A Critical Path Problem Using Intuitionistic. Trapezoidal Fuzzy Number

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1 pplied Mthemticl Sciences, Vol. 8, 0, no. 5, HIKRI Ltd, Criticl Pth Prolem Using Intuitionistic Trpezoidl Fuzzy Numer P. Jygowri Deprtment of Mthemtics Sudhrsn Engineering College nn University Chenni, Indi G. Geethrmni Deprtment of Mthemtics, BIT Cmpus, nn University Chenni, Indi Copyright 0 P. Jygowri nd G. Geethrmni. This is n open ccess rticle distriuted under the Cretive Commons ttriution License, which permits unrestricted use, distriution, nd reproduction in ny medium, provided the originl work is properly cited. strct Criticl pth method is network sed method premeditted for scheduling nd orgniztion of complex project in rel world ppliction. In this pper, novel pproch hs een mde to find the criticl pth in directed cyclic grph, whose ctivity time is uncertin. The indistinguishle prmeters in the network re represented y intuitionistic trpezoidl fuzzy numers, insted of crisp numers. new procedure is proposed to find the optiml pth, nd n illustrtive exmple is provided to vlidte the proposed pproch. Keywords: Intuitionistic trpezoidl fuzzy numer, Grded men integrtion representtion of intuitionistic trpezoidl fuzzy numer, Rnking of intuitionistic fuzzy numer, Criticl pth.. Introduction constructed network is n impertive tool in the development nd orgnizes definite project execution. Network digrm plys vitl role in formtive project-completion time. In rel life sitution vgueness my rise from numer

2 556 P. Jygowri nd G. Geethrmni of possile sources like: due dte my e distorted, cpitl my unville wether sitution my root severl impediments. Therefore the fuzzy set theory cn ply significnt role in this kind of prolems to hndle the miguity out the time durtion of deeds in project network. G.Ling nd T.C.Hn [6] proposed fuzzy criticl pth for project networks. Elizeth nd L.Sujth[5] discussed criticl pth prolem under fuzzy Environment. C.T Chen nd S.F Hung[] proposed new model tht comines fuzzy set theory with the PERT technique to determine the criticl degrees of ctivities nd pths, ltest nd erliest strting time nd flots. S.H. Nsution[8] proposed fuzzy criticl pth method y considering interctive fuzzy sutrction nd y oserving tht only the non-negtive prt of the fuzzy numers cn hve physicl work. This pper is orgnized s follows: In section, sic definitions of intuitionistic fuzzy set theory hve een reviewed. Section, give procedures to find out the intuitionistic fuzzy criticl pth using n illustrtive exmple.in section the otined results re discussed. Finlly, in section 5 some conclusions re drwn.. Preliminries. Definition: Intuitionistic Fuzzy Set Let X e n Universe of discourse, then n Intuitionistic fuzzy set(ifs) in X is given y = x, μ ( x ), γ ( x) /x X, where the function μ (x) :X 0, nd γ (x) : X 0, determine the degree of memership nd non-memership of the element x X, respectively nd for every x X, 0 μ (x) γ (x).. Definition: Trpezoidl Intuitionistic Fuzzy Numer n Intuitionistic fuzzy numer,,,,,, is sid to e trpezoidl intuitionistic fuzzy numer if its memership function nd nonmemership function re given y (x) (x ) ( ) (x ) ( ) x x x (x) (x ( (x ( ) ) ) ) x x x

