GHHAGA for Environmental Systems Optimization

Transcription

1 GHHAGA for Envronmental Systems Optmzaton X. H. Yang *, Z. F. Yang, and Z. Y. Shen State Key Laboratory of Water Envronment Smulaton, School of Envronment, Beng Normal Unversty, Beng 00875, Chna Key Laboratory for Water and Sedment Scences Mnstry of Educaton, School of Envronment, Beng Normal Unversty, Beng 00875, Chna ABSTRACT. The global optmzaton of complcated nonlnear systems s mathematcally ntractable and such an optmzaton extensvely exsts n scence and engneerng. Once an obectve functon has many local extreme ponts, the tradtonal optmzaton methods may not obtan the global optmzaton effcently. A genetc algorthm (GA) based on the genetc evoluton of a speces provdes a robust procedure to explore broad and promsng regons of solutons and to avod beng trapped at the local optmzaton. However, the computatonal amount s very large. To reduce computatons and to mprove the computatonal accuracy, a method based on the two-pont crossover and two-pont mutaton of the hybrd acceleratng genetc algorthm wth Hooke-Jeeves searchng operator s developed for systems optmzaton. Wth the shrnkng of searchng range, the method gradually drects to optmal result by the excellent ndvduals obtaned by Gray code genetc algorthm embeddng wth Hooke-Jeeves searchng operator and Hooke-Jeeves algorthm. The effcency of the new algorthm s verfed by applcaton of several test functons. The comparson of our GA wth sx other algorthms s presented. Ths algorthm overcomes the Hammng-clff phenomena n other exstng genetc methods, and t s very effcent for the gven envronmental systems optmzaton. Key words: Genetc algorthm, Gray code, Hooke-Jeeves algorthm, Global optmzaton, Envronmental systems * Correspondng author: yxh@sohu.com 转载

2 . Introducton The global optmzaton of complcated nonlnear systems s ntractable mathematcally and such an optmzaton extensvely exsts n scence and engneerng. Fndng the mnmum of a nonlnear functon of one varable, wthout any constrans, can be challengng, too. A severe dffculty arses f the functon n queston has many purely local mnma, because what one really wants s a global mnmum pont (Davd et al., 003). Once an obectve functon has many local extreme ponts, the tradtonal optmzaton methods may not obtan the global optmzaton effcently. A genetc algorthm (GA) based on the genetc evoluton of a speces was proposed by Holland (Holland 975). The detaled genetc algorthm and mplementaton were gven by Goldberg (Goldberg 989). De Jong showed that the standard bnary-encoded GA (SGA) could consttute an nterestng alternatve to perform the global optmzaton of a functon dependng on several contnuous varables (Jong, 975; Andre et al., 00). Ths algorthm provdes a robust procedure to explore broad and promsng regons of solutons, and to avod beng trapped at the local optmzaton (e.g.,yang et al.,004; Leung et al., 00). Expermental results show that GAs can solve a lot of dffcult problems (Bessaou et al., 00) through an approprate choce of representaton patterns of elements of the search space and operators. However, the computatonal amount s very large and premature convergence phenomena exst n SGA. To reduce computatons and mprove the computng precson, the bnary-encoded acceleratng genetc algorthm, real-encoded genetc algorthm and nteger-encoded genetc algorthm were developed (Jn et al., 000; Jankow et al., 99; Renders et al., 996). However, these genetc algorthms cannot be effectvely appled for the global optmzaton of contnuous varable n complcated nonlnear systems. The Hammng dstance between two closest ntegers n bnary code s very large. For nstance, ntegers 63 and 64 are expressed by the 00 and n bnary code, respectvely. All of the codes must be changed f we turn 63 nto 64 n bnary code. Ths operaton depresses the effcency of the genetc algorthms. Ths phenomenon s termed the Hammng clff. To solve the above Hammng clff problem, SGA was mproved wth gray encodng of parameters (e.g., Andre et al., 00; Yang et al., 003). The ntegers 63 and 64 are expressed by and n Gray code, respectvely. The Gray-encoded genetc algorthm (GGA) can partly overcome the Hammng clff of bnary code (Zhong et al., 00). But the Gray-encoded genetc algorthm needs approprate operators, and the mathematcal theory of ths algorthm has not been developed at

