic gates. Gates indi- cate relationships of events. While doing the system-design, fault tree (the logic diagram) is outlined for analysis of the potential factors in system failure; factors like hard-ware, soft- ware, environment, human factor. Based on the known combinations and probabilities of basic events, we calculate the probabilities of system failure. 70 F. ABBASI AND T. ALLAHVIRANLOO 3.2. Fuzzy Operators based on TA of Fault Tree Analysis During the fuzzy fault tree analysis, the probabilities of basic events are described as fuzzy numbers and the traditional logic gate operators are replaced by fuzzy logic gate operators to obtain the fuzzy probability of the top event. In this subsection, we present a new method for analyzing fuzzy system reliability based on TA, where the reliability of the components of a system is represented by pseudo- hexagonal fuzzy number. Lemma 3.2.1: Let A , A , ... , A be pseudo-hexagonal fuzzy numbers as follows: 1 2 n a + a ¯ A = (a , a , a , a ¯ , a , a , (l (x), l (x)), (r (x), r (x)), a = , (a > 0) i i1 i2 i i3 i4 A1i A2i A2i A1i i i with the following α-cut form: A = α..A ∪ α..A , i 11iα 22iα 1 1 α∈ 0, α∈ ,1 2 2 −1 −1 −1 −1 A = l (α), r (α) , A = l (α), r (α) , 11iα 22iα A1i A1i A2i A2i then, (1) n n n n A = α.. A ∪ α.. A , i 11iα 22iα 1 1 α∈ 0, α∈ ,1 2 2 i=1 i=1 i=1 i=1 ⎡ ⎤ ⎡ ⎤ −1 −1 −1 −1 ⎣ ⎦ ⎣ ⎦ A = l (α), r (α) , A = l (α), r (α) , 11iα n n 22iα n n A1i A1i A2i A2i i=1 i=1 i=1 i=1 i=1 where n−2 i a i=1,i=n−k i −1 −1 i=2 −1 l (α) = l (α) + l (α), A1(n−k) A11 k+1 n−1 A1i k=0 i=1 n−2 n i a i=1,i=n−k i −1 −1 −1 i=2 r (α)) = r (α) + r (α), A1(n−k) A11 k+1 n−1 2 2 A1i k=0 i=1 n−2 i a i=1,i=n−k i −1 −1 i=2 −1 l (α) = l (α) + l (α), A21 A2(n−k) n−1 k+1 A2i k=0 i=1 n−2 i a i=1,i=n−k i −1 −1 i=2 −1 r (α)) = r (α) + r (α). A2(n−k) A21 k+1 n−1 2 2 A2i k=0 i=1 1 − A = α · (1 − A) ) ∪ ( α.(1 − A) ), (1 − A) i 11iα 22iα 11iα 1 1 α∈ 0, α∈ ,1 2 2 −1 −1 −1 −1 = [l (α), r (α)], (1 − A) = [l (α), r (α)], 22iα (1−A)1i (1−A)1i (1−A)2i (1−A)2i FUZZY INFORMATION AND ENGINEERING 71 where 1 1 −1 −1 −1 −1 −1 l (α) = (2 − 3a + l (α)), r (α) = (2 − 3a + r (α)), l (α) i i (1−A)1i A1i (1−A)1i A1i (1−A)2i 2 2 1 1 −1 −1 −1 = (2 − 3a + l (α)), r (α) = (2 − 3a + r (α)). i i A2i A2i (1−A)2i 2 2 Proof: We have the above cases, by mathematical induction and according to the fuzzy arithmetic operations of TA on pseudo- hexagonal fuzzy numbers. Consider a serial system shown in Figure 6, where the reliability R of component x is i i represented by a pseudo- hexagonal fuzzy number defined in the universe of discourse [0, 1]: r + r¯ R = (r , r , r , r¯ , r , r , (l (x), l (x)), (r (x), r (x)), r = , (r > 0) i i1 i2 i i3 i4 R1i R2i R2i R1i i i or, −1 −1 R = α.R α.R , R = l (α), r (α) , i 11iα 22iα 11iα 1 1 R1i R1i α∈ 0, α∈ ,1 2 2 −1 −1 R = l (α), r (α) . 22iα R2i R2i Then, the reliability R of the serial system can be evaluated by the (3.1) lemma as follows: n n n n R = R · R ... R = R = α. R ∪ α. R , R 1 2 n i 11iα 22iα 11iα 1 1 α∈ 0, α∈ ,1 2 2 i=1 i=1 i=1 i=1 ⎡ ⎤ ⎡ ⎤ −1 −1 −1 −1 ⎣ ⎦ ⎣ ⎦ = l (α), r (α) , R = l (α), r (α) , n n 22iα n n R1i R1i R2i R2i i=1 i=1 i=1 i=1 i=1 where n−2 n i r i=1,i=n−k i −1 −1 −1 i=2 l (α) = l (α) + l (α), R1(n−k) R11 k+1 n−1 2 2 R1i k=0 i=1 n−2 n i r i=1,i=n−k i −1 −1 i=2 −1 r (α)) = r (α) + r (α), R1(n−k) R11 k+1 n−1 2 2 R1i k=0 i=1 n−2 i r i=1,i=n−k i −1 −1 i=2 −1 l (α) = l (α) + l (α), R2(n−k) R21 k+1 n−1 2 2 R2i k=0 i=1 n−2 i r i=1,i=n−k i −1 −1 i=2 −1 r (α)) = r (α) + r (α). R2(n−k) R21 k+1 n−1 2 2 R2i k=0 i=1 72 F. ABBASI AND T. ALLAHVIRANLOO Figure 6. Configuration of a serial system. Furthermore, consider the parallel system shown in Figure 7, where the reliability A of component x is represented by a pseudo- hexagonal fuzzy number defined in the universe of discourse [0, 1]: r + r¯ R = (r , r , r , r¯ , r , r , (l (x), l (x)), (r (x), r (x)), r = , i i1 i2 i i3 i4 R1i R2i R2i R1i i or, −1 −1 R = α.R ∪ α.R , R = l (α), r (α) , i 11iα 22iα 11iα 1 1 R1i R1i α∈ 0, α∈ ,1 2 2 −1 −1 R = l (α), r (α) . 22iα R2i R2i Then, the reliability R of the parallel system can be evaluated as follows: R = 1− (1 − R ) = α.R α ∪ α.R α , R α i 11 22 11 1 1 α∈ 0, α∈ ,1 2 2 i=1 ⎧ ⎫ ⎧ ⎫ ⎡ ⎤ n n ⎨ ⎬ ⎨ ⎬ 1 1 −1 −1 ⎣ ⎦ = 2 − 3 1 − r + l (α) , 2 − 3 1 − r + r (α) , R α i i 22 n n ⎩ ⎭ ⎩ ⎭ 2 2 (1−R)1i (1−R)1i i=1 i=1 i=1 i=1 ⎡ ⎧ ⎫ ⎧ ⎫ ⎤ n n ⎨ ⎬ ⎨ ⎬ 1 1 −1 −1 ⎣ ⎦ = 2 − 3 1 − r + l (α) , 2 − 3 1 − r + r (α) , i n i n 2⎩ ⎭ 2⎩ ⎭ (1−R)2i (1−R)2i i=1 i=1 i=1 i=1 where n n n (1 − R ) = α. (1 − R) ) ∪ α. (1 − R) ) , i 11iα 22iα 1 1 α∈ 0, α∈ ,1 2 2 i=1 i=1 i=1 ⎡ ⎤ −1 −1 ⎣ ⎦ (1 − R) = l (α), r (α) , 11iα n n (1−R)1i (1−R)1i i=1 i=1 i=1 ⎡ ⎤ −1 −1 ⎣ ⎦ (1 − R) = l (α), r (α) , 22iα n n (1−R)2i (1−R)2i i=1 i=1 i=1 n−2 (1 − r ) i (1 − r ) i=1,i=n−k i −1 −1 i=2 −1 l (α) = l (α) + l (α), (1−R)1(n−k) (1−R)11 k+1 n−1 2 2 (1−R)1i k=0 i=1 n−2 (1 − r ) i (1 − r ) i=1,i=n−k i −1 −1 i=2 −1 r (α)) = r (α) + r (α), (1−R)1(n−k) n−1 (1−R)11 k+1 (1−R)1i k=0 i=1 FUZZY INFORMATION AND ENGINEERING 73 Figure 7. Configuration of a parallel sysytem. n−2 n (1 − r ) i (1 − r ) i=1,i=n−k i −1 −1 i=2 −1 l (α) = l (α) + l (α), (1−R)21 (1−R)2(n−k) n−1 k+1 2 2 (1−R)2i k=0 i=1 n−2 (1 − r ) i (1 − r ) i=1,i=n−k i −1 −1 i=2 −1 (α)) = r (α) + r (α), (1−R)2(n−k) (1−R)21 k+1 n−1 2 2 (1−R)2i k=0 i=1 −1 −1 −1 −1 −1 1 1 l (α) = (2 − 3r + l (α)), r (α) = (2 − 3r + r (α)), l (α) i i (1−R)1i 2 R1i (1−R)1i 2 R1i (1−R)2i 1 1 −1 −1 −1 = (2 − 3r + l (α)), r (α) = (2 − 3r + r (α)). i i R2i (1−R)2i R2i 2 2 In the following, we use an example to illustrate the fuzzy system reliability analysis process. 