Omkar project report submitted m tec mechanical

ACKNOWLEDGEMENT It gives me a great pleasure in presenting the Dissertation on “Evolutionary Algorithms

for

Multi-Objective

Optimization:

Modelling

and

Comparative

Evaluation�. The project work has certainly rendered me a tremendous learning.

Apart from my efforts, the success of this project depends largely on the encouragement and guidelines of many others. I take this opportunity to express my gratitude to the people who have been instrumental in the successful completion of this project.

First and foremost, I would like to thank to my guide, Dr. Satish S. Chinchanikar (Professor, Department of Mechanical Engineering, VIIT, Pune) for his valuable guidance and advice. My special thanks to my Co-guide, Mr. Mahendra G. Gadge (Assistant Professor, Department of Mechanical Engineering, VIIT, Pune) for his valuable guidance and advice. He inspired me greatly to work in this project. His willingness to motivate me contributed tremendously to this project.

I would like to thank Dr. Atul P. Kulkarni (Associate Professor and Head, Department of Mechanical Engineering, VIIT, Pune) for allowing me to work on this project and giving me valuable guidance and advice in my Project.

I am highly grateful to Dr. Bilavari S. Karkare (Principal) and I would also thank our Institution, faculties and technical staff of mechanical engineering department who helped me directly or indirectly during this project work. I also extend my heartfelt thanks to my family, siblings, all my friends and well-wishers.

Omkar Mahesh Manav

i

LIST OF FIGURE Fig. N0 Name of Figures

Page No

1.1

Hierarchy of Computational Intelligence

2

1.2

Approaches to Computational Intelligence

3

1.3

Global Search Optimization Hierarchy

6

1.4

Synergies of Computational Intelligence

7

1.5

Applied Optimization and Learning Methodology

12

3.1

Workflow for chapter 3

28

3.2

Evolutionary Model

33

3.3

NSGA II Algorithm

41

3.4

Rank and Pareto for Ra (35HRC)

50

3.5

Pareto-front for Tf (35HRC)

50

3.6

Average distance between consecutive generations (35HRC)

50

3.7

Rank and Pareto for Ra (45HRC)

52

3.8

Pareto-front for Tf (45HRC)

52

3.9

Average distance between consecutive generations (45HRC)

52

3.10

SPEA2 Algorithm

54

3.11

Rank and Pareto for Ra (35HRC)

58

3.12

Pareto-front for Tf (35HRC)

58

3.13

Average distance between consecutive generations (35HRC)

58

3.14

Rank and Pareto for Ra (45HRC)

59

3.15

Pareto-front for Tf (45HRC)

59

ii

3.16

Average distance between consecutive generations (45HRC)

60

3.17

PSO Algorithm

66

3.18

Pareto spread surface roughness and cutting force 35HRC

73

3.19

3D surface plot of optimal Ra with best position 35HRC

73

3.20

3D surface plot of Tf with best position 35HRC

74

3.21

Depth of cut influence on cutting forces 35 HRC

74

3.22

Pareto spread surface roughness and cutting force 45HRC

75

3.23

3D surface plot of optimal Ra with best position 45HRC

75

3.24

3D surface plot of Tf with best position 45 HRC

76

3.25

Depth of cut influence on cutting forces 45 HRC

76

3.26

Solution Spectrum for 35 hrc NSGA II

77

3.27

Solution Spectrum for 45 hrc NSGA II

77

3.28

Solution Spectrum for 35 hrc PSO

77

3.29

Solution Spectrum for 45 hrc PSO

77

3.30

Solution Spectrum for 35 hrc SPEA 2

77

3.31

Solution Spectrum for 45 hrc SPEA2

77

4.1

Workflow for Chapter 4

80

4.2

Simple network

82

4.3

Multi-layer feed forward network structure

85

4.4

Feed forward neural network for AISI 4340 Hard turning

88

Plots for NN 35 HRC 4.5

Performance plot of Network

89

iii

4.6

Training state of Network at each epoch

89

4.7

Training error in Ra

90

4.8

Regression fit plot for Ra

90

4.9

Training error in Ft

90

4.10

Regression fit plot for Ft

90

4.11

Training error in Fa

91

4.12

Regression fit plot for Fa

91

4.13

Training error in Fr

91

4.14

Regression fit plot for Fr

91

4.15

Training error in Tf

91

4.16

Regression fit plot for Tf

91

Plots for NN 45 HRC 4.17

Performance plot of Network

92

4.18

Training state of Network at each epoch

92

4.19

Training error in Ra

92

4.20

Regression fit plot for Ra

92

4.21

Training error in Ft

92

4.22

Regression fit plot for Ft

92

4.23

Training error in Fa

93

4.24

Regression fit plot for Fa

93

4.25

Training error in Fr

93

4.26

Regression fit plot for Fr

93

iv

4.27

Training error in Tf

93

4.28

Regression fit plot for Tf

93

4.29

ANFIS two input model

95

4.30

Applied ANFIS grid partitioning architect

101

ANFIS Grid Partitioning Plots for 35 HRC 4.31

Training Error Plots for Ra (Target vs Output)

103

4.32

Testing Error Plots for Ra (Target vs Output)

103

4.33

Validation Error Plots for Ra (Target vs Output)

103

4.34

Regression Plots for Ra (Train /Test/Validate)

103

4.35

Response Surface Plot for Ra

104

4.36

Training Error Plots for Ft (Target vs Output)

104

4.37

Testing Error Plots for Ft (Target vs Output)

104

4.38

Validation Error Plots for Ft (Target vs Output)

105

4.39

Regression Plots for Ft (Train /Test/Validate)

105

4.40

Response Surface Plot for Ft

105

4.41

Training Error Plots for Fa (Target vs Output)

105

4.42

Testing Error Plots for Fa (Target vs Output)

105

4.43

Validation Error Plots for Fa (Target vs Output)

106

4.44

Regression Plots for Fa (Train /Test/Validate)

106

4.45

Response Surface Plot for Fa

106

4.46

Training Error Plots for Fr (Target vs Output)

106

4.47

Testing Error Plots for Fr (Target vs Output

106

v

4.48

Validation Error Plots for Fr (Target vs Output)

107

4.49

Regression Plots for Fr (Train /Test/Validate)

107

4.50

Response Surface Plot for Fr

107

4.51

Training Error Plots for Tf (Target vs Output)

108

4.52

Testing Error Plots for Tf (Target vs Output)

108

4.53

Validation Error Plots for Tf (Target vs Output)

108

4.54

Regression Plots for Tf (Train /Test/Validate)

108

4.55

Response Surface Plot for Tf

108

ANFIS Grid Partitioning Plots For 45 HRC 4.56

Training Error Plots for Ra (Target vs Output)

109

4.57

Testing Error Plots for Ra (Target vs Output)

109

4.58

Validation Error Plots for Ra (Target vs Output)

109

4.59

Regression Plots for Ra (Train /Test/Validate)

109

4.60

Response Surface Plot for Ra

109

4.61

Training Error Plots for Ft (Target vs Output)

110

4.62

Testing Error Plots for Ft (Target vs Output)

110

4.63

Validation Error Plots for Ft (Target vs Output)

110

4.64

Regression Plots for Ft (Train /Test/Validate)

110

4.65

Response Surface Plot for Ft

110

4.66

Training Error Plots for Fa (Target vs Output)

111

4.67

Testing Error Plots for Fa (Target vs Output)

111

4.68

Validation Error Plots for Fa (Target vs Output)

111

vi

4.69

Regression Plots for Fa (Train /Test/Validate)

111

4.70

Response Surface Plot for Fa

111

4.71

Training Error Plots for Fr (Target vs Output)

112

4.72

Testing Error Plots for Fr (Target vs Output

112

4.73

Validation Error Plots for Fr (Target vs Output)

112

4.74

Regression Plots for Fr (Train /Test/Validate)

112

4.75

Response Surface Plot for Fr

112

4.76

Training Error Plots for Tf (Target vs Output)

113

4.77

Testing Error Plots for Tf (Target vs Output)

113

4.78

Validation Error Plots for Tf (Target vs Output)

113

4.79

Regression Plots for Tf (Train /Test/Validate)

113

4.80

Response Surface Plot for Tf

113

4.81

Developed ANFIS (Subtractive Cluster)

115

ANFIS Subtractive Cluster plots For 35 HRC 4.82

Training Error Plots for Ra (Target vs Output)

117

4.83

Testing Error Plots for Ra (Target vs Output)

117

4.84

Validation Error Plots for Ra (Target vs Output)

117

4.85

Regression Plots for Ra (Train /Test/Validate)

117

4.86

Response Surface Plot for Ra

117

4.87

Training Error Plots for Ft (Target vs Output)

118

4.88

Testing Error Plots for Ft (Target vs Output)

118

4.89

Validation Error Plots for Ft (Target vs Output)

118

vii

4.90

Regression Plots for Ft (Train /Test/Validate)

118

4.91

Response Surface Plot for Ft

118

4.92

Training Error Plots for Fa (Target vs Output)

119

4.93

Testing Error Plots for Fa (Target vs Output)

119

4.94

Validation Error Plots for Fa (Target vs Output)

119

4.95

Regression Plots for Fa (Train /Test/Validate)

119

4.96

Response Surface Plot for Fa

119

4.97

Training Error Plots for Fr (Target vs Output)

120

4.98

Testing Error Plots for Fr (Target vs Output

120

4.99

Validation Error Plots for Fr (Target vs Output)

120

4.100

Regression Plots for Fr (Train /Test/Validate)

120

4.101

Response Surface Plot for Fr

120

4.102

Training Error Plots for Tf (Target vs Output)

121

4.103

Testing Error Plots for Tf (Target vs Output)

121

4.104

Validation Error Plots for Tf (Target vs Output)

121

4.105

Regression Plots for Tf (Train /Test/Validate)

121

4.106

Response Surface Plot for Tf

121

ANFIS Subtractive Cluster plots For 45 HRC 4.107

Training Error Plots for Ra (Target vs Output)

122

4.108

Testing Error Plots for Ra (Target vs Output)

122

4.109

Validation Error Plots for Ra (Target vs Output)

122

4.110

Regression Plots for Ra (Train /Test/Validate)

122

viii

4.111

Response Surface Plot for Ra

122

4.112

Training Error Plots for Ft (Target vs Output)

123

4.113

Testing Error Plots for Ft (Target vs Output)

123

4.114

Validation Error Plots for Ft (Target vs Output)

123

4.115

Regression Plots for Ft (Train /Test/Validate)

123

4.116

Response Surface Plot for Ft

123

4.117

Training Error Plots for Fa (Target vs Output)

124

4.118

Testing Error Plots for Fa (Target vs Output)

124

4.119

Validation Error Plots for Fa (Target vs Output)

124

4.120

Regression Plots for Fa (Train /Test/Validate)

124

4.121

Response Surface Plot for Fa

124

4.122

Training Error Plots for Fr (Target vs Output)

125

4.123

Testing Error Plots for Fr (Target vs Output

125

4.124

Validation Error Plots for Fr (Target vs Output)

125

4.125

Regression Plots for Fr (Train /Test/Validate)

125

4.126

Response Surface Plot for Fr

125

4.127

Training Error Plots for Tf (Target vs Output)

126

4.128

Testing Error Plots for Tf (Target vs Output)

126

4.129

Validation Error Plots for Tf (Target vs Output)

126

4.130

Regression Plots for Tf (Train /Test/Validate)

126

4.131

Response Surface Plot for Tf

126

4.132

Developed ANFIS Fuzzy C Mean Clustering architect

128

ix

ANFIS FCM plots For 35 HRC 4.133

Training Error Plots for Ra (Target vs Output)

130

4.134

Testing Error Plots for Ra (Target vs Output)

130

4.135

Validation Error Plots for Ra (Target vs Output)

130

4.136

Regression Plots for Ra (Train /Test/Validate)

130

4.137

Response Surface Plot for Ra

130

4.138

Training Error Plots for Ft (Target vs Output)

131

4.139

Testing Error Plots for Ft (Target vs Output)

131

4.140

Validation Error Plots for Ft (Target vs Output)

131

4.141

Regression Plots for Ft (Train /Test/Validate)

131

4.142

Response Surface Plot for Ft

131

4.143

Training Error Plots for Fa (Target vs Output)

132

4.144

Testing Error Plots for Fa (Target vs Output)

132

4.145

Validation Error Plots for Fa (Target vs Output)

132

4.146

Regression Plots for Fa (Train /Test/Validate)

132

4.147

Response Surface Plot for Fa

132

4.148

Training Error Plots for Fr (Target vs Output)

133

4.149

Testing Error Plots for Fr (Target vs Output

133

4.150

Validation Error Plots for Fr (Target vs Output)

133

4.151

Regression Plots for Fr (Train /Test/Validate)

133

4.152

Response Surface Plot for Fr

133

4.153

Training Error Plots for Tf (Target vs Output)

134

x

4.154

Testing Error Plots for Tf (Target vs Output)

134

4.155

Validation Error Plots for Tf (Target vs Output)

134

4.156

Regression Plots for Tf (Train /Test/Validate)

134

4.157

Response Surface Plot for Tf

134

ANFIS FCM plots For 45 HRC 4.158

Training Error Plots for Ra (Target vs Output)

135

4.159

Testing Error Plots for Ra (Target vs Output)

135

4.160

Validation Error Plots for Ra (Target vs Output)

135

4.161

Regression Plots for Ra (Train /Test/Validate)

135

4.162

Response Surface Plot for Ra

135

4.163

Training Error Plots for Ft (Target vs Output)

136

4.164

Testing Error Plots for Ft (Target vs Output)

136

4.165

Validation Error Plots for Ft (Target vs Output)

136

4.166

Regression Plots for Ft (Train /Test/Validate)

136

4.167

Response Surface Plot for Ft

136

4.168

Training Error Plots for Fa (Target vs Output)

137

4.169

Testing Error Plots for Fa (Target vs Output)

137

4.170

Validation Error Plots for Fa (Target vs Output)

137

4.171

Regression Plots for Fa (Train /Test/Validate)

137

4.172

Response Surface Plot for Fa

137

4.173

Training Error Plots for Fr (Target vs Output)

138

4.174

Testing Error Plots for Fr (Target vs Output

138

xi

4.175

Validation Error Plots for Fr (Target vs Output)

138

4.176

Regression Plots for Fr (Train /Test/Validate)

138

4.177

Response Surface Plot for Fr

138

4.178

Training Error Plots for Tf (Target vs Output)

139

4.179

Testing Error Plots for Tf (Target vs Output)

139

4.180

Validation Error Plots for Tf (Target vs Output)

139

4.181

Regression Plots for Tf (Train /Test/Validate)

139

4.182

Response Surface Plot for Tf

139

Comparison Error Plots of Neural Network and ANFIS Prediction FOR 35 HRC 4.183

Error Estimation Plots For R a

140

4.184

Error Estimation Plots For F t

140

4.185

Error Estimation Plots For F a

141

4.186

Error Estimation Plots For F r

141

For 45 HRC 4.187

Error Estimation Plots For R a

141

4.188

Error Estimation Plots For F t

141

4.189

Error Estimation Plots For F a

141

4.190

Error Estimation Plots For F r

141

Error Plots of ANFIS (Grid Partitioning Clustering) Results

xii

FOR 35 HRC 4.191

Error Estimation Plots For R a

142

4.192

Error Estimation Plots For F t

142

4.193

Error Estimation Plots For F a

142

4.194

Error Estimation Plots For F r

142

FOR 45 HRC 4.195

Error Estimation Plots For R a

142

4.196

Error Estimation Plots For F t

142

4.197

Error Estimation Plots For F a

143

4.198

Error Estimation Plots For F r

143

Error Plots of ANFIS (Subtractive Clustering) FOR 35 HRC 4.199

Error Estimation Plots For R a

143

4.200

Error Estimation Plots For F t

143

4.201

Error Estimation Plots For F a

143

4.202

Error Estimation Plots For F r

143

FOR 45 HRC 4.203

Error Estimation Plots For R a

144

4.204

Error Estimation Plots For F t

144

4.205

Error Estimation Plots For F a

144

4.206

Error Estimation Plots For F r

144

Error Plots of ANFIS (Fuzzy C-Mean Clustering))

xiii

FOR 35 hrc 4.207

Error Estimation Plots For R a

144

4.208

Error Estimation Plots For F t

144

4.209

Error Estimation Plots For F a

145

4.210

Error Estimation Plots For F r

145

FOR 45 HRC 4.211

Error Estimation Plots For R a

145

4.212

Error Estimation Plots For F t

145

4.213

Error Estimation Plots For F a

145

4.214

Error Estimation Plots For F r

145

5.1

Chapter 5 flow chart

147

5.2

Developed NSGA–NN architect

149

NSGA-NN 35HRC 5.3

Performance plot of Network

151

5.4

Training state of Network at each epoch

151

5.5

Training error in Ra

151

5.6

Regression fit plot for Ra

151

5.7

Training error in Ft

151

5.8

Regression fit plot for Ft

151

5.10

Training error in Fa

152

5.11

Regression fit plot for Fa

152

5.12

Training error in Fr

152

xiv

5.13

Regression fit plot for Fr

152

5.14

Training error in Tf

152

5.15

Regression fit plot for Tf

152

Plots for NSGA-NN 45 HRC 5.16

Performance plot of Network

153

5.17

Training state of Network at each epoch

153

5.18

Training error in Ra

153

5.19

Regression fit plot for Ra

153

5.20

Training error in Ft

153

5.21

Regression fit plot for Ft

153

5.22

Training error in Fa

154

5.23

Regression fit plot for Fa

154

5.24

Training error in Fr

154

5.25

Regression fit plot for Fr

154

5.26

Training error in Tf

154

5.27

Regression fit plot for Tf

154

5.28

. SI-NN collaborative combination

155

5.29

Applied SI-NN synergy architect

155

Results of NSGA-NN 35 HRC 5.30

Performance plot of Network

157

5.31

Training state of Network at each epoch

157

5.32

Training error in Ra

157

xv

5.33

Regression fit plot for Ra

157

5.34

Training error in Ft

157

5.35

Regression fit plot for Ft

157

5.36

Training error in Fa

158

5.37

Regression fit plot for Fa

158

5.38

Training error in Fr

158

5.39

Regression fit plot for Fr

158

5.40

Training error in Tf

158

5.41

Regression fit plot for Tf

158

5.42

GA based ANFIS (FCM) applied Strategy

159

ANFIS-GA FCM plots For 35 HRC 5.43

Training Error Plots for Ra (Target vs Output)

160

5.44

Testing Error Plots for Ra (Target vs Output)

160

5.45

Regression Plots for Ra (Train /Test/Validate)

161

5.46

Response Surface Plot for Ra

161

5.47

Training Error Plots for Ft (Target vs Output)

161

5.48

Testing Error Plots for Ft (Target vs Output)

161

5.49

Regression Plots for Ft (Train /Test/Validate)

162

5.50

Response Surface Plot for Ft

162

5.51

Training Error Plots for Fa (Target vs Output)

162

5.52

Testing Error Plots for Fa (Target vs Output)

162

5.53

Regression Plots for Fa (Train /Test/Validate)

163

xvi

5.54

Response Surface Plot for Fa

163

5.55

Training Error Plots for Fr (Target vs Output)

163

5.56

Testing Error Plots for Fr (Target vs Output

163

5.57

Regression Plots for Fr (Train /Test/Validate)

164

5.58

Response Surface Plot for Fr

164

5.59

Training Error Plots for Tf (Target vs Output)

164

5.60

Testing Error Plots for Tf (Target vs Output)

164

5.61

Regression Plots for Tf (Train /Test/Validate)

165

5.62

Response Surface Plot for Tf

165

FCM plots For 45 HRC 5.63

Training Error Plots for Ra (Target vs Output)

166

5.64

Testing Error Plots for Ra (Target vs Output)

166

5.65

Regression Plots for Ra (Train /Test/Validate)

166

5.66

Response Surface Plot for Ra

166

5.67

Training Error Plots for Ft (Target vs Output)

167

5.68

Testing Error Plots for Ft (Target vs Output)

167

5.69

Regression Plots for Ft (Train /Test/Validate)

167

5.70

Response Surface Plot for Ft

167

5.71

Training Error Plots for Fa (Target vs Output)

168

5.72

Testing Error Plots for Fa (Target vs Output)

168

5.73

Regression Plots for Fa (Train /Test/Validate)

168

5.74

Response Surface Plot for Fa

168

xvii

5.75

Training Error Plots for Fr (Target vs Output)

169

5.76

Testing Error Plots for Fr (Target vs Output

169

5.77

Regression Plots for Fr (Train /Test/Validate)

169

5.78

Response Surface Plot for Fr

169

5.79

Training Error Plots for Tf (Target vs Output)

170

5.80

Testing Error Plots for Tf (Target vs Output)

170

5.81

Regression Plots for Tf (Train /Test/Validate)

170

5.82

Response Surface Plot for Tf

170

5.83

PS0-ANFIS applied strategy

171

PSO based ANFIS (FCM) Plots For 35 HRC 5.84

Training Error Plots for Ra (Target vs Output)

172

5.85

Testing Error Plots for Ra (Target vs Output)

172

5.86

Regression Plots for Ra (Train /Test/Validate)

172

5.87

Response Surface Plot for Ra

172

5.88

Training Error Plots for Ft (Target vs Output)

173

5.89

Testing Error Plots for Ft (Target vs Output)

173

5.90

Regression Plots for Ft (Train /Test/Validate)

173

5.91

Response Surface Plot for Ft

173

5.92

Training Error Plots for Fa (Target vs Output)

174

5.93

Testing Error Plots for Fa (Target vs Output)

174

5.94

Regression Plots for Fa (Train /Test/Validate)

174

5.95

Response Surface Plot for Fa

174

xviii

5.96

Training Error Plots for Fr (Target vs Output)

175

5.97

Testing Error Plots for Fr (Target vs Output

175

5.98

Regression Plots for Fr (Train /Test/Validate)

175

5.99

Response Surface Plot for Fr

175

5.100

Training Error Plots for Tf (Target vs Output)

176

5.101

Testing Error Plots for Tf (Target vs Output)

176

5.102

Regression Plots for Tf (Train /Test/Validate)

176

5.103

Response Surface Plot for Tf

176

PSO based ANFIS (FCM) Plots FOR 45 HRC 5.104

Training Error Plots for Ra (Target vs Output)

177

5.105

Testing Error Plots for Ra (Target vs Output)

177

5.106

Regression Plots for Ra (Train /Test/Validate)

177

5.107

Response Surface Plot for Ra

177

5.108

Training Error Plots for Ft (Target vs Output)

178

5.109

Testing Error Plots for Ft (Target vs Output)

178

5.110

Regression Plots for Ft (Train /Test/Validate)

178

5.111

Response Surface Plot for Ft

178

5.112

Training Error Plots for Fa (Target vs Output)

179

5.113

Testing Error Plots for Fa (Target vs Output)

179

5.114

Regression Plots for Fa (Train /Test/Validate)

179

5.115

Response Surface Plot for Fa

179

5.116

Training Error Plots for Fr (Target vs Output)

180

xix

5.117

Testing Error Plots for Fr (Target vs Output

180

5.118

Regression Plots for Fr (Train /Test/Validate)

180

5.119

Response Surface Plot for Fr

180

5.120

Training Error Plots for Tf (Target vs Output)

181

5.121

Testing Error Plots for Tf (Target vs Output)

181

5.122

Regression Plots for Tf (Train /Test/Validate)

181

5.123

Response Surface Plot for Tf

181

Comparison Error Plots of NN-NSGA Prediction FOR 35 HRC 5.124

Error Estimation Plots For R a

182

5.125

Error Estimation Plots For F t

182

5.126

Error Estimation Plots For F a

183

5.127

Error Estimation Plots For F r

183

For 45 HRC 5.128

Error Estimation Plots For R a

183

5.129

Error Estimation Plots For F t

183

5.130

Error Estimation Plots For F a

183

5.131

Error Estimation Plots For F r

183

Error Plots of PSO-NN Results FOR 35 HRC 5.132

Error Estimation Plots For R a

184

5.133

Error Estimation Plots For F t

184

5.134

Error Estimation Plots For F a

184

xx

5.135

Error Estimation Plots For F r

184

Error Plots of ANFIS (FCM)-GA FOR 35 HRC 5.136

Error Estimation Plots For R a

184

5.137

Error Estimation Plots For F t

184

5.138

Error Estimation Plots For F a

185

5.139

Error Estimation Plots For F r

185

Error Plots of ANFIS (FCM)-GA FOR 45 HRC 5.140

Error Estimation Plots For R a

185

5.141

Error Estimation Plots For F t

185

5.142

Error Estimation Plots For F a

185

5.143

Error Estimation Plots For F r

185

Error Plots of ANFIS (FCM)-PSO FOR 35 HRC 5.144

Error Estimation Plots For R a

186

5.145

Error Estimation Plots For F t

186

5.146

Error Estimation Plots For F a

186

5.147

Error Estimation Plots For F r

186

Error Plots of ANFIS (FCM)-PSO FOR 45 HRC 5.148

Error Estimation Plots For R a

186

5.149

Error Estimation Plots For F t

186

5.150

Error Estimation Plots For F a

187

5.151

Error Estimation Plots For F r

187

xxi

LIST OF TABLES Table No`

Name of Table

Pg. No.

3.1

Machining Constraints

30

3.2

NSGA II Setting

41

3.3

Results of NSGA II family of best solution for AISI 4340 35 HRC Steel

48

3.4

Results of NSGA II family of best solution for AISI 4340 45 HRC Steel

50

3.5

SPEA2 Parameter Setting

54

3.6

Results of SPEA II family of best solution for AISI 4340 35 HRC Steel

57

3.7

Results of SPEA II family of best solution for AISI 4340 45 HRC Steel

58

3.8

PSO Setting

67

3.9

MOPSO family of optimal solutions for 35 HRC AISI 4340 steel

72

3.10

MOPSO family of optimal solutions for 35 HRC AISI 4340 steel

74

3.11

Diversity of Evolutionary Algorithm

78

4.2.4 (a)

Description of Neural network

88

4.2.4(b)

Calibrated weights and bias above Neural Network

89

4.3.4(a)

Grid Partioning Fuzzy Structure

102

4.3.4(b)

Statistical Results of ANFIS Grid Partioning Cluster for 35 HRC and 45HRC

102

4.3.16(a)

Subtractive Fuzzy Structure

115

4.3.16(b)

Statistical Error analysis of ANFIS Subtractive clustering for 35 HRC and 45 HRC

116

xxii

4.3.26 (a)

Fuzzy structure

127

4.3.26(b)

Statistical Results of ANFIS FCM for 35 HRC and 45HRC

129

4.4.1

Statistical Comparison of Neural Network and ANFIS Prediction Results with Experimental Statistics for AISI 4340 Steel 35hrc

140

5.2.1 (a)

Description of NSGA-NN

150

5.2.1 (b)

Calibrated weigths and bias for NSGA-NN 35 HRC and 45HRC Steel

150

5.3.1 (a)

Description of PSO-NN

156

5.3.2 (b) 5.4.1 (a)

Calibrated weigths and bias for 35 HRC and 45HRCSteel Statistical Error analysis of GA and PSO based ANFIS (FCM) for 35 HRC and 45HRC

156 159

5.6.1 (a)

Statistical Comparison of Prediction Results with Experimental Statistics for AISI 4340 Steel 35HRC and 45HRC

182

6.1

Tradeoffs among forces for Surface roughness and Tool life

190

6.2

Tradeoffs among forces for Surface roughness and Tool life

192

6.3

Tradeoffs among forces for Surface roughness and Tool life

193

6.4

Tradeoffs among forces for Surface roughness and Tool life

194

6.5

Mean error and Standard Deviation between Experimental and Predicted statistics

201

6.6

Table.6.6 MSE and RMSE between Experimental and Predicted statistics

201

6.7

Mean error and Standard Deviation between Experimental and Predicted statistics

205

6.8

Mean Square Error and Root Mean Square Error between Experimental and Predicted statistics

205

xxiii

NOMENCLATURE HRC

Rockwell C Hardness

V, (vc)

Cutting Speed

f

Feed Rate

d

Depth Of Cut

Ra

Surface Roughness

Ft

Tangential Force

Fa

Axial Force

Fr

Radial Force

Tf

Tool Life

AI

Artificial Intelligence

CI

Computational Intelligence

HC

Hard Computation

SC

Soft Computing

FS

Fuzzy Systems

NN

Neural Network

EA

Evolutionary Algorithm

GSO

Global Search Optimization

SI

Swarm Intelligence

GA

Genetic Algorithm

DE

Differential Evolution

CA

Culture Algorithm

PSO

Particle Swarm Optimization

FA

Fire Fly Algorithm

BBO

Biogeography Based Optimization

xxiv

W1/W2/W3

Weights

EA-NN

Evolutionary Based Neural Network

NN-EA

Neural Network Based Evolutionary Algorithm

NSGA

Non-Sorted Domination Based Genetic Algorithm

PESA

Pareto Envelope Based Selection Algorithm

SPEA

Strength Pareto Based Evolutionary Algorithm

MF

Membership Function

ANFIS

Adaptive Neuro-Fuzzy Interface System

ACO

Ant Colony Algorithm

HLGA

Hybrid Adaptive Learning Based Genetic Algorithm

Pmt

Probability of m bits in K sting length to Mutate

Stn.d , rt n,d X t , Vt

Random number generators Particle Position and Velocity

Gt

Global Best attractor

 n ', n

Local Guide attractor

Lt

nt ,d

Swarm Potential

Sij

Layers



Membership cluster vector

d ( xi , j )

Dissimilarity Function

J q ( , u)

Cluster Function

xxv

CONTENTS Sr.

Title

Pg. No

No Acknowledgement

i

List of Figures

ii

List of Tables

xix

Nomenclature Contents Abstract

xxi xxiii xxviii

1

INTRODUCTION

1.1

Computational Intelligence (CI)

1

1.2

Approaches to Computational Intelligence

2

1.2.1

Fuzzy logic

4

1.2.2

Neural Network

4

1.2.3

Evolutionary Computing

4

1.2.4

Learning Theory

5

1.2.5

Probabilistic Methods

5

1.2.6

Swarm Intelligence

6

1.2.7

Global Search Optimization

6

1.3

Synergies of Computational Intelligence Techniques

6

1.4

Applications of Computational Intelligence

8

1.4.1

Application of NN

8

1.4.2

Application of Evolutionary Systems

9

1.4.3

Application of Fuzzy system

9

1.5

Overview of the chapter

10

1.6

Problem Statement

10

1.7

Objectives

10

1.8

Methodology

11

1.9

Thesis Outline

12

2

LITERATURE REVIEW

xxvi

2.1

Literature Review

14

2.2

Literature Summary

27

3

GLOBAL SEARCH ALGORITHMS FOR MULTI-OBJECTIVE

3.1

Introduction

28

3.2

Multi-objective Optimization

29

3.3

Application of MOOPs to Machining system

30

3.3.1

Machining Model (Surface Roughness Cutting force components material

31

3.3.2

hardness: 35 HRC).

Machining Model (Surface Roughness Cutting force components work

31

3.3.3

material hardness: 45 HRC).

Tool life model for 35 HRC

32

3.4

Evolutionary Algorithms (EAs)

32

3.4.1

Mathematical Formulation of Evolutionary Algorithms

33

3.4.1.

Definition of Evolutionary systems

34

3.4.1.

1 3.4.1.

Convergence Analysis of Evolutionary Algorithm

34

Criteria for mutation

35

3 3.4.1.

Criteria for Crossover

37

4 points from convergence analysis Key

40

3.4.2 3.5

3.6

NSGA II Algorithm

41

3.5.1

Initialize Variables and Evaluate Objectives

41

3.5.2

Non_dominated_sort

42

3.5.3

Selection.

45

3.5.4

Genetic Operators

45

3.5.5

Recombination of parent and off springs

47

3.5.6

Plots for NSGA II results (35 HRC)

50

3.5.7

Plots for NSGA II results (45 HRC)

52

Strength Pareto Evolutionary Algorithm 2

52

3.6.1

Initialize Variables and Evalaute Objectives

54

3.6.2

Tournament Selection

55

3.6.3

Genetic Operator

55

3.6.4

Plots for SPEA 2 results( 35 HRC)

57

3.6.5

Plots for SPEA 2 results( 45 HRC)

59

xxvii

3.7

Swarm Intelligence

60

3.8

Mathematical Formulation of PSO Algorithm

60

3.8.1

Typical Initialization strategy

62

3.8.2

Topologies of PSO

62

3.8.3

Definitions of Swarm topology

63

3.8.4

Convergence criteria

63

3.8.5

Criteria for Inertia clamping and acceleration co-efficeint

64

3.9

3.10

4

PSO Algorithm

66

3.9.

Initialize population and Evaluate fitness

67

1 3.9.

Create Grid Index

68

2 3.9.

Select Leader

69

3 3.9.

Delete extra elements

70

3.9.

4

Swarm Movement

71

6 3.9.

Plots for MOPSO results (35 HRC)

73

73.9.

Plots for MOPSO results (45 HRC)

75

8 Comparison between EA and SI technique

76

3.10.

Comparison Based on Spectrum of solution space

77

1 3.10.

Comparison Based on Diversity in solution space.

