VAM (Vogel’s Approximation Method) is a heuristic algorithm used in transportation problems in operations research to find an initial feasible solution to the transportation problem, which involves minimizing the transportation cost of goods from several suppliers to several consumers.

Objective of the Transportation Problem:

The objective of a transportation problem is to determine the most cost-effective way of transporting goods from multiple suppliers to multiple consumers, while meeting demand and supply constraints.

VAM (Vogel’s Approximation Method) Overview:

VAM is a method to obtain an initial feasible solution by minimizing the penalty cost associated with transporting goods. The idea behind VAM is to calculate a “penalty” for each row and column of the transportation table, which represents the cost of not using the cheapest route in that row or column. The method helps in finding an approximate solution that can then be refined using other optimization techniques like MODI method.

Steps to Apply VAM:

1. Calculate the Penalties:
   For each row and each column in the transportation matrix, calculate the penalty cost, which is the difference between the two smallest costs in that row or column.

2. Find the Row/Column with the Highest Penalty:
   Identify the row or column with the highest penalty value. This indicates that the best cost-saving option comes from that row or column.

3. Allocate the Maximum Possible Units:
   In the row or column with the highest penalty, allocate as much as possible to the cell with the lowest cost. This allocation is subject to the supply and demand constraints (i.e., the supply of the supplier or the demand of the consumer must not be exceeded).

4. Adjust the Supply and Demand:
   After making the allocation, adjust the corresponding supply and demand by subtracting the allocated amount. If either the supply or demand for a row or column is completely satisfied (i.e., reduced to zero), cross out that row or column.

5. Repeat the Process:
   Repeat steps 1-4 until all the supply and demand constraints are satisfied (i.e., when all supplies and demands are zero). This will provide an initial feasible solution.

Example:

Let’s say there are 3 suppliers and 3 consumers with the following costs, supply, and demand:

Step 1: Calculate Penalties

• For Supplier 1:
  The two smallest costs are 6 and 8. The penalty is $8-6=2$.

• For Supplier 2:
  The two smallest costs are 4 and 7. The penalty is $7-4=3$.

• For Supplier 3:
  The two smallest costs are 2 and 4. The penalty is $4-2=2$.

• For Consumer 1:
  The two smallest costs are 4 and 6. The penalty is $6-4=2$.

• For Consumer 2:
  The two smallest costs are 5 and 6. The penalty is $6-5=1$.

• For Consumer 3:
  The two smallest costs are 2 and 4. The penalty is $4-2=2$.

Step 2: Find the Highest Penalty

The highest penalty is 3, which corresponds to Supplier 2.

Step 3: Allocate to the Lowest Cost Cell in Supplier 2

• The lowest cost in Supplier 2 is 4 (for Consumer 3), so allocate as much as possible to this cell. The demand of Consumer 3 is 35, and the supply of Supplier 2 is 30. Therefore, allocate 30 units to (Supplier 2, Consumer 3).

Step 4: Adjust the Supply and Demand

• After this allocation:
  • Supply for Supplier 2 becomes 0 (30 – 30 = 0).
  • Demand for Consumer 3 becomes 5 (35 – 30 = 5).

Step 5: Repeat the Process

• Now, repeat the process by recalculating penalties for the remaining unallocated cells, and then identify the next highest penalty.

Step 6: Calculate Total Cost.

• Calculate total cost by multiplying the fulfilled demand with fulfilled cost

Advantages of VAM:

• Efficient for Large Problems: VAM provides a quick and reasonably good initial solution for large transportation problems.
• Simple to Implement: The method is easy to understand and can be implemented with a few straightforward steps.
• Good Approximation: VAM generally provides a solution close to the optimal, which can then be refined using other methods like MODI.

Limitations of VAM:

• Not Always Optimal: VAM does not guarantee the optimal solution; it provides an initial feasible solution that can then be optimized.
• Heuristic Nature: VAM is a heuristic method, so the solution it provides may not always be the best possible one.

Conclusion:

Vogel’s Approximation Method (VAM) is a useful and efficient heuristic for finding an initial feasible solution to a transportation problem, helping to minimize the total transportation cost. Though it does not guarantee the optimal solution, it provides a good starting point for further optimization methods like the MODI method.