Chapter 11, Problem 95RE

To determine

Solve each linear programming problem.

Answer to Problem 95RE

  $\boxed{z=3at\left(1,0\right)}$

Explanation of Solution

Given information:

Solve each linear programming problem.

Minimize $z=3x+5y$ subject to $x\ge 0,y\ge 0,x+y\ge 1,3x+2y\le 12,x+3y\le 12$ .

Calculation:

Consider the function.

  $z=3x+5y$

To minimize the given objective function first graph the given inequalities and label the corner points.

Now evaluate the given function $z=3x+5y$ , and tabulate as follows,

  $\begin{array}{cc}\begin{array}{l}CornerPo\mathrm{int}\\ \left(x,y\right)\end{array}& \begin{array}{l}ValueOfTheObjectiveFunction\\ z=3x+5y\end{array}\\ \left(0,1\right)& z=3\left(0\right)+5\left(1\right)=5\\ \left(0,4\right)& z=3\left(0\right)+5\left(4\right)=20\\ \left(1,0\right)& z=3\left(1\right)+5\left(0\right)=3\\ \left(4,0\right)& z=3\left(4\right)+5\left(0\right)=12\\ \left(1.7,3.4\right)& z=3\left(1.7\right)+5\left(3.4\right)=22.1\end{array}$

The minimum value of the objective function is $3$ at the coordinates $\left(1,0\right)$ .

Hence $\boxed{z=3at\left(1,0\right)}$ .