Converting Z-number to Fuzzy Number using. Fuzzy Expected Value

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1 ISSN , Engand, UK Journa of Information and Computing Science Vo. 1, No. 4, 017, pp Converting Z-number to Fuzzy Number using Fuzzy Expected Vaue Mahdieh Akhbari * Department of Industria Engineering, E-Branch, Isamic Azad University, Tehran, Iran E-mai: m_akhbari@iauec.ac.ir. (Received March 0, 017, accepted May 11, 017) Abstract. In order to dea with uncertain information of rea-word, in 011 Zadeh suggested the concept of a Z-number, as an ordered pair of fuzzy numbers (A,B ) that describes the restriction and the reiabiity of the evauation. Due to the imitation of its basic properties, converting Z-number to cassica fuzzy number is rather significant for appication. In this paper, we wi cacuate fuzzy expected vaue of a Z-number with assuming uniform distribution and inear membership functions. This fuzzy expected vaue can be used instead of Z-number in appications. An Exampe is used to iustrate the procedure of the proposed approach. Keywords: Z-number; Fuzzy Expected vaue; Fuzzy Set 1. Introduction In the rea word, decisions are based on information usuay uncertain, imprecise and/or incompete, such information is often characterized by fuzziness, but it is not sufficient and the other important property of information is that information must be reiabe. Thus, fuzziness from the one side and reiabiity form the other side are strongy associated to each other. In order to take into account this fact, in 011 Zadeh proposed a concept, namey Z-number, which is an order pair of fuzzy numbers (A, B ). The first component, A, is a fuzzy restriction and the second component B is a reiabiity of the first component. Typicay, A and B are described in a natura anguage for exampe (about 5 minutes, very sure) [1,, 3]. In 01, Yager used Z-number to provide a simpe iustration of a Z-vauation (X, A, B ). He showed that these Z-vauations essentiay induce a possibiity distribution G(p) over probabiity distributions associated with an uncertain variabe X and used this representation to make decisions and answer questions. He suggested manipuation and combination of mutipe Z-vauations. He showed the reationship between Z-numbers and inguistic summaries and provided for a representation of Z-vauations in terms of Dempster Shafer beief structures, which made use of type- fuzzy sets [4,5].Yager found a fuzzy expected vaue of a Z-number whose probabiity density function is expressibe in terms of an exponentia distribution. Kang et a. [6] proposed a simpe method for converting Z-numbers with range [0, 1] to the cassica fuzzy numbers in 3 step: first convert the second part (reiabiity) into a crisp number and then cacuate the weighted Z-number by adding the weight of the second part (reiabiity) to the first part (restriction) and finay converting the weighted fuzzy number to norma fuzzy number. The benefit of the proposed method is represented by its ow anaytica and computationa compexity, hence their theorem for converting Z- number to cassica fuzzy number was used in some papers [7,8,9,10]. Gardashova [11] suggested an agorithm of decision making method using Z-numbers, in 5 steps: Construction of the fuzzy decision making matrix, transforming the inguistic vaue to numerica vaue, normaizing the fuzzy decision making matrix, Converting the Z-numbers to crisp number and Determining the priority weight of each aternative. He used a simpe way of the canonica representation of mutipication operation on trianguar fuzzy numbers [1] for the converting the Z-numbers to crisp number. In this study we find a fuzzy expected vaue of a Z-number whose probabiity density function is expressibe in terms of a uniform distribution. The resut can be used in many appications to converting Z- number to fuzzy number. The remainder of the paper is organized as foows: In Section we discuss the concept of Z-number, Z-number with the probabiity based on uniform distribution and the probabiity of a restriction (Ã) on the vaues of uncertain variabe (X) based on the parameters of uniform distribution. Section 3 expains the method of cacuating the maximum probabiity of Ã. Later in section 4, the membership function of the Pubished by Word Academic Press, Word Academic Union

