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Ranking

PHF-EDAS - Probabilistic Hesitant Fuzzy EDAS (Xu-Zhou PHFS reduction P(h)=Σγp; classical EDAS mechanics)

Probabilistic Hesitant outranking/ranking - Probabilistic Hesitant Fuzzy Element (PHFE: {γ|p} pairs), plain expected-value reduction (NOT ELCOWA/BUM)

Xu, Z., Zhou, W.2017doi:10.1007/s10700-016-9257-5 ↗

Overview

phf-edas extends EDAS to handle Probabilistic Hesitant uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Probabilistic Hesitant Fuzzy Element (PHFE: {γ|p} pairs) algebra. The final scores are defuzzified via E[PHFE] = Σ γ_k p_k before ranking.

Output: utility, higher is better
Data: Hesitant, uncertainty tuples complete
Weights: Needs a weight source
Size: 2+ alternatives, 3-10 criteria works best
Used for: Probabilistic Hesitant MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1
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.
$A V_{j} = ELCOWA (H_{S}^{1 j}, \dots, H_{S}^{mj}) = C_{m} (K \int_{\frac{i - 1}{m}}^{\frac{i}{m}} u (y) d y, H_{S}^{q (i)}; i = 1, \dots, m)$
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
Step 3 - Positive Distance from Average (PDA) using possibility-based priority scores P(·): deviation of alternative i's EHFLTS above AV_j (Eq.13).
$P D A_{ij} = \frac{max ( 0 , P ( H _{S}^{ij} ) - P ( A V _{j} ) )}{P ( A V _{j} )}$
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
Negative Distance from Average (NDA): deviation of AV_j above alternative i's EHFLTS (Eq.14).
$N D A_{ij} = \frac{max ( 0 , P ( A V _{j} ) - P ( H _{S}^{ij} ) )}{P ( A V _{j} )}$
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
Weighted sums SP_i and SN_i across criteria.
$S P_{i} = j = 1 \sum n w_{j} \cdot P D A_{ij}, S N_{i} = j = 1 \sum n w_{j} \cdot N D A_{ij}$
Feng-Wei-Liu 2018 (Int. J. Fuzzy Systems, DOI:10.1007/s40815-018-0504-5) Step 4
5
Normalize SP and SN by their maxima to obtain NSP and NSN.
$N S P_{i} = \frac{S P _{i}}{max _{k} S P _{k}}, N S N_{i} = 1 - \frac{S N _{i}}{max _{k} S N _{k}}$
Feng-Wei-Liu 2018 (Int. J. Fuzzy Systems, DOI:10.1007/s40815-018-0504-5) Step 5 (NSP/NSN embedded in AS formula)
6
Step 5-6 - Appraisal Score AS_i as average of NSP and NSN; rank descending. Paper result: A1 > A2 > A4 > A3.
$A S_{i} = \frac{1}{2} (\frac{S P _{i}}{max _{k} S P _{k}} + 1 - \frac{S N _{i}}{max _{k} S N _{k}}), 0 \leq A S_{i} \leq 1$
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)

Fits when / Look elsewhere when

Fits when

•Preserves hesitant uncertainty through the pipeline rather than premature crispification at elicitation
•Native group-decision support (multi-DM aggregation built into the pipeline)

Look elsewhere when

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

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

Limitations

•Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)

Edge cases and pitfalls

Value-space violation: ensure all entries satisfy PHFE: {γ_k|p_k} where γ_k ∈ [0,1], Σ p_k ≤ 1 before computation.
Defuzzification method affects ranking: E[PHFE] = Σ γ_k p_k is the canonical choice but alternatives exist.

Works with

Commonly takes its weights from

n_a

How to cite

Xu, Z.; Zhou, W. (2017). Consensus building with a group of decision makers under the hesitant probabilistic fuzzy environment. Fuzzy Optimization and Decision Making. https://doi.org/10.1007/s10700-016-9257-5

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

PHF-EDAS

Overview

Output

utility, higher is better

Data

Hesitant, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Probabilistic Hesitant MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

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.

A V_{j} = ELCOWA (H_{S}^{1 j}, \dots, H_{S}^{mj}) = C_{m} (K \int_{\frac{i - 1}{m}}^{\frac{i}{m}} u (y) d y, H_{S}^{q (i)}; i = 1, \dots, m)

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

Step 3 - Positive Distance from Average (PDA) using possibility-based priority scores P(·): deviation of alternative i's EHFLTS above AV_j (Eq.13).

P D A_{ij} = \frac{max ( 0 , P ( H _{S}^{ij} ) - P ( A V _{j} ) )}{P ( A V _{j} )}

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)

Negative Distance from Average (NDA): deviation of AV_j above alternative i's EHFLTS (Eq.14).

N D A_{ij} = \frac{max ( 0 , P ( A V _{j} ) - P ( H _{S}^{ij} ) )}{P ( A V _{j} )}

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)

Weighted sums SP_i and SN_i across criteria.

S P_{i} = j = 1 \sum n w_{j} \cdot P D A_{ij}, S N_{i} = j = 1 \sum n w_{j} \cdot N D A_{ij}

Feng-Wei-Liu 2018 (Int. J. Fuzzy Systems, DOI:10.1007/s40815-018-0504-5) Step 4

Normalize SP and SN by their maxima to obtain NSP and NSN.

N S P_{i} = \frac{S P _{i}}{max _{k} S P _{k}}, N S N_{i} = 1 - \frac{S N _{i}}{max _{k} S N _{k}}

Feng-Wei-Liu 2018 (Int. J. Fuzzy Systems, DOI:10.1007/s40815-018-0504-5) Step 5 (NSP/NSN embedded in AS formula)

Step 5-6 - Appraisal Score AS_i as average of NSP and NSN; rank descending. Paper result: A1 > A2 > A4 > A3.

A S_{i} = \frac{1}{2} (\frac{S P _{i}}{max _{k} S P _{k}} + 1 - \frac{S N _{i}}{max _{k} S N _{k}}), 0 \leq A S_{i} \leq 1

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)

Fits when / Look elsewhere when

Fits when

•Preserves hesitant uncertainty through the pipeline rather than premature crispification at elicitation
•Native group-decision support (multi-DM aggregation built into the pipeline)

Look elsewhere when

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

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

Limitations

•Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)