4. A Technical Example A marine power plant [4] has two generators G1 and G2 one located at the stern and the other at the bow. Each generator is connected to its respective micro switch board-1 and micro switch board-2. The distributive switchboard receives the supply from the switchboards through cables C1 and C2 and respective junction boxes D and E. The two micro switchboards are intercon- nected through a long cable C3 and the junction boxes A and B. The schematic diagram is shown in Figure 8. Let us assume that basic components subjected to failure are (a) Generators G1 and G2. (b) Microswitch board-1 (MSB-1) and Microswitch board-2 (MSB-2). (c) Interconnecting cable C3 and junction boxes A and B, all are treated as one unit. (d) Junction boxes D and E. (e) Distributive switchboard (DSB). 74 F. ABBASI AND T. ALLAHVIRANLOO Figure 8. Marine Power Plant. In this example, we show a failure of the marine power plant in the form of a fuzzy num- ber for the more comprehensive analysis to improve the educational process. A fault tree for the top event ‘failure of the marine power plant’ is shown in Figure 9. Due to a more accurate estimate of each failure event and generalized reliability anal- ysis of the system shown in Figure 9, let us assume the basic events of this fault tree have the following pseudo- hexagonal fuzzy number defined in the universe of discourse [0, 1]: r + r¯ R = (r , r , r , r¯ , r , r , (l (x), l (x)), (r (x), r (x)), r = , i i1 i2 i i3 i4 R1i R2i R2i R1i i or, −1 −1 R = α.R ∪ α.R , R = l (α), r (α) , i 11iα 22iα 11iα 1 1 R1i R1i α∈ 0, α∈ ,1 2 2 −1 −1 R = l (α), r (α) , i = 1, 2, ... , 22, 22iα R2i R2i where R , represents the unreliability of the distributive switchboard. R , represents the reliability of the event that no power is coming to distributive switch- board. R , represents the reliability of the event that there is no power supply from the junction box D. FUZZY INFORMATION AND ENGINEERING 75 Figure 9. Faulttreeofmarine powerplant. R , represents the reliability of the event that there is no power supply from the junction box E. R , represents the unreliability of the junction box D. R , represents the reliability of the event that there is no power supply to the junction box D. R , represents the unreliability of micro switchboard-1. R , represents the reliability of the event that there is no power supply to micro switch board-1. R , represents the unreliability of generator G1. R , represents the reliability of the event that there is no power supply through the junction boxes A and B. R , represents the unreliability of generator G2. R , represents the unreliability of the junction boxes A and B. R , represents the unreliability of micro switchboard-2. 13 76 F. ABBASI AND T. ALLAHVIRANLOO R , represents unreliability of the junction box E. R , represents the reliability of the event that there is no power supply to the junction box E. R , represents the unreliability of micro switchboard-2. R , represents the reliability of the event that there is no power supply to micro switchboard-2. R , represents the unreliability of generator G2. R , represents the reliability of the event that there is no power supply through the junction boxes D and E. R , represents the unreliability of generator G1. R , represents the unreliability of the junction boxes A and B. R , represents the unreliability of micro switchboard-1. Based on the previous discussion(the reliability of the serial and parallel systems), we get a failure of the marine power plant (R) as follows: R = 1 − [(1 − R )(1 − R )], 1 2 R = R · R , 2 3 4 where the calculation of R : R = 1 − [(1 − R )(1 − R )], 3 5 6 R = 1 − [(1 − R )(1 − R )], 6 7 8 R = R · R , 8 9 10 R = 1 − [(1 − R )(1 − R )(1 − R )], 10 11 12 13 the calculation of R : R = 1 − [(1 − R )(1 − R )], 4 14 15 R = 1 − [(1 − R )(1 − R )], 15 16 17 R = R · R , 17 18 19 R = 1 − [(1 − R )(1 − R )(1 − R )]. 19 20 21 22 Finally, we can calculate the system reliability R by the (3.2.1) lemma. If we required the system to have a fault probability of x as a limit, then, α ≥ α is necessary, where α = 0 0 0 inf {α|x ∈ / R }. In this case, we allow the system to be uncertain and flexible to an extent 0 α that, the fault probabilities be in the R . It is worth mentioning, the proposed model is applicable for every marine power plant with having the statistical data. 5. Conclusion and Future Research Fuzzy reliability is based on the concept of fuzzy set. When the failure rate is fuzzy, according to Zadeh’s extension principle, the reliability measure will be fuzzy as well. In this paper, the FUZZY INFORMATION AND ENGINEERING 77 use of the concept of pseudo-hexagonal fuzzy numbers and the component failure proba- bilities are considered as a new type of fuzzy number as pseudo-hexagonal to incorporate the uncertainties in the parameter, due to a more realistic estimate of them. We used the new TA-operations [1,2], because of smaller results, easier computations and some particular properties. The developed method has been used to analyze the fuzzy reliability of a marine power plant. The major advantage of using the pseudo-hexagonal fuzzy numbers and the new operations of transmission average (TA), is the smaller results support, easier calculations and special properties than fuzzy arithmetic operations based on the extension principle (in the domain of the membership function) and the interval arithmetic (in the domain of the α-cuts). The proposed methodology can be used for a more general problem when systems are distributed according to other fuzzy numbers. The future work of this study will focus on the fuzzy arithmetic operations based on TA for n-polygonal fuzzy numbers and its application in fuzzy system reliability analysis. Acknowledgements We are grateful to the referee for their valuable suggestions, which have improved this paper. Disclosure statement No potential conflict of interest was reported by the authors. Notes on