First Example: Modeling (1.4) and

Practical Optimization (3.5)

An agent wants to maximize (or minimize) a certain variable, which we call the target variable. He or she does this by choosing the value for some so-called choice variable. There may be more variables involved, we call these the other variables. Consider the following example:

A rectangular area of 5,000 square yards is to be build and fenced on three sides. The total length of the fence should be kept as small as possible.

Choice variable: x: the length of the unfenced side.

other variable: z: the length of the other side of the rectangle.

target variable: the length of the fence: f.

We need to express the target variable in terms of the choice and other variables.

in our example: f=x+2z

Now, since the target variable should become a function of the choice variable alone, every other variable occurring in the above equation must be expressed in terms of the choice variable and replaced. Usually we need as many equations between the variables as we have "other" (non-target and non-choice) variables. Ideally each such equation is just an equation between such an "other" variable and the choice variable, but it could be more complicated.

We know that the area equals 5000

xz=5000

z = 5000/x

Plugging this into the equation above yields f=x+10000/x

Now we have a functional relation between choice and target variable. We differentiate this function with respect to choice variable, set it equal to 0, find the critical points, and so on.

We have the function f(x)=x+10000/x

It's graph is shown to the right, and its minimum seems to be achieved around x = 100.

We differentiate to get

f'(x)=1-10000/x².

The equation f'(x)=0 leads to x²=10000 or x = 100.

Look for another example

Erich Prisner, November 2004