CHAPTER 8 MATHEMATICAL MODEL AND GENETIC ALGORITHM...

106

CHAPTER 8

MATHEMATICAL MODEL AND GENETIC ALGORITHM

8.1 INTRODUCTION

This chapter describes the development of mathematical model

between the input parameters and output function using SYSTAT software.

Models for all kinds of pistons are derived and the same is compared with

the respective experimental results. This chapter also deals with genetic

algorithm using MATLAB 7 to optimize the machining parameters for

various dipped in conditions. The optimized results by GA are compared

with the Taguchi’s robust design output values.

8.2 NEED FOR A MATHEMATICAL MODEL

Functional or empirically developed mathematical models

explicitly link a quantitative dependent variable to certain independent

variables. The test time approach to build such models from observed

quantitative data is known as regressions analysis. Empirical models are

developed to predict the cutting forces using SYSTAT software. Present

work uses regression analysis to develop a mathematical model using the

experimental results. Prediction of cutting forces is carried out using the

developed model. Typical comparison is presented in 8.5. From the

comparison it is understood that variation between the experimentally

observed values and the predicted values are minimum and it confirmed the

potential applicability of the mathematical model.

107

8.3 THE SYSTAT SOFTWARE

SYSTAT is a high value integrated desktop statistics and

graphics software package with features like Design of Experiments,

ANOVA, Multivariate Analysis (MANOVA, MANCOVA) Repeated

Measure Analysis, General Linear Models, Time series, Regression

(Linear, Non Linear and Logistic) Survival Analysis, Path Analysis and

Simulation techniques.

8.4 CREATION OF MODEL

In order to create the mathematical model, experimental results of

different time dipped in condition pistons are considered. The parameters

like speed, feed, and depth of cut are referred as independent function and

the cutting force is identified as dependent function. Regression analysis is

carried out on the obtained results. Separate mathematical model is created

for each case.

8.5 MODEL RESULTS

Using the SYSTAT, the regression model has been formulated

for all kinds of pistons (1½ minutes, 2minutes, 3 minutes, 4 minutes and 5

minutes dipped inserts). The model for each case is given separately.

108

8.5.1 Result on 1½ minutes dipped insert piston OLS Regression Dependent Variable CUTTINGFORCEN 9Multiple R 0.882Squared Multiple R 0.777Adjusted Squared Multiple R 0.644Standard Error of Estimate 9.865 Regression Coefficients B = (X'X)-1X'Y Effect CoefficientStandard ErrorStd.

CoefficientTolerance t p-value

CONSTANT 0.111 68.423 0.000 . 0.002 0.999 SPEED -0.020 0.040 -0.105 1.000 -0.4970.641 FEED 196.667 80.544 0.515 1.000 2.442 0.059 DEPTHOFCUT 270.000 80.544 0.708 1.000 3.352 0.020 Analysis of Variance Source SS df Mean Squares F-ratio p-valueRegression 1697.667 3 565.889 5.815 0.044Residual 486.556 5 97.311 WARNING Case 7 is an Outlier (Studentized Residual : -2.168) Durbin-Watson D Statistic 1.489First Order Autocorrelation0.205 Information CriteriaAIC 71.452AIC (Corrected)91.452Schwarz's BIC 72.438

Based on the given input model for 1½ minutes dipped insert is found as

CF = 0.111-0.020S+196.667F+270D Where

CF- Cutting Force; S- Speed; F- Feed; D –Depth of cut.

109

8.5.2 Result on 2 minutes dipped insert piston

OLS Regression

Dependent Variable CUTTINGFORCEN 9 Multiple R 0.941Squared Multiple R 0.885Adjusted Squared Multiple R0.816Standard Error of Estimate 10.096 Regression Coefficients B = (X'X)-1X'Y Effect CoefficientStandard ErrorStd.


CONSTANT 56.333 70.029 0.000 . 0.804 0.458 SPEED -0.072 0.041 -0.264 1.000 -1.7390.143 FEED 260.000 82.435 0.478 1.000 3.154 0.025 DEPTHOFCUT416.667 82.435 0.766 1.000 5.054 0.004 Analysis of Variance Source SS dfMean SquaresF-ratiop-valueRegression3926.3333 1308.778 12.840 0.009 Residual 509.667 5 101.933 WARNING Case 5 is an Outlier (Studentized Residual : -2.438) Case 9 is an Outlier (Studentized Residual : 2.093) Durbin-Watson D Statistic 2.318First Order Autocorrelation-0.311 Information CriteriaAIC 71.870AIC (Corrected)91.870Schwarz's BIC 72.856 Based on the input the model for 2 min dipped insert is found as CF = 56.333-0.072S+260F+416.667D

110

8.5.3 Result on 3 minutes dipped insert piston OLS Regression Dependent Variable CUTTINGFORCEN 9 Multiple R 0.966Squared Multiple R 0.933Adjusted Squared Multiple R0.892Standard Error of Estimate 6.998 Regression Coefficients B = (X'X)-1X'Y Effect CoefficientStandard ErrorStd.