Contributors Fazlollah Abbasi is Ph.D of Applied Mathematics. He is member of Department of Mathematics, Aya- tollah Amoli Branch, Islamic Azad University, Amol, Iran. His research interests are in the field of fuzzy mathematics. Tofigh Allahviranloo is a Senior Full Professor of Applied Mathematics at Bahcesehir International University (BAU). His research interests are in the field of uncertain mathematics, uncertain dynamical systems, decision science, fuzzy systems. References [1] Abbasi F, Allahviranloo T, Abbasbandy S. A new attitude coupled with fuzzy thinking to fuzzy rings and fields. J Intell Fuzzy Syst. 2015;29:851–861. [2] Abbasi F, Abbasbandy S, Nieto JJ. A new and efficient method for elementary fuzzy arithmetic operations on pseudo-geometric fuzzy numbers. J Fuzzy Set Valued Anal. 2016;2:156–173. [3] Ye F, Lin Q. Partner selection in a virtual enterprise: a group multiattribute decision model with weighted possibilistic mean values. Math Probl Eng. 2013;13:1–14. [4] Zhang XH, Xu XH, Tao L. Some similarity measures for triangular fuzzy numbers and their applications in multiple criteria groups decision making. J Appl Math. 2013;13:1–7. [5] Cheng CH, Mon DL. Fuzzy system reliability analysis by interval of confidence. Fuzzy Sets Syst. 1993;56:29–35. [6] Chen SM. Fuzzy system reliability analysis using fuzzy number arithmetic operations. Fuzzy Sets Syst. 1994;64:31–38. [7] Chen SM. Analyzing fuzzy system reliability using vague set theory. Int. J. Appl. SciEng. 2003;1(1):82–88. [8] Mon DL, Cheng CH. Fuzzy system reliability analysis for components with different membership functions. Fuzzy Sets Syst. 1994;64:145–157. [9] Onisawa T, Kacprzyk J. Reliability and safety analyses under fuzziness. Heidelberg: Physica- Verlag; 1995. 78 F. ABBASI AND T. ALLAHVIRANLOO [10] Wu HC. Fuzzy reliability analysis based on closed fuzzy numbers. Inf Sci (Ny). 1997;103:135–159. [11] Zadeh LA. The concept of a linguistic variable and its application to approximate reasoning – Part I, II, III. Inf Sci (Ny). 1975;9(1):43–80. [12] Fullór R. Fuzzy reasoning and fuzzy optimization. No. 9. Abo: Turku Centre for Computer Science; [13] Zimmermann HJ. Fuzzy set theory and its applications. New York (NY): Springer; 2001. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

The Fuzzy Arithmetic Operations of Transmission Average on Pseudo-Hexagonal Fuzzy Numbers and Its Application in Fuzzy System Reliability Analysis

Abbasi, F.; Allahviranloo, T.
Fuzzy Information and Engineering , Volume 13 (1): 21 – Jan 2, 2021
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FUZZY INFORMATION AND ENGINEERING 2021, VOL. 13, NO. 1, 58–78 https://doi.org/10.1080/16168658.2021.1915449 The Fuzzy Arithmetic Operations of Transmission Average on Pseudo-Hexagonal Fuzzy Numbers and Its Application in Fuzzy System Reliability Analysis a b F. Abbasi and T. Allahviranloo a b Department of Mathematics, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran; Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey ABSTRACT KEYWORDS Reliability analysis; fault tree In most natural science fields, triangular and trapezoidal fuzzy num- analysis; pseudo- hexagonal bers are commonly used while in engineering and social science fuzzy numbers; extension fields such as sociology and psychology while treating the uncer- principle (EP); transmission tainties these numbers are not applicable and fuzzy numbers with average (TA) more parameters and clear definitions of their arithmetic opera- tions are needed. In order to fill this gap in the literature, we pro- pose the new fuzzy arithmetic operations based on transmission average on pseudo-hexagonal fuzzy numbers, which was already implied in [1] in its rudimentary form and was finally presented in its fully fledged form in [2]. Several illustrative examples were given to show the accomplishment and ability of the proposed method. We present a new method for fuzzy system reliability analy- sis based on the fuzzy arithmetic operations of transmission average, where the reliabilities of the components of a system are repre- sented by pseudo- hexagonal fuzzy numbers defined in the uni- verse of discourse [0, 1]. The proposed method has the advantages of modeling and analyzing fuzzy system reliability in a more flex- ible and more intelligent manner. Finally, a marine power plant is considered in fuzzy environment. The reliability of components of the proposed model is considered as pseudo-hexagonal fuzzy numbers. 1. Introduction In most of the cases in social sciences triangular and trapezoidal fuzzy numbers [3,4]may not be enough to measure the attributes usually associated with opinions leading to an ordinal information which can be represented by more than four different points on the real line. Therefore, even the trapezoidal fuzzy numbers can not be enough to represent such cases arising from social science measurements. In this study, we propose the notion of pseudo-octagonal fuzzy numbers in order to fill this gap in literature. As regards fuzzy arithmetic operations using the extension principle (in the domain of the membership function) or the interval arithmetics (in the domain of the α- cuts), we have some problem in subtraction operator, division operator and obtaining the membership CONTACT F. Abbasia k.9121946081@gmail.com © 2021 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society of China & Operations Research Society of Guangdong Province. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons. org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 59 functions of operators. Although with the revised definitions on subtraction and division, usage of interval arithmetic for fuzzy operators have been permitted, because it always exists, but it’s not efficient, it means that result’s support is a major agent (dependence effect) and also complex calculations of interval arithmetic in determining the membership function of operators based on the extension principle, are not yet resolved. Therefore, we eliminated such deficiency with the new fuzzy arithmetic operations based on transitional spaces as TA [1,2]. In this paper, we propose the new fuzzy arithmetic operations based on TA on pseudo- hexagonal fuzzy numbers. The aim of this study is twofold: At first, providing the fuzzy arithmetic operations of transmission average on pseudo- hexagonal fuzzy numbers and then use such fuzzy num- bers with the proposed operations in fuzzy system reliability analysis. Reliability analysis is an important topic in engineering. Reliability is the probability that a component will not fail to perform within specified limits in a given time. In order to handle the insufficient infor- mation, the fuzzy approach is used to evaluate. Therefore fuzzy theory opened the way for facilitating the modeling and fuzziness aspect of system reliability. Several investigators pay attention to applying the fuzzy sets theory to reliability analysis [5–10]. In this paper, we have considered the failure distribution of the components with the pseudo- hexagonal fuzzy numbers based on the fuzzy arithmetic operations of TA. This is more flexible and more general than above mentioned methods including the interval arithmetic and α -cuts are used to evaluate fuzzy system reliability. The paper is organized as follows. In Section 2, we present the fuzzy arithmetic opera- tions of transmission average on pseudo- hexagonal fuzzy numbers. The analysis of fuzzy system reliability is investigated in Section 3. A technical example to illustrate applying the method and comparing the results of the new method with the previous methods are given in Section 4. Finally, conclusions and future research are drawn in Section 5. 2. The Fuzzy Arithmetic Operations of TA on Pseudo- Hexagonal Fuzzy Numbers At first, we propose the new fuzzy arithmetic operations based on transmission average on pseudo- hexagonal fuzzy numbers, which was already implied in [1] in its rudimentary form and was finally presented in its fully-fledged form in [2]. The properties of these proposed operations and their fundamental qualities are discussed. Several illustrative examples were given to show the accomplishment and ability of the proposed method. 2.1. Preliminaries and Notations In this subsection, some notations and background about the concept are brought. Definition 2.1.1: [11]Let X = X ×··· × X , be the Cartesian product of universes, and 1 n A = A ×··· × A be fuzzy sets in each universe respectively. Also let Y be another universe and 1 n B∈Y be a fuzzy set such that B = f(A × ··· × A ), where f : X → Y is a monotonic mapping. 1 n Then Zadeh’s EP is defined as follows: μ (y) = sup min (μ (x ), ... , μ (x )) B A 1 A 1 1 n (1) −1 (x × ··· × x ) ∈ f (y) 1 n −1 where f (y) is the inverse function of y = f (x × ··· × x ). 1 n 60 F. ABBASI AND T. ALLAHVIRANLOO Definition 2.1.2: [11](α -cut Representation) A fuzzy set, A can be represented (decom- posed) as, A = α.A , A = [A , A ]. (2) α α α α∈(0,1] α where A ={x|μ (x) ≥ α}, α A (3) α.A ={(x, α)|x ∈ A }. α α It is easy to check that the following holds: μ (x) = sup α. (4) x∈A Definition 2.1.3: [12] (Fuzzy number) A fuzzy set A in R is called a fuzzy number if it satisfies the following conditions: (i) A is normal, (ii) A is a closed interval for every α ∈ (0, 1], (iii) the support of A is bounded. According to the definition of fuzzy number mentioned above and our emphasis on pseudo-geometric fuzzy numbers, we define a pseudo- hexagonal fuzzy number as follows: Definition 2.1.4: (Pseudo- hexagonal fuzzy numbers) A fuzzy number à is called a Pseudo- hexagonal fuzzy number if its membership function μ (x) is given by l (x), a ≤ x ≤ a , 1 2 A1 l (x), a ≤ x ≤ a, ⎪ A2 1, a ≤ x ≤ a ¯, μ (x) = ⎪ (5) ⎪ ¯ r (x), a ≤ x ≤ a , ⎪ A2 r (x), a ≤ x ≤ a , 3 4 A1 0, otherwise. Where the pair of functions (l (x), l (x))and (r (x), r (x)) are nondecreasing and A1 A2 A2 A1 nonincreasing functions, respectively. Also, l (a ) = l (a ) = , 2 2 A1 A2 (6) r (a ) = r (a ) = . 3  3 A2 A1 The pseudo- hexagonal fuzzy number A is denoted by (a , a , a, a ¯, a , a , (l (x), l (x)), (r (x), r (x)), 1 2 3 4 A1 A2 A2 A1 FUZZY INFORMATION AND ENGINEERING 61 and the hexagonal fuzzy number by (a , a , a, a, a , a , (−, −), (−, −)), 1 2 3 4 that, (−, −), (−, −) means l (x), l (x), r (x) and r (x) are linear. A1 A2 A2 A1 Definition 2.1.5: [13] (Fuzzy arithmetic operations based on interval arithmetic(α-cut)) A popular way to carry out fuzzy arithmetic operations is by way of interval arithmetic. This is possible because any α- cut of a fuzzy number is always an interval. Therefore, any fuzzy number may be represented as a series of intervals. Let us consider two interval numbers [a,b] and [c,d] where a ≤ b and c ≤ d. Then the following arithmetic operations proceed as shown below: (i) addition: [a, b ] +[c,d] = [a + c, b +d], (ii) subtraction: [ a, b ]−[c,d] = [ a–d, b–c ], (iii) multiplication: [ a, b ] . [ c, d ] = [min { ac,ad, bc,bd },max { ac,ad, bc,bd } ], (iv) division: [ a, b ]/[ c, d ] = [a,b].[1/d,1/c]. 