78

2 PREDICTION MODELS FOR MACHINING SYSTEM

THROUGH INTELLIGENT LEARNING TECHNIQUES 4.1

Introduction

80

4.2

Neural Network

81

4.2.1

Feed forward Neural network

81

4.2.2

Mathematical background of neural network

82

4.2.2.1

Gradient Descent Approach

83

4.2.3

Key notes form feed forward analysis

87

4.2.4

Multi-layer Perceptron for Turning of AISI 4340 Steel

87

4.2.5

Results of perceptron for 35 HRC Steel

89

4.2.6

Results of perceptron for 45 HRC Steel

92

4.3

Adaptive Neuro-Fuzzy Interference System (ANFIS)

94

4.3.1

96

Hybrid learning in ANFIS

xxviii

4.3.2

Backpropogation Learning

97

4.3.3

Fuzzy clustering Algorithms

98

4.3.4

Grid Partition clustering based Adaptive Neuro-Fuzzy Interference System

101

4.3.5

ANFIS Grid Partioning Cluster Plots For Ra 35 HRC

103

4.3.6

ANFIS Grid Partioning Cluster Plots For Ft 35 HRC

104

4.3.7

ANFIS Grid Partioning Cluster Plots For Fa 35 HRC

105

4.3.8

ANFIS Grid Partioning Cluster Plots For Fr 35 HRC

106

4.3.9

ANFIS Grid Partioning Cluster Plots For Tf 35 HRC

108

4.3.10

ANFIS Grid Partioning Cluster Plots For Ra 45 HRC

109

4.3.11

ANFIS Grid Partioning Cluster Plots For Ft 45 HRC

110

4.3.12

ANFIS Grid Partioning Cluster Plots For Fa 45 HRC

111

4.3.13

ANFIS Grid Partioning Cluster Plots For Fr 45 HRC

112

4.3.14

ANFIS Grid Partioning Cluster Plots For Tf 45 HRC HRC

113

4.3.15

Subtractive Clustering

114

4.3.16

ANFIS Substractive Cluster Plots For Ra 35 HRC

117

4.3.17

ANFIS Substractive Cluster Plots For Ft 35 HRC

118

4.3.18

ANFIS Substractive Cluster Plots For Fa 35 HRC

119

4.3.19

ANFIS Substractive Cluster Plots For Fr 35 HRC

120

4.3.20

ANFIS Substractive Cluster Plots For Tf 35 HRC

121

4.3.21

ANFIS Substractive Cluster Plots For Ra 45 HRC

122

4.3.22

ANFIS Substractive Cluster Plots For Ft 45 HRC

123

4.3.23

ANFIS Substractive Cluster Plots For Fa 45 HRC

124

4.3.24

ANFIS Substractive Cluster Plots For Fr 45 HRC

125

4.3.25

ANFIS Substractive Cluster Plots For Tf 45 HRC

126

4.3.26

Fuzzy C Mean Clustering

127

4.3.27

ANFIS

130

4.3.28

ANFIS FCM Cluster Plots For Ft 35 HRC

131

4.3.29

ANFIS FCM Cluster Plots For Fa 35 HRC

132

4.3.30

ANFIS FCM Cluster Plots For Fr 35 HRC

133

4.3.31

ANFIS FCM Cluster Plots For Tf 35 HRC

134

4.3.32

ANFIS FCM Cluster Plots For Ra 45 HRC

135

FCM Cluster Plots For Ra 35 HRC

xxix

4.4

4.3.33

ANFIS FCM Cluster Plots For Ft 45 HRC

136

4.3.34

ANFIS FCM Cluster Plots For Fa 45 HRC

137

4.3.35

ANFIS FCM ve Cluster Plots For Fr 45 HRC

138

4.3.36

ANFIS FCM Cluster Plots For Tf 45 HRC

139

Comparison of Prediction Results with Experimental Statistics

140

4.4.2

Error Plots of Neural Network Prediction Results with Experimental

140

4.4.3

Statistics for AISI 4340 Steel 35hrc Error Plots of Neural Network Prediction Results with Experimental

141

4.4.4

Statistics for AISI 4340 Steel 45HRC Error Plots of ANFIS (Grid Partitioning Clustering) Results with

142

Experimental Statistics for AISI 4340 Steel 35hrc 4.4.5

Error Plots of ANFIS (Grid Partitioning Clustering) Results with

142

Experimental Statistics for AISI 4340 Steel 45hrc 4.4.6

Error Plots of ANFIS (Subtractive Clustering) Results with Experimental

143

Statistics for AISI 4340 Steel 35hrc 4.4.7

Error Plots of ANFIS (Subtractive Clustering) Results with Experimental

144

Statistics for AISI 4340 Steel 45hrc 4.4.8

Error Plots of ANFIS (Fuzzy C-Mean Clustering)) Results with

144

Experimental Statistics for AISI 4340 Steel 35hrc 4.4.9 4.5

5

Error Plots of ANFIS (Fuzzy C-Mean Clustering)) Results with

Experimental Statistics for AISI 4340 Steel 45hrc Conclusion

145 146

HYBRIDISATION OF C.I SYNERGIES

5.1

Introduction

147

5.2

EA-NN Synergism

148

5.2.1

NSGA combined Neural Network

149

5.2.2

Results of NSGA-NN 35 HRC Steel

151

5.2.3

Results of NSGA-NN for 45 HRC Steel

153

5.3

SI-NN synergism

154

5.3.1

PSO combined Neural Network

155

5.3.2

Results of PSO-NN for 35 HRC Steel

157

xxx

5.4

5.5

5.6

Synergies of EA and ANFIS

158

5.4.1

ANFIS GA

159

5.4.2

GA based ANFIS (FCM) Plots For Ra 35 HRC

169

5.4.3

GA based ANFIS (FCM) Plots For Ft 35 HRC

161

5.4.4

GA based ANFIS (FCM) Plots For Fa 35 HRC

162

5.4.5

GA based ANFIS (FCM) Plots For Fr 35 HRC

163

5.4.6

GA based ANFIS (FCM) Plots For Tf 35 HRC

164

5.4.7

GA based ANFIS (FCM) Plots For Ra 45 HRC

166

5.4.8

GA based ANFIS (FCM) Plots For Ft 45 HRC

167

5.4.9

GA based ANFIS (FCM) Plots For Fa 45 HRC

168

5.4.10

GA based ANFIS (FCM) Plots For Fr 45 HRC

169

5.4.11

GA based ANFIS (FCM) Plots For Tf 45 HRC

170

PSO based ANFIS (FCM) 35 HRC and 45HRC

171

5.5.1

PSO based ANFIS (FCM)

171

5.5.2

PSO based ANFIS (FCM) Plots For Ra 35 HRC

172

5.5.3

PSO based ANFIS (FCM) Plots For Ft 35 HRC

173

5.5.4

PSO based ANFIS (FCM) Plots For Fa 35 HRC

174

5.5.5

PSO based ANFIS (FCM) Plots For Fr 35 HRC

175

5.5.6

PSO based ANFIS (FCM) Plots For Tf 35 HRC

176

5.5.8

PSO based ANFIS (FCM) Plots For Ra 45 HRC

177

5.5.9 5.5.10

PSO based ANFIS (FCM) Plots For Ft 45 HRC PSO based ANFIS (FCM) Plots For Fa 45 HRC

178 179

5.5.11

PSO based ANFIS (FCM) Plots For Fr 45 HRC

180

5.5.12

PSO based ANFIS (FCM) Plots For Tf 45 HRC

181

Comparison of Prediction Results with Experimental Statistics

182

5.6.2

Error Plots of NSGA-NN Prediction Results with Experimental Statistics

182

5.6.3

for AISI 4340 Steel 35HRC Error Plots of NSGA-NN Prediction Results with Experimental Statistics

183

5.6.4

for AISI 4340 Steel 45HRC Error Plots of PSO-NN Results with Experimental Statistics for AISI 4340

184

5.6.5

Steel 35HRC Error Plots of PSO-NN Results with Experimental Statistics for AISI 4340

184

Steel 45HRC

xxxi

5.6.6

Error Plots of GA based ANFIS (Fuzzy C-Mean Clustering)) Results with

185

Experimental Statistics for AISI 4340 Steel 45HRC 5.6.7

Error Plots of PSO based ANFIS (Fuzzy C-Mean Clustering)) Results with

186

5.6.8

Experimental Statistics for AISI 4340 Steel 35HRC Error Plots of PSO based ANFIS Prediction Results with Experimental

186

5.7

Statistics for AISI 4340 Steel 45HRC Conclusion

6

RESULTS AND DISCUSSION

6.1

Global Search Optimization

189

6.1.1

189

6.1.2

6.1.3

6.1.4

6.1.5 6.2

6.1.1 (a)

For AISI 4340 35HRC

189

6.1.1(b)

For AISI 4340 45HRC

191

Results for SPEA 2

192

6.1.2 (a)

For AISI 4340 35HRC

192

6.1.2 (b)

For AISI 4340 45HRC

193

Results for PSO

194

6.1.3(a)

For AISI 4340 35HRC

194

6.1.3(b)

For AISI 4340 45HRC

195

Comparison between EA and SI

196

6.1.4 (a)

196

Comparison among the Solution spectrum

Evident from the literature

197

Intelligent Learning Techniques

198

6.2.1

Neural Network

198

6.2.2

Adaptive Neuro Fuzzy Interference Technique

199

6.2.2(a)

ANFIS Grid partition

199

6.2.2 (b)

ANFIS Subtractive Clustering

200

6.2.2 (c)

ANFIS Fuzzy C-mean Clustering

200

6.2.3 6.3

NSGAII

187

Comparative Evaluation of the predictive technique on Experimental statistics

200

Synergies of CI

202

6.3.1

EA-NN

203

6.3.2

SI-NN

203

xxxii

7

6.3.3

ANFIS Synergies

204

6.3.4

Comparative Evaluation of the predictive technique on Experimental statistics

204 207

CONCLUSION

Appendix- A: Conferences and Publications Appendix- B: PG-CON Certificate Appendix- C: Certificates

xxxiii

ABSTRACT Obtaining the process parameters to optimize machining performance is vital in machining execution since they significantly affect the productivity rate, cost and quality of machining operation. Although process parameters optimization has been widely investigated for conventional machining operation, very limited work is reported on optimization of hard turning using evolutionary algorithm. In this work multi-objective optimization of hard turning with evolutionary optimization technique is attempted (i.e, NSGA II, SPEA II, PSO) during hard turning of hardened AISI 4340 Steel at different hardness level (35 and 45 HRC) with experimental based multi regression models as objective functions. The process variables are cutting speed, feed rate, and depth of cut with appropriate constraints. Further -more different intelligent learning techniques (i.e, Neural network, Adaptive Neural network based fuzzy learning) were applied using (EA-NN and NN-EA) supportive combinations to recognize the pattern of optimal solution through learning This learnt prediction model is compared with experimental statistics a comparative evaluation is made which is in good agreement with experimental data.

Keywords: Multi-Objective Optimization, Adaptive Neuro-Fuzzy Optimization, Hard Turning;

xxxiv

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

Chapter-1 1. Introduction The complexity of design optimization of any dynamic system has many aspects among them the major facets are the ambiguity of objectives, conflicting nature of objectives and many possible solutions this brings an issue in characterizing the difficulty of design optimization task. Any design solution has combination of values for parameters of a solution and the challenge lies in identifying the solution. Considering second issue functionality of obtained solution, the solution should be practical enough, look appealing and have moderate cost. Third issue is contributing to the ambiguity of design optimization is conflicting objectives which inhibits unidirectional solution. To tackle these complexity several attempts are made through conventional methods but the solution in general are partially satisfactory To address this issue new computational approaches are followed which has multi-agent system each agent is defined by its behavior that are classified into various categories in Computational Intelligence.

1.1 Computational Intelligence (CI) Computational Intelligence deals with the design of intelligent agents which act intelligently for goal attainment in any circumstances, flexible enough to adopt changing environment and goals. Computational Intelligence has the ability to comprehend reason, learn, and simulate intelligent behavior in systems for complete knowledge formulation.

Much real-time system behavior cannot be captured exactly through classical mathematical description in spite of complex formulations; moreover complexity of mathematical description inhibits development of system model. Hence it is really advantageous to model a real time system with piece-wise linearity and non-linearity so that the highly complex and un-anticipatory models can be captured by intelligent agents.

Any real time problem has uncertainties involved in it with multiple objectives and the risk in decision making should be such that the performance criteria are maintained even in drastic change, this necessity of capturing the dynamic behavior of system is replacing conventional techniques with intelligent techniques. Computational Intelligent techniques VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 1

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

are thus an illustration of alternative methods to conventional technique when the system knowledge is highly important in system modeling and control. The structure of such systems is determined by experimental evidence where direct input-output response behavior is utilized to develop system model. Intelligent systems are meant for the processes that are not properly defined, complex and stochastic in nature, time varying.

The fundamental property of any intelligent system is that it must sense and reason without prior knowledge about the environment and adapt to the control action in a robust manner. Many attempts have been made to define by different researchers but the property of a system to be computationally intelligent is if it deals with numerical data and has ability of pattern recognition. CI is a subdivision of machine intelligence where subtle difference between the techniques lies in the type of computing. Machine intelligence has two constituents Artificial Intelligence based on hard computing (HC) and Computational Intelligence based on soft computing (SC) [Fuzzy Sets (FS), Neural Network (NN), Evolutionary Algorithms(EA)] (Fig. 1.1) distinguishes clearly the components of Machine intelligence and their components.

Fig. 1.1 Hierarchy of Computational Intelligence

1.2 Approaches to Computational Intelligence

VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 2

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

The core of the computational intelligence is designing process or system model which is not responsive to mathematical modeling since the process exhibits following attributes 

Too complex to represent mathematical model



Models difficult to Compute



Uncertainties in operations



Nonlinear, Stochastic and disturbed in nature

The system is capable of learning to adapt to unknown situations and is able to make predictions about the process status in future time step. CI is a combination of soft computing and numerical technique with methods involving adaptive control, (Fig. 1.2) optimal control, learning theory, fuzzy logic, neural-network, evolutionary computing.

All methods tuned to attain common goal set. There are five elemental methods to CI 

Fuzzy logic



Neural network

Fig. 1.2 Approaches to Computational Intelligence 

Evolutionary computing



Learning theory VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 3

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation



Probabilistic methods



Swarm Intelligence

1.2.1 Fuzzy logic In any real-time process, the measurement, process modeling, and control can never be exact to the theoretical definitions [1]. There is always a certain amount of uncertainties i.e., incompleteness, randomness of data. Fuzzy assimilates human experimental knowledge converts it into engineering model and control. Most process which are illdefined with nonlinearity and uncertainties. The fuzzy logic is more of reasoning and inference technique based on high level linguistic or semantic rules and operations.

1.2.2 Neural Network Neural network neural network is a technique adopted from the biological brain which involves a neuron as a fundamental building block [1]. These neurons receive signals from neighboring neurons through their cell body and transfer the results through a long fiber called an axon. The axons behave like signal conducting device. A similar electrical analogy of biological neural network is artificial neural network which is characterized by computational power learning of real-time data error tolerance, pattern recognition, and generalization capabilities, low-level computational algorithms which manifest good performance in numerical data processing. The learning is in different form supervised, unsupervised, and competitive and reinforcement learning.

1.2.3 Evolutionary Computing Evolutionary computing is the imitation of the process of natural selection in a search procedure based on evolutionary theory of Charles Darwin [1]. The species undergoes reproduction, gives birth to new offspring with features of combating adverse environment and survive The process of natural selection makes sure that the individuals with better fitness have opportunity to make most of the time, with expectations that the offsprings will have similar higher fitness levels. Evolutionary computation uses iterative

VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 4

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

progress and development in a population. This population is then selected in random search to achieve the desired population of solution.

1.2.4 Learning Theory Learning theory is based on the human learning capabilities without much effort in a conventional sense. The mechanism of learning in humans is the process of bringing together cognitive, emotional and environmental effects to acquire enhance or change knowledge, skills. In general, learning is characterized by how the information is input, processed and stored. Learning theories fall into three framework behaviorism cognitive theories and constructivism. Behaviorism is learning based on objectively observable feature learning Cognitive learning is how learning occurs in brain. Constructivism learning is a process in which permutation of existing idea builds a new idea. In most of the machine learning four basic forms of learning si adopted i.e., supervised learning where a mapping of input to desired output is done, unsupervised learning where a set of input feature is modeled and mapping of input to output is done with similar pattern.

Semi-supervised learning, combination of both learned and unlearned datasets are used to generate an appropriate classifier. Reinforcement learning involves decision making on given observation and feedback is taken from the consequence to supervise the learning process [1].

1.2.5 Probabilistic Methods Probabilistic theory is methodology which guides in dealing with the uncertainties and imprecisions. The probabilistic methods involve a space consisting probabilities of whole system. The uncertainties of complex dynamic system are calculated and combined behavior of system is analyzed for the degree of causticity. The chaotic behavior of system is estimated by the past. In general, the chaotic behavior of system grows exponentially with time [1].

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

1.2.6 Swarm Intelligence Swarm systems are based on behavior of school of birds, insects, fireflies where a flocks of birds twisting, V-shaped structure of migrating geese, winter birds hunting for food, the synchronized flashing of fireflies are tried to imitating. The well-choreographed collective behavior without any leader is adopted to search for optimal solutions for instance, ants living in colony, their behavior is driven by the goal of colony survival instead of individual survival, While searching for food ants initially explores surrounding nests in random manner. A similar behavior is observed with flocks of birds where a leader keeps guiding the flock to updated food location [1].

1.2.7 Global Search Optimization Both EA and SI together form a broader class of optimization driven search techniques defined as global search optimization technique as shown in Fig.1.3 below.

Fig. 1.3 Global Search Optimization Hierarchy 1.3 Synergies of Computational Intelligence Techniques The different combination of all the methodologies can be used to design intelligent systems. Though a particular technique might be excellent in approximate reasoning and modeling uncertainty but may not be so good at learning and adopting with experimental data. A combined approach with computational intelligence technique and their implementation can help in designing better intelligent agents.

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

Different forms of synergisms (Fig. 1.4) of fuzzy logic, neural networks and evolutionary algorithms the common forms of weakly coupled synergism of neural network and evolutionary algorithms include training designing, optimizing architecture and parameters of neural networks and feature selection scaling training data for neural network using evolutionary algorithms. A strongly coupled synergism between the two methodologies where genetic operators are represented in the form of neural network and the epochs are meant to be the generations of evolutions [1].

Synergisms of neural networks and fuzzy systems have proven to be very powerful for system modeling and learning. In weakly coupled synergism, neural networks and fuzzy system work independently towards a common goal where neural network assist fuzzy logic to form rules and tuning membership functions. In strongly coupled synergism fuzzy system assist neurons to assign weights to its membership functions where neural network learns data over the epoch. There is other synergism possible between swarm intelligence, fuzzy systems, evolutionary algorithms and neural network.

Fig. 1.4 Synergies of Computational Intelligence

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

1.4 Applications of Computational Intelligence The system designed on the basis of computational intelligence elucidates data irrespective of its orientation and it has upper hand over the classical domain based analysis where the data computing and processing becomes difficult. CI exhibits characteristic of domain independence where same methodology can be applied to different fields For instance both neural network and fuzzy logic can be applied to solve a problem the only difference would be in the performance.

Neural network can be applied in five ways i.e., data analysis, classifier, clustering, pattern recognition, control strategy neural network has been successfully applied in problems behaving non-linearly whereas fuzzy logic has been applied to appliances where a module control is required the most adaptive implementation is done on stabilizing an unsteady image. Fuzzy expert systems are applied to medical systems, diagnostic, scheduling, and financial systems.

1.4.1 Application of NN 1. In aerospace neural networks are applied to high performance autopilot flight path simulation, aircraft control systems, fault detection system.

2. In automotive neural networks are used for automatic guidance system.

3. In Banking, Financial and Business it is applied for document reading, credit application evaluation, credit, and activity.

4. In defense, it is used for weapon steering, target tracking, object discrimination, facial recognition.

5. In Industrial, Manufacturing and Electronics to control, process identification, machine diagnosis, quality inspection.

6. In Medical cancer cell analysis, EEG and ECG signal analysis, optimization of transplantation.

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

7. In speech, applied for speech recognition, compression, and text to speech synthesis.

8. Telecommunication applied for image and data compression, speech processing, realtime translation of spoken language.

1.4.2 Application of Evolutionary Systems 1. In automotive design, including research on composite material design and multiobjective design of automotive components for crashworthiness, weight savings and other characteristics.

2. In optimizing the structural and operational design of Industry and Manufacturing systems. For optimization of mechanical systems like heat exchanger, turbines, flywheel and computer-assisted engineering design.

3. In automotive design of mechatronic systems using bond graphs and industrial equipment design using catalogs of exemplar lever pattern.

4. In Travelling Salesman Problem (TSP) and sequences scheduling.

5. In Control-gas pipeline, pole balancing, missile evasion, design-semiconductor layout, aircraft design, keyboard configuration, communication network.

6. In Combinatorial Optimization, Scheduling applications, including job-shop scheduling with objective to schedule jobs in both sequence-dependent and non-sequence dependent for maximum production volume.

1.4.3 Application of Fuzzy system 1. Fuzzy systems are used in automobiles and vehicle subsystems such as automatic transmissions, ABS and cruise control.

2. In air conditioners, washing machines and other home appliances.

3. In digital image processing, such as edge detection and video gam artificial intelligence.

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

4. In Pattern recognition in remote sensing, microcontrollers and microprocessors.

6. In Hydrometeor classification for polarimetric weather radar.

1.5 Overview of the chapter In this chapter a complete overview of ideas and foundations of computational Intelligence and their methodologies are discussed (i.e., Evolutionary Systems, Neural networks and Fuzzy systems). The Possible interactions of these techniques and the type of synergism in which the limitations of one technique can be surpassed by combination of two or more CI techniques is discussed. Depending on the compatibility of individual methodologies, a better computational model can be built which could complement respective methodologies.

1.6 Problem Statement Computational Intelligence has captured the attention of major manufacturing segment for its robust and dynamic adaptability, flexibility, versatility in problem solving and decision making skill. Computational Intelligence has potential competency to capture and compare real time data. Application of Computational Intelligence is not yet explored in machining problems to its fullest potential Most of the machining problems are modeled and optimized through conventional techniques.

In the present work different soft computing technique is applied over machining system to optimize machining performance and recognize machining pattern with a case study from literature [2-9] in which extensive machinability aspects of AISI 4340 alloy steel with different machining characteristics during, hard turning. Operation with different faceted is discussed.

1.7 Objectives In the present work, an attempt is made to apply the Computational Techniques and their synergies with the objective to optimize and build prediction models for conventional machining system. Different Computational methods are applied over the machining

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

system for optimization and learning of machining data Further, the optimized and learnt data are compared with the result obtained from the literature 1. To optimize hard turning of AISI 4340 steel by applying evolutionary algorithm and swarm intelligence (i.e., Non-Dominated Sorting Genetic Algorithm, Strength Pareto, Particle Swarm Optimization technique) and compare optimized results with literature.

2. To develop prediction models for hard turning of AISI 4340 steel by applying intelligent learning techniques i.e., Neural network (NN) and Adaptive NeuroFuzzy systems (ANFIS). (Tagaki-Sugeno fuzzy based Neural network) and compare the prediction model with the experimental statistics.

3. To develop supportive-combination of evolutionary based prediction neural network, furthermore the predicted model will be tested with experimental data.

4. To develop Adaptive Neuro-Fuzzy based evolutionary estimator for predicting optimized parameters for AISI 4340 steel hard turning machining operation.

1.8 Methodology The adapted methodology is developed (Fig. 1.5) to achieve the above mentioned objective with focus to optimize and develop prediction models.

1. Methodology follows two directions where in one division optimization and prediction models are applied exclusively and another division supportive– combination of optimization-prediction models is applied.

2. After obtaining results from each technique comparative evaluation of respective techniques is made with the experimental statistics of hard turning operation.

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

Fig.1.5 Applied Optimization and Learning Methodology 1.9 Thesis Outline The present thesis comprises of six chapters, In chapter 1, a brief introduction of Computer Intelligence along with the various techniques and their combination is discussed, applications of CI techniques how CI can be applied to machining systems, problem statement, objectives, and methodology for the current work has been discussed.

In Chapter 2, a review of literature pertaining to problem and objectives is made and conclusion from literature is drawn.

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In Chapter 3, Optimization techniques is applied to machining model of hard turned AISI 4340 steel both evolutionary and swarm intelligence based algorithm is described in brief, mathematical aspects and the pseudo code along with their results is discussed.

In Chapter 4, applied predictive models i.e., neural network and adaptive neuro fuzzy based learning model are discussed. Mathematical aspects and pseudo code along with their results is discussed.

In Chapter 5, synergies of CI techniques along with their mathematical background and applied pseudo code is described the results of the applied techniques are discussed.

Finally, in Chapter 6 results of applied techniques relative comparison and evaluation with the experimental statistics is discussed. Conclusions of present research work and future scope is briefed.

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CHAPTER-2 2.1 LITERATURE REVIEW The literature was explored with an aim to gather research work of authors who utilized different evolutionary and learning techniques. In the literature authors have utilized optimization and predictive techniques exclusively; multi-objective machining systems were converted to single objective for performance evaluation. Many authors have optimized machining systems with unit control parameter to enhance machining performance avoiding the complex nature of conflicting objective. Literature review was emphasized on the process parameters and control parameters utilized to model machining system, applied optimization techniques and their degree of accuracy in comparison to conventional techniques.

Ansalam et al. [10] investigated on improvising machining performance of hard turning operation on SS420 Steel. Process parameters considered were cutting speed, feed rate, and depth of cut to model surface roughness. RSM based regression model was built for prediction and further improvised optimization technique was applied viz., Integrated GA (IGA) which was comprehended with Conventional GA (CGA). The IGA gave better results than CGA. Hesam et al. [11] executed EDM process on DIN 1.452 stainless steel in which surface roughness and white layers were control parameters. The machining model was built on Taguchi technique and NSGA II was applied for optimization which could produce convincing results.

Garg et al. [12] improvised machining turning operation of AISI 1040 Steel with surface roughness as a control parameter. Taguchi technique was applied to model surface roughness apart from that (Artificial Neural Network) ANN, (Support Vector Regression) SVR techniques were used to build regression. Genetic programming (C-GP) coupled with classifier was used as optimization technique. The results suggested that C-GP was on par with the ANN but SVR performed poorer than C-GP and ANN.

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

Khaider et al. [13] examined hard turning operation of AISI 52100 bearing steel with CBN (7020). The machining performance was measured in surface roughness, tool wear and material removal rate were modeled with Taguchi, RSM and grey-relation, these models were utilizes to optimize performance by applying GA and the results obtained from GA predicted parameters which gave better machining performance.

Ozel and Karpat [14] investigated on enhancing the performance of AISI H13 grade steel turning with CBN tool. Prediction model for surface roughness and tool wear was built on process parameters (cutting speed, feed rate, depth of cut). Experimental data of AISI52100 steel was referred form literature and further data experiment was performed on AISI H-13 steel these data was utilized to train neural network, Regression was also carried out, two feed-forward neural networks was modeled. In the first model the input layer constituted edge geometry, hardness, cutting speed, feed rate and depth of cut to predict tool wear and surface roughness, while in the latter network material hardness, cutting speed, feed rate, depth of cut and forces were utilized to model tool wear and surface roughness. The latter network performed well than the former.

Alhameri et al. [15] studied multi pass turning of austenitic AISI 302 Steel. Box-Behken design was utilized to develop model. Prediction models were also built by Regression analysis and NN to predict Tool life and machining economics with motive to minimize machine economics and maximize tool life.

Abbas et al. [16] carried out research on turning operation of J steel with TungstenCarbide insert. Models were built to predict Surface roughness and Material removal rate by applying Taguchi technique. The formulated regression equations were utilized as objectives with appropriate constraints in process parameters. Multi-objective EGO algorithm was implemented to optimize machining performance.

Zhenghua et al [17] investigated high speed milling aluminum alloy AlMN1CU with Carbide tipped tool. Both Linear and quadratic regression models were built and Bayesian Neural nets (BNN) was built using experimental data to predict surface roughness, the regression model built was utilized as objective functions with precise

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

constraints applied to process parameters. GA was applied to optimize parameters GA predicted optimized parameters for each surface roughness which was verified by BNN. Yunguang et al. [18] worked on micro-grinding of nickel based super alloy (DD98), surface roughness was modeled in linear and non-linear degree using CCD based RSM. GA was applied to predict control parameters for best machining performance. The results were verified experimentally and were found to show pretty good agreement.

Shaharam et al. [19] examined on cellular manufacturing systems with objective to minimize cellular movement distance and machine idle time. Regression models were developed and optimization techniques viz., NSGA II, lingo, Fuzzy-GA was applied and results concluded that NSGA II gives better results than Lingo and Fuzzy-GA. N.Alberti and Perrone [20] worked on multi pass turning operation to predict least power consumption, machine economics and surface roughness for which three different modelling approaches was adopted viz., deterministic model, possibilistic model, a fuzzy possibilistic-GA model with constrained and unconstrained search space. The results established that fuzzy-possibilistic model predicted most failures and fuzzy-possibilisticGA optimized objectives to practically feasible solution.

Garge et al. [21] experimented on EDM of Titanium and Inconel alloy in which surface roughness and cutting speed were control parameters modeled with process parameters.

NSGA II was applied to optimize performance.

Pramanic et al. [22] worked on EDM of ZrB2 where cutting speed, material removal rate, and surface roughness was modeled with process parameters by applying Taguchi technique and optimization based on Taguchi based grey relation. ANN was used to predict cutting speed and surface roughness, the predicted accuracy was checked with experimental statistics for confidence level and it gave the appreciative result.

Sahali et al. [23] worked on multi point turning operation, modeling machine economics with process parameters and constrains in surface roughness, chip-tool temperature, tool life and force was applied. Optimization techniques applied were viz., Deterministic

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

technique, probabilistic technique, probabilistic- NSGA II (P-NSGAII).Results concluded that P-NSGA II outperformed deterministic and probablisitc technique.

Dureja et al. [24] reviewed different optimization and modeling techniques used in hard turning operations viz., RSM, Taguchi, Regression analysis, NN, Fuzzy modeling, GA. Ganesan and kumar [25] investigated in performance enhancement of turning operation by predicting machining cost, machining time, tool wear with process parameters and GA was applied to optimize objective function.

Jawahir [26] presented an analytical and numerical solution to 2D and 3D chip formation, hybrid predictive model was developed to characterize and optimize chip breakability and chip curl geometry. Furthermore, GA was used to optimize chip formation so that machining happening according to desirability, the optimized results were verified with FEM simulation results.

Sundaraman et al. [27] in contrast worked on fixture design and layout of end milling; quadratic model was built using RSM and optimization done by GA and PSO. The model was built to predict work piece deformation with positions of clamp and location as parameters. The results of optimization suggested that RSM-PSO gave better solution than RSM-GA technique. Furthermore, these results were compared with FEM simulation of fixture layout.

Costa [28] investigated on multi-pass turning with the objective to minimize unit production cost. The objective was constituted of actual machining cost, machining idle cost, tool replacement cost. The characteristic equation was built on process parameters viz., cutting speed, depth of cut, feed rate both in rough and finish pass, operation constraints were applied in tool life, cutting forces, power and surface roughness. A novel hybrid technique in PSO was formulated for optimization. Furthermore, this solution was compared with other techniques suggested 2.035 unit production cost, while FEGA gave 2.3057 as optimal cost SA gave 2.29, MGA gave 2.30, HC gave 2.27 and ACO gave 2.25.The hybrid PSO technique could optimize solution superior than other techniques.

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Bharathi et al. [29] investigated on turning operation with diamond shape tungsten carbide tool on four different materials i.e., brass, aluminium, copper and mild steel.

Machining was characterized by forces, surface roughness with cutting speed, feed rate, and depth of cut as process parameters. Furthermore, these equations were optimized using PSO technique; the optimal solution suggested the trend that higher cutting speed, lower feed, and depth of cut gave better surface roughness. The optimal surface roughness obtained from PSO for brass was 0.07μm, for copper 0.08μm, for aluminium 0.08μm and mild steel 0.08μm.

Bharathi et al. [30] carried out investigation on milling operation of aluminum bar with carbide tool. Machining time and surface roughness were characterized with process parameters such as spindle speed, feed rate, and depth of cut. The characterized equations were optimized using PSO technique. The solution obtained from PSO was verified by conducting confirmation test. The solution trend showed that higher speed, lower depth of cut, lower feed rate gave better surface roughness and feed rate had a greater influence on surface roughness. The prediction ability of present approach was found to be 96 % for machining time and 85% for surface roughness.

Bharathi et al. [31] investigated on modeling and optimizing both turning and grinding operation. The turning operation was done on single and multi-pass while grinding was done in the single-pass. The performance of turning was measured on machining time while grinding was done on machining time and material removal rate. Optimization technique applied were PSO, GA, and SA whose optimal solutions were comprehensively evaluated. The computational time obtained by PSO in single and multi-pass turning was 11 sec and 12 sec respectively, while for grinding 4 sec. Similarly, results of GA gave 15 sec in both single and multi- pass turning, while in grinding it gave 6 sec as optimal computational time. Likewise in SA, it was 12, 13 and 5 respectively.

Optimal material removal rate in grinding was in the range of 0.17-0.44 μm, from the results it could be inferred that PSO proved to be better than GA and SA. Chandrasen et al. [32] reviewed different soft computing techniques that could be applied to machining performance prediction. Any machining system could be VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 18

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generalized by its corresponding inputs and outputs. Input in general are process parameters, material properties, sensory feed and output of the system are concerned about machining performance i.e., dimensional deviation, cutting forces and tool wear, after machining characterizations done various soft computing techniques are applied to optimize machine model. The review concluded that best strategy to predict performance is to couple fuzzy with a neural network. Likewise, to optimize precisely GA, PSO and similar heuristic techniques are the best technique.

Prabhakaran et al. [33] carried out work on machining fixture analysis where location and displacement of clamp and locator were objective functions. Regression models were developed for displacement and location and optimized using GA and ACA. Ant colony algorithm gave near optimal solution than GA. Farahnakian et al. [34] investigated end milling operation; performance was modeled in cutting force and surface roughness with process parameters as cutting speed, depth of cut, feed rate. The characterized equation was utilized to frame optimization problem, coupled PSO-NN technique was applied to optimize. The applied technique gave better Pareto-spread in solution space with good convergence.

Yang et al. [35] carried out worked on multi- pass face milling operation. Performance was characterized by unit production cost with process parameters such as number of pass, depth of cut, cutting speed, feed rate. Fuzzy based multi-objective PSO was applied to optimize process parameters which gave better solution with fast convergence.

Escamilla et al. [36] experimented end milling with a performance characterized in surface roughness with process parameters cutting speed, feed rate and depth of cut.

Taguchi based regression equation was utilized to formulate optimization. PSO was used to optimize parameters.

Li et al. [37] worked on in improving the performance of milling operation. The performance was measured such as cutting force, tool life, surface roughness and cutting power with process parameters spindle speed, feed rate, depth of cut. PSO technique was

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

applied to optimize process parameters to optimize process parameters. The Paretospread of optimal solutions was well spread and converged well.

Chen and Li [38] worked on optimizing grinding operation by maximizing material removal rate. PSO was applied to optimize material removal rate.

Sukla and Singh [39] investigated on Abrasive Water jet Machining (AJM) of Aluminum alloy with garnet abrasive particles. Machining model was built for kerf width and taper angle prediction with process parameters by applying Taguchi technique and Optimization techniques applied, PSO, Firefly, Simulated Annealing, Black Hole, BioGeographical, NSGA. PSO gave better results than other techniques.

Asilturk and Cunkas [40] carried experimental investigation on turning operation of AISI 1040 steel with Al2O3 coated carbide insert. Tool variable considered were tool material, nose radius, and rake angle cutting edge geometry while work piece variable considered material hardness with cutting conditions such as cutting speed, feed rate and depth of cut. With these variables, a multi-regression model was developed and full factorial experimental design was built Further, ANN was developed with back propagation training algorithm to predict surface roughness. ANN and multi-regression were had close estimated of surface roughness prediction. ANN performed better with 99% regression coefficient and regression with 97% regression coefficient.

Senthil et al. [41] predicted performance of cutting tool inserts using neural network.

Experiments were performed on workpiece with carbide inserts with process parameters such as cutting speed, feed rate, depth of cut, material hardness, and cutting insert shape (relief angle, nose radius) to model surface roughness and flank wear. The Taguchi based ANN model was built with these process parameters as input layer. Results predicted by neural network model were compared with experimental values which predicted values close to experimental statistics.

Miron et al. [42] worked on dynamic characterization and vibration analysis of lathe machining system by which the machine condition was determined. Modal analysis was

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

done to determine the natural frequencies, the frequency was compared with numerical model and validation experiment was performed.

Dilbag and Venkateshwara [43] developed analytical tool wear model while turning bearing steel with ceramic tool. The model incorporated abrasion, adhesion and diffusion wear mechanism further it was validated by conducting experiment. The analytical model had capability of predicting flank wear using cutting parameter and tool geometry.

Yahya et al. [44] worked on turning operation of steel at different conditions with P25 HSS tool at different working conditions. Surface roughness, flank wear, and crater wear were modeled with process parameters to determine machinability of tool steel. The relative degree of influence of each parameter on control parameters was quantified. This work can help in sorting the priority of objective functions and the contribution to overall machining performance.

