Q. An aeroplane can carry a maximum of 200 passengers. $A$ profit of $₹ 1000$ is made on each executive class ticket and a profit of $₹ 600$ is made on each economy class ticket. The airline reserves atleast 20 seats for executive class. However, atleast 4 times as many passengers prefer to travel by economy class than by the executive class. Then, to maximise the profit, number of tickets of executive class and economy class are respectively
Linear Programming
Solution:
Let $x$ passengers travel by executive class and $y$ passengers travel by economy class. We construct the following table
Classes
Number of tickets
Profit (in ₹)
Executive
x
1000x
Economy
y
600y
Total
x+y
1000x+600y
So, our problem is to maximise
$Z=1000 x+600 y$ ...(i)
Subject to the constraints are
$x+y \leq 200$...(ii)
$x \geq 20 $...(iii)
$y-4 x \geq 0 \Leftrightarrow y \geq 4 x $...(iv)
$x \geq 0, y \geq 0$...(v)
Firstly, draw the graph of the line $x+y=200$.
x
0
200
y
200
0
Putting $(0,0)$ in the inequality $x+y \leq 200$, we have
$0+0 \leq 200 \Rightarrow 0 \leq 200$(which is true)
So, the half plane is towards the origin.
Secondary, draw the graph of the line $y=4 x$.
x
0
20
y
0
80
Putting $(10,0)$ in the inequality $y \geq 4 x$, we have
$0 \geq 4 \times 10 \Rightarrow 0 \geq 40$(which is false)
So, the half plane is towards $X$-axis.
Thirdly, draw the graph of the line $x=20$.
Putting $(0,0)$ in the inequality $x \geq 20$, we have
$0 \geq 20$(which is false)
So, the half plane is away from the origin.
Since, $ x, y \geq 0$
So, the feasible region lies in the first quadrant.
On solving the equations, we get $A(20,80), B(40,160)$ and $C(20,180)$.
$\therefore$ Feasible region is $A B C A$.
The corner points of the teasible region are $A(20,80)$, $B(40,160)$ and $C(20,180)$.
The values of $Z$ at these points are as follows
Corner points
$1000 x+600 y$
$A(20,80)$
68000
$B(40,160)$
$136000 \rightarrow$ Maximum
$C(20,180)$
128000
Thus, the maximum value of 7 is 136000 at $R(40,160)$. Thus, 40 tickets of executive class and 160 tickets of economy class should be sold to maximise the profit and the maximum profit is ₹ 136000 .
| Classes | Number of tickets | Profit (in ₹) |
|---|---|---|
| Executive | x | 1000x |
| Economy | y | 600y |
| Total | x+y | 1000x+600y |
| x | 0 | 200 |
| y | 200 | 0 |
| x | 0 | 20 |
| y | 0 | 80 |
| Corner points | $1000 x+600 y$ |
|---|---|
| $A(20,80)$ | 68000 |
| $B(40,160)$ | $136000 \rightarrow$ Maximum |
| $C(20,180)$ | 128000 |