Transportation Problem

Unbalanced Transportation Problem

So far we have assumed that the total supply at the origins is equal to the total requirement at the destinations.
Specifically,
S_i = D_j
But in certain situations, the total supply is not equal to the total demand. Thus, the transportation problem with unequal supply and demand is said to be unbalanced transportation problem.

If the total supply is more than the total demand, we introduce an additional column, which will indicate the surplus supply with transportation cost zero. Similarly, if the total demand is more than the total supply, an additional row is introduced in the table, which represents unsatisfied demand with transportation cost zero. The balancing of an unbalanced transportation problem is illustrated in the following example.

Example

Solution:

The total demand is 1000, whereas the total supply is 800.
S_i < D_j
Total supply < total demand.
To solve the problem, we introduce an additional row with transportation cost zero indicating the unsatisfied demand.

Using matrix minimum method, we get the following allocations.

Plant	Warehouse			Supply
	W1	W2	W3
A		17		500
B	19			300
Unsatisfied demand		0	0	200
Demand	250	250	500	1000

Initial basic feasible solution

50 X 28 + 450 X 26 + 250 X 12 + 50 X 16 + 200 X 0 = 16900.