6. Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops, D, E and F whose requirements are 60, 50 and 40 quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table:
|Transportation cost per quintal (in Rs)|
How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost?
Let godown A supply x and y quintals of grain to shops D and E respectively. Then , (100-x-y) will be supplied to shop F. Requirements at shop D is 60 since godown A supply x .Therefore remaining (60-x) quintals of grain will be transported from godown B.
Similarly, (50-y) quintals and 40-(100-x-y)=(x+y-60) will be transported from godown B to shop E and F respectively. The problem can be represented diagrammatically as follows:
Total transportation cost z is given by ,
Mathematical formulation of given problem is as follows:
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 :
therefore 510 may or may not be minimum value of Z .
Hence , Z has miniimum value 510 at point