The Simplex Algorithm

The basic purpose of the simplex algorithm is to solve linear programming problems. In the following example, the function f (x, y) = x + y is to be maximized subject to the two inequalities shown. The function f (x, y) is the objective function, and the set of linear constraints is called the linear system.


$\blacktriangleright$ To enter a linear programming problem with two constraints

$\blacktriangleright$ Simplex + Maximize

$\left[\vphantom{
\begin{array}{c}
x+y \\
4x+3y\leq 6 \\
3x+4y\leq 4
\end{array}
}\right.$$\begin{array}{c}
x+y \\
4x+3y\leq 6 \\
3x+4y\leq 4
\end{array}$$\left.\vphantom{
\begin{array}{c}
x+y \\
4x+3y\leq 6 \\
3x+4y\leq 4
\end{array}
}\right]$, Maximum is at: $\left\{\vphantom{ x=%
\frac{12}{7},y=-\frac{2}{7}}\right.$x = ${\frac{{12}}{{7}}}$, y = - ${\frac{{2}}{{7}}}$$\left.\vphantom{ x=%
\frac{12}{7},y=-\frac{2}{7}}\right\}$


Of course, these are the same coordinates that minimize - x - y. In the following linear programming problem, click the matrix, and from the Simplex submenu choose Minimize.


$\blacktriangleright$ Simplex + Minimize

- x - y
4x + 3y≤6
3x + 4y≤4
, Minimum is at: $\left\{\vphantom{ y=-\frac{2}{7},x=\frac{12}{7}}\right.$y = - ${\frac{{2}}{{7}}}$, x = ${\frac{{12}}{{7}}}$$\left.\vphantom{ y=-\frac{2}{7},x=\frac{12}{7}}\right\}$