Mathematical Modelling and Optimization of Engineering Problems (eBook)

Preface 6
Contents 8
1 Heuristic Techniques for Real-Time Order Acceptance and Scheduling in Metal Additive Manufacturing 10
1.1 Introduction 10
1.2 Literature Review 11
1.3 Problem Statement 13
1.3.1 Assumptions 14
1.3.2 Notation 14
Decision Variables 15
Indicators 15
1.3.3 Basic Formulations 15
1.3.4 Objective Function 16
1.3.5 Constraints 16
1.4 Meta-heuristic Procedures 17
1.4.1 Generation of Feasible Solutions 17
1.4.1.1 Single Machine 17
1.4.1.2 Multiple Machines 19
1.4.2 Selection Rules 20
1.4.2.1 Stochastic Selection 21
1.4.2.2 Profit-Time Based Selection 21
1.4.2.3 Cost Benefit Based Selection 23
1.5 Computational Experiments 24
1.5.1 Data Generation 24
1.5.2 Experimental Results and Discussions 25
1.5.2.1 The Difference of Stochastic Results 25
1.5.2.2 Performance of Non-random Selection Rules 28
1.6 Conclusions and Future Research 30
References 32
2 Developing a Nationwide Energy Storage Policy by Optimal Size and Site Selection 34
2.1 Introduction 34
2.2 Optimization Models in Energy Economics 35
2.2.1 Economic Dispatch Model 35
2.2.2 Unit Commitment Model 36
2.2.3 Energy Storage System Modeling in UC 38
2.2.4 AC Optimal Power Flow Model 39
2.2.5 DC Optimal Power Flow Model 41
2.2.6 Optimal Energy Storage System Placement and Sizing Model 42
2.3 Optimization of the Nationwide Energy Storage System 43
2.3.1 The Maximum Sizing of Energy Storage Systems 44
2.3.2 Distribution of Energy Storage Systems Within the Network 45
2.3.3 The Proposed Bi-level Optimization for Sizing and Siting of Storage Units 46
2.4 Case Study: Power Systems in Turkey 47
2.5 Discussion on a Nationwide Energy Policy 53
2.6 Conclusion 56
Nomenclature 56
References 57
3 Pontryagin's Principle for a Class of Discrete Time Infinite Horizon Optimal Growth Problems 59
3.1 Introduction 59
3.2 One-Sector Optimal Growth Model 61
3.2.1 Social Planner's Problem of E 62
3.2.2 Necessary and Sufficient Conditions of Optimality 62
3.3 Optimal Growth Model with an Natural Exhaustible Resource 65
3.3.1 Management Problem of Eenr 66
3.3.2 Necessary and Sufficient Conditions of an Optimal Management of a Natural Resource 67
3.4 Optimal Growth Model of a Forest: An Optimal Management Model of Forestry 71
3.4.1 Planner's Management Problem 73
3.4.2 Necessary Conditions of Optimality 74
3.5 Conclusion 77
References 77
4 A Medical Modelling Using Multiple Linear Regression 78
4.1 Introduction 78
4.2 Materials and Methods 79
4.2.1 Study Samples 80
4.2.2 Multiple Linear Regression Analysis 81
4.2.3 Test for the Model 83
4.2.4 Residual Analysis 84
4.3 Building Regression Analysis Model 85
4.4 Discussion and Analysis 87
4.5 Conclusions and Recommendations 90
References 92
5 Lie Group Method Solution for Two-Dimensional Heat and Viscous Flow Driven by Injection Through a Deformable Rectangular Channel with Porous Walls 95
5.1 Introduction 95
5.2 Mathematical Modelling of the Problem 97
5.2.1 Problem Statement 97
5.2.2 Flow Configuration 98
5.2.3 Forces Affecting the Dynamics of the Flow 98
5.2.3.1 Surface Force 98
5.2.3.2 Body Forces 101
5.2.4 Derivation of Governing Equations 104
5.2.4.1 Conservation of Mass 104
5.2.4.2 Conservation of Momentum 104
5.2.4.3 Conservation of Energy 105
5.2.4.4 Boundary Conditions 105
5.3 Mathematical Representation of Problem 106
5.3.1 Governing Equations and Boundary Conditions 106
5.4 Solution of the Problem 108
5.4.1 Lie Group Analysis 108
5.4.2 Semi-Analytical Solution 111
5.5 Results and Discussion 113
5.5.1 Effects of Wall Dilation 113
5.5.2 Effects of Reynolds number inside the Filtration Chamber 114
5.5.3 Effects of Porosity Variable Inside the Filtration Chamber 116
5.5.4 Effects of Stuart Number Inside the Filtration Chamber 116
5.5.5 Temperature Distribution Inside the Chamber 116
5.6 Concluding Remarks 117
References 119
6 Optimal Siting of Wind Turbines in a Wind Farm 121
6.1 Introduction 121
6.2 Numerical Methods of the Present Study 124
6.2.1 Wake Model 124
6.2.2 Power Model 127
6.3 Methodology 127
6.3.1 Problem Formulation 127
6.3.2 Initial Population Based on Elevation Values 130
6.3.3 Genetic Algorithm for Optimization 132
6.3.3.1 Population Formation 132
6.3.3.2 Selection 134
6.3.3.3 Crossover 135
6.3.3.4 Mutation 136
6.3.3.5 Genetic Algorithm Parameters 136
6.4 Results and Discussion 139
6.5 Conclusion 141
References 142
7 RSM-Based Optimization of Excitation Capacitance and Speed for a Self-Excited Induction Generator 144
7.1 Introduction 144
7.2 Modelling of SEIG 146
7.3 Voltage Build-up process 147
7.4 Analysis 148
7.5 Response Surface Method 150
7.6 Results and Discussions 151
7.7 Conclusion 158
References 158
8 Distance-Constrained Vehicle Routing Problems: A Case Study Using Artificial Bee Colony Algorithm 161
8.1 Introduction 161
8.2 Research Background 162
8.3 Artificial Bee Colony (ABC) Algorithm 165
8.3.1 Initialization of the Population 166
8.3.2 Initialization of the Bee Phase 167
8.3.3 Onlooker Bee Phase 167
8.3.4 Scout Bee Phase 168
8.3.5 Stopping Phase 168
8.4 Case Study 168
8.5 Results and Discussion 171
8.6 Conclusion 175
References 175
9 Fractional Model for Type 1 Diabetes 178
9.1 Introduction 178
9.1.1 Some Concepts of Fractional Calculus 179
9.2 Description of the Model 180
9.3 Model Analysis 181
9.4 Global Stability of the Disease-Free Equilibrium 182
9.5 Numerical Results 184
9.6 Conclusion 186
References 187
10 Mathematical Modelling and Additive Manufacturingof a Gyroid 189
10.1 Infinite Periodic Minimal Surfaces (IPMS) Without Self-intersections: Gyroid 189
10.2 Additive Manufacturing Technology 190
10.3 3D Printing Process of an IPMS Gyroid 191
10.3.1 Creating the 3D Mathematical Model of the IPMS Gyroid with K3DSurf Program 191
10.3.2 Converting the CAD Model Data to ``.obj'' or ``.stl'' File Format 192
10.3.3 Generating a Solid, Thickened Shell or Hollow CAD Model 193
10.3.4 Slice the Model into Layers, Generate the Travel Movements and Support Structure 194
10.3.5 3D Printing of the Model 195
10.3.5.1 Fused Deposition Modelling 195
10.3.5.2 3D Printing of the IPMS Gyroid 196
10.3.6 Removing the Support Material If Any and Apply Finishing Process 196
10.4 Conclusion 196
References 197
Index 199