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Ranking

PF-VIKOR - Pythagorean extension of VIKOR

Pythagorean outranking/ranking - Pythagorean Fuzzy Number (PFN: μ, ν; μ²+ν² ≤ 1)

Abbas, S., Hussain, Z., Hussain, Z., Ali, I., Mudabar, S.M.2024doi:10.25102/fer.2024.02.03 ↗

Overview

pf-vikor extends VIKOR to handle Pythagorean uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Pythagorean Fuzzy Number (PFN: μ, ν; μ²+ν² ≤ 1) algebra. The final scores are defuzzified via score function S = μ² − ν² before ranking.

Output: utility, lower is better
Data: Pythagorean Fuzzy, uncertainty tuples complete
Weights: Needs a weight source
Size: 2+ alternatives, 3-10 criteria works best
Used for: Pythagorean Fuzzy MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1
Construct the Pythagorean fuzzy decision matrix X = [α_ij]_{m×n} where each entry α_ij = (μ_ij, ν_ij) is a PFN satisfying μ_ij²+ν_ij²≤1, with associated hesitancy π_ij² = 1−μ_ij²−ν_ij².
$α_{i} j = (μ_{i} j, ν_{i} j) \in Φ (U); μ_{i} j, ν_{i} j \in [0, 1]; μ_{i} j^{2} + ν_{i} j^{2} \leq 1; π_{i} j^{2} = 1 - μ_{i} j^{2} - ν_{i} j^{2} (A bba s §4 S t e p 1; Y a g er 2013 P F S d e f)$
Abbas 2024 §4 Step 1; Yager 2013 PFS definition
2
Weights w_j per criterion are either externally provided (e.g. from AHP, BWM, ENTROPY) or computed from PFN entropy. Abbas 2024 proposes E_SA-entropy then W_j = E_SA(C_j) / Σ_k E_SA(C_k). The PF-VIKOR algorithm proper is weight-source-agnostic; w_j with Σ w_j = 1 is the only requirement.
$w_{j} \geq 0, Σ_{j} w_{j} = 1. O pt i o na l : W_{j} = E_{S} A (C_{j}) / Σ_{k} E_{S} A (C_{k}) w h er e E_{S} A (C_{j}) = (1/ (2 m)) Σ_{i} [∣ (1 + μ_{i} j^{2} + π_{i} j^{2}) /2 - ν_{i} j^{2} ∣]^{(} 1/2) (A bba s 2024 E q . (5)) .$
Abbas 2024 §4 Steps 2-3; Eq.(5)
3
Determine the synthetic best PFN f*_j and worst PFN f^-_j per criterion. For benefit criterion j: f*_j = (max_i μ_ij, min_i ν_ij); f^-_j = (min_i μ_ij, max_i ν_ij). For cost criterion j: swap (f*_j and f^-_j roles inverted). The reference points are synthetic constructions; they need not equal any actual alternative.
$I f j b e n e f i t : f *_{j} = (ma x_{i} μ_{i} j, mi n_{i} ν_{i} j), f_{j}^{-} = (mi n_{i} μ_{i} j, ma x_{i} ν_{i} j) . I f j cos t : f *_{j} = (mi n_{i} μ_{i} j, ma x_{i} ν_{i} j), f_{j}^{-} = (ma x_{i} μ_{i} j, mi n_{i} ν_{i} j) . (A bba s 2024§4 S t e p 4)$
Abbas 2024 §4 Step 4
4
Compute per-cell normalised PF gap using PF distance. Abbas 2024 uses PF Hamming distance over squared components: d_H(α,β) = (|μ_α²−μ_β²| + |ν_α²−ν_β²| + |π_α²−π_β²|)/3. The normalised gap is Δ_ij = w_j · d_H(α_ij, f*_j) / d_H(f^-_j, f*_j). PF Euclidean distance d_E is a documented alternative.
$d_{H} (α, β) = (1/3) \cdot [∣ μ_{α}^{2} - μ_{β}^{2} ∣ + ∣ ν_{α}^{2} - ν_{β}^{2} ∣ + ∣ π_{α}^{2} - π_{β}^{2} ∣]; Δ_{i} j = w_{j} \cdot d_{H} (α_{i} j, f *_{j}) / d_{H} (f_{j}^{-}, f *_{j}) . (A bba s 2024§4 S t e p 5)$
Abbas 2024 §4 Step 5
5
Compute group utility S_i (L_1-aggregated) and individual regret R_i (L_∞-aggregated) over the normalised gap matrix Δ.
$S_{i} = Σ_{j} Δ_{i} j = Σ_{j} [w_{j} \cdot d_{H} (α_{i} j, f *_{j}) / d_{H} (f_{j}^{-}, f *_{j})]; R_{i} = ma x_{j} Δ_{i} j . (A bba s 2024§4 S t e p 5; O p r i co v i c 1998 V I K O R L_{1} / L_{\infty} a g g r e g a t i o n)$
Abbas 2024 §4 Step 5; Opricovic 1998
6
Compute the VIKOR compromise index Q_i as a convex combination of normalised S and R, weighted by strategy parameter β (default β=0.5; β>0.5 favours group utility, β<0.5 favours individual regret). S* = min_i S_i, S^- = max_i S_i; R* = min_i R_i, R^- = max_i R_i.
$Q_{i} = β \cdot (S_{i} - S *) / (S^{-} - S *) + (1 - β) \cdot (R_{i} - R *) / (R^{-} - R *) . (A bba s 2024§4 S t e p 7; O p r i co v i c 1998)$
Abbas 2024 §4 Step 7
7
Produce three rankings by sorting Q, S, R ascending. The alternative with the smallest Q is the candidate compromise solution.
$r ank in g_{Q} = a r g sor t_{a} sc (Q); r ank in g_{S} = a r g sor t_{a} sc (S); r ank in g_{R} = a r g sor t_{a} sc (R) . B es t a l t er na t i v e = r ank in g_{Q} [0] . (A bba s 2024§4 S t e p 8)$
Abbas 2024 §4 Step 8; Opricovic 1998 VIKOR
8
Apply the two-condition VIKOR compromise test. C1 (acceptable advantage): Q(A^(2)) − Q(A^(1)) ≥ DQ where DQ = 1/(m−1). C2 (acceptable stability in decision-making): A^(1) is also best ranked by S and/or R. If both hold, propose A^(1) as compromise solution. If C1 fails, return the longest prefix {A^(1)..A^(M)} satisfying Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}.
$C 1 : Q (A^{(} 2)) - Q (A^{(} 1)) \geq 1/ (m - 1) . C 2 : A^{(} 1) = r ank in g_{S} [0] O R A^{(} 1) = r ank in g_{R} [0] . co m p r o mi s e_{s} e t = i f C 1 \land C 2 t h e n A^{(} 1) e l se i f \neg C 1 t h e n p r e f i x (Q - Q^{(} 1) < D Q) e l se i f \neg C 2 t h e n A^{(} 1), A^{(} 2) . (O p r i co v i c 1998 V I K O R co m p r o mi se t es t)$
Opricovic 1998 VIKOR compromise test (inherited)

Fits when / Look elsewhere when

Fits when

•Preserves pythagorean uncertainty through the pipeline rather than premature crispification at elicitation
•Native group-decision support (multi-DM aggregation built into the pipeline)

Look elsewhere when

•Crisp data sufficient - use base VIKOR directly (avoid unnecessary uncertainty layer)
•Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Assumptions to verify

Decision matrix entries are valid Pythagorean Fuzzy numbers/tuples
Underlying crisp method's compensation assumption holds in uncertain space
All decision-maker(s) and experts use the same linguistic/uncertainty scale

Limitations

•Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)

Edge cases and pitfalls

•default β=0.5; β>0.5 favours group utility, β<0.5 favours individual regret). S* = min_i S_i, S^- = max_i S_i; R* = min_i R_i, R^- = max_i R_i.
•If both hold, propose A^(1) as compromise solution. If C1 fails, return the longest prefix {A^(1)..A^(M)} satisfying Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}.

Value-space violation: ensure all entries satisfy PFN: μ ∈ [0,1], ν ∈ [0,1], μ²+ν² ≤ 1 before computation.
Defuzzification method affects ranking: score function S = μ² − ν² is the canonical choice but alternatives exist.

Works with

Commonly takes its weights from

n_a

How to cite

Abbas, S.; Hussain, Z.; Hussain, Z.; Ali, I.; Mudabar, S.M. (2024). Advanced Entropy Models in Pythagorean Fuzzy Sets: Revolutionizing Multi-Criteria Decision Making with PF-VIKOR. Fuzzy Economic Review. https://doi.org/10.25102/fer.2024.02.03

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

PF-VIKOR

Overview

Output

utility, lower is better

Data

Pythagorean Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Pythagorean Fuzzy MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

Construct the Pythagorean fuzzy decision matrix X = [α_ij]_{m×n} where each entry α_ij = (μ_ij, ν_ij) is a PFN satisfying μ_ij²+ν_ij²≤1, with associated hesitancy π_ij² = 1−μ_ij²−ν_ij².

α_{i} j = (μ_{i} j, ν_{i} j) \in Φ (U); μ_{i} j, ν_{i} j \in [0, 1]; μ_{i} j^{2} + ν_{i} j^{2} \leq 1; π_{i} j^{2} = 1 - μ_{i} j^{2} - ν_{i} j^{2} (A bba s §4 S t e p 1; Y a g er 2013 P F S d e f)

Abbas 2024 §4 Step 1; Yager 2013 PFS definition

Weights w_j per criterion are either externally provided (e.g. from AHP, BWM, ENTROPY) or computed from PFN entropy. Abbas 2024 proposes E_SA-entropy then W_j = E_SA(C_j) / Σ_k E_SA(C_k). The PF-VIKOR algorithm proper is weight-source-agnostic; w_j with Σ w_j = 1 is the only requirement.

w_{j} \geq 0, Σ_{j} w_{j} = 1. O pt i o na l : W_{j} = E_{S} A (C_{j}) / Σ_{k} E_{S} A (C_{k}) w h er e E_{S} A (C_{j}) = (1/ (2 m)) Σ_{i} [∣ (1 + μ_{i} j^{2} + π_{i} j^{2}) /2 - ν_{i} j^{2} ∣]^{(} 1/2) (A bba s 2024 E q . (5)) .

Abbas 2024 §4 Steps 2-3; Eq.(5)

Determine the synthetic best PFN f*_j and worst PFN f^-_j per criterion. For benefit criterion j: f*_j = (max_i μ_ij, min_i ν_ij); f^-_j = (min_i μ_ij, max_i ν_ij). For cost criterion j: swap (f*_j and f^-_j roles inverted). The reference points are synthetic constructions; they need not equal any actual alternative.

I f j b e n e f i t : f *_{j} = (ma x_{i} μ_{i} j, mi n_{i} ν_{i} j), f_{j}^{-} = (mi n_{i} μ_{i} j, ma x_{i} ν_{i} j) . I f j cos t : f *_{j} = (mi n_{i} μ_{i} j, ma x_{i} ν_{i} j), f_{j}^{-} = (ma x_{i} μ_{i} j, mi n_{i} ν_{i} j) . (A bba s 2024§4 S t e p 4)

Abbas 2024 §4 Step 4

d_{H} (α, β) = (1/3) \cdot [∣ μ_{α}^{2} - μ_{β}^{2} ∣ + ∣ ν_{α}^{2} - ν_{β}^{2} ∣ + ∣ π_{α}^{2} - π_{β}^{2} ∣]; Δ_{i} j = w_{j} \cdot d_{H} (α_{i} j, f *_{j}) / d_{H} (f_{j}^{-}, f *_{j}) . (A bba s 2024§4 S t e p 5)

Abbas 2024 §4 Step 5

Compute group utility S_i (L_1-aggregated) and individual regret R_i (L_∞-aggregated) over the normalised gap matrix Δ.

S_{i} = Σ_{j} Δ_{i} j = Σ_{j} [w_{j} \cdot d_{H} (α_{i} j, f *_{j}) / d_{H} (f_{j}^{-}, f *_{j})]; R_{i} = ma x_{j} Δ_{i} j . (A bba s 2024§4 S t e p 5; O p r i co v i c 1998 V I K O R L_{1} / L_{\infty} a g g r e g a t i o n)

Abbas 2024 §4 Step 5; Opricovic 1998

Compute the VIKOR compromise index Q_i as a convex combination of normalised S and R, weighted by strategy parameter β (default β=0.5; β>0.5 favours group utility, β<0.5 favours individual regret). S* = min_i S_i, S^- = max_i S_i; R* = min_i R_i, R^- = max_i R_i.

Q_{i} = β \cdot (S_{i} - S *) / (S^{-} - S *) + (1 - β) \cdot (R_{i} - R *) / (R^{-} - R *) . (A bba s 2024§4 S t e p 7; O p r i co v i c 1998)

Abbas 2024 §4 Step 7

Produce three rankings by sorting Q, S, R ascending. The alternative with the smallest Q is the candidate compromise solution.

r ank in g_{Q} = a r g sor t_{a} sc (Q); r ank in g_{S} = a r g sor t_{a} sc (S); r ank in g_{R} = a r g sor t_{a} sc (R) . B es t a l t er na t i v e = r ank in g_{Q} [0] . (A bba s 2024§4 S t e p 8)

Abbas 2024 §4 Step 8; Opricovic 1998 VIKOR

Apply the two-condition VIKOR compromise test. C1 (acceptable advantage): Q(A^(2)) − Q(A^(1)) ≥ DQ where DQ = 1/(m−1). C2 (acceptable stability in decision-making): A^(1) is also best ranked by S and/or R. If both hold, propose A^(1) as compromise solution. If C1 fails, return the longest prefix {A^(1)..A^(M)} satisfying Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}.

C 1 : Q (A^{(} 2)) - Q (A^{(} 1)) \geq 1/ (m - 1) . C 2 : A^{(} 1) = r ank in g_{S} [0] O R A^{(} 1) = r ank in g_{R} [0] . co m p r o mi s e_{s} e t = i f C 1 \land C 2 t h e n A^{(} 1) e l se i f \neg C 1 t h e n p r e f i x (Q - Q^{(} 1) < D Q) e l se i f \neg C 2 t h e n A^{(} 1), A^{(} 2) . (O p r i co v i c 1998 V I K O R co m p r o mi se t es t)

Opricovic 1998 VIKOR compromise test (inherited)

Fits when / Look elsewhere when

Fits when

•Preserves pythagorean uncertainty through the pipeline rather than premature crispification at elicitation
•Native group-decision support (multi-DM aggregation built into the pipeline)

Look elsewhere when

•Crisp data sufficient - use base VIKOR directly (avoid unnecessary uncertainty layer)
•Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Assumptions to verify

Decision matrix entries are valid Pythagorean Fuzzy numbers/tuples

Underlying crisp method's compensation assumption holds in uncertain space

All decision-maker(s) and experts use the same linguistic/uncertainty scale

Limitations

•Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)

Edge cases and pitfalls

•default β=0.5; β>0.5 favours group utility, β<0.5 favours individual regret). S* = min_i S_i, S^- = max_i S_i; R* = min_i R_i, R^- = max_i R_i.

•If both hold, propose A^(1) as compromise solution. If C1 fails, return the longest prefix {A^(1)..A^(M)} satisfying Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}.

Value-space violation: ensure all entries satisfy PFN: μ ∈ [0,1], ν ∈ [0,1], μ²+ν² ≤ 1 before computation.

Defuzzification method affects ranking: score function S = μ² − ν² is the canonical choice but alternatives exist.

How to cite

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

PF-VIKOR