Z-MARCOS - Z-Number extension of MARCOS

Overview

z-marcos extends MARCOS to handle Z-Number uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Z-Number (Z = (A, B): restriction A, reliability B; both fuzzy) algebra. The final scores are defuzzified via convert to regular fuzzy: Ã = B·A, then centroid before ranking.

Output

utility, higher is better

Data

Z-Number, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Z-Number MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 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.

   $${\tilde{A}}_{ij}^{\prime }(x)=\sqrt{{\alpha }_{ij}}\cdot {\mu }_{A_{ij}}(x),{\alpha }_{ij}=\frac{\int x{\mu }_{B_{ij}}(x)dx}{\int {\mu }_{B_{ij}}(x)dx};X^{\ast }=[X^T;AAI;AI{]}^T,AA{I}_j=\min /\max ,A{I}_j=\max /\min$$

   Yazdani 2021 §3; Kang 2012; Stević 2020 Eq.(1)

2. 2

   Normalise vs ideal AI per criterion direction.

   $$n_{ij}=\{\begin{array}{ll}{x}_{ij}/x_{AI,j}& j\in J^{+}\\ x_{AI,j}/x_{ij}& j\in J^{-}\end{array}$$

   Stević 2020, p.6 Eq.(2)

3. 3

   Weighted normalised matrix v_ij = w_j · n_ij.

   $$v_{ij}=w_j{n}_{ij}$$

   Stević 2020, p.6 Eq.(3)

4. 4

   Sum each alternative's weighted matrix row: S_i = Σ v_ij.

   $$S_i=\sum _{j=1}^n{v}_{ij}$$

   Stević 2020, p.7 Eq.(4)

5. 5

   Utility ratios vs ideal K+ and anti-ideal K− rows.

   $$K_i^{+}=\frac{S_i}{S_{AI}},K_i^{-}=\frac{S_i}{S_{AAI}}$$

   Stević 2020, p.7 Eqs.(5)-(6)

6. 6

   Utility functions f(K+_i) and f(K−_i) of the utility ratios.

   $$f(K_i^{+})=\frac{K_i^{-}}{K_i^{+}+K_i^{-}},f(K_i^{-})=\frac{K_i^{+}}{K_i^{+}+K_i^{-}}$$

   Stević 2020, p.7 Eqs.(7)-(8)

7. 7

   Final utility f(K_i) and descending ranking.

   $$f(K_i)=\frac{K_i^{+}+K_i^{-}}{1+\frac{1-f(K_i^{+})}{f(K_i^{+})}+\frac{1-f(K_i^{-})}{f(K_i^{-})}}$$

   Stević 2020, p.7 Eq.(9)

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• Value-space violation: ensure all entries satisfy Z = (A, B) where A restricts values, B indicates reliability of A; both TFNs typically before computation.

• Defuzzification method affects ranking: convert to regular fuzzy: Ã = B·A, then centroid is the canonical choice but alternatives exist.

• If B (reliability) is identical in every cell, the sqrt(alpha) constant cancels exactly in the AI/AAI ideal-distance ratios (N_ij = X_ij/AI_j or AAI_j/X_ij); ranking AND scores stay unchanged. This is a mathematical property, not a defect. B changes the result only when reliability differs between alternatives.

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