IV-PROJECTION: Interval-Valued Intuitionistic Fuzzy Projection-based MCDM

Strengths

• •Captures both distance and angle to ideal solution (projection geometry)
• •Avoids defuzzification information loss (operates directly on IVIFS algebra)
• •Native support for group decision making via averaging over decision-maker matrices
• •Linear-time per criterion; lightweight compared with LP-based methods

Limitations

• •Requires IVIFS evaluations (often elicited via expert linguistic-to-IVIFS translation tables)
• •Sensitive to PIS construction choice (benefit ⟨[1,1],[0,0]⟩ vs. data-driven max)
• •Does not model preference structure explicitly (use UTA-family if preference-disaggregation needed)

Assumptions to verify

• •IVIFS axiom muU + nuU ≤ 1 holds for every cell
• •Criterion directions correctly labelled (max/min) before PIS construction
• •All experts use the same IVIFS-linguistic conversion table

When not to use

• •Crisp data sufficient: use base PROJECTION/TOPSIS directly
• •Preference-disaggregation needed: use UTA-family
• •Stochastic uncertainty dominates: use STOCHASTIC-UTA

Edge cases

• •Zero-norm alternative (all cells = ⟨[0,0],[0,0]⟩): NP undefined; degenerate alternative: flag and skip.
• •Identical alternatives: equal NP → tie-break by alternative_id argsort.
• •muU + nuU > 1 violation: rejected by E-2 (IVIFS axiom).

Common pitfalls

• •Treating IVIFS cells as 4 independent scalars and applying TOPSIS-style normalization (Yue 2019 does NOT divide cells by column norms).
• •Using unweighted inner product / norm: Rehber §2.3.5 explicitly weights both numerator and denominator with w_j (consistent NP).
• •Forgetting the cost-direction PIS flip: leads to A2-dominated cost criteria being penalized backwards.

Commonly paired with

• •ENTROPY + IV-PROJECTION (common)
• •AHP + IV-PROJECTION (common)

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