3 present. It was found that ths algorthm stll needs a large amount of computaton (Yang, 00). So GGA should be mproved and developed. In ths paper, a Gray-encoded, Hooke-Jeeves (Hooke et al., 96), hybrd acceleratng genetc algorthm (GHHAGA) s presented to reduce computatons and to mprove the computatonal precson. Ths approach wll be appled to fve nonlnear functons for verfcaton and two envronmental systems optmzaton.. The Descrptons of GHHAGA The GHHAGA starts wth an ntal populaton of n ndvduals : each ndvdual s composed of Gray code, respectvely assocated wth the varables of the obectve functon at hand. Then evoluton starts and genetc operatons are consttuted. The reproducton operators are appled to ths populaton; offsprng are created from parents. The new populaton s consttuted n selectng the best ndvduals. After two-pont crossover and two-pont mutaton, the new ndvduals are created. And output of the best pont s further found by the Hooke-Jeeves algorthm (Hooke et al., 96) around the above best ndvdual (pont). The new best pont nsde the offsprng wll be nserted to replace the worst one n the prevous phase. Repeat the above genetc operatons untl the evoluton tmes Q s met. The new ntervals are gotten from the varable ranges of n s -excellent ndvduals by Q -tmes of the Hooke-Jeeves evoluton, and then the whole process back to the Gray-encodng. Fnally, the most excellent chromosome currently s the optmum soluton of GHHAGA... Steps of GHHAGA Consder the followng nonlnear optmzaton problem: mn f ( x, x,, x n ) () s.t. where x { x, =,,..., } a p = n p, functon and f 0. x b, for =,,, p Steps of GHHAGA are gven as follows. Step. Gray encodng. n, x s a varable to be optmzed, f s an obectve 3

4 An e-bt Gray varable s used to represent one varable x. The nteger of the e Gray varable ranges from 0 to, and t can be mapped lnearly to the varable range a, b ]. The th varable range s the nterval a, b ], and then each nterval s [ [ e dvded nto sub- ntervals: x = a + I Δx () e where Δx = ( b a ) /( ). The Gray code array of the th varable s denoted by the grd ponts of { d, k) k,,..., e} ( = (Yang et al., 003): I e e m ( d(, k)), (3) m= k= m = Step. Generatng ntal father populaton. Intally, the chromosomes are generated at random n Gray-encoded genetc algorthm, and n-chromosomes n father populaton are: I e ( ) = nt( u(, ) ) for =,,..., n p ; =,,..., n, (4) where u (, ) s unformty random number, u(, ) [0,], I () s searchng locaton, nt ( ) s an nteger functon. From Eq.(3), the n-correspondng chromosomes are d (, k, ) for =,,..., n p ; k =,,..., e; =,,..., n. To cover homogeneously the whole soluton space and to avod the rsk of havng too much ndvduals n the same regon, a large unformty random populaton s selected n ths algorthm. Step 3. ftness evaluaton. The ftness functon F() of th chromosome s defned: F ( ) = [ f ( )] + (5) 0. Step 4. Selecton. The chromosomes n the ntal father populaton are selected by a known probablty p s () as s n p ( ) = F( ) / F( ), (6) = Such two groups of n -chromosomes are selected by the above probabltes. 4

