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 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) From/To A B D E F 6 3 2.50 4 2 3 How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost?

Answer 1

Let godown A transport x quintals of grain to shop D and godown B transport y quintals of grain to shop E.

The equations can be understood from figure given below,

The equations from the above diagram are given as,

x+y≤100 x≤60 y≤50 x+y≥60

We need to minimize the cost of transportation so we can use function which will minimize Z.

Minimize Z=6x+3y+2.50[ 100−x−y ]+4( 60−x )+2( 50−y )+3( x+y−60 ) Z=2.50x+1.50y+410

All constraints are,

x+y≤100 x+y≥60 y≤50,x≤60 x≥0,y≥0

x+y≤100

x	0	100
y	100	0

x+y≥60

x	0	60
y	60	0

The graph have the shaded region away from the origin.

Plot the points of all the constraint lines,

Substitute these points in the given objective function to find the maximum value of Z.

Corner points	Value of Z
( 10,50 )	510 (Minimum)
( 50,50 )	610
( 60,40 )	620
( 60,0 )	560

Thus, the amount of grain transported from A is 10,50,40 quintals to shops D,E,F respectively and from B is 50,0,0 quintals to shops D,E,F respectively and minimum cost is 510.