2.2. The new Fuzzy Arithmetic Operations based on TA As regards fuzzy arithmetic operations using of the extension principle (in the domain of the membership function) or the interval arithmetics (in the domain of the α- cuts), we have some problem in subtraction operator, division operator and obtaining the mem- bership functions of operators. Although with the revised definitions on subtraction and division, usage of an interval arithmetic for fuzzy operators have been permitted, because it always exists, but it is not efficient, it means that results support is major agent (dependence effect) and also complex calculations of interval arithmetic in determining the membership function of operators based on the extension principle, are not yet resolved. Therefore, we eliminated such deficiency with the fuzzy arithmetic operations based on TA [1]. We define fuzzy arithmetic operations based on TA for addition, subtraction, multiplica- tion and division on pseudo-hexagonal fuzzy numbers as follows: Consider two pseudo- hexagonal fuzzy numbers, a + a ¯ A = (a , a , a, a ¯, a , a , (l (x), l (x)), (r (x), r (x)), a = , 1 2 3 4 A1 A2 A2 A1 b + b ˜ ¯ B = (b , b , b, b, b , b , (l (x), l (x)), (r (x), r (x)), b = , 1 2 3 4 B1 B2 B2 B1 with the following α-cut forms: A = α.A α .A , 11α 22α 1 1 α∈ 0, α∈ ,1 2 2 −1 −1 −1 −1 A = [l (α), r (α)], A = [l (α), r (α)], 11α 22α A1 A1 A2 A2 −1 −1 B = α.B α B , B = [l (α), r (α)], B 11α 22α 11α 22α 1 1 B1 B1 α∈ 0, α∈ ,1 2 2 −1 −1 = [l (α), r (α)]. B2 B2 62 F. ABBASI AND T. ALLAHVIRANLOO Then, (i) addition, A + B = α.(A + B) α.(A + B) , (7) 1 11α 1 22α α∈ 0, α∈ ,1 2 2 where −1 −1 −1 −1 l (α) + l (α) r (α) + r (α) a + b a + b A1 A1 B1 B1 (8) (A + B) = + , + , 11α 2 2 2 2 −1 −1 −1 −1 l (α) + l (α) r (α) + r (α) a + b a + b A2 B2 A2 B2 (9) (A + B) = + , + , 22α 2 2 2 2 (ii) subtraction, Firstly, −B = α.(−B) α.(−B) , (10) 11α 22α 1 1 α∈ 0, α∈ ,1 2 2 where −1 −1 (−B) = [−2b + l (α), −2b + r (α)], (11) 11α B1 B1 −1 −1 (−B) = [−2b + l (α), −2b + r (α)], (12) 22α B2 B2 finally, A − B = A + (−B), A − B = α. (A − B) α.(A − B) , (13) 1 11α 1 22α α∈ 0, α∈ ,1 2 2 −1 −1 −1 −1 l (α) + l (α) r (α) + r (α) a − 3b a − 3b A1 B1 A1 B1 (14) (A − B) = + , + , 11α 2 2 2 2 −1 −1 −1 −1 l (α) + l (α) r (α) + r (α) a − 3b a − 3b A2 B2 A2 B2 (A − B) = + , + , (15) 22α 2 2 2 2 (iii) multiplication, A.B = α.(A − B) α.(A − B) , (16) 1 11α 1 22α α∈ 0, α∈ ,1 2 2 FUZZY INFORMATION AND ENGINEERING 63 where b a b a −1 −1 −1 −1 l (α) + l (α), r (α) + r (α) , a ≥ 0, b ≥ 0 A1 B1 A1 B1 ⎪ 2 2 2 2 b a b a −1 −1 −1 −1 ⎪ r (α) + l (α), l (α) + r (α) , a ≥ 0, b ≤ 0, A1 B1 A1 B1 2 2 2 2 (A.B) = 11α ⎪ b a b a ⎪ −1 −1 −1 −1 r (α) + r (α), l (α) + l (α) , a ≤ 0, b ≤ 0, ⎪ A1 B1 A1 B1 2 2 2 2 b a b a −1 −1 −1 −1 l (α) + r (α), r (α) + l (α) , a ≤ 0, b ≥ 0, A1 B1 A1 B1 2 2 2 2 (17) b a b a −1 −1 −1 −1 ⎪ l (α) + l (α), r (α) + r (α) , a ≥ 0, b ≥ 0 A2 B2 A2 B2 2 2 2 2 ⎪ b a b a ⎪ −1 −1 −1 −1 r (α) + l (α), l (α) + r (α) , a ≥ 0, b ≤ 0, A2 B2 A2 B2 2 2 2 2 (A.B) = 22α ⎪ b a b a ⎪ −1 −1 −1 −1 r (α) + r (α), l (α) + l (α) , a ≤ 0, b ≤ 0, ⎪ A2 B2 A2 B2 2 2 2 2 b a b a −1 −1 −1 −1 l (α) + r (α), r (α) + l (α) , a ≤ 0, b ≥ 0, A2 B2 A2 B2 2 2 2 2 (18) (iv) division, Firstly, −1 −1  −1 B = α.(B ) α.(B ) , (19) 1 1 11α 22α α∈ 0, α∈ ,1 2 2 where 1 1 −1 −1 −1 (B ) = l (α), r (α) , (20) 11α 2 B1 2 B1 b b 1 1 −1 −1 −1 (B ) = l (α), r (α) , (21) 22α 2 2 B2 B2 b b finally, −1 −1 −1 A.B = α.(A.B ) α.(A.B ) , (22) 1 11α 1 22α α∈ 0, α∈ ,1 2 2 1 a 1 a −1 −1 −1 −1 l (α) + l (α), r (α) + r (α) , a ≥ 0, b > 0 ⎪  2   2 A1 B1 A1 B1 ⎪ 2b 2b 2b 2b 1 a 1 a −1 −1 −1 −1 ⎪ r (α) + l (α), l (α) + r (α) , a ≥ 0, b < 0, A1 2 A1 2 B1 B1 2b 2b 2b 2b −1 (A.B ) = 11α ⎪ 1 a 1 a −1 −1 −1 −1 r (α) + r (α), l (α) + l (α) , a ≤ 0, b < 0, ⎪  2   2 A1 B1 A1 B1 ⎪ 2b 2b 2b 2b 1 a 1 a −1 −1 −1 −1 ⎩ l (α) + r (α), r (α) + l (α) , a ≤ 0, b > 0, 2 2 A1 B1 A1 B1 2b 2b 2b 2b (23) 64 F. ABBASI AND T. ALLAHVIRANLOO 1 a 1 a −1 −1 −1 −1 l (α) + l (α), r (α) + r (α) , a ≥ 0, b > 0 A2 2 B2 A2 2 B2 ⎪ 2b 2b 2b 2b 1 a 1 a −1 −1 −1 −1 ⎪ r (α) + l (α), l (α) + r (α) , a ≥ 0, b < 0, 2 2 A2 B2 A2 B2 2b 2b 2b 2b −1 (A.B ) = 22α ⎪ 1 a 1 a ⎪ −1 −1 −1 −1 r (α) + r (α), l (α) + l (α) , a ≤ 0, b < 0, ⎪ A2 2 B2 A2 2 B2 ⎪ 2b 2b 2b 2b 1 a 1 a −1 −1 −1 −1 ⎩ l (α) + r (α), r (α) + l (α) , a ≤ 0, b > 0, 2 2 A2 B2 A2 B2 2b 2b 2b 2b (24) Remark 2.2.1: The division operation on the pseudo-hexagonal fuzzy number 0 = (a , a , − a, a, a , a , l (x), l (x), r (x), r (x)), a > 0, 1 2 3 4 ˜ ˜ ˜ ˜ 01 02 02 01 is not able to be defined. Remark 2.2.2: Since the pseudo-hexagonal fuzzy numbers are a special case of pseudo- geometric fuzzy numbers, we have the lemma and theorems from the ref. [1] for the pseudo-hexagonal fuzzy numbers. 2.3. Numerical Examples In this subsection, we provided several numerical samples to illustrate the application of the proposed method on pseudo-hexagonal and hexagonal fuzzy numbers. We also compared the results of the new method with the previous methods. Example 2.3.1: In this example, we compare the results TA method with EP (α-cut) method. Let A = (1, 2, 4, 6, 7, 9, (−, −), (−, −)), B = (4, 5, 8, 9, 10, 12, (−, −), (−, −)), with the following α-cut forms: (See Figure 1) A = α.A α A , A = [2α + 1, −4α + 9], 11α 22α 11α 1 1 α∈ 0, α∈ ,1 2 2 A = [4α, −2α + 8], 22α B = α.B α.B , B = [2α + 4, −4α + 12], 11α 22α 11α 1 1 α∈ 0, α∈ ,1 2 2 B = [6α + 2, −2α + 11]. 22α Then using the elementary fuzzy arithmetic operations based on the EP (α -cut) and TA, we get: FUZZY INFORMATION AND ENGINEERING 65 1. Based on the EP (α -cut): A + B = (5, 7, 12, 15, 17, 21, (−, −), (−, −)), − B = (−12, −10, −9, −8, −5, −4, (−, −), (−, −)), A − B = (−11, −8, −5, −2, 2, 5, (−, −), (−, −)), −1 A.B = (4, 10, 32, 54, 70, 108(−, −), (−, −)), B 1 1 1 1 1 1 = , , , , , , (−, −), (−, −) , 12 10 9 8 5 4 1 2 4 6 7 9 −1 A.B = , , , , , , (−, −), (−, −) . 12 10 9 8 5 4 2. Based on the TA: 37 41 51 57 61 69 A + B = , , , , , , (−, −), (−, −) , − 4 4 4 4 4 4 B = (−13, −12, −9, −8, −7, −5, (−, −), (−, −)), A − B −31 −27 −17 −11 −7 −1 = , , , , , , (−, −), (−, −) , A.B 4 4 4 4 4 4 57 219 273 −1 = , 21, 37, 38, , (−, −), (−, −) , B 4 4 4 16 20 32 36 40 48 −1 = , , , , , , (−, −), (−, −) , A.B 2 2 2 2 2 2 17 17 17 17 17 17 57 84 148 192 219 273 = , , , , , , (−, −), (−, −) . 