Z-TOPSIS: Z-Number extension of TOPSIS

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

Edge cases

• •when needed.

Common pitfalls

• •Hatalı: 'Z-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Z-Number numbers/tuples
• •Hatalı: 'Z-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'Z-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: Z-TOPSIS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: Z-TOPSIS'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 the Z-number decision matrix where each element Z_ij = (A_ij, B_ij): A_ij is a fuzzy restriction on the criterion value and B_ij is a fuzzy measure of reliability. Convert each Z-number to an ordinary (type-1) fuzzy number Ã'_ij by Ã'_ij = α^(1/2)·A_ij with α = ∫μ_B(x)dx (Kang 2012); centroid defuzzification yields a crisp x'_ij when needed. Formül: Z_{ij} = (A_{ij}, B_{ij});\ \ \alpha = \int \mu_{B}(x)\, dx;\ \ \tilde{A}'_{ij}(x) = \sqrt{\alpha}\cdot \mu_{A_{ij}}(x) Anchor: Report §5.1 Steps 1-3; Gardashova 2018 Z-TOPSIS; Kang 2012 conversion
2. 2.Adım 2 (F2): Step 2: Apply classical TOPSIS normalisation to converted matrix Ã' (or crisp x') per criterion direction; produce weighted normalised matrix v_ij = w_j · r_ij. Formül: r_{ij} = \dfrac{\tilde{A}'_{ij}}{\sqrt{\sum_{i=1}^{m}\tilde{A}'^{2}_{ij}}};\ \ v_{ij} = w_{j}\cdot r_{ij} Anchor: Report §5.1 Steps 4-5: normalisation and weighting
3. 3.Adım 3 (F3): Step 3: Determine positive-ideal A⁺ and negative-ideal A⁻ per criterion direction on the converted (fuzzy or crisp) values. Formül: A^{+}_{j} = \begin{cases}\max_{i} v_{ij} & j\in J^{+}\\ \min_{i} v_{ij} & j\in J^{-}\end{cases};\ \ A^{-}_{j} = \begin{cases}\min_{i} v_{ij} & j\in J^{+}\\ \max_{i} v_{ij} & j\in J^{-}\end{cases} Anchor: Report §5.1 Step 6: PIS/NIS
4. 4.Adım 4 (F4): Step 4: Distance between Z-derived TFN values d(Z_1,Z_2) = √((1/3)·[(a'_1−a'_2)² + (b'_1−b'_2)² + (c'_1−c'_2)²]) for triangular fuzzy (a,b,c); compute D_i^+ and D_i^- by summing per criterion. Formül: d(Z_{1}, Z_{2}) = \sqrt{\dfrac{1}{3}\left[(a'_{1}-a'_{2})^{2} + (b'_{1}-b'_{2})^{2} + (c'_{1}-c'_{2})^{2}\right]};\ \ D_{i}^{\pm} = \sum_{j=1}^{n} d(v_{ij}, A^{\pm}_{j}) Anchor: Report §5.1 Formulas 4, 7: Z-distance and separation
5. 5.Adım 5 (F5): Step 5: Closeness coefficient CC_i = D_i^-/(D_i^+ + D_i^-); rank in descending order. Formül: CC_{i} = \dfrac{D_{i}^{-}}{D_{i}^{+}+D_{i}^{-}};\ \ \text{rank} = \text{argsort}_{\text{desc}}(CC_{i}) Anchor: Report §5.1 Steps 8-9: closeness coefficient

Commonly paired with

• •n_a + Z-TOPSIS (common)

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