Z-WASPAS: Z-Number extension of WASPAS

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

Edge cases

• •default λ=0.5) and descending ranking.

Common pitfalls

• •Hatalı: 'Z-WASPAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Z-Number numbers/tuples
• •Hatalı: 'Z-WASPAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'Z-WASPAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: Z-WASPAS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: Z-WASPAS'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 each Z-number to a regular 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; obtain crisp x'_{ij} by centroid defuzzification. Formül: \tilde{A}'_{ij}(x) = \sqrt{\alpha_{ij}}\cdot\mu_{A_{ij}}(x),\quad \alpha_{ij} = \frac{\int x\,\mu_{B_{ij}}(x)\,dx}{\int \mu_{B_{ij}}(x)\,dx} Anchor: Ghoushchi 2021 §3; Kang 2012
2. 2.Adım 2 (F2): Step 2: Linear normalisation on the converted matrix (max for benefit, min/x for cost). Formül: \bar{x}_{ij} = \begin{cases} x'_{ij}/\max_{i} x'_{ij} & j\in J^{+} \\ \min_{i} x'_{ij}/x'_{ij} & j\in J^{-} \end{cases} Anchor: Ghoushchi 2021 §3; Zavadskas 2012 Eq.(1)
3. 3.Adım 3 (F3): Step 3: Weighted Sum Model component Q^{(1)}_i on converted Z-values. Formül: Q^{(1)}_{i} = \sum_{j=1}^{n} w_{j}\,\bar{x}_{ij} Anchor: Ghoushchi 2021 §3 Eq.(WSM)
4. 4.Adım 4 (F4): Step 4: Weighted Product Model component Q^{(2)}_i on converted Z-values. Formül: Q^{(2)}_{i} = \prod_{j=1}^{n} (\bar{x}_{ij})^{w_{j}} Anchor: Ghoushchi 2021 §3 Eq.(WPM)
5. 5.Adım 5 (F5): Step 5: Joint WASPAS aggregation with λ∈[0,1] (default λ=0.5) and descending ranking. Formül: Q_{i} = \lambda\,Q^{(1)}_{i} + (1-\lambda)\,Q^{(2)}_{i} Anchor: Ghoushchi 2021 §3 Eq.(WASPAS)

Commonly paired with

• •n_a + Z-WASPAS (common)

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