SHORTEST PATH ARC LENGTH NETWORK USING TRIANGULAR INTUITIONISTIC FUZZY NUMBER

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1 e-issn : SHORES PAH ARC LENGH NEWORK USING RIANGULAR INUIIONISIC FUZZY NUMBER A.D.CHANDRASEKARAN #, K.GANESAN *, S.BALAMURALIHARAN #,, Faculty of Scence and Humantes, Department of Mathematcs, SRM Unversty, Kattankulathur 60 0, AMIL NADU, INDIA. Emal: adchandru977@gmal.com Emal: gansan_k@yahoo.com Emal: balamural.maths@gmal.com Abstract In ths paper, an ntutonstc fuzzy shortest path s presented to fnd the optmal path n a network whch a fuzzy number, nstead of a postve nteger s assgned to each arc length. he algorthm s based on the dea that frstly from all the shortest paths from source to destnaton, an arc wth shortest length s computed and then the Eucldean dstance s computed for all the paths wth the arc of mnmum dstance. Fnally an llustratve numercal example s gven to express the proposed work. Keywords: Intutonstc Fuzzy Set (IFS), Intutonstc Fuzzy Shortest Path, Intutonstc Fuzzy Number (IFN) I. INRODUCION he shortest path problem s the basc network problem, the real number s assgned to each edges. However, t seems that n the lterature there s lttle nvestgaton on aggregaton operators for aggregatng a collecton of papers [-]. he Fuzzy shortest path problem was frst analyzed by Dubos and Prade[], usng fuzzy number nstead of a real number s assgned to each edges. He used Floyds algorthm and Fords algorthm to treat the fuzzy shortest path problem. Although n ther method the shortest path length can be obtaned, maybe the correspondng path n the network doesnt exst. In ths paper we propose an ntutonstc fuzzy number nstead of fuzzy number. An algorthm s based on the dea that from all the shortest paths from source node to destnaton node, an edge wth shortest length s computed and the Eucldean dstance s computed for all the paths wth the edge of mnmum dstance s the shortest path for membershp and non-membershp values. Okada and Soper [5] developed an algorthm based on the multple labelng approach, by whch a number of non-domnated paths can be generated. Besdes, the multple labelng approaches are an exhaustve approach, and t needs to compare all the possble paths from the source node to the other nodes. Klen [] proposed a dynamcal programmng recursonbased fuzzy algorthm. zung-nan Chuang and Jung-Yuan Kung (006) [7] found the fuzzy shortest path length n a network by means of a fuzzy lnear programmng approach. Shu MH, et al. [5], proposed fuzzy shortest path length procedure that can fnd fuzzy shortest path length among all possble paths n a network. It s based on the dea that a crsp number s the mnmum f and only f any other number s larger than or equal to t [8]. he algorthm s based on the dea that shortest length s computed and then the Eucldean dstance s derved for all the paths wth the arc of optmal dstance []. hs paper s organzed as follows: In secton provdes prelmnary concepts requred for analyss. In secton, an ntutonstc fuzzy shortest path length procedure s gven. In secton and 5 ntutonstc fuzzy shortest path problem s explaned usng an llustratve example and concluson n secton 6. II. PRELIMINARY AND DEFINIIONS he concept of an ntutonstc fuzzy set was ntroduced by Atanassov [] to deal wth vagueness, whch can be defned as follows.. Intutonstc Fuzzy Set Let X s a unverse of dscourse. An Intutonstc Fuzzy Set (IFS) n X s gven by x, ( x), ( x)/ xx where ( x ): X [0,] and ( x ): X [0,] determne the degree of membershp and non-membershp of the element xx,0 ( x) ( x).. Intutonstc Fuzzy Graph Let X s a unverse, contanng fxed graph vertces and let V X be a fxed set. Construct the IFS V x, v( x), v( x)/ xxwhere the functons v ( x ): X [0,] and v( x ): X [0,] determne the degree of membershp and non-membershp to set V of the element (vertex) x X, such that 0 ( x) ( x). p-issn : 9-86 Vol 8 No Apr-May

