Solving the fuzzy shortest path problem on networks by a new algorithm

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1 Proceedgs of the 0th WSEAS Iteratoal Coferece o FUZZY SYSTEMS Solvg the fuzzy shortest path proble o etworks by a ew algorth SADOAH EBRAHIMNEJAD a, ad REZA TAVAKOI-MOGHADDAM b a Departet of Idustral Egeerg, Islac Azad Uversty, Kara Brach, Kara, Ira b Departet of Idustral Egeerg, Faculty of Egeerg, Uversty of Tehra, Tehra, Ira Correspodg author. Fax: 98 8 E-al addresses: Ibrahead@kau.ac.r E-al addresses: tavakol@ut.ac.r Abstract: I Ths paper we presetes the fuzzy shortest path (FSP proble to fd the best fuzzy path aog other fuzzy paths sets. To solve ths proble, we propose a ew ad effcet algorth, based o a ew defto of a deal fuzzy set (IFS, order to fd the shortest path. Further, we exted ths algorth for a fuzzy etwork proble cosstg of three crtera: te, cost, ad qualty rsk of actvtes. Soe uercal exaples are llustrated ad the experetal results are the copared agast the fuzzy u algorth wth referece to the ultple labelg algorth, based o the slarty degree, to show the effcecy of the proposed algorth. Our coputatoal results ad statstcal aalyses suggest that our proposed algorth outperfors the fuzzy u algorth. Keywords: Fuzzy shortest path proble; Mult-crtera etwork; Ideal fuzzy set.. Itroducto Due to coforty of the fuzzy shortest path (FSP proble real-world cases, soe researchers have focused o ths ssue. Dubos ad Prade [] frst aalyzed ths proble ad proposed a algorth to fd the shortest path. They reported soe solutos for the classcal FSP proble by the use of exteded su ad exteded ad ax operators. Kle [6] preseted a dyac prograg recurso algorth order to fd the path(s related to the threshold of a ebershp degree set by the decso aker. Chaas et al. [] proposed a approach based o the α-cut cocept. I addto, Furukawa [] troduced a approach based o paraetrc orders. Okada & Spore [] preseted a algorth based o the order relato for a fuzzy etwork proble wth -R fuzzy ubers. They defed odoated or Pareto optal paths fro the specfed ode to every other ode. Okada [] troduced the cocept of degree of possblty that a arc s o the shortest path ad defed a ew coparso dex betwee the su of fuzzy ubers, whch the teracto s cosdered aog fuzzy ubers. Okada's approach has a great depedece to the α-level ad the lower degree of possblty o a etwork path showg a great uber of o-doated paths. Chuag & Kug [] preseted aother algorth to fd the shortest path based o the dea of a u crsp uber, f ad oly f ay the other uber s greater tha or equal to ths uber, they developed ths dea to the fuzzy shortest path legth. Chuag & Kug [] developed ther algorth a dscrete ode ad proposed a ew algorth to fd the dscrete fuzzy shortest legth a etwork. Kug & Chuag [7] proposed a ew algorth to solve the shortest path proble wth dscrete fuzzy arc legths. They s developed a fuzzy shortest path legth procedure by a fuzzy u algorth. Moaze [8] represeted a lexcographc order relato aog fuzzy ubers. By usg ultple labelg ad Dkstra s shortest path algorths, she proposed a ew algorth to fd a set of o-doated paths, whch s related to the exteso prcple cocept. Okada & Ge [9] cosdered a shortest path proble wth a ew defto for order relato betwee tervals. Okada & Ge [0] cosdered a shortest path proble a etwork wth arcs represeted as tervals o real le. They defed a order relato betwee tervals by usg two paraeters ad proposed a algorth based o the Dkstra's ethod to solve the large-szed probles.. Defto of the deal fuzzy set The deal fuzzy set (IFS cossts of a set of shorter ad loger legths havg the axu ad u ISSN: ISBN:

