question_answer

An airline agrees to charter planes for a group. The group needs at least 160 first class seats and at least 300 tourist class seats. The airline must use at least two of its model 314 planes which have 20 first class and 30 tourist class seats. The airline will also use some of its model 535 planes which have 20 first class seats and 60 tourist class seats. Each flight of a model 314 plane costs the company Rs. 100000 and each flight of a model 535 plane costs Rs.150000. How many of each type of plane should be used to minimize the flight cost? Formulate this as a LPP.

Answer:

Suppose the airline uses x planes of model 314 and y planes of model 535.             \[\therefore \]    Cost\[=\text{ }100000\,\,x+\text{ }150000\text{ }y\]         Let Z denotes the profit. Then, \[Z=\text{ }100000x+\text{ }150000y\] and it is to be minimized. It is given that model 314 planes have 20 first class and 30 tourist class while model 535 have 20 first class and 60 tourist class seats. The group needs at least 160 first class seats and at least 300 tourist class seats. \[\therefore \]      \[20x+20y\ge 160\] \[\Rightarrow \] \[x+y\ge 8\] Also,     \[30x+60y\ge 300\] \[\Rightarrow \] \[x+2y\ge 10\] Finally, the number of planes cannot be negative. \[\therefore \]      \[x\ge 0,\] \[y\ge 0\] Thus, the mathematical formulation of the given LPP is as follows Minimize    \[Z=\text{ }100000x+\text{ }150000y\] Subject to constaints \[x+y\ge 8\] \[x+2y\ge 10\]             and       \[x,\,\,y\ge 0\]