Hamdi et al. [45] investigated behaviour of hard turning while machining AISI H11 Steel with CBN tool. Forces and surface profile were considered as process responses. A CCD based RSM was applied to build machining model furthermore a comprehensive analysis was done on influence of process parameters over machining quality.

Shihab et al. [46] conducted experiment on hard turning of AISI 52100 Steel alloy with coated carbide tool in which surface roughness and micro hardness were modelled and optimized utilizing CCD based RSM approach. The RSM based optimization technique gave satisfactory results but by reducing multi objective to single objective.

Waleed et al. [47] worked on hard turning of AISI 4340 Steel with CBN tool; in his work surface roughness and tool flank wear were modelled by Taguchi technique to form multi-regression equation. This equation was used as objective functions along with constraints in process parameters; the S/N ratio analysis was done on regression to optimize control parameters. saha et al. [48] experimented on EDM Hard facing on Nano- card-11.Mathematical model was built on material removal rate, cutting speed and machining time for both

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

brass wire and zinc coated brass wire by applying both RSM and multi criteria grey relation. RSM was utilized to optimize the objectives individually for both the wires.

Emeryl et al. [49] worked on hard turning of Ni-Steel alloy (62HRC) with CBN tool insert, the performance was modeled to predict cutting forces, surface roughness with process parameters by applying Taguchi method and optimization was done by Taguchi based S/N ratio. The results were in agreement with experimental data with good level of confidence.

Ilhan and asks [50] worked on hard turning of AISI 4140 (51HRC) steel by carbide tool coated with Al2O3 and TiC. A three-level full factorial with Taguchi based experimental design was applied to model surface roughness by applying cutting conditions and control factors. The process variability was measured by S/N ratio. Taguchi based S/N response suggested that larger difference in S/N ratio will have more significant effect on surface roughness. The result of process variables for optimum surface roughness was 120 m/min, 0.18 mm/rev and 0.4 mm cutting speed, feed rate and depth of cut respectively.

Gaurav and Choudhary [51] focused study on hard turning of EN31 bearing steel (5862HRC) with CBN tool insert. A three-level full factorial experimental design was developed; ANOVA was performed to find the relative contribution. RSM was utilized to build regression equation on cutting forces and surface roughness, further RSM optimization was done. Results showed that depth of cut had more influence on cutting forces and while speed had the least influence. Results also revealed that initially forces decreased with increasing speed later on increased along with speed due thermal softening of tool material.

Ashvin and Nanavati [52] enquired on turning operation of AISI 410 steel with carbide inserts of TNMG series differing in nose radius. A three-level full factorial experimental design was done, further, RSM was utilized to model and optimize surface roughness which suggested optimal solution 225 m/min,0.1 mm/rev, 0.3 mm, 0.12 mm for cutting speed, feed rate, depth of cut and tool radius respectively.

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Asilturk and Suleyman [53] investigated hard turning of AISI 304 austenite stainless steel with carbide inserts (SNMG series). A three-level full factorial Taguchi based experimental design was built. RSM based regression equation was modeled for surface parameters (Ra and Rz) using process variable, S/N ratio was determined then RSM based optimization was done. The optimized control factors setting for Ra was found to be cutting speed 50 m/min, feed rate 0.15 mm/rev, depth of cut 1.5mm and for Rz was cutting speed 150m/min, feed rate 0.15mm/rev, depth of cut 1mm. These authors applied Taguchi and RSM to model and optimize machining parameters. RSM relates response based input parameters by experimental statistics and applying regression. RSM consists of three stages design of experiment, regression, and optimization to find best optimal solution RSM is coupled with meta-heuristic technique.

Aggarwal and Singh [54] reviewed different machining modeling technique for conventional machining model and types of optimization methodology to optimize and characterize machine model.

Chinmaya et al. [55] experimented on hybrid machining where laser assisted machining (LAM) was coupled with turning operation. High strength alloy (Ti-6Al-4V) was machined with cobalt bound tungsten carbide, liquid nitrogen was used coolant. The LAM hybrid turning operation reduced specific cutting energy and improved surface roughness when compared to conventional machining.

Wang et al. [56] worked on multi-pass turning operation of AISI 1045 with a different set of tools (TNMG carbide inserts). A hybrid model was built to predict machining performance, surface roughness, forces, and chip breakability were characterized by process parameters with operation constraints such as surface roughness, forces, and tool life. RSM was utilized to optimize process parameter. The hybrid model developed could predict slip line field accurately, which was verified by Finite Element Modelling results.

Devender and Kumar [57] worked on turning of Aluminum matrix composites reinforced with SiC (Al 6061) using coated tungsten carbide tool. The effect of reinforcement on cutting forces was characterized to improve machining performance and VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 23

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

it was found that weight forces have maximum impact on the cutting forces. A quadratic model was developed using RSM. Optimal solution concluded that cutting forces was majorly affected by the type of reinforcement.

EranAlsan et al. [58] investigated hard turning of AISI 4140 steel with ceramic tool mixed with Al2O3 and TiCN. Machining performance was modeled using Taguchi based RSM on flank wear and surface roughness as control parameters with process variables.

The optimization results obtained from RSM technique suggested cutting speed 250m/min, feed rate 0.1mm and depth of cut 0.25-0.4 mm for surface roughness and flank wear.

Hashimoto et al. [59] identified the fundamental difference in the surface integrity of hard turning and ground surface, their subsequent impact on rolling contact fatigue life.

The work concluded that the mechanical deformation could play a large role during hard turning than grinding while the size effect in grinding introduced surface hardness furthermore the hard turned surface may have more than 100% longer fatigue life than a ground one with an equivalent surface finish due to very different characterization of surface integrity. The effect of turned or ground surface free of white layer was clarified a super finished turned surface may have twice a fatigue life than ground surface.

Ozel et al. [60] investigated on hard turning of AISI 4340 steel with uniform and variable edge PCBN insert where the forces and tool wear was measured and 3D finite model was utilized to predict chip formation, temperature and tool wear on both type of inserts the predicted tool wear and forces were compared with experimentation. The result showed that the variable edge tool insert has advantage of less tool wear and good temperature distribution profile.

Ravinder and Santram [61] investigated the effects of cutting parameters on surface roughness in turning of Al7075 hard ceramic composites and Al7075 hybrid composite using polycrystalline diamond tool (PCD) dry turning was conducted to examine the trend of roughness by using roughness tester for both composites. It was concluded that surface roughness of hybrid composite was lesser in all combination of experiment. Further VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 24

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RSM based artificial neural network was applied to validate the results obtained during experimentation and to protect the behavior of the system under any condition within the operation range.

Mia and Dhar [62] developed a predictive model of average tool-workpiece interface in hard turning of AISI 1060 steel by coated-carbide insert. Cutting condition used were cutting speed, feed rate, and depth of cut utilized to model temperature profile.

Experiment was conducted in both dry and high pressure coolant environment with fullfactorial design. Temperature was measured using tool-work thermocouple. Response Surface Methodology (RSM) and Artificial Neural Network (ANN) were employed to predict the temperature. The accuracy of both RSM and ANN model were in region of acceptance. The regression coefficient of ANN for both the environment was greater than 99.8%. ANN model demonstrated a higher accuracy which was found convincing if employed for controlling cutting temperature in turning of hardened steel.

Pontes et al. [63] worked on turning of AISI52100 hardened steel with multi-layered coating (Al2O3+Tic+TiN) chamfer edge. Experiments were conducted with training sets of different size to compare performance of best network in each experiment. Process parameters considered were cutting speed, feed rate, depth of cut to model performance in surface roughness. Radial base function (RBF) neural network was developed with the use of Taguchi’s orthogonal array as a tool to design parameter of network. The factors considered in designing RBF-NN were number of radial units, algorithm for selection of radial center and algorithm for selection of spread factor for evaluating performance of RBF-NN. The results revealed that algorithm for calculation of radial spread factor was most influencing among the three factors RBF-NN trained gave least mean standard deviation for worst trained case. The results suggested that DOE based RBF network are more efficient and effective than trail and error based NN architect.

Gaitonde et al. [64] investigated the influence of process parameters on machinability characteristics of turning AISI D2 (cold work) tool steel with different ceramic inserts. A multi-layer feed forward Neural was developed with inputs as ceramic insert grade, cutting speed, feed rate and machining time to predict specific cutting force, surface VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 25

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

roughness and tool wear. Statistical comparison was done between the predicted results and experimental results further more interaction effects among the process parameters were studied.

Wang [65] developed neural network based optimal estimator for predicting CBN tool wear during hard turning operation. Prediction model was based on fully forward connected neural network, with inputs as cutting condition and machining time and predicted output in tool flank wear. The feed forward fully connected neural network (FFCNN) based optimization was validated with experimental data. Comparison showed that the (FFCNN) estimated a close value to experimental tool wear and developed FFCNN model was found to be faster, accurate than other neural network approaches.

Umbrello et al. [66] developed predictive hybrid model based on neural network and finite element method with objective to predict residual stress profile in hard turning for different combination of material properties, cutting tool geometry and cutting condition.

A converse prediction of cutting conditions and geometry was made for a given residual stress profile which acted as constrained based process parameters determination.

Furthermore, this model was utilized as closed feedback where the predicted residual stress of ANN were applied to simulate cutting condition in FEA and vice-versa The results obtained from ANN based FE simulation gave practical results.

Ravinder and Santram [67] investigated the effect of cutting parameters (cutting speed, feed rate and approach angle) on roughness while turning Al 7075 hard ceramic based composite using polycrystalline tool diamond tool (PCD). The surface roughness was modeled by both RSM and ANN. Moreover, the influence of parameters on surface roughness was analyzed both RSM and ANN model correlated fairly to the experimental data.

Yildiz [68] presented a Hybrid Differential Evolution Algorithm (HEADA) for minimizing production cost in multi-pass turning operation the algorithm was illustrated two case studies. Taguchi based differential evolution was applied to solve machining economics problem. Further hybrid differential evolution based optimization technique VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 26

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

was compared with PSO, HEGA, Scattered Search (SS), Simulated Annealing (SA), Pattern Search (PS), Floating encoding genetic algorithm (FEGA) and Hybrid Harmony Search (HSS) HEDEA outperformed all other techniques.

Kara et al. [69] worked on turning of AISI 310L Stainless steel with both coated and uncoated cutting tool (TiCN+Al2O3+TiN). The cutting condition (cutting speed, feed rate, and depth of cut) were used to model tangential forces and feed force Prediction model was developed for both the responses with ANN. Two learning methods were deployed i.e., scaled conjugate learning and Lavenberg-Madquart learning. The predicted forces were accurate with error within 5%.

Sener Karabulut [70] worked on milling of metal matrix composite (Aluminum Alloy 7039/Al2O3 powder metallurgy) with CVD carbide tools the process parameters were material removal rate, cutting speed, feed rate and axial depth of cut to model machining performance such as surface roughness and cutting force ANN model was developed with cutting condition to predict performance. The predicted performance was compared with experimental model which gave close results with 99.8% regression.

2.2 Literature Summary Authors have applied Evolutionary techniques on different machining systems for optimization. From the literature it could be concluded that very few authors have applied swarm intelligence techniques for optimization. While concerning to regression based learning techniques, authors have applied neural network based prediction models and few authors have utilized hybrid learning based evolutionary optimization estimators.In the literature authors, have utilized optimization and predictive techniques exclusively; multi-objective machining systems were converted to single objective for performance evaluation. Many authors have optimized machining systems with unit control parameter to enhance machining performance avoiding the complex nature of conflicting objectives.

The literature motivates for research work in multi performance of machining system and multi-regression prediction models as literature lacks application of synergies of computational techniques for performance evaluation and prediction.

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

CHAPTER 3 GLOBAL SEARCH ALGORITHMS FOR MULTI-OBJECTIVE OPTIMIZATION 3.1Introduction In this chapter Multi-Objective Optimization problem (MOOPs) of machining system is solved by applying Evolutionary Algorithms (EA’s) and Swarm Intelligence (SI) which are Global Search based Optimization techniques (ref Fig.1.3). In the first section Evolutionary Based NSGA II (Non-Dominated Sort Genetic Algorithm) and SPEA 2 (Strength Pareto Evolutionary Algorithm) algorithms are applied to Machining MOOPs and a comparison of between both is made on basis of diversity of solution. In the second section Particle Swarm based Swarm Intelligence is applied to existing Machining MOOPs and optimized results are interpreted through, swarm surface and pareto plots. In the third section comparative evaluation of obtained results between EA and SI technique are analyzed and the difference in the nature of solution space between the two search optimization techniques is assessed. The workflow of this chapter is explained below,

Fig.3.1 Workflow for Chapter 3

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

3.2 Multi-objective Optimization Most of the practical problems are complex and their definition of optimality is not simple as they need to satisfy multiple competing objective functions at the same time.

Moreover, some of these objectives may have conflicting relations with others, which makes the optimization difficult. Problems requiring simultaneous optimization of more than one objective function are known as multi-objective optimization problems (MOOPs). They can be defined as problems consisting of multiple objectives, which are to be minimized or maximized while maintaining some constraints. Formally, they can be defined as: Minimize/maximize f (x) Subject to objectives and constraints

g j ( x)  0 j  1, 2,3... j hk ( x)  0,

k  1,2, 3,..., K

Here, the problem optimizes g j ( x) objectives and satisfies J inequality and hk ( x) equality constraints. This type of problem has no unique perfect solution. In traditional multi-objective optimization, it is very common to simply combine all the objectives together to form a single (scalar) fitness function. But the obtained solution using a single scalar is sensitive to the weight vector used in the scaling process. This requires knowledge about the underlying problem which is not known before in most cases.

Moreover, the objectives can interact or conflict with each other. Therefore, trade-offs exists when dealing with such MOOPs, rather than a single solution. Most MOOPs do not provide a single solution; rather, they offer a set of solutions. Such solutions are the ‘trade-offs’ or good compromises among the objectives. In order to generate these tradeoff solutions, an old notion of optimality called the ‘Pareto-optimum set’ is normally adopted.

In multi-objective optimization, the definition of quality of solution is more complex than for single-objective optimization problems. The main challenges in a multi- objective optimization are: converge as closely as possible to the Pareto-optimal front, and VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 29

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

maintain as diverse a set of solutions as possible. The first task ensures that the obtained set of solutions is near optimal, while the second task ensures that a wide range of tradeoff solutions is obtained.

3.3 Application of MOOPs to Machining system The Machining data from hard turning of AISI 4340 steel [2] is utilized as machining objectives in which machining is performed on two different hardness and regression equations were built using RSM with process variables involving in cutting speed, feed rate and depth of cut to model surface roughness, cutting forces and tool life. The machining constraints and objectives are as follows, Table 3.1 Machining Constraints [2] Parameter

Constraints

Process parameters

Velocity(m/min)

142

265

125

175

Feed rate(mm/rev)

0.15

0.25

0.15

0.25

Depth of cut(mm)

1

2

1

2

Tangential forces(N)

337

1197

492

1296

Axial forces(N)

219

605

298

663

Radial forces(N)

197

496

256

564

Control parameters

lower Higher lower Higher bound(35) bound(35) bound(45) bound(45)

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

3.3.1 Machining Model (Surface Roughness Cutting force components and Tool Life work material hardness: 35 HRC).

Ra  12.793  0.03118   v  – 28.8786   f  – 2.8599   d   0.0354   v    f   0.000236   v    d   11  f    d   0.00000381  v 2   32.039   f 2   0.2853   d 2 

 3.1

FT  373.0294  0.5308   v   788.39   f   697.2733   d   7.2420   v    f   1.9860   f    d   235   f    d   0.00075  v 2   6659.8  f 2   0.598  d 2 

 3.2 

Fa  375  2.971  v   360.24   f   76.68   d   7.9052   v    f   0.4   v    f  145   f    d   0.000398   v 2   1528.4   f 2   66.71  d 2 

 3.3

Fr  239.69  2.4094   v   755.0606   f   133.18   d   0.0559   v    f   0.2472   v    d   585   f    d   0.000415   v 2   2593.5   f 2   22.93   d 2 

 3.4 

3.3.2 Machining Model (Surface Roughness Cutting force components and Tool Life work material hardness: 45 HRC).

Ra  11.3037  0.0614   v   16.075  f   2.3075   d   0.0006   v    f   0.102   v    d   3.7   f    d   0.0000128   v 2   49   f 2   0.22   d 2 

 3.5

Ft  50.57  0.1484   v   3270   f   143.102   d   33.5   v    f 

 1.11  v    d   1175   f    d   0.01183   v 2   5909.091  f 2   38.9091 d 2 

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 3.6 

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

Fa  260.483  0.86523   v   287.159   f   113.8068   d   6.9   v    f   1.39   v    d   695   f    d   0.01210   v 2   3477.273   f 2   41.271  d 2 

 3.7 

Fr  86.3465  1.5970   v   948.0681  f   212.0113   d   7.9   v    f   1.19   v    d   95   f    d   0.01518   v 2   2695.4545   f 2   26.95   d 2 

 3.8

3.3.3 Tool life model for 35 HRC

Tf 

(v)

0.59

423  ( f )0.4697  (d )0.47

 3.9

Tool life model for 45 HRC

Tf 

(v)

0.59

23135.13  ( f )0.4697  (d )0.47

 3.10

3.4 Evolutionary Algorithms (EAs) Optimum seeking is one of the central issue in Manufacturing system. Every problem solved is outcome of best possible choice for which a variety of tools and techniques have been developed and applied to systems for optimum seeking.

Meanwhile optimum seeking in nature, biological and social systems takes place in a completely different way i.e., natural evolution they have adapted themselves to a constantly shifting and changing environment in order to survive. Those weaker and lesser fit members of species tend to die away leaving create stronger and fitter to mate create to create offspring and ensure the containing survival of species and it is upon this dictated idea

that evolutionary computing is based on. Evolutionary computing is

emulation of the process of natural selection in search procedure (as shown in Fig.3.1).

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

 ( p  1)  [ ( ( p))] Fig 3.2 Evolutionary Model The current EAs applied in Multi-Objective Optimization Problem (MOOPs) and the combination became known as a multi-objective evolutionary algorithm (MOEA). An MOEA will be considered good only if both the goals of convergence and diversity are satisfied simultaneously. The MOEA’s population-based approach helps to preserve and utilize the non-dominated diverse set of solutions in a population. The MOEA converges to a Pareto-optimal front with a good spread of solutions in some fixed number of generations. Most MOEAs use the concept of domination to attain the set of Paretooptimal solutions.

3.4.1 Mathematical Formulation of Evolutionary Algorithms Evolutionary algorithm is stochastic in nature. The probability of finding best solution or no solution is equally likely, if the parameters of genetic operator are not appropriate depending upon nature of problem in hand. A prior convergence analysis is essential for favorable working of Evolutionary algorithm So at each genetic operations the probability of obtaining best solution and its heritance in the subsequent operation is essential.

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

3.4.1.1 Definition of Evolutionary systems Considering a generic function f  x   0 on X. with no constraints imposed on X. further EA system which does not use any specific properties of the set X, the only condition is that the function f should be defined at every point of X i.e, X  R n . Then the problem of optimization can be defined as,

Max / Min xX  f ( x)

where

f : SD  R

f  x   0 X  Rn Let S be a space of binary string and C(X) be an encoding function: C : X  S and we discretize search space X to XD by a simple binary code as encoding function the optimization problem is converted to finite set SD  S , SD  C ( X ) .

max SD S f (s)

where

f : SD  R

There are Variety of Evolutionary system with different types of selection, crossover and mutation. This section discusses most generalized terms with no bias in different genetic operation.

3.4.1.2 Convergence Analysis of Evolutionary Algorithm Consider the following events in evolutionary system with their respective properties let P be a population and n be the size of population

P  A | H   Probability that the population does not contain solution after mutation provided it contained after crossover.





P A | H  Probability that the no solution is found after mutation and crossover



P H  Probability that solution will be found after a crossover.

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_ P  H   No solution after crossover.  

max (f(S)) where f  0, S  S S is a finite S  2m m

Encoding capacity (No. of bits).

3.4.1.3 Criteria for mutation * Let Pmt be the probability of mutation s  S be selected individual, then the probability

of individual not getting mutated be P S x  mut  PD 

 and not giving rise

S * in the

population  m / 2 integral part of quotient m/2. and P S  S n  is probability of mutation that S mutates to S * .

Let K be the no. of different bits in S & S *

 0  k  m

then the probability that S

mutates in S * is given by binomial theorem.

P

 P , P   PS  S   1 P  

k

k

m

mt

k

m

Pm  Pmt 

1  k mk P 1  Pmt   : Pmk is The probability of ‘K’ mutation k  m   Cm

1  Pmt 

mk

 is the probability that the remaining (m-k ) bits do not mutate

1  probability that precisely the necessary K-bits mutate but not other bits.

Cmk

Then P{s  s '  s*}  1  P{s  s*}  1 

m pmt is the probability that s does not mutate Cm[ m/2]

in S*

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

Now, the probability that best solution is obtained after mutation is



P S S

*



Pmtm 1 k 1 m mk  k Pmt 1  Pmt   k Pmt  ( m /2) Cm Cm Cm

For an individual to get mutated the minimum condition is that half of its bit length must participate in mutation and in contradictory assuming that mutation does not happen then,





P S *  mut  P   P{  S  S '  S *} SP

  Pmt m   P     P S  S  S *    1  ( m /2)   1  mmt/2  s p s p  Cm   Cm 



'



(3.2)

The above condition is valid when the probability of mutation pmt <0.5, if the probability of mutation exceeds 0.5 then the probability that the population does not contain solution S* is given as follows.

 (1  Pmt )  P S *  mut ( P)  1   Cmm /2  









P S *  mut ( P)

Now evaluating binary strings over the objective f  P k    max  ( f ( S )) if elite method  SSp   is utilized for selection then, f x  f ( Pk )  f ( Pk 1 )  ...........  f ( P0 ) Where f ( P K ) which are randomly calculated

and subsequently we get, f f p p 'f

 

0 f ( p kf )  f ( s ) f p p kf  Pf0

Re sults Pr obability

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

Here Pfk  Probability that f ( P k )  f after k th iteration and  Pfk  is the probability that f ( P k )  f * after the k th iteration now that the required expectation probability does not

decrease, E  P k    f .Pfk , k  0,1, 2..., E  P k   E  P k 1   E  P k 2  .......  E  P0 

Now if l  individual with fitness f *( x) in the general population S (i.e. optimization) problem has ‘l’ solution

S

 1

,........., Sl



if elistism is applied then the expectation is

unchanged i.e, E[ P K ]  f * Consider situation when A’ =A that no solution is found after the first iteration, consequently suppose that a hypothesis H stating at least one of the solutions results from  _  _ crossover, Then possible events P  A  P  A | H   P H   P  A | H   P  H     

3.4.1.4 Criteria for Crossover Estimating P{H} from the above which only differ from the pmt by the fact that population contains one solution before mutation, where as it does not contain any in second case. Applying from the above, PA | H  P

S .......S 

_   P A | H   P  

*

*

1

l

 P   mut cross( P 0 )  1  mmt/2   Cm 

S .......S  *

*

1

l





  Pmt m    1  m /2  C m 

n l

  Pmt m    1  m /2  C m 

 P   mut cross( P )  1  mmt/2   Cm 



0

While Estimating P  H  we say that pairs



 S1 , S2 

n

n l

n

is good if it yield solution after

crossover it can be concluded that a pair is good if both the individuals in the pair contains fragments of the same solution as sub-string.

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

Consider the following events B  Event of at least a good pair will be chosen.

C  Event that all the pair is chosen for crossover are good.

P H   P H | B P B  P H | B  P H | C This can be concluded from the fact that if all the pairs are good then the probability is maximum compared to other events, So the probability that a good pair (s1,s2) yields a solution .can be written as

q pc  pc q  m  1, m 1 P{s1  s2  s*}  1  pc

P{s1  s2  s*} 

In contrary P{H | C}, the probability that a solution does not arise after a crossover, provided that all pairs are good:



P{H | C} 

( S1 , S2 )Cross ( P )

P{s1  s2  s1* ,...sl }   (1  pc )  (1  pc ) n

P{H | C}  1  (1  pc ) n Assuming that atleast ‘n’ pairs take part in the crossover. Then probability of good pairs after crossover

P{H }  P{H | C}  1  P{H | C}  (1  (1  pc )n Now if Event A happens then the possibility of event A can be written as follows

P{A}  P{A | H }P{H }  P{A | H }P{H }  (1 

m n pmt pmt n n   p   ) (1 (1 ) ) (1 )n *1 c [ m /2] [ m /2] CM Cm

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S  (1 

m pmt )n (2  (1  pc ) n ) [ m /2] CM

(3.3)

Therefore probability in kth iteration can be written through mathematical induction as, P{ Ak } = P{AK | Ak 1}P{AK 1}  SS K 1  S K , where P{ AK | Ak 1} is the probability

That no solution arises after the kth iteration now from above frame works the expectation of solution after the kth iteration

E[ P K ]   fp kf  f * p kf  f * (1  S k )  f * With this following conclusions can be drawn from the expectation regarding the parameters which influence mean convergence rate 1. AS pc ( 0  pc  1 ) increases S also increases 2. As pmt (0  pmt  1) increases S decreases 3. As m increases S also increases 4. The dependence of S on n can be drawn from the below expression

(1 

m pmt )n  0 as (n infinity) [ m /2] CM

n and (2  (1  pc ) )  2 as (n infinity)

Minimum value of S =1 for n=0 In order extract the extreme limits of convergence we consider the extreme of the function S(n).

Finding optimal parameters for above events we use eq. m pmt Let a  1  [ m /2] CM

b  1  pc

S (n)  a n (2  bn ) VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 39

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

For extremum S ' (n) =0 S ' (n)  a n (2  bn ) ln a  bn ln b

The optimum n is given by n= log b

2log a ln ab

3.4.2 Key points from convergence analysis 

With increase in population size the string length increases, reaches optimum peek and then decreases. The least string length possible for encoding is 1 for zero size population.



From above result it can be concluded that n exists and is real and S attains its maximum i.e, as n increase, S first increases and then decrease.



For accelerating convergence pc should be minimum while pmt should be maximum but the drawback of having pc minimum is that the best solution are not inherited to next generation and high pmt destroys best solution space.



With elitism the expectation of solution getting transferred to next generation increases moreover if crossover probability is increased the density of best solution increases including good solution in every iteration.



If the mutation probability is decreased then the string length participating decrease which decreases the passivity of killing best solutions, but if mutation is accurately tuned then there is quite a possibility that worst strings could give good solution.



If the crossover size is increased the convergence decelerates but the chances of obtaining good solution increases. In contrast for accelerating convergence if the mutation rate is increased then good solutions are lost leading to no solution so a good balance between convergence time and crossover-mutation rate is essential.



It is always suggested that in order to obtain good solution if convergence rate is allowed to float freely whenever possible and when convergence rate is strict criteria then it is suggested that the string length is kept minimum so that humming effect and relative degrees of change in string character is merely small.

VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 40

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

With the key points in mind evolutionary algorithm is chosen with properties which could overcome these drawbacks A fast elite NSGA II and SPEA2 algorithms are utilized to optimize our machining system.

3.5 NSGA II Algorithm

Fig. 3.3 NSGA II Algorithm [69-70] Table 3.2 NSGA II Setting Population size Generation Crossover probability Crossover constant Mutation probability Mutation constant

1000 100 0.8 0.1 0.1 0.2

3.5.1 Initialize Variables and Evaluate Objectives Initialize Variables uses the bounds of ‘V’ decision variables and randomly generates N number of population over the bound ‘range’ and each objective ‘M’ is evaluated over this population ‘pop’ for fitness through Evaluate Objectives.

Initialize Variables (N, M, V, range) 1. For i :[1-N] 2. For j :[1-V] 3. V[i,j]= Rmin+ random(0,1)*Range VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 41

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

4. Pop: Initialize Variables 5. Evaluate Objectives [Pop, M, V, N] 6. For i :[1-N] 7. For j :[1-V] 8. Objectives[I,j]= f (v) 3.5.2 Non_dominated_sort Now the evaluated objective is sorted using Non_dominated_sort technique where population is sorted on the basis of domination front and over this front the population.

Procedure for Non-dominate sorting 1. For each individual p in main population P do the following 2. Initialize Sp = ∅. This set would contain all the individuals that is being dominated by p.

3. Initialize np = 0. This would be the number of individuals that dominate p.

4. for each individual q in P if p dominated q then add q to the set Sp i.e. Sp = Sp

∪ {q} else if q dominates p then

5. Increment the domination counter for p i.e. np = np + 1 6. If np= 0 i.e. number of individuals dominate p then p belongs to the first front; Set rank of individual p to one i.e. prank = 1. Update the first front set by adding p to front one i.e F1 = F1 ∪ {p}

This is carried out for all the individuals in main population P. 7. Initialize the front counter to one. i = 1 following is carried out while the ith front is nonempty i.e. F[ ƒ]= ∅.

8. Q = ∅. The set for storing the individuals for (i + 1)th front.

9. for each individual p in front Fi for each individual q in Sp (Sp is the set of individuals dominated by p).

10. nq = nq −1, decrement the domination count for individual q.

VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 42

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

11. if nq = 0 then none of the individuals in the subsequent fronts would dominate q. Hence set qrank = i + 1.

12. Update the set Q with individual q i.e. Q = Q â&#x2C6;Ş q.

13. Increment the front counter by one.

14. Now the set Q is the next front and hence Fi = Q. Non_dominated_sort (pop,M,V,N) 1. For i :[1-N]//Initialize domination set 2. Initiate Pop[i].domination set [] 3. Pop[i].domination count=0; 4. //Initialize empty front F[1]=[] 5. For i :[1-N] 6. For j :[i+1-N] 7. P=pop[i] q=pop[j] //Consequetive population 8. //Check for domination 9. If dominates(p,q) 10. d=dominates(p,q) 11. d=all(p<=q)&&any(p<q) 12. P.dominates set=[P.dominates set ,j] 13. q.dominates count=[q.dominated count +1] 14. if dominates (q.cost p.cost) 15. q.dominate set=[q.dominates ,i] 16. p.dominate set=p.dominated count+1 17. Exchange pop[i] with p and pop[j] with q 18. If pop[i].dominated count==0 19. F[i] = [F[i],i] &pop[i].rank=1 20. While (~front not empty) 21. //calculated the subsequent fronts 22. Exchange p=pop[F[i] & q=pop[j] 23. q.dominated count =q.dominated count-1 24. q.dominated count ==0 VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 43

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

25. Q=[q,j] 26. q.rank=pop(F[i])+1 27. exchange q with pop[j] 28. F[pop[i]+1]=Q Once the non-dominated sort is complete the crowding distance is assigned. Since the individuals are selected based on rank and crowding distance all the individuals in the population are assigned a crowding distance value. Crowding distance is assigned front wise and comparing the crowding distance between two individuals in different front is meaningless. The crowing distance is calculated as below 15. For each front Fi, n is the number of individuals.

16. Initialize the distance to be zero for all the individuals i.e. Fi (dj ) = 0, where j corresponds to the jth individual in front Fi. 17. for each objective function m 18. Sort the individuals in front Fi based on objective m i.e. I = Sort (Fi, m).

19. Assign infinite distance to boundary values for each individual in Fi i.e. I (d k ) = ∞ and I (dn) =∞ 20. for k = 2 to (n − 1) I (d k )  I (d k ) 

I (k  1).m  I ( K  1).m f mmax  f mmin

I(k).m is the value of the mth objective function of the kth individual in I Distance (M, V, N) 1. For front : [i-length[(F)]-1] //Calculate distance 2. //Initiate disyance d=0 3. //Index of fronts F[p,q] 4. S.up=S.up+1 5. S.down=S. up 6. Push[S,pop[i]] 7. I=sort(pop(push(S,pop[i]) VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 44

 3.4 

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

8. Set Fmax=sort(pop[I],first]=inf && Fmin=sort(pop[I],last)=inf I (d k )  I (d k ) 

I (k  1).m  I ( K  1).m f mmax  f mmin

9. Return fronts and distance 3.5.3 Selection Once the individuals are sorted based on non-domination and with crowding distance assigned, the selection is carried out using a crowded- comparison-operator Non-domination rank prank i.e. individuals in front Fi will have their rank Crowding distance Fi(dj ) 1. p <n q if 2. prank < qrank 3. or if p and q belong to the same front Fi then Fi(dp) > Fi(dq) i.e. the crowing distance should be more.

The individuals are selected by using a binary tournament selection with crowedcomparison-operator Tournament selection (pop, Toursize, V, M, N) 1. For i :[1-N] 2. For j :[1-Tour size] 3. I:random(N,Tour size) 4. Get [i1,i2]:I(j) 5. //Check rank and distance of candidate 6. [I_Rank, I_distance]=pop[min(pop(Rank),max(pop(distance))] 7. I_min : pop[find(min(pop(rank)&&max(pop(distance))] 3.5.4 Genetic Operators

VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 45

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

Real-coded GA’s use Simulated Binary Crossover (SBX) operator for crossover and polynomial mutation Simulated Binary Crossover. Simulated binary crossover simulates the binary crossover simulated.

Simulated Binary Crossover (SBX) c1, k  0.5 1   k

 p1, k

 1   k

 p2, k



c2, k  0.5 1   k  p1, k  1   k  p 2, k 

 3.5  3.6

Where ci, k is the ith child with kth component, pi,k is the selected parent and βk (≥ 0) is a sample from a random number generated having the density.

p( )  0.5(c  1)1/  c 2

if 0    1

p( )  0.5(c  1) c 2

if 0    1

p( )  0.5(c  1)1/  c 2

if  1

3.7  3.8 3.9

This distribution can be obtained from a uniformly sampled random number u between (0, 1). ηc is the distribution index for crossover

 (u)  (2u)1/( 1) c

1/(c 1)

 (u)  (1/ 2(1  u))

Crossover (parent pop, M, V, Rang, PC) 1. For i:[1-N] 2. If Random(0,1)<PC 3. //Child initiation child1 and child 2 4. Select parents 5. P1: round [N*random(0,1)] 6. P2: round[N*random(0,1)] 7. Parent 1=parent pop[P1,:] 8. Parent 2=parent pop[P2,:] 9. //Simulated Binary Crossover VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 46

3.10 3.11

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

10. Ui[i] : random(0,1) 11. If Uj[i] <0.5 1

  (2U i )1 Pc 12. Else



1 2(1  U i ) Pc 1

13. Evaluate objective(child1,M,V) 14. Evaluate objective(child2,M,V) 15. Polynomial Mutation ck  pk  ( p u k  pkl ) k

 k  (2rk )1/

m 1

1

if  0.5

 3.12 

1/m 1

 k  1  [2(1  rk )]

16. Mutate (parent pop, M, V, Range, Pm) 17. For i :[1-N] 18. P3=round[N*random(0,1)] 19. Child3=parentpop(P3,1) 20. m(i)=random(0,1) 21. if m(i)<0.5 1

 (i )  2 * m(i ) pm 1 else

 (i ) 

1 2(1  m(i )) pc 1

22. Child 3=child3[i]+δ(i) 23. evaluate objective(child 3,M,V,N) Where, rK is a uniformly sampled random number between (0, 1) and ηm is mutation rate.

