Question

A firm manufactures two products; the net profit on product 1 is Birr 3 per unit and Birr 5 per unit on product 2. The manufacturing process is such that each product has to be processed in two departments D1 and D2. Each unit of product1 requires processing for 1 minute at D1 and 3 minutes at D2; each unit of product 2 requires processing for 2 minutes at D1 and 2 minutes at D2. Machine time available per day is 860 minutes at D1 and 1200 minutes at D2. How much of product 1 and 2 should be produced every day so that total profit is maximum. (solve with graphical method) Q3. Solve the following LP problem by graphical method: Maximise Z = 300X1 + 700X2 Subject to the constraints: X1 + 4X2 ≤ 20 2X1 + X2 ≤ 30 X1 + X2 ≤ 8 And X1, X2 ≥ 0 Q4.

Answer 1

Solution:

Let $x_1$ and $x_2$ be levels of production of two products, then

$Z=3 x_1+5 x_2$ is profit function and it should be maximized.

Subject to the constraints:

$x_1+2 x_2\le860;\\ 3 x_1+2 x_2\le1200;\\$

$x_1\ge0,x_2\ge0$

Corners points are (0,430), (170,345), (400,0).

At (0,430), $Z=3(0)+5 (430)=2150$

At (170,345), $Z=3(170)+5 (345)=2235$(Maximum)

At (400,0), $Z=3(400)+5 (0)=1200$

Hence, the optimal solution to the given LP problem is : x₁=170,x₂=345 and max Z=2235.

2. Maximise $Z = 300x_1 + 700x_2$ Subject to the constraints:

$x_1 + 4x_2 ≤ 20 \\2x_1 + x_2 ≤ 30 \\x_1 + x_2 ≤ 8 \\x_1, x_2 ≥ 0$

Corner points are (0,5), (4,4), (8,0).

At (0,5), $Z = 300(0) + 700(5)=3500$

At (4,4), $Z = 300(4) + 700(4)=4000$(maximum)

At (8,0), $Z = 300(8) + 700(0)=2400$

Thus, at $x_1=4,x_2=4, Z=4000$ is the maximum value.