Optimization of JK Flip Flop Layout with Minimal Average Power of Consumption based on ACOR, Fuzzy-ACOR, GA, and Fuzzy-GA

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1 Journa of mathematcs and computer Scence 4 (05) - 5 Optmzaton of JK Fp Fop Layout wth Mnma Average Power of Consumpton based on ACOR, Fuzzy-ACOR, GA, and Fuzzy-GA Farshd Kevanan *,, A Yekta *,, Nasser Mehrshad *,3 * Department of Eectrca and Computer Engneerng, Unversty of Brjand, Iran, FarshdKevanan@brjand.ac.r A.Yekta@brjand.ac.r 3 NMehrshad@brjand.ac.r Artce hstory: Receved Juy 04 Accepted October 04 Avaabe onne November 04 Abstract The object of heurstc agorthms s to produce an optmum souton for sovng a probem. When the number of varabes n the probem s hgh the Heurstc Agorthms are used. In ths artce the goa s to fnd an optmum ayout for JK Fp Fop for mnmzng the average power. There are twenty MOSFETs wth dfferent channe wdths. They make a twenty dmensona search space whch are ndependent decson varabes. Motvated by the convergence of Ant Coony Optmzaton n rea doman (ACOR) and Genetc Agorthm (GA) and the nk of MATLAB wth HSPICE Software the optmzed ayout of JK Fp Fop s obtaned. Based on ACOR, Fuzzy-ACOR, GA, Fuzzy-GA agorthms the best resutng JK Fp Fop ayout n CMOS Technoogy wth suppy votage of 5v has the average power consumpton of.6 nw wth Fuzzy-ACOR. Keywords: Optmum ayout, JK Fp Fop, ACOR, Fuzzy-ACOR, GA, Fuzzy-GA.. Introducton Component ayout pays an mportant roe n the desgn and usabty of many engneerng products []. Ths artce descrbes ACOR, Fuzzy-ACOR, GA, and Fuzzy-GA agorthms to sove the probem of the channe wdth of MOSFETs n JK Fp Fop such that a set of objectves such as deay and average power can be optmzed. In ths artce the snge object of optmzaton of the average power s proposed. The goa s to mnmze the average power n JK Fp Fop ayout n CMOS Technoogy. The ssue of energy optmzaton has been proposed n other appcatons ke wreess sensor networks based on heurstc agorthms []. To avod a arge number of cacuatons for power consumpton of the crcut the heurstc agorthm s empoyed. Furthermore the performance of proposed heurstc agorthms ncudng ACOR, and GA has been tested over probems [3]. Genetc Agorthm (GA) s an Evoutonary Computaton agorthm and Ant Coony Optmzaton n contnuous space (ACOR) s a Swarm Integence agorthm whch both are used to acheve the optmum ayout desgn of JK Fp Fop and a of the resuts are ustrated and compared. The crcut of JK Fp Fop n gate desgn eve

2 F. Kevanan, A. Yekta, N. Mehrshad / J. Math. Computer Sc. 4 (05) - 5 s shown n Fgure, aso n the eve of transstor desgn s pctured n Fgure, n addton n ayout eve t s ustrated n Fgure 3. Fgure. JK fp fop desgn n gate eve Fgure. JK fp fop desgn n transstor eve, CMOS technoogy

3 F. Kevanan, A. Yekta, N. Mehrshad / J. Math. Computer Sc. 4 (05) - 5 Fgure 3. JK fp fop desgn n ayout eve, LEDIT ayout. Heurstc agorthms: ACOR and GA In the past few decades, there has been wdespread nteracton between researchers seekng varous evoutonary computaton methods to seek the best soutons to a gven functon. The Evoutonary Agorthms are deveoped by mmckng or smuatng processes found n nature and many ncudes Genetc Agorthms, Mmetc Agorthms, Partce Swarm Optmzaton, Ant Coony Optmzaton, and Shuffed Frog Leapng Agorthm. Optmzaton agorthms have consttuted most sgnfcant subjects n mathematcs and ndustry to conceve more accurate and expedtous soutons.. Ant coony optmzaton (ACOR) agorthm n rea doman to mnmze the average power n JK fp fop One of the frst attempts to appy an ant-reated agorthm to the contnuous optmzaton probems was Contnuous ACO (CACO) (Bechev and Parmee, 995) [3]. The extenson of ACO to the contnuous doman, ACOR whch s apped to the contnuous optmzaton probem n ths work s the one orgnay proposed by Schoa and further extended by Schoa and Dorgo [3]. The proposed ACOR n [3] compances wth ACO n dscrete doman as the process of pheromone s avaabe n both of them. In ths part the ACOR n [3] s carfed and used for desgnng optmum JK Fp Fop. The fgure 4, dagram 4- shows the ACO n dscrete doman ncudng severa ways that the ants can pass through them and there s no change around each way so there w not be any dstrbuton around each souton of a varabe. The more ants are passed or the better ants are moved through a way the more pheromone vaue s created. Meanwhe for ACOR the varabes are contnuous, and n ths work they are mted to the constrant from the mnmum vaue ( VarMn=0.µm ) to the maxmum vaue ( VarMax=µm ) for each channe wdth. 3

