Chapter 7, Problem 90RE

To determine

To find: plot graph and find minimum and maximum values of objective function and points where they occur subject to constraints.

Answer to Problem 90RE

The minimum value of $z$ is 0 and it occurs at when $x=0$ and $y=0$

The maximum value of $z$ is 60000 and it occurs at when $x=500$ and $y=500$

Explanation of Solution

Given:

Consider the objective function as

  $z=50x+70y$

Constraints

  $\begin{array}{l}x\ge 0\hfill \\ y\ge 0\hfill \\ x+2y\le 1500\hfill \\ 5x+2y\le 3500\hfill \end{array}$

Consider the objective function as

  $z=50x+70y$

Also, the above objective function subjects to the following constraints

  $\begin{array}{l}x\ge 0\hfill \\ y\ge 0\hfill \\ x+2y\le 1500\hfill \\ 5x+2y\le 3500\hfill \end{array}$

The inequalities $x\ge 0,y\ge 0$ are sign constraints which implies that the feasible region lies in

Quadrant |

Consider the equations

  $\begin{array}{l}x+2y=1500\\ 5x+2y=3500\end{array}$

Plot these equations using MAPLE

Step1: Write the equations in the rectangular box as follows

  $x=0,y=0,x+2y=1500and\text{ 5}x+\text{ 2}y=\text{ 3}500$

Step2: Right click on the equation, go to plot and click on 2-D- Implicit plot

The graphs of the equations $x=0,y=0,x+2y=1500and\text{ 5}x+\text{ 2}y=\text{ 3}500$ are shown below

  

The region bounded by the constraints is shown in the above plot.

Test objective function at each vertex.

At $\left(0,750\right)$

  $\begin{array}{l}z=\text{ 5}0\left(0\right)+70\left(750\right)\hfill \\ =52500\hfill \\ \hfill \end{array}$

At $\left(0,0\right)$

  $\begin{array}{l}z=\text{ 5}0\left(0\right)+70\left(0\right)\hfill \\ =0\hfill \\ \hfill \end{array}$

At $\left(500,500\right)$

  $\begin{array}{l}z=\text{ 5}0\left(500\right)+70\left(500\right)\hfill \\ =25000+35000\hfill \\ =60000\hfill \end{array}$

At $\left(700,0\right)$

  $\begin{array}{l}z=\text{ 5}0\left(700\right)+70\left(0\right)\hfill \\ =35000\hfill \\ \hfill \end{array}$

Hence, the minimum value of $z$ is 0 and it occurs at when $x=0$ and $y=0$

Also, the maximum value of $z$ is 60000 and it occurs at when $x=500$ and $y=500$