2 e-issn : Intutonstc Fuzzy Number Let x, ( x), ( x)/ xx be an IFS, then we call ( x), ( x) an Intutonstc Fuzzy Number (IFN). We denote t by x, yz,, lmn,, where xyz,, and lmn,, FI ( ), I0,, 0zn.. Intutonstc Fuzzy Number A trangular ntutonstc fuzzy number A s denoted by A A, A / x, where A and A are c trangular fuzzy numbers wth. Equvalently, A rangular Intutonstc Fuzzy Number (IFN) A s gven by A A A xyz,,, lmn,, wth l, mn, xyz,, that s ether l y, m z (or) m x, n yare membershp and non-membershp fuzzy numbers of A. he addtons of two IFN are as follows. For two trangular ntutonstc fuzzy numbers,, :,,, : and x, y, z :, l, m, n : xx y y zz Mn and, defne ll mm nn Max x y z l m n wth c,, :,,.,, :, III. PROCEDURE FOR AN INUIIONISIC FUZZY SHORES PAH LENGH In ths paper the arc length n a network s consdered to be an ntutonstc fuzzy number, namely trangular ntutonstc fuzzy number. he shortest path length procedure s based on the Chuang and Kung method. Step: Compute all the possble path lengths L from,,, n. Where L a, b, c, l, m, n Step: Intalze L mn abc,,, lmn,, L a, b, c, l, m, n Step: Intalze. Step:() Calculate abc,, for membershp values by, a mn a a ; b, f b a b bb aa f b a ( bb) ( aa) Step:() Calculate lmn,, for non-membershp values by, l mn l l ; Step:5 Set Lmn abc,,, lmn,, Step:6 Replace m, f m l m mm ll f m l ( mm) ( ll) and c c b mn,. and n n m as computed n Step () & () Step:7 f n, then go to Step () & () mn,. p-issn : 9-86 Vol 8 No Apr-May

3 e-issn : IV. NUMERICAL EXAMPLE In order to llustrate the above procedure consder a small network shown n fgure, where each arc length s represented as a trangular ntutonstc fuzzy number. Consderr a network wth the trangular ntutonstc fuzzy arc lengths as shown below. he arc lengths are assumed to be EUV 7,0,, 7,50,56 ; E UW 5,5, 9,7,5,56 ; EVW, 9,56, 5,6,69 ; EVX,,5,,55,6 ; E WY,,9,,,9 ; EXY,8,5,,5, 5 ; EXZ 0, 8,6, 0,0,9 ; E YZ,5,5,,55,65. Step: In the above network there are four possble paths n from source node U to destnaton node Z. he possble path lengths L for,,, are as follows: W : U V X Y Z; L 87,,,99, 66,,9 Step: Step:() Calculate Step:5 Step:6 Step:7 W : U V W : U V W : U W Y Z; L 6,9,0 Iteraton I: L a, b, c, lmn,, Intalze X Z; L 70,99,, 09,5,69 W Y Z; mn, a b, c, l, m, n 87,,99, 66,, 9 Step: Intalze Step:() Calculate abc,, for membershp values by a mn aa, mn 87, bb aa ()(99) (87)(70) f b a; b ( b b ) ( a a ) ( 99) (87 70) 9.95 and c mn c, b mn99, Replace f n 5, then go to Step () & () L 87,5,79, 56,97,9 lmn,, for non-membershp values by l mn l, l, 0,7,60. L mm ll f m l then m ( m m) ( ll) and n mn nm, mn 9,5 5. ()(5) (66)(09) 5. ( 5) (66 09) Set Lmn abc,,, lm,, n 70,9.95,99, 09,5.,5 as computed n Step () & () mn 66,09 09 ; p-issn : 9-86 Vol 8 No Apr-May

4 e-issn : Iteraton II: Step:() Calculate,, f b a ; then, abc for membershp values by a a a aa mn, mn 70,87 70 bb (9.95)(5) (70)(87) b 9. and ( bb ) ( aa ) (9.95 5) (70 87) c b c mn, mn 99,5 99. Step:() Calculate,, Step:5 lmn for non-membershp values by l l l and n n m f m l then m 5. Lmn a, b, c, l, m, n Set 70,9.,99, 09,5,5 Step:6 Replace Step:7 f n5, then go to Step () & () Iteraton III: Step:() Calculate,, mn, mn 09,56 09 ; mn, mn 5,97 5. as computed n Step () & () abc for membershp values by a a a bb aa b ( bb f b a; ) ( aa) (9.)(9) (70)(6) 79. (9. 9) (70 6) Step:() Calculate,, f mn, mn 70,6 6 and c c b mn, mn 99,9 9. lmn for non-membershp values by l l l mm ll then mm ll ml m ( ) ( ) (5.)