prepared by s. mustafa ali zaidi, 1032109...

LASER CUTTING OPTIMIZATION OF NON-METALLIC

MATERIALS OVERALL QUALITY

Prepared by

S. Mustafa Ali Zaidi, 1032109

Supervised by

Dr. Imran Amin (Head of Computer Science)

A Dissertation

Submitted to the

Faculty of Computing

of the

Shaheed Zulfikar Ali Bhutto Institute of Science and Technology

In partial fulfillment of the requirements for the degree of PhD

July 2nd , 2015

ii

CERTIFICATE OF APPROVAL

This is to certify that Syed Mustafa Ali Zaidi bearing registration no. 1032109 has

completed his dissertation, entitled LASER CUTTING OPTIMIZATION OF NON-

METALLIC MATERIALS OVERALL QUALITY in partial fulfillment of the

requirements for the degree of Doctor in Philosophy (Ph.D.) in the field of Computing,

under the supervision of Dr. Imran Amin (Associate Professor & Head of Computing).

The Dissertation meets the prescribed requirements and standard as set by Computer Science

Department, Shaheed Zulfiqar Ali Bhutto Institute of Science and Technology (SZABIST),

Karachi, Pakistan.

___________________________ ___________________________

Dr. Imran Amin

Associate Professor & Head of Computing

Supervisor

Dr. Husnain Mansoor

MS/PhD-Program Manager

Computing Department

___________________________

Dr. Mohammad Altaf Mukati

Dean (Computing & Engineering Sciences)

DECLARATION

I certify that this is my own research work. The work has not, in whole or in part, been

presented elsewhere for assessment. Where material has been used from other sources, it has

been properly acknowledged. If this statement is untrue and if I am found guilty of the

plagiarism, the punitive actions against me may be taken as per the SZABIST Anti-

Plagiarism Policy.

Signatures:_____________ Date:______________

Name of Student: Syed Mustafa Ali Zaidi

Registration No: 1032109

Degree: Ph.D

Program : Computing

DEDICATED

This Dissertation is dedicated to my Parents, wife and children

vi

ABSTRACT

As the global stock of natural resources depletes the need of electricity efficient processes

emerges. Laser cutting, an advance non-contact processing technique, outweighs the old

methods such as hotwire and milling due to the requirement of retightening and replacement

of cutting tools with time. Orthogonal array and Factorial design are selected as a design of

experiment for modelling and optimization of Laser cutting process. The range adjustment of

laser machine requires knowledge of experimental design, laser cutting process and material

properties, otherwise missing values generate due to unsuccessful cutting. For this reason,

many universities are unable to utilize these machines effectively. It is essential to formulate

a technique which allows modelling the data with some missing values, consequently, it

enhance the utilization of laser machines for research and other purposes. Initially, the

qualities of output characteristic were modelled by Statistical and Neural network without

missing values and then by supervised and novel Semi-supervised learning algorithms with

missing values. The Statistical modelling results using one and two way analysis of variance

with replication were better than other data mining techniques like linear and nonlinear

regression, however, it is difficult to use these methods with missing values. Therefore,

supervised neural network modelling is carried out and the effects of its parametric change

are observed along the datasets size to model the orthogonal array. The neural network

modelling results in edge quality and kerf width signal to noise ratio, it is acceptable, the edge

quality indicates that modelling improves by pre-normalization, further improvement was

made by increasing training data size to factorial design. It is observed that for the artificial

neural network, supervised learning is not sufficient associated to orthogonal array, only due

to edge quality mean modelling, average error were higher than the acceptable limit. The

average error with factorial design was under 10%. The vast modelling experience of

supervised learning engenders the development of novel Semi-Supervised learning algorithm.

Consequently, the average error was reduced by utilizing the systematic randomize

techniques to initialize the neural network weights and increase the number of initialization

by using orthogonal array design of experiment, with up to 22% missing values. This

algorithm reduces modelling time and cost thus reduces electricity consumption. The average

error in Perspex sheet did not exceed 8.0% and 11.5% for edge quality and kerf width

respectively.

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The overall quality was calculated by aggregation technique of data mining and a more

generous and better aggregation is carried out by the novel combination of Fuzzy logic which

provides overall quality for the customer while saving cost, time and Electricity.

viii

ACKNOWLEDGEMENT

Thanks to the ALLAH, the Almighty, for His countless blessings and bringing me to the

position where I am today. During the progress of my Doctor of Philosophy (Ph.D.) study at

Shaheed Zulfikar Ali Bhutto Institute of Science & Technology (SZABIST), Karachi,

Pakistan, I came across many persons and institutional entities that collaborated directly and

indirectly with my research work. Without having their support, it would have been

impossible for me to finish my Ph.D. work. Here I wish to acknowledge and recognize their

massive support.

I sincerely begin with acknowledging the huge support and guidance of my advisor, Dr.

Imran Amin who gave me the opportunity to conduct research under his worthy supervision.

I received motivation; encouragement and support from him during the entire period of my

PhD study. Under his supervision and guidance, I have learned writing papers for

conferences, journals and sharing my ideas to the public. I am also thankful for the

motivation, inspiration and support received from the faculty of SZABIST (Dr Husnain

Mansoor, Dr. Bushra Saeed, Dr. Faraz Junejo, Dr Faisal Ahmed Bukhari, Mr. Asim Riaz, Mr.

Asif Qazi, Wajeeh ul Hasan), University of Malaya (Professor Nukman Bin Yousuf,

Professor Imtiaz Ahmed and Mr. Tan See Bon), UTP (Dr Fawad Hasan Junejo), NED (Dr.

Najimi Ghani Haider) and Karachi University (Dr. Ejaz Ahmed).

The Scholarship from Higher Education Commission and research funding from SZABIST

Karachi and University of Malaya for the development of this MS/PhD research are also

sincerely acknowledged. In the end, I would like to thank my family Father: Mazahir Hussain,

Mother: Kaniz Fatima (Late), wife: Aelya and children: Baqar, Shaheer and Manahil for their

unconditional support, inspiration and love.

LIST OF PUBLICATION

JOURNAL PUBLICATIONS 1. M. Zaidi, et al., “Error Assessment of Laser cutting Predictions by Semi-Supervised

Learning,” Journal of Central South University, vol. 21, pp. 3736-3745, October

2014.

2. M. Zaidi et al., “Estimation of ANN Modelling of Laser Cutting with Missing Values.”

Pensee Journal, 75(11), 159-171, 2013.

3. M. Zaidi et al., “Experimental Data Mining Techniques (Using Multiple Statistical

Methods),” IJCSI International Journal of Computer Science Issues, Elsevier, vol. 9,

2012.

4. M. Zaidi, et a.l “Evaluation of Training Algorithms for the Laser Cutting Process,”

Journal of Independent Studies and Research-Computing (JISR), vol. 9, p. 1-13, 2011.

5. M. Zaidi, "Hybrid combination of Crisp and Fuzzy aggregation for Overall quality of

Laser Cutting Process, Submitted ISI indexed Journal for review.

ORAL PRESENTATIONS

International Conferences 1. M. Zaidi et al., “Laser Cutting problem modelling Using Statistical Tools,” Paper

presented in APIEMS: Asia Pacific Industrial Engineering and Management Systems

Conference, Melaka, Malaysia, 2010.

2. M. Zaidi, et al., “Laser Cutting Quality Control of Melamine Using Artificial Neural

Networks,” Paper presented in APIEMS: Asia Pacific Industrial Engineering and

Management Systems Conference, Melaka, Malaysia, 2010.

National Conferences 1. M. Zaidi et al. “Comparative Study of Customer Quality Function, Genetic Algorithm

and Fuzzy Logic in Laser Cutting Process,” Paper presented at the SZABIST 16th

National research conference (NRC) Karachi, Pakistan, 2010.

TABLE OF CONTENTS ABSTRACT .......................................................................................................................................................... VI

ACKNOWLEDGEMENT ...................................................................................................................................VIII

LIST OF PUBLICATION ..................................................................................................................................... IX

TABLE OF CONTENTS ........................................................................................................................................ X

LIST OF TABLES .............................................................................................................................................. XIV

LIST OF FIGURES ............................................................................................................................................. XX

LIST OF SYMBOLS AND ABBREVIATION ................................................................................................ XXIII

LIST OF APPENDICES .................................................................................................................................. XXVII

1. INTRODUCTION .................................................................................................................................... 1

1.2 PROBLEM STATEMENT ............................................................................................................................. 5

1.3 PURPOSE OF STUDY ................................................................................................................................... 6

1.4 CONTRIBUTION OF STUDY ....................................................................................................................... 7

1.5 OUTLINE ......................................................................................................................................................... 7

2. LASER CUTTING .................................................................................................................................. 10

2.2 LASER CUTTING TECHNIQUES ................................................................................................................ 11

2.2.1 VAPORIZATION CUTTINNG ................................................................................................................. 11

2.2.2 FUSION CUTTING ............................................................................................................................... 11

2.2.3 CHEMICAL DEGRADATION CUTTING .................................................................................................. 12

2.3 LOW POWER CO2 LASER............................................................................................................................ 12

3. DESIGN OF EXPERIMENT ..................................................................................................................... 21

3.1 GENICHI TAGUCHI’S METHODOLOGY .................................................................................................. 22

3.2 GENICHI TAGUCHI’S METHOD APPLICATIONS ................................................................................... 23

3.3 RESPONSE SURFACE METHODOLOGY .................................................................................................. 26

3.4 FACTORIAL DESIGN ................................................................................................................................... 27

3.5 SUMMARY .................................................................................................................................................... 31

4. NEURAL NETWORK MODELLING ......................................................................................................... 33

4.1 ARTIFICIAL NEURAL NETWORK ............................................................................................................. 33

4.2 FEED-FORWARD BACK-PROPAGATION NEURAL NETWORK .......................................................... 37

4.3 COMPARISON OF TRAINING ALGORITHMS.......................................................................................... 39

4.4 BACK PROPAGATION ALGORITHM ........................................................................................................ 40

4.4.1 GRADIENT DESCENT ALGORITHM ...................................................................................................... 41

4.4.2 GRADIENT DESCENT WITH MOMENTUM .......................................................................................... 42

4.5 FASTER TRAINING ALGORITHM ............................................................................................................. 43

4.5.2 Quasi Newton Algorithm.................................................................................................................... 44

4.6 LEVENBERG-MARQUARDT ALGORITHM ............................................................................................. 44

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4.7 OVERTRAINING OF DATA MODEL .......................................................................................................... 46

4.8 SUMMARY .................................................................................................................................................... 47

5. OVERALL QUALITY OPTIMIZATION ..................................................................................................... 49

5.1 GENETIC ALGORITHM ............................................................................................................................... 50

5.2 MULTI-OBJECTIVE STATISTICAL METHODS ....................................................................................... 51

5.3 FUZZY AGGREGATION .............................................................................................................................. 54

5.4 SUMMARY .................................................................................................................................................... 55

6. PROPOSED MODULAR RESEARCH METHODOLOGY ............................................................................. 59

6.1 PREPROCESSING OF PROCESS MODULE ............................................................................................... 61

6.2 EXPERIMENTAL DESIGN MODULE ......................................................................................................... 62

6.2.1 SINGLE PARAMETER CHANGE AT A TIME........................................................................................... 62

6.2.2 TAGUCHI METHOD ............................................................................................................................. 63

6.2.3 FRACTIONAL FACTORIAL DESIGN ....................................................................................................... 63

6.2.4 FACTORIAL DESIGN ............................................................................................................................ 63

6.2.5 RESPONSE SURFACE METHODOLOGY (RSM) ..................................................................................... 63

6.3 MODELING AND OPTIMIZATION MODULE ........................................................................................... 64

6.3.1 ANALYSIS OF VARIANCE ..................................................................................................................... 65

6.3.2 REGRESSION ANALYSIS ....................................................................................................................... 67

6.3.3 TAGUCHI, FRACTIONAL FACTORIAL AND FACTORIAL DESIGN ........................................................... 68

6.3.4 RESPONSE SURFACE METHODOLOGY (RSM) ..................................................................................... 69

6.3.5 ARTIFICIAL NEURAL NETWORK BY SUPERVISED LEARNING ............................................................... 70

6.3.6 NEURAL NETWORK BY SEMI-SUPERVISED LEARNING ........................................................................ 72

6.4 MULTI QUALITY OPTIMIZATION MODULE .......................................................................................... 75

6.4.1 SIMPLE AGGREGATION AND CUSTOMER QUALITY FUNCTION .......................................................... 77

6.4.2 GENETIC ALGORITHM ........................................................................................................................ 77

6.4.3 FUZZY AGGREGATION ........................................................................................................................ 78

6.5 SUMMARY .................................................................................................................................................... 80

7. EXPERIMENTAL DESIGN MODULE & SETUP ......................................................................................... 83

7.1 INTRODUCTION .......................................................................................................................................... 83

7.2 SCOPE AND LIMITATION .......................................................................................................................... 87

7.3 PLAN FOR EXPERIMENT DESIGN MODULE .......................................................................................... 88

7.3.1 PROCEDURE .............................................................................................................................................. 88

7.3.2 ORTHOGONAL ARRAYS AND FACTORIAL DESIGN ......................................................................... 90

7.4 EXPERIMENTAL SETUP ............................................................................................................................. 92

7.4.1 LASER MACHINE 500 WATTS ............................................................................................................. 92

7.4.2 MEASUREMENT TOOLS OF EXPERIMENT 1 and 2 .............................................................................. 94

7.4.3 PROPERTIES OF POLYSTYRENE FOAM ................................................................................................ 96

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7.4.4 PROPERTIES OF PERSPEX MATERIAL .................................................................................................. 96

7.5 DATA COLLECTION ................................................................................................................................... 97

7.5.1 POLYSTYRENE FOAM and PERSPEX SHEET ......................................................................................... 97

7.6 VERIFICATION OF SIMULATED DATA ................................................................................................... 99

7.7 SUMMARY .................................................................................................................................................... 99

8. DISCUSSION OF RESULTS AND ANALYSIS .......................................................................................... 102

8.1 PREPROCESSING OF EDGE QUALITY AND KERF-WIDTH DATA .................................................... 102

8.2 ANALYSIS OF VARIANCE ....................................................................................................................... 105

8.2.1 ONE WAY ANOVA WITHOUT REPLICATION...................................................................................... 106

8.2.2 ONE WAY ANOVA WITH REPLICATION ............................................................................................. 108

8.2.3 TWO WAY ANOVA WITH REPLICATION ............................................................................................ 109

8.3 REGRESSION ANALYSIS ......................................................................................................................... 112

8.3.1 LINEAR REGRESSION ANALYSIS ........................................................................................................ 112

8.3.2 MULTIPLE LINEAR REGRESSION ....................................................................................................... 121

8.3.3 NONLINEAR REGRESSION ANALYSIS ................................................................................................ 124

8.3.4 MULTIPLE NON-LINEAR REGRESSION .............................................................................................. 128

8.4 SUPERVISED LEARNING WITH MISSING VALUE ............................................................................... 129

8.4.1 EDGE QUALITY OF PERSPEX SHEET ................................................................................................. 131

8.4.2 KERF WIDTH QUALITY OF PERSPEX SHEET ....................................................................................... 138

8.5 SEMI-SUPERVISED ALGORITHM ........................................................................................................... 141

8.5.1 EDGE QUALITY OF PERSPEX SHEET .................................................................................................. 145

8.5.2 KW QUALITY OF PERSPEX SHEET ...................................................................................................... 146

8.6 FUZZY AGGREGATION ............................................................................................................................ 149

8.7 SUMMARY .................................................................................................................................................. 160

9. CONCLUSION .................................................................................................................................... 170

10. FUTURE DIRECTION .......................................................................................................................... 175

REFERENCE ..................................................................................................................................................... 178

A. DOE & STATISTICAL MODELLIN ......................................................................................................... 185

A.1 ONE WAY ANOVA WITHOUT REPLICATION ..................................................................................... 206

A.2 ONE WAY ANOVA WITH REPLICATION .............................................................................................. 207

A.3 TWO WAY ANOVA WITH REPLICATION ............................................................................................. 210

A.4 LINEAR REGRESSION ANALYSIS ........................................................................................................ 217

A.4.1 LASER POWER AND KERF WIDTH ..................................................................................................... 217

A.4.2 CUTTING SPEED AND KERF WIDTH .................................................................................................. 219

A.4.3 ASSIST GAS PRESSURE AND KERF-WIDTH ........................................................................................ 221

A.4.4 STANDOFF DISTANCE AND KERF WIDTH......................................................................................... 223

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A.5 NONLINEAR REGRESSION ANALYSIS ................................................................................................ 225

A.6 MULTIPLE NON-LINEAR REGRESSION ............................................................................................... 236

B. NEURAL NETWORK & OVERALL QUALITY .......................................................................................... 238

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LIST OF TABLES

TABLE 5-1: DIFFERENCE BETWEEN DMT AND TAGUCHI METHODS .................................................................................... 54

TABLE 7-1: FOUR VARIABLES WITH THREE LEVELS ORTHOGONAL DESIGN MATRIX ................................................................. 91

TABLE 7-2: CONTROLLABLE INPUT AND FACTORS LEVELS ................................................................................................. 91

TABLE 7-3: ZL1010 SPECIFICATION ............................................................................................................................ 93

TABLE 7-4: ZLX5 SPECIFICATION ................................................................................................................................ 93

TABLE 7-5: PROPERTIES OF POLYSTYRENE FOAM ........................................................................................................... 96

TABLE 7-6: PROPERTIES OF PERSPEX SHEET (CAST ACRYLIC) ............................................................................................ 97

TABLE 8-1: MEASUREMENT OF EDGE QUALITY OF POLYSTYRENE FOAM ........................................................................... 103

TABLE 8-2: MEASUREMENT OF KERF WIDTH OF POLYSTYRENE FOAM .............................................................................. 104

TABLE 8-3: OBSERVATIONS CONSIDER LASER POWER (A) .............................................................................................. 106

TABLE 8-4: SUMMARY OF DESCRIPTIVE STATISTICS ....................................................................................................... 106

TABLE 8-5: ANOVA FOR LASER POWER .................................................................................................................... 107

TABLE 8-6: ONE WAY ANOVA WITHOUT REPLICATION ................................................................................................. 108

TABLE 8-7: ONE WAY ANOVA WITH REPLICATION ...................................................................................................... 108

TABLE 8-8: INTERACTION BETWEEN LASER POWER AND CUTTING SPEED WITH REPLICATION ................................................ 109

TABLE 8-9: INTERACTION BETWEEN LASER POWER AND CUTTING SPEED 0.2 .................................................................... 109

ABLE 8-10: INTERACTION BETWEEN LASER POWER AND CUTTING SPEED 0.7 .................................................................... 110

TABLE 8-11: INTERACTION BETWEEN LASER POWER AND CUTTING SPEED 1.2 .................................................................. 110

TABLE 8-12: TOTAL INTERACTION BETWEEN LASER POWER AND CUTTING SPEED ............................................................... 110

TABLE 8-13: ANOVA OF INTERACTION BETWEEN LASER POWER AND CUTTING SPEED ....................................................... 111

TABLE 8-14: SUMMARY OF TWO ANOVA WITH REPLICATION ....................................................................................... 111

TABLE 8-15: DESCRIPTIVE STATISTICS OF LASER POWER AND KERF WIDTH ........................................................................ 113

TABLE 8-16: CORRELATION BETWEEN LASER POWER AND KERF WIDTH ............................................................................ 113

TABLE 8-17: REGRESSION STATISTICS ........................................................................................................................ 114

TABLE 8-18: REGRESSION BETWEEN LASER POWER AND KERF WIDTH ANOVA ................................................................. 115

TABLE 8-19: LINEAR REGRESSION LINE OF LASER POWER .............................................................................................. 116

TABLE 8-20: RESIDUAL OUTPUT ............................................................................................................................... 117

TABLE 8-21: LINEAR REGRESSION ANOVA ................................................................................................................ 121

TABLE 8-22: DESCRIPTIVE STATISTICS ........................................................................................................................ 122

TABLE 8-23: CORRELATIONS ................................................................................................................................... 122


TABLE 8-25: MULTIPLE LINEAR REGRESSION ANOVA .................................................................................................. 123

TABLE 8-26: LINEAR REGRESSION OF MULTIVARIABLE ................................................................................................... 123

TABLE 8-27: SUMMARY OF LINEAR REGRESSION .......................................................................................................... 123

TABLE 8-28: RESIDUAL OUTPUT............................................................................................................................... 124

TABLE 8-29: REGRESSION DATA WITHOUT REPLICATION FOR LASER POWER ...................................................................... 125

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TABLE 8-31: NONLINEAR REGRESSION ANOVA FOR LASER POWER AND KW WITHOUT REPLICATION ................................... 125

TABLE 8-32: NONLINEAR REGRESSION OF LASER POWER .............................................................................................. 126

TABLE 8-33: NON-LINEAR REGRESSION ANOVA ........................................................................................................ 126

TABLE 8-34: SUMMARY OF NON- LINEAR REGRESSION ................................................................................................. 127


TABLE 8-36: MULTIPLE NON-LINEAR REGRESSION ANOVA .......................................................................................... 128

TABLE 8-37: NONLINEAR REGRESSION OF MULTIVARIABLE ............................................................................................ 128

TABLE 8-38: RESIDUAL OUTPUT............................................................................................................................... 129

TABLE 8-39: PRELIMINARY TRAINING BY LEVENBERG MARQUARDT ................................................................................. 131

TABLE 8-40: ENRICHMENT TRAINING BY LEVENBERG MARQUARDT ................................................................................. 133

TABLE 8-41: EQ MEAN AND S/N RATIO WITH FACTORIAL DESIGN ................................................................................. 134

TABLE 8-42: AVERAGE PERCENT ERRORS .................................................................................................................... 148

TABLE 8-43: FOUR QUALITIES FACTORIAL DESIGN DATA ................................................................................................ 149

TABLE 8-44: QUALITY QUANTIFICATION ..................................................................................................................... 150

TABLE 8-45: COMPARE RESULTS OF Q AV, CQF, FL AV AND QFL AV ............................................................................... 151

TABLE 8-46: SORTED WITH CQF, QUANTIFIED FUZZY AGGREGATION AND QUANTIFIED NORMALIZED AGGREGATION ................. 155

TABLE 8-47: ENERGY CONSUMPTION QUALITY CALCULATION FOR FACTORIAL DESIGN .......................................................... 158

TABLE 8-48: POWER LEVELS CORRESPONDING TO LASER POWER AND CUTTING SPEED ......................................................... 159

TABLE 8-49: INCLUDING POWER CONSUMED AND THEN SORTED WITH CQF, Q AV AND QFL AV ........................................... 159

TABLE 8-50: SUMMARY OF LINEAR REGRESSION ANALYSIS ............................................................................................. 162

TABLE 8-51: SUMMARY OF NONLINEAR REGRESSION ................................................................................................... 162

TABLE A-1: FOUR VARIABLES WITH THREE LEVELS FACTORIAL DESIGN MATRIX ................................................................... 185

TABLE A-2: EDGE QUALITY MEAN AND S/N RATIO OF POLYSTYRENE FOAM SHEET (13MM) ................................................. 187

TABLE A-3: KERF WIDTH OBSERVATIONS OF POLYSTYRENE FOAM SHEET (13MM) .............................................................. 188

TABLE A-4: EDGE QUALITY OBSERVATIONS OF PERSPEX SHEET (3MM) ............................................................................. 188

TABLE A-5: KERF WIDTH OBSERVATIONS FOR OF PERSPEX SHEET (3MM) .......................................................................... 191

TABLE A-6: KERF WIDTH MEAN AND SIGNAL TO NOISE RATIO OF PERSPEX SHEET (3MM) ..................................................... 194

TABLE A-7: EDGE QUALITY OBSERVATIONS OF PERSPEX SHEET (5MM) ............................................................................. 197

TABLE A-8: OBSERVATION FOR KERF WIDTH OF PERSPEX SHEET OF 5MM ......................................................................... 200

TABLE A-9: KERF WIDTH MEAN AND SIGNAL TO NOISE RATIO OF PERSPEX SHEET OF 5MM ................................................... 203

TABLE A-10: OBSERVATIONS CONSIDER LASER POWER (A) ........................................................................................... 206

TABLE A-11: SUMMARY OF DESCRIPTIVE STATISTICS .................................................................................................... 206

TABLE A-12: ANOVA FOR LASER POWER .................................................................................................................. 206

TABLE A-13: OBSERVATIONS CONSIDER CUTTING SPEED .............................................................................................. 206


TABLE A-15: ANOVA FOR CUTTING SPEED ............................................................................................................... 206

TABLE A-16: OBSERVATIONS CONSIDER ASSIST GAS PRESSURE ....................................................................................... 207

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TABLE A-18: ANOVA FOR ASSIST GAS PRESSURE ....................................................................................................... 207

TABLE A-19: OBSERVATIONS CONSIDER STANDOFF DISTANCE ........................................................................................ 207


TABLE A-21: ANOVA FOR STANDOFF DISTANCE ........................................................................................................ 207

TABLE A-22: OBSERVATIONS CONSIDER LASER POWER WITH REPLICATION ....................................................................... 207


TABLE A-24: ONE WAY ANOVA WITH REPLICATION FOR LASER POWER .......................................................................... 208

TABLE A-25: OBSERVATIONS CONSIDER CUTTING SPEED WITH REPLICATION ..................................................................... 208


TABLE A-27: ONE WAY ANOVA WITH REPLICATION FOR CUTTING SPEED ........................................................................ 208

TABLE A-28: OBSERVATIONS CONSIDER ASSIST GAS PRESSURE WITH REPLICATION .............................................................. 208


TABLE A-30: ONE WAY ANOVA WITH REPLICATION FOR ASSIST GAS PRESSURE ............................................................... 209

TABLE A-31: OBSERVATIONS CONSIDER STANDOFF DISTANCE WITH REPLICATION .............................................................. 209


TABLE A-33: ONE WAY ANOVA WITH REPLICATION FOR STANDOFF DISTANCE ................................................................. 209

TABLE A-34: ONE WAY ANOVA WITH REPLICATION .................................................................................................... 209

TABLE A-35: INTERACTION BETWEEN LASER POWER AND CUTTING SPEED WITH REPLICATION .............................................. 210

TABLE A-36: INTERACTION BETWEEN LASER POWER AND CUTTING SPEED 0.2 .................................................................. 210



TABLE A-39: TOTAL INTERACTION BETWEEN LASER POWER AND CUTTING SPEED .............................................................. 210

TABLE A-40: ANOVA OF INTERACTION BETWEEN LASER POWER AND CUTTING SPEED ....................................................... 211

TABLE A-41: INTERACTION BETWEEN LASER POWER AND ASSIST GAS PRESSURE WITH REPLICATION ..................................... 211

TABLE A-42: INTERACTION BETWEEN LASER POWER AND ASSIST GAS PRESSURE 0.5 ......................................................... 211



TABLE A-45: TOTAL INTERACTION BETWEEN LASER POWER AND ASSIST GAS PRESSURE ...................................................... 212

TABLE A-46: ANOVA OF INTERACTION BETWEEN LASER POWER AND ASSIST GAS PRESSURE ............................................. 212

TABLE A-47: INTERACTION BETWEEN LASER POWER AND STANDOFF DISTANCE WITH REPLICATION ....................................... 212

TABLE A-48: INTERACTION BETWEEN LASER POWER AND STANDOFF DISTANCE 1 .............................................................. 212

TABLE A-49: INTERACTION BETWEEN LASER POWER AND STANDOFF DISTANCE 5 .............................................................. 212

TABLE A-50: INTERACTION BETWEEN LASER POWER AND STANDOFF DISTANCE 10 ............................................................ 213

TABLE A-51: TOTAL INTERACTION BETWEEN LASER POWER AND STANDOFF DISTANCE ....................................................... 213

TABLE A-52: ANOVA OF INTERACTION BETWEEN LASER POWER AND STANDOFF DISTANCE ................................................ 213

TABLE A-53: INTERACTION BETWEEN CUTTING SPEED AND ASSIST GAS PRESSURE WITH REPLICATION .................................... 213

TABLE A-54: INTERACTION BETWEEN CUTTING SPEED AND ASSIST GAS PRESSURE 0.5 ....................................................... 213

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TABLE A-57: TOTAL INTERACTION BETWEEN CUTTING SPEED AND ASSIST GAS PRESSURE.................................................... 214

TABLE A-58: ANOVA OF INTERACTION BETWEEN CUTTING SPEED AND ASSIST GAS PRESSURE ........................................... 214

TABLE A-59: INTERACTION BETWEEN CUTTING SPEED AND STANDOFF DISTANCE WITH REPLICATION ..................................... 214

TABLE A-60: INTERACTION BETWEEN CUTTING SPEED AND STANDOFF DISTANCE 1 ............................................................ 215

TABLE A-61: INTERACTION BETWEEN CUTTING SPEED AND STANDOFF DISTANCE 5 ............................................................ 215

TABLE A-62: INTERACTION BETWEEN LASER CUTTING SPEED AND STANDOFF DISTANCE 10 ................................................. 215

TABLE A-63: TOTAL INTERACTION BETWEEN CUTTING SPEED AND STANDOFF DISTANCE ..................................................... 215

TABLE A-64: ANOVA OF INTERACTION BETWEEN CUTTING SPEED AND STANDOFF DISTANCE............................................. 215

TABLE A-65: INTERACTION BETWEEN ASSIST GAS PRESSURE AND STANDOFF DISTANCE WITH REPLICATION ............................ 216

TABLE A-66: INTERACTION BETWEEN ASSIST GAS PRESSURE AND STANDOFF DISTANCE 1 ................................................... 216

TABLE A-67: INTERACTION BETWEEN ASSIST GAS PRESSURE AND ASSIST GAS PRESSURE 5 .................................................. 216

TABLE A-68: INTERACTION BETWEEN ASSIST GAS PRESSURE AND STANDOFF DISTANCE 10 ................................................. 216

TABLE A-69: TOTAL INTERACTION BETWEEN ASSIST GAS PRESSURE AND STANDOFF DISTANCE............................................. 216

TABLE A-70: ANOVA OF INTERACTION BETWEEN ASSIST GAS PRESSURE AND STANDOFF DISTANCE .................................... 217

TABLE A-71: DESCRIPTIVE STATISTICS ....................................................................................................................... 217

TABLE A-72: CORRELATION ..................................................................................................................................... 217

TABLE A-73: REGRESSION STATISTICS ....................................................................................................................... 218

TABLE A-74: REGRESSION BETWEEN LASER POWER AND KERF WIDTH ANOVA ................................................................. 218

TABLE A-75: LINEAR REGRESSION LINE OF LASER POWER .............................................................................................. 218

TABLE A-76: RESIDUAL OUTPUT ............................................................................................................................... 218




TABLE A-80: REGRESSION BETWEEN CUTTING SPEED AND KERF WIDTH ANOVA ............................................................... 220

TABLE A-81: LINEAR REGRESSION LINE OF CUTTING SPEED ............................................................................................. 220





TABLE A-86: REGRESSION BETWEEN ASSIST GAS PRESSURE AND KERF WIDTH ANOVA ..................................................... 222

TABLE A-87: LINEAR REGRESSION LINE OF ASSIST GAS PRESSURE.................................................................................... 222





TABLE A-92: REGRESSION BETWEEN STANDOFF DISTANCE AND KERF WIDTH ANOVA ....................................................... 224

xviii

TABLE A-93: LINEAR REGRESSION LINE OF STANDOFF DISTANCE ..................................................................................... 224


TABLE A-95: REGRESSION DATA WITHOUT REPLICATION FOR LASER POWER...................................................................... 225


TABLE A-97: NON-LINEAR REGRESSION ANOVA FOR LASER POWER AND KERF WIDTH WITHOUT REPLICATION ....................... 225

TABLE A-98: NONLINEAR REGRESSION OF LASER POWER .............................................................................................. 226


TABLE A-100: REGRESSION DATA WITH REPLICATION FOR LASER POWER ......................................................................... 226

TABLE A-101: REGRESSION STATISTICS ..................................................................................................................... 227

TABLE A-102: NON-LINEAR REGRESSION ANOVA FOR LASER POWER AND KERF WIDTH WITH REPLICATION........................... 227

TABLE A-103: NONLINEAR REGRESSION OF LASER POWER ............................................................................................ 227

TABLE A-104: RESIDUAL OUTPUT ............................................................................................................................ 228

TABLE A-105: REGRESSION DATA WITH REPLICATION FOR CUTTING SPEED ....................................................................... 228


TABLE A-107: NON-LINEAR REGRESSION ANOVA FOR CUTTING SPEED AND KERF WIDTH WITH REPLICATION ........................ 230

TABLE A-108: NONLINEAR REGRESSION OF CUTTING SPEED .......................................................................................... 230


TABLE A-110: REGRESSION DATA WITH REPLICATION FOR ASSIST GAS PRESSURE ............................................................... 231


TABLE A-112: NON-LINEAR REGRESSION ANOVA FOR ASSIST GAS PRESSURE AND KERF WIDTH WITH REPLICATION ................ 232

TABLE A-113: NONLINEAR REGRESSION OF ASSIST GAS PRESSURE .................................................................................. 232


TABLE A-115: REGRESSION DATA WITH REPLICATION FOR STANDOFF DISTANCE ................................................................ 233


TABLE A-117: NON-LINEAR REGRESSION ANOVA FOR STANDOFF DISTANCE AND KERF WIDTH WITH REPLICATION ................. 235

TABLE A-118: NONLINEAR REGRESSION OF STANDOFF DISTANCE ................................................................................... 235


TABLE A-120: MULTIPLE NON-LINEAR REGRESSION DATA WITH REPLICATION FOR FOUR INPUTS........................................... 236


TABLE A-122: MULTIPLE NON-LINEAR REGRESSION DATA WITH REPLICATION ANOVA ...................................................... 236

TABLE A-123: NONLINEAR REGRESSION OF MULTIVARIABLE .......................................................................................... 237


TABLE B-1: TRAINING ON FACTORIAL DESIGN .............................................................................................................. 238

TABLE B-2: EDGE QUALITY MEAN TRAINING USING FACTORIAL DATASETS .......................................................................... 239

TABLE B-3: EDGE QUALITY SIGNAL TO NOISE RATIO OF FACTORIAL DATASETS ..................................................................... 240

TABLE B-4: EDGE QUALITY MEAN OF NORMALIZED DATASET .......................................................................................... 240

TABLE B-5: KERF WIDTH MEAN TRAINING OF FACTORIAL DESIGN ..................................................................................... 241

TABLE B-6: SIMULATED DATA OF FACTORIAL DESIGN .................................................................................................... 246

xix

TABLE B-7: COMPARE RESULTS OF NORMALIZED AGGREGATION, CUSTOMER QUALITY FUNCTION AND FUZZY AGGREGATION ...... 249

TABLE B-8: SORTED WITH CQF, QUANTIFIED FUZZY AGGREGATION AND QUANTIFIED NORMALIZED AGGREGATION.................. 252

TABLE B-9: ENERGY CONSUMPTION QUALITY CALCULATION FOR FACTORIAL DESIGN ........................................................... 255

xx

LIST OF FIGURES

FIGURE 1-1: BLOCK DIAGRAM OF INPUT AND OUTPUT PARAMETERS ................................................................................... 2

FIGURE 1-2: PROBLEM STATEMENT SCHEMATIC VIEW ...................................................................................................... 5

FIGURE 2-1: VAPORIZATION CUTTING PROCESS ............................................................................................................. 11

FIGURE 2-2: FUSION CUTTING SCHEMATIC DIAGRAM BY C02 LASER ................................................................................... 12

FIGURE 2-3: LASER CUTTING PROCESS ......................................................................................................................... 13

FIGURE 2-4: CO2 LASER CUTTING SHEET IN SQUARE PROFILE ............................................................................................ 17

FIGURE 3-1: COMPARISON TAGUCHI’S OA AND FACTORIAL DESIGNS NUMBER OF RUNS ........................................................ 23

FIGURE 3-2: INCREASING TREND OF NUMBER OF RUNS AS FACTORS WITH TWO LEVELS .......................................................... 28

FIGURE 3-3: INCREASING TREND OF NUMBER OF RUNS AS FACTORS WITH THREE LEVEL ......................................................... 28

FIGURE 3-4: DIFFERENCE OF NUMBER OF RUNS IN 2 AND 3 LEVELS ................................................................................... 29

FIGURE 4-1: FEED-FORWARD BACK-PROPAGATION MODEL .............................................................................................. 38

FIGURE 4-2: SEARCHING OF GLOBAL MINIMA IN ERROR SPACE .......................................................................................... 42

FIGURE 6-1: PROPOSED FRAMEWORK MODULES FOR PROCESS IMPROVEMENT ................................................................... 60

FIGURE 6-2: PREPROCESSING OF PROCESS MODULE PART OF THE PROPOSED FRAMEWORK ................................................... 62

FIGURE 6-3: EXPERIMENT DESIGN MODULE OF THE PROPOSED FRAMEWORK ...................................................................... 62

FIGURE 6-4: MODELLING MODULE OF THE PROPOSED FRAMEWORK ................................................................................. 65

FIGURE 6-5: ANALYSIS OF VARIANCE ........................................................................................................................... 66

FIGURE 6-6: REGRESSION ANALYSIS ............................................................................................................................ 67

FIGURE 6-7: MODELLING AND VERIFICATION METHODOLOGY OF SUPERVISED LEARNING ...................................................... 71

FIGURE 6-8: MODELLING AND VERIFICATION METHODOLOGY OF SEMI-SUPERVISED LEARNING .............................................. 74

FIGURE 6-9: MULTI-QUALITY MODULE OF THE PROPOSED FRAMEWORK ............................................................................ 75

FIGURE 6-10: FUZZY AGGREGATION AS A MULTI QUALITY OPTIMIZATION .......................................................................... 79

FIGURE 7-1: GENERAL MODEL OF PROCESS ................................................................................................................... 83


FIGURE 7-3: TRAINING, TESTING AND SIMULATION PROCESS ............................................................................................ 86

FIGURE 7-4: PROCEDURE OF LASER CUTTING OF PERSPEX AND POLYSTYRENE FOAM ............................................................. 89

FIGURE 7-5: PICTURE OF ZECH LASER SYSTEM .............................................................................................................. 92


FIGURE 7-7: PICTURE OF DIGITAL CALIPER .................................................................................................................... 94

FIGURE 7-8: INNER AND OUTER SIDELINE LENGTH .......................................................................................................... 94

FIGURE 7-9: SCHEMATIC DIAGRAM OF LIGHT OPTICAL MICROSCOPE................................................................................... 95

FIGURE 7-10: SCHEMATIC DIAGRAM OF ICAMSCOPE ....................................................................................................... 95

FIGURE 7-11: PERSPEX SHEET EDGE QUALITY MEASURED BY MICROSCOPE .......................................................................... 95

FIGURE 7-13: VIEWS OF DEFECTIVE CUTS ..................................................................................................................... 98

FIGURE 7-12: MAXIMUM DEVIATION BETWEEN CUT EDGES ............................................................................................. 98

FIGURE 8-1: OUTLIER ANALYSIS OF EDGE QUALITY REPLICATION OF POLYSTYRENE FOAM ..................................................... 103

xxi

FIGURE 8-2: OUTLIER ANALYSIS OF KERF WIDTH REPLICATION OF POLYSTYRENE FOAM ........................................................ 104

FIGURE 8-3: INTERACTIVE GRAPH OF LASER POWER AND KERF WIDTH ............................................................................ 113

FIGURE 8-4: LASER POWER (A) LINE FIT PLOT ............................................................................................................. 117

FIGURE 8-5: INTERACTIVE GRAPH OF CUTTING SPEED AND KERF WIDTH .......................................................................... 118

FIGURE 8-6: CUTTING SPEED (B) LINE FIT PLOT ........................................................................................................... 118

FIGURE 8-7: INTERACTIVE GRAPH OF ASSIST GAS PRESSURE AND KERF WIDTH .................................................................. 119

FIGURE 8-8: ASSIST GAS PRESSURE (C) LINE FIT PLOT .................................................................................................. 119

FIGURE 8-9: INTERACTIVE GRAPH OF STANDOFF DISTANCE AND KERF WIDTH ................................................................... 120

FIGURE 8-10: STANDOFF DISTANCE (D) LINE FIT PLOT ................................................................................................. 120

FIGURE 8-11: COMPREHENSIVE INTERACTIVE GRAPH BETWEEN ALL PARAMETERS .............................................................. 121

FIGURE 8-12: QUADRATIC GRAPH OF LASER POWER WITHOUT REPLICATION ..................................................................... 124

FIGURE 8-13: COMPARISON OF EDGE QUALITY MEAN NETWORKS ERRORS ........................................................................ 136

FIGURE 8-14: EDGE QUALITY MEAN COMPARISON IN PERCENT ERROR FOR NORMALIZED DATASETS ....................................... 137

FIGURE 8-15: AVERAGE PERCENT ERRORS COMPARISON FOR EDGE QUALITY SIGNAL TO NOISE RATIO ..................................... 138

FIGURE 8-16: COMPARISON OF KERF WIDTH MEAN FACTORIAL DATASETS ........................................................................ 139

FIGURE 8-17: VARIATIONS IN 100, 500, 1000, 3000 RE-INITIALIZATIONS ..................................................................... 143

FIGURE 8-18: EFFECT OF WEIGHT INITIALIZATIONS ON AVERAGE ERROR............................................................................ 144

FIGURE 8-19: NEURON‘S VARIATIONS IN EQ MEAN ..................................................................................................... 145

FIGURE 8-20: NEURON'S VARIATIONS IN EQ S/N ........................................................................................................ 146

FIGURE 8-21: NEURON‘S VARIATIONS IN KW MEAN .................................................................................................... 147

FIGURE 8-22: NEURON'S VARIATIONS IN KW S/N ....................................................................................................... 147

FIGURE 8-23: VIEW OF ALL AGGREGATED VALUES ........................................................................................................ 152

FIGURE 8-24: COMPARISON OF QUANTIFIED NORMALIZED AGGREGATION WITH CQF ........................................................ 152

FIGURE 8-25: FUZZY INFERENCE SYSTEM FOR POLYSTYRENE SHEET CUTTING PROCESS ......................................................... 153

FIGURE 8-26: APPLICATION OF PARALLEL RULES WITH IMPLICATION ................................................................................ 154

FIGURE 10-1: BRIEF PROCEDURE TO SOLVE THE PROBLEM ............................................................................................. 175

FIGURE A-1: OUTLIER ANALYSIS OF EDGE QUALITY OF PERSPEX GLASS SHEET (3MM)........................................................... 191

FIGURE A-2: OUTLIER ANALYSIS OF KERF WIDTH OBSERVATIONS OF PERSPEX GLASS SHEET (3MM) ........................................ 194

FIGURE A-3: OUTLIER ANALYSIS OF EDGE QUALITY OBSERVATIONS OF PERSPEX GLASS SHEET (5MM) ..................................... 200

FIGURE A-4: OUTLIER ANALYSIS OF KERF WIDTH OBSERVATIONS OF PERSPEX GLASS SHEET (5MM) ........................................ 203

FIGURE A-5: INTERACTIVE GRAPH OF LASER POWER AND KERF WIDTH ............................................................................ 217

FIGURE A-6: LASER POWER (A) LINE FIT PLOT ............................................................................................................ 218

FIGURE A-7: INTERACTIVE GRAPH OF CUTTING SPEED AND KERF WIDTH .......................................................................... 219

FIGURE A-8: CUTTING SPEED (B) LINE FIT PLOT .......................................................................................................... 220

FIGURE A-9: INTERACTIVE GRAPH OF ASSIST GAS PRESSURE AND KERF WIDTH .................................................................. 221

FIGURE A-10: ASSIST GAS PRESSURE (C) LINE FIT PLOT ................................................................................................ 222

FIGURE A-11: INTERACTIVE GRAPH OF STANDOFF DISTANCE AND KERF WIDTH ................................................................. 223

FIGURE A-12: STANDOFF DISTANCE (D) LINE FIT PLOT ................................................................................................. 224

xxii

FIGURE A-13: QUADRATIC GRAPH OF LASER POWER WITHOUT REPLICATION .................................................................... 225

FIGURE A-14: QUADRATIC GRAPH OF LASER POWER WITH REPLICATION .......................................................................... 227

FIGURE A-15: QUADRATIC GRAPH OF CUTTING SPEED WITH REPLICATION ........................................................................ 229

FIGURE A-16: QUADRATIC GRAPH OF ASSIST GAS PRESSURE WITH REPLICATION ................................................................ 232

FIGURE A-17: QUADRATIC GRAPH OF STANDOFF DISTANCE WITH REPLICATION ................................................................. 234

FIGURE B-1: NO. OF TIMES INITIALIZATION COMPARISON OF 3MM SHEET EQ MEAN .......................................................... 242

FIGURE B-2: NO. OF TIMES INITIALIZATION COMPARISON OF 5MM SHEET EQ MEAN .......................................................... 242

FIGURE B-3: NO. OF TIMES INITIALIZATION COMPARISON OF 3MM SHEET EQ S/N ............................................................. 243

FIGURE B-4: NO. OF TIMES INITIALIZATION COMPARISON OF 5MM SHEET EQ S/N ............................................................. 243

FIGURE B-5: NO. OF TIMES INITIALIZATION COMPARISON OF 3MM SHEET KW MEAN ......................................................... 244

FIGURE B-6: NO. OF TIMES INITIALIZATION COMPARISON OF 5MM SHEET KW MEAN ......................................................... 244

FIGURE B-7: NO. OF TIMES INITIALIZATION COMPARISON OF 3MM SHEET KW S/N ............................................................ 245

FIGURE B-8: NO. OF TIMES INITIALIZATION COMPARISON OF 5MM SHEET KW S/N ............................................................ 245

FIGURE B-9: COMPARISON OF QUANTIFIED AGGREGATION WITH CUSTOMER QUALITY FUNCTION .......................................... 252

xxiii

LIST OF SYMBOLS AND ABBREVIATION

% Percent

ρ Density

A Laser power

AI Artificial Intelligence

ANN Artificial Neural Network

ANFIS Adapting network fuzzy inference system

ANOVA Analysis of variance

Av Aggregated value

Ave Average

B Cutting speed

C Assisting gas pressure

CCD Central Composite Design

CE Confirmation Experiment

CGA Common Genetic Algorithm

Class Response/ output characteristic

CNC Computer numerically controlled

CO2 Carbon dioxide

CPU Central Processing Unit

CQF Customer quality function

CW Continuous wave

D Standoff distance

DE Differential Evaluation

DM Data Mining

DMT Desirability, MSE and Transformation

DOE Design of experiment

DOEs Design of experiments

etc. etcetera

EPS Expanded Polystyrene

F Overall quality of flatness

FD Factorial Design

FIS Fuzzy inference system

FL Fuzzy logic

xxiv

GA Genetic algorithm

GD Gradient Descent method

GDM Gradient Descent with Momentum

GHz Giga hertz

GMDH Group method of data handling

GUI Graphical User interface

GRA Gray relational analysis

HAZ Heat Affected Zone

i.e. That is

kW Kilo Watt

ISL Inner scrap length

L9 9 runs orthogonal array



L9(34) LRun(Number of levelsNumber of variables)

LASER Light Amplification by Stimulated Emission of Radiation

LC Laser cutting

LBC Laser beam cutting

LCP Laser cutting process

LGP Light guide plate

LM Levenberg Marquardt

LMA Levenberg-Marquardt Algorithm

LTB Larger the better

Max Maximum

MDF Medium density fiber board

Millisecond ms

Millimetre mm

mW milli Watt

Min Minimum

MOGA Multi-Objective Genetic Algorithm

Mn-Mo Manganese- Molybdenum

MRR Material removal rate

MS Membership

xxv

MSE Mean square error

ns Nanosecond

ND:YAG Neodymium-doped Yttrium Aluminium garnet

NPN Noise performance measure

NTB Normal the better

OA Orthogonal Array

OFAT One factor at a time

PC Polycarbonate

PCA Principal Component Analysis

PCR Principal Component Regression

PE Polyethylene

PSO Particle swarm optimization

PMMA Polymethyl methacrylate

PMMA Plexiglass

POC Percent Overcut

PP Polypropylene

QC Quality characteristic

QFL Quantified fuzzy logic

QFN Quad Flat No-Lead

QNA Quasi Newton algorithm

QP Quadratic Programming

R2 Coefficient of determination

RF Radio frequency

RP Rapid Prototyping RP

RSM Response surface methodology

S/N Signal to noise

Sll Side line length

STB Smaller the better

SVM Support Vector Machine

SSLA Semi-supervised learning algorithm

Ra Surface roughness

Ta Kerf taper

TEM Transverse electromagnetic

xxvi

TM Taguchi Method

TPM Target performance measure

TQM Total Quality Method

V Volume

W Overall quality of waviness

XOR Exclusive OR (Logic gate)

xxvii

LIST OF APPENDICES

APPENDIX A DOE & STATISTICAL MODELLING

APPENDIX B NEURAL NETWORK & OVERALL QUALITY

CHAPTER 1

INTRODUCTION

1

1. INTRODUCTION Laser cutting is one of the latest technologies in machining materials which effectively cuts

and engraves materials through thermal cutting process [1]. Laser beam cutting and other

modern machining techniques such as ion beam, plasma beam, electron beam, electrical

discharge, electro-chemical, chemical, abrasive water jet and water jet machining are

increasingly being used as substitutes to the conventional machining methods[1, 2].

Machining by conventional methods has limits due to the irregular shapes of the work-piece

as well as its strict tolerances. In addition, new materials are introduced every day [1]. The

use of laser technology is justified because of its unaltered cutting quality, finishing and

reliability even with its high cost (however, cost is constantly falling) [2-4]. Laser cutting is

being used in a large variety of industries such as wood, metal, glass and plastic [1, 5-13]. It

is evident that the industries are increasingly adopting laser cutting (LC) techniques for

plastic or acrylic sheets to achieve precision and fine quality cuts of varying thickness by

adjusting laser power [1, 5, 13, 14]. Ahn and Tan [15] explained that the motivation of using

laser instead of hotwire cutting in the perspective of wire stiffness and quality variation which

are nonexistent in LC due to no-contact approach. Studies show that the effect of retightening

and replacement in hotwire and milling tool restricts quality management, which gives an

edge to LC over rest of the cutting methods. [1, 15-19].

The Proposed Framework can be applied on metallic and non-metallic materials but materials

for experimentation were selected keeping in view the low material cost. Therefore, Urea

Formaldehyde (Melamine), Polystyrene foam and Perspex glass sheet has been selected for

cutting process by CO2 laser. For these experimental investigations, one factor at a time

(OFAT) has been varied to analyze the effect of input process parameters on output quality

characteristics [13, 16, 18, 20-22] also mentioned that DOE is a better technique than OFAT,

its major benefit is that it describes the relationship between input parameters and the

responses as shown in Figure 1-1 [23]. Taguchi’s orthogonal array (OA) and Factorial

design (FD) were selected as an experiment design for the purpose of modelling and

verification [2-4, 11, 12, 24, 25]. Taguchi’s OA design is a robust, Statistically efficient and

less expensive method for high quality cutting systems. Its usage can reduce the number of

observations from 81 to 9 with three replications, hence it becomes energy efficient by

2

reducing time and cost. Factorial design allows to study Treatment and Interactions effects on

the response variables as both are significant factors in many industrial processes [23].

Input Parameters

Levels

Output parameters

Attributes

Laser Power Cutting speed Assist gas pressure

Stand off distance

100 300 500 0.2 0.7 1.2 0.5 2.5 4.5 1 5 10

Edge quality Mean

Edge quality S/N ratio

Kerf width Mean

Kerf width S/N ratio

Input/output Relations

Figure 1-1: Block Diagram of Input and Output parameters

The problem of modelling can be solved by the existing techniques such as transfer function,

curve fitting and Taguchi methods, or by training of experimental data to provide sufficient

knowledge of input parameters for LC machine for efficient cutting. To build a transfer

function is not an easy task because of variations in four variables. It would be difficult to

plot four inputs with one or more output quality parameters. The one and two way ANOVA

is able to model the problem with replication by considering interaction effects. The result of

nonlinear regression is comparatively worse with replication. The results of multi-linear

regression can be used for approximation [25]. However the data sets were without missing

values. Because of missing values unbalanced analysis tables generate different prediction by

Target Performance Measurement (TPM) and Noise Performance Measurement (NPM). So

NPM is preferred as a thumb rule for more robust solution. Modelling can be improved by

ANN as it models better than regression and other Statistical techniques [2, 4, 12, 26-35].

Sivarao, et al. [9, 33, 36-38] and other researchers used ANN and Fuzzy modelling with full

FD without any missing values, which is the area where improvements can be made. Pandey

et al. [39] used large size Taguchi method of L27 with two replications and L9. In their view,

interaction plays an important role and confidence interval is better in L9, This work has also

motivated us to perform this study with OA [3]. M. Zaidi et al. [3] used back-propagation

3

training algorithm for mapping Melamine cutting experiment using OA to expand the size of

data sets to FD and then utilize in aggregation for overall quality, the results were

encouraging. Researchers also claimed that ANN prediction of surface quality in laser cutting

process lessens the cutting cost by as much as 70% of the overall manufacturing cost [33]

Levenberg Marquardt (LM) training algorithms of feed forward back-propagation was

selected for trainings based on the previous study [4, 24] but verified with FD data. The

primary purpose is to select the appropriate training algorithms of neural networks. It is clear

that LM is the best choice but Gradient Descent with Momentum (GDM) cannot be ignored

even though its values were not promising and slow. The modelling of edge quality and Kerf

width S/N were better but the modelling of mean value was achieved by enlarging the span of

data. Small datasets and small span datasets modelling are difficult such as edge quality

mean. It was observed that ANN supervised learning is not sufficient with OA. After

increasing 80% dataset of FD, the average percent errors are acceptable. This study explains

the effects of changes in neural network parameters and the size of datasets for modelling

with supervised learning [4]. For this purpose, FD were selected, it estimates the prediction

errors considering by average, maximum, minimum error and MSE for each prediction. The

huge modelling experience gained in these studies which built the basis of Semi-Supervised

learning algorithm.[24] Semi-supervised learning shows more suitable modelling results than

supervised learning [4] with at least up to 22% missing values. The results depict a better

prediction on average by utilizing the systematic randomize techniques to initialize the neural

network weights and increase the number of initialization. Even with very small dataset of

OA by GDM and LM they were able to model the problem. This algorithm makes it easy for

the researchers having less insight in soft-computing and laser cutting. This achievement

encourages new researchers/engineers to utilize their expensive machines with the adjustment

of small size experimentation even with missing values due to inappropriate range

adjustments. The motivation of this study is to find minimum dataset sizes for modelling and

estimate the errors by changing neural network attributes to achieve better modelling.

Bahar and Golnabi [40] explained that most of the researchers optimize laser cutting process

for single quality while studying single input parameter at a time. Hence, utilizing large

experimental data and increasing cost and time. With reference to [3], the overall quality is

measured on the basis of very small data sets of OA based on experimental observation. The

ANN training was performed on L9 OA aggregated data and overall quality was predicted on

the basis of simulated data. In both papers [2, 3] OA design was used to take experimental

4

data. But in [25] normalized aggregation was used instead of simple aggregation on

simulated data. The aggregation function was further checked by customer quality function

[2]. Most of the researchers used aggregation methods for more than two inputs or more than

two quality parameters because of the constraint of plotting the data within 3 dimension.

Yilbas and Rashid [41] cut 800HT alloy by CO2 laser and performed experiment using

Factorial design (FD) for overall quality of flatness (F) and waviness (W). However, authors

used simple aggregation by converting the values in score i.e. 1, 2 and 3. The overall quality

is the summation of F and W. The calculation is simple and crisp in nature.

Pandey and Dubey [39] have used kerf width family as quality parameters. Overall quality is

achieved easily with the same settings for kerf width family quality parameters. Therefore,

there is no need of compromises or adjustments and can be solved with simple aggregation.