2 9 Mahdieh Akhbari: Converting Z-number to Fuzzy Number using Fuzzy Expected Vaue possibiity of the expected vaue of Z-number (µ) is proposed. Finay, the whoe presented materia is summed up in section 5.. The concept of Z-numbers Any Z-number is an ordered pair of fuzzy numbers (A, B ) where à is a restriction on the vaues of uncer tain variabe, X, and B is the eve of confidence in Ã. Both à and B are fuzzy sets. Zadeh[1] defines Z-vauat ion as an ordered tripe, (X,A,B ), which is equivaent to X is (A, B ). Moreover, Z-vauation can be defined as Prob(X is A ) is B (1) According to the Zadeh s equation, we express the probabiity that X is à as: Prob p (X is A ) = A(x)f(x)dx () R where A(x) is the membership function of à and f(x) is the probabiity density function of X on R. Now, we can get G(p), the degree to which p satisfies our Z-vauation, Prob _p (X is A ) is B, as: G(p) = B (Prob p (X is A)) = B( R A(x)f(x)dx) (3) Since the probabiity density function of X is unknown, according to Zadeh s simpifying assumption, a particuar set of parametric distributions (e.g. norma, exponentia and uniform distributions) can be appied based on avaiabe knowedge about the variabe [1]. But when the knowedge avaiabe is not sufficient, the uniform distribution is proposed for a poory known variabe [13]. This distribution is sometimes referred to as the no knowedge distribution. Hence, we used uniform distribution in the current study. Let (A,B ) be a Z-number,where à is a trianguar number, (c-, c, c+r), with a membership function as foows: 0 if x < c (x c + )/ if c x < c A(x) = (c + r x)/r if c x c + r (4) 0 if x > c + r On the other hand, B, confidence, coud be from the set Likey, Usuay, Sure } are modeed using the right hand fuzzy sets as foows (see Fig. 1) 0 P < 0.5 P 0.5 B L (P) = 0.5 P 0.6 (5) B U (P) = < P 0 P < P 0.75 P < P P < 0.8 P 0.8 B S (P) = 0.8 P < P (6) (7) Fig 1. Fuzzy sets of inguistic reiabiity vaues. Assuming that the probabiity density function, f(x), is uniform as: f(x) = 1, a x b. (8) b a JIC emai for contribution: editor@jic.org.uk

3 Category Journa1 Information and Computing Science, Vo. 1(017) No. 4, pp Therefore, based on equation (), the probabiity of à is: b a P = f(x)a(x)dx min (b,c+r) A(x)dx = 1 b a max (a,c ) Now, having equation (9), P (the probabiity of Ã) can be determined based on the positions of a and b reative to the fuzzy number à (Tabe 1). Tabe 1: The probabiity of à based on the positions of a and b reative to à (9) positions of a and b a c, b > c + r a c, c b c + r a < c, b < c c < a < c, c b < c + r c a c, b < c c a c + r, c < b < c + r c < a < c, c + r b c a c + r, c + b P 1 = P = P 3 = P 4 = P 5 = P 6 = P 7 = P 8 = + r (b a) Probabiity (Pi) (b c)(r + c b) + (b a) (b a)r (b c + ) (b a) (c a)( + a c) (b c)(r + c b) + (b a) (b a)r a + b c + r + c a b r (c a)( + a c) (b a) (r + c a) r(b a) r + (b a) 3. Expected vaue of Z-number We reca for the uniform distribution (4) that E(x) = μ = a+b. (10) As discussed earier, the parameters of the density function (a and b) are unknown and μ, the expected vaue can be determined based on the given Z-number. Assuming a particuar µ wi ead to various pairs of (a,b) with the mean vaue of µ which can then be used to cacuate probabiity of à according to Tabe 1. Exampe 1. Let a Z-number be ((4,30,36), Sure) which means fuzzy number (Ã) and confidence (B ) are (4,30,36) and Sure respectivey. By assuming uniform distribution and µ=9, Fig. depicts change in the probabiity of à for various pairs of (a,b)with the mean vaue of 9. JIC emai for subscription: pubishing@wau.org.uk

