IF-TODIM - Intuitionistic Fuzzy TODIM

Overview

IF-TODIM extends crisp TODIM to settings where each rating is an Intuitionistic Fuzzy Number (μ, ν). The algorithm (i) optionally aggregates K DMs via IFWA, (ii) complements cost columns to unify direction, (iii) picks the highest-weight criterion as reference and computes relative weights w_jr = w_j / w_r, (iv) computes per-criterion IF Euclidean distance d_j(A_i, A_k), (v) builds the per-criterion dominance contribution φ_j using prospect-theory loss aversion (gain branch + clipped, loss branch − amplified by 1/θ, sign chosen by Chen-Tan score), (vi) sums φ_j over criteria to δ(A_i, A_k), (vii) row-sums δ to ζ_i and min-max-normalises to global value ξ_i ∈ [0, 1], (viii) ranks descending. Higher ξ = better. Algorithm SKELETON is INVARIANT across pure IF-TODIM (Krohling 2013) and the IULS variant (Liu-Teng 2015) - only the distance metric, the underlying value-space arithmetic, and the optional group-aggregation step differ.

Output

utility, higher is better

Data

Intuitionistic Fuzzy, ifn tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Supplier selection under epistemic uncertainty (Krohling 2013 case), Investment evaluation with linguistic IF ratings (Liu-Teng 2015 case), MAGDM under epistemic uncertainty with loss-aversion modelling, Behavioural / prospect-theory-aware decision making

How it works

1. 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.

   $$R=(r_{ij}{)}_{m\times n},r_{ij}=({\mu }_{ij},{\nu }_{ij}),{\mu }_{ij}+{\nu }_{ij}\le 1;\text{IFWA aggregation if }K\ge 2$$

   Krohling-Pacheco-Siviero 2013 Step 1; Liu-Teng 2015 Step 1 (eq. 28); Xu 2007 IFWA

2. 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.

   $$j\in J_c:r_{ij}^{\ast }=({\nu }_{ij},{\mu }_{ij});j\in J_b:r_{ij}^{\ast }=r_{ij}$$

   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

   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.

   $$C_r=\arg \underset{j}{\max }{w}_j;{\bar{w}}_{jr}=\frac{w_j}{w_r}({\bar{w}}_{rr}=1)$$

   Krohling-Pacheco-Siviero 2013 Step 2 (relative weights); Liu-Teng 2015 Step 2 (eq. 29); Gomes-Lima 1992 (crisp TODIM reference weight)

4. 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.

   $$d_j(A_i,A_k)=\sqrt{\frac{({\mu }_{ij}^{\ast }-{\mu }_{kj}^{\ast }{)}^2+({\nu }_{ij}^{\ast }-{\nu }_{kj}^{\ast }{)}^2+({\pi }_{ij}^{\ast }-{\pi }_{kj}^{\ast }{)}^2}{2}}$$

   Szmidt-Kacprzyk 2000 (IF Euclidean distance); Krohling-Pacheco-Siviero 2013 Eq.(3) per-criterion distance

5. 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.

   $${\varphi }_j(A_i,A_k)=\{\begin{array}{ll}+\sqrt{\frac{{\bar{w}}_{jr}\cdot d_j(A_i,A_k)}{{\sum }_j{\bar{w}}_{jr}}},& s(r_{ij}^{\ast })>s(r_{kj}^{\ast })\\ 0,& s(r_{ij}^{\ast })=s(r_{kj}^{\ast })\\ -\frac{1}{\theta }\sqrt{\frac{({\sum }_j{\bar{w}}_{jr})\cdot d_j(A_i,A_k)}{{\bar{w}}_{jr}}},& s(r_{ij}^{\ast })<s(r_{kj}^{\ast })\end{array}$$

   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

   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.

   $$\delta (A_i,A_k)=\sum _{j=1}^n{\varphi }_j(A_i,A_k);\text{(group: }\delta (A_i,A_k)=\sum _k{\lambda }_k{\delta }^k(A_i,A_k)\text{)}$$

   Krohling-Pacheco-Siviero 2013 Eq.(5); Liu-Teng 2015 Eq.(31)-(32)

7. 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.

   $${\zeta }_i=\sum _{k=1}^m\delta (A_i,A_k);{\xi }_i=\frac{{\zeta }_i-{\min }_{i^{\prime }}{\zeta }_{i^{\prime }}}{{\max }_{i^{\prime }}{\zeta }_{i^{\prime }}-{\min }_{i^{\prime }}{\zeta }_{i^{\prime }}}\in [0,1]$$

   Krohling-Pacheco-Siviero 2013 Eq.(6) (global value normalisation); Liu-Teng 2015 Eq.(33); Gomes-Lima 1992 (crisp TODIM global value)

8. 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.

   $$\text{rank}(A_i)={\text{argsort}}_{\downarrow }({\xi }_i)$$

   Krohling-Pacheco-Siviero 2013 Step 4 (rank by ξ); Liu-Teng 2015 Step 7

Fits when / Look elsewhere when

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)

Edge cases and pitfalls

• •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.

• Confusing IFN (pair μ, ν) with TFN (triple a, b, c) or IULS quaternion ([s_θ, s_τ], (u, v)): IF-TODIM values MUST be IFN tuples satisfying μ+ν ≤ 1, NOT triangular fuzzy numbers and NOT IULS quaternions.

• Forgetting the cost-criterion complement in F2: cost columns must be replaced by (ν, μ) before computing F5 dominance; otherwise the gain/loss branch fires on the wrong side and the ranking is inverted.

• Using the IF distance directly as the dominance sign instead of comparing Chen-Tan scores: d_E is SYMMETRIC. The asymmetry in φ_j (which alternative dominates which) MUST come from the score comparison s(r_ij*) vs s(r_kj*). Using d_j as the sign produces undirected dominance - the prospect-theory loss aversion is lost.

• Setting θ too small (e.g. θ < 0.1): the loss branch is amplified by 1/θ which can dominate the gain branch by orders of magnitude - small ζ differences then translate to extreme ξ values. Calibrate θ to the data; Krohling 2013 used θ = 1 and noted θ ∈ [1, 2.5] is typical in TODIM literature.

• Forgetting to choose the reference criterion C_r = argmax_j w_j before computing w_jr: if w_jr is computed against an arbitrary criterion (or unitary), the asymmetry of the gain/loss branches is rescaled and the resulting ξ ordering can shift. The reference must be the maximum-weight criterion per Krohling 2013 Step 2 and Liu-Teng 2015 Step 2.

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