ISM - Interpretive Structural Modeling

Overview

Apply F.steps in order.

Output

weight or utility, 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

General MCDM

How it works

1. 1

   Binary relation matrix D = [π_ij] (1 if e_i affects e_j).

   $$D=[{\pi }_{ij}{]}_{n\times n},{\pi }_{ij}\in \{0,1\}$$

   Warfield 1973, p.405 Sec.2

2. 2

   Reachability matrix M* via Boolean powers until stable.

   $$M=D+I,M^{\ast }=M^k\text{ s.t. }{M}^k=M^{k+1}$$

   Warfield 1973, p.406 Eq.(1)

3. 3

   Reachability set R(t_i) and antecedent set A(t_i).

   $$R(t_i)=\{e_j\mid m_{ji}^{\ast }=1\},A(t_i)=\{e_j\mid m_{ij}^{\ast }=1\}$$

   Warfield 1973, p.406 Eqs.(2)-(3)

4. 4

   Level partition: e_i at current level iff R(t_i) ∩ A(t_i) = R(t_i).

   $$R(t_i)\cap A(t_i)=R(t_i)⟹e_i\in \text{level }l$$

   Warfield 1973, p.407 Eq.(4)

5. 5

   Build hierarchical digraph from level partition.

   $$\mathcal{G}=(V,E),V=\{e_1,\dots ,e_n\}$$

   Warfield 1973, p.407 Sec.4

Edge cases and pitfalls

• •if e_i affects e_j).

Works with

How to cite