INNOVIZATION-Innovative INNOVIZATION-Innovative solutions through Optimizationsolutions through Optimization

Prof. Kalyanmoy Deb & Aravind Srinivasan Prof. Kalyanmoy Deb & Aravind Srinivasan Kanpur Genetic Algorithm Laboratory (KanGAL) Kanpur Genetic Algorithm Laboratory (KanGAL)

Department of Mechanical Engineering Department of Mechanical Engineering Indian Institute of Technology Kanpur Indian Institute of Technology Kanpur

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InnovizationInnovization Identification of commonalities amongst optimal

solutions or Knowledge discovery.

– Optimal Solutions satisfy - KKT conditions.Optimal Solutions satisfy - KKT conditions.

– Single Objective optimizationSingle Objective optimization No global information about any property that the No global information about any property that the

optimal solutions may carry. optimal solutions may carry. No flexibility for the decision maker. No flexibility for the decision maker.

– Multi-Objective OptimizationMulti-Objective Optimization Need for Evolutionary Algorithms(GA)Need for Evolutionary Algorithms(GA) NSGA-2: Established Algorithm for EMONSGA-2: Established Algorithm for EMO

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EMOEMO

Principle:Principle: Find multiple Pareto-Find multiple Pareto-optimal solutions optimal solutions simultaneouslysimultaneously

Three main reasons:Three main reasons:For a better decision-For a better decision-makingmaking

For unveiling salient For unveiling salient optimality properties of optimality properties of solutionssolutions

For assisting in other For assisting in other problem solvingproblem solving

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PotentialsPotentials

• Better Understanding of the problem.Better Understanding of the problem.

• Reduces Cost.Reduces Cost.

• Eliminates the need for new optimization Eliminates the need for new optimization for small change in parameters. for small change in parameters.

• Deciphers innovative ideas for further Deciphers innovative ideas for further design.design.

• Benchmark Designs for industriesBenchmark Designs for industries.

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Innovization ProcedureInnovization Procedure

Choose two or more conflicting objectives Choose two or more conflicting objectives (e.g., size and power)(e.g., size and power)

Usually, a small sized solution is less poweredUsually, a small sized solution is less powered

Obtain Obtain Pareto-optimal solutionsPareto-optimal solutions using an using an EMOEMO

Investigate for any common properties Investigate for any common properties manually or automaticallymanually or automatically

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Multi-Disk Brake DesignMulti-Disk Brake Design• Minimize brake massMinimize brake mass• Minimize stopping timeMinimize stopping time• 16 non-linear constraints16 non-linear constraints• 5 variables: Discrete 5 variables: Discrete

(ri,ro,t,,F,Z) (ri,ro,t,,F,Z) • ri in 60:1:80,ri in 60:1:80, ro in 90:1:110 mmro 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 2:1:10 Z in 2:1:10

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Innovized Principles Innovized Principles t = 1.5 mmt = 1.5 mmF = 1,000 NF = 1,000 N

rroo-r-rii=20mm=20mm

Z = 3 till 9 (monotonic)Z = 3 till 9 (monotonic)

Starts with small rStarts with small r ii and and smallest rsmallest roo

Both increases with brake Both increases with brake massmass

rrii reaches max limit, r reaches max limit, roo increasesincreases

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Innovized Principles (cont.)Innovized Principles (cont.)

Surface area, Surface area, S=S=ΠΠ(r(roo

22-r-rii22)n)n

T ∞ 1/ST ∞ 1/S

May be intuitive, but May be intuitive, but comes out as an comes out as an optimal propertyoptimal property

r_i,max reduces the r_i,max reduces the gap, but same T-S gap, but same T-S relationshiprelationship

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Mechanical Spring DesignMechanical Spring Design

Minimize Minimize material volumematerial volumeMinimizeMinimize developed stressdeveloped stressThree variables: (d, D, N): discrete, real, Three variables: (d, D, N): discrete, real, integerintegerEight non-linear constraintsEight non-linear constraints

Solid length restrictionSolid length restrictionMaximum allowable deflection (P/k≤6in)Maximum allowable deflection (P/k≤6in)

Dynamic deflection (PDynamic deflection (Pmm-P)/k≥1.25in-P)/k≥1.25in

Volume and stress limitationsVolume and stress limitations

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Innovized PrinciplesInnovized PrinciplesPareto-optimal front Pareto-optimal front have niches with dhave niches with dOnly 5 (out of 42) Only 5 (out of 42) values of d (large values of d (large ones) are optimalones) are optimalSpring stiffness more Spring stiffness more or less identicalor less identical

