Question

A manufacturing company makes two models A and B of a product. Each piece of Model A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each piece of Model B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of Rs 8000 on each piece of model A and Rs 12000 on each piece of Model B. How many pieces of Model A and Model B should be manufactured per week to realise a maximum profit? What is the maximum profit per week?

Answer 1

Let x pieces of model A and y pieces of model B be manufactured.

$\mathrm{Clearly}x,y\ge 0$

The given information can be tabulated as follows:

	Fabricating(hrs)	Finishing(hrs)
Model A	9	1
Model B	12	3
Availability	180	30

Therefore, the constraints are

$$\begin{gathered} 9x+12y\le 180 \\ x+3y\le 30 \end{gathered}$$

The company makes a profit of Rs 8000 on each piece of model A and Rs 12000 on each piece of Model B. Therefore, profit gained from x pieces of model A and y pieces of model B be Rs 8000x and Rs 12000y respectively.
Total profit = Z = 8000x + 12000y

​Thus, the mathematical formulation of the given LPP is

Max Z = 8000x + 12000y
subject to

$$\begin{gathered} 9x+12y\le 180 \\ x+3y\le 30 \end{gathered}$$

First we will convert inequations into equations as follows:9
9x + 12y = 180, x + 3y = 30, x = 0 and y = 0

Region represented by 9x + 12y ≤ 180:
The line 9x + 12y = 180 meets the coordinate axes at A₁(20, 0) and $B_1\left(0,15\right)$ respectively. By joining these points we obtain the line 9x + 12y = 180. Clearly (0,0) satisfies the 9x + 12y = 180. So, the region which contains the origin represents the solution set of the inequation 9x + 12y ≤ 180.

Region represented by x + 3y ≤ 30:
The line x + 3y = 30 meets the coordinate axes at C₁(30, 0) and $D_1\left(0,10\right)$ respectively. By joining these points we obtain the line x + 3y = 30. Clearly (0,0) satisfies the inequation x + 3y ≤ 30. So,the region which contains the origin represents the solution set of the inequation x + 3y ≤ 30.

Region represented by x ≥ 0 and y ≥ 0:
Since, every point in the first quadrant satisfies these inequations. So, the first quadrant is the region represented by the inequations x ≥ 0, and y ≥ 0.
The feasible region determined by the system of constraints 9x + 12y ≤ 180, x + 3y ≤ 30, x ≥ 0 and y ≥ 0 are as follows.




The corner points are O(0, 0), D₁(0, 10), E₁(12, 6) and A₁(20, 0).

The value of the objective function at the corner points

Corner points	Z = 8000x + 12000y
O	0
D1	120000
E1	168000
A1	160000

The maximum value of Z is 168000 which is attained at E₁$\left(12,6\right)$.

Thus, the maximum profit is Rs 168000 which is attained when 12 units of model A and 6 units of model B.