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Analytic hierarchy process

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In the theory of <a href="Decision_making" class="mw-redirect" title="Decision making">decision making</a>, the analytic hierarchy process (AHP), also analytical hierarchy process,<a href="#cite_note-FormanGass-1">[1]</a> is a structured technique for organizing and analyzing <a href="MCDA" class="mw-redirect" title="MCDA">complex decisions</a>, based on <a href="Mathematics" title="Mathematics">mathematics</a> and <a href="Psychology" title="Psychology">psychology</a>. It was developed by <a href="Thomas_L._Saaty" title="Thomas L. Saaty">Thomas L. Saaty</a> in the 1970s; Saaty partnered with Ernest Forman to develop <a href="Expert_Choice" title="Expert Choice">Expert Choice</a> 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 <a href="Expert" title="Expert">experts</a>’ 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.

<div class="mw-heading mw-heading2"><h2 id="Uses_and_applications">Uses and applications</h2></div>

AHP is targeted at <a href="Group_decision_making" class="mw-redirect" title="Group decision making">group decision making</a>,<a href="#cite_note-GDM-2">[2]</a> and is used for <a href="Decision_making" class="mw-redirect" title="Decision making">decision situations</a>, in fields such as government, business, industry,<a href="#cite_note-3">[3]</a> 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.<a href="#cite_note-RACSAM-4">[4]</a>

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 <a href="https://en.wiktionary.org/wiki/priority" class="extiw external" title="wikt:priority">priority</a> 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.<a href="#cite_note-Bhushan,_2004-5">[5]</a>

Decision situations to which the AHP can be applied include:<a href="#cite_note-FormanGass-1">[1]</a>

<ul><li>Choice – The selection of one alternative from a given set of alternatives, usually where there are multiple decision criteria involved.</li>

<li><a href="Ranking" title="Ranking">Ranking</a> – Putting a set of alternatives in order from most to least desirable.</li>

<li>Prioritization – Determining the relative merit of members of a set of alternatives, as opposed to selecting a single one or merely ranking them</li>

<li><a href="Resource_allocation" title="Resource allocation">Resource allocation</a> – Apportioning resources among a set of alternatives</li>

<li><a href="Benchmarking" title="Benchmarking">Benchmarking</a> – Comparing the processes in one's own organization with those of other best-of-breed organizations</li>

<li><a href="Quality_management" title="Quality management">Quality management</a> – Dealing with the multidimensional aspects of quality and quality improvement</li>

<li><a href="Conflict_resolution" title="Conflict resolution">Conflict resolution</a> – Settling disputes between parties with apparently incompatible goals or positions<a href="#cite_note-GDM-2">[2]</a></li></ul>

The applications of AHP include <a href="Planning" title="Planning">planning</a>, <a href="Resource_allocation" title="Resource allocation">resource allocation</a>, priority setting, and selection among alternatives.<a href="#cite_note-Bhushan,_2004-5">[5]</a> Other areas have included <a href="Forecasting" title="Forecasting">forecasting</a>, <a href="Total_quality_management" title="Total quality management">total quality management</a>, <a href="Business_process_reengineering" class="mw-redirect" title="Business process reengineering">business process reengineering</a>, <a href="Quality_function_deployment" title="Quality function deployment">quality function deployment</a>, and the <a href="Balanced_scorecard" title="Balanced scorecard">balanced scorecard</a>.<a href="#cite_note-FormanGass-1">[1]</a> Other uses of AHP are discussed in the literature:

<ul><li>Deciding how best to reduce the impact of global <a href="Climate_change" title="Climate change">climate change</a> (Fondazione Eni Enrico Mattei)<a href="#cite_note-6">[6]</a></li>

<li><a href="Quantification_(science)" title="Quantification (science)">Quantifying</a> the overall <a href="Quality_(business)" title="Quality (business)">quality</a> of <a href="Software_system" title="Software system">software systems</a> (<a href="Microsoft_Corporation" class="mw-redirect" title="Microsoft Corporation">Microsoft Corporation</a>)<a href="#cite_note-MSDN-7">[7]</a></li>

<li>Selecting <a href="Faculty_(teaching_staff)" class="mw-redirect" title="Faculty (teaching staff)">university faculty</a> (<a href="Bloomsburg_University_of_Pennsylvania" class="mw-redirect" title="Bloomsburg University of Pennsylvania">Bloomsburg University of Pennsylvania</a>)<a href="#cite_note-8">[8]</a></li>

<li>Deciding where to locate <a href="Offshoring" title="Offshoring">offshore manufacturing plants</a> (<a href="University_of_Cambridge" title="University of Cambridge">University of Cambridge</a>)<a href="#cite_note-9">[9]</a></li>

<li>Assessing <a href="Risk" title="Risk">risk</a> in operating cross-country <a href="Pipeline_transport" class="mw-redirect" title="Pipeline transport">petroleum pipelines</a> (<a href="American_Society_of_Civil_Engineers" title="American Society of Civil Engineers">American Society of Civil Engineers</a>)<a href="#cite_note-10">[10]</a></li>

<li>Deciding how best to <a href="Watershed_management" title="Watershed management">manage U.S. watersheds</a> (<a href="U.S._Department_of_Agriculture" class="mw-redirect" title="U.S. Department of Agriculture">U.S. Department of Agriculture</a>)<a href="#cite_note-de_Steiguer_2003-11">[11]</a></li>

