A New Algorithm for Solving Shortest Path Problem on a Network with Imprecise Edge Weight

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1 Availabl at Appl Appl Math ISSN: Vol 6, Issu (Dcmbr 011), pp Applications and Applid Mathmatics: An Intrnational Journal (AAM) A Nw Algorithm for Solving Shortst Path Problm on a Ntwork with Imprc Edg Wight Amit Kumar and Manjot Kaur School of Mathmatics and Computr Applications Thapar Univrsity Patiala , India amit_rs_iitr@yahoocom;manjotthaparian@gmailcom Rcivd: August 0, 011; Accptd: Dcmbr 1, 011 Abstract Naym and Pal (Shortst path problm on a ntwork with imprc dg wight, Fuzzy Optimization and Dcion Making 4, 93-31, 005) proposd a nw algorithm for solving shortst path problm on a ntwork with imprc dg wight In th papr th shortcomings of th xting algorithm, (Naym and Pal, 005) ar pointd out and to ovrcom ths shortcomings a nw algorithm proposd To show th advantags of th proposd algorithm ovr xting algorithm th numrical xampls prsntd in (Naym and Pal, 005) ar solvd using th proposd algorithm and obtaind rsults ar dcussd Kywords: Fuzzy shortst path problm, Ranking function, Intrval numbrs, Triangular fuzzy numbrs MSC 010 No: 03E7, 74P10, 5B05 1 Introduction Th shortst path problm concntrats on finding th path with minimum dtanc To find th shortst path from a sourc nod to th othr nods a fundamntal mattr in graph thory In convntional shortst path problms, it assumd that dcion makr crtain about th paramtrs (dtanc, tim tc) btwn diffrnt nods But in th ral lif situations thr 60

2 AAM: Intrn J, Vol 6, Issu (Dcmbr 011) 603 always xts uncrtainty about th paramtrs of shortst path problms To dal with such typ of problms, th paramtrs of shortst path problms ar rprsntd by fuzzy numbrs (Zadh, 1965) Klin (1991) prsntd nw modls basd on fuzzy shortst paths and also givn a gnral algorithm basd on dynamic programming to solv th nw modls Lin and Chrn (1993) considrd th cas that th arc lngths ar fuzzy numbrs and proposd an algorithm for finding th singl most vital arc in a ntwork Okada and Gn (1994) dcussd th problm of finding th shortst paths from a fixd origin to a spcifid nod in a ntwork with arcs rprsntd as intrvals on ral lin Li t al (1996) introducd th nural ntworks for solving fuzzy shortst path problms Gnt t al (1997) invstigatd th possibility of using gntic algorithms to solv shortst path problms Shih and L (1999) invstigatd multipl objctiv and multipl hirarch minimum cost flow problms with fuzzy costs and fuzzy capacit in th arcs Okada and Sopr (000) concntratd on a shortst path problm on a ntwork in which a fuzzy numbr, instad of a ral numbr, assignd to ach arc lngth Okada (001) concntratd on a shortst path problm on a ntwork in which a fuzzy numbr, instad of a ral numbr, assignd to ach arc lngth Liu and Kao (004) invstigatd th ntwork flow problms in that th arc lngths of th ntwork ar fuzzy numbrs Sda (005) dald with th stinr tr problm on a graph in which a fuzzy numbr, instad of a ral numbr, assignd to ach dg Takahashi (005) dcussd th shortst path problm with fuzzy paramtrs H proposd a modification in Okada's (001) algorithm, using som proprt obsrvd by othr authors H also proposd a gntic algorithm to sk an approximatd solution for larg scal problms Chuang and Kung (005) rprsntd ach arc lngth as a triangular fuzzy numbr and proposd a nw algorithm to dal with th fuzzy shortst path problms Naym and Pal (005) considrd a ntwork with its arc lngths as imprc numbr, instad of a ral numbr, namly, intrval numbr and triangular fuzzy numbr Ma and Chn (005) proposd an algorithm for th on-lin fuzzy shortst path problms, basd on th traditional shortst path problm in th domain of th oprations rsarch and th thory of th on-lin algorithms Kung and Chuang (005) proposd a nw algorithm composd of fuzzy shortst path lngth procdur and similarity masur to dal with th fuzzy shortst path problm Gupta and Pal (006) prsntd an algorithm for th shortst path problm whn th connctd arcs in a transportation ntwork ar rprsntd as intrval numbrs Moazni (006) dcussd th shortst path problm from a spcifid nod to vry othr nod on a ntwork in which a positiv fuzzy quantity with finit support assignd to ach arc as its arc lngth Chuang and Kung (006) pointd out that thr ar svral mthods rportd to solv th kind of problm in th opn litratur In ths mthods, thy can obtain ithr th fuzzy shortst lngth or th shortst path In thir papr, a nw algorithm was proposd that can obtain both of thm Th dcrt fuzzy shortst lngth mthod proposd to find th fuzzy shortst lngth, and th fuzzy similarity masur utilizd to gt th shortst path Ji t al (007) considrd th shortst path problm with fuzzy arc lngths According to diffrnt dcion critria, th concpts of xpctd shortst path, a-shortst path and th shortst path in fuzzy nvironmnt ar originally proposd, and thr typs of modls ar formulatd In ordr to solv ths modls, a hybrid intllignt algorithm intgrating simulation and gntic algorithm providd and som numrous xampls ar givn to illustrat its ffctivnss

