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Ranking

PIF-ARTASI - Picture Fuzzy ARTASI (Kara et al. 2024)

Picture outranking/ranking - PiFS (μ, η, ν; μ+η+ν ≤ 1) + adaptive standardized intervals

Kara, K., Yalçın, G. C., Kaygısız, E. G., Simic, V., Örnek, A. Ş., Pamucar, D.2024doi:10.1016/j.asoc.2024.111826 ↗

Overview

PIF-ARTASI extends ARTASI (Kara et al. 2024) to Picture Fuzzy Sets. PFS-based qualitative criteria are aggregated via PFWA, converted to crisp via score function Sc, and combined with quantitative criteria before the ARTASI standardisation pipeline. Higher K is better.

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

How it works

1
Step 12: Construct per-expert PFS decision matrix W̃^(h)_{ij} for each expert h=1..d. Each cell is a PFS triple (μ, η, ν) with μ+η+ν ≤ 1.
$W_{t} i l d e_{ij}^{(h)} = (m u_{ij}^{(h)}, e t a_{ij}^{(h)}, n u_{ij}^{(h)}); co n s t r ain t m u + e t a + n u <= 1$
Kara et al. 2024, Step 12
2
Step 13 (Eq 15): Aggregate expert matrices via PFWA with expert weights w_h: PFWA(p_1..p_d) = ⟨1−∏(1−μ_h)^{w_h}, ∏ η_h^{w_h}, ∏(ν_h+η_h)^{w_h} − ∏ η_h^{w_h}⟩ (Wang 2017 form, preserves μ+η+ν ≤ 1).
$m u_{a} g g = 1 - p r o d_{h} (1 - m u_{h})^{w_{h}}; e t a_{a} g g = p r o d_{h} e t a_{h}^{w_{h}}; n u_{a} g g = p r o d_{h} (n u_{h} + e t a_{h})^{w_{h}} - p r o d_{h} e t a_{h}^{w_{h}}$
Kara et al. 2024, Eq (15); Wang et al. 2017 PFWA
3
Step 14 (Eq 16): Collapse PFS to crisp via score function U_{ij} = Sc(W̃_{ij}) = (μ + (1−η) + (1−ν))/3.
$U_{ij} = (m u_{ij} + (1 - e t a_{ij}) + (1 - n u_{ij})) /3$
Kara et al. 2024, Eq (16)
4
Step 15: Combine crisp qualitative U_{ij} (from PFS) with quantitative T_{ij} (crisp from start) into L_{ij}. If only qualitative criteria, L = U.
$L_{ij} = U_{ij} (q u a l i t a t i v e); L_{ij} = T_{ij} (q u an t i t a t i v e)$
Kara et al. 2024, Step 15
5
Step 16 (Eqs 17-18): Adaptive bounds per criterion j: S_j^max = max_i L_{ij} + (max_i L_{ij})^{1/m}; S_j^min = min_i L_{ij} − (min_i L_{ij})^{1/m}. m = number of alternatives.
$S_{j}^{m} a x = ma x_{i} L_{ij} + (ma x_{i} L_{ij})^{1/ m}; S_{j}^{m} in = mi n_{i} L_{ij} - (mi n_{i} L_{ij})^{1/ m}$
Kara et al. 2024, Eqs (17)-(18)
6
Step 17a-b (Eqs 19-20): Two-level standardization. First-level rate R_{ij} ∈ [0,1] direction-aware: for benefit R_{ij} = (L_{ij} − S_j^min)/(S_j^max − S_j^min); for cost R_{ij} = (S_j^max − L_{ij})/(S_j^max − S_j^min). Second-level scale C_{ij} = R_{ij}·(β^u − β^l) + β^l ∈ [β^l, β^u]; default (β^l, β^u) = (1, 100).
$R_{ij} = (L_{ij} - S_{j}^{m} in) / (S_{j}^{m} a x - S_{j}^{m} in) [b e n e f i t]; R_{ij} = (S_{j}^{m} a x - L_{ij}) / (S_{j}^{m} a x - S_{j}^{m} in) [cos t]; C_{ij} = R_{ij} * (b e t a_{u} - b e t a_{l}) + b e t a_{l}$
Kara et al. 2024, Eqs (19)-(20)
7
Step 18a-b (Eqs 21-22): Ideal/anti-ideal weighted utility via Tatar 2025 reversal-on-cost-only convention. Benefit: V_{ij}^+ = (C_{ij}/max_i C_{ij})·w_j·β^u; intermediate U_{ij} = (min_i C_{ij}/C_{ij})·w_j·β^u. Cost: swap roles of max and min. Then V_{ij}^- = −U_{ij} + max_i U_{ij} + min_i U_{ij}. (Tatar 2025 correction explicitly forbids the simple v_{ij}/v_j^- form.)
$V_{ij}^{+} = (C_{ij} / ma x_{i} C_{ij}) * w_{j} * b e t a_{u} [b e n e f i t]; V_{ij}^{+} = (mi n_{i} C_{ij} / C_{ij}) * w_{j} * b e t a_{u} [cos t]; U_{ij} = (mi n_{i} C_{ij} / C_{ij}) * w_{j} * b e t a_{u} [b e n e f i t] or (C_{ij} / ma x_{i} C_{ij}) * w_{j} * b e t a_{u} [cos t]; V_{ij}^{-} = - U_{ij} + ma x_{i} U_{ij} + mi n_{i} U_{ij}$
Kara et al. 2024, Eqs (21)-(22); Tatar 2025 §2 corrections
8
Step 19 (Eqs 24-25): Row sums N_i^+ = Σ_j V_{ij}^+; N_i^- = Σ_j V_{ij}^-.
$N_{i}^{+} = s u m_{j} V_{ij}^{+}; N_{i}^{-} = s u m_{j} V_{ij}^{-}$
Kara et al. 2024, Eqs (24)-(25)
9
Step 20 (Eq 26): Final utility K_i = (N_i^+ + N_i^-) · [ψ·f(N_i^+)^τ + (1−ψ)·f(N_i^-)^τ]^{1/τ}. Defaults ψ=0.5, τ=1 (Tatar 2025 application example), f = identity. Multiplicative juxtaposition between (N+ + N−) and bracket per math convention; higher K_i is better.
$K_{i} = (N_{i}^{+} + N_{i}^{-}) * (p s i * f (N_{i}^{+})^{t} a u + (1 - p s i) * f (N_{i}^{-})^{t} a u)^{(} 1/ t a u); f = i d e n t i t y; d e f a u l t s p s i = 0.5, t a u = 1$
Kara et al. 2024, Eq (26); Tatar 2025 application example

