Development of New Fuzzy Logic-based Ant Colony Optimization Algorithm for Combinatorial Problems

Transcription

1 roceengs of the 4 th Internatona Me East ower Systems Conference MECON, Caro Unversty, Egypt, December 9-,, aper ID 94 Deveopment of New Fuzzy Logc-base Ant Coony Optmzaton Agorthm for Combnatora robems Ahme Rabe Gn Gn Ahme M A M Kame Hassen Taher Dorrah Automatc Contro an System Engneerng Group Automatc Contro an System Engneerng Group Automatc Contro an System Engneerng Group Dept of Eectrc ower an Machnes Engneerng Dept of Eectrc ower an Machnes Engneerng Dept of Eectrc ower an Machnes Engneerng Facuty of Engneerng, Caro Facuty of Engneerng, Caro Facuty of Engneerng, Caro Unversty, Gza, Egypt Unversty, Gza, Egypt Unversty, Gza, Egypt ahmerabegn@yahoocom amaame@hotmacom orrahht@aocom Abstract - Ths paper s recte towars eveopng a new fuzzy-ogc base ACO agorthm The propose agorthm taes nto conseraton the uncertantes that can be foun n both the heurstc an the pheromone factors Ths s acheve by representng the parameters of the probem an the pheromone tras as a par of vaue an fuzzy eve The fuzzy eve s consere as an ncaton of the uncertanty n the corresponng parameter Durng the souton teratons the cacuatons are performe tang nto conseraton the fuzzy eves of the nvove parameters Hence, both pheromone upates an heurstc nformaton are cacuate wth ther corresponng fuzzy eves Consequenty, the probabtes of choosng the possbe oncomng step are aso cacuate wth ther corresponng fuzzy eves A stochastc-base technque s propose to enabe the artfca ant to choose the best oncomng step base on the vaues of the probabtes an ther corresponng fuzzy eves The propose Fuzzy Ant Coony Optmzaton gves the optma souton n a form of an optma vaue an ts corresponng fuzzy eve The fuzzy eve of the souton s nterprete as the uncertanty n the vaue of the optma resut ue to the uncertantes of the pheromone tras an the probem parameters The propose agorthm s teste usng the benchmar Quaratc Assgnment robem QA an Traveng Saesman robem TS The resuts ncate that the eveope has better vaues wth mprove performance Inex Terms Ant Coony Agorthm, Quaratc Assgnment robem, Traveng Saesman robem, Combnatora robems, Fuzzy Systems, Fuzzy-base Logc Agebra I INTRODUCTION Combnatora optmzaton probems nvove fnng vaues for screte varabes such that the optma souton wth respect to a gven obectve functon s foun Many optmzaton probems of practca an theoretca mportance are of combnatora nature Exampes are the shortest-path probems, as we as many other mportant rea-wor probems e fnng mnmum cost pan to ever goos to customers, an optma assgnment of empoyees to tass to be performe, a best routng scheme for ata pacet n the Internet, an optma sequence of obs whch are to be processe n a proucton ne, an aocaton of fght crews to arpanes, an many more Combnatora probems are ntrgung because they are often easy to state but very ffcut to sove Some of the best nown an wey appe metaheurstcs are smuate anneang, tabu search, evoutonary computaton, etc have been eveope to sove these probems One of the mportant casses of combnatora optmzaton probems s the quaratc assgnment probem QA In ths probem factes are to be assgne to ocatons to optmze certan cost functon Some fes of appcatons of ths probem are pannng, anayzng, sovng networs an contro of the reazaton of compex proects Severa methos such as partce swarm [], genetc agorthm [], an tabu search [3] have been eveope to sove these probems Another cass of the combnatora optmzaton probems s the cass of the traveng saesman probems TS In these probems, a saes man s requre to mae a compete tour through certan number of ctes such that the tour acheves certan obectve functon Severa methos such as genetc agorthm [4] an ant coony optmzaton [5] have been eveope to sove these probems Combnatora optmzaton probems can be moee by ether etermnstc or probabstc stochastc representatons Conventona methos are base on exhaustve search, that s, the enumeraton of a possbe soutons an the choce of the best one Unfortunatey, n most cases, such a natve