IF-TODIM: Intuitionistic Fuzzy TODIM

Overview

Prospect-theory pairwise dominance under Intuitionistic Fuzzy uncertainty. Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Prospect-theory pairwise dominance under Intuitionistic Fuzzy uncertainty
• •Preserves intuitionistic uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

• •Rank reversal known on alternative-set changes (ref: Inherited from crisp TODIM base (Gomes-Lima 1992): global value ξ_i is min-max normalized over row-sum dominance ζ_i = Σ_k δ(A_i, A_k); adding/removing an alternative rebalances every δ pair and may invert the ξ ordering.)
• •Assumes: Decision matrix entries are valid IFN tuples (μ, ν) with μ+ν ≤ 1
• •Assumes: All decision-makers use the same linguistic-to-IFN scale
• •Assumes: Criterion directions (benefit/cost) are explicitly labelled
• •Assumes: Criterion weights form a simplex; reference criterion is the maximum-weight one

Assumptions to verify

• •Decision matrix entries are valid IFN tuples (μ, ν) with μ+ν ≤ 1
• •All decision-makers use the same linguistic-to-IFN scale
• •Criterion directions (benefit/cost) are explicitly labelled
• •Criterion weights form a simplex; reference criterion is the maximum-weight one
• •Loss-aversion factor θ calibrated to the data (default 1.0; θ ∈ [1, 2.5] typical in TODIM literature)

When not to use

• •Compensatory utility ranking is preferred (use IF-EDAS or IF-TOPSIS)
• •Decision-maker is risk-neutral and prospect-theory framing is not desired (use IF-VIKOR or IF-TOPSIS)
• •Hesitation degree π must be modelled explicitly with μ+ν > 1 raw judgements: consider Pythagorean Fuzzy TODIM or q-ROF TODIM
• •Ratings are probability distributions rather than IFNs: see Lourenzutti-Krohling 2014 Hellinger-TODIM (separate manifest)

Edge cases

• •IF decision matrix R = (r_ij) where r_ij = (μ_ij, ν_ij) is the IFN rating of alternative A_i on criterion C_j. For group-decision settings (K≥2 DMs), collect K such matrices and aggregate cell-wise vi
• •IF complement r_ij* = (ν_ij, μ_ij); benefit criteria are unchanged. Converts the matrix to uniform benefit-direction representation so that the gain/loss branch of φ_j in F5 is governed by score diffe
• •IF distance. Compute the symmetric Szmidt-Kacprzyk Euclidean distance d_j(A_i, A_k) = d_E(r_ij*, r_kj*) = √[((μ_ij* − μ_kj*)² + (ν_ij* − ν_kj*)² + (π_ij* − π_kj*)²) / 2] ∈ [0, 1]. This is the canonica
• •zero branch (s(r_ij*) = s(r_kj*)) gives 0; loss branch (s(r_ij*) < s(r_kj*)) gives −(1/θ) · √((Σ_j w_jr) · d_j(A_i, A_k) / w_jr). The loss branch is asymmetric by a factor of (Σ_j w_jr / w_jr) / 1 = Σ
• •when K=1 or when IFWA-before-F2 aggregation was applied in F1.