The inclusion of edge quality raises the need of overall quality rather than only kerf width

family. Syn at el. [42] used FD with five levels and three variables. However, Mamdani

fuzzy logic results are considerably good with the current research as well as other mentioned

researcher works [9, 36, 39, 42-46]. Sivarao et al. [43] also built fuzzy modelling of surface

roughness for LC by pressure vessel of Mn-Mo 5mm. He used FD of 128 observations. But,

the accuracy is less than 90%. Sivarao et al. [45] developed Mamdani fuzzy inference system

FIS with GUI. The prediction error is quiet high, maximum error is 72% and minimum error

is 1%. In author’s view, this is due to the lack of ability of Mamdani algorithm to capture the

nonlinearity of pulsed ND:Yag LCP. Sivarao at el. [9] himself built GUI-ANFIS Sugeno FIS

model for the surface roughness and kerf width on similar data [45]. The accuracy improves

to 91% consequently, error decreases to a value of 9% as compared to experimental values

which is adequate for any modelling system.

Further improvement was carried out by applying the customer quality function based on

customer specifications for better overall quality. However, the model is calculated based on

crisp logic and is unable to predict quality variable values separately. The FIS is working

with the real values of quality characteristics which provide better understanding of data

handling than CQF and normalizing the data. FIS generates better overall quality in fuzzy

manner as the rules sets is applied in parallel with aggregation. It provides unmatched results

in comparison with others. The novel combination of customer quality function, quantified

fuzzy aggregation and quantified normalized aggregation suggest overall optimized datasets.

5

In addition, electricity-efficient solution could be provided which is the requirement of

present day.

1.2 PROBLEM STATEMENT Experimental analysis has been carried out to model and optimize the laser cutting overall

quality of non-metallic sheets based on proposed framework modules with electricity-

efficient process as it is the need of time without compromising on desired quality. The

overall quality achieved by Laser cutting has an advantage over other methods like hotwire

and milling due to requirement of retightening and replacement of cutting tool with time.

Orthogonal array and Factorial design are selected as a design of experiment for modelling

and optimization. The range adjustment of laser machine input parameters requires

knowledge of DOE, laser cutting process and material properties otherwise missing values

generate due to unsuccessful cutting. It is the reason many universities have been unable to

utilize these machines to their potential. The need of a modelling technique which can model

the data with some missing values will raise the utilization of laser machines for research as

shown in Problem Statement Schematic View Figure 1-2.

Figure 1-2: Problem Statement Schematic View

Initially output qualities characteristic were modelled by Statistical and neural network

without missing values and then with missing values by supervised and novel Semi-

supervised algorithm. The Statistical modelling results of one and two way ANOVA with

Needs Overall LASER cutting quality

Issues:• Electricity efficient Process solution• Domain Experts Shortage in Universities• Small datasets

Consequence:• Need consideration of Electricity with desirable

Quality• Missing values generated

Solution:• Systematic experimentation• Better modelling techniques• Better overall quality measurement technique

6

replication were better than other data mining techniques like linear and nonlinear regression

but it is difficult to use these method with missing values. Therefore, supervised neural

network modelling is done and observed the effects of change of its parameters and size of

datasets for modelling. The neural network modelling results of edge quality and kerf width

signal to noise ratio were acceptable, however edge quality mean modelling was improved by

normalization and further improvement was made by increasing training data size to factorial

design. It is observed that ANN supervised learning is not sufficient for L9(34-2) orthogonal

array. Although only edge quality mean modelling average error were higher than required i.e.

46%. The average error with factorial design was under 10% and maximum individual error

was 25%. The vast modelling experience of supervised learning served as basis for the novel

Semi-Supervised learning algorithm and the average error was reduced, by utilizing the

systematic randomize techniques to initialize the neural network weights and increase the

number of initialization, by using orthogonal array design of experiment with up to 22%

missing values. This algorithm reduces modelling time and cost. The average error in

Perspex sheet did not exceed 8.0% to 11.5% in edge quality and kerf width.

The overall quality was calculated by data mining technique of aggregation and improved by

aggregation after normalization for equalizing the contribution of all qualities and predicts

unknown machining input setting by neural network model. Further improvement was

achieved by applying the customer quality equation based on customer specifications.

However, the model is calculated based on crisp logic and is unable to predict quality

variable values separately. A more generous and better aggregation is carried out by the novel

combination of Fuzzy logic to provide overall quality for customer and an indication for

rework at initial stage for saving cost and time.

1.3 PURPOSE OF STUDY Our goal is to propose the solution by framework modules for Process improvement using the

modular approach. The problem solves by Preprocessing of process module, Experimental

design module, Modelling and optimization module and multi-quality optimization module as

shown in Figure 6.1. Final output generates multi-objective optimized results which reduce

the financial loss of discarded items with electricity efficient solution and of desired quality.

In addition to this, built modelling algorithm to handle missing values and utilized smallest

possible datasets such as Taguchi’s orthogonal array.

7

1.4 CONTRIBUTION OF STUDY We have proposed the general “Process improvement framework modules” for process

improvement, which covers preprocessing, design of experiment, modelling and overall

quality calculation. Initially, preprocessing standard techniques were applied and avoided one

factor at a time, OFAT, preferring DOE techniques. The size of small datasets of orthogonal

array of nine observations was selected for modelling. Factorial design of eighty one

observations was selected for the estimation of modelling error and for overall quality

measurement. Performed detailed study of Statistical tools, pointed out the limitations and

solved them with Soft-computing techniques. Modelling is achieved on such small datasets of

orthogonal array including the missing values of 11% and 22% datasets by Supervised

learning and novel Semi-Supervised learning algorithm. The overall quality of any process

for two variables is a difficult task rather than Separate quality parameter. However, this

method is capable to optimize two and more than two quality parameters. This study

developed a new combination of Crisp and fuzzy aggregation for overall quality.

1.5 OUTLINE Chapter 2 (LASER CUTTING), briefly explains LASER and its application with the

justification of use of machining, it also explains Laser Cutting Techniques. In particular it

provides literature review of CO2 laser cutting to give details of its applications, machine

adjustment parameters, desired quality parameters used. Also provides concise information of

applied modeling techniques, optimization techniques and Design of experiment.

Chapter 3 (DESIGN OF EXPERIMENT), mentions the research work utilization of one

factor at a time, OFAT, technique for experimentation and shows that although the size of

experimental runs are huge in number, still are unable to study combined effects of different

variables. To overcome this problem benefits of DOE are outlined. The review of DOE

techniques used in laser cutting process is briefly mentioned. Reviews and discusses

important DOE methods used in laser cutting process. Reviews and explains vital DOE

techniques such as Taguchi Method, Factorial design and response surface methodology.

In Chapter 4 (NEURAL NETWORK MODELLING), history of Artificial Neural Network is

briefly mentioned along with a comparison of the Von Neumann architecture and brain

computing capability. It emphasizes on ANN applications while discussing and comparing

with other data mining techniques. The survey of modelling of laser cutting process by ANN

8

was carried out while considering number of inputs / outputs variables, DOE method(Size of

datasets),training algorithm and modelling methods. Explains feed-forward back-propagation

neural networks along with explanation and comparison of important training algorithms, and

justify its utility.

Chapter 5 (OVERALL QUALITY OPTIMIZATION), states that researchers generally

optimize laser cutting process for single quality and also use OFAT technique. The literature

review was performed to investigate better overall quality techniques for more than two

quality variables optimization. For more than two variables optimization, suitable data

mining techniques are discussed such as simple aggregation, normalized aggregation, genetic

Statistical techniques, Algorithm (GA) or fuzzy logic (FL). The objective is to find out the

overall quality (edge quality, kerf width, overcut and material removal rate) with Electricity-

efficiency to predict optimized input datasets. Also explains that novel Fuzzy aggregation is a

far better option for overall quality prediction.

Chapter 6 (PROCESS IMPROVEMENT FRAMEWORK MODULES), proposes Process

improvement framework modules and explains each module, with its further details for better

understanding to solve the problem of laser cutting process from pre-processing to overall

quality.

Chapter 7 (EXPERIMENTAL DESIGN MODULE & SETUP), explains the LCP with the

help of standard DOE format and points out all possible process variables. Similarly, it

describes ANN modelling process and identifies its parameters to be varied for finer training

process of ANN modelling. The Plan, Procedure and setup of experimentation with selected

DOE based machine setting, table of observation were included. The instruments

specification and work-piece material properties are recorded in this chapter with some

important definitions, formulae of quality parameters are also defined.

Chapter 8 (DISCUSSION AND ANALYSIS), briefly discussed the preprocessing of edge

quality and kerf width data. A detailed analysis of variance One/Two ANOVA with and

without replication was performed and discussed. Similarly, Linear, multiple linear, nonlinear

and multiple non-linear regression analysis were carried out and the results were discussed.

Though the First dataset was without missing values, still there is a need to improve the

modeling. Therefore, ANN supervised learning was attempted to model the problem with

9

missing values datasets of Perspex sheet. The experience of supervised learning was

discussed and provided advantage in the preparation of Semi-Supervised learning algorithm.

A detailed solution of overall quality was discussed from simple aggregation to combination

of Fuzzy aggregation.

In Chapter 9 (CONCLUSION), conclusions are outlined.

In Chapter 10 (FUTURE DIRECTION), future directions are outlined.

CHAPTER 2

LASER CUTTING

10

2. LASER CUTTING LASER is an electromagnetic radiation with the property of coherent and monochromatic

beam that propagates with insignificant divergence in the same direction and having a broad

range of frequency, power (mW to kW), laser beam density in continuous or pulsed power

and hence it finds wide applications in normal to scientific use, in medical and defense areas.

The Laser application for metallic and non-metallic processes is widely categorized into four

areas such as machining, surface, welding and laser assisted forming. It can be divided into

change of phase or require high energy to induce the phase change. This study is concerned

with laser cutting of non-metallic sheets like Perspex, melamine and Polystyrene foam which

are application of material processing [7] with the objective of achieving quality in the

cutting process[3].

Many production based industries follow total quality management (TQM) philosophy to

establish the system for continual improvement in the whole system including process

control. In process control, improvement provides more robustness, reliability and less time

to complete the process [47] . It may be the reason of selection of Laser beam cutting (LBC)

popularity. Many alternatives to conventional machining techniques are used in industries

for example electrical discharge, laser beam, water jet, abrasive water jet, ion beam, electron

beam, plasma beam, chemical and electro chemical machining. The rising applications

material processing of laser is due to requirement of high productivity, quality, material

utilization, reduction of finishing operation, processing cost, heat affected zone, kerf width,

edge quality and non-contact processing of CO2 and Nd–YAG lasers [7]. Laser cutting is a

common industrial application for organic compounds, metals, ceramics, polymers

composites, radioactive fuel and wood materials irrespective of their hardness. Diamond

cutting is ten times slower than laser in case of Textiles and hard brittle ceramics. These

materials can be cut by the combination of fusion, vaporization and chemical degradation

processes [5]. In the process non-metallic materials like carbon, plastic and wood, the laser

beam boils the surface and creates a keyhole. It creates basis of fast absorption due to

multiple times beam reflection and starting point gets deeper resulting in cut [7]. However,

Laser beam cutting has a variety of applications in fine cutting of sheet, metallic and non-

metallic and with finer precision in process [13] as pointed out in the study that laser cutting

is now commonly used in most industries. Cutting, welding and surface treatment of both

11

metallic and nonmetallic materials is carried out by various kinds of lasers at varied operating

powers. It is also mentioned by [47] that many major areas of industries have laser

applications like cutting, drilling, welding, thermal treatment and marking.

2.2 LASER CUTTING TECHNIQUES Caiazzo et al. [5] indicated that in general, a very large number of organic materials are

present which have a tendency of high surface absorption for the wavelength (10.6µm)

typically of CO2 laser. Thus, a CO2 laser with a power as low as 100 to 500 Watts may be

sufficient to cut materials such as glass, plastic, ceramics, rubber, paper, cardboard, fabrics,

wood, leather and organic compound. These materials can be cut by the combination of

fusion, vaporization and chemical degradation processes.

2.2.1 VAPORIZATION CUTTINNG Plexiglass (PMMC) is the most commonly used polymer cut by vaporization by laser cutting

consequential produce admirable finish [5]. The glossy finish is acquired by proper

adjustment of assist gas pressure parameter of laser cutting machine. The higher pressure

leaves streaks on the face of cut and it is slow leaving molten material on the face of cut. The

optimum pressure at slower side produces better finish [48, 49]. Vaporization cutting process

is shown in Figure 2.1.

Figure 2-1: Vaporization cutting process

2.2.2 FUSION CUTTING This fusion cutting phenomenon is similar to that of metal cutting with inert gases, since the

laser beam produces fusion while the covering gas removes the molten material, thus

generating the actual severing of the piece. Fusion cutting Schematic diagram by C02 laser is

shown in Figure 2.2. The cut edge and faces are macroscopically smooth with some streaks

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which are produced by the melted material. If the cut edges of the finished product are to

acquire a glossy finish, the pressure and flow of the cutting gas (compressed air) must be

adequately low so as to make it possible for the residual molten material still present on the

cut faces and edges to solidify in a non turbulent manner [5]. Fusion cutting is mostly used

for thermoplastic polymers [50-53] such as polycarbonate (PC) and Polypropylene (PP),

polyethylene (PE) [48, 49].

Figure 2-2: Fusion cutting Schematic diagram by C02 laser

2.2.3 CHEMICAL DEGRADATION CUTTING Chemical degradation is used to cut thermosetting materials using higher power level and

temperature as compared to simple fusion cutting because of a three dimensional lattice.

Therefore, cutting speeds are generally lower for thermosetting materials as compared to

thermoplastics as reported by [15]. The schematic diagram of CO2 laser cutting shows

construction of nozzle which includes Laser beam, focusing lens, inlet of assist gas pressure

and nozzle with air and beam focused on work piece. In this process smoke is generated

which could be deposited on the cut surface [48, 50, 51] and there is need to take care of

safety issues [54].

2.3 LOW POWER CO2 LASER LASER, meaning light amplification by stimulated emission of radiation, emits infrared laser

radiations with a wavelength of 10.6 μm and posses overall efficiencies of approximately 10

to 13% mentioned. The laser-active medium in a CO2 laser is a mixture of 9.5% CO2, 13.5%

N2 and 77% He gases, where CO2 is the laser-active molecule. The stimulation of the laser-

active medium is attained by electrical discharge in the gas. There are different designs of

the CO2 laser such as: Transverse flow (cross-flow) laser, Fast-axial flow laser, Diffusion-

13

cooled slab laser and Sealed-Off laser. They can be driven in either the continues wave mode

or pulse mode as reported by [21]. LASER is applicable in ordinary to sophisticated devices,

in common use to scientific purposes, and in medical as well as defense as explained by [7].

Figure 2-3: Laser cutting process

Majumdar and Manna [7] define that the application of lasers for material processing are

broadly divided into four major categories; namely, laser-assisted forming, joining,

machining and surface engineering. The study focuses on machining process of laser cutting

of polystyrene foam and Perspex sheets which can be used as an initial work for

understanding cutting process optimization in other metallic and nonmetallic materials.

Laser can be of very low power mille watt to extremely high (1–100KW) focused power with

a high precision of spot size and delay time on to any kind of medium. Laser is better than

other electromagnetic radiations in terms of its spatial and temporal coherence, low

divergence, high continuous or pulsed power density, monochromic and propagation in a

straight line as reported by [7]. Laser material processing applications can be divided into two

classes

• Requires limited power and causes no change of phase.

• Requires high energy to induce phase change

High energy applications are cutting, welding, fusion, heat treatments on metallic and thick

materials. For limited power non-metallic materials are the area of application as well as thin

metal materials can be used. Laser power/efficiency and interaction-time are crucial as the

14

processes involves single or multiple phase changes within a very short time. In this class of

application CO2 and Nd–YAG lasers are practically used. Majumdar and Manna [7]

explained that CO2 laser is used in material machining; laser cutting is a common industrial

application for organic compounds, metals, ceramics, polymers composites and wood

materials irrespective of their hardness.

Many researchers are working in the field of machining of laser processing with different

angles. The relevant important researchers work is briefly discussed below.

Zhou and Mahdavian [13] developed the theoretical model to estimate the depth of cut with

cutting speed and laser power for different materials. They investigated theoretical and

experimental results for different materials. A semi-empirical equation is developed to

optimize or assist the cutting process and find out the laser power range selection. As CO2

laser wavelength absorption tendency for nonmetallic materials is high and often have a

characteristic of low thermal and thermal diffusion coefficients. Therefore, very low power of

60 watt of CO2 laser can cut non-metallic material like plastic board with good cutting quality

but releases toxic gases during the process which can be handled easily in glass covered table

based laser machines. Zhou and Mahdavian provide guidance for wood cutting and particle

board that keep cutting speed low for small Kerf width. The cutting of rubber material is

improved by assist gas.

However, it produces smoke in the cutting area. The theory and experiment shows that higher

energy is required for deeper cut. The theoretical model is improved by experimental result

by introducing constants in the equation by Zhou. Modeling is based on energy balance

method but ignores the loss of energy due to heat conduction in the surrounding. The

empirical equation is built on the basis of energy balance method but ignoring heat

conduction around the cutting area. Therefore, there is still need to build better modelling in

the laser cutting area. Caiazzo et al. [5] investigated plastic material cutting analytically and

systematically by CO2 laser. They used polyethylene, polypropylene and polycarbonate of

thickness ranging 2-10 mm in order to provide sufficient knowledge to the industry. He [5]

reported that the researchers conclude results after exhaustive examination of different

process parameters which is time consuming and cost ineffective. The exhaustive method is

used to optimize the input parameters. The cutting quality of polyethylene (PE) was low,

polypropylene (PP) was medium and polycarbonate (PC) was high.

15

Yusoff [10] used CO2 laser of 500 watt for the experiment of cutting Malaysian wood. He

studied wood characteristics like thickness, density and water content. The aim of analysis

was to obtain geometrical & dimensional accuracy, higher material removal rate and to avoid

wood burns, checking results of assist gas by using air or nitrogen. He also [10] concluded

that machine adjustments are dependent on the thickness, density and moisture content in

wood. The use of inert gas (nitrogen) generates better quality of kerf width. More thickness

or moisture consumed higher power. Almeida et al. [47] optimized the process parameters of

laser machine for the cutting of high cost and difficult to cut titanium material with pulsed

laser technique. They used alter six machine parameters and converted it into four level

ordinal variables and kept constant about seven parameters. They used factorial analysis. The

edge quality was improved but using large data set with high cost material. In my view the

size of experiment should be small which can apply novel semi supervised artificial neural

network or RSM.

Tan et al. [16] conducted experiments to find the optimum combination of parameters in laser

engraving process by CO2 laser machine on ceramic tile of 6 mm thickness. For optimization

Taguchi method was used for DOE. The engraving speed is the most significant factor for

depth of engrave, heat affected zone and material removal rate. However standoff distance

was most influential on kerf width. The study of M. Zaidi at al. [25] shows that interaction

effects may be possible in the laser cutting process over the selected ranges of input

parameters which is missing in the this study [16]. Nukman et al. [10] explained the effect of

machine parameters setting on the cutting quality of soft and hard wood. They demonstrated

the improvement in quality by varying assist gas pressure and used different nozzle design.

Also shows the vital importance of speed in achieving better productivity. He also discussed

noise or uncontrollable parameters such as shield gas pressure before empty cylinder. He also

explained the effect of standoff distance on beam width. The beam changes due to focal

point on the work piece which indeed affect the cutting quality [10].

Imtiaz at al. [1] performed an experimental investigation on polymeric materials and

developed a model equation relating input parameters with the output. The quality parameters

are heat affected zone, surface roughness and dimensional accuracy. Exhaustive experiments

were not conducted, only twelve sets of experiment were taken for each polymer material

using the technique of central composite design. The Response surface methodology was

16

used as a predictive model, further analysis was done by ANOVA. The analysis is based on

dimensional deviation and concludes that Polymethyl methacrylate PMMA has less heat

affected zone than PC and PP. In case of surface roughness PMMA has much better edge

quality than PC and PP. The response model can be used in the industry for production. With

reference to paper [1] laser cutting processing proved advantageous in industries and is able

to improve by increasing the ordinal variable beyond low or high. At least three numbers of

levels are more appropriate for empirical modelling and need better techniques to handle

higher polynomials or other non-linear modelling.

Berrie and Birketi [55] investigated the effects on Perspex sheet by varying lens position,

focal plane, cutting speed, applied power on the machining process of cutting and drilling

rates. The experimental results were compared with the developed theoretical models of the

thermal conductivity and the authors studied vapors removal. The results support the theories

of vapors removal. But they did not study the effects on kerf width or edge quality of Perspex

sheet.

Balkin and Lin [56] study the optimum level of the independent variables 4 to 9 and use only

2 variables with quadratic regression and ANN. The results were discussed for different

levels of input parameters. It shows that improvement is possible but number of observation

increased in L(Runs) (Levels) (Variables) . In the studied case only two variables with observation

for 4 levels will be 16 and similarly for others 25, 36, 49, 64 and 81 respectively. Therefore,

it is important to select less number of controllable parameters with optimum number of

levels.

M. Zaidi at al. [25] studied different statistical modelling techniques of Polystyrene cutting

process such as one and two way ANOVA, were able to model the problem with and without

replication. The model gives better results in case of replication showing suitable analysis

technique for given datasets. The interaction should be considered to get a better picture of

the process optimization. The 89.9% highly significant value of R2 encourages using the

multiple linear regression model because Kerf width variation can be explained by the

selected input variables. It can be used in rough modeling, simulation and optimization. The

results of nonlinear regression are worst compare to others and with replication become more

non-realistic due to increase in the number of observation over and above the fitted points

similar to linear regression. The average error reaches to 50%.

17

The best method was one way ANOVA with pooling but the current research shows that

there is one thing missing that is significantly participating in the variation of dependent

parameters i.e. interaction between two independent and one dependent parameters. The

discussion of interaction above shows that three combinations of interactions were

significantly participating in the variation of dependent variable.

Therefore, modelling can be improved with the combination of Treatments and Interaction

based design of experiment. The results of regression are highly inferior than ANN as

mentioned in [2, 25]. Many researchers handled missing value but did not apply ANN in the

Laser cutting process to solve the range adjustment problem. The modern techniques of

regression [25] had higher prediction errors. Therefore, it is better to use novel method of

Semi-supervised algorithm for process modelling [12]. The detailed experiment of Perspex

CO2cutting in square profile is shown in Figure 2.3

Figure 2-4: CO2 laser cutting Sheet in square profile

C.B. Yang, et al. [35] prepared a model of a CO2 laser cutting process of polymethyl

methacrylate PMMA using orthogonal array of L9 for four factors with three levels at the

first stage, which is similar to our modelling with novel semi-supervised learning algorithm

[12]but built empirical relationship between the input and response variables and increased

18

the data size for ANN. The modelling with L9 orthognal array in single stage is difficult

which has been done by M. Zaidi et al. in [12]. The result shows that thickness is a major

parameter in the modelling process. The increase in the size of data improves the ANN

accuracy.

Noor and Kadirgama [57] prepared a statistical linear model to predict surface roughness and

roughness height based on laser power, cutting speed and tip distance. The model is prepared

for acrylic (3mm thick) cutting by 30W pulse CO2 using the Box Behnken (RSM) design.

The first order model shows that laser power is the major factor. In the next study they [57,

58] prepared quadratic RSM model which also considered the interaction and quadratic

effects. The interaction effects are insignificant but quadratic is significant and also shows

that second order is more close to experimental values. In addition to these two studies they

[8] developed artificial intelligent (AI) model using partcle swarm optimization (PSO) to

predict the optimum surface roughness for 30W pulsed Nd-Yag cutting acrylic sheets while

using Box-Behnken Response surface method (RSM) to reduce the data size of experiments.

They also explained the effect of machine settings of cutting speed, material thickness, gap of

tip and power towards surface roughness.

Combination of slow cutting speed of cut, high laser power and optimum standoff distance

construct superior surface roughness. Some defects were found in the microstructure such as

burning, melting and wavy surface.

Meanwhile, Ciazzo et al. [5]found that all of the three materials generally follow the rule

(which the results of experiments on ferrous and nonferrous metals have already been amply

validated) according to which the value of Ra diminishes as cutting speed increases. However

Ra values are much lower compared to typical construction steel. His research showed that

the surface roughness is significantly affected by the tip distance followed by the power

requirement, cutting speed and material thickness.

2.4 SUMMARY Many researchers have modelled the laser cutting problem to build theoretical or

experimental modelling techniques [1-3, 13, 16, 25, 47, 55]. Basically the problems in laser

cutting process are

• Process is non-linear

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• Process is multivariable

• Process adjustment parameters are large in number

• Noise can disturb the quality due to human error or environment.

• Multi-output qualities

• Day to day new materials introduce

• Day to day new laser machines introduced

Many researchers and Yang et al. [35] quote that improper controllable machine parameters

will result in poor quality of laser cutting. However, numerous researchers are doing research

on the same direction as Yang et al. i.e. adjusts machine parameters properly. Therefore, it is

important to build easy to use, low cost, less time consuming and less complicated techniques

for old and new engineers and researchers. There is a need of Statistics expert for the

modelling and analysis of non-linear and multi-variable process. A number of researchers

have used regression, ANOVA, Taguchi, RSM, factorial analysis and multi-output

optimization techniques[25] .

There are a lot of researchers using AI techniques to model the laser cutting process [2-4, 12].

But, the need of expert does not diminish such as supervised algorithms in artificial neural

networks needs AI experts and experience counts in the results. There is a need to build

some semi-supervised algorithms based on his experience of supervised learning. Laser

cutting with missing values is also an important issue to resolve which can be resolved better

by ANN ( LM back-propagation algorithm) than statistical methods [4] for experienced

researcher and Semi-supervised is much better for very small data sets [12]. Another issue of

multi-output qualities is that the techniques are very complicated sometimes [24, 30, 59-63].

Some techniques are based on crisp logic and some are fuzzy logic based. The need of hybrid

technique is a requirement of time to solve the issue in fuzzy manner with some strict

boundary of acceptable quality. But, expertise of DOE is required to select the proper design

of experiment. Normally, researchers avoid this technique and solve their problems by

changing single variable at a time or without DOE in detail which increases the number of

experiments taking more time and cost [5]. It is also possible that interaction between the

variables is present and variation in a single variable is unable to be recorded and analyze

these issues.

20

It is necessary to further investigate the DOE, modeling and overall quality optimization.

Therefore, In the next Chapter 3, we will discuss DOE in detail for better understanding and

its appropriate selection for the LCP. In Chapter 4, we will discuss ANN modeling and in

Chapter 5 overall quality optimization. Finally, propose the Process improvement Framework

modules for solution approach.

CHAPTER 3

DESIGN OF EXPERIMENT

21

3. DESIGN OF EXPERIMENT In most of the experimental investigations of Laser beam cutting (LBC) process, one factor at

a time (OFAT) has been varied to analyze the effect of input process parameters on output

quality characteristics or responses [13, 16, 18, 20-22]. Bahar and Golnabi [40] also varied

only one parameter at a time in the study of the response on output quality for laser cutting

of steel and mild steel and have not adopted any DOE technique but relied on OFAT to

observe the effect on output quality. They analyzed the changes individually on output based

on

• Speed of cutting

• Applied laser power

• Thickness of sheet

• Oxygen gas pressure

The author performed 400 runs to understand the effects of input parameters. However, this

technique requires a large number of experimental runs as just one factor is changed in each

run, with all other factors remaining constant. In this technique, the interaction effects among

different input process parameters are not considered.

To overcome these problems, it would be better to use any scientific DOE which will save

time and cost. Viles et al. [64] and Sivarao et al. [23] highlighted the significance of

Statistical DOE application and discussed problems in the preliminary stages of DOE

recommending that the scientific community should write and present more DOE

applications focusing on the preliminary stages of the methodology. It would be helpful in

bridging the gap existing between research and industry.

They mainly focused on Taguchi method, Factorial Design and Response surface

methodology in laser cutting process. Sivarao et al. [23] also mentioned that DOE is a better

technique than OFAT and its major benefit is that it shows the relationship between input

parameters and responses. It can also show the interaction effects on responses, mathematical

model developed are normally used for prediction which can predict the possible optimize

response. It also saves time due to its well planed method. Sometimes experiments will have

some error, some of which might be predictable while others are just uncontrollable. DOE

22

allows us to handle these errors while still continuing with the analysis. DOE is an excellent

choice when it comes to predicting linear behavior. However, in nonlinear behavior, DOE

does not always give the best results [23]. It means there are chances of unsuccessful results.

Therefore, it is possible to select more appropriate DOE techniques for better results.

3.1 GENICHI TAGUCHI’S METHODOLOGY Taguchi’s orthogonal array (OA) design of experiments (DOE) is a robust, efficient

Statistical method for designing high quality systems at lesser expense for laser cutting

process. It designs a systematic and efficient way to optimize controllable parameters with

the assumption that there is no interaction among controllable factors and ignoring the

variation caused by uncontrollable factors. It is able to provide optimized robust input data

setting [2, 12, 16, 23].

Besseris [65] explained that Taguchi approach is drawn for a large number of the associating

parameters that changes the output quality parameters comparatively larger than other

parameters variations. Taguchi optimization of any process or product variables may be

analyzed in eight different steps.

1. Evaluate the importance of the given problem and set the required outcome which

needs to be achieved by analyzing and solving this issue.

2. Collection of implicating quality characteristics that need attention along with the

respective controlled and noise factors that may affect these quality factors.

3. Investigation and classification of a significant physical range the factors are allowed

to vary during experimentation.

4. Selection of the proper orthogonal array that will fit all the considered factors and

possible interactions along with the appropriate levels.

5. Execution of trial runs according to a randomized schedule dictated by the selected

orthogonal array.

6. Analysis of the experimental data with nonparametric tests for small samples.

7. Determination of optimum settings for all statistically significant factors and

interactions.

8. Confirmation of prediction outcomes by conduction of experimental runs at the

optimum level values obtained from the previous step.

23

Orthogonal array were discovered more than a hundred years back in 1988 by Ross, they are

still a dependable source of planning for experimenters. With reference to [66] Genichi

Taguchi’s approach to ensuring quality is based on building robust product and process

designs. He taught controlling process quality by design of product and their process instead

of inspection. He focuses on selecting factors levels of controllable parameters that reduce

variability, bringing the process on target which is the secondary objective. Taguchi’s method

is one of the better designs of experiment. As large number of variables with higher number

of levels makes Factorial design useless, however Taguchi’s method works as shown in

Figure 3-1.

Figure 3-1: Comparison Taguchi’s OA and factorial designs number of Runs

3.2 GENICHI TAGUCHI’S METHOD APPLICATIONS The aim of robust process design is to build a stable process that produces minimum

variation due to uncontrollable parameters [60]. J. Kunert et al. [67] compared Taguchi’s

combined and product array to explained robust parameters design, the experimenter looks

for settings of controllable factors close to the target, with little variation. The difference is

caused due to two sources noise and pure error. Taguchi method produces a robust design but

it is better to calculate the variation with the help of an outer array by identifying the possible

noise factors in outer array. The results of outer array are better than product array or simple

orthogonal array. However the results of Genetic algorithm are even better than outer array

Taguchi method. Chatsirirungruang and Masami Miyakawa [68] studied the usefulness of

genetic algorithm (GA) as compared to orthogonal array’s Taguchi method. They used signal

to noise ratio as a deciding factor as is a routine practice of Statisticians but sometimes this

9

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24

criteria is not true. Therefore, noise factor (outer array) is used to solve the problem more

efficiently in his all papers [68-70].

M. A. Wazed et al. [71] explained that the essential part of managing uncertainty is pointing

out maximum sources and factors of uncertainty. This article is based on detailed review on

manufacturing under uncertain conditions. Different sources and factors of uncertainty in

manufacturing/production setting have been pointed out and focus has been shifted on the

techniques and models used to deal with them. Uncertainty can also be from its sources as

follows:

• Physical deviation observed in a system because of natural uncertainty. It is also

known as physical randomness and inherent uncertainty.

• Many assumptions for simplification in prediction and analytical models are the

reason for model uncertainty.

• Measurement uncertainty due to limited capability of measurement techniques and

inferior ability of measurement instruments.

• Environment and operational uncertainty.

• Statistical uncertainty results due to incomplete Statistical data (missing value issue)

and use of improper sample size or inappropriate sample selection to approximate the

characteristics of these parameters.

• Subjective uncertainty resulted due to human errors in calculation, improper

parameter selection, judgment and fabrication.

All types of uncertainties are not present in all process systems at the same time. However,

sometimes one or more than one type of uncertainty can significantly participate in the

system variations.

W.Q. Meeker [72] explained the relationship between engineering quality, dependability and

role of Statistics, Statisticians and brief Statistical tools applied in the area of reliability.

Quality and reliability have both improved with the Statistical tools that were originally made

to control and improve quality such as process monitoring and designed experiments. Every

passing day brings advancement in scientific knowledge thus Statisticians will have an

important part to play in the area of reliability in future as well.

25

Many experimental studies used well known L9, L18 and L27 Taguchi’ orthogonal arrays [3,

12, 16, 28, 35, 39, 61] . The importance of the Taguchi method is demonstrated when many

researchers use Taguchi DOE with other methods to make hybrid combinations and produce

better results and in multi-quality cases they used Taguchi at the first stage for single and

grey relational analysis and fuzzy for multi- qualities. Following are some examples of

Taguchi DOE.

Kuo et al. [28] utilized slightly larger data sets of L18 orthogonal array DOE for the

modelling of laser processing (CO2) by back propagation with Levenberg-Marquard training

algorithm. They used 13 datasets for training and remaining 5 for validation. The average

percent error was 5% which is a better result with L18. C.B. Yang, et al. [35] prepared a

model of a CO2 laser cutting process of polymethyl methacrylate PMMA using orthogonal

array of L9 for four factors with three levels at the first stage and built factor response table

which shows that thickness is the major factor. In the second stage they used hybrid

technique with ANN. The new combined model is called progressive Taguchi neural

network model and its prediction accuracy will improve with small datasets by increasing the

size of data set with Taguchi and model with ANN. Ming-Fei et al. [61] applied the grey

relational analysis method to study the multi-quality optimization of two responses (surface

roughness and optical transmittance ratio) as a result of PMMA (thickness 6mm) CO2 laser

cutting. The orthogonal array of L9 Taguchi DOE was used for multi quality.

Pandey and Dubey [39] have used large size Taguchi method of L27 with two replications.

The use of L27 does give a better fuzzy model, so to achieve a better Taguchi model three

replications are an appropriate design. It is clear that Fuzzy model is better with L27 than L9.

However, the chance of achieving good results with L9 is the area where new researchers can

work.

Taguchi method used orthogonal arrays to solve the problems. This DOE has many qualities

such as reliability and small data size but its disadvantage is the assumption that all input

variables are independent in nature i.e. interaction effects of two or more than two variables

participates insignificantly in the output quality. M. Zaidi et al. [3, 4, 12, 24, 25] used L9

orthogonal array for Statistical and AI modelling purpose. M. Zaidi et al. [12] achieved better

modelling with L9 orthogonal array without the outer array. It is an achievement of M. Zaidi

26

et al. [12] to model with L9 array, having 12% to 22% missing values in the datasets, by

applying Semi-supervised algorithm.

3.3 RESPONSE SURFACE METHODOLOGY Response surface methodology (RSM) is an advanced mathematical and Statistical DOE

method which is useful for the modelling and optimization of manufacturing engineering

process or laser cutting process [1, 23, 57, 58]. RSM design of experiment optimizes the

output quality that is varying due to input parameters. In laser cutting process modelling the

data is collected based on selected DOE. It is used for modelling and optimizing of laser

cutting process and the size of data is two to three times that of the Taguchi method.

However, smaller than the standard Factorial design. [25]. The basic RSM used first order

linear model. However most of the laser cutting process is non-linear. Therefore, lack of fit of

first order models create the need of second order regression model [73]. Noor et al. [57]

employs the Box-Behnken design of RSM for the optimization of experimental data to

explain effect of selected significant input variables. This design of RSM is often selected

when experiment executes in non-sequential manner and the whole experiment is performed

in one session. First and second order coefficients produce better estimation. In these designs

lesser number of runs are required as compared to central composite design (CCD) with same

number of input factors. It does not contain axial points. Therefore, it is easy to set the

extreme ranges within operating range similar to factorial design as well as all factors are not

adjusted at maximum value simultaneously. Hence, the experiment is less expensive [57,

74-76].

Noor and Kadirgama [57] prepared a surface height and roughness model with input

variables, such as standoff distance, cutting speed and laser power. The work-piece is 3mm

acrylic sheet cut by pulse CO2 of 30 watts. The first order model shows that laser power is the

major factor. The analysis of variance shows P- value is not significant i.e. linear factor

model was not sufficient. In this work Surface height and roughness models show that laser

power and standoff distance are significant factors and both output qualities are directly

proportional.

It is observed that both Noor and Kadirgama et al. [57, 58] articles are similar with the

exception that the first article is linearly fit and second quadratic RSM model which checks

27

he interaction and quadratic effects. The interaction effects are insignificant but quadratic is

significant and also shows that second order is more close to experimental values.

Noor and Kadirgama eta al. [8] selected the same Box Behnken design of RSM used for

small experiment size, but changed modelling method from RSM to artificial intelligent (AI)

model. They made a combination of ANN and Particle swarm optimization (PSO). This

article shows that surface roughness is significantly affected by the priority of standoff

distance, laser power applied, speed of cut and thickness of work-piece.

Mathew et al, [14] used RSM central composite design of experiment for the modelling of

heat affected zone of Carbon Fiber reinforced plastic composites cutting by Nd:YAG

pulsed laser at the optimum process parameter ranges. The obtained results are in the

acceptable ranges.

Baş and Boyacı [77] studied the recent papers (since 2005) utilizing RSM and attempted to

find out some common mistakes and limitation of RSM. RSM model fits well but it is

possible that its prediction out of experimental data range is not predicting well. The size of

RSM observations is smaller than factorial design but it is capable to model the effects of

interactions which will produce better results than Taguchi method if interaction between the

variables is significant.

Balkin and Lin [56] studied the optimum level of the independent variables to understand the

quadratic banana function to predict Response surface methodology RSM that includes only

two variables X1 and X2. The results of the quadratic regression analysis and ANN were

discussed for the different levels of input parameters for 4, 5, 6, 7, 8 and 9 levels. The number

of observation for 4 levels will be 16 and similarly for others 25, 36, 49, 64 and 81

respectively. The results show that as the size of data increases ANN is incomparably better

than RSM. In addition to this ANN can model multiple output [32]. Dhupal, et al. [26] also

prepared ANN model to predict data on microgroove width and depth phenomena with RSM

design of experiment.

3.4 FACTORIAL DESIGN Factorial design gives the opportunity to study Treatment and interactions effects on response

variables because frequent interactions are significant factors in many industrial processes

28

[23]. It is one of the better designs of experiment. But large number of factors with higher

number of levels makes Factorial design useless.

Figure 3-2 shows that number of Runs increase as the number of factors increase when

number of levels are 2. Two levels are often used for the selection of factors which is called

screening, even though the increase in number of factors with two levels decreases the

usability of factorial design with 8 or more factors.

Figure 3-2: Increasing trend of number of Runs as factors with two levels

Figure 3-3: Increasing trend of number of Runs as factors with three level

4 8 1632

64

128

258

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100

150

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29

Figure 3-3 shows that the number of Runs also increases more as number of factors increases

when number of levels are 3. Three levels are often used for modelling and analysis. It shows

that increase in number of factors decreases the usability of factorial design with 5 or more

factors.

Figure 3-4: Difference of number of Runs in 2 and 3 levels

Figure 3-4 shows the difference between two and three levels of factorial design. It shows

that as the number of levels increase there is a remarkable increase in number of Runs which

shows that increase in levels increases number of Runs remarkably and decreases the

usability of Factorial design.

J. Ciurana, et al. [78] used factorial design of experiment for the preparation of many models

for laser milling process qualities dimensional, geometry and roughness properties. The

results of ANN with factorial designs are better than other researchers. They also compare the

results with multiple linear regressions and concluded that ANN is a better solution [4, 78].

The experiment is based on full factorial design which includes 27 observations. Two third

data is used for training and one third for testing of ANN model.

M. Zaidi et al. [4] used Full factorial design for the purpose of modelling and verification.

After getting data in the form of factorial design in the beginning, evaluate orthogonal array

which ignores the interaction effects. The results show that it is not at all as good as it sounds.

It has been proposed that the experimental design which is sequential in nature has greater

efficiency and accuracy but is avoided due high cost of experiment. The results were

4 8 16 32 64 1282589 27

81 243729

2187

6561

0

1000

2000

3000

4000

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6000

7000

2 3 4 5 6 7 8

Num

ber o

f Run

s

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2 Levels

3 Levels

30

encouraging but only performed on Perspex of 3mm edge quality with missing value problem

in design of experiment. The problem with OA was that it is unable to solve the problem

within the desired percent errors in prediction, because it utilizes 60% data for training, 20%

for verification and 10% for final testing. The modeling in case of supervised learning with

missing values is possible.

Sivarao, et al. [9, 33, 36-38] has presented many papers in conferences and published many

journals to model the laser cutting process with ANN and fuzzy logic but all of them utilize

full factorial design without any missing values which is the area where improvements can

be made. The studies show factorial design is appropriate for modelling with ANN.

Sivarao et al. [43] has been working in the area of fuzzy logic modelling and has been able to

prepare the laser cut surface roughness model for Mn-Mo 5mm pressure vessel plate based on

two level factorial design consisting of 128 observations. About five observations of

experimental results deviate much more than desired value. This could be a modelling error

except in these observations the error is still greater than 10%. The author’s expectation was

0.5% which is much lower than the actual results.

M.-J. Tsai et al. [34] built both artificial and multi-regression model for QFN cutting of 6

qualities and 3 input parameters. The results of LM-back-propagation neural network were

better than multi-regression because of the selection of full factorial design. The results of

multi-regression are also acceptable because the size of data sets is sufficient and the model

prepared based with the effects of Treatments and interactions. However, sufficient data size

provides lead to ANN. Hence, if it is possible to select factorial design and it will not effect

on cost and time then factorial design is the better approach.

Yilbas and Rashid [41] also selected factorial design for cutting 800HT alloy with CO2 laser

to investigate the optimum parameters and used very simple crisp logic to assign the score to

flatness and waviness. The overall quality is the aggregation result of both parameters scores.

The striation frequency is measured from surface roughness and they built the relationship

with the dross ejected from the kerf. However, it is better to apply AI techniques to build

model and overall quality tools.

31

Yilbas [79] studied CO2 laser cutting of Stainless steel to investigate the input parameters,

effects of speed of cut, pressure of assist gas, thickness of work piece and laser pulsing

frequency in order to monitor the variations in Waviness, flatness and metallurgical changes

at the cut surface and overall cut quality utilizing factorial design of experiment. Neural

network has been utilized successfully for classification of waviness patterns. He concludes

all parameters are statistically significant on quality responses. He used scores assignment

technique for overall quality. Better results can be produced because, all the required data is

provided through factorial design but overall quality can be improved by fuzzy aggregation.

He also discussed Artificial Intelligence (AI) based techniques such as artificial neural

network (ANN) and fuzzy logic which are widely used AI techniques. In my point of view

the design of experiment used by AI techniques need some modification in design of

experiment. They need some data set out of designed matrix for checking of generalization of

model and some for testing of the model. This issue can be resolved without investing on the

experimentation, by taking the observation taken for range adjustment before the actual

experimentation.

3.5 SUMMARY It was observed during literature review of LCP for DOE point view that many researchers

used OFAT technique. However, large size experimentation runs and ignorance of combined

effects of different input variables can be overcome by using DOE. In LCP, Scientists

focused essentially on TM, FD and RSM. Taguchi’s orthogonal array DOE is a robust,

efficient method for designing high quality stable process at lesser expense for LCP due to

minimum noise. However, outer array are better than simple OA but results of AI (Genetic

algorithm) are even better than outer array.

Different sources and factors of uncertainty are mentioned but Statistical uncertainty due

incomplete data (missing value issue) is present in our problem. Many experimental studies

used orthogonal array (L9, L18 and L27). Hybrid combination of Taguchi DOE with other

methods produced better results in modelling and multi-quality optimization. However, most

of the researchers used Larger size OA than L9. If they used L9 they expand the size of

datasets by using RSM model and then used larger data sets for training by ANN. But,

accuracy depends on the selection of RSM fitting function which is difficult. The LCP is non-

linear, therefore, lack of fit of first order models creates the need of second order regression

32

model or sometimes both are unable to model. Box-Behnken and Central composite design

are similar in size but Box-Behnken range contains axial points. The axial point extreme

values of input settings are sometimes harmful for Machine. RSM model fits well but it is

possible that its prediction out of experimental data range is not predicting well.

Factorial design gives the opportunity to study Treatment and interactions effects on response

variables. It is one of the better designs of experiment. The number of Runs increase with the

increase in the number of factors or levels or both which makes FD impractical L(Runs)

(Levels) (Variables) .

Many researchers used FD for LCP for ANN modeling, fuzzy for mapping and overall

quality. FD with 2 levels is used for screening. The number of Runs with three levels and

four factors is reasonably small datasets.

Many papers with ANN and fuzzy logic utilize FD without any missing value which is the

area with margin of improvement. The articles solve overall quality with crisp logic and

Statistical methods also used FD which can be improved by fuzzy aggregation.

We selected Factorial Design and tried to model L9 orthogonal array for high cost materials.

Nine observations are used for modeling and remaining for detail verification. It has been

proposed that the experimental design which is sequential in nature has greater efficiency and

accuracy but is avoided due high cost of experiment. The problem with OA was that it is

unable to solve the problem within the desired percent errors in prediction with supervised

learning but later achieved better modelling with L9 orthogonal array, without the outer array

having 12% to 22% missing values in the datasets, by applying Semi-supervised algorithm.

The next Chapter 4 provides the history of ANN and literature review to provide guidance for

the modeling of the problem with small datasets. Chapter 5 will provide the details of overall

quality optimization using the selected DOE.

CHAPTER 4

NEURAL NETWORK MODELLING

33

4. NEURAL NETWORK MODELLING

4.1 ARTIFICIAL NEURAL NETWORK Connectionist architectures originate from Psychology, Physiology and Computer Science.

McCulloch and Pitts (1943) [80] demonstrated artificial neural network model for binary

problems. The model is binary transfer function (OR, NOT and AND) therefore, its output is

only 0 or 1. Its functionality can be represented mathematically

𝑌𝑌𝑏𝑏 = 𝑓𝑓 ��𝑊𝑊𝑖𝑖𝑏𝑏𝑋𝑋𝑖𝑖𝑏𝑏 + 𝐵𝐵𝑖𝑖𝑏𝑏𝑖𝑖

� (4.1)

Where Yb is transfer function binary output, f is function either 0 or 1 for neuron, Wib is

neighbors weighted values of input Xib and Bib is intercept or bias of straight line model.

However, no weight adjustment algorithm was presented. Animal learning theories motivated

Hebb in 1949 towards Neural Networks. The basics of Artificial Neural Network (ANN) is

artificial neuron, network topology encoding scheme and learning algorithm [81].

Frank Rosenblatt in 1958 [82] made neural network linearly separable by Perceptron. His

major contribution is to provide iterative algorithm for the weight adjustment which were

missing in McCulloch and Pitts [80] binary model.

But Minsky and Papert, 1969 [83] point out that Perceptron model is unable to classify

Exclusive OR logic gate (XOR) problem. Due to this ANN’s reputation reduced. Addition of

hidden layers with the algorithm resolved this issue and again attained attention in mid

1980’s. Connectionist architecture of artificial neural network models have been utilized in a

broad variety of problems by Rumelhart and McClelland in 1986 [84], Anderson and

McNeill in 1990 [85] and Churchland and Sejnowski in 1992 [86].

Multilayered networks are computationally complete i.e. equivalent to Turing machine [81].

ANN used common error correction (back propagation) algorithm for training [85], the

increase in number of neuron and connection improve the networks to model the problem.

The increase in neuron gives the benefit up to some extent and begins to decrease the ability

of classification upon further increase, so it needs to be adjusted to maximum accuracy by

adjusting optimum number of neurons.

34

Biological networks work in parallel and ANN follow Von Neumann architecture [85]. A

neuron response time for 1 GHz CPU is more than biological single neuron time of 1x10-3

seconds because CPU executes one step of execution in nanosecond (ns). Even then brain is

far better than a computer because of the ability of billions of neuron computational ability in

parallel [87] such as humans perform exponentially better than computers e.g. recognizing a

face from thousands of familiar faces instantly. Some salient features of Artificial Neural

Networks are non-linearity, input- output mapping, and ability to adapt to environment,

evidential response, and fault tolerance, analogous to neurobiological systems.

Imran Amin et al. [88] also mentioned inspiration of human brain in two ways, an ANN

attains knowledge through learning and ANN's knowledge is stored within synaptic weights.

ANN is based on different mathematical models that follow some properties of the biological

nervous systems and adapt the similarities of biological learning. ANN model is comprised of

a large number of interconnected processing elements that are similar to neurons and are

connected together with weighted connections that are analogous to synapses. Similar to

nervous system neural network are processing in parallel.

The ANN function is built by the connections among the elements. The ANNs can be

differentiated by feed-forward and recurrent algorithms. The selection of network is based on

the data processing technique. They can also be distinguished by the method of training used,

some ANNs use supervised and others are unsupervised training. The decision of using

supervised or unsupervised network for its training depends on data to be processed. In the

process of supervised learning of ANNs, inputs are applied to obtain output response. These

outputs are compared with the target output i.e. the target response. If the output response

differs from the desired response, the error ‘e’ in equation (4.2) for a single training pattern is

the sum of square difference i.e.

2)(21

dd yte −= ∑

(4.2)

where,

td represents Target response for dth unit,

yd represents actual output produced response for dth unit.

35

The error “e” calculates the deviation from the desired training response. To minimize the

error, optimization should be made to the network’s synaptic weights. Contrary to supervised

learning, unsupervised learning does not require a teacher; i.e. no desired output is required.

It is typically used in the background of recurrent and competitive nets. In unsupervised

learning the training set is not separated into input and output pairs during training, the neural

net receives many different excitations, or input patterns, and it randomly classifies the

patterns into categories. Although unsupervised learning does not require a teacher, it needs

guidelines to form groups. Grouping may be based on the properties of data. The basic

accepted categories are supervised and unsupervised learning but Semi-supervised learning

algorithm is prepared based on the supervised learning experiences [4]. The results of this

algorithms are more encouraging than the typical supervised learning [12].

Chavez [87] explained the Blue Brain project and future direction in ANN. He also defines

that two major problems of ANN are teaching and designing networks [87] and better

teaching (training) is the major concern in modelling module while doing training, testing and

simulation process of ANN modelling.

Dhupal, et al. [26] utilizes RSM and ANN to build model for the prediction of microgroove

width and depth phenomenon. Author built RSM model for the optimization of the response

variables and verified the optimal results with ANN predictions and experimental data. The

results were satisfactory and also proved the ability of ANN to use this tool in other laser

cutting applications.

In the extrusion process of food material problem ANN modelling executed better than

Response surface methodology (RSM) [32]. In fact RSM uses predefined linear or quadratic

regression modelling to model the surface. M. J. Tsai, et al. [34] prepared multi-regression

analysis and artificial neural network model for Quad Flat No-Lead (QFN) cutting of 3 input

and 6 qualities.

The results of LM-back-propagation neural network were better than regression model. ANN

model was trained with full factorial design. Complex and non-linear problems model by

ANN definitely gives better approximation than regression. Hence implies that it is better

than RSM. But there is a need to verify that ANN, whether works better even with the

missing values or not.

36

M. Zaidi et al. applied that multi-regression, ANOVA models which are not sufficient for

solving this problem, as a matter of fact it was shown that average error reaches 50% which is

a huge error due to small dataset of OA and missing values in observation table. This is the

reason why neural network approach was adopted in [2, 4].

Sivarao, et al. [33] used ANN in the area of laser cutting without missing values. The process

design requires subject expert participation throughout the design of experiment. But

application of genetic algorithm reduces the dependency of subject expert. They used full

factorial design L128 which is a large data size compared to M. Zaidi et al. [3, 4, 12, 25]

works using quick back-propagation training algorithm. ANN modelling is often suitable for

large number of observations and non-linear data. ANN learns from the available experience

(input-output datasets) and captures the functional relationship between the input and output

parameters. It is quoted in different papers that any continuous function can approximate to

any required accuracy by ANN network [4, 12, 29]. Sivarao, et al. [33] claimed that the

ability of ANN prediction of surface quality in laser cutting process minimizes the cutting

cost by as much as 70% of the overall manufacturing because of rework.

C.B. Yang, et al. [35] also prepared a model of a CO2 laser cutting process of polymethyl

methacrylate PMMA using orthogonal array of L9 for four factors with three levels but

triplicated the data size by statistical modelling prediction and then prepared ANN model at

the second stage with large datasets which shows OA DOE is not sufficient for ANN

modelling and also in case of supervised learning by Zaidi and Amin et al. [4, 35].

The combined progressive model Taguchi and neural network prediction improved with

small datasets. Taguchi modelling is difficult with unbalanced matrix means with missing

values and need the experimentation in the supervision of domain expert. In this paper the

deficiency of ANN on small datasets such as L9 is overcome by increasing data by Taguchi

modelling. But in our study small datasets were modeled and moved a step ahead by handling

missing values. The author admits that the Taguchi modelling is inferior so increased data

sets are also not accurate and in case of interaction this method is not useful.

Zaidi and Amin et al. [3] developed a model of laser cutting process with missing values by

LM-feed forward neural network. The results were verified with Taguchi method results. But

37

Taguchi model itself not verified by experimental data. Therefore, more study is required to

verify the modelling technique. In supervised learning the quality of model vary with the

ability of modelling of supervisor. This can be further standardized by semi-supervised

learning.

This ANN approach is sometimes not the best however, it enables to produce a less precise

model in difficult scenarios such as non-linear problems and in case of missing values [4] to

give a rough estimate. The need of a program to replace supervised learning by semi-

supervised approach create networks, vary their values and initialize synaptic weights many

times automatically to find the global minima on the error surface. The Semi-supervised

algorithm search out the better network architecture, training algorithm and most probably

global minima based model. It is learned from neural networks that it is possible to train large

systems with many inputs on the basis of relatively small data sets [3]. The resulting systems

usually have a moderately nonlinear structure.

M. Zaidi and I. Amin et al. [4] utilized supervised learning and then established semi-

supervised technique for normal and missing values data for training and simulation [12]. In

this modelling, the benefit of ANN is in learning of nonlinear, multivariable datasets of

different DOE. The model is able to predict desired input values for the optimization. By

using the trained network the machine setting time and human error on the new work-piece

can be reduced [2].

4.2 FEED-FORWARD BACK-PROPAGATION NEURAL NETWORK The simple and effective nature of feed-forward back-propagation is most widely used in

artificial neural network structures, having been used effectively in many research works

such as [2-4, 10, 24, 29-31, 35, 89]. Figure 4.1 is a supervised training neural network,

containing “n” numbers of neurons interconnected to form an input layer, with hidden layers

and then to output layer. The input and output layer nodes work as a buffer for input and

output values of the predictive model and hidden layer forwarded input relations to be

represented in the output layer.

In the beginning of training session, inputs and target values are run through the network

with randomly chosen weights for the nodes, which makes the network analogous to

newborn's brain but without knowledge [88]. With reference to paper [27] he explains the

38

concept of back-propagation mathematically. Each hidden and output neuron processes its

inputs by multiplying each input by its weight, summing the product and then passing the

sum through a non-linear transfer function to produce a result. The S-shaped sigmoid curve is

commonly used as the transfer function. The neural network learns by modifying the weights

of the neurons in response to the errors between the actual output values and the target output

values. This is carried out through the gradient descent on the sum of squares of the errors for

all the training datasets. The changes in weights are in proportion to the negative of the

derivative in the error term. One pass through the set of training patterns along with the

updating of the weights is called a cycle or epoch. Training is carried out by repeatedly

presenting the entire set of training patterns (with the weights updated at the end of each

cycle) until the average sum squared error over all the training patterns are minimized and

within the tolerance specified for the problem or reach to the limit of number of epoch.