CONSTANT 52.444 48.543 0.000 . 1.080 0.329 SPEED -0.070 0.029 -0.285 1.000 -2.4500.058 FEED 293.333 57.142 0.596 1.000 5.133 0.004 DEPTHOFCUT346.667 57.142 0.704 1.000 6.067 0.002 Analysis of Variance Source SS dfMean SquaresF-ratiop-valueRegression3387.3333 1129.111 23.054 0.002 Residual 244.889 5 48.978 WARNING Case 2 is an Outlier (Studentized Residual : -3.379) Durbin-Watson D Statistic 2.863First Order Autocorrelation-0.458 Information CriteriaAIC 65.273AIC (Corrected)85.273Schwarz's BIC 66.259 Based on the input the model for 3 min dipped insert is found as CF = 52.444-0.070S+293.333F+346.66D

111

8.5.4 Result on 4 minutes dipped insert piston OLS Regression Dependent Variable CUTTINGFORCEN 9 Multiple R 0.959Squared Multiple R 0.919Adjusted Squared Multiple R0.870Standard Error of Estimate 7.695 Regression Coefficients B = (X'X)-1X'Y Effect CoefficientStandard ErrorStd.


CONSTANT -73.889 53.373 0.000 . -1.3840.225 SPEED 0.013 0.031 0.054 1.000 0.424 0.689 FEED 170.000 62.828 0.345 1.000 2.706 0.042 DEPTHOFCUT440.000 62.828 0.893 1.000 7.003 0.001 Analysis of Variance Source SS dfMean SquaresF-ratiop-valueRegression3348.1673 1116.056 18.849 0.004 Residual 296.056 5 59.211 WARNING Case 5 is an Outlier (Studentized Residual : -2.041) Case 7 is an Outlier (Studentized Residual : 4.283) Durbin-Watson D Statistic 1.749First Order Autocorrelation0.111 Information CriteriaAIC 66.981AIC (Corrected)86.981Schwarz's BIC 67.967 Based on the input the model for 4 min dipped insert is found as

C.F = -73.889+0.013S+170F+440D

112

8.5.5 Result on 5 minutes dipped insert piston

OLS Regression Dependent Variable CUTTINGFORCEN 9 Multiple R 0.952Squared Multiple R 0.906Adjusted Squared Multiple R0.850Standard Error of Estimate 7.191 Regression Coefficients B = (X'X)-1X'Y Effect CoefficientStandard ErrorStd.


CONSTANT -17.778 49.879 0.000 . -0.3560.736 SPEED -0.017 0.029 -0.078 1.000 -0.5680.595 FEED 236.667 58.715 0.553 1.000 4.031 0.010 DEPTHOFCUT330.000 58.715 0.771 1.000 5.620 0.002 Analysis of Variance Source SS dfMean SquaresF-ratiop-valueRegression2490.3333 830.111 16.053 0.005 Residual 258.556 5 51.711 WARNING Case 7 is an Outlier (Studentized Residual : -3.831) Durbin-Watson D Statistic 1.999First Order Autocorrelation-0.031 Information CriteriaAIC 65.762AIC (Corrected)85.762Schwarz's BIC 66.748 Based on the input the model for 5 min dipped insert is found as

CF =-17.778-0.017S+236.667F+330D

113

8.6 COMPARISON BETWEEN EXPERIMENT AND

MATHEMATICAL MODEL

It is essential to know the applicability of the model. By

comparing the experimental result and the mathematical model value one

can understand it. Comparison is made for each case and is presented in

Figure. 8.1 to Figure. 8.5 for visible reference.

COMPARISON CHART - 90 sec

0

20

40

60

80

100

1 2 3 4 5 6 7 8 9

Experiment No

Experiment valueModel value

Figure 8.1 Comparison chart for 1½ minutes dipped insert piston

Comparison Chart - 2 min

020406080

100120

1 2 3 4 5 6 7 8 9

Experiment No

ExperimentModel

Figure 8.2 Comparison chart for 2 minutes dipped insert piston

114

Comparison chart - 3 min

0

50

100

150

1 2 3 4 5 6 7 8 9

Experiment No

ExperimentModel


Comparison chart - 4 min

020406080

100

1 2 3 4 5 6 7 8 9

Experiment No

Experiment Model


115

Comparison chart- 5min

020406080

100

1 2 3 4 5 6 7 8 9

Experiment No

Cut

ting

forc

e (N

) Experiment Model


The close agreement with experiment and model values confirms

that the error associated with the developed model is minimal. Only minor

deviation is observed in Figure 8.4 for the first experimental run. This can

be attributed to experimental error. Also an experiment is carried out with

different parameters which are not in L9 matrix, and the experimental

results are verified with the mathematical model. It gives the satisfactory

result. The chart presented in Figure 8.6 reveals the applicability of the

developed model.

