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Weight Subjective

DEMATEL - Decision Making Trial and Evaluation Laboratory

Cause-effect influence network (total relation matrix) - produces prominence + relation for criteria weighting

Gabus, A., Fontela, E.1972

Overview

DEMATEL produces: (1) Weights from normalised prominence P_i; (2) Causal map - criteria with E_i > 0 belong to the cause group (drivers), those with E_i < 0 to the effect group (receivers). The causal map is the primary insight; weights are secondary.

Output: Weight, higher is better
Data: Crisp, complete numeric matrix
Weights: Derived internally, no weight source needed
Size: 2+ alternatives, 3-12 criteria works best
Used for: Causal analysis, criteria interdependence

How it works

1
Direct-relation matrix A from expert pairwise influence judgements.
$A = [a_{ij}]_{n \times n}, a_{ij} \in {0, 1, 2, 3, 4}$
Gabus-Fontela 1972, p.4
2
Normalised direct relation X = A / max(row sum, col sum).
$X = \frac{A}{max ( max _{i} \sum _{j} a _{ij} , max _{j} \sum _{i} a _{ij} )}$
Gabus-Fontela 1972, p.5 Eq.(1)
3
Total relation matrix T = X(I − X)^{−1}.
$T = X (I - X)^{- 1}$
Gabus-Fontela 1972, p.5 Eq.(2)
4
Row sums D and column sums R; prominence D+R, relation D−R.
$D_{i} = j \sum t_{ij}, R_{j} = i \sum t_{ij}; Prominence = D + R, Relation = D - R$
Gabus-Fontela 1972, p.5 Eqs.(3)-(4)
5
DEMATEL weights w_j = ((D+R)² + (D−R)²)^{1/2} normalised.
$w_{j} = \frac{( D _{j} + R _{j} ) ^{2} + ( D _{j} - R _{j} ) ^{2}}{\sum _{k} ( D _{k} + R _{k} ) ^{2} + ( D _{k} - R _{k} ) ^{2}}$
Gabus-Fontela 1972, p.6 Eq.(5)

Look elsewhere when

•No data variation (constant criterion). Weight degenerates.
•Expert judgment is the actual driver. Use subjective weighting.

Assumptions to verify

Decision matrix exists with measurable criteria
Sufficient inter-alternative variation per criterion

Edge cases and pitfalls

Spectral radius of X ≥ 1: (I−X)^{−1} diverges - reduce influence values or check data.

Works with

Commonly takes its weights from

DEMATEL

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

Gabus, A.; Fontela, E. (1972). World problems, an invitation to further thought within the framework of DEMATEL. Battelle Geneva Research Centre, Geneva, Switzerland.

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System ID, as it appears in reports and the API

DEMATEL

Overview

Output

Weight, higher is better

Data

Crisp, complete numeric matrix

Weights

Derived internally, no weight source needed

Size

2+ alternatives, 3-12 criteria works best

Used for

Causal analysis, criteria interdependence

How it works

Direct-relation matrix A from expert pairwise influence judgements.

A = [a_{ij}]_{n \times n}, a_{ij} \in {0, 1, 2, 3, 4}

Gabus-Fontela 1972, p.4

Normalised direct relation X = A / max(row sum, col sum).

X = \frac{A}{max ( max _{i} \sum _{j} a _{ij} , max _{j} \sum _{i} a _{ij} )}

Gabus-Fontela 1972, p.5 Eq.(1)

Total relation matrix T = X(I − X)^{−1}.

T = X (I - X)^{- 1}

Gabus-Fontela 1972, p.5 Eq.(2)

Row sums D and column sums R; prominence D+R, relation D−R.

D_{i} = j \sum t_{ij}, R_{j} = i \sum t_{ij}; Prominence = D + R, Relation = D - R

Gabus-Fontela 1972, p.5 Eqs.(3)-(4)

DEMATEL weights w_j = ((D+R)² + (D−R)²)^{1/2} normalised.

w_{j} = \frac{( D _{j} + R _{j} ) ^{2} + ( D _{j} - R _{j} ) ^{2}}{\sum _{k} ( D _{k} + R _{k} ) ^{2} + ( D _{k} - R _{k} ) ^{2}}

Gabus-Fontela 1972, p.6 Eq.(5)