VAM Methods

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Vogel’s Approximation Method (VAM) for Transportation Problems

Vogel’s Approximation Method (VAM) is a popular technique for finding an initial feasible solution to transportation problems. It is designed to reduce the total transportation cost by considering the penalty costs of not choosing the least cost routes, which helps in obtaining a solution that is closer to optimal.

Steps Involved in Vogel’s Approximation Method:

  • Calculate Penalties:
    • Penalties are calculated for each row and column. The penalty for a cell is the difference between the two smallest costs in that row or column. This represents the cost of not using the least-cost option.
    • For each row, calculate the penalty as follows: 
Penaltyrow=Cost​2nd lowest−Costlowest​
    • Similarly, calculate the penalty for each column.
  • Select the Highest Penalty:
    • Identify the row or column with the highest penalty. This indicates the most significant cost incurred by not selecting the least cost option in that row or column.
    • Allocate as much as possible to the cell with the lowest cost in this row or column while satisfying the supply and demand constraints.
  • Update the Matrix:
    • Adjust the supply and demand values based on the allocation.
    • If the supply or demand for a particular row or column is fully satisfied, delete that row or column from the matrix.
  • Repeat Until Complete:
    • Recalculate the penalties for the remaining rows and columns.
    • Repeat the process of selecting the highest penalty and allocating units until all supply and demand constraints are met.
  • Calculate Total Cost:
    • Compute the total transportation cost by multiplying the allocated units in each cell by the respective transportation cost and summing these values.

Example Problem and Solution:

Problem Statement:

  • Three sources: A, B, C with supplies of 50, 40, and 60 units respectively.
  • Three destinations: D, E, F with demands of 20, 95, and 35 units respectively.
  • The transportation costs are given in the matrix.

Transportation Cost Matrix:

D E F
A 3 4 2
B 3 1 4
C 4 6 5

Solution Steps Using VAM:

  • Calculate Penalties:
    • Row Penalties:
      • Row A: Penalty = 4 - 2 = 2
      • Row B: Penalty = 3 - 1 = 2
      • Row C: Penalty = 6 - 4 = 2
    • Column Penalties:
      • Column D: Penalty = 3 - 4 = -1
      • Column E: Penalty = 6 - 4 = 2
      • Column F: Penalty = 5 - 2 = 3
    • Highest Penalty: Column F (Penalty = 3)
  • Allocate Units Based on Highest Penalty:
    • Column F:
      • Allocate 35 units to cell AF (minimum cost in column F) to satisfy the demand for F. Now, supply at A is 15 units remaining, and demand at F is fully met.
    • Update Matrix:
D E F
A 3 4 2
B 3 1 4
C 4 6 5
    • Column F is now removed.
  • Recalculate Penalties for Remaining Rows and Columns:
    • Row Penalties:
      • Row A: Penalty = 4 - 2 = 2
      • Row B: Penalty = 3 - 1 = 2
      • Row C: Penalty = 6 - 4 = 2
    • Column Penalties:
      • Column D: Penalty = 3 - 4 = -1
      • Column E: Penalty = 6 - 4 = 2
    • Highest Penalty: Column E (Penalty = 2)
  • Allocate Units Based on Highest Penalty:
    • Column E:
      • Allocate 20 units to cell BE (minimum cost in column E). Now, supply at B is 20 units remaining, and demand at E is 75 units.
    • Update Matrix:
D E
A 3 4
B 3 1
C 4 6
    • Row A has been fully satisfied.
  • Continue Allocations:
    • Allocate remaining units in row B and column C, as the matrix is updated similarly.
    • Final Allocation:
D E
A 20 0
B 0 35
C 0 60
  • Calculate Total Cost:
  • Total Cost=(20×3)+(35×1)+(60×4)+(0×8) 
Total Cost=60+35+240+0=335

Key Points:

  • Penalties:
    • The penalty is the difference between the two smallest costs in each row or column, indicating the opportunity cost of not using the cheapest route.
  • Feasible Solution:
    • VAM provides an initial feasible solution, which is generally better than the North-West Corner Method or Least Cost Method.
  • Optimality:
    • VAM does not guarantee an optimal solution but is a good starting point for further optimization techniques.
  • Efficiency:
    • VAM efficiently handles the transportation problem by focusing on the cost differences and penalties, making it a practical choice for initial feasible solutions.

Vogel’s Approximation Method is advantageous for providing a better initial solution compared to simpler methods, helping in reducing the total cost of transportation while solving complex problems efficiently.