Question

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?

Answer 1

Let x passengers travel by executive class and y passengers travel by economy class. We construct the following table :

Class	Number of tickets	Profit(in Rs.)
Executive	x	1000x
Economy	y	600y
Total	x + y	1000x + 600y

So, our problem is to maximize Z = 1000x + 600y …(i) 

Subject to constraints x + y ≤ 200 …(ii) 

x ≥ 20 …(iii) y − 4x ≥ 0 ⇒ y ≥ 4x …(iv) 

x ≥ 0, y ≥ 0 …(v) 

Firstly, draw the graph of the line x + y = 200. 

Secondly, draw the graph of the line y = 4x 

Thirdly, draw the graph of the line x = 20 

On solving the equations, we get A(20, 80), B(40, 160) and C(20, 180). 

∴ Feasible region is ABCA.(See the below figure)

The corner points of the feasible region are A(20, 80), B(40, 160) and C(20, 180). The value of Z at these points are as follows:

Corner points	1000x + 600y
A(20, 80)	68000
B(40, 160)	136000 Maximum
C(20, 180)	128000

Thus, the maximum value of Z is 136000 at B(40, 160). 

Thus, 40 tickets of executive class and 160 tickets of economy class should be sold to maximize the profit and the maximum profit is Rs. 136000.