Common pitfalls

• •Hatalı: 'IF-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid IFN tuples (μ, ν) with μ+ν ≤ 1
• •Hatalı: 'IF-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-makers use the same linguistic-to-IFN scale
• •Hatalı: 'IF-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criterion directions (benefit/cost) are explicitly labelled
• •Hatalı: 'IF-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criterion weights form a simplex; reference criterion is the maximum-weight one
• •Hatalı: 'IF-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Loss-aversion factor θ calibrated to the data (default 1.0; θ ∈ [1, 2.5] typical in TODIM literature)
• •Hatalı: IF-TODIM'yi 'Compensatory utility ranking is preferred (use IF-EDAS or IF-TOPSIS)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: IF-TODIM'yi 'Decision-maker is risk-neutral and prospect-theory framing is not desired (use IF-VIKOR or IF-TOPSIS)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: IF-TODIM'yi 'Hesitation degree π must be modelled explicitly with μ+ν > 1 raw judgements' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Construct the IF decision matrix R = (r_ij) where r_ij = (μ_ij, ν_ij) is the IFN rating of alternative A_i on criterion C_j. For group-decision settings (K≥2 DMs), collect K such matrices and aggregate cell-wise via IFWA with DM weights λ_k before entering F2: r_ij = (1 − Π(1−μ_ij^{(k)})^{λ_k}, Π (ν_ij^{(k)})^{λ_k}). Krohling 2013 treats single-DM case; Liu-Teng 2015 (IULS variant) Step 5 aggregates per-DM dominance δ^k via λ-weighted sum AFTER F6: DecisionMind_v3 engine uses the IFWA-before-F2 convention for full pipeline reproducibility. Formül: R = (r_{ij})_{m \times n},\ r_{ij} = (\mu_{ij}, \nu_{ij}),\ \mu_{ij}+\nu_{ij} \le 1;\quad \text{IFWA aggregation if } K \ge 2 Anchor: Krohling-Pacheco-Siviero 2013 Step 1; Liu-Teng 2015 Step 1 (eq. 28); Xu 2007 IFWA
2. 2.Adım 2 (F2): Step 2: Cost-criterion adjustment: for j ∈ J_c (cost criteria) replace r_ij = (μ_ij, ν_ij) with its IF complement r_ij* = (ν_ij, μ_ij); benefit criteria are unchanged. Converts the matrix to uniform benefit-direction representation so that the gain/loss branch of φ_j in F5 is governed by score difference alone. Formül: j \in J_c: r_{ij}^* = (\nu_{ij}, \mu_{ij});\quad j \in J_b: r_{ij}^* = r_{ij} Anchor: Krohling-Pacheco-Siviero 2013 (cost complement convention from crisp TODIM); Liu-Teng 2015 eq. 28 (neg operator on IULS quaternion is the direct generalisation)
3. 3.Adım 3 (F3): Step 3: Reference criterion and relative weights. Pick reference criterion C_r = argmax_j w_j (the criterion carrying the highest crisp weight); compute relative weights w_jr = w_j / w_r for every j. By construction w_rr = 1 and Σ_j w_jr ≥ 1. The relative weighting makes the dominance contribution scale-invariant in the absolute weight magnitude. Formül: C_r = \arg\max_j w_j;\quad \bar{w}_{jr} = \frac{w_j}{w_r}\quad (\bar{w}_{rr} = 1) Anchor: Krohling-Pacheco-Siviero 2013 Step 2 (relative weights); Liu-Teng 2015 Step 2 (eq. 29); Gomes-Lima 1992 (crisp TODIM reference weight)
4. 4.Adım 4 (F4): Step 4: Per-criterion IF distance. Compute the symmetric Szmidt-Kacprzyk Euclidean distance d_j(A_i, A_k) = d_E(r_ij*, r_kj*) = √[((μ_ij* − μ_kj*)² + (ν_ij* − ν_kj*)² + (π_ij* − π_kj*)²) / 2] ∈ [0, 1]. This is the canonical default; alternative metrics (Szmidt-Kacprzyk Hamming, Hong-Choi accuracy-weighted) are selectable via D.distance_function. d_j is symmetric: the sign of the dominance contribution in F5 comes from the Chen-Tan score comparison, NOT from d_j. Formül: d_j(A_i, A_k) = \sqrt{\frac{(\mu_{ij}^* - \mu_{kj}^*)^2 + (\nu_{ij}^* - \nu_{kj}^*)^2 + (\pi_{ij}^* - \pi_{kj}^*)^2}{2}} Anchor: Szmidt-Kacprzyk 2000 (IF Euclidean distance); Krohling-Pacheco-Siviero 2013 Eq.(3) per-criterion distance
