SF-EDAS: Spherical extension of EDAS

Overview

Spherical outranking/ranking: Spherical Fuzzy Set (SFS: μ, ν, π; μ²+ν²+π² ≤ 1). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Spherical outranking/ranking: Spherical Fuzzy Set (SFS: μ, ν, π; μ²+ν²+π² ≤ 1)
• •Preserves spherical uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

• •Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)
• •Assumes: Decision matrix entries are valid Spherical Fuzzy numbers/tuples
• •Assumes: Underlying crisp method's compensation assumption holds in uncertain space
• •Assumes: All decision-maker(s) and experts use the same linguistic/uncertainty scale

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

When not to use

• •Crisp data sufficient: use base EDAS directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

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

Common pitfalls

• •Hatalı: 'SF-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Spherical Fuzzy numbers/tuples
• •Hatalı: 'SF-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'SF-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: SF-EDAS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: SF-EDAS'yi 'Aggregation operator (PFWA/PFOWA/etc.) not specified' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): 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. Formül: \widetilde{av}_j = \mathrm{SWAM}_{1/n}(\tilde{x}_{1j},\ldots,\tilde{x}_{mj}) = \Bigl(\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}}\Bigr) Anchor: Garg-Sharaf 2022, Step 3, Eq.(29); SWAM operator Eq.(7)
2. 2.Adım 2 (F2): 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↑). Formül: d_H(\tilde{x}_{ij},\widetilde{av}_j)=|\mu_{ij}-\mu_j^{av}|+|\upsilon_{ij}-\upsilon_j^{av}|+|\pi_{ij}-\pi_j^{av}|;\\ \mu'=\frac{|\mu_{ij}-\mu_j^{av}|}{d_H}\mathbb{1}[\mu_{ij}>\mu_j^{av}]+\frac{|\upsilon_{ij}-\upsilon_j^{av}|}{d_H}\mathbb{1}[\upsilon_{ij}<\upsilon_j^{av}]+\frac{|\pi_{ij}-\pi_j^{av}|}{d_H}\mathbb{1}[\pi_{ij}<\pi_j^{av}];\\ \upsilon'=\frac{|\mu_{ij}-\mu_j^{av}|}{d_H}\mathbb{1}[\mu_{ij}<\mu_j^{av}]+\frac{|\upsilon_{ij}-\upsilon_j^{av}|}{d_H}\mathbb{1}[\upsilon_{ij}>\upsilon_j^{av}];\\ \pi'=\frac{|\pi_{ij}-\pi_j^{av}|}{d_H}\mathbb{1}[\pi_{ij}>\pi_j^{av}] Anchor: Garg-Sharaf 2022, Sect.3.1, Eqs.(20)-(27); Step 4, Eq.(30)
3. 3.Adım 3 (F3): 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. Formül: \tilde{S}_{ij} = \tilde{w}_j \otimes \tilde{S}'_{ij} Anchor: Garg-Sharaf 2022, Step 5, Eq.(31)
4. 4.Adım 4 (F4): 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. Formül: \widetilde{as}_i = \mathrm{SFAgg}_{1/m}(\tilde{S}_{i1},\ldots,\tilde{S}_{im}) = \Bigl(\sqrt{\tfrac{1}{m}\sum_{j=1}^{m}\mu^2_{\tilde{S}_{ij}}},\; \sqrt{\tfrac{1}{m}\sum_{j=1}^{m}\upsilon^2_{\tilde{S}_{ij}}},\; \sqrt{\tfrac{1}{m}\sum_{j=1}^{m}\pi^2_{\tilde{S}_{ij}}}\Bigr) Anchor: Garg-Sharaf 2022, Step 6, Eqs.(28),(32); SFAgg Def.3.2.2
5. 5.Adım 5 (F5): 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. Formül: \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^2_{\widetilde{as}_i}+\upsilon^2_{\widetilde{as}_i}+\pi^2_{\widetilde{as}_i};\\ \text{rank descending by Score (Accuracy as tie-break)} Anchor: Garg-Sharaf 2022, Steps 7-8, Eqs.(4)-(5)

Commonly paired with

• •n_a + SF-EDAS (common)

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