Rough-TODIM - Rough extension of TODIM

Overview

rough-todim extends TODIM to handle Rough uncertainty. Decision matrix entries are rough numbers [L,U] representing customer preference scores. Normalization uses global-max (all bounds divided by max upper bound across matrix). The TODIM φ_c function computes crisp scalar distances d_c = |p^L_ic - p^L_jc| + |p^U_ic - p^U_jc| and applies importance-based scaling: full scale (α=β=1) for Important/Most-Important criteria, ×0.5 for Averagely Important, Rule 3 for Low Important. All dominance computations (δ, Σδ) are crisp. The final ξ scores are crisp [0,1] values computed directly from crisp dominance sums - NOT from midpoint defuzzification. CAUTION: paper does not specify how to compare rough intervals [L,U] for gain/loss determination; paper's Table 8 δ values cannot be reproduced from the described formula alone.

Output

utility, 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

Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid Rough numbers/tuples
• Underlying crisp method's compensation assumption holds in uncertain space
• All decision-maker(s) and experts use the same linguistic/uncertainty scale

Edge cases and pitfalls

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

• ROUGH-TODIM does NOT use midpoint defuzzification for final ranking. Distances d_c are crisp scalars; all dominance sums are crisp. The unresolved issue is how to compare rough intervals for gain/loss determination in φ_c: the paper never specifies this, so different comparison rules (midpoint, L-bound, comparison table) will yield different δ values. The paper-reported Table 8 δ values cannot be reproduced from the described algorithm.

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