ANP (Analytic Network Process)

ANP is a subjective weighting method for when the criteria or alternatives you hold are not fully independent of one another. Unlike AHP, it does not assume a strictly top-down, one-way ordering (goal, then criteria, then alternatives); it accepts that alternatives can influence how important criteria are judged to be, just as criteria influence alternatives, and that elements within a single set can influence one another. Its output is a priority (weight) vector summing to one, which can be computed for criteria and for alternatives alike. Saaty proposed it in 1996, as a generalisation that relaxes the one-way-hierarchy assumption of his own earlier AHP.

Picture AHP as a one-way staircase: the goal sits at the top, criteria in the middle, alternatives at the bottom, and influence flows only downward. ANP turns that staircase into a road network in which some roads run both ways. A criterion can influence an alternative, but the nature of that alternative can equally influence how important the criterion is judged to be; or two criteria within the same set can influence each other. ANP collects these mutual interactions through pairwise comparisons, then lets these effects propagate across the network until they settle into a stable equilibrium, by multiplying the same matrix by itself repeatedly.

The philosophical consequence of this idea is an acknowledgement that real decisions rarely fit a clean hierarchy. AHP assumes criteria are independent of one another and of the alternatives; where that assumption holds, AHP is enough and ANP's added complexity is unnecessary. But where a genuine interdependence exists between criteria, or between criteria and alternatives, ignoring it produces the wrong weights; ANP offers a more accurate picture in that situation.

The method proceeds through five steps.

First, building the unweighted supermatrix. Pairwise comparisons are made for every dependency relationship within and between the sets involved (goal, criteria, alternatives, and sub-criteria where relevant); the local priorities that emerge from these comparisons are gathered into a single large table, the supermatrix.

Second, building the weighted supermatrix. The relative importance of the sets themselves is likewise determined by pairwise comparison and multiplied into the first table; once this is done every column of the table sums to one, giving it a column-stochastic structure.

Third, finding the limit supermatrix. The weighted table is raised to successive powers, multiplied by itself again and again, until it reaches a stationary point. This represents the effects in the network propagating around its loops until they settle into equilibrium.

Fourth, reading off the priorities. The rows belonging to the set of interest (alternatives, say) are read from the stationary table; these rows now give that set's final priorities across the whole network.

Fifth, ranking from highest to lowest. The resulting priorities are ranked by size.

The formulas behind each step, the intermediate tables and the citation formats are given on the DecisionMind method page; this card carries no formulas.

The output is a priority vector summing to one; each figure shows that element's (criterion's or alternative's) relative share of influence within the defined network. This share is fed not only by direct comparisons, as in AHP, but by every indirect feedback loop in the network; so an element scoring low on direct comparisons need not end up with a low share across the network as a whole. Two elements can end up with equal priority; this does not mean the two are "the same," only that the defined network structure is not fine-grained enough to distinguish between them.

Thus instead of writing:

"ANP measured the true value of the alternatives"

the report should read:

"With this network structure and these pairwise comparisons, the alternatives' relative shares of influence are as follows; if the structure of the network, which set influences which, changes, the priorities change with it"

ANP works with crisp data, but that data is not a numerical measurement, it is a pairwise-comparison judgement, as in AHP, of how many times more important one element is than another. DecisionMind carries no extension of this method; the base method stands alone. You need: a network design stating which dependencies exist within and between the sets (who influences whom); pairwise-comparison judgements for each of these dependencies; and judgements on the relative importance of the sets themselves. A weighted supermatrix whose columns do not sum to one prevents the limit matrix from existing, or from being unique; the column sums should therefore be checked once the supermatrix is built. A minimum of two alternatives and two criteria is required; the comfortable working range for the number of criteria is three to twelve.

If your criteria are genuinely independent of one another and of the alternatives, that is, there is no feedback at all, AHP is enough and ANP's added complexity is unnecessary. If there is a genuine mutual interaction between criteria, or between criteria and alternatives, where a criterion's importance varies with the alternative, or criteria feed into one another, ANP is appropriate. The number of pairwise comparisons required grows quickly as the network becomes more complex; where collecting that many judgements from experts is not practical, methods such as DANP, which derive the network structure from influence relations rather than pairwise comparison, should be considered.

Criteria and alternatives independent of one another, a one-way hierarchy → AHP

Criteria and alternatives mutually interacting, feedback present → ANP

The network structure is to be derived from influence relations rather than pairwise comparison → DANP

What is needed is not a ranking but a resource allocation → portfolio methods such as HF-TRADEOFF-PORT

ANP's most important strength is that it models the mutual dependencies common in real decisions explicitly, instead of ignoring them; AHP is, in this respect, a special case of ANP (where every dependency is one-way, ANP reduces to AHP). Because it uses the same pairwise-comparison logic, it is a natural extension for decision-makers already familiar with AHP. Because the limit supermatrix carries every indirect interaction in the network to a mathematically consistent stationary point, it also accounts for indirect effects that looking at direct relationships alone would miss (Saaty, 1996; Saaty and Vargas, 2006).

ANP's fundamental limitation is that the number of pairwise comparisons required grows quickly as the network becomes more complex; this means asking the decision-maker for a great many judgements and raises the risk of inconsistent answers (Whitaker, 2007). A second limitation is the requirement that the columns of the weighted supermatrix sum to one, column-stochasticity; if this is not satisfied, the limit matrix either fails to exist or is not unique. Third, the structure of the network, which set influences which, depends on the analyst's own design decision; different analysts can build different network structures for the same problem, and this leads to different priorities (Sipahi and Timor, 2010). Fourth, ANP still carries AHP's subjective foundation; pairwise comparisons reflect the decision-maker's own judgement, not an objective reality.

The most common mistake is adding a mutual dependency to the network that does not actually exist, purely to make the method look more "advanced"; where no dependency exists, AHP is enough, and the extra comparison only adds noise. A second mistake is computing the limit matrix without ever checking the column sums of the weighted supermatrix; if a column does not sum to one, the result can come out meaningless. A third is presenting the weighted supermatrix, in its state before it has converged, directly as the result, without ever moving on to the limit matrix. A fourth is interpreting two elements' equal priority as meaning "the two are genuinely of equal value"; this usually shows only that the defined network is not detailed enough to distinguish between them. The governing principle is this:

ANP priorities are a reflection of the structure of the defined network and of the pairwise comparisons supplied; if the network structure or the comparisons change, the priorities change with them.