5. 5.Adım 5 (F5): Step 5: Per-criterion dominance contribution φ_j(A_i, A_k) with prospect-theory loss aversion. Three branches selected by Chen-Tan score comparison: gain branch (s(r_ij*) > s(r_kj*)) gives +√(w_jr · d_j(A_i, A_k) / Σ_j w_jr); zero branch (s(r_ij*) = s(r_kj*)) gives 0; loss branch (s(r_ij*) < s(r_kj*)) gives −(1/θ) · √((Σ_j w_jr) · d_j(A_i, A_k) / w_jr). The loss branch is asymmetric by a factor of (Σ_j w_jr / w_jr) / 1 = Σ_j w_jr / w_jr ≥ 1 RELATIVE to the gain branch: losses are amplified, modelling prospect-theory loss aversion with attenuation factor θ. Formül: \phi_j(A_i, A_k) = \begin{cases} +\sqrt{\frac{\bar{w}_{jr} \cdot d_j(A_i, A_k)}{\sum_j \bar{w}_{jr}}}, & s(r_{ij}^*) > s(r_{kj}^*) \\ 0, & s(r_{ij}^*) = s(r_{kj}^*) \\ -\frac{1}{\theta} \sqrt{\frac{(\sum_j \bar{w}_{jr}) \cdot d_j(A_i, A_k)}{\bar{w}_{jr}}}, & s(r_{ij}^*) < s(r_{kj}^*) \end{cases} Anchor: Krohling-Pacheco-Siviero 2013 Eq.(4) (IF-TODIM dominance); Liu-Teng 2015 Eq.(30) (IULS-TODIM dominance: identical skeleton); Gomes-Lima 1992 (crisp TODIM dominance); Kahneman-Tversky 1979 (prospect-theory loss aversion)
6. 6.Adım 6 (F6): Step 6: Aggregated dominance δ(A_i, A_k) = Σ_j φ_j(A_i, A_k) sums per-criterion contributions over j into a single signed scalar per ordered pair (A_i, A_k). δ is sign-reversing on pair swap (δ(A_i, A_k) and δ(A_k, A_i) carry opposite signs but DIFFERENT MAGNITUDES because the gain/loss asymmetry inflates whichever direction is losing): this asymmetry is the mechanical signature of prospect-theory in the TODIM family. δ(A_i, A_i) = 0 by construction. Group-DM extension (Liu-Teng 2015 Step 5): aggregate per-DM δ^k via λ-weighted sum δ(A_i, A_k) = Σ_k λ_k δ^k(A_i, A_k); skipped when K=1 or when IFWA-before-F2 aggregation was applied in F1. Formül: \delta(A_i, A_k) = \sum_{j=1}^{n} \phi_j(A_i, A_k);\quad \text{(group: } \delta(A_i, A_k) = \sum_k \lambda_k \delta^k(A_i, A_k)\text{)} Anchor: Krohling-Pacheco-Siviero 2013 Eq.(5); Liu-Teng 2015 Eq.(31)-(32)
7. 7.Adım 7 (F7): Step 7: Global value ξ_i. First compute the row-sum dominance ζ_i = Σ_k δ(A_i, A_k) ∈ ℝ; ζ_i measures the net dominance of A_i over the rest of the alternative set. Then min-max normalise ζ across the m alternatives to ξ_i = (ζ_i − min_i' ζ_{i'}) / (max_i' ζ_{i'} − min_i' ζ_{i'}) ∈ [0, 1]: the best alternative carries ξ = 1, the worst ξ = 0. Σ_i ξ_i is NOT a constant (unlike PROMETHEE II's Σ_i φ_i = 0); ξ values can be compared cardinally within a single run but not across reruns with different alternative sets. Formül: \zeta_i = \sum_{k=1}^{m} \delta(A_i, A_k);\quad \xi_i = \frac{\zeta_i - \min_{i'} \zeta_{i'}}{\max_{i'} \zeta_{i'} - \min_{i'} \zeta_{i'}} \in [0, 1] Anchor: Krohling-Pacheco-Siviero 2013 Eq.(6) (global value normalisation); Liu-Teng 2015 Eq.(33); Gomes-Lima 1992 (crisp TODIM global value)
8. 8.Adım 8 (F8): Step 8: Descending ranking by ξ_i. The alternative with the largest ξ is the most preferred; ties are broken deterministically by alternative_id (lexicographic). Unlike PROMETHEE, TODIM produces only a complete (linear) order: there is no partial-order analogue. Formül: \text{rank}(A_i) = \text{argsort}_{\downarrow}(\xi_i) Anchor: Krohling-Pacheco-Siviero 2013 Step 4 (rank by ξ); Liu-Teng 2015 Step 7

Commonly paired with

• •IF-AHP + IF-TODIM (common)
• •IF-ENTROPY + IF-TODIM (common)

How to cite