3 604 Amit Kumar and Manjot Kaur Hrnands t al (007) proposd an itrativ algorithm that assums a gnric ranking indx for comparing th fuzzy numbrs involvd in th problm, in such a way that ach tim in which th dcion-makr wants to solv a concrt problm (s)h can choos (or propos) th ranking indx that bst suits that problm Yu and Wi (007) proposd a simpl linar multipl objctiv programming to dal with th fuzzy shortst path problm Mahdavi t al (009) proposd a dynamic programming approach to solv th fuzzy shortst chain problm using a suitabl ranking mthod In th papr th shortcomings of th xting algorithm, (Naym and Pal, 005) ar pointd out and to ovrcom ths shortcomings a nw algorithm proposd To show th advantags of th proposd algorithm ovr xting algorithm th numrical xampls prsntd in (Naym and Pal, 005) ar solvd using th proposd algorithm and obtaind rsults ar dcussd Th papr organizd as follows: In Sction, som basic dfinitions, addition of intrval and triangular fuzzy numbrs and comparon mthods btwn such numbrs ar rviwd and also th notations usd throughout th papr ar prsntd In Sction 3, an algorithm proposd for finding th fuzzy shortst path and fuzzy shortst dtanc of ach nod from sourc nod In Sction 4, to illustrat th proposd algorithm and to point out th shortcomings of th xting algorithm (Naym and Pal, 005) numrical xampls prsntd in Naym and Pal (005) ar solvd by using th proposd algorithm In Sction 5, shortcomings of th xting comparon mthod (Naym and Pal, 005) ar pointd out In Sction 6, th comparon mthods proposd by Liou and Wang (199) ar rviwd In Sction 7, th comparon mthods, rviwd in Sction 6, ar usd to solv th numrical xampls prsntd in Sction 4 Th obtaind rsults ar dcussd in Sction 8 Th conclusions ar dcussd in Sction 9 Prliminar In th sction som basic dfinitions, addition of intrval numbrs and triangular fuzzy numbrs and comparon mthods of such numbrs ar rviwd Also th notations usd throughout th papr ar prsntd 1 Basic Dfinitions In th sction, som basic dfinitions ar rviwd Dfinition 1 (Naym and Pal, 005) An intrval numbr dfind as A [ al, ar ] { a : al a ar}, whr, a L and a R ar th ral numbrs calld th lft nd point and th right nd point of th intrval A Anothr way to rprsnt an intrval numbr in trms of midpoint and width a A m( A), w( A), whr m(a) midpoint of R a A L and w(a) half width of a R a A L