Back

Propagation

Feed Forward

A1 A2 A3

y1 y2 y3

………………………………………………….

………………………………………………….

………………………………………………….

Output layer

Hidden layer

Input layer

Input values

t1 t2 t3 Target values

Figure 4-1: Feed-forward back-propagation model

39

At the end of the training phase, the neural network should correctly reproduce the target

output values for the training data provided the errors are minimal, i.e. convergence occurs.

The associated trained weights of the neurons are then stored in the neural network memory.

In the next phase, the trained neural network is fed a separate set of data. In this testing phase,

the neural network predictions are compared with the target output values and evaluate the

dependability of the neural network to generalize correct responses for the testing datasets

that only broadly resemble the data in the training set. No additional learning or weight

adjustments occur during the testing phase. Once the training and testing phases are found to

be successful, the neural network is ready to be used in practical applications. The neural

network will produce almost instantaneous results of the output for the practical inputs

provided. The predictions should be reliable.

After experimental observations and calculations the edge quality, kerf width, data tables are

prepared based on the requirement of DOE. The mean and S/N values are calculated and final

observation tables prepared. The training data and this trained network is then able to predict

other unknown inputs. As a result, simulated data results are nearer to the desired results and

the tolerance range gives the inputs parameter values the range applied while training [27] .

In order to obtain global minima rather than local the supervised network teacher changes the

initial network weights several times on different architectures and training algorithms. At

times modifications are made in network settings such as changes in the quantity of hidden

layers, neurons and training algorithms to obtain minimum error from the desired output with

tolerance near 10% in order to have better supervised training. This process of modifications

is called "training". A sigmoid function in the hidden layer and a linear function in the output

layer are being used by the back propagation neural network, which is applied in this study.

4.3 COMPARISON OF TRAINING ALGORITHMS In ANN modelling and simulation process first create a network consisting of input, output

with selection of training algorithm, number of hidden layers, number of neurons and

selection of transfer function with the selection of feed forward back propagation and a

training algorithm. Create a network and export to workspace. Programming is carried out in

MATLAB (The Language of Technical Computing).

In ANN wide range of learning algorithms exist, the greater part of them including the

famous back-propagation learning algorithm, are of the gradient descent type [24, 90]. The

http://www.mathworks.com/products/matlab/�

40

network is prepared for training after initialization of network bias and weights. The network

mapped the input and output examples to get the approximate transfer function on the basis

of given datasets. The inputs and target output are iteratively minimizing the error. Some of

the training algorithms for feed forward back propagation are explained in the following

section. All the following training algorithms utilize the negative gradient of performance

function to set the weights to minimize error in desired and actual output. The negative

gradient is calculated by back propagation method.

4.4 BACK PROPAGATION ALGORITHM Even single hidden layer can be employed to approximate any nonlinear function [91] but

the selection of training algorithm is of vital importance. Some of the training algorithms for

feed forward back propagation are explained for this purpose. All these algorithms use

negative gradient and it is calculated by back propagation method. Back propagation

algorithm is simply an algorithm which changes weights and bias on the basis of previously

calculated mean squared error to reduce it. The algorithm is iterative in nature and can be

expressed in the form of a pseudo code as follows:

1. First randomly or by some systematic means initialize the weights

2. Get the output from the network with these weights

3. Calculate error by subtracting output from known input output mapping

4. Compute new weights or delta from output layer to hidden layers

5. Compute for all successive layers to input layer

6. Change the old weights with the new weights

7. Repeat from step 2 until network converges to desired error

Goh [27] model the two engineering problems to show the back-propagation neural networks

usability. These two problems are noisy data and significant participation of interaction

effects that complicates engineering problem. Both complex modelling method along with

supervisor intuition and experience are considered necessary for the forming of these

practical relationships. Fame of ANN has increased as compared to statistical methods for

nonlinear problems. The statistical methods are needed to identify significant factors first,

ANN gives low weightage to less significant variables without the interference of the

researcher. Goh [27] also mentioned that the inherent core property of ANN is that with

incomplete input information it can match correct output in pattern recognition. Hence, it can

41

work with missing values in experimental data models. This advantage increases the

utilization of ANN in both quantitative and qualitative problems.

4.4.1 GRADIENT DESCENT ALGORITHM Initialize the network with required number of layers with desired neurons and randomize the

weights. Apply input output for mapping on neural network. Now do the forward

computation of the network from book [92]. First calculate the induced level fields by the

following formula

VjL(m) = �Wji

L(m)Yi(L−1)

i

(m) (4.3)

Where

Yi(L−1)(m) is the ith neuron output signal function of preceding layer L-1at mth iteration

Wji(L)(m) is jth neuron synaptic weight in hidden layer L from ith neuron L-1(previous)

In case i=0

Yj0(L)(m) = Bj

(L)(m) is the bias applied to neuron j in layer L (hidden) and Y0(L−1)(m) =

+1 . Now apply the desired transfer function. The output signal of neuron j layer L is

Yj(L) = φj(Vj(m)).

If L=1 means neuron j is in the first hidden layer, set

yj(o)(m) = Xj(m) (4.4)

where

Xj(m) is the jth component of the input vector x(n). If jth neuron is in the output layer. i.e. L

= OL, apply

YjOL = Oj(m) (4.5)

The error signal is

Ej(m) = Dj(m) − Oj(m) (4.6)

Further do forward computation from book [92]. Calculate the local gradient (δs ) of the

network

Dj(m) is the jth element of the desired response vector D(m).

δjL(m) = �

EjOLφj

′ �VjOL (m)�

φj′(Vj

L(m))�δkL+1(m)Wkj

L+1(m)k

�

For jth neuron in output layer OL

For jth neuron in hidden layer L

(4.7)

Where

42

φj′(∗) is shows differentiation w.r.t the expression within brackets.

Fine-tune the synaptic weights of the neural network in layer L as per “generalized delta rule”.

The individual weight can be calculated by delta rule

WjiL(m + 1) = Wji

L(m) + [WjiL(m− 1)] + η δj

L(m)YiL−1(m) (4.8)

Where

η is the learning rate parameter.

Iteration keeps running until its stopping condition met. Iteration computation runs forward

and backward as shown in above derivation [92]. Batch gradient descent algorithm is not

very effective because learning rate is adjusted manually. If it is very small, then there is high

probability of being erroneous and getting trapped in a local minima, while a large learning

rate may cause training to become unstable due to overshooting. It is also called steepest

descent. The gradient descent algorithm often gets stuck in local minima in error surface as

shown in Figure 4.2.

Error Surface

Local Minima

Global Minima

Learning rate W1 old

W1 new

W1

Error

Figure 4-2: Searching of global minima in error space

4.4.2 GRADIENT DESCENT WITH MOMENTUM It is easy to understand that gradient decent algorithm learning can be time consuming

including the issue of local minima. The idea of momentum improves the local minima

43

concern and increases the rate of convergence considerably fast. This method resolves the

matter of batch gradient descent in case of small learning rate. The convergence speed

reduction and convergence trapped in the local minima is resolved by momentum. The

gradient descent with momentum works as a low pass filter. The momentum characteristic

ranges from 0 to 1. Zero means no momentum and 1 means high momentum. Gradient

descent with momentum (GDM) is a good algorithm as proposed in [93]; still it depends on

the user to tell it what the momentum term should be, none the less it has a probability of

finding the global minima. A mathematical representation would be:

WjiL(m + 1) = Wji

L(m) + α[WjiL(m − 1)] + η δj

L(m)YiL−1(m) (4.9)

Where α represents momentum parameter. However, probability to reach global minima

seems lesser than Quasi Newton and Levenberg Marquardt.

Ranaganth and Viswanath [31] prepared laser cutting process (2KW CO2) model of surface

roughness by Back propagation using gradient decent with momentum for mass production

and following optimization of cutting process. No specific design of experiment was used but

15 run were used for training and 5 for validation. The best selected learning rate 0.6 and

momentum 0.8 are required to achieve the desired error tolerance. The desired error goal was

met after forty one times initialize weights and 29653 iterations. They also used the technique

of weights randomization up to 41 times which shows that this is a difficult method to train

the model in supervised learning. In my point of view semi-supervised algorithm is a useful

solution for the more generalized way of modelling [12].

4.5 FASTER TRAINING ALGORITHM Gradient descent method (GD) is slower than faster training algorithms like variable learning

rate and resilient back propagation. Both of the faster algorithms are also heuristic. It is

difficult to adjust learning rate before the training. Advantage in these algorithms is that the

optimal training rate is changing during the training process as the algorithm moves in the

error space. Therefore, learning rate change during the training improves with the training

process. The dynamic and adaptive change in learning rate increases the size of learning rate

until the training process is unstable. The algorithm of variable learning with the feature of

momentum is a better choice.

44

4.5.2 Quasi Newton Algorithm The point at which the derivative of a function is zero is called the stationary point of that

particular function. Newton proposed a mathematical method to locate the stationary point on

a function, in our case error surface. The Quasi Newton Algorithm has its origins in that

function. Newton made an assumption that the area around the optimum can be approximated

to be quadratic in nature. The advantage of the Quasi Newton method is that it does not

require the computation of second derivate Hessian Matrix instead; one after another gradient

vectors are analyzed and used to update the Hessian Matrix. Quasi Newton is used as an

alternate to conjugate gradient methods and is mostly faster because there is no need to

calculate second derivatives. The modified Hessian matrix, at every iteration, is a function of

gradient. It can be explained mathematically as follows:

iiii glww 1

1−

+ −= (4.10)

Where li-1 is a Second derivative Hessian matrix of MSE index at present values of bias and

weights in equation (4.10).

But usually in the domain of neural networks this algorithm is used when training is of small

networks [93] and conjugate gradient algorithms are more suitable for large networks.

Therefore, Quasi Newton’s algorithm is better for the given case.

Secondly it must be noted that this algorithm requires a line which it uses to find the direction

in which it starts the descent, once that is set it goes into the depth of that line. The default

line function for Quasi Newton is “Charalambous’ method for 1-D minimization”.

4.6 LEVENBERG-MARQUARDT ALGORITHM The Levenberg-Marquardt training algorithm is repeatedly used as optimization algorithm. It

outperforms simple gradient descent and other conjugate gradient methods in a wide variety

of problems [94]. Utilization of Levenberg-Marquardt Algorithm (LMA) eliminates the

importance of Quasi-Newton, because it reaches to second order training pace without

calculating Hessian matrix. Therefore mean square error (MSE) is used as performance

criteria and Hessian matrix becomes the following equation; JJH T .= (4.11)

Where J is a Jacobian matrix which consists of all first order partial derivatives of network

∂ (error)/ ∂ (biases and weights) in equation (4.11) and the reason because of which partial

derivative is needed is that the error is partially dependent on many variables. As shown in

45

the equation we are taking partial change in error with respect to weights and biases. Whereas

gradient g is computed as

eJg T .= (4.12)

Where e is the error vector of the network in (4.12). The LMA can be expressed as

[ ] eJIJJww TTii

11

−

+ +−= µ

(4.13)

The intelligence lies in the µ symbol in equation (4.13), if µ is zero then it becomes equation

(4.10) i.e. Quasi Newton’s or if µ is very large then it becomes equation (4.9) i.e. Gradient

descent method with a smaller step size. As Newton’s method performs faster convergence

near error with more accuracy, therefore in LMA µ value reduces after each better iteration,

or increases in case of inferior iteration results on the basis of MSE. So that it does not get

stuck in a “local minima” and when it is away from minima it acts like a normal gradient

descent with momentum.

According to [93] the LMA is suitable for medium size datasets and fastest compared to other

training algorithms. The only drawback lies in the large memory it requires to handle the

matrices, but since our requirement is small size datasets and analysis is offline in laser

cutting process modelling. Therefore, LMA is the most suitable theoretically and the results

favor it from our modelling and simulations. Kuo et al. [28] model the Light guide plate

(LGP) manufactured by CO2 laser cutting using L18 Taguchi orthogonal array as DOE. Back

propagation neural network with Levenberg-Marquard training algorithm is used to model

the problem. They used 13 datasets for training and 5 for Test. The simulation error was

within 5%. The model is without missing values and size is double the size of L9.

J. Ciurana, et al. [78] also built up many models for laser milling process by evaluating the

roughness, geometric and dimensional characteristics. Results with ANN are better than

multiple linear regressions and it should be noted that one training algorithm was not applied

to all models. The results of resilient back-propagation algorithm predicting are optimum for

surface roughness and LM algorithm has performed better for predicting geometrical and

dimensional characteristics. The LM algorithm approach reduced the size of neural network

and produced better prediction fast conversion. The size is also three times the L9 orthogonal

array. The training is utilizing supervised learning in both the above studies. However, semi-

46

supervised learning does not need supervisor’s expertise and reduces the time consumption of

the researcher to select the best model.

4.7 OVERTRAINING OF DATA MODEL Robert P.W. Duin [95] raised the important issue that well trained networks generalize in the

sense that their outputs are (close to) correct for objects not available in the training set. If the

training set is noisy or contains errors (like overlapping classes in case of a classification

problem), a too heavily trained network may be adapted to the noise which does not

generalize. It is over-trained and produces random outputs for unseen input objects. Sivarao,

et al. [23] also mentioned that some datasets are required out of designed matrix for checking

of generalization of model and testing of the model. This disadvantage can be resolved by

modifying design of experiment or without investing on the experimentation by using the

observation obtained for range adjustment before the DOE based observation tables recording

or to rotate the some parts of the training dataset and the test dataset. Overtraining is

sometimes related to strong nonlinearity. This can be detected by the size of the weights.

Only for large weights the network is able to have a strong nonlinear functionality.

Overtraining [95] may be avoided by:

• Early stopping: Stopping the training in time, before an adaptation to the noise starts.

It is difficult to detect an optimal stopping moment without the use of a test set.

• Small networks: If the network has much less parameters than objects in the training

set, it cannot be over-trained.

• Pruning: reducing the size of a network after it is trained. In this way unimportant

neurons, causing noise, are deleted. It is necessary to retrain the network after

pruning.

• Large training sets: Networks of a fixed size cannot be over-trained for large

training sets.

• Data enrichment: artificially enlarging the training set by objects that ‘smooth out’

the noise.

• Regularization: One way to do this is to add a penalty term to the optimization

criterion for networks with large weights, as these are related to strong nonlinearity.

Over training can be avoided with other methods such as Z. Xu and Z. Mao [96] presented

support vector machine (SVM). SVM is also able to handle overtraining problem with small

47

datasets. Therefore, it can be used in laser cutting process modelling due to its ability to

model small datasets.

4.8 SUMMARY Biological networks work in parallel and ANN follow Von Neumann architecture. ANN is

capable to model non-linear problem and is analogous to neurobiological systems. ANN

attains knowledge through learning and knowledge is stored within synaptic weights. Often

ANN is distinguished into supervised and unsupervised training. The two major problems are

teaching and designing networks where better training is the major concern in modelling.

Sometimes RSM and ANN model have better results. The usage of test data for

generalization made ANN superior than RSM, regression and multi-regression，non-linear

regression and ANOVA models are not sufficient for solving this problem. However,

generally ANN is trained on FD for non-linear problems with different training algorithms

such as Quick back-propagation training algorithm was used for LCP with FD (L128) which

is large data size compared to OA. But it was claimed that ANN prediction reduces the need

of rework. With FD many researchers succeeded in modelling different materials in LCP.

However, many used Fractional factorial and L18 larger datasets without missing values.

They also used L9 to prepare a model of a CO2 LCP of PMMA sheets with L9 runs for four

factors with three levels similar to our data. But triplicate the data size by Statistical

modelling prediction and then prepared ANN model at the second stage with large datasets

which shows that they are able to model with L9 OA with ANN. Similarly, in combined

progressive model L9 was expanded with TM. Although it is difficult to model unbalanced

matrix and unable to handle interaction effects with TM.

Many researchers quoted that any continuous function can approximate to any required

accuracy by ANN network and even single hidden layer can be employed to approximate any

nonlinear function, but the selection of training algorithm is of vital importance. Regression

models are overstrained but ANN generalizes due to testing of model on unseen datasets.

Overtraining can be avoided by early stopping of training, network Pruning, reducing number

input parameters, large training data, data enrichment by duplicating datasets or adding

penalty terms. Feed-forward back-propagation ANN explained and compared the important

training algorithms and justifies its utility. Batch GD algorithm is not very effective as its

learning rate is adjusted manually. If its value is small then there is high probability of getting

48

trapped in local minima, while a large learning rate may cause training to become unstable.

But, the idea of momentum improves the local minima concern and works as a low pass filter,

it also increases the rate of convergence considerably swiftly. The LCP (CO2) model the

surface roughness by back-propagation using GDM training algorithm without using any

specific DOE using 18 runs. But, it required more time than LMA. The technique of weights

randomization used up to 41 times with GDM which indicates the importance of this

parameter. In Quasi Newton’s training algorithm, the point at which the derivative of a

function is zero is called the stationary point of that particular function. It assumed that the

area around the optimum can be approximated as quadratic in nature. Therefore, no need of

computation of second derivate (Hessian Matrix), instead one after another gradient vectors

are analyzed and used to update the Hessian Matrix. Quasi Newton algorithm is better to use

for small networks and conjugate gradient algorithms are more suitable for large networks.

Utilization of LMA eliminates the importance of Quasi-Newton, because it reaches to second

order training pace without calculating Hessian matrix. The intelligence lies in the µ symbol

in equation (4.13), if µ is zero then it becomes Quasi Newton’s and if µ is very large then it

becomes Gradient descent method with a smaller step size. As Newton’s method performs

faster convergence near error with more accuracy, therefore in LMA µ value reduces after

each better iteration or increases in case of inferior iteration results on the basis of MSE. So

that it does not get stuck in a local minima and when it is away from a minima it acts like a

normal GDM. LMA is suitable for medium size datasets and is fastest compared to others and

GDM training algorithms. It needs large memory but since our requirement is small size

datasets. GDM and LMA are more suitable theoretically and the results from our modelling

favor them, LMA is time efficient. In order to obtain global minima rather than local, the

supervised network teacher changes the initial network weights many times on different

architectures and training algorithms. At times modifications are made in network settings

such as changes in the quantity of hidden layers, neurons and training algorithms to obtain

minimum error within 10% in order to have better supervised training. Hence GDM and

LMA are the best training algorithm for the given problem.

CHAPTER 5

OVERALL QUALITY OPTIMIZATION

49

5. OVERALL QUALITY OPTIMIZATION Bahar and Golnabi [40] state that researchers usually optimize laser cutting process for single

quality and also use OFAT technique. In this dissertation Overall quality means considering

more than one quality for optimization. The solution is able to optimize two qualities at a

time. However, aggregation is better for more than two qualities such as three or four or more.

Some other significant multi-objective research has been mentioned earlier in introduction

and laser cutting literature review but some specialize advance Statistical methods are

discussed under the Multi-objective Statistical Methods heading. For more than two variables

optimization suitable data mining techniques are simple aggregation, normalized

aggregation, , Statistical methods, Genetic Algorithm (GA) or fuzzy logic (FL). The objective

is to provide a product or work-piece with better overall quality. The aim of this research is to

find out the overall quality (edge quality, kerf width, overcut and material removal rate) with

Electricity-efficient to predict optimize input datasets. The relationship between parameters

can be found by the modelling of the system by some Mathematical or Statistical methods or

Soft-computing techniques [2-4, 12, 25] . Overall quality utilizes large experimental data.

With reference to [3] the overall quality is measured on the basis of very small data sets of

OA based experimental observation. The artificial neural network (ANN) training was

performed on L9 orthogonal array aggregated data and overall quality was predicted on the

basis of simulated data. In both papers [2, 3] orthogonal array design was used to obtain

experimental data. But in [2] normalized aggregation was used instead of simple aggregation

on simulated data. The Aggregation function was further checked by customer quality

function [2]. Yilbas and Rashid [41] cuts 800HT alloy by CO2 laser and performed

experiment using Factorial design (FD) for overall quality of flatness (F) and waviness (D).

However they used simple aggregation by converting the values in score i.e. 1, 2 and 3. The

calculation is simple and crisp in nature. It was discussed in literature survey of [70] that GA

algorithm is also suitable for single response and multi-response problems. It was concluded

by Chatsirirungruang [70] that the application of GA and Taguchi method reduces average

loss and is less responding to noise i.e. improved robust system. Noor et al. [8] mentioned the

fact that the utilization of Particle swarm optimization (PSO) also has many benefits

compared to other evolutionary heuristic algorithms like GA such as faster convergence, less

number of parameters to adjust and easy application. Therefore, it is better to use partial

50

swarm or genetic algorithm in robust optimization mechanism. A combination of novel

Fuzzy aggregation is far better option for overall quality prediction.

5.1 GENETIC ALGORITHM The aim of robust process design is to build a stable process that produces minimum

variation due to uncontrollable parameters [60]. In recent times, Taguchi methodology has

become a practical technique for the improvement of productivity at low cost. Researchers

utilizing Taguchi methodology have experimentally studied the effect of input parameters on

kerf taper, kerf deviation, kerf width, heat affected zone and surface roughness in the process

of laser cutting. Taguchi methodology has been used for multi-objective optimization.

However, weight assignment is major task of multi-objective optimization. Therefore, many

researchers used hybrid techniques with Grey relational analysis [36, 61], fuzzy [39] etc. The

application of GA is also better technique and it provides better techniques of simultaneous

optimization of multiple dependent parameters [68-70] . The ability of genetic algorithm to

generate huge amount of possible population and selection on cost function basis made it an

important off line quality tool [60]. It is improbable to trap in local minima due to the

assumption of GA and does not depend on continuity, uni-modality or convexity of functions

[97]. GA using mutation, crossover and reproduction genetic operations then grow successive

generations and apply values to transfer function to predict the output quality for optimized

value [60].

Multi-objective problem can be solved by common genetic algorithm (CGA) by combining

both the quality parameters with or without weighting factor using averaging or aggregation

etc. It can be solved by multi-objective genetic algorithm (MOGA) [98]. It was concluded by

Fernandes and Vicente et. al. [98] that MOGA is probably faster and efficient. Orthogonal

array is used in this research but the selection of ordinal levels can be optimized using genetic

algorithm [99]. The study concludes that genetic algorithm can be applied at DOE level.

Chatsirirungruang and Masami Miyakawa [68] study the usefulness of GA as compared to

OA method. They used signal to noise ratio as a deciding factor as routine practice of

Statisticians but the criteria is not always true. Therefore, noise factors (outer array) are used

to solve the problem more efficiently in his all papers [68-70]. They apply GA to find

optimum controllable parameters with the support of outer array [68]. They also performed

more research by the same method including the handling of dynamic system [69].

51

The research shows that GA is useful for these problems to find optimum signal to noise ratio

and target value. The reason was consideration of interaction between the variables and more

number of levels used for continuous controllable parameters than OA to find better results

[69]. They also used the benefits of Taguchi and GA for unknown problem using the

simulations by GA [70]. Hence Genetic algorithm can be utilized for overall quality based on

the predefined cost function. But, it is preferable to use outer array at the time of DOE for

better results.

Pawar, et al. [62] applied Particle swarm optimization (PSO) to solve multi-objective

optimization problem of rough and finished grinding process parameters of three responses.

The method of constraint is handled by penalty methods as others. The PSO results are better

than GA, differential evaluation (DE) algorithm and Quadratic programming (QP). They also

studied the convergence rate, performance and accuracy of PSO algorithm. Thirty to forty

iterations are often required by PSO, which are comparatively quite lower than other

evolutionary computational methods. However, the GA has its own limitations such as risk of

replacement of a good parent string with the deteriorated child, less convergence speed, and

difficulty in selecting the controlling parameters such as population size, crossover rate, and

mutation rate. Also the results of GA presented by the authors are erroneous.

5.2 MULTI-OBJECTIVE STATISTICAL METHODS Some other significant multi-objective research has been mentioned earlier in introduction

and laser cutting literature review. But there is a need to further discuss some research. B. S.

Yilbas [79] investigated overall quality at the cut surface due to input attributes variation in

the process of CO2 laser cutting of stainless steel. The overall quality was calculated based on

cut surface flatness, material changes and Waviness. He analyzes the significance of input

attributes on mentioned qualities and found them to be significant. He utilizes scores as in

desirability method instead of weightage and sums the scores in crisp logic to calculate

overall quality. The scores are assigned in flexible range therefore crisp does not reflect the

true participation of qualities in the overall value. Fuzzy aggregation definitely improves the

results. As standard membership functions generate standard aggregation results with the

quality of applied rules and fuzzy aggregation.

52

Dubey and Vinod Yadava [59] discussed hybrid TM and RSM for the multi-response

optimization of laser cutting process. The weighted impacts of quality characteristics and cost

components are incorporated in this quality function.

Dubey [59] concluded on the basis of modelling and optimization results show significant

enhancement in kerf width and MRR qualities with hybrid TM and RSM technique. He found

that linear input parameters are significantly participated in both the output qualities. Cutting

speed and pulse frequency interaction effects are significant. Therefore, TM is not sufficient

to solve the issue. They were able to solve the problem because they used RSM with TM.

The ability to model interaction effect in RSM produces enhanced results. But, for analysis,

modelling and optimization it is a difficult method to utilize two different Statistical methods

and two different DOE. Dubey [59] shows that interaction is significant therefore it is better

to use straight RSM DOE and utilize for modelling and optimization.

Gani and I. A. Choudhury et al. [100] used TM optimization in case of end milling process

with P10 TiN coated carbide on AISI H13 Steel tool. The high speed cutting was done with

finishing and Semi-finishing conditions. They investigated interaction effect and found TM

sufficient for the problem. The main target is to find the condition to achieve good surface

roughness and with low cutting force . The optimization was achieved by using signal to

noise ratio and Pareto ANOVA methods for detailed analysis. They concluded that often high

cutting speed, low feed rate and low depth of cut produces good Surface roughness and

cutting force is kept low.

L. K. Pan et al. [30] provides optimizations of multiple qualities of laser welding of

Titanium with Nd:YAG laser. The data analysis techniques used for integrated performance

of overall quality is TM-based Grey analysis. They mentioned the presence of interactions

among pulse shape, shielding gas, welding speed, pulse frequency, laser focus and laser

energy in laser welding. The overall optimization was calculated by Grey relational grade but

the TM limitations reduce the best results. Since very complex interaction is present in this

process and modelling was done by TM. Hence the results are definitely not best. Some other

technique at the Modelling level can be used and need to change DOE technique as per

modelling and interaction requirements such as RSM.

53

Excellent work has been achieved by Phillips and Kim [63], they developed a statistical

method, the DMT method, to evaluate and optimize multiple quality characteristic problems.

The method incorporates desirability functions, a performance statistic based on the mean

squared error, and data-driven transformations to provide a systematic approach that is

adjustable to a variety of situations and easy for non experts to apply. They take weighted

geometric mean and MSE (mean square error), the DMT method allows the decision-maker

to alter the relative importance of the quality characteristics, mean, and variance to suit the

particular situation. Taguchi parameter design offers no similar flexibility. Once the data are

converted into individual desires, either Harrington’s (1965) simple geometric mean or

Derringer’s (1994) weighted geometric mean aggregation methods may be used to combine

the individual desires.

In this method if one of the variables is undesirably high and other is very low then the

aggregated suggest wrong prediction for overall quality. Table 5-1 summarizes the difference

in Taguchi and DMT method.

H. A. Eltawahni, et al. [6, 101] developed RSM (Box-Behnken design) based model of laser

cutting of MDF and AISI316L and solved the issue of overall quality in two steps by using

the desirability approach. RSM model established all factors effect on output depending on

priority and constrained based on cost and quality. Eltawahni, et al. [6, 101] is unable to

provide single optimize overall quality. The decision is based on operator to decide whether

quality is more important or cost. It is better to use fuzzy aggregation, as fuzzy can solve the

issue of quality and cost in a simpler manner than the complicated approach which the

operator used to investigate for optimization. The verification error was 17.4 to 10.04%.

However, in his second paper he was able to minimize the error below 10%. It is required

that they consider cost-effective solution while not compromising on quality.

Ming-Fei et al. [61] achieved multi-objective optimization for two qualities (optical

transmittance ratio and roughness) of CO2 laser cutting by applying grey relational analysis.

The method of grey relational analysis can be improved by considering interaction effect by

applying better techniques in the building block of measuring quality data.

Sharma and Yadava [36] developed second order RSM and Taguchi model for Aluminium

alloy sheet by ND:Yag pulsed laser . Weights of both qualities were measured by Taguchi-

54

based GRA with entropy measurement methodology. Taguchi-based GRA is an effective

approach to attain optimum values of both the responses only by orthogonal array and needs

Statistical expert. However, is unable to model with missing values because unbalanced

matrix calculation is a difficult task. M. Zaidi et al. [12] modelled L9 orthogonal array and

FD model with missing values by ANN.

Table 5-1: Difference between DMT and Taguchi methods

Method description Taguchi Method DMT Method

Design of orthogonal array Yes Yes

Noise Factors Yes Yes

Analyze Mean and variance Yes Yes

Signal to noise ratio Yes No

Desirability functions reduces multiple dimensions No Yes

Optimization of design factor levels by graph Yes Yes

5.3 FUZZY AGGREGATION Understanding of fuzzy logic in addition to classical crisp logic starts from the work of

Zadeh’s Sets theory paper which provides revolutionary ideas of fuzzy sets and

computational rule of inference [102]. The set (fuzzy) is a class of objects with scale of

number of grades of membership (MS). Fuzzy set is described by a MS function which

allocates to each object a score of MS ranging between 0 to 1. The idea of crisp sets is also

proved in fuzzy sets such as union, intersection, compliment and convexity etc. Fuzzy logic

deals with the classes of objects which don’t have accurately defined measure of MS [102]

such as class of beautiful girls and class of tall person. These sets or classes are not present in

crisp mathematics.

This concept is not only given to explain these fuzzy sets but to give a parallel framework

dealing like human for such problems [102]. Fuzzy logic is a very useful technique in the

areas of pattern recognition, information processing, classification and aggregation etc. The

concept of membership function raises the utility of fuzzy logic due to its wide variety of

functions and possibility of optimization of the shapes of the MS functions. Sivarao and

Brevern et. al [9] used fuzzy logic concept in the laser cutting process. Adapting network

fuzzy inference system ANFIS of MATLAB was selected for introducing to the researcher

55

working in the area of laser cutting modelling [9]. In the laser cutting machine higher number

of controllable variables has to be controlled as compare to other modern machine. It is more

scientific to model the problems such as kerf width and surface roughness.

With the increase in parameters, machining process increases and produces complexity which

introduces the usage of computer (neural networks, genetic algorithms and fuzzy logic) [9].

Sivarao [9] explained FIS system of MATLAB and identified the future work of comparative

study by changing the models based on FIS variables.

5.4 SUMMARY Researchers generally optimize laser cutting process for single quality and use OFAT

technique. For more than two variables optimization, suitable data mining techniques are

simple aggregation, normalized aggregation, Statistical methods, Genetic Algorithm (GA) or

fuzzy logic (FL). The aim of this research is to find out the best possible overall quality (edge

quality, kerf width, overcut and material removal rate) with Electricity-efficiency to predict

optimized input datasets.

Review shows that results of GA are better than TM and more superior with outer array.


both the quality parameters with or without weighting factor using averaging or aggregation.

It can be solved by multi-objective genetic algorithm (MOGA). It is probably faster and

efficient. The PSO results are better than GA, differential evaluation (DE) algorithm and

Quadratic programming (QP). They also studied the convergence rate, performance and

accuracy of PSO algorithm. Thirty to forty iterations are often required by PSO, which are

lower than other evolutionary computational methods. Hybrid TM has been used for multi-

objective optimization. However, weight assignment is a major task of multi-objective

optimization. Therefore, many researchers used hybrid techniques with Grey relational

analysis, it may be used with fuzzy aggregation. Some researchers used unnecessary hybrid

combination of TM and RSM for modelling and optimization results show significant

enhancement in kerf width and MRR qualities. They found that linear input and interaction

effects are significant in both the output qualities. It is difficult to utilize both Statistical

methods, even RSM DOE is capable to model the problem. Overall optimization of laser

welding of Titanium with Nd:YAG laser was performed with TM-based Grey analysis. They

mentioned the presence of interactions among pulse shape, shielding gas, welding speed,

56

pulse frequency, laser focus and laser energy in laser welding, but TM limitations reduces the

best results.

Desirability method, DMT, is commonly used to evaluate and optimize multiple quality

characteristic problems. The method incorporates desirability functions, a performance

statistic based on the mean squared error, and data-driven transformations to provide a

systematic approach that is adjustable to a variety of situations and easy for non experts to

apply. They take weighted geometric mean and MSE (mean square error), the DMT method

allows the decision-maker to alter the relative importance of the quality characteristics, mean,

and variance to suit the particular situation. Taguchi parameter design offers no similar

flexibility. Once the data are converted into individual desires simple geometric mean or

weighted geometric mean aggregation methods may be used to combine the individual

desires. In this method if one of the variables is undesirably high and other is very low range

then the aggregated suggest wrong prediction for overall quality.

RSM (Box-Behnken design) based model of laser cutting of MDF and AISI316L solved the

issue of overall quality in two steps by using the desirability approach. RSM model

established all factors effect on output depending on priority and constrained based on cost

and quality but is unable to provide single optimize overall quality. The decision is based on

operator to prioritize between quality and cost. It is better to use fuzzy aggregation, as fuzzy

can solve the issue of quality and cost in a simpler manner. Ming-Fei was able to achieve

multi-objective optimization for two qualities (optical transmittance ratio and roughness) of

CO2 laser cutting by applying grey relational analysis. The method of grey relational analysis

can be improved by considering interaction effect by applying better techniques in the

building block of measuring quality data. Sharma developed second order RSM and TM for

Al alloy sheet cutting by ND:Yag pulsed laser. The weights of both qualities were measured

by Taguchi-based GRA with entropy measurement methodology. It is an effective approach

to attain optimum values of both the responses only by OA and needs Statistical expert.

However, is unable to model with missing values because unbalanced matrix calculation is

not an easy task.

Understanding of fuzzy logic in addition to classical crisp logic starts from revolutionary

ideas of fuzzy sets and computational rule of inference by the work of Zadeh’s Sets theory

paper. Fuzzy set is described by an MS function which allocates to each object a score of MS

57

ranging between 0 to 1 with a wide variety of functions and possibility of optimization of the

shapes of MS functions. This concept is not only given to explain these fuzzy sets but to give

parallel framework dealing like human for such problems. Researchers used fuzzy logic for

expert systems and modeling in LCP. Adapting network fuzzy inference system ANFIS was

also workable in the area of laser cutting modelling. In the laser cutting machine higher

number of controllable variables have to be controlled as compare to other modern machine.


introduces the usage of computer (neural networks, genetic algorithms and fuzzy logic).

Earlier, we had not mapped input and output like others. Mapped input with aggregated

overall quality, which was aggregated after normalization on the basis L9 datasets. However

was unable to see individual values of each quality. Laser cutting of 800HT alloy by CO2

laser, experiment was performed using FD for overall quality of flatness and waviness.

However they used simple crisp aggregation by converting the values in score i.e. 1, 2 and 3.

They utilized scores as in desirability method instead of weightage and sum the scores in

crisp logic to calculate overall quality. The Aggregation function was further checked by

customer quality function used by the author. Fuzzy aggregation definitely improves the

results. As standard membership functions generate standard aggregation results with the

quality of applied rules and fuzzy aggregation. The solution with Fuzzy aggregation is further

explained in Chapter 6 and in complete detail in Chapter 8.

RSM (Box-Behnken design) based model of laser cutting of MDF and AISI316L solved the

issue of overall quality in two steps by using the desirability approach. RSM model

established all factors effect on output depending on priority and constrained based on cost

and quality but is unable to provide single optimize overall quality. The decision is based on

operator to prioritize between quality and cost. It is better to use fuzzy aggregation, as fuzzy

can solve the issue of quality and cost in a simpler manner. Ming-Fei was able to achieve

multi-objective optimization for two qualities (optical transmittance ratio and roughness) of

CO2 laser cutting by applying grey relational analysis. The method of grey relational analysis

can be improved by considering interaction effect by applying better techniques in the

building block of measuring quality data. Sharma developed second order RSM and TM for

Al alloy sheet cutting by ND:Yag pulsed laser. The weights of both qualities were measured

by Taguchi-based GRA with entropy measurement methodology. It is an effective approach

to attain optimum values of both the responses only by OA and needs Statistical expert.

58

However, is unable to model with missing values because unbalanced matrix calculation is

not an easy task.

Understanding of fuzzy logic in addition to classical crisp logic starts from revolutionary

ideas of fuzzy sets and computational rule of inference by the work of Zadeh’s Sets theory

paper. Fuzzy set is described by an MS function which allocates to each object a score of MS

ranging between 0 to 1 with a wide variety of functions and possibility of optimization of the

shapes of MS functions. This concept is not only given to explain these fuzzy sets but to give

parallel framework dealing like human for such problems. Researchers used fuzzy logic for

expert systems and modeling in LCP. Adapting network fuzzy inference system ANFIS was

also workable in the area of laser cutting modelling. In the laser cutting machine higher

number of controllable variables have to be controlled as compare to other modern machine.


introduces the usage of computer (neural networks, genetic algorithms and fuzzy logic).

Earlier, we had not mapped input and output like others. Mapped input with aggregated

overall quality, which was aggregated after normalization on the basis L9 datasets. However

was unable to see individual values of each quality. Laser cutting of 800HT alloy by CO2

laser, experiment was performed using FD for overall quality of flatness and waviness.

However they used simple crisp aggregation by converting the values in score i.e. 1, 2 and 3.

He utilizes scores as in desirability method instead of weightage and sums the scores in crisp

logic to calculate overall quality. The Aggregation function was further checked by customer

quality function used by the author. Fuzzy aggregation definitely improves the results. As

standard membership functions generate standard aggregation results with the quality of

applied rules and fuzzy aggregation. The solution with Fuzzy aggregation is explained in

Chapter 6 and in complete detail in Chapter 8.

CHAPTER 6

PROPOSED MODULAR RESEARCH

METHODOOLOGY

59

6. PROPOSED MODULAR RESEARCH METHODOLOGY Instead of difficult mathematical proof technique, an empiricism methodology is used to

solve the problem by Statistical and soft computing techniques. Propose the solution by

framework modules for Process improvement using the modular approach. The problem is

solved by the following modules as shown in Figure 6.1.

• Preprocessing of process module

• Experimental design module

• Modelling and optimization module

• Multi-quality optimization module

An empiricism research methodology is adopted to solve the problem by Statistical and soft

computing algorithms using proposed framework major focus on Computing rather than

Material Engineering. It is not as difficult as mathematical proof techniques and this

methodology is applied only on experimental data. Hence proposed framework does not

cover the theoretical/mathematical modelling problems.

In the process of laser cutting modelling or optimization of single or multiple qualities the

first module is used by every researcher as per their knowledge and demand. There are two

categories of researchers one who follow one factor at a time (OFAT) others who use design

of experiment (DOE). In both cases selection of material, laser cutting machine is necessary.

But proper screening of variables is not performed. The researchers normally need to inspect

the results carefully as part of procedure does not follow all the time. There are tools to

examine the observed data such as outlier analysis.

In the process of laser cutting parameter optimization the first vital stage is “Preprocessing of

Process Module”. In this stage first select the material to prove our framework workability. It

is easy to select non-metallic material because there is margin of research [39] in the area

with low power laser cutting machines and with low cost experimentation due to low material

cost. Selection of Laser cutting machine is normally based on the material properties. The

laser cutting machines are classified into low and high power. After the selection of material

and Laser cutting machine the next step is to study the possible parameters that include

controllable and uncontrollable parameters and constant parameters for the desired response

variables. This can be done by simple statistical analysis by screening or references and

domain experts’ opinion. This module is shown in Figure 6.1 and separately in Figure 6.2.

60

A. Preprocessing of Process Module

Selection of Material Selection of Laser MachineCO2 or ND:Yag

Screening controllable and uncontrollable parameter for the Desired qualities

Selection of controllable parameters for the Desired Quality

B Experiment Design ModuleTaguchi

Method L9/L18/L27

Factorial design

Fractional Factorial RSM CCD RSM BBD RSM BBD

C Modelling Module

Regression Taguchi Factorial Fractional Factorial RSM CCD RSM BBD ANN

D Multi-quality Optimization Module

RSMMulti-

objective optimization

Crisp logic aggregation

Genetic Algorithm

Fuzzy aggregation

Prediction error

Acceptable?

Sel

ect b

est

Optimize

Sel

ect b

est

Decide the fitness rules

Select best

Produce optimized results

Figure 6-1: Proposed Framework modules for Process Improvement

61

In the next stage of “Experimental design module” which is being used by the researchers

who follow DOE techniques, as shown in Figure 6.1 and Figure 6.3. The selection of DOE is

based on linear, non-linear, multi-input, single output, multi-output, number of variables and

the cost of material. The selection of DOE is a technical job based on some preliminary

analysis and experience. If the selected DOE is identified on the first stage that it will not

work then it is better to reselect another DOE. There are a lot of researchers using design of

experiment such as Taguchi’s orthogonal array, response surface methodology and factorial

design [1, 4, 12, 25, 47] .

Based on selected DOE the modelling technique is chosen from the modelling and

optimization module which is basically modelling module. But for the improvement of single

quality, it can work for optimization such as neural networks, regression analysis, Taguchi

method and factorial analysis etc. The adequacy of model is checked and it is needed to

decide whether to improve by replacing the modelling technique or reject it and go back to

DOE stage and select different DOE and perform the experiment again. If model is adequate

and requirement of one by one quality improvement is required then provide the machine

settings at this stage for production. Otherwise go to the next stage of multi-quality

optimization module which is shown in Figure 6.1 and Figure 6.9. Select the appropriate tool

for overall quality and predict best results based on fitness rules and suggest machine setting

for overall quality. It is better to used electricity-efficient solution with desired quality.

6.1 PREPROCESSING OF PROCESS MODULE For the purpose of optimization of the laser cutting parameters the essential initial phase is

“Preprocessing of Process Module”. Selection of material in order to prove framework

workability is the first step. Non-metallic materials provide a broader margin of research [39]

when it comes to low power laser cutting machines and are inexpensive as well keeping the

cost of experimentation low. Material properties provide the basis of selection of Laser

cutting machine. The laser cutting machines are classified into low and high power.

After the selection of material and Laser cutting machine the next step is to study the possible

parameters that includes controllable and uncontrollable parameters and constant parameters

for the desired response variables. The experts’ papers, experts themselves and statistical data

mining tools can identify the important parameters. This process is called screening of the

parameters. This module is shown in Figure 6.1 and separately in Figure 6.2.

62

A. Preprocessing of Process Module

Selection of Material Selection of Laser MachineCO2 or ND:Yag

Screening controllable and uncontrollable parameter for the Desired qualities

Selection of controllable parameters for the Desired Quality

Figure 6-2: Preprocessing of Process Module part of the Proposed Framework

6.2 EXPERIMENTAL DESIGN MODULE

6.2.1 SINGLE PARAMETER CHANGE AT A TIME In most of the experimental investigations of Laser beam cutting (LBC) process, one factor at

a time (OFAT) has been varied to analyze the effect of input process parameters on output

quality characteristics or responses [13, 16, 18, 20-22]. It is proved that the one parameter

change at a time process is very lengthy, difficult to perform and high cost for large number

of treatments. It is also unable to show the interaction effects of multiple parameters on

response variables. Hence, it does not always ensure the perfect modelling. Therefore, Design

of experiment (DOE) is a vital step in the study of product process quality. There are

numerous design of experiment available in research papers, some of the important DOEs are

discussed below. The analysis of DOE based data explains the relationship that the selected

DOE is appropriate otherwise revision of the DOE is required as explained in the Figure 6.1

and Figure 6.3.

B Experiment Design Module

Taguchi Method L9/

L18/L27

Factorial Design

Fractional Factorial RSM CCD RSM BBD RSM BBD

Figure 6-3: Experiment design module of the Proposed Framework

63

6.2.2 TAGUCHI METHOD The most simple and useful DOE is Taguchi Method (TM). Its small size decreases the

experimentation time and cost. It is derived from factorial design. The orthogonal array

L9 3 (4-2) has been derived from Factorial Design L(81) 3 (4). It shows that 9 runs are derived

from 81 runs of four variables with three levels. There are a number of orthogonal designs

available such as L4, L6, L8, L9, L18, L27 [103] and a lot more depending on the number of

treatments. This experiment is robust, small in size, low cost and produces excellent results

[2, 12, 16, 23] if interaction effect among the variables is negligible.

6.2.3 FRACTIONAL FACTORIAL DESIGN The fractional factorial design is also derived from factorial design, it normally reduces the

size of experiment by 1/3 or 1/2. This model results are closer than Taguchi method. It can

explain the interaction effect and produces better model than orthogonal array [73, 104].

6.2.4 FACTORIAL DESIGN The most established and useful design is Factorial Design (FD) which has the ability to

study the variation caused by independent and interaction effects on the response qualities.

The size of runs increases as the number of input parameters increases. The feasible size is 3

at the most 7 variables with two levels and 3 to 4 variables with three levels. It can be written

as L(Runs) (Levels) (Variables) therefore, in case L(81) 3 (4) means 81 runs with four variables

contains three levels. It is very important to understand that the data converted to ordinal data

of two, three or five levels definitely loses the information during the modelling based on this

ordinal data. However, Factorial design is closer to the actual model than the extracted or

derived DOEs such as Taguchi and fractional factorial design. While factorial design

experiment are unable to search optimize quality parameter directly, but are able to produce

basis for the results. However, RSM does better optimization [73, 104] .

6.2.5 RESPONSE SURFACE METHODOLOGY (RSM) Response surface methodology RSM is a statistical and mathematical method. It is helpful in

modelling and optimizing industrial machining processes which are normally two to three

times larger datasets than Taguchi method, but less than half observation as compared to

factorial design and is able to study low level interaction effects [23].

The size of RSM observations is smaller than factorial design but it is capable to model the

effects of interactions which will produce better results than Taguchi method if interaction

64

between the variables is significant [77]. Finally identify the optimize region of single quality

on three-dimensional diagram. Baş and Boyacı [77] utilize RSM and try to find out some

common mistakes and limitations of RSM. RSM model fits well but it is possible that its

prediction out of experimental data range does not predict well.

Noor and Kadirgama [57, 58] prepared a statistical model to predict surface roughness and

roughness height based on laser power, cutting speed and tip distance. The model is prepared

for acrylic (3mm thick) cutting by 30W pulse CO2 using the Box Behnken (RSM) design of

first order model. However it is better to apply quadratic model based on analysis results.

Noor and Kadirgama used Box Behnken design of experiment and applied Particle swarm

optimization in their second paper for multiple output qualities. They also defend the use of

Box Behken design of experiment. Its experiment size is smaller than Central composite

design of experiment because, axial points are absent and provide safe operating boundaries.

BB design is capable to bifurcate the order of significance and appropriate for optimization of

process by RSM and capable of linear and quadratic order modelling.

6.3 MODELING AND OPTIMIZATION MODULE Selection of modeling technique is based on the response of dependent variable trend.

Statistical modelling techniques are applied by many researchers and nowadays soft-

computing techniques are more popular. The statistical concept of orthogonal array and

factorial design is not ideal for the artificial neural network case. However, it is suited for

statistical modelling to predict better values though it is not perfect. As sampling data is never

equivalent to population data and also conversion of fraction data into three level ordinal data

loses data knowledge for perfect modelling as shown in Figure 6.4. The issue of sampling

size can be improved by selecting FD instead of OA [4]. There are a number of statistical and

soft-computing techniques available at this stage to understand these methods as well as to

compare them for better application of available tools. The possible methods are:

• One Way ANOVA

• Two Way ANOVA

• Single Variable Linear Regression Analysis

• Multivariable Regression Analysis

• Nonlinear Regression Analysis

• Multivariable nonlinear Regression Analysis

65

• Artificial neural network (Supervised learning)

• Artificial neural network (Semi-supervised learning)

The above problems can be solved by using excel. However, it is sometimes easier to handle

Statistical Software SPSS, Minitab et cetera.

Optimize the modelling results

C Modelling Module

Regres-sion Taguchi Factorial Fractional

Factorial

RSM CCD/BBD

ANN Supervised

ANN Semi-Supervised

Figure 6-4: Modelling module of the Proposed Framework

Figure 1.2 represents the four controllable input parameters with their value divided into

three stages to understand the effect of input parameters on the output parameters [93]. The

relationship between the parameters can be found by the modelling of the system by some

Mathematical, Statistical method and Soft-computing mapping. The model has been

developed for simulation and expansion of the size of datasets.

6.3.1 ANALYSIS OF VARIANCE M. Zaidi et al. [25] modeled the laser cutting data with the statistical techniques as explained

in Figure 6.5 flow chart. The data was forwarded from experimental design module based on

DOE observation tables. Perform “One way ANOVA” to understand the significance of

controllable parameters such as

• Laser Power (A)

• Cutting Speed (B)

• Assist Gas Pressure(C)

• Standoff Distance (D).

This portion is divided into two parts, with replication and without replication. The meaning

of replication is to take observations more than once to reduce random error. In the focused

66

experiment polystyrene foam was cut with replication three times. In the analysis of variance

these three assumptions are used:

• The response parameter is normally distributed.

• The variance of the response parameters is same.

• The response parameters are independent.

Start

Input Experimental data for Kerf-Width of Polystyrene foam

Conclude the Analysis

Perform Analysis of Variance

Perform One Way ANOVA to understand the significance of controllable parameters (A/B/C/D)

Perform Two Way ANOVA to understand the significance of interaction between controllable parameters (A/B/C/D)

Analyzing the significance of each parameter and also interaction effects between them

End

Figure 6-5: Analysis of variance

It is possible that the interaction effects are present. Therefore, in the second stage perform

“Two way ANOVA” to understand the significance of interaction between controllable

parameters (A/B/C/D). Analyzing the significance of each (A/B/C/D) parameter and

interaction effects between them (A and B, A and C, A and D, B and C, B and D, C and D).

67

The results indicate the real contributors of the output quality and hence the quality can be

improved by using the analysis results.

6.3.2 REGRESSION ANALYSIS

Start

Input Experimental data for Kerf-Width of Polystyrene foam

Conclude the Analysis

Draw Scatter Diagram

Calculate Correlation Coefficient to find the significance of relationship

Search the best fit to find the regression equation between two variables

Perform regression analysis for multivariable

End

If coefficient of correlation is significant

No

Yes

Figure 6-6: Regression Analysis

M. Zaidi et al. [25] modeled the laser cutting data with the Statistical techniques through

regression analysis as explained in Figure 6.6. The data was forwarded from experimental

and design module based on selected DOE observation tables. Draw Scatter plot, the scatter

68

plot shows the nature of relation between the variables. The Figure 6.6 may be Positive

linear, negative linear or curvilinear relationships or discernible. The flow chart shows that

the next step is to calculate coefficient of correlation “r” to see the significance of correlation

between the variables. If the value of “r” is not significant then predicting dependent

parameter values by regression equation is a useless practice. If “r” is significant then

determine regression equation by least square method. The purpose of regression analysis is

to see the relationship between dependent and independent parameters and predict unknown

values as and when required to reduce the time and cost of experimentation and design.

6.3.3 TAGUCHI, FRACTIONAL FACTORIAL AND FACTORIAL DESIGN Taguchi’s orthogonal array (OA) L9 was selected from experimental and design module. The

factorial design is selected for verification. However, after adequacy test the size of data set

increased from OA to fractional factorial design or factorial design observation tables.

Taguchi’ OA is one of the better methods to reduce cost of experiment, get better quality and

minimize design and development interval [3]. Dr. Taguchi had designed a method based on

OA experiments resulting in controlling variations for the experiment with optimum settings

of input controllable variables [2, 16]. OA gives sets of well balanced (minimum)

experiments. In this experiment four-three level orthogonal array L9(34) i.e. a full factorial

design (3x3x3x3) of 81 sets of experiments was reduced to 9 runs i.e. L9(34-2).

Similarly L9(34-1) is fractional factorial design which gives a much better representation as

sampling data. The size of fractional factorial design for the given matrix size is 27 data sets

which are 3 times greater than the orthogonal array of L9. At last the best possible

representation is factorial design i.e. L9(34). The size is 81 datasets. The fractional factorial

explain the interaction effect and produces better model than orthogonal array [73, 104] and

factorial design results are much better. However, Taguchi assumed independent variable

concept.

Taguchi method uses the Signal to Noise (S/N) ratio of output quality as quality parameter

for optimization with the property of robustness. It is utilized in data variation and prediction

of optimum results, these signal to noise ratio expressions are used in [2, 3, 16] and shown in

equation (7.1), (7.2) and (7.3)

.

69

The robustness reduced the noise/uncontrollable parameters effects on desired quality

parameter of product or process. Target performance (TPM), noise performance measurement

(NPM) and analysis of variance (ANOVA) explain the significance of controllable and

unknown parameter in term of percentage. The highest significant factor has higher

weighting effect on the cutting quality. The screening procedure needs to run again if the

analysis of variance results show that unknown factors participation is more than 50% in the

variation of cutting quality. It means the selected major controllable factor or factors are

missing in the list of controllable factors. M. Zaidi et al. [25] took three replications to reduce

uncontrolled error factors and human error by taking the arithmetic mean.

6.3.4 RESPONSE SURFACE METHODOLOGY (RSM) The more expressive and complicated methods is Response surface methodology (RSM). The

results of RSM technique are encouraging with small data sets which reduces the size of

experimental runs for the modelling. The surface and contour graphs are utilized to explain

the relationship between the dependent and independent parameters.

Response surface methodology RSM is a statistical and mathematical method. It is helpful in

modelling and optimizing industrial machining processes which is able to study low level

interaction effects [23]. RSM produces much better results than TM if interaction between the

variables is significant [77].

Mathew, et al. [14] response surface methodology central composite design of experiment

was used to model Carbon fiber reinforced plastic composites cutting by a pulsed Nd:YAG

laser at the optimum process parameter ranges. Predictive models have been developed for

heat affected zone (HAZ) and the taper of the cut surface. It shows RSM is capable to model

and optimize problems with Central composite design (CCD) of RSM.

Baş and Boyacı [77] mentioned that RSM model fits well but it is possible that its prediction

out of experimental data range is not predicting well. Dhupal et al. used ANN for RSM which

is capable to predict. Noor and Kadirgama [57, 58] fit first order and quadratic but quadratic

results are better for acrylic (3mm thick) cutting by 30W pulse CO2 using the Box Behnken

(RSM) design. There are many examples available to prove the adequacy of RSM. Therefore,

it can be used for optimization.

70

6.3.5 ARTIFICIAL NEURAL NETWORK BY SUPERVISED LEARNING M. Zaidi et al. [4] model laser cutting of Perspex sheet by artificial neural network with

supervised learning. The model requires preparation of training, test and Simulation datasets

for training and simulation purpose. Appropriate training algorithm is selected and utilizes

supervised learning technique.

After the selection of training algorithm, select the constant parameters for training

environment for the systematic training. Start training and observe the changes in number of

neurons and hidden layers. The results were recorded and used for analysis and discussion to

conclude the effectiveness of ANN with the given datasets with missing values replace by 0

i.e. uncut.

The results were evaluated based on novel assessment technique by minimum, maximum and

average percent error with mean square error. The model adequacy check results show that

supervised learning techniques are not sufficient for missing values with orthogonal array of

L9 observations.

Therefore, there is a need to increase the size of data for modelling or prepare some other

modelling technique and by increasing the size of datasets the model adequacy is achieved

with 60% data sets for training purpose. Essentially, training data sets size is 5 observation

only, 2 for verification and 2 for test. Therefore, there is need to increase the size of data sets.