<li>More Effectively Define and Evaluate <a href="SAP_ERP" title="SAP ERP">SAP</a> Implementation Approaches (<a rel="nofollow" class="external text" href="http://sapexperts.wispubs.com/IT/Articles/Use-the-AHP-Methodology-to-More-Effectively-Define-and-Evaluate-Your-SAP-Implementation-Approach?id=52B9059CD94C40C4998639C32D1468B9">SAP Experts</a>)</li>

<li>Integrated evaluation of a community's sustanaibility in terms of environment, economy, society, institution, and culture.<a href="#cite_note-MDPI_2017-12">[12]</a></li>

<li><a href="Accelerated_Bridge_Construction" class="mw-redirect" title="Accelerated Bridge Construction">Accelerated Bridge Construction</a> Decision Making Tool to assist in determining the viability of accelerated bridge construction (ABC) over traditional construction methods and in selecting appropriate construction and contracting strategies on a case-by-case basis.<a href="#cite_note-13">[13]</a></li></ul>

AHP is sometimes used in designing highly specific procedures for particular situations, such as the rating of buildings by historical significance.<a href="#cite_note-14">[14]</a> It was recently applied to a project that uses <a href="Video" title="Video">video</a> footage to assess the condition of highways in <a href="Virginia" title="Virginia">Virginia</a>. <a href="Highway_engineer" class="mw-redirect" title="Highway engineer">Highway engineers</a> first used it to determine the optimum scope of the <a href="Project" title="Project">project</a>, and then to justify its <a href="Budget" title="Budget">budget</a> to <a href="Lawmaker" class="mw-redirect" title="Lawmaker">lawmakers</a>.<a href="#cite_note-15">[15]</a>

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.<a href="#cite_note-MDPI_2017-12">[12]</a><a href="#cite_note-16">[16]</a>

<div class="mw-heading mw-heading2"><h2 id="Education_and_scholarly_research">Education and scholarly research</h2></div>

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<a href="#cite_note-17">[17]</a> and <a href="Business_school" title="Business school">graduate schools of business</a>.<a href="#cite_note-18">[18]</a> It is a particularly important subject in the <a href="Quality_(business)" title="Quality (business)">quality</a> field, and is taught in many specialized courses including <a href="Six_Sigma" title="Six Sigma">Six Sigma</a>, <a href="Lean_Six_Sigma" title="Lean Six Sigma">Lean Six Sigma</a>, and <a href="Quality_function_deployment" title="Quality function deployment">QFD</a>.<a href="#cite_note-19">[19]</a><a href="#cite_note-20">[20]</a><a href="#cite_note-21">[21]</a>

The <a href="International_Symposium_on_the_Analytic_Hierarchy_Process" title="International Symposium on the Analytic Hierarchy Process">International Symposium on the Analytic Hierarchy Process</a> (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".<a href="#cite_note-22">[22]</a>

At the 2007 meeting in <a href="Valpara%C3%ADso%2C_Chile" class="mw-redirect" title="Valparaíso, Chile">Valparaíso, Chile</a>, 90 papers were presented from 19 countries, including the US, Germany, Japan, Chile, Malaysia, and Nepal.<a href="#cite_note-ISAHP2007-23">[23]</a> A similar number of papers were presented at the 2009 symposium in <a href="Pittsburgh%2C_Pennsylvania" class="mw-redirect" title="Pittsburgh, Pennsylvania">Pittsburgh, Pennsylvania</a>, when 28 countries were represented.<a href="#cite_note-ISAHP2009-24">[24]</a> 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.

As it can be seen in the material that follows, using the AHP involves the mathematical synthesis of numerous judgments about the decision problem at hand. It is not uncommon for these judgments to number in the dozens or even the hundreds. While the math can be done by hand or with a calculator, it is far more common to use one of several computerized methods for entering and synthesizing the judgments. The simplest of these involve standard spreadsheet software, while the most complex use custom software, often augmented by special devices for acquiring the judgments of decision makers gathered in a meeting room.

The procedure for using the AHP can be summarized as:

<ol><li>Model the problem as a hierarchy containing the decision goal, the alternatives for reaching it, and the criteria for evaluating the alternatives.</li>

<li>Establish priorities among the elements of the hierarchy by making a series of judgments based on pairwise comparisons of the elements. For example, when comparing potential purchases of commercial real estate, the investors might say they prefer location over price and price over timing.</li>

<li>Synthesize these judgments to yield a set of overall priorities for the hierarchy. This would combine the investors' judgments about location, price and timing for properties A, B, C, and D into overall priorities for each property.</li>

<li>Check the consistency of the judgments.</li>

<li>Come to a final decision based on the results of this process.<a href="#cite_note-DMFL-25">[25]</a></li></ol>

These steps are more fully described below.

<div class="mw-heading mw-heading3"><h3 id="Model_the_problem_as_a_hierarchy">Model the problem as a hierarchy</h3></div>

The first step in the analytic hierarchy process is to model the problem as a hierarchy. In doing this, participants explore the aspects of the problem at levels from general to detailed, then express it in the multileveled way that the AHP requires. As they work to build the hierarchy, they increase their understanding of the problem, of its context, and of each other's thoughts and feelings about both.<a href="#cite_note-DMFL-25">[25]</a>

<div class="mw-heading mw-heading4"><h4 id="Hierarchies_defined">Hierarchies defined</h4></div>

A hierarchy is a stratified system of ranking and organizing people, things, ideas, etc., where each element of the system, except for the top one, is subordinate to one or more other elements. Though the concept of hierarchy is easily grasped intuitively, it can also be described mathematically.<a href="#cite_note-Principia-26">[26]</a> Diagrams of hierarchies are often shaped roughly like pyramids, but other than having a single element at the top, there is nothing necessarily pyramid-shaped about a hierarchy.

