Supplementary Crossover Operator for Genetic Algorithms based on the Center-of-Gravity Paradigm

Transcription

1 Published in Cntrl and Cybernetics, vl. 30, N, 00, pp , ISSN: Supplementary Crssver Operatr fr Genetic Algrithms based n the Center-f-Gravity Paradigm Plamen Angelv Department f Civil and Building Engineering, University f Lughbrugh Lughbrugh, Leicestershire LE 3TU, UK tel.: +44 (509) 3 774; fax: +44 (509) 3 98; Abstract A supplementary crssver peratr fr genetic algrithms (GA) is prpsed in the paper. It perfrms specific breeding between the tw fittest parental chrmsmes. The new child chrmsme is based n the center f gravity (CG) paradigm, taking int accunt bth the parental weight (measured by their fitness) and their actual value. It is designed t be used in cmbinatin with ther crssver and mutatin peratrs (it applies t the best fitted tw parental chrmsmes nly) bth in binary and real-valued (evlutinary) GA. Analytical prf f its ability t imprve the result is prvided fr the simplest case f ne variable and when elitist selectin strategy is used. The new peratr is validated with a number f usually used numerical test functins as well as with a practical example f supply air temperature and flw rate scheduling in a hllw cre ventilated slab thermal strage system. The tests indicate that it imprves results (the speed f cnvergence as well as the final result) withut significant increasing cmputatinal expenses. Key wrds: Genetic algrithms; crssver, mutatin, selectin peratrs; center f gravity. Intrductin Recently, GA have been widely applied t different cntrl and ptimizatin prblems due t their rbustness, success in dealing with multi-mdal and cmplex prblems (Pal and Wang, 996; Angelv and Guthke, 997, Onnen et.al., 997). The main specific f the GA as an ptimizatin methd is their implicit parallelism, which is a result f the evlutin and the hereditary-like prcess (Michalewicz, 996). GA is, in fact, a driven stchastic search technique, which cmbine stchastic (represented by mutatin peratr) and 'lgical' search (represented by crssver f parental chrmsmes and survival f the fittest by apprpriate selectin). These characteristics f GA ffers pssibilities fr their imprvement by apprpriate balance between explratin (pssible because f the diversity in the ppulatin) and explitatin (due t the

2 preservatin f the search lgic). Initially, imprvements f GA has been sught in the ptimal prprtin and adaptatin f the main parameters f the GA, namely prbability f mutatin, prbability f crssver, ppulatin size (Davis, 989) (Grefenstette, 986). Mre recently, the attentin has been shifted t the breeding (prcess f frming new trial chrmsmes at each epch) (Muhlenbein and Schlierkamp-Vsen, 993). In this paper a supplementary crssver peratr is intrduced, which is mre infrmative than mutatin and mre innvative than crssver itself. It increases diversity by creating a new chrmsme different t the previus ppulatin elements and in the same time preserves the 'search lgic' by accumulating weighted infrmatin abut parental ppulatin. It is designed t be used in additin (in cmbinatin) with ther crssver and mutatin peratrs bth in binary and real-valued GA. Supplementary crssver peratr is applied t the best fitted tw parental chrmsmes nly. The rest f the chrmsmes in the ppulatin are prduced by applying any ther mutatin and crssver peratrs as usual. The new child chrmsme, very ften have better fitness as it represents a CG f the tw best parental chrmsmes and, thus, imprves the search capability f the whle algrithm. The supplementary crssver peratr has been tested with a number f cmmnly used in the literature test functins. A practical prblem f scheduling f the supply air temperature and flw rate t a ventilated slab thermal strage system is als presented. The results demnstrate its superirity cmpared with the case when it is nt used. Basic cncepts f GA GA mimics the prcess f natural selectin where "the fittest survives". The basic difference between the GA and all ther ptimizatin techniques is its implicit parallelism, a result f the evlutin and hereditary. The GA explre a set f trial pints (chrmsmes) frming a ppulatin at each iteratin (epch). A gene (g) in the chrmsme represents binary encded prblem variable (x) in the 'standard' binarycded GA (Gldberg, 989) r directly its value in the real-value cded GA (Michalewicz, 996).