But, Ranaganth and Viswanath [31] also model laser cutting process for surface roughness

with small datasets of fifteen runs and for testing 5 runs without missing values. They model

the process after 41 times weights randomization which shows this randomization improved

the model. This evidence motivated to prepare Semi-supervised model with small data sets of

OA by utilizing the randomization technique systematically even the limitation of missing

values.

71

Start

Define ProblemDesign of Experiment for verification

Prepare Procedure of Experimentation

Performed Experiment

Check experimental

tables

Calculate edge quality and kerf width Prepare training data

Prepare Test dataPrepare Simulation data

Compare Training AlgorithmSelect constants such as

Network type (feed forward back propagation)Training Algorithm(Levenberg Marquardt)

Performance (Mean Square Error) Hidden layers: Tangent Sigmoid

Output layers: Pure Linear

Change No. of neurons and hidden layer and prepare network

Performed Training, Testing and Simulation

Record data for discussion and analysis

End

False

Yes

Check model adequacy

False

Yes

Used for optimization or next step

Figure 6-7: Modelling and Verification Methodology of Supervised learning

72

6.3.6 NEURAL NETWORK BY SEMI-SUPERVISED LEARNING The data was forwarded from DOE observation tables for modelling. M. Zaidi et al. [12]

modelled laser cutting of Perspex sheet by artificial neural network with Semi-Supervised

learning algorithm. The model requires preparation of training, test and Simulation datasets

for training and simulation purpose. The FD has been selected for verification and orthogonal

array L9 for training in experimental and design module. After collection of experimental data

prepare training, testing and simulation data. The edge quality and kerf width data was

normalized and then applied for the training process. This technique provides better results in

the modelling of many problems for the stretching of the training data [4, 28, 105].

Mathematical expression used for Normalization is

Xnor =

(Xact − Xmin )(Xmax − Xmin )

(Nmax − Nmin ) + Dmin (6.1)

Similarly, the parameters have been put into two categories, controllable and constant for

training. Thus, an algorithm is prepared for Semi-Supervised training, simulation and

evaluation for the modelling of LCP with the exception that it handles very small size

datasets and functions even in the presence of missing values as shown in Figure 6.8.

In this algorithm some parameters have been taken as constants based on the previous study

[2-4] experience such as training algorithm, network type (feed forward back propagation)

Performance criteria, hidden layer transfer function(Tangent Sigmoid), output layer (pure

linear). However, these items can be selected for more than one setting e.g. network type can

be Gradient Descent with Momentum & learning rate, Levenberg Marquardt, and different

number of neurons can be selected.

In the process of creating networks for modelling the training data and testing data sets are

prepared. N. Qazi and H. Yeung [105] utilize stacked neural network and apply

normalization to stretch the span of applied data, using single response parameter for better

modelling. Similar experience was gained by M. Zaidi et al. in [4].

Normalization and single response model strategy is adopted from both the work experiences

[4, 105]. N. Qazi and H. Yeung [105] use Principal component regression (PCR) and P. Xiao,

et al. [106] utilize Particle swarm optimization (PSO) for the selection of initial weights for

global minimum error. However, initial weights can be adjusted by random points and may

be needed to search larger area for global minima. Even then it does not guarantee to be able

73

to reach global minima point in error space. B.J. Ranaganth and G. Viswanath [31] used

weights randomization technique and initialized about 41 times. This is difficult with

supervised learning and does not necessarily build a good model with 41 times with simple

randomization.

Similar technique of initial weight randomization up to 41 times were used by Sivarao and

Peter Brevern et al. [37]. They [38] used Dither Randomization for re-initialization of neural

network weights. However, for the selection of initial weights proven simple randomization

can be improved by Nguyen and Widrow methods [91] and initialize weights thousands of

times in semi-supervised method provide excellent models. This algorithm is also suitable for

small data sets size of OA DOE based experiments.

Kondo and Ueno [107] presented the idea to change the attributes of the feedback looped

GNDH type neural network architecture without supervisor intervention for example number

of inputs, hidden layers and neurons in each layer to improve the model accuracy based on

Akaike’s Information Criterion. Therefore, Semi-supervised algorithm automate the

supervised learning algorithm to semi-automatic and creates multiple networks by changing

number of neurons, learning rates, momentum and training algorithm with all possible

combinations, and for every combination network weights are reinitialized for 100, 500, 1000

or 3000 networks based on Nguyen-Widrow method [108]. This method recommends

applying small initial weight values in neural networks. The small values are selected by

randomization. Afterwards, weights are changed in such a way that the region of vital

importance is divided into small intervals. By assigning the initial weights in first layer, every

node is able to pick different interval from the beginning of the training session. Therefore,

every node in the hidden layer has a free will to fine-tune its size of interval and location in

error space in the process of training. Often these interval sizes are small and most of the

weight changes freeze. A famous example is that of "Truck‐Backer‐Upper" training, in it the

time of training is dramatically reduced from 2 days to a mere 4 hours [91].

The weight re-initialization value based on the error value improvement and number of

neurons in hidden layer was calculated by the following formula [35, 109]:

( )2

noninh

nnn +=

(6.2)

74

Semi Supervised Neural network modeling

Define ProblemDesign of Experiment for verification

Prepare Procedure of Experimentation

Performed Experiment

Check experimental

tables

Calculate edge quality and kerf width Prepare training, Test and Simulation data

Select constants such asNetwork type (feed forward back propagation)

Training Algorithm(Gradient Descent with Momentum & Levenberg Marquardt)

Performance (Mean Square Error) Hidden layers: Tangent Sigmoid

Output layers: Pure Linear


End

Initialize weights 100, 500, 1000, 3000 times based on Wind Nguyen-Widrow method instead of Random

Semi supervised Training, Testingand Simulation

Create network based on Change No. of neurons, learning rates, momentum

and training algorithm

Measure MSE, minimum, maximum and average error for Testing and factorial prediction

Select three best networks for new created network architecture

False

True

Figure 6-8: Modelling and Verification Methodology of Semi-Supervised learning

75

Where nnh is number of neurons in hidden layer, nni number of neuron in input layers and nno

number of neurons in output layers.

Finally, the best three networks are selected from all the networks, which were created

through the above process. The output is not just the best network but it is also the best

network architecture. If this architecture is used for further training, the results have a higher

probability of improving. The idea of selecting best networks comes from Genetic algorithm

elite values concept, Particle swarm optimization and firefly algorithms. This idea combined

with the systematic randomization method of Nguyen and Widrow [91] ensures better results.

This elite value changes on every result based on the error value comparison, thus, the

chances of losing values closer to the global minima diminishes.

6.4 MULTI QUALITY OPTIMIZATION MODULE

D Multi-quality Optimization Module

RSM Multi-objective optimization

Crisp logic aggregation

Genetic Algorithm

Fuzzy Aggregation

Sele

ct b

est

Decide the fitness rules

Produce optimized results

Figure 6-9: Multi-quality module of the Proposed Framework

Some of the other significant multi-objective research has been mentioned earlier in the

introduction and laser cutting literature review. But there is a need to discuss some research

Dubey and Vinod Yadava [59] discussed hybrid TM and RSM for the multi-response

optimization of laser cutting process. The weighted impacts of quality characteristics and cost

76

components are incorporated in this quality function. They mentioned that interaction is

significant therefore it is better to use straight RSM DOE and modelling for optimization.

L. K. Pan et al. [30] provides optimizations of multiple qualities of laser welding of

Titanium with Nd:YAG laser. The overall optimization was calculated by Grey relational

grade but complex interaction is significant in this process and modeling was done by TM.

Hence the results are definitely not best.

However Gani and I. A. Choudhury et al. [100] used Tagchi’s method (TM) and

optimization in case of end milling process with P10 TiN coated carbide on AISI H13 Steel

tool. The interaction effect is absent. Therefore, aim of optimization to provide solution was

achieved by using signal to noise ratio and Pareto ANOVA methods for detailed analysis.

H. A. Eltawahni, et al. [6, 101] developed RSM (Box-Behnken design) based model of laser

cutting of MDF and AISI316L and solved the issue of overall quality in two steps by using

the desirability approach. RSM model established all factors effect on output depending on

priority and constrained based on cost and quality. Eltawahni, et al. [6, 101] is unable to

provide single optimize overall quality. The decision is based on operator to decide whether

quality is more important or cost. It is better to use fuzzy aggregation

Ming-Fei et al. [61] achieved multi-objective optimization for two qualities (optical

transmittance ratio and roughness) of CO2 laser cutting by applying grey relational analysis.

The method of grey relational analysis can be improved by considering interaction effect by

applying better techniques in the building block of measuring quality data.

Sharma and Yadava [36] developed second order RSM and Taguchi model for Aluminium

alloy sheet by ND:Yag pulsed laser . Weights of both qualities were measured by Taguchi-

based GRA with entropy measurement methodology. Taguchi-based GRA is an effective

approach to attain optimum values of both the responses only by orthogonal array and needs

Statistical expert. However, it is unable to model with missing values because unbalanced

matrix calculation is a difficult task. M. Zaidi et al. [12] modelled L9 orthogonal array and

FD model with missing values by ANN. Therefore, it is better to model the problem with

soft-computing techniques. They are capable to handle the issue of missing values in fuzzy

aggregation by applying rules to handle it.

77

6.4.1 SIMPLE AGGREGATION AND CUSTOMER QUALITY FUNCTION Yilbas and Rashid [41] cut 800HT alloy by CO2 laser and performed experiment using

Factorial design (FD) for overall quality of flatness (F) and waviness (D). But used simple

aggregation by converting the values in score i.e. 1, 2 and 3. The overall quality = F+D. The

calculation is simple and crisp in nature. In the next work B. S. Yilbas [79] investigated

overall quality at the cut surface due to input attributes variation in the process of CO2 laser

cutting of stainless steel. The similar method applied for overall quality. He utilizes scores as

in desirability method instead of weightage and sums the scores in crisp logic to calculate

overall quality. The scores are assigned in flexible range therefore crisp does not reflect the

true participation of qualities in the overall value.

The aggregation function is used in data mining techniques to evaluate overall effects or

quality. But, there are ways to give priority to any quality in the total aggregated value by

assigning higher weigtage or converting the values into score and select the score criteria

which automatically gives higher priority to selected quality. Normalization is the method

which equalizes the priority of every participant in final aggregation value [2, 3, 11]. The

normalized aggregation is calculated by equation ( 8.16) and ( 8.17).

The customer quality function can be calculated after normalization and converted into

ordinal values using Table 8-44 by applying equation (8.18), which simply used the max

function among all qualities and assign maximum value to customer quality function. This

rule is completely in the favour of the customer. Therefore, Fuzzy aggregation is definitely

the superior method of aggregation due to many standard membership functions available and

rules applied in parallel. So, it produces balanced results from both manufacturer and

customer point of view and provides the indication of rework.

6.4.2 GENETIC ALGORITHM The aim of robust process design is to build a stable process that produces minimum

variation due to uncontrollable parameters [60]. Taguchi’s approach is unable to do multi-

objective optimization and constraints handling. Genetic algorithm (GA) provides the

techniques of simultaneous optimization of multiple dependent parameters. The ability of

genetic algorithm to generate huge amount of possible population and selection on cost

function basis made it an important off line quality tool [60]. It is improbable to trap in local

minima due to the assumption of GA and does not depend on continuity, uni-modality or

78

convexity of functions [97] . GA using mutation, crossover and reproduction genetic

operations then grow successive generations and apply values to transfer function to predict

the output quality for optimized value [60]. It is a better method to solve with outer array in

orthogonal array to handle noise factors [68-70]provided that the transfer function is available

for making cost function. Chatsirirungruang and Masami Miyakawa [68] study the

usefulness of GA as compared to OA method and concluded that GA produces good results.


both the quality parameters with or without weighting factor using averaging or aggregation

etc. It can be solved by multi-objective genetic algorithm (MOGA) [98]. It was concluded by

Fernandes and Vicente et. al. [98] that MOGA is probably faster and efficient. Pawar, et al.

[62] applied Particle swarm optimization (PSO) to solve multi-objective optimization

problem of rough and finished grinding process parameters of three responses. The method of

constraint is handled by penalty methods as others. The PSO results are better than GA,

differential evaluation (DE) algorithm and Quadratic programming (QP). They also studied

the convergence rate, performance and accuracy of PSO algorithm.

6.4.3 FUZZY AGGREGATION M. Zaidi et al. [110] performed this work to solve the issue of overall quality of laser cutting

work-piece. The data was collected from modelling and optimization module of best

predicted datasets. Instead of difficult mathematical proof technique an empiricism

methodology is used to solve the problem by soft computing techniques. Recently, many

researchers have applied intelligent algorithms such as Genetic Algorithm and fuzzy logic for

overall quality in [9, 60, 68-70, 98, 99, 111, 112]. The comparative study leads to find better

techniques as in iterative methodology. The methodology is explained briefly in the flow

chart as shown in Figure 6.10. The definitions and formulae are given below[2].

The input parameters with properties are explained in Figure 1.2. The primary output

qualities are edge quality and kerf width. The percent overcut and material removal rate can

be calculated by the side line length and kerf width. The definitions and formulae [2] can be

seen from equation (7.4), (7.5), and (7.6). The adequacy of overall quality is checked based

on selection of DOE, modelling results and the problem statement. After the collection of

data normalized aggregation values are calculated using equation ( 8.16), ( 8.17) and

customer quality function calculated by (8.18).

79

Start

Define Problem

Gather useful data of Polystyrene foam modelling

Suitability of GA for application

Study Normalized Aggregation functionStudy customer quality function

Study Genetic Algorithm GA for overall qualityStudy Fuzzy Logic FL for overall quality

True

False

Use Fuzzy inference system (FIS) Connect input/output with FIS

Select membership function on simulated quality datasets for all I/P and O/P

Fine tune membership functionDefine defuzzification

Apply simulated data for evaluation


End

Suitability of FIS/FL for application

False

True

Analyze results and predict better input data sets based on novel combination of CQF, Quantified fuzzy aggregation and Quantified

normalized aggregation. Calculate Energy consumption quality for factorial design with the help of Eltawahni et al. [106] Formulas of cutting

cost and flow rate. Including power consumed in factorial table and again

sorted with Similar novel combination.

Predict machine setting for overall quality with energy efficient solution

Figure 6-10: Fuzzy Aggregation as a Multi Quality Optimization

80

The selection criteria and reasoning has been discussed in the discussion part of overall

quality. If selected technique is fuzzy aggregation, then use Matlab’s Fuzzy inference system

(FIS) and connect laser cutting qualities values as a input and normalized aggregation values

with output for the mapping. Connect input with appropriate membership function and

optimize its range and available parameters which changes the membership function shapes.

The rules are applied by the theme of customer quality function (CQF) to get fuzzy

aggregation value. Fine-tune FIS based on mapping data. Record the fuzzy aggregation

values in observation tables. Utilize quality quantification information by Table 8-44 for the

conversion of normalized and fuzzy aggregation values into quantified normalized and fuzzy

aggregation for the selection of best overall quality. As we know the tags

• Excellent 0<X≤0.25

• Desired 0.25<X≤0.5

• Worst >0.5

Finally, the decision is made based on novel combination of quantified customer quality

function, fuzzy aggregation and quantified normalized aggregation. However, the concept of

electricity –efficient solution is considered along with the desired quality.

6.5 SUMMARY Framework modules proposed for Process improvement using four modular approaches as

shown in Figure 6.1.

• Preprocessing of process module

• Experimental design module

• Modelling and optimization module

• Multi-quality optimization module

An empiricism research methodology is adopted to solve the problem by Statistical and soft

computing algorithms using proposed framework, focusing mainly on Computing rather than

Material Engineering. This Framework is applied only on experimental data in the process of

LC of single or multiple qualities modeled by “Preprocessing of Process Module” as per their

knowledge and demand. There are two categories of methodology which are followed OFAT

and DOE. In both cases, it is easy to select non-metallic material because there is margin of

81

research in the area with low power laser cutting machines and with low cost experimentation

to prove the workability of the framework. Selection of Laser cutting machine is normally

based on the material properties. The laser cutting machines are classified into low and high

power. Screening of parameters was performed for the selection of controllable and

uncontrollable parameters and constant parameters for the desired response variables. This

can be done by screening or references and domain experts’ opinion.

In the next stage of “Experimental design module” selection of DOE is based on linear, non-

linear, multi-input, single output, multi-output, number of variables and the cost of material.

If the selected DOE will not work then it is better to reselect another DOE. Many researchers

use Single parameter change at a time, TM, Fractional Factorial design, FD and RSM

techniques which are briefly explained in this chapter. The orthogonal array L9 3 (4-2) has been

derived from Factorial Design L(81) 3 (4). It shows that 9 runs are derived from 81 runs of four

variables with three levels. There are a number of orthogonal designs available such as L4,

L6, L8, L9, L18, L27 and a lot more depending on the number of treatments. The fractional

factorial design L27 3 (4-1) is also derived from FD, it normally reduces the size of experiment

by 1/3 or 1/2. The feasible size is 3, at the most 7 variables with two levels and 3 to 4

variables with three levels. It can be written as L(Runs) (Levels) (Variables) therefore, in case L(81)

3 (4) means 81 runs with four variables contains three levels. As sampling data is never

equivalent to population data and data converted to ordinal data of two, three or five levels

loses the information during the modelling based on this ordinal data. However, Factorial

design is closer to the actual model than the extracted or derived DOEs such as Taguchi and

fractional factorial design. RSM is helpful in modelling and optimizing industrial machining

processes which are normally two to three times larger datasets than TM, but has less than

half observations as compared to FD and is able to study low level interaction effects. The

size of RSM is smaller than FD but it is capable to model the effects of interactions which

will produce better results than TM. Finally, identify the optimize region of single quality on

three-dimensional diagram. RSM model fits well but it is possible that its prediction out of

experimental data range does not predict well.

Based on selected DOE, the modelling technique is chosen from the “Modelling and

optimization module” for the improvement of single quality, it can work for optimization

such as neural networks, regression analysis, Taguchi method and factorial analysis etc. The

adequacy of model is checked and it is needed to decide whether to improve by replacing the

82

modelling technique or reject it and go back to DOE stage and select different DOE and

perform the experiment again. If model is adequate then go to the next stage of multi-quality

optimization module. The issue of sampling size can be improved by selecting FD instead of

OA. The applied methods/Algorithms flowcharts of “One Way ANOVA, Two Way ANOVA,

Single Variable Linear Regression Analysis, Multivariable Regression Analysis, Nonlinear

Regression Analysis, Multivariable nonlinear Regression Analysis, ANN (Supervised

learning) and ANN (Semi-supervised learning)” modelling techniques are explained for

implementation. The model has been developed for simulation and expansion of the size of

datasets.

Some of the other significant multi-objective research has been mentioned earlier in the

introduction and laser cutting literature review. But there is a need to discuss some research

using hybrid Statistical Techniques for the multi-response optimization of laser cutting

process such as RSM with TM, TM with Grey relational grade, TM with Pareto ANOVA,

grey relational analysis, Taguchi-based GRA with entropy measurement methodology, RSM

(Box-Behnken design) with desirability approach and the decision is based on operator to

prioritize between quality or cost. However, these methods do not attempt to solve with

missing values because unbalanced matrix calculation is a difficult task. The applied novel

Fuzzy aggregation combination is capable to handle the issue of missing values.

The chapter also explains the following techniques

• Simple aggregation and customer quality function

• Genetic algorithm

• Fuzzy aggregation

Finally, the decision is made based on novel combination of quantified customer quality

function, fuzzy aggregation and quantified normalized aggregation. However, the concept of

electricity –efficient solution is considered along with the desired quality.

CHAPTER 7

EXPERIMENTAL DESIGN MODULE & SETUP

83

7. EXPERIMENTAL DESIGN MODULE & SETUP

7.1 INTRODUCTION In the laser cutting experiment of X1, X2, X3…..Xn controllable independent parameters, the

Y1, Y2, Y3…..Yn dependent parameters have to be focused for the improvement of the sheet

cutting quality. The laser cutting data is designed for the estimation of the errors of training,

testing and simulation of small datasets in the presence of n% missing values in orthogonal

array. Like any other process as shown in Figure 7.1, a well planned experiment is performed

to identify the controllable variable effect on output response to investigate the reasons for

changes in the output [73].

Training and Simulation/Laser cutting ProcessInput Output

Con

trolla

ble

fact

ors

Unc

ontro

llabl

e fa

ctor

s

X1 X2 X3 ………….Xi

U1 U2 U3 ………….Xi

Y1 Y2 Y3 ………….Yi

Figure 7-1: General model of process

The constant factors have been selected based on the M. Zaidi et al., [2, 4] work in order to

reduce the complexity of the modelling process. The factorial design based inputs were

applied on selected network and calculated overall mean square error, maximum percent

error, minimum percent error and average percent error for training and simulation intended

for the selection of best model. The simulated results are compared with each other based on

average percent error and only three elite class models were selected. The best were selected

on the basis of sorting on average, mean square error and maximum percent error. This novel

error assessment method reduces human involvement and is able to evaluate different

84

combinations with thousands of times re-initialization of network weights instead of 10 to 12

times in supervised learning. The semi supervised algorithm is able to apply ANN far better

than supervised learning.

Experiments are carried out by researchers as a rule to determine something about a

particular system or process. The meaning of an experiment is a test. A planned experiment is

a test or set of tests conducted to identify the controllable variable effect on output response

to investigate the reasons for changes in the output and its main contributors [2, 4, 73] . A

systematic experimental process is called Design of Experiment (DOE). In the design of

experiment there are a number of factors which affect the output quality. The laser cutting

process modelling by data mining techniques such as ANN, under investigation can be

represented by the model proposed by Montgomery [73] as shown in Figure 7.3. In this

model X1, X2, X3,,..Xn are controllable parameters for example laser cutting machine has

many variables that can be controlled.

Process

Training, testing and Simulation

InputWork-piece

OutputKerf width, Edge

quality etc.

Controllable factorPower, cutting speed, frequency, standoff distance, pressure etc.

Uncontrollable factorTemperature, humidity, varing cylinder gas pressure, voltage

fluctuation etc.

ConstantNozzle diameter, delay time,

thickness of work-piece, assist gas pressure etc.


The following parameters are generally taken into consideration:

Controllable parameters

• Power (Watt)

• Cutting speed (mm/ min)

• Frequency (Hz)

• Standoff Distance (mm)

• Focal Distance (mm)

• Pressure (Bar)

• Duty cycle (%)

85

The output qualities generally under investigation by many researchers include:

Output quality parameters

• Measure the exhaust gas by gas chromatography

• Kerf Width

• Surface Roughness

• Heat affected zone

• Electrical energy consumption

• Edge quality

• Percent overcut, ought to be calculated by using kerf width value

• Material removal rate, ought to be calculated by using kerf width value

The variables U1, U2, U3,…….Ui are uncontrollable parameters which may be controlled. In

addition to these controllable factors, some other factors have been taken as constant to

reduce the complexity of problem. They include:

Constant parameters:

• Nozzle Diameter: 2mm or less

• Delay Time: 2 seconds

• Thickness of work piece: mm

• Assist Gas: Compressed Air/Nitrogen/Oxygen

• Corner Power: 70%

• Actual Sideline Length: 40 mm

• Operating mode is Continuous wave

Some unknown or noise factors may include:

• Temperature, Season dependent can be control by robustness

• Humidity, Season dependent can be control by robustness

• CO2 gas pressure in cylinder from start to end

• Backash in mechanical machine gears

• Fluctuation in electrical voltages causes variation in parameters

• Effect of controllable factors on each other

• Inherent errors in laser cutting system

• Variation in sheet due variation in thickness and material

86

The statistical analysis in [93] shows that output quality factors are not independent variables

and calculation of Percent over cut (POC) and Material removal rate (MRR) shows that they

are calculated by side line length and cutting speed. It is concluded that it is sufficient to train

and simulate only kerf width and edge quality of polystyrene foam. Therefore POC and MRR

will be calculated by the existing data which will reduce the work of training and simulation

of these two output quality factors. The Taguchi method (orthogonal array) was selected as

DOE and only 9 runs were performed with three time replications.

In 2nd design of experiment based study, laser cutting process and ANN training process are

the major processes. The process of laser cutting has been explained in the above discussion

as shown in Figure 7.1. But, this time Factorial design was selected as DOE and Perspex

glass sheet was chosen as material. The target of the second experiment is to verify the

simulation results with better modelling by ANN as shown in Figure 7.3.

Process

Training, testing and Simulation

InputExperimental

data

OutputCalculation, &

Novel error assessment

Controllable factorNo. neurons, learning rate,

momentum, no. of initialization, training algorithm etc

Uncontrollable factorInitial weights, Human mistakes in

observation etc

ConstantNetwork type, Training

Algorithms, Performance method, Hidden layers & output

layers transfer functions

Figure 7-3: Training, testing and simulation process

Similarly, training, testing and simulation on experimental data are assumed as a process.

There are a number of factors which affect the output quality of the model. This process, can

be represented by the model proposed in [73]. In this model X1, X2, X3,……..Xn are

controllable parameters. They include:

• Number of neurons in hidden layer

• Number of hidden layers

• Performance function

87

• Number of epochs

• Training algorithm

• Leaning rate

• Momentum

• Number of times initialization performed etc.

In addition to controllable factors, some other factors have been taken as constant to reduce

the complexity of problem such as:

• Network type (feed forward back propagation)

• Training Algorithm(Levenberg Marquardt)

• Performance (Mean Square Error)

• Transfer function (Hidden layers): Tangent Sigmoid

• Transfer function (Output layers): Pure Linear

• The process of training is supervised learning

The variables U1, U2, U3,….Ui are uncontrollable parameters which may be controlled. Some

unknown or noise factors may include:

• Initial weights

• Human mistakes in observation etc.

The variation in output variables were measured by the novel error assessment method which

reduces human involvement.

7.2 SCOPE AND LIMITATION The scope of research is to select the small size DOE, improve modelling and optimization

techniques based on data mining techniques such as Statistical or Applied Soft-Computing

focusing mainly on the Computing aspect. Therefore, the whole focus is on data mining

techniques and laser cutting process is selected for this purpose. The scope of the proposed

framework is experimental work i.e. empiricism research methodology.

88

Proposed framework scope does not cover the theoretical/mathematical modelling problems.

The model was trained on orthogonal array and accuracy of modelling was measured on

factorial design. The supervised and semi-supervised algorithm was used for modelling in the

special case of missing value.

The Laser cutting machine used in our experiment was Zech laser machine which is mainly

used for low power applications and its max power is 500 watt, which is sufficient for plastic,

polymeric and organic material. The machine is installed in University of Malaya (UM),

Malaysia and the experiment was conducted in University of Malaya in co-ordination with

Professor, Dr. Nukman Bin Yusoff. There is a huge distance between UM and SZABIST

therefore experiment was performed less number of times. Also cost is also a constraint

regarding material selection. After adding more data during the industrial work, the ANN

model can be improved progressively.

7.3 PLAN FOR EXPERIMENT DESIGN MODULE The laser cutting of Polystyrene foam and Perspex glass sheets materials was selected for the

experiment for improvement of machining process. The procedure is explained in brief and

safety of personnel was taken into consideration which includes protection from released

fumes before performing the experiment. The precautionary steps are taken based on the

selection material and type of machines used in the process [113]. The next step is to find out

the process parameters.

7.3.1 PROCEDURE The Procedure of Laser cutting of Perspex and polystyrene foam is shown in Figure 7.4. In

the first experiment Polystyrene foam sheets were selected as a material. CO2 Laser cutting

machine of 500 watts was used for this experiment. Taguchi Method (orthogonal array) was

selected as a DOE with only 9 observations. In M. Zaidi [2] ANN modelling based on only

nine observations were used for training, testing and Simulation. However, verification was

done by comparing with statistical results. The statistical modelling was performed with

possible suitable methods on such a small data sets without the missing values in order to do

better modelling and study the importance of interaction effects.

89

Laser cutting process Start

Selected: Materials are Polystyrene foam and Perspex glass sheets Cutting profiles is square Constant, input and output parameters list down Nozzle diameter

Prepared: Table for observation based on design of experiment

Measured: Average thickness of work piece by digital caliper

Machine Startup:Fit selected NozzleApply constant values to the experimentation processMachine mode ContinuousZech Carbon Dioxide Laser Machine switched onSupply of gas 1 and gas 2 to be switched on.Place work piece on the platformFocused nozzle onto work-piece in z-axis direction.To order the model from nest open C-CUT program.

Start Experimentation: Adjust input parameters rangeCut the material in three replications / repetitions.

Measure the Quality parameters by digital caliper and scope

Store Data: Store data in worksheets Store Pictures

Perform: AnalysisModellingOptimizationValidation

Report writing

Figure 7-4: Procedure of Laser cutting of Perspex and polystyrene foam

90

Similarly, in the 2nd experiment Perspex glass sheet were selected as a material. CO2 Laser

cutting machine of 500 watts was used for this experiment. Factorial design was selected as a

DOE with only 81 observations. In my previous work ANN modelling based on only nine

observations were not verified experimentally. Therefore, Factorial design is selected for

verification and also for the best possible smallest size for modelling by supervised neural

network.

A Semi-Supervised neural network algorithm was built to improve the modelling on

orthogonal array based DOE. This small size data was successfully utilized by Semi-

Supervised neural network algorithm.

7.3.2 ORTHOGONAL ARRAYS AND FACTORIAL DESIGN Dr. Genichi Taguchi’s quality engineering design of experiments (DOE) is a robust, efficient

statistical method of total quality method (TQM) for designing high quality systems at lesser

expense for laser cutting process [3]. M. Zaidi and I. Amin [3] mentioned that the concept of

design of experiment (DOE) method reduces cost of experiment, provides better quality and

minimizes design and development interval. It designs a systematic and efficient way to

optimize controllable parameter as explained in [3]. Orthogonal Arrays (OA) provide a set of

well balanced experiments. In this experiment four-three level orthogonal array L9(34) whose

equivalent full factorial design (3x3x3x3) i.e. 81, reduce in size to 9 test sets. It converts test

range into factors and levels. DOE using Orthogonal Array resulted in efficient and to the

point test set without losing information. Many researchers used the following Signal to

Noise (S/N) ratios of Taguchi's method [2, 3, 16].

)/(log10 2210 σyNTB = (7.1)

)(log10 2210 ySTB +−= σ

(7.2)

++

−= 2

2

210311log10yy

LTB σ

(7.3)

Where NTB is normal the better in equation (7.1), STB smaller the better in (7.2) and LTB is

Larger the better in (7.3).Where y� is mean of output quality replicated values and σ2 is

variance between the replicated values.

91

Table 7-1: Four variables with three levels Orthogonal design matrix

Run Input Variable Measurement(Units) Laser Power

A

Cutting speed

B

Assist gas Pressure

C

Standoff Distance

D

Replication Mean S/N Ratio R1 R2 R3

1 100 0.2 0.5 1 2 100 0.7 2.5 5 3 100 1.2 4.5 10 4 300 0.2 2.5 10 5 300 0.7 4.5 1 6 300 1.2 0.5 5 7 500 0.2 4.5 5 8 500 0.7 0.5 10 9 500 1.2 2.5 1

Table 7-2: Controllable input and factors levels

Input Factors Level 1 Level 2 Level 3 Units Laser Power (LP) 100 300 500 Watt Cutting speed(CS) 0.2 0.7 1.2 m/min Assist gas Pressure (AGP) 0.5 2.5 4.5 Bars Standoff Distance (SD) 1 5 10 Mm

Table 7-1 shows the Experiment orthogonal design matrix which includes four input

controllable parameters with three stage ordinal data type in orthogonal array pattern and two

quality parameters for the measurement point of view for both the experiments performed for

Polystyrene foam and Perspex sheet. Appendix A (Table A-1) shows the experimental four

variables with three levels Factorial design matrix and two quality parameters for the

measurement point of view for both the experiments performed for Polystyrene foam and

Perspex sheet. The repeatability is achieved by three time replication. As the experiment is

performed with the same method, same researcher, same measurement tools, repetition over a

short time, same lab and same maintained conditions in lab.

Table 7-2 explains input factors three levels to cover the whole range of input data and also

shows their units for Polystyrene foam and Perspex sheet. The available laser cutting

92

machines are 0 to100 and 0 to 500 watts. Therefore, the whole work statements and claimes

based on the limitation of maximum possible laser power of 500 Watts.

7.4 EXPERIMENTAL SETUP In this section the experimental setup of both Polystyrene foam and Perspex sheet cutting is

explained which includes:

• Laser Machine

• Measurement tools

• Work piece material

• Data collection

• Verification of simulation data

7.4.1 LASER MACHINE 500 WATTS Both the experiments were conducted in Malaya University. Zech Laser Austria system (CO2

laser cutting machine) was used for cutting Polystyrene foam thickness of 13 mm and

Perspex sheet of thickness of 3 mm and 5 mm using the laser cutting machine as shown in

Figure 7.5 and Figure 7.6. It shows the laser cutting workstation, the model is ZL1010

specifications are given in Table 7-3 and Laser generation system ZLX5 specifications in

Table 7-4 using AutoCAD, C-Cut® for Laser cutting and Zechlaser for automatic machining

mode software.

\

Figure 7-5: Picture of Zech Laser System

93

Table 7-3: ZL1010 Specification

S. No. Parameter Ranges

1 Max. Cutting Speed 7500 mm/min

2 Maximum Rapid Power 10,000 mm/min

3 Precision Of Position +/- 0.10 mm

4 Precision Of Outline +/-0.10 mm

5 Weight Of Machine 550 Kilogram

6 Height Of Working Table 840 mm

7 X-Axis 1000 mm

8 Y-Axis 1000 mm

9 Z-Axis 100 mm

10 Max. Weight Of Working Piece 120 Kilogram

11 Size Of Machine

Length, Breadth, Height

(5280 m3)

2000 mm,1660 mm,1600 mm

Table 7-4: ZLX5 Specification

S. No. Parameter Ranges

1 Weight 630 Kilogram

2 Cooling Power 5 kW

3 Water Cooling 16 -18 ºC , 12 liters/min

4 Electrical Power 400V, 50Hz, 16A

5 Gas Mixture Co2 -N2 –He 7-28-65 (%)

6 Mode Structure Gaussian modes ,TEM01

7 Beam Divergence 2 mrad

8 Beam Diameter 14 mrad

9 Power Output CO2 laser up to 500 Watt

10 Laser system Continuous wave CO2 laser

94


7.4.2 MEASUREMENT TOOLS OF EXPERIMENT 1 and 2 For the selection of measurement tool, first identify the type, possible range and accuracy. In

this experiment digital caliper and microscope were used. The digital Vernier is easier for

measurement and reading in digital format. Routine calibration is used to avoid errors in

measurement, it is less prone to human error as compared to analog vernier caliper. It can

measure internally and externally in the range of 0 to 150mm thickness and least count

0.01mm as shown in Figure 7.7 . In these experiments, it is used to measure Polystyrene foam

and Perspex sheet thickness, standoff distance side line lengths are also shown in Figure 7.8 .

Figure 7-8: Inner and outer sideline length

Figure 7-7: Picture of digital caliper

Kerf width

Lin

Sheet

Lout

Work-piece

95

Light optical microscope (icamcope) were used to zoom in and show the geometry of edge

quality as shown in Figure 7.11 . The edge surfaces of specimens were examined using these

instruments. The results were taken by magnification of 40x and 100x in “icamscope”.

Figure 7-9: Schematic diagram of light optical microscope

Figure 7-10: Schematic diagram of icamscope

Figure 7-11: Perspex sheet edge quality measured by microscope

Edge Quality

96

7.4.3 PROPERTIES OF POLYSTYRENE FOAM Polystyrene foam sheet is used as work piece material for this experiment. It is usually white

and made of expanded polystyrene beads. Polystyrene's chemical formula is (C8H8)n, it

contains the chemical elements carbon and hydrogen. Expanded polystyrene (C8H8)n is a

generic term for polystyrene and styrene copolymers. Expanded Polystyrene is supplied to

molders in the form of a polystyrene bead. Normally used in packing, models and building

insulation etc. Some of the important properties of work piece material are listed in

Table 7-5.

Table 7-5: Properties of Polystyrene foam

Properties Range

Density 1.05 g/cm3

Density of EPS 25–200 kg/m3

Dielectric constant 2.4–2.7

Electrical conductivity (s) 10−16 S/m

Thermal conductivity (k) 0.08 W/(m·K)

Melting point 240 °C

Linear expansion coefficient (a) 8 10−5 /K

Specific heat (c) 1.3 kJ/(kg·K)

The thermal conductivity and specific heat are considered to be temperature dependent. The

convection coefficient and density are considered to be temperature independent also

mentioned by [15] . The cushioning is dependent on bead shape and size as well as density.

The strength increases with density. A key benefit of Expanded Polystyrene is that it is

recyclable and its disadvantage is that it is not consumed by time even buried in earth.

7.4.4 PROPERTIES OF PERSPEX MATERIAL Perspex sheet is used as work piece material for this experiment. The material properties of

Perspex sheet is shown in Table 7-6 [114].

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97

Table 7-6: Properties of Perspex sheet (Cast Acrylic)

Property setting Units Value

Physical property

Relative Density g/cm³ 1.19

Water Absorption % 0.2

Mechanical property

Tensile Strength at yield 5mm/min MPa 75

Elongation at break 5mm/min % 4

Flexural Modulus 2mm/min MPa 3210

Property setting Units Value

Flexural Strength at yield 2mm/min MPa 116

Rockwell Hardness M Scale 102

Thermal property

Vicat Softening Temperature ˚C >110

Coefficient of Thermal Expansion mm/m°C 0.077

Optical property

Light Transmission 3mm sheet % >92

Refractive Index 1.49

Electrical property

Dielectric Strength kV/mm-1 15

Surface Resistivity Ω m-2 >1014

7.5 DATA COLLECTION

7.5.1 POLYSTYRENE FOAM and PERSPEX SHEET Laser cutting of Polystyrene foam(13mm sheet), Perspex glass sheet 3mm and 5 mm data

was observed and collected in the observation tables of edge quality are shown in Appendix

A (Table A-2, Table A-4 and Table A-7) respectively.

98

EDGE QUALITY: Is a distance between two maximum effective parallel straight lines

(tangents) after cutting straight profile. It also includes perpendicular and flatness deviation

as shown in Figure 7.12.

KERF Width:

Laser cutting of Polystyrene foam 13mm Perspex glass sheets 3mm and 5mm data was

observed and collected in the given table format and complete observation tables of Kerf

width are shown in Appendix A (Table A-3, Table A-5, Table A-6 and Table A-9

respectively.

Figure 7-13: Views of defective cuts

The width of the path of laser beam which is moving through the work piece is defined as a

kerf. Measurement for kerf width was performed at the straight section cut by the laser. The

kerf width is a distance of the cut surfaces at the upper edge of cut or with existing melting of

top edge immediately below, as caused by the laser beam as shown in Figure 7.8.

𝐾𝐾𝐾𝐾𝐾𝐾𝑓𝑓 𝑊𝑊𝑖𝑖𝑊𝑊𝑊𝑊ℎ =(𝑊𝑊𝑊𝑊𝐾𝐾𝑊𝑊 𝑝𝑝𝑖𝑖𝑝𝑝𝐾𝐾 𝐿𝐿𝑊𝑊𝑜𝑜𝑊𝑊 −𝑊𝑊𝑊𝑊𝐾𝐾𝑊𝑊 𝑝𝑝𝑖𝑖𝐾𝐾𝑝𝑝𝐾𝐾 𝐿𝐿𝑖𝑖𝑖𝑖 )

2

(7.4)

Edge quality

Sheet 40 mm

Figure 7-12: Maximum deviation between cut edges

99

PERCENT OVER-CUT: Is explained in terms of percentage over-cut of the measured

length of work piece length by the following simple formula.

Percent Overcut =

(Measured 𝐿𝐿𝑊𝑊𝑜𝑜𝑊𝑊 − Actual 𝐿𝐿𝑊𝑊𝑜𝑜𝑊𝑊 ) Actual 𝐿𝐿𝑊𝑊𝑜𝑜𝑊𝑊

× 100 (7.5)

Negative sign in percent over-cut shows that the measured side line length is smaller than

actual side line length and positive is vice versa as shown in Figure 7.8.

MATERIAL REMOVAL RATE (MRR): In laser cutting experiment it is a function of

kerf width, cutting speed and work piece thickness. 𝑀𝑀𝑀𝑀𝑀𝑀 = 𝑇𝑇ℎ𝑖𝑖𝑝𝑝𝑊𝑊𝑖𝑖𝐾𝐾𝑖𝑖𝑖𝑖 × 𝐶𝐶𝑜𝑜𝑊𝑊𝑊𝑊𝑖𝑖𝑖𝑖𝐶𝐶 𝑖𝑖𝑝𝑝𝐾𝐾𝐾𝐾𝑊𝑊 × 𝐾𝐾𝐾𝐾𝐾𝐾𝑓𝑓 𝑤𝑤𝑖𝑖𝑊𝑊𝑊𝑊ℎ (7.6)

7.6 VERIFICATION OF SIMULATED DATA Factorial Design was selected for the Confirmation experiment (CE) of orthogonal array

based modelling. The statistical and artificial neural network based prediction modelling

ability was confirmed in the second experiment of Perspex glass sheet. It is necessary to

prove the simulation ability. Therefore, experiment was performed on Perspex sheet and the

results were ensured.

7.7 SUMMARY LCP is explained pictorially based on DOE literature including list of possible parameters of

Controllable, quality, Constant and uncontrollable or noise. These parameters have to be

focused for the improvement of the sheet cutting quality. The laser cutting data is designed

for the estimation of the errors of training, testing and simulation of small datasets in the

presence of n% missing values in OA or FD. Like any other process as shown in Figure 7.1

and Figure 7.2, a well planned experiment is performed to identify the controllable variable

effect on output response to investigate the reasons for changes in the output and finally

expand the size of existing datasets. The constant factors have been selected based on the

previous experience in order to reduce the complexity of the modelling process.

Similarly, training, testing and simulation on experimental data are assumed as a process and

mentioned all parameters in the list and also shown pictorially. This model again shows list of

possible parameters of Controllable variables, quality parameter, Constant parameter and

uncontrollable parameters. These parameters have to be focused for the improvement of the

modelling and change each parameter as per algorithm shown in Figure 6.8.

100

In the earlier work FD based inputs were applied on selected network and calculated overall

mean square error, maximum percent error, minimum percent error and average percent error

for training and simulation intended for the selection of best model. The simulated results are

compared with each other based on average percent error and only three elite class models

were selected. The best were selected on the basis of sorting on average, mean square error

and maximum percent error. This novel error assessment method reduces human involvement

and is able to evaluate different combinations with thousands of times re-initialization of

network weights instead of 10 to 12 times in supervised learning. The Material engineering

formulas show that output quality factors like Percent over cut (POC) and Material removal

rate (MRR) are not independent, they can be calculated by side line length and cutting speed.

It is concluded that it is sufficient to train and simulate only Kerf width and Edge quality of

polystyrene foam and other parameters can be calculated. The first experiment observation

tables are based on OA and in 2nd FD is selected for experimentation of Perspex sheet. The

target of the second experiment is to verify the simulation results with better modelling by

ANN.

The Plan, Procedure and setup of experimentation with selected DOE based machine setting,

table of observation were mentioned in this chapter. The laser cutting of Polystyrene foam

and Perspex glass sheets materials is explained in brief and safety of personnel was taken into

consideration which includes protection from released fumes before performing the

experiment. CO2 Laser cutting machine of 500 watts was used for both experiments. TM

(OA) was selected as a DOE with only 9 observations. The experimental verification was not

done with Polystyrene data so it was verified with TM results. Similarly, in the 2nd

experiment of Perspex glass sheet FD was selected as a DOE with only 81 observations. FD

is selected for verification and also for the best possible smallest size for modelling by

supervised neural network.

Many researchers used Signal to Noise (S/N) ratios of TM. The observation table is prepared

from experimental data and Controllable input and Appendix A (Table A-1) shows the

experimental four variables with three levels FD matrix and two quality parameters for the

measurement point of view for both the experiments performed for Polystyrene foam and

Perspex sheet.

101

In the end experimental setup of both materials sheet cutting is explained which includes

Laser Machine specification, Measurement tools, Work piece material properties, Data

collection and Verification of simulation data. Both the experiments were conducted in

Malaya University. Zech Laser Austria system (CO2 laser cutting machine) was used for

cutting Polystyrene foam thickness of 13 mm and Perspex sheet of thickness of 3 mm and 5

mm using the laser cutting machine. In this experiment digital caliper and microscope were

used. Light optical microscope (icamcope) was used to show the geometry of edge quality.

The edge surfaces of specimens were examined using these instruments. The results were

taken by magnification of 40x and 100x in “icamscope”.

Polystyrene foam and Perspex sheet is used as work piece material for these experiments.

The material properties of both the materials are mentioned in the chapter.

Laser cutting of Polystyrene foam (13mm sheet), Perspex glass sheet 3mm and 5 mm data

was observed and collected in the observation tables for edge quality and Kerf width. Design

was selected for the Confirmation experiment (CE) of orthogonal array based modelling. The

statistical and artificial neural network based prediction modelling ability was confirmed in

the second experiment of Perspex glass sheet. It is necessary to prove the simulation ability.

Therefore, experiment was performed on Perspex sheet and the results were ensured.

CHAPTER 8

DISCUSSION OF RESULTS AND ANALYSIS

102

8. DISCUSSION OF RESULTS AND ANALYSIS The problem is solved using the Processed Framework for the Improvement of process. The

whole effort is to improve the techniques or algorithm to improve the process optimization.

The chapter discussed the preprocessing of process and after selection of material, machine

and screening they demonstrate and discussed the outlier analysis. Modelling and

optimization module is vital part of this dissertation. A detailed analysis of variance One/Two

ANOVA with and without replication was performed and discussed. Similarly, Linear,

multiple linear, nonlinear and multiple non-linear regression analysis were carried out and the

results were discussed. Though the First dataset was without missing values, still there is a

need to improve the modeling. Therefore, ANN supervised learning was attempted to model

the problem with missing values datasets of Perspex sheet. The experience of supervised

learning was discussed and provided advantage in the preparation of Semi-Supervised

learning algorithm. A detailed solution of overall quality was discussed from simple

aggregation to combination of Fuzzy aggregation.

8.1 PREPROCESSING OF EDGE QUALITY AND KERF-WIDTH DATA Experiments of laser cutting on polystyrene foam sheet had been performed in order to

understand the relation between cutting parameters and the resulting cutting quality. This was

achieved using Taguchi’s method. Some variable parameters were given higher attention

during the laser cutting process such as edge quality, kerf width, percent overcut and material

removal rate as it affected the quality of laser cutting process. CO2 laser machine with

maximum continuous wave output power of 500W was employed for carrying out the

experiment. Polystyrene foam sheets with 13 mm thickness were used as work pieces. To

focus the laser beam, a ZnSe lens of 127mm focal length was installed in the lens holder.

Compressed air was introduced co-axially with the laser beam through a converging conical

nozzle of about 2 mm exit diameter. The cutting output quality parameters included laser

power (A), cutting speed (B), assisting gas pressure (C) and standoff distance (D). A/ B/ C/ D are the controllable parameters of the experiment. The selection of input test sets

is based on orthogonal array. To avoid uncontrollable and human factor errors, data was

replicated three times and reduced by averaging. To avoid the following human errors outlier

analysis is used. The possible errors can be detected by outlier analysis for example:

• Wrong measurement once in three times • Missing values

103

• Miss matched reading etc.

Table 8-1: Measurement of Edge Quality of Polystyrene foam

Run Laser

Power

A

Cutting

Speed

B

A. Gas

Pressure

C

Standoff

Distance

D

Edge Quality Edge

Quality Mean Replication

R1 R2 R3

1 100 0.2 0.5 1 3.5 4 3.5 3.667

2 100 0.7 2.5 5 2.2 3 3 2.733

3 100 1.2 4.5 10 1.5 1.5 2 1.667

4 300 0.2 2.5 10 1.5 1.5 2 1.667

5 300 0.7 4.5 1 1 1 1 1.000

6 300 1.2 0.5 5 2.5 2.5 3 2.667

7 500 0.2 4.5 5 1.5 2 3 2.167

8 500 0.7 0.5 10 1.5 2 2 1.833

9 500 1.2 2.5 1 1.5 2.5 2 2.000

Figure 8-1: Outlier analysis of Edge Quality replication of Polystyrene foam

Figure 8.1 shows that the replicate values are reasonably closed to each other except run

number 7 which is also acceptable which shows better repeatability. Choua at al. [115]

defined that before the calculation of signal to noise ratio in case of 0 value in algorithm used

zero value directly as in run 5.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 1 2 3 4 5 6 7 8 9

Repl

icat

ion

Run

Replication 1

Replication 2

Replication 3

104

Table 8-2: Measurement of Kerf Width of Polystyrene foam

Run LP A

CS B

AGP C

SD D

Kerf Width Kerf Width Mean (y1)

Replication 1 2 3

1 100 0.2 0.5 1 1.58 1.615 1.525 1.573

2 100 0.7 2.5 5 1.73 1.255 1.48 1.488

3 100 1.2 4.5 10 1.66 1.695 1.86 1.738

4 300 0.2 2.5 10 1.94 1.885 1.915 1.913

5 300 0.7 4.5 1 1.765 1.775 1.985 1.842

6 300 1.2 0.5 5 1.655 1.715 1.855 1.742

7 500 0.2 4.5 5 2.01 2.04 1.855 1.968

8 500 0.7 0.5 10 1.975 1.94 2.29 2.068

9 500 1.2 2.5 1 1.79 1.89 2.08 1.920

Figure 8-2: Outlier analysis of Kerf width replication of Polystyrene foam

Figure 8.2 shows that the replicate values are reasonably close to each other except run

number 2 and 8 which shows better repeatability. The idea is to use these input parameters

and output quality parameter for training of data by neural network, to be able to predict

output variable against the given input parameters so as to complete a full factorial table

without experiment.

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 1 2 3 4 5 6 7 8 9

Repl

icat

ion

Run

Replication 1

Replication 2

Replication 3

105

Similarly, experiments of laser cutting on Perspex glass sheet had been performed using FD

DOE. Selected quality parameters were given higher attention during the LCP as it affected

the quality of laser cutting process. CO2 laser machine with maximum continuous wave

output power of 500W was employed for carrying out the experiment. Perspex glass sheet

with 3 mm and 5 mm thickness were used as work pieces and other machine parameters were

remain same.

In the process of Perspex glass sheet (3mm) cutting, some of the machine settings are unable

to cut at 21, 24 and 27th “run”s as shown in Appendix A (Figure A-1, Table A-4, Figure A-2

and Table A-6) due to mediocre range adjustments of input parameters in Appendix A (Table

A-3). Similarly, In the process of Perspex glass sheet (5mm) cutting, some machine settings

are unable to cut at 11, 12, 15, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 48, 49, 50,51, 53, 54, 74,

75, 77, 78, 80 and 81 “runs” as shown in Appendix A (Figure A-3, Table A-7, Figure A-4

and Table A-8) due to mediocre range adjustments of input parameters of Appendix A (Table

A-3). Therefore, researcher is unable to measure edge quality and kerf width of Perspex sheet

qualities in uncut condition and assigned 0 value in observation tables.

M. Zaidiet al. [25] discussed the possible solutions of non-linear multivariable by

experimental data mining techniques using polystyrene foam cutting data [2]. Taguchi

method is a very useful technique to reduce the time and cost of the experiment but it ignores

all kind of interaction effects. The results are not much encouraging and motivated to study

Laser cutting process of non-linear multivariable modeled by one and two way analysis of

variance, also linear and nonlinear regression analysis.

Four controllable input parameters value range converted to three stages ordinal type variable

to understand the effect of input parameters on the output parameters [2]. The relationship

between the parameters can be found by the modelling of the system by some mathematical

or Statistical methods.

8.2 ANALYSIS OF VARIANCE In the beginning one way ANOVA was performed to understand the significance of

controllable parameters (Laser power (A), cutting speed (B), assist gas pressure(C) and

standoff distance (D) [2]. This analysis is divided into two parts with and without replication.

In the focused experiment polystyrene foam was cut, replicated three times as shown in Table

106

8-3 Appendix A (Table A-10 to Table A-21). Kerf width is selected for analysis and

modeling as it is the easier than edge quality and without any unsuccessful cutting.

8.2.1 ONE WAY ANOVA WITHOUT REPLICATION In one way ANOVA the effects of input parameters are analyzed one by one on laser cutting

quality of kerf width. Three replicated observations were including in the calculation in case

of with replication but for without replication only one set of replication is used. Smaller kerf

width means better cutting quality therefore, target performance measurement (TPM) is

smaller the better and signal to noise ratio is larger the better. The continuous input variables

were transformed in three levels and built an input table based on orthogonal array (OA),

which reduced the number of observations and time of experiment [18].

Table 8-3: Observations consider Laser Power (A)

100 300 500

1.573 1.913 1.968

1.488 1.842 2.068

1.738 1.742 1.920

Table 8-4: Summary of descriptive Statistics

Groups Count Sum Average Variance

100 3 4.800 1.600 0.016

300 3 5.497 1.832 0.007

500 3 5.957 1.986 0.006

The analysis of individual input parameters on Kerf Width Mean were done based on the data

recorded in Appendix A (Table A-3). Following formulas from equation (8.1) to (8.7) were

used for one way ANOVA with or without replication analysis shown in Table 8-3 to Table

8-21 by using SPSS Software.