4 AAM: Intrn J, Vol 6, Issu (Dcmbr 011) 605 Dfinition (Naym and Pal, 005) Two intrval numbrs ar said to b non-dominating if i m A mb and ii w w A B A, m w and B m, w A A B B Dfinition 3 (Naym and Pal, 005) A triangular fuzzy numbr rprsntd by a triplt A m,, with th mmbrship function m x 1 x m ( x) 1 A 0 for m x m for m x m othrw whr m R and, 0 Dfinition 4 (Naym and Pal, 005) Two triangular fuzzy numbrs B b,, ar said to b non-dominating if A a,, and i a b and ii or but, not both simultanously Addition of Intrval Numbrs and Triangular Fuzzy Numbrs (Kaufmann and Gupta, 1985) Th addition of two intrval numbrs A a L, a ] and B b, b ] L givn by A B a b, a b ] [ L L R R [ R [ R Altrnatly, in man-width notations, if A m, w and B m, w thn, 1 1 A B m 1 m, w1 w Lt A m1, 1, 1 and B m,, b two triangular fuzzy numbrs thn, A B m 1 m, 1, 1

5 606 Amit Kumar and Manjot Kaur 3 Comparon of Intrval Numbrs (Naym and Pal, 005) Naym and Pal (005) usd accptability indx (A-indx) to th proposition A infrior to B m m1 as A ( A B) w w 1 In connction with th accptability indx, Naym and Pal (005) dfind th total dominanc and partial dominanc of two intrval numbrs m, w 1 1 B m, w on ovr anothr as follow: i If A ( A B) 1 thn, A said to b totally dominating ovr B in th sns of minimization and B said to b totally dominating ovr A in th sns of maximization W dnot th by A B, i, minimum { A, B} A ii If 0 A ( A B) 1 thn, A said to b partially dominating ovr B in th sns of minimization and B said to b partially dominating ovr A in th sns of maximization W dnot th by A P B, i, minimum { A, B} A iii But whn, A ( A B) 0 i, m1 m thn w may not gt an ordr rlation from th abov cass Thn w may mphasiz on th widths of th intrval numbrs A and B If w1 w thn th lft nd point of A lss than that of B and on finding a minimum dtanc, thr a chanc that th dtanc may li on A But at th sam tim, sinc th right nd point of A gratr than that of B, if on prfrs A to B in minimization thn in worst cas, h may b loosr than on who prfrs B to A Thus in such a situation an optimtic dcion-makr would prfr A to B whras a pssimtic dcion-makr would do th convrs 4 Comparon of Triangular Fuzzy Numbrs (Naym and Pal 005) Th accptability indx (A-indx) to th proposition A a,, prfrrd to B b,, b a givn by A ( A B ) Using th A-indx Naym and Pal (005) dfind th following ranking ordrs i If A ( A B) 1 thn, A said to b totally dominating ovr B in cas of minimization and th cas convrs in cas of maximization and th dnotd by A B, i, minimum { A, B} A ii If 0 A ( A B) 1 thn, A said to b partially dominating ovr B in th sns of minimization and B said to b partially dominating ovr A in th sns of maximization Th dnotd by A P B, i, minimum { A, B} A

6 AAM: Intrn J, Vol 6, Issu (Dcmbr 011) Notation In th sction th notation that will b usd throughout th papr ar prsntd N { 1,,, n} : Th st of all nods in a ntwork Np ( j) : Th st of all prdcssor nods of nod j : Th dtanc btwn nod i and first (sourc) nod i : Th dtanc btwn nod i and j ij i : Th fuzzy dtanc btwn nod i and first (sourc) nod : Th fuzzy dtanc btwn nod i and j ij Rmark 1 A nod i said to b prdcssor nod of nod j if (i) Nod i dirctly connctd to nod j (ii) Th dirction of path, conncting nod i and j, from i to j 3 Proposd Algorithm In th sction a nw algorithm proposd for finding th fuzzy shortst path and fuzzy shortst dtanc of ach nod from sourc nod Th stps of th algorithm ar summarizd as follows: Stp1 Assum 0,0, 0 (or 0, 0 intrval numbr) and labl th sourc nod (say nod 1) as 1 [ 0,0,0, ] (or [ 0,0, ]) Stp Find minimum{ / i Np( j)}; j 1, j,3,, n j i ij Stp 3 If minimum occurs corrsponding to uniqu valu of i i, i r thn labl nod j as [ j, r] If minimum occurs corrsponding to mor than on valus of i thn it rprsnts that thr ar mor than on fuzzy path btwn sourc nod and nod j but fuzzy dtanc along all paths j, so choos any valu of i Stp 4 Lt th dstination nod (nod n ) b labld as [ n, l], thn th fuzzy shortst dtanc btwn sourc nod and dstination nod n

