The Vogel's Approximation Method
Vogel's Approximation Method tackles the problem of finding a good initial solution by taking into account the costs associated with each alternative route, which is something that Northwest Corner Rule did not do.
To apply VAM, we must first compute for each row and column the penalty faced if the second best route is selected instead of the least cost route.
To illustrate the same, we will look at the Bengal Plumbing transportation problem.
We will take the following steps to find an initial solution via VAM method.
In the above table, the numbers at the heads of the columns and to the right of the rows represent the differences between two lowest shipping costs for the respective row or column.
For example in the Column E the three transportation costs are Rs.50, Rs.80 & Rs.90. The lowest costs are Rs.50 & Rs.80. The difference between these two is Rs.30.
Hence Rs.30 is the Opportunity Cost for Column E.
VAM Step 2 : Identify the row or column with the greatest Opportunity Cost, or difference. In this case the row or column selected is Column E with an Opportunity Cost of 30.
VAM Step 3 : Assign as many units as possible to the lowest cost route (A  E) with a per unit shipping cost of Rs.50. In this case 100 units are assigned to that square as anything more would exceed Plant (A)'s availability.
VAM Step 4 : Eliminate any row or column that has been completely satisfied by the assignment just made. This can be done by placing Xs in each appropriate square as shown below.
VAM Step 5 : Recompute the cost differences for the transportation table, excluding the rows or columns left satisfied or crossed out in the preceding step as shown below.
As a result of the elimination of the first row the Opportunity Costs for the rest of the rows and columns have changed.
We will now recompute the Opportunity Costs and assign as many units as possible to the row or column with the lowest unit shipping cost having the largest Opportunity Cost, and will cross out the unnecessary squares.
The largest Opportunity Cost in the current transportation table is 50 for Row B. The lowest per unit shipping cost is Rs.30. We can only assign 100 units to the route (B  G) to satisfy the capacity constraint of Plant (B). This is illustrated in the following table.
VAM Step 7 : The final two allocation (C  E) and (C  G) can be made by inspecting the supply restrictions (in rows) and demand requirements (in columns). We see that an assignment of 200 units has been agreed to route (C  E) and another assignment of 100 units has been agreed to (C  G) route, satisfying all the demands and supply constraints.
The total cost of this initial solution is Rs.39000.00 which is less than both the Northwest Corner Rule and the Intuitive Lowest Cost Method.
In addition to the North West Corner and Intuitive Lowest Cost Methods for setting an initial solution to transportation problems, we can use another important technique  Vogel's Approximation Method (VAM).
Though VAM is not quite as simple as Northwest Corner approach, but it facilitates a very good initial solution, one that is often the optimal solution.
Vogel's Approximation Method tackles the problem of finding a good initial solution by taking into account the costs associated with each alternative route, which is something that Northwest Corner Rule did not do.
To apply VAM, we must first compute for each row and column the penalty faced if the second best route is selected instead of the least cost route.
To illustrate the same, we will look at the Bengal Plumbing transportation problem.
Transportation Matrix for Bengal Plumbing
 
From \ To

Warehouse E

Warehouse F

Warehouse G
 
Plant A

Rs.50

Rs.40

Rs.30

100
 
Plant B

Rs.80

Rs.40

300
 
Rs.90

Rs.70

Rs.50

300
 
Warehouse Requirement

300

200

200

700

We will take the following steps to find an initial solution via VAM method.
VAM Step 1 : For each row and column of the transportation table, find the difference between the two lowest unit shipping costs as follows.
These figures represent the difference between the distribution cost of the best route and the second best route, or the per unit penalty for choosing the second best route instead of the lowest shipping route and hence are called the Opportunity Cost.
30

0

0


From \ To

Warehouse E

Warehouse F

Warehouse G

Factory Capacity


Plant A

Rs.50

Rs.40

Rs.30

100

10


Plant B

Rs.80

Rs.40

Rs.30

300

10


Plant C

Rs.90

Rs.70

Rs.50

300

20


Warehouse Requirement

300

200

200

700

In the above table, the numbers at the heads of the columns and to the right of the rows represent the differences between two lowest shipping costs for the respective row or column.
For example in the Column E the three transportation costs are Rs.50, Rs.80 & Rs.90. The lowest costs are Rs.50 & Rs.80. The difference between these two is Rs.30.
Hence Rs.30 is the Opportunity Cost for Column E.
VAM Step 2 : Identify the row or column with the greatest Opportunity Cost, or difference. In this case the row or column selected is Column E with an Opportunity Cost of 30.
VAM Step 3 : Assign as many units as possible to the lowest cost route (A  E) with a per unit shipping cost of Rs.50. In this case 100 units are assigned to that square as anything more would exceed Plant (A)'s availability.
VAM Step 4 : Eliminate any row or column that has been completely satisfied by the assignment just made. This can be done by placing Xs in each appropriate square as shown below.
30