4 F. Kevanan, A. Yekta, N. Mehrshad / J. Math. Computer Sc. 4 (05) - 5 Fgure 4. Dagram 4- s dscrete ACO, dagram s ACO R, and dagram 4-3 s the nature of contnuty n ACO R Based on dagram 4-a, each channe wdth or varabe has dfferent soutons and there are pheromone functons whch are smar to norma dstrbutons and the pheromone vaues are dstrbuted over the dfferent soutons. In dagram 4-a, the contnuous varabe or channe wdth n horzonta axe has three possbe vaues and norma dstrbutons around them. The better souton or vaue has the more amount of pheromone around t, n comparson wth other soutons. Aso as t s seen n dagram 4-b the pheromone functons have been merged or added together and the resutng pheromone functon s more ntensfed around the better souton for the probem. As t s shown n dagram 4-3, naturay n a group of ants the second ant does not exacty pass through the way of the frst ant, so there s a change around the exact way or specfc souton whch means there s neary a norma dstrbuton over each souton. And the centers of these norma dstrbutons are the good soutons n comparson wth the other soutons. So the archve of good soutons w be made and updated unt they become the best soutons for the probem. Based on tabe the best souton whch s found for the channe wdth of MOSFET Q5 ( W5 ) s µm. The nature of ants and ther pass ways are shown n fgure 5, graph 5-. The varabe W5 stands for the channe wdth of MOSFET Q5 n fgure 5-. The fgure 5-3 shows the good soutons, for the varabe of W5. Each of norma dstrbutons as t s seen n graph 5-3 has ts own probabty of seecton P. The probabty of seecton s based on Rouette Whee Seecton. Aso these good soutons have ther own cost functon. The better souton has the ower cost functon vaue and the hgher probabty of seecton. In ths artce each ftness vaue or cost vaue s determned by the HSPICE Software. In fact HSPICE s the ftness evauator of ths probem snce t cacuates the average power of JK fp fop crcut n each teraton n ACOR n MATLAB software. Each of dstrbuton s norma dstrbuton wth ts mean and varance. The mean vaue s the good souton and the varance s the paths around the good way as t s seen n fgure 5- and 5-3. The probabtes of seecton P, P*, and Pj are ndependent of each other. Based on the prncpes of the probabty, the ndependent probabtes are apped to the equatons of and, and based on them the formua of 3 s obtaned snce the probabtes of seecton are mutped by norma dstrbutons and add together, so the resutng dstrbuton s Gaussan Dstrbuton of P(x 5) whch s shown n equaton 3. It s used for the tota dstrbuton over the random varabe x. Tabe. The best soutons n ACO R for W to W 0 W W W3 W4 W5 W6 W7 W8 W9 W W W W3 W4 W5 W6 W7 W8 W9 W