Human organizations are often structured as hierarchies, where the hierarchical system is used for assigning responsibilities, exercising leadership, and facilitating communication. Familiar hierarchies of "things" include a desktop computer's tower unit at the "top", with its subordinate monitor, keyboard, and mouse "below."

In the world of ideas, we use hierarchies to help us acquire detailed knowledge of complex reality: we structure the reality into its constituent parts, and these in turn into their own constituent parts, proceeding down the hierarchy as many levels as we care to. At each step, we focus on understanding a single component of the whole, temporarily disregarding the other components at this and all other levels. As we go through this process, we increase our global understanding of whatever complex reality we are studying.

Think of the hierarchy that medical students use while learning anatomy—they separately consider the musculoskeletal system (including parts and subparts like the hand and its constituent muscles and bones), the circulatory system (and its many levels and branches), the nervous system (and its numerous components and subsystems), etc., until they've covered all the systems and the important subdivisions of each. Advanced students continue the subdivision all the way to the level of the cell or molecule. In the end, the students understand the "big picture" and a considerable number of its details. Not only that, but they understand the relation of the individual parts to the whole. By working hierarchically, they've gained a comprehensive understanding of anatomy.

Similarly, when we approach a complex decision problem, we can use a hierarchy to integrate large amounts of information into our understanding of the situation. As we build this information structure, we form a better and better picture of the problem as a whole.<a href="#cite_note-DMFL-25">[25]</a>

<div class="mw-heading mw-heading4"><h4 id="Hierarchies_in_the_AHP">Hierarchies in the AHP</h4></div>

An AHP hierarchy is a structured means of modeling the decision at hand. It consists of an overall goal, a group of options or alternatives for reaching the goal, and a group of factors or criteria that relate the alternatives to the goal. The criteria can be further broken down into subcriteria, sub-subcriteria, and so on, in as many levels as the problem requires. A criterion may not apply uniformly, but may have graded differences like a little sweetness is enjoyable but too much sweetness can be harmful. In that case, the criterion is divided into subcriteria indicating different intensities of the criterion, like: little, medium, high and these intensities are prioritized through comparisons under the parent criterion, sweetness.

Published descriptions of AHP applications often include diagrams and descriptions of their hierarchies; some simple ones are shown throughout this article. More complex AHP hierarchies have been collected and reprinted in at least one book.<a href="#cite_note-Hierarchon-27">[27]</a> More complex hierarchies can be found on a special talk page for this article.

The design of any AHP hierarchy will depend not only on the nature of the problem at hand, but also on the knowledge, judgments, values, opinions, needs, wants, etc. of the participants in the decision-making process. Constructing a hierarchy typically involves significant discussion, research, and discovery by those involved. Even after its initial construction, it can be changed to accommodate newly-thought-of criteria or criteria not originally considered to be important; alternatives can also be added, deleted, or changed.<a href="#cite_note-DMFL-25">[25]</a>

To better understand AHP hierarchies, consider a decision problem with a goal to be reached, three alternative ways of reaching the goal, and four criteria against which the alternatives need to be measured.

Such a hierarchy can be visualized as a diagram like the one immediately below, with the goal at the top, the three alternatives at the bottom, and the four criteria in between. There are useful terms for describing the parts of such diagrams: Each box is called a node. A node that is connected to one or more nodes in a level below it is called a parent node. The nodes to which it is so connected are called its children.

Applying these definitions to the diagram below, the goal is the parent of the four criteria, and the four criteria are children of the goal. Each criterion is a parent of the three Alternatives. Note that there are only three Alternatives, but in the diagram, each of them is repeated under each of its parents.

To reduce the size of the drawing required, it is common to represent AHP hierarchies as shown in the diagram below, with only one node for each alternative, and with multiple lines connecting the alternatives and the criteria that apply to them. To avoid clutter, these lines are sometimes omitted or reduced in number. Regardless of any such simplifications in the diagram, in the actual hierarchy each criterion is individually connected to the alternatives. The lines may be thought of as being directed downward from the parent in one level to its children in the level below.

<div class="mw-heading mw-heading3"><h3 id="Evaluate_the_hierarchy">Evaluate the hierarchy</h3></div>

Once the hierarchy has been constructed, the participants analyze it through a series of pairwise comparisons that derive numerical scales of measurement for the nodes. The criteria are pairwise compared against the goal for importance. The alternatives are pairwise compared against each of the criteria for preference. The comparisons are processed mathematically, and priorities are derived for each node.

Consider the "Choose a Leader" example above. An important task of the decision makers is to determine the weight to be given each criterion in making the choice of a leader. Another important task is to determine the weight to be given to each candidate with regard to each of the criteria. The AHP not only lets them do that, but it lets them put a meaningful and objective numerical value on each of the four criteria.

