Vogel’s Approximation Method

Vogel’s Approximation Method (VAM) is one of the methods used to calculate the initial basic feasible solution to a transportation problem. However, VAM is an iterative procedure such that in each step, we should find the penalties for each available row and column by taking the least cost and second least cost. In this article, you will learn how to find the initial basic feasible solution to a transportation problem such that the total cost is minimized.

​​Vogel’s Approximation Method Steps

Below are the steps involved in Voge’s approximation method of finding the feasible solution to a transportation problem.

Step 1: Identify the two lowest costs in each row and column of the given cost matrix and then write the absolute row and column difference. These differences are called penalties.

Step 2: Identify the row or column with the maximum penalty and assign the corresponding cell’s min(supply, demand). If two or more columns or rows have the same maximum penalty, then we can choose one among them as per our convenience.

Step 3: If the assignment in the previous satisfies the supply at the origin, delete the corresponding row. If it satisfies the demand at that destination, delete the corresponding column.

Step 4: Stop the procedure if supply at each origin is 0, i.e., every supply is exhausted, and demand at each destination is 0, i.e., every demand is satisfying. If not, repeat the above steps, i.e., from step 1.

The above procedure can be understood in a better way with the help of a solved example given below.

Vogel’s Approximation Method Solved Example

Question:

Solve the given transportation problem using Vogel’s approximation method.

Solution:

For the given cost matrix,

Total supply = 50 + 60 + 25 = 135

Total demand = 60 + 40 + 20 + 25 = 135

Thus, the given problem is balanced transportation problem.

Now, we can apply the Vogel’s approximation method to minimize the total cost of transportation.

Step 1: Identify the least and second least cost in each row and column and then write the corresponding absolute differences of these values. For example, in the first row, 2 and 3 are the least and second least values, their absolute difference is 1.

These row and column differences are called penalties.

Step 2: Now, identify the maximum penalty and choose the least value in that corresponding row or column. Then, assign the min(supply, demand).

Here, the maximum penalty is 3 and the least value in the corresponding column is 2. For this cell, min(supply, demand) = min(50, 40) = 40

Allocate 40 in that cell and strike the corresponding column since in this case demand will be satisfied, i.e., 40 – 40 = 0.

Step 3: Now, find the absolute row and column differences for the remaining rows and columns. Then repeat step 2.

Here, the maximum penalty is 3 and the least cost in that corresponding row is 3. Also, the min(supply, demand) = min(10, 60) = 10

Thus, allocate 10 for that cell and write down the new supply and demand for the corresponding row and column.

Supply = 10 – 10 = 0

Demand = 60 – 10 = 50

As supply is 0, strike the corresponding row.

Step 4: Repeat the above step, i.e., step 3. This will give the below result.

In this step, the second column vanishes and the min(supply, demand) = min(25, 50) = 25 is assigned for the cell with value 2.

Step 5: Again repeat step 3, as we did for the previous step.

In this case, we got 7 as the maximum penalty and 7 as the least cost of the corresponding column.

Step 6: Now, again repeat step 3 by calculating the absolute differences for the remaining rows and columns.

Step 7: In the previous step, except for one cell, every row and column vanishes. Now, allocate the remaining supply or demand value for that corresponding cell.

Total cost = (10 × 3) + (25 × 7) + (25 × 2) + (40 × 2) + (20 × 2) + (15 × 3)

= 30 + 175 + 50 + 80 + 40 + 45

= 420

Vogel’s Approximation Method Problems

1. Consider the transportation problem given below. Solve this problem by Vogel’s approximation method.

2. Find the solution for the following transportation problem using VAM.

3. Use Vogel’s Approximation Method to find a basic feasible solution for the following.

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