5 F. Kevanan, A. Yekta, N. Mehrshad / J. Math. Computer Sc. 4 (05) - 5 Fgure 5. The nature of dfferent soutons n the behavor of ants n 5-, MOSFET Q5 n 5-, norma dstrbutons over soutons for the channe wdth of Q5, W5 n 5-3 P( j) P( ) P( j) P( j) P( ) P( j) P( x 5) P( ) N( W,δ ) P(*) N( W*,δ* ) P( j) N( Wj,δ j ) () () (3) G 5 P ( x 5 ) (4) -μ ) x N( x;μ,δ ).exp( π δ δ (5) The graph 5-3 shows three spread paces of pheromones n back, bue, and pnk coors that are dstrbuted around the centers W, Wj, and W* and the percentage of pheromones are determned by P, Pj and P*. By the equaton 3 the probabstc mode of P(x 5) for the channe wdth of W5 ( x5 ) s estabshed. The ants w pay more attenton to the pace or center of W* where P(x) w be hghy assgned for t. And the best souton n yeow coor n fgure 5-3 for W5 s obtaned from equaton 3. It ncudes more pheromone than others around W*. P(x) s separatey cacuated for each of twenty varabes (twenty channe wdths) n ths probem ke P(x ) P(x 5) P(x 0). The aternatve names for P(x ) s G such that the equvaent name of G 5 s used for P(x 5) n the next parts and fgures. Aso the equaton 5 shows the norma dstrbuton functon for each varabe wth mean and varance. P(x 5) n equaton 3 can produce new dfferent soutons for the random varabe of x5 so the random vaues can be produced for the channe wdth of Q5 ( W5 ) by means of equaton 3. Each souton of the souton archve has ts own sgma δ. If a sgma n equaton 3 are zero then the exact soutons of souton archve w be produced but f a sgma are non-zero wth dfferent vaues then the oca search w be happened around the soutons. Whatever the sgma s coser to zero further expotaton s done. When the sgma (δ) becomes great then the exporaton becomes hgh and t ncreases the dversty of soutons. In ths work the probem ncudes twenty channe wdths ( W W0 ) that have dstnct probabty dstrbutons ( P(x) P(x0) ). So by controng the parameter of sgma, δ the dversty of channe wdths w be ncreased whch w be carfed n part.. Whatever the produced pheromone of a way or varabe s more the probabty vaue s more. In fgure 5, graph 5-3, the 5

6 F. Kevanan, A. Yekta, N. Mehrshad / J. Math. Computer Sc. 4 (05) - 5 smpe souton archve of W, W*, and Wj create the probabstc mode of P(x5) whch ts work s to generate new random soutons or new popuaton for the channe wdth of Q5 ( W5 ). Then the probabstc mode w be samped and the new popuaton w be produced and they w be compared wth the prevous popuaton n souton archve. By means of ths competton the souton archve w be updated and the worse soutons w be emnated. Fgure 6. Trade off cyce between souton archve and probabstc mode Fgure 7. The th souton vector wth dfferent soutons from to npop, reated souton weghts, reated probabtes of seecton and norma dstrbutons The fgure 6 shows a trade-off cyce between souton archve and probabstc mode. Ths cyce cause the popuaton and probabstc mode to be graduay better. The souton archve ncudes souton vectors. Each souton vector of S ncudes soutons of S S npop and each souton has ts own souton weght. The souton weghts w be sorted based on the reaton 6 and the better souton has more souton weght the fgure 7 shows a souton vector wth reatve souton weght, probabty of seecton, and norma dstrbuton. Every souton has ts own specfc dstrbuton. The notaton for souton archve sze s npop whch s set to 30 n ths work and ths parameter s smar to the parameter of popuaton sze n Genetc Agorthm n part 3.. As t s shown n the reaton 7 the random vector of x s obtaned from the Jont Gaussan Mxture of G. w w... w... w npop x ~ G ( G, G,..., G 0 ) (6) (7) 6

7 F. Kevanan, A. Yekta, N. Mehrshad / J. Math. Computer Sc. 4 (05) - 5 The th weght, w, s obtaned from equaton 8. As t s smar to Norma Dstrbuton t s expressed by Norma Dstrbuton, N, n equaton 8. The souton archve sze s npop and stands for th souton weght and q s a postve parameter whch affects on the varance. w q. npop..exp( π.( ) ) N(,μ,δ q. npop ( q. npop) ) (8) The norma dstrbuton of N wth the mean µ and the standard devaton of δ s shown n equaton 0. The δ s defned q.npop n ths artce. The parameter q contros the souton seecton, f q s zero then ony the best member of souton archve w nfuence on producng new soutons whe f q s hgh the soutons around the good soutons n souton archve w hghy nfuence on producng new soutons. The roe of parameter q n equaton 8 s smar to the roe of parameter α n Dscrete Ant Coony Optmzaton (ACO) whch s shown n equaton 9 [4]. k pj ( t) k 0 α [τ ( t)] [η j [τ k aowed k ( t)] α j ] β [η k ] β f j aowed otherwse k (9) The parameter α n [ j ( t)] s the soca ntegence factor n dscrete ACO [4] aso here n ACO R the parameter q s the soca ntegence, of course /q s assocated wth α. Based on equaton 0, the probabty, p s the probabty of seecton of th souton among the npop soutons for the random varabe of x or th channe wdth. Ths probabty s assocated wth the reated souton weght of th souton. The souton weghts are normazed and p s obtaned. Ths probabty s used n the equaton and the resutng Gaussan Dstrbuton, G n fact s the probabstc mode of th channe wdth whch can produce the new other soutons for ths varabe. Whatever the varance δ n the equaton s hgh, the dversty of soutons becomes hgh. The varance δ n ACO R determnes the step of search n agorthm and t s caed step sze. p m w w m, m :... npop (0) G P. N( S,δ ), :... npop () Based on fgure 8 the standard devaton δ 4 s more than δ, δ 4 > δ. Because the average dstance for fourth souton of S 4 s more than other soutons, ts hgh standard devaton causes more search space. 7

