Chapter 11.8, Problem 16AYU

In Problems 9-18, solve each linear programming problem.

Maximize $z=2x+3y$ subject to $x\ge 0,y\ge 0,x+y\ge 3,x+y\le 9,x+3y\ge 6$

To determine

To solve: The given linear programming problem.

Answer to Problem 16AYU

Solution:

The minimum value of $z$ is $7.5$, and it occurs at the point $\left(3/2,3/2\right)$.

Explanation of Solution

Given:

$\mathrm{minimize}z=2x+3y$

Subject to

$x\ge 0$

$y\ge 0$

$x+y\ge 3$

$x+y\le 9$

$x+3y\ge 6$

Calculation:

The graph of this system (the set of feasible points) is shown as shaded region in figure below. The corner points have also been plotted.

The below table lists the corner points and the corresponding values of objective function.

Corner points are	Value of objective function
$\left(0,3\right)$	9
$\left(6,0\right)$	12
$\left(9,0\right)$	18
$\left(0,9\right)$	27
$\left(3/2,3/2\right)$	$15/2=7.5$

From the table, the minimum value of $z$ is $7.5$, and it occurs at the point $\left(3/2,3/2\right)$.