Analytic hierarchy process
In the theory of decision making, the analytic hierarchy process (AHP), also analytical hierarchy process,[1] is a structured technique for organizing and analyzing complex decisions, based on mathematics and psychology. It was developed by Thomas L. Saaty in the 1970s; Saaty partnered with Ernest Forman to develop Expert Choice software in 1983, and AHP has been extensively studied and refined since then. It represents an accurate approach to quantifying the weights of decision criteria. Individual experts’ experiences are utilized to estimate the relative magnitudes of factors through pair-wise comparisons. Each of the respondents compares the relative importance of each pair of items using a specially designed questionnaire. The relative importance of the criteria can be determined with the help of the AHP by comparing the criteria and, if applicable, the sub-criteria in pairs by experts or decision-makers. On this basis, the best alternative can be found.[2]
AHP is targeted at group decision making,[3] and is used for decision situations, in fields such as government, business, industry,[4] healthcare and education.
Rather than prescribing a "correct" decision, the AHP helps decision makers find the decision that best suits their goal and their understanding of the problem. It provides a comprehensive and rational framework for structuring a decision problem, for representing and quantifying its elements, for relating those elements to overall goals, and for evaluating alternative solutions.
Users of the AHP first decompose their decision problem into a hierarchy of more easily comprehended sub-problems, each of which can be analyzed independently. The elements of the hierarchy can relate to any aspect of the decision problem—tangible or intangible, carefully measured or roughly estimated, well or poorly understood—anything at all that applies to the decision at hand.
Once the hierarchy is built, the decision makers evaluate its various elements by comparing them to each other two at a time, with respect to their impact on an element above them in the hierarchy. In making the comparisons, the decision makers can use concrete data about the elements, and they can also use their judgments about the elements' relative meaning and importance. Human judgments, and not just the underlying information, can be used in performing the evaluations.[5]
The AHP converts these evaluations to numerical values that can be processed and compared over the entire range of the problem. A numerical weight or priority is derived for each element of the hierarchy, allowing diverse and often incommensurable elements to be compared to one another in a rational and consistent way. This capability distinguishes the AHP from other decision making techniques.
In the final step of the process, numerical priorities are calculated for each of the decision alternatives. These numbers represent the alternatives' relative ability to achieve the decision goal, so they allow a straightforward consideration of the various courses of action.
While it can be used by individuals working on straightforward decisions, the Analytic Hierarchy Process (AHP) is most useful where teams of people are working on complex problems, especially those with high stakes, involving human perceptions and judgments, whose resolutions have long-term repercussions.[6]
Decision situations to which the AHP can be applied include:[1]
The applications of AHP include planning, resource allocation, priority setting, and selection among alternatives.[6] Other areas have included forecasting, total quality management, business process reengineering, quality function deployment, and the balanced scorecard.[1] Other uses of AHP are discussed in the literature:
AHP is sometimes used in designing highly specific procedures for particular situations, such as the rating of buildings by historical significance.[15] It was recently applied to a project that uses video footage to assess the condition of highways in Virginia. Highway engineers first used it to determine the optimum scope of the project, and then to justify its budget to lawmakers.[16]
The weights of the AHP judgement matrix may be corrected with the ones calculated through the Entropy Method. This variant of the AHP method is called AHP-EM.[13][17]
Education and scholarly research[edit]
Though using the analytic hierarchy process requires no specialized academic training, it is considered an important subject in many institutions of higher learning, including schools of engineering[18] and graduate schools of business.[19] It is a particularly important subject in the quality field, and is taught in many specialized courses including Six Sigma, Lean Six Sigma, and QFD.[20][21][22]
The International Symposium on the Analytic Hierarchy Process (ISAHP) holds biennial meetings of academics and practitioners interested in the field. A wide range of topics is covered. Those in 2005 ranged from "Establishing Payment Standards for Surgical Specialists", to "Strategic Technology Roadmapping", to "Infrastructure Reconstruction in Devastated Countries".[23]
At the 2007 meeting in Valparaíso, Chile, 90 papers were presented from 19 countries, including the US, Germany, Japan, Chile, Malaysia, and Nepal.[24] A similar number of papers were presented at the 2009 symposium in Pittsburgh, Pennsylvania, when 28 countries were represented.[25] Subjects of the papers included Economic Stabilization in Latvia, Portfolio Selection in the Banking Sector, Wildfire Management to Help Mitigate Global Warming, and Rural Microprojects in Nepal.
