Z-TOPSIS - Z-Number extension of TOPSIS

Overview

z-topsis extends TOPSIS 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

   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.

   $$Z_{ij}=(A_{ij},B_{ij});\alpha =\int {\mu }_B(x)dx;{\tilde{A}}_{ij}^{\prime }(x)=\sqrt{\alpha }\cdot {\mu }_{A_{ij}}(x)$$

   Report §5.1 Steps 1-3; Gardashova 2018 Z-TOPSIS; Kang 2012 conversion

2. 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.

   $$r_{ij}=\frac{{\tilde{A}}_{ij}^{\prime }}{\sqrt{{\sum }_{i=1}^m{\tilde{A}}_{ij}^{\prime 2}}};v_{ij}=w_j\cdot r_{ij}$$

   Report §5.1 Steps 4-5 - normalisation and weighting

3. 3

   Determine positive-ideal A⁺ and negative-ideal A⁻ per criterion direction on the converted (fuzzy or crisp) values.

   $$A_j^{+}=\{\begin{array}{ll}{\max }_i{v}_{ij}& j\in J^{+}\\ {\min }_i{v}_{ij}& j\in J^{-}\end{array};A_j^{-}=\{\begin{array}{ll}{\min }_i{v}_{ij}& j\in J^{+}\\ {\max }_i{v}_{ij}& j\in J^{-}\end{array}$$

   Report §5.1 Step 6 - PIS/NIS

4. 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.

   $$d(Z_1,Z_2)=\sqrt{\frac{1}{3}\left[(a_1^{\prime }-a_2^{\prime }{)}^2+(b_1^{\prime }-b_2^{\prime }{)}^2+(c_1^{\prime }-c_2^{\prime }{)}^2\right]};D_i^{\pm }=\sum _{j=1}^{n}d(v_{ij},A_j^{\pm })$$

   Report §5.1 Formulas 4, 7 - Z-distance and separation

5. 5

   Closeness coefficient CC_i = D_i^-/(D_i^+ + D_i^-); rank in descending order.

   $$C{C}_i=\frac{D_i^{-}}{D_i^{+}+D_i^{-}};\text{rank}={\text{argsort}}_{\text{desc}}(C{C}_i)$$

   Report §5.1 Steps 8-9 - closeness coefficient

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

• •when needed.

• 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 column-wise vector-norm normalisation (R_ij = x'_ij / ||column||); ranking AND scores stay unchanged. This is a mathematical property of the Kang (2012) conversion, not a defect of the fix. B changes the result only when reliability differs between alternatives on the same criterion.

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