2 Proceedgs of the 0th WSEAS Iteratoal Coferece o FUZZY SYSTEMS degree of ebershp, respectvely. Because we are terested fdg the shortest path, the path legth s the ost sgfcat factor the gve proble. Hece, the axu ad u ebershp degrees are assged to shorter ad loger legths of the IFS, respectvely... Coputato of the IFS If x s the shortest path legth the frst eber of ths fuzzy path, based o the above defto of the IFS, the ost possble ebershp s assged to x, where {µ(x =. x s also the logest legth the last eber of the fuzzy path, whch has the lowest ebershp. However, ths ebershp caot be zero, where {µ(x >0. Because ths s a dscrete set, we assg zero ebershp to the ext legth (.e., x, where {µ(x =0. Thus, x gas the lowest feasble ebershp. Eq. ( defes the ebershp fucto of x x the IFS. μ ( x = ( x x Fg. depcts a dscrete IFS where the ebershp of shorter legths tha x s set to 0. Because x s the shortest path legth the IFS ad there s o shorter legth tha all legths ths set. Assue that,.,..., are fuzzy etwork paths k defed as follows: = {( x, μ ( x x X ( =,,..., k where, X represets the set of legths of path. Followg s the IFS of ths etwork. μ( α μ( γ 0 I = {,,...,, ( α α γ γ where, γ = Max{Max { x, α = M{M { x ad x s eber of set. Its degree of ebershp s coputed by usg Eq. (. α x μ( x = ( γ α Dstace [α, γ] s the referece set for the etwork paths, whose ebers s uo of all ebers of etwork paths, ad the shortest path of the etwork ust be a subset of ths dstace. μ(x μ(x μ(x x 0 x x X x Fg. Dscrete deal fuzzy set.. Defto of the optal set The optal set s a subset of the referece set of etwork paths, whch the ebers of the fuzzy shortest path are detered. The optal set [ α, β ] s coputed by Eq. (. α = M{M { x ( β = M{Max { x Paths,,, k ca be coo ay dstace pots (.e., [α, β]. A pot, sayδ, ca be two or ore ebershp degrees, whch each ebershp degree belogs to a path. We wat to fd out whch ebershp degree s desrable to defe the dfferet pot (x... Idfferet pot (x A dfferet pot belogs to the optal set, x [ α, β ], whch s ot eber of ay shortest ad logest legth set. I other words, ths pot s eutral relato to the prevous ad ext pots. For stace, f we wat to dvde the dstace [ α, β ] to two equal parts of shortest ad logest legths, ths dstace wll be α β α β α β [ α,,(, β ]. Hece, pot does ot belog to oe of these two dstaces. If the dstaces α β α β are dvded to [ α, ],[, β ], due to pot α β belogg to two sub-dstaces, the ths aout s ot cosdered as the dfferet pot. Based o the above defto, ths pot s the ea of optal set, α x = β. ISSN: ISBN:

3 Proceedgs of the 0th WSEAS Iteratoal Coferece o FUZZY SYSTEMS.. Deterg the ebershp degree of set legths By usg Proposto, we descrbe the ethod to detere the ebershp degree of the shortest path set. Deftos: Cosder the fuzzy paths,,.,..., k where, = {( x, μ ( x x X ; =,,..., k We also cosder I as the IFS. x : Idfferet pot μ ( x : I Mebershp of deal set at pot. μ ( : Mebershp of path at pot. x μ ( x : Mebershp of the selected shortest path at pot. δ (x : Postve devato betwee ebershp of path fro the deal set at pot, where (x x. δ ( x : Negatve devato betwee ebershp of path fro the deal set at pot, where (x x. ε (x : Postve devato betwee ebershp of path fro the deal set at pot, where (x >x. ε (x : Negatve devato betwee ebershp of path fro the deal set at pot, where (x > x. δ ( x 0 ; =,., k ; =,,..., ; x x µ I% X = δ ( x < 0 = Max {,,..., k = Max {,,..., k µ (X - ( δ δ δ δ δ δ δ δ ε ( x 0 ; =,., k ; =,,..., ; x > x µ ( X - µ ( X = I% ε ( x < 0 ε = M { ε, ε,..., ε k ε = M { ε, ε,..., ε k Proposto : If x x ad μ μ ( x - μ ( x = δ δ. I% = ( x Max{ μ ( x, the Otherwse, x > x ad μ ( x M { μ (, = the μ ( x - μi ( x = ε ε. Proof: (by cotradcto x Assue that μ ( x I μ ( x = η η > δ δ The, we have μ ( x = Max { μ ( x, whch does ot support the assupto. The, the proposto s proved. Slarly, for x > x we have μ x ( μ The, we have μ I ( x = ζ ζ < ε ε ( x = M { μ ( x, whch does ot support the assupto. The, the proposto s proved.. The Proposed algorth Followg are the