A Sweeping Cones Technique for Post-Pareto Analysis

1. A Sweeping Cones Techniquefor Post-Pareto Analysis Presented by: Victor Carrillo Computational Science Program Advisor: Dr Heidi Taboada Industrial, Manufacturing and Systems Engineering The University of Texas at El Paso 2013 Industrial and Systems Engineering Research Conference

2. Outline • Introduction • Problem description • Previous work • Proposed method: Sweeping cones heuristic for Post Pareto optimality • Numerical example • Conclusions 2013 Industrial and Systems Engineering Research Conference

3. A Multiobjective Optimization Problem 2013 Industrial and Systems Engineering Research Conference

4. Pareto front • Most of evolutionary methods obtain as a solution a set of Pareto optimal solutions based in the Pareto dominance concept .(Figure source Kalyanmoy. Deb Web page) 2013 Industrial and Systems Engineering Research Conference

5. Pareto fronts NSGA-II • Figures source Carlos. Coello Web page 2013 Industrial and Systems Engineering Research Conference

6. Previous work • CompromiseProgramming Approach Yu (1973); Zeleny (1973) • MarginalRate of Substitution Approach Miettinen (1999) • Pseudo-Weight Vector Approach Deb (2004) • Greedy Reduction (GR) algorithm Vandana, et al (2004) 2013 Industrial and Systems Engineering Research Conference

7. Previous work (Cont,d…) • Pruning using k-meansclustering Taboada & Coit (2006) • Non numerical Ranking preferences method Taboada & Coit (2007) • Visualization technique for Pareto Front Blasco, et al (2008). • Local Search with ASF (Achievement Scalarizing function) Padhye, et al (2009) • A Clustering Method Based on Dynamic Self Organizing Trees for Post-Pareto Optimality Analysis Aguirre & Taboada(2011) 2013 Industrial and Systems Engineering Research Conference

8. Sweeping Cones Heuristic Our Method is a combination of: • The non-numerical ranking preferences method • A non-uniform weight generator algorithm • Pruning procedure using sweeping cones 2013 Industrial and Systems Engineering Research Conference

9. Non-numerical ranking preferencesmethod • This is a post-Pareto optimality method proposed and developed by Taboada and Coit (2008). • The Decision maker ranks the objective component functions . • Afterwards based on the objective functions preferences a random weight collection 0< 2013 Industrial and Systems Engineering Research Conference

10. Non-numerical ranking preferencesmethod (Cont, d….) • Optimize the weighting function • 3-dimensional probability density function is developed to obtain the weight values. • ,,)= 2013 Industrial and Systems Engineering Research Conference

11. 3-dimensional probability density function is developed to obtain the weight values. ,,)= 2013 Industrial and Systems Engineering Research Conference

12. Sweeping Cones Heuristic • Normalize as • Normalize as • and are points over the unit sphere surface.(Unit Sphere Wikipedia) 2013 Industrial and Systems Engineering Research Conference

13. Sweeping Cones Heuristic Pseudo code Select a lower boundary  = 0.742≈ b) Randomly generate a collection of weights <1 , “The weight vector is the axis of the cone” c) Compute ) d) Choose such that satisfy α e) Increase gradually the threshold value  (reduce the cone angle) and repeat steps a, b, c and d, until there is no that satisfy α 2013 Industrial and Systems Engineering Research Conference

14. Sweeping Cones HeuristicExample: • MORAP Multiple Objective Redundancy Allocation Problem (Taboada, Coit ,Baheranuala,Wattanapongsakorn.(2007). "Practical solutions for multi-objective optimization: An application to system reliability design problems ",Reliability Engineering and System Safety ,92, 314-322) • The use of redundancy improves system reliability but adds to the system cost, weight, etc • Max reliability; min cost; min weight • Optimal component choices. 2013 Industrial and Systems Engineering Research Conference

15. RAP(*) Pareto front data (NSGA-II) Figure 0 Pareto Front 75 points (*)Taboada, Coit ,Baheranuala,Wattanapongsakorn.(2007). "Practical solutions for multi-objective optimization: An application to system reliability design problems ",Reliability Engineering and System Safety ,92, 314-322 2013 Industrial and Systems Engineering Research Conference

16. Sweeping Cones Heuristic Results 2013 Industrial and Systems Engineering Research Conference

17. Final Post-Pareto Analysis Figure 1.Pareto Front 75 points Figure 2. Table 1 data 2013 Industrial and Systems Engineering Research Conference

18. Results Figure 3. Table 2 data Figure 5. Table 4 data Figure 4. Table 3 data Figure 6. Table 5 data Figure 7. Table 6 data Figure 8. Table 7 data 2013 Industrial and Systems Engineering Research Conference

19. Summary 2013 Industrial and Systems Engineering Research Conference

20. Conclusions • The Sweeping cones method is a double filtering procedure , a sort of “min-max” screening that uses twice the non-numerical preferences method. One for pruning with lower boundaries and second for shrinking the first pruned subsets. • The method reduced in a 99.91% the 75 points RAP Pareto set in an acceptable execution time of one minute per subset, considering that each run was of one million of iterations. • Future work has to be done to evaluate the quality of the solutions obtained. 2013 Industrial and Systems Engineering Research Conference

21. Future work • Test the method in larger multiple optimization problems • Compare algorithm performance against other post-Pareto approaches 2013 Industrial and Systems Engineering Research Conference

22. Questions? 2013 Industrial and Systems Engineering Research Conference