3.5.5 Recombination of parent and off springs

VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 47

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

The best parents from former generations and off springs in current generations are combined to preserve parents so that the parents are conserved in the consecutive generations Replace pop (Intermediate pop,M,V,N) 1. For i : [1-N] 2. Sort pop 3. Max_rank=Intermediate pop[find(maxRank)] 4. For i: 1 to max_rank 5. J=max(find[sorted pop(max>rank)==i) 6. If (j>N) 7. //sorted with rank 8. //find the number of individuals with current rank 9. k=j-N 10. p=sorted_pop(k:N) 11. //Sort according to distance 12. For j : [1-N] 13. F[N+K:]=p(N:j) 14. Elseif j<N 15. F[N:j]=sorted_pop[j:N] 16. For i :[1-Kmax] 17. Pool=round(Np/2),tour=2 18. Parent pop=Tournament selection (pop, Tour) 19. [Child 1,Child 2]=Crossover (Parent pop, M, V, Range, PC) 20. [Child 3]=Mutate(Parent pop, M, V, Range, Pm) 21. Offspring pop=[Child 1, Child 2, Child 3] 22. Intermediate pop =[pop, offspring] 23. Replace pop(Intermediate pop, M, V, N) Table.3.3 Results of NSGA II family of best solution for AISI 4340 35 HRC Steel Vc

f

d

Ra

Ft

Fa

Fr

Tf

VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 48

R

D

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

265

0.15

2

1.440575

689.4132

257.6872

376.0769

27.67777

1

65535

265

0.15

1

1.732035

481.3859

108.6272

196.3489

38.33675

1

65535

265

0.25

2

2.972375

889.7312

341.015

436.8659

21.77353

1

65535

169.434

0.15

1

4.023594

412.8885

152.2218

231.225

49.91373

1

65535

261.1212

0.15

1

1.823688

475.9385

108.9812

196.2864

38.67171

1

65535

265

0.25

2

2.972375

889.7312

341.015

436.8659

21.77353

1

65535

264.9902

0.25

2

2.972568

889.7437

341.0119

436.8632

21.774

1

65535

264.9983

0.241889

1.927966

2.753395

851.9404

321.1578

419.3945

22.49811

1

0.02106

231.6619

0.150082

1

2.523533

442.0419

115.6496

199.9704

41.49104

1

0.017777

256.2502

0.157696

1.086001

1.906649

498.2758

125.4584

216.8083

36.74274

1

0.016073

265

0.241178

1.922463

2.735899

848.8689

319.5894

417.9773

22.55941

1

0.015979

255.922

0.160701

1.078365

1.924486

500.1209

127.8311

218.4533

36.56701

1

0.01586

264.9996

0.246009

1.921864

2.81867

861.1237

322.9102

422.1392

22.35356

1

0.015844

185.7026

0.15

1

3.628576

414.8737

139.6661

219.9264

47.28547

1

0.015625

177.0656

0.15

1

3.838041

413.3254

146.0696

225.6509

48.63301

1

0.015374

265

0.197867

1.329979

1.873269

621.6293

191.4162

291.9741

29.43819

1

0.014759

264.961

0.161601

1.172733

1.681103

531.3475

137.4695

234.1206

34.35057

1

0.014673

218.6432

0.150107

1

2.834868

431.2042

120.7764

203.8858

42.92803

1

0.014661

264.9999

0.187048

1.311578

1.792259

598.7628

178.8963

278.8227

30.42464

1

0.014372

264.8573

0.241097

1.982864

2.795903

862.7214

331.3484

426.7589

22.24438

1

0.014369

174.3167

0.15

1

3.904826

413.0673

148.2322

227.6029

49.08405

1

0.014283

265

0.244146

1.891249

2.756463

849.2464

315.7876

416.0992

22.60344

1

0.014065

265

0.198594

1.349524

1.881922

627.2379

194.4536

295.5071

29.18677

1

0.014015

225.3303

0.150002

1

2.674819

436.3252

117.9001

201.6127

42.18555

1

0.013776

185.282

0.15

1

3.638763

414.7724

139.9641

220.1908

47.34876

1

0.013629

186.0766

0.15

1

3.619519

414.966

139.4022

219.6925

47.22937

1

0.013486

264.9959

0.240879

1.953051

2.760167

855.148

325.316

422.2024

22.40593

1

189.9865

0.15

1

3.524895

416.0567

136.7103

217.3171

46.65346

1

0.013037

264.9613

0.150352

1.247266

1.609129

533.0857

133.4604

236.7962

34.52022

1

0.01294

264.5977

0.171423

1.103768

1.747466

529.6774

141.0782

232.7329

34.4049

1

0.01288

232.9983

0.150418

1

2.491644

443.7511

115.4963

199.9651

41.30708

1

0.012815

265

0.246117

1.759588

2.664722

823.9152

293.6823

399.6592

23.29502

1

0.012699

208.8492

0.150014

1

3.070018

424.5623

125.443

207.6562

44.11734

1

0.01264

Table 3.3 consists optimized

results for 35 HRC process parameters in first three

columns (vc,f,d) and latter columns contains optimized results for objective functions (Ra VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 49

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

,Ft ,Fa ,Fr ,Tf ).Out of the 1000 chromosome solutions few best chromosomes

are listed in the above table.

3.5.6 Plots for NSGA II results (35 HRC)

Fig. 3.4 Rank and Pareto for Ra(35HRC)

Fig. 3.5 Pareto-front for Tf (35HRC)

Fig. 3.6 Average distance between consecutive generations (35HRC) Table.3.4 Results of NSGA II family of best solution for AISI 4340 45 HRC Steel Vc

f

d

Ra

Ft

Fa

Fr

Tf

R

D

175

0.2500

2

5.4703

1.1135e+03

576.9927

494.4395

8.9403

1

65535

135.9606

0.1500

1

3.9895

519.5461

331.3059

282.9869

28.2786

1

65535

130.8263

0.1500

1

4.0725

524.2045

331.6248

282.5873

30.0149

1

65535

VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 50

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

175

0.1500

1

3.5798

504.5262

349.7478

312.2032

19.1319

1

65535

175

0.2500

2

5.4703

1.1135e+03

576.9927

494.4395

8.9403

1

65535

171.7381

0.1500

1

3.5991

504.4006

346.7949

307.9906

19.6973

1

65535

170.6747

0.1512

1

3.6100

505.0240

346.2216

307.2963

19.8126

1

0.0195

175.0000

0.2487

1.9979

5.4454

1.1089e+03

574.4013

492.7024

8.9698

1

0.0172

160.3800

0.1500

1.0066

3.6897

508.6927

339.0356

296.4326

21.7907

1

0.0168

174.9624

0.2458

1.9982

5.3980

1.1015e+03

569.8832

489.6478

9.0210

1

0.0160

175

0.2373

2

5.2635

1.0800e+03

556.7698

480.8408

9.1639

1

0.0156

171.7067

0.1500

1.0037

3.6009

505.9857

346.9955

308.2302

19.6489

1

0.0154

167.9552

0.1514

1.0066

3.6322

508.2227

344.5538

304.7724

20.2004

1

0.0154

168.5030

0.1500

1.0865

3.6577

541.3019

350.1705

311.2217

19.0637

1

0.0146

175

0.2306

2

5.1613

1.0631e+03

546.5998

474.0485

9.2885

1

0.0145

175

0.1857

1.4158

4.0521

716.6107

404.1508

368.3416

13.3300

1

0.0144

136.3303

0.1500

1

3.9838

519.2347

331.3076

283.0465

28.1599

1

0.0143

174.4113

0.1512

1.1067

3.6398

551.0508

356.2463

320.0333

17.7614

1

0.0142

174.8377

0.1511

1.0501

3.6089

526.7210

352.9048

316.2096

18.4048

1

0.0141

145.7619

0.1500

1

3.8500

512.3850

332.4684

285.9718

25.3898

1

0.0141

174.9083

0.1500

1

3.5803

504.5192

349.6613

312.0804

19.1474

1

0.0140

175

0.2393

2

5.2960

1.0853e+03

559.9760

482.9890

9.1264

1

0.0139

174.9987

0.2313

1.5566

4.6773

856.1857

468.7186

420.8659

11.1902

1

0.0137

174.9734

0.2064

1.1378

4.0238

611.7845

388.5914

358.3491

14.9399

1

0.0137

171.7164

0.1528

1.0761

3.6445

538.6585

352.6485

315.3194

18.4855

1

0.0136

175

0.1504

1.0131

3.5873

510.3848

350.6039

313.3299

18.9233

1

0.0136

175

0.2458

2

5.4006

1.1023e+03

570.2226

489.8743

9.0119

1

0.0135

175.0000

0.2028

1.1931

4.0322

634.4821

392.0288

360.3824

14.5323

1

0.0135

174.9971

0.2481

2

5.4377

1.1083e+03

573.8392

492.3111

8.9735

1

0.0133

175

0.1500

1.0286

3.5935

516.9707

351.4236

314.2841

18.7316

1

0.0132

168.5690

0.1500

1.0653

3.6481

532.3890

348.6946

309.5442

19.3321

1

0.0131

175

0.2295

1.0450

4.1928

593.8836

396.2472

370.1817

15.1377

1

0.0131

Table 3.4.consists optimized

results for 45 HRC process parameters in first three

columns (vc ,f, d) and latter columns contains optimized results for objective functions (Ra ,Ft , Fa , Fr , Tf ).Out of the 1000 chromosome solutions few best chromosomes are listed in the above table.

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3.5.7 Plots for NSGA II results (45 HRC)

Fig. 3.7 Rank and Pareto for Ra (45HRC)

Fig. 3.8 Pareto-front for Tf (45HRC)

Fig. 3.9 Average distance between consecutive generations (45HRC) 3.6 Strength Pareto Evolutionary Algorithm (Type 2) Initialize population structure with fields in position cost fitness variables, dominance field and cumulative fitness. Then objectives are evaluated over random pop. Position with initial fitness, pop.cost. A fitness subset â&#x20AC;&#x2DC;archiveâ&#x20AC;&#x2122; for best individuals is initiated, first, all non-dominated population members are copied to the archive; any dominated individuals or duplicates are removed from the archive during this update operation. If the size of the updated archive exceeds a predefined limit, further archive members are deleted by a clustering technique which preserves the characteristics of the nonVISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 52

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dominated front. Afterwards, fitness values are assigned to both archive and population members: Each individual i in the archive is assigned a strength value S(i) ∈ [0, 1), which at the same

time represents its fitness value F (i).and S(i) is the number of population members ‘j’

that are dominated by or equal to i with respect to the objective values, divided by the population size plus one.

The fitness F (j) of an individual j in the population is calculated by summing the strength values S(i) of all archive members i that dominate or are equal to j, and adding one at the end. To avoid the situation that individuals dominated by the same archive members have identical fitness values, for each individual both dominating and dominated solutions are taken into account. Each individual i in the archive P t and the population Pt is assigned a strength value S(i), representing the number of solutions it dominates.

S  i   {j j  P.t  Pt  i  j} | R(i) 



S ( j)

j Pt  Pt , j i

On the basis of the S values, the raw fitness R(i) of an individual i is calculated: That is the raw fitness is determined by the strengths of its dominators in both archive and population. fitness is to be minimized here, i.e., R(i) =0 corresponds to a nondominated individual, while a high R(i) value means that i is dominated by many individuals.

Additional density information is incorporated to discriminate between individuals having identical raw fitness values.

The density estimation utilizes kth nearest neighbor method, where the density at any point is a (decreasing) function of the distance to the kth nearest data point. Here the inverse of the distance to the kth nearest neighbor is used for as density estimate. For each individual i the distances (in objective space) to all individuals j in archive and population VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 53

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

are calculated and stored in a list. After sorting the list in increasing order, the kth element gives the distance sought, denoted as σk. The kth nearest parameter is defined as square root of the sample size,

NN

D(i) 

and density distribution as 1 k  2

(3.13)

F (i)  R(i)  D(i)

(3.14)

In the denominator, two is added to ensure that its value is greater than zero and that D(i) < 1. Finally, adding D(i) to the raw fitness value R(i) of an individual i yields its fitness F (i).

Fig. 3.10 SPEA2 Algorithm[71] Table 3.5 SPEA2 Parameter Setting Population size Generation Archive size

Crossover probability Crossover constant Mutation probability Mutation constant

1000 100 300 0.7 0.1 0.1 0.2

3.6.1 Initialize Variables and Evalaute Objectives Initialize {pop.position, pop.cost, pop.S, pop.R, pop.σ, pop.D, pop.F} 1. For i : [1-N] VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 54

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2. Pop.position :random(range,N) 3. Pop.cost=evalautefunction(pop.position) 4. //Initialize archive 5. Archive={ } 3.6.2 Tournament Selection The tournament selection is similar to that applied in NSGA II Binary tournament selection(archive,[archive.F],N) 6. I=random(N,2) 7. I1=I(1) 8. I2=I(2) 9. If F(i1)<F(i2) 10. P=pop[i1] 11. Else 12. P=pop[i2]

3.6.3 Genetic Operator Genetic operator applied is similar to that of NSGA II with slight variation in mutation technique.

Crossover (p1,p2,crossover parameters) 13. Parametrs :(Υ,range) 14. α:random(-Υ,1+Υ,N) y1   * p1  (1   )* p 2

15. y 2   * p 2  (1   )* p1 16. Y1=min[max(y1,range)] 17. Y2=min[max(y2,range)] 18. Mutate (p3,mutation parameters, range) 19. Parameters : , range 20. Rmin=min(range)

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21. Rmax=max(range) 22. dr=Rmax-Rmin 23. σ: *dr 24. y=p1+σ*random(Nm) 25. y=min(max(y,range)) 26. Main learning 27. Do until( max iteration IT) 28. P=[pop,archive] 29. //check for domination 30. [dom,p.S]=dominates (p[i]) 31. S=[p.S] 32. P[I].R=sum(S[dom]) 33. Q=[p.cost] 34. σ : Euclidiean distance[q] 35. σ : sort[σ] 36. p[i].σ =σ 37. p[i].σ[k]=p[i].σ[k]

p[i].D 

1 p[i]. [k  2]

38.

39. //Fitness Calculation

40. P[i].F=p[i].R+p[I].D 41. Fit =sum(find(p.R==0)) 42. P.F=[fit] 1. Archive =p[size[p.R]] 2. While[min(σ)==max(σ(k)&&k<size(σ))] 3. Pareto front=archive[archive.R==0] 4. [p1,p2]=binary tournament selection(archive,[archive.F],N) 5. Popc.cost=evalautefitness[child1,child2] 6. [child 1 child2]=crossover(p1,p2,crossoverparameters) 7. [p3]=binarytournamentselection(archive,[archive.F],N)

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8. [child 3]=mutate(p3,mutation parameters,N) 9. Popm.cost=evalautefitness[child3] 10. pop=[popc.cost popm.cost] Table 3.6 Results of SPEA II family of best solution for AISI 4340 35 HRC Steel

Position

Cost

S

R

[259.15 0.17 1.07]

[1.88 514.11 136.22 225.95 35.51]

[9.00]

[0.00]

0.375 [0.32] [0.32]

[265.00 0.16 1.84]

[1.60 674.93 238.28 350.68 27.63]

[16.00] [0.00]

0.3563 [0.32] [0.32]

[265.00 0.24 1.81]

[2.57 813.37 295.99 398.32 23.38]

[17.00] [0.00]

0.2863 [0.34] [0.34]

[265.00 0.20 1.96]

[2.05 757.72 292.73 391.58 24.60]

[9.00]

[0.00]

0.305 [0.33] [0.33]

[264.60 0.15 1.94]

[1.51 683.95 250.23 366.11 27.68]

[6.00]

[0.00]

0.4035 [0.32] [0.32]

[264.83 0.17 1.74]

[1.64 661.12 224.65 335.27 28.00]

[26.00] [0.00]

0.3147 [0.33] [0.33]

[264.98 0.21 1.77]

[2.11 737.27 265.23 367.09 25.17]

[26.00] [0.00]

0.2558 [0.34] [0.34]

[265.00 0.19 1.79]

[1.89 706.15 252.72 356.92 26.15]

[20.00] [0.00]

0.3483 [0.33] [0.33]

[264.23 0.21 1.92]

[2.25 776.56 295.04 392.75 24.13]

[11.00] [0.00]

0.2511 [0.34] [0.34]

[264.85 0.22 1.47]

[2.09 689.39 226.10 330.51 26.92]

[34.00] [0.00]

0.3906 [0.32] [0.32]

[265.00 0.24 1.69]

[2.57 802.55 280.54 388.16 23.84]

[16.00] [0.00]

0.2971 [0.34] [0.34]

[264.99 0.18 1.88]

[1.81 711.36 262.25 367.37 26.09]

[17.00] [0.00]

0.3296 [0.33] [0.33]

[264.72 0.25 1.93]

[2.91 874.97 328.11 427.52 22.13]

[6.00]

[0.00]

0.2971 [0.34] [0.34]

[256.63 0.16 1.09]

[1.90 505.03 129.90 221.26 36.21]

[13.00] [0.00]

0.3225 [0.33] [0.33]

[264.97 0.19 1.45]

[1.83 636.05 200.63 303.73 28.72]

[39.00] [0.00]

0.3818 [0.32] [0.32]

[263.42 0.20 1.31]

[1.91 616.67 188.86 289.03 29.75]

[30.00] [0.00]

0.368 [0.32] [0.32]

[264.62 0.21 1.91]

[2.16 763.63 289.27 387.95 24.44]

[12.00] [0.00]

0.2511 [0.34] [0.34]

[264.57 0.23 1.41]

[2.16 695.37 225.90 331.84 26.94]

[30.00] [0.00]

0.3986 [0.32] [0.32]

[265.00 0.21 1.97]

[2.28 789.31 306.06 402.04 23.74]

[8.00]

0.2477 [0.34] [0.34]

[0.00]

sigmaK D

F

Table 3.6 has results of optimized solution obtained from SPEA2 algorithm for 35HRC position matrix represents the process parameters (vc,f,d), cost matrix represents optimized solution for objectives (Ra ,Ft ,Fa ,Fr ,Tf ). and the remaining cells correspond to SPAE2 parameters. Out of the 300 solutions in archive only few are listed in the above table.

3.6.4 Plots for SPEA 2 results( 35 HRC)

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Fig. 3.12 Pareto-front for Tf (35HRC)

Fig. 3.11 Rank and Pareto for Ra (35HRC)

Fig. 3.13 Average distance between consecutive generations (35HRC) Table 3.7 Results of SPEA II family of best solution for AISI 4340 45 HRC Steel

Position

Cost

S

R

sigmaK D

[175.00 0.22 1.90]

[4.92 997.97 515.49 452.74 9.82]

[14.00] [0.00]

0.238 [0.35]

[0.35]

[174.72 0.23 1.54]

[4.67 849.67 466.88 419.61 11.30]

[10.00] [0.00]

0.271 [0.34]

[0.34]

[174.92 0.18 1.34]

[3.93 674.57 389.76 355.44 14.19]

[29.00] [0.00]

0.427 [0.31]

[0.31]

[174.83 0.23 1.70]

[4.80 920.84 489.93 435.20 10.54]

[14.00] [0.00]

0.247 [0.35]

[0.35]

[174.85 0.18 1.55]

[4.14 773.13 418.81 380.47 12.54]

[21.00] [0.00]

0.299 [0.33]

[0.33]

[175.00 0.16 1.33]

[3.83 656.51 380.43 346.35 14.81]

[26.00] [0.00]

0.482 [0.30]

[0.30]

[174.99 0.19 1.70]

[4.37 852.70 448.58 404.23 11.40]

[21.00] [0.00]

0.253 [0.34]

[0.34]

[175.00 0.19 1.44]

[4.14 737.76 413.56 376.50 12.91]

[24.00] [0.00]

0.373 [0.32]

[0.32]

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F

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

[174.99 0.22 1.47]

[4.45 793.05 442.97 401.34 11.96]

[11.00] [0.00]

0.278 [0.34]

[0.34]

[175.00 0.21 1.84]

[4.64 934.49 483.32 430.12 10.45]

[20.00] [0.00]

0.238 [0.35]

[0.35]

[175.00 0.22 1.51]

[4.54 819.43 453.20 409.14 11.62]

[13.00] [0.00]

0.26 [0.34]

[0.34]

[173.87 0.19 1.34]

[4.06 693.36 401.38 366.13 13.76]

[26.00] [0.00]

0.435 [0.31]

[0.31]

[175.00 0.21 1.40]

[4.23 738.85 419.98 382.61 12.77]

[23.00] [0.00]

0.348 [0.33]

[0.33]

[175.00 0.24 1.29]

[4.51 735.25 436.39 400.33 12.70]

[7.00]

[0.00]

0.346 [0.33]

[0.33]

[168.21 0.16 1.02]

[3.67 519.42 348.53 310.79 19.45]

[15.00] [0.00]

1.069 [0.33]

[0.33]

[175.00 0.24 1.19]

[4.48 688.80 426.53 395.02 13.40]

[8.00]

[0.00]

0.425 [0.31]

[0.31]

[174.95 0.23 1.26]

[4.33 698.11 419.46 385.62 13.28]

[16.00] [0.00]

0.347 [0.33]

[0.33]

[175.00 0.22 1.70]

[4.68 901.42 478.44 426.62 10.73]

[18.00] [0.00]

0.266 [0.34]

[0.34]

[175.00 0.24 1.82]

[5.07 1000.50 524.52 459.35 9.80]

[11.00] [0.00]

0.353 [0.32]

[0.32]

Table 3.7 has results of optimized solution obtained from SPEA2 algorithm for 45HRC position matrix represents the process parameters, (vc, f, d) cost matrix represents optimized solution for objectives (Ra ,Ft ,Fa , Fr , Tf ) and the remaining cells correspond to SPAE2 parameters Out of the 300 solutions in archive only few are listed in the above table 3.6.5 Plots for SPEA 2 results( 45 HRC)

Fig. 3.14 Rank and Pareto for Ra (45HRC)

Fig. 3.15 Pareto-front for Tf (45HRC)

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Fig. 3.16 Average distance between consecutive generations (45HRC)

3.7 Swarm Intelligence Swarm systems are based on behavior of school of birds, insects, fireflies where a flocks of birds twisting ,V-shaped structure of migrating geese, winter birds hunting for food, the synchronized flashing of fireflies are tried to imitating. The well-choreographed collective behavior without any leader is adopted to search for optimal solutions. For instance Ants living in colony, their behavior is driven by the goal of colony survival instead of individual survival, while searching for food ants initially explores surrounding nests. In random manner A similar behavior is observed with flocks of birds where a leader keeps guiding the flock to updated food location.

3.8

Mathematical Formulation of PSO Algorithm

Assuming a swarm ‘S’ of N(n) particle moves through dimension ‘D” in search space RD Let ‘f’ be objectives of our optimization problem the function ‘f’ is defined over the discretized space as f : R D  R over the space ‘R’. The definition of swarm can be condensed to S   St tN  X t ,Vt , Lt , Gt tN  [ X 0 ,V0 , L0 , G0  ,  X1 ,V1 , L1 , G1  ...............  X n , Vn , Ln , Gn ] o o

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Where X t  X tn,d 1  n  Nt  & 1  d  D 

(dth coordinate of the velocity of particle ‘n’ after time step ‘t’) Vt  Vt n,d 1  n  Nt  & 1  d  D 

(dth coordinate of the velocity of particle ‘n’ after time step ‘t’) Lt  Lnt ,d 1  n  Nt  & 1  d  D 

(dth coordinate of local best (attraction) of particle ‘n’ after time step ‘t’) Gt  Gtn,d 1  n  Nt  & 1  d  D 

(dth coordinate of global best (attraction) of particle ‘n’ after time step ‘t’) n 1 N ,d l ,d Furthermore Gtn ,d = Gt if (n<N) and satisfying initial and final conditions Gt  Gt 1

with given distribution for initial position and velocity (x0,v0) Initial grid index G0I is determined by minimum argument of function.

G0l  arg min1  n  N{ f ( X 0n )}atl0  x0

St 1   X t 1 ,Vt 1 , Lt 1 , Gt 1 which is determined by movement equation.







vtn,1d  vtn,1d  C1.r1n,d Lnt ,d  X tn,d  C2 .S1n,d Gtn,d  X tn,d



(3.15)

Where  C1 & C2  control the influence of personal best of particle and the common knowledge of swarm also known as acceleration co-efficient.

Stn.d & rtn,d are randomness which are drawn uniformly at random [0,1]

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If a particle velocity component exceed a certain interval [-Vmax, Vmax]. It is set back to interval found. The movement equation is altered by clamping inertia.







vtn,d  1.vtn,d  C1.rt n,d Lnt ,d  X tn,d  C2 .Stn,d Gtn,d  X tn,d



and position is altered as

X tn,1d  X tn,d  vtn,1d   0,1   0.72 C1  C2  1.49

3.8.1

Typical Initialization strategy

1) Random initiation of position, velocity 2) Initiate velocity with zero matrix movement v n,d   .vn,d  C1rand (0,1)  Ln,d  X n,d   C2 rand (0,1) G n,d  X n,d 

3) Update position along with the personal and global best X n,d  X n,d  v n,d

If f  X n   L  Ln  then Ln  X n

 

f X n  L  G  then G  X n

3.8.2 Topologies of PSO To better predict social learning process the global best particle (Gd) is replaced by the local guide particle (Ld) Topology is typically represented as graph whose nodes are particles and edges connect neighboring particles. The edge connections between any two grid points n1 & n2 as a particle for its own local guide, the edge connections are determined by minimum argument of function Pt n  Euclidean(arg min f ( x)

 n ', n

where

x  [ Lt

| n'  N (n) |]

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 n ', n

Lt

 is local guide attraction of particle ‘n’ at the time step ‘t’ when particle ‘n’

makes its move.  n ', n

L

  n ' n ',n  Lt    n ' L If n’  n  L t 1 

Attempts to form neighborhood topology depend on Inclined distance of particle in search space. In general neighborhood topology is choose independent of particle position in search space.

3.8.3 Definitions of Swarm topology For a Swarm, definition of topology and its potential drives solution space, leader of swarm both locally and global depends on the topology, though topology remains static the co-ordinates of particle attractors monitors the distribution of swarm. In general the global and local attractor is determined by fitness augments i.e.

G0l  arg min1  n  N{ f ( X 0n )}atl0  x0 Lnt  arg min[ X tn1 , Lnt ] t ,d For a given swarm S in stochastic process  X t ,Vt , Lt , Gt  the potential function n in

dimension d is determines the swarm fitness level at interval time step t. It is determined by global best and personal best of swam at all interval.

nt ,d 

n 1

 ( | vtn ,d |  | Gtn,1d  X tn ,d |)  '

n' 1

'

n N

 ( | v

n' , d t 1

'

|  | Gtn,1d  X tn1,d |)

(3.16)

n'  n 1

3.8.4 Convergence criteria It is important to control swarm topology for determining the desired solution space which depends on the movement constants applied on swarm so for a swarm to converge

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the inertia damping co –efficient which keeps the swarm in bound and the acceleration co–efficient keep drives the co-ordinates of local and global attractors.

3.8.5 Criteria for Inertia clamping and acceleration co-efficeint At the topological development both global and local attractors are bound to same value

X t1,1  X 01,1  v1,1s For consecutive iterations t

X t1,1  X 01,1   v1,1 s t 0

t 1

 X 01,1  v0   s t 0

Lt s  X

1,1 t

 Lt s  [ X

1,1 0

v

t 1



s

]

0 t 0

1 ]  X 01,1  v0 [ 1 

So for swarm to be in bounds the inertia damping co-efficient should be between [0<  <1]. Now for acceleration co-efficient the movement equation is analyzed

vtn,1d  1.vtn,d  C1.rt n ,d  Lnt ,d  X tn ,d   C2 .Stn,1d  Gtn ,d  X tn ,d  X tn,1d  X tn,d  vtn,1d [ X tn,1d , vtn ,d ]tN The consecutive velocities cab be relates as follows

vtn,1d  X tn,1d  X tn ,d vtn ,d  X tn ,d  X tn,1d vtn,1d  vtn ,d  [ X tn,1d  X tn,1d ] and position can be re written as

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X tn,1d  [1    (C1.rtn1,d  C2 .Stn,1d ) X tn,d   X tn,1d  C1.rtn,d Lnt ,d  C2 .Stn,1d Gtn,d ] Now to calculate the error in position between the expected and attained the expentancy operator is applied on position vector.

E[ X tn,1d ]  E[ X tn,d ][1    0.5*(C1  C2 )]   E[ X tn,1d ]  0.5*[C1.Lnt ,d  C2 .Gtn,d ]

(3.17)

Assuming the expectance to be  , now for a particle in swarm it is expecte that the calculated and expected positions are equal so for optimum goal attainment the expectancy is equal to zero which converts the position expectancy equation to

 2  [1    0.5*(C1  C2 )]   =0 Finding roots of expectancy

1,2  0.5*([1    0.5*(C1  C)]  [1    0.5*(C1  C2 )]2  4 Now for expectancy is always a positive real

[1    0.5*(C1  C2 )]2  4.  0

(3.18)

We arrive at optimal co-efficient criteria for better convergence.

0  (C1  C2 )  4(1   )

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3.9 PSO Algorithm

Fig.3.17 PSO Algorithm[74] Table 3.8 PSO Setting MOPSO Definition Stopping/convergence

100

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Total particle population Max no of repository elements Inertia weight ( w) Inertia weight damping rate (w.damp) Personal learning co efficient (a1) Global learning co-efficient (a2) No of grid in each dimension Inflation rate ( á˝°) Leader selection pressure ( ) Deletion selection pressure ( ) Mutation rate (mu)

1000 500 0.5 0.99 1 2 7 0.1 2 2 0.1

3.9.1 Initialize population and Evaluate fitness Initialize population structure with fields Particle. Evaluate fitness over particle position and determine the dominance levels of each particle by dominates function //Initialize particle structure {Particle.Position,

Particle.Velocity,

Particle.Cost,

Particle.Best

Position,

Particle.Bestcost, Particle.Is Dominated, Particle.Grid Index, Particle.Grid Subindex} //Evaluate particle.position and cost 1. For i :[1-N] 2. Pop[i].position=random[Range,N] 3. Pop[i].velocity : zeros[N] 4. Pop[i].cost :evaluate (pop[i].position) 5. //update personal best 6. Exchange pop[i].best position with pop[i].position 7. Exchange popi[i].best cost with pop[i].cost Initiate repository element which is subset of all particles with best position and cost then select leader for swarm at every iteration through select leader function and i=update the particle structure for current leader swarm. //determine domination level VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 67

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Pop : domination(pop,N) 1. For i:[1-N] 2. For j:[i+1-N] 3. If dominates (pop[i], pop[j]) 4. True(pop[j] Is dominated) 5. Else if dominates(pop[j],pop[i]) 6. True(pop[i] Is dominated) 7. b=dominates(pop[i],pop[j]) 8. b= all(x<y)&&any(x<y) 9. //Initiate repository element 10. Rep=pop[~dominated pop) Apply mutation operator on the updated particle structure then calculate the dominance level for current structure. Create neighborhood for swarm by initiating grid topology for swarm. Update the swarm in the repository element with current dominance level.

3.9.2 Create Grid Index Now the topology is built for swarm which was initialized through particle structure topology is static and remains unchanged at every generation. Naumann /Grid based topology is built which utilizes Euclidean co-ordinates based on position of each swarm //create grid index Grid=create grid(rep,ngrid,Îą) 1. P=[pop.cost] 2. Rmin=min(p,[],2) 3. Rmax=max(p,[],2) 4. dr=Rmax-Rmin

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Rmin  Rmin   * dr 5. Rmax  Rmin   * dr 6. //Initiate grid index 7. Grid.L : [] Grid.H :[] 8. Object : size[p,1] 9. For j: [1-object] 10. p.object[equal spacing in R] 11. Grid[j].L =[-∞, p.object] 12. Grid[j].H=[p.object, ∞] 13. For i :[1-size(Rep)] 14. Rep[i]=find Grid index (Rep[i],Grid) 15. Obj=size(particle.cost) 16. Grid size=size(Grid.L) 17. Particle.Grid sub Index :zeros(p.object) 18. For i : [1-obj] 19. Particle.Grid sub Index[i]=find(particle.cost[i]<grid] 20. Particle.Grid Index=N*Grid size*particle grid index+ particle grid sub index 21. //Initiate repository element 22. Rep=pop[~dominated pop)

3.9.3 Select Leader After building topology each particle in repository is recognized with its position and velocity co–ordinates the identity of each particle is recognized through this co –ordinate Swarm is lead most fittest particle/particles At each generation the swarm changes its leader according to the swarm velocities and position evaluated through swarm movement equation. //select leader Select leader(rep, )

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1. //Grid Index of all repository 2. I=[rep.Grid Index] 3. //occupied cells 4. C=select unique cells(I) 5. Q=find(I==C) 6. //selection probability 7. p=exp(- *N)

P

p

p 8.

9. s=select(P>random(0,1)) 10. //select cell 11. Sc=unique (s) 12. Find(I==s) 13. Leader=Rep(sc) 14. Mutation(pop,pm,Range) 15. If pm<random[0,1] 16. Pop[i]=R+dr*R 17. dr=pm*(Rmax-Rmin) 3.9.5 Delete extra elements Excess particles in the repository are either deleted or replaced by better fit particles in each generation if repository exceed its size then the interia damping factor reduces the velocity of swarm resulting in poor convergence hence at aevery generation repository is checked for its size. //Delete extra elements Delete Rep member(Rep,Υ) [1] Grid Index =(Rep.Grid Index) [2] Deletion=select leader (Rep,Υ)

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[3] Deletion={ } 3.9.6 Swarm Movement Now that the swarm and its topology is built the swarm is allowed to move over the topology with governing movement equation followed by a slight mutation in fit particles which would accelerate motion of swarm by randomly changing the velocity and position coordinates.

Main Learning Do until max (IT) For i : [1-N] //Select Leader 1. leader=Select leader(rep, )

pop  i  .Velocity  w * pop  i  .Velocity c1* rand VarSize  . *  pop  i  .Best.Position  pop  i  .Position   c 2* rand VarSize  . *  leader.Position  pop  i  .Position  pop  i  .Position  pop  i  .Position  pop  i  .Velocity pop  i  .Position  max  pop  i  .Position, VarMin  pop  i  .Position  min  pop  i  .Position, VarMax  pop  i  .Cost  Evaluate  pop  i  .Position  2. Newpop=mutatute(pop,pm,Range) 3. newpop.cost=evaluate(newpop.position) 4. Determine domination(Rep) 5. If dominates(New.pop.position, pop.position) 6. True(Is dominated pop.position) 7. Else if dominates(pop.position, New.pop.position) 8. True(Is dominated New.pop.position) 9. Grid=Create Grid(Rep, grid size,α) 10. Check if resize>maxrep

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11. Rep=delete rep members (Rep,Î¥).

