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\title{Math 242, Lecture 17}
\maketitle

\section{ Linear programming in applied problems}  

Linear programming, and our steps for finding maxima and minima,
are applicable in a number of kinds of applied
optimization problems.      As we practice these problems, we will
see that setting them up accurately can be more difficult than solving the
mathematics problem which arises,    which for bounded regions
is pretty straightforward given the steps we outlined last lecture.   

\begin{example}
You want to invest up to \$10,000 in the stock market.  Dorf shares sell for
\$50 each, yield a dividend of 5\% and have a risk index of 2.0 each.    
Cubik shares sell for \$25 each, yield a dividend of 8\% and have a risk index
of 3.0 each.      How many shares of each stock should you buy if you want
your dividends to be at least \$500 and you want to minimize your total risk
index?    
\end{example}

The mathematics of investments and finance can become highly complicated.  

\section{ Unbounded feasibility regions}   

As stated when we gave the steps for solving linear programming problems,
if the feasibility region is unbounded we need to check the values
of the objective function at boundary points which ``go to infinity''.   
More formally, we systematically approximate the unbounded feasibility
region by bounded feasibility regions which grow to fill the original
(as is best seen in  a picture).   

\begin{example}
Find the maxima and minima, if they exist, 
of the function $F(x,y) = 2x + 4y$ over the constraint region 
$\begin{cases} x + y \geq 2 \\ y \geq 0 \\ -x  + 2y \leq 4 \end{cases}$.  
Repeat the problem for the function $F(x,y) = 4x + 4y$.  
\end{example}

In the second half of this example, we see that 
%(for both unbounded and bounded feasibility regions)
an optimum value
can occur along all of the boundary points in a boundary line when,
the boundary line is parallel to the lines of constant value for the 
objective function.   

\begin{example}
Suppose that in a production facility the proportion of two components
is never allowed to be more than 2 to 1, so they are always ordered
together in less than 2 to 1 proportions to one another.     Suppose the first 
component costs \$20 each and the second \$30 each, and orders must be at
least \$500.  What is the minimum number of 
components which can be purchased?  
\end{example}


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