Z-WASPAS - Z-Number extension of WASPAS

Overview

z-waspas extends WASPAS 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 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.

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

   Ghoushchi 2021 §3; Kang 2012

2. 2

   Linear normalisation on the converted matrix (max for benefit, min/x for cost).

   $${\bar{x}}_{ij}=\{\begin{array}{ll}{x}_{ij}^{\prime }/{\max }_i{x}_{ij}^{\prime }& j\in J^{+}\\ {\min }_i{x}_{ij}^{\prime }/x_{ij}^{\prime }& j\in J^{-}\end{array}$$

   Ghoushchi 2021 §3; Zavadskas 2012 Eq.(1)

3. 3

   Weighted Sum Model component Q^{(1)}_i on converted Z-values.

   $$Q_i^{(1)}=\sum _{j=1}^n{w}_j{\bar{x}}_{ij}$$

   Ghoushchi 2021 §3 Eq.(WSM)

4. 4

   Weighted Product Model component Q^{(2)}_i on converted Z-values.

   $$Q_i^{(2)}=\prod _{j=1}^n({\bar{x}}_{ij}{)}^{w_j}$$

   Ghoushchi 2021 §3 Eq.(WPM)

5. 5

   Joint WASPAS aggregation with λ∈[0,1] (default λ=0.5) and descending ranking.

   $$Q_i=\lambda Q_i^{(1)}+(1-\lambda )Q_i^{(2)}$$

   Ghoushchi 2021 §3 Eq.(WASPAS)

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

• •default λ=0.5) and descending ranking.

• 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.

• The Q_WSM component (max/min-ratio SAW normalisation) cancels the sqrt(alpha) constant when B is identical in every cell; the Q_WPM component, however, raises the unnormalised crisp values to signed weight exponents, so a uniform B still rescales Q_WPM and the final Q. A uniform B therefore changes the scores but not the ranking. B only changes the ranking when reliability differs between alternatives.

Works with

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