Z-MARCOS: Z-Number extension of MARCOS

Overview

Z-Number outranking/ranking: Z-Number (Z = (A, B): restriction A, reliability B; both fuzzy). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Z-Number outranking/ranking: Z-Number (Z = (A, B): restriction A, reliability B; both fuzzy)
• •Preserves z_number 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 Z-Number 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 Z-Number 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 Z-MARCOS-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'Z-MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Z-Number numbers/tuples
• •Hatalı: 'Z-MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'Z-MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: Z-MARCOS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: Z-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: Validate Z-number decision matrix Z_{ij}=(A_{ij},B_{ij}); convert to TFN via Kang 2012 \tilde{A}'_{ij}(x)=\sqrt{\alpha_{ij}}\cdot\mu_{A_{ij}}(x) with \alpha_{ij}=\int x\mu_{B_{ij}}(x)dx/\int\mu_{B_{ij}}(x)dx and centroid-defuzzify to crisp x_{ij}; then form extended matrix X^{*} by appending AAI (anti-ideal) and AI (ideal) rows. Formül: \tilde{A}'_{ij}(x) = \sqrt{\alpha_{ij}}\cdot\mu_{A_{ij}}(x),\ \alpha_{ij} = \frac{\int x\mu_{B_{ij}}(x)dx}{\int \mu_{B_{ij}}(x)dx};\quad X^{*} = [X^{T}\,;\,AAI\,;\,AI]^{T},\quad AAI_{j}=\min/\max,\ AI_{j}=\max/\min Anchor: Yazdani 2021 §3; Kang 2012; Stević 2020 Eq.(1)
2. 2.Adım 2 (F2): Step 2: Normalise vs ideal AI per criterion direction. Formül: n_{ij} = \begin{cases} x_{ij}/x_{AI,j} & j\in J^{+} \\ x_{AI,j}/x_{ij} & j\in J^{-} \end{cases} Anchor: Stević 2020, p.6 Eq.(2)
3. 3.Adım 3 (F3): Step 3: Weighted normalised matrix v_ij = w_j · n_ij. Formül: v_{ij} = w_{j}\,n_{ij} Anchor: Stević 2020, p.6 Eq.(3)
4. 4.Adım 4 (F4): Step 4: Sum each alternative's weighted matrix row: S_i = Σ v_ij. Formül: S_{i} = \sum_{j=1}^{n} v_{ij} Anchor: Stević 2020, p.7 Eq.(4)
5. 5.Adım 5 (F5): Step 5: Utility ratios vs ideal K+ and anti-ideal K− rows. Formül: K^{+}_{i}=\dfrac{S_{i}}{S_{AI}},\quad K^{-}_{i}=\dfrac{S_{i}}{S_{AAI}} Anchor: Stević 2020, p.7 Eqs.(5)-(6)
6. 6.Adım 6 (F6): Step 6: Utility functions f(K+_i) and f(K−_i) of the utility ratios. Formül: f(K^{+}_{i})=\dfrac{K^{-}_{i}}{K^{+}_{i}+K^{-}_{i}},\quad f(K^{-}_{i})=\dfrac{K^{+}_{i}}{K^{+}_{i}+K^{-}_{i}} Anchor: Stević 2020, p.7 Eqs.(7)-(8)
7. 7.Adım 7 (F7): Step 7: Final utility f(K_i) and descending ranking. Formül: f(K_{i})=\dfrac{K^{+}_{i}+K^{-}_{i}}{1+\dfrac{1-f(K^{+}_{i})}{f(K^{+}_{i})}+\dfrac{1-f(K^{-}_{i})}{f(K^{-}_{i})}} Anchor: Stević 2020, p.7 Eq.(9)

Commonly paired with

• •n_a + Z-MARCOS (common)

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