5 Step 5. Two-pont crossover. For two-pont crossover, two crossng ponts are randomly chosen, and two ndvduals d(, k, ), d (, k, ) are gotten by the crossng probablty p c. In order to enhance the dversty of populaton, the crossng probablty s set as Step 6. Two-pont mutaton. p c =. For two-pont mutaton, two mutatng ponts are randomly chosen, and a new offsprng d 3 (, k, ) can be computed by a mutatng probablty p m (Yang et al., 003). Step 7. Hooke-Jeeves evoluton. The Hooke-Jeeves algorthm s a useful, local descent algorthm, whch does not make use of the obectve functon dervatves (Hooke et al., 96). The best pont n the prevous phase becomes a new ntal soluton n the Hooke-Jeeves algorthm, and then a new best pont s obtaned by ths Hooke-Jeeves algorthm. The new best pont nsde the offsprng wll be nserted to replace the worst one n the prevous phase. Repeat step3 to step 7 untl the evoluton tmes Q or termnaton crtera s met. Step 8. Acceleratng cycle. The varable ranges of n s -excellent ndvduals obtaned by Q -tmes of the Hooke-Jeeves evoluton become the new ranges of the varables, and then the whole process back to the Gray-encodng. The computaton process s over untl the obectve functon value gets to an expected value, or algorthm runnng tmes gets to the desgn T tmes. Heren, the most excellent chromosome currently s the optmum soluton of GHHAGA. A pseudo code of GHHAGA s gven n Fgure. ================================================ Begn For t=0 to T (acceleraton cycle tmes t) t t Gve varable nterval [ a, b ] Gray encodng q=0 Intalze populaton Pop(q,t) Whle (q Q) do (evoluton tmes q) For = to n do Evaluate ftness of Pop (q,t) 5

6 Endfor For = to n do Select operaton to Pop(q,t) Endfor For = to n do Two-pont crossover operaton to Pop(q,t) Endfor For = to n do Two-pont Mutaton operaton to Pop(q,t) Endfor Hooke-Jeeves evoluton Endwhle t+ t+ Get new varable nterval [ a, b ] from the varable ranges of n s -excellent ndvduals n Pop (,t), Pop (,t),, Pop (Q,t) Endfor End ================================================= Fg.. Pseudo code for GHHAGA.. GHHAGA Theory We now turn to the analyss of the behavor of our algorthm when t s appled to problem (). The GHHAGA s convergent (Yang, 00). The global optmzaton of the GHHAGA s not only accurate but also stable. Let the Hooke-Jeeves evoluton tmes be Q, the number of excellent ndvduals be n s,the number of optmzed varable be p and the tmes of acceleratng cycle be T, the probablty p 0 of Qns excellent ndvduals surround the optmum pont s p0 ( ) pt =. The GHHAGA s global convergence wth probablty p 0 = when n p =0, t=5, n s =0, Q=5; n p =30, t=0, n s =0, Q=5, etc. 6

7 3. Expermentaton 3.. Crtera Three man crtera are very mportant when tryng to determne the performances of an algorthm: convergence, speed and robustness (Andre et al., 00). The parameters of the GHHAGA are selected as follows: The length e =0, populaton sze n = 300, the number of excellent ndvduals n = 0, the tmes of Hooke-Jeeves evoluton Q=5, the crossover probablty p c =.0 s, the mutaton probablty p =0.5,and the tmes of Hooke-Jeeves searchng m 300. m The global optmzaton of fve test functons (Andre et al., 00) s accomplshed by usng the followng methods: Standard bnary-encoded GA (SGA) (Andre et al., 00) and GHHAGA. To compare wth the global optmzaton ablty of the above algorthms obectvely, one of the three followng termnaton crtera s used for ensurng the optmzaton precson and avodng algorthm nvaldaton. Crtera one: The relatve error E rel between the result f lg calculated by the a o algorthm and the optmum value f exact of each test functon s used each tme f t s possble: E rel = fa lg o f f exact exact (7) Crtera two: When the optmum value s 0, t s no longer possble to use ths expresson, so we calculate the absolute error E abs as follows: E abs = f f (8) a lg o exact Here we let the absolute error or relatve error n neghbor generatons be less than or equal 0 -. Crtera three: The total computaton number for the functons s less than or equal Expermentaton and Result To test our GHHAGA, the followng fve analytcal test functons were used. 7

