The third method for determining an initial solution, Vogel’s approximation model(VAM) is based on the concept of penalty cost or regret.

A penalty cost is the difference between the largest and next largest cell cost in a row. VAM allocates as much as possible to the minimum cost cell in the row or column with the largest penalty cost.

After the initial allocation is made, all of the penalty costs must be recomputed. In some cases the penalty costs will change; in other cases they will not change. For example, the penalty cost for column C in Table B-7 changed from $1 to $2 (because cell 2C is no longer considered in computing penalty cost), and the penalty cost in row 2 was eliminated altogether (because no more allocations are possible for that row).

Next, we repeat the previous step and allocate to the row or column with the highest penalty cost, which is now column B with a penalty cost of $3 (see Table B-7). The cell in column B with the lowest cost is 3B, and we allocate as much as possible to this cell, 100 tons. This allocation is shown in Table B-8.

Table B-9 also shows the recomputed penalty costs after the third allocation. Notice that by now only column C has a penalty cost. Rows 1 and 3 have only one feasible cell, so a penalty does not exist for these rows. Thus, the last two allocations are made to column C. First, 150 tons are allocated to cell 1C because it has the lowest cell cost. This leaves only cell 3C as a feasible possibility, so 150 tons are allocated to this cell. Both of these allocations are shown in Table B-10.

The total cost of this initial Vogel’s approximation model solution is $5,125, which is not as high as the northwest corner initial solution of $5,925. It is also not as low as the mini- mum cell cost solution of $4,550. Like the minimum cell cost method, VAM typically results in a lower cost for the initial solution than does the northwest corner method.

The steps of Vogel’s approximation model can be summarized in the following list.

1.Determine the penalty cost for each row and column by subtracting the lowest cell cost in the row or column from the next lowest cell cost in the same row or column.

2.Select the row or column with the highest penalty cost (breaking ties arbitrarily or choosing the lowest-cost cell).

3.Allocate as much as possible to the feasible cell with the lowest transportation cost in the row or column with the highest penalty cost.

4.Repeat steps 1, 2, and 3 until all rim requirements have been met.