• (k=560 lb/in)(k=560 lb/in)– 559.005, 559.877, 559.005, 559.877,

559.998 lb/in559.998 lb/in

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Optimal Springs, Optimal RecipeOptimal Springs, Optimal Recipe

k=559.9 lb/ink=559.9 lb/in

k=559.0 lb/ink=559.0 lb/in

k=559.5 lb/ink=559.5 lb/in

k=559.6 lb/ink=559.6 lb/in

k=560.0 lb/ink=560.0 lb/in

d=0.283 ind=0.283 in

d=0.331 ind=0.331 in

d=0.394 ind=0.394 in

d=0.4375 ind=0.4375 in

d=0.5 ind=0.5 in

Incr

ease

d v

olu

me

Incr

ease

d v

olu

me In

crease

d stre

ssIn

crease

d stre

ss

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Innovized Principles (cont.)Innovized Principles (cont.)Investigation reveals: Investigation reveals: S∞1/(kVS∞1/(kV0.50.5))Two constraints Two constraints reveal: 50≤k≤560 reveal: 50≤k≤560 lb/inlb/inLargest allowable k Largest allowable k attains optimal attains optimal solutionsolutionDynamic deflection Dynamic deflection constraint activeconstraint active

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Higher-Level InnovizationsHigher-Level InnovizationsAll optimal solutions All optimal solutions have identical spring have identical spring constantconstantConstraint g_6 is active:Constraint g_6 is active:

(P_max-P)/k ≥ (P_max-P)/k ≥ δδww

k=(p_max-P)/k=(p_max-P)/δδww

k=(1000-300)/1.25 or k=(1000-300)/1.25 or 560 lb/in560 lb/in

Change Change δδww

k values changek values change

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Welded-Beam DesignWelded-Beam Design

Minimize cost Minimize cost and deflectionand deflectionFour variables Four variables and four and four constraintsconstraints

Shear stressShear stressBending stressBending stressb≥hb≥hBuckling loadBuckling load

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InnovizationsInnovizations

Two propertiesTwo propertiesVery small cost Very small cost solutions solutions behave behave differently than differently than rest optimal rest optimal solutionssolutions

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Innovizations (cont.)Innovizations (cont.)

All solutions make shear stress constraint All solutions make shear stress constraint activeactiveMinimum deflection at Minimum deflection at t=10, b=5 (upper t=10, b=5 (upper bounds)bounds)Transition when Transition when buckling constraint is buckling constraint is activeactiveMinimum cost when all Minimum cost when all four are activefour are active

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Variations in VariablesVariations in VariablesSmall-cost: t reduces, b, l, h increases Small-cost: t reduces, b, l, h increases Otherwise: t constant, b reduces, Otherwise: t constant, b reduces,

l increases, h reducesl increases, h reduces

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Reliability of this procedureReliability of this procedure

• Confidence in the obtained Pareto frontConfidence in the obtained Pareto front– Benson’s method, Normal Constraint Benson’s method, Normal Constraint

method, KKT conditions.method, KKT conditions.

• Confidence in the obtained principles.Confidence in the obtained principles.– KKT AnalysisKKT Analysis

– Big proof and Benchmark results.Big proof and Benchmark results.

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Higher Level InnovizationHigher Level Innovization

• Innovization principles for Innovization principles for – Robust OptimizationRobust Optimization– Reliability Based OptimizationReliability Based Optimization

• Innovization principles consideringInnovization principles considering– Different pairs of objectives.Different pairs of objectives.

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Further Challenges: Automated Further Challenges: Automated InnovizationInnovization

Find principles from Pareto-optimal dataFind principles from Pareto-optimal dataObjectives and decision variablesObjectives and decision variables

A complex data-mining taskA complex data-mining taskClustering cum concept learningClustering cum concept learning

Rule extractionRule extraction

DifficultiesDifficultiesMultiple relationshipsMultiple relationships

Relationships span over a partial setRelationships span over a partial set

Mathematical forms not known a-prioriMathematical forms not known a-priori

Dealing with inexact dataDealing with inexact data

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Thank YouThank You Questions and suggestions are welcomeQuestions and suggestions are welcome