3 Criticl pth prolem 557. Definition: Grded Men Integrtion Representtion for Trpezoidl Intuitionistic Fuzzy Numer The memership nd non-memership function of trpezoidl intuitionistic fuzzy numers re defined s follows. L μ (x) L (x) Then x x w ; L nd R w ; L μ (h) ( L γ (h) ( x x & R (x) & R (x) x - x w ; x w ; x re inverse functions of functions L nd R respectively, ) h/w ) h/w & R μ (h) ( & R γ (h) ( ) h/w ) h/w Then the grded men integrtion representtion of memership function nd nonmemership function re, Pμ () 6 () & P () 6 ().5 Definition: Rnking of Intuitionistic Trpezoidl Fuzzy Numer[0] Let α nd β e ny α-cut nd β-cut set of n trpezoidl intuitionstic fuzzy numer ~ (,,c, d,,c, d ) respectively. Then the vlue of the memership nd non-memership function re defined s follows. c d c d c V ( ) nd V ( ) () 6 6 ~ ~ (,, c d,, c d nd B (,, c d, c d,,,,, e two trpezoidl intuitionstic fuzzy numers. If >d then >B..If Vµ()>Vµ(B) then >B. If Vµ()<Vµ(B) then < B. If Vµ()=Vµ(B) then equivlent B. Intuitionistic Fuzzy Criticl Pth Method The following is the procedure for finding intuitionistic fuzzy criticl pth.. Nottions N= the set of ll nodes in project network.

4 558 P. Jygowri nd G. Geethrmni EST = Erliest Strting Time EFTµij, EFTγij = Erliest Finishing Time for memership nd non-memership function. LFTµij, LFTγij = Ltest Finishing Time for memership nd non-memership function LSTµij, LSTγij = Ltest Strting Time for memership nd non-memership function TF = Totl Flot & Tij = The intuitionistic fuzzy ctivity time.. Forwrd Pss Clcultion: Forwrd pss clcultions re employed to clculte the Erliest Strting Time (EST) in the project network E E j j Mx Min i E i( ) t i j, i numer of preceding nodes E ( ) t i j, i numer of preceding nodes () i i Erliest Finishing Time for memership function EFT EST ( ) Intuitionisticfuzzyctivity time μ i j μ i j Erliest Finishing Time for non memership function EFT EST ( ) Intuitionisticfuzzyctivity time (5) γi j γi j. Bckwrd Pss Clcultion: Bckwrd pss clcultions re employed to clculte the Ltest Finishing Time (LFT) in the project network L L i j LST μ i LST γi Min Mx j i L j( ) t i j, j numer of succeeding nodes L ( ) t i j, j numer of succeeding nodes (6) i LtestStrting Time for memership function LFT ( ) Intuitionisticfuzzyctivity time j μ i j Ltest Finishing time for non memership function LFT ( ) Intuitionisticfuzzyctivity time j γi j (7). Totl Flot(TF) TF LFT TF LFT i j i j EFT EFT i j i j or TF LST or TF LST i j i j EST EST i j i j (8)

5 Criticl pth prolem 559. Procedure to Find Intuitionistic Fuzzy Criticl Pth Step: Construct network G(V,E) where V is the set of vertices nd E is the set of edges. Here G is n cyclic digrph nd rc length or edge weight re tken s intuitionistic trpezoidl fuzzy numers. Step : Expected time in terms of intuitionistic trpezoidl fuzzy numers re defuzzified using eqution nd in the network digrm. Step: Clculte erliest strting time for memership nd non-memership functions ( ESTµij nd ESTγij respectively) ccording to forwrd pss clcultion given in eqution () Step : Clculte erliest finishing time for memership nd non-memership functions (EFTµij nd EFTγij respectively ) using eqution (5). Step 5: Clculte ltest finishing time- LFTµij nd LFTγij - ccording to ckwrd pss clcultion given in eqution(6) Step 6: Clculte ltest strting time -LSTµij nd LSTγij- using eqution (7) Step 7: Clculte totl flot- TFµ nd TFγ -using eqution (8) Step 8: In ech ctivity using (8) whenever one get 0, such ctivities re clled s intuitionistic fuzzy criticl ctivities nd the corresponding pths intuitionistic criticl pths..5 Illustrtive exmple: Consider smll network with 5 vertices nd 6 edges shown in figure, where ech rc length is represented s trpezoidl Intuitionistic fuzzy numer Fig

