Question

Two godowns A and B have grains capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops D, E, and F whose requirement are 60, 50 and 40 quintals respectively. The cost of transportation per quintal from the godowns to the shops is given in the following table; Transportation cost per quintal(in Rs.)

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

Answer 1

Express the problem diagrammatically as shown above. The total transportation cost is given by

Z = 6x + 3y + 2.5{100 – (x + y)} + 4(60 – x) + 2(50 – y) + 3(-60 + (x + y))

⇒ Z = 2.5x + 1.5y + 410

100 – (x + y) ≥ 0 ⇒ x + y ≤ 100

60 – x ≥ 0 ⇒ x ≤ 60

50 – y ≥ 0 ⇒ y ≤ 50 – 60 + x + y ≥ 0 ⇒ x + y ≥ 60

Then the given LPP is

Minimise; Z = 2.5x + 1.5y + 410

x + y ≤ 100, x + y ≥ 60

0 ≤ x ≤ 60, 0 ≤ y ≤ 50

In the figure the shaded region ABCD is the feasible region. Here the region is bounded. The corner points are

A(60, 0), B(60, 40), C(50, 50), D(10, 50).

Given; Z = 2.5x + 1.5y + 410

Corner points	Value of Z
A	Z = 2.5(60) + 1.5(0) +410 = 560
B	Z = 2.5(60) + 1.5(40) + 410 = 620
C	Z = 2.5 (50) +1.5(50) +410 = 610
D	Z = 2.5(10) + 1.5(50) +410 = 510

Since minimum value of Z occurs at D, the soluion is Z = 510.