Transportation model optimality testing: After you build an initial feasible solution, which methods are widely used to test and improve a transportation solution toward optimality?

Introduction / Context:
Transportation Problems are often approached in two phases: first, find an initial basic feasible solution; second, test for optimality and iterate if improvements exist. Two classic improvement tests—Stepping Stone and MODI (Modified Distribution)—provide systematic ways to evaluate reduced costs and adjust allocations to reduce total cost.

Given Data / Assumptions:

• An initial solution exists (e.g., by Northwest Corner, Least Cost, or Vogel’s Approximation).
• Costs and supplies/demands are fixed and nonnegative.
• Goal is minimum total transportation cost.

Concept / Approach:
The Stepping Stone Method evaluates the opportunity cost of introducing a shipment into an unoccupied cell by tracing a closed path (alternating plus/minus) and computing net cost change. The MODI Method calculates row and column potentials (u_i, v_j) to find reduced costs efficiently for all empty cells. If any reduced cost is negative, you can pivot along a loop to improve the solution until no further improvement is possible.

Step-by-Step Solution:

Verification / Alternative check:
Compare total costs after each iteration; they must monotonically decrease or remain unchanged once optimal.

Why Other Options Are Wrong:

• Slack variable method: General LP concept; not the standard named test for transportation optimality.
• None: Incorrect because both listed methods are standard.

Common Pitfalls:
Forgetting degeneracy handling (m + n − 1 basic variables); mis-tracing loops; stopping before all reduced costs are nonnegative.

Final Answer:
Both of the above