# Vogel Approximation Method

The Vogel approximation method (Unit cost penalty method) is an iterative procedure for computing a basic feasible solution of a transportation problem.

This method is preferred over the two methods discussed in the previous sections, because the initial basic feasible solution obtained by this method is either optimal or very close to the optimal solution.

This method is a little complex than the previously discussed methods. So go slowly and reread the explanation atleast twice.

## Steps in Vogel Approximation Method (VAM)

The standard instructions are paraphrased below:

1. Identify the boxes having minimum and next to minimum transportation cost in each row and write the difference (penalty) along the side of the table against the corresponding row.
2. Identify the boxes having minimum and next to minimum transportation cost in each column and write the difference (penalty) against the corresponding column
3. Identify the maximum penalty. If it is along the side of the table, make maximum allotment to the box having minimum cost of transportation in that row. If it is below the table, make maximum allotment to the box having minimum cost of transportation in that column.
4. If the penalties corresponding to two or more rows or columns are equal, you are at liberty to break the tie arbitrarily.
5. Repeat the above steps until all restrictions are satisfied.

Example-1, Example-2

#### Example 1

Consider the transportation problem presented in the following table:

Destination
Origin 1 2 3 4 Supply
1 20 22 17 4 120
2 24 37 9 7 70
3 32 37 20 15 50
Demand 60 40 30 110 240

Solution.

Calculating penalty for table 1

17 - 4 = 13, 9 - 7 = 2, 20 - 15 = 5
24 - 20 = 4, 37 - 22 = 15, 17 - 9 = 8, 7 - 4 = 3

Table 1

On small screens, scroll horizontally to view full calculation

Destination
Origin 1 2 3 4 Supply Penalty
1 20 17 4 120  80 13
2 24 37 9 7 70 2
3 32 37 20 15 50 5
Demand 60 40 30 110 240
Penalty 4 15 8 3

The highest penalty occurs in the second column. The minimum cij in this column is c12 (i.e., 22). So x12 = 40 and the second column is eliminated. The new reduced matrix is shown below:

Now again calculate the penalty.

Table 2

Origin 1 3 4 Supply Penalty
1 20 17 80 13
2 24 9 7 70 2
3 32 20 15 50 5
Demand 60 30 110
Penalty 4 8 3

The highest penalty occurs in the first row. The minimum cij in this row is c14 (i.e., 4). So x14 = 80 and the first row is eliminated. The new reduced matrix is shown below:

Table 3

Origin 1 3 4 Supply Penalty
2 24 7 70 2
3 32 20 15 50 5
Demand 60 30 30
Penalty 8 11 8

The highest penalty occurs in the second column. The minimum cij in this column is c23 (i.e., 9). So x23 = 30 and the second column is eliminated. The reduced matrix is given in the following table.

Table 4

Use Horizontal Scrollbar to View Full Table Calculation

Origin 1 4 Supply Penalty
2 40 17
3 15 50 17
Demand 60 30
Penalty 8 8

The following table shows the computation of penalty for various rows and columns.

### Final table of Vogel Approximation Method

Destination
Origin 1 2 3 4 Supply Penalty
1 20 17 120 13 13 - - - -
2 37 70 2 2 2 17 24 24
3 37 20 15 50 5 5 5 17 32 -
Demand 60 40 30 110 240
Penalty 4 15 8 3
4 - 8 3
8 - 11 8
8 - - 8
8 - - -
24 - - -

#### Initial basic feasible solution

22 X 40 + 4 X 80 + 24 X 10 + 9 X 30 + 7 X 30 + 32 X 50 = 3520.