Least Cost and VAM Method

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Here’s a breakdown of the Least Cost Method (LCM) and Vogel’s Approximation Method (VAM) used to find an initial basic feasible solution for transportation problems in linear programming:

Least Cost Method (LCM):

This method prioritizes assigning shipments to cells with the lowest transportation cost, aiming to find a more cost-effective initial solution compared to the North-West Corner Method. Here’s how it works:

  1. Identify the Lowest Cost Cell: Find the cell in the transportation table with the absolute minimum transportation cost.

  2. Allocate Maximum: Allocate the maximum possible value to this cell, considering the limitations:

    • Supply available at the current row.
    • Demand required at the current column.

  3. Update Supply and Demand:

    • Supply: Subtract the allocated value from the available supply at the current row. If the supply becomes zero at this location, remove the entire row from further consideration.
    • Demand: Subtract the allocated value from the demand required at the current column. If the demand becomes zero at this location, remove the entire column from further consideration.

  4. Repeat Until Completion: Continue by finding the cell with the lowest cost among the remaining non-zero cells in the table. Repeat steps 2 and 3 until all supply and demand requirements are fulfilled.

Advantages of Least Cost Method:

  • Generally Lower Initial Cost: Compared to the North-West Corner Method, LCM often leads to a solution with a lower initial total transportation cost, bringing you closer to the optimal solution.
  • Relatively Simple to Implement: The method follows a clear logic of prioritizing the lowest cost cell, making it easier to understand than VAM.

Disadvantages of Least Cost Method:

  • May Not Be Optimal: The solution obtained might not be the absolute optimal solution, but it’s usually a good starting point for further optimization using the simplex method.

Vogel’s Approximation Method (VAM):

VAM is a more sophisticated approach that aims to find a better initial solution than both the North-West Corner Method and the Least Cost Method. It considers both the transportation cost and the potential opportunity cost associated with not choosing the lowest cost cell in a particular row or column.

  1. Calculate Penalty Costs:

    • For each row, find the difference between the lowest and the second-lowest transportation cost. This difference represents the penalty cost of not assigning to the cell with the lowest cost in that row.
    • Similarly, calculate the penalty cost for each column.

  2. Identify Maximum Penalty Row/Column: Find the row or column with the highest penalty cost. This indicates where the initial allocation might have the most significant impact on the overall solution.

  3. Allocate Maximum: Allocate the maximum possible value to the cell with the lowest cost within the chosen row (with the highest penalty cost) considering supply and demand limitations (same as Least Cost Method).

  4. Update and Repeat: Update supply, demand, and remove rows or columns if necessary. Then, recalculate penalty costs and repeat steps 2 and 3 until all supply and demand requirements are fulfilled.

Advantages of Vogel’s Approximation Method:

  • Closer to Optimal Solution: VAM often leads to an initial solution with a lower total transportation cost compared to the other two methods, providing a better starting point for optimization.
  • Considers Opportunity Cost: By incorporating penalty costs, VAM tries to avoid making sub-optimal choices early on.

Disadvantages of Vogel’s Approximation Method:

  • More Complex to Apply: Calculating penalty costs and choosing the row/column with the highest penalty adds complexity compared to LCM.

Choosing Between the Methods:

  • If you prioritize simplicity and a quick initial solution, the Least Cost Method is a good choice.
  • If you want a better initial solution and are willing to invest some extra time in calculations, Vogel’s Approximation Method is recommended.

Remember, all these methods provide starting points for the simplex method, which refines the solution to find the true optimal allocation with minimum transportation cost.