2 2 2 2 2 2 17 17 17 17 17 17 The graphical comparison is shown in Figures 2–5. Example 2.3.2: Let A= α.A α.A , 11α 22α 1 1 α∈ 0, α∈ ,1 2 2 √ √ √ √ A = 2 − 4 4 − 4α,4 + 16 − 16α , A = 2 − 8 2(1 − α),4 − 4 2(α − 1) , 11α 22α B= α.B α.B , 11α 22α 1 1 α∈ 0, α∈ ,1 2 2 5 33 7 1 2 2 B = α − 1, − α − , B = [(α + 1) − 2, (3 − α) + 2]. 11α 22α 2 4 2 2 Then using the elementary fuzzy arithmetic operations based on the TA, we get: A + B = α.(A + B) α.(A + B) , (25) 11α 22α 1 1 α∈ 0, α∈ ,1 2 2 66 F. ABBASI AND T. ALLAHVIRANLOO Figure 1. The hexagonal fuzzy numbers of example 2.3.1. Figure 2. The red graph based on the extension principle (α-cut). The black graph is based on the transmission average. where 2−4 4−4α+ α−1 3 + 4 3 + 4 (A + B) = + , 11α 2 2 2 (26) 33 7 1 4 + 16 − 16α + − α − 4 2 2 + , 2 FUZZY INFORMATION AND ENGINEERING 67 Figure 3. The red graph based on the extension principle (α-cut). The black graph is based on the transmission average. Figure 4. The red graph based on the extension principle (α-cut). The black graph is based on the transmission average. 3 + 4 2−8 2(1−α)+(α+1) −2 3 + 4 (A + B) = + , 22α 2 2 2 (27) 4 − 4 2(α − 1) + (3 − α) + 2 + . −B = α.(−B) α.(−B) , (28) 1 11α 1 22α α∈ 0, α∈ ,1 2 2 where 5 33 7 1 (−B) = −2 × 4 + α − 1, −2 × 4 + − α − , (29) 11α 2 4 2 2 68 F. ABBASI AND T. ALLAHVIRANLOO Figure 5. The red graph based on the extension principle (α-cut). The black graph is based on the transmission average. 2 2 (−B) = [−2 × 4 + (α + 1) − 2, −2 × 4 + (3 − α) + 2]. (30) 22α A − B = A + (−B), A − B = α.(A − B) α.(A − B) , (31) 11α 22α 1 1 α∈ 0, α∈ ,1 2 2 where 2 − 4 4 − 4α + α − 1 3 − 3 × 4 3 − 3 × 4 (A − B) = + , 11α 2 2 2 33 7 1 4 + 16 − 16α + − α − 4 2 2 + , (32) 3 − 3 × 4 2 − 8 2(1 − α) + (α + 1) − 2 3 − 3 × 4 (A − B) = + , 22α 2 2 2 √ (33) 4 − 4 2(α − 1) + (3 − α) + 2 + . A.B = α.(A.B) α.(A.B) , (34) 1 11α 1 22α α∈ 0, α∈ ,1 2 2 where √ √ 4 3 5 4 (A.B) = 2 − 4 4 − 4α + α − 1 , 4 + 16 − 16α 11α 2 2 2 2 3 33 7 1 + − α − (35) 2 4 2 2 FUZZY INFORMATION AND ENGINEERING 69 √ √ 4 3 4 (A.B) = 2 − 8 2(1 − α) + ((α + 1) − 2), 4 − 4 2(α − 1) 22α 2 2 2 + ((3 − α) + 2) (36) −1 −1 −1 B = α..(B ) α..(B ) , (37) 1 11α 1 22α α∈ 0, α∈ ,1 2 2 where 1 5 1 33 7 1 −1 (B ) = α − 1 , − α − , (38) 11α 2 2 4 2 4 4 2 2 1 1 −1 2 2 (B ) = ((α + 1) − 2), ((3 − α) + 2) , (39) 22α 2 2 4 4 −1 −1 −1 A.B = α.(A.B ) α.(A.B ) , (40) 11α 22α 1 1 α∈ 0, α∈ ,1 2 2 −1 (A.B ) 11α √ √ 1 3 5 1 = 2 − 4 4 − 4α + α − 1 , 4 + 16 − 16α 2 × 4 2 × 4 2 2 × 4 3 33 7 1 + − α − (41) 2 × 4 4 2 2 −1 (A.B ) 22α 1 3 1 = 2 − 8 2(1 − α) + ((α + 1) − 2), 2 × 4 2 × 4 2 × 4 × 4 − 4 2(α − 1) + ((3 − α) + 2) (42) 2 × 4 3. Reliability Analysis of Fuzzy System Using TA-based Arithmetic Operations Using TA-based fuzzy number arithmetic operations, a new procedure for analyzing fuzzy system reliability is shown in this section; the reliability of each system component is denoted by a pseudo- hexagonal fuzzy number. This is a more flexible and more generic method than all the aforementioned methods (including the interval arithmetic), and α-cuts are utilized for assessing fuzzy system reliability. 3.1. Fault Tree Analysis A fault tree usually includes the top event, the basic events and the logic gates. Gates indi- cate relationships of events. While doing the system-design, fault tree (the logic diagram) is outlined for analysis of the potential factors in system failure; factors like hard-ware, soft- ware, environment, human factor. Based on the known combinations and probabilities of basic events, we calculate the probabilities of system failure. 70 F. ABBASI AND T. ALLAHVIRANLOO 3.2. Fuzzy Operators based on TA of Fault Tree Analysis During the fuzzy fault tree analysis, the probabilities of basic events are described as fuzzy numbers and the traditional logic gate operators are replaced by fuzzy logic gate operators to obtain the fuzzy probability of the top event. In this subsection, we present a new method for analyzing fuzzy system reliability based on TA, where the reliability of the components of a system is represented by pseudo- hexagonal fuzzy number. Lemma 3.2.1: Let A , A , ... , A be pseudo-hexagonal fuzzy numbers as follows: 1 2 n a + a ¯ A = (a , a , a , a ¯ , a , a , (l (x), l (x)), (r (x), r (x)), a = , (a > 0) i i1 i2 i i3 i4 A1i A2i A2i A1i i i with the following α-cut form: A = α..A ∪ α..A , i 11iα 22iα 1 1 α∈ 0, α∈ ,1 2 2 −1 −1 −1 −1 A = l (α), r (α) , A = l (α), r (α) , 11iα 22iα A1i A1i A2i A2i then, (1) n n n n A = α.. A ∪ α.. A , i 11iα 22iα 1 1 α∈ 0, α∈ ,1 2 2 i=1 i=1 i=1 i=1 ⎡ ⎤ ⎡ ⎤ −1 −1 −1 −1 ⎣ ⎦ ⎣ ⎦ A = l (α), r (α) , A = l (α), r (α) , 11iα n n 22iα n n A1i A1i A2i A2i i=1 i=1 i=1 i=1 i=1 where n−2 i a i=1,i=n−k i −1 −1 i=2 −1 l (α) = l (α) + l (α), A1(n−k) A11 k+1 n−1 A1i k=0 i=1 n−2 n i a i=1,i=n−k i −1 −1 −1 i=2 r (α)) = r (α) + r (α), A1(n−k) A11 k+1 n−1 2 2 A1i k=0 i=1 n−2 i a i=1,i=n−k i −1 −1 i=2 −1 l (α) = l (α) + l (α), A21 A2(n−k) n−1 k+1 A2i k=0 i=1 n−2 i a i=1,i=n−k i −1 −1 i=2 −1 r (α)) = r (α) + r (α). A2(n−k) A21 k+1 n−1 2 2 A2i k=0 i=1 1 − A = α · (1 − A) ) ∪ ( α.(1 − A) ), (1 − A) i 11iα 22iα 11iα 1 1 α∈ 0, α∈ ,1 2 2 −1 −1 −1 −1 = [l (α), r (α)], (1 − A) = [l (α), r (α)], 22iα (1−A)1i (1−A)1i (1−A)2i (1−A)2i FUZZY INFORMATION AND ENGINEERING 71 where 1 1 −1 −1 −1 −1 −1 l (α) = (2 − 3a + l (α)), r (α) = (2 − 3a + r (α)), l (α) i i (1−A)1i A1i (1−A)1i A1i (1−A)2i 2 2 1 1 −1 −1 −1 = (2 − 3a + l (α)), r (α) = (2 − 3a + r (α)). i i A2i A2i (1−A)2i 2 2 Proof: We have the above cases, by mathematical induction and according to the fuzzy arithmetic operations of TA on pseudo- hexagonal fuzzy numbers. Consider a serial system shown in Figure 6, where the reliability R of component x is i i represented by a pseudo- hexagonal fuzzy number defined in the universe of discourse [0, 1]: r + r¯ R = (r , r , r , r¯ , r , r , (l (x), l (x)), (r (x), r (x)), r = , (r > 0) i i1 i2 i i3 i4 R1i R2i R2i R1i i i or, −1 −1 R = α.R α.R , R = l (α), r (α) , i 11iα 22iα 11iα 1 1 R1i R1i α∈ 0, α∈ ,1 2 2 −1 −1 R = l (α), r (α) . 