Experienced practitioners know that the best way to understand the AHP is to work through cases and examples. Two detailed case studies, specifically designed as in-depth teaching examples, are provided as appendices to this article:
Some of the books on AHP contain practical examples of its use, though they are not typically intended to be step-by-step learning aids.[26][31] One of them contains a handful of expanded examples, plus about 400 AHP hierarchies briefly described and illustrated with figures.[28] Many examples are discussed, mostly for professional audiences, in papers published by the International Symposium on the Analytic Hierarchy Process.[32][33][34][35][36]
The AHP is included in most operations research and management science textbooks, and is taught in numerous universities; it is used extensively in organizations that have carefully investigated its theoretical underpinnings.[1] The method does have its critics.[8]
In the early 1990s a series of debates between critics and proponents of AHP was published in Management Science[37][38][39][40] and The Journal of the Operational Research Society,[41][42][43] two prestigious journals where Saaty and his colleagues had considerable influence. These debates seem to have been settled in favor of AHP:
A 1997 paper examined possible flaws in the verbal (vs. numerical) scale often used in AHP pairwise comparisons.[45] Another from the same year claimed that innocuous changes to the AHP model can introduce order where no order exists.[46] A 2006 paper found that the addition of criteria for which all alternatives perform equally can alter the priorities of alternatives.[47]
In 2021, the first comprehensive evaluation of the AHP was published in a book authored by two academics from Technical University of Valencia and Universidad Politécnica de Cartagena, and published by Springer Nature. Based on an empirical investigation and objective testimonies by 101 researchers, the study found at least 30 flaws in the AHP and found it unsuitable for complex problems, and in certain situations even for small problems.[48]
Rank reversal[edit]
Decision making involves ranking alternatives in terms of criteria or attributes of those alternatives. It is an axiom of some decision theories that when new alternatives are added to a decision problem, the ranking of the old alternatives must not change — that "rank reversal" must not occur.
There are two schools of thought about rank reversal. One maintains that new alternatives that introduce no additional attributes should not cause rank reversal under any circumstances. The other maintains that there are some situations in which rank reversal can reasonably be expected. The original formulation of AHP allowed rank reversals. In 1993, Forman[49] introduced a second AHP synthesis mode, called the ideal synthesis mode, to address choice situations in which the addition or removal of an 'irrelevant' alternative should not and will not cause a change in the ranks of existing alternatives. The current version of the AHP can accommodate both these schools—its ideal mode preserves rank, while its distributive mode allows the ranks to change. Either mode is selected according to the problem at hand.
Rank reversal and AHP are extensively discussed in a 2001 paper in Operations Research,[1] as well as a chapter entitled Rank Preservation and Reversal, in the current basic book on AHP.[31] The latter presents published examples of rank reversal due to adding copies and near copies of an alternative, due to intransitivity of decision rules, due to adding phantom and decoy alternatives, and due to the switching phenomenon in utility functions. It also discusses the Distributive and Ideal Modes of AHP.
A new form of rank reversal of AHP was found in 2014[50] in which AHP produces rank order reversal when eliminating irrelevant data, this is data that do not differentiate alternatives.
There are different types of rank reversals. Also, other methods besides the AHP may exhibit such rank reversals. More discussion on rank reversals with the AHP and other MCDM methods is provided in the rank reversals in decision-making page.
Non-monotonicity of some weight extraction methods[edit]
Within a comparison matrix one may replace a judgement with a less favorable judgment and then check to see if the indication of the new priority becomes less favorable than the original priority. In the context of tournament matrices, it has been proven by Oskar Perron[51] that the principal right eigenvector method is not monotonic. This behaviour can also be demonstrated for reciprocal n x n matrices, where n > 3. Alternative approaches are discussed elsewhere.[52][53][54][55]