HF-MARCOS: Hesitant Fuzzy MARCOS

Overview

Hesitant Fuzzy compromise ranking: MARCOS extended via Hesitant Fuzzy Elements (HFE). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Hesitant Fuzzy compromise ranking: MARCOS extended via Hesitant Fuzzy Elements (HFE)
• •Preserves hesitant 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 MARCOS family; cf. Stević et al. (2020) Sec. 3.3 on AI/AAI sensitivity to alternative set composition.)
• •Assumes: Each cell is a non-empty HFE with values in [0,1]
• •Assumes: Compensation across criteria is acceptable (MARCOS is fully compensatory)
• •Assumes: All experts share the same scale/interpretation of HFE membership degrees
• •Assumes: Criterion weights are externally sourced (HF-MARCOS does not produce weights)

Assumptions to verify

• •Each cell is a non-empty HFE with values in [0,1]
• •Compensation across criteria is acceptable (MARCOS is fully compensatory)
• •All experts share the same scale/interpretation of HFE membership degrees
• •Criterion weights are externally sourced (HF-MARCOS does not produce weights)

When not to use

• •Crisp data sufficient: use base MARCOS directly
• •Non-compensatory or outranking-style ranking is required (use ELECTRE/PROMETHEE family)
• •Probability distributions are available: prefer probabilistic linguistic / interval-valued methods

Edge cases

• •zero via epsilon_zero.

Common pitfalls

• •Hatalı: 'HF-MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Each cell is a non-empty HFE with values in [0,1]
• •Hatalı: 'HF-MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Compensation across criteria is acceptable (MARCOS is fully compensatory)
• •Hatalı: 'HF-MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All experts share the same scale/interpretation of HFE membership degrees
• •Hatalı: 'HF-MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criterion weights are externally sourced (HF-MARCOS does not produce weights)
• •Hatalı: HF-MARCOS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: HF-MARCOS'yi 'Non-compensatory or outranking-style ranking is required (use ELECTRE/PROMETHEE family)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: HF-MARCOS'yi 'Probability distributions are available' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Validate inputs: per-expert HFE decision matrices {H^(k)}_{k=1..K}, criterion weights w, criterion directions, expert weights λ. Formül: Check E-1..E-4; abort on error. Anchor: Li-Geng-Yuan 2023, §3.2
2. 2.Adım 2 (F2): Step 2: Aggregate K expert HFE matrices into a single HFE matrix H via HFWA: h_ij = HFWA_λ(h_ij^(1),…,h_ij^(K)) = ∪_{γ_k ∈ h_ij^(k)} { 1 − ∏_{k=1}^K (1 − γ_k)^{λ_k} }. (K=1 → identity) Formül: h_ij = HFWA_lambda(h_ij^(1),...,h_ij^(K)) = union_{gamma_k in h_ij^(k)} { 1 - prod_{k=1}^K (1 - gamma_k)^{lambda_k} } Anchor: Xia-Xu 2011 Def. 5; Li-Geng-Yuan 2023 §3.2 Eq.(7)
3. 3.Adım 3 (F3): Step 3: Compute HFE score x_ij = S(h_ij) = (1/|h_ij|) Σ_{γ ∈ h_ij} γ for every cell. Formül: x_ij = S(h_ij) = (1/|h_ij|) * sum_{gamma in h_ij} gamma Anchor: Xia-Xu 2011 Def. 6; Li-Geng-Yuan 2023 §3.2 Eq.(8)
4. 4.Adım 4 (F4): Step 4: Construct AI (ideal) and AAI (anti-ideal) per criterion: benefit C_j: AI_j = max_i x_ij, AAI_j = min_i x_ij; cost C_j: AI_j = min_i x_ij, AAI_j = max_i x_ij. Form extended matrix X_ext of shape (m+2)×n with rows [AAI; A_1; …; A_m; AI]. Formül: AI_j = max_i x_ij (benefit) | min_i x_ij (cost); AAI_j = min_i x_ij (benefit) | max_i x_ij (cost) Anchor: Stević 2020 Eq.(1); Li-Geng-Yuan 2023 §3.3 Eq.(9)
5. 5.Adım 5 (F5): Step 5: Normalise X_ext against AI: benefit C_j: n_ij = x_ij / AI_j; cost C_j: n_ij = AI_j / x_ij. Guard against division by zero via epsilon_zero. Formül: n_ij = x_ij / AI_j (benefit) | AI_j / x_ij (cost); replace x with max(x, epsilon_zero) for cost denominators Anchor: Stević 2020 Eq.(2); Li-Geng-Yuan 2023 §3.3 Eq.(10)
6. 6.Adım 6 (F6): Step 6: Weighted normalised matrix v_ij = w_j · n_ij; row-sum to get S_i = Σ_j v_ij for each row of X_ext (including AAI and AI rows). Formül: v_ij = w_j * n_ij; S_i = sum_j v_ij Anchor: Stević 2020 Eqs.(3)-(4); Li-Geng-Yuan 2023 §3.3 Eqs.(11)-(12)
7. 7.Adım 7 (F7): Step 7: Utility ratios: K^+_i = S_i / S_AI; K^-_i = S_i / S_AAI. Utility functions: f(K^+_i) = K^-_i / (K^+_i + K^-_i); f(K^-_i) = K^+_i / (K^+_i + K^-_i). Final utility f(K_i) = (K^+_i + K^-_i) / (1 + (1 − f(K^+_i))/f(K^+_i) + (1 − f(K^-_i))/f(K^-_i)). Rank by f(K_i) descending. Formül: Kp_i = S_i/S_AI; Km_i = S_i/S_AAI; f(Kp_i) = Km_i/(Kp_i+Km_i); f(Km_i) = Kp_i/(Kp_i+Km_i); f(K_i) = (Kp_i+Km_i) / (1 + (1-f(Kp_i))/f(Kp_i) + (1-f(Km_i))/f(Km_i)) Anchor: Stević 2020 Eqs.(5)-(9); Li-Geng-Yuan 2023 §3.3 Eqs.(13)-(17)

Commonly paired with

• •BWM + HF-MARCOS (common)
• •AHP + HF-MARCOS (common)

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