3.2: Graphical Solutions

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Before giving a more general algorithm for handling this problem and problems like it, we note that when the number of variables is small (preferably 2), a graphical technique can be used.

Inequalities, such as the four given in Pablo's problem, are often called $\textit {constraints}$, and values of the variables that satisfy these constraints comprise the so-called $\textit {feasible region}$. Since there are only two variables, this is easy to plot:

Example $\PageIndex{1}$: Constraints and feasible region

Pablo's constraints are

\begin{eqnarray*}
&x\geq 5&\\
&y\geq 7&\\[2mm]
&15\leq x+y\leq25\, .&
\end{eqnarray*}

Plotted in the $(x,y)$ plane, this gives:

$Figure 1.png$

You might be able to see the solution to Pablo's problem already. Oranges are very sugary, so they should be kept low, thus $y=7$. Also, the less fruit the better, so the answer had better lie on the line $x+y=15$. Hence, the answer must be at the vertex $(8,7)$. Actually this is a general feature of linear programming problems, the optimal answer must lie at a vertex of the feasible region. Rather than prove this, lets look at a plot of the linear function $s(x,y)=5x+10y$.

Example $\PageIndex{2}$: The sugar function

Plotting the sugar function requires three dimensions:

$Figure 2.png$

The plot of a linear function of two variables is a plane through the origin. Restricting the variables to the feasible region gives some lamina in 3-space. Since the function we want to optimize is linear (and assumedly non-zero), if we pick a point in the middle of this lamina, we can always increase/decrease the function by moving out to an edge and, in turn, along that edge to a corner. Applying this to the above picture, we see that Pablo's best option is 110 grams of sugar a week, in the form of 8 apples and 7 oranges.

It is worthwhile to contrast the optimization problem for a linear function with the non-linear case you may have seen in calculus courses:

$linvsnonlin.jpg$

Here we have plotted the curve $f(x)=d$ in the case where the function $f$ is linear and non-linear. To optimize $f$ in the interval $[a,b]$, for the linear case we just need to compute and compare the values $f(a)$ and $f(b)$. In contrast, for non-linear functions it is necessary to also compute the derivative $\frac{\mathrm{d} f}{\mathrm{d} x}$ to study whether there are extrema inside the interval.

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David Cherney, Tom Denton, and Andrew Waldron (UC Davis)

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Example \(\PageIndex{1}\): Constraints and feasible region

Example \(\PageIndex{2}\): The sugar function

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