Osyczka Andrzej Krenich Stanislaw

Habel Jacek 

Department of Mechanical Engineering, Cracow University of Technology,

31-864 Krakow, Al. Jana Pawla II 37, Poland email: osyczka@mech.pk.edu.pl,

krenich@mech.pk.edu.pl, habel@mech.pk.edu.pl

Contents   Introduction

General description of EOS

Some methods from EOS    

Bicriterion method

Constraint tournament method for single and multicriteria optimization

Indiscernibility interval method

Features of EOS

Running EOS

Applications examples

Spring design automation

Robot gripper mechanism design

Network optimization

Evolutionary Optimization System (EOS) is designed to solve single and multicriteria optimization problems for nonlinear programming problems, i.e. for the problems formulated as follows:  

find x* = [x1*, x2

*, ..., xI*] which will satisfy the K inequality constraints

g k (x) 0 for k = 1, 2, …, K (1)

and the M equality constraints

h m (x) = 0 for m = 1, 2, …, M (2)

and optimize the vector of objective functions:

f(x*) = [f1(x), f2(x),...,fN(x)] (3)

where: x = [x1,x2,...,xI] is the vector of decision variables,

 For single criterion optimization problems instead of the vector function f(x) we have the scalar function f(x) which is to be minimized.

The system is coded in the ANSI C language.

Introduction

General description of EOS

The main idea of the proposed method consists in transforming the single criterion optimization problem into the bicriterion optimization problem with the following objective functions:

Some methods from EOS - Bicriterion method

K

kkk

M

mm gGhf

11xxx 1

where: Gk is the Heaviside operator such that

Gk= -1 dla gk (x) 0, Gk= 0 dla gk (x) 0.

f2(x) = f(x) - the objective function that is to be minimized

The minimum of f1(x) is known and equals zero.

The function f1(x) will achieve its minimum for any solution that is in the feasible

region.

Some methods from EOS - Bicriterion method

Sets of Pareto solutions for a numerical example

0,00

1,00

2,00

3,00

4,00

5,00

6,00

0,0000 0,0001 0,0010 0,0100 0,1000 1,0000 10,0000 100,0000

f1(X)

f2(X

)

30

50

100

1000

Minimum

If both chromosomes are not in the feasible region the one which is closer to the feasible region is taken to the next generation. The values of the objective function are not calculated for either of chromosomes.

If one chromosome is in the feasible region and the other one is out of the feasible region the one which is in the feasible region is taken to the next generation. The values of the objective function are not calculated for either chromosome.

If both chromosomes are in the feasible region, the values of the objective function are calculated for both chromosomes and the one, which has a better value of the objective function is taken to the next generation.

In this method the tournament between two chromosomes is carried out in the following way:

Some methods from EOS - Constraint Tournament Method for Single Criterion Optimization

The constraint violation function can be evaluated as follows:

M

m

K

kkkm gGh

1 1

22 xxx

where: Gk is the Heaveside operator such that Gk =0 for

and Gk =1 for

0xkg

0xkg

x 1

x 2The feasible domain

(x1,t) < (x2,t)

6

(x6,t)=0

(x5,t)=0

5

(x3,t)

3

(x2,t)

2

(x1,t)

1

(x4,t)=0

4

5

41

Feasiblesolution

f(x5,t) < f(x6,t)

Some methods from EOS - Constraint Tournament Method for Single Criterion Optimization

Some methods from EOS - Constraint Tournament Method for Multicriteria Optimization

x 1

x 2The feasible domain

(x1,t) < (x2,t)

6

(x6,t)=0

(x5,t)=0

5

(x3,t)

3

(x2,t)

2

(x1,t)

1

(x4,t)=0

4

5

41

Random lychosen

6

f2

f1

f1

f2

Feasiblesolution

The steps of the method are as follows: Step 1. Set t = 1, where t is the number of the currently run generation. Step 2. Generate the set of Pareto optimal solutions using any evolutionary algorithm method. Step 3. Is the criterion for filtration the set of Pareto solutions satisfied? If yes, select the representative subset of Pareto solutions using the indiscernibility interval method and go to step 4. Otherwise, go straight to step 4. Step 4. Set t = t + 1 and if t T, where T is the assumed number of generations, go to step 2. Otherwise, terminate the calculations.

Some methods from EOS - Indiscernibility interval method

The idea of the method consists in reducing the set of Pareto optimal solutions using indiscernibility interval method after running a certain number of generations.

Graphical illustration of the indiscernibility interval method

Some methods from EOS - Indiscernibility interval method

Features of EOS

For both single and multicriteria optimization methods the following models can be solved:• with continuous decision variables,• with integer decision variables,• with discrete decision variables, • with mixed continuous – integer decision variables,• with mixed continuous – discrete decision variables.

In EOS chromosomes can have:• binary representation,• real number representation,• Grey coding representation.

Crossover operations can be performed as follows:• one point crossover, • two point crossover,• variable point crossover.

Mutation operations can be performed as follows:• uniform mutation,• non-uniform mutation.

