Evolutionary Computation: Solution Representation. Set Covering Problem. Set Covering Problem. Set Covering Problem 2/28/2008.

Transcription

1 Evolutionary Computation: Solution Representation Andrew Kusiak 9 Seamans Center Iowa City, Iowa - Tel: 9-9 Fax: andrew-kusiak@uiowa.edu Set Covering Problem Min Σ cjxj Scheduling Process planning Reference Σ aij >= for all lli, j xj = 0, for all j Set Covering Problem [x] x x x x x [aij] [c] 8 Decision variable Matrix Cost Set Covering Problem [x] x x x x x [aij] [c] 8 Mathematical programming representation x =, x = 0, x =, x =, x = 0

2 Set Covering Problem [x] x x x x x [aij] Placing n objects into at most n bins [c] 8 Genetic representation [, 0,,, 0] Container packing Space shuttle spatial layout CD ROM packing Min Σ yj Σwjxij <= cyi all i Σxij = all j yi = 0, xij = 0, wj = weight of object j c = bin capacity xij = if object j assigned to bin i, = 0 otherwise yi = if bin i is used, = 0 otherwise Bin-based representation 6 Bin Chromosome in Bin, in Bin, in Bin,..., 6 in Bin

3 Bin-based Representation Equal length chromosomes; use of standard generic operators Bin number Item No. 6 Redundant representation, e.g., and are identical solutions (for equal capacity containers) In both cases items and 6 are stored in one container; -based representation Bin Bin Bin 6 Chromosome Bin - the first three objects Bin - the next three objects Bin - the last object -based Representation Equal length chromosomes; use of standard generic operators Redundant representation, e.g., 6, 6, and 6 are identical solutions Group-based representation 6 A D B C E B Bin Chromosome Using the above equivalence scheme, the chromosome ADBCEB implies A = {}, B = {, 6}, C = {}, D = {}, and E ={}

4 Group-based Representation Group-based Representation ADBCEB implies A = {}, B = {, 6}, C = {}, D = {}, and E ={}, represented as Most suitable of all three methods Genes represent both objects and groups (bins),6 B E C D A Bin Chromosomes of variable length Clustering Problem Clustering Problem Features p Group Feature Group Multitude of formulations, ranging from binary matrix to mathematical programming implies Many possible solution representations in genetic programming

5 Order-based Representation () Consider matrix with three clusters, and seven features (or objects) Note: number of object clusters = number of feature clusters [ 6 ] Chromosome Order-based Representation () Mutation: Random permutation Crossover*: Partially mapped crossover (PMX) Order crossover (OX) Cycle crossover (CX) Recombination crossover (ER) * Reference Project Scheduling () Project Scheduling () Position: Activity ID 6 Value: Priority of activity Project Activity Network

6 Project Scheduling () Project Scheduling () Topologically sorted graph Level : Level :,, Level :, 6 Level : Triangularization algorithm Triangularization algorithm Project Scheduling () Project Scheduling 6() Topologically sorted graph Topologically sorted graph 6

7 Project Scheduling () Position-based Crossover Parent Child Parent Random genes from one parent are transferred to a child Missing genes are filled left-to-right from the other parent Topologically sorted graph Parent Swap Mutation 6 Local Search-based Mutation Parent chromosome 6 Pivot gene 6 (, ) Child 6 Neighborhood 6 (, ) (, ) Pairwise gene exchanges of the parent chromosome 6 (, 6)

8 Fitness Function ive function = Min finish time of the last network activity Min problem transformed to a Max problem so that the fitter individuals correspond higher value of the fitness function Fitness Function fmax, fmin = max, min value of the objective function in the current generation f = objective function value in the current generation γ = real positive number in (0, ) Fitness function g g = (fmax - f + γ)/(fmax - fmin + γ) The selection method based on the fitness function g changes with γ from proportional to random References Ref. : M. Gen and R. Cheng, Genetic Algorithms and Engineering Optimization, John Wiley, New York, 000. Ref. : J. Wang and A. Kusiak, Computational Intelligence in Manufacturing Handbook, CRC Press, 00. Ref. : W. Banzhaf, P. Nordin, R.E. Keller, and Frank D. Framcone, Genetic Programming, Morgan Kaufmann, San Francisco, CA,