7 608 Amit Kumar and Manjot Kaur Stp 5 Sinc dstination nod labld as [ n, l] So, to find th fuzzy shortst path btwn sourc nod and dstination nod, chck th labl of nod l Lt it b [ l, p], now chck th labl of nod p and so on Rpat th sam procdur until nod 1 obtaind Stp 6 Now th fuzzy shortst path can b obtaind by combining all th nods obtaind by th Stp 5 Rmark If thr no uncrtainty about any paramtr thn th proposd algorithm also a nw algorithm for finding th optimal solution for convntional shortst path problms 4 Illustrativ Exampls To show th advantags of th proposd algorithm ovr th xting algorithm (Naym and Pal, 005), th numrical xampls prsntd in Naym and Pal (005) ar solvd by using th proposd algorithm and th rsults of xting and th proposd algorithms ar compard Exampl 1 (Naym and Pal, 005) Th problm to find th shortst path btwn sourc nod (say nod 1) and dstination nod (say nod 6) on th ntwork consts of 6 vrtics { 1,,3,4,5,6} and 11 dgs { 1, 13, 14, 3, 4, 34, 35, 36, 45, 46, 56} th arc lngths of th ntwork, shown in Figur 1 ar all intrval numbrs and givn by 1 [ 10,1], 13 [5,8], 14 [19,0], 3 [0,1], 4 [30,35], 34 [65,75], 35 38,40], [43,44], [35,40], [49,51], [1,13] [ Figur 1 A ntwork Solution Th man-width notations of intrval numbrs as follow:

8 AAM: Intrn J, Vol 6, Issu (Dcmbr 011) ,1, 65,15, 195,05, 05,05, 35,5, 7,05, ,1, 435,05, 375,5, 50,1, 15, Sinc nod 6 th dstination nod, so n 6 46 Assum 0, 0 and labl th sourc nod (say nod 1) as [ 0,0, ], th valus of j ; j,3,4,5,6 can b obtaind as follows: Itration 1 Sinc only nod 1 th prdcssor nod of nod, so putting i 1 and j in Stp of th proposd algorithm, th valu of 0,0 11,1 11, 1 minimum{ 1 1} minimum Sinc minimum occurs corrsponding to i 1, so labl nod as [ 11,1,1] Itration Th prdcssor nods of th nod 3 ar nod 1 and, so putting i 1, and j 3 in stp of th proposd algorithm, th valu of 3 minimum{ 1 13, 3} 3 minimum 0,0 65,15, 11,1 05,05 minimum 65,15, 315, A 65,15 315, Using Sction 3, 65,15, 315,15 minimum = 6 5,1 5 i, 65,1 5 3 Sinc minimum occurs corrsponding to i 1, so labl nod 3 as [ 65,15,1] Itration 3 Th prdcssor nod of th nod 4 nod 1, and 3, so putting i 1,, 3 and j 4 in stp of th proposd algorithm, th valu of 4 4 minimum{ 1 14, 4, 3 34}