8 Goldprce: f ( x, y) = [ + ( x + y + ) (9 4x + 3x 4y + 6xy + 3y [30 + (x 3y) (8 3x + x + 48y 36xy + 7y where x, y )] )] Hartman : f ( x) = 4 c exp[ 6 = = a ( x p ) ] where 0 x, for =,,6 wth x=(x,,x 3 ), p = p,..., p ), a = ( a,..., a ) ( 6 6 a c p Hosc45: f ( x) = 0 = n! x where x = x,..., x ), and 0 x,n=0 ( 0 Brown : 3 0( x x + ) f ( x) = [ ( x 3)] + [0 ( x 3) ( x x+ ) + e ] J J where J={,3,...,9}, x 4,for 0 and x = x,..., x ] [ 0 T F5n: 9 f x) = 0. sn (3π x ) + [( x ) = ( + sn (3πx + ))] + 0.( x0 ) [ + sn (πx )] ( 0 where 0 x 0,for 0 and x = x,..., x ] [ 0 T 8

9 Ths set of classcal test functons, very often used n the lterature (Andre et al., 00;Bessaou et al., 00), ncludes some functons havng the followng features: Contnuous/dscontnuous; Convex/non-convex; Unmodal/multmodal; Quadratc/non-quadratc; Low dmenson/hgh dmenson. The effcency of the algorthm s quantfed by: The rate of success; The number of evaluatons of the obectve functon; The error between the calculated soluton and the global optmum value of the functon. Because of the stochastc nature of GAs, the dscusson of results derved from one sngle executon of the algorthm s meanngless (Bessaou et al., 00). So all results reported n ths secton are obtaned by averagng the results from 00 executons per functon. The computaton results of the fve nonlnear test functons are gven n Table wth the GHHAGA. It s obvously observed that the GHHAGA s the best one both n effcency (see success rate and number of evaluaton of the functons n Table ) and n accuracy (see mnmum found n Table ) compared wth exstng algorthms. The results gven n Table show that the global optmum value can be gotten for the fve test functons and the Hammng clff phenomena are avoded n GHHAGA. Table. Results wth the SGA (Andre et al., 00) and the GHHAGA Name of the Theoretcal Mnmum found Number of evaluaton of Success rate % functons mnmum the functons SGA GHHAGA SGA GHHAGA SGA GHHAGA Goldprce Hartman Hosc Brown

10 F5n We have performed a comparson of our GA wth sx other methods lsted n Table : pure random search (PRS) ( Anderssen et al., 97), multstart (MS) (Rnnoy et al., 987), smulated dffuson (SD) (Aluff et al., 985), smulated annealng (SA) (Andre et al., 00), tabu search (TS) (Cvovc et al., 995) and bnary-encoded genetc algorthm (GA) (Andre et al., 00). The effcency was qualfed by use of the number of functon evaluatons to fnd the global optmum. Each program was stopped as soon as the relatve error between the best pont found and the known global optmum was less than %. The number of functon evaluatons used by the varous algorthms to optmze test functons s lsted n Table. The results of our GA are satsfed (see the numbers of functon evaluatons n Table ). In addton, our results were the average outcome of 00 ndependent runs; for some publshed methods, the tmes of runs was equal to 4 or unspecfed (Andre et al., 00). Table. Number of functon evaluatons on global optmzaton of two functons wth the seven dfferent methods Method Name The number of functon evaluaton Goldprce Hartman PRS Pure random search MS Multstart SD Smulated dffuson SA Smulated annealng TS Tabu search GA Bnary-encoded genetc algorthm GHHAGA Gray-encoded, Hooke-Jeeves, acceleratng hybrd genetc algorthm 0

