# About the Analytic Network Process (ANP)

The Analytic Network Process (ANP) for decision-making with dependence and feedback is a comprehensive framework for the analysis of societal, governmental and corporate decisions. It is a process that allows one to include all the factors and criteria, tangible and intangible, which have a bearing on making a decision. The ANP allows both interaction and feedback within clusters of elements (inner dependence) and between clusters (outer dependence). Such feedback can capture the complex effects of interplay in human society, and this is especially important when risk and uncertainty are involved. Applying the ANP to a decision includes analysis of benefits, opportunities, costs and risks and combining their results by weighting them with respect to strategic criteria. A simple ANP model may consist of a single network, such as those used to estimate market share. But complex decisions invariably require the use of networks and sub-networks concerned with the following types of evaluation: a set of strategic criteria, in terms of which the outcomes to be described next are rated one at a time and the resulting weights used to synthesize these outcomes to determine the best one. Four major divisions follow this, each with several networks.

The first is concerned with the benefits (B) of the decision, the second with its opportunities (O), the third with its costs (C) and the fourth with its risks (R); these are called the BOCR merits of a decision. Each of these has a control hierarchy (or network) of criteria and sub-criteria. These are known as control criteria. They are used to facilitate thinking about influence according to the meaning of each; for example, economic influences are different from political influences and must be sorted out that way and examined separately in a decision network and then combined. If such control criteria interact, then a simple network structure is needed for them also instead of a hierarchy. Under each control criterion there is a network of influences among the elements and clusters of the decision problem, which must include a cluster of the alternatives of that decision. These influences are determined through paired comparisons that lead to priority vectors included as the columns of a matrix of interactions among the elements of two clusters (or the same cluster) in which the interactions take place.

These matrices comprise the entries of a supermatrix that is raised to powers to capture the transitivity of influence among all the elements and determine the overall priorities of all the elements in the network particularly those of the alternatives. For the supermatrix to converge one needs to determine, each time using a paired comparisons matrix of judgments, the influence of the clusters on a given cluster with respect to the control criterion. The vector of priorities derived from the comparisons is then used to weight the corresponding column of matrix entries of the supermatrix corresponding to the influenced cluster. One number multiplies an entire matrix entry of the supermatrix. The number is the corresponding component of the vector of cluster priorities. Each column of the resulting supermatrix now sums to one. It is known as a column stochastic matrix.

The priorities of the alternatives thus derived are then put in ideal form by dividing each value by the largest value among them, and then weighted by the priority of their control criterion. The weighted priorities of the alternatives under the several control criteria under benefits are then summed to obtain the overall outcome with respect to benefits. The same is done for each of the other three merits B, C and R each time leading to an overall vector for that merit. These four outcomes are each rated with respect to the strategic criteria and the ratings normalized and used to weight the four vectors and sum the results for the benefits and opportunities and then also for the costs and risks and subtract the second sum from the first to obtain the final overall outcome. The resulting priorities for some or for all the alternatives can have negative values.

Sometimes, when all the values are negative one chooses the alternative with smallest negative value for the decision as the best of the worst possible choices, which can happen often, for example, in case of badly injured soldiers in a war. The ANP has been applied to a large variety of decisions: marketing, medical, political, social, forecasting and prediction and many others. Its accuracy is impressive in predicting economic trends, winners in sports and other events for which the outcome later became known. Detailed case studies of applications are included in the ANP software manual and in the Encyclicon. Both books are available from RWS Publications, 4922 Ellsworth Avenue, Pittsburgh, PA 15213 USA; phone number: 412-414-5984; fax: 412-681-4510. The process is briefly described with some of its underlying mathematics and then illustrated with several application examples.