a steps our proposed algorth for a fuzzy sgle-crtero etwork. Step Detere the optal set (.e., ebers of the shortest path set Step Detere the dfferet pot, α x = β Step Detere the ebershp degree of the ebers of the shortest path set =,,..., If x x the μ ( x { ( x S {(, ( = Max μ = x μ x x x Else x > x the μ ( x { ( x S {( x, ( x x x = M μ = μ > Step fd the shortest path the fuzzy etwork, % = S % U S % Our proposed algorth ay fd a fuzzy shortest path that s ot cluded aog fuzzy paths a etwork. The, by usg the slarty degree relato, each par of paths legth s coputed ad the bggest slarty degree represets the fuzzy shortest path that s avalable etwork paths... The geeralzato of proposed algorth We also apply our proposed algorth for a fuzzy ultcrtera etwork. As etoed above, the fuzzy shortest. ISSN: ISBN:

4 Proceedgs of the 0th WSEAS Iteratoal Coferece o FUZZY SYSTEMS path s selected fro aog all paths a fuzzy etwork based o te crtero. Further, f ths etwork based o cost crtero, the fuzzy shortest path wll be ters of costs. The shortest path the frst etwork s ot ecessarly equal to the secod oe. The a purpose of the proposed algorth s to fd the shortest path a etwork cosstg of ult crtera, such as te, cost, rsk, ad dstace... Score of rak for etwork paths (S r The weght of every crtero (w s cosdered accordg to the udget of decso aker. We rak path k ters of te ad cost crtera. Table shows the related raks to the etwork paths based o te ad cost crtera, as gve Exaple. Also, Table coputes the score of ay rak, whch raks are frst versed ad the ultpled wth the axu rak (s r =/r {ax r, as show bellow. For stace, the scores of rak for path ters of cost ad te crtera o are ad.66, respectvely. Crtera /rak Table. Score of raks (S r Paths/ te crtero Score value (te = =. =.66 =. = Paths/ cost crter o Score value (cost = =... Crtero atrx (M The dagoal atrx related to each crtero cosders the ubers of etwork paths (M = [a kr ], k=r, whch eleets of the dagoal atrx for each crtero are as follows. Λ (M = dag [w s,w s,w s,,w s ] The dagoal eleets show weghted score for each crtero. For exaple, the su of atrx of te crtero (M t ad atrx of cost crtero (M C s as follows: If we assue w t = w c = 0. The M t (a M C (a = 00. =. =.66 =. = If path k crtero has rak r, we wll have M = w s r, cosderg that every crtero of the kr etwork has oe exclusve atrx. Followg s the atrx of crtero, where w s the weght of crtero ad s r s the score of rak r crtero. Table shows the etwork atrx that s equal to the su of crtera atrces of the sae etwork (.e., M Network = M, where s the uber of = crtera the etwork. M etwork Table. Network atrx = Coparso of etwork paths Network paths are copared by usg the etwork atrx. The ext step s to copute the rak of paths the etwork based o the su of crtera scores. To specfy the fal rak of paths, the score of each path should be coputed. For stace, the score of path k s equal to the su of the row arrays (.e., = related to path k. Thus, there s a order of paths prorty fro a path wth ore score to a path wth the lowest score. I other words, a path wth the hghest score s the shortest path the etwork. =., =, =. 0.8 =.08, =. 0.8 =., =.. =.7 the we have : > > > > We coclude that path s the shortest path, ad the sequece of paths are as follows: > > > > k a kr r=.. Ma steps of the proposed algorth Ths secto presets a steps of the geeralzed algorth that s proposed for the fuzzy shortest path a fuzzy ult-crtera etwork. Notatos ad steps used ths algorth are as follows: r ( k : rak of path k crtero, where =,,, ad k=,,, K ISSN: ISBN:

