A manufacturing unit makes two models, 'classic' and 'supreme' of the scooter. Each piece of the classic model requires 9 labour hours for assembling and 1 labour hour for finishing. Each piece of supreme model requires 12 labour hours for assembling and 3 labour hour for finishing. For assembling and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of Rs. 10000 on each piece of the classic model and Rs. 15000 on each piece of the supreme model. Which of the following options describes the given linear Programming Problem (LPP) to maximize the profit Z (Max Z) (where x and y are the number of pieces of the classic model and the supreme model respectively)?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → Max \(Z = 10000x + 15000y\), subject to following constraints \(3x + 4y \le 60, x + 3y \le 30, x \ge 0, y \ge 0\).
With x classic and y supreme scooters, the profit in rupees is
\(Z = 10000x + 15000y\)
The hours used cannot exceed the hours available. Assembling (9 and 12 hours per piece, 180 hours available):
\(9x + 12y \le 180\)
Dividing both sides by 3:
\(3x + 4y \le 60\)
Finishing (1 and 3 hours per piece, 30 hours available):
\(x + 3y \le 30\)
The numbers of scooters cannot be negative, so \(x \ge 0,\ y \ge 0\). This is Option (4).
Why the other options are incorrect
Option (1) Max \(Z = 10x + 15y\), subject to following constraints \(3x + 4y \le 180, x + 3y \le 30, x \ge 0, y \ge 0\). — Incorrect. The profit is Rs. 10000 and Rs. 15000 per piece, not 10 and 15; and when \(9x + 12y \le 180\) is divided by 3, the right side becomes 60, not 180.
Option (2) Max \(Z = 10000x + 15000y\), subject to following constraints \(9x + y \le 180, 12x + 3y \le 30, x \ge 0, y \ge 0\). — Incorrect. The coefficients are mixed up: assembling needs 9x + 12y hours and finishing x + 3y hours.
Option (3) Max \(Z = 10000x + 15000y\), subject to following constraints \(3x + 4y \ge 180, 12x + 3y \ge 30, x \ge 0, y \ge 0\). — Incorrect. The hours are maximum limits, so the constraints must be “≤”; the right side 180 and the finishing coefficients are also wrong.