ANP - Analytic Network Process (AHP with feedback and interdependences)

Overview

ANP extends AHP by allowing feedback loops and interdependences between criteria clusters. The supermatrix captures all pairwise influences; the limit supermatrix (W̄^∞) gives stable long-run priorities. ANP is much more complex than AHP - use only when criteria have genuine mutual influence (e.g. political feasibility influences technical performance and vice versa).

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

   Construct unweighted supermatrix W from cluster/criterion pairwise comparisons.

   $$W=[w_{ij}{]}_{N\times N},N=\sum _c{n}_c$$

   Saaty 1996, p.137 Eq.(1)

2. 2

   Weighted supermatrix W' = W · C where C is the cluster weight matrix; columns stochastic.

   $$W^{\prime }=W\cdot C,\sum _i{W}_{ij}^{\prime }=1\forall j$$

   Saaty 1996, p.139 Eq.(2)

3. 3

   Limit supermatrix W^∞ = lim_{k→∞} (W')^k (powers converge to steady state).

   $$W^{\infty }=\underset{k\to \infty }{\lim }(W^{\prime }{)}^k$$

   Saaty 1996, p.140 Eq.(3)

4. 4

   Read alternative priorities from W^∞ rows in the alternatives cluster.

   $$p_i=W_{i,\text{any col in alt cluster}}^{\infty }$$

   Saaty 1996, p.141

5. 5

   Descending ranking by priorities.

   $$\text{rank}(A_i)\propto -p_i$$

   Saaty 1996, Ch.4

Fits when / Look elsewhere when

Assumptions to verify

• Domain experts available
• Experts can express consistent comparisons

Edge cases and pitfalls

• •ties from W^∞ rows in the alternatives cluster.
• •ties.

• The weighted supermatrix must be column-stochastic (each column sums to 1) for the limit to exist and be unique. Verify column sums after assembly.

Works with

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