(7) (09)(0) 5.9 (5. 7) (09 0) Step:5 Set Lmn abc,,, lmn,, as computed n Step () & () Step:6 Replace 5 Step:7 f n5, SOP the procedure. mn, mn 09,0 0 ; and n n m mn, mn 5,7 7. Lmn a, b, c, l, m, n Fnally we obtan an Intutonstc Shortest Path Length 6, 79., 9, 0,5.9,7. V. SHORES PAH ALGORIHM Our am s to determne an ntutonstc fuzzy shortest path length Lmn abc,,, lmn,, and the shortest needed to traverse from source to destnaton. By combnng an ntutonstc fuzzy shortest length method wth smlarty measure, an algorthm s as follows. An algorthm for ntutonstc fuzzy shortest path: Step: Fnd out all the possble paths from source node U to destnaton node Z and calculate the correspondng path lengths L for,,, n. Step: Determne Lmn abc,,, lmn,, by usng an ntutonstc fuzzy shortest path length procedure. Step: Compute the Eucldean dstance d for,,, n. between all the possble path and Lmn a, b, c, l, m, n. Step: Conclude the shortest path wth path havng mnmum Eucldean dstance. Applyng ths algorthm to the numercal example network. he frst two steps have been already S L L calculated n secton.hen step s to fnd out the smlarty degree mn, p-issn : 9-86 Vol 8 No Apr-May

5 e-issn : between Lmn and L for,,, n. by means of smlarty measure SA B M Ax ( k) Bx ( k),. In m order to get the accuracy n smlarty degree, we should let a generc element U denoted by for,,, n. Step: Step: Step: mn mn mn Step: mn From Secton, already we have four possble path lengths are as follows: W : U V X Y Z; L 87,,99, 66,, 9 L 70, 99,, 09,5,69 L 87,5,79, 56,97, 9 L 6, 9,0, 0,7,60. W : U V X Z; W : U V W Y Z; W : U W Y Z; From Secton, also we have computed mnmum path length s Lmn a, b, c, l, m, n 6, 79., 9, 0,5.9,7. Determne the Eucldean dstance between all the four path lengths L for,,, and Lmn a, b, c, l, m, n 6, 79., 9, 0,5.9, , 8.5, , , , , , , Conclude the shortest path wth the path havng mnmum Eucldean dstance for member and nonmember by examnng the Eucldean dstance d between L for,,, and Lmn a, b, c, l, m, n. K p-issn : 9-86 Vol 8 No Apr-May

6 e-issn : VI. CONCLUSION In ths paper we nvestgated an algorthm for solvng shortest path problem on a network wth fuzzy arc lengths. he algorthm can be useful to decson makers. We have developed an example wth the help of a mnmum Eucldean dstance. From the above computatons we can observe that Path W : U W Y Z has the shortest Eucldean dstance for membershp and non-membershp. Hence the shortest path from source node U to destnaton node Z s W : U W Y Z. REFERENCES [] K.Atanassov (986), Intutonstc Fuzzy Sets, Fuzzy sets and System, volume 0, No., pp [] D.Dubos and H.Prade(980), Fuzzy Sets and Systems:heory and Applcatons, Academc Press, New York. [] A.Kran Yadav, B.Ranjt Bswas (009), On Searchng Fuzzy Shortest Path n a Network, Internatonal Journal of Recent rends n Engneerng,Vol.,No.. [] V.Lakshman gomath Nayagam, G.Venkateshwar and Geetha Svaraman (008), Rankng of Intutonstc Fuzzy Numbers, IEEE Internatonal Conference on Fuzzy Systems. [5] Shu MH, Cheng CH and Chang JR (006), Usng ntutonstc fuzzy sets for fault-tree analyss on prnted crcut board assembly, Mcro-electron Relablty Vol.6(), pp [6] ShuE.Q.V. Martns, On a multcrtera shortest path problem, Eur. J. Operat. Res. 6 (98), 6 5. [7] zung-nan Chuang and Jung-Yuan Kung (006), A new algorthm for the dscrete fuzzy shortest path problem n a network, Appled Mathematcs and Computaton 7, pp [8] Xnfan Wang (008), Fuzzy Number Intutonstc Fuzzy Arthmetc Aggregaton Operators, Internatonal Journal of Fuzzy systems, Volume0, No., pp -. [9] P. Jenson, J. Barnes, Network Flow