Rough-DRSA - Rough extension of DRSA

Overview

DRSA builds dominance cones from ordinal/cardinal criteria (F1), computes lower approximations {x: D_P^+(x) ⊆ Cl_t^≥} and upper approximations {x: D_P^-(x) ∩ Cl_t^≥ ≠ ∅} of upward unions of decision classes (F2), measures quality of approximation γ_P (F3), induces certain decision rules from lower approximations (F4), and assigns each alternative to a class (F5). Output is a SORTING (certain-good / boundary / certain-bad), not a numeric ranking. For ranking display a weighted-dominance surrogate score is used (Σ_j w_j · |D_P^-(x)|), not midpoint defuzzification.

Output

class label, higher is better

Data

Rough Number, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Rough MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Build decision table T: rows = examples, columns = condition criteria + 1 decision class column (e.g., low/medium/high).

2. 2

   For criteria set P ⊆ C, compute dominance: x D_P y ⟺ ∀q ∈ P: f(x,q) ≥ f(y,q) [or ≤ for min criteria]. Build P-dominating set D_P^+(x) and P-dominated set D_P^-(x) for each example.

3. 3

   Define class unions: upward Cl_t^≥ = ∪_{s≥t} Cl_s (e.g., 'medium or high'); downward Cl_t^≤ = ∪_{s≤t} Cl_s ('medium or low').

4. 4

   Compute P-lower approximation underline{P}(Cl_t^≥) = {x: D_P^+(x) ⊆ Cl_t^≥} (every x that dominates x is also in Cl_t^≥). Compute P-upper overline{P}(Cl_t^≥) = {x: D_P^-(x) ∩ Cl_t^≥ ≠ ∅}. Boundary BN = overline ∖ underline = inconsistent zone.

5. 5

   Quality of approximation γ_P = |∪_t underline{P}(Cl_t^≥)| / |U|. If γ_P = 1, P perfectly discriminates the classes.

6. 6

   Induce CERTAIN rules from lower approximations (form: ⋀ f(x,q) ≥ r_q ⇒ x ∈ Cl_t^≥). DOMLEM/VC-DomLEM find minimal cover. Kujawinska 2016 obtained 125 reliable rules from 866 records using VC-DomLEM, 4 attributes (Mn,Si,Ni,Cu).

7. 7

   Apply rule set to a new alternative: match premises, assign to class(es). Conflict-resolution (voting, strength, min-loss) if multiple rules trigger different classes. Evaluate via 10-fold cross validation accuracy + per-class sensitivity/precision (Kujawinska: 78% accuracy, sensitivity {0.95, 0.6, 0.56} for classes {1, 2, 3}).

Look elsewhere when

Assumptions to verify

• Class profiles or reference examples available in same uncertainty space
• Linguistic categories map to Rough tuples

Edge cases and pitfalls

• Value-space violation: ensure all entries satisfy Rough: L ≤ U; approximations defined by equivalence classes before computation.

• DRSA does NOT use midpoint defuzzification - it produces class assignments (sorting), not numeric scores. The weighted-dominance surrogate displayed for ranking is an implementation convenience, not a DRSA output. Boundary objects (S2/S3/S5 in the example) are genuinely unclassified - do not interpret surrogate scores as DRSA confidence.

Works with

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