Let required number of first class and economy class tickets be x and y respectively.
Each ticket of first class and economy class make profit of Rs 400 and Rs 600 respectively.
So, x ticket of first class and y tickets of economy class make profit of Rs 400x and Rs 600y respectively.
Let total profit be Z = 400x + 600y
Given, aeroplane can carry a minimum of 200 passengers, so,
x + y ≤ 200
Given airline reserves at least 20 seats for first class, so,
x ≥ 20
Also, at least 4 times as many passengers prefer to travel by economy class to the first class, so
y ≥ 4x
Hence the mathematical formulation of the LPP is
Max Z = 400x + 600y
Subject to constraints
[since seats in both the classes can not be zero]
Region represented by x + y ≤ 200: the line x + y = 200 meets the axes at A(200, 0), B(0, 200). Region containing origin represents x + y ≤ 200 as (0, 0) satisfies x + y ≤ 200.
Region represented by x ≥ 20: line x = 20 passes through (20,0) and is parallel to y axis. The region to the right of the line x = 20 will satisfy the inequation x ≥ 20
Region represented by y 4x: line y = 4x passes through (0,0). The region above the line y = 4x will satisfy the inequation y 4x
Region x, y ≥ 0: it represents the first quadrant.
The corner points are C(20, 80), D(40, 160), E(20, 180).
The values of Z at these corner points are as follows:
The maximum value of Z is attained at E(20, 180).
Thus, the maximum profit is Rs 116000 obtained when 20 first class tickets and 180 economy class tickets are sold.