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On Extended Compact Genetic Algorithm

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In this study we present a detailed analysis of the extended compact genetic algorithm (ECGA). Based on the analysis, empirical relations for population sizing and convergence time have been derived ...

In this study we present a detailed analysis of the extended compact genetic algorithm (ECGA). Based on the analysis, empirical relations for population sizing and convergence time have been derived and are compared with the existing relations. We then apply ECGA to a non-azeotropic binary working fluid power cycle optimization problem. The optimal power cycle obtained improved the cycle efficiency by 2.5% over that existing cycles, thus illustrating the capabilities of ECGA in solving real-world problems.

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    On Extended Compact Genetic Algorithm On Extended Compact Genetic Algorithm Presentation Transcript

    • On Extended Compact Genetic Algorithm (ECGA) Kumara Sastry David E. Goldberg Illinois Genetic Algorithms Laboratory University of Illinois at Urbana-Champaign http://www-illigal.ge.uiuc.edu Genetic and Evolutionary Computation Conference July 8-12, 2000 Las Vegas, Nevada
    • On ECGA 1 Outline • Motivation • Overview of ECGA • Test problems: Onemax, Trap, Folded Trap • Results on test problems • A simple power cycle • Results on power cycle optimization • Conclusions Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 8-12, 2000 University of Illinois at Urbana-Champaign K. Sastry, D.E. Goldberg Urbana, IL 61801. USA.
    • On ECGA 2 Motivation • Background – Increased interest in competent GAs – An interesting competent GA is Harik’s ECGA ∗ builds models of good data as linkage groups – Under-investigated ∗ Does it match GA theory? Not tried on real problems. • Rhetorical Purpose – Overview of ECGA – Investigate connection to quality-duration theory – Apply to practical problem Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 8-12, 2000 University of Illinois at Urbana-Champaign K. Sastry, D.E. Goldberg Urbana, IL 61801. USA.
    • On ECGA 3 Overview of ECGA • Key Idea: Good probability distribution ≡ Linkage learning • Probability distribution: Marginal Product Models (MPM) • Quantified based on Minimum Description Length (MDL) • MDL Concept: Simpler distributions are better Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 8-12, 2000 University of Illinois at Urbana-Champaign K. Sastry, D.E. Goldberg Urbana, IL 61801. USA.
    • On ECGA 4 Flowchart of ECGA Np = Population size Initialize Evaluate Perform s = Tournament size population fitness of tournament randomly Pc = Crossover probability chromosomes selection Create Np*Pc chromosomes by crossover Has Build MPM No population model using converged MDL Transfer Np*(1-Pc) best individuals to next generation Yes End Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 8-12, 2000 University of Illinois at Urbana-Champaign K. Sastry, D.E. Goldberg Urbana, IL 61801. USA.
    • On ECGA 5 Marginal Product Models • Product of marginal distributions on a partition of genes • Similar to CGA (Harik et. al;1998) and PBIL(Baluja;1994) • Represent more than one gene in a partition • Make exposition simpler • Gene partition maps to linkage groups Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 8-12, 2000 University of Illinois at Urbana-Champaign K. Sastry, D.E. Goldberg Urbana, IL 61801. USA.
    • On ECGA 6 Minimum Description Length Models • Hypothesis: Good distributions are those for which – Representation of the distribution is minimum (model complexity, Cm ). – Representation of population compressed is minimum (compressed population complexity, Cp ). • Penalize complex models • Penalize inaccurate models • Combined complexity, Cc = Cm + Cp Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 8-12, 2000 University of Illinois at Urbana-Champaign K. Sastry, D.E. Goldberg Urbana, IL 61801. USA.
    • On ECGA 7 Building MPM using MDL Uses a steepest ascent search: 1. Assume all the genes to be independent ([1],[2],· · ·,[L]) and compute Cc . 2. Form all possible combinations (Nbb (Nbb -1)/2) of merging two subsets. eg., ([1,2],[3],· · ·,[L]), · · ·, ([1],[2],[3],· · ·,[L-1,L]). 3. Select the set with minimum combined complexity (Cc ). 4. If Cc > Cc go to step 6. 5. Use the set with Cc as the current MPM and go to step 2. 6. Merging is not possible, exit with set from step 2 as MPM. Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 8-12, 2000 University of Illinois at Urbana-Champaign K. Sastry, D.E. Goldberg Urbana, IL 61801. USA.
    • On ECGA 8 Generation of New Population • Transfer Np *(1-Pc ) best individuals to the next generation • The rest Np *Pc individuals are generated as follows: – Take each of the subset of MPM from one of the individuals. – Similar to multiple point crossover. – Number of crossover points = Nbb . – Instead of two parents we have Nbb parents. Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 8-12, 2000 University of Illinois at Urbana-Champaign K. Sastry, D.E. Goldberg Urbana, IL 61801. USA.