3 chrmsme chrmsme chrmsme ps g g g n g ps g g n g ps ps g g n Table : Ppulatin f Individual Chrmsmes In binary cded GA each gene is represented by a number f bits, which have value 0 r. Fr example: g i = {;0;;;;0;0}; i=,,...,n In real-value cded GA each gene is a real value representing certain variable g i = x i. The measure f the quality f a chrmsme (candidate slutin) is its fitness functin (F), which shuld be maximized. All individual chrmsmes are evaluated (by calculating their fitness) at each epch. A part f the chrmsmes frm the current epch (parental chrmsmes) are selected fr mating and reprductin, based n their fitness values. Crssver and mutatin are then applied fr prducing new (child) trial pints (chrmsmes). An example f a tw-pint crssver is represented as: chrm ={ child g ; g ;...; g ; g ; g ;...; g ; g ; g ;...; g ; g } xver xver xver + xver xver xver + chrm ={ child g ; g ;...; g ; g ; g ;...; g ; g ; g ;...; g ; g } xver where dentes the first parent; dentes the secnd parent; xver xver + xver dentes the first crssver pint xver dentes the secnd crssver pint xver xver xver + Mutatin in binary-cded GA is a triggering frm 0 t and vice versa. In realcded GA, generally, unifrm and nn-unifrm mutatin exist (Michalewicz, 996) with alteratin f the mutated gene t a randm value (in the range f feasible values) in the frmer ne and alteratin f the mutated gene with a certain value (added r subtracted depending n the 0 r value f a randm number) in the later ne. A tw-pint mutatin peratr, generally, culd be represented as: mut mut chrm child = {g ; g ;... ; g i ; g i ; g i + ; ; g i ; g i ; g i + ;...;g n } mut mut where g i and g i are mutated genes. n n n n

4 A GA culd be represented by the fllwing pseud-cde: Prgram GA Begin Number_f_epchs = 0; Set the initial ppulatin 0 Determine fitness functin value F; While (Number_f_epchs < Maximal_number_f_epchs) d Number_f_epchs = Number_f_epchs + ; Selectin; Crssver; Mutatin; Fitness evaluatin; end End. In this general scheme the main bjects (Selectin, Crssver and Mutatin) culd vary depending n the specific type chsen ( Michalewicz, 996). 3 New supplementary crssver peratr 3. New peratr - definitin The peratr prpsed in the paper is designed t be used in cmbinatin with (in additin t) the usually used crssver peratrs bth in binary- and real-cded GA. It cnsiders ne f the child chrmsmes t be prduced by a special breeding f the tw best fitted parental chrmsmes (called chrm prime and chrm secnd ), while all ther (ps- ) child chrmsmes are prduced in an usual way. The last place in the ppulatin is preserved fr this special chrmsme, which represents the center f gravity f chrm prime and chrm secnd frm the previus ppulatin (Table ): chrmsme i chrmsme i CG i- Table Ppulatin i Fr the fittest chrmsme we have: F(chrm prime ) F(chrm j ); j=,...,ps

