Transportation Problem

Modified Distribution Method (MODI) or (u - v) method

The modified distribution method, also known as MODI method or (u - v) method provides a minimum cost solution to the transportation problem. In the stepping stone method, we have to draw as many closed paths as equal to the unoccupied cells for their evaluation. To the contrary, in MODI method, only closed path for the unoccupied cell with highest opportunity cost is drawn.

MODI method is an improvement over stepping stone method.

The method, in outline, is :

Steps

1. Determine an initial basic feasible solution using any one of the three methods given below:

• North West Corner Rule
• Matrix Minimum Method
• Vogel Approximation Method

2. Determine the values of dual variables, u_i and v_j, using u_i + v_j = c_ij

3. Compute the opportunity cost using c_ij – ( u_i + v_j ).

4. Check the sign of each opportunity cost. If the opportunity costs of all the unoccupied cells are either positive or zero, the given solution is the optimal solution. On the other hand, if one or more unoccupied cell has negative opportunity cost, the given solution is not an optimal solution and further savings in transportation cost are possible.

5. Select the unoccupied cell with the smallest negative opportunity cost as the cell to be included in the next solution.

6. Draw a closed path or loop for the unoccupied cell selected in the previous step. Please note that the right angle turn in this path is permitted only at occupied cells and at the original unoccupied cell.

7. Assign alternate plus and minus signs at the unoccupied cells on the corner points of the closed path with a plus sign at the cell being evaluated.

8. Determine the maximum number of units that should be shipped to this unoccupied cell. The smallest value with a negative position on the closed path indicates the number of units that can be shipped to the entering cell. Now, add this quantity to all the cells on the corner points of the closed path marked with plus signs, and subtract it from those cells marked with minus signs. In this way, an unoccupied cell becomes an occupied cell.

"A man has a burger and you give him one burger more, that's addition." -Vinay Chhabra & Manish Dewan

9. Repeat the whole procedure until an optimal solution is obtained.