    • On ECGA 9 Test Problems Trap Folded Trap I Folded Trap II One-Max 5 5 5 5 4 4 4 4 Obj Value Obj Value Obj Value Obj Value 3 3 3 3 2 2 2 2 1 1 1 1 1234 1 234 1 2345 6 1 2345 No. of ones No. of ones No. of ones No. of ones • Uniformly scaled, building blocks (BBs) are known • Trap, FT-I, FT-II: 10 BBs, Onemax: 40,50,60 bits • Tournament sizes: 2-20, Population Size = 50-25000 Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 8-12, 2000 University of Illinois at Urbana-Champaign K. Sastry, D.E. Goldberg Urbana, IL 61801. USA.
    • On ECGA 10 Results: Building Block Identification 16 Trap 23 Fold Trap I Fold Trap II 14 21-22 No. of generations for an optimal chromosome 20 12 11 19 10 18 Generation 9 17 Generation 8 15-16 7 12-14 6 11 x = y line 4 10 5 4-9 2 1 2 3 456 7 8 9 10 1 2 3 456 7 8 9 10 0 0 1 2 3 4 5 6 7 8 9 10 Building Block Building Block No. of generations for identifying all BBs • BB identification is necessary • BB identification modes: Flash and Sequential • Flash: smaller BB size, large tournament sizes • Sequential: larger BB size, small tournament sizes Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 8-12, 2000 University of Illinois at Urbana-Champaign K. Sastry, D.E. Goldberg Urbana, IL 61801. USA.
    • On ECGA 11 Results: Onemax BB Identification 4 4 x 10 3 L = 40 Cp,ideal−Cp,actual L = 50 C −C L = 60 m,actual m,ideal 3.5 2.5 Ideal and actual metric value difference s = 20 3 2 Average BB size 2.5 1.5 s = 20 s = 10 2 s = 10 1 1.5 0.5 s=2 s=2 1 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Population size, N Population size, Np p (a) (b) • Apparent linkage or hitchhiking • A good random number generator reduces linkage Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 8-12, 2000 University of Illinois at Urbana-Champaign K. Sastry, D.E. Goldberg Urbana, IL 61801. USA.
    • On ECGA 12 Results: Convergence Time 30 1 Calc 40 Calc 50 Calc 60 Onemax 40 Trap 4 0.9 Onemax, s=5 25 Onemax 50 Trap, s=5 Trap 5 Fold Trap I, s=5 Fold trap II Onemax, s=10 Onemax 60 Scaled average fitness (f/fmax) Trap, s=10 Trap 6 0.8 Fold Trap I, s=10 Fold trap I Onemax, s=20 20 Trap, s=20 Generation Fold Trap I, s=20 0.7 15 0.6 10 0.5 0.4 5 0 2 4 6 8 10 12 14 2 4 6 8 10 12 14 16 18 20 Generation Tournament Size log2 L + log2 (log2 L) tconv = 1 + kc (L < 100) log2 s √ πL tconv = (L > 100) 2I Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 8-12, 2000 University of Illinois at Urbana-Champaign K. Sastry, D.E. Goldberg Urbana, IL 61801. USA.
    • On ECGA 13 Results: Population Sizing 9000 4 1200 x 10 4.5 Onemax 40 pred lbb = 1 Onemax 50 pred BB size 4 l =4 bb Onemax 60 pred 8000 BB size 5 l =5 Onemax 40 expt bb 4 BB size 6 l =6 1000 Onemax 50 expt Trap 4 bb Onemax 60 expt Trap 5 7000 Trap 6 3.5 Fold trap II Fold trap I 6000 800 3 Model Complexity, Cm p Population Size, N Population size, Np 5000 2.5 600 4000 2 3000 400 1.5 2000 1 200 1000 0.5 0 0 0 0 0.5 1 1.5 2 2.5 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 Population Size, N 4 Tournament Size, s Tournament size, s p x 10 Nbb 2lbb − 1 min {0.1515, c1 s + c2 } Np = = log(2) 3.5 + 0.5 (lbb − 1) Similar to equations by (Goldberg et. al;1992, Pelikan et. al; 2000) Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 8-12, 2000 University of Illinois at Urbana-Champaign K. Sastry, D.E. Goldberg Urbana, IL 61801. USA.
