DANP - DEMATEL-based ANP

Overview

DANP needs a cluster partition and exactly one supermatrix source: enter the cluster-partitioned ANP supermatrix together with an explicit cluster-level DEMATEL route, or enter the criteria-level DEMATEL influence matrix (0-4) and let the engine derive the supermatrix as W=(T_C^alpha)' with the cluster matrix T_D as the block average of T_C. A single direct-relation matrix is never used as a supermatrix.

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. 1

   Direct-influence matrix A from experts (criteria pairwise influence on 0-4 scale).

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

   Tzeng-Huang 2011, Ch.7 Eq.(7.1)

2. 2

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

   $$X=\frac{A}{\max ({\max }_i{\sum }_j{a}_{ij},{\max }_j{\sum }_i{a}_{ij})}$$

   Tzeng-Huang 2011, p.140 Eq.(7.2)

3. 3

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

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

   Tzeng-Huang 2011, p.140 Eq.(7.3)

4. 4

   Normalise T column-wise to form unweighted supermatrix W.

   $$w_{ij}=\frac{t_{ij}}{{\sum }_i{t}_{ij}}\text{(column-stochastic supermatrix)}$$

   Tzeng-Huang 2011, p.141 Eq.(7.4)

5. 5

   Limit supermatrix W^∞ = lim_{k→∞} W^k (powers converge to steady state). Use Cesaro sum if cyclic.

   $$W^{\infty }=\underset{k\to \infty }{\lim }{W}^k\text{or}\underset{N\to \infty }{\lim }\frac{1}{N}\sum _{r=1}^N{W}^r$$

   Tzeng-Huang 2011, p.141 Eq.(7.5)

6. 6

   Final criterion weights from any column of W^∞ (DEMATEL-derived dependency baked in).

   $$\mathbf{w}=W_{:,k}^{\infty }\text{(any column)}$$

   Tzeng-Huang 2011, p.142 Sec.7.5

Look elsewhere when

Assumptions to verify

• Domain experts available
• Experts can express consistent comparisons

Edge cases and pitfalls

• •if cyclic.

• T is not the unweighted ANP supermatrix.

• Equation (9) weights ANP blocks by Tz; it is not ordinary same-size matrix multiplication.

• A singular I-D is an error, not permission to substitute D or A.

• In the derived cluster route the block of rows of cluster j and columns of cluster i takes t_D^alpha_ij, the ROW-normalised influence of i on j (Peng & Tzeng 2013 Eq. 13): Tz is the transpose of the row-normalised T_D, not a column normalisation.

Works with

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