PHF-EDAS: Extended Hesitant Fuzzy Linguistic EDAS (EHFL-EDAS)

Overview

Probabilistic Hesitant outranking/ranking: Probabilistic Hesitant Fuzzy Element (PHFE: {γ|p} pairs). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Probabilistic Hesitant outranking/ranking: Probabilistic Hesitant Fuzzy Element (PHFE: {γ|p} pairs)
• •Preserves hesitant 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 Probabilistic Hesitant 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 Probabilistic Hesitant 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

• •See F.steps and D.parameters for PHF-EDAS-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'PHF-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Probabilistic Hesitant numbers/tuples
• •Hatalı: 'PHF-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'PHF-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: PHF-EDAS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PHF-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 2: Compute the Average Solution (AV) per criterion using the ELCOWA operator (Eq.11): order alternatives by possibility degree p_E, then apply convex combination with BUM weights. Formül: AV_j = \mathrm{ELCOWA}(H^{1j}_S, \ldots, H^{mj}_S) = C_m\!\left(K\!\int_{\frac{i-1}{m}}^{\frac{i}{m}}\!u(y)\,dy,\; H^{q(i)}_S;\; i=1,\ldots,m\right) Anchor: Feng-Wei-Liu 2018 (Int. J. Fuzzy Systems, DOI:10.1007/s40815-018-0504-5) Step 2, Eq.(11); ELCOWA Def.4.4; BUM function K=1/\int_0^1 u(y)dy; permutation q s.t. p_E(H^{q(i)}_S \geq H^{q(j)}_S) \geq 0.5 for i \leq j
2. 2.Adım 2 (F2): Step 3: Positive Distance from Average (PDA) using possibility-based priority scores P(·): deviation of alternative i's EHFLTS above AV_j (Eq.13). Formül: PDA_{ij} = \frac{\max\!\bigl(0,\;P(H^{ij}_S) - P(AV_j)\bigr)}{P(AV_j)} Anchor: Feng-Wei-Liu 2018 (Int. J. Fuzzy Systems, DOI:10.1007/s40815-018-0504-5) Step 3, Eq.(13); P(H) = priority weight via pairwise p_E comparisons (Eq.5-6)
3. 3.Adım 3 (F3): Step 3: Negative Distance from Average (NDA): deviation of AV_j above alternative i's EHFLTS (Eq.14). Formül: NDA_{ij} = \frac{\max\!\bigl(0,\;P(AV_j) - P(H^{ij}_S)\bigr)}{P(AV_j)} Anchor: Feng-Wei-Liu 2018 (Int. J. Fuzzy Systems, DOI:10.1007/s40815-018-0504-5) Step 3, Eq.(14); P(H) = priority weight via pairwise p_E comparisons (Eq.5-6)
4. 4.Adım 4 (F4): Step 4: Weighted sums SP_i and SN_i across criteria. Formül: SP_i = \sum_{j=1}^{n} w_j \cdot PDA_{ij}, \quad SN_i = \sum_{j=1}^{n} w_j \cdot NDA_{ij} Anchor: Feng-Wei-Liu 2018 (Int. J. Fuzzy Systems, DOI:10.1007/s40815-018-0504-5) Step 4
5. 5.Adım 5 (F5): Step 5: Normalize SP and SN by their maxima to obtain NSP and NSN. Formül: NSP_i = \frac{SP_i}{\max_k SP_k}, \quad NSN_i = 1 - \frac{SN_i}{\max_k SN_k} Anchor: Feng-Wei-Liu 2018 (Int. J. Fuzzy Systems, DOI:10.1007/s40815-018-0504-5) Step 5 (NSP/NSN embedded in AS formula)
6. 6.Adım 6 (F6): Step 5-6: Appraisal Score AS_i as average of NSP and NSN; rank descending. Paper result: A1 > A2 > A4 > A3. Formül: AS_i = \frac{1}{2}\!\left(\frac{SP_i}{\max_k SP_k} + 1 - \frac{SN_i}{\max_k SN_k}\right), \quad 0 \le AS_i \le 1 Anchor: Feng-Wei-Liu 2018 (Int. J. Fuzzy Systems, DOI:10.1007/s40815-018-0504-5) Step 5, Eq.(15); paper ranking: A1>A2>A4>A3 (AS: 0.6424>0.5551>0.5085>0.2545)

Commonly paired with

• •n_a + PHF-EDAS (common)

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