8 F. Kevanan, A. Yekta, N. Mehrshad / J. Math. Computer Sc. 4 (05) - 5 Fgure 8. The soutons n souton archve for the frst souton vector S, : npop If the search space s mted ony to the th souton vector, then the dstance of the souton S from the other soutons of S r (r : npop) s cacuated by equaton. Aso the number of other soutons except the souton of S s npop, so the equaton shows the average dstance of th souton to the other soutons. When the average dstance of the souton from the other soutons s so hgh, then the varance of the norma dstrbuton around that souton shoud be greater than others to ncrease the search capabty around t. AverageDs r S S r npop, r :... npop, r () As t s shown n equaton 3 the standard devaton s proportona to the average dstance. The parameter zeta s a postve contro parameter whch w be controed n fuzzy-aco R n part.. If th souton s far away than the other soutons then the amount of Sgma Σ and subsequenty the amount of standard devaton δ becomes hgh and t causes the coarse search step whe f zeta s ow then the search step becomes fne. δ ζ r S - S r npop -, r :... npop, r (3) The equaton 3 determnes the standard devaton around each souton whch s shown n fgure 8 and t s used n the power of two for the varance n the Norma Dstrbuton n equaton. The east average power of JK Fp Fop s converged to 8. nw whch s shown n fgure 9. The Pseudo Code of ACO R s shown n box. Fgure 9. Convergence of average power to 8.nw based on ACOR 8

9 F. Kevanan, A. Yekta, N. Mehrshad / J. Math. Computer Sc. 4 (05) - 5 Box. ACO R Pseudo-Code n ths artce Defne the probem of JK Fp Fop Set the Souton Archve Sze to npop, the Sampe Sze to nsampe, Seecton Pressure to q Set the devaton dstance rato to zeta Generate Inta Soutons, Evauate, Sort, and pace them n the Souton Archve Cacuate the Souton Weghts W and the Seecton Probabtes P Do Whe termnaton condtons do not meet Create probabstc mode G for each varabe separatey, Compute random sampes based on probabstc mode and create new popuaton (newpop) Merge New Popuaton (Sampes) and Man Popuaton (Archve), Sort, and Deete Extra Members Update the best souton ever found and Store Best Cost End Do Pot Resuts. Fuzzy ant coony optmzaton n rea doman (Fuzzy-ACOR) for the optmzaton of JK fp fop As noted prevousy by controng the parameter of zeta the speed of convergence and the performance of ACO R w be mproved. When s hgh the exporaton n search process becomes hgh whe when s ow the expotaton becomes hgh aso the search step w be fne or sma and the search space s more effectvey searched. At the begnnng of search process when the cost functon s hgh, zeta must be hgh. Aso the sze of new popuaton (Sampe Sze), nsampe n ths artce s another parameter that s controed and by ncreasng t the dversty of search w be ncreased. The fgure 0 shows the bock dagram of fuzzy-aco R whch s used n ths work. Based on the fgure 0 both HSPICE and MATLAB software run together smutaneousy. In each teraton the software of MATLAB w produce ayout of JK fp fop, and HSPICE w cacuate ts average power unt the best average power s obtaned. The Pseudo Code for Fuzzy-ACO R s ustrated n Box. Fgure 0. The bock dagram of Fuzzy-ACOR system whch s defned n ths artce 9

10 F. Kevanan, A. Yekta, N. Mehrshad / J. Math. Computer Sc. 4 (05) - 5 Box. Fuzzy-ACO R Pseudo-Code n ths artce Defne the probem of JK Fp Fop Set the Archve Sze to npop, the Sampe Sze to nsampe, Seecton Pressure to q Set the devaton dstance rato to zeta Generate Inta Soutons, Evauate, Sort, and pace them n the Souton Archve Cacuate the Souton Weghts W and the Seecton Probabtes P Do Whe termnaton condtons do not meet Create probabstc mode G for each varabe separatey, Compute random sampes based on probabstc mode and create new popuaton (newpop) Merge New Popuaton (Sampes) and Man Popuaton (Archve), Sort, and Deete Extra Members Update the best souton and the worst souton that ever found and Store Best Cost and Worst Cost Normaze tnormazed = t / Maxt, and Bestnormazed = [ WorstCost - BestCost(t) ] / WorstCost Read Fuzzy Inference System Fe and Fuzzy Rues (Fuzzy_ACOR_FIS.fs) Defne Input Varabes for FIS and Fre the Rues End Do Pot Resuts Tabe shows the Fuzzy-ACO R rues. The optmum wdth channes are shown n tabe 3. The fgure shows the east average power of.6 nw. Tabe. The fuzzy rues n Fuzzy-ACOR If Itnormazed s Hgh and BestCostnormazed s Hgh Itnormazed s Hgh and BestCostnormazed s Low Itnormazed s Low and BestCostnormazed s Hgh Itnormazed s Medum and BestCostnormazed s Medum Then zeta s Hgh and nsampe s Hgh zeta s Low and nsampe s Low zeta s Hgh and nsampe s Hgh zeta s Medum and nsampe s Medum Tabe 3. The best soutons n Fuzzy-ACOR for W W 0 W W W3 W4 W5 W6 W7 W8 W9 W W W W3 W4 W5 W6 W7 W8 W9 W

11 F. Kevanan, A. Yekta, N. Mehrshad / J. Math. Computer Sc. 4 (05) - 5 Fgure. The convergence of the best average power to.6nw for JK fp fop based on Fuzzy- ACOR The membershp functons are n trapmf type as t s shown n the Fgure. Fgure. Membershp functons for normazed teraton, best cost normazed, zeta, and nsampe.3 Genetc agorthm to mnmze the average power n proposed JK fp fop Genetc Agorthm as another heurstc agorthm s used, the Pseudo Code for GA for ths artce s n Box 3, the optmum ayout obtaned by GA s shown n tabe 4, and the resutng convergence s shown n Fgure 3. Box 3. GA Pseudo-Code n ths artce Defne the probem of JK Fp Fop and Upper band and Lower band of Varabes Determne GA Parameters (Popuaton Sze, Crossover Percentage, and Mutaton Percentage) Set the crossover percentage to pc and mutaton percentage to pm Generate Inta Popuaton, Evauate, Sort, and pace them n the Best Soutons Do Whe termnaton condtons do not meet Cacuate Seecton Probabtes Seect Parents ndces and appy crossover and evauate off sprngs, n Crossover Operator Seect Parents ndces and appy mutaton and evauate Mutant, n Mutaton Operator Create Merged Popuaton, sort popuaton Update and store the best souton and Cost that ever found End Do Pot Resuts

12 F. Kevanan, A. Yekta, N. Mehrshad / J. Math. Computer Sc. 4 (05) - 5 Tabe 4. The best soutons n GA for W to W 0 W W W3 W4 W5 W6 W7 W8 W9 W W W W3 W4 W5 W6 W7 W8 W9 W Fgure 3. The convergence of the best average power to.6nw for JK fp fop based on Fuzzy-ACOR.4 Fuzzy genetc agorthm (Fuzzy-GA) for optmzaton of JK fp fop ayout The bock dagram of Fuzzy-GA s shown n Fgure 4 [5]. The parameters of mutaton percentage ( pm ), crossover percentage ( pc ), and the number of popuaton ( npop ) are optmay controed to get the better resuts. The fuzzy-ga rues are ustrated n tabe 5. The pseudo code of Fuzzy-GA s shown n Box 4 [6]. The optmum channe wdths are shown n tabe 6. Aso the east average power after 00 teratons based on Fuzzy-GA s shown n Fgure 5. The membershp functons are aso n trapmf type, the same as Fuzzy-ACOR [7]. Fgure 4. Bock dagram of Fuzzy-GA system whch s defned n ths artce Tabe 5. Fuzzy rues n Fuzzy-GA If Itnormazed s Hgh and BestCostnormazed s Hgh Itnormazed s Hgh and BestCostnormazed s Low Itnormazed s Low and BestCostnormazed s Hgh Itnormazed s Medum and BestCostnormazed s Medum Then pm s Hgh and pc s Low and npop s Hgh pm s Low and pc s Hgh and npop s Low pm s Hgh and pc s Medum and npop s Hgh pm s Medum and pc s Medum and npop s Medum