4 94 Mahdieh Akhbari: Converting Z-number to Fuzzy Number using Fuzzy Expected Vaue Fig. Changes in probabiity for µ = 9 Fuzzy membership degrees of each (a,b) can be computed based on the membership function of confidence, B, in the Z-number. Since the eve of confidence was set at Sure, membership vaues of ordered pairs (a,b), with the mean vaue of 9, was determined based on (7). Then an appropriate S-norm, here maximum, can be used to obtain the membership function G over the space µ R : G(μ) = maxb(p(a, b)) a + b = μ}, μ R, (11) (a,b) Since higher probabiity of (a,b) increases their membership degree in the confidence function, the (11) can be rewritten as: G(μ) = B(max P(a, b) a + b = μ}), μ R (1) (a,b) Then we can first find the maximum probabiity of à with the particuar μ and then compute its possibiity by using equation (1). For exampe, In Exampe 1, the maximum probabiity of à with µ = 9 is According to equation (7), the membership vaue of this event in the Sure is 0.3. In order to find the maximum probabiity of à for different vaues of µ, we evauated it in four intervas (Tabe ). Tabe - Four intervas of µ to find the maximum PROBABILITY of à State Interva 1 μ c a + b c c / < μ c c < a + b c 3 c < μ c + r/ c < a + b c + r 4 c + r < μ c + r < a + b If a and b fa in any of the eight categories summarized in Tabe 1, for particuar μs, the maximum probabiity of à in the above-mentioned Intervas wi be as foows: 3.1 First Interva (µ c - ) If a and b faing in the first category of Tabe 1, the maximum probabiity of à for µ c - is obtained as foows (Fig. 3): a c b > c + r P 1 = +r = +r (13) (b a) 4(b μ) dp 1 = (+r) dp 1 < 0 maxp db 4(b μ) db 1 = +r (14) 4(c+r μ) JIC emai for contribution: editor@jic.org.uk

5 Journa1 Information and Computing Science, Vo. 1(017) No. 4, pp Fig 3. The area invoved in cacuation of the maximum probabiity (P1) when µ c - / When a and b beong to the second category of Tabe 1, the maximum probabiity of à with µ c - is obtained as: a c c b c + r P = r+(b c)(r+c b) (15) (b a)r dp = 0 db b = (c + r μ) r r + μ (16) In other words, if no restriction is imposed, the maximum of P based on b wi occur at a point ike b *. Theorem 1. If µ c -, then b* c Proof by contradiction: If b * < c, then b < c (c + r μ) r r + μ < c (c + r μ) r r < (c μ) c < μ Which is a contradiction since we had an initia assumption of µ c -. Therefore, b* c is correct. Now, the maximum of P based on b and the position of b * can be determined as (Fig. 4): maxp = r+(b c)(r+c b ) 4(b μ)r +r (c+r μ) c < b < c + r c + r b (17) Fig 4.The area invoved in cacuation of the maximum probabiity (P) withµ c - / If a and b fit in the third category of Tabe 1, the maximum probabiity of à with µ c - can be cacuated as (Fig. 5): a < c (b c + ) P b < c 3 = 4(b μ) JIC emai for subscription: pubishing@wau.org.uk

6 96 Mahdieh Akhbari: Converting Z-number to Fuzzy Number using Fuzzy Expected Vaue dp 3 db = ( a + c )(b c + ) 4(b μ) a<c ( a+c )>0 b>c (b c+)>0 maxp 3 = (c c+) 4(c μ) = 4(c μ) dp 3 db > 0 (18) Fig 5. The area invoved in cacuation of the maximum probabiity (P3) when µ c - / With a and b ying in the fourth category of Tabe 1, the maximum probabiity of à with µ c - can be computed as: c < a < c c b < c + r c < a + b < c + r c / < μ P 4 = 0 (19) The maximum probabiity of à with µ c - when a and b beong to the fifth category of Tabe 1 is determined as (Fig. 6): c a c P b < c 5 = a+b c+ = μ c+ maxp 5 = μ c+ (0) P 5 =cte Fig 6. The area invoved in cacuation of the maximum probabiity (P5) when µ c - / The maximum probabiity of à with µ c - when a and b fa into the sixth-eighth categories of Tabe 1 can be defined as: c r < a < c c < a + b P c + r b 7 = 0 (1) a+b c r c a c + r c + r < a + b P c + r b 8 = 0 () a+b c r Therefore, according to equations (14),0, 错误! 未找到引用源, and (0), when µ c -, the maximum probabiity of à can be stated as: maxp(μ) = maxmaxp 1, maxp, maxp 3, maxp 5 } (3) And based on 0: JIC emai for contribution: editor@jic.org.uk

