1. A Genetic Algorithm Approach to Multiple Response Optimization Francisco Ortiz Jr. James R. Simpson Joseph J. Pignatiello, Jr. Alejandro Heredia-Langner

2. Department of Industrial Engineering Florida A&M and Florida State University Many industrial problems involve many response solutions –“It is not unusual to find RSM applications with 20 or more responses…” (Montgomery, JQT, 1999) Current methods don’t always work Work by Heredia-Langner, and suggestions from Carlyle, Montgomery and Runger (JQT, 2000) Motivation

3. Department of Industrial Engineering Florida A&M and Florida State University Composites Production INPUTS (Factors)OUTPUTS (Responses) PROCESS: Resin Transfer Molding Resin Flow Rate Mold Complexity Gate Location Type of Resin Fiber Weave Fiber Weight Resin Flow Properties Tensile Strength Fiber Permeability Dynamic Mechanical Analysis Compression Strength Flexural Strength Acoustic Properties

4. Department of Industrial Engineering Florida A&M and Florida State University Charge Control Agents Charge Voltage Colorants Release Agents Transfer Voltage Surface Additives Laser Toner Development INPUTS (Factors)OUTPUTS (Responses) PROCESS: Toner Transfer Background Developer Roll Mass/Area Spitting Powder Flow Isopel Optical Density Toner-to-Cleaner Film Onset  24 responses

5. Department of Industrial Engineering Florida A&M and Florida State University As the number of response and decision variables increases –The combined response function typically can be highly nonlinear, multi-modal and heavily constrained –Conventional optimization methods can get trapped at a local optimum and even fail to find feasible solutions Challenges for Current Multiple Response Optimization Methods

6. Department of Industrial Engineering Florida A&M and Florida State University This paper develops and evaluates a multiple response solution technique using a genetic algorithm in conjunction with an unconstrained desirability function Purpose

7. Department of Industrial Engineering Florida A&M and Florida State University A desirability function for the genetic algorithm “Robust-ising” the genetic algorithm for multiple response optimization Performance evaluation and comparison Topics to Cover

8. Department of Industrial Engineering Florida A&M and Florida State University Most methods convert multiple responses with different units of measurements into a single commensurable objective Distance or Loss Functions –Khuri and Conlon, 1981 –Pignatiello, 1993 –Vining, 1998 Desirability Methods –Derringer and Suich, 1980 –Del Castillo, Montgomery, and McCarville, 1996 Desirability approach converts individual response ŷ i into individual desirability d i (ŷ i ) Used in combination with optimization algorithm to locate a single solution Combining Responses

9. Department of Industrial Engineering Florida A&M and Florida State University For example, target as the goal response Constraint developed based on response’s utility 0 < d i (ŷ i ) < 1 Overall desirability Optimization using Nelder-Mead simplex or GRG Desirability Approach sisi titi

10. Department of Industrial Engineering Florida A&M and Florida State University A genetic algorithm (GA) is a search technique that is based on the principles of natural selection –“survival of the fittest” Uses the objective function magnitude directly in the search The GA generates and maintains a population pool of solutions throughout It then evaluates the quality of each individual chromosome using a fitness function –In this case the desirability function = fitness function Genetic Algorithm

11. Department of Industrial Engineering Florida A&M and Florida State University A designed experiment produces an empirical system model A coding scheme widely used is to transform the natural variables into coded variables that fall between -1 and 1 Here we have multiple fitted responses Using the GA in Response Surface Studies Chromosome Gene

12. Department of Industrial Engineering Florida A&M and Florida State University Multiplicative desirability functions –Overall desirability is –If any d i (x) = 0, D(x) = 0 –The GA is unable to compare infeasible solutions –Perhaps we could extend the D(x) function Limitations of Current Desirability for the GA

13. Department of Industrial Engineering Florida A&M and Florida State University GA can be designed to handle constraint violations by using a penalty method where Formulating a Desirability Function

14. Department of Industrial Engineering Florida A&M and Florida State University Every fitted response has an individual desirability d i (ŷ) and a penalty p i (ŷ) where c is a small constant (e.g. c= 0.0001) Formulating a Desirability Function

15. Department of Industrial Engineering Florida A&M and Florida State University Hence, the overall unconstrained desirability function is The penalty function enables the GA to find feasible solutions After feasible solutions are found, P(x) = 0, and no effect on the D*(x) Formulating a Desirability Function

