The MODI (Modified Distribution) Method is an optimization technique used in linear programming and operations research to improve the efficiency of an initial feasible solution for transportation problems. This method is part of the iterative improvement algorithms designed to find the optimal solution by systematically modifying the allocations in the transportation tableau.
Steps in the MODI Method:
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| Steps in the MODI Method |
Compute the U and V Values:
Calculate the Ui values for each row and Vi values for each column in the cost matrix. These values help determine the shadow prices associated with each row and column, indicating the marginal cost of transportation.
Compute the Cij - Ui - Vj Differences:
Calculate the differences between the cost values (Cij) and the sum of the corresponding Ui and Vj values for each cell (i, j) in the tableau. These differences represent the reduced costs associated with each cell.
Identify the Cell with the Largest Reduced Cost:
Identify the cell (i, j) with the largest reduced cost. This cell represents the most promising candidate for adjustment.
Create a Closed Loop:
Create a closed loop by following the allocation path from the identified cell. The loop should consist of cells with allocations equal to 1, forming a cycle within the tableau.
Adjust the Allocations:
Adjust the allocations along the closed loop by increasing or decreasing values in the cells. Maintain the feasibility of the solution by ensuring that the supply and demand constraints are not violated.
Recalculate Ui and Vj Values:
Recalculate the Ui values for rows and Vj values for columns based on the adjusted allocations. This step ensures that the updated solution continues to satisfy the constraints.
Repeat the Process:
Repeat the process by identifying the cell with the next largest reduced cost and adjusting allocations along a closed loop. Continue iterating until no negative reduced costs remain in the tableau.
Optimality Check:
Check whether the obtained solution is optimal. This involves verifying that all reduced costs are non-negative. If the solution is not optimal, further iterations may be necessary.
Example of Optimal Solution MODI Method:
Consider the following transportation tableau:
| C1 | C2 | C3 | Supply
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S1 | 3 | 1 | 4 | 30
S2 | 2 | 6 | 8 | 50
S3 | 5 | 7 | 2 | 20
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Demand | 20 | 60 | 30 |
After applying the MODI Method:
- Calculate Ui and Vi values.
- Compute the Cij - Ui - Vj differences.
- Identify the cell with the largest reduced cost, let's say (S2, C3).
- Create a closed loop by following the allocation path.
- Adjust allocations along the closed loop.
- Recalculate Ui and Vi values.
- Repeat the process until all reduced costs are non-negative.
Advantages of Modi Method:
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| Advantages of Modi Method |
Systematic Improvement:
The MODI Method provides a systematic approach for improving the initial feasible solution obtained from heuristic methods, leading to the optimal solution.
Efficient Algorithm:
MODI is an efficient algorithm for solving transportation problems, especially when compared to exhaustive search methods.
Flexibility:
The method is flexible and applicable to various transportation problems, making it suitable for a wide range of scenarios.
Balancing Effect:
MODI tends to balance the allocations during the iterative process, considering the interdependence of rows and columns.
Widely Used:
The MODI Method is widely used in practice due to its effectiveness in finding optimal solutions in transportation and distribution problems.
In summary, the MODI Method is a powerful algorithm for refining initial feasible solutions in transportation problems, leading to the optimal solution by iteratively adjusting allocations based on reduced costs.
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