9 610 Amit Kumar and Manjot Kaur minimum 0,0 195,05, 11,1 35,5, 65,15 7,05 195,0 5 Sinc minimum occurs corrsponding to i 1, so labl nod 4 as [ 195,05,1] Itration 4 Th prdcssor nods of th nod 5 ar nod 3 and 4, so putting i 3, 4 and j 5 in Stp of th proposd algorithm, th valu of 5 minimum{ 3 35, 4 45} 5 minimum 65,15 39,1, 195,05 375,5 minimum 655,5, 57,3 57, 3 Sinc minimum occurs corrsponding to i 4, so labl nod 5 as [ 57,3,4] Itration 5 Th prdcssor nods of th nod 6 ar nod 3, 4 and 5, so putting i 3, 4, 5 and j 6 in stp of th proposd algorithm, th valu of 6 minimum{ 3 36, 4 46, 5 56} 6 minimum 65,15 435,05, 195,05 50,1, 57,3 15,05 minimum 70,, 695,15, 695,35 695,15 or 695,35 6 Sinc minimum occurs corrsponding to i 4, 5 so w can labl nod 6 as [ 695,15,4] or [ 695,35,5], if w labl nod 6 as [ 695,15,4] thn th corrsponding shortst dtanc 695 Now th fuzzy shortst path btwn nod 1 and nod 6 can b obtaind by using th following procdur: Sinc nod 6 labld by [ 695,15,4], which rprsnts that w ar coming from nod 4 Nod 4 labld by [ 195,05,1], which rprsnts that w ar coming from nod 1 Now th fuzzy shortst path btwn nod 1 and nod 6 obtaind by joining all th obtaind nods Hnc th fuzzy shortst path and in th scond cas if w labl nod 6 as [ 695,35,5] thn th corrsponding shortst dtanc sam i, 695 but th shortst path

10 AAM: Intrn J, Vol 6, Issu (Dcmbr 011) 611 Exampl (Naym and Pal, 005) Lt us considr th sam ntwork, shown in Fig 1, with its arc lngths as triangular fuzzy numbrs givn by 1 (10,11,1), (5,7,8), (19,0,), 3 (0,1,1), 4 (30,34,35), ( 65,7,8 ), 35 (30,30,3), 36 (43,44,45), 45 (39,40,40), 46 (49,50,5), (9,9,10) and w ar intrstd to find th fuzzy shortst path and fuzzy shortst path btwn th nods 1 and 6 Solution: m,,, i, in trms of man and th lft- Th triangular fuzzy numbrs in th form of sprads and right-sprads ar as follow: 1 11,1,1, 7,,1, 0,1,, 1,1,0, ,4,1, 34 7,05,1, 35 30,0,, 36 44,1,1, 45 40,1, 0, 46 50,1,, 9,0, 1 56 Sinc nod 6 th dstination nod, so n 6 Assum 1 0,0, 0 and labl th sourc nod (say nod 1) as [ 0,0,0, ], th valus of j ; j,3,4,5,6 can b obtaind as follows: Itration 1 Sinc only nod 1 th prdcssor nod of nod, so putting i 1 and j in Stp of th proposd algorithm, th valu of 0,0,0 11,1,1 11,1, 1 minimum{ 1 1} minimum Sinc minimum occurs corrsponding to i 1, so labl nod as [ 11,1,1,1] Itration Th prdcssor nods of th nod 3 ar nod 1 and, so putting i 1, and j 3 in Stp of th proposd algorithm, th valu of 3 minimum{, } minimum 0,0,0 7,,1, 11,1,1 1,1,0 minimum 7,,1, 3,,1 Sinc A a,, 7,, 1 and B b,, 3,, 1

11 61 Amit Kumar and Manjot Kaur 3 7 A ( A B) So using Sction 3, 7,,1, 3,,1 7,, 1 minimum, i, 3 7,, 1 Sinc minimum occurs corrsponding to i 1, so labl nod 3 as [ 7,,1,1] Similarly, 4 0,1,, labl nod 4 as [ 0,1,,1 ], 5 57,,3, labl nod 5 as [ 57,,3,3 ], 6 66,,4, labl nod 6 as [ 66,,4,5 ] Sinc nod 6 th dstination nod of th givn ntwork, so th fuzzy shortst dtanc btwn nod 1 and 6 66,, 4 and th fuzzy shortst path Shortcomings of Exting Comparon Mthods (Naym and Pal, 005) To show th advantags of th proposd algorithm ovr xting algorithm (Naym and Pal, 005) th numrical xampls prsntd in Naym and Pal (005) ar solvd by th proposd algorithm and it found that th rsults of xting and proposd algorithm ar sam, whil th xting algorithm vry confusing to undrstand and to apply for finding th optimal solution compar to th proposd algorithm For solving th numrical xampls th comparon mthods prsntd in Naym and Pal (005) ar usd but thr ar th following shortcomings in ths comparon mthods: (i) To show that th xting comparon mthod (Naym and Pal, 005) can t b usd for finding th fuzzy shortst path of ral lif problms th fuzzy shortst path and fuzzy shortst dtanc btwn nod 1 and 4 of th ntwork, shown in Fig, obtaind by th xting comparon mthod (Naym and Pal, 005) and th obtaind rsults ar as follows: (,1,1) (,,3) 1 4 (15,05,35) 3 (5,5,35) Figur A ntwork