7 Journa1 Information and Computing Science, Vo. 1(017) No. 4, pp If c + r b *, then: max maxp(μ) = +r,, μ c+ 4(c+r μ) 4(c μ) max r+(b c)(r+c b ) 4(b μ)r } c + r b, μ c+ } c < b < c + r μ c+ = 4(c μ)(μ c+) = ( (c μ)) > 0 > μ c+ 4(c μ) 4(c μ) 4(c μ) 4(c μ) and 4(c μ) + r 4(c + r μ) = r 4 ( c + μ) (c + r μ)(c μ) Now, two cases may occur: c + r + μ 0 1 μ c c + r μ > 0 c μ > 0 4(c μ) < + r 4(c + r μ) c + + μ > 0 c < μ c / c + r μ > 0 c μ < 0 4(c μ) < + r 4(c + r μ) Hence, in both cases, the foowing is true: Based on (6)-(7), it can be concuded that: Or: max μ c / 4(c μ) < +r 4(c+r μ) +r,, μ c+ 4(c+r μ) 4(c μ) } = +r 4(c+r μ) c + r b maxp(μ) = maxp = maxp 1 = +r 4(c+r μ) If c < b * < c + r, then equation (17) entais that: r + (b c)(r + c b ) μ c + 4(b > μ)r So μ c c < b < c + r maxp(μ) = maxp = r+(b c)(r+c b ) (9) 4(b μ)r Finay, under the first condition, µ c -, equations (7) and (9) indicate that: μ c maxp(μ) = maxp (30) 3. Second Interva (c - < µ c) According to equation (14), the maximum probabiity of à with c - <µ c when a and b beong to the first category of Tabe 1 is (Fig. 7): a c b > c + r maxp 1 = +r (31) 4(c+r μ) (4) (5) (6) (7) (8) Fig 7. The area invoved in cacuation of the maximum probabiity (P1) when c - /< µ c With a and b fitting into the second category of Tabe 1, the maximum probabiity of à with c - <µ c can be obtained based on equations (15) and (17): JIC emai for subscription: pubishing@wau.org.uk

8 98 Mahdieh Akhbari: Converting Z-number to Fuzzy Number using Fuzzy Expected Vaue a c c b c + r P r+br+bc b cr c = (3) 4(b μ)r dp = 0 db b = (c + r μ) r r + μ (33) Which eads to three distinct conditions: b c dp < 0 maxp db = 4(c μ) c < b < c + r maxp = r+(b c)(r+c b ) (34) c + r b dp db > 0 maxp = 4(b μ)r +r 4(c+r μ) Fig 8. The area invoved in cacuation of the maximum probabiity (P) when c - /< µ c When a and b fa into the third category of Tabe 1, the maximum probabiity of à with c - <µ c can be determined as: a < c a + b < c μ < c / maxp b < c 3 = 0 (35) The maximum probabiity of à with c - <µ c when a and b beong to the fourth category of Tabe 1 wi be: c < a < c c b < c + r P 4 = (c a)(+a c) + (b c)(r+c b) 4(b μ) 4(b μ)r (36) Fig 9. The area invoved in cacuation of the maximum probabiity (P4) when c - /< µ c With a and b ying in the fifth category of Tabe 1, the maximum probabiity of à with c - <µ c can be cacuated as: c a c maxp b < c 5 = μ c+ (37) JIC emai for contribution: editor@jic.org.uk

9 Journa1 Information and Computing Science, Vo. 1(017) No. 4, pp Fig 10. The area invoved in cacuation of the maximum probabiity (P5) when c - /< µ c Theorem. P5> P4 is aways true. Proof: P 5 P 4 = r(b c) + (b c) 4(b μ)r 0 P 5 P 4 Finay, the maximum probabiity of à with c - <µ c when a and b beong to the sixth, seventh and eighth categories of Tabe 1 can be stated as: c a c + r c < b < c + r c < a + b P 6 = 0 (38) a+b c c r < a < c c < a + b P c + r b 7 = 0 (39) a+b c c a c + r c + r < a + b P c + r b 8 = 0 (40) a+b c According to the equation (34): And based on Theorem : maxp maxp 1 (41) maxp 5 maxp 4 (4) Therefore, when c - <µ c, equations (41) and (4) ead to the foowing equation: Theorem 3. If c - <µ c, then P5>P is aways true. Proof: P P 5 = 4r(μ a /)(μ c + /) (b c) (b a)r P 5 > P maxp(μ) = maxmaxp, maxp 5 } (43) P P 5 < 0 c /<μ c c <μ / 0<μ c+/ a<c 0<μ a Based on Theorem 3 if c- <µ c, then maxp 5>maxP is aways true. JIC emai for subscription: pubishing@wau.org.uk

10 300 Mahdieh Akhbari: Converting Z-number to Fuzzy Number using Fuzzy Expected Vaue Hence, when c - < µ c, it can be concuded that: c < μ c maxp(μ) = maxp 5 (44) 3.3 Third Interva (c < μ c + r/) With cacuations Simiar to second interva, we have: c < μ c + r maxp(μ) = maxp 6 = r+c μ 3.4 Forth Interva (c + r/ < μ) r (45) With cacuations Simiar to first interva, we have: c + r < μ maxp(μ) = maxp 7 (46) Maximum of P 7 can be computed as: r + (c a )( + a c) 4(μ a maxp 7 = ) + r 4(c μ) c < a < c a < c Where a = (μ c + ) r + μ Now, according to the equations (30) and (44)-(46), it can be inferred that: maxp μ c / maxp maxp(μ) = 5 c / < μ c maxp 6 c < μ c + r/ maxp 7 c + r/ < μ Theorem 4. Minimum of maxp 5 equas 0.5. Proof: maxp 5 = μ c+ Theorem 5. Minimum of maxp 6 equas 0.5. Proof: Simiar to Theorem 4. dmaxp 5 dμ (47) c > 0 c+ = 0.5 (48) μ=c Fig 11. The maximum probabiity curve for the constant mean (µ) for Exampe 1 After cacuating the maximum probabiity of Ã, the membership function of the µ at three eves of the possibiity (L, U, and S) can be computed using equations (5)-(7). JIC emai for contribution: editor@jic.org.uk

11 Journa1 Information and Computing Science, Vo. 1(017) No. 4, pp Cacuating the membership function of the expected vaue of Z-number (µ) By repacing probabiity vaues (P), obtained from equation (47), in reevant equation of (5)-(7), the membership function of the µ can be determined. G μ L(p) = G μ U(p) = G μ S(p) = 0 p < c 0.5 p c c 0.5 p < c c 0.4 p < c + 0.4r c+0.5r p 0.1r c + 0.4r p < c + 0.5r 0 c + 0.5r p 0 p < c 0.35 p c c 0.35 p < c c 0.5 p < c + 0.5r c+0.35r p 0.1r c + 0.5r p < c r 0 c r p 0 p < c 0. p c+0. 0 c + 0.r p Thus, the µ in L, U, and S eves of the confidence are trapezoida fuzzy numbers shown as: 0.1 c 0. p < c c 0.1 p < c + 0.1r c+0.r p 0.1r c + 0.1r p < c + 0.r (49) (50) (51) μ L = (c 0.5, c 0.4, c + 0.4r, c + 0.5r) (5) μ U = (c 0.35, c 0.5, c + 0.5r, c r) (53) This resut for Exampe1 is μ S = (c 0., c 0.1, c + 0.1r, c + 0.r) (54) G μ (p) = 0 p < 8.8 p p < p < p p < p Or the fuzzy expected vaue is: μ = (8.8, 9.4, 30.6, 31.). We can compare this resut with fuzzy expectation proposed by Kang et a [6]. In their method first, the reiabiity of Z-number (B ) is converted to crisp number with the center of gravity method. α = xμ S (x)dx μ S (x)dx = 0.9 Then the weighted Z-number is obtained by adding the α to the first part (Ã). (Ã, α) =((4,30,36);0.9). Finay, weighted Z-number is converted to reguar fuzzy number: μ = ( 0.9 4, , ) = (8.8, 5.76, 5.76) (55) JIC emai for subscription: pubishing@wau.org.uk