5 If it take the last place in the ppulatin, then the secnd best chrmsme (chrm secnd ) culd be determined as: F(chrm secnd ) F(chrm j ); j=,...,ps - Using these ntatins, the genes f the child chrmsme are determined as CG f the parental nes: g CG i prime prime sec nd sec nd gi * F( chrm ) + gi * F( chrm ) = ; i=,,n prime sec nd F( chrm ) + F( chrm ) where g prime dentes a gene frm the chrm prime ; g secnd dentes a gene frm the chrm secnd. In the case f binary GA the supplementary crssver peratr is applied after selectin and decding f the genes values frm binary int decimal numbers tgether with the main crssver peratr. Each gene f the resulting child (CG-based) chrmsme is then encded int binary bits in a similar way as the rest (ps-) child chrmsmes (see Table ) and, finally, mutatin and reprductin are applied t the whle new ( i+ ) ppulatin. 3. New peratr - hw it wrks Let us cnsider a simple example t illustrate the new peratr. The example ppulatin cnsists f chrmsmes having 4 real-cded genes each. Let the best chrmsme at the i-th epch be: etc. chrm prime = {; 3; 4; } Let its fitness is F(chrm prime ) = Further, let the chrm secnd and its fitness be respectively: chrm secnd ={8; 6; 5; 3}; F(chrm secnd ) = 0.75 Then the CG-based child chrmsme will be: CG={4.8; 4.4; 4.47;.47}, because CG g = CG g = * *0.75 =4.8; * *0.75 =4.4,

6 3.3 New peratr - why it wrks Thugh, it is difficult t prve strngly that sme new peratr in GA is better even fr sme class f prblems because f the prbabilistic nature f the GA (Michalewicz, 996), we can expect that in many cases CG chrmsme will have better fitness. This culd easily be illustrated with the very simple example f ne variable functin F(x)= e x 0.( ) x (Fig.). Figure : Hw Supplementary CG Crssver Wrks - simple ne variable example Let the i-th ppulatin be i ={; ; 3; 4; 5; 6; 7; 8}. The chrm prime and chrm secnd will bviusly be respectively: chrm prime = 3; F(chrm prime ) = e = ; chrm secnd = 4; F(chrm secnd ) = e = ; The CG-based new child chrmsme/gene (in the case f ne variable the chrmsme is equivalent t the gene) then will be: CG i 3exp( ) + 4exp( 0.563) = = 3.47; F(3.47) = exp( ) + exp( 0.563) It is easy t see that it is much clser t the real maximum ( ). In the next ppulatin CG-based child chrmsme is cnsidered. Nte that the last gene (3) is due t the elitist strategy applied in additin t CG in this case: i+ i+ i+ i+ ={ gene ; gene ;...; gene ; 3.47; 3} 6

7 The ther 6 chrmsmes are prduced by crssver and mutatin f the parental chrmsmes frm the previus epch (generatin) as usual. Analytically it is pssible t prve that imprvements will ccur fr the simplest case with ne variable and cnvex in the interval (x - ;x + ) fitness functin (F) when: x - x* x + where x* = {x F(x*) = max(f)} x - = min (crm prime, chrm secnd ) x + = max (crm prime, chrm secnd ) Real situatins, hwever, are mre cmplex, but as the test results indicate imprvements ften ccur. This culd be explained with the fact that CG-based child chrmsme is prduced by the tw best parent individuals incrprating als infrmatin abut their fitness. By differ frm the simple hill climbing it determines the new (ften better) value f variables (x) directly (withut using an estimatin f the gradient and a step, which is usually cmputatinally expensive, prblem-dependent and a surce f subjectivity). 4 Test examples The new supplementary CG-based crssver peratr has been applied t five different cmmnly used numerical test prblems (Michalewicz, 996). Each test is perfrmed with the same GA parameters (prbabilities f crssver, mutatin, ppulatin size). Fr cnsistency f the results, the same randm number sequence is als used in bth cases (with and withut applying the new supplementary crssver peratr). 4. Numerical test functins Tw series f 30 runs has been carried ut with all f the five test functins: i) search stps after a value f the bjective functin (J*) clse t the theretical ptimum (J ptimal ) is reached (J* J ptimal ). The number f epchs needed is recrded fr bth cases: using CG-based supplementary crssver (N COG ); withut using CG-based supplementary crssver (N cnv ).

