Modified Distribution Method (MODI) Method

The Modified Distribution Method (MODI) is another iterative procedure used to improve an initial feasible solution for the transportation problem. It is a more efficient version of the Stepping Stone Method as it avoids the need to evaluate all possible paths for transferring units of shipment between cells. Instead, it uses a set of reduced costs and a matrix of dual variables to identify the optimal path.

The MODI method works as follows:

Begin with an initial feasible solution obtained using a heuristic method such as the North-West Corner Method, Least Cost Method, or Vogel’s Approximation Method.

Calculate the reduced cost for each unoccupied cell in the transportation table. The reduced cost is the difference between the cell’s actual cost and the sum of the dual variables for its row and column.

Select an unoccupied cell with the largest negative reduced cost and trace a closed path that starts and ends in the same row or column.

Calculate the net change in the reduced cost if one unit of shipment is moved along the closed path. The net change is the difference between the reduced cost of the unoccupied cell and the reduced cost of the cell that releases a unit of shipment.

Repeat step 4 for all other closed paths containing the unoccupied cell.

Select the closed path that results in the largest reduction in the reduced cost, and adjust the allocation by transferring one unit of shipment along the closed path.

Recalculate the reduced costs and the dual variables for the transportation table.

Repeat steps 3 to 7 until no further improvement can be made.

The MODI method is faster and more efficient than the Stepping Stone Method as it focuses on evaluating closed paths with negative reduced costs. It is also guaranteed to find an optimal solution for the transportation problem. However, like the Stepping Stone Method, the MODI method assumes that the cost matrix is linear and that there are no constraints on the amount of shipment that can be transferred along a path