Answer to Question #98124 in Operations Research for Ahamd Rajpoot

Question

Suppose PIA is offering a new route: from Lahore to New York. The aircraft on this flight has a total of 280 seats. These seats can be converted into two categories: Business Class or Economy class, before flight schedule, depending on passengers buying any particular class of ticket. For the Lahore-New York flight to be profitable, PIA must sell a minimum of 80 Business class tickets and a minimum of 100 Economy class tickets. However, PIA does not want to have more than 150 seats in economy class to promote business class travelling. The airline earns a profit of $150 for each Business Class ticket and $100 for each Economy class ticket. How many of each category of ticket should be sold in order to MAXIMIZE total profit from of a flight. Use linear programming in following steps:
1. Prepare a mathematical model for this problem
2. Plot the problem conditions on a GRAPH PAPER accurately.

Answer 1

Solution:

First make up the function of the goal

$$f(x,y)=150x+100y→max$$

f(x,y)=150x+100y→max

Where x - Business class tickets,

           y- Economy class tickets.

Then a system of restrictions is compiled according to the submitted task text:

$$x≥80$$

$$100≤y≤150$$

$$x+y=280$$

$$x≥0$$

$$y≥0$$

Find the gradient of the objective function

$$(grad f(x,y) ) ⃗=(∂f/∂x;∂f/∂y)=(150;100)$$

 Graphic image for this task:

The solution to the problem will be the coordinates of the rightmost point of the hatched area through which the level line passes, perpendicular to the line containing the gradient of the function.

Answer: 180 - Business class tickets,

           100 - Economy class tickets.