8 They make pssible t calculate the rate f cnvergence (rate cnv = N N CG cnv ) which illustrates the effect f the new peratr in saving cmputatinal time. Number f flating pint peratins fr bth cases is als registered and respective rate is calculated. In additin the number f imprvements (N impr ) is registered as the number f epchs in which CG i >F(chrm i+ j ); j=,, ps; i=,,,n epchs. This indicates the number f cases in which the imprvements in the fitness functin are due t the new peratr (in all ther cases nrmally used crssver and mutatin leads t the fitness imprvement); ii) search stps after a pre-specified number f epchs (max_n epch ). Then bjective functin values and the number f imprvements (N impr ) are J recrded. The rate f bjective is calculated (rate bj = J CG cnv ), which illustrates imprvements in precisin. 4.. De Jng's functin The simplest test functin is the s-called De Jng's functin. It is cntinus, cnvex and unimdal functin: n x i i= f i (x)= The glbal minimum (J ptimal ) f f(x) = 0 is at x i = 0. The test with n = 30 variables and parameters f GA p c = 0.6, p m = 0.03, ps = 30 has been carried ut fr 60 different runs. The results are shwn in Table 3 and a typical plt f the cnvergence is given n Fig. (In this as well as in all next diagrams slid line represents the case when the new peratr is used, while the dash-dtted line - the case when it is nt used). Frm the results ne can cnclude that the new peratr allws t find the same slutin (J*=0.) fr almst 5% less epchs in average. In 5.8% f epchs imprvement f the fitness is due t the new peratr. Similarly, fr a fixed number (3000) f epchs the result is clser t the glbal minimum f the bjective functin Run Results after J* 0. is reached Results after max_n epchs =3000

9 N N CG N cnv N impr rate cnv rate flps J CG J cnv rate bj N impr Avrg Table 3 DeJng's functin. Tw series f 30 runs Fig. A typical cnvergence in DeJng's functin minimizatin

10 (J ptimal =0) - the value f the bjective functin is almst times less in average with imprvements in 4.% f epchs due t the new peratr. It shuld be mentin that the rate f elementary flating pint peratins is practically the same as the rate f cnvergence, which means that the additinal cmputatinal effrt due t the applicatin f the new peratr is negligible. 4.. Rastrigin's functin Rastrigin's functin has many lcal minima as it uses csine mdulatin. Althugh, the test functin is highly multi-mdal, the minima are regularly distributed. n f i (x)=0n+ ( x i 0 cs(π x)) i= The glbal minimum (J ptimal ) f f(x)=0 is at x i =0. GA parameters used were p m =0.005, p c =0.8, number f bits = 0, ppulatin size =30. Test results fr n=30 variables fr tw series f 30 runs are given in Table 4. A typical cnvergence in bth cases (with and withut using the new supplementary CG-based crssver) are shwn n Fig.3. Similar cnclusins culd be made that 3 times faster cnvergence take place t reach J*=00 with practically n additinal cmputatin effrt; in % f cases imprvement is due t the new peratr and.54 times better result is achieved fr the fixed number (0000) f epchs and imprvements ccur in the same prprtin (.9%) f epchs Sum f different pwers The sum f different pwers is usually used in unimdal tests: n f i (x)= i= i+ x i The glbal minimum (J ptimal ) f f(x)=0 is at x i =0. The test results fr n=30 variables and the fllwing GA parameters (p m =0.0, p c =0.6, ps=60, bits=0) are given in Table 5 and a typical cnvergence is depicted n Fig.4. In this test the advantage f the additin f the new peratr is mst bvius: while J*=0.0 is reached fr 8 epchs in average when it is used, 78 epchs are necessary fr the case when it is nt used, which means

11 Run Results after J* 00 is reached Results after max_n epchs = 3000 N N CG N cnv N impr rate cnv rate flps J CG J cnv rate bj N impr Av Table 4 Rastrigin's functin. Tw series f 30 runs Fig. 3: Rastrigin's Functin Minimizatin

12 Run Results after J* 0.0 is reached Results after 3000 epchs *0-0 N N CG N cnv N impr rate cnv rate flps rate bj J CG J cnv N impr Av Table 5 Different pwers functin. Tw series f 30 runs Fig. 4: Different pwers functin minimizatin