5 Proceedgs of the 0th WSEAS Iteratoal Coferece o FUZZY SYSTEMS w : weght of crtero s ( : score of rak r crtero for path k. r k k : score of path k Step 0 Select the shortest path the etwork for every crtero, accordg to Secto. Step Detere the score of all raks for paths (.e., s ( r K = {ax r (. r ( K K Step Assg the weght for each crtero (w the etwork based o the DM's prorty. Step Fd the shortest path the etwork for each crtero by the proposed algorth, ad detere the rak of each etwork path for each crtero r ( k by the use of the slarty degree. Step Copute the score of each etwork path (.e., k = wsr ( k. = Step Rak paths wth ore score showg the shortest path..6. Coplexty of the algorths I ths paper, we copare the results obtaed by our proposed algorth wth the fuzzy u ad ultple labelg algorths. Fro the pot of vew of coputatoal coplexty theory, we dscuss the fuzzy shortest path proble. Due to the NP-hardess of the gve proble, the te coplexty of our proposed algorth steps s calculated as follows. I step,we have the axu k coparsos for each of α, β, where k s the uber of etwork paths. The, step we sort the uber of optal set aog the uber of etwork paths the axu coparsos, where s the axu uber of ebers of the fuzzy shortest path. I step, to detere the ebershp degree of the obtaed set fro step, we have the axu k coparsos. Cosequetly, t has a coplexty of O(k O( O(k =O(k. Whereas, the te coplexty of the fuzzy u algorth s O(. The te coplexty of the ultple labelg algorth s O( V ax l, where s the uber of etwork odes, V ax s the axu uber of addresses each ode,ad l s the axu of the uber of ebers of each address each ode. Coparso of these three algorths ters of the te coplexty shows that the te coplexty of the proposed algorth s ore sutable rather tha two algorths.. Coputatoal results I ths secto, we preset a exaple for solvg the fuzzy shortest path sgle ad ult-crtera etworks. The coputatoal results obtaed by ths algorth are copared wth two kow algorths take fro the lterature. Fally, the statstcal aalyss s dscussed to show the effcecy of our proposed algorth... Exaple Fg. depcts a classc etwork, aely Network, whose arc legths are fuzzy. A B E C F 6 I H Fg. Network Accordg to Fg., followg are the legth of arcs o Network A = {,, B = {,,,, C{,,, K = {, D = {,,, E = {,, F = {,,, = {,, G = {,,, H = {,, I = {,, J = {,, The legth of each arc shows the task durato o Network. Ths etwork has fve paths whose each legth s gve as follows: : ACGK, : ACGJ, : ADH, : BEH, : BFI. p S( A, B = = p μ ( x μ ( x A A B = = {,,,,,, = {,,,,,,,, = {,,,,,, = {,,,,,,, = {,,,,,,,, B μ ( x μ ( x G 7 8 (6 J K 9 ISSN: ISBN:

6 Proceedgs of the 0th WSEAS Iteratoal Coferece o FUZZY SYSTEMS By the use of the proposed algorth, the fuzzy shortest path of ths etwork s foud as follows: Step α = M{8,8,7,6, = the β = M{, 6,,, = Optal set = [, ] Step x = = 9 Step x= < 9 μ( x = Max{0. = 0. x= 6 < 9 μ( x = Max{0.,0. = 0. x= 7 < 9 μ( x = Max{0.,0.,0. = 0. x= 8 < 9 μ( x = Max{0.,0.,0.6,0.,0.7 = 0.7 x= 9 μ( x = Max{0.,0.,0.7,0.7,0.7 = S % = {,,,, x6= 0 > 9 μ( x6 = M{0.,0.,0.6,0.7,0.6 = 0. x7= > 9 μ( x7 = M{0.6,0.7,0.,0.6,0.6 = 0. x8= > 9 μ( x8 = M{0.,0.6,0.,0.6,0.6 = 0. x9= > 9 μ( x9 = M{0.,0.,0.,0.,0. = S % = {,,, 0 Step % = {,,,,,,,, Karokaplds & Papps [] suggested equato (6 for the slarty degrees. S(, = 0.969, S(, = 0.879, S(, = 0.887, S(, = 0.6, S(, = 0.69 As a result, we have: S(, > S(, > S(, > S(, > S(,. So, the shortest path ad other paths are gve order, as follows: BFI < ADH < BEH < ACGK < ACGJ By usg the proposed algorth, we ca specfy the shortest fuzzy path as ult-crtera the etwork. If we suppose crtera rakg accordg to table the, the shortest etwork path for te, cost ad qualty rsk of actvty wll be as follows. Step Detere the score of raks for paths. w = = w = = w = =. w = = Step Assue the followg weghts of crtera arbtrarly. W t =0., W c =0., W r =0. Step Rak paths respect to three crtera, as show Table. Table. Paths-crtera atrx Crtera Paths Te rak W t = 0. Step Copute the score of paths. = ( = =.7 = ( = =.6 = ( = =.698 = ( = =. Step Sort o-decreasg paths. > > > < < < We