SF-EDAS - Spherical extension of EDAS

Overview

sf-edas extends EDAS to handle Spherical uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Spherical Fuzzy Set (SFS: μ, ν, π; μ²+ν²+π² ≤ 1) algebra. The final scores are defuzzified via score function S = μ² − ν² before ranking.

Output

utility, higher is better

Data

Spherical Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Spherical Fuzzy MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Step 3 - Average Solution: aggregate each criterion column using SWAM with equal weights (w_i = 1/n) to obtain the SFS average solution ãv_j.

   $${\widetilde{av}}_j={\mathrm{SWAM}}_{1/n}({\tilde{x}}_{1j},\dots ,{\tilde{x}}_{mj})=(\sqrt{1-\prod _{i=1}^m(1-{\mu }_{ij}^2{)}^{1/m}},\prod _{i=1}^m{\upsilon }_{ij}^{1/m},\sqrt{\prod _{i=1}^m(1-{\mu }_{ij}^2{)}^{1/m}-\prod _{i=1}^m(1-{\mu }_{ij}^2-{\pi }_{ij}^2{)}^{1/m}})$$

   Garg-Sharaf 2022, Step 3, Eq.(29); SWAM operator Eq.(7)

2. 2

   Step 4 - SF Difference Matrix: compute spherical fuzzy difference S̃'_ij between each x̃_ij and ãv_j. For cost criteria apply conjugate (υ,μ,π) first. dH = |μ_ij−μ_j^av|+|υ_ij−υ_j^av|+|π_ij−π_j^av|. Then μ' = (advantage from μ↑, υ↓, π↓), υ' = (disadvantage from μ↓, υ↑), π' = (hesitancy from π↑).

   $$\begin{gathered} d_H({\tilde{x}}_{ij},{\widetilde{av}}_j)=|{\mu }_{ij}-{\mu }_j^{av}|+|{\upsilon }_{ij}-{\upsilon }_j^{av}|+|{\pi }_{ij}-{\pi }_j^{av}|; \\ {\mu }^{\prime }=\frac{|{\mu }_{ij}-{\mu }_j^{av}|}{d_H}1[{\mu }_{ij}>{\mu }_j^{av}]+\frac{|{\upsilon }_{ij}-{\upsilon }_j^{av}|}{d_H}1[{\upsilon }_{ij}<{\upsilon }_j^{av}]+\frac{|{\pi }_{ij}-{\pi }_j^{av}|}{d_H}1[{\pi }_{ij}<{\pi }_j^{av}]; \\ {\upsilon }^{\prime }=\frac{|{\mu }_{ij}-{\mu }_j^{av}|}{d_H}1[{\mu }_{ij}<{\mu }_j^{av}]+\frac{|{\upsilon }_{ij}-{\upsilon }_j^{av}|}{d_H}1[{\upsilon }_{ij}>{\upsilon }_j^{av}]; \\ {\pi }^{\prime }=\frac{|{\pi }_{ij}-{\pi }_j^{av}|}{d_H}1[{\pi }_{ij}>{\pi }_j^{av}] \end{gathered}$$

   Garg-Sharaf 2022, Sect.3.1, Eqs.(20)-(27); Step 4, Eq.(30)

3. 3

   Step 5 - Weighted SF Differences: multiply each SF difference S̃'_ij by criterion weight w̃_j using SFS product (⊗) to obtain weighted matrix S̃_ij. If weights are crisp scalars, use scalar multiplication.

   $${\tilde{S}}_{ij}={\tilde{w}}_j\otimes {\tilde{S}}_{ij}^{\prime }$$

   Garg-Sharaf 2022, Step 5, Eq.(31)

4. 4

   Step 6 - Appraisal Scores via SFAgg: aggregate weighted differences across all criteria for each alternative using the proposed SFAgg function (balanced aggregation, equal weights w_j = 1/m), yielding SFS appraisal tuple ãs_i.

   $${\widetilde{as}}_i={\mathrm{SFAgg}}_{1/m}({\tilde{S}}_{i1},\dots ,{\tilde{S}}_{im})=(\sqrt{\frac{1}{m}\sum _{j=1}^m{\mu }_{{\tilde{S}}_{ij}}^2},\sqrt{\frac{1}{m}\sum _{j=1}^m{\upsilon }_{{\tilde{S}}_{ij}}^2},\sqrt{\frac{1}{m}\sum _{j=1}^m{\pi }_{{\tilde{S}}_{ij}}^2})$$

   Garg-Sharaf 2022, Step 6, Eqs.(28),(32); SFAgg Def.3.2.2

5. 5

   Steps 7-8 - Defuzzify and Rank: apply score function to each ãs_i; if tie, apply accuracy function. Rank alternatives in descending order - highest score is best.

   $$\begin{gathered} \mathrm{Score}({\widetilde{as}}_i)=({\mu }_{{\widetilde{as}}_i}-{\pi }_{{\widetilde{as}}_i}{)}^2-({\upsilon }_{{\widetilde{as}}_i}-{\pi }_{{\widetilde{as}}_i}{)}^2; \\ \mathrm{Accuracy}({\widetilde{as}}_i)={\mu }_{{\widetilde{as}}_i}^2+{\upsilon }_{{\widetilde{as}}_i}^2+{\pi }_{{\widetilde{as}}_i}^2; \\ \text{rank descending by Score (Accuracy as tie-break)} \end{gathered}$$

   Garg-Sharaf 2022, Steps 7-8, Eqs.(4)-(5)

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid Spherical Fuzzy 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

• •If weights are crisp scalars, use scalar multiplication.
• •if tie, apply accuracy function. Rank alternatives in descending order - highest score is best.

• Value-space violation: ensure all entries satisfy SFS: μ,ν,π ∈ [0,1]; μ²+ν²+π² ≤ 1 before computation.

• Defuzzification method affects ranking: score function S = μ² − ν² is the canonical choice but alternatives exist.

Works with

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