13 50 times mre epchs (!). In each frth epch an imprvement ccur due t the additin f the new peratr. 4 billin times (!!!) better result is achieved fr the same fixed number f epchs (3000) Schwefel's functin Schwefel's functin (Schwefel, 98) determines a gemetrically distant minimum (at x i = ; J ptimal = *n) frm the next best lcal minima. Therefre, the search algrithms ften cnverge in a wrng directin. n f i (x)=0n+ x i sin( x i ) i= The test results fr n=0 variables and p m =0.0, p c =0.6, ps=30, bits=0 are represented in Table 6 and a typical cnvergence is given in Fig.5. The results are quite bvius: the GA which des nt uses the new peratr much mre ften falls int a lcal extremum. In average 5 times less epchs are necessary t reach J*=-6000 and value f the bjective functin better with 56% in average is reached fr the same fixed number f epchs (000). Imprvements ccur in 88% f epchs (!) due t the new peratr. It is interesting t mentin that the standard deviatin in results is 5 times higher fr the case when the new peratr is nt used than when it is used. This culd be explained by the fact that in the case when it is nt used the algrithm relatively ften falls int a lcal extremum Griewangk's functin Griewangk's functin is similar t Rastrigin's functin and has many regularly distributed lcal minimuma which are spread ver the search space. The glbal ne (J ptimal ) is at x i =0; f(x)=0. n n xi xi f i (x)= cs( ) + i= 4000 i= i The test results fr n=30, p m =0.0, p c =0.7, ps=30, bits=0 variables are given in Table 7 and a typical cnvergence is represented n Fig.6. Again, times faster cnvergence with practically n additinal cmputatin effrt has place. In 4% f the epchs imprvements are due t the applicatin f the new peratr. Fr a fixed number f

14 Run Results after J* is reached Results after 000 epchs N N CG N cnv N impr rate cnv rate flps J CG J cnv rate bj N impr Av Table 6 Schwefel's functin. Tw series f 30 runs Fig. 5: Schwefel's Functin Minimizatin

15 Run Results after J* 000 is reached Results after 3000 epchs N N CG N cnv N impr Rate cn rate flps J CG J cnv rate bj rate flps N impr Av Table 7 Griewangk's functin. Tw series f 30 runs Fig. 6: Griewangk's Functin Minimizatin

16 epchs (3000) times better result (in average) is registered and in 7% f epchs imprvements are due t the new peratr. It is interesting t mentin that the standard deviatin f the results f the 30 runs with the new peratr are mre than 3 times smaller (0.09 instead f 0.066) which means that the rle f mutatin and the randmness is less imprtant in this case than when it is nt used. 4. Air-cnditining system ptimizatin The last example represents an practical ptimizatin prblem: t minimize the energy csts in a hllw cre ventilated slab system used as a thermal strage during the night and ff-peak electricity tariff perids such that nt t cmprmise the cmfrt f the ccupants (Ren, 997). The results f applicatin f the new CG-based supplementary crssver peratr tgether with the GA, which des nt uses it are depicted n Fig.7-Fig.0 (all ther parameters, including randm generatr are the same). Supplementary CG-based crssver imprves significantly the cnvergence (Fig.7) as well as the final result: while the thermal cmfrt is practically nt changed, the mre effective (with 6%) slutin is achieved (nrmalized value f the csts is instead f 0.956). The ptimal prfiles f the fan pwer, supply air temperature and flw rate are given n Fig.8 - Fig.0 respectively. The effect is achieved by lwering the supply air flw rate during the mrning precling (Fig.0) while fan is switched n an hur earlier (Fig.9) with lwer pwer. Supply air-temperature during the mrning pre-cling and after wrking evening hurs is lwer (Fig.9). Fig. 7 Ttal csts f energy used Fig. 8 Fan Pwer Prfile