approach becomes rapy nfeasbe because the number of possbe soutons grows exponentay wth the probem The wor of metaheurstcs s rch an mutfacete Severa characterstcs mae ACO a unque approach: t s a constructve, popuaton-base metaheurstc whch expots an nrect form of memory of prevous performance Ths combnaton of characterstcs s not foun n any of other metaheurstcs Consequenty, ACO technques are one of the recent an most successfu heurstc methos [6, 7, 8] Unfortunatey, these agorthms have been eveope for etermnstc representaton of the combnatora probems However, the practca systems, a nee for eveopng effcent ACO technque that s abe to represent a the uncertantes of the parameters of the system n a way sutabe for performng cacuatons an gvng optma souton n reasonabe tme arses [9,, ] Ths paper ntrouces the eveopment of such technque 83

2 II ANT COLONY OTIMIZATION ALGORITHM ACO s an evoutonary metaheurstc agorthm base on a graph representaton that has been appe successfuy to sove varous har combnatora optmzaton probems The man ea of ACO s to moe the probem as the search for a mnmum cost path n a graph Artfca ants wa through ths graph, oong for goo paths Each ant has a rather smpe behavor so that t w typcay ony fn rather poorquaty paths on ts own Better paths are foun as the emergent resut of the goba cooperaton among ants n the coony The behavor of artfca ants s nspre from rea ants They ay pheromone tras on the graph eges an choose ther path wth respect to probabtes that epen on pheromone tras an these pheromone tras progressvey ecrease by evaporaton Ants prefer to move to noes, whch are connecte by short eges wth a hgh amount of pheromone In aton, artfca ants have some extra features that o not fn ther counterpart n rea ants In partcuar, they ve n a screte wor an ther moves consst of transtons from noes to noes as shown n fg Aso, they are usuay assocate wth ata structures that contan the memory of ther prevous acton In most cases, the amount of pheromone eposte s usuay a functon of the quaty of the path Fnay, the probabty for an artfca ant to choose an ege often epens not ony on pheromone, but aso on some probem-specfc oca heurstcs Fg robabty of rea ant to choose the path At each generaton, each ant generates a compete tour by choosng the noes accorng to a probabstc state transton rue Every ant seects the noes n the orer n whch they w appear n the permutaton For the seecton of a noe, an ant uses a heurstc factor as we as a pheromone factor The heurstc factor, enote byη, an the pheromone factor, enote byτ, are ncators of how goo t seems to have noe at noe of the permutaton The heurstc vaue s generate by some probem epenent heurstcs whereas the pheromone factor stems from former ants that have foun goo soutons η s the vsbty functon whch s usuay seecte as the nverse of the weght for to, Lower ns are mae more esrabe, represents the weght between ns, then ant prefers seecton of the shortest ns e: η = / To stuy the whoe space of soutons, the pheromones shou evaporate If the coeffcent of evaporaton s enote by ρ [,], the upate rue for the pheromones taes the form Δτ t = Δτ Q / L,, + t The pheromone vaue on eges s taen to be equa to a sma postve numberτ When the tour s compete, the K th ant ays own on the ege, the pheromone vaue Q / L, f, T Δτ, = 3, f, T where L t s the souton of the ant m at teraton t ftness vaue of souton, T t s the tour chosen by the ant an Q > s an austabe parameter If the amount of pheromone eposte s nversey proportona to the quaty of the souton, then the arger L t that s, the worse the constructe souton, the smaer / L t, hence the ess the amount of pheromone eposte on the n Thus, a ong path causes a the ns of that path to become ess esrabe as a component of the fna souton Ths s the case for any quaty measure that nees to be mnmze To stuy the whoe space of soutons, the pheromones shou evaporate If the coeffcent of evaporaton s enote by ρ [,], consequenty the crsp vaue for the pheromone upate s τ t + = ρ τ + Δτ, 4 The constant ρ specfes the rate at whch pheromones evaporate, causng ants to forget prevous ecsons The tota number of ants n the coony remans constant The operatona mechansm of basc ant coony agorthm s base on the combnaton of postve feebac prncpe an a certan heurstc search technque It can be brought to ght from the transton probabty formua, formuate as foows: = [ τ ] [ η ] / [ τ ] [ η ] 5 J where α an β are austabe parameters escrbng the weghts of the pheromone tra an vsbty when choosng the route To choose these parameters, the foowng observatons can be obtane Hgh vaues of α an ow vaues of β w ea to ba soutons ue to premature convergence because the goba ntegence s over emphasze an the 83

