Fuzzy WASPAS - Fuzzy extension of WASPAS

Overview

fuzzy-waspas extends WASPAS to handle Fuzzy uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Triangular Fuzzy Number (TFN: l, m, u) algebra. The final scores are defuzzified via centroid (l+m+u)/3 before ranking.

Output

utility, higher is better

Data

Fuzzy (TFN), uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Fuzzy (Triangular) MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Form fuzzy decision-making matrix (FDMM). Performance values x̃ij = (xijα, xijβ, xijγ) and fuzzy weights w̃j = (wjα, wjβ, wjγ) are TFNs.

   $$\tilde{X}=[\tilde{x}ij{]}_{m\times n},\tilde{x}ij=(xij\alpha ,xij\beta ,xij\gamma );\tilde{W}=[\tilde{w}j],\tilde{w}j=(wj\alpha ,wj\beta ,wj\gamma )$$

   Turskis 2015, Eq.(15)

2. 2

   Normalize FDMM. Benefit: divide each TFN component by scalar max_i(xijγ). Cost: multiply by scalar min_i(xijα), divide TFN components. Cost: x̄̃ij = (min_i(xijα)/xijγ, min_i(xijα)/xijβ, min_i(xijα)/xijα)

   $$Benefit:\widetilde{\stackrel{ˉ}{x}}ij=(xij\alpha /ma{x}_i(xij\gamma ),xij\beta /ma{x}_i(xij\gamma ),xij\gamma /ma{x}_i(xij\gamma ))$$

   Turskis 2015, Eq.(16)

3. 3

   Step 3a - WSM weighted matrix: x̂̃ij = x̄̃ij ⊗ w̃j (component-wise TFN multiplication). Step 3b - WPM weighted matrix: x̄̄̃ij = x̄̃ij^w̃j (TFN power, Eq.13: (α^γ, β^β, γ^α) for values in [0,1]). WPM: x̄̄̃ij = (x̄̃ijα^w̃jγ, x̄̃ijβ^w̃jβ, x̄̃ijγ^w̃jα) [Eq.18, Eq.13]

   $$WSM:\tilde{\hat{x}}ij=(\widetilde{\stackrel{ˉ}{x}}ij\alpha \cdot \tilde{w}j\alpha ,\widetilde{\stackrel{ˉ}{x}}ij\beta \cdot \tilde{w}j\beta ,\widetilde{\stackrel{ˉ}{x}}ij\gamma \cdot \tilde{w}j\gamma )[Eq.17]$$

   Turskis 2015, Eq.(17-18)

4. 4

   Compute Q̃i (WSM) and P̃i (WPM) as TFN sums/products, then defuzzify via centroid COA. P̃i = Πj x̄̄̃ij = (Πj x̄̄̃ijα, Πj x̄̄̃ijβ, Πj x̄̄̃ijγ) [Eq.20] Qi = (Qiα+Qiβ+Qiγ)/3 [Eq.21] Pi = (Piα+Piβ+Piγ)/3 [Eq.22]

   $$\tilde{Q}i=\Sigma j\tilde{\hat{x}}ij=(\Sigma j\tilde{\hat{x}}ij\alpha ,\Sigma j\tilde{\hat{x}}ij\beta ,\Sigma j\tilde{\hat{x}}ij\gamma )[Eq.19]$$

   Turskis 2015, Eq.(19-22)

5. 5

   Compute adaptive λ from defuzzified scores, then integrated utility Ki = λ·Qi + (1−λ)·Pi. Ki = λ·Qi + (1−λ)·Pi, λ∈[0,1] [Eq.23]

   $$\lambda =\Sigma iPi/(\Sigma iQi+\Sigma iPi)[Eq.24]$$

   Turskis 2015, Eq.(23-24)

6. 6

   Rank alternatives by Ki descending. Highest Ki = best alternative.

   $$rank=argsort(K,descending=True)$$

   Turskis 2015, Section 2.3

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid Fuzzy (Triangular) 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 TFN: l ≤ m ≤ u, all ≥ 0 before computation.

• Defuzzification method affects ranking: centroid (l+m+u)/3 is the canonical choice but alternatives exist.

Works with

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