Chapter1 - Experimental Optimum Engineering Design Course...

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Unformatted text preview: Experimental Optimum Engineering Design Course Notes Copyright 2004 Raphael T. Haftka Department of Mechanical and Aerospace Engineering University of Florida, Gainesville, Florida 1 2 Chapter 1 Formulation of Optimization Problems 1.1 Elementary Concepts Design problems are translated into mathematical optimization problems by defining several elements of the mathematical problem. The first is to specify design variables , which are variables that the designer can change in order to optimize his design. The second is objective functions which are figures of merit to be minimized or maximized. The third is constraint functions , which specify limits that must be satisfied by the design. We will use a simple example problem the design of a soft drink can to illustrate the concepts associated with the translation of a design problem into an optimization problem. Example 1.1.1 Soft-Drink Can Design: A soft-drink company has a new drink that they plan to market, and their preliminary research has determined that the cost to produce and distribute a cylindrical can is approximately given as C = 0 . 8 V o + 0 . 1 S , (1.1.1) where C is the cost in cents, V o is the volume in fluid ounces, and S is the surface area of the can in square inches.They also determined that they can sell the drink in cans ranging from 5 to 15 ounces, and then the price P in cents that can be charged for a can is estimated as P = 2 . 5 V o- . 02 V 2 o . (1.1.2) Based on their past experience they will consider only a can with diameter D between 1.5 and 3.5 inches, and their market research has shown that soft-drink cans have to have an aspect ratio of at least 2.0 to be easy to drink from. That is, the height H of the can has to be at least twice the diameter. However, they also found that cans with an aspect ratio of at least 2.2 look more sophisticated, and they can charge extra 5 cents per can. The company would like to maximize their profit from the sales of the soft drink, and they consider two measures of profit. One is the profit per can, and the other is the profit per ounce. The first measure is more useful if we assume that consumers will buy a fixed number of cans, and the other if we expect consumers to buy a fixed volume of drink. Let us consider the three elements that go into formulating this design example as a mathematical optimization problem. First the design variables, which should specify the shape of the can. In this 3 CHAPTER 1. FORMULATION OF OPTIMIZATION PROBLEMS case, the diameter D and the height H are natural design variables. However, we could use the radius of the can instead of the diameter, and conceivably even use the volume and surface area as design variables and calculate from them the diameter and aspect ratio of the can. In addition we have one binary design variable, indicating whether the can is made extra long or not. We can denote this variable by l (for long can), and have it assume the value of 1 if the can needs to be long, and 0 if it need not.long, and 0 if it need not....
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Chapter1 - Experimental Optimum Engineering Design Course...

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