𝑋𝑋�𝑗𝑗 = �𝑋𝑋𝑖𝑖𝑗𝑗 /𝑖𝑖𝑗𝑗

𝑖𝑖𝑗𝑗

𝑖𝑖=1

( 8.1)

𝑖𝑖𝑗𝑗2 = �(𝑋𝑋𝑖𝑖𝑗𝑗 − 𝑋𝑋�𝑗𝑗 )2

𝑖𝑖𝑗𝑗

𝑖𝑖

𝑖𝑖=1

( 8.2)

𝑆𝑆𝑆𝑆𝑇𝑇𝑀𝑀 = �𝑖𝑖𝑗𝑗 �𝑋𝑋�𝑗𝑗+𝑋𝑋��

𝑊𝑊

𝑗𝑗=1

2 ( 8.3)

107

𝑀𝑀𝑆𝑆𝑇𝑇𝑀𝑀 = �𝑆𝑆𝑆𝑆𝑇𝑇𝑀𝑀𝑊𝑊 − 1

𝑊𝑊

𝑗𝑗=1

( 8.4)

𝑆𝑆𝑆𝑆𝑆𝑆 = ��𝑖𝑖𝑗𝑗 − 1�𝑖𝑖𝑗𝑗 2𝑊𝑊

𝑗𝑗=1

( 8.5)

𝑀𝑀𝑆𝑆𝑆𝑆 = 𝑆𝑆𝑆𝑆𝑆𝑆𝑖𝑖𝑇𝑇 − 𝑊𝑊

��𝑖𝑖𝑗𝑗 − 1�𝑖𝑖𝑗𝑗 2𝑊𝑊

𝑗𝑗=1

( 8.6)

𝐹𝐹 =𝑀𝑀𝑆𝑆𝑇𝑇𝑀𝑀𝑀𝑀𝑆𝑆𝑆𝑆

( 8.7)

Where

𝑋𝑋�𝑗𝑗 𝐾𝐾𝐾𝐾𝑝𝑝𝐾𝐾𝐾𝐾𝑖𝑖𝐾𝐾𝑖𝑖𝑊𝑊𝑖𝑖 𝑆𝑆𝑆𝑆𝑆𝑆𝑝𝑝𝑆𝑆𝐾𝐾 𝑆𝑆𝐾𝐾𝑆𝑆𝑖𝑖 𝑓𝑓𝑊𝑊𝐾𝐾 𝑊𝑊𝐾𝐾𝐾𝐾𝑆𝑆𝑊𝑊𝑆𝑆𝐾𝐾𝑖𝑖𝑊𝑊 𝑗𝑗

𝑋𝑋𝑖𝑖𝑗𝑗 𝐾𝐾𝐾𝐾𝑝𝑝𝐾𝐾𝐾𝐾𝑖𝑖𝐾𝐾𝑖𝑖𝑊𝑊𝑖𝑖 𝑉𝑉𝑆𝑆𝑆𝑆𝑜𝑜𝐾𝐾 𝑊𝑊𝑓𝑓 𝑊𝑊𝑏𝑏𝑖𝑖𝐾𝐾𝐾𝐾𝑜𝑜𝑆𝑆𝑊𝑊𝑖𝑖𝑊𝑊𝑖𝑖 𝑓𝑓𝑊𝑊𝐾𝐾 𝑊𝑊𝐾𝐾𝐾𝐾𝑆𝑆𝑊𝑊𝑆𝑆𝐾𝐾𝑖𝑖𝑊𝑊 𝑗𝑗

𝑖𝑖𝑗𝑗 𝐾𝐾𝐾𝐾𝑝𝑝𝐾𝐾𝐾𝐾𝑖𝑖𝐾𝐾𝑖𝑖𝑊𝑊𝑖𝑖 𝑖𝑖𝑜𝑜𝑆𝑆𝑏𝑏𝐾𝐾𝐾𝐾 𝑊𝑊𝑓𝑓 𝑊𝑊𝑏𝑏𝑖𝑖𝐾𝐾𝐾𝐾𝑜𝑜𝑆𝑆𝑊𝑊𝑖𝑖𝑊𝑊𝑖𝑖 𝑓𝑓𝑊𝑊𝐾𝐾 𝑊𝑊𝐾𝐾𝐾𝐾𝑆𝑆𝑊𝑊𝑆𝑆𝐾𝐾𝑖𝑖𝑊𝑊 𝑗𝑗

𝑖𝑖𝑗𝑗2 𝐾𝐾𝐾𝐾𝑝𝑝𝐾𝐾𝐾𝐾𝑖𝑖𝐾𝐾𝑖𝑖𝑊𝑊𝑖𝑖 𝑖𝑖𝑆𝑆𝑆𝑆𝑝𝑝𝑆𝑆𝐾𝐾 𝑜𝑜𝑆𝑆𝐾𝐾𝑖𝑖𝑆𝑆𝑖𝑖𝑝𝑝𝐾𝐾 𝑓𝑓𝑊𝑊𝐾𝐾 𝑊𝑊𝐾𝐾𝐾𝐾𝑆𝑆𝑊𝑊𝑆𝑆𝐾𝐾𝑖𝑖𝑊𝑊 𝑗𝑗

𝑋𝑋� 𝐾𝐾𝐾𝐾𝑝𝑝𝐾𝐾𝐾𝐾𝑖𝑖𝐾𝐾𝑖𝑖𝑊𝑊𝑖𝑖 𝑂𝑂𝑜𝑜𝐾𝐾𝐾𝐾𝑆𝑆𝑆𝑆𝑆𝑆 𝑖𝑖𝑆𝑆𝑆𝑆𝑝𝑝𝑆𝑆𝐾𝐾 𝑆𝑆𝐾𝐾𝑆𝑆𝑖𝑖/𝐴𝐴𝑜𝑜𝐾𝐾𝐾𝐾𝑆𝑆𝐶𝐶𝐾𝐾

𝑆𝑆𝑆𝑆𝑇𝑇𝑀𝑀 𝐾𝐾𝐾𝐾𝑝𝑝𝐾𝐾𝐾𝐾𝑖𝑖𝐾𝐾𝑖𝑖𝑊𝑊𝑖𝑖 𝑆𝑆𝑜𝑜𝑆𝑆 𝑊𝑊𝑓𝑓 𝑖𝑖𝑠𝑠𝑜𝑜𝑆𝑆𝐾𝐾𝐾𝐾 𝑊𝑊𝑜𝑜𝐾𝐾 𝑊𝑊𝑊𝑊 𝑊𝑊𝐾𝐾𝐾𝐾𝑆𝑆𝑊𝑊𝑆𝑆𝐾𝐾𝑖𝑖𝑊𝑊

𝑀𝑀𝑆𝑆𝑇𝑇𝑀𝑀 𝐾𝐾𝐾𝐾𝑝𝑝𝐾𝐾𝐾𝐾𝑖𝑖𝐾𝐾𝑖𝑖𝑊𝑊𝑖𝑖 𝑀𝑀𝐾𝐾𝑆𝑆𝑖𝑖 𝑖𝑖𝑠𝑠𝑜𝑜𝑆𝑆𝐾𝐾𝐾𝐾 𝑊𝑊𝑜𝑜𝐾𝐾 𝑊𝑊𝑊𝑊 𝑊𝑊𝐾𝐾𝐾𝐾𝑆𝑆𝑊𝑊𝑆𝑆𝐾𝐾𝑖𝑖𝑊𝑊

𝑆𝑆𝑆𝑆𝑆𝑆 𝐾𝐾𝐾𝐾𝑝𝑝𝐾𝐾𝐾𝐾𝑖𝑖𝐾𝐾𝑖𝑖𝑊𝑊𝑖𝑖 𝑆𝑆𝑜𝑜𝑆𝑆 𝑊𝑊𝑓𝑓 𝑖𝑖𝑠𝑠𝑜𝑜𝑆𝑆𝐾𝐾𝐾𝐾 𝑊𝑊𝑜𝑜𝐾𝐾 𝑊𝑊𝑊𝑊 𝐾𝐾𝐾𝐾𝐾𝐾𝑊𝑊𝐾𝐾

𝑀𝑀𝑆𝑆𝑆𝑆 𝐾𝐾𝐾𝐾𝑝𝑝𝐾𝐾𝐾𝐾𝑖𝑖𝐾𝐾𝑖𝑖𝑊𝑊𝑖𝑖 𝑀𝑀𝐾𝐾𝑆𝑆𝑖𝑖 𝑖𝑖𝑠𝑠𝑜𝑜𝑆𝑆𝐾𝐾𝐾𝐾 𝑊𝑊𝑜𝑜𝐾𝐾 𝑊𝑊𝑊𝑊 𝐾𝐾𝐾𝐾𝐾𝐾𝑊𝑊𝐾𝐾

𝐹𝐹 𝐾𝐾𝐾𝐾𝑝𝑝𝐾𝐾𝐾𝐾𝑖𝑖𝐾𝐾𝑖𝑖ts 𝐹𝐹 𝑜𝑜𝑆𝑆𝑆𝑆𝑜𝑜𝐾𝐾 𝑓𝑓𝑊𝑊𝐾𝐾 𝐹𝐹 𝑊𝑊𝐾𝐾𝑖𝑖𝑊𝑊

Table 8-5: ANOVA for Laser Power

Source of Variation SS df MS F P-value F critical

Between Groups 0.226 2 0.113 11.568 0.009 5.143

Within Groups 0.059 6 0.010

Total 0.285 8

The Table 8-3, Table 8-4 and Table 8-5 shows the analysis between Laser Power and Kerf

Width Mean as a sample. The F critical value shows that Laser Power is significantly

participating in the variation in kerf width quality.

With reference to Table 8-5 and Table 8-6 input controllable parameter Laser power’s P and

F values reject null hypothesis Ho. Similarly, Appendix A (Table A-10 to Table A-21) shows

the analysis between Laser power, cutting speed, assist gas pressure and standoff distance

108

with kerf width mean one by one. Therefore population means of different groups are not

equal. The Table 8-6, P and F values accept Ho in case of cutting speed, assist gas pressure

and standoff distance means they are insignificantly participating in the variation of kerf

width quality.The one way ANOVA without replication analysis shows that better kerf width

predicted inputs are Laser power (100 watt), Cutting speed (0.7 m/s), Assist gas pressure (2.5

bar) and Standoff distance (5 mm).

Table 8-6: One way ANOVA without replication

Treatments F P-value F critical

Laser Power 11.568 0.009 5.143 Cutting Speed 0.007 0.993 5.143 Assist gas pressure 0.0997 0.907 5.143 Standoff distance 0.620 0.569 5.143

8.2.2 ONE WAY ANOVA WITH REPLICATION With reference to Appendix A (Table A-22, Table A-23, Table A-24) and Table 8-7 Laser

power is significant. Table 8-7 shows that P and F values accept Ho in case of cutting speed,

assist gas pressure and standoff distance are insignificantly participating in the variation of

kerf width quality but F value is more than 1, also indicating the standoff distance

significance as compared to cutting speed and assist gas pressure. The results are improved

with replication. The analysis show that kerf width predicted the same input data set and

similar prediction on the variance and mean point of view but in case of without replication

variance results are not acceptable.

Similarly, Appendix A (Table A-25 to Table A-34) show the analysis between cutting speed, assist gas pressure and standoff distance with kerf width mean one by one.

Table 8-7: One way ANOVA with replication

Treatments F P-value F critical

Laser Power 16.066 0.000 3.403 Cutting Speed 0.021 0.979 3.403 Assist gas pressure 0.285 0.755 3.403 Standoff distance 1.691 0.206 3.403

109

The analysis of individual input parameters on Kerf Width Mean were done based on the data

recorded in Appendix A (Table A-3). Formulas (equation (8.1) to (8.7)) were used for one

way ANOVA with or without replication analysis such as Table 8-3 to Table 8-21.

In one way analysis of variance results with replication and without replication support the

benefit of replication. The results of one way ANOVA show that the use of replication

improves the F value. It means replication improves the ability of bifurcation between

controllable and uncontrollable variations.

8.2.3 TWO WAY ANOVA WITH REPLICATION In Table 8-8 to Table 8-13 the analysis shows the means and variance due to interaction

between the two parameters ignoring other variables for each Laser Power and at cutting

Speed of 0.2, 0.7 and 1.2 in separate tables.

Table 8-8: Interaction between Laser Power and Cutting Speed with replication

A/B 100 300 500

0.2 1.580 1.940 2.010

1.615 1.885 2.040

1.525 1.915 1.855

0.7 1.730 1.765 1.975

1.255 1.775 1.940

1.480 1.985 2.290

1.2 1.660 1.655 1.79

1.695 1.715 1.89

1.860 1.855 2.08

Table 8-9: Interaction between Laser Power and Cutting speed 0.2

100/0.2 300/0.2 500/0.2 Total

Count 3 3 3 9

Sum 4.720 5.740 5.905 16.365

Average 1.573 1.913 1.968 1.818

Variance 0.002 0.001 0.010 0.037

T

110

able 8-10: Interaction between Laser Power and Cutting speed 0.7

100/0.7 300/0.7 500/0.7 Total

Count 3 3 3 9

Sum 4.465 5.525 6.205 16.195

Average 1.488 1.842 2.068 1.799

Variance 0.056 0.015 0.037 0.091

Table 8-11: Interaction between Laser Power and Cutting speed 1.2

100/1.2 300/1.2 500/1.2 Total

Count 3 3 3 9

Sum 5.215 5.225 5.760 16.200

Average 1.738 1.742 1.920 1.800

Variance 0.011 0.011 0.022 0.019

Table 8-12: Total Interaction between Laser Power and Cutting speed

Count 9 9 9

Sum 14.400 16.490 17.870

Average 1.600 1.832 1.986

Variance 0.030 0.012 0.021

The reason to perform the analysis is to understand how the Kerf Width behaves when

subjected to combination of parameters. With reference to Table 8-13,”Sample” is used for

Cutting Speed. The F value is less than the F critical value and P value is not less than 0.05,

which indicates that the there is no significant difference between Cutting Speed’s Means,

concluding that it will not play an important role in the variation of the Kerf Width.

“Columns” mean Laser power. The F value is greater than F critical value and P value is less

than 0.05 indicating that there is significant difference between Laser Power Means

concluding that it will play an important role in the variation of the Kerf Width.

“Interaction” stands for effect of Laser Power and Cutting Speed on Kerf Width. The F value

is not greater than F critical value and P value is not less than 0.05 showing that there is no

significant difference between means which is shown in Table 8-13.

111

Table 8-13: ANOVA of Interaction between Laser Power and Cutting speed

Source of Variation SS df MS F P-value F critical

Sample 0.002 2 0.001 0.057 0.945 3.555

Columns 0.678 2 0.339 18.457 4.37x10-5 3.555

Interaction 0.174 4 0.043 2.365 0.092 2.928

Within 0.331 18 0.018

Total 1.185 26

The summary of interaction result of six combinations of two way ANOVA with replication

is listed in Table 8-14. Similarly, detail analysis is indicated in Appendix A (Table A-35 to

Table A-70). The Table 8-14 shows interaction significance in the variation of the Kerf

Width.

Table 8-14: Summary of Two ANOVA with replication

S. No. Treatments F P-value F critical

1 Laser Power & Cutting speed 2.365 0.092 2.928 2 Laser Power & A. gas pressure 2.020 0.135 2.928 3 Laser power and standoff distance 0.402 0.805 2.928 4 Assist gas pressure and Cutting speed 11.220 9x10-5 2.928 5 Cutting speed and Standoff distance 9.602 0.0002 2.928 6 Assist gas pressure and standoff distance 9.257 0.0003 2.928

1. Interaction between laser power and cutting speed is insignificantly participating in

the variation of kerf width.

2. Interaction between laser power and assist gas pressure is insignificantly participating

in the variation of kerf width.

3. Interaction between laser power and standoff distance is insignificantly participating


4. Interaction between assist gas pressure and cutting speed is significantly participating


5. Interaction between cutting speed and standoff distance is significantly participating


6. Interaction between assist gas pressure and standoff distance is significantly

participating in the variation of kerf width.

112

The interaction 4, 5 and 6 are significant but had been ignored at the time and added in

pooled error. For better optimization considering all the above interaction in [2, 3, 16] will

give better results or apply soft-computing techniques. It is not a good practice to assume

without analysis that input parameters are independent from each other. In most of the

experimental investigations of Laser beam cutting, one factor at a time has been varied to

analyze the effect of input process parameters on output quality characteristics or responses

[13, 16, 18, 20-22] which shows inferior analysis results.

8.3 REGRESSION ANALYSIS In Appendix A (Table A-3), the relationship can be studied by collecting the experimental

data and afterwards drawing scatter plot. The scatter plot shows the nature of relation

between the variables. They may be positive linear, negative linear or curvilinear

relationships. Then calculate coefficient of correlation “r” to see the significance of

correlation between the variables. If the value of r is not significant then predicting dependent

parameter values by regression equation is a useless practice. If r is significant then determine

regression equation by least square method as shown in equation (8.8).

𝑌𝑌𝑖𝑖 = 𝑏𝑏1𝑋𝑋𝑖𝑖 + 𝑏𝑏0 (8.8)

Where

b1 is slope of the regression line or coefficient of independent parameter

b0 is y intercept

The regression analysis shows the relationship between dependent and independent

parameters and predicts unknown values as and when required to reduce the time and cost of

experimentation and design.

8.3.1 LINEAR REGRESSION ANALYSIS Figure 8.3 shows the scatter plot between Kerf Width Mean and Laser power. It shows the

positive linear regression relationship, coefficient of correlation r is positive, +0.885 is

significant between laser power and kerf width. It also shows the curve fitting and 95%

confidence interval. R2 also shows 78.3% variation i.e. significant. F and P values shows

significant role. In linear regression coefficient of line were calculated.

113

Figure 8-3: Interactive graph of Laser Power and Kerf Width

In Table 8-15 description of the dependent (Kerf width) and independent (Laser power)

variables are calculating Mean and Standard deviation by SPSS Statistics Software. The units

of Laser power is Watts and Kerf width mean is mm.

Table 8-15: Descriptive Statistics of Laser power and Kerf width

Parameters Mean Std. Deviation N

Kerf Width Mean (dependent) 1.806 0.189 9

Laser Power (Independent) 300.000 173.205 9

In Table 8-16, Pearson Correlation coefficient r is shown and its significance is calculated by

SPSS using the formula in the following equation (8.9).

Table 8-16: Correlation between Laser power and Kerf width

Kerf Width Laser Power

Pearson Correlation Kerf Width Mean 1.000 0.885

Laser Power 0.885 1.000

Sig. (1-tailed) Kerf Width Mean

Laser Power

. .001

.001 .

N Kerf Width Mean 9 9

Laser Power 9 9

114

r =

∑ XY − ∑ Xni=1 ∑ YXn

i=1N

ni=1

��∑ X2ni=1 −

(∑ X2ni=1 )

N � �∑ Y2ni=1 −

(∑ Y2ni=1 )

N �

(8.9)

Where

r is the Coefficient Correlation and is the correlation between the measured and

predicted values of dependent parameter.

X is independent/input variable

Y is dependent/output variable

n is number of observations

Table 8-17: Regression Statistics

R Square 0.783

Adjusted R Square 0.752

Standard Error 0.094

Observations 9

In Table 8-17 R2, adjusted R2 and standard error have been shown to explain the goodness of

fit. R2 is the variance of the dependent parameter that can be explicated by the input

parameters. It measures the degree of association between input and output parameters. It

can also be explained by the following equation (8.10).

𝑀𝑀2 =𝑆𝑆𝑆𝑆𝑀𝑀𝑆𝑆𝑆𝑆𝑇𝑇

(8.10)

Where

R2 is Coefficient of Determination

SSR is Sum of Square due to variation in Regression

SST is Sum of Square due to Total variation including error / undefined.

Adjusted R2 explains the variation in the dependent by independent parameter. In formula of

Adjusted R2 account for the number of independent parameters and sample size. Therefore,

adjusted R2 is generally considered to be a more accurate goodness-of-fit measure than R2.

Mathematically shown below:

𝐴𝐴𝑊𝑊𝑗𝑗𝑜𝑜𝑖𝑖𝑊𝑊𝐾𝐾𝑊𝑊 𝑀𝑀2 = 1 − (1 − 𝑀𝑀2) �

(𝑖𝑖 − 1)(𝑖𝑖 − 𝑝𝑝 − 1)�

( 8.11)

Where

115

Adjusted R2 is Adjusted R Square

n is Sample size

p is no of independent parameters in the model

The standard error is similar to root mean square error for sampling distribution. In Table

8-17 standard deviation of dependent variable shows that change in input setting by Laser

Power causes change in dependent variable. Therefore there is a need to find, through

correlation and regression analysis, the significance of change. The calculated results of

coefficient of correlation in Table 8-16 shows the positive and significant relation in single

tailed test basis. The results show that laser power effect on Kerf Width is significant even

when the value of alpha is 1%.

In Table 8-17 R2 value shows that the variation in Kerf width is 78.3% due to Laser Power

showing that it is a highly significant parameter in controlling the quality compared to other

parameters. The adjusted R2 is smaller than R2 due to small sample size. In Table 8-17

analysis of variance results also shows the significant role of Laser power by F value and

significance of F.

Table 8-18: Regression between Laser power and Kerf width ANOVA

d.f. SS MS F Significance F

Regression 1 0.223 0.223 25.280 0.0015

Residual 7 0.062 0.009

Total 8 0.285

In Table 8-18 analysis of variance is performed. Regression, Residual, Total are the breakup

of variance in output quality, The “Total” variance is divided into two parts, variance which

can be associated or related to independent parameter like in regression and the variance

which is not associated with independent parameter like in error. Sum of Squares is related to

the three Total, Model and Residual sources of variance. The Total variance is split in the

variance clarified by independent variables i.e. Regression and the variance clarified by

independent variables i.e. Residual. Degree of freedom is related with the causes of

variance. The Total variance has N-1, Regression number of coefficients (including

intercept) anticipated minus 1 and the Error total minus the degree of freedom of the model.

Mean Square is explained as the Sum of Squares divided by their degree of freedom. The F-

116

test is the Mean Square Regression divided by the Mean Square error as explained earlier in

equation ( 8.7). The p-value is compared to tolerance value alpha which is by default 0.05

used to test the null hypothesis. The coefficients or intercept mean square is 0.

Table 8-19: Linear regression line of Laser Power

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 1.517 0.065 23.163 7.09 x10-8 1.362 1.672

A 0.001 0.000 5.028 1.52 x10-3 0.001 0.001

In Table 8-19 coefficient 1.517 means the y intercept of a least square line and 0.001 is a

coefficient of dependent variable A, which is a slope of a line. Laser Power (A) is the

independent parameter in this analysis using equation (8.8) of linear regression. The standard

errors are

𝜎𝜎𝑏𝑏0 = �1𝑁𝑁 +

𝑌𝑌�2

∑ (𝑌𝑌𝑖𝑖 − 𝑌𝑌�)ni=1

2

𝜎𝜎𝜀𝜀

( 8.12)

𝜎𝜎𝑏𝑏1 = �𝑌𝑌�2

∑ (𝑌𝑌𝑖𝑖 − 𝑌𝑌�)ni=0

2

𝜎𝜎𝜀𝜀

( 8.13)

Where 𝜎𝜎𝜀𝜀 = 𝑆𝑆𝑆𝑆𝑆𝑆𝑁𝑁−2

SSE is sum of squared residual

is the ith value of output paramaeters

Y is output parameter ith value

Y� is output parameters mean value

𝜎𝜎𝑏𝑏0 , 𝜎𝜎𝑏𝑏1 are standard error of coefficients

𝑊𝑊 =

𝑝𝑝𝑊𝑊𝐾𝐾𝑓𝑓𝑓𝑓𝑖𝑖𝑝𝑝𝑖𝑖𝐾𝐾𝑖𝑖𝑊𝑊𝑆𝑆𝑊𝑊𝑆𝑆𝑖𝑖𝑊𝑊𝑆𝑆𝐾𝐾𝑊𝑊 𝐾𝐾𝐾𝐾𝐾𝐾𝑊𝑊𝐾𝐾

( 8.14)

The value of t statistic is measured by equation 8.14. The value of t statistic gives the p- value

i.e. probability value in t-distribution. If the value of p is less than tolerance (alpha) value

then null hypothesis is rejected i.e. significant difference caused by the independent

parameter on dependent variable. In default case of 95% confidence lower and upper limit of

the y intercept and slope of line is mentioned in the last two columns. It is related to p value

and shows the tolerance space in numerical and on graph by placing two lines under and over

117

the fit line as shown in Figure 8.3. In Table 8-19 y intercept and slope of the line are

calculated. T test value and p value rejected the null hypothesis that Mean values are not

equal. The Laser Power causes significant variation in Kerf width.

Table 8-20: Residual output

Observation Predicted Kerf width Mean Residuals

1 1.613 -0.040

2 1.613 -0.125

3 1.613 0.125

4 1.806 0.107

5 1.806 0.036

6 1.806 -0.064

7 1.999 -0.030

8 1.999 0.070

9 1.999 -0.079

In Table 8-20 and Figure 8.4 predicted Kerf Width is measured by regression line fitting

equation (8.9) and residual is measured from difference between experimental value and the

predicted value. The standard residuals are measured by experimental value minus predicted.

The regression line model is based on laser power and predicted minimum, maximum, and

average percent errors are 5.17%, 21.55% and 12.95% respectively. The values show that the

average error is considerably high i.e. more than 10%.

Figure 8-4: Laser Power (A) Line fit Plot

0.000

0.500

1.000

1.500

2.000

2.500

100 100 100 300 300 300 500 500 500

Expe

rim

enta

l and

Pred

icte

d Ke

rf W

idth

Laser Power

Kerf Width

Predicted Kerf Width

118

Figure 8-5: Interactive graph of Cutting Speed and Kerf Width

Figure 8.5 shows r is -0.041 which is insignificant between cutting speed and kerf width. R2

also shows 0.2% variation i.e. insignificant. F and P values shows insignificant role. In linear

regression coefficient of line were calculated. The detailed analysis are attached in Figure 8.6

and Appendix A (Table A-77 to Table A-82).

Figure 8-6: Cutting speed (B) Line fit Plot

0.000

0.500

1.000

1.500

2.000

2.500

0.2 0.7 1.2 0.2 0.7 1.2 0.2 0.7 1.2

Expe

rim

enta

l and

Pred

icte

d Ke

rf W

idth

Cutting Speed

Kerf Width


119

Figure 8-7: Interactive graph of Assist Gas Pressure and Kerf Width

Scatter plot in Figure 8.7 shows r is positive 0.026 and is insignificant between assist gas

pressure and kerf width. R2 also shows 1.6% variation i.e. insignificant. F and P values also

shows insignificant role. In linear regression coefficient of line were calculated. The detailed

analysis are attached in Figure 8.8 and Appendix A (Table A-83 to Table A-88).

Figure 8-8: Assist gas Pressure (C) Line Fit Plot

0

0.5

1

1.5

2

2.5

0.5 2.5 4.5 2.5 4.5 0.5 4.5 0.5 2.5

Expe

rim

enta

l and

Pred

icte

d Ke

rf W

idth

Assist Gas Pressure

TPM

Predicted TPM

120

Figure 8-9: Interactive graph of Standoff Distance and Kerf Width

Scatter plot in Figure 8.9 shows r is positive 0.312 and is insignificant between standoff

distance and kerf width. R2 shows that the 9.8% variation in kerf width due to standoff

distance is significant than two insignificant factors. F value shows that it is insignificant and

P value of linear regression coefficient of line shows insignificance. The detailed analysis are

attached in Figure 8.10 and Appendix A (Table A-89 to Table A-94).

Figure 8-10: Standoff Distance (D) Line Fit Plot

0

0.5

1

1.5

2

2.5

1 5 10 10 1 5 5 10 1

Expe

rim

enta

l and

Pred

icte

d Ke

rf W

idth

Standoff Distance

Kerf Width


121

Table 8-21: Linear Regression ANOVA

Laser

Power

Cutting

Speed

Assist gas

pressure

Standoff

Distance

R +0.885 -0.041 +0.126 0.312

Significant r Significant Insignificant Insignificant Insignificant

F 25.280 0.012 0.113 0.758

Sig. F 0.0015 0.914 0.746 0.413

Intercept 1.517 1.819 1.772 1.725

Coefficient X 0.001 -0.018 0.014 0.015

P-value 1x10-3 0.0914 0.746 0.413

Min error 5.17% 6.2% 1.38% 6.38%

Max. error 21.55% 54.83% 54.83% 53.97%

Average error 12.95% 25.77% 25.07% 25.94%

The Table 8-21 shows that laser power causes significant variation in dependent variable

(Kerf width mean). But cutting speed, assist gas pressure, standoff distance causes

insignificant role in the variation of dependent variable. The maximum error model based on

significant factor is 21.55% which is just above acceptable limits but insignificant factor

results more than 50% which is completely unacceptable.

8.3.2 MULTIPLE LINEAR REGRESSION

Figure 8-11: Comprehensive Interactive graph between all parameters

Scatter plots are drawn in Figure 8.11 to observe relationship between the parameters. It

shows the linear regression relationship drawn 95% confidence interval lines below and

122

above if it is possible with the current scale of the plot. The data points lie in between them.

The relation between Kerf width and input parameters are laser power positive, cutting speed

slightly negative, assist gas pressure slightly positive and standoff distance is positive.

Table 8-22: Descriptive Statistics

Mean Std. Deviation N

Kerf Width Mean 1.806 0.1887 9

Laser Power 300.00 173.205 9

Cutting Speed 0.700 0.4330 9

Assist Gas pressure 2.500 1.7321 9

Stand of distance 5.33 3.905 9

Table 8-23: Correlations

Kerf Width Mean

Laser Power

Cutting Speed

Assist Gas pressure

Standoff Distance

Pearson Correlation Kerf Width Mean 1.000 0.885 -0.041 0.126 0.312

Laser Power 0.885 1.000 0.000 0.000 0.000 Cutting Speed -0.041 0.000 1.000 0.000 0.000 Assist Gas pressure 0.126 0.000 0.000 1.000 0.000 Stand of distance 0.312 0.000 0.000 0.000 1.000

Kerf Width Mean

Laser Power

Cutting Speed

Assist Gas pressure

Standoff Distance

Sig. (1-tailed) Kerf Width Mean . 0.001 .458 0.373 0.207 Laser Power 0.001 . .500 0.500 0.500 Cutting Speed 0.458 0.500 . 0.500 0.500 Assist Gas pressure 0.373 0.500 0.500 . 0.500 Stand of distance 0.207 0.500 0.500 0.500 . N Kerf Width Mean 9 9 9 9 9 Laser Power 9 9 9 9 9 Cutting Speed 9 9 9 9 9 Assist Gas pressure 9 9 9 9 9 Stand of distance 9 9 9 9 9


The coefficient of correlation r is 0.948 and significant on single tailed test. R2 value shows

that the variation in kerf width is 89.9% it can be explained by independent parameter which

Multiple R 0.948

R Square 0.899



Observations 9.000

123

is highly significant in controlling the quality as compared to the variation due to

uncontrollable parameters.

Table 8-25: Multiple linear Regression ANOVA

D.o.f SS MS F Significance F

Regression 4 0.256 0.064 8.858 0.0288

Residual 4 0.029 0.007

Total 8 0.285

Table 8-26: Linear regression of multivariable

Coefficients Standard Error t Stat P-value Lower 95%

Intercept 1.4147 0.097 14.566 0.0001 1.1450

( A ) 0.0010 0.000 5.557 0.0051 0.0005

( B ) -0.0183 0.069 -0.264 0.8047 -0.2110

( C ) 0.0138 0.017 0.793 0.4723 -0.0344

( D ) 0.0151 0.008 1.963 0.1212 -0.0063

The mathematical expression of multiple linear regression as shown in equation (8.15). 𝑌𝑌𝑖𝑖 = 𝑏𝑏1𝑋𝑋1𝑖𝑖 + 𝑏𝑏2𝑋𝑋2𝑖𝑖 + 𝑏𝑏1𝑋𝑋3𝑖𝑖 + 𝑏𝑏1𝑋𝑋4𝑖𝑖 + 𝑏𝑏0 (8.15)

In Table 8-27 Laser Power is a significant parameter in the variation of kerf width and at a

low level so is standoff distance. The value of R2 encourages to use the regression modelling

techniques but their maximum residual and average residual errors do not allow to

recommend the method to be used as shown in Appendix A (Table A-76, Table A-82, Table

A-88 and Table A-94).

Table 8-27: Summary of linear Regression

The overall coefficient of correlation is improved to positive 0.984 and is significant. R2

encourages using the model because kerf width is 89.9% due to independent parameters

R2 F sig. P , t test Max. error Average error Remarks

A 78.3% 0.001 0.001 21.55% 12.95% Significant

B 0.2% 0.914 0.091 54.83% 25.77% Insignificant

C 1.6% 0.74 0.746 54.83% 25.07% Insignificant

D 9.8% 0.412 0.413 53.97% 25.94% Insignificant

124

which is a highly significant parameter in controlling the quality as compared to the

variation due to uncontrollable parameters.

Analysis of variance results show the significant role of independent controllable parameters

by F-value, t-test and p-value. Kerf width residual predicted value shows that the minimum,

maximum, and average percent errors are 0.71%, 20.66%, 7.99% respectively. The values

show that the error is considerably low i.e. around 8%. Therefore, this model can be used to

see an approximate trend of idea dependent variable, hence can be used as an empirical

formula. However, the results are good without missing value data.

Table 8-28: Residual Output

Observation Predicted Kerf width Mean Residuals

1 1.5294 0.0440

2 1.6081 -0.1198

3 1.7019 0.0364

4 1.8856 0.0278

5 1.7680 0.0737

6 1.7642 -0.0226

7 2.0303 -0.0620

8 2.0417 0.0267

9 1.9241 -0.0041

8.3.3 NONLINEAR REGRESSION ANALYSIS

Figure 8-12: Quadratic graph of Laser Power without replication

y = -1E-06x2 + 0.001x + 1.454R² = 0.794

0.000

0.500

1.000

1.500

2.000

2.500

0 100 200 300 400 500 600

Kerf

Wid

th M

ean

Laser Power

Kerf Width

Poly. (Kerf Width)

125

Regression analysis between Laser power and Kerf width without replication is considered.

For the analysis in the beginning Scatter plot is drawn in Figure 8.12 showing the nonlinear

regression quadratic equation without replication. The coefficient of correlation r is 0.891

(positive), significant and R2 value shows that the variation in Kerf width is 79.4% and F-

value due to Laser Power that shows that it is a highly significant parameter in controlling the

quality compared to other parameters.

Table 8-29: Regression data without replication for Laser Power

S. No. Laser Power A A2 Kerf Width Mean

1 100 10000 1.573

2 100 10000 1.488

3 100 10000 1.738

4 300 90000 1.913

5 300 90000 1.842

6 300 90000 1.742

7 500 250000 1.968

8 500 250000 2.068

9 500 250000 1.920


Multiple R 0.891

R Square 0.794



Observations 9

Table 8-31: Nonlinear Regression ANOVA for Laser Power and KW without replication


Regression 2 0.2261 0.1130 11.5684 0.0087

Residual 6 0.0586 0.0098

Total 8 0.2847

Y intercept and equation shown on Figure 8.12 is based on nonlinear regression of laser

power. T-test and p-value accepted null hypothesis. The Laser Power causes no significant

126

variation in Kerf width in nonlinear model i.e. is unable to model the problem. The

prediction error is more than 50%.

Table 8-32: Nonlinear regression of Laser Power

Coefficients Standard Error t Stat P-value

Intercept 1.454 0.130 11.154 3.10E-05

A 0.002 0.001 1.457 0.195

A2 -9x10-7 1x10-6 -0.564 0.593

Scatter plots of laser cutting, cutting speed, assist pressure and standoff distance with

replication shown in Appendix A (Table A-100 to Table A-119, Figure A-14, Figure A-15,

Figure A-16 and Figure A-17) and detail parameters such as values of r, R2, F-value, T-test

and p-value are shown in Table 8-33. This information shows that laser power is a significant

parameter based on r, R2 and F-values. But, finally T-test and p-values show that laser power

is an insignificant parameter and unable to model the problem. The prediction error is more

than 50%. The cutting speed, assist gas pressure and standoff distance are insignificantly

participating in the variation of kerf width based on r, R2, F-value, T- test and p-values. Kerf

Width residual values in predicted value tables shows that the minimum, maximum, and

average percent errors are 1.628%, 23.85%, 11.94% respectively. The values shows that the

error is high i.e. more than 10%.

Table 8-33: Non-linear Regression ANOVA

A Without

Rep.

A With

Rep.

B With

Rep.

C With

Rep.

D With

Rep.

R +0.891 +0.757 +0.042 +0.152 +0.351

Sig. r Significant Significant Insignificant Insignificant Insignificant

F 11.568 16.066 0.021 0.285 1.691

Sig. F 0.0087 3x 10-5 0.979 0.755 0.206

T test X 1.457 1.7169 -0.140 -0.403 -0.865

P-value 0.195 0.0989 0.889 0.691 0.395

T test X2 -0.564 -0.665 0.107 0.536 1.206

P-value 0.593 0.5124 0.915 0.597 0.240

127

The Null hypothesis is rejected in Linear Regression, ANOVA analysis of Laser Power effects

on Kerf Width and in multivariable case also rejects the null hypothesis. The data is

nonlinear but regression accepts the null hypothesis due to data points’ pattern and error

squared in X2 values.

The value of curve fitting parameters like coefficient of correlation and R2 are better in

nonlinear case but null hypothesis is accepted which is not desirable in this case. The residual

values are also not better than the single and multi linear regression.

Table 8-34: Summary of Non- linear Regression

Input variable % R2 F sig. P of t test H0

Without replication

Laser cutting 79.4 0.009 0.195 Accepted

With replication

Laser cutting 57.2 3x10-5 0.989 Accepted

Cutting speed 0.2 0.979 0.889 Accepted

Assist gas Pressure 2.3 0.755 0.691 Accepted

Standoff distance 12.4 0.206 0.395 Accepted

The results of nonlinear regression show that Laser power is the most important parameter.

Its R2 value decreases with the replication. The pattern of the data is above or below the least

square point. In case of without replication the model touches only one point out of 3 while in

case of replication it only touches one point out of 9 points. The error is increased due to

replication and it will remain increasing if more observations are considered as in factorial

design [47]. The null hypothesis H0 is accepted for all variables which show that nonlinear

regression with or without replication cannot explain the variations in dependent variables.

The residual error due to predicted values increased in nonlinear case rather than in linear

prediction. N. Yusoff et al. [10] explain many nonlinear relations by using only one

independent and one dependent variable and kept others constant but in our case three other

independent variables are changing along with the considered variable. Therefore, this

modelling technique cannot be recommended on the basis of the above Table 8-34 results.

The detailed analysis is shown in Appendix A (Table A-95 to Table A-119).

128

8.3.4 MULTIPLE NON-LINEAR REGRESSION Regression analysis between inputs with kerf width is explained. r is positive 0.849 which

shows it is significant and R2 shows that the variation in kerf width is 72.1% due to

independent parameters which is a sufficiently significant parameter in controlling the quality

compared to the variation due to unknown variables. Analysis of variance shows the

significant role of independent controllable parameters by F value.


Multiple R 0.849 R Square 0.721 Adjusted R Square 0.597 Standard Error 0.136 Observations 27

Nonlinear regression of multivariable calculates the coefficient of quadratic equations. The

hypothesis by T test value and p value of A, B, C and D and their square values are accepted

i.e. the independent parameters cause insignificant variation in kerf width.

Table 8-36: Multiple Non-linear Regression ANOVA


Regression 8 0.854 0.107 5.811 0.0010

Residual 18 0.331 0.018

Total 26 1.185

Table 8-37: Nonlinear Regression of multivariable

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 1.498 0.160 9.357 2.46 x10-8 1.162 1.834

A 0.002 0.001 1.840 0.082 0.000 0.003

A2 -9.86 x10-7 1.38 x10-6 -0.713 0.485 -3.89 x10-6 1.92 x10-6

B -0.073 0.316 -0.230 0.821 -0.738 0.592

B2 0.039 0.221 0.176 0.863 -0.426 0.504

C -0.046 0.071 -0.652 0.522 -0.195 0.103

C2 0.012 0.014 0.868 0.397 -0.017 0.041

D -0.042 0.032 -1.328 0.201 -0.109 0.025

D2 0.005 0.003 1.850 0.081 -0.001 0.011

129

Kerf Width predicted residual shows that the minimum, maximum, and average percent

errors are 1.82%, 59.48%, 55.81% respectively. The values show that the error is more than

50%. Detail analysis is attached with Appendix A (Table A-120 to Table A-124).

Table 8-38: Residual Output

Observation

Predicted KERF WIDTH MEAN

Residuals

1 1.573 0.007 2 1.573 0.042 3 1.573 -0.048 4 1.488 0.242 5 1.488 -0.233 6 1.488 -0.008 7 1.738 -0.078 8 1.738 -0.043 9 1.738 0.122 10 1.913 0.027 11 1.913 -0.028 12 1.913 0.002 13 1.842 -0.077 14 1.842 -0.067 15 1.842 0.143 16 1.742 -0.087 17 1.742 -0.027 18 1.742 0.113 19 1.968 0.042 20 1.968 0.072 21 1.968 -0.113 22 2.068 -0.093 23 2.068 -0.128 24 2.068 0.222 25 1.920 -0.130 26 1.920 -0.030 27 1.920 0.160

8.4 SUPERVISED LEARNING WITH MISSING VALUE Statistical work shows that the modelling is possible with it. However, the modelling with

missing value DOE based data analysis is challenging. Normally, there are methods to

130

complete the missing table with interpolation, experienced guess etc. In laser cutting process

during the DOE based experiment sometimes the metallic or nor metallic sheets are cut

incompletely [25]. Therefore, the value of kerf width or edge quality measurement is not

possible. The observation data was unbalanced. In case of unbalanced data matrix the

statistical analysis is very difficult and the results are awkward. The ability of recognition of

patterns with incomplete information motivated to utilize the ANN to handle missing values.

Often larger size datasets rather than simple orthogonal array are used by many researchers

without the missing values.

Orthogonal array based experimental data were trained by Artificial Neural Network (ANN)

for the modelling of laser cutting process of Perspex glass sheet. The simulation results were

compared with factorial design experimental observations. The factorial design is selected for

estimation of error on each dataset of observation table with missing values. The ANN

simulation evaluation results are expected to depict better generalization based on very small

training datasets by applying feed forward back-propagation. The network parameters

variation shows the benefits and disadvantages on neural network model. The increment in

size of training datasets based on factorial design generates better accuracy in simulations as

compared to orthogonal array modelling. ANN modelling possibly will be utilized by the new

researcher without much trouble in case of missing values in observation tables.

The problem solution methodology is shown in Figure 6.7. M. Zaidi et al. [4] modeled laser

cutting of Perspex sheet by Artificial neural network with supervised learning. The model

requires preparation of training, test and Simulation datasets for training and simulation

purpose.

Training, testing and simulation on experimental data are assumed as a process. There are a

number of factors which affect the output quality of the model. This process, can be

represented by the model proposed in [73]. Constant factors are selected in the beginning for

the sake of simplification of modelling on the basis of theoretical and literature review such

as:

• Network type (feed forward back propagation)

• Training Algorithm (Levenberg Marquardt)

• Performance (Mean Square Error)

131

• Transfer function (Hidden layers): Tangent Sigmoid

• Transfer function (Output layers): Pure Linear

• The process of training is supervised learning

• Regression coefficient > 0.99

8.4.1 EDGE QUALITY OF PERSPEX SHEET Levenberg Marquardt LM algorithm was selected for training and explained under the

heading 4.3 to 4.6 in detail and an independent study was also performed by [4, 12, 24] which

shows LM training algorithm is potentially better than other back-propagation algorithms.

This training was performed with two output parameters (mean and signal to noise ratio with

three replications) of single quality (edge quality) on orthogonal array consisting of only nine

observations with missing value. The statistical modelling utilizes all data for modelling. But,

MATLAB’s Neural Networks Toolbox by default utilizes 60% data for training, 20% for

validation and 20% for ultimate testing of generalization. The limitation of orthogonal dataset

is that only five datasets were used for training, two for validation and two for testing. As

discussed earlier, Perspex sheet 3mm was cut by CO2 laser cutting machine in factorial

design to provide sufficient data to verify simulation design based on orthogonal array. For

the purpose of error estimation many training sessions were carried out but some selected

sessions results were recorded in Table 8-39. The average percent errors are 6.8% and 7.8%

(mean and signal to noise ratio), enrichment technique is applied for improvement with

duplicating the datasets for training of models [2, 3, 95].

Table 8-39: Preliminary training by Levenberg Marquardt

S. No. No. of

Datasets

No. of

neurons

Max.

% error

Min.

% error

Average

% error Results

1 1 10 114.85 3.610-10 41.263 Training

33.505 5.64E-12 7.578

2 1 20-20 98.282 1.086 26.744 Training

64.143 0.867 15.631

3 1 9-9 14.642 0.006 6.844 Training

23.184 0.005 7.807

132

In the beginning a model for training is prepared for both the output (mean and signal to

noise ratio) of edge quality of 3 millimeter thick Perspex sheet based on orthogonal array.

With reference to Table 8-39 (serial number 1) the input layer with four input, hidden layer

with 10 neurons and output layer with two output, the training results show very high level of

error on average 41% and 7% (edge quality mean and signal to noise ratio respectively)

however the regression coefficient and performance of three training, validation and test data

sets are sufficiently better. The single hidden layer is a universal approximate [95, 116]. It

was the better result among the other network settings of different number of neurons in

hidden layer.

With reference to serial number 2 the number of layers is three with two hidden layers with

20-20 neurons in each hidden layer. The training was worse and error on average was 26%

and 15% (edge quality mean and signal to noise ratio respectively). In the process of training

the weights are initialized many times to get the global minima. Similarly, with reference to

serial number 3 only number of neurons in hidden layer was changed from 20-20 to 9-9. Now

the results are comparatively better even though the results are not good for simulation. The

training error on average is 6% and 7% (edge quality mean and signal to noise ratio

respectively) and corresponding maximum percent error in the training process is 14% and

23%, which is higher than expected.

It is concluded from Table 8-39 that the weights are initialized many times to get the global

minima. The model consists of four inputs and two outputs. With reference to serial number 3,

number of neurons in two hidden layer was 9-9. Now the results are comparatively better

even though the results are not good for simulation. The training error on average is 6% and

7% (edge quality mean and signal to noise ratio respectively) and corresponding maximum

percent error in the training process is 14% and 23%, which is higher than expected which

clearly shows the disadvantage of small dataset available for training and so there is less

generalization.

With reference to Table 8-40 the enriched datasets provide improvement in training up to

4.7% and 3%. Hence, enrichment improves the training. This technique is also used in

training models of Melamine and Polystyrene foam materials and similarly in current

experiment of Perspex sheet cutting. However, the average simulation percent error of mean

and S/N ratio is 19% and 24% which is not up to international standards. The basic reason is

133

very small datasets with the presence of missing values and the data is multi-variable

nonlinear. The whole model is based on five datasets, which is not sufficient to predict the

population data.

For the purpose of improvement enrichment technique with two times all datasets are used in

training models of Melamine and Polystyrene foam materials[2, 3] and similarly in current

experiment of Perspex sheet cutting. With reference to Table 8-40, serial number 1 the

number of layers are three i.e. two hidden layers with 9-9 neurons and training results show

improvement by the enrichment of data. The error on average is 4% and 3% (edge quality

mean and signal to noise ratio respectively) with the regression coefficient and performance

of three training, validation and test data sets are sufficiently better. The simulation based on

factorial datasets show that the average error is 19%, 24% for mean and signal to noise ratio

respectively and the corresponding maximum errors are at very high level i.e. 83%, 109% in

comparison with the experimental results. The training and simulation model of edge quality

signal to noise ratio is better than edge quality mean based on average error because of its

larger range of values trained. One of the major deficiencies in the model is the size of the

dataset.

Table 8-40: Enrichment Training by Levenberg Marquardt

S.

No.

No. of

neurons

Max.

% error

Min.

% error

Average

% error

Results

1 9-9 14.643 0.006 4.78 Training

12.685 0.002 3.169

83.741 0.0047 19.33 Simulation

109.517 0.0020 24.03

2 9-9 2.027 0.077 0.59 Training

1.105 0.045 0.37

108.744 0.000 23.79 Simulation

104.671 0.000 25.54

3 9-9 2.966 0.033 1.07 Training

1.105 0.045 0.51

77.256 0.517 18.28 Simulation

74.196 0.065 18.80

134

With reference to serial number 2 after extensive search of minimum error by initializing

weight of network many times, the training results are improved and error on average is

0.59% and 0.36% (edge quality mean and signal to noise ratio respectively). The simulation

of factorial input table results show the average errors are 23%, 25% and corresponding

maximum errors are at a higher level i.e. 108%, 104% in comparison with the experimental

values. Overtraining reduces the generalization in model.

Similarly, with reference to serial number 3 try to find better training and simulation, the

average training error are 1% and 0.5% (edge quality mean and signal to noise ratio

respectively) and corresponding maximum error is 1.9% and 0.5% which are better results.

The simulation based on factorial datasets show the average errors are 18% and 18%. The

corresponding maximum errors are at a higher level i.e. 77%, 74% in comparison with the

experimental values. The results show better generalization in model. Simulation with two

hidden layers improved.

Similarly, with reference to Table 8-40, serial number 1, 2 and 3 try to find better training

and simulation model. The average training error are 1% and 0.5% (edge quality mean and

signal to noise ratio respectively) and corresponding maximum error is 1.9% and 0.5% which

are better results. The simulation based on factorial datasets show better generalization in this

model and reaches to average errors 18%, 18%. The corresponding maximum errors are at a

higher level i.e. 77%, 74% in comparison with the experimental values. The simulation with

two hidden layers improved.

Table 8-41: EQ Mean and S/N Ratio with Factorial Design

S.

No.

No. of

Neurons

Max.%

error

Min.%

error

Average

% error Results Rem.

1

20.20.20

12.37 0.0003 0.663 Train

Factorial

design

2.2 0.0001 0.186

12.37 0.0003 14.135 Test

2.2 0.0001 9.115

81.00 11.000 46.000 Simulation

0.29 0.000 0.123

135

Simulation is improved in orthogonal design training model but not up to the desired level.

The results show that the size of training model of orthogonal is not sufficient for Perspex

sheet process cutting model. In MATLAB 2010 the training is performed on five datasets,

two datasets are used for validation and two for test datasets. It shows that the whole model is

based on the five datasets which are not sufficient to predict the population. It is decided on

the basis of above trainings that training will be performed on 74 datasets with 7 data sets

used exclusively for testing the model and then performing the factorial simulation. The

results were observed on the basis of training, testing and simulation data.

With reference to Table 8-41, serial number 1 the number of hidden layers increases to three

with 20-20-20 neurons. The factorial design training results are improved and error on

average is 0.66% and 0.18% (edge quality mean and signal to noise ratio respectively). After

training 7 Test datasets are applied for prediction on the given model and compared with the

experimental results. The errors on average are 12%, 2% and maximum errors are level i.e.

14%, 9%. The test results are encouraging. The simulation results show that the average

errors are 46%, 0.12% (edge quality mean and signal to noise ratio respectively) and

corresponding maximum errors are at a higher level i.e. 81%, 0.29%, in comparison with the

experimental results the training of edge quality mean is clearly inferior than signal to noise

ratio of edge quality. The results can be improved by changing number of neurons. With

reference to Table 8-41(EQ Mean and S/N Ratio with FD) and Appendix B (Table B-1), the

simulation results of 30-30-30 neurons in each layer and model is better by decreasing the

number of neurons to 20-20-20 which shows excellent results for the signal to noise ratio of

edge quality i.e. 0.29, upon further decreasing the number of neurons to 10-10-10 and 5-5-5

degrades the results. Hence, better results are 20-20-20 and 30-30-30 neurons as compared to

other models in this table. The major problem to resolve the issue of modelling is edge

quality mean. It is also better to reduce the number of three hidden layers to two or one. The

results also show that modelling both output parameters separately may provide better results.

The models were made for outputs edge quality mean values and signal to noise ratios with

controllable input variables. The results seem suitable but the maximum value in terms of

error percent is much higher than expected. The idea to improve the modelling and

simulation single output is mapped with input variables i.e. Edge quality mean [105].

136

Figure 8-13: Comparison of edge quality mean networks errors

In the beginning a model was prepared for edge quality of 3 mm thick Perspex sheet using

Factorial design. With reference to Figure 8.13 and Appendix B (Table B-2 serial number 1,

2, 3, 9 and 10 the number of layers are three i.e. two hidden layer with number of neurons 5-

5, 10-10, 30-30, 40-40 and 50-50. The training results show percent error on average is high

with 5-5 neurons, higher in 10-10, similarly increasing with 30-30 and improved with 40-40

neurons and further decreased with 50-50 neurons also shown in Figure 8.13 serial 10. With

reference to serial number 4, 5, 6, 7 and 8 the number of layers are four i.e. three hidden

layers. The training and simulation results show percent error on average low with 10-10-10

number of neurons in each hidden layer, similarly increasing with 20-20-20 and 30-30-30

even with 40-40-40 neurons and again decrease with 50-50-50 neurons.

The results demonstrate that minimum training error does not guaranty the best

generalization. If early stopping of training is avoided the percent errors in training reduces

but the simulation results will be inferior due to over training and reduction in the

generalization. The results of “Test errors” show the possible best generalization. Therefore,

serial number 1 is the best possible percent average errors of training, test and simulation of

mean edge quality are 7.69%, 9.87% and 17.47% respectively. The maximum percent errors

of training, test and simulation are 28.76%, 25.5 and 53.41 respectively. However, 10-10

neurons in hidden layer simulation errors are improved even Test errors are inferior than 5-5

neuron. Therefore, better model is probably the Test error based. But, the simulation results

0

10

20

30

40

50

60

Ave

rage

% e

rror

Neurons in hidden layers

Training

Testing

Simulation

137

are 14.3% i.e. little high due to missing values. Hence, this model needs to be improved. It is

better to normalize the data for spreading the range of edge quality mean data.

Figure 8-14: Edge quality mean comparison in percent error for normalized datasets

The models were made for both mean values and signal to noise ratios. The results of signal

to noise ratios are already better, but mean data training results can be improved by

normalization to stretch the range of mean data. The results are looking suitable even at

maximum error.

Normalization is applied on the training data of edge quality mean and found better results

with 10-10-10 neurons in Figure 8.14 and Appendix B (Table B-4). The dataset is same as

earlier i.e. factorial design. The average simulation errors are encouraging. The early stopping

of training provides better generalization models. Therefore, the training and testing on

average percent errors are higher than previous case but simulation results are better and

average errors is 6.9% which is definitely better, without normalizing the dataset the error

was 14.3%. Hence, this training technique improved results by normalization (stretching data

sets) of edge quality mean data. The ANN modelling shows small error with hidden layers

10-10-10 and 20-20 neurons. The training, testing and simulations error are 2%, 9% and 3%,

but maximum possible individual error is 25% due to the fact of missing values. The results

are better for modelling. The models were made for both mean values and signal to noise

ratios. The results of signal to noise ratios are already better, but mean data training results

improved by normalization. The results are looking suitable even at maximum error.

0

5

10

15

20

25

30

10-10 20-20 30-30 10-10-10 20-20-20 30-30-30 40-40-40

Ave

rage

% e

rror


Training

Testing

Simulation

138

Figure 8-15: Average percent errors comparison for edge quality signal to noise ratio

Now study the signal to noise ratio model training of edge quality, of 3mm thickness Perspex

sheet on Factorial design array. With reference to Figure 8.15 and Appendix B (Table B-3)

the best training datasets of are 20 and 40-40 neurons in hidden layer in Figure 8.15. The

training dataset size is same. Average simulation errors are low. The early stopping of

training provides better generalization model therefore the training and testing on average

percent errors are higher than previous case but results are better and simulation on average

errors is 3.3% which is also better. Hence, this technique is sufficient for the training.

Modelling of signal to noise ratio of edge quality of Perspex sheet using Factorial design

array produces better results with 10, 20 and 6 neurons with single hidden layer. The

training results of hidden layer containing 20 neuron average errors are lower than 10 and 6

neurons. The average percent errors of training, testing and simulations are 6%, 3%, and 6%

but maximum possible individual error is 25%. The results are better for modelling. The 40-

40 neurons in hidden layer average percent errors of training, testing and simulations are

2%, 9%, and 3% and maximum possible individual error is 25%. The results are better with

20 and 40-40 neurons in hidden layers as shown in Figure 8.15.

8.4.2 KERF WIDTH QUALITY OF PERSPEX SHEET The models were made for both mean values and signal to noise ratios of dependent variables

kerf width. The results of signal to noise ratios are already better, but mean data training

results can be checked without normalization. The results are looking suitable even at

maximum error. With reference to Figure 8.16 and Appendix B (Table B-5) serial number 1

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and 2 the number of layers is two i.e. one hidden layer with 10 and 20 neurons. The training

results show percent error on average are better. With reference to Figure 8.16 and Appendix

B (Table B-5), serial number 3, 4 and 5 the number of layers is three i.e. two hidden layer

with 10-10, 20-20 and 30-30 neurons. The training results show percent error on average has

not improved. Therefore, further training is performed and with reference to serial number 1

and 2 of Figure 8.16 and Appendix B (Table B-5) the number of layers is four i.e. three

hidden layer with 10-10-10 and 20-20-20 neurons. The training results show percent error on

average are improved in 10-10-10 neurons. The better average percent errors of training,

testing and simulations are 3%, 16%, and 4% and maximum possible individual error is 35%.

The results are better for modeling. Therefore, in the above table results of serial 6 modelling

results are better than others as shown in Figure 8.16 even with 20-20-20 neurons.

Figure 8-16: Comparison of Kerf width mean factorial datasets

The best neural network model contains 20, 10-10 and 10-10-10 neurons in hidden layers as

shown in Figure 8.16. The average simulation errors are 3.84, 3.13 and 4.39. The results

show that on average simulation errors are acceptable, however better models can be

prepared by single, double and triple hidden layers. Hence the idea of single hidden layer

network architecture is successful as many researchers mentioned that single layer hidden

layer architecture is capable to approximate any arbitrary function [91]. It can be improved

by normalization. Hence, this training technique is sufficient for training and improvement,

which is possible to obtain by normalization of kerf width mean. The kerf width signal-to-

noise ratio range is sufficiently large. Therefore, its training is not an issue and can be

performed easily.

