IF-PROMETHEE - Intuitionistic Fuzzy PROMETHEE

Overview

IF-PROMETHEE extends crisp PROMETHEE 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) computes pairwise signed deviation via the IF score function s = μ − ν, (iv) applies a preference function P_j (default Type V linear with thresholds q_j, p_j), (v) builds the multi-criteria preference index π(a, b) = Σ w_j P_j, (vi) derives leaving/entering flows φ⁺, φ⁻, (vii) the PROMETHEE II net flow φ = φ⁺ − φ⁻ ∈ [−1, 1]. Higher φ = better. Algorithm is INVARIANT across pure IF-PROMETHEE application papers (Liao-Xu 2014, Krishankumar et al. 2017); the preference function family and thresholds (q, p) are the parametric choices.

Output

preference flow, 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 (Krishankumar 2017 case), Alternative energy selection (Liao-Xu 2014 case), MAGDM under epistemic uncertainty with linguistic terms, Expert-driven evaluation with IFN scales

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}).

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

   Liao-Xu 2014 Step 1; Krishankumar 2017 Steps 1-2; 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. This converts the matrix to a uniform benefit-direction representation for pairwise comparison.

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

   Liao-Xu 2014 Step 2; Krishankumar 2017 cost-complement convention

3. 3

   Pairwise signed deviation per criterion: d_j(a, b) = s(r_aj*) − s(r_bj*) where s(·) is the IF score function (default Chen-Tan 1994: s(α) = μ − ν ∈ [−1, 1]). A positive d_j(a, b) means a outranks b on criterion j; a negative value means b outranks a. The deviation lives in [−2, 2].

   $$d_j(a,b)=s(r_{aj}^{\ast })-s(r_{bj}^{\ast })=({\mu }_{aj}^{\ast }-{\nu }_{aj}^{\ast })-({\mu }_{bj}^{\ast }-{\nu }_{bj}^{\ast })$$

   Liao-Xu 2014 Step 3 (signed IF distance); Krishankumar 2017 V function deviation; Chen-Tan 1994 score

4. 4

   Apply per-criterion preference function P_j(d) to the signed deviation. Default Type V (linear with thresholds): P_j(d) = 0 if d ≤ q_j; P_j(d) = (d − q_j)/(p_j − q_j) if q_j < d < p_j; P_j(d) = 1 if d ≥ p_j. The output P_j(d_j(a, b)) ∈ [0, 1] is the strength of preference of a over b on criterion j.

   $$P_j(d)=\{\begin{array}{ll}0,& d\le q_j\\ \frac{d-q_j}{p_j-q_j},& q_j<d<p_j\\ 1,& d\ge p_j\end{array}$$

   Liao-Xu 2014 Step 4 (Type V); Krishankumar 2017 V function; Brans-Vincke 1985 Type V

5. 5

   Multi-criteria preference index: π(a, b) = Σ_j w_j · P_j(d_j(a, b)) ∈ [0, 1]. π is asymmetric (π(a, b) and π(b, a) are independent) and represents how strongly a outranks b across all criteria, weighted by w_j.

   $$\pi (a,b)=\sum _{j=1}^n{w}_j\cdot P_j(d_j(a,b)),\pi (a,b)\in [0,1]$$

   Liao-Xu 2014 Step 5; Krishankumar 2017 IFPR aggregation; Brans-Mareschal 2002 Eq.(5)

6. 6

   Outranking flows. Leaving (positive) flow φ⁺(a) = (1/(m−1)) Σ_{b ≠ a} π(a, b) measures how much a outranks the rest of the set. Entering (negative) flow φ⁻(a) = (1/(m−1)) Σ_{b ≠ a} π(b, a) measures how much a is outranked by the rest. Both lie in [0, 1].

   $${\varphi }^{+}(a)=\frac{1}{m-1}\sum _{b\ne a}\pi (a,b);{\varphi }^{-}(a)=\frac{1}{m-1}\sum _{b\ne a}\pi (b,a)$$

   Liao-Xu 2014 Step 6; Krishankumar 2017 φ⁺/φ⁻ flows; Brans-Mareschal 2002 Eqs.(6)-(7)

7. 7

   Net outranking flow φ(a) = φ⁺(a) − φ⁻(a) ∈ [−1, 1]. By PROMETHEE II convention this is the single defuzzified score per alternative. Σ_a φ(a) = 0 always holds (algebraic identity).

   $$\varphi (a)={\varphi }^{+}(a)-{\varphi }^{-}(a),\varphi (a)\in [-1,1],\sum _a\varphi (a)=0$$

   Liao-Xu 2014 Step 7; Krishankumar 2017 net flow; Brans-Mareschal 2002 Eq.(8)

8. 8

   PROMETHEE II complete ranking: sort alternatives in descending order of φ(a); the largest φ(a) is the best. PROMETHEE I partial ranking (a P b iff φ⁺(a) ≥ φ⁺(b) and φ⁻(a) ≤ φ⁻(b) with at least one strict; incomparability otherwise) is available as a secondary output.

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

   Liao-Xu 2014 Step 8; Krishankumar 2017 ranking; Brans-Mareschal 2002 PROMETHEE I/II

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
• Per-criterion thresholds (q_j, p_j) calibrated to the data's deviation scale

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. This converts the matrix to a uniform benefit-direction representation for pairwise comparison.
• •default Chen-Tan 1994: s(α) = μ − ν ∈ [−1, 1]). A positive d_j(a, b) means a outranks b on criterion j; a negative value means b outranks a. The deviation lives in [−2, 2].
• •Default Type V (linear with thresholds): P_j(d) = 0 if d ≤ q_j; P_j(d) = (d − q_j)/(p_j − q_j) if q_j < d < p_j; P_j(d) = 1 if d ≥ p_j. The output P_j(d_j(a, b)) ∈ [0, 1] is the strength of preference

• Confusing IFN (pair μ, ν) with TFN (triple a, b, c): IF-PROMETHEE values MUST be IFN tuples satisfying μ+ν ≤ 1, NOT triangular fuzzy numbers.

• Forgetting the cost-criterion complement in F2: cost columns must be replaced by (ν, μ) before pairwise comparison; otherwise the direction is inverted and the ranking is wrong.

• Using the absolute deviation |d_j(a, b)| in F3 instead of the signed difference: PROMETHEE's preference function P_j is designed to be zero for non-positive deviations. Taking the absolute value destroys the asymmetry of π(a, b) vs π(b, a).

• Choosing p_j too small (e.g. p_j = q_j): the linear ramp collapses and P_j becomes a step function - every preferring pair gets weight 1 regardless of magnitude. Calibrate p_j to the actual deviation scale of your data (e.g. ~max d_j observed).

• Reporting only PROMETHEE II net flow when partial-order PROMETHEE I disagrees: φ_net can rank two alternatives but the φ⁺/φ⁻ pair may flag them as incomparable. Surface both views in the engine output.

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