MODI Method

Modified Distribution Method (MODI Method)

The Modified Distribution Method (MODI Method) is used to determine the optimal solution for a transportation problem once an initial feasible solution is available. It refines the solution by checking for improvements that reduce the total transportation cost.

Steps to Apply the MODI Method:

• Initial Basic Feasible Solution:
  • Begin with an initial feasible solution, often derived using methods like the North-West Corner Rule, Least Cost Method, or Vogel’s Approximation Method.
• Calculate $u_i$ and $v{}_{\mathrm{j }}$Values:
  • These values are calculated using the equation $u_i+v{}_j=C_{ij}$​, where $C_{ij}$ is the cost associated with cell (i,j).
  • Set $u_i or v{}_j$ to zero (typically $u_i$​) as a reference point.
  • Solve for the remaining $u_i and v{}_j$​ values using the available cost data and the established equations.
• Example Calculation: Given costs and assignments:
  • $u_1+v{}_1=C_{11}$​
  • $u_1+v{}_2=C_{12}$​
  • $u_2+v{}_2=C_{22}$
  • $u_3+v{}_2=C_{32}$
  • $u_3+v{}_3=C_{33}$
• Solve these equations step-by-step to determine the values for $u_i$​ and $v{}_j$.
• Calculate Opportunity Costs:
  • Opportunity cost for an unoccupied cell (i,j) is calculated using: 

$\mathrm{Opportunity} Cos{t}_{ij}=C_{ij}-(u_i+v{}_j)$

• This represents the potential benefit of using the cell if it were to be occupied.

• Example Calculation: For an unoccupied cell (i,j):

$\mathrm{Opportunity} Cos{t}_{ij}=C_{ij}-(u_i+v{}_j)$

• Select and Adjust Based on Largest Opportunity Cost:
  • Identify the unoccupied cell with the highest positive opportunity cost.
  • Create a closed path starting from this cell, alternating the signs of the cells involved. Assign positive signs to cells where the allocation is increased and negative signs where it is decreased.
  • Adjust the allocations along this path by adding units to cells with positive signs and subtracting from those with negative signs. Ensure that the adjustments satisfy the supply and demand constraints.
• Repeat Until Optimality:
  • After adjusting the matrix, recalculate the opportunity costs for the unoccupied cells.
  • Continue the process of selecting the highest positive opportunity cost and adjusting the allocations until no further improvements can be made.
• Determine Optimal Solution:
  • The process concludes when all opportunity costs are zero or negative, indicating that no further improvements can be made.

Example Walkthrough:

• Initial Basic Feasible Solution: Assume you have a feasible solution with allocated units and initial costs.
• Calculate $u_i$​ and $v{}_j$ Values:
  • For given costs and assignments, solve the system of equations to find $u_i$ and $v{}_j$
• Calculate Opportunity Costs:
  • For each unoccupied cell, compute:

$\mathrm{Opportunity} Cos{t}_{ij}=C_{ij}-(u_i+v{}_j)$

• Select Largest Opportunity Cost:
  • Suppose the largest positive opportunity cost is for cell (i,j). Create a closed loop around this cell with alternating signs.
• Adjust Allocations:
  • Increase allocations in cells with positive signs and decrease in those with negative signs, ensuring all constraints are satisfied.
• Repeat Process:
  • Recalculate and adjust until no positive opportunity costs remain.

Key Points:

• Efficiency:
  • The MODI Method is more efficient than the Stepping Stone Method, especially for larger problems, as it reduces the number of computations required to find the optimal solution.
• Optimal Solution:
  • MODI ensures that the solution is optimal by iteratively improving the initial feasible solution.
• Application:
  • Suitable for large-scale transportation problems, where manual adjustments would be impractical.
• Complexity Reduction:
  • MODI simplifies the process by directly focusing on the opportunity costs and systematically refining the solution, reducing computational complexity.

The MODI Method is a powerful tool in operations research for optimizing transportation costs, ensuring that solutions are both feasible and cost-effective.