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The different studies [117, 118] of machining with different methods are utilizing ANN since

a long time for the relationship between the two variables which is very simple and between

multi-input/output problems. However, they do not always define each and every aspect of

distribution of datasets for training, validation and testing. The calculation of signal to noise

ratio will be able to provide robust modelling solution. In case of training based on

orthogonal design L9 or fractional factorial or major part of the factorial design dataset, if

the prediction are made on full factorial dataset then the chance of maximum error in all

prediction raises because missing values are around 11% in OA and 1.3% in factorial

design. In this study, we have learned from OA and predicted the full factorial design values.

The average error is 19 to 24% which is reasonably close due to these reasons. The actual

size of OA used for training is 5 which should be 9 for better ANN mapping. However, any

missing value will increase smaller error on average and larger near the missing input

dataset. Prediction is not tested only on the known input with missing values in the

observation tables.


computing techniques are more popular. The statistical concept of orthogonal array and

factorial design is not ideal for the artificial neural network case. However, it is suited for

Statistical modelling to predict better values which is not perfect. As sampling data is never

equivalent to population data and also conversion of fraction data into three level ordinal

data loses data knowledge for perfect modelling. The issue of sampling size can be improved

by selecting FD instead of OA.

The ANN technique is better than regression model. The results of ANN with factorial

designs are better than other researchers. They also compare the results with multiple linear

regressions and concluded that ANN is a better solution [4, 78]. M.-J. Tsai et al. [34] also

built both artificial and multi-regression model for QFN cutting of 6 qualities and 3 input

parameters. The results of LM-back-propagation neural network were better than multi-

regression because of the selection of full factorial design. The regression model just fits the

curve on the experimental data by least square method and then predicts data based on the

fitted model which can be over fitted and losses generalization. Sometimes the results of ANN

modelling are not of high-quality. As the ANN is similar to regression but regression utilizes

the whole data for curve fitting and ANN uses 60% data for training and rest for validation

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and testing. For medium size datasets ANN results are better but for small datasets like OA

the rest of the data, after reserving for validation and testing, is insufficient for training.

Therefore, on the small datasets like OA it is recommended to utilize nine datasets for

training and other four data sets for validation and testing so they may be added for ANN

modelling to generate even better results than statistical techniques in special case of small

datasets. This modelling work shows the effect of early stopping, enrichment techniques,

normalization, change in number of hidden layers, change in number of neurons, size of

datasets, training algorithm and effect of missing values in the training datasets, which is

very difficult to handle with statistics and has not been addressed in earlier studies of ANN

due to unease with Statistics. It is also important to mention that design of experiment for

ANN is different than the routine statistical modelling tools because the extra datasets need

to be noted for validation, testing as well as for prediction on unseen examples. For extra

datasets there is no need to experiment more because the data observed during range

adjustment may be noted and utilized for this purpose.

8.5 SEMI-SUPERVISED ALGORITHM There are different methods for resolving the matter of missing value. Few of the methods are

mentioned here of which some are modern and some traditional [119]. The missing values

can be generated due to some of following reasons:

• Uncontrollable factors / Random Error

• Human errors

• Forget to note down

• Range adjustment

The range adjustment of laser cutting is difficult for a new researcher. Applying DOE based

machine adjustment and work-piece is unable to cut every set of inputs, then these values are

considered as missing values. The Statistical traditional methods try to measure the missing

values however errors are higher due to covariance. Sometimes Statisticians analyze data

without using missing values if the data is more related with less variance and is a larger size

datasets. In our view for small data sets like L9 it is difficult to make models with missing

values. Some of the methods are [119]:

• Deletion List Wise

• Deletion Pair-Wise

• Weighted Arithmetic

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• Normal Ratio Method

• Inverse Distance Method

Modern methods are built for more accuracy and some of these methods are [119]:

• Time Series Analysis and Regression

• ANN

• Stochastic Techniques of Interpolation

• Maximum Likelihood

• Multiple Imputations

The results of regression are quite inferior to ANN that are mentioned in [2-4, 25, 34, 78].

Many researchers handled missing value but did not apply ANN in the Laser cutting process

to solve the range adjustment problem. The modern techniques of regression [25] had higher

prediction errors. Therefore, it is better to use novel method of Semi-supervised algorithm for

process modelling.

For the purpose of simplification, some parameters were kept constant based on experience

of supervised learning work [4]. The best possible training algorithms were discussed under

the heading 4.1, 4.2, 4.3 and 4.4 in detail and Levenberg Marquardt algorithm and Gradient

Descent with Momentum were selected. An independent study was performed by M. Zaidiet

al. [4, 24] to select the training algorithm. This training was performed with two output

parameters (mean and signal to noise ratio with three replications) of single quality (edge

quality) on orthogonal array consisting of nine observations preprocessed data with missing

values. The training set needed validation and test datasets. Therefore, nine datasets were

used for training, three for validation and three for testing.

M. Zaidi [4] discussed some recorded results of edge quality mean and signal to noise ratio

modelling for the purpose of modelling and verification. Different training sessions were run

with supervised learning mode and measure of average percent errors in normal training were

6.8% and 7.8%, which improves by applying enrichment technique (duplicating the dataset)

on training data (Melamine and Polystyrene foam materials) [2, 95]. It improves up to 4.7%

and 3%, but simulation average percent errors are 19% and 24% [4], which indicate higher

error level because of small datasets, nonlinear multi-variables and insufficient number of re-

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initializations of neural networks weights, as it is a human limitation because of the training

being supervised. The previous research has increased the size of dataset from orthogonal to

factorial design for the training and testing which improves the results but the option of re-

initialization had not been tested by the author.

Figure 8-17: Variations in 100, 500, 1000, 3000 re-initializations

Therefore, there is a need to increase the number of re-initialization and necessitate a semi-

supervised algorithm to decrease the human load. ANN Training and simulations were

performed by fixing number of neurons to 4 and changing the number of initializations in the

algorithm and have observed the standard deviation of average percent error on factorial

input predictions in Figure 8.17 which is probabilistic in nature. Hence shows that there is a

chance of improvement by changing the number of re-initialization from 100 to 500, 1000

and 3000 times. The small dataset OA is utilized for all models of edge quality/kerf widths of

3mm or 5mm Perspex sheets. Increase in number of re-initializations of neural network

weights will generate better results as shown in Figure 8.18. One of the major lead of Semi-

supervised on supervised learning is a huge number of re-initializations of the network

weights such as 100 or 1000 or 3000 times but in other cases it is 5 to 10 times.

Therefore, it can be concluded that Semi Supervised learning is required to produce better

models. Because in supervised learning it is difficult to do this job even for 100 times. The

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use of higher speed processors as Intel Core I3 based systems to perform the Semi-supervised

study to observe clearly produces noteworthy improvements.

Figure 8-18: Effect of weight initializations on average error

As in supervised learning the edge quality model of 3mm were studied and the average

percent error were 19% (mean) and 24% (S/N) but in case of Semi-supervised they were not

more than 9.5% and 11%. These results encourage utilizing the Semi-supervised algorithm

technique based on supervised learning experiences. The error at most is 12.0 with 22%

missing values in 5mm models as shown in Figure 8.18. The average percent error in case of

edge quality and kerf width for 3mm and 5mm was calculated by changing number of re-

initializations of weights of the network. The mean and standard deviation was calculated for

each case. The results are depicted in Figure 8.17 and Figure 8.18 respectively which shows

that this parameter can be utilized for the improvement of neural network modelling and is

also capable of modelling small datasets such as orthogonal array.

In the [4] previous study only 3mm material was used but in this study an additional 5mm

work piece is also used for experimentation, training & modelling. The number of missing

values doubled with the same input parameters ranges. But, the modelling error increased no

more than 2.5%. The modelling of missing data encourages the cutting of new materials with

small experiment sets even when missing values exist where more time, cost and domain

expert are needed to adjust the range to perform experiment without missing values. The

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results of single output quality were better than multiple outputs [4]. Figure 8.18 shows that

3000 time re-initialization will generate better results.

The modelling is improved by running the program for GDM and LM training algorithms.

Different learning rates (0.01 0.1 0.2) and Momentum (0.01 0.1 0.2 0.3) values are the

manual input requirement of GDM. Both the training algorithms are run repeatedly for each

number of neuron.

8.5.1 EDGE QUALITY OF PERSPEX SHEET Edge quality EQ mean with 3mm Perspex sheet average percent error (9.9%) is minimum

with 5 neurons with the selection of LM algorithm as shown in Figure 8.19. Similarly, with

5mm Perspex sheet average percent error (11.5%) is minimum with 2 neurons with the

selection of LM algorithm as shown in Figure 8.19. The increase in error is 1.6% due to the

increase in number of missing values from 11% to 22% in training data. The error reduced

by increasing the number of re-initialization of Appendix B (Figure B-1 and Figure B-2)

showing that 12000 time re-initialization will generate better results than 3000 times. After

3000 times re-initialization the process was repeated for 12000 times and improvement was

observed in edge quality mean value of 3 and 5mm. Therefore, the result shown in Figure

8.19 was based on 12000 time re-initialization.

Figure 8-19: Neuron‘s variations in EQ mean

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Edge quality EQ S/N with 3mm Perspex sheet average percent error (9.1%) is minimum with

5 neurons with the selection of LM algorithm as shown in Figure 8.20. Similarly, with 5mm

Perspex sheet average percent error (10.8%) is minimum with 2 neurons with the selection of

LM algorithm as shown in Figure 8.20.

The increase in error is 1.7% due to the increase in number of missing values from 11% to

22% in training data. Appendix B (Figure B-3 and Figure B-4) shows edge quality S/N values

of 3 and 5mm sheet cutting process model 12000 time re-initialization and is able to generate

better results than 3000 times. After 3000 times re-initialization the process was repeated for

12000 times and the improvement was observed in edge quality S/N value of 3 and 5mm.

Therefore, the result shown in Figure 8.20 was based on 12000 time re-initialization.

Figure 8-20: Neuron's variations in EQ S/N

8.5.2 KW QUALITY OF PERSPEX SHEET Kerf width KW mean with 3mm Perspex sheet average percent error (4.7%) is minimum with

8 neurons with the selection of GDM algorithm as shown in Figure 8.21.

The minimum error of KW Mean with LM was 5.0%, which is only 0.3 percent higher. The

difference is not remarkable and can be omitted or reduced by more re-initializations while

saving training time in hours. Similarly, with 5mm Perspex sheet average percent error

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(8.0%) is minimum with 3 neurons with the selection of LM algorithm. The increase in error

is 3.3% due to the increase in number of missing values from 11% to 22% in training data.

Figure 8-21: Neuron‘s variations in KW mean

Edge quality KW S/N with 3mm Perspex sheet average percent error (5.3%) is minimum

with 5 neurons with the selection of LM algorithm as shown in Figure 8.22. Similarly, with

5mm Perspex sheet average percent error (8.7%) is minimum with 3 neurons with the

selection of LM algorithm. When the missing values double it increases by 3.4% only.

Figure 8-22: Neuron's variations in KW S/N

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The different studies [73, 118] of machining with different methods have been utilizing ANN

since a long time for the relationship between two variables and between multi-input/output

problems. However, they do not always define each and every aspect of distribution of

datasets for training, generalization and testing. The optimized datasets are normally

predicted which are in case of Taguchi method, more close to the desired value and also

more robust in nature. Therefore, the predictions mostly produce better results.

However, in case of training based on orthogonal design L9, fractional factorial or major

part of the factorial design dataset, if the predictions are made on full factorial dataset then

the chance of maximum error in all prediction raises. In this study, we have learned from OA

and predicted the full factorial design values. The average percent error in supervised

learning and semi-supervised are listed in Table 8-42. The results show remarkable

improvement with 11% (3mm) and 22% (5mm) missing values yet error increases in small

amount. The error is calculated against the whole factorial inputs.

Table 8-42: Average percent errors

Supervised Semi Supervised

Edge quality Edge quality Kerf Width

Mean S/N Mean S/N Mean S/N

3mm 19% 24% 9.9% 9.1% 4.7% 5.3%

5mm - - 11.5% 10.8% 8.0% 8.7%

To understand the missing values, assume a collection of points, which are to be used for the

formation of a regression line. With more points the regression will have a higher probability

of being accurate. Similarly, in our case, data used for checking generalization and testing

gives smaller error on average but larger error near the missing points.


computing techniques are used as well. The statistical concept of orthogonal array and

factorial design is not suitable for the artificial neural network case because by nature the

orthogonal and factorial design can only represent approximate population values. It loses

knowledge because of less number of samples for the representation of a population. In other

words it only represents the generalization aspect of a sample while for neural networks such

a sample can be used for training but as far as validation and testing is concerned, it results

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in a regression model which has higher probability of having errors. Conversion of fractional

data into ordinal data also loses a lot of information that the artificial neural network can be

modeled with, using real non-reprocessed data. There is a similar loss of information in

factorial design due to usage of ordinal three level datasets. The factorial design is better

than orthogonal design because of its larger size than OA.

The ANN technique is better than regression model. The regression model just fits the curve

on the experimental data by least square method and then predicts data based on the fitted

model. The results of ANN are better because the training data works similar to regression,

but the results are degraded because of decrease in size of datasets due to need of datasets

for validation and testing. Using these extra datasets will improve generalization and

accuracy. Hence, provide better results than statistical modelling techniques.

8.6 FUZZY AGGREGATION Table 8-43: Four qualities factorial design data

Run A B C D Simulated results

EQ KW POC MRR

1 100 0.2 0.5 1 3.670 1.552 2.356 6.74E-08

2 100 0.2 0.5 5 2.729 1.842 3.089 6.98E-08

3 100 0.2 0.5 10 2.090 1.882 3.868 6.80E-08

4 100 0.2 2.5 1 2.833 1.440 2.413 6.31E-08

5 100 0.2 2.5 5 2.991 1.547 3.207 6.37E-08

. . . . . . . . .

. . . . . . . . .

75 500 1.2 0.5 10 2.305 1.962 4.168 3.24E-07

76 500 1.2 2.5 1 1.756 1.920 4.031 4.85E-07

77 500 1.2 2.5 5 2.383 1.935 4.118 4.85E-07

78 500 1.2 2.5 10 1.624 1.950 4.109 4.82E-07

79 500 1.2 4.5 1 0.625 1.977 3.842 4.85E-07

80 500 1.2 4.5 5 1.125 1.957 3.933 4.85E-07

81 500 1.2 4.5 10 1.688 1.892 4.049 4.82E-07

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The data of Table 8-43 is taken from Polystyrene foam experimental data as shown in

Appendix B (Table B-6). It contains the unique key of serial number or run which can be

used as a substitute of four inputs (Laser power (A), cutting speed (B), assist gas pressure (C)

and standoff distanced (D)) and output datasets in further representation for ease.

For equal participation of all four quality factors values normalized between 0 to 1. For

simplicity, the fraction (continuous) data is converted to ordinal type data i.e. 1, 2 and 3 else

0 (Unsuccessful cutting) as shown in Table 8-44.

Table 8-44: Quality quantification

Value Quality Customer Range

1 Excellent 0<X≤0.25

2 Desired 0.25<X≤0. 5

3 Worst 0.5<X≤1

Often the industry sends the cutting quality specifications along with the acceptable

tolerances. Therefore, good or desired quality is also called as per specification of the

customer. Excellent means better then desired quality and worst means unacceptable for the

customer as shown in Table 8-44. The conversion of variable at this stage does not harm the

optimization objective. The aggregation was performed using the following formula between

the range 0 to 4 [2] .

Av = Nor(EQ) + Nor(KW) + Nor(POC) + (1 − Nor(MRR)) ( 8.16)

Where Av is Aggregated value, Nor(EQ), Nor(KW), Nor(POC) and Nor(MRR) are

normalized edge quality, kerf width, percent over cut and material removal rate respectively.

Nor(Av) = (V − Vmin col ) Vmax col ⁄ ( 8.17)

Where Nor(Av) is Normalized Aggregated value, V is current value, Vmin col is minimum

column value and Vmax col is maximum column value.

It will help to perform customer quality function in a more simplified way in equation (8.18)

than M. Zaidi et al. used in [2].

𝐶𝐶𝐶𝐶𝐹𝐹 = Max�VEQ or VKW or VPOC or VMRR � (8.18)

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Where CQF is customer quality function, VEQ, VEQ, VEQ and VEQ are edge quality, kerf width,

percent over cut and material removal rate quantified ordinal values.

The customer quality function performs by crisp logic and aggregation function performed

with fuzzy logic. The existing problem is solved by Matlab’s FIS toolbox. The results are

shown in Table 8-45 and complete table is in Appendix B (Table B-7) which consists of

quantified normalized aggregated values (Q.Av), customer quality function (CQF),

defuzzified fuzzy aggregation value (FL Av) and quantified fuzzy aggregation value

(QFL Av).

Table 8-45: Compare results of Q Av, CQF, FL Av and QFL Av

Run Ordinal outputs Overall quality

EQ KW POC MRR Q AV CQF FL Av QFL Av

1 2 2 1 1 1 2 0.45 2

2 1 2 1 1 1 2 0.48 2

3 2 1 1 2 1 2 0.48 2

4 3 2 1 1 2 3 0.46 2

. . . . . . . . .

. . . . . . . . .

81 2 3 3 3 3 3 0.48 2

The overall quality was measured by the aggregation function of fuzzy logic. This addition

will prepare novel combination with crisp and simple data mining aggregation technique. The

normalized aggregation was performed with equation ( 8.16) after normalization of each

output parameter by equation (8.17). The simple aggregation was performed by equation

( 8.17). The results show that normalized aggregation value shows a better picture of overall

quality as compared to other two as shown in Appendix B (Table B-7). But CQF shows the

real picture of the overall quality as shown in Figure 8.23. However, quantified fuzzy

aggregation shows a more appropriate picture of the overall quality.

Normalized aggregation is unable to reflect worst quality any one of the aggregated quality

all the time. However, customer quality can do this by quantifying every quality before

aggregation and accept maximum value as CQF value, shown in Table 8-45 of Run 4 that

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normalized aggregation value is 2 which shows that the overall quality is desirable but CQF

shows 3 i.e. worst quality, which protects the customer to accept overall desired quality

because one of the qualities is worst. It also secures the product quality inspection as shown

in Figure 8.24. This function is also based on crisp logic and it is completely in favor of the

customer rights and is damaging to the manufacturer.

Figure 8-23: View of all aggregated values

Figure 8-24: Comparison of quantified Normalized aggregation with CQF

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Fuzzy aggregation Quantified Fuzzy Aggregation

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Aggregation

Customer quality function

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The defuzzified fuzzy aggregation FL Av has a benefit of member ship function and applies

parallel rule which can protect the rights of customer and lower aggregation value shows that

minor repair can save a lot of revenue of the product cost which eventually saves the

customer’s cost and time, as shown in run 4 that edge quality is worst but by repairing it

product can be acceptable.

Instead of using the process of normalization and aggregation equation, fuzzy logic is directly

applied to output qualities. In Fuzzy logic there are many possible membership functions

other than normalized function. Therefore, the possibility to transform, stretch or compress

the simulated data is much higher. Fuzzy logic is applied using Matlab (2010a) software

utility called “Fuzzy Inference System”.

Figure 8-25: Fuzzy inference system for polystyrene sheet cutting process

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The following steps have been taken

• Define input variables function and range. Four input simulated quality parameters

were created.

• Define output variables function and range. Fuzzy aggregation function was created.

• Link inputs and output with model name of Polystyrene sheet laser cutting process as

“Poly2” using Mamdani algorithm which consists of rules to be followed for

aggregation.

• The rules apply in parallel as shown in Figure 8.26.

• Implication method min is used which just truncated output on the basis of logic

operators on right most column and first four columns are inputs.

Figure 8-26: Application of parallel rules with implication

In this problem following membership functions are used

• “zmf” Opened from left side polynomial curve asymmetrical.

• “gaussmf” Gaussian membership function

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• “sigmf” Sigmoid membership function

• In crisp logic consequent may be true or false based on Antecedent, but in fuzzy it

might be partially true or false. If-then rule in fuzzy solves first antecedent which

means fuzzifying the four inputs and applying fuzzy operators and assign results to

consequent.

• The results are aggregated and then defuzzified i.e. giving single number.

The fuzzy logic results are closer to the CQF but not its replacement. The fuzzy rules are

applied simultaneously and cater for the effect of participation of simulated results in fuzzy

manner, it shows the degree of Excellent, Good or Worst based on the inputs value, its

member ship function and applied rules. The results of FIS cannot be utilized in place of CQF

but its property of using any membership function and considering the degree of participation

in aggregation gives the idea to utilize this function for optimization and rework function

with CQF and normalized aggregation.

Table 8-46: Sorted with CQF, Quantified fuzzy aggregation and quantified normalized

aggregation

Run EQ KW POC MRR Agg. CQF FL AV QFL AV

1 2 2 1 1 1 2 0.45 2

2 1 2 1 1 1 2 0.48 2

3 2 1 1 2 1 2 0.48 2

10 2 2 1 2 2 2 0.44 2

8 2 2 1 1 2 2 0.45 2

6 2 2 2 2 2 2 0.48 2

7 1 3 1 2 2 3 0.36 2

14 2 3 1 2 2 3 0.38 2

18 1 3 2 3 2 3 0.55 3

64 3 2 3 3 3 3 0.82 3

65 1 3 3 3 3 3 0.83 3

73 3 3 3 2 3 3 0.86 3

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For this purpose the result is sorted in novel order of Customer quality function, quantified

fuzzy aggregation and quantified normalized aggregation values. CQF has the highest

priority because if any one of the output quality is worst i.e. 3 it is not acceptable for any

customer. The quantified fuzzy aggregation has second priority because it aggregates the

value based on simple aggregation with selected membership functions and also with the

rules applied in CQF but in fuzzy manner. The normalized aggregation function helped to

achieve a more comparative list for better optimization. A part of this list is shown in Table

8-46 and complete table in Appendix B (Table B-8).

The sorted Table 8-46 shows overall input data setting of Run 1, 2 and 3. In addition to this

Run 7 and 14 can be utilized by rework because CQF value is 3 but QFL and Normalized

aggregated (AV) values are 2. These settings results show that the quantified value of Kerf

width is worst but other qualities are excellent and desired showing that overall quality is a

low degree of worst. However, run 18 and up to end are shown in Table 8-46 and the full

table in Appendix B (Table B-8) has QFL value at 3 which shows that the product quality is

worst to the high degree so it is better to reject rather than rework. This rework function can

guide us as to whether the product is suitable for rework and save the cost of the loss of

whole product. In many cases the result from FIS is desirable/good/2 but in CQF the result is

worst/useless/3. The overall quality fuzzy model can be improved by fine-tuning of

membership function’s boundaries, intersection points, width and shape. These parameters of

FIS system can be fine tuned by genetic algorithm.

In overall quality problem genetic algorithm is learned and studied for the implementation

point of view. A detailed study was performed for the implementation of genetic algorithm

[60, 68-70, 98, 99, 111] and it was concluded that for the following reasons GA is not

suitable.

• It is an experimental study based on OA without outer array.

• Factorial design dataset is filled with neural network modelling with quality

parameters consisting of three levels (1-3).

• GA is applied instead of experimentation and the initial population is generated on

random basis and selection is based on cost function. Therefore, transfer function is

necessary.

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• Therefore, GA is not suitable for multi-objective overall quality calculation at this

stage.

The genetic algorithm can be applied in our problem but it is out of the scope at the moment.

It is possible to apply genetic algorithm by performing the following tasks.

• Select appropriate DOE for GA.

• Model the laser cutting process for all qualities.

• Use ANN trained model for simulation.

• Prepare cost function for minimum the better.

• Generate initial population of input datasets and predict quality parameter of the

applied input datasets.

• The quality functions values were applied in cost function and selected the population

based on cost function.

• In the end, there are a number of optimized datasets that will be gathered by the

utilization of genetic algorithm.

The overall quality was solved by combination of customer quality function, quantified fuzzy

aggregation and quantified normalized aggregation. Neural network helps in modelling of

laser cutting process. The process industry usually needs to improve multiple quality

parameters which are resolved by novel combination of Fuzzy aggregation. Consumption of

Electricity is also an important factor there is a pressing need to improve energy efficient

processes which lead to reduction in the manufacturing cost by increasing cutting speed and

decrease the utilization of laser power per meter cutting. Therefore, the study adds to the

electricity-efficient solution and shows the new quality parameter calculation which will

include in the overall quality as shown in Table 8-47 and possible energy levels in three

levels factorial design are 9 as shown in Table 8-48. With reference to Table 8-49 the results

are appropriate even for energy point of view based on the assumption that laser power is the

major contributor and it is more energy efficient if assumed that change in cutting speed

causes change in energy consumption. Therefore, the units are watts . m/hr.

Eltawahni et al. [101] looked into the CO2 laser cutting variables of MDF and explained the

cutting cost.

𝐶𝐶𝑜𝑜𝑊𝑊𝑊𝑊𝑖𝑖𝑖𝑖𝐶𝐶 𝑝𝑝𝑊𝑊𝑖𝑖𝑊𝑊 =

2.654 + 1.376 × 𝐴𝐴 + 1.3718 × 10−5 × 𝐹𝐹0.051 × 𝐵𝐵

( 8.19)

𝐹𝐹𝑆𝑆𝑊𝑊𝑤𝑤 𝐾𝐾𝑆𝑆𝑊𝑊𝐾𝐾 = 𝐹𝐹(1 ℎ⁄ ) = 492 × 𝑊𝑊2(C + 1) ( 8.20)

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Where,

A is the power (KW)

F is flow rate (1/h)

B is cutting speed (mm/min)

d is nozzle diameter (mm)

C is assist gas pressure (bar).

Table 8-47: Energy consumption quality calculation for factorial design

Run Laser Power

Cutting Speed

A.Gas pressure

Standoff distance

Energy Consumed

1 100 0.2 0.5 1 8.33 . . . . . . 9 100 0.2 4.5 10 8.33 10 100 0.7 0.5 1 2.38 . . . . . .

18 100 0.7 4.5 10 2.38 19 100 1.2 0.5 1 1.39 . . . . . .

27 100 1.2 4.5 10 1.39 28 300 0.2 0.5 1 25.00 . . . . . .

36 300 0.2 4.5 10 25.00 37 300 0.7 0.5 1 7.14 . . . . . .

45 300 0.7 4.5 10 7.14 46 300 1.2 0.5 1 4.17 . . . . . .

54 300 1.2 4.5 10 4.17 55 500 0.2 0.5 1 41.67 . . . . . .

63 500 0.2 4.5 10 41.67 64 500 0.7 0.5 1 11.90 . . . . . .

72 500 0.7 4.5 10 11.90 73 500 1.2 0.5 1 6.94 . . . . . .

81 500 1.2 4.5 10 6.94

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Table 8-48: Power levels corresponding to Laser power and cutting speed

Levels Power levels Laser Power Cutting speed

1 1.39 100 1.2

2 2.38 100 0.7

3 8.33 100 0.2

4 6.94 300 1.2

5 7.14 300 0.7

6 8.33 300 0.2

7 11.90 500 1.2

8 25.00 500 0.7

9 41.67 500 0.2

Table 8-49: Including power consumed and then sorted with CQF, Q AV and QFL AV Run A B C D EQ KW POC MRR Q AV CQF FL AV QFL AV Power

1 100 0.2 0.5 1 2 2 1 1 1 2 0.45 2 8.33

2 100 0.2 0.5 5 1 2 1 1 1 2 0.48 2 8.33

3 100 0.2 0.5 10 2 1 1 2 1 2 0.48 2 8.33

10 100 0.7 0.5 1 2 2 1 2 2 2 0.44 2 2.38

8 100 0.2 4.5 5 2 2 1 1 2 2 0.45 2 8.33

6 100 0.2 2.5 10 2 2 2 2 2 2 0.48 2 8.33

7 100 0.2 4.5 1 1 3 1 2 2 3 0.36 2 8.33

14 100 0.7 2.5 5 2 3 1 2 2 3 0.38 2 2.34

77 500 1.2 2.5 5 2 3 3 3 3 3 0.79 3 6.94

64 500 0.7 0.5 1 3 2 3 3 3 3 0.82 3 11.90

65 500 0.7 0.5 5 1 3 3 3 3 3 0.83 3 11.90

73 500 1.2 0.5 1 3 3 3 2 3 3 0.86 3 11.90

The above equations indicate that laser power (A) is the major contributor in cost. They also

identified that the power consumed by motion controller does not differ with different speed

[101]. It means difference in power consumption with 0.2 m/min or 0.7 m/min or 1.2 m/min

is negligible. So, assumed constant power consumed in motion controller. It means Flow rate

of assist gas pressure is also another parameter which affects the cost per unit length.

Reducing power and increasing speed results in the reduction of cost. Therefore, for the sake

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of simplicity we calculate cost based on laser power only. As the cost of chiller electrical

power, compressor power, exhaust system power and manpower are constant. With reference

to Table 8-49 the results are appropriate even for cost point of view.

H. A. Eltawahni, et al [6, 101] applied limitation on input, output, overall quality and cost

using desirability approach. In overall quality they are unable to give single optimized results,

entrusting the operator to decide whether money is important or quality. They mentioned that

they were able to reduce the operating cost of cutting up to 71%. But need to compromise on

quality due to cost-effective solution. In our case electricity-effective solution was provided

without compromising on quality. The overall quality can be compromised in some other

cases as many researchers face in cost or other quality parameters [6, 101].

8.7 SUMMARY Experiments of laser cutting non-metallic sheets, based on proposed framework modules,

have been performed in order to model and optimize the laser cutting quality. This was

initially achieved by using orthogonal array as DOE. Some variable parameters were focused

during the laser cutting process such as edge quality, kerf width, percent overcut and material

removal rate as it affected the quality of laser cutting process. However, percent over cut and

material removal rate can be calculated by the kerf width. The combination of input sets is

based on Taguchi’s orthogonal array design. To avoid uncontrollable and human errors, data

was replicated three times and normalized by averaging.

In the work of M. Zaidi et al. [3] data response tables were made without using

preprocessing techniques. Therefore, one value has been taken zero which causes difficulties

in the calculations of target (TPM) and noise performance measurement (NPM) consequently,

unbalancing the analysis table. Therefore, the TPM and NPM recommendations differ and

NPM was preferred as thumb rule of Statisticians[66].

Further, M. Zaidi et al. [25] discussed the possible solutions of non-linear multivariable by

experimental data mining techniques using polystyrene foam cutting data [2]. The results are

not encouraging to study laser cutting process of non-linear multivariable modeled by one

and two way analysis of variance, also linear and nonlinear regression analysis. This detailed

study was started by M. Zaidi et al. [3] by predicting best input data sets using TPM and

NPM. Applying sorting, curve fitting and aggregation technique to solve the overall quality

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of multi-input output problem. This work was presented in international conference and it

was realized that there is a need of systematic approach to handle the whole problem in

modular form as shown in Figure 6.1.

After data collection “Modulling and optimization Module” was focused for the improvement

of laser cutting process using Statistical methods. The one way ANOVA shows in Table 8-5

and Table 8-6 that feeds controllable parameter Laser power’s P and F values reject null

hypothesis Ho. The Table 8-6 shows that cutting speed, assist gas pressure and standoff

distance insignificantly participating in the variation of kerf width quality. The analysis also

shows that better kerf width predicted as laser power (100 watt), cutting speed (0.7 m/s),

assist gas pressure (2.5 bar) and standoff distance (5 mm).

In comparison, the results of “one way analysis of variance” with both type either replication

or without replication favor the replication. It improves the F value. It means, replication

improves the ability of bifurcation between controllable and uncontrollable variations.

The results of “two way ANOVA” show the interaction results between the two variables and

ignoring among three variables as they were insignificant. The results are concluded in Table

8-14.

The interaction 4, 5 and 6 were significant and was ignored as they will be considered as

uncontrollable variables or pooled error. For better optimization considering all the above

interaction in [2, 3, 16] will give better results or apply Semi-supervised learning algorithm

for modelling. It is not a good practice to assume without analysis that input parameters are

independent from each other in Statistical analysis. One factor at a time has been varied to

analyze the effect of input process parameters on output quality characteristics or responses

[13, 16, 18, 20-22] which shows inferior analysis results due to wrong assumption.

The Table 8-50 shows that the laser power causes significant variation on Kerf width mean.

However, the parameters such as cutting speed, assist gas pressure, standoff distance causes

insignificant role in the variation of dependent variable. The average error model based on

significant factor (approximately 12.95%) which is barely acceptable but insignificant factor

having the value of more than 25% is absolutely unacceptable. In addition to this Kerf width

modelling is easier than edge quality mean and also the data is without any missing value.

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Table 8-50: Summary of linear regression analysis

Laser

Power

Cutting

Speed

Assist gas

pressure

Standoff

distance

R +0.885 -0.041 +0.126 0.312

Significant r Significant Insignificant Insignificant Insignificant

Min error 5.17% 6.2% 1.38% 6.38%

Max. error 21.55% 54.83% 54.83% 53.97%

Average error 12.95% 25.77% 25.07% 25.94%

The overall coefficient of correlation is improved to positive 0.984 and it is significant. R2

favors the model usage as the kerf width is 89.9% due to planned input variations Kerf width

residual predicted value shows that the minimum, maximum, and average percent errors are

0.71%, 20.66%, 7.99% respectively. Hence, it can be used as an empirical formula without

missing value data.

Table 8-51: Summary of nonlinear Regression

Input Factors % R2 F sig. P of t test Max. error Average error H0

Without replication

Laser power 79.4 0.009 0.195 23.85% 11.94% Insignificant

With replication

Laser power 57.2 3x10-5 0.989 59.48% 55.81% Insignificant

Cutting speed 0.2 0.979 0.889 93.84% 84.75% Insignificant

Assist gas pressure 2.3 0.755 0.691 89.46% 8.89% Insignificant

Standoff distance 12.4 0.206 0.395 82.37% 76.14% Insignificant

The results of nonlinear regression Table 8-51 show that Laser power is the key parameter.

Its R2 value decreases with the replication. The error is increased due to replication and it will

remain increasing if more observations are considered as in factorial design [47]. The null

hypothesis H0 is accepted for all variables which show that nonlinear regression with or

without replication cannot explain the variations in output quality. The residual error due to

predicted values increases in nonlinear case rather than in linear prediction.

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Kerf Width predicted residual shows that the minimum, maximum, and average percent

errors are 1.82%, 59.48%, 55.81% respectively. The values show that the error is more than

50%. This modelling technique cannot be recommended on the basis of the above Table 8-4

results.

It is concluded that one way ANOVA is able to model the problem with and without

replication as shown in Table 8-6 and Table 8-7. The model gives better results in the case of

replication which shows suitable analysis technique for given datasets. It is also observed that

interaction should be considered to get the better picture of the process optimization as shown

in Table 8-50 [25]. R2 encourages using multiple linear regression model as the Kerf width

variation is 89.9% which is greatly significant. It can be used in rough modeling, simulation

and optimization. The results of nonlinear regression are worst compared to the others

moreover, with replication it becomes more non-realistic because of large number of

observations over and above the fitted points similar to linear regression. The average error

reaches to 50%. The best method was one way ANOVA with pooling but the current research

shows that there is one thing missing that is significantly participating in the variation of

dependent parameters i.e. interaction between two independent and one dependent

parameters. The literature survey shows that ANN model better than regression and other

Statistical techniques [2, 4, 12, 26-35], but need to verify for missing value.

To solve the issue of missing value, the Proposed Framework modules provide guideline to

apply Preprocessing of process module as shown in Figure 6.1 and Figure 6.2. The outlier

analysis and visual inspection of observation tables clearly show that the missing values are

present and the video and picture gallery also show that the sheet was uncut a few times.

The studies show factorial design is appropriate for modelling with ANN. But in this study a

full factorial design is used for L9 orthogonal array and also for larger size of datasets for the

improvement in modelling and verification with supervised learning algorithm of ANN.

This research encourages new researchers to use supervised learning of ANN in case of

missing values. Improvements in the edge quality mean value modelling could be achieved

by normalization. It enlarges the range in case of small range of datasets. The huge modelling

experience gained in this study built the basis of Semi-supervised learning algorithm. This

research verified the results of OA by datasets of FD. The modelling of edge quality signal to

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noise ratio and kerf width S/N were better but the modelling of kerf width and edge quality

mean can be achieved by normalization of quality parameters. Kerf width mean modelling is

easier. However, edge quality mean is difficult and requires larger datasets for modelling.

Other researchers used 9 datasets for modelling and verified with one or two runs. But, in this

study the prediction was verified by the experimental values of full factorial of 81 runs. So,

the modelling capability of ANN with some missing values was never studied before by

predicting on the whole FD input datasets. The OA data utilizes 5 runs for training, 4 runs for

validation, testing and single missing value rest for the verification. Even though the best

possible results of edge quality mean and S/N ratios in terms of average percent error are

46% and 0.123%. Therefore, this modelling does not meet the desire requirement but S/N

ratio can be utilized. After increasing dataset size as factorial design then edge quality mean

modelling average percent error is 6.9% and signal to noise ratio is 3% which is really

encouraging. The kerf width mean average error is also acceptable i.e. 6.4%. The average

percent error in simulation was under 10% and maximum possible individual percent error is

25%. The process of training explains the effects of change in number of neurons, number of

hidden layers, selection of suitable training algorithm, benefit of enrichment, early stopping,

re-initialization of weights, performance measurement and size of datasets It is better to use

Semi-supervised ANN learning algorithm to model with OA (smaller dataset size).

Previous research suggests larger data set size from orthogonal to factorial design for the

training and testing consequently improves the results, however, the option of re-initialization

had not been tested by the authors. Increase in number of re-initializations of neural network

weights will generate better results as shown in Figure 8.18. One of the major advantages of

Semi-supervised on supervised learning is a huge number of re-initializations of the network

weights such as 100 or 1000 or 3000 times however, in other cases it is 5 to 10 times.

Therefore, it can be concluded that the Semi-Supervised learning is indispensible to produce

better models. As in supervised learning it is difficult to do this job even for 100 times.

In the supervised learning, the edge quality model of 3mm were studied and the average

percent error were 19% (mean) and 24% (S/N) but in case of Semi-supervised the same

parameters were not more than 9.5% and 11%. These results encourage utilizing the Semi-

supervised algorithm technique based on supervised learning experiences. The error at most

is 12.0 with 22% missing values in 5mm models as shown in Figure 8.18. The average

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percent error in case of edge quality and kerf width for 3mm and 5mm was calculated by

changing number of re-initializations of weights of the network. The mean and standard

deviation were calculated for each case. The results are depicted in Figure 8.17 and Figure

8.18 respectively which shows that this is also an important parameter that can be utilized for

the improvement of neural network modelling. It is also capable of modelling small datasets

such as orthogonal array.

Earlier [4] only 3mm material was used but later an additional 5mm work piece is also used

for experimentation, training & modelling. The number of missing values doubled with the

same input parameters ranges. But, the modelling error increased no more than 2.5%. The

modelling of missing data encourages the cutting of new materials with small experimental

data sets even when missing values exist. However, it is time consuming, expensive and

needs domain expert to adjust the range for performing experiments without missing values.

The results of single output quality were better than multiple outputs [4]. Figure 8.18 shows

that 3000 time re-initialization will generate better results.

The modelling is improved by running GDM and LM training algorithms. Both the training

algorithms are run repeatedly for each number of neuron.To understand the missing values,

assume a collection of points, which are to be used for the formation of a regression line.

With more points the regression will have a higher probability of being accurate. Similarly, in

our case, data used for checking generalization and testing gives small errors on average

results but larger error near the missing points.

Conversion of fractional data into ordinal data also loses a lot of information that the artificial

neural network can be modeled with, using real non-reprocessed data. There is a similar loss

of information in factorial design due to usage of ordinal three level datasets. The factorial

design is better than orthogonal design because of its larger size than OA.

M. Zaidi et al. [12] in the process of laser cutting modelling, orthogonal array based

experimental data was trained by Semi-supervised learning algorithm of ANN, using GDM

and LM training algorithm. In only one model out of eight GDM predicted better than LM.

The GDM based model average percent error was 4.7% and second lowest was 5%, which

was trained by LM. These results also conclude that utilization of LM training algorithm

reduce training time to hours instead of days. It is true that the re-initialization of neural

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network initial weights is an important factor. The new idea of Semi-supervised learning

definitely improves the modelling accuracy containing up to 22% missing values. This

algorithm will make it easy for the researchers having less insight in soft-computing and laser

cutting. This work reduces modelling time, cost and encourages utilizing existing machines

for the cutting of new materials. The average percent error in 3mm and 5mm Perspex sheet

did not exceed 11.5% and 10.8% in edge quality. Similarly, it also did not exceed 8.0% and

8.7% in case of kerf width. The major achievement is to model orthogonal array with missing

values in a very short time. Therefore, this modelling technique can be used for difficult and

high cost materials for the purpose of modelling and optimization.

M. Zaidi et al. [110] solve the issue of overall quality of laser cutting work-piece using

multi-quality optimization module. The overall quality was calculated by aggregation and

improved it by normalization for equalizing the contribution of all qualities and predicts

unknown machining input setting by neural network model [3]. The normalization was

estimated by equation (8.21) and (8.22). Further improvement was achieved by applying the

customer quality equation based on customer specifications. However, the model is

calculated based on crisp logic and is unable to predict quality variable values separately [2].

The predicted values’ trend was verified by the OA datasets trend. The solution was so

simple and unable to bifurcate, that the total aggregated value rose due to individual factor

domination or it is the accumulated effect of all qualities. Based on customer requirements,

weightage is given to the output quality and the range of the quality tag can be set. Many

researchers have applied intelligent algorithms for overall quality in [9, 11, 60, 68-70, 98, 99,

111, 112] and some others applied data mining techniques [39, 41] and several other

references mentioned in literature review and discussion.

Sum of all qualities = Nor. (Edge quality) + Nor. (Kerf width)

+ Nor. (Overcut) + (1- Nor. (Material removal rate))

(8.21)

Normalized Aggregation = Normalized (Sum of all qualities) (8.22)

The overall quality is based on ANN therefore, the issue of interaction does not effect on the

quality of modelling. However, Ming-Fei et al. [61] and Sharma et al. [36] used statistical

techniques in the modelling, they ignored the interaction effects which generates inferior

results. Semi-supervised learning algorithm overcomes both the issues of missing values and

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interaction effects. M. Zaidi et al. [12] prepared a more generous and better aggregation,

carried out by the novel combination of Fuzzy logic to provide overall quality for customer

and an indication for rework at initial stage for saving cost and time, by the novel

combination of CQF, Q FL AV and Q AV.

The fuzzy logic results are closer to the CQF. The optimizations of membership function

generate closest result to CQF but not as Crisp aggregation results. The fuzzy rules are

applied simultaneously and cater for the effect of participation of simulated results in fuzzy

manner, it shows the degree of Excellent, Good or worst based on the inputs value, its

member ship function and applied rules. The results of FIS cannot be utilize in place of CQF

but its property of using any membership function and considering the degree of participation

in aggregation gives the idea to utilize this function for optimization and rework function

with CQF and quantified normalized aggregation. For this purpose the result is sorted in

novel combination of Customer quality function, quantified fuzzy aggregation and quantified

normalized aggregation values. CQF has the highest priority because if any one of the output

quality is worst i.e. 3 it is not acceptable for any customer. The quantified fuzzy aggregation

has second priority because it aggregates the value based on simple aggregation with selected

membership functions and also with the rules applied in CQF but in fuzzy manner. The

normalized aggregation function helped to achieve a more comparative list for better

optimization. A part of this list is shown in Table 8-46 and complete table in Appendix B

(Table B-8). The sorted Table 8-46 shows overall input data setting of Run 1, 2 and 3. In

addition, Run 7 and 14 can be utilized by rework because CQF value is 3 but QFL AV and

Normalized aggregated (AV) values are 2. These settings results show that the quantified

value of Kerf width is worst but other qualities are excellent and desired, it shows that overall

quality is a low degree of worst. However, run 77 and up to end are shown in Table 8-46 and

the full table in Appendix B (Table B-8) has QFL AV value at 3 which shows that the product

quality is worst to a high degree so it is better to reject rather than rework. Sometimes, one of

the qualities is worst such as Kerf width while other qualities are excellent and desired,

showing that overall quality is a low degree of worst. In this case, rework is suitable which

saves the cost and time of manufacturing organization. It is also possible to sell the product to

the customers with higher tolerances.

The overall quality fuzzy model can be improved by fine-tuning of membership function’s

boundaries, intersection points, width and shape. These parameters of FIS system can be fine

168

tuned by genetic algorithm.In overall quality problem, genetic algorithm is learned and

studied for the implementation point of view. A detailed study was performed for the

implementation of genetic algorithm [60, 68-70, 98, 99, 111] and it was concluded that for

the following reasons GA is not suitable for our study.

• It is an experimental study based on OA without outer array.

• GA is applied instead of experimentation and the initial population is generated on

random basis and selection is based on cost function. Therefore, cost function is

required.

The genetic algorithm can be applied in our problem however ANN trained model for

simulation is used. Prepare cost function for minimum the better for KW, EQ and POC and

maximum the better for MRR.

The process industry usually needs to improve multiple quality parameters which are

resolved by novel combination of Fuzzy aggregation. It is also an important factor to improve

energy efficient processes which lead to reduction in the manufacturing cost by increasing

cutting speed and decreasing the utilization of laser power per meter cutting. Therefore, the

study adds an energy efficient solution and shows the new quality parameter calculation

which will include in the overall quality as shown in Table 8-47 and possible levels in three

levels factorial design are 9. With reference to Table 8-49 the results are appropriate even

from energy point of view. Assuming that changes in cutting speed causes change in energy

consumption. Therefore, the units are watts. m/hr. Eltawahni et al. [101] looked into the CO2

laser cutting variables of MDF and explained the cutting cost. The equation ( 8.19) and ( 8.20)

indicate that laser power (A) is the major contributor in cost. They also identified that the

power consumed by motion controller does not differ with different speed [101]. It means

that the difference in power consumption with 0.2 m/min or 0.7 m/min or 1.2 m/min is

negligible. So, it is assumed constant power is consumed in motion controller. It means Flow

rate of assist gas pressure is also another parameter which affects the cost per unit length.

Reducing power and increasing speed results in the reduction of cost. Therefore, for the sake

of simplicity, we calculate cost based on the laser power only. As the cost of chiller electrical

power, compressor power, exhaust system power and manpower are constant. With reference

to Table 8-49 the results are appropriate even from cost point of view.

169

H. A. Eltawahni, et al [6, 101] applied limitation on input, output, overall quality and cost

using desirability approach. In overall quality they are unable to give single optimized results,

entrusting the operator to decide whether money is important or quality. In our case

electricity-effective solution was provided without compromising on quality.

CHAPTER 9

CONCLUSION AND FUTURE DIRECTION

170

9. CONCLUSION Laser cutting experiment on non-metallic sheets, based on proposed framework modules,

have been performed in order to model and optimize the laser cutting quality. Edge quality,

kerf width, percent overcut and material removal rate were focused as it affected the quality

of laser cutting process. In the initial work, calculations of TPM and NPM were unbalanced

due to missing values [3]. Therefore, TPM and NPM recommendations were different and

NPM was preferred as a thumb rule of Statisticians. It was realized that there is a need of

systematic approach to handle the whole problem in modular form as shown in Figure 6.1.

Further, detailed Statistical techniques were applied to model the problem. The one way

ANOVA in Table 8-5 and Table 8-6 shows that Laser power’s is a significant factor while,

cutting speed, assist gas pressure and standoff distance insignificantly participate in the

variation of kerf width quality. The results are favorable with or without replication in one

way and two way ANOVA. However, the results of replications are best. The analysis shows

that better kerf width predicted at laser power (100 watt), cutting speed (0.7 m/s), assist gas

pressure (2.5 bar) and standoff distance (5 mm). The results of “two way ANOVA” show the

interaction 4, 5 and 6 were significant between the two variables as shown in Table 8-50. It

was ignored as their error will be added with noise or pooled error.

The Table 8-51 linear regression summary shows that the laser power is significant and

cutting speed, assist gas pressure, standoff play an insignificant role on Kerf width mean. The

average error model based on significant factor (approximately 12.95%) is acceptable but

other factors having the value of more than 25% is absolutely unacceptable. Kerf width

residual predicted value of multiple linear regression shows that the minimum, maximum,

and average percent errors are 0.71%, 20.66%, 7.99% respectively. Hence, it can be used as

an empirical formula without missing value data. The results in Table 8-52 show that

nonlinear regression with or without replication cannot explain the variations in output

quality and results of replication were more inferior. Kerf Width predicted residual shows

that the minimum, maximum, and average percent errors are 1.82%, 59.48%, 55.81%

respectively.

It is concluded that one way ANOVA results are best among the Statistical models with

replication as shown in Table 8-7 provided that missing value do not exist. However, results

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show that ANN models better than regression and other Statistical techniques. [2, 4, 12, 26-

35]. However, it needs to be verified for missing value problem.

To solve the issue of missing value apply Preprocessing module (outlier analysis) as shown

in Figure 6.1 and Figure 6.2. The studies show factorial design is appropriate for modelling

with ANN. But, unable to model supervised learning model with OA. The supervised

learning modelling of edge quality signal to noise ratio and kerf width S/N were better than

kerf width mean and edge quality mean. Kerf width mean modelling is easier than edge

quality mean which is difficult and requires larger datasets for modelling. The modelling

capability of ANN with some missing values was never studied before by predicting on the

whole FD input datasets. The OA data is utilized 5 runs for training, 4 runs for validation,

testing and single missing value, remaining is for verification. Even though the best possible

results of edge quality mean and S/N ratios in terms of average percent error are 46% and

0.123%. Therefore, this modelling does not meet the desired requirement but S/N ratio can be

utilized.

After increasing dataset size as factorial design then edge quality mean modelling average

percent error is 6.9% and signal to noise ratio is 3% which is really encouraging. The kerf

width mean average error is also acceptable i.e. 6.4%. The average percent error in simulation

was under 10% and maximum possible individual percent error is 25%. The process of

training explains the effects of change in number of neurons, number of hidden layers,

selection of suitable training algorithm, benefit of enrichment, early stopping, re-initialization

of weights, performance measurement and size of datasets. Increase in number of re-

initializations of neural network weights will generate better results as shown in Figure 8.18.

In the supervised learning, the edge quality model of 3mm were studied and the average

percent error were 19% (mean) and 24% (S/N) but in case of Semi-supervised the same

parameters were not more than 9.5% and 11%. These results encourage utilizing the Semi-

supervised algorithm technique based on supervised learning experiences. The error at most

is 12.0 with 22% missing values in 5mm models as shown in Figure 8.18.

M. Zaidi et al. [12] used GDM and LM training algorithm in ANN. In only one model out of

eight GDM predicted better than LM. The GDM based model average percent error was 4.7%

and second lowest was 5%, which was trained by LM. These results also conclude that

172

utilization of LM training algorithm reduce training time to hours instead of days. The

number of missing values doubled with the same input parameters ranges. But, the modelling

error increased no more than 2.5%. The modelling of missing data encourages the cutting of

new materials with small experimental data sets even when missing values exist. However, it

is time consuming, expensive and needs domain expert to adjust the range for performing

experiments without missing values. The results of single output quality were better than

multiple outputs [4]. Figure 8.18 shows that 3000 time re-initialization will generate better

results.

This algorithm will make it easy for the researchers having less insight in soft-computing and

laser cutting. This work reduces modelling time, cost and encourages utilizing existing

machines for the cutting of new materials. It is concluded that Semi-supervised learning

algorithm is capable of modelling small datasets such as OA.

M. Zaidi et al. [110] solved the issue of overall quality of laser cutting work-piece using

multi-quality optimization module. It was calculated by aggregation and improved by

normalization for equalizing the contribution of all qualities and predicts unknown machining

input setting by neural network model [3]. The normalization was estimated by equation

(8.21) and (8.22). Further improvement was achieved by applying the customer quality

equation based on customer specifications. However, the model is calculated based on crisp

logic and is unable to predict quality variable values separately [2]. The predicted values’

trend was verified by the OA datasets trend. The solution was so simple and unable to

bifurcate, that the total aggregated value rose due to individual factor domination or it is the

accumulated effect of all qualities. Based on customer requirements, weightage is given to

the output quality and the range of the quality tag can be set. The overall quality is based on

ANN (Semi-supervised learning algorithm) therefore, the issue of interaction does not effect

the quality of modelling. M. Zaidi et al. [12] prepared a more generous and better aggregation,

carried out by the novel combination of Fuzzy logic to provide overall quality for customer

and an indication for rework at initial stage for saving cost and time, by the novel

combination of CQF, Q FL AV and Q AV.

The fuzzy logic results are closer to the CQF. The fuzzy rules are applied simultaneously and

cater for the effect of participation of simulated results in fuzzy manner, it shows the degree

of Excellent, Good or worst based on the inputs value, its member ship function and applied

173

rules. The property of using any membership function and considering the degree of

participation in aggregation gives the idea to utilize this function for optimization and rework

function with CQF and quantified normalized aggregation. For this purpose the result is

sorted in novel combination. CQF has the highest priority because if any one of the output

quality is worst i.e. 3 it is not acceptable for any customer. The quantified fuzzy aggregation

has second priority because it aggregates the value based on simple aggregation with

selected membership functions and also with the rules applied in CQF but in fuzzy manner.

The normalized aggregation function helped to achieve a more comparative list for better

optimization. A part of this list is shown in Table 8-46 and complete table in Appendix B

(Table 12-8). The sorted Table 8-46 shows overall input data setting of Run 1, 2 and 3.

In addition, Run 7 and 14 can be utilized by rework because CQF value is 3 but QFL AV and

Normalized aggregated (AV) values are 2. These settings results show that the quantified

value of Kerf width is worst but other qualities are excellent and desired, it shows that overall

quality is a low degree of worst. However, run 77 and up to end are shown in Table 8-46 (full

table in Appendix B (Table B-8) having QFL AV value at 3 which shows that the product

quality is worst to the high degree so it is better to reject rather than rework. Sometimes, one

of the qualities is worst such as Kerf width is worst but other qualities are excellent and

desired, showing that overall quality is a low degree of worst. In this case, rework is suitable

which saves the cost and time of manufacturing organization. It is also possible to sell the

product to the customers with higher tolerances.

Electricity is also an important factor to improve energy efficient processes which lead to

reduction in the manufacturing cost by increasing cutting speed and decreasing the utilization

of laser power per meter cutting as shown in Table 8-49 the results are appropriate even from

energy point of view. Assuming that changes in cutting speed causes change in energy

consumption. Therefore, the units are watts. m/hr. The equation ( 8.19) and ( 8.20) indicate

that laser power (A) is the major contributor in cost and power consumed by motion

controller which does not differ with different speed [101]. It means Flow rate of assist gas

pressure is also another parameter which affects the cost per unit length. Reducing power and

increasing speed results in the reduction of cost. Therefore, for the sake of simplicity, we

calculate cost based on the laser power only. As the cost of chiller electrical power,

compressor power, exhaust system power and manpower are constant. With reference to

Table 8-49 the results are appropriate even from cost point of view. Eltawahni, et al [6, 101]

174

used desirability approach for overall quality but were unable to give single optimized results,

entrusting the operator to decide whether money is important or quality. In our case

electricity-effective solution was provided without compromising on quality.

CHAPTER 10

FUTURE DIRECTION

175

10. FUTURE DIRECTION

In my view the design of experiment used by AI techniques requires some modifications.

Some data set out of designed matrix are required for checking of generalization of model

and some for testing of the model. This issue can be resolved without investing on

experimentation, by taking the observation taken for range adjustment before the actual

experimentation. Range adjustment will be carried out systematically and this observation

table will be used in ANN modelling.

