PiF-MARCOS: Picture extension of MARCOS

Overview

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

Strengths

• •Method-specific: Picture outranking/ranking: Picture Fuzzy Set (PiFS: μ, η, ν; μ+η+ν ≤ 1)
• •Preserves picture 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 base; cf. Belton-Gear 1983, Wang-Luo 2009)
• •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 MARCOS directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •See F.steps and D.parameters for PIF-MARCOS-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'PIF-MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Picture Fuzzy numbers/tuples
• •Hatalı: 'PIF-MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'PIF-MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: PIF-MARCOS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PIF-MARCOS'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 1: Form Picture Fuzzy decision matrix X = [r_ij] with r_ij = (μ_ij, η_ij, ν_ij), μ_ij+η_ij+ν_ij ≤ 1. Formül: X = [r_{ij}]_{m×n}, r_{ij} = (μ_{ij}, η_{ij}, ν_{ij}), μ_{ij}+η_{ij}+ν_{ij} ≤ 1 Anchor: Cuong 2013, Definition 1
2. 2.Adım 2 (F2): Step 2: Cost-complement normalize: for cost criteria, (μ,η,ν) → (ν,η,μ); benefit criteria unchanged. Result: R = [r̂_ij] all in benefit direction. Formül: r̂_{ij} = (ν_{ij}, η_{ij}, μ_{ij}) if j∈Cost; r̂_{ij} = (μ_{ij}, η_{ij}, ν_{ij}) if j∈Benefit Anchor: Tatar & Ayvaz 2024, §3.3 step 2; Cuong 2013 complement
3. 3.Adım 3 (F3): Step 3: Extend R with anti-ideal AAI and ideal AI rows: AAI_j = argmin_i s(r̂_ij), AI_j = argmax_i s(r̂_ij), where s(α) = (1+μ−ν)/2 is the score function. Formül: AAI_j = r̂_{i*j} \text{ where } i* = \arg\min_i s(r̂_{ij}); \quad AI_j = r̂_{i**j} \text{ where } i** = \arg\max_i s(r̂_{ij}); \quad s(α) = (1+μ-ν)/2 Anchor: Stević 2020 Eq.(1); Tatar & Ayvaz 2024 §3.3 step 3
4. 4.Adım 4 (F4): Step 4: Per-row PFWA weighted aggregation: S_i = PFWA(r̂_i1,…,r̂_in; w). Apply to all m alternative rows plus AAI and AI rows. Formül: S_i = \text{PFWA}(r̂_{i1},…,r̂_{in}; w) = \left(1 - \prod_{j=1}^{n}(1-μ_{ij})^{w_j},\; \prod_{j=1}^{n} η_{ij}^{w_j},\; \prod_{j=1}^{n} ν_{ij}^{w_j}\right) Anchor: Wei 2017 Eq.(8); Tatar & Ayvaz 2024 §3.3 step 4
5. 5.Adım 5 (F5): Step 5: Defuzzify via score function: s_i = s(S_i) = (1+μ_i−ν_i)/2 for each alternative and for AAI, AI rows. Formül: s_i = (1 + μ_i - ν_i)/2; \quad s_{AAI} = (1 + μ_{AAI} - ν_{AAI})/2; \quad s_{AI} = (1 + μ_{AI} - ν_{AI})/2 Anchor: Cuong 2013 score function; Stević 2020 Eq.(2)-(4) crisp analogue
6. 6.Adım 6 (F6): Step 6: Utility ratios vs ideal and anti-ideal: K_i^+ = s_i / s_AI, K_i^- = s_i / s_AAI. Formül: K_i^+ = s_i / s_{AI}; \quad K_i^- = s_i / s_{AAI} Anchor: Stević 2020 Eqs.(5)-(6); Tatar & Ayvaz 2024 §3.3 step 5
7. 7.Adım 7 (F7): Step 7: Utility functions: f(A_i^+) = K_i^- / (K_i^+ + K_i^-), f(A_i^-) = K_i^+ / (K_i^+ + K_i^-). Formül: f(A_i^+) = K_i^- / (K_i^+ + K_i^-); \quad f(A_i^-) = K_i^+ / (K_i^+ + K_i^-) Anchor: Stević 2020 Eqs.(7)-(8); Tatar & Ayvaz 2024 Table 2 confirms constants f(A^-)=0.4634, f(A^+)=0.5366 in their example
8. 8.Adım 8 (F8): Step 8: Final utility f(K_i) = (K_i^+ + K_i^-) / (1 + (1−f(A_i^+))/f(A_i^+) + (1−f(A_i^-))/f(A_i^-)); rank alternatives in descending order of f(K_i). Formül: f(K_i) = \frac{K_i^+ + K_i^-}{1 + \frac{1-f(A_i^+)}{f(A_i^+)} + \frac{1-f(A_i^-)}{f(A_i^-)}} Anchor: Stević 2020 Eq.(9); Tatar & Ayvaz 2024 Table 2

Commonly paired with

• •n_a + PIF-MARCOS (common)

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