16. Department of Industrial Engineering Florida A&M and Florida State University Overall GA desirability function for a case with one response and linear weights (s 1 = t 1 = 1) for the d(y 1 ) Unbounded Desirability Function D*D*

17. Department of Industrial Engineering Florida A&M and Florida State University Proposed desirability function D*(x) vs. D DS (x) Desirability Function Investigation

18. Department of Industrial Engineering Florida A&M and Florida State University Proposed desirability function D*(x) vs. D DS (x) GA Performance Investigation

19. Department of Industrial Engineering Florida A&M and Florida State University The GA can be sensitive to the many parameter choices that must be made for its application The GA parameters considered for this study are –Population size –Parent-to-Offspring ratio –Selection type –Mutation type –Mutation rate –Crossover rate Tuning the GA Parameters

20. Department of Industrial Engineering Florida A&M and Florida State University Done by incorporating four problem environment parameters incorporated into a designed experiment framework Considered noise variables for the purposes of robust design Creating Multiple Response Problem Scenarios

21. Department of Industrial Engineering Florida A&M and Florida State University Certain rules were used to ensure that the problems generated mimic what typically is found in a real world situation Specifications –The decision variables that appear in each response are chosen randomly –For second order models, ½ of the terms are linear (main effects), ¼ of the terms are interactions, ¼ of the terms are pure quadratic –The exact values of the regression coefficients are selected from a uniform probability distribution U(5, 20) –Model hierarchy is always maintained Creating Multiple Response Problem Scenarios

22. Department of Industrial Engineering Florida A&M and Florida State University For example, a case with 4 responses and 4 decision variables, 75% of the responses are second order models Creating Multiple Response Problem Scenarios

23. Department of Industrial Engineering Florida A&M and Florida State University Robust Designed Experiment Robust parameter design using a combined array 2 10-4 resolution IV (80 runs includes 16 pseudo-centers)

24. Department of Industrial Engineering Florida A&M and Florida State University Robust Designed Experiment Results GA performance metrics were the number of evaluations until –Feasible –Within 10% of optimal, or D*(x) > 0.90 Response model of the form Significant effects included –Noise factor main effects –Control x control interactions –Control x noise interactions

25. Department of Industrial Engineering Florida A&M and Florida State University Robust Designed Experiment Results Response: Achieving D*(x) > 0.90

26. Department of Industrial Engineering Florida A&M and Florida State University Robust GA Parameter Settings All but one determined by response surface models

27. Department of Industrial Engineering Florida A&M and Florida State University Performance Evaluation Evaluate and compare proposed GA method Considered the multiple response problem scenarios –Dropped percentage of second order models –2 3 factorial plus a center point

28. Department of Industrial Engineering Florida A&M and Florida State University Results of investigation using 30 starting point locations GRG Performance

29. Department of Industrial Engineering Florida A&M and Florida State University Performance of proposed GA using four replicates GA Performance Investigation

30. Department of Industrial Engineering Florida A&M and Florida State University Current study shows consistent, effective performance Can be effectively combined with direct search methods (e.g. GA with GRG) to improve run-time performance Flexible to handle a host of objective functions –Distance or loss functions can all be applied as long as covariance information is available Able to perform reasonable mapping of response function over design space Why Use the GA for Multiple Response Optimization?

31. Department of Industrial Engineering Florida A&M and Florida State University 2.4 Genetic Algorithm –Coding 00001010000110000000011000101010001110111011 is partitioned into 2 halves 0000101000011000000001 and 1000101010001110111011 these strings are then converted from base 2 to base 10 to yield: x 1 = 165377 and x 2 = 2270139 The following is an example of real-value coding 0.125 -1.000 0.525

32. Department of Industrial Engineering Florida A&M and Florida State University Recombination –Exchange information/genes between parent chromosome to make new offspring Parent 1 250085 1780 103 1200 95 1900 98 2500 168 500 40 Parent 2 185053 2200 99 785 67 1000 102 750 75 900 69 Offspring 2500 85 1780 103 1200 95 1900 98 750 75 900 69 Crossover Point GA cannot rely on recombination alone.

33. Department of Industrial Engineering Florida A&M and Florida State University Mutation –Uniform mutation –Multiple uniform mutation –Guassian mutation All entities of the chromosome are mutated such that the resulting chromosome lies somewhere within the neighborhood of it parent. X1X1 X2X2