Future research can be focused on running process industry to improve the quality and

reduction in cost by optimizing the use of Electricity. The modelling and optimization can be

applied in the light of “Proposed Framework Module”.

Figure 10-1: Brief Procedure to solve the problem

Titanium Tungsten Inert Gas (TIG) welding is more complex as compared to other welding

processes applied in process industry. Welding guidelines are available however “experience

and practice is the trainer”. The cost of Titanium pipe is very high and in case of rejection

more pipe is required which can take up to 6 months to procure. Therefore, bad weld costs

money and time. A 10% Radiography (RT) rejection in welding is considered acceptable.

Smooth movement of welder’s hand, maintaining uniform speed and constant distance from

•Understand the problem and aim

•Screening of variables for this purpose

•Select the data for modelling

Preprocessing & Experimental data

•Select the modelling method •Model the problem by Semi-

supervised method or more appropriate

Modelling•Apply appropriate

techhnique• If apply Fuzzy agregation

then optimize memebership functions

Optimization

176

work-piece joint will produce better results. But due to human error it is difficult to maintain

consistency. The cost of labor, material, workshop capital and workshop running are

expensive. There is a need to develop a setting to reduce the rejections.

Normally vendor documents suggest the welding parameters, however recommended values

are only guidelines, quality also relies on skill and experience. Hence our proposed

Framework can be applied in this problem for consistent and improved results. Evaluation of

the problem shows that following are the constant parameters

1. Temperature < 26 °C

2. Relative Humidity < 30%

3. Air tight Environment

4. Stable power supply

5. Metal Thickness 0.1875”

Following are the controllable parameters

1. Shielding Gases for TIG

Argon

Helium

Mixtures

2. Tungsten Electrode diameter

0.125” to 0.156”

3. Current in Ampere

180 to 225

4. Distance between Electrode and Work-piece

5mm to 50 mm

Speed of Electrode on joint in first pass

Quality will be assessed by the

1. RT

2. Heat affected zone

3. Joint roughness.

For high accuracy in measuring, instead of using a Caliper other measuring tools could be

used for small least count and better repeatability.

177

A mechanical structure will be prepared around the pipe where TIG electrode can move with

different speed settings and the distance between the electrode and joint can be adjusted.

There are four controllable variables and each have 3 ordinal levels. The output quality will

be assessed by the three variables. The DOE is L9 orthogonal array and range adjustment

observation will be used to model the problem by Semi-supervised model. The Fuzzy

aggregation is more appropriate to resolve the issue of overall optimization because quality

parameters are more than 3. The overall quality fuzzy model can be improved by fine-tuning

of membership function’s boundaries, intersection points, width and shape. These parameters

of FIS system can be fine tuned by genetic algorithm.

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[116] K. Funahashi, "On the Approximate Realization of Continuous Mappings by Neural Networks," Neural Networks, vol. 2, pp. 183-192, 1989.

[117] F. J. Pontes, et al., "Artificial neural networks for machining processes surface roughness modeling," The International Journal of Advance Manufacturing Technology, vol. 49, pp. 879–902, 2010.

[118] A. M. Zain, et al., "Review of ANN Technique for Modeling Surface Roughness Performance Measure in Machining Process," presented at the 2009 Third Asia International Conference on Modelling & Simulation, 2009.

[119] H. K. Ghuge and D. G. Regulwar, "Artificial Neural Network Method for Estimation of Missing Data," International Journal of Advanced Technology in Civil Engineering, vol. 2, 2013.

APPENDIX A

185

A. DOE & STATISTICAL MODELLIN Table A-1: Four variables with three levels Factorial design matrix

Run Input variable Measurement

Laser Power

A

Cutting speed

B

Assist gas Pressure

C

Standoff Distance

D

Mean S/N Ratio

1 100 0.2 0.5 1 2 100 0.2 0.5 5 3 100 0.2 0.5 10 4 100 0.2 2.5 1 5 100 0.2 2.5 5 6 100 0.2 2.5 10 7 100 0.2 4.5 1 8 100 0.2 4.5 5 9 100 0.2 4.5 10

10 100 0.7 0.5 1 11 100 0.7 0.5 5 12 100 0.7 0.5 10 13 100 0.7 2.5 1 14 100 0.7 2.5 5 15 100 0.7 2.5 10 16 100 0.7 4.5 1 17 100 0.7 4.5 5 18 100 0.7 4.5 10 19 100 1.2 0.5 1 20 100 1.2 0.5 5 21 100 1.2 0.5 10 22 100 1.2 2.5 1 23 100 1.2 2.5 5 24 100 1.2 2.5 10 25 100 1.2 4.5 1 26 100 1.2 4.5 5 27 100 1.2 4.5 10 28 300 0.2 0.5 1 29 300 0.2 0.5 5 30 300 0.2 0.5 10

186

Run Input variable Measurement

Laser Power

A

Cutting speed

B

Assist gas Pressure

C

Standoff Distance

D

Mean S/N Ratio

31 300 0.2 2.5 1 32 300 0.2 2.5 5 33 300 0.2 2.5 10 34 300 0.2 4.5 1 35 300 0.2 4.5 5 36 300 0.2 4.5 10 37 300 0.7 0.5 1 38 300 0.7 0.5 5 39 300 0.7 0.5 10 40 300 0.7 2.5 1 41 300 0.7 2.5 5 42 300 0.7 2.5 10 43 300 0.7 4.5 1 44 300 0.7 4.5 5 45 300 0.7 4.5 10 46 300 1.2 0.5 1 47 300 1.2 0.5 5 48 300 1.2 0.5 10 49 300 1.2 2.5 1 50 300 1.2 2.5 5 51 300 1.2 2.5 10 52 300 1.2 4.5 1 53 300 1.2 4.5 5 54 300 1.2 4.5 10 55 500 0.2 0.5 1 56 500 0.2 0.5 5 57 500 0.2 0.5 10 58 500 0.2 2.5 1 59 500 0.2 2.5 5 60 500 0.2 2.5 10 61 500 0.2 4.5 1 62 500 0.2 4.5 5 63 500 0.2 4.5 10 64 500 0.7 0.5 1 65 500 0.7 0.5 5 66 500 0.7 0.5 10

187

Run Input variable Measurement Laser Power

A

Cutting speed

B

Assist gas Pressure

C

Standoff Distance

D

Mean S/N Ratio

67 500 0.7 2.5 1 68 500 0.7 2.5 5 69 500 0.7 2.5 10 70 500 0.7 4.5 1 71 500 0.7 4.5 5 72 500 0.7 4.5 10 73 500 1.2 0.5 1 74 500 1.2 0.5 5 75 500 1.2 0.5 10 76 500 1.2 2.5 1 77 500 1.2 2.5 5 78 500 1.2 2.5 10 79 500 1.2 4.5 1 80 500 1.2 4.5 5 81 500 1.2 4.5 10

Table A-2: Edge quality mean and S/N ratio of Polystyrene foam Sheet (13mm)

Run Input Variable Measurement Laser Power

A

Cutting speed

B

Assist gas Pressure

C

Standoff Distance

D

Replication Edge quality Mean

S/N Ratio R1 R2 R3

1 100 0.2 0.5 1 3.50 4.00 3.50 3.67 -11.312 2 100 0.2 0.5 5 2.20 3.00 3.00 2.73 -8.856 3 100 0.2 0.5 10 1.50 1.50 2.00 1.67 -4.565 4 100 0.2 2.5 1 1.50 1.50 2.00 1.67 -4.565 5 100 0.2 2.5 5 1.00 1.00 1.00 1.00 0.000 6 100 0.2 2.5 10 2.50 2.50 3.00 2.67 -8.570 7 100 0.2 4.5 1 1.50 2.00 3.00 2.17 -7.225 8 100 0.2 4.5 5 1.50 2.00 2.00 1.83 -5.371 9 100 0.2 4.5 10 1.50 2.50 2.00 2.00 -6.284

188

Table A-3: Kerf width observations of Polystyrene foam Sheet (13mm)


A

Cutting speed

B

Assist gas Pressure

C

Standoff Distance

D

Replication Edge quality Mean

S/N Ratio R1 R2 R3

1 100 0.2 0.5 1 1.58 1.62 1.53 1.573 -3.940 2 100 0.7 2.5 5 1.73 1.26 1.48 1.488 -3.563 3 100 1.2 4.5 10 1.66 1.70 1.86 1.738 -4.819 4 300 0.2 2.5 10 1.94 1.89 1.92 1.913 -5.637 5 300 0.7 4.5 1 1.77 1.78 1.99 1.842 -5.324 6 300 1.2 0.5 5 1.66 1.72 1.86 1.742 -4.834 7 500 0.2 4.5 5 2.01 2.04 1.86 1.968 -5.893 8 500 0.7 0.5 10 1.98 1.94 2.29 2.068 -6.350 9 500 1.2 2.5 1 1.79 1.89 2.08 1.920 -5.692

Table A-4: Edge quality observations of Perspex Sheet (3mm)


A

Cutting speed

B

Assist gas Pressure

C

Standoff Distance

D

Replication Edge qualityMean

S/N Ratio

R1 R2 R3

1 100 0.2 0.5 1 0.14 0.15 0.15 0.147 16.67

2 100 0.2 0.5 5 0.1 0.1 0.09 0.097 20.28

3 100 0.2 0.5 10 0.06 0.06 0.06 0.060 24.44

4 100 0.2 2.5 1 0.14 0.14 0.14 0.140 17.08

5 100 0.2 2.5 5 0.07 0.07 0.07 0.070 23.10

6 100 0.2 2.5 10 0.06 0.05 0.05 0.053 25.41

7 100 0.2 4.5 1 0.18 0.18 0.18 0.180 14.89

8 100 0.2 4.5 5 0.06 0.06 0.06 0.060 24.44

9 100 0.2 4.5 10 0.12 0.11 0.11 0.113 18.90

10 100 0.7 0.5 1 0.29 0.29 0.29 0.290 10.75

11 100 0.7 0.5 5 0.12 0.12 0.12 0.120 18.42

12 100 0.7 0.5 10 0.09 0.09 0.09 0.090 20.92

13 100 0.7 2.5 1 0.21 0.23 0.23 0.223 13.01

14 100 0.7 2.5 5 0.07 0.06 0.07 0.067 23.49

15 100 0.7 2.5 10 0.08 0.08 0.08 0.080 21.94

16 100 0.7 4.5 1 0.29 0.29 0.29 0.290 10.75

17 100 0.7 4.5 5 0.11 0.1 0.1 0.103 19.70

18 100 0.7 4.5 10 0.18 0.14 0.14 0.153 16.19

189


A

Cutting speed

B

Assist gas Pressure

C

Standoff Distance

D


S/N Ratio

R1 R2 R3

19 100 1.2 0.5 1 0.24 0.28 0.28 0.267 11.45

20 100 1.2 0.5 5 0.13 0.13 0.13 0.130 17.72

21 100 1.2 0.5 10 Unsuccessful cutting

22 100 1.2 2.5 1 0.17 0.17 0.17 0.170 15.39

23 100 1.2 2.5 5 0.11 0.11 0.11 0.110 19.17


25 100 1.2 4.5 1 0.25 0.25 0.25 0.250 12.04

26 100 1.2 4.5 5 0.1 0.12 0.1 0.107 19.39

27 100 1.2 4.5 10 Unsuccessful

28 300 0.2 0.5 1 0.15 0.15 0.15 0.150 16.48

29 300 0.2 0.5 5 0.09 0.11 0.09 0.097 20.23

30 300 0.2 0.5 10 0.09 0.13 0.11 0.110 19.03

31 300 0.2 2.5 1 0.07 0.07 0.07 0.070 23.10

32 300 0.2 2.5 5 0.03 0.04 0.04 0.037 28.61

33 300 0.2 2.5 10 0.05 0.05 0.05 0.050 26.02

34 300 0.2 4.5 1 0.14 0.15 0.14 0.143 16.87

35 300 0.2 4.5 5 0.09 0.08 0.08 0.083 21.56

36 300 0.2 4.5 10 0.08 0.07 0.07 0.073 22.67

37 300 0.7 0.5 1 0.11 0.11 0.11 0.110 19.17

38 300 0.7 0.5 5 0.11 0.11 0.11 0.110 19.17

39 300 0.7 0.5 10 0.07 0.07 0.07 0.070 23.10

40 300 0.7 2.5 1 0.17 0.17 0.17 0.170 15.39

41 300 0.7 2.5 5 0.1 0.09 0.09 0.093 20.58

42 300 0.7 2.5 10 0.1 0.1 0.1 0.100 20.00

43 300 0.7 4.5 1 0.22 0.22 0.22 0.220 13.15

44 300 0.7 4.5 5 0.08 0.09 0.08 0.083 21.56

45 300 0.7 4.5 10 0.11 0.11 0.11 0.110 19.17

46 300 1.2 0.5 1 0.23 0.23 0.23 0.230 12.77

47 300 1.2 0.5 5 0.08 0.08 0.08 0.080 21.94

48 300 1.2 0.5 10 0.1 0.1 0.1 0.100 20.00

49 300 1.2 2.5 1 0.21 0.21 0.21 0.210 13.56

50 300 1.2 2.5 5 0.07 0.07 0.07 0.070 23.10

190


A

Cutting speed

B

Assist gas Pressure

C

Standoff Distance

D


S/N Ratio

R1 R2 R3

51 300 1.2 2.5 10 0.11 0.11 0.11 0.110 19.17

52 300 1.2 4.5 1 0.29 0.29 0.29 0.290 10.75

53 300 1.2 4.5 5 0.11 0.11 0.11 0.110 19.17

54 300 1.2 4.5 10 0.1 0.09 0.09 0.093 20.58

55 500 0.2 0.5 1 0.12 0.12 0.15 0.130 17.64

56 500 0.2 0.5 5 0.08 0.08 0.08 0.080 21.94

57 500 0.2 0.5 10 0.09 0.09 0.09 0.090 20.92

58 500 0.2 2.5 1 0.11 0.11 0.11 0.110 19.17

59 500 0.2 2.5 5 0.07 0.08 0.07 0.073 22.67

60 500 0.2 2.5 10 0.06 0.06 0.06 0.060 24.44

61 500 0.2 4.5 1 0.14 0.14 0.14 0.140 17.08

62 500 0.2 4.5 5 0.12 0.12 0.12 0.120 18.42

63 500 0.2 4.5 10 0.11 0.11 0.11 0.110 19.17

64 500 0.7 0.5 1 0.17 0.17 0.17 0.170 15.39

65 500 0.7 0.5 5 0.1 0.12 0.1 0.107 19.39

66 500 0.7 0.5 10 0.16 0.13 0.13 0.140 17.01

67 500 0.7 2.5 1 0.11 0.11 0.11 0.110 19.17

68 500 0.7 2.5 5 0.06 0.06 0.06 0.060 24.44

69 500 0.7 2.5 10 0.13 0.13 0.13 0.130 17.72

70 500 0.7 4.5 1 0.21 0.21 0.21 0.210 13.56

71 500 0.7 4.5 5 0.13 0.1 0.1 0.110 19.07

72 500 0.7 4.5 10 0.12 0.12 0.12 0.120 18.42

73 500 1.2 0.5 1 0.15 0.13 0.13 0.137 17.26

74 500 1.2 0.5 5 0.1 0.1 0.1 0.100 20.00

75 500 1.2 0.5 10 0.08 0.08 0.08 0.080 21.94

76 500 1.2 2.5 1 0.11 0.11 0.11 0.110 19.17

77 500 1.2 2.5 5 0.08 0.08 0.08 0.080 21.94

78 500 1.2 2.5 10 0.1 0.1 0.1 0.100 20.00

79 500 1.2 4.5 1 0.19 0.19 0.19 0.190 14.42

80 500 1.2 4.5 5 0.09 0.08 0.08 0.083 21.56

81 500 1.2 4.5 10 0.1 0.1 0.1 0.100 20.00

191

Figure A-1: Outlier analysis of edge quality of Perspex glass sheet (3mm)

Table A-5: Kerf width observations for of Perspex Sheet (3mm)

Input Variable Measurements Run Laser

Power (watts)

Cutting Speed

(m/min)

Assist Gas

pressure (bar)

Stand off

distance (mm)

Inner side line length (mm)

Replications

Outer scrap length (mm)

Replications Lin1 Lin2 Lin3 Lout1 Lout2 Lout3

1 100 0.2 0.5 1 20.76 20.77 20.79 19.67 19.66 19.66 2 100 0.2 0.5 5 20.82 20.82 20.82 19.5 19.52 19.5 3 100 0.2 0.5 10 21.3 21.35 21.35 19.23 19.23 19.23 4 100 0.2 2.5 1 20.75 20.76 20.76 19.65 19.64 19.65 5 100 0.2 2.5 5 20.88 20.82 20.87 19.49 19.49 19.49 6 100 0.2 2.5 10 21.24 21.24 21.26 19.04 19.03 19.04 7 100 0.2 4.5 1 20.7 20.71 20.76 19.78 19.78 19.78 8 100 0.2 4.5 5 20.82 20.89 20.89 19.59 19.61 19.6 9 100 0.2 4.5 10 21.28 21.22 21.24 19.19 19.2 19.21 10 100 0.7 0.5 1 20.57 20.6 20.63 19.65 19.68 19.69 11 100 0.7 0.5 5 20.68 20.72 20.72 19.6 19.61 19.6 12 100 0.7 0.5 10 20.89 20.94 20.93 19.41 19.42 19.42 13 100 0.7 2.5 1 20.49 20.52 20.54 19.78 19.79 19.78 14 100 0.7 2.5 5 20.72 20.79 20.73 19.63 19.63 19.63

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79

Edge

qua

lity

obse

rvat

ions

Runs

Replicate 3

Replicate 2

Replicate 1

192


Power (watts)

Cutting Speed

(m/min)

Assist Gas

pressure (bar)

Stand off

distance (mm)


Replications



15 100 0.7 2.5 10 21.11 21.11 21.15 19.37 19.39 19.38 16 100 0.7 4.5 1 20.64 20.64 20.64 19.81 19.79 19.79 17 100 0.7 4.5 5 20.69 20.71 20.73 19.76 19.75 19.75 18 100 0.7 4.5 10 21.12 21.07 21.05 19.31 19.29 19.29 19 100 1.2 0.5 1 20.52 20.5 20.5 19.87 19.88 19.87 20 100 1.2 0.5 5 20.63 20.6 20.7 19.71 19.71 19.7 21 100 1.2 0.5 10 Unsuccessful cutting 22 100 1.2 2.5 1 20.56 20.52 20.53 19.9 19.93 19.89 23 100 1.2 2.5 5 20.65 20.68 20.69 19.75 19.74 19.73 24 100 1.2 2.5 10 Unsuccessful cutting 25 100 1.2 4.5 1 20.49 20.49 20.49 19.8 19.81 19.81 26 100 1.2 4.5 5 20.63 20.62 20.59 19.79 19.74 19.73 27 100 1.2 4.5 10 Unsuccessful cutting 28 300 0.2 0.5 1 21 21 21 19.37 19.37 19.37 29 300 0.2 0.5 5 21.18 21.18 21.18 19.13 19.12 19.12 30 300 0.2 0.5 10 21.47 21.45 21.42 18.89 18.89 18.9 31 300 0.2 2.5 1 20.93 20.91 20.99 19.56 19.55 19.57 32 300 0.2 2.5 5 21.17 21.16 21.18 19.26 19.22 19.22 33 300 0.2 2.5 10 21.67 21.65 21.63 18.74 18.74 18.73 34 300 0.2 4.5 1 20.84 20.85 20.86 19.48 19.51 19.48 35 300 0.2 4.5 5 21.21 21.21 21.21 19.38 19.38 19.38 36 300 0.2 4.5 10 21.53 21.54 21.54 18.85 18.85 18.85 37 300 0.7 0.5 1 20.8 20.83 20.79 19.58 19.58 19.58 38 300 0.7 0.5 5 20.97 20.99 21 19.38 19.38 19.38 39 300 0.7 0.5 10 21.27 21.25 21.26 19.02 19.05 19.02 40 300 0.7 2.5 1 20.82 20.79 20.81 19.62 19.62 19.62 41 300 0.7 2.5 5 21 20.98 20.98 19.41 19.4 19.41 42 300 0.7 2.5 10 21.43 21.44 21.44 18.97 18.97 18.97 43 300 0.7 4.5 1 20.82 20.82 20.8 19.63 19.63 19.63 44 300 0.7 4.5 5 20.94 20.95 20.99 19.44 19.44 19.44 45 300 0.7 4.5 10 21.35 21.33 21.32 19.02 19.03 19.03 46 300 1.2 0.5 1 20.81 20.82 20.82 19.67 19.67 19.67 47 300 1.2 0.5 5 20.85 20.85 20.85 19.46 19.46 19.46 48 300 1.2 0.5 10 21.09 21.11 21.09 19.32 19.32 19.32 49 300 1.2 2.5 1 20.71 20.69 20.67 19.72 19.72 19.72 50 300 1.2 2.5 5 20.91 20.9 20.89 19.4 19.42 19.4 51 300 1.2 2.5 10 21.34 21.34 21.34 19.08 19.08 19.08

193


Power (watts)

Cutting Speed

(m/min)

Assist Gas

pressure (bar)

Stand off

distance (mm)


Replications



52 300 1.2 4.5 1 20.68 20.68 20.68 19.74 19.74 19.74 53 300 1.2 4.5 5 20.91 20.93 20.91 19.47 19.47 19.47 54 300 1.2 4.5 10 21.07 21.07 21.07 19.11 19.11 19.11 55 500 0.2 0.5 1 21.14 21.14 21.14 19.31 19.31 19.31 56 500 0.2 0.5 5 21.31 21.34 21.34 19.05 19.05 19.05 57 500 0.2 0.5 10 21.54 21.54 21.54 17.75 17.78 17.76 58 500 0.2 2.5 1 20.98 20.98 20.97 19.41 19.41 19.41 59 500 0.2 2.5 5 21.31 21.31 21.31 19.11 19.11 19.11 60 500 0.2 2.5 10 21.7 20.69 21.7 18.92 18.91 18.92 61 500 0.2 4.5 1 21.06 21.06 21.06 19.49 19.51 19.51 62 500 0.2 4.5 5 21.21 21.21 21.21 19.24 19.24 19.24 63 500 0.2 4.5 10 21.63 21.65 21.65 18.72 18.71 18.71 64 500 0.7 0.5 1 20.87 20.87 20.87 19.49 19.51 19.49 65 500 0.7 0.5 5 21 21.03 21.03 19.28 19.28 19.28 66 500 0.7 0.5 10 21.4 21.4 21.4 18.94 18.94 18.94 67 500 0.7 2.5 1 20.89 20.93 20.91 19.57 19.59 19.57 68 500 0.7 2.5 5 21.09 21.09 21.1 19.31 19.33 19.33 69 500 0.7 2.5 10 21.52 21.52 21.52 18.85 18.85 18.85 70 500 0.7 4.5 1 20.84 20.84 20.84 19.49 19.51 19.49 71 500 0.7 4.5 5 21.04 21.04 21.04 19.43 19.43 19.43 72 500 0.7 4.5 10 21.38 21.41 21.4 18.94 18.99 18.98 73 500 1.2 0.5 1 20.81 20.81 20.76 19.59 19.61 19.6 74 500 1.2 0.5 5 20.98 20.98 20.98 19.44 19.44 19.45 75 500 1.2 0.5 10 21.26 21.22 21.26 19.19 19.2 19.18 76 500 1.2 2.5 1 20.79 20.81 20.81 19.62 19.62 19.59 77 500 1.2 2.5 5 20.95 20.92 20.93 19.44 19.44 19.44 78 500 1.2 2.5 10 21.41 21.46 21.45 18.98 18.97 18.98 79 500 1.2 4.5 1 20.73 20.72 20.71 19.72 19.72 19.72 80 500 1.2 4.5 5 21.09 21 21.05 19.38 19.38 19.38 81 500 1.2 4.5 10 21.29 21.31 21.3 19.1 19.11 19.09

194

Figure A-2: Outlier analysis of Kerf width observations of Perspex glass sheet (3mm)

Table A-6: Kerf width mean and signal to noise ratio of Perspex Sheet (3mm)

Input Variable Measurements

Run Laser Power (watts)

Cutting Speed (m/min)

Assist Gas pressure (bar)

Standoff distance (mm)

Mean S/N Ratio

1 100 0.2 0.5 1 0.56 5.11 2 100 0.2 0.5 5 0.66 3.65 3 100 0.2 0.5 10 1.05 -0.44 4 100 0.2 2.5 1 0.56 5.11 5 100 0.2 2.5 5 0.68 3.30 6 100 0.2 2.5 10 1.11 -0.87 7 100 0.2 4.5 1 0.47 6.52 8 100 0.2 4.5 5 0.63 3.96 9 100 0.2 4.5 10 1.02 -0.20 10 100 0.7 0.5 1 0.46 6.68 11 100 0.7 0.5 5 0.55 5.16 12 100 0.7 0.5 10 0.75 2.48 13 100 0.7 2.5 1 0.37 8.71 14 100 0.7 2.5 5 0.56 5.06

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79

Kerf

wid

th o

bser

vati

ons

Runs

Replication 1

Replication 2

Replication 3

195






Mean S/N Ratio

15 100 0.7 2.5 10 0.87 1.19 16 100 0.7 4.5 1 0.42 7.50 17 100 0.7 4.5 5 0.48 6.40 18 100 0.7 4.5 10 0.89 1.00 19 100 1.2 0.5 1 0.32 9.99 20 100 1.2 0.5 5 0.47 6.57 21 100 1.2 0.5 10 0.00 0.00 22 100 1.2 2.5 1 0.32 10.02 23 100 1.2 2.5 5 0.47 6.62 24 100 1.2 2.5 10 0.00 0.00 25 100 1.2 4.5 1 0.34 9.33 26 100 1.2 4.5 5 0.43 7.33 27 100 1.2 4.5 10 0.00 0.00 28 300 0.2 0.5 1 0.82 1.78 29 300 0.2 0.5 5 1.03 -0.24 30 300 0.2 0.5 10 1.28 -2.12 31 300 0.2 2.5 1 0.69 3.20 32 300 0.2 2.5 5 0.97 0.28 33 300 0.2 2.5 10 1.46 -3.27 34 300 0.2 4.5 1 0.68 3.35 35 300 0.2 4.5 5 0.92 0.77 36 300 0.2 4.5 10 1.34 -2.56 37 300 0.7 0.5 1 0.61 4.24 38 300 0.7 0.5 5 0.80 1.90 39 300 0.7 0.5 10 1.12 -0.95 40 300 0.7 2.5 1 0.59 4.53 41 300 0.7 2.5 5 0.79 2.05 42 300 0.7 2.5 10 1.23 -1.82 43 300 0.7 4.5 1 0.59 4.56 44 300 0.7 4.5 5 0.76 2.38 45 300 0.7 4.5 10 1.15 -1.24 46 300 1.2 0.5 1 0.57 4.83 47 300 1.2 0.5 5 0.70 3.16 48 300 1.2 0.5 10 0.89 1.03 49 300 1.2 2.5 1 0.49 6.28 50 300 1.2 2.5 5 0.75 2.54 51 300 1.2 2.5 10 1.13 -1.06 52 300 1.2 4.5 1 0.47 6.56

196






Mean S/N Ratio

53 300 1.2 4.5 5 0.72 2.81 54 300 1.2 4.5 10 0.98 0.18 55 500 0.2 0.5 1 0.92 0.77 56 500 0.2 0.5 5 1.14 -1.14 57 500 0.2 0.5 10 1.89 -5.52 58 500 0.2 2.5 1 0.78 2.12 59 500 0.2 2.5 5 1.10 -0.83 60 500 0.2 2.5 10 1.22 -1.99 61 500 0.2 4.5 1 0.78 2.18 62 500 0.2 4.5 5 0.99 0.13 63 500 0.2 4.5 10 1.47 -3.32 64 500 0.7 0.5 1 0.69 3.26 65 500 0.7 0.5 5 0.87 1.21 66 500 0.7 0.5 10 1.23 -1.80 67 500 0.7 2.5 1 0.67 3.52 68 500 0.7 2.5 5 0.89 1.06 69 500 0.7 2.5 10 1.34 -2.51 70 500 0.7 4.5 1 0.67 3.46 71 500 0.7 4.5 5 0.81 1.88 72 500 0.7 4.5 10 1.21 -1.68 73 500 1.2 0.5 1 0.60 4.48 74 500 1.2 0.5 5 0.77 2.29 75 500 1.2 0.5 10 1.03 -0.24 76 500 1.2 2.5 1 0.60 4.48 77 500 1.2 2.5 5 0.75 2.54 78 500 1.2 2.5 10 1.23 -1.81 79 500 1.2 4.5 1 0.50 6.02 80 500 1.2 4.5 5 0.83 1.58 81 500 1.2 4.5 10 1.10 -0.83

197

Table A-7: Edge quality observations of Perspex Sheet (5mm)


A

Cutting speed

B

Assist gas Pressure

C

Standoff Distance

D


1 100 0.2 0.5 1 0.2 0.19 0.19 0.19 14.27

2 100 0.2 0.5 5 0.08 0.08 0.09 0.08 21.56

3 100 0.2 0.5 10 0.13 0.1 0.1 0.11 19.07

4 100 0.2 2.5 1 0.11 0.11 0.11 0.11 19.17

5 100 0.2 2.5 5 0.09 0.09 0.09 0.09 20.92

6 100 0.2 2.5 10 0.11 0.11 0.11 0.11 19.17

7 100 0.2 4.5 1 0.14 0.14 0.14 0.14 17.08

8 100 0.2 4.5 5 0.07 0.07 0.07 0.07 23.10

9 100 0.2 4.5 10 0.09 0.09 0.09 0.09 20.92

10 100 0.7 0.5 1 0.2 0.2 0.18 0.19 14.26



13 100 0.7 2.5 1 0.23 0.21 0.23 0.22 13.01

14 100 0.7 2.5 5 0.06 0.06 0.06 0.06 24.44


16 100 0.7 4.5 1 0.21 0.19 0.19 0.20

17 100 0.7 4.5 5 0.09 0.09 0.09 0.09











28 300 0.2 0.5 1 0.14 0.16 0.14 0.15 16.65

29 300 0.2 0.5 5 0.11 0.1 0.1 0.10 19.70

30 300 0.2 0.5 10 0.14 0.14 0.16 0.15 16.65

31 300 0.2 2.5 1 0.09 0.09 0.09 0.09 20.92

198


A

Cutting speed

B

Assist gas Pressure

C

Standoff Distance

D


32 300 0.2 2.5 5 0.05 0.05 0.05 0.05 26.02

33 300 0.2 2.5 10 0.06 0.06 0.06 0.06 24.44

34 300 0.2 4.5 1 0.07 0.07 0.07 0.07 23.10

35 300 0.2 4.5 5 0.07 0.07 0.07 0.07 23.10

36 300 0.2 4.5 10 0.06 0.06 0.06 0.06 24.44

37 300 0.7 0.5 1 0.14 0.14 0.14 0.14 17.08

38 300 0.7 0.5 5 0.09 0.09 0.09 0.09 20.92

39 300 0.7 0.5 10 0.11 0.11 0.11 0.11 19.17

40 300 0.7 2.5 1 0.19 0.19 0.19 0.19 14.42

41 300 0.7 2.5 5 0.07 0.07 0.07 0.07 23.10

42 300 0.7 2.5 10 0.1 0.1 0.1 0.10 20.00

43 300 0.7 4.5 1 0.07 0.07 0.07 0.07 23.10

44 300 0.7 4.5 5 0.05 0.05 0.05 0.05 26.02

45 300 0.7 4.5 10 0.08 0.08 0.08 0.08 21.94

46 300 1.2 0.5 1 0.24 0.24 0.24 0.24 12.40



49 300 1.2 2.5 1 0.27 0.27 0.27 0.27 11.37

50 300 1.2 2.5 5 0.09 0.09 0.09 0.09 20.92


52 300 1.2 4.5 1 0.15 0.15 0.15 0.15 16.48



55 500 0.2 0.5 1 0.12 0.13 0.12 0.12 18.17

56 500 0.2 0.5 5 0.1 0.1 0.1 0.10 20.00

57 500 0.2 0.5 10 0.15 0.13 0.14 0.14 17.06

58 500 0.2 2.5 1 0.12 0.12 0.12 0.12 18.42

59 500 0.2 2.5 5 0.08 0.08 0.08 0.08 21.94

60 500 0.2 2.5 10 0.07 0.07 0.07 0.07 23.10

61 500 0.2 4.5 1 0.11 0.11 0.11 0.11 19.17

62 500 0.2 4.5 5 0.08 0.08 0.08 0.08 21.94

63 500 0.2 4.5 10 0.1 0.1 0.1 0.10 20.00

199


A

Cutting speed

B

Assist gas Pressure

C

Standoff Distance

D


64 500 0.7 0.5 1 0.17 0.16 0.16 0.16 15.73

65 500 0.7 0.5 5 0.08 0.08 0.07 0.08 22.28

66 500 0.7 0.5 10 0.11 0.11 0.11 0.11 19.17

67 500 0.7 2.5 1 0.18 0.18 0.18 0.18 14.89

68 500 0.7 2.5 5 0.07 0.07 0.07 0.07 23.10

69 500 0.7 2.5 10 0.11 0.11 0.11 0.11 19.17

70 500 0.7 4.5 1 0.1 0.09 0.09 0.09 20.58

71 500 0.7 4.5 5 0.08 0.08 0.08 0.08 21.94

72 500 0.7 4.5 10 0.12 0.12 0.12 0.12 18.42

73 500 1.2 0.5 1 0.2 0.21 0.19 0.20 13.97



76 500 1.2 2.5 1 0.17 0.17 0.17 0.17 15.39



79 500 1.2 4.5 1 0.13 0.12 0.12 0.12 18.17



200

Figure A-3: Outlier analysis of Edge quality observations of Perspex glass sheet (5mm)

Table A-8: Observation for Kerf width of Perspex Sheet of 5mm



Cutting Speed

(m/min)

Assist Gas

pressure (bar)

Stand off

distance (mm)


Replications

Outer scrap length (mm) Replications

Lin1 Lin2 Lin3 Lout1 Lout2 Lout3 1 100 0.2 0.5 1 20.73 20.75 20.72 19.63 19.62 19.62 2 100 0.2 0.5 5 20.82 20.81 20.84 19.41 19.39 19.38 3 100 0.2 0.5 10 21.1 21.11 21.12 19.15 19.17 19.18 4 100 0.2 2.5 1 20.68 20.72 20.71 19.7 19.71 19.73 5 100 0.2 2.5 5 20.8 20.83 20.82 19.49 19.47 19.46 6 100 0.2 2.5 10 21.21 21.23 21.21 19.12 19.1 19.14 7 100 0.2 4.5 1 20.71 20.73 20.7 19.75 19.73 19.73 8 100 0.2 4.5 5 20.89 20.9 20.87 19.57 19.56 19.59 9 100 0.2 4.5 10 21.21 21.19 21.2 19.13 19.1 19.11 10 100 0.7 0.5 1 20.46 20.48 20.45 19.89 19.86 19.88 11 100 0.7 0.5 5 Unsuccessful cutting 12 100 0.7 0.5 10 Unsuccessful cutting 13 100 0.7 2.5 1 20.57 20.55 20.59 19.9 19.89 19.87

0

0.05

0.1

0.15

0.2

0.25

0.3

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79

Edge

qua

lity

obse

rvat

ions

Runs

Replication 1

Replication 2

Replication 3

201



Cutting Speed

(m/min)

Assist Gas

pressure (bar)

Stand off

distance (mm)


Replications


Lin1 Lin2 Lin3 Lout1 Lout2 Lout3 14 100 0.7 2.5 5 20.65 20.67 20.68 19.76 19.76 19.76 15 100 0.7 2.5 10 Unsuccessful cutting 16 100 0.7 4.5 1 20.55 20.59 20.57 19.8 19.82 19.81 17 100 0.7 4.5 5 20.7 20.68 20.71 19.83 19.86 19.84 18 100 0.7 4.5 10 Unsuccessful cutting 19 100 1.2 0.5 1 Unsuccessful cutting 20 100 1.2 0.5 5 Unsuccessful cutting 21 100 1.2 0.5 10 Unsuccessful cutting 22 100 1.2 2.5 1 Unsuccessful cutting 23 100 1.2 2.5 5 Unsuccessful cutting 24 100 1.2 2.5 10 Unsuccessful cutting 25 100 1.2 4.5 1 Unsuccessful cutting 26 100 1.2 4.5 5 Unsuccessful cutting 27 100 1.2 4.5 10 Unsuccessful cutting 28 300 0.2 0.5 1 20.93 20.95 20.96 19.36 19.34 19.33 29 300 0.2 0.5 5 21 21.03 21 19.14 19.11 19.15 30 300 0.2 0.5 10 21.46 21.45 21.48 18.71 18.69 18.67 31 300 0.2 2.5 1 21 20.98 21.03 19.57 19.56 19.54 32 300 0.2 2.5 5 21.11 21.13 21.11 19.25 19.23 19.21 33 300 0.2 2.5 10 21.56 21.55 21.54 18.69 18.69 18.71 34 300 0.2 4.5 1 20.97 20.99 20.96 19.42 19.4 19.43 35 300 0.2 4.5 5 21.26 21.23 21.24 19.15 19.11 19.13 36 300 0.2 4.5 10 21.56 21.57 21.59 18.6 18.65 18.63 37 300 0.7 0.5 1 20.77 20.78 20.79 19.57 19.59 19.6 38 300 0.7 0.5 5 20.86 20.88 20.88 19.43 19.46 19.47 39 300 0.7 0.5 10 21.14 21.11 21.1 19.21 19.23 19.2 40 300 0.7 2.5 1 20.76 20.77 20.79 19.74 19.71 19.72 41 300 0.7 2.5 5 20.93 20.91 20.9 19.57 19.53 19.55 42 300 0.7 2.5 10 21.38 21.36 21.35 19.17 19.15 19.16 43 300 0.7 4.5 1 20.74 20.77 20.76 19.56 19.54 19.54 44 300 0.7 4.5 5 20.95 20.94 20.92 19.4 19.42 19.43 45 300 0.7 4.5 10 21.3 21.35 21.38 19.09 19.07 19.06 46 300 1.2 0.5 1 20.6 20.61 20.63 19.78 19.79 19.81 47 300 1.2 0.5 5 Unsuccessful cutting 48 300 1.2 0.5 10 Unsuccessful cutting 49 300 1.2 2.5 1 20.69 20.7 20.7 19.85 19.86 19.88

202



Cutting Speed

(m/min)

Assist Gas

pressure (bar)

Stand off

distance (mm)


Replications


Lin1 Lin2 Lin3 Lout1 Lout2 Lout3 50 300 1.2 2.5 5 20.85 20.87 20.87 19.64 19.67 19.66 51 300 1.2 2.5 10 Unsuccessful cutting 52 300 1.2 4.5 1 20.72 20.71 20.74 19.9 19.87 19.88 53 300 1.2 4.5 5 Unsuccessful cutting 54 300 1.2 4.5 10 Unsuccessful cutting 55 500 0.2 0.5 1 21.13 21.11 21.1 19.22 19.24 19.25 56 500 0.2 0.5 5 21.28 21.29 21.31 18.97 18.99 18.99 57 500 0.2 0.5 10 21.58 21.56 21.59 18.6 18.62 18.64 58 500 0.2 2.5 1 21.12 21.14 21.15 19.37 19.39 19.4 59 500 0.2 2.5 5 20.84 20.83 20.86 19.63 19.6 19.61 60 500 0.2 2.5 10 21.53 21.51 21.5 18.67 18.69 18.68 61 500 0.2 4.5 1 21.05 21.02 21.03 19.32 19.3 19.3 62 500 0.2 4.5 5 21.26 21.28 21.29 19.09 19.08 19.06 63 500 0.2 4.5 10 21.64 21.65 21.67 18.59 18.6 18.6 64 500 0.7 0.5 1 20.87 20.85 20.89 19.45 19.43 19.48 65 500 0.7 0.5 5 21.02 21.04 21.01 19.36 19.38 19.39 66 500 0.7 0.5 10 21.25 21.24 21.22 19.04 19.06 19.08 67 500 0.7 2.5 1 20.85 20.83 20.86 19.72 19.74 19.76 68 500 0.7 2.5 5 20.9 20.92 20.93 19.48 19.44 19.46 69 500 0.7 2.5 10 21.33 21.31 21.34 18.94 18.95 18.96 70 500 0.7 4.5 1 20.8 20.82 20.84 19.5 19.52 19.53 71 500 0.7 4.5 5 21.09 21.11 21.09 19.27 19.24 19.25 72 500 0.7 4.5 10 21.43 21.41 21.4 19.04 19.02 19.01 73 500 1.2 0.5 1 20.76 20.78 20.79 19.75 19.74 19.74 74 500 1.2 0.5 5 Unsuccessful cutting 75 500 1.2 0.5 10 Unsuccessful cutting 76 500 1.2 2.5 1 20.72 20.74 20.75 19.73 19.76 19.75 77 500 1.2 2.5 5 Unsuccessful cutting 78 500 1.2 2.5 10 Unsuccessful cutting 79 500 1.2 4.5 1 20.78 20.79 20.76 19.67 19.68 19.7 80 500 1.2 4.5 5 Unsuccessful cutting 81 500 1.2 4.5 10 Unsuccessful cutting

203

Figure A-4: Outlier analysis of Kerf width observations of Perspex glass sheet (5mm)

Table A-9: Kerf width mean and signal to noise ratio of Perspex Sheet of 5mm



Cutting Speed

(m/min)

Assist Gas pressure

(bar)

Standoff distance

(mm)

Mean S/N Ratio

1 100 0.2 0.5 1 0.56 5.11 2 100 0.2 0.5 5 0.72 2.91 3 100 0.2 0.5 10 0.97 0.25 4 100 0.2 2.5 1 0.50 6.11 5 100 0.2 2.5 5 0.67 3.45 6 100 0.2 2.5 10 1.05 -0.41 7 100 0.2 4.5 1 0.49 6.22 8 100 0.2 4.5 5 0.66 3.65 9 100 0.2 4.5 10 1.04 -0.37 10 100 0.7 0.5 1 0.29 10.64 11 100 0.7 0.5 5 Unsuccessful cutting 12 100 0.7 0.5 10 Unsuccessful cutting 13 100 0.7 2.5 1 0.34 9.32 14 100 0.7 2.5 5 0.45 6.87

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1 4 7 101316192225283134374043464952555861646770737679

Ker

wid

th o

bser

vati

ons

Runs

Replication 1

Replication 2

Replication 3

204



Cutting Speed

(m/min)

Assist Gas pressure

(bar)

Standoff distance

(mm)

Mean S/N Ratio

15 100 0.7 2.5 10 Unsuccessful cutting 16 100 0.7 4.5 1 0.38 8.40 17 100 0.7 4.5 5 0.43 7.39 18 100 0.7 4.5 10 Unsuccessful cutting 19 100 1.2 0.5 1 Unsuccessful cutting 20 100 1.2 0.5 5 Unsuccessful cutting 21 100 1.2 0.5 10 Unsuccessful cutting 22 100 1.2 2.5 1 Unsuccessful cutting 23 100 1.2 2.5 5 Unsuccessful cutting 24 100 1.2 2.5 10 Unsuccessful cutting 25 100 1.2 4.5 1 Unsuccessful cutting 26 100 1.2 4.5 5 Unsuccessful cutting 27 100 1.2 4.5 10 Unsuccessful cutting 28 300 0.2 0.5 1 0.80 1.92 29 300 0.2 0.5 5 0.94 0.55 30 300 0.2 0.5 10 1.39 -2.84 31 300 0.2 2.5 1 0.72 2.81 32 300 0.2 2.5 5 0.94 0.51 33 300 0.2 2.5 10 1.43 -3.09 34 300 0.2 4.5 1 0.78 2.18 35 300 0.2 4.5 5 1.06 -0.48 36 300 0.2 4.5 10 1.47 -3.37 37 300 0.7 0.5 1 0.60 4.49 38 300 0.7 0.5 5 0.71 2.97 39 300 0.7 0.5 10 0.95 0.43 40 300 0.7 2.5 1 0.53 5.59 41 300 0.7 2.5 5 0.68 3.33 42 300 0.7 2.5 10 1.10 -0.84 43 300 0.7 4.5 1 0.61 4.36 44 300 0.7 4.5 5 0.76 2.38 45 300 0.7 4.5 10 1.14 -1.10 46 300 1.2 0.5 1 0.41 7.74 47 300 1.2 0.5 5 Unsuccessful cutting 48 300 1.2 0.5 10 Unsuccessful cutting 49 300 1.2 2.5 1 0.42 7.60 50 300 1.2 2.5 5 0.60 4.39 51 300 1.2 2.5 10 Unsuccessful cutting

205



Cutting Speed

(m/min)

Assist Gas pressure

(bar)

Standoff distance

(mm)

Mean S/N Ratio

52 300 1.2 4.5 1 0.42 7.53 53 300 1.2 4.5 5 Unsuccessful cutting 54 300 1.2 4.5 10 Unsuccessful cutting 55 500 0.2 0.5 1 0.94 0.55 56 500 0.2 0.5 5 1.16 -1.25 57 500 0.2 0.5 10 1.48 -3.40 58 500 0.2 2.5 1 0.88 1.16 59 500 0.2 2.5 5 0.62 4.22 60 500 0.2 2.5 10 1.42 -3.03 61 500 0.2 4.5 1 0.86 1.28 62 500 0.2 4.5 5 1.10 -0.83 63 500 0.2 4.5 10 1.53 -3.68 64 500 0.7 0.5 1 0.71 3.00 65 500 0.7 0.5 5 0.82 1.69 66 500 0.7 0.5 10 1.09 -0.74 67 500 0.7 2.5 1 0.55 5.14 68 500 0.7 2.5 5 0.73 2.75 69 500 0.7 2.5 10 1.19 -1.50 70 500 0.7 4.5 1 0.65 3.72 71 500 0.7 4.5 5 0.92 0.71 72 500 0.7 4.5 10 1.20 -1.55 73 500 1.2 0.5 1 0.52 5.73 74 500 1.2 0.5 5 Unsuccessful cutting 75 500 1.2 0.5 10 Unsuccessful cutting 76 500 1.2 2.5 1 0.49 6.11 77 500 1.2 2.5 5 0.00 0.00 78 500 1.2 2.5 10 0.00 0.00 79 500 1.2 4.5 1 0.55 5.24 80 500 1.2 4.5 5 Unsuccessful cutting 81 500 1.2 4.5 10 Unsuccessful cutting

206

A.1 ONE WAY ANOVA WITHOUT REPLICATION Kerf width observations of Polystyrene foam Sheet (13mm) used for analysis Table A-3. Table A-10: Observations consider Laser Power (A)

100 300 500 1.573 1.913 1.968 1.488 1.842 2.068 1.738 1.742 1.920

Table A-11: Summary of descriptive Statistics

Groups Count Sum Average Variance 100 3 4.800 1.600 0.016 300 3 5.497 1.832 0.007 500 3 5.957 1.986 0.006

Table A-12: ANOVA for Laser Power Source of Variation SS df MS F P-value F crit Between Groups 0.226 2 0.113 11.568 0.009 5.143 Within Groups 0.059 6 0.010 Total 0.285 8

Table A-13: Observations consider Cutting speed

0.2 0.7 1.2 1.573 1.488 1.738 1.913 1.842 1.742 1.968 2.068 1.920

Table A-14: Summary of descriptive Statistics

Groups Count Sum Average Variance 0.2 3 5.455 1.818 0.046 0.7 3 5.398 1.799 0.085 1.2 3 5.400 1.800 0.011

Table A-15: ANOVA for Cutting Speed Source of Variation SS df MS F P-value F critical Between Groups 0.001 2 0.000 0.007 0.993 5.143 Within Groups 0.284 6 0.047 Total 0.285 8

207

Table A-16: Observations consider Assist gas pressure 0.5 2.5 4.5

1.573 1.488 1.738 1.742 1.913 1.842 2.068 1.920 1.968

Table A-17: Summary of Descriptive Statistics


Table A-18: ANOVA for Assist Gas Pressure Source of Variation SS df MS F P-value F critical Between Groups 0.009 2 0.005 0.0997 0.907 5.143 Within Groups 0.276 6 0.046 Total 0.285 8

Table A-19: Observations consider Standoff Distance

1 5 10 1.573 1.488 1.738 1.842 1.742 1.913 1.920 1.968 2.068



Table A-21: ANOVA for Standoff Distance Source of Variation SS df MS F P-value F critical Between Groups 0.049 2 0.024 0.620 0.569 5.143 Within Groups 0.236 6 0.039 Total 0.285 8

A.2 ONE WAY ANOVA WITH REPLICATION Table A-22: Observations consider Laser Power with replication

S. No. 100 300 500 S. No. 100 300 500 1 1.580 1.940 2.010 6 1.480 1.985 2.290 2 1.615 1.885 2.040 7 1.660 1.655 1.790 3 1.525 1.915 1.855 8 1.695 1.715 1.890 4 1.730 1.765 1.975 9 1.860 1.855 2.080 5 1.255 1.775 1.940

208

Table A-23: Summary of Descriptive Statistics Groups Count Sum Average Variance

100 9 14.400 1.600 0.030 300 9 16.490 1.832 0.012 500 9 17.870 1.986 0.021

Table A-24: One way ANOVA with replication for Laser Power Source of Variation SS Df MS F P-value F critical Between Groups 0.678 2 0.339 16.066 0.000 3.403 Within Groups 0.507 24 0.021 Total 1.185 26

Table A-25: Observations consider Cutting Speed with replication

S. No. 0.2 0.7 1.2 S. No. 0.2 0.7 1.2 1 1.580 1.730 1.660 6 1.915 1.985 1.855 2 1.615 1.255 1.695 7 2.010 1.975 1.790 3 1.525 1.480 1.860 8 2.040 1.940 1.890 4 1.940 1.765 1.655 9 1.855 2.290 2.080 5 1.885 1.775 1.715



Table A-27: One way ANOVA with replication for Cutting Speed Source of Variation SS Df MS F P-value F critical Between Groups 0.002 2 0.001 0.021 0.979 3.403 Within Groups 1.183 24 0.049

Table A-28: Observations consider Assist gas pressure with replication

S. No. 0.5 2.5 4.5 S. No. 0.5 2.5 4.5 1 1.580 1.730 1.660 6 1.855 1.915 1.985 2 1.615 1.255 1.695 7 1.975 1.790 2.010 3 1.525 1.480 1.860 8 1.940 1.890 2.040 4 1.655 1.940 1.765 9 2.290 2.080 1.855 5 1.715 1.885 1.775

209

Table A-29: Summary of Descriptive Statistics Groups Count Sum Average Variance

0.5 9 16.150 1.794 0.060 2.5 9 15.965 1.774 0.066 4.5 9 16.645 1.849 0.019

Table A-30: One way ANOVA with replication for Assist Gas Pressure Source of Variation SS Df MS F P-value F critical Between Groups 0.027 2 0.014 0.285 0.755 3.403 Within Groups 1.157 24 0.048 Total 1.185 26

Table A-31: Observations consider Standoff Distance with replication

S. No. 1 5 10 S. No. 0.5 2.5 4.5 1 1.580 1.730 1.660 6 1.985 1.855 1.915 2 1.615 1.255 1.695 7 1.790 2.010 1.975 3 1.525 1.480 1.860 8 1.890 2.040 1.940 4 1.765 1.655 1.940 9 2.080 1.855 2.290 5 1.775 1.715 1.885



Table A-33: One way ANOVA with replication for Standoff Distance Source of Variation SS df MS F P-value F crit Between Groups 0.146 2 0.073 1.691 0.206 3.403 Within Groups 1.039 24 0.043 Total 1.185 26

Table A-34: One way ANOVA with replication Treatments F P-value F critical Laser Power 16.066 0.000 3.403 Cutting Speed 0.021 0.979 3.403 Assist gas pressure 0.285 0.755 3.403 Standoff distance 1.691 0.206 3.403

210

A.3 TWO WAY ANOVA WITH REPLICATION Table A-35: Interaction between Laser Power and Cutting Speed with replication

A/B 100 300 500 0.2 1.580 1.940 2.010

1.615 1.885 2.040 1.525 1.915 1.855

0.7 1.730 1.765 1.975 1.255 1.775 1.940 1.480 1.985 2.290

1.2 1.660 1.655 1.79 1.695 1.715 1.89 1.860 1.855 2.08

Table A-36: Interaction between Laser Power and Cutting speed 0.2 100/0.2 300/0.2 500/0.2 Total Count 3 3 3 9 Sum 4.720 5.740 5.905 16.365 Average 1.573 1.913 1.968 1.818 Variance 0.002 0.001 0.010 0.037



Table A-39: Total Interaction between Laser Power and Cutting speed Count 9 9 9 Sum 14.400 16.490 17.870 Average 1.600 1.832 1.986 Variance 0.030 0.012 0.021

211

Table A-40: ANOVA of Interaction between Laser Power and Cutting speed Source of Variation SS df MS F P-value F crit

Sample 0.002 2 0.001 0.057 0.945 3.555 Columns 0.678 2 0.339 18.457 4.37x10-5 3.555 Interaction 0.174 4 0.043 2.365 0.092 2.928 Within 0.331 18 0.018 Total 1.185 26

Table A-41: Interaction between Laser Power and Assist Gas Pressure with replication

A/C 100 300 500 0.5 1.580 1.655 1.975

1.615 1.715 1.940 1.525 1.855 2.290

2.5 1.730 1.940 1.790 1.255 1.885 1.890 1.480 1.915 2.080

4.5 1.660 1.765 2.010 1.695 1.775 2.040 1.860 1.985 1.855

Table A-42: Interaction between Laser Power and Assist Gas Pressure 0.5

100/0.5 300/0.5 500/0.5 Total Count 3 3 3 9 Sum 4.720 5.225 6.205 16.150

Average 1.573 1.742 2.068 1.794 Variance 0.002 0.011 0.037 0.060


100/2.5 300/2.5 500/2.5 Total Count 3 3 3 9 Sum 4.465 5.740 5.760 15.965



100/4.5 300/4.5 500/4.5 Total Count 3 3 3 9 Sum 5.215 5.525 5.905 16.645


212

Table A-45: Total Interaction between Laser Power and Assist Gas Pressure Count 9 9 9 Sum 14.400 16.490 17.870 Average 1.600 1.832 1.986 Variance 0.030 0.012 0.021

Table A-46: ANOVA of Interaction between Laser Power and Assist Gas Pressure Source of Variation SS df MS F P-value F crit Sample 0.027 2 0.014 0.747 0.488 3.555 Columns 0.678 2 0.339 18.457 4.37 x10-5 3.555 Interaction 0.148 4 0.037 2.020 0.135 2.928 Within 0.331 18 0.018 Total 1.185 26

Table A-47: Interaction between Laser Power and Standoff Distance with replication

A/D 100 300 500 1 1.580 1.765 1.790 1.615 1.775 1.890 1.525 1.985 2.080 5 1.730 1.655 2.010 1.255 1.715 2.040 1.480 1.855 1.855

10 1.660 1.940 1.975 1.695 1.885 1.940 1.860 1.915 2.290

Table A-48: Interaction between Laser Power and Standoff Distance 1 100/1 300/1 500/1 Total Count 3 3 3 9 Sum 4.720 5.525 5.760 16.005 Average 1.573 1.842 1.920 1.778 Variance 0.002 0.015 0.022 0.035


213


Table A-51: Total Interaction between Laser Power and Standoff Distance Count 9 9 9 Sum 14.400 16.490 17.870 Average 1.600 1.832 1.986 Variance 0.030 0.012 0.021

Table A-52: ANOVA of Interaction between Laser Power and Standoff Distance Source of Variation SS df MS F P-value F crit Sample 0.146 2 0.073 3.982 0.037 3.555 Columns 0.678 2 0.339 18.457 0.000 3.555 Interaction 0.030 4 0.007 0.402 0.805 2.928 Within 0.331 18 0.018 Total 1.185 26

Table A-53: Interaction between Cutting speed and Assist Gas Pressure with replication C/B 0.2 0.7 1.2 0.5 1.580 1.975 1.655 1.615 1.940 1.715 1.525 2.290 1.855 2.5 1.940 1.730 1.790 1.885 1.255 1.890 1.915 1.480 2.080 4.5 2.010 1.765 1.660 2.040 1.775 1.695 1.855 1.985 1.860

Table A-54: Interaction between Cutting Speed and Assist Gas Pressure 0.5 0.2/0.5 0.7/0.5 1.2/0.5 Total Count 3 3 3 9 Sum 4.720 6.205 5.225 16.150 Average 1.573 2.068 1.742 1.794 Variance 0.002 0.037 0.011 0.060

214



Table A-57: Total Interaction between Cutting Speed and Assist Gas Pressure Count 9 9 9 Sum 16.365 16.195 16.200 Average 1.818 1.799 1.800 Variance 0.037 0.091 0.019

Table A-58: ANOVA of Interaction between Cutting Speed and Assist Gas Pressure

Source of Variation SS df MS F P-value F crit Sample 0.027 2 0.014 0.747 0.488 3.555 Columns 0.002 2 0.001 0.057 0.945 3.555 Interaction 0.825 4 0.206 11.220 9.58 x10-5 2.928 Within 0.331 18 0.018 Total 1.185 26

Table A-59: Interaction between Cutting speed and Standoff Distance with replication B/D 0.2 0.7 1.2 1 1.580 1.765 1.790 1.615 1.775 1.890 1.525 1.985 2.080 5 2.010 1.730 1.655 2.040 1.255 1.715 1.855 1.480 1.855 10 1.940 1.975 1.660 1.885 1.940 1.695 1.915 2.290 1.860

215

Table A-60: Interaction between Cutting Speed and Standoff Distance 1 0.2/1 0.7/1 1.2/1 Total Count 3 3 3 9 Sum 4.720 5.525 5.760 16.005 Average 1.573 1.842 1.920 1.778 Variance

0.002 0.015 0.022 0.035

Table A-61: Interaction between Cutting Speed and Standoff Distance 5 0.2/5 0.7/5 1.2/5 Total Count 3 3 3 3 Sum 5.905 4.465 5.225 15.595 Average 1.968 1.488 1.742 1.733 Variance 0.010 0.056 0.011 0.062

Table A-62: Interaction between Laser Cutting Speed and Standoff Distance 10 0.2/10 0.7/10 1.2/10 Total Count 3 3 3 9 Sum 5.740 6.205 5.215 17.160 Average 1.913 2.068 1.738 1.907 Variance 0.001 0.037 0.011 0.033

Table A-63: Total Interaction between Cutting Speed and Standoff Distance Count 9 9 9 Sum 16.365 16.195 16.200 Average 1.818 1.799 1.800 Variance 0.038 0.091 0.019

Table A-64: ANOVA of Interaction between Cutting Speed and Standoff Distance Source of Variation SS df MS F P-value F crit Sample 0.146 2 0.073 3.982 0.0370 3.555 Columns 0.002 2 0.001 0.057 0.9451 3.555 Interaction 0.706 4 0.176 9.602 0.0002 2.928 Within 0.331 18 0.018 Total 1.185 26

216

Table A-65: Interaction between Assist Gas Pressure and Standoff Distance with replication C/D 0.5 2.5 4.5

1 1.580 1.790 1.765 1.615 1.890 1.775 1.525 2.080 1.985 5 1.655 1.730 2.010 1.715 1.255 2.040 1.855 1.480 1.855

10 1.975 1.940 1.660 1.940 1.885 1.695 2.290 1.915 1.860

Table A-66: Interaction between Assist Gas Pressure and Standoff Distance 1 0.5/1 2.5/1 4.5/1 Total Count 3 3 3 9 Sum 4.720 5.760 5.525 16.005 Average 1.573 1.920 1.842 1.778 Variance 0.002 0.022 0.015 0.035

Table A-67: Interaction between Assist Gas Pressure and Assist Gas Pressure 5 0.5/5 2.5/5 4.5/5 Total Count 3 3 3 9 Sum 5.225 4.465 5.905 15.595 Average 1.742 1.488 1.968 1.733 Variance 0.011 0.056 0.010 0.062

Table A-68: Interaction between Assist Gas Pressure and Standoff Distance 10 0.5/10 2.5/10 1.2/4.5 Total Count 3 3 3 9 Sum 6.205 5.740 5.215 17.160 Average 2.068 1.913 1.738 1.907 Variance 0.037 0.001 0.011 0.033

Table A-69: Total Interaction between Assist Gas Pressure and Standoff Distance Count 9 9 9 Sum 16.150 15.965 16.645 Average 1.794 1.774 1.849 Variance 0.060 0.066 0.019

217

Table A-70: ANOVA of Interaction between Assist Gas Pressure and Standoff Distance Source of Variation SS df MS F P-value F crit

Sample 0.146 2 0.073 3.982 0.0370 3.555 Columns 0.027 2 0.014 0.747 0.4877 3.555 Interaction 0.680 4 0.170 9.257 0.0003 2.928 Within 0.331 18 0.018 Total 1.185 26

A.4 LINEAR REGRESSION ANALYSIS

A.4.1 LASER POWER AND KERF WIDTH

Figure A-5: Interactive graph of Laser Power and Kerf Width

Table A-71: Descriptive Statistics

Parameters Mean Std. Deviation N Kerf Width Mean(dependent) 1.806 0.189 9 Laser Power (Independent) 300.000 173.205 9

Table A-72: Correlation

Kerf Width Laser Power Pearson Correlation Kerf Width Mean 1.000 0.885 Laser Power 0.885 1.000 Sig. (1-tailed) Kerf Width Mean

Laser Power . .001

.001 . N Kerf Width Mean 9 9 Laser Power 9 9

218

Table A-73: Regression Statistics R Square 0.783 Adjusted R Square 0.752 Standard Error 0.094 Observations 9

Table A-74: Regression between Laser Power and Kerf width ANOVA d.f. SS MS F Significance F Regression 1 0.223 0.223 25.280 0.0015 Residual 7 0.062 0.009 Total 8 0.285

Table A-75: Linear regression line of Laser Power Coefficients Standard Error t Stat P-value Lower

95% Upper 95%

Intercept 1.517 0.065 23.163 7.09 x10-8 1.362 1.672 A 0.001 0.000 5.028 1.52 x10-3 0.001 0.001

Table A-76: Residual output

Observation Predicted KERF WIDTH MEAN Residuals 1 1.613 -0.040 2 1.613 -0.125 3 1.613 0.125 4 1.806 0.107 5 1.806 0.036 6 1.806 -0.064 7 1.999 -0.030 8 1.999 0.070 9 1.999 -0.079

Figure A-6: Laser Power (A) Line fit Plot

0.000

0.500

1.000

1.500

2.000

2.500

100 100 100 300 300 300 500 500 500

Expe

rim

enta

l and

Pred

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d Ke

rf W

idth

Laser Power

Kerf Width


219

A.4.2 CUTTING SPEED AND KERF WIDTH

Figure A-7: Interactive graph of Cutting Speed and Kerf Width

Table A-77: Descriptive Statistics Mean Std. Deviation N Kerf Width Mean 1.80578 .188667 9 Cutting Speed .700 .4330 9


Table A-79:

Regression Statistics Multiple R 0.042 R Square 0.002 Adjusted R Square -0.141 Standard Error 0.202 Observations 9.000

Kerf Width Cutting Speed Pearson Correlation Kerf Width Mean 1.000 -0.041 Cutting Speed -0.041 1.000 Sig. (1-tailed)

Kerf Width Mean Cutting Speed

. .458 .458 .