Unlike most surveys which adopt the five point <a href="Likert_scale" title="Likert scale">Likert scale</a>, AHP's questionnaire is 9 to 1 to 9.<a href="#cite_note-28">[28]</a>

<div class="mw-heading mw-heading3"><h3 id="Establish_priorities">Establish priorities</h3></div>

This section explains priorities, shows how they are established, and provides a simple example.

<div class="mw-heading mw-heading4"><h4 id="Priorities_defined_and_explained">Priorities defined and explained</h4></div>

Priorities are numbers associated with the nodes of an AHP hierarchy. They represent the relative weights of the nodes in any group.

Like probabilities, priorities are <a href="Dimensionless_quantity" title="Dimensionless quantity">absolute numbers</a> between zero and one, without units or dimensions. A node with priority .200 has twice the weight in reaching the goal as one with priority .100, ten times the weight of one with priority .020, and so forth. Depending on the problem at hand, "weight" can refer to importance, or preference, or likelihood, or whatever factor is being considered by the decision makers.

Priorities are distributed over a hierarchy according to its architecture, and their values depend on the information entered by users of the process. Priorities of the Goal, the Criteria, and the Alternatives are intimately related, but need to be considered separately.

By definition, the priority of the Goal is 1.000. The priorities of the alternatives always add up to 1.000. Things can become complicated with multiple levels of Criteria, but if there is only one level, their priorities also add to 1.000. All this is illustrated by the priorities in the example below.

Observe that the priorities on each level of the example—the goal, the criteria, and the alternatives—all add up to 1.000.

The priorities shown are those that exist before any information has been entered about weights of the criteria or alternatives, so the priorities within each level are all equal. They are called the hierarchy's default priorities. If a fifth Criterion were added to this hierarchy, the default priority for each Criterion would be .200. If there were only two Alternatives, each would have a default priority of .500.

Two additional concepts apply when a hierarchy has more than one level of criteria: local priorities and global priorities. Consider the hierarchy shown below, which has several Subcriteria under each Criterion.

The local priorities, shown in gray, represent the relative weights of the nodes within a group of siblings with respect to their parent. The local priorities of each group of Criteria and their sibling Subcriteria add up to 1.000. The global priorities, shown in black, are obtained by multiplying the local priorities of the siblings by their parent's global priority. The global priorities for all the subcriteria in the level add up to 1.000.

The rule is this: Within a hierarchy, the global priorities of child nodes always add up to the global priority of their parent. Within a group of children, the local priorities add up to 1.000.

So far, we have looked only at default priorities. As the Analytical Hierarchy Process moves forward, the priorities will change from their default values as the decision makers input information about the importance of the various nodes. They do this by making a series of pairwise comparisons.

<div class="mw-heading mw-heading2"><h2 id="Practical_examples">Practical examples</h2></div>

Experienced practitioners know that the best way to understand the AHP is to work through cases and examples. Two detailed <a href="Case_study" title="Case study">case studies</a>, specifically designed as in-depth teaching examples, are provided as appendices to this article:

<ul><li>Simple step-by-step example with four Criteria and three Alternatives: Choosing a leader for an organization.</li>

<li>More complex step-by-step example with ten Criteria/Subcriteria and six Alternatives: <a href="Analytic_hierarchy_process_%E2%80%94_Car_example" class="mw-redirect" title="Analytic hierarchy process — Car example">Buying a family car</a> and Machinery Selection Example.<a href="#cite_note-29">[29]</a></li></ul>

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.<a href="#cite_note-DMFL-25">[25]</a><a href="#cite_note-SaatyFundamentals-30">[30]</a> One of them contains a handful of expanded examples, plus about 400 AHP hierarchies briefly described and illustrated with figures.<a href="#cite_note-Hierarchon-27">[27]</a> Many examples are discussed, mostly for professional audiences, in papers published by the <a href="International_Symposium_on_the_Analytic_Hierarchy_Process" title="International Symposium on the Analytic Hierarchy Process">International Symposium on the Analytic Hierarchy Process</a>.<a href="#cite_note-31">[31]</a><a href="#cite_note-32">[32]</a><a href="#cite_note-33">[33]</a><a href="#cite_note-34">[34]</a><a href="#cite_note-35">[35]</a>

<div class="mw-heading mw-heading2"><h2 id="Criticisms">Criticisms</h2></div>

The AHP is included in most <a href="Operations_research" title="Operations research">operations research</a> and <a href="Management_science" title="Management science">management science</a> textbooks, and is taught in numerous universities; it is used extensively in organizations that have carefully investigated its theoretical underpinnings.<a href="#cite_note-FormanGass-1">[1]</a> The method does have its critics.<a href="#cite_note-MSDN-7">[7]</a>

In the early 1990s a series of debates between critics and proponents of AHP was published in Management Science<a href="#cite_note-36">[36]</a><a href="#cite_note-37">[37]</a><a href="#cite_note-38">[38]</a><a href="#cite_note-39">[39]</a> and The Journal of the Operational Research Society,<a href="#cite_note-40">[40]</a><a href="#cite_note-41">[41]</a><a href="#cite_note-42">[42]</a> two prestigious journals where Saaty and his colleagues had considerable influence. These debates seem to have been settled in favor of AHP:

<ul><li>An in-depth paper was published in Operations Research in 2001.</li>

<li>A 2008 Management Science paper reviewing 15 years of progress in all areas of <a href="Multiple-criteria_decision_analysis" title="Multiple-criteria decision analysis">Multicriteria Decision Making</a></li>