3 oca heurstc s severey scounte The ant behavours are hghy affecte by pheromone experences an reach convergent behavor qucy after ony a few teratons Low vaues of α an hgh vaues of β can obtan above-average quaty soutons an the resuts wth fferent runs are more consstent Among these settngs, the expermenta resuts wth α =, β = 5 or α =, β = 5 has the best performance 3 When ether α = or β =, the performance s sgnfcanty eterorate As for the case of α =, no pheromone nformaton s use, e prevous search experence s negecte The search then egraes to a stochastc greey search an ACO s reuce to a greey heurstc agorthm whch consers ony the vaue ofη The ants ust gnore +, the goba ntegence represente as pheromone tras an o not now about the quaty of the prevousy constructe soutons, so there s no communcaton happenng between the ants As for the case of β =, ACO becomes an expotatonprone search metho whch ntensfes the search wthn a sma neghbourhoo of the best souton observe so far an has a very ow probabty of exporng new regons of the souton space an the attractveness of moves s negecte The heurstc nformaton as an expct bas towars the most attractve soutons, an s therefore a probemepenent functon III DEFINITION OF ROOSED FUZZY LOGIC ARITHMETIC RERESENTATION As ths new approach s base on assgnng a certan fuzzy eve to each parameter an coeffcent, the stuy w present frst the efnton of the fuzzy ogc arthmetc representaton Then, the agebra of these fuzzy representatons s gven for the scaar forms as represente by Gabr an Dorrah [] Let s a genera scaar parameter comprsng two man components; as foows: = + 6 f where s the etermnstc equvaence an f s the fuzzy equvaence representng a sma uncertanty or vaue toerance n the parameter = 7 +, = f / The propose fuzzy ogc arthmetc representaton s expresse by repacng each parameter wth a par of parentheses, the frst s the actua vaue an the secon s corresponng fuzzy eve, that s Vaue, Fuzzy Leve Ths s smar to vector representaton of parameters, where vectors are not ae recty A summary of the man fuzzyogc base agebrac operatons are presente n Tabe I TABLE I SUMMARY OF MOST COMMON FUZZY LOGIC ARITHMETIC ALGEBRA Symboc Name of Basc Operaton Representat on of Resutng Vaues an Fuzzy Leves from Orgna Operaton Operaton Mutpcaton Y Z Y Z = Y Z, + + Dvson Aton an Subtracton Other oynomas Y/Z -Y +Z Y Y / Z = Y / Z, + Y Z W = Y + Z, an = Y + Z W m n / n m Y o, y Z / Y + Z = an y = n / m x IV THE DEVELOED FUZZY LOGIC-BASED ACO ALGORITHM Conventona ant coony agorthm uses the parameters of the probem, the heurstc nformaton an the pheromone tras to sove the combnatora optmzaton probems In most of the practca probems, the parameters cannot be consere as crsp because they usuay have fferent types of uncertantes In ths paper, a new fuzzy ogc-base ant coony optmzaton s presente to sove the combnatora probems wth uncertantes to ts parameters These uncertantes shou affect both the heurstc nformaton an pheromone upate factors of the agorthms The uncertantes of the heurstc nformaton an pheromone upate factors are represente as fuzzy eves Ths means that Heurstc nformaton w be eveope as foows The heurstc nformaton s η = /, Appyng the rues of fuzzy ogc arthmetc representaton, then η, η = /, 3 where η s the crsp vaue of heurstc nformaton, s η the fuzzy eve for the vaue of heurstc nformaton represents the weght between two noes an represents the fuzzy eve for the weght between two noes When the tour s compete, the Kth ant ays own on the ege, the pheromone vaue Q / L, L, f, T t Δτ,, Δ = 4 τ, t, f, T where Δτ, s the crsp vaue of quantty of pheromone eposte by an ant at teraton t, s the fuzzy eve Δτ, t for the quantty of pheromone of the ant at teraton t, s the crsp vaue for functon souton eposte by L t Y Z 833