22iα R2i R2i Then, the reliability R of the serial system can be evaluated by the (3.1) lemma as follows: n n n n R = R · R ... R = R = α. R ∪ α. R , R 1 2 n i 11iα 22iα 11iα 1 1 α∈ 0, α∈ ,1 2 2 i=1 i=1 i=1 i=1 ⎡ ⎤ ⎡ ⎤ −1 −1 −1 −1 ⎣ ⎦ ⎣ ⎦ = l (α), r (α) , R = l (α), r (α) , n n 22iα n n R1i R1i R2i R2i i=1 i=1 i=1 i=1 i=1 where n−2 n i r i=1,i=n−k i −1 −1 −1 i=2 l (α) = l (α) + l (α), R1(n−k) R11 k+1 n−1 2 2 R1i k=0 i=1 n−2 n i r i=1,i=n−k i −1 −1 i=2 −1 r (α)) = r (α) + r (α), R1(n−k) R11 k+1 n−1 2 2 R1i k=0 i=1 n−2 i r i=1,i=n−k i −1 −1 i=2 −1 l (α) = l (α) + l (α), R2(n−k) R21 k+1 n−1 2 2 R2i k=0 i=1 n−2 i r i=1,i=n−k i −1 −1 i=2 −1 r (α)) = r (α) + r (α). R2(n−k) R21 k+1 n−1 2 2 R2i k=0 i=1 72 F. ABBASI AND T. ALLAHVIRANLOO Figure 6. Configuration of a serial system. Furthermore, consider the parallel system shown in Figure 7, where the reliability A of component x is represented by a pseudo- hexagonal fuzzy number defined in the universe of discourse [0, 1]: r + r¯ R = (r , r , r , r¯ , r , r , (l (x), l (x)), (r (x), r (x)), r = , i i1 i2 i i3 i4 R1i R2i R2i R1i i or, −1 −1 R = α.R ∪ α.R , R = l (α), r (α) , i 11iα 22iα 11iα 1 1 R1i R1i α∈ 0, α∈ ,1 2 2 −1 −1 R = l (α), r (α) . 22iα R2i R2i Then, the reliability R of the parallel system can be evaluated as follows: R = 1− (1 − R ) = α.R α ∪ α.R α , R α i 11 22 11 1 1 α∈ 0, α∈ ,1 2 2 i=1 ⎧ ⎫ ⎧ ⎫ ⎡ ⎤ n n ⎨ ⎬ ⎨ ⎬ 1 1 −1 −1 ⎣ ⎦ = 2 − 3 1 − r + l (α) , 2 − 3 1 − r + r (α) , R α i i 22 n n ⎩ ⎭ ⎩ ⎭ 2 2 (1−R)1i (1−R)1i i=1 i=1 i=1 i=1 ⎡ ⎧ ⎫ ⎧ ⎫ ⎤ n n ⎨ ⎬ ⎨ ⎬ 1 1 −1 −1 ⎣ ⎦ = 2 − 3 1 − r + l (α) , 2 − 3 1 − r + r (α) , i n i n 2⎩ ⎭ 2⎩ ⎭ (1−R)2i (1−R)2i i=1 i=1 i=1 i=1 where n n n (1 − R ) = α. (1 − R) ) ∪ α. (1 − R) ) , i 11iα 22iα 1 1 α∈ 0, α∈ ,1 2 2 i=1 i=1 i=1 ⎡ ⎤ −1 −1 ⎣ ⎦ (1 − R) = l (α), r (α) , 11iα n n (1−R)1i (1−R)1i i=1 i=1 i=1 ⎡ ⎤ −1 −1 ⎣ ⎦ (1 − R) = l (α), r (α) , 22iα n n (1−R)2i (1−R)2i i=1 i=1 i=1 n−2 (1 − r ) i (1 − r ) i=1,i=n−k i −1 −1 i=2 −1 l (α) = l (α) + l (α), (1−R)1(n−k) (1−R)11 k+1 n−1 2 2 (1−R)1i k=0 i=1 n−2 (1 − r ) i (1 − r ) i=1,i=n−k i −1 −1 i=2 −1 r (α)) = r (α) + r (α), (1−R)1(n−k) n−1 (1−R)11 k+1 (1−R)1i k=0 i=1 FUZZY INFORMATION AND ENGINEERING 73 Figure 7. Configuration of a parallel sysytem. n−2 n (1 − r ) i (1 − r ) i=1,i=n−k i −1 −1 i=2 −1 l (α) = l (α) + l (α), (1−R)21 (1−R)2(n−k) n−1 k+1 2 2 (1−R)2i k=0 i=1 n−2 (1 − r ) i (1 − r ) i=1,i=n−k i −1 −1 i=2 −1 (α)) = r (α) + r (α), (1−R)2(n−k) (1−R)21 k+1 n−1 2 2 (1−R)2i k=0 i=1 −1 −1 −1 −1 −1 1 1 l (α) = (2 − 3r + l (α)), r (α) = (2 − 3r + r (α)), l (α) i i (1−R)1i 2 R1i (1−R)1i 2 R1i (1−R)2i 1 1 −1 −1 −1 = (2 − 3r + l (α)), r (α) = (2 − 3r + r (α)). i i R2i (1−R)2i R2i 2 2 In the following, we use an example to illustrate the fuzzy system reliability analysis process. 4. A Technical Example A marine power plant [4] has two generators G1 and G2 one located at the stern and the other at the bow. Each generator is connected to its respective micro switch board-1 and micro switch board-2. The distributive switchboard receives the supply from the switchboards through cables C1 and C2 and respective junction boxes D and E. The two micro switchboards are intercon- nected through a long cable C3 and the junction boxes A and B. The schematic diagram is shown in Figure 8. Let us assume that basic components subjected to failure are (a) Generators G1 and G2. (b) Microswitch board-1 (MSB-1) and Microswitch board-2 (MSB-2). (c) Interconnecting cable C3 and junction boxes A and B, all are treated as one unit. (d) Junction boxes D and E. (e) Distributive switchboard (DSB). 74 F. ABBASI AND T. ALLAHVIRANLOO Figure 8. Marine Power Plant. In this example, we show a failure of the marine power plant in the form of a fuzzy num- ber for the more comprehensive analysis to improve the educational process. A fault tree for the top event ‘failure of the marine power plant’ is shown in Figure 9. Due to a more accurate estimate of each failure event and generalized reliability anal- ysis of the system shown in Figure 9, let us assume the basic events of this fault tree have the following pseudo- hexagonal fuzzy number defined in the universe of discourse [0, 1]: r + r¯ R = (r , r , r , r¯ , r , r , (l (x), l (x)), (r (x), r (x)), r = , i i1 i2 i i3 i4 R1i R2i R2i R1i i or, −1 −1 R = α.R ∪ α.R , R = l (α), r (α) , i 11iα 22iα 11iα 1 1 R1i R1i α∈ 0, α∈ ,1 2 2 −1 −1 R = l (α), r (α) , i = 1, 2, ... , 22, 22iα R2i R2i where R , represents the unreliability of the distributive switchboard. R , represents the reliability of the event that no power is coming to distributive switch- board. R , represents the reliability of the event that there is no power supply from the junction box D. FUZZY INFORMATION AND ENGINEERING 75 Figure 9. Faulttreeofmarine powerplant. R , represents the reliability of the event that there is no power supply from the junction box E. R , represents the unreliability of the junction box D. R , represents the reliability of the event that there is no power supply to the junction box D. R , represents the unreliability of micro switchboard-1. R , represents the reliability of the event that there is no power supply to micro switch board-1. R , represents the unreliability of generator G1. R , represents the reliability of the event that there is no power supply through the junction boxes A and B. R , represents the unreliability of generator G2. R , represents the unreliability of the junction boxes A and B. R , represents the unreliability of micro switchboard-2. 