Perturbation-stable failure and weighted dual repair for harmonic-mean transportation allocations

Harmonic-mean transportation rules can generate feasible schedules without providing a cost guarantee. This paper studies the rule that selects the active row or column of largest harmonic mean and allocates to its cheapest cell. An explicit positive-cost family has unbounded approximation ratio, and a perturbed family shows that the failure persists throughout a full-dimensional neighbourhood with strict deciding choices. For every route-cost perturbation of magnitude at most 0.01, the ratio is bounded below by \((M+11.97)/13.03\) for \(M\geq32\). A weighted potential-correction problem then computes the strongest transportation dual bound obtainable by downward movement of supplied candidate potentials. Its bipartite packing dual supplies an independent repair certificate, and the resulting bound dominates a scalar correction. Computation includes 300 assignment instances, 600 perturbation tests and 120 rectangular transportation instances. The weighted correction improves the scalar bound on all rectangular cases, with a mean correction-loss ratio of approximately 0.372. The worst-case theorem concerns the specified initial rule; it does not apply to subsequent exact improvement. The study combines a perturbation-stable negative result with an auditable certificate framework. All numerical data are synthetic, and the underlying LP duality is established.