A consultancy firm is preparing an office trip for its $400$ employees. The bus rental company has $2$ options for the buses that they can offer along with a maximum of $9$ drivers. The capacity and rental costs are shown below.
|
Bus type |
Maximum capacity |
Rental cost (in ₹) |
|
Large |
$50$ |
$3000$ |
|
Medium |
$40$ |
$2000$ |
Determine the number of buses that the consultancy firm should book to minimise the total rental cost. Sketch the feasible region and show your steps.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → $x = 4$ Large, $y = 5$ Medium ##
Let the number of large buses be $x$ and medium buses be $y$. We write the LPP as:
Minimise $Z = 3000x + 2000y$
Subject to the constraints:
$\begin{aligned} 50x + 40y \ge 400 \\ x + y \le 9 \\ x \ge 0, y \ge 0 \end{aligned}$
The objective function value at the $3$ corners of the feasible region is:
|
Vertex |
Objective function value $Z=3000x+2000y$ |
|
$(4, 5)$ |
$₹22000$ (Minimum) |
|
$(8, 0)$ |
$₹24000$ |
|
$(9, 0)$ |
$₹27000$ |
Conclude that the consultancy firm should book $4$ large buses and $5$ medium buses to minimize cost.