Chapter 7, Problem 88RE

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 88RE

Minimum value of $z$ is 600 and it occurs at when $x=25$ and $y=50$

There is no maximum value of $z$ .

Explanation of Solution

Given:

Consider the objective function as

  $z=10x+7y$

Constraints

  $\begin{array}{l}x\ge 0\hfill \\ y\ge 0\hfill \\ 2x+y\ge 100\hfill \\ x+y\ge 75\hfill \end{array}$

Consider the objective function as

  $z=10x+7y$

Also, the above objective function subjects to the following constraints

  $\begin{array}{l}x\ge 0\hfill \\ y\ge 0\hfill \\ 2x+y\ge 100\hfill \\ x+y\ge 75\hfill \end{array}$

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

Quadrant |

Consider the equations

  $\begin{array}{l}2x+y=100\hfill \\ x+y=75\hfill \end{array}$

Plot these equations using MAPLE

Step1: Write the equations in the rectangular box as follows

  $x=0,y=0.2x+y=100andx+y=75$

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

The graphs of the equations $x=0,y=0.2x+y=100andx+y=75$ are shown below

  

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

Test objective function at each vertex.

At $\left(75,0\right)$

  $\begin{array}{l}z=10\left(75\right)+7\left(0\right)\hfill \\ =750+0\hfill \\ =750\hfill \end{array}$

At $\left(25,50\right)$

  $\begin{array}{l}z=10\left(25\right)+7\left(50\right)\hfill \\ =250+350\hfill \\ =600\hfill \end{array}$

At $\left(0,100\right)$

  $\begin{array}{l}z=10\left(0\right)+7\left(100\right)\hfill \\ =700\hfill \end{array}$

Hence, the minimum value of $z$ is 600 and it occurs at when $x=25$ and $y=50$

Also, for the above unbounded region, there is no maximum value of $z$ .