Both the Stepping Stone Method and the MODI Method (u-v method) serve as powerful tools to find the optimal solution (minimum total transportation cost) in transportation problems. They come into play after you’ve established an initial feasible solution using methods like the North-West Corner Method or Least Cost Method. Here’s a comprehensive comparison to help you choose the most suitable approach:
Stepping Stone Method: A Visual Path to Optimality
The Stepping Stone Method is a widely used technique known for its clear visualization and methodical approach. It involves a series of adjustments based on the concept of opportunity costs.
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Evaluating Unassigned Cells: The method starts by identifying all the unoccupied cells (zeros) in the transportation table. These represent potential allocation opportunities.
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Opportunity Cost Calculations: For each unoccupied cell, the opportunity cost is calculated. This metric essentially reflects the potential cost reduction if a value were assigned to that cell. It’s determined by subtracting the sum of the minimum row cost (excluding the occupied cell in that row) and the minimum column cost (excluding the occupied cell in that column) from the transportation cost of the unoccupied cell itself.
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Finding the Best Candidate: The core step involves selecting the unoccupied cell with the highest positive opportunity cost. A positive opportunity cost signifies that assigning a value to this cell has the most significant potential to reduce the overall transportation cost.
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Closed Loop – A Visual Journey: Once the champion cell is chosen, a closed loop path is traced through the table. This path incorporates only occupied cells and right-angle turns, starting and ending at the chosen unoccupied cell. The key restrictions are that horizontal movements can only switch between occupied and unoccupied cells, and vice versa, while vertical movements follow the same rule.
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Minimum Allocation and Adjustments: After establishing the closed loop path, the minimum allocation value is identified among all the occupied cells encountered in the loop. Here’s where the magic happens:
- The value of the initially chosen unoccupied cell is increased by this minimum allocation value.
- For each occupied cell encountered in the closed loop path (excluding the starting cell), the value is decreased by the minimum allocation value. This maintains the overall supply and demand constraints.
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Reaching the Summit of Optimality: The process iterates through steps 1 to 5 until no unoccupied cell has a positive opportunity cost. This indicates that you’ve reached the summit – the optimal solution with the minimum total transportation cost.
Advantages of the Stepping Stone Method:
- Intuitive Visualization: The closed-loop path concept makes it easier to understand the logic behind the method, especially for beginners.
- Systematic Procedure: The step-by-step approach ensures a clear and structured path towards the optimal solution.
Disadvantages of the Stepping Stone Method:
- Potential for Tedium: For large transportation problems with numerous unoccupied cells, the calculations and closed-loop tracing can become cumbersome and time-consuming.
- Suboptimal Path Selection: The method might not always identify the most efficient path to the optimal solution. The selection process relies on finding the highest positive opportunity cost, which might lead to unnecessary iterations before reaching the optimal solution.
MODI Method: A More Direct Approach
The MODI Method (u-v method) offers a more analytical approach compared to the Stepping Stone Method. It focuses on directly identifying the cell that will yield the most significant cost reduction, potentially leading to a faster path to the optimal solution.
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Introducing u and v Values: The method starts by assigning initial values of zero to all rows (u values) and columns (v values). Then, for each unoccupied cell, the difference between its transportation cost and the minimum cost of its corresponding row (excluding the occupied cell in that row) is calculated. This difference is then added to the u value of that row. A similar calculation is done for columns, subtracting the difference between the cell’s cost and the minimum cost of its column (excluding the occupied cell) from the corresponding v value.
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Calculating the Reduced Cost: For each unoccupied cell, a reduced cost is computed. This metric combines the cell’s inherent transportation cost (c_ij) with the row penalty (u_i) and the column penalty (v_j). The reduced cost is calculated as c_ij + u_i – v_j. A negative reduced cost indicates that assigning a value to this cell will reduce the overall cost.
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Identifying the Champion: The unoccupied cell with the most negative reduced cost is chosen as the champion for allocation. Since a negative reduced cost signifies the most significant cost reduction potential, this approach aims for a more targeted selection.