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?

Answer 1

HINT: - In linear programming, we try to get a required region that is enclosed by the constraints that we form by reading the question and then try to maximize a statement that we only form by reading the question itself.
It is a mathematical tool to improve the production by industrial units and to extract more profits.

Complete step-by-step solution -
Let the number of executive class tickets sold be x and the number of economy class tickets sold be y.
We have to maximize the profit for the airline.
Clearly,
$x\ge 0\ and\ y\ge 0$
(This is because the number of both the tickets sold cannot be negative)

As mentioned in the question, we have only 12 hours of machine, we will use the following constraints:-
$x+y\le 200~~\ldots \left( a \right)$
(Because the maximum number of passengers in the airlines can be maximum 200)

& 4x\le y~ \\\ & y-4x\ge 0\ \ \ \ \ldots \left( b \right) \\\ \end{aligned}$$ (As mentioned in the question) $$x\ge 20\ \ \ \ \ ...(c)$$ (Because at least 20 tickets are reserved for executive class) The profit on the sale of an economy ticket is Rs.600 per ticket sold and the profit on the sale of an executive ticket is Rs. 1000 per ticket sold. We need to maximize 1000x+600y. Hence, z=1000x+600y (This is the maximizing statement) We can see that the applicable region is bounded and in the first quadrant. On solving the equations, we get the corner points as following which are the intersection points of the equations that we have formed using the information that has been provided in the question. Therefore, required points are the corner points. A (20, 180) B (40, 160) C (20,80)  Hence, on plugging in these points in the maximizing statement, we get that Maximum profit will be Rs.136000, when number of executive class tickets sold is 40 and number of economy class tickets sold is 160. Because the value of z that we will get on plugging the different points, we get A (20, 180) $$\begin{aligned} & z=1000\times 20+600\times 180 \\\ & z=20000+108000=128000 \\\ \end{aligned}$$ B (40, 160) $$\begin{aligned} & z=1000\times 40+600\times 160 \\\ & z=40000+96000=136000 \\\ \end{aligned}$$ C (20,80) $$\begin{aligned} & z=1000\times 20+600\times 80 \\\ & z=20000+48000=68000 \\\ \end{aligned}$$ NOTE: - The students can make a mistake if he or she is not aware of the information about linear programming that in linear programming, we try to get a required region that is enclosed by the constraints that we form by reading the question and then try to maximize a statement that we only form by reading the question itself.