Methods for Initial Solution

The transportation problem aims to minimize the cost of transporting goods from suppliers (sources) to consumers (destinations) while satisfying supply and demand constraints. Several methods can be used to find an initial feasible solution to the transportation problem. The most common methods include:

1. Northwest Corner Method

• Procedure:
  • Start at the "northwest" (top-left) corner of the cost matrix.
  • Allocate as much as possible to satisfy the supply of the row or the demand of the column.
  • Adjust the supply and demand by subtracting the allocated quantity.
  • Move to the next cell to the right if the row's supply is exhausted or downward if the column's demand is exhausted.
  • Repeat the process until all supply and demand requirements are satisfied.
• Pros and Cons:
  • Easy to apply manually.
  • Ignores the costs, which may result in a non-optimal initial solution.

2. Least Cost Method (Minimum Cost Method)

• Procedure:
  • Choose the cell with the smallest transportation cost.
  • Allocate as much as possible to satisfy the supply or demand of that cell.
  • Adjust the supply and demand accordingly.
  • Cross out the row or column where the supply or demand is met, and repeat the process with the remaining cells.
  • If there is a tie in costs, choose arbitrarily among the tied cells.
• Pros and Cons:
  • Takes transportation costs into account, potentially resulting in a better initial solution than the Northwest Corner Method.
  • More complex than the Northwest Corner Method, but still manageable manually for small problems.

3. Vogel's Approximation Method (VAM)

• Procedure:
  • For each row and column, calculate the penalty, which is the difference between the two smallest costs in that row or column.
  • Identify the row or column with the highest penalty.
  • Allocate as much as possible to the cell with the minimum cost in the selected row or column.
  • Adjust the supply and demand, then cross out the row or column that has been satisfied.
  • Recalculate penalties for the remaining rows and columns, and repeat the process.
• Pros and Cons:
  • Generally provides a better initial feasible solution compared to the Northwest Corner and Least Cost Methods.
  • More complex to implement manually, but very effective for larger problems.

Each of these methods helps in finding an initial feasible solution to the transportation problem, which can then be optimized further using techniques like the Modified Distribution Method (MODI) or the stepping-stone method to find the optimal solution.