A Directional Reading of the Ratio-Based Similarity Measure for Soft Sets under Containment

Similarity measures for soft sets are almost always symmetric, treating a pair of soft sets as interchangeable. In this paper we fix the two penalty coefficients of the ratio-based similarity measure $S_{R}$ as unequal constants $\alpha>\beta$, a setting under which $S_{R}$ becomes directional, and use it to model the containment of one soft set in another. We first give a condition on the coefficients that makes $S_{R}$ symmetric and classify the directional measure as a quasi-similarity. We then characterize the direction of a pair: at a shared parameter with positive agreement, the forward score dominates the backward score if and only if the overshoot of the first soft set does not exceed that of the second, equivalently when the first approximate set is no larger than the second, a comparison independent of $\alpha$ and $\beta$. Soft containment is the special case in which the first overshoot vanishes, so the difference between the forward and backward scores carries a sign that identifies the direction of inclusion. The same characterization delimits the reading: a parameter with empty agreement is blind to direction, and the sign records the larger residue rather than containment itself. Worked examples illustrate the method, including a screening problem in which the directed measure separates objects that exceed a target profile from those that fall short of it, a distinction that a symmetric measure cannot express, together with a sensitivity analysis over the penalties whose outcome the characterization predicts.