4 an ant at teraton t, an s the fuzzy eve for the L t functon souton of the ant at teraton t Thus, the crsp vaue of quantty of pheromone on the path s Δτ t = Δτ Q / L 5,, + t Then, the fuzzy eve for the quantty of pheromone on the path w be = Δτ, Δ Δτ, t τ, t Δτ, + Q / L Q L t / L / Consequenty, the crsp vaue for the pheromone upate s τ, t 6 t + = ρ τ + Δτ 7 At the eary stage of the optmzaton process, the pheromone vaue on eges s taen to be equa to a sma postve numberτ The fuzzy eve for the pheromone upate equaton s τ, / t + = ρ τ τ t + Δτ Δτ, t 8 ρ τ + Δτ, In the conventona ACO agorthm, each ant uses the pheromone tra to cacuate the probabty of transferrng from ts current poston to each of ts neghbor noes In the propose technque, each probabty has ts own fuzzy eve Now, the ant has to mae a ecson usng fuzzy probabtes Fuzzy probabtes mean that when the ant cacuates the probabtes of choosng noe x n as epcte n Fg, ths vaue has a corresponng fuzzy eve accorng to Consequenty, the crsp vaue for probabty equaton s = [ ] [ η ] / [ τ ] [ η ] J τ 9 Then the fuzzy eve for the probabty equaton = [ α / J τ [ τ ] + [ β η ] ] [ η ] J [ α τ ] + [ β η ][ τ p t ] [ η ] As every eement treate as a crsp vaue an ts corresponng fuzzy eve, thus the ant can choose the next noe accorng to t = +, t Ths means that the probabty has ts fuzzy eve of x as epcte n Fg The ecson mang agorthm x uner these fuzzy contons s base on assumng that each of the cacuate probabtes has a probabty ensty functon DF of certan type sprea between an + Then, the ecson of the ant w be base on the average of the DF The DF may be Gaussan, Beta, near, nonnear, etc However, n ths approach, the near probabty ensty functon can be consere Ant Fg The probabty between the noe an the other noes The probabty ensty functon of system fuzzness s assume to be trapezoa as epcte n Fg 3 It s efne n the nterva, ] The reatonshp between [ h, h,, an can be gven by h + h =, Ths s a rect resut as the area uner the DF shou equa to, h Fg 3 The probabty ensty functon of system fuzzness s trapezoa The equaton of the ncne ne s gven by h h y = x + h 3 Usng the above equaton, the average vaue can be cacuate as foows Average vaue= h y h h = 6, x x h, x n x n h, x x x + h xx, x < 4 + h 5 6 The speca cases of are gven beow:- For h = h The DF of the system fuzzness s unform Then, the average vaue s gven by + λ, where λ =/ x + h x x x n, h h x 834

5 For h =, an from 7 h = / The DF of the system fuzzness s trange wth negatve sope an consequenty the probabty between two noes that s use n the ecson mang = + λ, where λ =/3 3 For h =, an from 7 h = / The DF of the system fuzzness s trange wth postve sope an consequenty the probabty between two noes that s use n the ecson mang = + λ, where λ =/3 The above scusson reveas that the probabty between two noes s gven by + λ, where λ [ / 3, / 3] From the prevous arguments, t s to be note that when the probabty ensty functon s assume to be near wth snge an/or oube rues, the average vaue that can be use by the ant to etermne the next noe has the range one thr to two thrs of the fuzzy eve Thus, the genera form of the probabtes that s appe n the eveope agorthm s as foows = + λ 6 t where λ [ / 3, / 3] V THE ALICATION OF THE DEVELOED FUZZY BASED ANT COLONY OTIMIZATION TO TRAVELLING SALESMAN ROBLEM The TS can be state as the probem of fnng a mnma tme requre to go through a tour whch nvoves traffcs, accentsetc wth constrants that vsts at each town must be once Ths probem wth uncertantes to ts parameters can be sove by appyng the eveope agorthm The frst step s to prepare the nput ata The tme taen to trave from town to town s be gven by Each tme s then assgne a fuzzy eve Ths fuzzy eve epens on the nature of the traffcs foun on the path from town to town as we as other aspects such as the statstcs of the accents on the path The secon step s to cacuate the heurstc nformaton The resut, of course, w have a crsp vaue an a corresponng fuzzy eve The thr step s to etermne the nta vaues of the pheromone tra at each branch These vaues w be n pars Each par has a crsp vaue an a corresponng fuzzy eve Durng the constructon of the souton, the ecson of an ant to trave from ts current poston to ts next poston epens on the propose technque That s to say, the probabtes of gong from the current poston to ts neghbours are cacuate wth ther fuzzy eves Then, the averages are cacuate usng 6 wth the gven vaue of λ The next poston of the ant s the poston that has maxmum average At the en of teraton, the pheromone tras an ther corresponng fuzzy eves are upate Ths s one n two steps The frst s the pheromone eposte as gven n 5 an ts corresponng fuzzy eve accorng to 6 The secon s the pheromone evaporaton accorng to 7 an ts corresponng fuzzy eve accorng to 8 The teratons contnue unt stoppng crteron s acheve At the en of the teraton, the tour tme of each ant an ts corresponng fuzzy eve are cacuate Then, the mnmum tour tme s the best of ths teraton The optma tour tme s the mnmum of the best tour tme of the current teraton an the mnmum tour tme of a teratons The path whch gves the tour tme s gven n 7 an ts corresponng fuzzy eve s gven n 8 n The tota tme L = π π π n π = The fuzzy eve for the tota tme = t n = n = π π + π π + π n π π n π π π π n π L / 8 where s the crsp vaue for the tme between town an, π gves the tme of town n the current souton π S n an S s the canate souton The term escrbes π π + the tme contrbutons of smutaneousy choosng the path between town an an represents the fuzzy eve π π + of the choosng the path between town an The parameters of the agorthm are chosen to be α =, β =5, m=3, ρ = an Q= In aton, the parameter λ shou scusse n the prevous secton s change from to The Over3 benchmar TS whch has 3 ctes an escrbe n [] s use The best tme an the average tme when λ =, at each teraton, are shown n Fg 4 From the Fgure, the tme of the shortest tour s 4359 mn an ts corresponng fuzzy eve equas to 38 It s notce that the corresponng fuzzy eve epen on the cacuatons Fg 4 Best an average tme at each teraton, where λ = It s note that, n the prevous resuts, λ s taen to be zero Ths means that the choce of the ant epens ony on the cacuate probabty Hence, n orer to compete the resuts, 835

6 λ shou tae vaues other than zero an the parameters of the agorthm are chosen to be α =, β =5, m=3, ρ = an Q= as mentone n case λ = That s to say, the performance of the propose technque shou be teste when the choce of the ant epens on the cacuate probabty an ts corresponng fuzzy eve As scusse n the above secton, f the probabty ensty functon s tranguar wth negatve sope, the vaue of the parameter λ w be /3 When ths vaue of λ s taen nto conseraton, the propose agorthm gves the resuts shown n Fg 5 As epcte n the Fgure, the souton s mprove to 45 wth corresponng fuzzy eve equas to 6797 The Fgure shows that the response of the souton aso s mprove That s to say, the optma souton s reache n a reatvey sma number of teratons Aso, the average tme of the tour at each teraton has ess fuctuatons than the prevous case whch s shown n Fg 3 If the probabty ensty functon s trange wth postve sope, the vaue of λ s /3 When ths vaue of λ s taen nto conseraton, the propose agorthm gves the resuts shown n fg 7 As epcte n the fgure, the souton s mprove to wth corresponng fuzzy eve equas to 6 Aso, the response of the souton s mprove Aso, the fuctuatons of the average tme of the tour at each teraton are ecrease Fg 7 Best an average tme at each teraton where λ = 667 Tabe II an Fg 8 summarze the effect of the vaue of λ on the souton of TS wth uncertanty to ts parameters TABLE II EFFECT OF ON THE SOLUTION OF TS WITH UNCERTAINTIES TO ITS ARAMETERS Fg 5 Best an average tme at each teraton, where λ = 333 If the probabty ensty functon s unform, the vaue of λ s / When ths vaue of λ s taen nto conseraton, the propose agorthm gves the resuts shown n Fg 6 As epcte n the Fgure, the souton s mprove to 4988 wth corresponng fuzzy eve equas to 55 Aso, the response of the souton s mprove Aso, the fuctuatons of the average tme of the tour at each teraton are ecrease λ Vaue Mn Fuzzy Leve ACO λ = λ =33 λ =5 λ =67 λ =8 λ = Fg 8 Effect of on the souton of tsp wth uncertantes to ts parameters Fg 6 Best an average tme at each teraton where λ = 5 836