11 4. Applcaton Example. The wastewater treatment cost tme seres as a dynamcal system optmzaton s consdered to satsfy at y( t) = A K /( + be ) (9), where y (t) s the computed value of wastewater treatment cost n the tth year, unt: 0 4 US$. A, K, b, a are optmzaton parameters. The observaton data can be seen lterature (Jn, 000). Ths obectve functon s as follows: h = n = ( y( t ) y ), (0) where y s the observaton value of the wastewater treatment cost n the t th year, unt: 0 4 US$, n s the total number of observaton data. The least resdual square sum h s wth GHHAGA. GHHAGA runs second for optmzaton of ths model only. The computatonal results of the above model are gven n Table 3. For the GHHAGA, the evaluaton number of the obectve functon s 400. For adaptve acceleratng genetc algorthm (AAGA) (Yang, 00), the evoluton number s For bnary-encoded acceleratng genetc algorthm (BAGA) (Jn, 000), the evoluton number s The comparson of the above methods can be observed n Table 3. The results of our GHHAGA are satsfyng both n effcency and n accuracy. Table 3. The calculatng result wth several methods for the dynamcal optmzaton problem Method Evaluaton Parameters number for h A K b a Least mean square sum h Intal nterval 0 [500,700] [00,300] [5, 8] [0, ] GHHAGA AAGA BAGA Example. The forecast problem of the sedment concentraton n Gongzu reservor as a system optmzaton s consdered to satsfy

12 u s ( t) = c + t () cx( t) + cx( t) c3x( ) where u s (t) s the smulated value of the sedment concentraton n Gongzu reservor n the tth year, Qn ( t) Qup ( t) x( t) =, Q up (t) s the runoff of upstream reservor and 000 Q n (t) s the nflow nto Gongzu reservor n the tth year, c 0, c, c, c3 are optmzaton parameters. The observaton data can be seen n Jn s lterature (Jn, 000). The amount of the sedment concentraton nto Gongzu reservor s one of the most mportant elements n decdng the runnng of the Gongzu reservor. The obectve functon s gven as follows: mn f = n = [( us u ) ( u + ) + ψ ( us )], () u and 0, us 0 ψ ( us ) = (3) 500, us < 0 where u and u s are the observaton value and the smulated value of the sedment concentraton n Gongzu reservor n the t th year, respectvely; n s the number of observaton data; ψ u ) s a penalty functon. ( s The obectve functon f s wth GHHAGA. For the GHHAGA, the evaluaton number of the obectve functon s 800. For bnary-encoded acceleratng genetc algorthm (BAGA) (Jn, 000), the evaluaton number of the obectve functon s 0500, and the obectve functon f s 33.. The result of GHHAGA s gven n Table 4. From Table 4, we can conclude that our GHHAGA s hghly effectve n optmzaton. From above two examples, we fnd that the number of obectve functon of GHHAGA s less than those obtaned by the other GAs technque. The precson of GHHAGA has been mproved by Gray encodng. The convergence of GHHAGA has also been consderably mproved by provdng new ntal nterval nformaton n acceleratng cycle. And ts computatonal amount s very small. Ths GHHAGA has overcome the Hammng-clff phenomena n exstng genetc methods.