<li>in 2008, the major society for operations research, the <a href="Institute_for_Operations_Research_and_the_Management_Sciences" title="Institute for Operations Research and the Management Sciences">Institute for Operations Research and the Management Sciences</a> formally recognized AHP's broad impact on its fields.<a href="#cite_note-43">[43]</a></li></ul>

A 1997 paper examined possible flaws in the verbal (vs. numerical) scale often used in AHP pairwise comparisons.<a href="#cite_note-44">[44]</a> Another from the same year claimed that innocuous changes to the AHP model can introduce order where no order exists.<a href="#cite_note-45">[45]</a> A 2006 paper found that the addition of criteria for which all alternatives perform equally can alter the priorities of alternatives.<a href="#cite_note-46">[46]</a>

In 2021, the first comprehensive evaluation of the AHP was published in a book authored by two academics from <a href="Technical_University_of_Valencia" title="Technical University of Valencia">Technical University of Valencia</a> and <a href="Universidad_Polit%C3%A9cnica_de_Cartagena" class="mw-redirect" title="Universidad Politécnica de Cartagena">Universidad Politécnica de Cartagena</a>, and published by <a href="Springer_Nature" title="Springer Nature">Springer Nature</a>. 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.<a href="#cite_note-47">[47]</a>

<div class="mw-heading mw-heading2"><h2 id="Rank_reversal">Rank reversal</h2></div>

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 "<a href="Rank_reversals_in_decision-making" title="Rank reversals in decision-making">rank reversal</a>" 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<a href="#cite_note-48">[48]</a> 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,<a href="#cite_note-FormanGass-1">[1]</a> as well as a chapter entitled Rank Preservation and Reversal, in the current basic book on AHP.<a href="#cite_note-SaatyFundamentals-30">[30]</a> The latter presents published examples of rank reversal due to adding copies and near copies of an alternative, due to <a href="Transitive_relation" title="Transitive relation">intransitivity</a> 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<a href="#cite_note-49">[49]</a> 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 <a href="Rank_reversals_in_decision-making" title="Rank reversals in decision-making">rank reversals in decision-making</a> page.

<div class="mw-heading mw-heading2"><h2 id="Non-monotonicity_of_some_weight_extraction_methods">Non-monotonicity of some weight extraction methods</h2></div>

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 <a href="Oskar_Perron" title="Oskar Perron">Oskar Perron</a><a href="#cite_note-LandauEdmund-50">[50]</a> 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.<a href="#cite_note-Zermelo-51">[51]</a><a href="#cite_note-Hasse-52">[52]</a><a href="#cite_note-:0-53">[53]</a><a href="#cite_note-54">[54]</a>

<ul><li><a href="Analytic_hierarchy_process_%E2%80%93_car_example" title="Analytic hierarchy process – car example">Analytic hierarchy process – car example</a></li>

<li>Analytic hierarchy process – leader example</li>

<li><a href="Analytic_network_process" title="Analytic network process">Analytic network process</a></li>

<li><a href="Arrow's_impossibility_theorem" title="Arrow's impossibility theorem">Arrow's impossibility theorem</a></li>

<li><a href="Decision_making" class="mw-redirect" title="Decision making">Decision making</a></li>

<li><a href="Decision-making_paradox" title="Decision-making paradox">Decision-making paradox</a></li>

<li><a href="Decision-making_software" title="Decision-making software">Decision-making software</a></li>

<li><a href="Hierarchical_decision_process" title="Hierarchical decision process">Hierarchical decision process</a></li>

<li><a href="L._L._Thurstone" class="mw-redirect" title="L. L. Thurstone">L. L. Thurstone</a></li>

<li><a href="Law_of_comparative_judgment" title="Law of comparative judgment">Law of comparative judgment</a></li>

<li><a href="Multi-criteria_decision_analysis" class="mw-redirect" title="Multi-criteria decision analysis">Multi-criteria decision analysis</a></li>

<li><a href="Pairwise_comparison_(psychology)" title="Pairwise comparison (psychology)">Pairwise comparison</a></li>

<li><a href="Preference" title="Preference">Preference</a></li>

<li><a href="Principal_component_analysis" title="Principal component analysis">Principal component analysis</a></li>

<li><a href="Rank_reversals_in_decision-making" title="Rank reversals in decision-making">Rank reversals in decision-making</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-Principia-26"><a href="#cite_ref-Principia_26-0">^</a> <cite id="CITEREFSaaty2010" class="citation book cs1"><a href="Thomas_L._Saaty" title="Thomas L. Saaty">Saaty, Thomas L.</a> (2010). Principia Mathematica Decernendi: Mathematical Principles of Decision Making. Pittsburgh, Pennsylvania: RWS Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-888603-10-1</bdi>.</cite>

</li>

<li id="cite_note-Hierarchon-27">^ <a href="#cite_ref-Hierarchon_27-0">a</a> <a href="#cite_ref-Hierarchon_27-1">b</a> <cite id="CITEREFSaatyErnest_H._Forman1992" class="citation book cs1"><a href="Thomas_L._Saaty" title="Thomas L. Saaty">Saaty, Thomas L.</a>; Ernest H. Forman (1992). The Hierarchon: A Dictionary of Hierarchies. Pittsburgh, Pennsylvania: RWS Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-9620317-5-5</bdi>.</cite> 496 pages, spiral bound. Each entry includes a description and diagram of an AHP model; the models are grouped in categories: educational, government/public policy, government public/strategy, health military, non-profit, personal, planning, political, etc.