7 It s to be note that the above resuts are better than those n [3], where genetc agorthms were appe to sove the Over3 probem; they cou fn a tour of ength mn The same resut was often obtane by ant-cyce [4], whch aso foun a tour of ength 4374 mn VI THE ALICATION OF THE DEVELOED FUZZY LOGIC- BASED ANT COLONY OTIMIZATION TO QUADRATIC ASSIGNMENT ROBLEM To appy the metaheurstc to assgnment probems, a frst step s to map the probem on a constructon graph G C = C, L, where C s the set of components usuay the components conssts of a ocatons an a the factes an L s the set of connectons that fuy connects the graph Transtons are from factes to ocatons an vce versa Typcay, an ant frst chooses facty, then a ocaton to whch to assgn the facty, then another facty, an so forth, unt a factes have been assgne Factes an ocatons are chosen from the feasbe neghbouhoo, that s, from factes ocatons not sgne yet These constrants can be easy enforce n the ants wa by bung ony coupng between st unsgne factes an ocatons The functon can be escrbe n 9 an ts corresponng fuzzy eve s gven n 3 f n n π b aπ = = = 9 n n n n f π = b aπ b a b + a / π π = = = = 3 where an, b s the crsp vaue for the fow between factes a s the crsp vaue for the tme between ocatons an, π gves the ocaton of facty n the current souton π S n an S s the canate souton The term b a π π escrbes the cost contrbutons of smutaneousy assgnng facty to ocaton π an facty to ocaton π an b + represents the fuzzy eve of aπ the cost contrbutons of smutaneousy assgnng facty to ocaton π an facty to ocaton π Ths probem wth uncertantes to ts parameters can be sove by appyng the eveope agorthm The frst step s to prepare the nput ata The nput parameters are the tme between ocatons, the fow between factes an the parameters of Ant Coony Technque Each tme an each fow s then assgne a fuzzy eve The secon step s to cacuate the heurstc nformaton The resut, of course, w have a crsp vaue an a corresponng fuzzy eve Cacuatng ths heurstc nformaton on the potenta gooness of an assgnment s as foows Two vectors an f are cacuate n whch the th components represent respectvey the sum of stances from ocaton to a other ocatons, an the sum of the fows from facty to a other factes For exampe, the ower, the tme potenta of ocaton, the more centra the ocaton, the hgher f, the fow potenta of facty, the more mportant s the facty Next a coupng T matrx E = f s cacuate, whose eements are e = f Then, the heurstc esrabty of assgnng facty to s gven by η = / e The motvaton for usng ths type of heurstc nformaton s that, ntutvey, goo soutons w pace factes wth hgh fow potenta on ocatons wth ow tme potenta n = D = = n 3 The corresponng fuzzy eve for the sum of tmes from ocaton to a other n n = D D D = n / 3 = = The corresponng fuzzy eve for the eements of coupng matrx s = + 33 e f The heurstc nformaton for the coupng matrx s η = / e, e 34 In ths case, a rea assgnment probem wth arge sze s taen nto conseraton The probem s the optmum aocaton of servces n the offces of a mutnatona company ocate n Man, Itay, as escrbe n [4]The sze of the probem s 33 The probem s then sove usng fferent vaues of λ The resuts are shown n Tabe III an are rawn n Fg 9 λ Vauemansecon per wee TABLE III EFFECT OF ON THE SOLUTION OF QA WITH UNCERTAINTIES TO ITS ARAMETERS ACO λ = λ =3 3 λ = 5 λ =6 7 λ =8 λ = Fuzzy Leve The above resuts are better than those foun n [4], where ACO were appe to sove ths probem; the vaue of the obectve functon s man-secons per wee The probem s then sove usng fferent vaues of λ The resuts are shown n Tabe III It s note that the propose agorthm gves better vaues for TS an QA, than those gven by the conventona ACO Furthermore, the range of the 837