13 F. Kevanan, A. Yekta, N. Mehrshad / J. Math. Computer Sc. 4 (05) - 5 Box 4. Fuzzy-GA Pseudo-Code Defne the probem of JK Fp Fop and Upper band and Lower band of Varabes Determne GA Parameters (Popuaton Sze, Crossover Percentage, and Mutaton Percentage Set the crossover percentage to pc and mutaton percentage to pm Generate Inta Popuaton, Evauate, Sort, and pace them n the Best Soutons Do Whe termnaton condtons do not meet Cacuate Seecton Probabtes Seect Parents ndces and appy crossover and evauate off sprngs, n Crossover Operator Seect Parents ndces and appy mutaton and evauate Mutant, n Mutaton Operator Create Merged Popuaton, sort popuaton Update the best souton and the worst souton that ever found and Store Best Cost and Worst Cost Normaze tnormazed = t / Maxt, and Bestnormazed = [ WorstCost - BestCost(t) ] / WorstCost Read Fuzzy Inference System Fe and Fuzzy Rues (Fuzzy_ACOR_FIS.fs) Defne Input Varabes for FIS and Fre the Rues End Do Pot Resuts Tabe 6. The best soutons n Fuzzy-GA for W to W 0 W W W3 W4 W5 W6 W7 W8 W9 W W W W3 W4 W5 W6 W7 W8 W9 W Fgure 5. Convergence of the best average power to 9.5nw for JK fp fop based on Fuzzy- GA 3

14 F. Kevanan, A. Yekta, N. Mehrshad / J. Math. Computer Sc. 4 (05) Concuson By mpementaton of heurstc agorthms such as ACOR, Fuzzy-ACOR, GA, Fuzzy-GA the optmum JK fp fop s obtaned. The best ayout consumes ony.6nw based on Fuzzy-ACOR. The comparsons are ustrated n tabe 7 and fgure 6. Aso the fuzzy-ga and fuzzy-acor showed the better resuts snce the fuzzy rues has ncreased the performance of agorthm and fnay the most optmum JK Fp Fop wth the east average power consumpton s obtaned by Fuzzy-ACOR. Tabe 7. The east average power consumpton are compared Heurstc Agorthms ACO R Fuzzy-ACO R GA Fuzzy-GA The best ftness vaue or owest average power whch s obtaned 8. nw.6 nw 9 nw 9.5 nw Average Power (nw) 5 0 ACOR Fuzzy ACOR GA Fuzzy GA Fgure 6. The comparson of convergences of dfferent heurstc agorthms: ACOR, Fuzzy- ACOR, GA, and Fuzzy-GA References [] I. Mhajovc, Z. Zvkovc, N. Strbac, D. Zvkovc, A. Jovanovc, Usng Genetc Agorthms To Resove Facty Layout Probem, Serban Journa of Management () (007) [] M. Esam, J. Vahd, M. Askarzadeh, Desgnng and Impementng a Dstrbuted Genetc Agorthm for Optmzng Work Modes n Wreess Sensor Network, Journa of Mathematcs and Computer Scence (04) [3] K. Socha, M. Dorgo, Ant coony optmzaton for contnuous domans, European Journa of Operatona Seacrh 85 (008), pp [4] Dorgo M. and G. D Caro, The Ant Coony Optmzaton Meta-Heurstc In D. Corne, M. Dorgo and F. Gover, edtors, New Ideas n Optmzaton, McGraw-H, -3, (999). 4

15 F. Kevanan, A. Yekta, N. Mehrshad / J. Math. Computer Sc. 4 (05) - 5 [5] Kevanan, F., Mehrshad, N., Zahr, S. H. Optmum Layout of Mutpexer wth Mnma Average Power based on IWO, Fuzzy-IWO, GA, and Fuzzy GA, ACSIJ Advances n Computer Scence : an Internatona Journa, Vo. 3 (04), Issue 5, No.. [6] S. M. Kaam Hers, PhD of Contro Engneerng, K. N. Toos Unversty of Technoogy, Tutora webste of [7] Mtche, Meane, An Introducton to Genetc Agorthms, Cambrdge, MA: MIT Press. ISBN , (996). 5