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

Overview

Picture outranking/ranking: PiFS (μ, η, ν; μ+η+ν ≤ 1) + adaptive standardized intervals. Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Picture outranking/ranking: PiFS (μ, η, ν; μ+η+ν ≤ 1) + adaptive standardized intervals
• •Preserves picture uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

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

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

When not to use

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

Edge cases

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

Common pitfalls

• •Hatalı: 'PIF-ARTASI bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Picture Fuzzy numbers/tuples
• •Hatalı: 'PIF-ARTASI bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'PIF-ARTASI bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: PIF-ARTASI'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PIF-ARTASI'yi 'Aggregation operator (PFWA/PFOWA/etc.) not specified' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): 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. Formül: W_tilde^{(h)}_{ij} = (mu^{(h)}_{ij}, eta^{(h)}_{ij}, nu^{(h)}_{ij}) ; constraint mu+eta+nu <= 1 Anchor: Kara et al. 2024, Step 12
2. 2.Adım 2 (F2): 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). Formül: mu_agg = 1 - prod_h (1 - mu_h)^{w_h} ; eta_agg = prod_h eta_h^{w_h} ; nu_agg = prod_h (nu_h + eta_h)^{w_h} - prod_h eta_h^{w_h} Anchor: Kara et al. 2024, Eq (15); Wang et al. 2017 PFWA
3. 3.Adım 3 (F3): Step 14 (Eq 16): Collapse PFS to crisp via score function U_{ij} = Sc(W̃_{ij}) = (μ + (1−η) + (1−ν))/3. Formül: U_{ij} = (mu_{ij} + (1 - eta_{ij}) + (1 - nu_{ij})) / 3 Anchor: Kara et al. 2024, Eq (16)
4. 4.Adım 4 (F4): 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. Formül: L_{ij} = U_{ij} (qualitative) ; L_{ij} = T_{ij} (quantitative) Anchor: Kara et al. 2024, Step 15
5. 5.Adım 5 (F5): 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. Formül: 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} Anchor: Kara et al. 2024, Eqs (17)-(18)
6. 6.Adım 6 (F6): 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). Formül: R_{ij} = (L_{ij} - S_j^min)/(S_j^max - S_j^min) [benefit] ; R_{ij} = (S_j^max - L_{ij})/(S_j^max - S_j^min) [cost] ; C_{ij} = R_{ij}*(beta_u - beta_l) + beta_l Anchor: Kara et al. 2024, Eqs (19)-(20)
7. 7.Adım 7 (F7): 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.) Formül: V_{ij}^+ = (C_{ij}/max_i C_{ij})*w_j*beta_u [benefit] ; V_{ij}^+ = (min_i C_{ij}/C_{ij})*w_j*beta_u [cost] ; U_{ij} = (min_i C_{ij}/C_{ij})*w_j*beta_u [benefit] or (C_{ij}/max_i C_{ij})*w_j*beta_u [cost] ; V_{ij}^- = -U_{ij} + max_i U_{ij} + min_i U_{ij} Anchor: Kara et al. 2024, Eqs (21)-(22); Tatar 2025 §2 corrections
8. 8.Adım 8 (F8): Step 19 (Eqs 24-25): Row sums N_i^+ = Σ_j V_{ij}^+; N_i^- = Σ_j V_{ij}^-. Formül: N_i^+ = sum_j V_{ij}^+ ; N_i^- = sum_j V_{ij}^- Anchor: Kara et al. 2024, Eqs (24)-(25)
9. 9.Adım 9 (F9): 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. Formül: K_i = (N_i^+ + N_i^-) * ( psi * f(N_i^+)^tau + (1 - psi) * f(N_i^-)^tau )^(1/tau) ; f = identity ; defaults psi=0.5, tau=1 Anchor: Kara et al. 2024, Eq (26); Tatar 2025 application example

Commonly paired with

• •n_a + PIF-ARTASI (common)

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