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Optimization Solutions

Linear programming is the mathematical approach that optimizes a function.
Optimization is finding the minimum and maximum values of a function, which allows you to analyze problems and scenarios.
Specifically, the objective function $f (x, y) = x + y$ is used to determine the minimum and/or maximum values within the feasibility region.
The feasibility region is another way to reference the region where all possible solutions exist for a system of linear inequalities.
The x– and y–values that are substituted into the objective function come from the ordered pairs that form the vertices of the feasibility region on the graph.
A vertex is a point where two or more lines intersect.

Reading graphs and creating graphs accurately on the coordinate plane is critical for linear programming problems.

Example 1

Lucy is given the following graph and asked to find the minimum and maximum values using the objective function $f (x, y) = 3 x - y$ .

Plan
Name the vertices.

Substitute each vertex into the objective function and evaluate.

Determine the minimum and maximum values.

Implement
There are 4 vertices, so there will be 4 equations to evaluate using $f (x, y) = 3 x - y .$

$f (0, 0) = 3 (0) - (0) f (0, 0) = 0$

$f (0, 5) = 3 (0) - (5) f (0, 5) = - 5 minimum$

$f (6.5, 3.5) = 3 (6.5) - (3.5) f (6.5, 3.5) = 16 maximum$

$f (4, 1) = 3 (4) - (1) f (4, 1) = 11$

Note

You can use a calculator to evaluate the objective function so that more time can be spent on analyzing the values.

The vertices are named in a clockwise order in this level when possible.

Explain

The vertex with the minimum value is (0, 5) . The vertex with a maximum value is (6.5, 3.5) .

Example 2

Write a system of inequalities given the graph. Then use the objective function to find the minimum and maximum values.

Objective function: $f (x, y) = y - x$

Plan
Mark and name all of the vertices.

Find the slope and y-intercepts.

Determine if the shading is above(↑) or below(↓) the line.

Write an inequality for each of the 4 sides of the figure.

Note

Labeling the vertices with capital letters is not required. However, it may be helpful when determining all of the equations on a given coordinate plane.

Implement

Vertices: A (1, 1) B (0, 5)
C (2, 5) D (4, 4)

Note

The vertices can also be written directly on the coordinate plane.

Lines:

$A B m = - 4, b = 5, ↑ y \geq - 4 x + 5$

$B C m = 0, b = 5, ↓ y \leq 5$

$C D m = - \frac{1}{2}, b = 6, \begin{matrix} ↓ \end{matrix} y \leq - \frac{1}{2} x + 6$

$A D m = 1, b = 0, ↑ y \geq x$

Function: $f (x, y) = y - x$

$f (1, 1) = 1 - 1 f (1, 1) = 0 minimum$

$f (0, 5) = 5 - 0 f (0, 5) = 5 maximum$

$f (2, 5) = 5 - 2 f (2, 5) = 3$

$f (4, 4) = 4 - 4 f (4, 4) = 0 minimum$

Explain

This problem has two minimums at (1, 1) and (4, 4), and a maximum at (0, 5).