6 560 P. Jygowri nd G. Geethrmni Tle: Results of the network ctivi ty Intuitionistic Fuzzy ctivity time <5,0,5,0> <,,6,8> <,6,8,0> <,6,9,> <8,,6,0> <,5,6,7> <0,5,8,> <0,,,6> <5,,5,> <,,,6> 5 <,6,0,> <8,0,,> Defuzzified ctivity TFµ TFγ time using eqution nd for memership nd non-memership functions <.5,5> 0.8 <7,7.> <,5.> 0 0 <6.,> 0.8 <.,.5> <8,0.5> 0 0 Tle: Rnk vlue of totl slck intuitionstic fuzzy time of ll possile pths Pths IFCPM(pµk) k= to m Rnk vlue Using eqution Rnk IFCPM(pγk) k= to m Rnk vlue Using eqution 5 <7,,5,65> 6.7 III <0,6,,5> 8.5 II 5 <0,8,6,> I <0,5,8,8> 5.7 I 5 <,,,55> 8. II <,9,5,8>. III Rnk. Results nd Discussion This pper proposes n lgorithm to tckle the prolem in intuitionistic fuzzy environment. In this pper the trpezoidl intuitionistic fuzzy numer is defuzzified using grded men integrtion representtion. Now the intuitionistic fuzzy numer is converted to crisp numer. Then pplying the proposed lgorithm we find the criticl pth. The pth in intuitionistic fuzzy project network re 5, 5nd 5.The criticl pth for intuitionistic fuzzy network for oth memership nd non-memership function re 5. Hence the procedure developed in this pper form new methods to get criticl pth, in intuitionistic fuzzy environment.

7 Criticl pth prolem Conclusion new nlyticl method for finding criticl pth in n intuitionistic fuzzy project network hs een proposed. We hve used new defuzzifiction formul for trpezoidl fuzzy numer nd pplied to the flot time for ech ctivity in the intuitionistic fuzzy project network to find the criticl pth. In generl intuitionistic fuzzy models re more effective in determining criticl pths in rel project networks. This pper, use the grded men integrtion representtion the procedure to find the optiml pth in n intuitionistic fuzzy weighted grph hving help decision mkers to decide on the est possile criticl pth in intuitionistic fuzzy environments. References [] K.tnssov, Intuitionistic Fuzzy sets, nd systems, volume 0, No.(986), [] C.T Chen nd S.F. Hung, pplying fuzzy methods for mesuring criticlity in project network, Informtion sciences, 77(007)8-58. [] S.Chns P. nd Zielinski, Criticl Pth nlysis in the Network with Fuzzy ctivity Times, Fuzzy sets nd Systems, (00), [] D. Duois nd H. Prde, Possiility Theory: n pproch to Computerized Processing of Uncertinty, New York: Plenum, 988. [5] S. Elizeth nd L.Sujth, Fuzzy Criticl Pth Prolem For Project Network, Interntionl Journl of Pure nd pplied Mthemtics Volume 85 No. 0, -0. [6] G.S. Ling nd T.C.Hn, Fuzzy Criticl Pth for Project Network, Informtion nd Mngement Sciences, 5, No. (00), 9-0. [7] Mhdvi. I.. Nourifr, R., Heidrzde.. nd miri. N. M. (009). Dynmic Progrmming pproch for Finding Shortest Chins in Fuzzy Network, pplied Soft Computing, Vol.9. No..pp [8] S.H. Nsution, Fuzzy criticl pth method, IEEE Trns. Systems Mn Cyer net, (99) 8-57.

8 56 P. Jygowri nd G. Geethrmni [0]Slim rezvni, Rnking method of Trpezoidl Intuitionistic Fuzzy, nnls of Fuzzy Mthemtics nd Informtics, Vol-5.No:,(My0) p.p [].I Slyeptsov, T.. Tyshchuk, Fuzzy Criticl Pth Method for Project Network Plnning nd Control, Cyer net. System nl. (997) Received: Mrch 7, 0