Table 3.9 MOPSO family of optimal solutions for 35 HRC AISI 4340 steel Grid Position

Velocity

[265.00 0.16

Cost

Best Position

Best Cost

[1.55 702.05 266.48 379.44

[263.49 0.22

[2.01 595.25 181.77 277.57

2.00]

[1.62 -0.06 1.07]

26.93]

1.01]

31.89]

[212.34 0.15

[-43.26 -0.04 -

[2.99 426.70 123.66 206.18

[255.60 0.19

[2.03 659.89 217.50 320.77

1.00]

0.97]

43.69]

1.59]

28.39]

[193.76 0.15

[137.29 -0.02 -

[3.43 417.33 134.23 215.15

[193.76 0.15

[3.43 417.33 134.23 215.15

1.00]

0.35]

46.12]

1.00]

46.12]

[174.19 0.15

[-85.70 -0.01 -

[3.91 413.06 148.34 227.70

[174.19 0.15

[3.91 413.06 148.34 227.70

1.00]

0.24]

[265.00 0.15

49.11]

1.00]

49.11]

[1.45 652.07 221.19 340.52

[265.00 0.15

[1.45 652.07 221.19 340.52

1.82]

[7.93 -0.01 0.20]

28.93]

1.82]

28.93]

[174.72 0.15

[-32.71 -0.00 -

[3.89 413.10 147.91 227.31

[207.43 0.15

[3.10 427.11 127.30 209.77

1.00]

0.01]

49.02]

1.01]

44.08]

[215.30 0.15

[-30.50 -0.03 -

[2.92 428.66 122.23 205.01

[215.30 0.15

[2.92 428.66 122.23 205.01

1.00]

1.23]

[211.21 0.15

43.34]

1.00]

43.34]

[3.01 425.98 124.22 206.64

[211.21 0.15

[3.01 425.98 124.22 206.64

1.00]

[-1.61 -0.04 -0.73]

43.83]

1.00]

43.83]

[194.42 0.15

[-14.35 -0.05 -

[3.42 417.57 133.80 214.78

[194.42 0.15

[3.42 417.57 133.80 214.78

1.00]

0.63]

[265.00 0.20

46.02]

1.00]

46.02]

[1.95 694.04 238.22 342.10

[265.00 0.20

[1.95 694.04 238.22 342.10

1.66]

[37.22 -0.00 0.20]

26.52]

1.66]

26.52]

[171.25 0.15

[-32.00 -0.03 -

[3.98 412.91 150.71 229.85

[203.25 0.18

[3.13 673.17 231.63 322.24

1.00]

0.62]

[265.00 0.15

49.60]

1.60]

33.15]

[1.66 509.07 120.80 217.67

[265.00 0.15

[1.66 509.07 120.80 217.67

1.13]

[32.08 -0.01 0.13]

36.15]

1.13]

36.15]

[265.00 0.15

[26.00 -0.05 -

[1.49 603.83 180.32 296.71

[265.00 0.15

[1.49 603.83 180.32 296.71

1.59]

0.03]

[249.95 0.16 1.00]

[8.34 -0.09 -0.99]

[265.00 0.16 2.00]

[32.06 0.00 1.15]

[265.00 0.16 1.55]

[2.84 -0.08 0.06]

[265.00 0.15 1.62]

[1.51 -0.02 0.03]

[265.00 0.20

30.84]

1.59]

30.84]

[2.10 480.93 125.34 211.22

[249.95 0.16

[2.10 480.93 125.34 211.22

37.96]

1.00]

37.96]

[1.50 696.28 262.59 377.89

[265.00 0.16

[1.50 696.28 262.59 377.89

27.25]

2.00]

27.25]

[1.57 608.71 184.51 296.17

[265.00 0.24

[2.38 749.62 248.99 359.27

30.28]

1.49]

25.43]

[1.48 610.04 185.18 302.22

[265.00 0.15

[1.49 603.83 180.32 296.71

30.57]

1.59]

30.84]

[1.96 698.30 241.07 344.82

[265.00 0.20

[1.96 698.30 241.07 344.82

1.68]

[10.04 0.01 0.04]

26.38]

1.68]

26.38]

[265.00 0.17

[125.17 -0.00 -

[1.74 530.80 141.76 233.24

[265.00 0.17

[1.74 530.80 141.76 233.24

1.10]

0.30]

34.33]

1.10]

34.33]

VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 72

Index

GridSubIndex

10668

[2,6,6,7,3]

33631

[6,2,2,2,7]

40282

[7,2,3,3,7]

46844

[8,2,3,3,8]

9850

[2,5,5,6,4]

46844

[8,2,3,3,8]

27070

[5,2,2,2,7]

33631

[6,2,2,2,7]

40282

[7,2,3,3,7]

17139

[3,6,5,6,3]

46844

[8,2,3,3,8]

14684

[3,3,2,3,5]

9031

[2,4,4,5,4]

21236

[4,3,2,2,5]

10659

[2,6,6,6,3]

9031

[2,4,4,5,4]

9031

[2,4,4,5,4]

17139

[3,6,5,6,3]

15494

[3,4,3,3,5]

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

[265.00 0.16

[1.61 708.49 270.60 381.20

[265.00 0.15

[1.49 603.83 180.32 296.71

2.00]

[0.68 0.01 0.47]

26.59]

1.59]

30.84]

[265.00 0.24

[28.76 -0.01 -

[2.70 846.43 323.67 419.78

[265.00 0.24

[2.70 846.43 323.67 419.78

1.96]

0.04]

[265.00 0.15 2.00]

[3.61 0.00 0.67]

[250.88 0.17 1.00]

[-7.54 -0.02 -0.11]

22.55]

1.96]

22.55]

[1.48 694.34 261.24 377.37

[265.00 0.15

[1.48 694.34 261.24 377.37

27.37]

2.00]

27.37]

[2.09 486.86 128.60 214.65

[250.88 0.17

[2.09 486.86 128.60 214.65

37.52]

1.00]

37.52]

10668

[2,6,6,7,3]

31169

[5,7,7,8,2]

10659

[2,6,6,6,3]

21326

[4,3,3,3,5]

Table 3.9 has results obtained from MOPSO for 35 HRC the position, Best position matrix represent process parameters (vc,f,d),. and Cost, Best Cost represents objective fitness (Ra ,Ft ,Fa ,Fr ,Tf ) and the latter columns are corresponding to grid index and grid sub index of PSO topology Out of the 500 repository elements few are listed in above table.

3.9.7 Plots for MOPSO results (35 HRC)

Fig 3.18 Pareto spread surface roughness and

Fig 3.19 3D surface plot of optimal Ra with

cutting force 35HRC

best position 35HRC

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Fig 3.20 3D surface plot of Tf with best

Fig 3.21. Depth of cut influence on cutting

position 35HRC

forces 35 HRC

Table 3.10 MOPSO family of optimal solutions for 35 HRC AISI 4340 steel Table 1 Position

Repository elements Solution for Population Velocity

[131.18 0.15 1.00]

[-38.59 -0.02 -0.73]

[145.95 0.15 1.00]

[-29.05 -0.08 -1.19]

[175.00 0.24 2.00]

[2.94 0.05 0.95]

[175.00 0.15 1.52]

[18.14 -0.00 0.06]

[175.00 0.19 2.00]

[2.26 0.01 0.10]

[157.18 0.15 1.00]

[-17.82 -0.04 -1.28]

[175.00 0.19 2.00]

[56.89 -0.05 0.12]

[175.00 0.24 1.68]

[4.45 -0.01 0.22]

[132.82 0.15 1.00]

[-53.49 -0.04 -0.47]

[149.38 0.15 1.00]

[-7.05 -0.03 -1.52]

[154.57 0.15 1.00]

[-20.43 -0.12 -0.45]

[162.55 0.15

Cost

Best Position

Best Cost

[4.07 523.87 331.58 282.59

[169.76 0.15

[3.91 730.05 393.60 357.11

29.89]

1.55]

14.41]

[3.85 512.27 332.51 286.06

[145.95 0.15

[3.85 512.27 332.51 286.06

25.34]

1.00]

25.34]

[5.24 1075.52 554.08 479.04

[172.92 0.18

[4.53 953.91 482.02 431.27

9.20]

2.00]

10.62]

[3.88 720.39 390.67 356.83

[175.00 0.15

[3.88 720.39 390.67 356.83

13.99]

1.52]

13.99]

[4.67 977.77 494.94 440.17

[175.00 0.19

[4.67 977.77 494.94 440.17

10.11]

2.00]

10.11]

[3.72 506.91 336.75 293.12

[157.18 0.15

[3.72 506.91 336.75 293.12

22.59]

1.00]

22.59]

[4.62 969.60 489.97 436.99

[136.93 0.15

[4.01 617.69 362.52 315.19

10.21]

1.26]

23.55]

[4.99 944.81 506.19 447.74

[175.00 0.24

[4.99 944.81 506.19 447.74

10.29]

1.68]

10.29]

[4.04 522.32 331.42 282.65

[132.82 0.15

[4.04 522.32 331.42 282.65

29.32]

1.00]

29.32]

[3.80 510.32 333.49 287.81

[149.38 0.15

[3.80 510.32 333.49 287.81

24.44]

1.00]

24.44]

[3.75 507.89 335.50 291.14

[154.57 0.15

[3.75 507.89 335.50 291.14

23.19]

1.00]

23.19]

Grid Index

GridSubIndex

20510

[4,2,2,2,8]

13948

[3,2,2,2,7]

45011

[7,8,7,7,2]

15501

[3,4,3,4,3]

31070

[5,7,6,6,2]

13947

[3,2,2,2,6]

31070

[5,7,6,6,2]

36911

[6,6,6,7,2]

20510

[4,2,2,2,8]

13947

[3,2,2,2,6]

13947

[3,2,2,2,6]

[3.67 505.40 339.86 297.86

[166.04 0.15

[3.64 504.79 342.25 301.40

1.00]

[-3.48 -0.02 -0.06]

21.45]

1.00]

20.75]

7395

[2,2,2,3,6]

[173.21 0.15

[0.62 -0.06 -0.09]

[3.64 549.52 354.83 317.92

[172.59 0.20

[4.02 635.55 389.86 357.26

8204

[2,3,3,3,5]

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1.10] [175.00 0.19 1.42]

[4.58 0.01 0.22]

[175.00 0.23 1.57]

[1.33 -0.02 -0.43]

[175.00 0.23 1.61]

[0.30 0.01 0.40]

[175.00 0.20 1.21]

[0.75 -0.05 -0.42]

[175.00 0.24 2.00]

[1.67 0.02 1.18]

[175.00 0.15 1.10]

[2.22 -0.01 0.10]

[131.18 0.15 1.00]

[-38.59 -0.02 -0.73]

[145.95 0.15 1.00]

[-29.05 -0.08 -1.19]

[175.00 0.24 2.00]

[2.94 0.05 0.95]

18.04]

1.19]

14.92]

[4.07 721.13 405.90 369.87

[175.00 0.19

[4.07 721.13 405.90 369.87

13.24]

1.42]

13.24]

[4.70 864.69 471.99 423.23

[175.00 0.23

[4.70 864.69 471.99 423.23

11.10]

1.57]

11.10]

[4.70 876.32 474.30 424.45

[175.00 0.23

[4.70 876.32 474.30 424.45

10.97]

1.61]

10.97]

[4.04 639.51 392.88 360.94

[175.00 0.20

[4.04 639.51 392.88 360.94

14.45]

1.21]

14.45]

[5.28 1082.79 558.47 481.98

[172.40 0.25

[4.61 740.79 441.72 405.77

9.14]

1.25]

13.04]

[3.63 548.62 356.03 319.83

[175.00 0.15

[3.63 548.62 356.03 319.83

17.79]

1.10]

17.79]

[4.07 523.87 331.58 282.59

[169.76 0.15

[3.91 730.05 393.60 357.11

29.89]

1.55]

14.41]

[3.85 512.27 332.51 286.06

[145.95 0.15

[3.85 512.27 332.51 286.06

25.34]

1.00]

25.34]

[5.24 1075.52 554.08 479.04

[172.92 0.18

[4.53 953.91 482.02 431.27

9.20]

2.00]

10.62]

22143

[4,4,4,4,3]

36822

[6,6,5,6,3]

36822

[6,6,5,6,3]

14854

[3,3,4,4,4]

45020

[7,8,7,8,2]

8204

[2,3,3,3,5]

20510

[4,2,2,2,8]

13948

[3,2,2,2,7]

45011

[7,8,7,7,2]

Table 3.10 has results obtained for MOPSO for 45 HRC the position, Best position matrix represent process parameters (vc,f,d),. and Cost, Best Cost represents objective fitness (Ra ,Ft ,Fa ,Fr ,Tf ) and the latter columns are corresponding grid index and gris sub index of PSO topology Out of the 500 repository elements few are listed in above table.

3.9.8 Plots for MOPSO results (45 HRC)

Fig 3.22 Pareto spread surface roughness Fig 3.23 and cutting force 45HRC

3D surface plot of optimal Ra

with best position 45HRC

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Fig.3.24 3D surface plot of Tf with best

Fig.3.25 Depth of cut influence on cutting

position 45 HRC

forces 45 HRC

3.10 Comparison between EA and SI technique

Fig.3.26. Solution Spectrum for 35 hrc Fig.3.27. Solution Spectrum for 45 hrc

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Fig.3.28 Solution Spectrum for 35 hrc Fig.3.29 Solution Spectrum for 45 hrc PSO

Fig 3.30. Solution Spectrum for 35 hrc Fig. 3.31 Solution Spectrum for 45 hrc

3.10.1 Comparison Based on Spectrum of solution space 1. The search exploration for solution in both EA and SI varies in demography of population size and density which can be observed from the spectrum distribution of solution space.

2. The spectrum of solution space reveals the demographic changes in solution space at each generation hence it is crucial to compare saturation levels in solution spectrum.

3. Solution Spectrum in NSGA II :

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3.1 The solution spectrum distribution in NSGA II (as shown in Fig. (3.26-3.27)) has attained uniform amplitude with periodic crusts and troughs in all the objectives exhibiting quit a good saturation level in the demography in solution space for both hardness levels.

3.2 Solution Spectrum in SPEA 2 While in SPEA 2 the demography of solution space (as shown Fig. (3.30-3.31)) is quite different from the NSGA II as the solution space is built on niche Pareto sharing 3.3 The spectrum is pretty disturbed with low levels of saturation and crusts and troughs varying throughout the wavelength of the data. for both the hardness levels.

4.

Solution Spectrum in PSO

4.1 The trend in PSO shows moderate level of disturbance in spectrum of solution space (as shown Fig. (3.28-3.29)) but the level of saturation is appreciable when compared to the SPEA 2 and change in demography is not as amplifying as SPEA 2 while in comparison with the NSGA II it is inferior in terms of saturation levels.

From the nature of spectrum in solution space conclusion can be condensed to as follows.

NSGA II relatively better compared to PSO and SPAE 2, PSO is better compared to SPEA 2.

3.10.2 Comparison Based on Diversity in solution space.

1. The diversity of solution space is evaluated through the average Pareto spread in each generation, diversity measures exploration potential in search space.

2. Higher diversity in generations gives greater chances of solutions getting retained from varying locals of search space hence increasing the strength of solution.

Table 3.11Diversity of Evolutionary Algorithm Evolutionary Algorithm NSGA II SPEA 2

Diversity [0.01-1] [0.27-0.37]

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3 The diversity of NSGA II is superior when compared to SPEA 2 the range of search exploration between the generations in NSGA II varied from Euclidean spread of [0.01 to 1] (Fig.3.6 and Fig. 3.9) which shows that the exploration happened between two extremum and in mid generations solution spread curled towards mean solution and drifted away from the mean solution.

4 While in the SPEA 2 the Euclidean spread was in a short range of 0.27-0.37 (as shown in (Fig.3.26 and Fig. 3.27). with all the equal dominating and non-dominated solutions in the spread, the diversity oscillated in short range at each generation with most of the generations between average spread [0.3-0.35].

5 The overall analysis between EA and PSO suggests that NSGA II performs well in terms of diverse solution while SPEA 2 and PSO performs well when the solutions spread is short In NSGA II more diverse solutions are preserved while in PSO and SPEA 2 neighborhood solutions are preserved.

Further potentials of both EA and SI are explored with synergism with prediction models in chapter 5 where EA and PSO are utilized to enhance learning in prediction models.

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CHAPTER 4 PREDICTION MODELS USING INTELLIGENT LEARNING TECHNIQUES 4.1 Introduction This chapter illustrates the development of prediction modeling using intelligent learning techniques

on machining system. Learning algorithm with their

mathematical framework is extensively discussed and applied to existing machining system. In the first segment prediction model by applying neural network is developed for both steels. In the second segment adaptive learning techniques are developed and the third section developed models are analyzed over machining statistics. Further extensive statistical analysis is done between the experimental da ta and the prediction results for evaluating the accuracy among the developed models.

The objective of this chapter is explained through the following work flow.

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4.2 Neural Network Neural network consists of nodes connected by direct links each link has numeric weight wij associated with it which determines the strength and configuration between links. A threshold function ‘activation function’ f(.) is applied to model which alters the topology of links, these connection between nodes forms layer pattern so called network architecture. Depending on the direction of propagation of weights in the link neural model is classified into feed forward and feed backward /recurrent network.

4.2.1 Feed forward Neural network Feed forward networks are arranged in layers such that each unit receives input only from the units in the preceding layer. The architect of the feed forward network has following composition. 

First layer is input layer receiver of data or input from the external stimuli, the incoming data is then sent to the next layer where the number of layers can be more than one.



Second layer consists of hidden layer in which the number of nodes depends on the complexity and non-linearity of data to be handled with weights defining connections between node and bias at each node A single hidden layer constituents a network activated by threshold /activation function which takes augment of weights and bias matrix from the net This augmented net is propagated to the subsequent layers depending number of hidden layers.



Data processed in hidden layers are routed to the output layer. This layer plays a role in determining the validity of data that are analyzed based on the existing limits in the activation function.



Neural network runs training examples through the net one at a time, adjusting the weights slightly after each example to reduce the error. Each cycle through the examples called epoch.

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4.2.2 Mathematical background of neural network Since the essential elements of neural network are discussed now the working matrix by which each layer learns is discussed through simple network.

Fig.4.2 Simple network Let X be input vector and Y be output layer vector with weight matrix mapping between the input and output layer. Then the neural network is characterized by the learning model n

Yi   Wx Xi  bi i 1

Where the input layer and output layers are defined by the vectors

X  {x1 , x2 , x3 ,......xl } Y  { y1 , y2 , y3 ,....... yl } Now for training and mapping between the input and output layer learning law which describes that prediction accuracy to increase the weights of nodes should be correlated to attain minimum error in predictor consequently to store a prototype

( xi , yi ) .The weights are altered by weight matrix

w  .yi x iT Where

is learning factor and is generally kept positive, elements of weight matrix w

starts from zero to a perfectly associative neuron weight. And inverse mapping is possible at any instance of learning stage by recalling the weight matrix as

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L

W   Yi .Xi T i 1

W . X K  YK

K  1, 2...l

Now that the weight matrix is defined and one to one mapping between the input and output layer is established the next step is to minimize the error of mapping in weight matrix by applying gradient descent approach.

4.2.2.1 Gradient Descent Approach According to gradient descent the mean square E(w) associated gradient of expected error .The error gradient ( points in the direction in which E(W) will decrease at fastest possible rate w  k  1  w  k   (E)

α arbitrary constant similarly the lest mean square error for predicted output and weight is calculated as 1 ek [w(k), y(k)]  [| w(k).y(k) | w  k .y  k ] 2

To minimize this error the error gradient showed be headed in the direction where error is least ek  w  k  y  k   w  k 



1  y  k  .Tb  w  k   .y  k   y  k  2

w  k  1

From extremum of error gradient



Tb  w  k    1  1

0

we get the if

w(k).y(k)  0

if

w(k).y(k)  0

By fixed incrementing weights at every epoch we get

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w  k  1  w  k  

 y  k   y  k  Tb  w  k   y  k  2





Now validating weights for desired output d(k), the augments w(k) and y(k)determines the rate at which the desired output is approached . ek  d  k   w  k  * y  k 

Evaluating least squared errors for the error function.

2 T e2k  d  k   w  k  .y  k   d  k   w  k  .y  k   

2

To minimize the least square error expectation operator E[ek2:]is applied over the squared gradient of error k 

e2k  2e k y  k  w

 

E k  2E  ek y  k   E(K )  2E  y  k  .yT  k  .w  k   d  k  .y  k 

By estimating the mean of the gradient the direction of least error propagation can be known K  2E  y  k  .yT  k  .w  k   d  k  .y  k 

the expectancy of gradient is expected to be zero and with few manipulations weight matrix which has least error propagation is obtained.as K  0 P  y  k  .yT  k  .

Q  d  k  .y  k  W*  Q1.P

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Fig.4.3 Multi-layer feed forward network structure Now that the weights matrix for a single layer is established an extension of weight matrix for consecutive layers are evaluated for the above multi-layer network.

E k 

2 1 m   y j  k   d  k  2 j1

N

ET   E  k  k 1

Where ET is the expectation of all the layers similarly evaluating the gradients of expectation E  k  dy j

 yj  dj

s j   yi wil   j

The activation function or transfer function characterizes the input output relationship

y j  f j (s j ) Most common choice of activation function is sigmoidal function which satisfies the continuous differential function and continuous everywhere.

yj 

1  1  exp( s j )

1 n

1  exp[ ( wij yi(1)   j )] i 1

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Now for the above function the error propagation for every input to layer can be written by chain rule as E  k  s j



E  k  y j  y j s j

After few assumptions in the activation function and applying the above chain rule following error propagation in unit layer j can be expressed as

dyi 1 d ( )  y j (1  y j )  ds j ds j 1  exp( s j ) E (k )  ( y j  d j ) y j (1  y j ) s j Where j are the threshold value generally referred as bias and yj in the consecutive layers is determined by transfer function.so at each layer the targets changes with the weights which are expected to approach to desired matrix. Thus the error gradient at each layer with respect to the weights can be written by chain rule as E  k  w m



E  k  yl si   yl sl w ml

Summing up all errors in all the layers by chain rule we arrive at E  k  yi

 j

E  k  s j

w ij

There are two approaches to apply gradient descent method to the training method of a multi-layer feed forward neural network .The first is based on periodic updating and second is based on continuous updating. In both the cases the weights are repeatedly monitored either sequentially or randomly until the convergence criteria is satisfied ET E(k) E(k) E(k) E(k) T [ , , .... ] w w1 w 2 w 3 w m VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 86

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With m donating the number of weights in the network and weights are updated only once every epoch after all the training patterns are evaluated the weights get updated by the generalized fixed increment/decrement rule. w new  w old  n

E  k  w

Where η is small constant referred to as learning rate. Wnew and wold are weights vector at epoch k+1 and k respectively.

4.2.3 Key notes form feed forward analysis In order to build accurate prediction model sufficient input and output vectors is necessary to be included in the network.

A reasonably sufficient amount of exemplars is essential for a prediction model to work accurately. There is no hard rule for selecting number of nodes and layers it is purely a trail error based mapping technique. If the developed model is accurate enough for a given set of nodes and layers and satisfies the stopping criteria then the network hypothesis is acceptable.

4.2.4 Multi-layer Perceptron for Turning of AISI 4340 Steel With the above fundamentals a prediction model for machining system [1] is built for both the steels i.e., 35 HRC and 45HRC with process parameters as input vectors and cutting speed, forces and tool life as output vectors. In the Fig a schematic description of developed prediction model is explained in detail. This model is adopted for predicting both the steels.

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Fig.4.4 Feed forward neural network for AISI 4340 Hard turning Table 4.2.4 (a) Description of Neural network Neural Network Type Training function No of neurons in Hidden layer No of neurons in output

Feed forward neural network Levenberg-Marqaurdt 10

Weights in hidden layer Weights in output layer Training samples

5 30 [3×10] 50[5×10] [700 3]

Testing samples

[150 3]

Validation samples Transfer function Training performance

[150 3] Tan-sigmoid function 2.861*10-4 (35HRC) NN

Testing performance Validation performance

4.147*10-4 (35HRC) NN 1.777*10-4 (35HRC) NN

7.930*104 (45HRC) NN 0.00106(45HRC) 0.00118(45HRC) NN

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Table 4.2.4 (b) Calibrated weights and bias above Neural Network Calibrated weights and bias for 35 HRC Steel Hidden layer Definition(sij)

Output layer Definition(sij)

Bias(i)

W1(vc)

W2(f)

W3(d)

bias(i)

W1(Ra)

W2(Ft)

W3(Fa)

W4(Fr)

W5(Tf)

0.660 -0.660

-0.135

-0.086

0.005

4.388

-3.260

-2.039

-8.270

-6.538

-4.691

0.0024

-0.034

0.1812

-0.80

1.545

-3.425

3.573

2.873

3.957

0.661

0.002

0.148

-0.099

11.57

4.809

-0.977

1.910

-5.394

0.541

0.901

-0.710 0.002

0.535 -0.180

0.138 -0.119

10.89 12.74

0.429 6.030

0.058 0.239

-0.07 0.002

0.042 -3.204

-0.035 3.619

-0.677 0.167

-0.307

-0.020

0.295

1.690

0.597

0.981

-0.136

-0.025

-0.180

-0.441

0.334

0.086

0.489

0.539

-0.648

0.390

-0.115

-0.672

-0.134 -0.255 -0.465

-0.005 -0.0165 0.351

-0.07 0.2469 0.0912

-0.067 1.143 0.6855

-6.242 2.986 1.169

6.290 4.917 -1.4074

6.749 -0.703 0.850

10.459 0.274 -0.113

-0.99003 -1.376

Caliberated weigths and bias for 45 HRC Steel 2.991 0.541 1.972 -2.656 0.245 0.276 0.134 0.323 -2.418 2.11 1.853 1.524

0.746 0.635 0.974

0.176 -0.127 0.200

0.0013 -1.326 0.0059

-0.007 0.707 0.0128

-0.027 1.233 0.0091

-0.017 -1.894 -0.010

0.210

0.180

0.099

0.428

0.620

0.860

2.276

0.0923

0.0118

0.566

0.519

-2.88

-1.639

-0.557

0.261

0.676

-0.0322

-0.0670

-0.039

-0.100

0.615 -1.914

2.144 -0.230

1.797 -0.261

1.122 -2.013

0.728 0.198

-0.0492 -0.063

-0.088 0.0148

-0.0488 0.0102

-0.106 0.072

0.493 6.36 0.733

0.646

-0.628

0.776

-0.298

0.0085

-0.029

-0.188

-0.062

-0.051 -1.295

-0.0027 -1.324

-0.004 -1.956

-0.011 -1.324

-0.08 -0.01

3.857359 0.268918

1.379878 -0.332

2.528731 -0.4

4.2.4 Results of perceptron for 35 HRC Steel

Fig.4.5 Performance plot of Network

Fig.4.6 Training state of Network at each

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Fig.4.7 Training error in Ra

Fig.4.8 Regression fit plot for Ra

Fig.4.9 Training error in Ft

Fig.4.10 Regression fit plot for Ft

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Fig.4.11 Training error in Fa

Fig.4.12 Regression fit plot for Fa

Fig.4.13 Training error in Fr Fig.4.14 Regression fit plot for Fr

Fig.4.15 Training error in Tf Fig.4.16 Regression fit plot for Tf

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4.2.6 Results of perceptron for 45 HRC Steel

Fig.4.17 Performance plot of Network

Fig.4.18 Training state of Network at each epoch

Fig.4.19 Training error in Ra

Fig.4.20 Regression fit plot for Ra

Fig.4.21 Training error in Ft

Fig.4.22 Regression fit plot for Ft

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Fig.4.23 Training error in Fa

Fig.4.25 Training error in Fr

Fig.4.27 Training error in Tf

Fig.4.24 Regression fit plot for Fa

Fig.4.26 Regression fit plot for Fr

Fig.4.28 Regression fit plot for Tf

The hypothesis of network for both the steels were quite accurate with the man square error approaching to order of 10 -3.Error gradients and mean error gradients in VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 93

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learning stage is found to converge to minimum criteria. Error analysis for each target output is evaluated at learning stages i.e., training testing and validating and targets are fir to linear regression model with regression coâ&#x20AC;&#x201C;efficient approach to 1.

4.3 Adaptive Neuro-Fuzzy Interference System (ANFIS) A neuro-fuzzy interference system utilizes the human ability of recognizing pattern in modeling information either numeric or linguistic by employing fuzzy membership function and fuzzy if-then rules combined with neural network architect.

ANFIS shares methodology of fuzzy sets and neural network for building learning model by interpreting the fuzzy system in terms of neural nodes with slight modification in the fuzzy rule base which is replaced by weights instead of linguistic rules.

There are two ways of implying fuzzy neural systems. In the first method the fuzzy rules are modified with no change in the input and output membership functions. In the second method fuzzy neural systems with learning algorithms such as backpropagation or hybrid learning are applied to learn and adjust the membership function parameters.

Different combinations of fuzzy neural systems are possible with varying input output membership functions. The applied adaptive neuro fuzzy interference system is explained through a two input-single output model utilizing Sugeno-type fuzzy system also known as Takagi-Sugeno-Kang type fuzzy system where rule base is replaced by neural network weights and output membership are defined by linear function instead of fuzzy linguistic model.

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Fig.4.29 ANFIS two input model Layer 1 Every node in this layer is defined by membership function where each node gives membership values after evaluating inputs over the membership function. The applied membership function can be linear or exponential with each input defined by desired subsets in membership function In most cases Gaussian membership function is applied

 A,i  exp[(

x  ci 2 ) ] 2ai

O1,i   A,i ( x),

i  1, 2..k

O1,i   B ,i ( y ),

i  1, 2..k

Layer 2 In this layer the node is fixed and takes the fuzzified value as input from the layer 1. The output of this node is the result of fuzzy multiplication of membership function which goes into the next node. Each node represents the firing strength of each rule in the second layer. The T-norm operator with and operation is applied to obtain the output. This layer is known as andecent layer

O2,i  wi   A,i ( x)* B,i ( y)

i  1, 2..k

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Layer 3 In this layer the normalized weight for each firing strength with respect to all the cumulative fire strength is calculated. The resulting weights is called normalized firing strength

O3,i  wi 

wi  wi

Layer 4 Every node in this layer is adaptive node known as consequent layer and gives output with node function defined as O4,i  wi fi

O4,i = wi ( pi x  qi y  ri ) Layer 5 Single node fixed node that calculates overall output of from consequent layer O5,i   wi fi

4.3.1 Hybrid learning in ANFIS The adaptive layers i.e., first and fourth layer contain parameters which can be modified at every iteration. The antecedent and consequent parameters can be updated through learning method. There are two paths of learning forward and backward path. In the forward path recursive least square method is used to alter consequent parameters. While in the backward path the antecedent parameters are changed through gradient descent method.at each iteration which is also called epochs.

Forward Learning In the forward learning consequent parameters are adjusted f  w1 f1  w2 f 2  w1 ( p1 x  q1 y  r1 )  w2 ( p2 x  q2 y  r2 )  ( w1 x) p1  ( w1 y )q1  w1r1  ( w2 x) p2  ( w2 y )q2  w2 r2

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( w1 x)1 p1  ( w1 y )1 q1  ( w1r1 )1  ( w2 x)2 p2  ( w2 y )2 q2  ( w2 r2 )3  f1 ( w1 x)n p1  ( w1 y) n q1  ( w1r1 ) n  ( w2 x) n p2  ( w2 y ) n q2  ( w2 r2 ) n  f n The above equation is simplified and expressed in matrix form A  y

Where  is the vector M×1 M is the number of elements that the consequent parameters is set and A is the vector P* M where P is the number of N data training provided to the adaptive network and y is the output vector P*1 whose elements are N number of output data of an adaptive network. The optimum solution for  is defined as

 *  ( AT A)1. AT y T 1 Where AT is the inverse of A and if not singular, ( A A) is pseudo inverse of A by

using recursive LSE method then

i 1  i  Pi 1ai 1 ( yiT1  aiT1i ) Where a a row is vector of matrix A and Pi sometimes called a covariance matrix and is defined by

Pi  ( AT A)1 4.3.2 Back propogation Learning The parameters in Gaussian parameters are trained for minimizing error For a given adaptive network where the network consists of five layers and has total of N (L) node in layer L then the square error in the L layer to n data is 1  n  N the error at each node can be written as N (l )

En   d k  X kL,n k 1

Where dk is the k-th component of the vector of the desired output while X kL,n is k-th component of the vector of actual output generated by adaptive network with input VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 97

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from the input vector n. The main aim of adaptive learning system is to reduce error that occurs .i.e.,

 L ,i 

En  2(di ,n  X iL,n )  iL,n

Applying chain rule for consecutive layers for error propagation we get l 1 N ( l 1) En En X m, p   l 1 l 1 X l ,i m 1 X m , n X m , n

With 0  l  L  1 internal node error is a cumulative node error in the layer l+1. For a specific node in adaptive layer the error rate corresponding to parameter

is given as

En E x  n  xS x 

Where S is the set of nodes containing the parameter

the error specific to this

parameters is given as

En En =  n 1  N



4.3.3 Fuzzy clustering Algorithms Fuzzy clustering algorithms are utilized to discretize the membership function into subsets in the input vectors so that each input vector is sub divided into topologies defined by densities of points in respective region. Building a fuzzy set requires following key points.

Selection of inputs-outputs vectors choice of specific type of fuzzy interference system the no of mf’s function and subsets in mf’s Generating the antecedent and consequent rule a)

Choosing an appropriate family of parameters mf’s. To illustrate fuzzy

clustering a fuzzy set with following definition is declared. Let u(t), y(t), x(t) denote the input, output and state of a system S at time t

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X (t  1)  f ( x(t ), u (t )) Y (t  1)  f ( x(t ), u (t )) f &g f : X *U  X G : X *U  Y

  [m1T , m2T ,.....mmT ]T b) Clustering approaches: A clustering approach can be applied to estimate the data distribution and resulting clustering which produces the membership function c) Clustering in mf’s In any clustering technique the goal is to estimate  that characterizes the best cluster for input vector X. The parameter vector  is sensitive to the shape of clusters. To define the topology of clusters a set of m points mi in the l-dimensional space is required which corresponds to a cluster

  [m1T , m2T ,.....mmT ]T / C (c1T , r1 , c2T , r2 ,.......cmT , rm ) d) Definition of cluster Let X be data set, for which m clustering is defined in R partition of X into m sets so that the following three conditions are met

X  {x1 , x2 ,......xn }

  [m1T , m2T ,.....mmT ]T Conditions for clustering

1.Ci   , i  1,........m m

2. Ci  X i 1

3.Ci  C j   , i, j  1,......, m The alternative definition is in terms of fuzzy sets is characterized by m functions u j where VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 99

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u j : X  [0,1], J  1,....m And m

 u j ( xi )  1, 2,.....N , j 1

N

0   u j ( xi )  N , j  1, 2,....m i 1

These are called membership functions. The value of fuzzy membership function is a mathematical characterization of clusters which is not precisely defined and each vector x belongs to more than one cluster simultaneously.

X  {x1 , x2 ,......xn } Ci   , i  1,........m m

Ci  X

i 1

Ci  C j   , i, j  1,......, m u j : X  [0,1], J  1,....m m

 u ( x )  1, 2,.....N , j 1

j

i

N

0   u j ( xi )  N , j  1, 2,....m i 1

e) Proximity measures: This parameter measure will quantify the similarities and dissimilarities between the two clusters and within the clusters with no bias in selected clusters that each cluster should contributed equally with no domination among each other. The proximity measure has two property functions which measure dissimilarity and similarity between two vectors.

Dissimilarity measure

d : X *X R d 0  R :   d 0  d ( x, )

,

x, 

d ( xi , j ) f) Clustering Algorithm Having adopted proximity measure clustering criteria is applied to choose specific algorithm scheme that forms the clustering structure Most of the fuzzy clustering algorithm are derived by minimizing functions of form

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N

m

J q ( , u )   uijq d ( xi , j ) t 1 j 1

Where J q ( , u) is clustering structure and uijq is membership function for each input vector and d ( xi , j ) is the proximity measure between the clusters.