</li>

<li id="cite_note-28"><a href="#cite_ref-28">^</a> <cite id="CITEREFLiChauZeng2019" class="citation journal cs1">Li, Rita Yi Man; Chau, Kwong Wing; Zeng, Frankie Fanjie (2019). <a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fsu11102863">"Ranking of Risks for Existing and New Building Works"</a>. Sustainability. 11 (10): 2863. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2019Sust...11.2863L">2019Sust...11.2863L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fsu11102863">10.3390/su11102863</a>.</cite>

</li>

<li id="cite_note-29"><a href="#cite_ref-29">^</a> <cite id="CITEREFPerez-RodriguezRojo-Alboreca,_Alberto2012" class="citation journal cs1">Perez-Rodriguez, Fernando; Rojo-Alboreca, Alberto (2012-01-12). <a rel="nofollow" class="external text" href="https://doi.org/10.5424%2Ffs%2F2012213-02641">"Forestry application of the AHP by use of MPC© software"</a>. Forest Systems. 21 (3): 418–425. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5424%2Ffs%2F2012213-02641">10.5424/fs/2012213-02641</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10347%2F21679">10347/21679</a>.</cite>

</li>

<li id="cite_note-SaatyFundamentals-30">^ <a href="#cite_ref-SaatyFundamentals_30-0">a</a> <a href="#cite_ref-SaatyFundamentals_30-1">b</a> <cite id="CITEREFSaaty2001" class="citation book cs1"><a href="Thomas_L._Saaty" title="Thomas L. Saaty">Saaty, Thomas L.</a> (2001). Fundamentals of Decision Making and Priority Theory. Pittsburgh, Pennsylvania: RWS Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-9620317-6-2</bdi>.</cite>

</li>

<li id="cite_note-31"><a href="#cite_ref-31">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.isahp.org/2001Proceedings/index.html">"Proceedings of the 6th International Symposium on the AHP"</a>. ISAHP Web Site. ISAHP. August 2001. Retrieved 2009-04-03.</cite>

</li>

<li id="cite_note-32"><a href="#cite_ref-32">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.isahp.org/2003Proceedings/index.htm">"Proceedings of the 7th International Symposium on the AHP"</a>. ISAHP Web Site. ISAHP. August 2003. Retrieved 2009-04-03.</cite>

</li>

<li id="cite_note-33"><a href="#cite_ref-33">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.isahp.org/2005Proceedings/index.html">"Proceedings of the 8th International Symposium on the AHP"</a>. ISAHP Web Site. ISAHP. August 2005. Retrieved 2009-04-03.</cite>

</li>

<li id="cite_note-34"><a href="#cite_ref-34">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.isahp.org/2007Proceedings/Index.htm">"Proceedings of the 9th International Symposium on the AHP"</a>. ISAHP Web Site. ISAHP. August 2007. Retrieved 2009-04-03.</cite>

</li>

<li id="cite_note-35"><a href="#cite_ref-35">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.isahp.org/2009Proceedings/index.htm">"Proceedings of the 10th International Symposium on the AHP"</a>. ISAHP Web Site. ISAHP. August 2009. Retrieved 2011-01-05.</cite>

</li>

<li id="cite_note-36"><a href="#cite_ref-36">^</a> Dyer, J. S. (1990): Remarks on the Analytic Hierarchy Process. In: Management Science, 36 (3), S. 249-258.

</li>

<li id="cite_note-37"><a href="#cite_ref-37">^</a> M. V. Mikhalevic "Remarks on the Dyer-Saaty controversy" Cybernetics and Systems Analysis, Volume 30, Number 1 / January, 1994

</li>

<li id="cite_note-38"><a href="#cite_ref-38">^</a> Patrick T. Harker, Luis G. Vargas, "Reply to 'Remarks on the Analytic Hierarchy Process' by J. S. Dyer", Management Science, Vol. 36, No. 3 (Mar., 1990), pp. 269-273

</li>

<li id="cite_note-39"><a href="#cite_ref-39">^</a> Dyer, J.S. (1990b), "A clarification of ‘Remarks on the analytic hierarchy process’", Management Science, Vol. 36 No.3, pp.274-5.

</li>

<li id="cite_note-40"><a href="#cite_ref-40">^</a> Holder, R.D., Some Comment on the Analytic Hierarchy Process, Journal of the Operational Research Society, 1990, 41, 11 1073-1076.

</li>

<li id="cite_note-41"><a href="#cite_ref-41">^</a> Thomas L. Saaty "Response to Holder's Comments on the Analytic Hierarchy Process" The Journal of the Operational Research Society, Vol. 42, No. 10 (Oct., 1991), pp. 909-914

</li>

<li id="cite_note-42"><a href="#cite_ref-42">^</a> R. D. Holder "Response to Holder's Comments on the Analytic Hierarchy Process: Response to the Response" The Journal of the Operational Research Society, Vol. 42, No. 10 (Oct., 1991), pp. 914-918

</li>

<li id="cite_note-43"><a href="#cite_ref-43">^</a> The <a href="Institute_for_Operations_Research_and_the_Management_Sciences" title="Institute for Operations Research and the Management Sciences">Institute for Operations Research and the Management Sciences</a> *In 2008, Thomas L. Saaty received the INFORMS Impact Prize for his development of the Analytic Hierarchy Process.

