Step-by-step explanation:
The Vogel's Approximation Method (VAM) is a method for obtaining an initial basic feasible solution to a transportation problem. The basic idea of the method is to determine the most critical cell, that is, the cell corresponding to the largest difference in the demand and supply of a row or a column, and allocate as much as possible to this cell. The allocation is then repeated for the next most critical cell until the supply and demand are balanced.
Here's the step-by-step process to obtain the initial basic feasible solution using Vogel's Approximation Method:
Calculate the difference between the largest and the second largest supply and demand values in each row and column.
For Row A: 5-3 = 2
For Row B: 5-3 = 2
For Row C: 6-4 = 2
For Row D: 4-2 = 2
For Column D1: 5-3 = 2
For Column D2: 3-4 = -1
For Column D3: 5-4 = 1
For Column D4: 4-3 = 1
Identify the row or column with the largest difference and allocate the maximum amount possible to the cell with the largest cost in that row or column.
The largest difference is 2 which is in Row A, B, C and D. Choose any one of them randomly, say Row A. The largest cost in Row A is in Column D1 with 5 units.
Repeat the process until all demand is met.
Now, we update the demand and supply:
Row A: 5 units (34-5=29)
Column D1: 3 units (21-3=18)
Next largest difference:
For Row B: 5-3 = 2
For Row C: 6-4 = 2
For Row D: 4-2 = 2
For Column D2: 4-3 = 1
For Column D3: 5-4 = 1
For Column D4: 4-3 = 1
The largest difference is 2 which is in Row B, C, and D. Choose any one of them randomly, say Row B. The largest cost in Row B is in Column D3 with 4 units.
Now, we update the demand and supply:
Row B: 4 units (15-4=11)
Column D3: 4 units (17-4=13)
And so on, until all the demand is met.
The final allocation will be the initial basic feasible solution.