12 AAM: Intrn J, Vol 6, Issu (Dcmbr 011) 613 In th ntwork shown in Figur, thr ar two possibl paths 1 4 and btwn nod 1 and 4 Using th xting comparon mthod (Naym and Pal, 005) th dtanc btwn nod 1 and nod 4 along th first path i 1 4 (4,3,4) whil along th scond path i th dtanc btwn th nod 1 and nod 4 (4,3,7) It obvious from dfinition 4 that th dtancs (4,3,4) and (4,3,7) ar non-dominating and it not possibl to find th minimum btwn ths dtancs so according to xting comparon mthod (Naym and Pal, 005) th dcion makr can choos ithr 1 4 or i using th xting comparon mthod it not possibl to choos th bst from 1 4 and But it obvious from th valus of th dtancs of paths that a dcion makr will choos th path 1 4 Sinc along th path th travld dtanc will b btwn 1 unit and 8 unit and th maximum possibility that it will b 4 unit whil along th scond path th travld dtanc will b btwn 1 and 11 unit and th maximum possibility that it will b 4 unit Hnc it can b concludd that xting comparon mthod Naym and Pal (005) should not b usd to compar th fuzzy numbrs for solving ral lif problms (ii) Naym and Pal (005) hav pointd out that thir mthod for comparon of diffrnt numbrs particular cas of Okada and Sopr (000) mthod but from th ntwork, shown in Fig 3, it clar that th rsults ar diffrnt using both xting mthods (3,1,1) (8,1,3) 1 4 (4,3,1) 3 (9,,) Figur 3 A ntwork According to comparon mthod prsntd in Okada and Sopr (000) th fuzzy shortst path and fuzzy shortst dtanc ar and ( 13,5,3) rspctivly, whil using th comparon mthod prsntd in Naym and Pal (005) th fuzzy shortst path and fuzzy shortst dtanc ar 1 4 and ( 11,,4) rspctivly, i, according to Okada and Sopr (000) th fuzzy shortst path whil according to Naym and Pal (005) th fuzzy shortst path 1 4 (iii) (iv) Both th xting algorithms (Okada and Sopr, 000; Naym and Pal, 005) ar vry difficult and confusing to undrstand and to apply for a nw dcion makr, for finding th fuzzy optimal solution of shortst path problms occurring in ral lif problms In th ral lif problms, it rquird to compar mor than two fuzzy numbrs (or intrval numbrs) simultanously But it vry difficult to compar a larg numbr of

13 614 Amit Kumar and Manjot Kaur fuzzy numbrs simultanously using th xting comparon mthod (Naym and Pal, 005) For xampl in 3 rd and 5 th itration of Exampl 1 and, it rquird to calculat th accptability indx of ach pair i, it not asy to find th minimum of thr numbrs To ovrcom th abov shortcomings th xting comparon mthod (Liou and Wang, 199) usd for solving th Exampls 1 and 6 Comparon of Intrval and Triangular Fuzzy Numbrs (Liou And Wang, 199) Du to th shortcomings of th xting comparon mthods (Naym and Pal, 005) it bttr to us th following comparon mthod (Liou and Wang, 199) 61 Comparon of Triangular Fuzzy Numbrs Lt (i) (ii) (iii) A m,, ) and B ( m,, ) b two triangular fuzzy numbr thn ( A B if ( A ) ( B ) A B if ( A ) ( B ) A B if ( A ) ( B ), 1 1 whr ( A ) m1 ( 1 1) and ( B ) m ( ) Comparon of Intrval Numbrs Lt A a, a ] and B b, b ] L b two intrval numbrs thn th symmtric triangular fuzzy [ L R [ R numbrs A and B corrsponding to A and B ar givn by al ar ar al ar a A,, L, bl br br bl br b B,, L and a ( ) L a A R, b ( ) L br B (i) A B if a L a R bl br (ii) A B if a L a R bl br