</li>

<li id="cite_note-44"><a href="#cite_ref-44">^</a> Mari A. Pöyhönen, Raimo P. Hämäläinen, Ahti A. Salo "An Experiment on the Numerical Modelling of Verbal Ratio Statements" Journal of Multi-Criteria Decision Analysis, vol 6, no 1, ppg 1-10, 1997

</li>

<li id="cite_note-45"><a href="#cite_ref-45">^</a> Stan Schenkerman "Inducement of nonexistent order by the analytic hierarchy process", Decision Sciences, Spring 1997

</li>

<li id="cite_note-46"><a href="#cite_ref-46">^</a> Perez et al. "Another Potential Shortcoming of AHP" TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Volume 14, Number 1 / June, 2006, Springer Berlin/Heidelberg

</li>

<li id="cite_note-47"><a href="#cite_ref-47">^</a> <cite id="CITEREFMunier2021" class="citation book cs1">Munier, Nolberto (2021). <a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-030-60392-2">Uses and Limitations of the AHP Method A Non-Mathematical and Rational Analysis</a>. Management for Professionals. Switzerland: <a href="Springer_Nature" title="Springer Nature">Springer Nature</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-030-60392-2">10.1007/978-3-030-60392-2</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-030-60392-2</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:241759250">241759250</a>.</cite>

</li>

<li id="cite_note-48"><a href="#cite_ref-48">^</a> Forman, Ernest H., "Ideal and Distributed Synthesis Modes for the Analytic Hierarchy Process" presented at the International Federation of Operations Research, Lisbon Portugal, July 1993.

</li>

<li id="cite_note-49"><a href="#cite_ref-49">^</a> <cite id="CITEREFArroyoTommeleinBallard2015" class="citation journal cs1">Arroyo, P.; Tommelein, I. D.; Ballard, G. (January 2015). "Comparing AHP and CBA as Decision Methods to Resolve the Choosing Problem in Detailed Design". Journal of Construction Engineering and Management. 141 (1): 04014063. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1061%2F%28ASCE%29CO.1943-7862.0000915">10.1061/(ASCE)CO.1943-7862.0000915</a>.</cite>

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<li id="cite_note-LandauEdmund-50"><a href="#cite_ref-LandauEdmund_50-0">^</a> Landau, E. (1914). <a rel="nofollow" class="external text" href="http://iris.univ-lille1.fr/jspui/bitstream/1908/2031/1/AP16_63.pdf#page=196">" Über Preisverteilung bei Spielturnieren</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200313112126/https://iris.univ-lille.fr/jspui/bitstream/1908/2031/1/AP16_63.pdf#page=196">Archived</a> 2020-03-13 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Zeitschrift für Mathematik und Physik, 63 band (1914), p. 192

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<li id="cite_note-Zermelo-51"><a href="#cite_ref-Zermelo_51-0">^</a> Zermelo, E. (1928). <a rel="nofollow" class="external text" href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002370808">Die Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung</a>, Mathematische Zeitschrift 29, 1929, S. 436–460

</li>

<li id="cite_note-Hasse-52"><a href="#cite_ref-Hasse_52-0">^</a> <cite id="CITEREFHasse1961" class="citation journal cs1">Hasse, M (1961). "Über die Behandlung graphentheoretischer Probleme unter Verwendung der Matrizenrechnung". Wiss. Zeit. Tech. Univ. Dresden. 10: 1313–6.</cite>

</li>

<li id="cite_note-:0-53"><a href="#cite_ref-:0_53-0">^</a> <cite id="CITEREFRamanujacharyulu1964" class="citation journal cs1">Ramanujacharyulu, C (1964). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131216232325/http://library.isical.ac.in/jspui/simple-search?query=%2BAnalysis%2Bpreferential%2Bexperiments&submit=Go">"Analyses of preferential experiments"</a>. Psychometrika. 29 (3): 257–261. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02289722">10.1007/bf02289722</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121033891">121033891</a>. Archived from <a rel="nofollow" class="external text" href="http://library.isical.ac.in/jspui/simple-search?query=%2BAnalysis%2Bpreferential%2Bexperiments&submit=Go">the original</a> on 2013-12-16.</cite>

</li>

<li id="cite_note-54"><a href="#cite_ref-54">^</a> Salavati, A., Haghshenas, H., Ghadirifaraz, B., Laghaei, J., & Eftekhari, G. (2016). Applying AHP and Clustering Approaches for Public Transportation Decisionmaking: A Case Study of Isfahan City. Journal of Public Transportation, 19(4), 3.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li>Saaty, Thomas L. Decision Making for Leaders: The Analytical Hierarchy Process for Decisions in a Complex World (1982). Belmont, California: Wadsworth. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-534-97959-9</bdi>; Paperback, Pittsburgh: RWS. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-9620317-0-4</bdi>. "Focuses on practical application of the AHP; briefly covers theory."</li>

<li>Saaty, Thomas L. Fundamentals of Decision Making and Priority Theory with the Analytic Hierarchy Process (1994). Pittsburgh: RWS. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-9620317-6-3</bdi>. "A thorough exposition of the theoretical aspects of AHP."</li>

<li>Saaty, Thomas L. Mathematical Principles of Decision Making (Principia Mathematica Decernendi) (2009). Pittsburgh: RWS. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-888603-10-0</bdi>. "Comprehensive coverage of the AHP, its successor the <a href="Analytic_network_process" title="Analytic network process">ANP</a>, and further developments of their underlying concepts."</li>

