In the context of the Transportation Problem in Operations Research, “degeneracy” refers to a situation where the basic feasible solution contains one or more zero-valued variables (non-basic variables) that should technically have a positive value to satisfy the constraints.

Degeneracy:

Degeneracy can occur when the number of allocated cells in the transportation tableau (i.e., the number of basic variables) is less than the total number of rows and columns (m + n – 1), where m is the number of sources and n is the number of destinations. When degeneracy occurs, the solution may not reach optimality using traditional methods such as the Simplex Method.

MODI (Modified Distribution) Method:

The MODI method is an optimization technique used to improve an existing feasible solution for the transportation problem, especially when degeneracy is present. Here’s an outline of the MODI method:

1. Calculate the Opportunity Cost (u-v values):
   • Calculate the dual variables
     
     ${}_{}$

     ui​ and vj
     $$vj for each row and column in the transportation tableau using any of the available methods (e.g., the simplex method).

2. Calculate the Modified Costs (Cij):
   • For each empty cell (non-basic variable) in the transportation tableau, calculate the modified cost
     
     ${}_{}$

     Cij​=cij​−(ui​+vj​), where
     $$

     cij​ is the original cost.

3. Determine the Cell with the Most Negative Modified Cost:
   • Identify the empty cell (non-basic variable) with the most negative modified cost. This cell will be the candidate for entering the solution.
4. Perform Cycle Test:
   • Use the cycle test to determine if adding the selected cell to the current solution will result in an improved solution.
5. Update Basic and Non-Basic Variables:
   • If the cycle test confirms an improvement, update the basic and non-basic variables accordingly.
6. Repeat Steps 2-5:
   • Repeat the process until all modified costs are non-negative, indicating that the current solution is optimal.

The MODI method is effective in handling degeneracy and improving the efficiency of the transportation problem solution. It helps to identify and rectify suboptimal allocations by considering the opportunity costs associated with the transportation constraints.