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New Multi-parent Crossover based on Crossing Two Segments Bounded by Selected ParentsAtthaphon Ariyarit, Kanazaki MasahiroDepartment of Aerospace Engineering, Graduate School of System Design, Tokyo Metropolitan University6 2014/03/071OutlineIntroductionObjectiveOverview of Popular Crossover OperatorsBlended Crossover (BLX)Unimodal Normal Distribution Crossover (UNDX)New Multi-parent Crossover MethodTest ProblemsSingle-objective optimization problemsMulti-objective optimization problemsMulti-objective airfoil optimization problemResultsConclusions22IntroductionThe Genetic Algorithms(GA) is popular optimization method to solve single-objective and multi-objective optimization problemGA have three main operator, such as, Selection, Crossover and mutationCrossover operator is the main operator for GA performanceThe popular Crossover operator is BLX and UNDX Algorithm[1]The GA is popular in aerospace engineering, such as, to increase the efficiency of aircraft, to used for the navigation of aircraft, or to reduce the weight of the aircraft weight, etc.[1] Hajime K, Isao O and Shigenobu K. Theoretical Analysis of the Unimodal Normal Distribution Crossover for Real-coded Genetic Algorithms. Transactions of the Society of Instrument & Control Engineers, 2002, 2(1), 187-19433IntroductionProcedure of Genetic AlgorithmsGenetic operator such as selection, crossover and mutation are applied to the parent to create the offspringBLX and UNDX cannot always maintain high diversityFor real world problem, the algorithm that can maintain higher diversity, good convergence rate while it shows

44Non-dominated Sorting Genetic Algorithm (NSGA-II)Step 1: Each individual is compared with another randomly selected individual(niche comparison)The copy of the winner is placed in the mating poolStep 2: Apply crossover rate for each individual in a mating pool and select a parentStep 3: All non-dominated fronts of Pt and Qt are copied to the parent population rank by rankStep 4: Stop adding the individuals in the rank when the size of parent population is larger than the population size

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Non-dominated Sorting Genetic Algorithm (NSGA-II)6

ObjectiveDevelopment efficient crossover operator for real world problemInvestigation of the proposed crossover operator by solving the single-objective optimization problem, multi-objective optimization problem and for the real world problem7BLX OperatorCenter pointOffspring1Offspring2Parent1Parent2Possible crossover region8d8UNDX operatorUNDX crossover is a crossover operator for real code Genetic AlgorithmsUNDX is a multi-parent crossover operatorUNDX is operator based on the normal distribution99Algorithm of the UNDX10dD10New Multi-parent Crossover OperatorThe disadvantage of the UNDX is very hard to find the optimal solution close to the boundary and low diversityThe advantage of the UNDX is good convergence rateMaintenance of diversity by the UNDX result is requiredDefinition of proper discover area can maintain high diversity1111Algorithm of the New Multi-parent Crossover 1212Single-Objective Test Function13

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Single-Objective Test Function1414Multi-objective Test Function15

15Multi-objective Test Function16For Multi-Objective optimization use 300 population and 200 generation

16Airfoil Optimization Problem 1717NACA 4 digit Airfoil18

Results-Sphere ProblemSphere ProblemThis graph show the best solution in each generationThe results show proposed method is better solution than BLX and UNDX

1919Results-Sphere ProblemThis graph show the average solution in each generationThis graph can explained the proposed method can maintain higher diversity than BLX and UNDX

2020Results-Sphere21This graph show the design parameter of the best solution in each generation that can show the convergence rateResults-Sphere22This graph show the close up view (from generation 1 to 200) of design parameter of the best solution in each generation that can show the convergence rate

Results-Rastrigin ProblemRastrigin ProblemThis graph show the best solution in each generationThe results show proposed method is better solution than BLX and UNDX

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Results-Rastrigin ProblemThis graph show the average solution in each generationThis graph can explained the proposed method can maintain higher diversity than BLX and UNDX

2424Results-Rastrigin25This graph show the design parameter of the best solution in each generation that can show the convergence rateResults-Rastrigin26This graph show the close up view (from generation 1 to 200) of design parameter of the best solution in each generation that can show the convergence rate

Results-Rosenbrock ProblemRosenbrock ProblemThis graph show the best solution in each generationThe results show proposed method is better solution than BLX and UNDX

2727Results-Rosenbrock ProblemThis graph show the average solution in each generation

2828Results-Rosenbrock29This graph show the design parameter of the best solution in each generation that can show the convergence rateResults-Rosenbrock30This graph show the close up view (from generation 1 to 200) of design parameter of the best solution in each generation that can show the convergence rate

Results-ZDT1

This graph show the Pareto solution of ZDT1 problemThis Pareto solution show the proposed method can get better solution than BLX and UNDX

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Results-ZDT1Metrices of ZDT1These graphs show the proposed method have good convergence rate for ZDT1 problem32HypervolumeMaximum SpreadSpacingResults-ZDT2

This graph show the Pareto solution of ZDT2 problemThis Pareto solution show the proposed method can get better solution than BLX and UNDX

3333Results-ZDT2Metrices of ZDT2These graphs show the proposed method have good convergence rate for ZDT1 problem

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HypervolumeMaximum SpreadSpacing

Results-Airfoil Optimization Problem

This graph show the Pareto solution of airfoil optimization problemThese Pareto solution shows the proposed method can find the optimal solution similar to the BLX algorithm and better than UNDX

3530th generation70th generation35Results-Airfoil Optimization ProblemMetrices of Airfoil Optimization Problem36

HypervolumeMaximum SpreadSpacingResults-Airfoil Optimization Problem37PCP plot of the Pareto solution at 70th generationThe plot shown the propose method have the solution similar to BLX Results-Airfoil Optimization Problem

Left figure show the airfoil comparison and the right figure show the Cp(Pressure Distribution) comparison at design pointCl=0.8 at 70th generationThe plot shown the propose method have the solution similar to BLX

3838ConclusionSuccessful to development of new multi-parent crossover methodThe proposed method successfully find out the better solution of every test function compared with existent crossover methodThe multi-objective test function are better compared with other crossover methodFor the airfoil optimization problem the searching areas are similar because the parameterizations is relatively smile so the investigation by more complex problem is need This method can maintain higher diversity3939