What variable is used to balance an unbalanced transportation problem?

Balancing Unbalanced Transportation Problems: The Role of Dummy Variables

In transportation problems, where the task is to optimize the distribution of goods from sources to destinations with minimum cost, it is often encountered that the total supply does not match the total demand. This creates an imbalance in the problem, which requires special treatment to find a feasible solution.

To resolve this imbalance, the concept of dummy variables is introduced. A dummy variable represents an artificial entity that either receives a surplus of supply or demands an equivalent amount. This allows the problem to be balanced and subsequently solved using standard techniques.

When the total supply exceeds the total demand, a dummy column is introduced. This column represents a fictitious destination that has a demand equal to the excess supply. The transportation cost associated with this dummy column is set to zero, as it does not represent any actual cost.

Conversely, if the total demand exceeds the total supply, a dummy row is introduced. This row represents a fictitious source that has a supply equal to the excess demand. Again, the transportation cost associated with this dummy row is set to zero.

The introduction of dummy variables essentially balances the problem by creating an artificial entity that absorbs the excess supply or demand. This allows the problem to be formulated as a standard transportation problem, which can be solved using a variety of algorithms, such as the North-West Corner Method, Vogel’s Approximation Method, or the Hungarian Method.

Once the original transportation problem has been balanced, the dummy variable can be removed from the solution. The resulting solution will represent the optimal distribution of goods, taking into account the initial imbalance between supply and demand.

In summary, dummy variables play a crucial role in balancing unbalanced transportation problems. By introducing an artificial entity with an offsetting supply or demand and assigning a zero transportation cost, the problem can be transformed into a standard form that can be solved using established techniques. This approach ensures that the solution obtained is feasible and optimal, even for unbalanced problems.