14 AAM: Intrn J, Vol 6, Issu (Dcmbr 011) 615 (iii) A B if a L a R b L b R 7 Illustrativ Exampls Using Exting Comparon Mthod (Liou and Wang, 199) In Sction 4, to solv th numrical xampls th proposd algorithm usd with xting comparon mthod (Naym and Pal, 005) but du to shortcomings in th xting comparon mthod in th sction th sam numrical xampls ar solvd using th proposd algorithm with xting comparon mthod (Liou and Wang, 199) Exampl 3 Lt th arc lngths of th ntwork shown in Fig 1 b all intrval numbrs and b givn by [10,1], ], 1 13 [5,8 [19,0 14 ], ], 3 [0,1 [30,35 4 ], ], 34 [65,75 35 [38,40], 36 [43,44], 45 [35,40], 46 [49,51], 56 [1,13] thn w hav to find out th shortst path btwn th vrtics 1 and 6 Solution: Sinc nod 6 th dstination nod, so n 6 and labl th sourc nod (say nod 1) as 0,0], Assum [0,0 1 ] j,3,4,5,6 can b obtaind as follows: [, th valus of j ; Itration 1 Sinc only nod 1 th prdcssor nod of nod, so putting i 1 and j in stp of th proposd algorithm, th valu of [0,0] [11,1] [11,1] minimum{ 1 1} minimum Sinc minimum occurs corrsponding to 1 i, so labl nod as 11,1],1 [ Itration Th prdcssor nods of th nod 3 ar nod 1 and, so putting i 1, and j 3 in Stp of th proposd algorithm, th valu of 3 minimum{ 1 13, 3} 3 minimum [0,0] [5,8],[10,1] [0,1] minimum [5,8],[30,33]

15 616 Amit Kumar and Manjot Kaur Sinc minimum, i, 3 [5,8] So using Sction 6, [5,8],[30,33] [5,8] Sinc minimum occurs corrsponding to 1 Similarly, 4 [19,0], labl nod 4 as 19,0],1 5 [54,60] i, so labl nod 3 as 5,8],1 [,, labl nod 5 as 54,60],4 [ [ 6 [68,71] or [66,73], Sinc minimum occurs corrsponding to i 4, 5 so w can labl nod 6 as [ 68,71],4or [66,73],5, if w labl nod 6 as [ 68,71],4 thn th corrsponding shortst dtanc 695 and path and in th scond cas if w labl nod 6 as [ 66,73],5 thn th corrsponding shortst dtanc sam i, 695 but th shortst path Exampl 4 (Naym and Pal, 005) Lt us considr th sam ntwork with its arc lngths as triangular fuzzy numbrs as shown in Exampl Solution: Sinc nod 6 th dstination nod, so n 6 Assum 0,0, 0 and labl th sourc nod (say nod 1) as [ 0,0,0, ], th valus of j ; 1 j,3,4,5,6 can b obtaind as follows: Itration1 Sinc only nod 1 th prdcssor nod of nod, so putting i 1 and j in Stp of th proposd algorithm, th valu of 0,0,0 11,1,1 11,1, 1 minimum{ 1 1} minimum Sinc minimum occurs corrsponding to i 1, so labl nod as [ 11,1,1,1] Itration Th prdcssor nods of th nod 3 ar nod 1 and, so putting i 1, and j 3 in Stp of th proposd algorithm, th valu of 3 minimum{, }