Comparsion chart

0

20

40

60

80

100

120

1 2 3 4 5 6 7 8 9Experim ental Run

Experiment valueModel value

Figure 8.6 Comparison between Experiment and model for a random

test

116

8.7 GENETIC ALGORITHM

8.7.1 Introduction

Genetic algorithms (GA) are computerized search procedures

based on the mechanics of natural genetics and natural selection that can be

used to obtain global and robust solutions to optimization problems. GAs

are computational optimization schemes with a nontraditional approach. It

was developed by John Holland and his colleagues at the University of

Michigan. The algorithm solves optimization problems imitating nature in

the way it has been working millions of years on the evolution of life. GA

combine the survival of the fittest among string structures, yet randomized

information exchange to form a search algorithm with some of the

innovative flair of human search. In every generation, a new set of artificial

strings are created using bits and pieces of the fittest of the world; an

occasional new part is tried for the good measure. This work concentrates

on simple GA with reproduction with crossover and mutation operators.

Though these operators look very simple at first sight, combined action is

much of GA’s power from a computer implementation point of view, they

involve only random number generation, string copying, partial string

swapping and bit conversion (Franci Cus et al 2003).

8.7.2 Genetic Algorithm Vs Traditional Methods

Genetic algorithms differ from conventional optimizations and

search procedures in several fundamental ways. These are summarized as

follows:

117

GA’s work with coding of solution set, not the solutions

themselves.

GA’s search from a population of solution, not a single

solution.

GA’s use pays of information (fitness function) of derivatives

or auxiliary knowledge.

GA’s probabilistic transition rules, not deterministic rules.

8.7.3 Genetic operators

The mechanics of simple GAs are surprisingly simple. A simple

GA that yields good results in many practical problems is composed with

these operators:

Reproduction

Cross over

Mutation

Fitness function

With the initial population, the population for the next generation is to be

generated which is the offspring of the current generation. The reproduction

operator selects the fit individual from the current population and places

them in a mating pool where as the less fit ones get fewer copies.

The worst fit individuals die of eventually. The factor (F/avg F) is

calculated for all individuals. This factor is expected count of individual in

the mating pool. It is then converted into an actual count by appropriately

rounding off, so that individuals get copies in the mating pool proportional

to their fitness. This process of reproduction confirms the Darwinian

principles of the survival of fittest.

118

In the crossover, set of crossover parameters are generated again

randomly. The first set step in the crossover is finding a match for

individuals. Once the pairs are decided it is necessary to find the crossover

sites, the substrings between the crossover sites are swapped from one

individual in the pair to the other.

The GA repeat the same process of the generation of new

population and evaluating its fitness. Proceeding with more generation,

there may not be much improvement in the population fitness bearing a few

because of the mutation operation and the best individual may not change

for subsequent population. As the generation advances, the population gets

filled, more fit individuals with only slight deviation from the fitness of the

best individual so far found and the average fitness comes very close to the

fitness of the best individual

8.7.3.1 Reproduction

Reproduction is the process in which individual strings are

copied according to their objective function values. Copying strings

according to their fitness values means the strings with a higher fitness

value have a higher probability of contributing one or more off spring in the

next generation. This operator, of course is an artificial version of natural

selection, a Darwin’s survival of the fittest among string creatures. It is

important to note that no new string is formed in the reproduction phase.

119

8.7.3.2 Cross over

After reproduction, simple single point cross over may proceed in

two steps. First, two individual strings are selected at random from the

mating pool generated by the reproduction operator. Next, a cross over site

is selected at a random along the string length and the binary digits are

swapped between the two strings following the cross over sites. For

example if two design vectors (parents), each with a string length of 1 0,

are given by

(Parent 1) x1= {01010-11011}

(Parent 2) x2= {10001-11100}

The result of the cross over, when the cross over site is 5, is given by

(Off spring 1) x 3= {01010-11100}

(Off spring 2) x 4= {10001-11011}

The new strings (off springs) obtained from cross over are placed in the

new population and the process is continued. A cross operator is mainly

responsible for the search of new strings. Above explained cross over

operator like edge recombination cross over, simulated binary cross over

etc. In edge recombination cross over, one parent gives one child as

explained below.

Parent 248395-1607

The result of cross over, when the cross over sight is 6, is given by

Child 248395-7061

120

8.7.3.3 Mutation

The mutation operator is applied to a new string with a specified

mutation. Probability mutation is the occasional random alteration of the

binary digit. Thus in a mutation 0 is changed to 1, and vice versa, at random

location.

