Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops D, E, and £ whose requirements are 60, 50 and 40 quintals respectively. The costs of transportation per quintal from the godowns to the shops are given in the following table:

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How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost?

Question

Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops D, E, and £ whose requirements are 60, 50 and 40 quintals respectively. The costs of transportation per quintal from the godowns to the shops are given in the following table:

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How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost?

Accepted Answer

.: Let 'x' quintals of grain be transported from Godown A to shop D, ‘y’ quintals to shop E.

Then, $100 - (x + y)$ quintals will be transported to shop F.

Thus, we have the table :

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Minimize : $Z = \cfrac{5}{2}x + \cfrac{3}{2}y + 410$

subject to constrains : $60 - x \ge 0 \Leftrightarrow x \le 60$ ….(1)

$50 - y \ge 0 \Leftrightarrow y \le 50$ ….(2)

$100 - (x + y) \ge 0 \Leftrightarrow x + y \le 100$ ….(3)

$x + y - 60 \ge 0 \Leftrightarrow x + y \ge 60$ ….(4)

And $x,y \ge 0$ ….(5)

${l_1}:x + y = 100;{l_2}:x + y = 60;{l_3}:x = 60;{l_4}:y = 50$

Let us graph the inequalities (1) to (5).

The shaded portion represents feasible region which is bounded.

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Let us evaluate Z at the corner points C(66, 0), E(60, 40), F(50, 50), G(10, 50).

Hence, minimum cost = Rs. 510 when from godown

A : 10 quintals ofgrain are sent to shop D,50

quintals to shop E and 40 quintals to shop F and from godown

B : 50 quintals are sent to shop D.

Linear Programming — Class 12 Maths Solution