Fuzzy Delphi - Expert consensus with triangular fuzzy opinions and defuzzification

Fuzzy expert elicitation - TFN Delphi with centroid defuzzification

Kaufmann, A., Gupta, M. M.1988doi:10.2307/1268889 ↗

Overview

Output: Weight, higher is better
Data: Crisp, expert input required
Weights: Derived internally, no weight source needed
Size: 2+ alternatives, 3-12 criteria works best
Used for: Expert-driven decision making, MAGDM

How it works

1
Collect K expert TFN assessments ẽjk=(ejkα, ejkβ, ejkγ) for each criterion j=1..n.
$\tilde{e}_{j k} = (e_{j k}^{α}, e_{j k}^{β}, e_{j k}^{γ}), k = 1, \dots, K, j = 1, \dots, n$
2
Aggregate K expert TFNs per criterion: lower bound = min, middle = arithmetic mean, upper = max.
$\tilde{E}_{j} = (k min e_{j k}^{α}, \frac{1}{K} k = 1 \sum K e_{j k}^{β}, k max e_{j k}^{γ})$
3
Consensus check: compute spread Δj = max_k(ejkγ) − min_k(ejkα); if Δj ≤ T (threshold), consensus reached; otherwise iterate.
$Δ_{j} = k max e_{j k}^{γ} - k min e_{j k}^{α}; if Δ_{j} \leq T \Rightarrow consensus, else iterate$
4
Defuzzify consensus TFN via COA: Cj = (Ejα + Ejβ + Ejγ) / 3; normalise weights wj = Cj / Σj Cj.
$C_{j} = \frac{E _{j}^{α} + E _{j}^{β} + E _{j}^{γ}}{3}; w_{j} = \frac{C _{j}}{\sum _{j = 1}^{n} C _{j}}$

Look elsewhere when

•No experts available. Use objective weighting.
•High inconsistency. Discard and re-elicit.

Assumptions to verify

Domain experts available
Experts can express consistent comparisons

Edge cases and pitfalls

•if Δj ≤ T (threshold), consensus reached; otherwise iterate.

Applying FUZZY-DELPHI without verifying this assumption.
Requirement: Domain experts available
Applying FUZZY-DELPHI without verifying this assumption.
Requirement: Experts can express consistent comparisons
Using FUZZY-DELPHI when: No experts available → use objective weighting.
An alternative method is recommended in this situation.
Using FUZZY-DELPHI when: High inconsistency → discard and re-elicit.
An alternative method is recommended in this situation.

Works with

Commonly takes its weights from

FUZZY-DELPHI

Its derived weights can feed

AHPSORTAPLOCO ARAS AROMAN ARTASI COBRA COCOSO CODAS COMPROMISE-PROGRAMMING COPRAS CRADIS DEA DEA-CROSSEFF DNMA EDAS ELECTRE ELECTRE-I ELECTRE-II ELECTRE-III ELECTRE-IVELECTRE-TRIELECTRE-TRI-BEVAMIX EXPROM-I EXPROM-II FMEA FUCA GOAL-PROGRAMMING GRA HELLWIG IDOCRIW LINMAP LMAW MABAC MAIRCA MARA MARCOS MAUT MOORA MULTIMOORA NAIADE OCRA ORESTE PAM PAMSSEM-I PAMSSEM-II PROMETHEE PROMETHEE-I PROMETHEE-III PSI QUALIFLEX RAFSI RAPS RAT RAWEC REGIME RIM ROV SAW SECA SIR SMART SPOTIS TAXONOMY TODIM TOPSISTOPSIS-SORTUTAUTADISUTASTAR VIKOR WASPAS WEBIRA WEDBA WISP WPM

How to cite

Kaufmann, A.; Gupta, M. M. (1988). Fuzzy Mathematical Models in Engineering and Management Science. Elsevier Science Publishers, Amsterdam. https://doi.org/10.2307/1268889

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

FUZZY-DELPHI

How it works

Collect K expert TFN assessments ẽjk=(ejkα, ejkβ, ejkγ) for each criterion j=1..n.

\tilde{e}_{j k} = (e_{j k}^{α}, e_{j k}^{β}, e_{j k}^{γ}), k = 1, \dots, K, j = 1, \dots, n

Aggregate K expert TFNs per criterion: lower bound = min, middle = arithmetic mean, upper = max.

\tilde{E}_{j} = (k min e_{j k}^{α}, \frac{1}{K} k = 1 \sum K e_{j k}^{β}, k max e_{j k}^{γ})

Consensus check: compute spread Δj = max_k(ejkγ) − min_k(ejkα); if Δj ≤ T (threshold), consensus reached; otherwise iterate.

Δ_{j} = k max e_{j k}^{γ} - k min e_{j k}^{α}; if Δ_{j} \leq T \Rightarrow consensus, else iterate

Defuzzify consensus TFN via COA: Cj = (Ejα + Ejβ + Ejγ) / 3; normalise weights wj = Cj / Σj Cj.

C_{j} = \frac{E _{j}^{α} + E _{j}^{β} + E _{j}^{γ}}{3}; w_{j} = \frac{C _{j}}{\sum _{j = 1}^{n} C _{j}}

Edge cases and pitfalls

•if Δj ≤ T (threshold), consensus reached; otherwise iterate.

Applying FUZZY-DELPHI without verifying this assumption.

Requirement: Domain experts available

Applying FUZZY-DELPHI without verifying this assumption.

Requirement: Experts can express consistent comparisons

Using FUZZY-DELPHI when: No experts available → use objective weighting.

An alternative method is recommended in this situation.

Using FUZZY-DELPHI when: High inconsistency → discard and re-elicit.

An alternative method is recommended in this situation.