Z-VIKOR: Z-Number extension of VIKOR

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 VIKOR directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •If C1 fails, return the maximum prefix A^(1)..A^(M) for which Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}.

Common pitfalls

• •Hatalı: 'Z-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Z-Number numbers/tuples
• •Hatalı: 'Z-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'Z-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: Z-VIKOR'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: Z-VIKOR'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: Construct Z-number decision matrix Z_ij = (A_ij, B_ij); convert via Ã'(x) = √α · μ_A(x) with α = ∫μ_B(x)dx (Kang 2012) or apply the Shen 2018 comprehensive weighted Z-distance directly. Determine best Z_j^* and worst Z_j^- per criterion direction in the converted space. Formül: Z_{ij} = (A_{ij}, B_{ij});\ \ \tilde{A}'(x) = \sqrt{\alpha}\cdot \mu_{A}(x),\ \alpha = \int \mu_{B}(x)\, dx;\ \ Z_{j}^{*} = \arg\max/\min_{i} Z_{ij},\ Z_{j}^{-} = \arg\min/\max_{i} Z_{ij} Anchor: Report §5.2 Formulas 1-2; Shen 2018 Z-VIKOR
2. 2.Adım 2 (F2): Step 2: Apply classical VIKOR utility S_i and regret R_i on the converted Z-values: S_i = Σ_j w_j·d(Z_j^*, Z_ij)/d(Z_j^*, Z_j^-) (L_1-metric); R_i = max_j [w_j · d(Z_j^*, Z_ij)/d(Z_j^*, Z_j^-)] (L_∞-metric), with d the comprehensive Z-distance. Formül: S_{i} = \sum_{j=1}^{n} w_{j}\,\dfrac{d(Z_{j}^{*}, Z_{ij})}{d(Z_{j}^{*}, Z_{j}^{-})},\quad R_{i} = \max_{j}\left[w_{j}\,\dfrac{d(Z_{j}^{*}, Z_{ij})}{d(Z_{j}^{*}, Z_{j}^{-})}\right] Anchor: Report §5.2 Formula 2: utility and regret with Z-distance
3. 3.Adım 3 (F3): Step 3: VIKOR index Q_i as convex combination of normalised S and R, with compromise coefficient v (classical VIKOR aggregation). Formül: Q_{i} = v\,\dfrac{S_{i}-S^{*}}{S^{-}-S^{*}} + (1-v)\,\dfrac{R_{i}-R^{*}}{R^{-}-R^{*}},\ S^{*}=\min_{i}S_{i},\ S^{-}=\max_{i}S_{i},\ R^{*}=\min_{i}R_{i},\ R^{-}=\max_{i}R_{i} Anchor: Report §5.2 Formula 3: VIKOR index (classical)
4. 4.Adım 4 (F4): Step 4: Propose A^(1) (the lowest-Q alternative) as compromise solution iff both C1 (acceptable advantage) and C2 (acceptable stability) hold. If C1 fails, return the maximum prefix A^(1)..A^(M) for which Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}. Formül: DQ = \dfrac{1}{m-1};\quad C1: Q(A^{(2)}) - Q(A^{(1)}) \ge DQ;\quad C2: A^{(1)} \text{ is best in } S \text{ or in } R Anchor: Opricovic & Tzeng 2004, §2 Eqs.(6)-(7)

Commonly paired with

• •n_a + Z-VIKOR (common)

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