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# An aeroplane can carry a maximum of 200 passengers. A profit of Rs 1000 is made on each executive class ticket and a profit of Rs 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximise the profit for the airline. What is the maximum profit?

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Toolbox:
• First formulate the objective function and identify the constraints from the problem statement, To solve a Linear Programming problem graphically, first plot the constraints for the problem. This is done by plotting the boundary lines of the constraints and identifying the points that will satisfy all the constraints.
• Let $R$ be the feasible region for a linear programming problem and let $Z=ax+by$ be the objective function.When $z$ has an optimum value (maximum or minimum),where variables $x$ and $y$ are subject to constraints described by linear inequalities,this optimum value must occur at a corner point of the feasible region.
• If R is bounded then the objective function Z has both a maximum and minimum value on R and each of these occur at corner points of R
Step 1:
Let the executive class air tickets and economy class tickets sold be $x$ and $y$
Now as the seating capacity of the aeroplane is 200,so $x+y \leq 200$
As 20 tickets for executive class are to be resolved ,So we have $x\geq 20$
And as the number of tickets of economy class should be at least 4 times that of executive class $y \geq 4x$
Profit on sale of $x$ tickets of executive class and $y$ tickets of economy class $Z=1000x+600y$
Therefore LPP is (i.e) maximize $Z=1000x+600y$ subject to constraints $x+y \leq 200,x \geq 20,y\geq 4x$ and $x,y\geq 0$
Step 2:
Now let us plot the lines on the graph .
$x=y=200,x=20$ and $y=4x$
The region satisfying the inequalities $x+y\leq 200,x\geq 20$ and $y\geq 4x$ is ABC and it is shown in the figure as the shaded portion.
Step 3:
$Z=100x+600y$
The corner points of the feasible region $a(20,180),B(40,160),C(20,80)$
The values of the objective function at these points are as follows:
At the Points $(x,y)$ the value of the objective function subject to $z=1000x+600y$
At $A(20,180)$,value of the objective function $Z=1000x+600y\Rightarrow 1000\times 20+600\times 180=20000+108000=128000$
At $A(40,160)$,value of the objective function $Z=1000x+600y\Rightarrow 1000\times 40+600\times 160=40000+96000=136000$
At $A(20,80)$,value of the objective function $Z=1000x+600y\Rightarrow 1000\times 20+600\times 80=20000+48000=68000$
Step 4:
It is clear that at $B(40,160)$ $Z$ has the maximum value.
Hence $x=40,y=160$
This implies 40 tickets of executive class and 160 of economy class should be sold to get the maximum profit of Rs.136000.
answered Aug 28, 2013