<li>Saaty, Thomas L., with Ernest H. Forman. The Hierarchon: A Dictionary of Hierarchies. (1992) Pittsburgh: RWS. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-9620317-5-5</bdi>. "Dozens of illustrations and examples of AHP hierarchies. A beginning classification of ideas relating to planning, conflict resolution, and decision making."</li>

<li>Saaty, Thomas L., with Luis G. Vargas The Logic of Priorities: Applications in Business, Energy, Health, and Transportation (1982). Boston: Kluwer-Nijhoff. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-89838-071-5</bdi> (Hardcover) <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-89838-078-2</bdi> (Paperback). Republished 1991 by RWS, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-888603-07-0</bdi>.</li>

<li>Kardi Teknomo. <a rel="nofollow" class="external text" href="http://people.revoledu.com/kardi/tutorial/AHP/purchase.html">Analytic Hierarchy Process Tutorial</a> (2012). Revoledu.</li>

<li>Kearns, Kevin P.; Saaty, Thomas L. Analytical Planning: The Organization of Systems (1985). Oxford: Pergamon Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-08-032599-8</bdi>. Republished 1991 by RWS, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-888603-07-0</bdi>.</li>

<li>with Joyce Alexander. Conflict Resolution: The Analytic Hierarchy Process (1989). New York: Praeger. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-275-93229-X</bdi></li>

<li>Vargas, Luis L.; Saaty, Thomas L. Prediction, Projection and Forecasting: Applications of the Analytic Hierarchy Process in Economics, Finance, Politics, Games and Sports (1991). Boston: Kluwer Academic. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7923-9104-7</bdi></li>

<li>Vargas, Luis L.; Saaty, Thomas L. Decision Making in Economic, Social and Technological Environments (1994). Pittsburgh: RWS. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-9620317-7-1</bdi></li>

<li>Vargas, Luis L.; Saaty, Thomas L. Models, Methods, Concepts & Applications of the Analytic Hierarchy Process (2001). Boston: Kluwer Academic. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7923-7267-0</bdi></li>

<li>Peniwati, Kirti; Vargas, Luis L. Group Decision Making: Drawing Out and Reconciling Differences (2007). Pittsburgh: RWS. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-888603-08-9</bdi></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://www.ijahp.org/index.php/IJAHP/index">International Journal of the Analytic Hierarchy Process</a> – An online journal about <a href="Multi-criteria_decision_analysis" class="mw-redirect" title="Multi-criteria decision analysis">multi-criteria decision making</a> using the AHP.</li>

<li><a rel="nofollow" class="external text" href="http://www.easyahp.com/">easyAHP Online tool to make collaborative decisions using AHP</a> easyAHP is a free online tool to make decisions in a collaborative or individual way. easy AHP uses AHP methodology: Analytic hierarchy process.</li>

<li><a rel="nofollow" class="external text" href="https://www.youtube.com/user/BPMSG#p/u/0/18GWVtVAAzs">AHP video. (9:17 YouTube clip)</a> Very thorough exposition of AHP by Dr. Klaus Göpel</li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20181024054632/http://waqqasfarooq.com/waqqasfarooq/index.php?option=com_content&view=article&id=46:analytical-hierarchy-process&catid=34:statistics&Itemid=53">Analytic Hierarchy Process (AHP) Example with Simulations using Matlab</a> – Waqqas Farooq – AHP example for college selection using matlab.</li>

<li><a rel="nofollow" class="external text" href="http://www.wiso.boku.ac.at/fileadmin/data/H03000/H73000/H73500/ahptutorial.pdf">An illustrated guide (pdf)</a> – Dr. Oliver Meixner University of Wien – "Analytic Hierarchy Process", a very easy to understand summary of the mathematical theory</li>

<li><a rel="nofollow" class="external text" href="https://archive.today/20110901225642/http://waqqasfarooq.com/ahp">AHP example with Matlab implementation</a> – AHP explanation with an example and matlab code.</li>

<li><a rel="nofollow" class="external text" href="http://ipub.com/ahp-package">R ahp package</a> – An AHP open source package.</li>

<li><a rel="nofollow" class="external text" href="https://pypi.org/project/ahpy/1.2/">AHPy</a> - An open source Python implementation of AHP with an optimal solver for missing pairwise comparisons</li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120506011344/http://www.business.pitt.edu/faculty/papers/saaty-into-to-ahp-mathematics.pdf">Introductory Mathematics of the Analytic Hierarchy Process</a> – An introduction to the mathematics of the Analytic Hierarchy Process.</li></ul>

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<ul><li><a rel="nofollow" class="external text" href="http://www.transparentchoice.com/jbrown/value-based-project-prioritization-webinar/view">How to use AHP for Project Prioritization by Dr. James Brown (webinar)</a></li>

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<li><a rel="nofollow" class="external text" href="http://sapexperts.wispubs.com/IT/Articles/Use-the-AHP-Methodology-to-More-Effectively-Define-and-Evaluate-Your-SAP-Implementation-Approach?id=52B9059CD94C40C4998639C32D1468B9">Use the AHP Methodology to More Effectively Define and Evaluate Your SAP Implementation Approach</a> by Jeetendra Kumar</li></ul></div><div><div class="zim-footer">

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