ca coclude the order of paths as follows: CGI < ADH < BFI < BEH Thus, path s (.e., CGI the shortest path etwork wth three crtera... Statstcal aalyss Cost rak W c = 0. Rak of qualty rsk of actvtes W r = 0. We radoly geerate etworks, whose fuzzy arc legths are dscrete ad rado. The, we copare the slarty degrees betwee our proposed algorth ad the labelg algorth agast the fuzzy u ad the labelg algorths. Table llustrates the coputatoal results, whch the statstcal hypotheses are as follows: H 0 : μ - μ = 0 H : μ - μ > 0 Where, paraeters μ ad μ are the ea slarty degrees for Colus ad, respectvely. It s clear that f μ or μ equal s oe, the both algorths gve Colu or are copletely the sae. The ISSN: ISBN:

7 Proceedgs of the 0th WSEAS Iteratoal Coferece o FUZZY SYSTEMS coputatoal results of the hypothess tests are obtaed bellow, whch the sgfcat level of the test s % Dfferece = μ (c - μ (c Estated for dfferece: % upper boud for dfferece: T-Test of dfferece = 0 (vs <: T value = 0., P Value = 0.9, DF = 6. Table. Slarty degree betwee our proposed algorthad the labelg algorth μ (c Slarty degree betwee the labelg ad the u fuzzy algorths μ (c We coclude that H 0 s accepted ad two eas of μ ad μ are ot dfferet sgfcatly. Nevertheless, estato reported by the MINITAB software package for the dfferece s ad ths shows that μ s ore tha μ. Thus, related to our proposed algorth s close to the labelg algorth agast the fuzzy u algorth.. Cocluso I ths paper, we proposed a ovel, effcet algorth for the fuzzy shortest path proble the sgle ad ult-crtera etwork wth dscrete fuzzy arcs. Our proposed algorth s based o our defto of the deal fuzzy set (IFS order to heretly select shorter ad loger legths wth the axu ad u ebershp, respectvely. Further, we coputed the slarty degrees betwee our proposed algorth ad the labelg algorth, ad betwee the fuzzy u algorth ad the labelg algorth order to show the effcecy of the proposed algorth ters of the te coplexty. The related statstcal aalyss llustrates the fuzzy shortest path obtaed by our proposed algorth respect to the labelg algorth s closer tha the fuzzy u algorth. We geeralzed our proposed algorth to fd the fuzzy shortest path the ultcrtera etwork. The a advatage of ths algorth s to cosder the fuzzy ult-crtera etwork by rakg paths for each crtero ad sug the weght of raks order to fd the fuzzy shortest path. []Chuag, T. N., & Kug, J.Y. (00. The fuzzy shortest path legth ad the correspodg shortest path a etwork. Coputers & Operatos Research,, []Chuag, T. N., & Kug, J.Y. (006. A ew algorth for the dscrete fuzzy shortest path proble a etwork. Appled Matheatcs &Coputato, 7, []Dubos, D., & Prade, H. (980. Theory ad applcatos, Acadec Press, New York. []Furukawa, N. (99. A paraetrc total order o fuzzy ubers ad a fuzzy shortest rout proble. Optzato, 0, [6]Kle, C. M. (99. Fuzzy shortest paths. Fuzzy Sets ad Systes, 9, 7-. [7]Kug, J. Y., & Chuag, T. N. (00. The shortest path proble wth dscrete fuzzy arc legths. Coputers & Matheatcs wth Applcatos, 9, 6-70 [8]Moaze, S. (006. Fuzzy shortest path proble wth fte fuzzy quattes. Appled Matheatcs ad Coputato, 8, [9]Okada, S. & Ge, M. (99. Order relato betwee tervals ad ts applcato to shortest path proble. Coputers & Idustral Egeerg, (-,7-0. [0]Okada, S. & Ge, M. (99. Fuzzy shortest path proble. Coputers & Idustral Egeerg, 7(-, []Okada, S.,& Soper, T. (000. A shortest path proble o a etwork wth fuzzy arc legth. Fuzzy Set ad Systes, 09, 9-0. []Okada, S. (00. Fuzzy shortest path probles corporatg teractvty aog paths, Fuzzy Sets ad Systes,, -7. []Papps, C. P., & Karacaplds, N. I. (99. Slarty easure betwee fuzzy sets ad betwee eleets. Fuzzy Set ad Systes, 6, 7-7. Refereces []Chaas, S., Delgado, M., Verdegay, J.., & Vla, M. A. (99. Fuzzy optal flow o precse structures. Europea Joural of Operato Research, 8, ISSN: ISBN: