On possibilistic mean value and variance of fuzzy numbers
|
|
- Jasper Robbins
- 5 years ago
- Views:
Transcription
1 On possibilistic mean value and variance of fuzzy numbers Supported by the E-MS Bullwhip project TEKES 4965/98 Christer Carlsson Institute for Advanced Management Systems Research, Robert Fullér Institute for Advanced Management Systems Research, TUCS Turku Centre for Computer Science TUCS Technical Report No 99 August 1999 ISBN ISSN
2 Abstract Dubois and Prade introduced the mean value of a fuzzy number as a closed interval bounded by the expectations calculated from its upper and lower distribution functions. In this paper introducing the notations of lower possibilistic and upper possibilistic mean values we definine the interval-valued possibilistic mean and investigate its relationship to the interval-valued probabilistic mean. We also introduce the notation of crisp possibilistic mean value and crisp possibilistic variance of continuous possibility distributions, which are consistent with the extension principle. We also show that the variance of linear combination of fuzzy numbers can be computed in a similar manner as in probability theory. Keywords: Fuzzy number, possibility distribution, mean value, variance TUCS Research Group Institute for Advanced Management Systems Research
3 Keywords: Fuzzy numbers; Possibility theory 1 Introduction In 1987 Dubois and Prade [] defined an interval-valued expectation of fuzzy numbers, viewing them as consonant random sets. They also showed that this expectation remains additive in the sense of addition of fuzzy numbers. In this paper introducing the notations of lower possibilistic and upper possibilistic mean values we definine the interval-valued possibilistic mean, crisp possibilistic mean value and crisp (possibilistic) variance of a continuous possibility distribution, which are consistent with the extension principle and with the well-known defintions of expectation and variance in probability theory. The theory developed in this paper is fully motivated by the principles introduced in [] and by the possibilistic interpretation of the ordering introduced in [3]. A fuzzy number A is a fuzzy set of the real line R with a normal, fuzzy convex and continuous membership function of bounded support. The family of fuzzy numbers will be denoted by F. Aγ-level set of a fuzzy number A is defined by [A] γ = {t R A(t) γ} if γ> and [A] γ = cl{t R A(t) > } (the closure of the support of A) if γ =. It is well-known that if A is a fuzzy number then [A] γ is a compact subset of R for all γ [, 1]. Let A and B F be fuzzy numbers with [A] γ = [a 1 (γ),a (γ)] and [B] γ = [b 1 (γ),b (γ)], γ [, 1]. In 1986 Goetschel and Voxman introduced a method for ranking fuzzy numbers as [3] A B γ(a 1 (γ)+a (γ)) dγ γ(b 1 (γ)+b (γ)) dγ (1) As was pointed out by Goetschel and Voxman this definition of ordering given in (1) was motivated in part by the desire to give less importance to the lower levels of fuzzy numbers. Possibilistic mean value of fuzzy numbers We explain now the way of thinking that has led us to the introduction of notations of lower and upper possibilitistic mean values. First, we note that from the equality M(A) := γ(a 1 (γ)+a (γ))dγ = γ a1(γ)+a (γ) dγ, () γ dγ it follows that M(A) is nothing else but the level-weighted average of the arithmetic means of all γ-level sets, that is, the weight of the arithmetic mean of a 1 (γ) and a (γ) is just γ. 1
4 Second, we can rewrite M(A) as M(A) = γ(a 1 (γ)+a (γ))dγ = = 1 = 1 γa 1 (γ)dγ + γa 1 (γ)dγ + 1 γa 1 (γ)dγ + γdγ γa (γ)dγ γa (γ)dγ 1 γa (γ)dγ. γdγ Third, let us take a closer look at the right-hand side of the equation for M(A). The first quantity, denoted by M (A) can be reformulated as M (A) = = = where Pos denotes possibility, i.e. γa 1 (γ)dγ = γa 1 (γ)dγ γdγ Pos[A a 1 (γ)]a 1 (γ)dγ Pos[A a 1 (γ)]dγ Pos[A a 1 (γ)] min[a] γ dγ, Pos[A a 1 (γ)]dγ Pos[A a 1 (γ)] = Π((,a 1 (γ]) = sup A(u) =γ. u a 1 (γ) (since A is continuous!) So M (A) is nothing else but the lower possibilityweighted average of the minima of the γ-sets, and it is why we call it the lower possibilistic mean value of A. In a similar manner we introduce M (A), the upper possibilistic mean value of A,
5 as M (A) = = = where we have used the equality γa (γ)dγ = γa (γ)dγ γdγ Pos[A a (γ)]a (γ)dγ Pos[A a (γ)]dγ Pos[A a (γ)] max[a] γ dγ, Pos[A a (γ)]dγ Pos[A a (γ)] = Π([a (γ, )) = Let us introduce the notation M(A) =[M (A),M (A)]. sup A(u) =γ. u a (γ) that is, M(A) is a closed interval bounded by the lower and upper possibilistic mean values of A. Definition.1. We call M(A) the interval-valued possibilistic mean of A. If A is the characteristic function of the crisp interval [a, b] then M((a, b,, )) = [a, b], that is, an interval is the possibilistic mean value of itself. We will now show that M is a linear function on F in the sense of the extension principle. Theorem.1. Let A and B be two non-interactive fuzzy numbers and let λ R be a real number. Then M(A + B) =M(A)+M(B), M(λA) =λm(a), i.e. M (A + B) =M (A)+M (B), M (A + B) =M (A)+M (B), and [M (λa),m (λa)] = { [λm (A), λm (A)] if γ [λm (A), λm (A)] if γ< where the addition and multiplication by a scalar of fuzzy numbers is defined by the sup-min extension principle [4]. 3
6 Proof. Really, from the equation [A + B] γ =[a 1 (γ)+b 1 (γ),a (γ)+b (γ)], we have and M (A + B) = = M (A + B) = γ(a 1 (γ)+b 1 (γ))dγ γa 1 (γ)dγ + = M (A)+M (B), = γ(a (γ)+b (γ))dγ γa (γ)dγ + = M (A)+M (B), γb 1 (γ)dγ γb (γ)dγ furthermore, from [λa] γ = λ[a] γ = λ[a 1 (γ),a (γ)] = for λ we get and for γ< M (λa) = M (λa) = M (λa) = M (λa) = Which ends the proof. γ(λa 1 (γ))dγ =λ γ(λa (γ))dγ =λ γ(λa (γ))dγ =λ γ(λa 1 (γ))dγ =λ { [λa1 (γ), λa (γ)] if γ [λa (γ), λa 1 (γ)] if γ< γa 1 (γ)dγ = λm (A). γa (γ)dγ = λm (A). γa (γ)dγ = λm (A). γa 1 (γ)dγ = λm (A). We introduce the crisp possibilistic mean value of A by () as the arithemtic mean of its lower possibilistic and upper possibilistic mean values, i.e. M(A) = M (A)+M (A). The following theorem shows two very important properties of M : F R. 4
7 Theorem.. Let [A] γ =[a 1 (γ),a (γ)] and [B] γ =[b 1 (γ),b (γ)] be fuzzy numbers and let λ R be a real number. Then M(A + B) = M(A)+ M(B), and M(λA) =λ M(A). Proof. First we find M(A + B) = γ(a 1 (γ)+a (γ))dγ + and for λ we get M(λA) = γ(a 1 (γ)+b 1 (γ)+a (γ)+b (γ))dγ = γ(λa 1 (γ)+λa (γ))dγ = λ and, finally, for γ< we have M(λA) = Which ends the proof. γ(λa (γ)+λa 1 (γ))dγ = λ γ(b 1 (γ)+b (γ))dγ = M(A)+ M(B), γ(a 1 (γ)+a (γ))dγ = λ M(A). γ(a 1 (γ)+a (γ))dγ = λ M(A). Example.1. Let A =(a, α, β) be a triangular fuzzy number with center a, leftwidth α> and right-width β> then a γ-level of A is computed by that is, and therefore, and, finally, M(A) = [A] γ =[a (1 γ)α, a + (1 γ)β], γ [, 1], M (A) = M (A) = M(A) = γ[a (1 γ)αdγ = a α 3, γ[a + (1 γ)β]dγ = a + β 3, [ a α 3,a+ β ], 3 γ[a (1 γ)α + a + (1 γ)β]dγ = a + β α 6. Specially, when A =(a, α) is a symmetric triangular fuzzy number we get M(A) = a. If A is a symmetric fuzzy number with peak [q,q + ] then the equation always holds. M(A) = q + q +. 5
8 3 Relation to upper and lower probability mean values We show now an important relationship between the interval-valued expectation E(A) = [E (A),E (A)] introduced in [] and the interval-valued possibilistic mean M(A) =[M (A),M (A)] for LR-fuzzy numbers with strictly decreasing shape functions. An LR-type fuzzy number A F can be described with the following membership function [1] ( ) q u L if q α u q α 1 if u [q,q + ] A(u) = ( ) u q+ R if q + u q + + β β otherwise where [q,q + ] is the peak of A; q and q + are the lower and upper modal values; L, R: [, 1] [, 1], with L() = R() = 1 and L(1) = R(1) = are nonincreasing, continuous mappings. We shall use the notation A =(q,q +, α, β) LR. The closure of the support of A is exactly [q α, q + + β]. If L and R are strictly decreasing functions then we can easily compute the γ-level sets of A. That is, [A] γ =[q αl 1 (γ),q + + βr 1 (γ)], γ [, 1]. Following [] (page 93) the lower and upper probability mean values of A F are computed by E (A) =q α L(u)du, E (A) =q + + β R(u)du, (note that the support of A is bounded) and the lower and upper possibilistic mean values are obtained as M (A) = M (A) = γ(q αl 1 (γ))dγ = q α γ(q + + βr 1 (γ))dγ = q + + β Therefore, we can state the following lemma. γl 1 (γ)dγ γr 1 (γ)dγ Lemma 3.1. If A F is a fuzzy number of LR-type with strictly decreasing (and continuous) shape functions then its interval-valued possibilistic mean is a proper subset of its interval-valued probabilistic mean, E(A) M(A). 6
9 Proof. From the relationships L(u)du = L 1 (γ)dγ and R(u)du = R 1 (γ)dγ. we get γl 1 (γ)dγ < L(u)du and γr 1 (γ)dγ < R(u)du. Which ends the proof. Lemma 3.1 reflects on the fact that points with small membership degrees are considered to be less important in the definition of lower and upper possibilistic mean values than in the definition of probabilistic ones. In the limit case, when A =(q,q +,, ), the possibilistic and probablistic mean values are equal, and the equality E(A) =M(A) =[q,q + ] holds. Example 3.1. Let A =(a, α, β) be a triangular fuzzy number with center a, leftwidth α> and right-width β> then M(A) = [ a α 3,a+ β ] [ E(A) = a α 3,a+ β ] and M(A) =a + β α 6 Ē(A) =E (A)+E (A) However, when A is a symmetric fuzzy number then the equation = a + β α 4. M(A) =Ē(A). always holds. 7
10 4 Variance of fuzzy numbers We introduce the (possibilistic) variance of A F as ( 1 [a1 (γ)+a (γ) Var(A) = Pos[A a 1 (γ)] ( 1 [a1 (γ)+a (γ) + Pos[A a (γ)] ([ ] a1 (γ)+a (γ) = γ a 1 (γ) [ ] a1 (γ)+a (γ) ) + a (γ) dγ = 1 γ ( a (γ) a 1 (γ) ) dγ. ] ) a 1 (γ) dγ ] ) a (γ) dγ The variance of A is defined as the expected value of the squared deviations between the arithmetic mean and the endpoints of its level sets, i.e. the lower possibility-weighted average of the squared distance between the left-hand endpoint and the arithmetic mean of the endpoints of its level sets plus the upper possibilityweighted average of the squared distance between the right-hand endpoint and the arithmetic mean of the endpoints their of its level sets. The standard deviation of A is defined by σ A = Var(A). For example, if A =(a, α, β) is a triangular fuzzy number then Var(A) = 1 γ ( a + β(1 γ) (a α(1 γ)) ) (α + β) dγ =. 4 especially, if A =(a, α) is a symmetric triangular fuzzy number then Var(A) = α 6. If A is the characteristic function of the crisp interval [a, b] then Var(A) = 1 ( ) b a γ(b a) dγ = that is, σ A = b a, a + b M(A) =. In probability theory, the corresponding result is: if the two possible outcomes of a probabilistic variable have equal probabilities then the expected value is their arithmetic mean and the standard deviation is the half of their distance. 8
11 We show now that the variance of a fuzzy number is invariant to shifting. Let A F and let θ be a real number. If A is shifted by value θ then we get a fuzzy number, denoted by B, satisfying the property B(x) =A(x θ) for all x R. Then from the relationship we find Var(B) = 1 [B] γ =[a 1 (γ)+θ, a (γ)+θ] = 1 γ ( (a (γ)+θ) (a 1 (γ)+θ) ) dγ γ ( a (γ) a 1 (γ) ) dγ = Var(A). The covariance between fuzzy numbers A and B is defined as Cov(A, B) = 1 γ(a (γ) a 1 (γ))(b (γ) b 1 (γ))dγ The covariance measures how much the endpoints of the γ-level sets of two fuzzy numbers move in tandem. Let A =(a, α) and B =(b, β) be symmetric trianglar fuzzy numbers. Then Cov(A, B) = αβ 6. The following theorem shows that the variance of linear combinations of fuzzy numbers can easily be computed (in the same manner as in probability theory). Theorem 4.1. Let λ, µ R and let A and B be fuzzy numbers. Then Var(λA + µb) =λ Var(A)+µ Var(B)+ λµ Cov(A, B) where the addition and multiplication by a scalar of fuzzy numbers is defined by the sup-min extension principle. Proof. Suppose λ< and µ<. Then we find Var(λA + µb) = 1 λ 1 1 γ ( λa 1 (γ)+µb 1 (γ) λa (γ) µb (γ) ) dγ = γ ( λ(a 1 (γ) a (γ)) + µ(b 1 (γ) b (γ)) ) dγ = γ(a 1 (γ) a (γ)) dγ + µ 1 λµ 1 γ(b 1 (γ) b (γ)) dγ+ γ(a 1 (γ) a (γ))(b 1 (γ) b (γ))dγ = 9
12 λ Var(A)+µ Var(B)+λµ 1 γ(a (γ) a 1 (γ))(b (γ) b 1 (γ))dγ = λ Var(A)+µ Var(B)+λµCov(A, B) = λ Var(A)+µ Var(B)+ λµ Cov(A, B). Similar reasoning holds for the case λ> and µ>. Suppose now that λ< and γ>. Then we get Var(λA + µb) = 1 λ 1 1 γ ( λa 1 (γ)+µb (γ) λa (γ) µb 1 (γ) ) dγ = γ ( λ(a 1 (γ) a (γ)) + µ(b (γ) b 1 (γ) ) dγ = γ(a 1 (γ) a (γ)) dγ + µ 1 λµ 1 λ Var(A)+µ Var(B) λµ 1 Which ends the proof. γ(b (γ) b 1 (γ)) dγ+ γ(a 1 (γ) a (γ))(b (γ) b 1 (γ))dγ = γ(a (γ) a 1 (γ))(b (γ) b 1 (γ))dγ = λ Var(A)+µ Var(B)+ λµ Cov(A, B). As a special case of Theorem 4.1 we get Var(λA) =λ Var(A) for any λ R and Var(A + B) = Var(A) + Var(B) + Cov(A, B). Let A =(a, α) and B =(b, β) be symmetric triangular fuzzy numbers and λµ be real numbers. Then Var(λA + µb) =λ α 6 β + µ 6 + λµ αβ 6 ( λ α + µ β) =, 6 which coincides with the variance of the symmetric triangular fuzzy number λa + µb =(λa + µb, λ α + µ β). Another important question is the relationship between the subsethood and the variance of fuzzy numbers. One might expect that A B (that is A(x) B(x) for all x R) should imply the relationship Var(A) Var(B) because A is considered a stronger restriction than B. The following theorem shows that subsethood does entail smaller variance. Theorem 4.. Let A, B F with A B. Then Var(A) Var(B). 1
13 Proof. From A B it follows that b 1 (γ) a 1 (γ) a (γ) b (γ), for all γ [, 1]. That is, for all γ [, 1], and therefore, a (γ) a 1 (γ) b (γ) b 1 (γ) Var(A) = 1 1 Which ends the proof. γ ( a (γ) a 1 (γ) ) dγ γ ( b (γ) b 1 (γ) ) dγ = Var(B). Remark 4.1. Alternatively, we could also introduce the variance of A F as Var (A) = = γ ( [ M(A) a 1 (γ)] +[ M(A) a (γ)] ) dγ γ(a 1(γ)+a (γ))dγ E (A), i.e. the possibility-weighted average of the squared distance between the expected value and the left hand and right hand endpoints of its level sets; and the covariance as Cov (A, B) = γ ( [ M(A) a 1 (γ)][ M(B) b 1 (γ)] + [ M(A) a (γ)][ M(B) b (γ)] ) dγ Then the following theorem would hold Theorem 4.3. Let λ, µ R such that λµ > and let A and B be fuzzy numbers. Then Var (λa + µb) =λ Var (A)+µ Var (B)+λµCov (A, B) where the addition and multiplication by a scalar of fuzzy numbers is defined by the sup-min extension principle. Proof. Suppose λ< and µ<. Using the linearity of the expected value we 11
14 find Var (λa + µb) = λµ γ ( (E(λA+µB) λa (γ) µb (γ)) +(E(λA+µB) λa 1 (γ) µb 1 (γ)) ) dγ = γ ( λ M(A)+µ M(B) λa (γ) µb (γ) ) dγ + γ ( λ M(A)+µ M(B) λa 1 (γ) µb 1 (γ) ) dγ = γ ( λ( M(A) a (γ)) + µ( M(B) b (γ)) ) dγ + γ ( λ( M(A) a 1 (γ)) + µ( M(B) b 1 (γ)) ) dγ = λ γ ( [ M(A) a 1 (γ)] +[ M(A) a (γ)] ) dγ + µ γ ( [ M(B) b 1 (γ)] +[ M(B) b (γ)] ) dγ + γ ( [ M(A) a 1 (γ)][ M(B) b 1 (γ)]+[ M(A) a (γ)][ M(B) b (γ)] ) dγ = λ Var (A)+µ Var (B)+λµCov (A, B). Similar reasoning holds for the case λ> and µ>. Which ends the proof. However, nothing can be said about Var (λa + µb) if λµ <, and it is not clear if subsethood entails smaller variance. However, if A =(a, α, β) is a triangular fuzzy number then Var (A) = α + β + αβ, 18 and, therefore, any triangular subset of A will have smaller variance. 5 Summary We have introduced the notation of interval-valued possibilistic mean of fuzzy numbers and investigated its relationship to interval-valued probabilistic mean. We have proved that the proposed concepts behave properly (in a similar way as their probabilistic counterparts). 6 Acknowledgment The authors are thankful to Didier Dubois for his comments and suggestions on the earlier versions of this paper. 1
15 References [1] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York, 198). [] D. Dubois and H. Prade, The mean value of a fuzzy number, Fuzzy Sets and Systems 4(1987) [3] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18(1986) [4] L.A. Zadeh, Fuzzy Sets, Information and Control, 8(1965)
16
17
18 Turku Centre for Computer Science Lemminkäisenkatu 14 FIN-5 Turku Finland University of Turku Department of Mathematical Sciences Åbo Akademi University Department of Computer Science Institute for Advanced Management Systems Research Turku School of Economics and Business Administration Institute of Information Systems Science
Approximation of Multiplication of Trapezoidal Epsilon-delta Fuzzy Numbers
Advances in Fuzzy Mathematics ISSN 973-533X Volume, Number 3 (7, pp 75-735 Research India Publications http://wwwripublicationcom Approximation of Multiplication of Trapezoidal Epsilon-delta Fuzzy Numbers
More informationAN APPROXIMATION APPROACH FOR RANKING FUZZY NUMBERS BASED ON WEIGHTED INTERVAL - VALUE 1.INTRODUCTION
Mathematical and Computational Applications, Vol. 16, No. 3, pp. 588-597, 2011. Association for Scientific Research AN APPROXIMATION APPROACH FOR RANKING FUZZY NUMBERS BASED ON WEIGHTED INTERVAL - VALUE
More informationBest proximation of fuzzy real numbers
214 (214) 1-6 Available online at www.ispacs.com/jfsva Volume 214, Year 214 Article ID jfsva-23, 6 Pages doi:1.5899/214/jfsva-23 Research Article Best proximation of fuzzy real numbers Z. Rohani 1, H.
More informationSolution of Rectangular Interval Games Using Graphical Method
Tamsui Oxford Journal of Mathematical Sciences 22(1 (2006 95-115 Aletheia University Solution of Rectangular Interval Games Using Graphical Method Prasun Kumar Nayak and Madhumangal Pal Department of Applied
More informationVesa Halava Tero Harju. Walks on Borders of Polygons
Vesa Halava Tero Harju Walks on Borders of Polygons TUCS Technical Report No 698, June 2005 Walks on Borders of Polygons Vesa Halava Tero Harju Department of Mathematics and TUCS - Turku Centre for Computer
More informationFuzzy multi objective linear programming problem with imprecise aspiration level and parameters
An International Journal of Optimization and Control: Theories & Applications Vol.5, No.2, pp.81-86 (2015) c IJOCTA ISSN:2146-0957 eissn:2146-5703 DOI:10.11121/ijocta.01.2015.00210 http://www.ijocta.com
More informationA Compromise Solution to Multi Objective Fuzzy Assignment Problem
Volume 113 No. 13 2017, 226 235 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu A Compromise Solution to Multi Objective Fuzzy Assignment Problem
More informationA New and Simple Method of Solving Fully Fuzzy Linear System
Annals of Pure and Applied Mathematics Vol. 8, No. 2, 2014, 193-199 ISSN: 2279-087X (P), 2279-0888(online) Published on 17 December 2014 www.researchmathsci.org Annals of A New and Simple Method of Solving
More informationREVIEW OF FUZZY SETS
REVIEW OF FUZZY SETS CONNER HANSEN 1. Introduction L. A. Zadeh s paper Fuzzy Sets* [1] introduces the concept of a fuzzy set, provides definitions for various fuzzy set operations, and proves several properties
More informationRanking of fuzzy numbers, some recent and new formulas
IFSA-EUSFLAT 29 Ranking of fuzzy numbers, some recent and new formulas Saeid Abbasbandy Department of Mathematics, Science and Research Branch Islamic Azad University, Tehran, 14778, Iran Email: abbasbandy@yahoo.com
More informationTriangular Approximation of fuzzy numbers - a new approach
Volume 113 No. 13 017, 115 11 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu Triangular Approximation of fuzzy numbers - a new approach L. S. Senthilkumar
More informationAn Appropriate Method for Real Life Fuzzy Transportation Problems
International Journal of Information Sciences and Application. ISSN 097-55 Volume 3, Number (0), pp. 7-3 International Research Publication House http://www.irphouse.com An Appropriate Method for Real
More informationInformation Granulation and Approximation in a Decision-theoretic Model of Rough Sets
Information Granulation and Approximation in a Decision-theoretic Model of Rough Sets Y.Y. Yao Department of Computer Science University of Regina Regina, Saskatchewan Canada S4S 0A2 E-mail: yyao@cs.uregina.ca
More informationAcyclic fuzzy preferences and the Orlovsky choice function: A note. Denis BOUYSSOU
Acyclic fuzzy preferences and the Orlovsky choice function: A note Denis BOUYSSOU Abstract This note corrects and extends a recent axiomatic characterization of the Orlovsky choice function for a particular
More informationFuzzy Inventory Model without Shortage Using Trapezoidal Fuzzy Number with Sensitivity Analysis
IOSR Journal of Mathematics (IOSR-JM) ISSN: 78-578. Volume 4, Issue 3 (Nov. - Dec. 0), PP 3-37 Fuzzy Inventory Model without Shortage Using Trapezoidal Fuzzy Number with Sensitivity Analysis D. Dutta,
More informationLecture 5: Duality Theory
Lecture 5: Duality Theory Rajat Mittal IIT Kanpur The objective of this lecture note will be to learn duality theory of linear programming. We are planning to answer following questions. What are hyperplane
More informationNotes on Fuzzy Set Ordination
Notes on Fuzzy Set Ordination Umer Zeeshan Ijaz School of Engineering, University of Glasgow, UK Umer.Ijaz@glasgow.ac.uk http://userweb.eng.gla.ac.uk/umer.ijaz May 3, 014 1 Introduction The membership
More informationA Comparative Study of Defuzzification Through a Regular Weighted Function
Australian Journal of Basic Applied Sciences, 4(12): 6580-6589, 2010 ISSN 1991-8178 A Comparative Study of Defuzzification Through a Regular Weighted Function 1 Rahim Saneifard 2 Rasoul Saneifard 1 Department
More informationA comparison between probabilistic and possibilistic models for data validation
A comparison between probabilistic and possibilistic models for data validation V. Köppen, H.-J. Lenz Freie Universität Berlin, Germany Institute of Production, Information Systems and Operations Research
More informationEpimorphisms in the Category of Hausdorff Fuzzy Topological Spaces
Annals of Pure and Applied Mathematics Vol. 7, No. 1, 2014, 35-40 ISSN: 2279-087X (P), 2279-0888(online) Published on 9 September 2014 www.researchmathsci.org Annals of Epimorphisms in the Category of
More informationFuzzy Transportation Problems with New Kind of Ranking Function
The International Journal of Engineering and Science (IJES) Volume 6 Issue 11 Pages PP 15-19 2017 ISSN (e): 2319 1813 ISSN (p): 2319 1805 Fuzzy Transportation Problems with New Kind of Ranking Function
More informationInverse and Implicit functions
CHAPTER 3 Inverse and Implicit functions. Inverse Functions and Coordinate Changes Let U R d be a domain. Theorem. (Inverse function theorem). If ϕ : U R d is differentiable at a and Dϕ a is invertible,
More informationLecture 15: The subspace topology, Closed sets
Lecture 15: The subspace topology, Closed sets 1 The Subspace Topology Definition 1.1. Let (X, T) be a topological space with topology T. subset of X, the collection If Y is a T Y = {Y U U T} is a topology
More informationAlgebraic Structure Of Union Of Fuzzy Sub- Trigroups And Fuzzy Sub Ngroups
Algebraic Structure Of Union Of Fuzzy Sub- Trigroups And Fuzzy Sub Ngroups N. Duraimanickam, N. Deepica Assistant Professor of Mathematics, S.T.E.T Women s College, Mannargudi, Tamilnadu. India-Pin-614016
More informationOn Generalization of Fuzzy Concept Lattices Based on Change of Underlying Fuzzy Order
On Generalization of Fuzzy Concept Lattices Based on Change of Underlying Fuzzy Order Pavel Martinek Department of Computer Science, Palacky University, Olomouc Tomkova 40, CZ-779 00 Olomouc, Czech Republic
More informationNumerical Optimization
Convex Sets Computer Science and Automation Indian Institute of Science Bangalore 560 012, India. NPTEL Course on Let x 1, x 2 R n, x 1 x 2. Line and line segment Line passing through x 1 and x 2 : {y
More informationA fuzzy subset of a set A is any mapping f : A [0, 1], where [0, 1] is the real unit closed interval. the degree of membership of x to f
Algebraic Theory of Automata and Logic Workshop Szeged, Hungary October 1, 2006 Fuzzy Sets The original Zadeh s definition of a fuzzy set is: A fuzzy subset of a set A is any mapping f : A [0, 1], where
More informationFUNDAMENTALS OF FUZZY SETS
FUNDAMENTALS OF FUZZY SETS edited by Didier Dubois and Henri Prade IRIT, CNRS & University of Toulouse III Foreword by LotfiA. Zadeh 14 Kluwer Academic Publishers Boston//London/Dordrecht Contents Foreword
More informationON DECOMPOSITION OF FUZZY BԐ OPEN SETS
ON DECOMPOSITION OF FUZZY BԐ OPEN SETS 1 B. Amudhambigai, 2 K. Saranya 1,2 Department of Mathematics, Sri Sarada College for Women, Salem-636016, Tamilnadu,India email: 1 rbamudha@yahoo.co.in, 2 saranyamath88@gmail.com
More informationRough Approximations under Level Fuzzy Sets
Rough Approximations under Level Fuzzy Sets W.-N. Liu J.T. Yao Y.Y.Yao Department of Computer Science, University of Regina Regina, Saskatchewan, Canada S4S 0A2 E-mail: [liuwe200, jtyao, yyao]@cs.uregina.ca
More informationStrong Chromatic Number of Fuzzy Graphs
Annals of Pure and Applied Mathematics Vol. 7, No. 2, 2014, 52-60 ISSN: 2279-087X (P), 2279-0888(online) Published on 18 September 2014 www.researchmathsci.org Annals of Strong Chromatic Number of Fuzzy
More informationMATH 423 Linear Algebra II Lecture 17: Reduced row echelon form (continued). Determinant of a matrix.
MATH 423 Linear Algebra II Lecture 17: Reduced row echelon form (continued). Determinant of a matrix. Row echelon form A matrix is said to be in the row echelon form if the leading entries shift to the
More informationd(γ(a i 1 ), γ(a i )) i=1
Marli C. Wang Hyperbolic Geometry Hyperbolic surfaces A hyperbolic surface is a metric space with some additional properties: it has the shortest length property and every point has an open neighborhood
More informationEC 521 MATHEMATICAL METHODS FOR ECONOMICS. Lecture 2: Convex Sets
EC 51 MATHEMATICAL METHODS FOR ECONOMICS Lecture : Convex Sets Murat YILMAZ Boğaziçi University In this section, we focus on convex sets, separating hyperplane theorems and Farkas Lemma. And as an application
More informationBipolar Fuzzy Line Graph of a Bipolar Fuzzy Hypergraph
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 13, No 1 Sofia 2013 Print ISSN: 1311-9702; Online ISSN: 1314-4081 DOI: 10.2478/cait-2013-0002 Bipolar Fuzzy Line Graph of a
More information1 Elementary number theory
Math 215 - Introduction to Advanced Mathematics Spring 2019 1 Elementary number theory We assume the existence of the natural numbers and the integers N = {1, 2, 3,...} Z = {..., 3, 2, 1, 0, 1, 2, 3,...},
More informationON THE STRONGLY REGULAR GRAPH OF PARAMETERS
ON THE STRONGLY REGULAR GRAPH OF PARAMETERS (99, 14, 1, 2) SUZY LOU AND MAX MURIN Abstract. In an attempt to find a strongly regular graph of parameters (99, 14, 1, 2) or to disprove its existence, we
More informationOn JAM of Triangular Fuzzy Number Matrices
117 On JAM of Triangular Fuzzy Number Matrices C.Jaisankar 1 and R.Durgadevi 2 Department of Mathematics, A. V. C. College (Autonomous), Mannampandal 609305, India ABSTRACT The fuzzy set theory has been
More informationPreprint Stephan Dempe, Alina Ruziyeva The Karush-Kuhn-Tucker optimality conditions in fuzzy optimization ISSN
Fakultät für Mathematik und Informatik Preprint 2010-06 Stephan Dempe, Alina Ruziyeva The Karush-Kuhn-Tucker optimality conditions in fuzzy optimization ISSN 1433-9307 Stephan Dempe, Alina Ruziyeva The
More informationReview of Fuzzy Logical Database Models
IOSR Journal of Computer Engineering (IOSRJCE) ISSN: 2278-0661, ISBN: 2278-8727Volume 8, Issue 4 (Jan. - Feb. 2013), PP 24-30 Review of Fuzzy Logical Database Models Anupriya 1, Prof. Rahul Rishi 2 1 (Department
More informationOptimization of fuzzy multi-company workers assignment problem with penalty using genetic algorithm
Optimization of fuzzy multi-company workers assignment problem with penalty using genetic algorithm N. Shahsavari Pour Department of Industrial Engineering, Science and Research Branch, Islamic Azad University,
More informationSolving Fuzzy Sequential Linear Programming Problem by Fuzzy Frank Wolfe Algorithm
Global Journal of Pure and Applied Mathematics. ISSN 0973-768 Volume 3, Number (07), pp. 749-758 Research India Publications http://www.ripublication.com Solving Fuzzy Sequential Linear Programming Problem
More informationPantographic polygons
203 Pantographic polygons John Miller and Emanuel Strzelecki Abstract Necessary and sufficient conditions are given for a polygon to be pantographic. The property is held by all regular polygons and by
More informationFuzzy interpolation and level 2 gradual rules
Fuzzy interpolation and level 2 gradual rules Sylvie Galichet, Didier Dubois, Henri Prade To cite this version: Sylvie Galichet, Didier Dubois, Henri Prade. Fuzzy interpolation and level 2 gradual rules.
More informationFuzzy Generalized γ-closed set in Fuzzy Topological Space
Annals of Pure and Applied Mathematics Vol. 7, No. 1, 2014, 104-109 ISSN: 2279-087X (P), 2279-0888(online) Published on 9 September 2014 www.researchmathsci.org Annals of Fuzzy Generalized γ-closed set
More informationTHREE LECTURES ON BASIC TOPOLOGY. 1. Basic notions.
THREE LECTURES ON BASIC TOPOLOGY PHILIP FOTH 1. Basic notions. Let X be a set. To make a topological space out of X, one must specify a collection T of subsets of X, which are said to be open subsets of
More informationAN ARITHMETIC OPERATION ON HEXADECAGONAL FUZZY NUMBER
AN ARITHMETIC OPERATION ON HEXADECAGONAL FUZZY NUMBER Dr.A.Sahaya Sudha 1 and R.Gokilamani 2 1 Department of Mathematics, Nirmala College for Women, Coimbatore 2 Department of Mathematics, Sri Ramakrishna
More informationROUGH MEMBERSHIP FUNCTIONS: A TOOL FOR REASONING WITH UNCERTAINTY
ALGEBRAIC METHODS IN LOGIC AND IN COMPUTER SCIENCE BANACH CENTER PUBLICATIONS, VOLUME 28 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1993 ROUGH MEMBERSHIP FUNCTIONS: A TOOL FOR REASONING
More informationOptimization with linguistic variables
Optimization with linguistic variables Christer Carlsson christer.carlsson@abo.fi Robert Fullér rfuller@abo.fi Abstract We consider fuzzy mathematical programming problems (FMP) in which the functional
More informationOrdering of Generalised Trapezoidal Fuzzy Numbers Based on Area Method Using Euler Line of Centroids
Advances in Fuzzy Mathematics. ISSN 0973-533X Volume 12, Number 4 (2017), pp. 783-791 Research India Publications http://www.ripublication.com Ordering of Generalised Trapezoidal Fuzzy Numbers Based on
More informationPartition of a Nonempty Fuzzy Set in Nonempty Convex Fuzzy Subsets
Applied Mathematical Sciences, Vol. 6, 2012, no. 59, 2917-2921 Partition of a Nonempty Fuzzy Set in Nonempty Convex Fuzzy Subsets Omar Salazar Morales Universidad Distrital Francisco José de Caldas, Bogotá,
More informationA Decision-Theoretic Rough Set Model
A Decision-Theoretic Rough Set Model Yiyu Yao and Jingtao Yao Department of Computer Science University of Regina Regina, Saskatchewan, Canada S4S 0A2 {yyao,jtyao}@cs.uregina.ca Special Thanks to Professor
More informationOn Fuzzy Topological Spaces Involving Boolean Algebraic Structures
Journal of mathematics and computer Science 15 (2015) 252-260 On Fuzzy Topological Spaces Involving Boolean Algebraic Structures P.K. Sharma Post Graduate Department of Mathematics, D.A.V. College, Jalandhar
More informationCollege of Computer & Information Science Fall 2007 Northeastern University 14 September 2007
College of Computer & Information Science Fall 2007 Northeastern University 14 September 2007 CS G399: Algorithmic Power Tools I Scribe: Eric Robinson Lecture Outline: Linear Programming: Vertex Definitions
More informationParallel and perspective projections such as used in representing 3d images.
Chapter 5 Rotations and projections In this chapter we discuss Rotations Parallel and perspective projections such as used in representing 3d images. Using coordinates and matrices, parallel projections
More informationDifferent strategies to solve fuzzy linear programming problems
ecent esearch in Science and Technology 2012, 4(5): 10-14 ISSN: 2076-5061 Available Online: http://recent-science.com/ Different strategies to solve fuzzy linear programming problems S. Sagaya oseline
More informationA C 2 Four-Point Subdivision Scheme with Fourth Order Accuracy and its Extensions
A C 2 Four-Point Subdivision Scheme with Fourth Order Accuracy and its Extensions Nira Dyn School of Mathematical Sciences Tel Aviv University Michael S. Floater Department of Informatics University of
More informationA Novel Method to Solve Assignment Problem in Fuzzy Environment
A Novel Method to Solve Assignment Problem in Fuzzy Environment Jatinder Pal Singh Neha Ishesh Thakur* Department of Mathematics, Desh Bhagat University, Mandi Gobindgarh (Pb.), India * E-mail of corresponding
More informationSolving Fuzzy Travelling Salesman Problem Using Octagon Fuzzy Numbers with α-cut and Ranking Technique
IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 239-765X. Volume 2, Issue 6 Ver. III (Nov. - Dec.26), PP 52-56 www.iosrjournals.org Solving Fuzzy Travelling Salesman Problem Using Octagon
More informationSolution of m 3 or 3 n Rectangular Interval Games using Graphical Method
Australian Journal of Basic and Applied Sciences, 5(): 1-10, 2011 ISSN 1991-8178 Solution of m or n Rectangular Interval Games using Graphical Method Pradeep, M. and Renukadevi, S. Research Scholar in
More information742 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 13, NO. 6, DECEMBER Dong Zhang, Luo-Feng Deng, Kai-Yuan Cai, and Albert So
742 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL 13, NO 6, DECEMBER 2005 Fuzzy Nonlinear Regression With Fuzzified Radial Basis Function Network Dong Zhang, Luo-Feng Deng, Kai-Yuan Cai, and Albert So Abstract
More informationA Study on Triangular Type 2 Triangular Fuzzy Matrices
International Journal of Fuzzy Mathematics and Systems. ISSN 2248-9940 Volume 4, Number 2 (2014), pp. 145-154 Research India Publications http://www.ripublication.com A Study on Triangular Type 2 Triangular
More informationOn Binary Generalized Topological Spaces
General Letters in Mathematics Vol. 2, No. 3, June 2017, pp.111-116 e-issn 2519-9277, p-issn 2519-9269 Available online at http:// www.refaad.com On Binary Generalized Topological Spaces Jamal M. Mustafa
More informationA Generalized Model for Fuzzy Linear Programs with Trapezoidal Fuzzy Numbers
J. Appl. Res. Ind. Eng. Vol. 4, No. 1 (017) 4 38 Journal of Applied Research on Industrial Engineering www.journal-aprie.com A Generalized Model for Fuzzy Linear Programs with Trapezoidal Fuzzy Numbers
More informationSome fixed fuzzy point results using Hausdorff metric in fuzzy metric spaces
Annals of Fuzzy Mathematics and Informatics Volume 13, No 5, (May 017), pp 641 650 ISSN: 093 9310 (print version) ISSN: 87 635 (electronic version) http://wwwafmiorkr @FMI c Kyung Moon Sa Co http://wwwkyungmooncom
More informationComputational Intelligence Lecture 10:Fuzzy Sets
Computational Intelligence Lecture 10:Fuzzy Sets Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2011 arzaneh Abdollahi Computational Intelligence Lecture
More informationA method for solving unbalanced intuitionistic fuzzy transportation problems
Notes on Intuitionistic Fuzzy Sets ISSN 1310 4926 Vol 21, 2015, No 3, 54 65 A method for solving unbalanced intuitionistic fuzzy transportation problems P Senthil Kumar 1 and R Jahir Hussain 2 1 PG and
More informationOperations on Intuitionistic Trapezoidal Fuzzy Numbers using Interval Arithmetic
Intern. J. Fuzzy Mathematical Archive Vol. 9, No. 1, 2015, 125-133 ISSN: 2320 3242 (P), 2320 3250 (online) Published on 8 October 2015 www.researchmathsci.org International Journal of Operations on Intuitionistic
More informationOperations in Fuzzy Labeling Graph through Matching and Complete Matching
Operations in Fuzzy Labeling Graph through Matching and Complete Matching S. Yahya Mohamad 1 and S.Suganthi 2 1 PG & Research Department of Mathematics, Government Arts College, Trichy 620 022, Tamilnadu,
More informationHAAR HUNGARIAN ALGORITHM TO SOLVE FUZZY ASSIGNMENT PROBLEM
Inter national Journal of Pure and Applied Mathematics Volume 113 No. 7 2017, 58 66 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu HAAR HUNGARIAN
More informationFuzzy Sets and Systems. Lecture 1 (Introduction) Bu- Ali Sina University Computer Engineering Dep. Spring 2010
Fuzzy Sets and Systems Lecture 1 (Introduction) Bu- Ali Sina University Computer Engineering Dep. Spring 2010 Fuzzy sets and system Introduction and syllabus References Grading Fuzzy sets and system Syllabus
More informationAdvanced Operations Research Techniques IE316. Quiz 1 Review. Dr. Ted Ralphs
Advanced Operations Research Techniques IE316 Quiz 1 Review Dr. Ted Ralphs IE316 Quiz 1 Review 1 Reading for The Quiz Material covered in detail in lecture. 1.1, 1.4, 2.1-2.6, 3.1-3.3, 3.5 Background material
More informationK-Means and Gaussian Mixture Models
K-Means and Gaussian Mixture Models David Rosenberg New York University June 15, 2015 David Rosenberg (New York University) DS-GA 1003 June 15, 2015 1 / 43 K-Means Clustering Example: Old Faithful Geyser
More informationLecture 3. Corner Polyhedron, Intersection Cuts, Maximal Lattice-Free Convex Sets. Tepper School of Business Carnegie Mellon University, Pittsburgh
Lecture 3 Corner Polyhedron, Intersection Cuts, Maximal Lattice-Free Convex Sets Gérard Cornuéjols Tepper School of Business Carnegie Mellon University, Pittsburgh January 2016 Mixed Integer Linear Programming
More informationOpen and Closed Sets
Open and Closed Sets Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points i.e., if x S : ɛ > 0 : B(x, ɛ) S. (1) Theorem: (O1) and X are open sets.
More informationApplication of Shortest Path Algorithm to GIS using Fuzzy Logic
Application of Shortest Path Algorithm to GIS using Fuzzy Logic Petrík, S. * - Madarász, L. ** - Ádám, N. * - Vokorokos, L. * * Department of Computers and Informatics, Faculty of Electrical Engineering
More informationEpistemic/Non-probabilistic Uncertainty Propagation Using Fuzzy Sets
Epistemic/Non-probabilistic Uncertainty Propagation Using Fuzzy Sets Dongbin Xiu Department of Mathematics, and Scientific Computing and Imaging (SCI) Institute University of Utah Outline Introduction
More informationComparison of Fuzzy Numbers with Ranking Fuzzy and Real Number
Journal of mathematics and computer Science 12 (2014) 65-72 Comparison of Fuzzy Numbers with Ranking Fuzzy and Real Number M. Yaghobi 1, M. Rabbani 2*, M. Adabitabar Firozja 3 J. Vahidi 4 1 Department
More informationAMS : Combinatorial Optimization Homework Problems - Week V
AMS 553.766: Combinatorial Optimization Homework Problems - Week V For the following problems, A R m n will be m n matrices, and b R m. An affine subspace is the set of solutions to a a system of linear
More informationRanking Fuzzy Numbers Using Targets
Ranking Fuzzy Numbers Using Targets V.N. Huynh, Y. Nakamori School of Knowledge Science Japan Adv. Inst. of Sci. and Tech. e-mail: {huynh, nakamori}@jaist.ac.jp J. Lawry Dept. of Engineering Mathematics
More informationAddress for Correspondence Department of Science and Humanities, Karpagam College of Engineering, Coimbatore -32, India
Research Paper sb* - CLOSED SETS AND CONTRA sb* - CONTINUOUS MAPS IN INTUITIONISTIC FUZZY TOPOLOGICAL SPACES A. Poongothai*, R. Parimelazhagan, S. Jafari Address for Correspondence Department of Science
More informationaxiomatic semantics involving logical rules for deriving relations between preconditions and postconditions.
CS 6110 S18 Lecture 18 Denotational Semantics 1 What is Denotational Semantics? So far we have looked at operational semantics involving rules for state transitions, definitional semantics involving translations
More informationCOMBINATION OF ROUGH AND FUZZY SETS
1 COMBINATION OF ROUGH AND FUZZY SETS BASED ON α-level SETS Y.Y. Yao Department of Computer Science, Lakehead University Thunder Bay, Ontario, Canada P7B 5E1 E-mail: yyao@flash.lakeheadu.ca 1 ABSTRACT
More informationMulti objective linear programming problem (MOLPP) is one of the popular
CHAPTER 5 FUZZY MULTI OBJECTIVE LINEAR PROGRAMMING PROBLEM 5.1 INTRODUCTION Multi objective linear programming problem (MOLPP) is one of the popular methods to deal with complex and ill - structured decision
More informationWhat is a vector in hyperbolic geometry? And, what is a hyperbolic linear transformation?
What is a vector in hyperbolic geometry? And, what is a hyperbolic linear transformation? Ken Li, Dennis Merino, and Edgar N. Reyes Southeastern Louisiana University Hammond, LA USA 70402 1 Introduction
More informationModified Procedure to Solve Fuzzy Transshipment Problem by using Trapezoidal Fuzzy number.
International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 4767 P-ISSN: 2321-4759 Volume 4 Issue 6 August. 216 PP-3-34 Modified Procedure to Solve Fuzzy Transshipment Problem by
More informationMATH3016: OPTIMIZATION
MATH3016: OPTIMIZATION Lecturer: Dr Huifu Xu School of Mathematics University of Southampton Highfield SO17 1BJ Southampton Email: h.xu@soton.ac.uk 1 Introduction What is optimization? Optimization is
More informationRank correlation between Two Interval Valued Intuitionistic fuzzy Sets
Rank correlation between Two Interval Valued Intuitionistic fuzzy Sets Abstract: Dr. Mary JansiRani 1, S. RethinaKumar 2, K.Abinaya Priya 3,J. Princivishvamalar 4 1 Head, PG & Research Department of Mathematics,
More informationSaturated Sets in Fuzzy Topological Spaces
Computational and Applied Mathematics Journal 2015; 1(4): 180-185 Published online July 10, 2015 (http://www.aascit.org/journal/camj) Saturated Sets in Fuzzy Topological Spaces K. A. Dib, G. A. Kamel Department
More informationMTAEA Convexity and Quasiconvexity
School of Economics, Australian National University February 19, 2010 Convex Combinations and Convex Sets. Definition. Given any finite collection of points x 1,..., x m R n, a point z R n is said to be
More informationOn Some Properties of Vague Lattices
Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 31, 1511-1524 On Some Properties of Vague Lattices Zeynep Eken Akdeniz University, Faculty of Sciences and Arts Department of Mathematics, 07058-Antalya,
More informationTree of fuzzy shortest paths with the highest quality
Mathematical Sciences Vol. 4, No. 1 (2010) 67-86 Tree of fuzzy shortest paths with the highest quality Esmaile Keshavarz a, Esmaile Khorram b,1 a Faculty of Mathematics, Islamic Azad University-Sirjan
More informationM3P1/M4P1 (2005) Dr M Ruzhansky Metric and Topological Spaces Summary of the course: definitions, examples, statements.
M3P1/M4P1 (2005) Dr M Ruzhansky Metric and Topological Spaces Summary of the course: definitions, examples, statements. Chapter 1: Metric spaces and convergence. (1.1) Recall the standard distance function
More informationOptimizing Octagonal Fuzzy Number EOQ Model Using Nearest Interval Approximation Method
Optimizing Octagonal Fuzzy Number EOQ Model Using Nearest Interval Approximation Method A.Farita Asma 1, C.Manjula 2 Assistant Professor, Department of Mathematics, Government Arts College, Trichy, Tamil
More information1.7 The Heine-Borel Covering Theorem; open sets, compact sets
1.7 The Heine-Borel Covering Theorem; open sets, compact sets This section gives another application of the interval halving method, this time to a particularly famous theorem of analysis, the Heine Borel
More informationA NEW APPROACH FOR FUZZY CRITICAL PATH METHOD USING OCTAGONAL FUZZY NUMBERS
Volume 119 No. 13 2018, 357-364 ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu A NEW APPROACH FOR FUZZY CRITICAL PATH METHOD USING OCTAGONAL FUZZY NUMBERS D. STEPHEN DINAGAR 1 AND
More informationSub-Trident Ranking Using Fuzzy Numbers
International Journal of Mathematics nd its pplications Volume, Issue (016), 1 150 ISSN: 7-1557 vailable Online: http://ijmaain/ International Journal 7-1557 of Mathematics pplications nd its ISSN: International
More informationSome Properties of Interval Valued Intuitionistic Fuzzy Sets of Second Type
Some Properties of Interval Valued Intuitionistic Fuzzy Sets of Second Type K. Rajesh 1, R. Srinivasan 2 Ph. D (Full-Time) Research Scholar, Department of Mathematics, Islamiah College (Autonomous), Vaniyambadi,
More informationUsing Ones Assignment Method and. Robust s Ranking Technique
Applied Mathematical Sciences, Vol. 7, 2013, no. 113, 5607-5619 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.37381 Method for Solving Fuzzy Assignment Problem Using Ones Assignment
More informationSome Inequalities Involving Fuzzy Complex Numbers
Theory Applications o Mathematics & Computer Science 4 1 014 106 113 Some Inequalities Involving Fuzzy Complex Numbers Sanjib Kumar Datta a,, Tanmay Biswas b, Samten Tamang a a Department o Mathematics,
More information