Programmng, John Wley and Sons, New York, 980. [0] E. Lawler, Combnatonal Optmzaton; Networks and Matrods, Holt, Renehart, New York, 976. [] R. Yager, Paths of least resstance on possblstc producton systems, Fuzzy Sets Syst., 9(986). [] J. Broumbaugh-Smth, D. Sher, An emprcal nvestgaton of some bcrteron shortest Path algorthm, Eur. J. Operatons Research (989) 6. [] C.M. Klen, Fuzzy shortest paths, Fuzzy Sets Syst. 9 (99) 7. [] K. Ln, M. Chen, he fuzzy shortest path problem and ts most vtal arcs, Fuzzy Sets Systems, 58,(99) 5. [5] S. Okada,. Soper, A shortest path problem on a network wth fuzzy lengths, Fuzzy Sets Systems,09 (000)9 0. [6] A. Kaufmann, Introducton to Fuzzy Arthmetc, Van Nostrand Renhold, New York, 985. [7] C. Murthy, S.K. Pal, D. Dutta Majumder, Correlaton between two fuzzy membershpfunctons, Fuzzy Sets Systems 7 (985) 8. [8] G.C. Oden, Integraton of fuzzy lngustc nformaton n language comprehenson, Fuzzy Sets and Systems, (98) 9. [9] X. Lu, Entropy, length measure and smlarty measure of fuzzy sets and ther relatons, Fuzzy Sets and Systems, 5 (99) [0]. Gerstenkorn, J. Man_ko, Correlaton of ntutonstc fuzzy sets, Fuzzy Sets Syst. (99)9. [] C.P. Papps, N.I. Karacaplds, A comparatve assessment of measures of smlarty of Fuzzy values, Fuzzy Sets Syst. 56 (99) 7 7. [] L.K. Hyung, Y.S. Song, K.M. Lee, Smlarty measure between fuzzy sets and betweenelements, Fuzzy Sets Syst. 6 (99) 9 9. [] W.J. Wang, New smlarty measures on fuzzy sets and on elements, Fuzzy Sets Syst. 85 (997) AUHOR PROFILE Author:A.D. Chandrasekaran s currently Assstant Professor, Department of Mathematcs, Faculty of Scence and Humantes, SRM Unversty. Currently he s pursung Ph.D. at SRM Unversty under the gudance of Dr.K.Ganesan. He receved hs M. Phl Mathematcs from Madura Kamaraj Unversty. Hs research area ncludes Fuzzy optmzaton, Intutonstc Fuzzy Graphs and Fuzzy Non-lnear equatons. He has presented hs research artcles n two nternatonal conferences. Hs academc experence s about 8 years n teachng. He s currently workng n the area of rangular Intutonstc Fuzzy Numbers. Author:Dr. K. Ganesan s currently Professor, Department of Mathematcs, Faculty of Scence and Humantes, SRM Unversty. He receved hs Ph.D. n Mathematcs from the Unversty of Madras, Chenna. Hs academc experence s about 6 years n teachng. Hs research nterest ncludes Fuzzy optmzaton and Decson makng, Interval Mathematcs, Intutonstc Fuzzy Numbers, Intutonstc Fuzzy Graphs, Operaton research, Dfferental equatons etc. He publshed more than 60 research papers n nternatonal journals and conferences. He s a regular revewer for varous nternatonal journals publshed by Elsever Scence, Sprnger, World Scentfc Publcatons Company Prvate Ltd etc. He has organzed many natonal conferences and workshops. He s recently successfully completed a BRNS funded research project for Department of Atomc energy, Government of Inda. He s one of the authors of more 0Mathematcal books. He s gudng more than 0 students n ths area. Author: Dr. S. Balamuraltharan was born on 0 H May 979, n hrunelvel, aml Nadu. He receved Ph.D degree n Mathematcs from B.S. Abdur Rahman Unversty n the year 0. He s workng as an Assstant Professor n SRM Unversty, kattankulathur-60 0 n the Department of Mathematcs from June 0 to tll date. Hs area of nterests ncludes Bology and nonlnear Dfferental Equaton. He has publshed 0 papers n the Internatonal Journals, presented 0 papers n the Internatonal Conferences (ncludng Malaysa & Chna) and presented papers n the Natonal Conferences. he e-mal ID s: balamural.maths@gmal.com and the contact number s: He s gudng one student n hs area. p-issn : 9-86 Vol 8 No Apr-May