Fits when / Look elsewhere when

Fits when

•Preserves picture 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 ARTASI directly (avoid unnecessary uncertainty layer)
•Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Assumptions to verify

Decision matrix entries are valid Picture 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

Edge cases and pitfalls

•If only qualitative criteria, L = U.
•default (β^l, β^u) = (1, 100).

Applying PIF-ARTASI without verifying this assumption.
Requirement: Decision matrix entries are valid Picture Fuzzy numbers/tuples
Applying PIF-ARTASI without verifying this assumption.
Requirement: Underlying crisp method's compensation assumption holds in uncertain space
Applying PIF-ARTASI without verifying this assumption.
Requirement: All decision-maker(s) and experts use the same linguistic/uncertainty scale
Using PIF-ARTASI when: Crisp data sufficient.
An alternative method is recommended in this situation.
Using PIF-ARTASI when: Aggregation operator (PFWA/PFOWA/etc.) not specified.
An alternative method is recommended in this situation.

Works with

Commonly takes its weights from

n_a

How to cite

Kara, K.; Yalçın, G. C.; Kaygısız, E. G.; Simic, V.; Örnek, A. Ş.; Pamucar, D. (2024). A picture fuzzy CIMAS-ARTASI model for website performance analysis in human resource management. Applied Soft Computing. https://doi.org/10.1016/j.asoc.2024.111826

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

PIF-ARTASI

Overview

Output

utility, higher is better

Data

Picture Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

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

How it works

Step 12: Construct per-expert PFS decision matrix W̃^(h)_{ij} for each expert h=1..d. Each cell is a PFS triple (μ, η, ν) with μ+η+ν ≤ 1.

W_{t} i l d e_{ij}^{(h)} = (m u_{ij}^{(h)}, e t a_{ij}^{(h)}, n u_{ij}^{(h)}); co n s t r ain t m u + e t a + n u <= 1

Kara et al. 2024, Step 12

Step 13 (Eq 15): Aggregate expert matrices via PFWA with expert weights w_h: PFWA(p_1..p_d) = ⟨1−∏(1−μ_h)^{w_h}, ∏ η_h^{w_h}, ∏(ν_h+η_h)^{w_h} − ∏ η_h^{w_h}⟩ (Wang 2017 form, preserves μ+η+ν ≤ 1).

m u_{a} g g = 1 - p r o d_{h} (1 - m u_{h})^{w_{h}}; e t a_{a} g g = p r o d_{h} e t a_{h}^{w_{h}}; n u_{a} g g = p r o d_{h} (n u_{h} + e t a_{h})^{w_{h}} - p r o d_{h} e t a_{h}^{w_{h}}

Kara et al. 2024, Eq (15); Wang et al. 2017 PFWA

Step 14 (Eq 16): Collapse PFS to crisp via score function U_{ij} = Sc(W̃_{ij}) = (μ + (1−η) + (1−ν))/3.

U_{ij} = (m u_{ij} + (1 - e t a_{ij}) + (1 - n u_{ij})) /3

Kara et al. 2024, Eq (16)

Step 15: Combine crisp qualitative U_{ij} (from PFS) with quantitative T_{ij} (crisp from start) into L_{ij}. If only qualitative criteria, L = U.

