IF-PROMETHEE: Intuitionistic Fuzzy PROMETHEE

Overview

Pairwise outranking under Intuitionistic Fuzzy uncertainty. Output typically preference_flow (higher value = preferred).

Strengths

• •Method-specific: Pairwise outranking 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 PROMETHEE base (Brans-Vincke 1985; Wang-Luo 2009): outranking flows are pairwise normalised by (m−1), so adding/removing an alternative rebalances every φ⁺/φ⁻ and may invert the net-flow order.)
• •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

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

When not to use

• •Small dataset (m<3): outranking machinery underutilised; use IF-EDAS or IF-TOPSIS instead
• •Compensatory utility ranking is preferred (use IF-EDAS or IF-TOPSIS)
• •Hesitation degree π must be modelled explicitly with μ+ν > 1 raw judgements: consider Pythagorean Fuzzy PROMETHEE or q-ROF PROMETHEE

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. 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

Common pitfalls

• •Hatalı: 'IF-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid IFN tuples (μ, ν) with μ+ν ≤ 1
• •Hatalı: 'IF-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-makers use the same linguistic-to-IFN scale
• •Hatalı: 'IF-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criterion directions (benefit/cost) are explicitly labelled
• •Hatalı: 'IF-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criterion weights form a simplex
• •Hatalı: 'IF-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Per-criterion thresholds (q_j, p_j) calibrated to the data's deviation scale
• •Hatalı: IF-PROMETHEE'yi 'Small dataset (m<3)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: IF-PROMETHEE'yi 'Compensatory utility ranking is preferred (use IF-EDAS or IF-TOPSIS)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: IF-PROMETHEE'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}). 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: Liao-Xu 2014 Step 1; Krishankumar 2017 Steps 1-2; 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. This converts the matrix to a uniform benefit-direction representation for pairwise comparison. Formül: j \in J_c: r_{ij}^* = (\nu_{ij}, \mu_{ij});\quad j \in J_b: r_{ij}^* = r_{ij} Anchor: Liao-Xu 2014 Step 2; Krishankumar 2017 cost-complement convention
3. 3.Adım 3 (F3): Step 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]. Formül: d_j(a, b) = s(r_{aj}^*) - s(r_{bj}^*) = (\mu_{aj}^* - \nu_{aj}^*) - (\mu_{bj}^* - \nu_{bj}^*) Anchor: Liao-Xu 2014 Step 3 (signed IF distance); Krishankumar 2017 V function deviation; Chen-Tan 1994 score
4. 4.Adım 4 (F4): Step 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. Formül: P_j(d) = \begin{cases} 0, & d \le q_j \\ \dfrac{d - q_j}{p_j - q_j}, & q_j < d < p_j \\ 1, & d \ge p_j \end{cases} Anchor: Liao-Xu 2014 Step 4 (Type V); Krishankumar 2017 V function; Brans-Vincke 1985 Type V
5. 5.Adım 5 (F5): Step 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. Formül: \pi(a, b) = \sum_{j=1}^{n} w_j \cdot P_j(d_j(a, b)),\quad \pi(a, b) \in [0, 1] Anchor: Liao-Xu 2014 Step 5; Krishankumar 2017 IFPR aggregation; Brans-Mareschal 2002 Eq.(5)
6. 6.Adım 6 (F6): Step 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]. Formül: \phi^+(a) = \frac{1}{m-1}\sum_{b \ne a} \pi(a, b);\quad \phi^-(a) = \frac{1}{m-1}\sum_{b \ne a} \pi(b, a) Anchor: Liao-Xu 2014 Step 6; Krishankumar 2017 φ⁺/φ⁻ flows; Brans-Mareschal 2002 Eqs.(6)-(7)
7. 7.Adım 7 (F7): Step 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). Formül: \phi(a) = \phi^+(a) - \phi^-(a),\quad \phi(a) \in [-1, 1],\quad \sum_{a} \phi(a) = 0 Anchor: Liao-Xu 2014 Step 7; Krishankumar 2017 net flow; Brans-Mareschal 2002 Eq.(8)
8. 8.Adım 8 (F8): Step 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. Formül: \text{rank}(A_i) = \text{argsort}_{\downarrow}(\phi(A_i)) Anchor: Liao-Xu 2014 Step 8; Krishankumar 2017 ranking; Brans-Mareschal 2002 PROMETHEE I/II

Commonly paired with

• •IF-AHP + IF-PROMETHEE (common)
• •IF-SWARA + IF-PROMETHEE (common)

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