7.An oil company has two depots A and B with capacities of 7000 L and 4000 L respectively. The company is to supply oil to three petrol pumps, D, E and F whose requirements are 4500L, 3000L and 3500L respectively. The distances (in km) between the depots and the petrol pumps is given in the following table:
Distance in (km.) | ||
From/To | A | B |
D | 7 | 3 |
E | 6 | 4 |
F | 3 | 2 |
Assuming that the transportation cost of 10 litres of oil is Re 1 per km, how should the delivery be scheduled in order that the transportation cost is minimum? What is the minimum cost?
Let x and y litres of oil be supplied from A to petrol pump,D and E. Then , (7000-x-y) will be supplied from A to petrol pump F.
Requirements at petrol pump D is 4500 L. since x L A are transported from depot A,remaining 4500-x L will be transported from petrol pump B
Similarly, (3000-y)L and 3500-(7000-x-y)=(x+y-3500) L will be transported from depot B to petrol E and F respectively.
The problem can be represented diagrammatically as follows:
and
and
Cost of transporting 10 L petrol =Re 1
Cost of transporting 1 L petrol
Total transportation cost z is given by ,
Mathematical formulation of given problem is as follows:
Minimize :
Subject to constraint ,
The feasible region determined by constraints is as follows:
The corner points of feasible region are
The value of Z at corner points is as shown :
corner points | ||
5000 | ||
5300 | ||
5550 | ||
4400 | minimum | |
5450 |
|
Hence , Z has miniimum value 4400 at point