17 Fig. 9 Supply Air Temperature Fig. 0 Supply Air Flw Rate All final results are given in Table 8 with GA parameters used. They illustrate efficiency f the prpsed new CG-based supplementary crssver peratr. Test functins Imprvements effect GA parameters rate cnv N imprve rate bj N imprve p m p c ps De Jng's % % Rastrigin's % % Schwefel's % % Griewangk's % % different pwers % 7*0-8.9% Air cnditining Cst CG = Cst cnv = Table 8 Imprvements effect (summarizing averages frm all tests) 5 Cnclusins The new supplementary crssver peratr is prpsed and tested in the paper. It perfrms specific breeding between the tw fittest parental chrmsmes prducing a new child chrmsme, which is based n the center f gravity f the parental nes. The test results indicate that it leads t -3 times better results than when it is nt used. Withut significant increasing cmputatinal expenses the speed f cnvergence as well as the final result in tests is significantly imprved. A limited prf f its efficiency is

18 prvided fr the case f ne variable and using elitists selectin strategy in cmbinatin with the new peratr. The new peratr culd be used bth in binary as well as in realcded GA. A number f numerical tests as well as a practical example f supply air temperature and flw rate scheduling in a ventilated slab thermal strage system are presented, which prves the viability f the prpsed new peratr. 5 Acknwledgements The authr wuld like t acknwledge the supprt f the EPSRC (by grant GR/M9799) as well as the helpful cmments f Dr. J.A.Wright and Dr. R. Farmani. 6 References. P.P.Angelv, R. Guthke, A GA-based Apprach t Optimizatin f Biprcesses Described by Fuzzy Rules, Jurnal f Biprcess Engineering(997) v.6, pp L.Davis, Adapting Operatr Prbabilities in Genetic Algrithms, Prc. f the Internatinal Cnference n Genetic Algrithms ICGA89 (989) T. C. Fgarty, Varying the Prbability f Mutatin in The Genetic Algrithm, Prc. f the Internatinal Cnference n Genetic Algrithms ICGA89 (989) D.E.Gldberg, Genetic Algrithms in Search, Optimizatin and Machine Learning, Addisn-Wesley, Reading, MA (989) 5. J.J.Grefenstette, Optimizatin f Cntrl Parameters fr Genetic Algrithms, IEEE Transactins. n Systems Man and Cybernetics, 6 () (986) P.Hajela, J.Y, Cnstraint Handling in Genetic Search - A Cmparative Study, Prc. f the American Institute f Aernautics and Astrnautics (995) Matlab, High Perfrmance Numeric Cmputatin and Visualizatin Sftware, MathWrks Inc. (994) 8. Z. Michalewicz, Genetic Algrithms + Data Structures = Evlutin Prgrams, Springer Verlag, Berlin, nd editin (996) 9. H.Muhlenbein, D.Schlierkamp-Vsen, Predictive Mdels fr the Breeder Genetic Algrithm. I. Cntinuus Parameter Optimizatin, Evlutinary Cmputatin, () (993) L.M.Patanik, S.Mandavali, Adaptatin in Genetic Algrithms, In: Genetic Algrithms fr Pattern recgnitin, S.K.Pal, P.P.Wang, Eds.,CRC Press, Bca Ratn, FL (996), 45-64

19 . S.K.Pal, P.P.Wang,Genetic Algrithms fr Pattern recgnitin, CRC Press, Bca Ratn, FL (996). C.Onnen et.al., Genetic algrithms fr Optimizatin in Predictive Cntrl, Cntrl Engineering Practice, 5 (0) (997) M.J.Ren, Optimal Predictive Supervisry Cntrl f Fabric Thermal Strage Systems, Ph.D. Thesis, Lughbrugh University, Lughbrugh, UK (997) 4. Schwefel, H.-P., Numerical Optimizatin f Cmputer Mdels, Jhn Wiley and Sns, Chichester (98) 5. R.Yager, D. Filev, Essentials f Fuzzy Mdeling and Cntrl, Jhn Willey and Sns, NY (994)