Now that the fuzzy clustering is defined the clusters are applied to adaptive layer of neuro fuzzy interference model.

4.3.4 Grid Partition clustering based Adaptive Neuro-Fuzzy Interference System Grid portioning is morphological clustering technique where the membership is sub divided into grid elements. Depending on the fuzzy rules R a fuzzy portioning is done on input membership function with cluster definitions N

m

J q ( , u )   uijq d ( xi , j ) t 1 j 1

Fig.4.30 Applied ANFIS grid partitioning architect VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 101

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Table 4.3.4 (a)Fuzzy Structure ANFIS type Fuzzy system And method Or method Defuzzification method Implication method Aggregation method Membership function Fuzzy rule Input Mf Type Output Mf type Max Epoch Error Goal Initial Step Step size Decrease rate Step size Increament rate

Grid Partitioning Sugeno Prod Max Weigthed average Prod Max Five 125 Gaussmf Linear 200 0 0.01 0.9 1.1

Table 4.3.4 b)Statistical Results of ANFIS Grid Partioning Cluster for 35 HRC and 45HRC Statistical Results of ANFIS Grid Partioning Cluster for 35 HRC ANFIS Grid35

Error mean(µ)

Results of ANFIS Grid Partioning Cluster for 45 HRC

Error STD(σ)

MSE

RMSE

Error mean(µ)

Error STD(σ)

MSE

RMSE

Surface roughness (Ra) Train Ra (µm)

7.74*10-4

6.41*10-4

4.11*10-7

6.41*10-4

5.49*10-7

6.185*10-4

3.82*10-7

6.181*10-4

Test Ra (µm)

1.74*10-4

0.0020

3.99*10-6

0.0020

1.09*10-4

0.0019

3.44*10-6

0.0019

Validation Ra (µm)

2.63*10

-5

9.74*10

-4

9.84*10

-7

9.74*10

-4

1.56*10

-5

9.17*10

-4

8.41*10

-7

9.17*10-4

Tangential force (Ft) 0.1352

0.0183

0.1351

5.69*10-5

0.0788

0.0062

0.0787

-0.0501

0.5101

0.2610

0.5109

7.612*10-4

0.3501

0.128

0.3148

-0.0023

0.714

0.41

0.57

1.62*10-4

0.153

0.0235

0.153

Axial force (Fa) Train Fa(N)

1.54*10-5

0.0847

0.0072

0.0847

3.76*10-5

0.0675

0.0045

0.0674

Test Fa(N)

0.0203

0.3453

0.118

0.3448

0.0055

0.1399

0.0195

0.1396

Validation Fa(N)

0.0031

0.1547

0.0239

01547

8.58*10-4

0.0824

0.0068

0.0824

2.4*10--5

0.082

0.0823

3.59*10-5

0.0654

0.00043

0.0654

Train Ft(N)

4.49*10

Test Ft(N) Validation Ft(N)

-5

Radial force(Fr) Train Fr(N)

0.0068

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Test Fr(N)

0.0567

0.3802

0.1468

0.3831

-0.0207

0.2173

0.0473

0.0654

Validation Fr(N)

0.0085

0.1665

0.0278

0.1666

-0.0031

0.1036

0.0107

0.1036

Tool Life (Tf) Train Tf (min)

7.56*10-6

0.0120

1.43*10-4

0.0120

4.44*10-6

0.0074

5.409*10-5

0.0074

Test Tf (min)

-6.69*10-4

0.0271

7.28*10-4

0.0270

0.0014

0.0235

5.487*10-4

0.0234

Validation Tf (min)

-9.34*10-4

0.0152

2.313*10-4

0.015

2.206*10-4

0.011

1.282*10-4

0.0113

4.3.5 ANFIS Grid Partioning Cluster Plots For Ra 35 HRC

Fig.4.31 Training Error Plots for Ra (Target vs

Fig.4.32 Testing Error Plots for Ra (Target vs Output)

Fig.4.33 Validation Error Plots for Ra (Target vs

Fig.4.34 Regression Plots for Ra (Train

Output)

/Test/Validate)

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Fig.4.35 Response Surface Plot for Ra 4.3.6 ANFIS Grid Partioning Cluster Plots For Ft 35 HRC

Fig.4.36 Training Error Plots for Ft (Target vs

Fig.4.37 Testing Error Plots for Ft (Target vs

Output)

Output)

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Fig.4.38 Validation Error Plots for Ft (Target vs

Fig.4.39 Regression Plots for Ft (Train /Test/Validate)

Output)

Fig.4.40 Response Surface Plot for Ft 4.3.7 ANFIS Grid Partioning Cluster Plots For Fa 35 HRC

Fig.4.41 Training Error Plots for Fa (Target vs

Fig.4.42 Testing Error Plots for Fa (Target vs

Output)

Output)

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Fig.4.43 Validation Error Plots for Fa (Target vs

Fig.4.44 Regression Plots for Fa (Train

Output)

/Test/Validate)

Fig.4.45 Response Surface Plot for Fa 4.3.8 ANFIS Grid Partioning Cluster Plots For Fr 35 HRC

Fig.4.46 Training Error Plots for Fr (Target vs Output)

Fig.4.47 Testing Error Plots for Fr (Target vs Output)

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Fig.4.48 Validation Error Plots for Fr (Target vs Fig.4.49 Regression Plots for F (Train /Test/Validate) r Output)

Fig.4.50 Response Surface Plot for Fr

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4.3.9 ANFIS Grid Partioning Cluster Plots For Tf 35 HRC

Fig.4.51 Training Error Plots for Tf (Target vs

Fig.4.52 Testing Error Plots for Tf (Target

Output)

vs Output)

Fig.4.53 Validation Error Plots for Tf (Target vs

Fig.4.54 Regression Plots for Tf (Train

Output)

/Test/Validate)

Fig.4.55 Response Surface Plot for Tf VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 108

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4.3.10 ANFIS Grid Partioning Cluster Plots For Ra 45 HRC

Fig.4.56 Training Error Plots for Ra (Target

Fig.4.57 Testing Error Plots for Ra (Target

vs Output)

vs Output)

Fig.4.58 Validation Error Plots for Ra (Target vs Output)

Fig.4.59 Regression Plots for Ra (Train /Test/Validate)

Fig.4.60 Response Surface Plot for Ra

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4.3.11 ANFIS Grid Partioning Cluster Plots For Ft 45 HRC

Fig.4.61 Training Error Plots for Ft (Target vs

Fig.4.62 Testing Error Plots for Ft

Output)

(Target vs Output)

Fig.4.63 Validation Error Plots for Ft (Target vs Fig.4.64 Regression Plots for Ft (Train Output

/Test/Validate)

Fig.4.65 Response Surface Plot for Ft a

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4.3.12 ANFIS Grid Partioning Cluster Plots For Fa 45 HRC

Fig.4.66 Training Error Plots for Fa (Target

Fig.4.67 Testing Error Plots for Fa

vs Output)

(Target vs Output)

Fig.4.68 Validation Error Plots for Fa (Target

Fig.4.69 Regression Plots for Fa (Train

vs Output

/Test/Validate)

Fig.4.70 Response Surface Plot for Fa a VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 111

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4.3.13 ANFIS Grid Partioning Cluster Plots For Fr 45 HRC

Fig.4.71 Training Error Plots for Fr (Target

Fig.4.72 Testing Error Plots for Fr (Target

vs Output)

vs Output)

Fig.4.73 Validation Error Plots for Fr (Target vs Output

Fig.4.74 Regression Plots for Fr (Train /Test/Validate)

Fig.4.75 Response Surface Plot for Fr

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4.3.14 ANFIS Grid Partioning Cluster for TF Plots For Tf 45 HRC

Fig.4.76 Training Error Plots for Tf (Target vs

Fig.4.77 Testing Error Plots for Tf

Output)

(Target vs Output)

Fig.4.78 Validation Error Plots for Tf (Target

Fig.4.79 Regression Plots for Tf (Train

vs Output)

/Test/Validate)

Fig.4.80 Response Surface Plot for Tf VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 113

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4.3.15 Subtractive Clustering In subtractive clusetering each point in membership function forms ac luster center and the point with higher influence on any cluster center will take in the that cluster.

Recalling the cluster function N

m

J q ( , u )   uijq d ( xi , j ) t 1 j 1

For subtractive clustering the dissimilarity function is d ( xi , j ) is defined by exponential function N

d ( xi , j )   exp[ J 1

|| xi  x j ||

 j (ra / 2)2

]

Where  j (ra / 2) is the radius of xi & x j are membership function points if a point has many other points surrounded around itself then point has highest density point.

The highest density point is as first cluster center xc and in the consecutive the iteration the density measure of each point is obtained by subtracting cluster points by applying the following equation.

N

di  di  d . exp[ J 1

|| xi  x j ||

 j (ra / 2)2

]

This is continued until the points in membership functions are exhausted.

After the calculation of the dissimilarity at each point first cluster center is identified as the point having highest density.

Eliminate all points in the vicinity around the first cluster center of its defined radius value. For the next iteration update the dissimilarity and apply the cluster function.

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Fig.4.81 Developed ANFIS (Subtractive Cluster) for machining system Table 4.3.15 (a) Fuzzy Structure ANFIS type

Substractive Clustering

Fuzzy system And method Or method Defuzzification method Implication method Aggregation method Cluster Radius No of clusters Max Epoch Error Goal Initial Step Step size Decrease rate Step size Increament rate

Sugeno Prod Probor Weigthed average Prod Max 0.5 12 100 0 0.01 0.9 1.1

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Table 4.3.15 (b) Statistical Error analysis of ANFIS Subtractive clustering for 35 HRC and 45 HRC Statistical Results of ANFIS Subtractive Cluster for 35 HRC ANFIS Grid35

Error mean(µ)

Error STD(σ)

MSE

RMSE

Results of ANFIS Subtractive Cluster for 45 HRC Error Error MSE RMSE mean(µ) STD(σ)

Surface roughness (Ra) Train Ra (µm)

1.0458*10-7

0.0044

1.922*10-5

0.0044

6.64*10-6

4.32*10-4

2.63*10-7

5.12*10-4

Test Ra (µm)

-2.82*10-5

0.0052

2.86*10-5

0.0052

1.165*10-5

0.0039

3.48*10-5

5.89*10-3

0.0045

2.036*10-5

0.0045

5.58*10-5

0.0035

1.25*10-5

0.0035

Validation Ra -4.121*10-6 (µm) Tangential force (Ft) Train Ft(N)

-4.28*10-6

0.5928

0.3510

0.5924

-1.29*10-6

0.4917

0.2415

0.4914

Test Ft(N)

-0.0135

0.6604

0.433

0.658

0.0128

0.6335

0.3988

0.6315

Validation -0.0020 Ft(N) Axial force (Fa) Train Fa(N) 2.98*10-6

0.6031

0.3633

0.6028

0.0019

0.515

0.2651

0.5149

0.3191

0.1017

0.318

3.621*10-6

0.2806

0.07806

0.2805

Test Fa(N)

0.4351

0.1886

0.434

0.0414

0.4176

0.1749

0.4183

0.3389

0.1147

0.3387

0.0062

0.305

0.0931

0.3051

-0.0233

Validation -0.35 Fa(N) Radial force(Fr) Train Fr(N)

-3.86*10-6

0.3432

0.1176

0.3430

4.85*10-6

0.3020

0.0911

0.3013

Test Fr(N)

-0.0484

0.380

0.1458

0.3818

-0.0051

0.352

0.1233

0.3512

Validation Fr(N) Tool Life (Tf)

-0.0073

0.3849

0.121

0.349

-7.62*10-4

0.3099

0.096

0.3098

Train Tf (min)

1.49*10-6

0.0325

0.011

0.0325

7.78*10-4

0.0249

6.19*10-4

0.0249

Test Tf (min)

0.0048

0.0358

0.0013

0.0360

0.0018

0.0283

7.96*10-4

0.0282

Validation Tf (min)

7.24*10-4

0.0311

0.0011

0.0311

2.776*10-4

0.0254

6.45*10-4

0.0254

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4.3.16 ANFIS Subtractive clustering Plots For Ra 35 HRC

Fig.4.82 Training Error Plots for Ra (Target

Fig.4.83 Testing Error Plots for Ra (Target

vs Output)

vs Output)

Fig.4.84 Validation Error Plots for Ra

Fig.4.85 Regression Plots for Ra (Train

(Target vs Output)

/Test/Validate)

Fig.4.86 Response Surface Plot for Ra VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 117

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4.3.17 ANFIS Subtractive clustering Plots For Ft 35 HRC

Fig.4.87 Training Error Plots for Ft (Target vs Fig.4.88 Testing Error Plots for Output)

(Target vs Output)

Fig.4.89 Validation Error Plots for Ft (Target vs

Fig.4.90 Regression Plots for Ft

Output)

(Train /Test/Validate

Fig.4.91 Response Surface Plot for Ft

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Ft

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4.3.18 ANFIS Subtractive clustering Plots For Fa 35 HRC

Fig.4.92 Training Error Plots for Fa (Target vs

Fig.4.93 Testing Error Plots for Fa

Output)

(Target vs Output)

Fig.4.94 Validation Error Plots for Fa (Target vs Output)

Fig.4.95 Regression Plots for (Train /Test/Validate)

Fig.4.96 Response Surface Plot for Fa

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Fa

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4.3.19 ANFIS Subtractive clustering Plots For Fr 35 HRC

Fig.4.97 Training Error Plots for Fr (Target vs Output)

Fig.4.99 Validation Error Plots for Fr (Target vs Output)

Fig.4.98 Testing Error Plots for

Fr

(Target vs Output)

Fig.4.100 Regression Plots for (Train /Test/Validate)

Fig.4.101 Response Surface Plot for Fr

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Fr

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

4.3.20 ANFIS Subtractive clustering Plots For Tf 35 HRC

Fig.4.102 Training Error Plots for Tf (Target

Fig.4.103 Testing Error Plots for Tf

vs Output)

(Target vs Output)

Fig.4.104 Validation Error Plots for Tf (Target

Fig.4.105 Regression Plots for Tf (Train

vs Output)

/Test/Validate)

Fig.4.106 Response Surface Plot for Tf

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4.3.21 ANFIS Subtractive clustering Plots For Ra 45 HRC

Fig.4.107 Training Error Plots for Ra

Fig.4.108 Testing Error Plots for Ra (Target

(Target vs Output)

vs Output)

Fig.4.109 Validation Error Plots for Ra

Fig.4.110 Regression Plots for Ra (Train

(Target vs Output)

/Test/Validate)

Fig.4.111 Response Surface Plot for Ra

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4.3.22 ANFIS Subtractive clustering Plots For Ft 45 HRC

Fig.4.112 Training Error Plots for Ft

Fig.4.113 Testing Error Plots for Ft

(Target vs Output)

(Target vs Output)

Fig.4.114 Validation Error Plots for Ft

Fig.4.115 Regression Plots for Ft (Train

(Target vs Output)

/Test/Validate)

Fig.4.116 Response Surface Plot for Ft VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 123

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4.3.23 ANFIS Subtractive clustering Plots For Fa 45 HRC

Fig.4.117 Training Error Plots for Fa (Target

Fig.4.118 Testing Error Plots for Fa (Target vs

vs Output)

Output)

Fig.4.119 Validation Error Plots for Fa (Target

Fig.4.120 Regression Plots for Fa (Train

vs Output)

/Test/Validate)

Fig.4.121 Response Surface Plot for Fa

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4.3.24 ANFIS Subtractive clustering Plots For Fr 45 HRC

Fig.4.122 Training Error Plots for Fr (Target vs

Fig. 4.123 Testing Error Plots for Fr (Target

Output)

vs Output)

Fig.4.124 Validation Error Plots for Fr (Target vs Output)

Fig.4.125 Regression Plots for /Test/Validate)

Fig.4.126 Response Surface Plot for Fr VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 125

Fr (Train

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4.3.25 ANFIS Subtractive clustering Plots For Tf 45 HRC

Fig.4.127 Training Error Plots for

Tf

(Target vs Output)

Fig.4.129 Validation Error Plots for TF (Target vs Output)

Fig.4.128 Testing Error Plots for Tf (Target vs Output)

Fig.4.130 Regression Plots for (Train /Test/Validate

Fig.4.131 Response Surface Plot for Tf VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 126

Tf

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4.3.26 Fuzzy C Mean Clustering

Fuzzy C-Mean clustering is another circular invariant clustering technique the radius of clusters is calculated by membership functions. Recall the cluster function N

m

J q ( , u )   uijq d ( xi , j ) t 1 j 1

N



Compute the cluster mean i 

u j 1 M

m

ij

 (u j 1

* xj

ij

)m



Compute dissimilarity function d ( xi , j ) || xi   j ||



Update the member partition matrix uij by uij 

1 M

k 1



dik

(d

) 2/ m1

kj

Evaluate cluster function J q ( , u) and do until the cluster criteria is reached Table 4.3.26 a) Fuzzy structure

ANFIS type Fuzzy system And method Or method Defuzzification method Implication method Aggregation method No of clusters Partition matrix Exponenet Maximum iteration Improvemenet level Max Epoch Error Goal Initial Step Step size Decrease rate Step size Increament rate

Fuzzy C Mean Sugeno Prod Probor Weigthed average Prod Max 15 2 200 1*10-5 200 0 0.01 0.9 1.1

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Fig.4.132 Developed ANFIS Fuzzy C Mean Clustering architect

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Table 4.3.26 b) Statistical Results of ANFIS FCM for 35 HRC and 45 HRC Statistical Results of ANFIS Fuzzy C -Mean for 35 HRC

Results of ANFIS Fuzzy C -Mean Cluster

ANFIS Grid-35

for 45 HRC Error Error mean(µ) STD(σ)

Error mean(µ)

Error STD(σ)

MSE

RMSE

MSE

RMSE

Surface roughness (Ra) Train (µm)

Ra 7.014*10-8 0.0046

2.077*10-5

0.0046

5.58*10-8

0.0033

1.076*1 0-5

0.0033

Test Ra (µm)

1.165*10-4 0.0053

2.78*10-5

0.0053

-0.0032

0.0457

0.0021

0.0456

Validation Ra (µm)

1.753*10-5 0.0047

2.182*10-5

0.0047

2.64*10-5

8.89*10-

7.34*10-

8.56*10-

4

7

4

Tangential force (Ft) Train Ft(N)

-1.02*10-

1.2296

1.5101

1.228

-2.93*10-6

1.2581

1.581

1.2574

Test Ft(N)

-0.0224

1.897

3.577

1.893

-0.1034

1.4454

2.085

1.44

1.3498

1.8202

1.3491

-0.0155

1.287

1.656

1.287

5

Validation -0.0034 Ft(N) Axial force (Fa) Train Fa(N)

-3.10*10-

0.7534

0.5669

0.753

6.36*10-6

0.466

0.2172

0.4661

Test Fa(N)

0.0835

0.9753

0.953

0.976

-0.0626

0.59206

0.3500

0.5916

Validation Fa(N)

0.0125

0.7908

0.625

0.7905

-0.0094

0.4871

0.2371

0.487

6

Radial force(Fr) Train Fr(N)

-2.149*10-

0.863

0.744

0.862

-3.36*10-7

2.12*10-4

0.0041

0.064

Test Fr(N)

0.1296

0.935

0.886

0.941

-0.0112

0.559

0.3132

0.559

Validation Fr(N)

0.0194

0.875

0.765

0.875

-0.0017

0.559

0.3132

0.5597

0.0027

0.0524

1.347*10-6

0.0460

0.0021

0.046

0.0053

0.0729

-0.0032

0.0457

0.0021

0.0456

0.0031

00560

-4.80*10-4

0.0459

0.0021

0.0459

6

Tool Life (Tf) Train Tf 2.648*10-6 (min) Test Tf (min) 0.0058 Validation Tf (min)

8.8668*10-4

0.052 9 0.072 9 0.056 0

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4.3.27 ANFIS Fuzzy C -Mean Clustering Plots For Ra 35 HRC

Fig. 4.133 Training Error Plots for Ra

Fig.4.134 Testing Error Plots for Ra

(Target vs Output)

(Target vs Output)

Fig.4.135 Validation Error Plots for Ra (Target vs Output)

Fig.4.136 Regression Plots for Ra (Train /Test/Validate

Fig.4.137 Response Surface Plot for

Ra

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4.3.28 ANFIS Fuzzy C -Mean Clustering Plots For Ft 35 HRC

Fig.4.138 Training Error Plots for Ft (Target

Fig.4.139 Testing Error Plots for Ft

vs Output)

(Target vs Output)

Fig.4.140 Validation Error Plots for Ft (Target vs Output)

Fig.4.141 Regression Plots for (Train /Test/Validate

Fig.4.142 Response Surface Plot for

Ft

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4.3.29 ANFIS Fuzzy C -Mean Clustering Plots For Fa 35 HRC

Fig.4.143 Training Error Plots for

Fa

(Target vs Output)

Fig.4.145 Validation Error Plots for Fa (Target vs Output)

Fig.4.144 Testing Error Plots for

Fa

(Target vs Output)

Fig4.146. Regression Plots for (Train /Test/Validate

Fig.4.147 Response Surface Plot for Fa

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4.3.30 ANFIS Fuzzy C -Mean Clustering Plots For Fr 35 HRC

Fig.4.148 Training Error Plots for

Fr

(Target vs Output)

Fig.4.150 Validation Error Plots for Fr (Target vs Output)

Fig.4.149 Testing Error Plots for

Fr

(Target vs Output)

Fig.4.151 Regression Plots for (Train /Test/Validate

Fig.4152 Response Surface Plot for Fr

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4.3.31 ANFIS Fuzzy C -Mean Clustering Plots For Tf 35 HRC

Fig.4.153 Training Error Plots for

Tf

(Target vs Output)

Fig.4.155 Validation Error Plots for Tf (Target vs Output)

Fig.4.154 Testing Error Plots for

Tf

(Target vs Output)

Fig.4.156 Regression Plots for (Train /Test/Validate)

Fig.4.157 Response Surface Plot for Tf

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4.3.32 ANFIS Fuzzy C -Mean Clustering Plots For Ra 45 HRC

Fig.4.158 Training Error Plots for Ra

Fig.4.159 Testing Error Plots for

Ra

(Target vs Output)

(Target vs Output)

Fig.4.160 Validation Error Plots for Ra

Fig.4.161 Regression Plots for Ra (Train

(Target vs Output)

/Test/Validate

Fig.4.162 Response Surface Plot for Ra

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4.3.33 ANFIS Fuzzy C -Mean Clustering Plots For Ft 45 HRC

Fig.4.163 Training Error Plots for Ft

Fig.4.164 Testing Error Plots for

Ft

(Target vs Output)

(Target vs Output)

Fig.4.165 Validation Error Plots for Ft

Fig.4.166 Regression Plots for Ft (Train

(Target vs Output)

/Test/Validate

Fig.4.167 Response Surface Plot for

Ft

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4.3.34 ANFIS Fuzzy C -Mean Clustering Plots For Fa 45 HRC

Fig.4.168 Training Error Plots for

Fa (Target

Fig.4.169 Testing Error Plots for

vs Output)

Output)

Fig.4.170 Validation Error Plots for Fa (Target vs Output)

Fa (Target vs

Fig.4.171 Regression Plots for Fa (Train /Test/Validate)

Fig.4.172 Response Surface Plot for Fa

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4.3.35 ANFIS Fuzzy C -Mean Clustering Plots For Fr 45 HRC

Fig.4.173 Training Error Plots for Fr

Fig.4.174 Testing Error Plots for Fr

(Target vs Output)

(Target vs Output)

Fig.4.175 Validation Error Plots for Fr

Fig.4.176 Regression Plots for Fr (Train

(Target vs Output)

/Test/Validate)

Fig.4.177 Response Surface Plot for Fr

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4.3.36 ANFIS Fuzzy C -Mean Clustering Plots For Tf 45 HRC

Fig.4.178 Training Error Plots for

Tf

Fig.4.179 Testing Error Plots for

(Target vs Output)

(Target vs Output)

Fig.4.180 Validation Error Plots for Tf

Fig.4.181 Regression Plots for Tf

(Target vs Output)

(Train /Test/Validate)

Fig.4.182 Response Surface Plot for Tf

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4.4 Comparison of Prediction Results with Experimental Statistics 4.4.1 Statistical Comparison of Neural Network and ANFIS Prediction Results with Experimental Statistics for AISI 4340 Steel 35hrc NN35 Error Error MSE RMSE NN VS Experimental 35 HRC STD(σ) Ra 0.0442 1.0814 1.1130 1.0550 Ft (N) 0.186.6 126.4 594.81 24.38 4 Fa (N) 103.98 33.77 1.897*10 109.07 Fr (N) 1.1318 93.95 93.95 9.69 ANFIS grid partitioning VS Experimental 35 HRC Ra -0.2308 0.5964 0.3912 0.6255 4 Ft (N) 17.49 106.9 1.116*10 105.66 Fa (N) 157 80 3.09*104 176 Fr (N) 32.57 41.29 2.6*103 51.78 ANFIS Subtractive VS Experimental 35 HRC Ra -0.226 0.5964 0.3875 0.6225 Ft (N) 15.29 105.36 1.07*104 103 4 Fa (N) 157.79 82.3 3.133*10 177.016 Fr (N) 27.89 40.81 2.3604*103 48.58 ANFIS FCM VS Experimental 35 HRC Ra -0.2339 0.5964 0.392 0.6266 Ft (N) 18.19 104.9 1.08*104 103.9 Fa (N) 157 79.42 3.073*104 175.32 Fr (N) 32.23 42.20 2.73*103 52.25

Error Error MSE RMSE NN VS Experimental STD(σ) 45 HRC -0.1700 0.2769 0.1018 0.3190 -57.23 128.3 1.8929*104 137.5 54.67 33.57 4.0607*103 63.72 0.415 14.0720 188.29 13.72 ANFIS grid partitioning VS Experimental -0.9278 0.5964 1.198 1.0949 3 -22.35 77.58 6.128*10 78.85 -7.5709 40.08 1.5841*103 39.8 -7.69 38.35 1.457*103 38.17 ANFIS Subtractive VS Experimental 45 -0.947 0.594 1.23 1.109 -23.106 76.64 6.1143*103 78.19 -6.9912 39.58 1.537*103 39.213 -8.525 36.57 1.343*103 36.65 ANFIS FCM VS Experimental 45 HRC -0.856 0.596 1.0714 1.0351 -21.7 78.8 6.38*103 79.8 -6.8003 38.45 6.36*10-6 0.4664 -6.63 36.89 1.33*103 36.56

4.4.2 Error Plots of Neural Network Prediction Results with Experimental Statistics for AISI 4340 Steel 35hrc

Fig.4.183 Error Estimation Plots For R a

Fig4.184.Error Estimation Plots For F t

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Fig.4.185.Error Estimation Plots For F a

Fig.4.186 Error Estimation Plots For F r

4.4.3 Error Plots of Neural Network Prediction Results with Experimental Statistics for AISI 4340 Steel 45hrc

Fig.4.187.Error Estimation Plots For Ra

Fig.4.188 Error Estimation Plots For F t

Fig.4.189 Error Estimation Plots For F a

Fig.4.190 Error Estimation Plots For F r

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4.4.4 Error Plots of ANFIS (Grid Partitioning Clustering) Results with Experimental Statistics for AISI 4340 Steel 35hrc

Fig.4.191 Error Estimation Plots For R a

Fig.4.193 Error Estimation Plots For F a

Fig.4.192 Error Estimation Plots For F t

Fig.4.194 Error Estimation Plots For F r

4.4.5 Error Plots of ANFIS (Grid Partitioning Clustering) Results with Experimental Statistics for AISI 4340 Steel 45hrc

Fig.4.195 Error Estimation Plots For R a

Fig.4.196 Error Estimation Plots For F t

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Fig.4.197 Error Estimation Plots For F a

Fig.4.198 Error Estimation Plots For F r

4.4.6 Error Plots of ANFIS (Subtractive Clustering) Results with Experimental Statistics for AISI 4340 Steel 35hrc

Fig.4.199 Error Estimation Plots For R a

Fig.4.200 Error Estimation Plots For F t

Fig.4.201 Error Estimation Plots For F a

Fig.4.202 Error Estimation Plots For F r

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4.4.7 Error Plots of ANFIS (Subtractive Clustering) Results with Experimental Statistics for AISI 4340 Steel 45hrc

Fig.4.203 Error Estimation Plots For R a

Fig.4.204 Error Estimation Plots For F t

Fig.4.205 Error Estimation Plots For F a

Fig.4.206 Error Estimation Plots For F r

4.4.8 Error Plots of ANFIS (Fuzzy C-Mean Clustering)) Results with Experimental Statistics for AISI 4340 Steel 35hrc

Fig.4.207 Error Estimation Plots For R a

Fig.4.208 Error Estimation Plots For F t

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Fig.4.209 Error Estimation Plots For F a

Fig.4.210 Error Estimation Plots For F r

4.4.9 Error Plots of ANFIS (Fuzzy C-Mean Clustering)) Results with Experimental Statistics for AISI 4340 Steel 45hrc

Fig.4.211 Error Estimation Plots For R a

Fig.4.213 Error Estimation Plots For F a

Fig.4.212 Error Estimation Plots For F t

Fig.4.214 Error Estimation Plots For F r

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4.5 Conclusion 

From the Comparison PLOTS and table the mean error and RMS of neural network was found to be lower compared to ANFIS models



Though the errors in neural network was less the prediction curve for surface roughness should poor match with the experimental curve while in ANFIS the curve match for surface roughness was better in comparison to neural network.

 With this results learning techniques for were further attempted to improve with synergies.

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CHAPTER 5 HYBRIDIZATION OF C.I SYNERGIES 5.1 Introduction In the previous chapters optimization and prediction techniques were applied exclusively and the results obtained from them were found to be pretty convincing when compared with the experimental statistics In this chapter hybridization of these techniques is applied to our current machining problem with objective to improve the ability of techniques through mutual assistance and improve prediction and optimization ability of exclusive techniques. A exposition of adapted synergies is illustrated in brief. In the first section the combinations of Neuro-Evolutionay and Neuro-Swarm techniques is implemented , in the second section combination of Evolutionary - Neuro fuzzy and Swarm-Neuro fuzzy is exercised and in the third segment a .comparison is made between the predicted results obtained from synergism and experimental statistics. The objective of this chapter is depicted through the flow chart.

Fig.5.1 Chapter flow chart

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5.2 EA-NN Synergism Various types of EAs and NNs synergies are possible which can be broadly classified in three combinations supportive combination, collaborative combination and amalgamated combination 

EA uses a population of entire solution space of optimization problem and in contrast NN uses these optimized results as exemplars for training and the learning then converges depending on the learning parameters and topology of NN. The performance of both EA and NN can be improved by accelerating the convergence if an appropriate population of data sets and strategic learning parameters is applied.



In a supportive combination EAs and NNs are used sequentially where one is primary problem solver and other is secondary.



In collaborative combination they are used simultaneously where both EAs and NNs solve the problem together and in amalgamated combination the EA search technique and NN as pattern model.



Collaborative Combination In collaborative learning both EAs and NNs are used simultaneously using the result of one to prepare data set for other .In other words one technique plays primary role of solving problem and the other technique is a supportive to solve the problem.

Finding an appropriate topology of NN for a given problem is trial and error task.

Synergies between EAs and NN assists in determining optimal network architecture and then evaluate neural network.

A typical procedure for EA-NN synergy can be outlines as below 1. Create an initial population of individuals which would go into Evolutionary algorithm and generate a optimal population which will be data set for Neural Network architect.

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2. Set up the training data for neural network as received from the EA results by permuting train and target sets and dividing them for training testing and validation 3. Apply learning criteria for network and evaluate the weights and bias for the targets. Test the targets against the expected outputs 4. Evaluate the training errors and fitness of network for current learning, transfer back weights and bias along with the targets to EA. Repeat the step2-4 until the convergence or max generation is reached.

5.2.1 NSGA combined Neural Network

Fig.5.2 Developed NSGAâ&#x20AC;&#x201C;NN architect

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Table 5.2.1 (a) Description of NSGA-NN

Population size Generation Crossover probability Crossover constant Mutation probability Mutation constant Neural Network Type Training function No of neurons in Hidden layer No of neurons in output layer Weights in hidden layer Weights in output layer Training samples Testing samples Validation samples Transfer function Training performance Testing performance Validation performance

1000 100 0.8 0.1 0.1 0.2 Feed forward neural network Levenberg-Marqaurdt 10 5 30 [3×10] 50[5×10] [700 3] [150 3] [150 3] Tan-sigmoid function 2.861e-04(35HRC) 1.861e-04(45HRC) 4.147e-04(35HRC) 5.847e-04(45HRC) 1.777e-04(35HRC) 8.777e-04(45HRC)

5.2.1 (b) Calibrated weigths and bias for NSGA-NN architect Calibrated weights and bias for 35 HRC Steel Hidden layer Definition(sij) Bias( W1(v W2(f) W3(d -0.771

0.242

-0.284

0.444

Output layer Definition(sij) bias W1(R W2(F W3(F W4(Fr  3.444 -2.004 0.010 0.068 -0.154

-0.410

-0.040

-0.039

-0.248

1.881

0.511

-1.764

0.516

-4.511

1.559

-0.734

0.065

0.155

-0.224

1.784

-1.875

2.474

1.788

1.7166

0.607

-0.279

-0.208

0.206

0.0684

3.254

3.329

2.564

0.611

-0.616

0.637

-1.215

-0.633

0.895

0.274

3.678

0.772

0.0842

-0.040

0.0041

-0.136



W5(Tf -0.138

0.746

0.164

0.173

-0.102

0.5492

0.377

-2.206

-0.811

-0.746

0.895

-0.051

-0.037

-0.306

-3.277

-0.713

-3.158

-2.335

-0.383

0.870

-0.0149 0.164

0.227 -0.16

0.050 0.2133

-1.800 1.23

-1.466 2.071

3.607 1.67

-1.017 -2.687

-2.811 0.373

-0.23

-0.21

-1.002

1.107

-0.036

0.1082

-0.077

0.5352

0.55 -1.672

Calibrated weights and bias for 45 HRC Steel -0.810

-0.856

-2.308

1.099

0.874

-0.651

-2.535

-0.896

-0.856

-2.308

1.099

1.1200

0.528

-0.651

-2.535

-0.896

-1.368

-0.039

-0.020

2.240 0.017 4 2.414

-1.434

0.0990

0.0157

0.731

-0.103

0.478

0.099

0.0157

-0.0209

-0.0158

-0.0248

-0.793

3.064

-0.1033

0.4782

-0.793

1.200

-0.135

1.867

-3.515

1.823

2.756

3.370

1.867

1.823

2.7567

2.046

0.589

2.358 -0.733

0.005

-0.001

0.002

0.0051

-0.001

0.0002

0.0021

0.0335

-0.013

-0.773

0.496

-0.013

-0.773

0.496

1.115

0.933

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-6.767 1.321 0.118

-0.0140

0.024

-0.020

-0.014

0.0242

-0.0207

-0.009

0.261

-0.444

-0.415

-1.545

-0.692

-1.112

-0.749

0.275 0.111

-0.109

0.164

0.697

-0.938

0.5015

1.499

-0.848

-0.583

5.2.2 Results of NSGA-NN 35 HRC Steel

Fig.5.3 Performance plot of Network

Fig.5.4 Training state of Network at each epoch

Fig.5.5 Training error in Ra

Fig.5.6 Regression fit plot for Ra

Fig.5.7 Training error in Ft

Fig.5.8 Regression fit plot for Ft

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Fig.5.10 Training error in Fa

Fig.5.11 Regression fit plot for Fa

Fig.5.12 Training error in Fr

Fig.5.13 Regression fit plot for Fr

Fig.5.14 Training error in Tf

Fig.5.15 Regression fit plot for Tf

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5.2.3 Results of NSGA-NN for 45 HRC Steel

Fig.5.16 Performance plot of Network

Fig.5.17 Training state of Network at each epoch

Fig.5.18 Training error in Ra

Fig.5.19 Regression fit plot for Ra

Fig.5.20 Training error in Ft

Fig.5.21 Regression fit plot for Ft

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Fig.5.22 Training error in Fa

Fig.5.24 Training error in Fr

Fig.5.26 Training error in Tf

Fig.5.23 Regression fit plot for Fa

Fig.5.25 Regression fit plot for Fr

Fig.5.27 Regression fit plot for Tf

5.3 SI-NN synergism Similar to EA-NN synergies, SI-NN is has three class of synergies which differ in the degree of coupling and interdependency in working towards solution. The collaborative

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combination strategy is analogues to EA-NN strategy. The figure below illustrates the collaborative combination of SI and NN.