Running EOS – The Main Control Window

Running EOS – The User Function File Window

Running EOS – The Output File Window

1) Spring design automation

2) Robot gripper mechanism design

3) Network optimization

Applications examples

Examples of Spring Design Automation - Helical Spring Design

x2

x1

x3

Scheme of the spring

The vector of decision variables is

x1 – wire diameter of the spring [mm]

x2 – meancoil diameter of the spring [mm]

x3 – length of the spring [mm]

x4 – number of active coils [–]

T4321 x,x,x,xx

The objective function is the volume of the spring

221

223

24

22

221 xx

4

πxxxx

4

πf x

The constraints are:

1) shear stress constraint,2) stiffness of the spring constraint, 3) clearance between coils constraint , 4) buckling constraint, 5) geometric constraints,

•

The optimization model is considered a discrete type, with the following sets of possible values:X1 = {0.5, 0.63, 0.8,..., 6.3, 8.0, 10.0 }, X2 = {1,2,3,4,...,60,61,62,...,300},

X3 = {1,2,3,...,50,51,52,...,600}, X4 = {1.5,2.5,...,49.5}

Helical Spring Design - Numerical Results

  Example no. 1

Example no.2

Material of the spring

45S 70S3

Compression force P [N]

1850 580.8

Stiffness of the spring s[N/mm]/Deflection of the

spring d[mm]

20.55  

  62

Type of the spring Non running

running

  The best results obtained using the automation design

method

Results obtained using a general

design procedure

f(x) 84 989.59 100 023.18

x1 8.0 8.0

x2 63.0 60.0

x3 159. 0 196. 0

x4 7.5 9.5

Table 2. Results of automation of design of the spring: example1

Table 1. Input data of the spring design problem

Optimization model of the robot gripper

Fk

Fk

y

z

f

e

l

ab

cVector of decision variables: x = [ a, b, c, e, f, l, ]T,

where a, b, c, e, f, l, are dimensions of the gripper and is the angle between the elements b and c.

Constraints:1. On the basis of the geometrical dependencies and dependencies between the forces

several constraints are built and used. 2. They depend also on the stages of the optimization process.

Objective functions: 1. f1(x) - the difference between maximum and minimum griping forces for the assumed

range of the gripper ends displacement,2. f2(x) - the force transmission ratio between the gripper actuator and the gripper ends,

3. f3(x) - the shift transmission ratio between the gripper actuator and the gripper ends,

4. f4(x) - the length of all the elements of the gripper,

5. f5(x) - the maximal force in the joints,

6. f6(x) - the efficiency of the gripper mechanism.

Multistage process of the robot gripper optimization

P

zFf

kz

),(min)(2

xx

minmax

minmax3

,,)(

ZZ

ZyZyf

xx

x

L

iilf

14 )(x

jj

Rf max)(5 x

max

06 ),(),()(

Z

z

Tk

BTk zFzFf xxx

zFzF f kz

kz

1 ,min,max xxx

Ordering of the criteria:

Constraints:

model basic the in as g do g xx 111

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0,00 1,00 2,00 3,00 4,00 5,00 6,00

Stałość siły na szczękach

Prz

eło

żen

ie s

iło

we

Stage 1:

zFzF f kz

kz

1 ,min,max xxx

P

zFf

kz

),(min)(2

xx

The set of Pareto optimal solutions

model basic the in as g do g xx 111

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0,5 1 1,5 2 2,5 3 3,5

Przełożenie przemieszczeniowe

Prz

eło

żen

ie s

iło

we

Multistage process of the robot gripper optimization

P

zFf

kz

),(min)(2

xx

minmax

minmax3

,,)(

ZZ

ZyZyf

xx

x

L

iilf

14 )(x

jj

Rf max)(5 x

max

06 ),(),()(

Z

z

Tk

BTk zFzFf xxx

Constraints:

Stage 2:

0,min,max-4 g12 zFzF kz

kz

xxx

P

zFf

kz

),(min)(2

xx

The set of Pareto optimal solutions

minmax

minmax3

,,)(

ZZ

ZyZyf

xx

x

model basic the in as g do g xx 111

152,20

152,40

152,60

152,80

153,00

153,20

153,40

153,60

153,80

154,00

90,6 90,8 91 91,2 91,4 91,6 91,8

Maksymalna siła w parach kinematycznych

Str

aty

na

ta

rcie

Multistage process of the robot gripper optimization

jj

Rf max)(5 x

max

06 ),(),()(

Z

z

Tk

BTk zFzFf xxx

Constraints:

Stage 5:

0,min,max-4 g12 zFzF kz

kz

xxx

05,0),(min

)(13 P

zFg

kz

xx

The set of Pareto optimal solutions

05,1

,,)(

minmax

minmax14

ZZ

ZyZyg

xxx 0600)(

115

L

iilg x

jj

Rf max)(5 x

max

06 ),(),()(

Z

z

Tk

BTk zFzFf xxx

Zmienne decyzyjne Wartości kryteriów

a b c e f l

f1(x)= 3.046

f2(x)= 0.500

f3(x)= 1.677

f4(x)= 499.81

f5(x)= 91.590

f6(x)= 152.400

135.00 90.53 102.20 0.00 1.28 170.80 1.57

Tabela 7.22. Przykładowe rozwiązania ze zbioru Pareto uzyskane w etapie 5 optymalizacji

Network optimization

An example of network which has no Markow property

An example of network which has Markow property

Osyczka Andrzej

Krenich Stanislaw

Habel Jacek

Department of Mechanical Engineering

Cracow University of Technology, 2002

Thank You for Your Attention