Using Genetic Algorithms to Design Experiments: A Review

Transcription

1 Using Genetic Algorithms to Design Experiments: A Review C. Devon Lin Department of Mathematics and Statistics, Queen s University Joint work with Christine M. Anderson-Cook, Michael S. Hamada, Lisa M. Moore, Randy R. Sitter Design and Analysis of Experiments (DAE) Oct 8, 202

2 Outline Background on genetic algorithms (GAs) Challenges of a good implementation Two examples Discussion 2

3 What we re not talking about Theoretical foundation (Schema Theorem) Theoretical properties of GAs General issues of GAs 3

4 Genetic algorithms (J. Holland, 975) are search and optimization techniques based on Darwin s Principle of Natural Selection. Select The Best, Discard The Rest 4

5 Standard applications in DoE Paper Problem Criterion Approach/Gene Notes [] Safadi&Wang (99) mixed-level OA [2] Govaerts & Sanchez-Rubal (992) Number of unbalanced level pairs column permutation of elements RSM D run crossover exchange, mutation SA exchange, candidate list 6 articles since 990 s Create and select different optimal experiments - response surface models - robust parameter designs - mixed-level OA and D-optimal designs - mixture experiments

6 Specialized applications Paper Problem Notes [7] Cela et al. (2000) Supersaturated experiments [8] Bashir & Simpson (2002) Supersaturated experiments E(S2), n0&m0 criterion, small even run size designs, select columns from balanced 2-level columns E(S2) criterion, select subsets of columns from halffraction of Hadamard matrix 5 articles since 990 s supersaturate experiments computer experiments follow-up design multi-stage experiments degradation tests assembled products fmri experiments microarrays 6

7 Why discuss GAs? Outperform other traditional methods in many problems Flexible implementation (no mathematical analysis is required) When considering a large, complex, non-smooth, poorly-understood problem Alternatives Exchange algorithms Simulated annealing algorithm Tabu search Particle swam optimization No Free Lunch Theorem (Wolpert and Macready,997) 7

8 GA algorithm Initialize population Evaluate fitness Yes Meet stopping criterion No Selection Crossover Mutation New Population Output results 8

9 Key elements in implementing a GA Fitness function Representation Selection Crossover Mutation 9

10 Fitness criterion Problem-specific D,A,G-optimality, orthogonality, Bayesian EIG 0

11 Representation The chromosome represents an individual design and the genes represent runs (columns, blocks) or factor levels Run-based, column-based Should complement the criterion for which the design is being optimized

12 Parents selection Better individuals have larger chance to be selected Roulette wheel selection Elitist selection Tournament selection Scaling selection Rank selection Generational selection Hierarchical selection 2

13 Crossover (for genetic diversity) N-point crossover 3

14 Mutation A mechanism for local search A fine-tuning stage that makes small adjustments around good solutions Use SA, k-exchange, DETMAX Mutation with punctuated equilibrium 4

15 Punctuated equilibrium: periodical mutation rate exp(-mu*mod(g,00))

16 Performance Paper Comparison Competitors Results [] NA OA(2,3^ 2^4) NA [2] NA 9-point exact D-optimal design NA [3] NA D(n,7^ 6^2 5^ 3^2), 25 <=n<=30 Yes [4] MFA Similar design efficiency as MFA, but faster, no new result Yes Time Efficiency many papers show that GA s can nearly achieve or provide modest improvement over the known optimal design or best existing one most of papers address performance but only a few address time efficiency not enough details to reproduce the results 6

17 7 An example from Hamada et al. (200) Consider a three-factor quadratic response model Maximizes Bayesian expected information gain (EIG) Prior specification dy d X y f X y X U ) ( ), ( )], ( log[ ) ( 00 6, ),, ( ~ ),, ( ~ R N IG ), ( ~, I X X X X N X y i i i j i j i ij i i i

18 GA specification Run-based GA Initial population: random uniform numbers Elitist selection, -point crossover to runs Apply mutation to each factor of each run Employ punctuated equilibrium in batches of 00 n 20, p 3, 0.0, M 0 Stop at 900th generations 8

19 Figure. EIG trace for Hamada et al. (200) Example 3 over 900 generations 9

20 Design EIG Points best 900 generations GA design near -.67, 0 and.67 optimal design and -.67 for each factor on the boundary best of 8,00 random

21 2 (Nearly) orthogonal arrays Use J2 optimality (Xu, 2002) m k jk ik k j i n j i j i d d w D D D J, 2, 2 ), ( ) (, )] ( [ ) (

22 Comparisons Row-based GA Row-order-based GA Column-based GA Safadi-Wang (99) Xu (2002) Random balanced designs

23 Row-order-based GA (base-s representation) combine the parent designs and order the combined vector take the rows of odd indexes and even indexes Apply mutation to each factor of each run Stop at 500th generations

24 An example of row-order-based GA D D

25 Relative efficiency comparison for OAs n m s a(i) a(ii) b(i) b(ii) c(i) c(ii) d(i) d(ii) Random Xu (0.23) (0.64).000 (0.2).000 (0.97).000 (0.99).000 (.00).000 (.00) (0.00) (0.04) (0.50).000 (0.59).000 (0.42).000 (0.43) (0.29).000 (0.33).000 (0.99).000 (0.99) a: row-based GA; b: row-order-based GA; c: column-based GA d: Safadi-Wang (99); (I): without punctuated equilibrium (II): with punctuated equilibrium; random: random balanced designs 25

26 Comparison for nearly OAs n m s a(i) a(ii) b(i) b(ii) c(i) c(ii) d(i) d(ii) Random Xu a: row-based GA; b: row-order-based GA; c: column-based GA d: Safadi-Wang (99); (I): without punctuated equilibrium (II): with punctuated equilibrium; random: random balanced designs 26

27 Observations For the criterion, row-based and row-order-based GAs are not natural and column-based GA is more natural. GA is not much better than random search and performs disappointingly so GA is not a panacea Xu s is the best and Safadi-Wang does not perform well Crossover is more random and mutation is more systematic. Punctuated equilibrium does not necessarily improve the performance of GA when the number of generations is small.

28 Fig. 3: J2 of 5-level designs of 00 runs with 20 factors obtained by row-based GA, row-ordered-based GA, column-based GA without punctuated equilibrium (I) and with punctuated equilibrium (II) with the mu =

29 Elements for publications Details about the implementation Stopping rule Comparison of existing designs or those generated by variants of GAs and other competitors 29

30 Concluding remarks Review the use of GAs in DoE Challenges of a good implementation Elusive issues: a representation that achieves the intent of crossover and accounts for isomorphism quantify the separate benefits of crossover and mutation The effect of fitness functions Think hard, Data structure! 30

31 Thank You!

32 Fig. 4: J2 of 5-level designs of 00 runs with 20 factors obtained by row-based GA, row-ordered-based GA, column-based GA without punctuated equilibrium (I) and with punctuated equilibrium (II) with the mu =