115
CHAPTER 19
Summary and Future Directions
19.1 SUMMARY
The object of this text is to generate interest in geometric programming amongst manufacturing
engineers,design engineers,manufacturing technologists, cost engineers, project managers,industrial
consultants and ﬁnance managers by illustrating the procedure for solving certain industrial and
practical problems. The various case studies were selected to illustrate a variety of applications as
well as a set of different types of problems from diverse ﬁelds. Several additional problems were added
focusing on proﬁt maximization and additional problems with degrees of difﬁculty. In addition, the
methods of dimensional analysis and the constrained derivative approach have been presented in
detail. Table 19.1 is a summary of the case studies presented in this text giving the type of problem,
degrees of difﬁculty, and other details.
The metal removal economics example also had variable exponents in the general solution.
The problems were worked in detail so general solutions could be obtained and also to show that
the dual and primal solutions were identical. The problems were selected to illustrate a variety of
types and also to show the use of the primal-dual relationships to determine the equations for the
primal variables. It is by showing the various types of applications in detailed examples that others
can follow the procedure and develop new applications.
19.2 FUTURE DIRECTIONS
The author is hopeful that others will communicate him additional examples to illustrate new appli-
cations that can be included in future editions. New applications will attract new practitioners to this
fascinating area of geometric programming. It is believed that the scope of geometric programming
will expand with new applications.
The author would like to include some software for different applications in geometric pro-
gramming in the future and would welcome contributions.
19.3 DEVELOPMENT OF NEW DESIGN RELATIONSHIPS
There are many different types of problems that can be solved by geometric programming and one
of the signiﬁcant advantages of the method is that it is possible in many applications to develop
general design relationships. The general design relationships can save considerable time and effort
in instances where the constants are changed.
116 19. SUMMARY AND FUTURE DIRECTIONS
Although geometric programming was ﬁrst presented nearly 50 years ago, the applications
have been rather sparse compared to that of linear programming. One goal is that as researchers
take advantage of the potential to develop design relationships that new applications will rapidly
occur. The development of new design relationships can signiﬁcantly reduce the development time
and cost for new products and this is essential for companies to remain competitive in the global
economy.
19.3. DEVELOPMENT OF NEW DESIGN RELATIONSHIPS 117
Table 19.1: Summary of Case Study Problem
Chapter Case Study
Degrees
of
Difficulty
Number
of
Constraints
Number
of
Variables
Variable Description
Number
of
Solutions
Special
Characteristics
4 The Optimal Box Design 0 1 3 Height, Width, Length 1
5 Trash Can 0 1 2 Height, Diameter 1
6 Open Cargo Shipping Box 0 1 3 Height, Width, Length 1 Classical Problem
7 Metal Casting Cylindrical Riser 0 1 2 Height, Diameter 1
8 Inventory Model 0 0 1 Lot Size 1
9 Process Furnace Design 0 1 3 Temperature, Length,
Height
1 Dominant Equation
Negative Dual Variable
Dominant Equation
10 Gas Transmission Pipe Line 0 1 4 Length, Diameter, Flow
Length
Pressure Ratio Factor
1 Four Variables
11 Profit Maximization 0 0 3
1 Cobb-Douglas Profit Function
12 Material Removal/Metal Cutting 0 1 2 Feed Rate, Cutting Speed 1 Fractional Exponents
13 Journal Bearing Design 1 1 2 Journal Radius,
Bearing Half-Length
1 Dimensional Analysis Approach
14 Metal Casting Hemispherical Top
Riser
2 1 2 Height, Diameter 1 Dimensional Analysis Approach
15 Liquefied Petroleum(LPG) Cylinder 1 1 2 Height, Diameter 2 Multiple Solutions
Dimensional Analysis Approach
16 Metal Cutting Economics-
2 Constraints
1 2 2 Feed Rate, Cutting Speed 3 Multiple Solutions
Derivative Approach
17 Open Cargo Shipping Box with Skids 1 0 3 Height, Width, Length 1 Derivative Approach
Dimensional Analysis Approach
Condensation of Terms Approach
18 Profit Maximization with Decreasing 1 0 2 Lot Size, Demand 1 Dimensional Analysis Approach
Cost Functions 1 1 3
Transformed Dual Approach

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