Sub-Trident Ranking Using Fuzzy Numbers

Transcription

1 International Journal of Mathematics nd its pplications Volume, Issue (016), ISSN: vailable Online: International Journal of Mathematics pplications nd its ISSN: International Journal of Mathematics nd its pplications Research rticle Praveen Prakash 1 and MGeethaLakshmi 1 Department of Mathematics, Hindustan University, Padur, Chennai, Tamilnadu, India Department of Mathematics, KCG College of Technology, Karapakkam, Chennai, Tamilnadu, India bstract: This paper deals with the new ranking technique by using Sub-Trident Form along with the help of fuzzy numbers Here the Sub-Trident Form is obtained from Fuzzy Sub-Triangular Form using fuzzy numbers The average Sub-Trident Form for each possible path is determined and the minimum of it gives the ranking for the shortest path Keywords: Sub-Trident, Ranking, verage, Fuzzy Number, Pascal s Triangle c JS Publication 1 Introduction In the year 1965, LotfiZadeh introduced the Fuzzy Set Theory [1] Dubois and Prade introduced the Shortest Path Problem in the year 1980 [] They analyzed this problem using Floyd s algorithm Okada and Gen introduced the same problem with the help of Dijkstra s algorithm in the year 199 [] Chen and Hsieh introduced the Graded Mean Integration Representation for fuzzy numbers [5] Okada and Soper worked on the shortest path problems using fuzzy numbers [] In this paper, the new ranking technique is introduced using Sub-Trident Form with the help of fuzzy numbers This ranking technique is called the Sub-Trident RankingThis paper consists of six sections: preliminaries in the first section, Graded Mean Integration Representation together with Pascal s Triangle Graded Mean in the second section, explaining the proposed method in the third section, lgorithm for our study in the fourth section, explaining the Sub-Trident Ranking Technique with a suitable numerical example in the fifth section and finally the conclusion based on our study Preliminaries 1 Basic Definitions Definition 1 fuzzy set in X is characterized by a membership function µ (x) represents grade of membership of x µ (x) More general representation for a fuzzy set is given by {( ) } = x, µ (x) /x X Definition The α-cut of a fuzzy set of the Universe of discourse X is defined as = α [0, 1] { } x X/µ (x) α, where geetharamon@gmailcom 1

2 Definition fuzzy set defined on the set of real numbers R is said to be a fuzzy number if its membership function : R [0, 1] has the following characteristics: a) is convex if µ (λx 1 + (1 λ)x ) min{µ (x 1), µ (x )}, x 1, x X, λ [0, 1] b) is normal if there exists an x R such that if max µ (x) = 1 c) µ (x) is piecewise continuous [10] Representation of Generalized (Trapezoidal) Fuzzy Number In general, a generalized fuzzy number is described at any fuzzy subset of the real line R, whose membership function µ satisfies the following conditions: 1) µ is a continuous mapping from R to [0, 1], ) µ (x) = 0, x c, ) µ (x) = L(x) is strictly increasing on [c, a] ) µ (x) = w, a x b, 5) µ (x) = R(x) is strictly decreasing on [b, d], 6) µ (x) = 0, d x, where 0 w 1 and a, b, c and d are real numbers Here denote this type of generalized fuzzy number as = (c, a, b, d; w) LR When w = 1, denote this type of generalized fuzzy number as = (c, a, b, d) LR When L(x) and R(x) are straight line, then is Trapezoidal fuzzy number and denote it as (c, a, b, d) Different Forms of Fuzzy Numbers The Following are the different forms of fuzzy numbers defined by its membership functions: 1 Triangular Fuzzy Number: Triangular fuzzy number is defined as = (a 1, a, a ), where all a 1, a, a are real numbers and its membership function is given below: µã(x) = 0, for x < a 1; x a 1 a a 1, for a 1 x a ; a x a a, for a x a ; 0, for x > a Trapezoidal Fuzzy Number: Trapezoidal fuzzy number is defined as = (a 1, a, a, a ), where all a 1, a, a, a are real numbers and its membership function is given below: µã(x) = 0, for x < a 1; x a 1 a a 1, for a 1 x a ; 1, for a x a ; a x a a, for a x a ; 0, for x > a 1

3 Praveen Prakash and MGeethaLakshmi Pentagonal Fuzzy Number: Pentagonal fuzzy number is defined as = (a 1, a, a, a, a 5), where all a 1, a, a, a, a 5 are real numbers and its membership function is given below: µã(x) = 0, for x < a 1; x a 1 a a 1, for a 1 x a ; x a a a, for a x a ; 1, for x = a ; a x a a, for a x a ; a x a a, for a x a 5; 0, for x > a 5 This can also be extended to any fuzzy number Graded Mean pproach 1 Graded Mean Integration Representation In 1998, Chen and Hsieh [6] and [7] proposed graded mean integration representation for representing generalized fuzzy number Suppose L 1, R 1 are inverse functions of L and R respectively, and graded mean h-level value of generalized fuzzy number = (c, a, b, d; w) LR is h[l 1 (h) + R 1 (h)]/ Then the graded mean integration representation of generalized fuzzy number based on integral value of graded mean h-level is P () = where h is between 0 and w, 0 < w 1 w h 0 ( L 1 (h)+r 1 (h) w hdh 0 ) dh = c + a + b + d, 6 Pascal s Triangle Graded Mean pproach The Graded Mean Integration Representation for generalized fuzzy number by Chen and Hsieh [5 7] Later SkKadhar Babu and BRajesh nand introduces Pascal s triangle graded mean in statistical optimization [9] But the present approach is very simple for analyzing fuzzy variables to get the optimum shortest path This procedure is taken from the following Pascal s triangle Here the coefficients of fuzzy variables as Pascal s triangle numbers Then just add and divide by the total of Pascal s number and call it as Pascal s Triangle Graded Mean pproach Figure 1 Pascal s Triangle 15

4 Proposed Method (a) Fuzzy Sub-Triangular Form of Fuzzy Numbers: The Fuzzy Sub-Triangular form for different fuzzy number is given below: () Triangular Fuzzy Numbers: The Pascal s Triangle for Triangular Fuzzy Number is given in Figure and the Sub -Triangles for Triangular Fuzzy Number is given in Figure (Set: I ) as follows: P 1 = P () = P = P () = P = P () = a1 + a, q 1 = P (B) = a1 + a, q = P (B) = a1 + a, q = P (B) = a1 + a, r 1 = P (C) = a1 + a, r = P (C) = a1 + a, r = P (C) = The Fuzzy Sub-Triangular Form for Triangular Fuzzy Number is given by a1 + a a1 + a a1 + a F ST f = (p p, q q, r r), where p p = p1 + p + p, q q = q1 + q + q r1 + r + r, r r = (B) Trapezoidal Fuzzy Numbers: The Pascal s Triangle for Trapezoidal Fuzzy Number is given in Figure and the Sub-Triangles for Trapezoidal Fuzzy Number is given in Figure 5 (Set: II ) as follows: P 1 = P () = P = P () = P = P () = a1 + a + a, q 1 = P (B) = a1 + a + a, q = P (B) = a1 + a + a, q = P (B) = 6 a1 + a + a, r 1 = P (C) = a1 + a + a, r = P (C) = 7 a1 + a + a, r = P (C) = 7 The Fuzzy Sub-Triangular Form for Trapezoidal Fuzzy Number is given by a1 + a + a a1 + a + a 6 a1 + a + a 16 F ST f = (p p, q q, r r), where p p = p1 + p + p, q q = q1 + q + q r1 + r + r, r r =

5 Praveen Prakash and MGeethaLakshmi (C) Pentagonal Fuzzy Numbers: The Pascal s Triangle for Pentagonal Fuzzy Number is given in Figure 6 and the Sub-Triangles for Pentagonal Fuzzy Number is given in Figure 7 (Set: III ) as follows: P 1 = P () = P = P () = P = P () = a1 + a + a + a, q 1 = P (B) = a1 + a + a + a, q = P (B) = a1 + a + a + a, q = P (B) = 10 a1 + a + a + a, r 1 = P (C) = 8 a1 + a + 6a + a, r = P (C) = 15 a1 + 6a + a + a, r = P (C) = 15 a1 + a + a + a a1 + a + a + a 10 a1 + a + a + a The Fuzzy Sub-Triangular Form for Pentagonal Fuzzy Number is given by F ST f = (p p, q q, r r), where p p = p1 + p + p, q q = q1 + q + q r1 + r + r, r r = Similarly we can extend this to different fuzzy numbers (b) Sub-Trident Form of Fuzzy Numbers: The Sub-Trident Form of Fuzzy Number is given by ST ri = ] [p 1 p + q q 1 + r 1 r, where p p, q q, r r is the Graded Means of the Pascal s Triangle from the Fuzzy Triangular Form 1 (c) Sub-Trident Ranking Technique: The Sub-Trident ranking is given by ST rirank = min(vg ST ri) in order to give the ranking for the Shortest path 5 lgorithm The working rule for the Sub-Trident Form to find the Shortest Path and Optimum Solution is given by the following algorithm: Step: 1 Start by choosing all the possible paths whose and edge weights as fuzzy numbers Step: Obtain the Fuzzy Sub Triangular form (F ST f ) for all fuzzy number Step: Obtain the Sub Trident Form (ST ri) for all fuzzy numbers Step: Obtain verage Sub Trident form (vg ST ri) in each possible path Step: 5 Minimum of verage Sub Trident Form gives the Sub Trident Ranking (ie) ST rirank = min (vg ST ri) 17

6 6 Numerical Example The numerical example is given below to illustrate the above procedure whose edge length is represented as Trapezoidal Fuzzy Number [8]: Figure 8 The Fuzzy numbers for the edges of Figure 8 is given below in the Table: 1 Edge Triangular Trapezoidal Pentagonal (1, )=a (0, 0, 06) (01, 0, 0, 0) (01, 0, 05, 07, 09) (1, )=b (0, 05, 07) (0, 0, 06, 08) (0, 0, 06, 08, 1) (1, )=c (01, 0, 0) (01, 0, 05, 07) (01, 0, 0, 0, 05) (, 5)=d (0, 05, 06) (0, 0, 08, 1) (0, 0, 0, 05, 06) (, 5)=e (0, 0, 08) (01, 0, 06, 08) (0, 06, 07, 09, 1) (, 6)=f (01, 0, 05) (0, 05, 07, 08) (05, 06, 07, 08, 09) (, 7)=g (0, 06, 09) (0, 06, 08, 1) (0, 0, 05, 06, 07) (, 7)=h (05, 07, 09) (05, 06, 07, 08) (0, 0, 05, 07, 09) (5, 8)=i (01, 0, 06) (0, 0, 05, 06) (01, 0, 05, 06, 07) (6, 8)=j (01, 0, 07) (0, 06, 08, 1) (0, 06, 08, 09, 1) (7, 8)=k (0, 0, 0) (0, 05, 07, 09) (0, 05, 06, 09, 1) Table 1 Sub-Trident form of fuzzy numbers is given below in the Tables,, and Edge p p q q r r F ST f = (p p, q q, r r) Sub-Trident Form ) (p 1 p + q 1 q + r r 1 ST ri = 1 (1, )=a (0111, 0, 0889) 0669 (1, )=b (0111, 0, 0889) 0768 (1, )=c (01557, 015, 01) 051 (, 5)=d (0556, 05, 0) 0766 (, 5)=e (0111, 0, 0889) 0669 (, 6)=f (0667, 05, 0) 0698 (, 7)=g (0667, 05, 0) 0766 (, 7)=h (06111, 06, 05889) (5, 8)=i (0667, 05, 0) 05 (6, 8)=j (0667, 05, 0) 05 (7, 8)=k (0, 0, 0) 0669 Table Triangular Fuzzy Numbers 18

7 Praveen Prakash and MGeethaLakshmi Edge p p q q r r F ST f = (p p, q q, r r) Sub-Trident Form ) (p 1 p + q q 1 + r 1 r ST ri = 1 (1, )=a (0111, 0, 01889) 0587 (1, )=b (0, 0, 0778) 0766 (1, )=c (0, 0, 0778) 0669 (, 5)=d (05000, 0508, 0) 079 (, 5)=e (0611, 07, 0055) (, 6)=f (095, 076, 089) (, 7)=g (06, 06, 05778) 08 (, 7)=h (06111, 06, 05889) 08 (5, 8)=i (0111, 0, 0889) 0768 (6, 8)=j (06, 06, 05778) 08 (7, 8)=k (05, 05, 0778) 0796 Table Trapezoidal Fuzzy Numbers Edge p p q q r r F ST f = (p p, q q, r r) Sub-Trident Form ) (p 1 p + q q 1 + r 1 r ST ri = 1 (1, )=a (0, 0, 0667) 076 (1, )=b (05, 05, 0667) 079 (1, )=c (066705, 0) 0698 (, 5)=d (0667, 05, 0) 0706 (, 5)=e (06567, 067, 059) 0856 (, 6)=f (06667, 065, 06) 0866 (, 7)=g (0667, 05, 0) 0766 (, 7)=h (05, 015, 0967) 0797 (5, 8)=i (08, 05, 0) 070 (6, 8)=j (070, 0687, 0667) (7, 8)=k (05867, 059, 051) Table Pentagonal Fuzzy Numbers 7 Sub-Trident Ranking Technique The Sub-Trident Ranking Technique for fuzzy numbers is given by the following Tables 6, 7 and 8 Possible paths verage Sub-Trident Form vg ST ri Ranking 1 a d 5 i b e 5 i b f 6 j b g 7 k c h 7 k Table 5 Ranking for Triangular Fuzzy Number 19

8 Possible paths verage Sub-Trident Form vg ST ri Ranking 1 a d 5 i b e 5 i b f 6 j b g 7 k c h 7 k Table 6 Ranking for Trapezoidal Fuzzy Number Possible paths verage Sub-Trident Form vg ST ri Ranking 1 a d 5 i b e 5 i b f 6 j b g 7 k c h 7 k Table 7 Ranking for Pentagonal Fuzzy Number 8 Conclusion This paper concludes from the Sub-Trident ranking that, the path 1 fuzzy numbers and the path 1 a d 5 b f 6 j 8as the shortest path for triangular i 8 as the shortest path for trapezoidal, pentagonal fuzzy numbers References [1] LZadeh, Fuzzy Sets, Information and Control, 8(1965), 8-5 [] DDubois and HPrade, Fuzzy sets and Systems, cademic press, New York, (1980) [] SOkada and MGen, Fuzzy shortest path problem, Computers and Industrial Engineering, 7(1-)(199), [] SOkada and TSoper, Shortest Path Problem on a network with Fuzzy rc Lengths, Fuzzy Sets and Systems, 109(1)(000), [5] Shan-Huo Chen, Operations on Fuzzy Numbers with Functions Principle, Tamkang J Management Sci, 6(1), 1-5 [6] Shan-Huo Chen and Chin Hsun Hseih, Graded Mean Integration Representation of Generalized Fuzzy Number, Journal of Chinese Fuzzy System ssociation, (5)(000), 1-7 [7] Shan-Huo Chen and Chin Hsun Hseih, Representation, Ranking, Distance and Similarity of L-R type fuzzy number and applications, ustralian Journal of Intelligent Information Processing Systems, 6()(000), 17-9 [8] HTaha, Operation Research Introduction, Prentice Hall of India Pvt Ltd, New Delhi, (00) [9] SKKhadar Babu and BRajesh nand, Statistical optimization for Generalised Fuzzy Number, International Journal of Modern Engineering Research, (01), [10] Felix and Victor Devadoss, New Decagonal Fuzzy Number underuncertain Linguistic Environment, International Journal of Mathematics nd its pplication, (1)(015),