ABSTRACT
Geometric programming is used for design and cost optimization, the development of generalized
design relationships, cost ratios for specific problems, and profit maximization. The early pioneers of
the process - Zener, Duffin, Peterson, Beightler, Wilde, and Phillips – played important roles in the
development of geometric programming. There are three major areas: 1) Introduction, History, and
Theoretical Fundamentals,2) Applications with Zero Degrees of Difficulty,and 3) Applications with
Positive Degrees of Difficulty. The primal-dual relationships are used to illustrate how to determine
the primal variables from the dual solution and how to determine additional dual equations when
the degrees of difficulty are positive. A new technique for determining additional equations for
the dual, Dimensional Analysis, is demonstrated.The various solution techniques of the constrained
derivative approach,the condensation of terms, and dimensional analysis are illustrated with example
problems.The goal of this work is to have readers develop more case studies to further the application
of this exciting tool.
KEYWORDS
posynominials, primal, dual, cost optimization, design optimization, generalized de-
sign relationships, cost ratios, profit maximization, constrained derivatives, dimensional
analysis, condensation of terms
vii
Contents
Preface .................................................................. xi
PART I Introduction, History, and Theoretical
Fundamentals of Geometric Programming ........1
1
Introduction ..............................................................3
1.1 Optimization and Geometric Programming ................................ 3
1.1.1 Optimization .................................................... 3
1.1.2 Geometric Programming .......................................... 3
1.2 Evaluative Questions .................................................... 4
References ............................................................. 4
2
Brief History of Geometric Programming ....................................5
2.1 Pioneers of Geometric Programming ...................................... 5
2.2 Evaluative Questions .................................................... 6
References ............................................................. 6
3
Theoretical Considerations .................................................7
3.1 Primal and Dual Formulation ............................................ 7
3.2 Evaluative Questions ................................................... 10
References ............................................................ 10
PART II Geometric Prog ramming Applications with
Zero Degrees of Difficulty .....................11
4
The Optimal Box Design Case Study ...................................... 13
4.1 The Optimal Box Design Problem ....................................... 13
4.2 Evaluative Questions ................................................... 16
viii
5
Trash Can Case Study .................................................... 17
5.1 Introduction .......................................................... 17
5.2 Problem Statement and General Solution ................................. 17
5.2.1 Example ....................................................... 17
5.3 Evaluative Questions ................................................... 20
6
The Open Cargo Shipping Box Case S tudy ................................. 21
6.1 Problem Statement and General Solution ................................. 21
6.2 Evaluative Questions ................................................... 24
References ............................................................ 25
7
Metal Casting Cylindrical Riser Case Study ................................ 27
7.1 Introduction .......................................................... 27
7.2 Problem Formulation and General Solution ............................... 29
7.3 Example ............................................................. 30
7.4 Evaluative Questions ................................................... 31
References ............................................................ 31
8
Inventory Model Case Study .............................................. 33
8.1 Problem Statement and General Solution. ................................. 33
8.2 Example ............................................................. 35
8.3 Evaluative Questions ................................................... 36
References ............................................................ 36
9
Process Furnace Design Case Study ....................................... 37
9.1 Problem Statement and Solution ......................................... 37
9.2 Evaluative Questions ................................................... 41
References ............................................................ 41
10
Gas Transmission Pipeline Case Study ..................................... 43
10.1 Problem Statement and Solution ......................................... 43
10.2 Evaluative Questions ................................................... 46
References ............................................................ 46
ix
11
Profit Maximization Case Study ........................................... 47
11.1 Profit Maximization and Geometric Programming ......................... 47
11.2 Profit Maximization using the Cobb-Douglas Production Function ........... 47
11.3 Evaluative Questions ................................................... 49
References ............................................................ 50
12
Material Removal/Metal Cutting Economics Case Study .................... 51
12.1 Introduction .......................................................... 51
12.2 Problem Formulation................................................... 51
12.3 Evaluative Questions ................................................... 54
References .............................................................. 55
PART III Geometric Prog ramming Applications with
Positive Degrees of Difficulty .................57
13
Journal Bearing Design Case Study ........................................ 59
13.1 Introduction .......................................................... 59
13.2 Primal and Dual Formulation of Journal Bearing Design .................... 59
13.3 Dimensional Analysis Technique for Additional Equation ................... 63
13.4 Evaluative Questions ................................................... 64
References ............................................................ 65
14
Metal Casting Hemispherical Top Cylindrical Side Riser Case Study .......... 67
14.1 Introduction .......................................................... 67
14.2 Problem Formulation................................................... 67
14.3 Dimensional Analysis Technique for Additional Two Equations .............. 72
14.4 Evaluative Questions ................................................... 75
References ............................................................ 75
15
Liquefied Petroleum Gas (LPG) Cylinders Case Study ....................... 77
15.1 Introduction .......................................................... 77
15.2 Problem Formulation................................................... 77
15.3 Dimensional Analysis Technique for Additional Equation ................... 82
15.4 Evaluative Questions ................................................... 83
References ............................................................ 84
x
16
Material Removal/Metal Cutting Economics with Two Constraints ........... 85
16.1 Introduction .......................................................... 85
16.2 Problem Formulation................................................... 85
16.3 Problem Solution ...................................................... 87
16.4 Example Problem ...................................................... 91
16.5 Evaluative Questions ................................................... 91
References ............................................................ 92
17
The Open Cargo Shipping Box with S kids .................................. 95
17.1 Introduction .......................................................... 95
17.2 Primal-Dual Problem Formulation ....................................... 95
17.3 Constrained Derivative Approach ........................................ 97
17.4 Dimensional Analysis Approach for Additional Equation.................... 98
17.5 Condensation of Terms Approach ....................................... 100
17.6 Evaluative Questions .................................................. 102
References ........................................................... 102
18
Profit Maximization Considering Decreasing Cost Functions of Inventory
Polic y ..................................................................103
18.1 Introduction ......................................................... 103
18.2 Model Formulation ................................................... 103
18.3 Example ............................................................. 107
18.4 Transformed Dual Approach ........................................... 109
18.5 Evaluative Questions .................................................. 112
References ........................................................... 112
19
Summary and Future Directions ......................................... 115
19.1 Summary ............................................................ 115
19.2 Future Directions ..................................................... 115
19.3 Development of New Design Relationships .............................. 115
20
Thesis and Dissertations on Geometric Progr amming ...................... 119
Author’s Biography ..................................................... 125
Index ................................................................. 127

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