13 Table 4. The calculatng result for the sedment concentraton n Gongzu reservor wth GHHAGA Evaluaton Number for f Parameters c 0 c c c 3 Obectve functon f 0 [-5, 5] [-5, 5] [-5, 5] [-5, 5] Conclusons In ths paper, a new method, GHHAGA s proposed to solve the nonlnear optmzaton problems. The correspondng global convergence of the new genetc algorthm s analyzed. Because the steps of Gray-encodng, Hooke-Jeeves hybrd searchng operator and acceleratng cycle are adopted, the effcency and accuracy of the new algorthm are very hgh compared to exstng algorthms. GHHAGA has been appled to fve nonlnear test functons and two envronmental systems optmzaton. Based on the comparsons and the applcaton, the proposed algorthm shows good performances for contnuous varable optmzaton of nonlnear systems. It overcomes the Hammng-clff phenomena n other exstng genetc methods. Our avenue for the future s adaptng the algorthm to perform a mult-obectve optmzaton for envronmental systems. Acknowledgement. Ths work was supported by Foundaton tem: Natonal Key Proect for Basc Research, No. 003CB4504. References Davd, K. and Ward, C. (003). Numercal analyss: mathematcs of scentfc computng. Chna Machne. Holland, J.H. (975). Adaptaton n natural and artfcal systems. Unversty of Mchgan. Goldberg, D, E. (989). Genetc algorthms: search, optmzaton and machne learnng. Addson-Wesley. Jong, D. (975). An analyss of the behavor of a class of genetc adaptve systems. Ph.D. Dssertaton, Unversty of Mchgan, Ann Arbor, MI.,USA. Andre, J., Sarry, P. and Dognon, T. (00). An mprovement of the standard genetc algorthm fghtng premature convergence n contnuous optmzaton. Advances 3

14 n Engneerng Software. 3, Bessaou, M., Sarry, P.(00). A new tool n electrostatcs usng a really-coded multpopulaton genetc algorthm tuned through analytcal test problem. Advances n Engneerng Software. 3, Yang, X.H., Yang, Z.F., Shen, Z.Y. and L, J.Q.(004). An deal nterval method of mult-obectve decson-makng for comprehensve assessment of water resources renewablty. Scence n Chna seres E-engneerng & materals scence. 47 (Suppl. S), Leung, Y. W. and Wang, Y. P. (00). An orthogonal genetc algorthm wth quantzaton for global numercal optmzaton, IEEE Trans, On Evolutonary Computaton. 5(), Jankow, C. Z. and Mchalewcz, Z. (99). An expermental comparson of bnary and floatng pont representaton n genetc algorthms. Proceedngs of the Fourth Internatonal Conference on Genetc Algorthms, San Francsco, pp Renders, J. M. and Flasse, S. P. (996). Hybrd methods usng genetc algorthms for global optmzaton. IEEE Trans Systems, Man Cybernetcs part B: Cybernetcs. 6(), Yang, L. N., We, G. M., Guo, K. and Luo, J. S. (003). Applcaton genetc algorthms wth ntal group floatng for mxed nteger nonlnear programmng. Computng technques for geophyscal and geochemcal exploraton. 5(3), Yang, X. H.,Lu, G. H.,Yang, Z. F. and L, J. Q. (003). Gray codng based acceleratng genetc algorthm and ts theory. Theory and Practce for System Engneer. 3(3), Zhong, M. and Sun, S. D. (00). Genetc algorthms: theory and applcatons. Beng Defense Industry. Yang, X. H. (00). Study on Parameter Optmzaton Algorthm and ts Applcaton n Hydrologcal Model. Ph.D. Dssertaton, School of Water Resources and Envronment, Hoha Unversty, Nanng, Chna. Hooke, R. and Jeeves, T. A. (96). Drect search soluton of numercal and statstcal problems. J. Ass. Comput. Mach.. 8, -9. Anderssen, R. S., Jennngs, L.S. and Ryan, D. M. (97). Optmzaton. Unversty of Queensland. Rnnoy, K. and Tmmer, G.T. (987). Stochastc global optmzaton methods. Mathematcal Programmng. 39,

15 Aluff, P. F., Pars, V and Zlll, F. (985). Global optmzaton and stochastc dfferental equatons. J. Opt. Theor. Appl.. 47, -6. Jn, J. L., Dng, J. (000). Genetc algorthm and ts applcatons to water scence. Schuan: Schuan unversty. Cvovc, D. and Klnowsk, J. (995). Taboo search: an approach to the multple mnma problem. Scence. 67,