8 parameter λ that gves better vaues s / 3 λ / 3 Beyon ths range, the souton eterorates Fg 9 Effect of on the souton of QA wth uncertantes to ts parameters Tang λ greater that /3 means that the assume probabty ensty functon covers a range from to greater than +, but accorng to the efnton of the fuzzy eve, a probabty of vaue an a fuzzy eve of cannot tae vaues beyon + Therefore, any probabty ensty functon that covers range from to greater than + s not reasonabe VIII CONCLUSIONS In ths paper, a new technque has been eveope The man avantage of the propose technque s ts abty to represent the uncertantes of the parameters of both the optmzaton probem an the metaheurstc agorthm n a fuzzy ogc-base form Consequenty, the propose has the abty to gve the optma souton n a form of an optma vaue an ts corresponng fuzzy eve The fuzzy eve of the souton s shown to be nterprete as the uncertanty n the vaue of the optma souton The propose technque has been teste usng two casses of combnatora probems The resuts have been compare to other technques foun n the terature Ths comparson ncates that the eveope gves better optma vaues Furthermore, the propose technque acheves the optma souton n number of tras ess than those requre by the conventona technques Ths means that the propose technque has mprove the quaty of the souton an ecrease ts tme It s seen that the new concept has an unmte scope of generazatons an extensons to many casses of probems an systems n varous scpnes A bref st of the areas of further research s presente as foows Stuyng of usng nonnear probabty ensty functons on the performance of ACO technque Appyng the eveope technque to other ACO agorthm such Etst ant system, Ran base ant system an MA-MIN ant system Appyng the new fuzzy ogc base representaton to other combnatora probems v Appyng the new fuzzy ogc base representaton to other metaheurstc optmzaton technques as partce swarm, tabu search etc REFERENCES [] H Lu, A Abraham, an J Zhang, A artce Swarm Approach to Quaratc Assgnment robems, Soft Computng n Inustra Appcatons, ASC 39, pp 3, 7 [] Long Choong Yeun, Wan Rosmanra Isma an Moura Zrour, Appcaton of Genetc Agorthm n a Speca Quaratc Assgnment robem, receng of the n IMT-GT Regona Conference on Mathematcs, Statstcs an Appcatons Unverstes Sans Maaysa, enang, pp3-5, June 6 [3] Tabtha James, César Rego, an Fre Gover, Mutstart Tabu Search an Dversfcaton Strateges for the Quaratc Assgnment robem, IEEE Transactons On Systems, Man, An Cybernetcs art A: Systems An Humans, Vo 39, No 3, May 9 [4] M Dorgo, V Manezzo, an A Coorn, "Ant system: optmzaton by a coony of cooperatng agents," IEEE Transacton on Systems, Man, an Cybernetcs, art B, vo 6, no, pp 9-4, 996 [5] M Dorgo, an L M Gambarea, "Ant Coones for the Traveng Saesman robem," Bosystems, vo 43, no, pp 73-8, Juy997 [6] M H Afshar, artay Constrane Ant Coony Optmzaton Agorthm for the Souton of Constrane Optmzaton robems: Appcaton to Storm Water Networ Desgn, Avances n Water Resources, vo 3, no 4, pp , Apr7 [7] M Dorgo, G D Caro, an L M Gambarea, Ant Agorthms for Dscrete Optmzaton," Artfca Lfe, vo 5, no, pp 37-7, 999 [8] Dorgo Marco an Stutze Thomas, Ant Coony Optmzaton Brafor Boo, 4 [9] Kahraman, Cengz an Toga, Ethem, Data Deveopment Anayss Usng Fuzzy Concept, IEEE aper No 95-63/98, pp , 998 [] Kanasamy, W B Vasantha, Smaranache, Forentn, an Ianthenra K,Eementary Fuzzy Matrx Theory an Fuzzy Moes for Soca Scentsts Automaton, Los Angees, USA, 7 [] Waaa Ibrahm Gabr an Hassen Taher Dorrah, New Fuzzy Logcbase Arthmetc an Vsua Representatons for Systems Moeng an Optmzaton IEEE Internatona Conference on Robotcs an Bommetcs, Bango, Thaan, aper No 8, December 4-7, 8 [] TSLIB: A Traveng Saesman robem Lbrary heeberge/groups/comopt/ software/tslib95/ [3] D Whtey, T Starweather, D Fuquay, "Scheung robems an Traveng Saesman: the Genetc Ege Recombnaton Operator," roceengs of the Thr Internatona Conference on Genetc Agorthms, Morgan Kaufmann, 989 [4] Vttoro Manezzo an Aberto Coorn, The Ant System Appe to the Quaratc Assgnment robem, IEEE Transactons on Knowege an Data Engneerng, Vo, No 5, September/October