How many of circuits of Type A and of Type B, should be produced by the manufacturer so as to maximize his profit?

Question

A manufacturer of electronic circuits has a stock of 200 resistors, 120 transistors and 150 capacitors and is required to produce two types of circuits A and B. Type A requires 20 resistors, 10 transistors and 10 capacitors. Type B requires 10 resistors, 20 transistors and 30 capacitors. If the profit on type A circuit is Rs 50 and that on type B circuit is Rs 60, formulate this problem as a LPP so that the manufacturer can maximize his profit.

How many of circuits of Type A and of Type B, should be produced by the manufacturer so as to maximize his profit? Determine the maximum profit.

Answer 1

Let x units of type A and y units of type B electric circuits be produced by the manufacturer.

From the given information the below table is constructed:

Now, the total profit function in rupees Z = 50x + 60y is to be maximized with subject to the constraints

20x + 10y ≤ 200 … (i); 10x + 20y ≤ 120 … (ii)

10x + 30y ≤ 150 … (iii); x ≥ 0, y ≥ 0 … (iv)

Therefore, the required LPP is

Maximize Z = 50x + 60y subject to the constraints

20x + 10y ≤ 200 2x + y ≤ 20;

10x + 20y ≤ 120 x + 2y ≤ 12 and

10x + 30y ≤ 150 x + 3y ≤ 15, x ≥ 0, y ≥ 0.

Maximize Z = 50x + 60y subject to the constraints

20x + 10y ≤ 200 2x + y ≤ 20 … (i)

10x + 20y ≤ 120 x + 2y ≤ 12 … (ii)

10x + 30y ≤ 150 x + 3y ≤ 15 … (iv)

x ≥ 0, y ≥ 0 … (iv)

Now, let’s construct a constrain table for the above

Table for (i)

Next, solving equations (i) and (ii) we get,

x = 28/3, y = 4/3

So, the corner point is B(28/3, 4/3).

Solving equations (ii) and (iii) we get,

x = 6, y = 3 and the corner point is C(6, 3)

Lastly, solving equations (i) and (iii) we get,

x = 9, y = 2 (not included in the feasible region)

Here, OABCD is the feasible region.

Hence, the corner points are O(0, 0), A(10, 0), B(28/3, 4/3), C(6, 3) and D(0, 5).

Let us evaluate the value of Z

So here, the maximum profit is Rs 546.6 which is not possible for number of items in fraction.

Therefore, the maximum profit for the manufacture is Rs 480 at (6, 3) i.e. Type A = 6 and Type B = 3.