16 AAM: Intrn J, Vol 6, Issu (Dcmbr 011) 617 minimum 0,0,0 7,,1, 11,1,1 1,1,0 minimum 7,,1, 3,,1 Sinc A 7,, 1 and B 3,, 1, using Sction 61, ( A ) 6 75 and ( B ) Sinc ( A ) ( B), so minimum 7,,1, 3,,1 7,, 1, i, 3 7,, 1 Sinc minimum occurs corrsponding to i 1, so labl nod 3 as [ 7,,1,1] Similarly, 4 0,1,, labl nod 4 as [ 0,1,,1 ], 5 57,,3, labl nod 5 as [ 57,,3,3 ], 6 66,,4, labl nod 6 as [ 66,,4,5 ] Sinc nod 6 th dstination nod of th givn ntwork, so th fuzzy shortst dtanc btwn nod 1 and 6 66,, 4 and th fuzzy shortst path Advantags of Exting Comparon Mthod (Liou and Wang, 199) In th sction, it shown that if w apply th proposd algorithm with xting comparon mthod (Liou and Wang, 199) to solv th fuzzy shortst path problms thn it ovrcoms all th shortcomings dscribd in Sction 5 (i) Using th proposd algorithm with xting comparon mthod (Liou and Wang, 199) th fuzzy shortst path and fuzzy shortst dtanc btwn nod 1 and 4, of th ntwork shown in Fig, ar 1 4 and (4,3,4) rspctivly (ii) Th proposd algorithm vry asy to undrstand and to apply for a nw dcion makr, for finding th fuzzy shortst path problms (iii) It vry asy to compar mor thn two fuzzy numbrs (intrval numbrs) simultanously 8 Rsults and Dcussion To compar th proposd algorithm with xting algorithm (Naym and Pal, 005) th numrical xampls prsntd in Naym and Pal (005) ar solvd using th proposd algorithm and th following rsults ar obtaind (i) If th proposd algorithm applid with xting comparon mthod (Naym and Pal, 005) thn th obtaind shortst path and shortst dtanc ar sam as obtaind by th xting algorithm (Naym and Pal, 005) but th xting algorithm vry confusing to undrstand and to apply for finding th optimal solution of shortst path problms for a nw dcion makr whil th proposd algorithm vry asy to undrstand and to apply for th sam

17 618 Amit Kumar and Manjot Kaur (ii) If th proposd algorithm applid with th xting comparon mthod (Liou and Wang, 199) thn it ovrcoms all th shortcomings, dscribd in Sction 5 and th shortst path and shortst dtanc ar sam as obtaind by th xting algorithm On th bas of abov rsults it can b suggstd that it bttr to us th proposd algorithm with xting comparon mthod (Liou and Wang, 199) compar to xting mthod (Naym and Pal, 005) for finding th fuzzy shortst path and fuzzy shortst dtanc of fuzzy shortst path problms occurring in ral lif situations 9 Conclusions Th shortcomings of th xting algorithm for finding th fuzzy shortst path and fuzzy shortst dtanc of any nod from sourc nod ar pointd out and to ovrcom ths shortcomings a nw algorithm proposd for th sam To show th advantag of th proposd algorithm ovr xting algorithm th rsults of som fuzzy shortst path problms, obtaind by using th xting algorithm and proposd algorithm ar compard Acknowldgmnts Th authors would lik to thank to th Editor-in-Chif Dr A M Haghighi and anonymous rfrs for various suggstions which hav ld to an improvmnt in both th quality and clarity of th papr First author want to acknowldg th adolscnt innr blssings of Mhar H bliv that Mhar an angl for him and without Mhar s blssing it was not possibl to think th ida proposd in th papr Mhar a lovly daughtr of Parmprt Kaur (Rsarch Scholar undr h suprvion) Th authors also acknowldg th financial support givn by th Univrsity Grant Commsion, Govt of India for complting th Major Rsarch Projct (39-40/010(SR)) REFERENCES Chuang, T N and Kung, J Y (005) Th fuzzy shortst path lngth and th corrsponding shortst path in a ntwork, Computrs and Oprations Rsarch, Vol 3, No6, pp Chuang, T N and Kung, J Y (006) A nw algorithm for th dcrt fuzzy shortst path problm in a ntwork, Applid Mathmatics and Computation, Vol 174, No 1, pp Gnt, M, Chng, R and Wang, D (1997) Gntic algorithms for solving shortst path problms, Procdings of th IEEE Intrnational Confrnc on Evolutionary Computation, pp Gupta, A S and Pal, T K (006) Solving th shortst path problm with intrval arcs, Fuzzy Optimization and Dcion Making, Vol 5, No 1, pp Hrnands, F, Lamata, M T, Vrdgay, J L and Yamakami, A (007) Th shortst path problm on ntworks with fuzzy paramtrs, Fuzzy Sts and Systms, Vol 158, No 14, pp

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