The *Analytic Hierarchy Process* (AHP) is a much popular and general purpose methodology for (hierarchic) problem structuring, (relative) measurement of "objects" with respect to tangible or intangible/subjective properties, and synthesis of results throughout the hierarchy. It has been applied to a wide spectrum of Multicriteria Decision Making (MCDM) problems.

In general, MCDM problems are most usefully represented by a set of "solution" alternatives and a set of "objectives" (instead of a single objective) along which these solutions should be evaluated. Solution of this kind of problem typically involves the 3 already mentioned steps: structuring, measurement, and synthesis.

Definition of alternatives and objectives comprises the structuring phase. The AHP implicitly assumes that a hierarchy is probably the most useful, natural representation format people adopt in order to deal with complex problems. In the measurement phase, the AHP uses paired comparisons of the hierarchical factors, made available by the decision makers (DMers), to derive (rather than assign) ratio scale measures that can be interpreted as local ranking priorities (or weights). Thus, the AHP converts individual preferences into ratio scale weights that are combined into linear additive weights for the associated alternatives. These global weights synthesize the multitude of factors in the hierarchy, and are used to rank the alternatives, therefore assisting people in their complex decision problems.

The vast majority of reported uses of the AHP is for the solution of choice problems (selecting one among a set of competing alternatives) in a multiobjective environment. However, the AHP is more than just a methodology for choice situations, being best explained by its 3 basic functions. Unlike other MCDM methods, this kind of description considerably enlarges the scope of application of the AHP, which is then ultimately (and carefully) delimited by its axiomatic foundation.

In this context, it's more convenient to interpret the typical AHP structuring phase as the decomposition of the original problem into a hierarchy of clusters (subclusters, subsubclusters, and so on) of homogeneous elements (rather than simply by alternatives and objectives). In the measurement phase, people have to perform pairwise comparisons of all combinations of elements in a cluster with respect to its parent, which are then used to derive "local" weights of that elements. Finally, in the synthesis phase, the local weights of the elements in a cluster must be multiplied by the weight of the parent element, thus producing global weights throughout the hierarchy.

In making the pairwise comparisons, it's most common to use a fundamental scale consisting of verbal judgements ranging from "equal" to "extreme" ("equal", "moderately more", "strongly more", "very strongly more" and "extremely more"). Corresponding to the verbal judgements, there are numerical judgements represented by the (absolute) numbers 1, 3, 5, 7, and 9; compromises between these values (i.e. rational numbers) can also be used in order to define finer differences between the elements being compared. In making a judgement, the DMer has to take the smaller element in each pair as a unit of measurement, and then pick a number from the fundamental scale to represent the relative importance between the 2 elements, with respect to the specific property defined by the parent element.

The AHP is based on 4 relatively simple axioms: (1) the reciprocal axiom ("if A is considered by the DMer to be 3 times more important than B, with respect to property C, then B should be considered 1/3 times more important than B, with respect to the same property"), (2) the homogeinety axiom (elements in a cluster should not differ by more than an order of magnitude, according to the parent element), (3) the synthesis axiom (weights of the elements in the hierarchy should not be influenced by lower level elements), and (4) the expectation axiom (individuals who have reasons for their beliefs about the elements in the hierarchy should make sure their ideas are adequately represented in the system for the outcome to match these expectations).

This set of axioms is based on well-known mathematical results involving consistent matrices of pairwise comparison judgements ("if A is 3 times more important than B, and 9 times more important than C, with respect to property D, then B should be 3 times more important than C, according to the same property") and their associated right-eigenvector's ability to generate the true or approximate value ("weights") of the underlying elements. There are several other methods for deriving weights from a comparison matrix; all of them yield the same results when the matrix is (perfectly) consistent. However, so long as some (limited) level of (human) inconsistency is tolerated (or even desired, from a "creative" perspective), dominance is the basic theoretical concept for deriving a ratio scale and no other method qualifies.

Broad areas in which the AHP has been succesfully employed include: (general) choice/prioritization problems, resource allocation, forecasting, quality management ("quality" is multidimensional by nature), benchmarking of key business processes, public policy decisions and strategic planning. Each problem domain requires the adoption of a particular perspective, so, unlike other MCDM methods, there's no strong point in formulating a "fool-proof" way of using the AHP for a specific domain. The AHP is rarely used in isolation; rather, it's used along with or in support of other methodologies.