Chapter 2.7, Problem 7CFU

To determine

To find: the production number of each model for maximizing revenue, and if not possible to find then state whether this situation is infeasible, has alternate optimal solutions, or is unbounded.

Answer to Problem 7CFU

225 Explorers and 0 Grande expeditions

Explanation of Solution

Given information:

Cost of Explorer model =$250

Cost of Grande Expedition model =$350

Number of frames available =375

Time required for painting Explorer and Grande Expedition is 2 hours and 3 hours respectively.

Time available for production = 450 hours

Calculation:

Let the number of explorers and Grande expeditions be x and y respectively. The inequalities and revenue function derived from the data are as follows:

  $\begin{array}{l}x+y\le 375\\ 2x+3y\le 50\\ x\ge 0\\ y\ge 0\\ f\left(x,y\right)=250x+350y\end{array}$

The graph for the given inequalities is as follows:

  

The vertices of the polygon boundary are $\left(0,150\right),\left(225,0\right)$ and $\left(0,0\right)$ . Put these points in the $f\left(x,y\right)$ and check the minimum value as follows:

  $\begin{array}{c}f\left(0,150\right)=250\left(0\right)+350\left(150\right)\\ =52500\\ f\left(225,0\right)=250\left(225\right)+350\left(0\right)\\ =56250\\ f\left(0,0\right)=250\left(0\right)+350\left(0\right)\\ =0\end{array}$

On comparing these three values we can conclude that revenue is maximum when $x=225$i.e. 225 Explorers and $y=0$ , i.e. 0 Grande expeditions