1. The W, C, & H Methods 4 Generating Pareto Solutions Prepared by: MA A-Sultan March 20, 2006 Reference: Entropic Vector Optimization and Simulated Entropy: Theory & Applications; Ph.D. Thesis, Liverpool University, 1991. http://library.liv.ac.uk/search/a?searchtype=t&searcharg=Entropic+Vector+Optimization&searchscope=0&S ORT=&SUBMIT=Search

2. MO Optimization MO Optimization Methods The Weighting Method A 1 st standard technique for MOO is to optimize a positively weighted convex sum of the objectives, that is, Optimize Non-ConvexWHY? Not adequate alone! Non-Convex Pareto solutions cant be generated; WHY? bi-objective For bi-objective example, will create a table like this: 0.8, 0.2 1.0, 0.0 0.0, 1.0 0.2, 0.8 0.35, 0.65 0.225, 0.775 Etc. Cause Headache? What if you have 3, 5, … 25 objs.?

3. Convexity & Concavity F2(X*) F1(X*)

4. Convexity & Concavity

5. The Weighting Method -w1/w2 Weights dont reflect objective importance at all!

6. MO Optimization MO Optimization Methods The Constraint Method A 2 nd standard technique for MOO is to optimize one objective while setting remaining ones to variable feasible bounds, that is: Optimize S.t: Values must be chosen so that feasible solutions to the resulting single- objective problem exist. Cause Headache? What if you have 3, 5, … 25 objs.? weights Virtually, there are weights here too!

7. The Constraint Method Effec. Cost Effec Cost Parerto Specifying a bound on Effec., defines a new feasible region which does allow Cost to be minimized to find a Parerto point.

8. Generation of Pareto Solutions by Entropy-based Methods Reference: Entropic Vector Optimization and Simulated Entropy: Theory & Applications; pp.(115 – 118), Ph.D. Thesis, Liverpool University, 1991. http://library.liv.ac.uk/search/a?searchtype=t&searcharg=Entropic+Vector+Optimization&searchscope=0&SORT=&SUBMIT=Search

9. The Entropy-based Weighting Method The Weighting Method The Entropy-based Weighting Method V = U = We have proved that We have proved that: where is an optimal solution Reference: Entropic Vector Optimization and Simulated Entropy: Theory & Applications; pp.(115 – 118), Ph.D. Thesis, Liverpool University, 1991. http://library.liv.ac.uk/search/a?searchtype=t&searcharg=Entropic+Vector+Optimization&searchscope=0&SORT=&SUBMIT=Search Weights follow Boltzmann Distribution!

10. The Entropy-based Constraint Method The Constraint Method The Entropy-based Constraint Method Reference: Entropic Vector Optimization and Simulated Entropy: Theory & Applications; pp.(115 – 118), Ph.D. Thesis, Liverpool University, 1991. http://library.liv.ac.uk/search/a?searchtype=t&searcharg=Entropic+Vector+Optimization&searchscope=0&SORT=&SUBMIT=Search Weights follow Boltzmann Distribution!

11. Bi-Obj Optimization Bi-Obj Optimization Example 1 CONCAVE No more than 2 solutions can be generated since those in- between are CONCAVE! Better Representative Pareto Set Reference: Entropic Vector Optimization and Simulated Entropy: Theory & Applications; pp.(115 – 118), Ph.D. Thesis, Liverpool University, 1991. http://library.liv.ac.uk/search/a?searchtype=t&searcharg=Entropic+Vector+Optimization&searchscope=0&SORT=&SUBMIT=Search

12. Bi-Obj Optimization Bi-Obj Optimization Example 2 GAP Grouped Pareto set with a large GAP between the two! Better Representative Pareto Set Reference: Entropic Vector Optimization and Simulated Entropy: Theory & Applications; pp.(119 – 122), Ph.D. Thesis, Liverpool University, 1991. http://library.liv.ac.uk/search/a?searchtype=t&searcharg=Entropic+Vector+Optimization&searchscope=0&SORT=&SUBMIT=Search

13. Bi-Obj Optimization Bi-Obj Optimization Example 3 MISSING Large portion of Pareto set is MISSING! shapewhole The shape of the whole Pareto set is generated Reference: Entropic Vector Optimization and Simulated Entropy: Theory & Applications; pp.(123 – 128), Ph.D. Thesis, Liverpool University, 1991. http://library.liv.ac.uk/search/a?searchtype=t&searcharg=Entropic+Vector+Optimization&searchscope=0&SORT=&SUBMIT=Search

14. Bi-Obj Optimization Bi-Obj Optimization Example 4 1) A to B Pareto set has no GAP 2) B to C Pareto set has large GAP! 1)B to C Pareto set is better representative! Reference: Entropic Vector Optimization and Simulated Entropy: Theory & Applications; pp.(130 – 134), Ph.D. Thesis, Liverpool University, 1991. http://library.liv.ac.uk/search/a?searchtype=t&searcharg=Entropic+Vector+Optimization&searchscope=0&SORT=&SUBMIT=Search A B C A B C

15. MOPGP94 Report Generation of Pareto Solutions by Entropy- based Methods (MOPGP94 Report) Portsmouth, UK, 1994 * http://www.amazon.com/gp/product/3540606629/qid=1143034880/sr=1-5/ref=sr_1_5/002-0382939-5305662?s=books&v=glance&n=283155 pp. 164 - 194http://www.amazon.com/gp/product/3540606629/qid=1143034880/sr=1-5/ref=sr_1_5/002-0382939-5305662?s=books&v=glance&n=283155 interesting The authors present in this paper a method to compute the set of Pareto points for some multi-valued optimization problems. The paper is interesting and it must be published* since: new idea 1) The Authors apply the entropy method to the study of Pareto points which is a new idea used also by other authors to the study of scalar optimization problems; important aspect 2) The method presented by the authors can be used to compute not only a Pareto point but the set of Pareto points, which is an important aspect; interesting 3) The applications to engineering problems are interesting.

16. Hybrid Method Generation of Pareto Solutions by the Hybrid Method

17. Hybrid Method The Hybrid Method Optimize Subject to:

18. W, C, & H Methods The Entropy-based W, C, & H Methods W-Method Concave The W-Method finds NO Concave solutions C-Methods Convex & Concave The C-Methods finds Convex & Concave solutions Both Need to be used! H-Method powerful yet rarely addressed! The H-Method is far more powerful than the W-Method and C- Method; yet rarely addressed! 3 MethodsNO Problem All 3 Methods have NO Problem with # of objectives

19. How the W, C, & H Methods were Incorporated into GAME ACSOM Prepared by: Michael M Sultan March 28, 2006

20. MO Optimization MO Optimization Methods The Weighting Method The Weighting Method A 1st standard technique for MOO is to optimize a positively weighted convex sum of the objectives, that is: The Constraint Method The Constraint Method A 2nd standard technique for MOO is to optimize one objective while setting remaining ones to variable feasible bounds, that is: OptimizeOptimize

21. The Weigh & Constrain Method

22. How The Weigh & Constrain Method Work?