1. INNOVIZATION-Innovative solutions through Optimization Prof. Kalyanmoy Deb & Aravind Srinivasan Kanpur Genetic Algorithm Laboratory (KanGAL) Department of Mechanical Engineering Indian Institute of Technology Kanpur

2. March 10, KanGAL2 Innovization Identification of commonalities amongst optimal solutions or Knowledge discovery. –Optimal Solutions satisfy - KKT conditions. –Single Objective optimization  No global information about any property that the optimal solutions may carry.  No flexibility for the decision maker. –Multi-Objective Optimization  Need for Evolutionary Algorithms(GA)  NSGA-2: Established Algorithm for EMO

3. March 10, KanGAL3 EMO Principle: Find multiple Pareto- optimal solutions simultaneously Three main reasons: For a better decision- making For unveiling salient optimality properties of solutions For assisting in other problem solving

4. March 10, KanGAL4 Potentials Better Understanding of the problem.Better Understanding of the problem. Reduces Cost.Reduces Cost. Eliminates the need for new optimization for small change in parameters.Eliminates the need for new optimization for small change in parameters. Deciphers innovative ideas for further design.Deciphers innovative ideas for further design. Benchmark Designs for industriesBenchmark Designs for industries.

5. March 10, KanGAL5 Innovization Procedure Choose two or more conflicting objectives (e.g., size and power) Usually, a small sized solution is less powered Obtain Pareto-optimal solutions using an EMO Investigate for any common properties manually or automatically

6. March 10, KanGAL6 Multi-Disk Brake Design Minimize brake massMinimize brake mass Minimize stopping timeMinimize stopping time 16 non-linear constraints16 non-linear constraints 5 variables: Discrete (ri,ro,t,,F,Z)5 variables: Discrete (ri,ro,t,,F,Z) ri in 60:1:80,ri in 60:1:80, ro in 90:1:110 mm ro in 90:1:110 mm t in 1:0.5:3 mm,t in 1:0.5:3 mm, F in 600:10:1000 NF in 600:10:1000 N Z in 2:1:10Z in 2:1:10

7. March 10, KanGAL7 Innovized Principles t = 1.5 mm F = 1,000 N r o -r i =20mm Z = 3 till 9 (monotonic) Starts with small r i and smallest r o Both increases with brake mass r i reaches max limit, r o increases

8. March 10, KanGAL8 Innovized Principles (cont.) Surface area, S=Π(r o 2 -r i 2 )n T ∞ 1/S May be intuitive, but comes out as an optimal property r_i,max reduces the gap, but same T-S relationship

9. March 10, KanGAL9 Mechanical Spring Design Minimize material volume Minimize developed stress Three variables: (d, D, N): discrete, real, integer Eight non-linear constraints Solid length restriction Maximum allowable deflection (P/k≤6in) Dynamic deflection (P m -P)/k≥1.25in Volume and stress limitations

10. March 10, KanGAL10 Innovized Principles Pareto-optimal front have niches with d Only 5 (out of 42) values of d (large ones) are optimal Spring stiffness more or less identical (k=560 lb/in)(k=560 lb/in) –559.005, 559.877, 559.998 lb/in

11. March 10, KanGAL11 Optimal Springs, Optimal Recipe k=559.9 lb/in k=559.0 lb/in k=559.5 lb/in k=559.6 lb/in k=560.0 lb/in d=0.283 in d=0.331 in d=0.394 in d=0.4375 in d=0.5 in Increased volume Increased stress

12. March 10, KanGAL12 Innovized Principles (cont.) Investigation reveals: S∞1/(kV 0.5 ) Two constraints reveal: 50≤k≤560 lb/in Largest allowable k attains optimal solution Dynamic deflection constraint active

13. March 10, KanGAL13 Higher-Level Innovizations All optimal solutions have identical spring constant Constraint g_6 is active: (P_max-P)/k ≥ δ w k=(p_max-P)/δ w k=(1000-300)/1.25 or 560 lb/in Change δ w k values change

14. March 10, KanGAL14 Welded-Beam Design Minimize cost and deflection Four variables and four constraints Shear stress Bending stress b≥h Buckling load

15. March 10, KanGAL15 Innovizations Two properties Very small cost solutions behave differently than rest optimal solutions

16. March 10, KanGAL16 Innovizations (cont.) All solutions make shear stress constraint active Minimum deflection at t=10, b=5 (upper bounds) Transition when buckling constraint is active Minimum cost when all four are active

17. March 10, KanGAL17 Variations in Variables Small-cost: t reduces, b, l, h increases Otherwise: t constant, b reduces, l increases, h reduces l increases, h reduces

18. March 10, KanGAL18 Reliability of this procedure Confidence in the obtained Pareto frontConfidence in the obtained Pareto front –Benson’s method, Normal Constraint method, KKT conditions. Confidence in the obtained principles. Confidence in the obtained principles. – KKT Analysis –Big proof and Benchmark results.

19. March 10, KanGAL19 Higher Level Innovization Innovization principles forInnovization principles for – Robust Optimization – Reliability Based Optimization Innovization principles considering Innovization principles considering – Different pairs of objectives.

20. March 10, KanGAL20 Further Challenges: Automated Innovization Find principles from Pareto-optimal data Objectives and decision variables A complex data-mining task Clustering cum concept learning Rule extraction Difficulties Multiple relationships Relationships span over a partial set Mathematical forms not known a-priori Dealing with inexact data

21. March 10, KanGAL21 Thank You Questions and suggestions are welcome Questions and suggestions are welcome