13 76 F. ABBASI AND T. ALLAHVIRANLOO R , represents unreliability of the junction box E. R , represents the reliability of the event that there is no power supply to the junction box E. R , represents the unreliability of micro switchboard-2. R , represents the reliability of the event that there is no power supply to micro switchboard-2. R , represents the unreliability of generator G2. R , represents the reliability of the event that there is no power supply through the junction boxes D and E. R , represents the unreliability of generator G1. R , represents the unreliability of the junction boxes A and B. R , represents the unreliability of micro switchboard-1. Based on the previous discussion(the reliability of the serial and parallel systems), we get a failure of the marine power plant (R) as follows: R = 1 − [(1 − R )(1 − R )], 1 2 R = R · R , 2 3 4 where the calculation of R : R = 1 − [(1 − R )(1 − R )], 3 5 6 R = 1 − [(1 − R )(1 − R )], 6 7 8 R = R · R , 8 9 10 R = 1 − [(1 − R )(1 − R )(1 − R )], 10 11 12 13 the calculation of R : R = 1 − [(1 − R )(1 − R )], 4 14 15 R = 1 − [(1 − R )(1 − R )], 15 16 17 R = R · R , 17 18 19 R = 1 − [(1 − R )(1 − R )(1 − R )]. 19 20 21 22 Finally, we can calculate the system reliability R by the (3.2.1) lemma. If we required the system to have a fault probability of x as a limit, then, α ≥ α is necessary, where α = 0 0 0 inf {α|x ∈ / R }. In this case, we allow the system to be uncertain and flexible to an extent 0 α that, the fault probabilities be in the R . It is worth mentioning, the proposed model is applicable for every marine power plant with having the statistical data. 5. Conclusion and Future Research Fuzzy reliability is based on the concept of fuzzy set. When the failure rate is fuzzy, according to Zadeh’s extension principle, the reliability measure will be fuzzy as well. In this paper, the FUZZY INFORMATION AND ENGINEERING 77 use of the concept of pseudo-hexagonal fuzzy numbers and the component failure proba- bilities are considered as a new type of fuzzy number as pseudo-hexagonal to incorporate the uncertainties in the parameter, due to a more realistic estimate of them. We used the new TA-operations [1,2], because of smaller results, easier computations and some particular properties. The developed method has been used to analyze the fuzzy reliability of a marine power plant. The major advantage of using the pseudo-hexagonal fuzzy numbers and the new operations of transmission average (TA), is the smaller results support, easier calculations and special properties than fuzzy arithmetic operations based on the extension principle (in the domain of the membership function) and the interval arithmetic (in the domain of the α-cuts). The proposed methodology can be used for a more general problem when systems are distributed according to other fuzzy numbers. The future work of this study will focus on the fuzzy arithmetic operations based on TA for n-polygonal fuzzy numbers and its application in fuzzy system reliability analysis. Acknowledgements We are grateful to the referee for their valuable suggestions, which have improved this paper. Disclosure statement No potential conflict of interest was reported by the authors. Notes on Contributors Fazlollah Abbasi is Ph.D of Applied Mathematics. He is member of Department of Mathematics, Aya- tollah Amoli Branch, Islamic Azad University, Amol, Iran. His research interests are in the field of fuzzy mathematics. Tofigh Allahviranloo is a Senior Full Professor of Applied Mathematics at Bahcesehir International University (BAU). His research interests are in the field of uncertain mathematics, uncertain dynamical systems, decision science, fuzzy systems. References [1] Abbasi F, Allahviranloo T, Abbasbandy S. A new attitude coupled with fuzzy thinking to fuzzy rings and fields. J Intell Fuzzy Syst. 2015;29:851–861. [2] Abbasi F, Abbasbandy S, Nieto JJ. A new and efficient method for elementary fuzzy arithmetic operations on pseudo-geometric fuzzy numbers. J Fuzzy Set Valued Anal. 2016;2:156–173. [3] Ye F, Lin Q. Partner selection in a virtual enterprise: a group multiattribute decision model with weighted possibilistic mean values. Math Probl Eng. 2013;13:1–14. [4] Zhang XH, Xu XH, Tao L. Some similarity measures for triangular fuzzy numbers and their applications in multiple criteria groups decision making. J Appl Math. 2013;13:1–7. [5] Cheng CH, Mon DL. Fuzzy system reliability analysis by interval of confidence. Fuzzy Sets Syst. 1993;56:29–35. [6] Chen SM. Fuzzy system reliability analysis using fuzzy number arithmetic operations. Fuzzy Sets Syst. 1994;64:31–38. [7] Chen SM. Analyzing fuzzy system reliability using vague set theory. Int. J. Appl. SciEng. 2003;1(1):82–88. [8] Mon DL, Cheng CH. Fuzzy system reliability analysis for components with different membership functions. Fuzzy Sets Syst. 1994;64:145–157. [9] Onisawa T, Kacprzyk J. Reliability and safety analyses under fuzziness. Heidelberg: Physica- Verlag; 1995. 78 F. ABBASI AND T. ALLAHVIRANLOO [10] Wu HC. Fuzzy reliability analysis based on closed fuzzy numbers. Inf Sci (Ny). 1997;103:135–159. [11] Zadeh LA. The concept of a linguistic variable and its application to approximate reasoning – Part I, II, III. Inf Sci (Ny). 1975;9(1):43–80. [12] Fullór R. Fuzzy reasoning and fuzzy optimization. No. 9. Abo: Turku Centre for Computer Science; [13] Zimmermann HJ. Fuzzy set theory and its applications. New York (NY): Springer; 2001.

Journal

Fuzzy Information and Engineering – Taylor & Francis

Published: Jan 2, 2021

Keywords: Reliability analysis; fault tree analysis; pseudo- hexagonal fuzzy numbers; extension principle (EP); transmission average (TA)

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