What is the method of solving unbalanced transportation problem?

Solving Unbalanced Transportation Problems: A Step-by-Step Guide

In transportation problems, it is often encountered that the total supply from origins exceeds the total demand at destinations. This imbalance can hinder the optimal allocation of resources and lead to incorrect solutions. To address this, the concept of dummy demand is employed.

Dummy Demand: A Balancing Act

A dummy demand column is a fictitious destination that absorbs the excess supply. By assigning demand to this column, we effectively balance the total supply and total demand, allowing us to solve the transportation problem using standard techniques. The demand assigned to the dummy column is the difference between the total supply and total demand.

Step-by-Step Solution

To solve an unbalanced transportation problem, follow these steps:

1. Calculate the Excess Supply: Determine the difference between the total supply and total demand.

2. Create a Dummy Demand Column: Add a dummy column to the destination table.

3. Assign Demand to Dummy Column: Set the demand in the dummy column equal to the excess supply calculated in Step 1.

4. Solve the Transportation Problem: Use any standard transportation algorithm (e.g., Vogel's Approximation Method, North-West Corner Method) to find an initial feasible solution.

5. Adjust for Dummy Demand: The optimal solution obtained may include shipments to the dummy column. Remove these shipments from the solution.

6. Recalculate Optimal Solution: Use the remaining shipments to calculate a new optimal solution. This solution represents the optimal distribution of resources without any surplus or shortage.

Example

Consider an unbalanced transportation problem with the following data:

Total Supply: 80 Total Demand: 60

Excess Supply: 80 - 60 = 20

We create a dummy demand column with a demand of 20. The adjusted demand table becomes:

Solving the transportation problem using Vogel's Approximation Method, we obtain the optimal solution:

Adjusting for dummy demand, we remove the shipments to the dummy column, resulting in the final optimal solution:

Conclusion

Using dummy demand is an effective method for solving unbalanced transportation problems. By balancing the total supply and total demand, we can obtain an optimal solution that allocates resources efficiently and prevents any surpluses or shortages. This technique is applicable in various real-world scenarios, such as production planning, inventory management, and distribution logistics.