Operations Research — Transportation models: What do we call a Transportation Problem in which the total supply available at all origins exactly equals the total demand required at all destinations (so no dummy source or dummy destination is needed)?
Correct Answer: Balanced Transportation Problem
Introduction / Context:In Operations Research, the Transportation Problem allocates shipments from multiple origins (suppliers) to multiple destinations (customers) at minimum cost while respecting supply and demand constraints. A key diagnostic before solving is whether the problem is balanced or unbalanced, because that choice determines if we must add a dummy source or destination and which methods apply directly without modification.
Given Data / Assumptions:
- Total supply across all origins may equal or differ from total demand across all destinations.
- Costs per unit are known and nonnegative; capacities (supply/demand) are fixed.
- We are identifying the correct term when total supply equals total demand, exactly.
Concept / Approach:A Balanced Transportation Problem is defined by the equality: sum of supplies = sum of demands. When this holds, the model can be solved directly by methods like Northwest Corner, Least Cost, or Vogel’s Approximation to get a starting solution, and then optimized with Stepping Stone or MODI without introducing dummy rows/columns. If totals differ, the model is Unbalanced and requires a dummy source or destination to reconcile the totals before proceeding.
Step-by-Step Solution:
Compute total_supply = sum of all origin supplies. Compute total_demand = sum of all destination demands. Compare totals: if total_supply == total_demand, the problem is balanced. Conclude the correct term is Balanced Transportation Problem.Verification / Alternative check:Attempt to set up the linear program. If all constraints can be written without slack from a dummy node and the equality condition holds, the instance is balanced by definition.
Why Other Options Are Wrong:
- Degenerate Solution: Refers to basic feasible solutions with fewer positive allocations than m + n − 1, not the supply-demand equality condition.
- Unbalanced Transportation Problem: Applies when total supply ≠ total demand.
- All/None: Incorrect because exactly one term fits the definition.
Common Pitfalls:Forgetting to add a dummy node when the problem is unbalanced; misclassifying degeneracy as balance/unbalance.
Final Answer:Balanced Transportation Problem