    • On ECGA 14 A Simple Power Cycle ∆ T = 10 x = 0.992 Tg,in = 455 K F wf = 0.827 kg/s Fg C pg= 10 kW/K Turbine T2 = 363.4 K T1 = 445 K η = 0.9 P = 9.38 bar T w,out = 308.6 K Annual profit = $143,319 2 P1 = 25.98 bar Cycle Efficiency = 9.9% Annual profit (Ibrahim,1996)= $99,163 Heater + Boiler Condenser Cycle Efficiency + (Ibrahim, 1996) = 9.7% Superheater T w,in = 286 K T0 = 296.9 K Fw C pw = 50 kW/K T g,out = 328.5 K P0 = 25.98 bar Pump η = 0.7 T3 = 296.2 K P3 = 9.38 bar Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 8-12, 2000 University of Illinois at Urbana-Champaign K. Sastry, D.E. Goldberg Urbana, IL 61801. USA.
    • On ECGA 15 Power Cycle Problem Formulation ˙ ˙ ˙ • Maximize profit: Ct Wt − Cp Wp − Cw Qc • 2nd law of thermodynamics: Tcold ≤ Thot − ∆Tmin • Fluid in/out of turbine must be a vapor • Fluid out of condenser and in/out of pump must be a liquid • Constraints are handled by penalty method • Highly nonlinear & complex thermodynamics (Ibrahim and Klein; 1993) Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 8-12, 2000 University of Illinois at Urbana-Champaign K. Sastry, D.E. Goldberg Urbana, IL 61801. USA.
    • On ECGA 16 Results:Power Cycle Optimization ∆ T = 10 x = 0.992 Tg,in = 455 K F wf = 0.827 kg/s Fg C pg= 10 kW/K Turbine T2 = 363.4 K T1 = 445 K η = 0.9 P = 9.38 bar T w,out = 308.6 K Annual profit = $143,319 2 P1 = 25.98 bar Cycle Efficiency = 9.9% Annual profit (Ibrahim,1996)= $99,163 Heater + Boiler Condenser Cycle Efficiency + (Ibrahim, 1996) = 9.7% Superheater T w,in = 286 K T0 = 296.9 K Fw C pw = 50 kW/K T g,out = 328.5 K P0 = 25.98 bar Pump η = 0.7 T3 = 296.2 K P3 = 9.38 bar Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 8-12, 2000 University of Illinois at Urbana-Champaign K. Sastry, D.E. Goldberg Urbana, IL 61801. USA.
    • GECCO, July 8-12, 2000 17 K. Sastry, D.E. Goldberg ££££££££££££££££££££§£§£££§£§£££§£§£££§ §§§§§§§§§§§§§§§§§§§§ §§ §§ §§ 45 45 55 45 Results:Power Cycle Optimization  ¡ 3 T 67 50 67 ¥¦ 67 ¥¦ ¥¦ ¥¦ ¥¦ ¥¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ££££££¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥££¥£¥£££¥£¥£££¥£¥£££¥£ ££££££££££££££££££¥¦£££¥¦£¥¦£££¥¦£¥¦£££¥¦£¥¦££¥¦ ¦¦ ¦¦ ¦ 23 45 23 ¨©© 23 ¨¨ 40 1 T 01 01 35 &'01 On ECGA ££££££££££££££££££¢¤£¢¤£££¢¤£¢¤£££¢¤£¢¤£££¢¤£¢¤£¢¤ ££££££¢£¢£¢£¢£¢£¢£¢£¢£¢£¢£¢£¢£££¢£¢£££¢£¢£££¢£¢£££ ¢¤ &' ¢¤ ¢¤ ¢¤ ¢¤ ¢¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤¤ ¤¤ ¤¤ &' BC£BC £ 30 Bit no. 2 P () () 25 $%() @A£@A £ $%() $% 20 "# $% 0 P "# 15 "#   ED ED University of Illinois at Urbana-Champaign ££££££F££F££F££F££  £  F££F£FG£F££FG£DE FG£DE F££FG£FG£££FG£FG££FG £££££££FG££FG££FG££F£FG££FG££££FG££££F£££F£F£££F£ FG  FG FG FG FG FG G G G G G G G G GG GG G Illinois Genetic Algorithms Laboratory   R£ S£RS Q£Q Department of General Engineering ! 89 x 10 P£HIP ! Urbana, IL 61801. USA.     5  wf F TU   0 1 30 20 10 5 21-29 11-18 Generation
    • On ECGA 18 Conclusions • Important insights into working of ECGA • Empirical relations for convergence time and population sizing • Effective in solving practical problem Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 8-12, 2000 University of Illinois at Urbana-Champaign K. Sastry, D.E. Goldberg Urbana, IL 61801. USA.
    • On ECGA 19 Acknowledgments • Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grants F49620-94-1-0103, F49620-95-1-0338, F49620-97-1-0050, and F49620-00-0163. • National Science Foundation under grant DMI-9908252. • U. S. Army Research Laboratory under the Federated Laboratory Program, Cooperative Agreement DAAL01-96-2-0003. Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 8-12, 2000 University of Illinois at Urbana-Champaign K. Sastry, D.E. Goldberg Urbana, IL 61801. USA.