12 30 Mahdieh Akhbari: Converting Z-number to Fuzzy Number using Fuzzy Expected Vaue Fig 1.The membership function of the possibiity ofµ As you can see in equations (5)-(54), in this approach the more reiabiity resuts smaer support set of the obtained fuzzy expected vaue. Aso the center of the first part of Z-number (Ã) doesn t change in this approach, however using the Kang et a [6] method, if the first part of Z-number (Ã) is positive, the more reiabiity resuts the smaer fuzzy expectation, without any reason. 5. Concusion The main contribution of the present paper is converting a Z-number to a fuzzy number which is very significant for appication. Hence the probabiity density function of uncertain variabe, X, in Z-vauation (X, A, B ) is unknown, according to Zadeh s simpifying assumption, a particuar set of parametric distributions can be appied based on avaiabe knowedge about the variabe. But when the knowedge avaiabe is not sufficient, the uniform distribution is proposed for a poory known variabe. Therefore, in practice, most appications use inear membership functions to describe fuzzy sets, in this study inear righthand membership functions are used for reiabiity vaues. With above assumptions, that is proved that the fuzzy expected vaue of a Z-number is a trapezoida fuzzy number. 6. References A. Zadeh, Lotfi "A note on Z-numbers." Information Sciences (011): A. Zadeh, Lotfi "The concept of a Z-number-A new direction in uncertain computation." Information Reuse and Integration (IRI), 011 IEEE Internationa Conference on. IEEE, 011. A. Aiev, Rafik, V. Akif Aizadeh, and Oeg H. Huseynov. "The arithmetic of discrete Z-numbers." Information Sciences 90 (015): R. Yager, Ronad "On a view of Zadeh s Z-numbers." Internationa Conference on Information Processing and Management of Uncertainty in Knowedge-Based Systems. Springer Berin Heideberg, 01. R. Yager, Ronad "On Z vauations using Zadeh's Z numbers." Internationa Journa of Inteigent Systems 7.3 (01): Kang, Bingyi, Daijun Wei, Ya Li, and Yong Deng. "A method of converting Z-number to cassica fuzzy number." Journa of Information and Computationa Science 9.3 (01): Azadeh, A., Kokabi, R., Saberi, M., Hussain, F.K. and Hussain, O.K., "Trust prediction using z-numbers and artificia neura networks." 014 IEEE Internationa Conference on Fuzzy Systems (FUZZ-IEEE). IEEE, 014. Azadeh, Ai, and Reza Kokabi. "Z-number DEA: A new possibiistic DEA in the context of Z-numbers." Advanced Engineering Informatics 30.3 (016): Yaakob, Abdu Maek, and Aexander Gegov. "Fuzzy rue based approach with z-numbers for seection of aternatives using TOPSIS." Fuzzy Systems (FUZZ-IEEE), 015 IEEE Internationa Conference on. IEEE, 015. Azadeh, A., Saberi, M., Atashbar, N.Z., Chang, E. and Pazhoheshfar, P. "Z-AHP: a Z-number extension of fuzzy anaytica hierarchy process." 013 7th IEEE Internationa Conference on Digita Ecosystems and Technoogies (DEST). IEEE, 013. Gardashova, Latafat A. "Appication of operationa approaches to soving decision making probem using Z- numbers." Appied Mathematics 5.09 (014): 133. Chou, Ch-Ch. "The canonica representation of mutipication operation on trianguar fuzzy numbers." Computers JIC emai for contribution: editor@jic.org.uk

13 Journa1 Information and Computing Science, Vo. 1(017) No. 4, pp & Mathematics with Appications (003): Seier, F. A., & Avarez, J. L. On the seection of distributions for stochastic variabes. Risk Anaysis16(1) (1996): JIC emai for subscription: