Complement of Type-2 Fuzzy Shortest Path Using Possibility Measure

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1 Intern. J. Fuzzy Mathematcal rchve Vol. 5, No., 04, 9-7 ISSN: (P, (onlne Publshed on 5 November 04 Internatonal Journal of Complement of Type- Fuzzy Shortest Path Usng Possblty Measure V. nusuya* and R.Sathya PG and Research Department of Mathematcs, Seethalakshm Ramaswam College Truchrappall 0 00, Taml Nadu, Inda E-mal :anusrctry@gmal.com *Correspondng uthor Department of Mathematcs, K.S.Rangasamy College of rts & Scence Truchengode 37 5, Taml Nadu, Inda E-mal: satmepr@gmal.com Receved November 04; accepted November 04 bstract. Ths paper presents a complement fuzzy shortest path problem on a network by possblty measure usng extenson prncple. proposed algorthm gves the complement shortest path and complement shortest path length. Complement type- fuzzy number have been taken for each edge whch have been assgned by type- fuzzy number. n llustratve example also ncluded to demonstrate our proposed algorthm. Keywords: Type- fuzzy set, complement type- fuzzy set, possblty measure, type- fuzzy number, extenson prncple MS mathematcs Subect Classfcaton (00: 94D05. Introducton In graph theory, the shortest path problem s the problem of fndng a path between source node to destnaton node on a network. It has applcatons n varous felds lke transportaton, communcaton, routng and schedulng. In real world problem the arc length of the network may represent the tme or cost whch s not stable n the entre stuaton, hence t can be consdered to be a fuzzy set. The fuzzy shortest path problem was frst analyzed by Dubos and Prade [4]. Okada and Soper [] developed an algorthm based on the multple labelng approach, by whch a number of nondomnated paths can be generated. Type- fuzzy set was ntroduced by Zadeh [5] as an extenson of the concept of an ordnary fuzzy set. Many researchers have studed on shortest path n fuzzy network [0], and one may refer the Book by Klr et al. [5], operatons on type- fuzzy sets [,,7,8,9].The type- fuzzy logc has ganed much attenton recently due to ts 9

2 V. nusuya and R.Sathya ablty to handle uncertanty, and many advances appeared n both theory and applcatons. complement of type- fuzzy set s also consdered as a type- fuzzy set havng complement for fuzzy membershp functon only. The fuzzy measures was ntroduced by sugeno [3], Good surveys of varous types of measures subsumed under ths broad concept were prepared by Dubos and Prade [3], Bacon [] and Werzchon [4]. There were many researches usng dfferent approaches, but we wll manly focus on an approach based on possblty theory proposed by Okada []. Ths paper s organzed as follows: In Secton, we have some basc concepts requred for analyss. Secton 3, gves an algorthm to fnd the complement shortest path and shortest path length wth complement type- fuzzy number. Secton 4 gves the network termnology. To llustrate the proposed algorthm the numercal example s solved n secton 5. Fnally n secton, concluson s ncluded.. Prelmnares.. Type- fuzzy set Type- fuzzy set denoted ɶ, s characterzed by a Type- membershp functon µ ( x, u ɶ where x X and u J x [0,]. e., ɶ = { ((x,u, µ ( x, u ɶ / x X, u J x [0,] } n whch 0 ( x, u ɶ. ɶ can be expressed as ɶ = µ ( x, u /( x, u x X u J x µ ɶ J x [0,], where denotes unon over all admssble x and u. For dscrete unverse of dscourse s replaced by... Complement of type- fuzzy set The complement of the type- fuzzy set havng a fuzzy membershp functon gven n the followng formula. ɶ = f x ( u / ( u / x x X u J x.3. Type- fuzzy number Let ɶ be a type- fuzzy set defned n the unverse of dscourse R. If the followng condtons are satsfed:. ɶ s normal,. ɶ s a convex set, 3. The support of ɶ s closed and bounded, then ɶ s called a type- fuzzy number. 0

3 Complement of Type- Fuzzy Shortest Path Usng Possblty Measure.4. Dscrete type- fuzzy number The dscrete type- fuzzy number ɶ can be defned as follows: ɶ = µ (x / x ɶ where µ (x fx(u / u ɶ wherej x s the prmary membershp. x X = u J x.5. Extenson prncple Let,,...., r be type- fuzzy sets n X, X,...,.X r, respectvely. Then, Zadeh s Extenson Prncple allows us to nduce from the type- fuzzy sets,,.., r a type- fuzzy set B on Y, through f,.e,b = f(,...., r, such that µ B( y = sup mn{ µ ( x,..., µ ( x ( n n ff y φ x, x,... xn f ( y 0, f ( y = φ.. ddton on type- fuzzy numbers Let ɶ and B ɶ be two dscrete type- fuzzy number be ɶ = µ ( x / x ɶ and B ɶ = µ B ɶ (y / y where µ ( x = f (u / u ɶ x and µ ( x = g (w / w Bɶ y. The addton of these two types- fuzzy numbers ɶ B ɶ s defned as µ (z = ( µ (x µ (y ɶ Bɶ ɶ Bɶ z= x+ y = (( f (u / u ( g (w / w x y z= x+ y µ ɶ Bɶ == x g y z= x+ y, (z (( (f (u (w / (u w.7. Relatve possblty Let ɶ and B ɶ be two dscrete type- fuzzy number then the relatve possblty for av( ɶ = x and av( B ɶ = y can be defned as s( av( ɶ = x av(bɶ = y= u. f ( u. w. g( w where av( ɶ s the actual value of fuzzy set ɶ and Bɶ g y (w / w/ y. = ɶ = f x (u / u/ x.8. Possblty measure Let ɶ and B ɶ be two dscrete type- fuzzy number then the possblty that ɶ < Bɶ s defned as and

4 s( ɶ < Bɶ = ( x, y Ω ( ɶ < Bɶ ( x, y Ω (T V. nusuya and R.Sathya = x av( Bɶ = y = x av( Bɶ = y where Ω ( ɶ < B ɶ s the set of all possble stuatons where av( ɶ < av ( B ɶ. Ω (T = Ω (ɶ < B ɶ + Ω (ɶ = B ɶ + Ω (ɶ > B. ɶ Smlarly s( ɶ > Bɶ = s( ɶ = Bɶ = ( x, y Ω ( ɶ > Bɶ ( x, y Ω (T ( x, y Ω ( ɶ = Bɶ = x av( Bɶ = y = x av( Bɶ = y and = x av( Bɶ = y = x av( Bɶ = y ( x, y Ω (T..9. Degree of possblty The Degree of possblty D p for the path P s defned as Dp = mn{ s( p ( } p + s p = p p p. 3. lgorthm Step : Complement type- fuzzy number Form the complement type- fuzzy number usng defnton. for each edge. Step : Computaton of possble paths Form the possble paths from startng node to destnaton node and compute the correspondng path lengths, L ɶ =,,....n for possble n paths. Step 3: Computaton of relatve possbltes ɶ µ ( x /x and B ɶ µ (y /y = ɶ = ɶ B Compute relatve possblty s ( av( pɶ ( x av( pɶ (x for all stuatons where x k s the actual value of the path pɶ and = +. Step 4: Computaton of possblty measure = = + Compute s ( pɶ < pɶ usng defnton.7 v Wrte s ( pɶ < pɶ v Put = + v If n then go to ( k k

5 Complement of Type- Fuzzy Shortest Path Usng Possblty Measure v Put = + v If <n then go to ( x Compute the possblty of s ( pɶ = pɶ Step 5: Computaton of degree of possblty D( pɶ = mn{ s ( pɶ < pɶ + (/ s ( pɶ = pɶ } where =,,...n and = +. Step : Shortest path and shortest path length The path whch s havng hghest degree of possblty s the Shortest path and the correspondng path length s the Shortest path Length for the complement type- fuzzy number. 4. Network technology Consder a drected network G(V,E consstng of a fnte set of nodes V = {,,...n} and a set of m drected edges E VXV. Each edge s denoted by an ordered par (,, where, V and. In ths network, we specfy two nodes, denoted by s and t, whch are the source node and the destnaton node, respectvely. We defne a path P as a sequence P = { =, (,,,...., l-, ( l-, l, l = } of alternatng nodes and edges. The exstence of at least one path P s n G(V,E s assumed for every node V {s}. d ɶ denotes a Type- Fuzzy Number assocated wth the edge (,, correspondng to the length necessary to transverse (, from to. The fuzzy dstance along the path P s denoted as dɶ ( P s defned as dɶ ( P = dɶ (, P 5. Numercal example The problem s to fnd the shortest path and shortest path length between source node and destnaton node n the network havng vertces and 7edges wth complement type- fuzzy number Fgure 5.: 7

6 V. nusuya and R.Sathya Soluton: The edge Lengths are = (0.5/0.+0.4/0.3/ + (0.4/0./3 = (0.3/ /0.3/ + (0./0.8/3 = (0.7/0./ + (0.9/ /0.5/4 3 = (0./0./4 4 = (0.9/ /0.5/3 + (0.4/0.7/5 5 = (0.8/ /0.5/ = (0./0.4/ +(0.7/ /0./4 7 Step : Complement type- fuzzy number The complement of type- fuzzy numbers are = (0.5/ /0.7/ + (0.4/0.8/3 = (0.3/ /0.7/ + (0./0./3 3 = (0.7/0.8/ + (0.9/ /0.5/4 4 = (0./0.8/4 5 = (0.9/ /0.5/3 + (0.4/0.3/5 = (0.8/ /0.5/ 7 = (0./0./ +(0.7/ /0.4/4 Step : Computaton of Possble Paths Form the possble paths from startng node to destnaton node and compute the correspondng path lengths, L ɶ =,,....n for possble n paths. pɶ = = - 4- pɶ = = 3-4- pɶ 3 = = 3-5- L ɶ = (0.5/ /0.5/+(0.4/ /0.5/7+ (0.5 / /0.5/8 + (0.4/0.+0.4/0.5/9 L ɶ = (0./ /0.5/7 + (0./0./9 L ɶ 3 = (0./ /0.5/ + ( 0./0.5+0./0.4/8 + (0./0.3/0 + (0./0./ Step 3: Computaton of Relatve Possbltes ɶ µ ( x /x and = ɶ 4

7 Complement of Type- Fuzzy Shortest Path Usng Possblty Measure = ɶ B ɶ µ (y /y B Compute relatve possblty s ( av( pɶ ( x av( pɶ (x for all stuatons where x k s the actual value of the path pɶ and = +. ɶ = pɶ = = 0.34 s p p s( p 7 = = 0.33 s p p3 s( p k 5 ( ɶ = ɶ = 9 = 0.0 ( ɶ = ɶ = 8 = = = s p p3 s( p 0 ɶ = pɶ = = 0.97 s p p s( p 7 7 = = 0.38 s p p3 s( p 7 = = s p p3 s( p 7 0 ɶ = pɶ = = 0.34 s p p s( p 8 7 = = 0.33 s p p3 s( p 8 = = 0.08 s p p3 s( p 8 0 ( ɶ = ɶ = = 0.0 ( ɶ = 7 ɶ = 9 = 0.09 ( ɶ = 7 ɶ = 8 = k ( ɶ = 7 ɶ = = 0.09 ( ɶ = 8 ɶ = 9 = 0.0 ( ɶ = 8 ɶ = 8 = ɶ = pɶ = = 0.78 s p p s( p 9 7 = = s p p3 s( p 9 = = 0.04 s p p3 s( p 9 0 ɶ = pɶ 3 = = s p p3 s( p 7 ɶ = pɶ 3 = = s p p3 s( p 7 0 ɶ = pɶ 3 = = 0.04 s p p3 s( p 9 ɶ = pɶ 3 = = s p p3 s( p 9 0 Step 4: Computaton of Possblty Measure = = + Compute s ( pɶ < pɶ usng defnton.7 v Wrte s ( pɶ < pɶ v Put = + v If n then go to ( v Put = + v If <n then go to ( x Compute the possblty of s ( pɶ = pɶ ( ɶ = 8 ɶ = = 0.0 ( ɶ = 9 ɶ = 9 = 0.07 ( ɶ = 9 ɶ = 8 = ( ɶ = 9 ɶ = = 0.07 ( ɶ = 7 ɶ = 8 = 0. ( ɶ = 7 ɶ = = ( ɶ = 9 ɶ = 8 = ( ɶ = 9 ɶ = = 0.00

8 V. nusuya and R.Sathya s ( pɶ < pɶ = s3 ( pɶ < pɶ 3 = s ( pɶ = pɶ = 0.08 s3 ( pɶ = pɶ 3 = 0. s( pɶ < pɶ = 0.09 s( pɶ = pɶ = 0.08 s3 ( pɶ < pɶ 3 = 0.04 s3 ( pɶ = pɶ 3 = 0 s3( pɶ 3 < pɶ =0.5 s3 ( pɶ 3 < pɶ = 0.09 Step 5: Computaton of Degree of Possblty D( pɶ = mn{ s ( pɶ < pɶ + (/ s ( pɶ = pɶ } where =,,...n and = +. D( pɶ = mn{0.47, 0.09} = 0.47 D( pɶ = mn{ 0.0, 0.04} = 0.04 D( pɶ 3 = mn{0.370, 0.09} = 0.09 Step : Shortest Path and Shortest Path Length The path whch s havng hghest degree of possblty s the Shortest path and the correspondng path length s the Shortest path Length for the complement type- fuzzy number. Snce hghest degree s pɶ e.,0.47. We conclude that the Shortest Path s 4 and the shortest length s L ɶ =(0.5/ /0.5/+(0.4/ /0.5/7+(0.5/0.+0.5/0.5/8+(0.4/0.+0.4/0.5/9. Concluson In ths paper we have developed an algorthm for fndng complement shortest path and shortest path length usng possblty measure wth complement type- fuzzy number. The order of gettng the degree of possblty for possble paths for the gven edge weghts n a network s gettng reversed whle we use the complement of the edge weghts n a same network. We conclude that the complement shortest path s not at all a shortest path for the gven edge weght but t s the shortest path for complement of gven type- fuzzy number. REFERENCES. V.nusuya and R.Sathya, Type- fuzzy shortest path, Internatonal Journal of Fuzzy Mathematcal rchve, ( G.Bacon, Dstncton between several subsets of fuzzy measures, Fuzzy Sets and Systems, 5 ( D.Dubos and H.Prade, Fuzzy sets and systems: Theory and applcatons, cademc Press, New York.

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