L_{ij} = U_{ij} (q u a l i t a t i v e); L_{ij} = T_{ij} (q u an t i t a t i v e)

Kara et al. 2024, Step 15

Step 16 (Eqs 17-18): Adaptive bounds per criterion j: S_j^max = max_i L_{ij} + (max_i L_{ij})^{1/m}; S_j^min = min_i L_{ij} − (min_i L_{ij})^{1/m}. m = number of alternatives.

S_{j}^{m} a x = ma x_{i} L_{ij} + (ma x_{i} L_{ij})^{1/ m}; S_{j}^{m} in = mi n_{i} L_{ij} - (mi n_{i} L_{ij})^{1/ m}

Kara et al. 2024, Eqs (17)-(18)

Step 17a-b (Eqs 19-20): Two-level standardization. First-level rate R_{ij} ∈ [0,1] direction-aware: for benefit R_{ij} = (L_{ij} − S_j^min)/(S_j^max − S_j^min); for cost R_{ij} = (S_j^max − L_{ij})/(S_j^max − S_j^min). Second-level scale C_{ij} = R_{ij}·(β^u − β^l) + β^l ∈ [β^l, β^u]; default (β^l, β^u) = (1, 100).

R_{ij} = (L_{ij} - S_{j}^{m} in) / (S_{j}^{m} a x - S_{j}^{m} in) [b e n e f i t]; R_{ij} = (S_{j}^{m} a x - L_{ij}) / (S_{j}^{m} a x - S_{j}^{m} in) [cos t]; C_{ij} = R_{ij} * (b e t a_{u} - b e t a_{l}) + b e t a_{l}

Kara et al. 2024, Eqs (19)-(20)

Step 18a-b (Eqs 21-22): Ideal/anti-ideal weighted utility via Tatar 2025 reversal-on-cost-only convention. Benefit: V_{ij}^+ = (C_{ij}/max_i C_{ij})·w_j·β^u; intermediate U_{ij} = (min_i C_{ij}/C_{ij})·w_j·β^u. Cost: swap roles of max and min. Then V_{ij}^- = −U_{ij} + max_i U_{ij} + min_i U_{ij}. (Tatar 2025 correction explicitly forbids the simple v_{ij}/v_j^- form.)

V_{ij}^{+} = (C_{ij} / ma x_{i} C_{ij}) * w_{j} * b e t a_{u} [b e n e f i t]; V_{ij}^{+} = (mi n_{i} C_{ij} / C_{ij}) * w_{j} * b e t a_{u} [cos t]; U_{ij} = (mi n_{i} C_{ij} / C_{ij}) * w_{j} * b e t a_{u} [b e n e f i t] or (C_{ij} / ma x_{i} C_{ij}) * w_{j} * b e t a_{u} [cos t]; V_{ij}^{-} = - U_{ij} + ma x_{i} U_{ij} + mi n_{i} U_{ij}

Kara et al. 2024, Eqs (21)-(22); Tatar 2025 §2 corrections

Step 19 (Eqs 24-25): Row sums N_i^+ = Σ_j V_{ij}^+; N_i^- = Σ_j V_{ij}^-.

N_{i}^{+} = s u m_{j} V_{ij}^{+}; N_{i}^{-} = s u m_{j} V_{ij}^{-}

Kara et al. 2024, Eqs (24)-(25)

Step 20 (Eq 26): Final utility K_i = (N_i^+ + N_i^-) · [ψ·f(N_i^+)^τ + (1−ψ)·f(N_i^-)^τ]^{1/τ}. Defaults ψ=0.5, τ=1 (Tatar 2025 application example), f = identity. Multiplicative juxtaposition between (N+ + N−) and bracket per math convention; higher K_i is better.

K_{i} = (N_{i}^{+} + N_{i}^{-}) * (p s i * f (N_{i}^{+})^{t} a u + (1 - p s i) * f (N_{i}^{-})^{t} a u)^{(} 1/ t a u); f = i d e n t i t y; d e f a u l t s p s i = 0.5, t a u = 1

Kara et al. 2024, Eq (26); Tatar 2025 application example

Fits when / Look elsewhere when

Fits when

•Preserves picture 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 ARTASI directly (avoid unnecessary uncertainty layer)
•Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Edge cases and pitfalls

•If only qualitative criteria, L = U.

•default (β^l, β^u) = (1, 100).

Applying PIF-ARTASI without verifying this assumption.

Requirement: Decision matrix entries are valid Picture Fuzzy numbers/tuples

Applying PIF-ARTASI without verifying this assumption.

Requirement: Underlying crisp method's compensation assumption holds in uncertain space

Applying PIF-ARTASI without verifying this assumption.

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

Using PIF-ARTASI when: Crisp data sufficient.

An alternative method is recommended in this situation.

Using PIF-ARTASI when: Aggregation operator (PFWA/PFOWA/etc.) not specified.

An alternative method is recommended in this situation.

How to cite

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

PIF-ARTASI