0

0


From \ To

Warehouse E

Warehouse F

Warehouse G

Factory Capacity


Plant A

100

Rs.50

X

Rs.40

X

Rs.30

100

10

Plant B

Rs.80

Rs.40

Rs.30

300

10


Plant C

Rs.90

Rs.70

Rs.50

300

20


Warehouse Requirement

300

200

200

700

VAM Step 5 : Recompute the cost differences for the transportation table, excluding the rows or columns left satisfied or crossed out in the preceding step as shown below.
10

30

20


From \ To

Warehouse E

Warehouse F

Warehouse G

Factory Capacity


Plant A

100

Rs.50

X

Rs.40

X

Rs.30

100


Plant B

Rs.80

Rs.40

Rs.30

300

10


Plant C

Rs.90

Rs.70

Rs.50

300

20


Warehouse Requirement

300

200

200

700

As a result of the elimination of the first row the Opportunity Costs for the rest of the rows and columns have changed.
For example the Opportunity Cost for Column E has changed from 30 to 10.
VAM Step 6 : Return to Step 2 and repeat the steps until an initial feasible solution which satisfies the demands and supplies of all the rows and columns are obtained.
At the current scenario the largest Opportunity Cost for the transportation table is 30 for Column F. We will assign as many units as possible to the lowest cost route (B  E) with a per unit shipping cost of Rs.40. We will assign 200 units satisfying Warehouse (F)'s demand.
And will cross out the leftover square in Column F as it will surpass Warehouse (F)'s demand.
10

30

20


From \ To

Warehouse E

Warehouse F

Warehouse G

Factory Capacity


Plant A

100

Rs.50

X

Rs.40

X

Rs.30

100


Plant B

Rs.80

200

Rs.40

Rs.30

300

10


Plant C

Rs.90

X

Rs.70

Rs.50

300

20


Warehouse Requirement

300

200

200

700

We will now recompute the Opportunity Costs and assign as many units as possible to the row or column with the lowest unit shipping cost having the largest Opportunity Cost, and will cross out the unnecessary squares.
The largest Opportunity Cost in the current transportation table is 50 for Row B. The lowest per unit shipping cost is Rs.30. We can only assign 100 units to the route (B  G) to satisfy the capacity constraint of Plant (B). This is illustrated in the following table.
10

20


From \ To

Warehouse E

Warehouse F

Warehouse G

Factory Capacity


Plant A

100

Rs.50

X

Rs.40

X

Rs.30

100


Plant B

X

Rs.80

200

Rs.40

100

Rs.30

300

50

Plant C

Rs.90

X

Rs.70

Rs.50

300

40


Warehouse Requirement

300

200

200

700

VAM Step 7 : The final two allocation (C  E) and (C  G) can be made by inspecting the supply restrictions (in rows) and demand requirements (in columns). We see that an assignment of 200 units has been agreed to route (C  E) and another assignment of 100 units has been agreed to (C  G) route, satisfying all the demands and supply constraints.
From \ To

Warehouse E

Warehouse F

Warehouse G

Factory Capacity


Plant A

100

Rs.50

X

Rs.40

X

Rs.30

100

Plant B

X

Rs.80

200

Rs.40

100

Rs.30

300

Plant C

200

Rs.90

X

Rs.70

100

Rs.50

300

Warehouse Requirement

300

200

200

700

The total computed shipping cost according to the Vogel's Approximation Method.
Initial Feasible Transportation Route Selection based on VAM
Method


From

To

Units Shipped

Cost per unit

Total Cost

Plant A

Warehouse E

100

Rs.50

Rs.5000

Plant B

Warehouse F

200

Rs.40

Rs.8000

Plant B

Warehouse G

100

Rs.30

Rs.3000

Plant C

Warehouse E

200

Rs.90

Rs.18000

Plant C

Warehouse G

100

Rs.50

Rs.5000

Total Shipping Cost

Rs.39000

The total cost of this initial solution is Rs.39000.00 which is less than both the Northwest Corner Rule and the Intuitive Lowest Cost Method.
Even though VAM takes many more calculations to find an initial solution as compared to the other two methods, it almost produces a much better initial solution.
Hence VAM tends to minimize the total number of computations needed to reach an optimal solution.