N Kerf Width Mean 9 9 Cutting Speed 9 9

220

Table A-80: Regression between Cutting speed and Kerf width ANOVA Df SS MS F Significance F Regression 1 0.001 0.001 0.012 0.9144 Residual 7 0.284 0.041 Total 8 0.285

Table A-81: Linear regression line of cutting speed Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 1.819 0.133 13.642 0.0000 1.504 2.134 ( B ) -0.018 0.165 -0.111 0.0914 -0.407 0.371

Table A-82: Residual output Observation Predicted KERF

WIDTH MEAN Residual

s Observation Predicted KERF

WIDTH MEAN Residuals

1 1.815 -0.242 6 1.797 -0.055 2 1.806 -0.318 7 1.815 0.153 3 1.797 -0.058 8 1.806 0.262 4 1.815 0.098 9 1.797 0.123 5 1.806 0.036

Figure A-8: Cutting Speed (B) Line Fit Plot

0.000

0.500

1.000

1.500

2.000

2.500

0.2 0.7 1.2 0.2 0.7 1.2 0.2 0.7 1.2

Expe

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enta

l and

Pred

icte

d Ke

rf W

idth

Cutting Speed

Kerf Width


221

A.4.3 ASSIST GAS PRESSURE AND KERF-WIDTH

Figure A-9: Interactive graph of Assist Gas Pressure and Kerf Width

Table A-83: Descriptive Statistics Mean Std. Deviation N Kerf Width Mean 1.806 0.189 9 Assist Gas pressure 2.500 1.732 9


Table A-85: Regression Statistics Multiple R 0.126 R Square 0.016 Adjusted R Square -0.125 Standard Error 0.2 Observations 9

Kerf Width Assist Gas Pressure Pearson Correlation Kerf Width Mean 1.000 0.126 Assist Gas pressure 0.126 1.000 Sig. (1-tailed)

Kerf Width Mean Assist Gas pressure

. 0.373 0.373 .

N Kerf Width Mean 9 9 Assist Gas pressure 9 9

222

Table A-86: Regression between Assist Gas Pressure and Kerf Width ANOVA df SS MS F Significance F Regression 1 0.005 0.005 0.113 0.7462 Residual 7 0.280 0.040 Total 8 0.285

Table A-87: Linear regression line of Assist Gas Pressure Coefficients Standard Error t Stat P-value Lower

95% Upper 95%

Intercept 1.772 0.122 14.527 0.0000 1.483 2.060 ( C ) 0.014 0.041 0.337 0.746 -0.083 0.110

Table A-88: Residual output Observation Predicted KERF WIDTH MEAN Residuals

1 1.778 -0.205 2 1.806 -0.318 3 1.833 -0.095 4 1.806 0.107 5 1.833 0.008 6 1.778 -0.037 7 1.833 0.135 8 1.778 0.290 9 1.806 0.114

Figure A-10: Assist gas Pressure (C) Line Fit Plot

0

0.5

1

1.5

2

2.5

0.5 2.5 4.5 2.5 4.5 0.5 4.5 0.5 2.5

Expe

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enta

l and

Pred

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d Ke

rf W

idth

Assist Gas Pressure

TPM

Predicted TPM

223

A.4.4 STANDOFF DISTANCE AND KERF WIDTH

Table A-89: Descriptive Statistics Mean Std. Deviation N Kerf Width Mean 1.806 0.189 9 Standoff distance 5.33 3.905 9

Figure A-11: Interactive graph of Standoff Distance and Kerf Width


Table A-91: Regression Statistics Multiple R 0.313 R Square 0.098 Adjusted R Square -0.031 Standard Error 0.192 Observations 9.000

Kerf Width Standoff Distance Pearson Correlation Kerf Width Mean 1.000 0.312 Standoff Distance 0.312 1.000 Sig. (1-tailed)

Kerf Width Mean Standoff Distance

. 0.207 0.207 .

N Kerf Width Mean 9 9 Standoff Distance 9 9

224

Table A-92: Regression between Standoff Distance and Kerf Width ANOVA

df SS MS F Significance F Regression 1 0.028 0.028 0.758 0.4128 Residual 7 0.257 0.037 Total 8 0.285

Table A-93: Linear regression line of Standoff Distance Coefficients Standard

Error t Stat P-value Lower

95% Upper 95%

Intercept 1.725 0.112 15.350 0.000 1.460 1.991 ( D ) 0.015 0.017 0.871 0.413 -0.026 0.056

Table A-94: Residual output

Observation Predicted TPM Residuals 1 1.740 -0.167 2 1.801 -0.313 3 1.876 -0.138 4 1.876 0.037 5 1.740 0.101 6 1.801 -0.059 7 1.801 0.167 8 1.876 0.192 9 1.740 0.180

Figure A-12: Standoff Distance (D) Line Fit Plot

0

0.5

1

1.5

2

2.5

1 5 10 10 1 5 5 10 1

Expe

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l and

Pred

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d Ke

rf W

idth

Standoff Distance

Kerf Width


225

A.5 NONLINEAR REGRESSION ANALYSIS Table A-95: Regression data without replication for Laser Power

S. No. Laser Power A A2 Kerf Width Mean 1 100 10000 1.573 2 100 10000 1.488 3 100 10000 1.738 4 300 90000 1.913 5 300 90000 1.842 6 300 90000 1.742 7 500 250000 1.968 8 500 250000 2.068 9 500 250000 1.920

Figure A-13: Quadratic graph of Laser Power without replication

Table A-96: Regression Statistics Multiple R 0.891 R Square 0.794 Adjusted R Square 0.725 Standard Error 0.099 Observations 9 Table A-97: Non-linear Regression ANOVA for Laser Power and Kerf width without replication d.f. SS MS F Significance F Regression 2 0.2261 0.1130 11.5684 0.0087 Residual 6 0.0586 0.0098 Total 8 0.2847

y = -1E-06x2 + 0.001x + 1.454R² = 0.794

0.000

0.500

1.000

1.500

2.000

2.500

0 200 400 600

Kerf

Wid

th M

ean

Laser Power

Kerf Width

Poly. (Kerf Width)

226

Table A-98: Nonlinear regression of Laser Power Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 1.454 0.130 11.154 3.10 x10-5 1.135 1.773 A 0.002 0.001 1.457 0.195 -0.001 0.004 A2 -9.86 x10-7 1.75 x10-6 -0.564 0.593 -5.26 x10-6 3.29 x10-6

Table A-99: Residual output Observation Predicted Kerf width Mean Residuals

1 1.600 -0.027 2 1.600 -0.112 3 1.600 0.138 4 1.832 0.081 5 1.832 0.009 6 1.832 -0.091 7 1.986 -0.017 8 1.986 0.083 9 1.986 -0.066

Table A-100: Regression data with replication for Laser Power

S. No. Laser Power A A2 Kerf Width Replications 1 100 10000 1.58 2 100 10000 1.615 3 100 10000 1.525 4 100 10000 1.73 5 100 10000 1.255 6 100 10000 1.48 7 100 10000 1.66 8 100 10000 1.695 9 100 10000 1.86 10 300 90000 1.94 11 300 90000 1.885 12 300 90000 1.915 13 300 90000 1.765 14 300 90000 1.775 15 300 90000 1.985 16 300 90000 1.655 17 300 90000 1.715 18 300 90000 1.855 19 500 250000 2.01 20 500 250000 2.04 21 500 250000 1.855

227

S. No. Laser Power A A2 Kerf Width Replications 22 500 250000 1.975 23 500 250000 1.94 24 500 250000 2.29 25 500 250000 1.79 26 500 250000 1.89 27 500 250000 2.08

Figure A-14: Quadratic graph of Laser Power with replication

Table A-101: Regression Statistics Multiple R 0.757 R Square 0.572 Adjusted R Square 0.537 Standard Error 0.145 Observations 27 Table A-102: Non-linear Regression ANOVA for Laser Power and Kerf width with replication d.f. SS MS F Significance F Regression 2 0.678 0.339 16.066 3.73E-05 Residual 24 0.507 0.021 Total 26 1.185

Table A-103: Nonlinear regression of Laser Power Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 1.4543 0.1106 13.145 1.85 x10-12 1.2260 1.6826 A 0.0016 0.0009 1.7169 0.0989 -0.0003 0.0034 A2 -9.86 x10-7 1.48 x10-6 -0.665 0.5124 -4.05 x10-6 2.07 x10-6

y = -1E-06x2 + 0.001x + 1.454R² = 0.572

0

0.5

1

1.5

2

2.5

0 200 400 600

Kerf

Wid

th

Laser Power

Kerf Width

Poly. (Kerf Width )

228

Table A-104: Residual Output

Observation Predicted Kerf width Mean

Residuals

1 1.600 -0.020 2 1.600 0.015 3 1.600 -0.075 4 1.600 0.130 5 1.600 -0.345 6 1.600 -0.120 7 1.600 0.060 8 1.600 0.095 9 1.600 0.260 10 1.832 0.108 11 1.832 0.053 12 1.832 0.083 13 1.832 -0.067 14 1.832 -0.057 15 1.832 0.153 16 1.832 -0.177 17 1.832 -0.117 18 1.832 0.023 19 1.986 0.024 20 1.986 0.054 21 1.986 -0.131 22 1.986 -0.011 23 1.986 -0.046 24 1.986 0.304 25 1.986 -0.196 26 1.986 -0.096 27 1.986 0.094

Table A-105: Regression data with replication for Cutting Speed

Cutting Speed B B2 Kerf width Mean 1 0.2 0.04 1.58 2 0.2 0.04 1.615 3 0.2 0.04 1.525 4 0.2 0.04 1.94 5 0.2 0.04 1.885 6 0.2 0.04 1.915 7 0.2 0.04 2.01 8 0.2 0.04 2.04 9 0.2 0.04 1.855

229

Cutting Speed B B2 Kerf width Mean 10 0.7 0.49 1.73 11 0.7 0.49 1.255 12 0.7 0.49 1.48 13 0.7 0.49 1.765 14 0.7 0.49 1.775 15 0.7 0.49 1.985 16 0.7 0.49 1.975 17 0.7 0.49 1.94 18 0.7 0.49 2.29 19 1.2 1.44 1.66 20 1.2 1.44 1.695 21 1.2 1.44 1.86 22 1.2 1.44 1.655 23 1.2 1.44 1.715 24 1.2 1.44 1.855 25 1.2 1.44 1.79 26 1.2 1.44 1.89 27 1.2 1.44 2.08

Figure A-15: Quadratic graph of Cutting Speed with replication


y = 0.038x2 - 0.072x + 1.831R² = 0.001

0

0.5

1

1.5

2

2.5

-0.3 0.2 0.7 1.2

Kerf

WId

th

Cutting Speed

Ker Width

Poly. (Ker Width)

230

Table A-107: Non-linear Regression ANOVA for Cutting Speed and Kerf width with replication

d.f. SS MS F Significance F Regression 2 0.002 0.001 0.021 0.979 Residual 24 1.183 0.049 Total 26 1.185

Table A-108: Nonlinear regression of Cutting Speed Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 1.831 0.145 12.658 4.11 x10-12 1.533 2.130 B -0.073 0.518 -0.140 0.889 -1.142 0.997 B2 0.039 0.363 0.107 0.915 -0.709 0.787


Observation Predicted Kerf width Mean

Residuals

1 1.818 -0.238 2 1.818 -0.203 3 1.818 -0.293 4 1.818 0.122 5 1.818 0.067 6 1.818 0.097 7 1.818 0.192 8 1.818 0.222 9 1.818 0.037 10 1.799 -0.069 11 1.799 -0.544 12 1.799 -0.319 13 1.799 -0.034 14 1.799 -0.024 15 1.799 0.186 16 1.799 0.176 17 1.799 0.141 18 1.799 0.491 19 1.800 -0.140 20 1.800 -0.105 21 1.800 0.060 22 1.800 -0.145 23 1.800 -0.085 24 1.800 0.055 25 1.800 -0.010 26 1.800 0.090 27 1.800 0.280

231

Table A-110: Regression data with replication for Assist Gas Pressure

S. No. Assist Gas Pressure(C) C2 Kerf Width Mean 1 0.5 0.25 1.58 2 0.5 0.25 1.615 3 0.5 0.25 1.525 4 0.5 0.25 1.655 5 0.5 0.25 1.715 6 0.5 0.25 1.855 7 0.5 0.25 1.975 8 0.5 0.25 1.94 9 0.5 0.25 2.29 10 2.5 6.25 1.73 11 2.5 6.25 1.255 12 2.5 6.25 1.48 13 2.5 6.25 1.94 14 2.5 6.25 1.885 15 2.5 6.25 1.915 16 2.5 6.25 1.79 17 2.5 6.25 1.89 18 2.5 6.25 2.08 19 4.5 20.25 1.66 20 4.5 20.25 1.695 21 4.5 20.25 1.86 22 4.5 20.25 1.765 23 4.5 20.25 1.775 24 4.5 20.25 1.985 25 4.5 20.25 2.01 26 4.5 20.25 2.04 27 4.5 20.25 1.855

232

Figure A-16: Quadratic graph of Assist Gas Pressure with replication


Table A-112: Non-linear Regression ANOVA for Assist Gas Pressure and Kerf width with replication d.f. SS MS F Significance F Regression 2 0.027 0.014 0.285 0.755 Residual 24 1.157 0.048 Total 26 1.185

Table A-113: Nonlinear regression of Assist Gas Pressure Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 1.815 0.111 16.281 1.8 x10-14 1.585 2.045 C -0.046 0.115 -0.403 0.691 -0.284 0.191 C2 0.012 0.022 0.536 0.597 -0.034 0.058

y = 0.012x2 - 0.046x + 1.814R² = 0.023

0

0.5

1

1.5

2

2.5

-0.5 0.5 1.5 2.5 3.5 4.5

Kerf

WId

th

Asssit Gas Pressure

Ker Width

Poly. (Ker Width)

233

Table A-114: Residual Output Observation Predicted Kerf Width Mean Residuals

1 1.794 -0.214 2 1.794 -0.179 3 1.794 -0.269 4 1.794 -0.139 5 1.794 -0.079 6 1.794 0.061 7 1.794 0.181 8 1.794 0.146 9 1.794 0.496 10 1.774 -0.044 11 1.774 -0.519 12 1.774 -0.294 13 1.774 0.166 14 1.774 0.111 15 1.774 0.141 16 1.774 0.016 17 1.774 0.116 18 1.774 0.306 19 1.849 -0.189 20 1.849 -0.154 21 1.849 0.011 22 1.849 -0.084 23 1.849 -0.074 24 1.849 0.136 25 1.849 0.161 26 1.849 0.191 27 1.849 0.006

Table A-115: Regression data with replication for Standoff Distance

Standoff Distance (D) D2 Kerf Width Mean 1 1 1 1.58 2 1 1 1.615 3 1 1 1.525 4 1 1 1.765 5 1 1 1.775 6 1 1 1.985 7 1 1 1.79 8 1 1 1.89 9 1 1 2.08

234

Standoff Distance (D) D2 Kerf Width Mean 10 5 25 1.73 11 5 25 1.255 12 5 25 1.48 13 5 25 1.655 14 5 25 1.715 15 5 25 1.855 16 5 25 2.01 17 5 25 2.04 18 5 25 1.855 19 10 100 1.66 20 10 100 1.695 21 10 100 1.86 22 10 100 1.94 23 10 100 1.885 24 10 100 1.915 25 10 100 1.975 26 10 100 1.94 27 10 100 2.29

Figure A-17: Quadratic graph of Standoff Distance with replication

Table A-116: Regression Statistics Multiple R 0.351 R Square 0.124 Adjusted R Square 0.050 Standard Error 0.208 Observations 27

y = 0.005x2 - 0.042x + 1.815R² = 0.123

0

0.5

1

1.5

2

2.5

0 5 10

Kerf

WId

th

Standoff Distance

Ker Width

Poly. (Ker Width)

235

Table A-117: Non-linear Regression ANOVA for Standoff Distance and Kerf width with replication


Table A-118: Nonlinear regression of Standoff Distance Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 1.815 0.103 17.686 2.86 x10-15 1.604 2.027 D -0.042 0.049 -0.865 0.395 -0.143 0.058 D2 0.005 0.004 1.206 0.240 -0.004 0.014

Table A-119: Residual Output Observation Predicted Kerf Width Mean Residuals

1 1.778 -0.198 2 1.778 -0.163 3 1.778 -0.253 4 1.778 -0.013 5 1.778 -0.003 6 1.778 0.207 7 1.778 0.012 8 1.778 0.112 9 1.778 0.302 10 1.733 -0.003 11 1.733 -0.478 12 1.733 -0.253 13 1.733 -0.078 14 1.733 -0.018 15 1.733 0.122 16 1.733 0.277 17 1.733 0.307 18 1.733 0.122 19 1.907 -0.247 20 1.907 -0.212 21 1.907 -0.047 22 1.907 0.033 23 1.907 -0.022 24 1.907 0.008 25 1.907 0.068 26 1.907 0.033 27 1.907 0.383

236

A.6 MULTIPLE NON-LINEAR REGRESSION Table A-120: Multiple Non-linear Regression data with replication for four inputs

A A2 B B2 C C2 D D2 Kerf Width 1 100 10000 0.2 0.04 0.5 0.25 1 1 1.58 2 100 10000 0.2 0.04 0.5 0.25 1 1 1.615 3 100 10000 0.2 0.04 0.5 0.25 1 1 1.525 4 100 10000 0.7 0.49 2.5 6.25 5 25 1.73 5 100 10000 0.7 0.49 2.5 6.25 5 25 1.255 6 100 10000 0.7 0.49 2.5 6.25 5 25 1.48 7 100 10000 1.2 1.44 4.5 20.25 10 100 1.66 8 100 10000 1.2 1.44 4.5 20.25 10 100 1.695 9 100 10000 1.2 1.44 4.5 20.25 10 100 1.86 10 300 90000 0.2 0.04 2.5 6.25 10 100 1.94 11 300 90000 0.2 0.04 2.5 6.25 10 100 1.885 12 300 90000 0.2 0.04 2.5 6.25 10 100 1.915 13 300 90000 0.7 0.49 4.5 20.25 1 1 1.765 14 300 90000 0.7 0.49 4.5 20.25 1 1 1.775 15 300 90000 0.7 0.49 4.5 20.25 1 1 1.985 16 300 90000 1.2 1.44 0.5 0.25 5 25 1.655 17 300 90000 1.2 1.44 0.5 0.25 5 25 1.715 18 300 90000 1.2 1.44 0.5 0.25 5 25 1.855 19 500 250000 0.2 0.04 4.5 20.25 5 25 2.01 20 500 250000 0.2 0.04 4.5 20.25 5 25 2.04 21 500 250000 0.2 0.04 4.5 20.25 5 25 1.855 22 500 250000 0.7 0.49 0.5 0.25 10 100 1.975 23 500 250000 0.7 0.49 0.5 0.25 10 100 1.94 24 500 250000 0.7 0.49 0.5 0.25 10 100 2.29 25 500 250000 1.2 1.44 2.5 6.25 1 1 1.79 26 500 250000 1.2 1.44 2.5 6.25 1 1 1.89 27 500 250000 1.2 1.44 2.5 6.25 1 1 2.08 Table A-121: Regression Statistics Multiple R 0.849 R Square 0.721 Adjusted R Square 0.597 Standard Error 0.136 Observations 27 Table A-122: Multiple Non-linear Regression data with replication ANOVA


237

Table A-123: Nonlinear Regression of multivariable Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 1.498 0.160 9.357 2.46 x10-8 1.162 1.834 A 0.002 0.001 1.840 0.082 0.000 0.003 A2 -9.86 x10-7 1.38 x10-6 -0.713 0.485 -3.89 x10-6 1.92 x10-6 B -0.073 0.316 -0.230 0.821 -0.738 0.592 B2 0.039 0.221 0.176 0.863 -0.426 0.504 C -0.046 0.071 -0.652 0.522 -0.195 0.103 C2 0.012 0.014 0.868 0.397 -0.017 0.041 D -0.042 0.032 -1.328 0.201 -0.109 0.025 D2 0.005 0.003 1.850 0.081 -0.001 0.011


Observation Predicted Kerf width Mean Residuals 1 1.573 0.007 2 1.573 0.042 3 1.573 -0.048 4 1.488 0.242 5 1.488 -0.233 6 1.488 -0.008 7 1.738 -0.078 8 1.738 -0.043 9 1.738 0.122 10 1.913 0.027 11 1.913 -0.028 12 1.913 0.002 13 1.842 -0.077 14 1.842 -0.067 15 1.842 0.143 16 1.742 -0.087 17 1.742 -0.027 18 1.742 0.113 19 1.968 0.042 20 1.968 0.072 21 1.968 -0.113 22 2.068 -0.093 23 2.068 -0.128 24 2.068 0.222 25 1.920 -0.130 26 1.920 -0.030 27 1.920 0.160

APPENDIX B

238

B. NEURAL NETWORK & OVERALL QUALITY Table B-1: Training on Factorial design

S. No. No. of neurons Max. % error Min. % error Average % error Results

1 20-20-20 12.374 0.0003 0.663 Train

2.2 0.0001 0.186

14.135 0.0003 12.37 Test

9.115 0.0001 2.2

81.000 11.000 46.000 Simulation

0.290 0.000 0.123

2 10-10-10 66.376 0.247 16.440 Train

9.533 0.002 1.272

66.376 0.002 25.173 Test

16.437 0.247 9.533

28.608 0.000 18.323 Simulation

83.740 0.005 21.338

3 5-5-5 51.685 0.527 19.819 Train

67.427 0.224 21.509

51.685 0.224 14.201 Test

67.427 0.527 16.673

109.517 0.002 26.268 Simulation

108.743 0.000 26.133

4 30-30-30 12.143 0.002 0.931 Train

2.764 0.0002 0.393

16.275 0.0002 12.143 Test

13.192 0.002 2.764

104.671 0.000 28.513 Simulation

77.256 0.517 19.343

5 30-30-30 10.220 0.0002 0.318 Train

3.088 2.2e-05 0.145

15.140 2.20e-5 10.220 Test

239

Table B-2: Edge quality mean training using factorial datasets

S. No. No. of

Neurons Max.% error

Min.% error

Average % error

Overall Regression

Results

1 5-5 28.757 0.017 7.691 0.910 Train

25.499 2.601 9.869 Test

57.414 0.049 17.472 Simulation

2 10-10 54.752 0.066 11.899 0.902 Train

53.614 0.447 20.029 Test

61.376 0.337 14.305 Simulation

3 30-30 45.797 0.001 4.215 0.911 Train

29.776 8.080 15.987 Test

67.506 0.213 17.732 Simulation

4 10-10-10 34.691 0.007 3.059 0.984 Train

42.789 1.695 14.067 Test

80.650 0.656 17.894 Simulation

5 20-20-20 12.309 0.002 0.667 0.994 Train

31.960 2.262 17.052 Test

83.781 0.432 18.819 Simulation

6 30-30-30 21.789 0.004 1.489 0.994 Train

18.301 1.138 11.822 Test

72.636 0.099 18.265 Simulation

7 40-40-40 20.361 0.060 1.965 0.986 Train

23.958 0.702 11.657 Test

90.140 0.246 18.495 Simulation

8 65-65-65 71.222 0.005 6.162 0.922 Train

54.716 1.047 19.281 Test

103.241 0.592 22.471 Simulation

9 40-40 1.587 0.000 0.115 0.999 Train

14.564 1.852 8.960 Test

83.677 0.372 17.660 Simulation

10 50-50 14.995 0.000 0.678 0.900 Train

87.975 21.624 57.457 Test

92.221 0.104 21.184 Simulation

240

Table B-3: Edge quality signal to noise ratio of factorial datasets

S. No. No. of

Neurons Max.% error

Min.% error

Average %error

Overall Regression

Results

1 10 26.066 0.016 4.431 0.963 Train 19.470 1.290 9.868 Test 26.066 0.016 4.901 Simulation 2 20 25.010 0.002 6.424 0.922 Train 12.830 0.147 3.784 Test 25.010 0.002 6.196 Simulation 3 6 23.623 0.063 6.950 0.913 Train 16.044 1.256 9.640 Test 26.066 0.016 4.901 Simulation 4 10-10 25.567 0.051 1.840 0.969 Train 23.367 0.215 15.292 Test 25.567 0.051 3.003 Simulation 5 20-20 25.679 0.013 1.678 0.949 Train 55.579 2.029 22.590 Test 55.579 0.013 3.485 Simulation 6 10-10-10 24.781 0.047 2.672 0.962 Train 21.998 1.136 9.546 Test 24.781 0.047 3.266 Simulation 7 20-20-20 29.493 0.257 6.985 0.943 Train 15.159 0.509 7.706 Test 29.493 0.257 7.047 Simulation

Table B-4: Edge quality mean of normalized dataset

S. No. No. of

Neurons Max.% error

Min.% error

Average % error

Overall Regression

Results

1 10-10 29.778 0.027 6.482 0.953 Train 39.168 4.180 17.639 Test 35.254 0.573 9.892 Simulation 2 20-20 29.272 0.000 2.242 0.954 Train 61.590 3.893 16.318 Test 29.759 0.006 6.723 Simulation 3 30-30 66.666 0.134 4.886 0.884 Train 69.056 9.207 23.385 Test 44.857 0.157 7.701 Simulation 4 10-10-10 14.891 0.085 3.299 0.957 Train 39.119 2.200 12.234 Test 31.493 0.019 6.953 Simulation 5 20-20-20 17.088 0.003 1.963 0.977 Train

241

S. No. No. of

Neurons Max.% error

Min.% error

Average % error

Overall Regression

Results

59.009 1.929 22.556 Test 31.892 0.079 7.398 Simulation 6 30-30-30 26.804 0.000 0.664 0.983 Train 76.089 1.852 21.426 Test 27.018 0.011 6.232 Simulation 7 40-40-40 24.207 0.004 1.434 0.966 Train 48.022 8.332 27.453 Test 47.686 0.034 7.707 Simulation

Table B-5: Kerf width mean training of factorial design S. No. No. of

Neurons Max.% error

Min.% error

Average % error

Overall Regression

Results

1 10 43.005 0.040 4.201 0.9861 Train 57.876 3.687 15.697 Test 57.876 0.040 5.194 Simulation 2 20 48.976 0.000 2.518 0.9767 Train 42.996 0.993 17.762 Test 48.976 0.000 3.835 Simulation 3 10-10 20.876 0.000 0.887 0.9631 Train 59.445 3.544 26.821 Test 59.445 0.000 3.128 Simulation 4 20-20 110.949 0.000 5.848 0.9986 Train 72.252 7.785 35.447 Test 110.949 0.000 8.406 Simulation 5 30-30 48.116 0.633 11.506 0.8802 Train 31.662 0.762 16.352 Test 48.116 0.633 11.925 Simulation 6 10-10-10 35.301 0.006 3.257 0.9614 Train 32.369 1.423 16.321 Test 35.301 0.006 4.386 Simulation 7 20-20-20 61.063 0.000 5.778 0.9722 Train 25.190 1.146 13.453 Test

61.063 0.000 6.441 Simulation

242

Figure B-1: No. of times initialization comparison of 3mm sheet EQ Mean

Figure B-2: No. of times initialization comparison of 5mm sheet EQ Mean

9.5

10.0

10.5

11.0

11.5

12.0

2 3 4 5 6 7 8 9 10

Ave

rage

per

cent

err

or

Number of neurons

3000 times initialize


11.0

11.5

12.0

12.5

13.0

13.5

14.0

14.5

15.0

15.5

16.0

2 3 4 5 6 7 8 9 10 11

Ave

rage

per

cent

err

or

Number of Neuron



243

Figure B-3: No. of times initialization comparison of 3mm sheet EQ S/N

Figure B-4: No. of times initialization comparison of 5mm sheet EQ S/N

8

8.5

9

9.5

10

10.5

11

11.5

2 3 4 5 6 7 8 9 10

Ave

rage

per

cent

err

or

Number of neuron



10.0

10.5

11.0

11.5

12.0

12.5

13.0

13.5

14.0

14.5

2 3 4 5 6 7 8 9 10 11

Ave

rage

per

cent

err

or

Number of neuron



244

Figure B-5: No. of times initialization comparison of 3mm sheet KW Mean

Figure B-6: No. of times initialization comparison of 5mm sheet KW Mean

4.0

4.5

5.0

5.5

6.0

6.5

7.0

2 3 4 5 6 7 8 9 10

Ave

rage

per

cent

err

or

Number of neuron



7.5

8.0

8.5

9.0

9.5

10.0

2 3 4 5 6 7 8 9 10 11

Ave

rage

per

cent

err

or

Number of neuron



245

Figure B-7: No. of times initialization comparison of 3mm sheet KW S/N

Figure B-8: No. of times initialization comparison of 5mm sheet KW S/N

5.06.07.08.09.0

10.011.012.013.014.015.016.017.018.0

2 3 4 5 6 7 8 9 10 11

Ave

rage

per

cent

err

or

Number of neuron



7.0

8.0

9.0

10.0

11.0

2 3 4 5 6 7 8 9 10 11

Ave

rage

per

cent

err

or

Number of neuron



246

Table B-6: Simulated data of factorial design

Run LP

A

CS

B

AGP

C

SD

D

Simulated results

EQ KW POC MRR

1 100 0.2 0.5 1 3.670 1.552 2.356 6.74E-08

2 100 0.2 0.5 5 2.729 1.842 3.089 6.98E-08

3 100 0.2 0.5 10 2.090 1.882 3.868 6.80E-08

4 100 0.2 2.5 1 2.833 1.440 2.413 6.31E-08

5 100 0.2 2.5 5 2.991 1.547 3.207 6.37E-08

6 100 0.2 2.5 10 2.086 1.878 3.799 6.62E-08

7 100 0.2 4.5 1 1.846 1.485 2.434 6.56E-08

8 100 0.2 4.5 5 1.545 1.500 3.259 7.98E-08

9 100 0.2 4.5 10 1.264 1.837 3.736 6.61E-08

10 100 0.7 0.5 1 3.968 1.489 2.228 7.21E-08

11 100 0.7 0.5 5 2.584 1.563 2.696 2.53E-07

12 100 0.7 0.5 10 1.484 1.877 3.871 7.76E-08

13 100 0.7 2.5 1 2.926 1.485 2.236 8.64E-08

14 100 0.7 2.5 5 2.730 1.488 2.763 2.25E-07

15 100 0.7 2.5 10 1.211 1.832 3.888 6.60E-08

16 100 0.7 4.5 1 1.406 1.636 2.220 4.05E-07

17 100 0.7 4.5 5 1.934 1.641 2.801 4.57E-07

18 100 0.7 4.5 10 1.539 1.694 3.832 4.40E-07

19 100 1.2 0.5 1 2.869 1.507 2.304 4.79E-07

20 100 1.2 0.5 5 2.146 1.484 2.589 4.35E-07

21 100 1.2 0.5 10 1.400 1.827 3.513 3.42E-07

22 100 1.2 2.5 1 2.348 1.623 2.249 4.85E-07

23 100 1.2 2.5 5 2.215 1.624 2.612 4.84E-07

24 100 1.2 2.5 10 1.078 1.680 3.627 4.45E-07

25 100 1.2 4.5 1 1.669 1.735 2.196 4.78E-07

26 100 1.2 4.5 5 2.211 1.735 2.586 4.58E-07

27 100 1.2 4.5 10 1.670 1.738 3.664 4.56E-07

28 300 0.2 0.5 1 2.988 1.632 3.519 8.01E-08

247

Run LP

A

CS

B

AGP

C

SD

D

Simulated results

EQ KW POC MRR

29 300 0.2 0.5 5 2.175 1.862 3.789 2.77E-07

30 300 0.2 0.5 10 1.542 1.917 3.924 1.30E-07

31 300 0.2 2.5 1 2.743 1.644 3.524 8.66E-08

32 300 0.2 2.5 5 2.302 1.643 3.764 8.10E-08

33 300 0.2 2.5 10 1.670 1.913 3.821 6.56E-08

34 300 0.2 4.5 1 2.405 1.748 3.504 4.20E-07

35 300 0.2 4.5 5 2.265 1.683 3.710 1.06E-07

36 300 0.2 4.5 10 1.276 1.862 3.766 6.57E-08

37 300 0.7 0.5 1 3.836 1.649 3.051 2.70E-07

38 300 0.7 0.5 5 2.795 1.650 3.732 3.05E-07

39 300 0.7 0.5 10 1.669 1.913 4.100 3.19E-07

40 300 0.7 2.5 1 2.212 1.805 3.091 4.85E-07

41 300 0.7 2.5 5 2.896 1.718 3.719 4.20E-07

42 300 0.7 2.5 10 1.948 1.870 4.034 6.95E-08

43 300 0.7 4.5 1 1.000 1.842 3.129 4.82E-07

44 300 0.7 4.5 5 1.135 1.778 3.662 4.57E-07

45 300 0.7 4.5 10 1.375 1.759 3.942 4.37E-07

46 300 1.2 0.5 1 3.045 1.829 2.731 4.85E-07

47 300 1.2 0.5 5 2.670 1.772 3.590 4.53E-07

48 300 1.2 0.5 10 2.014 1.877 4.098 3.22E-07

49 300 1.2 2.5 1 2.109 1.902 2.681 4.85E-07

50 300 1.2 2.5 5 2.273 1.827 3.648 4.85E-07

51 300 1.2 2.5 10 1.278 1.774 4.087 4.57E-07

52 300 1.2 4.5 1 0.888 1.837 2.680 4.85E-07

53 300 1.2 4.5 5 1.714 1.782 3.623 4.81E-07

54 300 1.2 4.5 10 1.718 1.757 4.028 4.57E-07

55 500 0.2 0.5 1 2.107 1.672 4.274 9.25E-08

56 500 0.2 0.5 5 1.599 1.715 4.036 3.18E-07

57 500 0.2 0.5 10 1.200 1.905 3.925 3.17E-07

58 500 0.2 2.5 1 2.051 1.870 4.1372 1.12E-07

248

Run LP

A

CS

B

AGP

C

SD

D

Simulated results

EQ KW POC MRR

59 500 0.2 2.5 5 1.658 1.895 3.869 3.24E-07

60 500 0.2 2.5 10 1.081 2.057 3.804 7.26E-08

61 500 0.2 4.5 1 2.353 1.972 3.916 4.81E-07

62 500 0.2 4.5 5 2.170 1.968 3.768 8.55E-08

63 500 0.2 4.5 10 1.095 2.084 3.763 6.51E-08

64 500 0.7 0.5 1 3.204 1.856 4.221 3.62E-07

65 500 0.7 0.5 5 2.793 1.862 4.108 3.22E-07

66 500 0.7 0.5 10 1.830 2.068 4.125 3.14E-07

67 500 0.7 2.5 1 1.970 1.911 4.115 4.85E-07

68 500 0.7 2.5 5 2.747 1.928 3.927 4.45E-07

69 500 0.7 2.5 10 2.330 2.095 4.015 3.43E-07

70 500 0.7 4.5 1 0.865 1.977 3.903 4.85E-07

71 500 0.7 4.5 5 1.047 1.964 3.787 4.84E-07

72 500 0.7 4.5 10 1.335 1.925 3.929 4.36E-07

73 500 1.2 0.5 1 3.356 1.884 4.117 4.85E-07

74 500 1.2 0.5 5 3.094 1.886 4.294 4.49E-07

75 500 1.2 0.5 10 2.305 1.962 4.168 3.24E-07

76 500 1.2 2.5 1 1.756 1.920 4.031 4.85E-07

77 500 1.2 2.5 5 2.383 1.935 4.118 4.85E-07

78 500 1.2 2.5 10 1.624 1.950 4.109 4.82E-07

79 500 1.2 4.5 1 0.625 1.977 3.842 4.85E-07

80 500 1.2 4.5 5 1.125 1.957 3.933 4.85E-07

81 500 1.2 4.5 10 1.688 1.892 4.049 4.82E-07

249

Table B-7: Compare results of Normalized aggregation, Customer quality function and fuzzy

aggregation

Run Ordinal outputs EQ KW POC MRR Aggregation Customer

quality function

Fuzzy aggregation

Quantified fuzzy

aggregation 1 2 2 1 1 1 2 0.45 2

2 1 2 1 1 1 2 0.48 2

3 2 1 1 2 1 2 0.48 2

4 3 2 1 1 2 3 0.46 2

5 3 1 1 2 2 3 0.42 2

6 2 2 2 2 2 2 0.48 2

7 1 3 1 2 2 3 0.36 2

8 2 2 1 1 2 2 0.45 2

9 2 1 1 3 2 3 0.49 2

10 2 2 1 2 2 2 0.44 2

11 2 1 3 3 2 3 0.39 2

12 3 1 1 3 2 3 0.48 2

13 3 1 1 3 2 3 0.47 2

14 2 3 1 2 2 3 0.38 2

15 1 2 3 3 2 3 0.48 2

16 3 1 2 3 2 3 0.49 2

17 2 3 3 1 2 3 0.61 3

18 1 3 2 3 2 3 0.55 3

19 2 3 2 2 2 3 0.48 2

20 1 3 3 1 2 3 0.49 2

21 2 2 3 1 2 3 0.48 2

22 2 2 3 2 2 3 0.49 2

23 1 3 3 1 2 3 0.49 2

24 1 3 3 3 2 3 0.49 2

25 1 3 3 2 2 3 0.49 2

26 1 3 3 1 2 3 0.50 2

27 1 2 3 2 2 3 0.49 2

250


quality function

Fuzzy aggregation

Quantified fuzzy

aggregation 28 2 2 3 3 2 3 0.49 2

29 1 3 3 2 2 3 0.68 3

30 3 3 2 2 2 3 0.48 2

31 2 3 3 1 2 3 0.49 2

32 3 1 1 3 3 3 0.49 2

33 2 3 3 2 3 3 0.49 2

34 3 2 3 2 3 3 0.63 3

35 3 1 1 3 3 3 0.49 2

36 2 2 3 2 3 3 0.49 2

37 3 2 3 2 3 3 0.59 3

38 1 3 3 2 3 3 0.63 3

39 2 3 3 1 3 3 0.48 2

40 1 3 3 2 3 3 0.63 3

41 3 1 1 3 3 3 0.63 3

42 3 3 3 2 3 3 0.48 2

43 2 3 3 2 3 3 0.63 3

44 3 2 3 3 3 3 0.63 3

45 3 2 3 2 3 3 0.63 3

46 1 3 3 3 3 3 0.57 3

47 2 3 3 2 3 3 0.49 2

48 3 1 2 3 3 3 0.48 2

49 2 3 3 2 3 3 0.53 3

50 1 3 3 3 3 3 0.51 3

51 3 3 3 1 3 3 0.49 2

52 2 2 3 3 3 3 0.52 3

53 2 2 3 3 3 3 0.63 3

54 3 2 2 3 3 3 0.49 2

55 1 3 3 3 3 3 0.49 2

56 2 3 3 3 3 3 0.62 3

251


quality function

Fuzzy aggregation

Quantified fuzzy

aggregation 57 2 3 3 3 3 3 0.63 3

58 3 3 3 1 3 3 0.49 2

59 2 3 3 3 3 3 0.62 3

60 2 3 3 3 3 3 0.49 2

61 2 3 3 3 3 3 0.63 3

62 3 2 3 3 3 3 0.49 2

63 3 3 3 1 3 3 0.49 2

64 3 2 3 3 3 3 0.82 3

65 1 3 3 3 3 3 0.83 3

66 3 3 2 3 3 3 0.48 2

67 1 3 3 3 3 3 0.63 3

68 3 3 3 2 3 3 0.63 3

69 3 3 3 3 3 3 0.54 3

70 1 3 3 3 3 3 0.63 3

71 3 3 3 2 3 3 0.63 3

72 3 3 3 2 3 3 0.63 3

73 3 3 3 2 3 3 0.86 3

74 2 3 3 3 3 3 0.73 3

75 2 3 3 3 3 3 0.48 2

76 2 3 3 3 3 3 0.68 3

77 2 3 3 3 3 3 0.79 3

78 2 3 3 3 3 3 0.48 2

79 2 3 3 3 3 3 0.63 3

80 3 3 3 3 3 3 0.67 3

81 2 3 3 3 3 3 0.48 2

252

Figure B-9: Comparison of quantified aggregation with customer quality function

Table B-8: Sorted with CQF, Quantified fuzzy aggregation and Quantified Normalized

aggregation

RUN EQ KW POC MRR Q AV CQF FL AV QFL AV

1 2 2 1 1 1 2 0.45 2

2 1 2 1 1 1 2 0.48 2

3 2 1 1 2 1 2 0.48 2

10 2 2 1 2 2 2 0.44 2

8 2 2 1 1 2 2 0.45 2

6 2 2 2 2 2 2 0.48 2

7 1 3 1 2 2 3 0.36 2

14 2 3 1 2 2 3 0.38 2

11 2 1 3 3 2 3 0.39 2

5 3 1 1 2 2 3 0.42 2

4 3 2 1 1 2 3 0.46 2

13 3 1 1 3 2 3 0.47 2

1

2

3

4

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79

Aggregation

Customer quality function

253


12 3 1 1 3 2 3 0.48 2

15 1 2 3 3 2 3 0.48 2

19 2 3 2 2 2 3 0.48 2

21 2 2 3 1 2 3 0.48 2

30 3 3 2 2 2 3 0.48 2

9 2 1 1 3 2 3 0.49 2

16 3 1 2 3 2 3 0.49 2

20 1 3 3 1 2 3 0.49 2

22 2 2 3 2 2 3 0.49 2

23 1 3 3 1 2 3 0.49 2

24 1 3 3 3 2 3 0.49 2

25 1 3 3 2 2 3 0.49 2

27 1 2 3 2 2 3 0.49 2

28 2 2 3 3 2 3 0.49 2

31 2 3 3 1 2 3 0.49 2

26 1 3 3 1 2 3 0.5 2

39 2 3 3 1 3 3 0.48 2

42 3 3 3 2 3 3 0.48 2

48 3 1 2 3 3 3 0.48 2

66 3 3 2 3 3 3 0.48 2

75 2 3 3 3 3 3 0.48 2

78 2 3 3 3 3 3 0.48 2

81 2 3 3 3 3 3 0.48 2

32 3 1 1 3 3 3 0.49 2

33 2 3 3 2 3 3 0.49 2

35 3 1 1 3 3 3 0.49 2

36 2 2 3 2 3 3 0.49 2

47 2 3 3 2 3 3 0.49 2

51 3 3 3 1 3 3 0.49 2

54 3 2 2 3 3 3 0.49 2

55 1 3 3 3 3 3 0.49 2

254


58 3 3 3 1 3 3 0.49 2

60 2 3 3 3 3 3 0.49 2

62 3 2 3 3 3 3 0.49 2

63 3 3 3 1 3 3 0.49 2

18 1 3 2 3 2 3 0.55 3

17 2 3 3 1 2 3 0.61 3

29 1 3 3 2 2 3 0.68 3

50 1 3 3 3 3 3 0.51 3

52 2 2 3 3 3 3 0.52 3

49 2 3 3 2 3 3 0.53 3

69 3 3 3 3 3 3 0.54 3

46 1 3 3 3 3 3 0.57 3

37 3 2 3 2 3 3 0.59 3

56 2 3 3 3 3 3 0.62 3

59 2 3 3 3 3 3 0.62 3

34 3 2 3 2 3 3 0.63 3

38 1 3 3 2 3 3 0.63 3

40 1 3 3 2 3 3 0.63 3

41 3 1 1 3 3 3 0.63 3

43 2 3 3 2 3 3 0.63 3

44 3 2 3 3 3 3 0.63 3

45 3 2 3 2 3 3 0.63 3

53 2 2 3 3 3 3 0.63 3

57 2 3 3 3 3 3 0.63 3

61 2 3 3 3 3 3 0.63 3

67 1 3 3 3 3 3 0.63 3

68 3 3 3 2 3 3 0.63 3

70 1 3 3 3 3 3 0.63 3

71 3 3 3 2 3 3 0.63 3

72 3 3 3 2 3 3 0.63 3

79 2 3 3 3 3 3 0.63 3

255


80 3 3 3 3 3 3 0.67 3

76 2 3 3 3 3 3 0.68 3

74 2 3 3 3 3 3 0.73 3

77 2 3 3 3 3 3 0.79 3

64 3 2 3 3 3 3 0.82 3

65 1 3 3 3 3 3 0.83 3

73 3 3 3 2 3 3 0.86 3

Table B-9: Energy consumption quality calculation for factorial design

Run Laser Power

Cutting Speed

A.Gas pressure

Standoff distance

Energy Consumed

1 100 0.2 0.5 1 8.33 2 100 0.2 0.5 5 8.33 3 100 0.2 0.5 10 8.33 4 100 0.2 2.5 1 8.33 5 100 0.2 2.5 5 8.33 6 100 0.2 2.5 10 8.33 7 100 0.2 4.5 1 8.33 8 100 0.2 4.5 5 8.33 9 100 0.2 4.5 10 8.33 10 100 0.7 0.5 1 2.38 11 100 0.7 0.5 5 2.38 12 100 0.7 0.5 10 2.38 13 100 0.7 2.5 1 2.38 14 100 0.7 2.5 5 2.38 15 100 0.7 2.5 10 2.38 16 100 0.7 4.5 1 2.38 17 100 0.7 4.5 5 2.38 18 100 0.7 4.5 10 2.38 19 100 1.2 0.5 1 1.39 20 100 1.2 0.5 5 1.39 21 100 1.2 0.5 10 1.39 22 100 1.2 2.5 1 1.39 23 100 1.2 2.5 5 1.39 24 100 1.2 2.5 10 1.39 25 100 1.2 4.5 1 1.39 26 100 1.2 4.5 5 1.39 27 100 1.2 4.5 10 1.39

256

Run Laser Power

Cutting Speed

A.Gas pressure

Standoff distance

Energy Consumed

28 300 0.2 0.5 1 25.00 29 300 0.2 0.5 5 25.00 30 300 0.2 0.5 10 25.00 31 300 0.2 2.5 1 25.00 32 300 0.2 2.5 5 25.00 33 300 0.2 2.5 10 25.00 34 300 0.2 4.5 1 25.00 35 300 0.2 4.5 5 25.00 36 300 0.2 4.5 10 25.00 37 300 0.7 0.5 1 7.14 38 300 0.7 0.5 5 7.14 39 300 0.7 0.5 10 7.14 40 300 0.7 2.5 1 7.14 41 300 0.7 2.5 5 7.14 42 300 0.7 2.5 10 7.14 43 300 0.7 4.5 1 7.14 44 300 0.7 4.5 5 7.14 45 300 0.7 4.5 10 7.14 46 300 1.2 0.5 1 4.17 47 300 1.2 0.5 5 4.17 48 300 1.2 0.5 10 4.17 49 300 1.2 2.5 1 4.17 50 300 1.2 2.5 5 4.17 51 300 1.2 2.5 10 4.17 52 300 1.2 4.5 1 4.17 53 300 1.2 4.5 5 4.17 54 300 1.2 4.5 10 4.17 55 500 0.2 0.5 1 41.67 56 500 0.2 0.5 5 41.67 57 500 0.2 0.5 10 41.67 58 500 0.2 2.5 1 41.67 59 500 0.2 2.5 5 41.67 60 500 0.2 2.5 10 41.67 61 500 0.2 4.5 1 41.67 62 500 0.2 4.5 5 41.67 63 500 0.2 4.5 10 41.67 64 500 0.7 0.5 1 11.90 65 500 0.7 0.5 5 11.90 66 500 0.7 0.5 10 11.90 67 500 0.7 2.5 1 11.90

257

Run Laser Power

Cutting Speed

A.Gas pressure

Standoff distance

Energy Consumed

68 500 0.7 2.5 5 11.90 69 500 0.7 2.5 10 11.90 70 500 0.7 4.5 1 11.90 71 500 0.7 4.5 5 11.90 72 500 0.7 4.5 10 11.90 73 500 1.2 0.5 1 6.94 74 500 1.2 0.5 5 6.94 75 500 1.2 0.5 10 6.94 76 500 1.2 2.5 1 6.94 77 500 1.2 2.5 5 6.94 78 500 1.2 2.5 10 6.94 79 500 1.2 4.5 1 6.94 80 500 1.2 4.5 5 6.94 81 500 1.2 4.5 10 6.94

prepared by s. mustafa ali zaidi, 1032109...

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