# How to solve this linear programming problem question?

We are started Linear programming problem question. Questions provided in examples are easy. And in exercise starting questions are easy to solve. As we have given conditions to form equations and solve them.

But this question little difficult.

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 maximize the profit for the airline. What is the maximum profit?

I tried it as let x passenger travel in executive class then 200-x travel in economy class.

But after then I am confused what to do. Can anyone please provide solution for this with explanation.

Let the airlines sells $x$ ticket for executive class and $y$ tickets for economy class.

Therefore we have to maximize $$Z=1000x+600y.....(1)$$ subject to the constraints,

$$x+y<=200.......(2)$$ $$x>=20.......(3)$$ $$y-4x>=0.......(4)$$ $$x,y>=0.......(5)$$ The feasible region by the constraints is as follows.

and the corner points are:: $A(20,80),B(40,160),C(20,180)$

• Can you please provide explanation – user404716 Jan 22 '17 at 12:27
• what explanation do you want? – Pushkar Soni Jan 22 '17 at 12:38
• which part is not clear to you? – Pushkar Soni Jan 22 '17 at 12:38
• How you find constraints – user404716 Jan 22 '17 at 14:11

Let the number of executive and economy class tickets sold be $x$ and $y$ respectively. Now as the seating capacity is $200$, so $x+y\leq 200$. Also, $x\geq 20$ as $20$ seats are reserved. And as the number of tickets of economy class should be at least $4$ times that of executive class, we have $y\geq 4x$.

Thus, profit on sale of $x$ tickets of executive class and $y$ tickets of economy class is $P=1000x+600y$

Therefore we maximize $$P=1000x+600y$$ subject to $$x+y\leq 200; x\geq 20; y\geq 4x; x,y\geq 0$$ Can you take it from here? Hope it helps.