Fig.5.28. SI-NN collaborative combination 5.3.1 PSO combined Neural Network

Fig.5.29 Applied SI-NN synergy architect

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Table 5.3.1 (a) Description of PSO-NN Total particle population Max no of repository elements Inertia weight ( w) Inertia weight damping rate (w.damp) Personal learning co efficient (a1) Global learning co-efficient (a2) No of grid in each dimension Inflation rate ( ὰ) Leader selection pressure ( ) Deletion selection pressure ( ) Mutation rate (mu) Neural Network Type Training function No of neurons in Hidden layer No of neurons in output layer Weights in hidden layer Weights in output layer Training samples Testing samples Validation samples Transfer function Training performance Testing performance Validation performance

1000 500 0.5 0.99 1 2 7 0.1 2 2 0.1 Feed forward neural network Levenberg-Marqaurdt 10 5 30 [3×10] 50[5×10] [700 3] [150 3] [150 3] Tan-sigmoid function 1.5224e-05 2.089e-05 2.03e-05

5.3.2 (b) Calibrated weights and bias for 35 SI-NN Hidden layer Definition(sij) Bias(i) W1(vc) W2(f) -0.904 1.656 1.671 -3.013 2.294 -1.05 0.829 -0.131 0.344 -3.295 1.724 1.051 -1.695 0.330 0.163 -0.369 -0.369 0.143 2.0164 -2.643 3.987 -1.248 2.022 -0.81 -0.629 0.403 0.346 -0.705 3.124 -1.47 -0.633 -0.304 -0.02 0.3125 0.390 -0.23 0.896 2.234 -1.43 -0.362 -0.010 -0.09 1.834 -0.298 -0.63

W3(d) -0.155 0.092 0.001 0.0081 -0.869 0.166 -0.614 3.412 -0.012 1.981 -0.185 0.075 1.056 -0.21 -1.286

Output layer Definition(sij) bias(i) W1(Ra) W2(Ft) 1.100 -0.027 -0.001 2.257 -0.121 -0.006 1.922 -0.200 -0.453 0.426 0.111 0.061 0.227 -0.292 0.017 3.723 2.250 0.001 0.0006 0.001 -0.001 0.44 0.556 -0.018 -0.002 0.833 -2.606 0.375 -0.425 0.087 0.010 0.540 -0.184 -0.166 0.0019

W3(Fa) -0.003 0.006 0.908 0.057 0.243 0.860 -0.0001 -0.003 0.616 0.0009 1.609 1.281 -0.005 -1.76 -0.0005

W4(Fr) 0.0011 -0.005 -0.629 0.049 0.167 -1.183 0.00007 -0.0002 0.436 0.0004 0.649 -1.364 -0.008 -3.427 -0.0023

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W5(Tf) 0.004 -0.03 -1.07 -0.01 0.2031 0.1944 0.0015 0.0024 0.122 -0.001 2.208 0.150 0.007 0.285 -0.002

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-4.050 -0.983 1.326 1.975 3.971

-4.123 -0.091 2.797 -0.015 1.211

2.584 -0.05 -1.62 0.364 -1.51

1.262 0.440 -0.057 0.611 -0.028

-1.35 -0.911 0.107 -1.441 0.4813

0.284 0.0689 0.0142 -0.074 -1.663

0.0736 1.9151 -0.0097 0.12064 -0.3358

-0.914 1.440 -0.029 0.341 -0.469

-2.361 -0.087 -0.024 -0.802 0.126

5.3.2 Results of PSO-NN for 35 HRC Steel

Fig.5.30 Performance plot of Network

Fig.5.31 Performance plot of Network

Fig.5.32 Training error in Ra

Fig.5.33 Regression fit plot for Ra

Fig.5.34 Training error in Ft

Fig.5.35 Regression fit plot for Ft

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Fig.5.36 Training error in Fa

Fig.5.38 Training error in Fr

Fig.5.40 Training error in Tf

Fig.5.37 Regression fit plot for Fa

Fig.5.39 Regression fit plot for Fr

Fig.5.41 Regression fit plot for Tf

5.4 Synergies of EA and ANFIS The synergism between EA and ANFIS is strongly coupled here the EA technique is toapplied on the membership function which optimizes fitness value of fuzzy output. The VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 158

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below flow chart outlines the applied strategy in synergism .Membership function are clustered through Fuzzy C-mean clustering and the fuzzy structure utilized is same as previously applied (refer chapter 4 table (4.3.27 (a) )). A detailed description is given in appendix.

5.4.1 ANFIS GA

Fig.5.42 GA based ANFIS (FCM) applied Strategy [72] 5.4.1 (a) Statistical Error analysis of GA based ANFIS (FCM) for 35 HRC and 45HRC Results of GA based ANFIS (FCM) for 35 HRC ANFIS Grid-

Error

Error

35

mean(µ)

STD(σ)

MSE

Results of GA based ANFIS (FCM) for 45 HRC RMSE

Error

Error

mean(µ)

STD(σ)

MSE

RMSE

Surface roughness (Ra) Train

Ra

Test Ra (µm)

-1.5*10-15

0.1016

0.0103

0.1015

3.14*10-16

0.0682

0.0046

0.0681

-0.0020

0.1004

0.0100

0.1002

0.0014

0.069

0.0048

0.0692

Tangential force (Ft) 9.22*10-3

Train Ft(N)

Test Ft(N) -0.3057 Axial force (Fa)

23.7

561.42

23.69

-2.69*10-13

18.31

335.08

18.30

25.68

657.6

25.64

1.3611

19.58

384

19.59

Train Fa(N)

1.21*10-13

11.57

133.7

11.56

-1.69*10-13

8.42

70.8

8.4176

Test Fa(N)

0.0194

10.79

116.09

10.77

1.681

9.79

98.45

9.92

Radial force(Fr) VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 159

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Train Fr(N) Test Fr(N)

7.568 7.90

57.2 62.314

7.56 7.894

-4.3*10-13 1.082

6.266 7.41

38.70 55.97

6.22 7.481

Tool Life (Tf) Train Tf (min)

-3.06*10-14

0.7850

0.6153

0.7844

2.69*10-14

0.6304

0.39

0.62

Test Tf (min)

0.034

0.80

0.6431

0.0802

-0.0157

0.16

0.38

0.618

Results of PSO based ANFIS (FCM) (35 HRC)

Results of PSO based ANFIS (FCM) 45 HRC

Surface roughness (Ra) Train Ra (µm) -8.43*10-17 Test Ra (µm) -4.79*10-4 Tangential force (Ft) Train Ft (N) -3.3*10-13 Test Ft (N) -2.214 Axial force (Fa) 1.02*10-13

Train Fa (N)

Test Fa (N) -0.2655 Radial force(Fr) Train Fr (N) -2.36*1013

1.4*10-15 1.4*10-15

1.4*10-15 1.4*10-15

0.0988 0.1010

0.2168 0.0102

0.4656 0.1008

0.0042 0.0048

0.0681 0.0692

23.87

589.7

23.86

-4.98*10-13

18.57

344.4

23.79

569.38

23.86

1.178

18.90

357.6

10.98

120.46

10.96

-5.59*10-13

8.78

77.08

8.78

11.98

143.22

11.96

0.080

9.045

81.54

9.03

7.57

57.22

7.56

-3.23*10-13

6.57

42.79

6.54

12.55

Test Fr (N) Tool Life (Tf) Train Tf (min)

0.221

7.71

59.42

7.708

-0.1446

6.73

45.25

6.72

3.03*10-14

0.796

0.6335

0.795

2.90*10-14

0.6186

45.25

6.72

Test Tf (min)

0.0042

0.88

0.774

0.8817

-0.0036

0.64

0.416

0.645

5.4.2 GA based ANFIS (FCM) Plots For Ra 35 HRC

Fig.5.43 Training Error Plots for Ra (Target vs Output)

Fig.5.44 Testing Error Plots for Ra (Target vs Output)

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Fig.5.45 Regression Plots for Ra (Train /Test/Validate)

Fig.5.46 Response Surface Plot for Ra 5.4.3 GA based ANFIS (FCM) Plots For Ft 35 HRC

Fig.5.47 Training Error Plots for Ft (Target vs Output)

Fig.5.48 Testing Error Plots for Ft (Target vs Output)

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Fig.5.49 Regression Plots for Ft (Train /Test/Validate)

Fig.5..50 Regression Plots for Ft (Train /Test/Validate) 5.4.4 GA based ANFIS (FCM) Plots For Fa 35 HRC

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Fig.5.51 Training Error Plots for Fa (Target vs Output)

Fig.5.52 Testing Error Plots for Fa (Target vs Output)

Fig.5.53 Testing Error Plots for Fa (Target vs Output)

Fig.5.54 Response Surface Plot for Fa 5.4.5 GA based ANFIS (FCM) Plots For Fr 35 HRC

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Fig.5.55 Training Error Plots for Fr (Target vs Output)

Fig.5.56 Testing Error Plots for Fr (Target vs Output)

Fig.5.57 Regression Plots for Fr (Train /Test/Validate)

Fig.5.58 Response Surface Plot for Fr 5.4.6 GA based ANFIS (FCM) Plots For Tf 35 HRC

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Fig.5.59 Training Error Plots for Tf (Target Fig.5.60 Testing Error Plots for Tf (Target vs Output) vs Output)

Fig.5.61 Regression Plots for Tf (Train /Test)

Fig.5.62 Response Surface Plot for Tf

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5.4.7 GA based ANFIS (FCM) Plots For Ra 45 HRC

Fig.5.63 Training Error Plots for Ra (Target vs Output)

Fig.5.64 Testing Error Plots for Ra (Target vs Output)

Fig.5.65 Regression Plots for Ra (Train /Test)

Fig.5.66 Response Surface Plot for Ra

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5.4.8 GA based ANFIS (FCM) Plots For Ft 45 HRC

Fig.5.67 Training Error Plots for Ft (Target vs Output)

Fig.5.68 Training Error Plots for Ft (Target vs Output)

Fig.5.69 Regression Plots for Ft (Train /TesT

Fig.5.70 Response Surface Plot for Ft a VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 167

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5.4.9 GA based ANFIS (FCM) Plots For Fa 45 HRC

Fig.5.71 Training Error Plots for Fa (Target vs Output)

Fig.5.72 Testing Error Plots for Fa (Target vs Output)

Fig.5.73 Regression Plots for Fa (Train /Test)

Fig.5.74 Response Surface Plot for Fa a

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5.4.10 GA based ANFIS (FCM) Plots For Fr 45 HRC

Fig.5.75 Training Error Plots for Fr (Target vs Output)

Fig.5.76 Testing Error Plots for Fr (Target vs Output)

Fig.5.77 Regression Plots for Fr (Train /Test/Validate)

Fig.5.78 Response Surface Plot for Fr a VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 169

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5.4.11 GA based ANFIS (FCM) Plots For Tf 45 HRC

Fig.5.79 Training Error Plots for Tf (Target vs Output)

Fig.5.80 Testing Error Plots for Tf (Target vs Output)

Fig.5.81 Regression Plots for Tf (Train /Test)

Fig.5.82 Response Surface Plot for Tf a VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 170

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5.5 PSO based ANFIS (FCM) 35 HRC and 45HRC Steel The synergism between SI and ANFIS is also strongly coupled where the SI technique is applied on the membership function to optimize fitness value of fuzzy output. The below flow chart outlines the applied strategy in synergism .Membership function are clustered through Fuzzy Cmean clustering and the fuzzy structure utilized is same as previously applied (refer chapter 4 table (4.3.27 (a) )). A detailed description is given in appendix.

5.5.1 PSO based ANFIS (FCM)

Fig.5.83 PS0-ANFIS applied strategy [72]

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5.5.2 PSO based ANFIS (FCM) Plots For Ra 35 HRC

Fig.5.84 Training Error Plots for Ra (Target vs Output)

Fig.5.85 Testing Error Plots for Ra (Target vs Output)

Fig.5.86 Regression Plots for Ra (Train /Test)

Fig.5.87 Response Surface Plot for Ra

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5.5.3 PSO based ANFIS (FCM) Plots For Ft 35 HRC

Fig.5.88 Training Error Plots for Ft (Target vs Output)

Fig.5.89 Testing Error Plots for Ft (Target vs Output)

Fig.5.90 Regression Plots for Ft (Train /TesT)

Fig.5.91 Response Surface Plot for Ft

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5.5.4 PSO based ANFIS (FCM) Plots For Fa 35 HRC

Fig.5.92 Training Error Plots for Fa (Target vs Output)

Fig.5.94 Regression Plots for

Fig.5.93 Testing Error Plots for Fa (Target vs Output)

Fa (Train /Test)

Fig.5.95 Response Surface Plot for Fa

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5.5.5 PSO based ANFIS (FCM) Plots For Fr 35 HRC

Fig.5.96 Training Error Plots for Fr (Target vs Output)

Fig.5.97 Testing Error Plots for (Target vs Output)

Fig.5.98 Regression Plots for Fr

(Train /Test/)

Fig.5.99 Response Surface Plot for Fr

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Fr

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5.5.6 PSO based ANFIS (FCM) Plots For Tf 35 HRC

Fig.5.100 Training Error Plots for Tf (Target vs Output)

Fig.5.101 Testing Error Plots for Tf (Target vs Output)

Fig.5.102 Regression Plots for Tf (Train /Test)

Fig.5.103 Response Surface Plot for Tf VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 176

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5.5.7 PSO based ANFIS (FCM) Plots For Ra 45 HRC

Fig.5.104 Training Error Plots for Ra (Target vs Output)

Fig.5.105 Testing Error Plots for Ra (Target vs Output)

Fig.5.106 Regression Plots for Ra (Train /Test)

Fig.5.107 Response Surface Plot for Ra VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 177

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5.5.8 PSO based ANFIS (FCM) Plots For Ft 45 HRC

Fig.5.108 Training Error Plots for Ft (Target vs Output)

Fig.5.109 Testing Error Plots for Ft (Target vs Output)

Fig.5.110 Regression Plots for Ft (Train /Test)

Fig.5.111 Response Surface Plot for Ft VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 178

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5.5.9 PSO based ANFIS (FCM) Plots For Fa 45 HRC

Fig.5.112 Training Error Plots for Fa (Target vs Output)

Fig.5.113 Testing Error Plots for Fa (Target vs Output)

Fig.5.114 Regression Plots for Fa (Train /Test)

Fig.5.115 Response Surface Plot for Fa VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 179

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5.5.10 PSO based ANFIS (FCM) Plots For Fr 45 HRC

Fig.5.116 Training Error Plots for (Target vs Output)

Fr

Fig.5.118 Regression Plots for

Fig.5.117 Testing Error Plots for Fr (Target vs Output)

Fr (Train /Test/Validate

Fig.5.119 Response Surface Plot for Fr VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 180

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5.5.11 PSO based ANFIS (FCM) Plots For Tf 45 HRC

Fig.5.120 Training Error Plots for vs Output)

Tf (Target

Fig.5.121 Testing Error Plots for Tf (Target vs Output)

Fig.5.122 Regression Plots for Tf (Train /Test

Fig.5.123 Response Surface Plot for Tf VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 181

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5.6 Comparison of Prediction Results with Experimental Statistics 5.6.1 NSGA-NN 5.6.1 (a) Statistical Comparison of

Prediction Results with Experimental

Statistics for AISI 4340 Steel 35HRC and 45HRC NN35 Error Error MSE NSGA-NN VS Experimental STD(σ) 35 HRC Ra --0.221 0.979 0.960 Ft (N) 42.67 86.21 8.88*103 Fa (N) 156.2 49.56 2.675*104 Fr (N) -0.723 14.77 206.55 PSO NN VS Experimental 35 HRC Ra 1.749 0.732 3.56 Ft (N) -53.25 104.6 1.32*14 Fa (N) 138.22 40.44 2.06*104 Fr (N) 40.08 64.63 5.57*103 ANFIS GA VS Experimental 35 HRC Ra -0.1572 0.4507 0.2177 Ft (N) 17.6 57.06 3.4*103 Fa (N) 136.152 51.87 2.109*104 Fr (N) 11.065 20.54 523 ANFIS PSO VS Experimental 35 HRC Ra 0.1578 0.4494 0.2168 Ft (N) 17.13 57.16 3.398*103 Fa (N) 135.9 52 2.11*104 Fr (N) 10.89 70.79 517

RMSE Error Error MSE RMSE NSGA-NN VS Experimental STD(σ) 45 HRC 0.980 -0.355 0.378 0.262 0.512 94.24 -57.23 128.36 1.89*104 137.58 114.49 -13.571 38.046 1.55*103 39.488 14.37 -11.74 35.99 1.36*103 36.99 PSO NN VS Experimental 45 HRC 1.88 115.04 43.73 74.66 ANFIS GA VS Experimental 45 HRC 0.466 0.0371 0.2796 0.075 0.2751 4 58.3 -21.84 107.08 1.64*10 107.76 145.2 -45.95 41.46 3.74*103 61.19 22.87 -0.91 36.86 1.29*103 35.93 ANFIS PSO VS Experimental 45 HRC 0.4656 0.0372 0.2798 0.075 0.275 58.29 -27.34 106.19 1.14*104 107.05 145.4 -46.83 41.55 3.83*103 61.91 22.75 -0.747 36.77 1.28*103 35.84

5.6.2 Error Plots of NSGA-NN Prediction Results with Experimental Statistics for AISI 4340 Steel 35hrc

Fig.5.124 Error Estimation Plots For R a

Fig.5.125 Error Estimation Plots For F t

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Fig.5.126 Error Estimation Plots For F a

Fig.5.127 Error Estimation Plots For Fr

5.6.3 Error Plots of NSGA-NN Prediction Results with Experimental Statistics for AISI 4340 Steel 45hrc

Fig.5.128 Error Estimation Plots For R a

Fig.5.129 Error Estimation Plots For F t

Fig.5.130 Error Estimation Plots For F a

Fig.5.131 Error Estimation Plots For F r

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5.6.4 Error Plots of PSO-NN Results with Experimental Statistics for AISI 4340 Steel 35 HRC

Fig.5.132 Error Estimation Plots For R a

Fig.5.133 Error Estimation Plots For F t

Fig.5134 Error Estimation Plots For F a

Fig.5.135 Error Estimation Plots For F r

5.6.5 Error Plots of GA based ANFIS (Fuzzy C-Mean Clustering)) Results with Experimental Statistics for AISI 4340 Steel 45hrc

Fig.5.136 Error Estimation Plots For R a

Fig.5.137 Error Estimation Plots For F t

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Fig.5.138 Error Estimation Plots For F a

Fig.5.139 Error Estimation Plots For F r

5.6.6 Error Plots of GA based ANFIS (Fuzzy C-Mean Clustering)) Results with Experimental Statistics for AISI 4340 Steel 45hrc

Fig.5.140 Error Estimation Plots For R a

Fig.5.141 Error Estimation Plots For F t

Fig.5.142 Error Estimation Plots For F a

Fig.5.143 Error Estimation Plots For F r

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5.6.7 Error Plots of PSO based ANFIS (Fuzzy C-Mean Clustering)) Results with Experimental Statistics for AISI 4340 Steel 35hrc

Fig.5.144 Error Estimation Plots For R a

Fig.5.145 Error Estimation Plots For F t

Fig.5.146 Error Estimation Plots For F a

Fig.5.147 Error Estimation Plots For F r

5.6.8 Error Plots of PSO based ANFIS Prediction Results with Experimental Statistics for AISI 4340 Steel 45hrc

Fig.5.148 Error Estimation Plots For R a

Fig.5.149 Error Estimation Plots For F t

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Fig.5.150 Error Estimation Plots For F a

Fig.5.151 Error Estimation Plots For F r

5.7 Conclusion An extensive statistical error analysis of developed prediction model through possible synergies is performed and at all the three learning stages the errors were monitored i.e, while training testing and validating. The learning converged well for all applied techniques.

Furthermore the predicted results are tested against experimental statistics for evaluating the prediction accuracy of each learning model. The prediction models demonstrated relatively varying accuracy results which are tabulated in each sections.

Following crucial observations can be made through statistical error analysis.

1. The results of EA â&#x20AC;&#x201C;NN were more accurate than SI-NN and less accurate when compare to the synergies of ANFIS. 2. EA-NN accuracy was convincing but the accuracy of ANFIS-EA and ANFIS-SI were better than the EA-NN and SI-NN prediction models which can be clearly commented from the statistical error table.

3. Between the ANFIS â&#x20AC;&#x201C;EA and ANFIS-SI the relative difference of accuracy is negligible as both the techniques demonstrated almost similar results in prediction 4. The Adaptive neuro fuzzy combination proved to be better than the neuro computing combination this difference is possible due to the adaptive layers introduced in the neuro-fuzzy inference. Though in both the techniques back VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 187

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propagation learning algorithm is applied for error minimization in learning, the adaptive layers and clustering technique in membership function of fuzzy structure improved the learning ability of prediction model.

5. However the synergism of neural network gave better accuracy than the exclusive techniques exhibiting improvement in learnability of pattern

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CHAPTER 6 RESULTS AND DISCUSSION The results from optimization and predictive techniques are elaborately discussed and comparison with experimental statistics is made.

6.1 Global Search Optimization Global Search optimization technique are explained extensively in chapter 3 and implemented strategy is shown in fig.3.1 both techniques from EA and SI were utilized to optimize machining performance 6.1.1 NSGAII NSGA II utilized non-domination technique for its population selection and mating, the fitness of individuals were determined by two criteria (a) Pareto-Individual rank (b) Pseudo Euclidean distance According to criteria an individual with least rank and maximum Euclidean distance is the fittest and these individual/individuals represents globally optimized solutions. The degree of optimality depends on their relative function of rank and distance which represents diversity of individuals in each generation. The results of both the steels AISI 4340 35HRC and 45HRC had initial population of 1000 and elitism was applied at each generation.

The Table 3.3 and Table 3.4 records results of family of optimized individuals for both steels after hundred generations arranged in descending order of fronts and fitness levels which are calculated by rank and distance. Only the first 20 solutions are listed out of the 1000 population in the tables.

6.1.1 (a) For AISI 4340 35HRC From Table 3.3 the first front has maximum distance and minimum rank the solutions and these solutions shows a good tradeoffs between the surface roughness and tool life among them fifth individual with fitness which gives both good surface finish and VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 189

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maximum tool life .This individual corresponds to process parameters (cutting speed ,feed rate and depth of cut) 170m/min, 0.15mm/rev, 1mm respectively with machining objectives(Ra, Ft, Fa, Fr, Tf) 4.0µm, 412N, 152N, 231N and 50 minutes. The subsequent individual with process parameter (cutting speed, feed rate and depth of cut) 231m/min,0.15mm/rev,1mm corresponding to machining performance (Ra, Ft, Fa, Fr, Tf) 2.5µm, 442N, 115N, 199N and 42.49 minutes can be picked out as optimal solution.

Though the fitness of first front is fittest among others, the individuals from subsequent front can also be chosen from the solution space with permitted tradeoffs.

The Fig.3.3-3.5 constitutes plots of NSGA II for 35HRC, in the Fig.3.3 the first subplot shows the rank of individual at each generation, from the figure it can be inferred that the rank converges to one for most of the population as non-dominated technique is applied for sorting, in the second subplot of Fig 3.3 Pareto plots between the objectives are plotted to evaluate relative tradeoffs between two objectives.

Pareto front depicts plots for elite members which can be between two or three objectives that are non-inferior. The subplots 2, 3 and 4 in the figure 3.3 are pareto plots between force and surface roughness, the second subplot represents pareto-front between Ra-Ft, in third and fourth sub plot pareto front between Ra-Fa and Ra-Fr. The Fig 3.4 contains pareto-fronts between tool life and forces; first subplot is paretofront between Tf -Ft, second subplot between Tf-Fa and third sub plot between Tf-Fr. From the pareto fronts for minimum surface roughness and maximum tool life following region range of forces were found to be favorable (approximately) optimal surface profile.

Table.6.1 Tradeoffs among forces for Surface roughness and Tool life Forces Tangential Forces Axial Forces Radial forces

Surface Roughness(1-2.5 µm) 800-600 N 300-200 400-300

Tool life (Tf>40 mins) <470N <160N <250N

In fig. 3.5 average pareto diversity plots in the consecutive generations is plotted which is determined by average pareto (Euclidean) distance among individuals The diversity of

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population varied from 1-0.001 The population in overall generations exhibited vivid diversity. Initially the average distance between the generations was maximum after 20 generations a shift in distance was observed with distance spread to 0.6 in mid generation and as final generations are approached the distance between generations converges to 0.001 showing a good migration of individuals across generations.

6.1.1 (b) For AISI 4340 45HRC For AISI 4340 45 HRC steel the family of solutions is listed in table 3.4 out of the 1000 elite solutions only 20 solutions are tabulated. The best individual in the first front is corresponding to process parameters (cutting speed, feed rate and depth of cut) 130.8m/min, 0.15 mm/rev, 1mm and objective fitness (Ra, Ft, Fa, Fr, Tf) 4.0 Âľm, 542N, 331N, 282N, 30 minutes is fittest. Similarly preceding individual with process parameters (cutting speed, feed rate and depth of cut) 135m/min, 0.15mm/rev, 1mm corresponding to objective fitness (Ra, Ft, Fa, Fr, Tf) 3.98, 519N, 331N, 282N, 28 minutes can be picked as best solution. Likewise favorable solutions can be chosen with the degree of tradeoffs permitted among the objectives.

The Fig.3.6 - 3.8 are plots of NSGA II for 45HRC, in the Fig.3.6 the first subplot depicts rank of individual at each generation and the rank converges to one for most of the population, in the second subplot of Fig.3.7 Pareto plots between the objectives are graphed and in Fig.3.8 diversity plot at each generation is drawn.

The subplots 2, 3 and 4 in the figure 3.6 are Pareto plots between surface roughness and force, the second subplot represents pareto-front between Ra-Ft, the third and fourth sub plot shows pareto front between Ra-Fa and Ra-Fr. The Fig.3.7 contains pareto-fronts between tool life and forces; first subplot is paretofront between Tf-Ft, second subplot between Tf-Fa and third sub plot between Tf-Fr. From the pareto fronts for minimum surface roughness and maximum tool life following region range of forces were found to be favorable (approximately) for optimal solution in 45HRC machining.

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Table.6.2 Tradeoffs among forces for Surface roughness and Tool life Forces Tangential Forces Axial Forces

Surface Roughness(3.5-4µm) 600-400 N 360-300N

Tool life (Tf>10 mins) <500N <160N

Radial forces

300-330N

<350N

In Fig.3.8 diversity of generations is plotted, at initial generations the average pareto distance was maximum latter on at mid generations a sharp drop is observed between 2535 generations and then at final generations the distance converged to as low as 0.001 showing good migration ability and diversity.

6.1.2 Results for SPEA 2 6.1.2 (a) For AISI 4340 35 HRC Since SPEA 2 is a combination of pareto envelope and niche–pareto the rank and fitness strength is calculated to both best elements in archive and individuals in population through dominance level. The proposition rule of determining best solution is by calculating effective fitness value which is sum of raw fitness and density distribution about individuals. Table 3.6 holds optimal solution for AISI 4340 steel 35 HRC which contains 300 archive elements. The matrix F contains the fitness strength of each individual which calculated by summation of R (rank) and D (distance /density function) matrix (in table 3.6), the distance D is calculated by K nearest cluster algorithm .The matrix S is strength value which is evaluated by dominance count. The position matrix represents process parameters and the cost matrix represents machining objectives.

The individual with least strength value represents the fittest individual i.e.,

the

individual in first front with strength fitness S 0.32 can be picked as best individual with position (process parameters :cutting speed, feed rate and depth of cut) 259m/min, 0.17mm/rev, 1.07mm corresponding to machining objectives (Ra, Ft, Fa, Fr, Tf) 1.88µm, 514N, 136N, 225N, 35.51 minutes.

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In Fig.3.10 subplots have individual ranks and pareto fronts between surface roughnessforces, the subplots 2, 3, 4 depicts the pareto front between Ra-Ft, Ra-Fa, Ra-Fr, which are archive element containing only fittest individual. Similarly the pareto fronts between tool life and forces are plotted in fig.3.11 the subplots 1,2,3 are pareto fronts for Tf- Ft, Tf- Fa, Tf- Fr are represented. Though the solutions suggested by SPEA 2 is different individuals from the NSGA II the force constraint obtained is same as NSGA II for minimum surface roughness and maximum tool life Table.6.3 Tradeoffs among forces for Surface roughness and Tool life Forces Tangential Forces Axial Forces Radial forces

Surface Roughness(1-2.5µm) 780-590 N 300-200N 300-330N

Tool life (Tf>35 mins) <500N <160N <250N

In Fig.3.12 diversity among generation is plotted the average pareto distance is different from the NSGA II and the spread is limited between 0.37-0.27 the spread is random representing a good mix in individual among generations but the combination is restricted between limited distance.

6.1.2 (b) For AISI 4340 45 HRC The results of 45 HRC steel is recorded in the Table 3.7 with same definition of elements.

From the table the strength fitness is least for third individual with F 0.31 the position (process parameters: cutting speed, feed rate and depth of cut) of this individual is 174 m/min,0.23 mm/rev, 1.7 mm with machining performance (Ra, Ft, Fa, Fr, Tf) 3.93µm, 674N, 389N, 355N, 14.19 minutes and next fittest individual is in consequent front with strength fitness S 0.33 with process parameters168m/min,0.16mm/rev,1.02mm for machining performance 3.67µm,519.42N,384N,310N,19 minutes.

Pareto fronts for 45 HRC objectives are in Fig.3.13 and Fig.3.14,in fig 3.13 the sub plot 2,3,4 represents the pareto front between Ra-Ft, Ra-Fa, Ra-Fr, and subplots in Fig.3.14 shows subplots of Tf-Ft, Tf - Fa, Tf- Fr respectively for minimum surface roughness and maximum tool life the force constraints should be as follows

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Table.6.4 Tradeoffs among forces for Surface roughness and Tool life Forces Tangential Forces Axial Forces Radial forces

Surface Roughness(3.5-4Âľm) 600-400 N 360-300N 300-330N

Tool life (Tf>10 mins) <500N <160N <350N

In Fig.3.15 diversity among generation is determined the average pareto distance is limited between 0.4-0.26 the spread is random representing a good combinations in individual among generations but this trend is limited.

6.1.3 Results for PSO 6.1.3 (a) For AISI 4340 35HRC The PSO algorithm utilizes swarm movement to find optimal solution. The applied swarm constitutes of particle structure with position, cost and their best position, best cost associated with it a pseudo velocity and acceleration associated for each particle. Table 3.9 comprises of optimal solution for AISI 4340 steel where the position matrix holds process parameters cost holds the machining performance for each position best cost ,best position is defined for each position. The velocity matrix is which correspond to the position matrix is utilized to move swarm in search space and changes at each transition.

The Grid Index matrix contains the topology of swarm at each transition and grid sub index contains the neighborhood topology of grid index. The table consists of 500 swarm particles which are in repository element out of 1000 swarm.

In Fig.3.17 pareto front between the surface roughness and tangential force is graphed which gives similar results as pareto front of Evolutionary Techniques.

Fig.3.18-3.19 are swam surfaces for surface roughness and tool life respectively which shows the potential of swarm. Each particle on the swarm surface is defined by particle position in Fig.3.18 the particles at the declination of surface are fittest particles which converged to minimum surface roughness values and in Fig.3.19 the particles at the projection of swarm surface represents the optimal tool life showing convergence at the foot of swarm surface.

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In Fig.3.20 the influence of depth of cut on force component on optimal particles is plotted. The subplots 1, 2 and 3 show the force gradients with respect to depth of cut for best particles. The change in force about the mean line is shows that best particles are about the mean line presenting favorable solutions for surface roughness and tool life.

From the table 3.9 optimal solution is determined by leaders among swarm the first leader corresponding is to gird index 40282 with best position 193.76m/min,0.15mm/rev, 1 mm with machining performance (Ra, Ft, Fa, Fr, Tf) 3.43 Âľm, 417N, 134N, 215N, 46.12 minutes the subsequent solution is corresponding to particle with grid index 46844 similarly other solutions are equally fit depending on the permitted tradeoffs solutions can be chosen.

6.1.3 (b) For AISI 4340 45HRC The solution for AISI 4340 45 HRC steel has same entities which are listed in table 3.10.The pareto-fronts and swarm surface are plotted in fig 3.21 and fig 3.22-3.23 respectively. The swarm surface of 45HRC showed similar trends as to 35 HRC steel the 45HRC steel shows similar trends in swarm surface. In Fig.3.21 the pareto front between surface roughness and tangential force is drawn which gives force constraints similar to NSGA II 45 HRC fronts .In fig 3.22 and fig 3.23 the swarm surface for surface roughness and tool life is drawn.

In fig 3.24 the influence of depth of cut on force component on optimal particles is plotted. The subplots 1, 2 and 3 show the force gradients with respect to depth of cut for best particles.. From the table 3.10 optimal solution is determined by swarm leaders among them the first leader corresponds to gird index 20510 with best position 136 m/min, 0.15mm/rev, 1mm with machining performance (Ra, Ft, Fa, Fr, Tf) 4.04 Âľm, 522N, 134N, 331N, 29.32 minutes the subsequent solution is corresponding to particle with grid index 31070 similarly other solutions are equally fit depending on the allowed tradeoffs solutions can be chosen.3

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6.1.4 Comparison between EA and SI The Fig.3.25 - Fig.3.30 are plots of solution spectrum obtained from applied global search optimization technique. Each plot has solutions for five objectives i.e, Surface roughness, tool life, radial force, axial force and tangential force. In fig 3.25 and fig 3.26 the solution space obtained from NSGA II for 35 HRC and 45 HRC steels is drawn.

Clearly distinction can be made the graphs that in 35 HRC steel radial forces are dominating axial force while in 45 HRC steel axial force is dominating radial force. A similar trend can be inferred from solutions of SPEA2 and PSO. Fig 3.27 and Fig 3.28 are solution space obtained from the PSO technique for 35 HRC and 45 HRC steel respectively and Fig.3.29 and Fig.3.30 are solution space obtained from the SPEA2 for 35HRC and 45HRC respectively.

For 35 HRC the sequence of solution for each objectives are represented in following order surface roughness, tool life, axial force, radial force and tangential force ,while for 35 HRC the solution for each objective is in order of surface roughness, tool life ,radial force, axial force and tangential force.

6.1.4 (a) Comparison among the Solution spectrum The solution space of NSGA II (fig 3.25 & 3.26) has thousand populations which is a combination of parents and offspring formed by tournament selection. The solution space is distributed with constant amplitude in solution space. While the solution space of PSO and SPEA2 (fig.3.27-fig.3.30) vary in amplitude in solution space at each generation.

The solution space of PSO (Fig 3.27 & fig 3.28) has five hundred best solutions which are elite solutions. The solution trend is random when compared to NSGAII showing varying amplitude across wavelength with immediate change in crest and troughs in local and global minima and maxima.

However the solution space of SPEA2 has three hundred best elite solutions which are highly disturbed amplitude when compared to NSGAII and PSO the jumps in maxima and minima is uneven exhibiting unsaturation in local and global minima .

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From the solution space analysis the NSGAII showed a better saturation in local global maxima minima while PSO should a moderate saturation and SPEA2 exhibited lower saturation levels in maxima and minima. The saturation levels also depicts tradeoffs in objectives from which an following inference can be made.

The solution obtained from the NSGAII solution gave better tradeoffs among objectives which can be observed from each front in solution space. While PSO gave a moderate trade off with the solution space was favoring lower surface roughness by reducing tool life for few swarm particles though the overall trade off was found to be good enough.

The solution space of SPEA2 gave unbalance tradeoff between objectives favoring surface roughness and reducing tool life for many individuals. From above discussion it can be concluded that the solution space provided by NSGAII is better than other two while the solution space provided by the PSO is better than SPEA2.

6.1.5 Evident from the literature The solutions obtained by fittest individuals in NSGAII and PSO keeps the flank wear of tool under working limit. The tool used for current hard turning was multi-layer coated carbide inserts and the cutting speed suggested by both the optimization techniques suggested cutting speed below 200 m/min and feed rate, depth of cut in between LFLD (low federate low depth of cut) and HLHD (high feed high depth of cut) at this condition the flank wear is less than 0.15 mm for tool life greater than 40 minutes [2]. This cutting condition keeps flank wear under appreciable level, restricting sharp rise in cutting forces due to high flank wear rate [3]. If the flank wear is inhibited above this then machining chatter due to excessive forces is controlled and better surface finish is obtained.

In contrast SPEA2 suggests cutting speed close to 260m/min (250-260 m/min) and feed rate, depth of cut near (HLHD condition) for which flank wear about 0.2mm for time cutting greater than 35 minutes.[2]. At this flank wear the forces tend to increase sharply which may increase machining vibrations resulting in poor surface finish. The tradeoff between the surface roughness and tool life should be picked wisely for a successful hard turning.

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6.2 Intelligent Learning Techniques Intelligent learning systems were applied to recognize machining pattern sequence which is discernible in the machining statistics. In fig 4.1 the adopted strategy for chapter 4 is illustrated and accordingly the results were discussed. Learned networks were utilized to predict machining performance on experimental runs and obtained results were analyzed for statistical error. An extensive mathematical framework is discussed for the all the applied learning techniques.

6.2.1 Neural Network The applied network architect for both steels is illustrated in fig 4.4. A multi-layer feed forward perceptron type network was used to for both steels which require many exemplars for mapping multiple vectors. From the regression model thousand machining data were generated for each vector model for better interpolation of target vectors in learning space. These data sets were randomly permutated and split for training, testing and validating samples. Out of the thousand sets 700 samples were utilized for training 150 sample for training and 150 for validation.

The description of applied neural network for AISI 4340 steel 35 HRC is listed in table 4.2.4 (a) and the calibrated weights and bias for the targets is given in table 4.2.4 (b). The network performance plot in Fig.4.5 shows drop in mean square error from 104 to 10-3 in 1000 epoch while learning (training, testing and validating stage). The error gradient Fig.4.6 between the targets and output converged to 10-2 at 1000 epoch from gradient of 105. For each objective an error histogram and regression plots were drawn in (Fig. 3.7Fig. 3.16) in each error plot the mean error was close enough to the zero error line depicting that the error minimized to its least possible value. Linear regression fit was obtained between targets and output while learning each objective with regression co efficient close to 1.

Same network was utilized for leaning machine statistics of 45HRC steel. The network description and calibrated weights and bias are listed in Table 4.2.5(c) and 4.2.5(d).The mean square error while learning converged to similar order (Fig.5.17) to that of 35HRC. VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 198

Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation

While in training state the error gradient gave sharp slopes though at final epoch the gradient converged to 1.125 with zero validation fails. The error histogram and regression plot for each vector is in (Fig.4.19-Fig.4.28).The error for each machining vector was in order of 10-2.

6.2.2 Adaptive Neuro Fuzzy Interference Technique In this network two adaptive layers with learnable parameters were used viz., antecedent and consequent layer. Three different clustering techniques were applied on membership function i.e., Grid portioning clustering, Subtractive clustering and Fuzzy C- mean clustering. Each cluster techniques had different weights (connections) and the fuzzy structure of each technique varied. The analytical description of hybrid neuro-fuzzy techniques is discussed in brief. For each machining vector fuzzy structure learns machining data sequentially. From the RSM model thousand machining data for learning were generated. These vectors were randomly permutated and split into training (850 samples), testing (150 samples) and validated on complete machining vector.

6.2.2 (a) ANFIS Grid partition The applied structure is illustrated in fig.4.30 and fuzzy structure implemented is tabulated in table 4.3.4 (a) fuzzy structure. For each machining vector following plots were drawn. (i)

Simultaneous plots between expected target and output.

(ii)

Errors between the target-output.

(iii)

Normal density fit between mean error and standard deviation.

At each epoch the error between targets and output while learning was calculated and error gradients were minimized through hybrid leaning and back propagation algorithm.

In table 4.3.4 (b) statistical errors in training, testing and validating for each objective in 35HRC is listed and mean error for each vector were as low of order 10-5 these statistical results are plotted in Fig.4.36-Fig.4.60.

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Same architect and fuzzy structure is utilized for learning machining pattern for 45 HRC. The statistical error in training, testing and validation for each objective is tabulated in table 4.3.4 (c) the error for was as low as low as 10-5 and plots for each objective are illustrated in between Fig.4.61-Fig.4.85.

6.2.2 (b) ANFIS Subtractive Clustering The developed structure for subtractive clustering is illustrated in fig.4.86, fuzzy structure implemented is tabulated in table 4.3.16 (a) and in table 4.3.16 (b) &4.3.16 (C) statistical errors in training, testing and validating for objective in 35 HRC and 45HRC respectively is listed and these statistical results are plotted for each objective.

In comparison to grid partition clustering the mean error in subtractive clustering is reduced to order of 10-7.for 35 HRC and 10-6 in 45 HRC showing improvement in leaning ability. The plots for each objective and their corresponding figures for 35HRC and 45HRC steel acquired from ANFIS subtractive Clustering are illustrated in figures Fig.4.87-Fig.4.136.

6.2.2 (c) ANFIS Fuzzy C-mean Clustering The developed structure for FCM clustering is shown in fig.4.137 and fuzzy structure implemented is tabulated in table 4.3.27 (a). In table 4.3.27 (b) & 4.3.27 (C) statistical errors in training, testing and validating for both steels is listed. The learning error in FCM was further reduced to order of 10-8 in both the steels though not for all vectors The mean error in all vectors was of similar order (10-6) showing similar results as that of subtractive clustering. The plots for each objectives and their corresponding figures for 35HRC and 45HRC steel obtained from ANFIS FCM are shown in Fig.4.138-Fig.4.187.

6.2.3 Comparative Evaluation of the predictive technique on Experimental statistics The developed predictive model was tested on Experimental statistics and a statistical analysis of the errors in predictions was made for each model. Comparison graphs for each objective in each technique were plotted (fig.4.188-fig.4.219). The Mean errors and

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their standard deviations of each objectives in both the steel are listed in table (6.5) and the means squared error (MSE), root mean squared error (RMSE) is tabulated in table.

Table.6.5 Mean error and Standard Deviation between Experimental and Predicted statistics Machining Objective

Neural Network

ANFIS Grid

Error Measure

ANFIS Sub

ANFIS FCM

Neural Network

Mean Error

ANFIS Grid

ANFIS Sub

ANFIS FCM

Standard Deviations

Comparison Errors for 35 HRC AISI 4340 Steel 0.0442 -0.2308 -0.226 -0.2339 1.0814 Ra(µm)

0.5964

0.5964

0.5964

Ft (N)

0.186.6

17.49

15.29

18.19

126.4

106.9

105.36

104.9

Fa (N)

103.98

157

157.79

157

33.77

80

82.3

79.42

Fr (N)

1.1318

32.57

27.89

32.23

93.95

41.29

40.81

42.20

Comparison Errors for 45 HRC AISI 4340 Steel Ra(µm)

-0.1700

-0.9278

-0.947

-0.856

0.2769

0.5964

0.594

0.596

Ft (N)

-57.23

-22.35

-23.106

-21.7

128.3

77.58

76.64

78.8

Fa (N)

54.67

-7.5709

-6.9912

-6.8003

33.57

40.08

39.58

38.45

Fr (N)

0.415

-7.69

-8.525

-6.63

14.0720

38.35

36.57

36.89

Table.6.6 MSE and RMSE between Experimental and Predicted statistics Machini ng Objectiv e Error Measure

Neural Networ k

ANFIS

ANFIS

ANFIS

Grid

Sub

FCM

Mean Square Error

Neural Network

ANFIS

ANFIS

ANFIS

Grid

Sub

FCM

Root Mean Square Error

Comparison Errors for 35 HRC AISI 4340 Steel Ra(µm) 1.1130 0.3912 0.3875 0.392

1.0550

0.6255

0.6225

0.6266

Ft (N)

594.81

1.116*104

1.07*104

1.08*104

24.38

105.66

103

103.9

Fa (N)

1.89*10

3.09*104

3.13*104

3.07*104

109.07

176

177.016

175.32

4

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Fr (N)

2.6*103

93.95

2.36*103

2.73*103

9.69

51.78

48.58

52.25

Comparison Errors for 45 HRC AISI 4340 Steel Ra(Âľm)

0.1018

1.198

1.23

1.0714

0.3190

1.0949

1.109

1.0351

Ft (N)

1.89*10

6.128*103

6.11*103

6.38*103

137.5

78.85

78.19

79.8

Fa (N)

4.06*10

1.58*103

1.53*103

6.36*10-6

63.72

39.8

39.213

0.4664

Fr (N)

188.29

1.45*103

1.33*103

1.33*103

13.72

38.17

36.65

36.56

4

3

Though the statistical errors in neural network was relatively in comparison to ANFIS models the relative change in error at each prediction points in neural network is higher than the ANFIS models and the curve traced by neural network in both the steels did not match well when compared to curved traced by ANFIS models. This observation can be inferred by analyzing corresponding comparison graphs for each objectives in specific techniques. The error plots for each technique are drawn in Fig.4.188-4.219.

The results obtained from the prediction models were accurate enough to predict around the experimental statistics, though the relative degree of accuracy varied for different learning technique. The mean error of neural network was less than the neuro-fuzzy but the prediction curve could not trace well with the experimental curve in converse the mean error of ANFIS was comparatively larger but the prediction curve traced well with the experimental curve.

6.3 Synergies of CI The combination of optimization techniques were applied to prediction model to further improve results in pattern learning of current machining statistics The combination of synergies applied is shown in Fig.5.1 results of each combination is discussed individually.

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6.3.1 EA-NN For EA-NN combination NSGA II optimization technique was applied to optimize weights. The applied architect is shown in Fig.5.2. The network description, calibrated weights of combined network is listed in table 5.2.2 (a) and 5.2.3 (b).The optimized populations from NSGAII were given input to neural network at each epoch and depending on drop in error gradients the weights were adjusted and fed to NSGA II for optimization. The optimized solution space (1000 population) of NSGA II are permutated randomly and split into training (700) testing (150) and validating samples. These samples would adjust weights accordingly at every epoch.

The mean square error for network converged to order of 10-3 just in 300 epoch exhibiting good convergence in network error. The plot for network performance is in Fig.5.3.The training state plots is illustrated in Fig.5.4 the error gradient dropped to 0.194 which is lesser than the neural network. Likewise the error plots and regression for each vector is in Fig.5.5-Fig.5.15.

Same architect is utilized for 45 HRC steel the learning performance was better than the exclusive neural network with mean square error as low as 10-4 in training, testing, validation stages. The performance plot is drawn in Fig.5.16 and the training state plots are in Fig.5.17, error gradients were also found to be minimized to gradient as low as 10 -, error and regression plots for each vector are in Fig.5.16-Fig.5.41.

6.3.2 SI-NN Applied architect for PSO-NN combination is illustrated in Fig.5.29. The best solutions in archive elements of PSO were utilized for network learning. The archive had 500 elements out of which 350 samples were utilized for training, 75 samples for testing and 75 samples for validation. The network description and calibrated weights are tabulated in table 5.3.1 (a) and table.5.3.1 (b). Two hidden layers were defined for network to work accurately since the examples provided by PSO were less compared to NSGA II. For each input and output vectors 20 weights were utilized for mapping and 10 bias were added at each hidden layer.

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The network performance of combined SI-NN was better than NN with mean square error dropping as low as 10-4 in all the learning state and error gradient going as low as 10-4 at final epoch. The performance plot of network is illustrated in Fig.5.30 and training state plots are in Fig.5.31.The overall network performance was better than the exclusive neural network.

6.3.3 ANFIS Synergies To enhance ANFIS, GA and PSO are applied to optimize membership function, FCM structure from previously developed model was utilized for combination. The applied strategy utilized for ANFIS-GA is explained in Fig.5.42, the architect utilized is same as that of ANFIS-FCM. Thousand learning samples were generated from RSM model these exemplars were randomly permuted and used for leaning. Out of thousand exemplars 700 were split for training sets and 300 for testing. Since fuzzy maps only one vector per training the multi objective NSGA was reduced to single objective GA with no change in genetic and selection operators.

From the error table 5.3.1 (a) the mean error for each vector of 35HRC dropped to order of 10-13 which is way better compared to exclusively applied ANFIS (10 -6) demonstrating excellent improvement in pattern learning ability. The error plots for each vector are drawn in Fig.5.44-Fig.5.64.Similarly for 45 HRC from the Table.5.3.7 mean error for each vector dropped to order of 10-13 exhibiting better leaning than exclusive ANFIS. Developed strategy for ANFIS-PSO technique is shown in Fig.5.84, the thousand samples employed in ANFISâ&#x20AC;&#x201C;GA is used for ANFIS-PSO learning. From table 5.4.7 the learning error was reduced to order of 10-14 exhibiting better learning ability The error plots for both the techniques is represented in Fig.5.44-Fig.5.125.

6.3.4 Comparative Evaluation of the predictive technique on Experimental statistics The developed predictive model was tested on Experimental statistics and a statistical analysis of the errors in predictions was made for each model. Comparison graphs for each objective in each technique were plotted (fig.4.188-fig.4.219).The Mean errors and

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their standard deviations of each objectives in both the steel are listed in table 6.7 and the means squared error(MSE), root mean squared error (RMSE) is tabulated in table 6.8 Table.6.7 Mean error and Standard Deviation between Experimental and Predicted statistics Machinin g Objective Error Measure

NSG A-NN

PSO-NN

ANFISGA

ANFISPSO

NSGA -NN

Mean Error

PSO-NN

ANFIS ANFIS-GA PSO

Standard Deviation

Comparison Errors for 35 HRC AISI 4340 Steel -0.221 1.749 -0.1572 0.1578 Ra(µm) 42.67 -53.25 17.6 17.13 Ft (N) 156.2 138.22 136.152 135.9 Fa (N) -0.723 40.08 11.065 10.89 Fr (N) Comparison Errors for 45 HRC AISI 4340 Steel -0.355 0.0371 0.0372 Ra(µm) -57.23 -21.84 -27.34 Ft (N) -13.57 -45.95 -46.83 Fa (N) -11.74 -0.91 -0.747 Fr (N)

0.979 86.21 49.56 14.77

0.732 104.6 40.44 64.63

0.4507 57.06 51.87 20.54

0.4494 57.16 52 70.79

0.378 128.36 38.046 35.99

0.4507 57.06 51.87 20.54

0.2796 107.08 41.46 36.86

0.2798 106.19 41.55 36.77

Table.6.8 Mean Square Error and Root Mean Square Error between Experimental and Predicted statistics Machini ng Objectiv e Error Measure

NSGANN

PSO-NN

ANFISGA

ANFIS-PSO

NSGANN

Mean Square Error

Comparison Errors for 35 HRC AISI 4340 Steel 3.56 0.2177 0.2168 Ra(µm 0.960 ) 8.88*103 1.32*104 3.4*103 3.398*103 Ft (N) 4 4 4 2.06*10 2.109*10 2.11*104 Fa (N) 2.67*10 5.57*103 523.

517 Fr (N) 206.55 Comparison Errors for 45 HRC AISI 4340 Steel 0.075 0.075 Ra(µm 0.262 ) 1.64*104 1.14*104 Ft (N) 1.89*104 3.74*103 3.83*103 Fa (N) 1.55*103 3 1.29*103 1.28*103 Fr (N) 1.36*10

PSONN

ANFISGA

ANFI S-PSO

Root Mean Square Error

0.980

1.88

0.466

0.4656

94.24 114.49 14.37

115.04 43.73 74.66

58.3 145.2 22.87

58.29 145.4 22.75

0.512

0.2751

0.275

137.58 39.488 36.99

107.76 61.19 35.93

107.05 61.91 35.84

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From table 6.7 the values of mean error and standard deviations of ANFIS combinations were less than the Neural Network synergies. Similarly (table 6.8) the mean square error and root mean square error for ANFIS synergies were lesser compared to Neural Network Synergies. The amplitude of error between experimental and predicted statistics was shorter compared to Neural Network.

The ANFIS combination performed better than the NN combination this enhancement is possibly due to the optimization of adaptive layers. Though the statistical error is not very large when compared in magnitude the curve traced by ANFIS synergies were found to better than the curve traced by NN synergies. The error plots for each technique are illustrated in Fig.5.126-Fig.5.153.

In brief the combinations of ANFIS performed better than the combinations of Neural Network. However the developed prediction techniques were accurate enough to learn experimental machining statistics.

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CHAPTER 7 CONCLUSION On an overview the work dealt with optimizing machining performance in hard turning of AISI 4340 steel and building accurate predictive models by applying intelligence learning techniques.

7.1 Optimization Trends 

From the results of optimization the best tradeoffs among machining objectives was obtained by NSGA II solution space, followed by PSO while solution space of SPEA2 was biased toward surface roughness recommending lower tool life in most of its best solution.



The difference in tradeoffs recommended by algorithms is due to the degree of elitism demonstrated by algorithms Elitism is similar to a pseudo memory that is associated to algorithm by which it recognizes the best individual so that it doesn’t search for same individual in again in consecutive reducing exploration time for searching best individual.



The elitism in NSGA II was implemented by tournament selection where consecutive best individuals selected at each tournament are preserved by replacement of chromosome in intermediate solution. Elitism was performed on single set of population i.e. both fittest individuals and intermediate chromosomes were members of chromosome. Though achieving elitism by single set of chromosome slowed down the convergence while recursion. This drawback was overcome by providing randomness in genetic operator. The mutation operator acted as agent of entropy in solution space when elite individuals enveloped to local minimums.



While in PSO the elitism was carried out by introducing pbest and gbest population sets where the local minimum were stored in pbest and global minimum in gbest.

The repository element in PSO was elite members which contained gbest populations. The elite mechanism in PSO was dynamic compared to NSGA II in VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 207

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PSO, the gbest and pbest attractor in neighborhood topology acted as initiator for gathering elite members. The problem of local minimum was overcome by the swarm movement equation Unlike NSGA, PSO doesn’t have genetic operator so when the size of elite members exceeded inertia damping co efficient controlled it, for better convergence a pseudo mutation on particles were introduced. 

Similarity in solution space in NSGA II and PSO could be justified by finding analogies between both algorithms. The best parents in NSGA acted as pseudo particle attractor in recognizing elite members which is similar to the gbest and pbest attractor in grid topology and the genetic operator is similar to that of swarm movement operator Since the elite members in both the algorithms were same (i.e., 500) which justly supports analogy.



But when solution space of SPEA 2 is compared the fittest individuals is quite different from NSGA and PSO algorithm it shows inclination towards one vector weakening other vector even though elitism is fairly applied. This behavior is drawn from the niche behavior of fit individuals in archive (elite members).

7.2 Prediction Trends 

Two variants of learning techniques were applied for recognizing pattern in machining statistics. Initially neural network and ANFIS were applied exclusively for learning machining examples. The accuracy of learnt networks were tested on experimental statistics both techniques gave similar results in prediction. Neural network gave relatively lower errors in comparison to ANFIS.



Among the different clustering techniques used in ANFIS the FCM clustering technique gave relatively lesser errors in learning hence demonstrating better leaning ability than among three techniques. To further improve the learning ability combination of optimization and learning techniques were utilized Both EA and SI coupled learning techniques were applied.



For both the leaning techniques (NN & ANFIS) collaborative combination was utilized. The optimization technique was secondary technique while the leaning technique was primary. In neural network the NSGA II and PSO optimized were

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used to optimize the weights in hidden layer and reduce the error between targets and outputs the prediction results obtained from the NSGA-NN were better in comparison to PSO-NN. ď&#x201A;ˇ

The ANFIS combination Technique exhibited better learning trends than the ANFIS applied exclusively. Both the combinations ANFIS-GA and ANFIS-PSO gave better results in comparison to experimental statistics. The accuracy of both techniques was similar with minor difference in learning error. In comparison to neural network synergies the ANFIS synergies gave better results on experimental statistics and illustrated enhanced learning.

ď&#x201A;ˇ

To summarize the combined predictive models performed better in comparison to the exclusive techniques and in optimization techniques the NSGA II and PSO algorithm gave relatively good tradeoffs in MOOPS when compared to SPEA2 algorithm.

Future Scope The unexplored Meta heuristic techniques can be applied for much better tradeoffs in MOOPS and further enhancement in learning technique is possible by applying further introducing stronger coupling among the prediction and optimization techniques.

The applied strategy for current machining system can be generalized to other conventional and non-conventional machining systems.

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ed., John Wiley & Sons, Chichester. Deb, K., Pratap, A., Agarwal, S. and Meyarivan, T. (2002) A fast and elitist multi-objective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6(2), 182–197.

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S.Milad.Nayyer Sabeti 1 and MR.Deevband 2 Hybrid Evolutionary Algorithms

based on PSO-GA for Training ANFIS Structure International Journal of Computer Science Issues, Volume 12, Issue 5, September 2015.

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Yosra JARRAYA, Souhir BOUAZIZ, Adel M. ALIMI Ajith ABRAHAM Fuzzy

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APPENDIX: A CONFERENCES AND PUBLICATION Conferences [1] Presented Paper in “VISHWACON 2016-2017” at Vishwakarma Institute of Information Technology, Pune on 17th Feb. 2017.

[2] Presented Paper in “ICMMM-2017” at VIT-University, Vellore on 10th March 2017.

[3] Presented Paper in “MECHPGCON-2017” held at Zeal College of Engineering and Research, Pune on 20th June 2017.

[4] Presented Paper in “ICMTS-2017” held at Indian Institute of Technology –Madras, Chennai on 7th July 2017.

[5] Paper presented in “IconAMMA-2017” held at Amrita Vishwa Vidyapeetham University on Aug 2017.

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Selected Paper for conferences and publication [1] A research paper selected at “IconAMMA-2017” held at Amrita Vishwa Vidyapeetham University on Aug 2017 and will be published in Materials Today: Proceedings.

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APPENDIX: B CERTIFICATES [1] Certificate of presented paper in “MECHPGCON-2017” held at Zeal College of Engineering and Research, Pune on 20th June 2017.

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Reviewer & Evaluation Report MECHPGCON-2017

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[2] Certificate of presented paper in “VISHWACON 2016-2017” at Vishwakarma Institute of Information Technology, Pune on 17th Feb. 2017.

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[3] Certificate of presented paper in “ICMMM-2017” at VIT-University, Vellore on 10th March 2017.

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[4] Certificate of presented paper in “ICMTS-2017” held at Indian Institute of Technology –Madras, Chennai on 7th July 2017.

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[5] Certificate of poster presentation in “AVISHKAR-2016” held at Vishwakarma Institute of Information Technology, Pune on 2nd December 2016.

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Appendix

ANFIS Algorithm //Load data [Inputs ,Targets]=Load Machining data //Shuffle data S=randompermutation([data]) [Inputs, Targets]=[Inputs(S,:),Targets(S,:)] Train inputs=[Inputs, Targets] Train(Inputs, Targets), Test(Inputs, outputs)]=[Input(S,:) Targets(S,: )] //Generate ANFIS structure Do ï&#x20AC;¢ Training objectives fis =Create Initial Fis (data) Create Initial fis(data) Switch case Case 1 Grid Partitioning ANFIS Params No of mf nmfs : 5 Input mf type Gauss mf Output mf type : Linear Fis structure= genfis(Train Input , Train targets, nmfs, gauss, Linear) Data size =size(data,1) VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 227

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In_n =size(data, 2)-1 Input mfs_Type= In_mfs type Rule_n=prod(nmfs) Fis.name=anfis Fis.and method=’prod’ Fis.or method=’ max’ Fis.defuzzification method=’weight average’ Fis.implication method=’prod’ Fis.agrregation=’max’ Case 2 Cluster method Influence Radius Fis=generate fis (Train inputs, Train targets ,Radius) Case 3 C-mean cluster method FCM option structure No of clusters (ncluster) Portioning matrix component Maximum no of iteration Fis = generate fis(Train input, Train target ,ncluster, FCM options,Optimization method) Training params [max epoch, error goal,initial step size, step size decreament rate, step size increament rate] Output =evaluate fis (Inputs,fis)

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Train output= output [Train(Inputs)] Test output=output[Test (Inputs)] //Statistical Analysis Train error =Train targets-Train outputs Train MSE =Mean(Train error) Train RMSE =

MSE

Train Mean error=Mean(Train error) Train Error STD= STD [Train error] Test Error calculation Test error = Test targets- Test outputs Test MSE =Mean(Test error) Test RMSE =

MSE

Test Mean error=Mean(Test error) Test Error STD= STD [Test error]

ANFIS-GA/PSO Algorithm ANFIS GA/PSO //Load data [Inputs ,Targets]=Load Machining data //Shuffle data S=randompermutation([data]) [Inputs, Targets]=[Inputs(S,:),Targets(S,:)] Train inputs=[Inputs, Targets] VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 229

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Train(Inputs, Targets), Test(Inputs, outputs)]=[Input(S,:) Targets(S,: )] //Generate ANFIS structure fis =Create Initial Fis (data) Switch case Case 1 Train ANFIS using GA (fis,data) Case 2 Train ANFIS using PSO(fis ,data) Train output: evaluate fis(data.train inputs, fis) Test output : evaluate fis(data.test inputs, fis) Create Initial fis(data) Switch case Case 1 Grid Partitioning ANFIS Params No of mf nmfs : 5 Input mf type Gauss mf Output mf type : Linear Fis structure= genfis(Train Input , Train targets, nmfs, gauss, Linear) Data size =size(data,1) In_n =size(data, 2)-1 Input mfs_Type= In_mfs type Rule_n=prod(nmfs) VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 230

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Fis.name=anfis Fis.and method=’prod’ Fis.or method=’ max’ Fis.defuzzification method=’weight average’ Fis.implication method=’prod’ Fis.agrregation=’max’ Case 2 Cluster method Influence Radius Fis=generate fis (Train inputs, Train targets ,Radius) Case 3 C-mean cluster method FCM option structure No of clusters (ncluster) Portioning matrix component Maximum no of iteration Fis = generate fis(Train input, Trin target ,ncluster, FCM options) Train fis cost(x,fis,data) P0=Get fis params(fis) P=x*p0 Fis =Set fis params(fis,p) X=data.Train inputs t=data.Train targets VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 231

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Y=evaluate(x,fis) // calculate Statistical error Error :[t]-[y] MSE : mean(Error) Get fis params(fis) P=[] N=size[fis.input] For i [1-N] Nmfs =size[fis.Input[i].mf] For j [1-nmfs] P=[p, fis.Input[j].mf.params] Noutput=size[fis.output] For i [1-noutputs] Nmfs =size[fis.output[i].mfs.params] For j [1-nmfs] P=[p, fis.Input[j].mf.params] Set fis params(fis,p) P0=Get fis parms P0=[] X= data.Train Inputs Y=data.Train Targets Y=evaluate fis (x, fis) Train Anfis using GA(fis, data) VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 232

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P0=Get fis params(fis) Evalaute fitness function= Train fis cost(x,fis,data) Nvar=size(p0) Range [min , max] Max it=50 N=25 // no of pop Pc=[0.4-0.8] //crossover percentage Nc=2*round(pc*N/2) Pm=[0.2-0.4] //mutation percentage Nm =round(pm*N) ÎĽ=0.7 Mu=0.15 //mutation rate //Initialize population Pop.position=[] Pop.cost=[] For i [1-N] If it>1 Pop[i].position =random(range ,N) //Evalute Pop[i].cost =Train fis cost(x,fis, data) //sort Population [cost ,sort order]= sort[pop.cost] Pop=pop[sort order] VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 233

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//Best cost Best pop=pop[1] Worst pop=pop[end] Main Loop Do until (max it) p  exp( P

  * best cos t ) worst cos t

p N

p n 1

[p1,p2]=roulette wheel selection(P) [child 1 child 2]=crossover (p1,p2, crossover params, N) //Evaluate off spring Popc=Train fis cost (child1, fis, data) //Mutate P3=roulette wheel selection(P) Child 3=muatate (p3,mutation params, N) Popm =Train fis cost (child 3, fis, data) Pop=[pop,popc,popm) Crossover (p1,p2,crossover parameters) Parametrs :(Υ,range) α:random(-Υ,1+Υ,N) y1   * p1  (1   )* p 2 y 2   * p 2  (1   )* p1

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Y1=min[max(y1,range)] Y2=min[max(y2,range)] Mutate(p3,mutation parameters, range) Parameters : , range Rmin=min(range) Rmax=max(range) dr=Rmax-Rmin Ď&#x192;: *dr y=p1+Ď&#x192;*random(Nm) y=min(max(y,range)) Train anfis using PSO (fis ,data) P0=Get fis params(fis) Evaluate fitness function = Train fis cost (x, fis ,data) nvar =size(p0,1) range(min, max) max it =50 N=25 //no of pop W=1, Wdamp=0.99, C1=1, c2 =2 //Initialize Particle structure {Particle.Position, Particle.Velocity, Particle.Cost, Particle.Best Position, Particle.Bestcost, Particle.Is Dominated, Particle.Grid Index, Particle.Grid Subindex.} //Evaluate particle.position and cost For i :[1-N]

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Pop[i].position=random[Range,N] Pop[i].velocity : zeros[N] Pop[i].cost :Train fis cost (pop[i].position) //update personal best Exchange pop[i].best position with pop[i].position Exchange popi[i].best cost with pop[i].cost //determine domination level Pop : domination(pop,N) For i:[1-N] For j:[i+1-N] If dominates (pop[i], pop[j]) True(pop[j] Is dominated) Else if dominates(pop[j],pop[i]) True(pop[i] Is dominated) b=dominates(pop[i],pop[j]) b= all(x<y)&&any(x<y) Do until max (IT) For i : [1-N] //Select Leader leader=Select leader(rep, )

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Evolutionary Algorithms for Multi-Objective Optimization: Modelling and Comparative Evaluation pop  i  .Velocity  w * pop  i  .Velocity c1* rand VarSize  . *  pop  i  .Best.Position  pop  i  .Position   c 2* rand VarSize  . *  leader.Position  pop  i  .Position  pop  i  .Position  pop  i  .Position  pop  i  .Velocity pop  i  .Position  max  pop  i  .Position, VarMin  pop  i  .Position  min  pop  i  .Position, VarMax  pop  i  .Cost  Train fis cost  pop  i  .Position 

New pop=mutatute(pop,pm,Range) New pop.cost= Train fis cost (newpop.position) Determine domination(pop) If dominates(New.pop.position, pop.position) True(Is dominated pop.position) Else if dominates(pop.position, New.pop.position) True(Is dominated New.pop.position) Pop[i].Position = max(particle[i].Position,Rmin); pop[i].Position = min(particle[i].Position,Rmax); if particle(i).Cost<particle(i).Best.Cost pop(i).Best.Position=particle(i).Position; pop(i).Best.Cost=particle(i).Cost; if pop(i).Best.Cost<BestSol.Cost BestSol=pop(i).Best

Algorithm for Neural Network Neural network //Load data [Train Inputs Target Inputs]= Load data VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 237

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//Define network structure Def Train function type :Train func Hidden layers //network type Net=fitnet (hidden layer, size, Train func) Fitnet (hidden layer ,size,Train func) nnetparamsInfo.hiddensize nnetparamsInfo.hidden layers nnetparamsInfo.nntype nnetparamsInfo.Train func net.input.process func : {mapinput vectors} net.output processfunc: {mapoutput vectors} net.divide params.train ratio=0.7 net.divide params.validate ratio=0.15 net.divide params.testratio=0.15 //performance function Net.perform fucn=â&#x20AC;&#x2122;MSEâ&#x20AC;&#x2122; [net ,train]=train(net, x,t) //test network Y=net(x) E=[t]-[y] Performance =perform(net ,t, y) EANNHRC(x) VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 238

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//Input layer 1 X1_step1_xoffset=[Input] X1_step1_gain=[‘e’] X1_step1_ymin=-1 //Layer 1 B1=[bais for each neuron] IW1=[weights for each neouron] //Layer2 B2=[bais for each neuron] IW2=[weights for each neuron] //output layer Y1_step1_ymin=-1 Y1_step1_gain=[] Y1_step1_xoffset=[outputs] //simulation For i:[1-size(output)] //Input1 Xp1=mapminmax_apply([x1,i],x1_step1_gain, x1_step1_xoffset,x1_step1_ymin) //Layer 1 A1=tansig_apply((b1,1,q)+IW1*xp1) //layer 2 A2=[b1,1,q]+lw1*a1 //output 2 VISHWAKARMA INSTITUTE OF INFORMATION TECHNOLOGY, PUNE M.E. (Mechanical) (Design Engineering) 239

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Y(1,t,s)=mapminmax(a2,y1_step1_gain,y1_step1_xoffset,y1_step1_ymin) Mapminmax_apply(x,settings_gain,settings_xoffset,setting_ymin) Y=bsxfun(@minus,x,setting_x_offset) Y=bsxfun(@times,y,setting_gain) Y=bsx(@plus,y,setting_ymin) //sigmoid symmetric transfer function Tansig_apply(x) //mapminimumand maximum output Mapminmax_reverse(y,setting_gain,setting_xoffset,setting_ymin) X=bsxfun(@minus,y,setting_x_offset) x=bsxfun(@times,x,setting_gain) x=bsx(@plus,x,setting_ymin)

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