These operators are simple and straight forward. The

reproduction operator selects good strings and cross over operators

recombines good strings together to hopefully create a better string. Even

though none of these clients are guaranteed and /or tested while creating a

string, it is expected that if bad strings are created, the reproduction

operator will eliminate them in the next generation and if good strings are

created, they will be increasingly emphasized.

8.7.3.4 Fitness function

GAs mimic the survival of the fittest principle, so naturally they

are suitable to solve maximization problems. Minimization problems are

usually transformed to maximization problems by suitable transformation.

A fitness function F(x) is derived from the objective function and used in

successive genetic operations. For maximization problems, fitness

functions can be considered the same as the objective function. For

minimization problems it is an equivalent maximization problem, chosen

such that the optimum point remains unchanged. A number of such

transformations are possible. The fitness function often used as F(x) =

1/(1+f(x)). This transformation doesn’t alter the location of the minimum

but converts the minimization problem into an equivalent maximization

problem.

121

8.8 PSEUDO CODE FOR GENETIC ALGORITHM

Step 1: Choose a Coding Scheme to represent the decision variables.

Choose appropriate Reproduction, Crossover and Mutation

Operators

Choose the Population Size, Crossover Probability,

Mutation Probability and the Termination Criterion

Step 2: Generate the Initial Population and evaluate the fitness

Values

Do while Termination Criterion is not met

Step 3: Perform Reproduction to create intermediate mating

Pool

Step 4: Perform Crossover to create off springs

Step 5: Perform Mutation on every string of the intermediate

Population

Step 6: Evaluate the strings in the new population

End do

Decode the best string in the final population to get the solution.

122

8.9 STEPS IN GENETIC ALGORITHM

The step by step procedure for implementing GA optimization

process is shown in Figure 8.7.

Figure 8.7 GA Flow Chart

Initialize parameters

Generate initial population

Evaluate fitness function for all population

Evaluate population statistics

Generate parents

Perform crossover

Generate offspring with mutation

Is optimal solution obtained

Terminate

Update population

123

8.10 USAGE OF GA TOOLBOX IN MATLAB

The mathematical model created using SYSTAT software to

relate cutting force and process parameters was encoded in MATLAB

software in M-FILE format. Later, using GA Toolbox available in the start

menu the input to the fitness function is given as @ file name. Then number

of variables is given as 3 and other GA functionalities were chosen

accordingly to obtain best fitness value in random process.

Population size 20

Scaling function rank

Selection function roulette

Reproduction

Elite count 2

Cross over function 0.8

Mutation

Function uniform

Rate 0.1

Crossover function two point

The GA program was then run for 100 generations and after the

termination of the process, the final point is displayed at the bottom of the

window which is the optimized parameters for the obtained best value.

124

8.11 GENETIC ALGORITHM RESULTS

8.11.1 GA result for 1½ minutes dipped insert piston

The equation formulated with the SYSTAT is named as case 1

and it is given as input to GA. The Figure 8.8 gives the result of 1½

minutes dipped in condition insert piston.

Figure 8.8 GA Result for 1½ minutes piston

125

8.11.2 GA result for 2 minutes dipped insert piston


and it is given as input to GA. The Figure 8.9 gives the result of 2 min

dipped in condition insert piston.

Figure 8.9 GA Result for 2 minutes dipped insert piston

126






127






128






129

8.12 COMPARISON BETWEEN TAGUCHI RESULT AND GA

OUTPUT

A comparison table is made to compare the GA output with

Taguchi robust design technique and is presented in the Table 8.1. From the

results, it can be understood that GA and Taguchi outputs are comparable

and they agree with each other.

Table 8.1 Comparison between Taguchi and GA

Condition Parameters Taguchi GA output

1½ minutes

dipped

Speed (m/min) 512 506

Feed (mm/rev) 0.15 0.1517

Depth of cut (mm) 0.15 0.1532

2 minutes

dipped


Feed (mm/rev) 0.20 0.1757


3 minutes

dipped

Speed(m/min) 512 508

Feed (mm/rev) 0.15 0.1509

Depth of cut(mm) 0.15 0.1504

4 minutes

dipped


Feed (mm/rev) 0.15 0.1530


5 minutes

dipped


Feed (mm/rev) 0.15 0.1578


130

8.13 SUMMARY

Mathematical model is developed and the output of mathematical

model is compared with the experimental values, suitability of the model

for predicting the cutting force is also assessed.

Genetic Algorithm is developed using MATLAB. The outputs of

GA (the optimized value) are compared with the output of Taguchi method.

The close agreement with the predicted and experimental values

of cutting force confirms the potential applicability of developed model.

The matching of GA and Taguchi’s design approach validates out method

of approach.

CHAPTER 8 MATHEMATICAL MODEL AND GENETIC ALGORITHM...

Documents

Transcript of CHAPTER 8 MATHEMATICAL MODEL AND GENETIC ALGORITHM...