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Analytic hierarchy process

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A simple AHP hierarchy, with final priorities. The decision goal is to select the most suitable leader from a field of three candidates. Factors to be considered are age, experience, education, and charisma. According to the judgments of the decision makers, Dick is the most suitable candidate, followed by Tom and Harry.

The Analytic Hierarchy Process (AHP) is a structured technique for dealing with complex decisions. Rather than prescribing a "correct" decision, the AHP helps the decision makers find the one that best suits their needs and their understanding of the problem.

Based on mathematics and psychology, it was developed by Thomas L. Saaty in the 1970s and has been extensively studied and refined since then. The AHP 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. It is used around the world in a wide variety of decision situations, in fields such as government, business, industry, healthcare, and education.

Several firms supply computer software to assist in using the process.

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 systematically evaluate its various elements by comparing them to one another two at a time. In making the comparisons, the decision makers can use concrete data about the elements, or they can use their judgments about the elements' relative meaning and importance. It is the essence of the AHP that human judgments, and not just the underlying information, can be used in performing the evaluations.[1]

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.

The diagram above shows a simple AHP hierarchy at the end of the decision making process. Numerical priorities, derived from the decision makers' input, are shown for each item in the hierarchy. In this decision, the goal was to choose a leader based on four specific criteria. Dick was the preferred alternative, with a priority of 0.450. He was preferred twice as strongly as Harry, whose priority was 0.225. Tom fell somewhere in between. Experience was the most important criterion in making the decision, followed by Age, Charisma and Education. These factors were weighted .400, .300, .200 and .100, respectively.

Uses and applications

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.[2] It has unique advantages when important elements of the decision are difficult to quantify or compare, or where communication among team members is impeded by their different specializations, terminologies, or perspectives.

Decision situations to which the AHP can be applied include:[3]

  • Choice - The selection of one alternative from a given set of alternatives, usually where there are multiple decision criteria involved.
  • Ranking - Putting a set of alternatives in order from most to least desirable
  • Prioritization - Determining the relative merit of members of a set of alternatives, as opposed to selecting a single one or merely ranking them
  • Resource allocation - Apportioning resources among a set of alternatives
  • Benchmarking - Comparing the processes in one's own organization with those of other best-of-breed organizations
  • Quality management - Dealing with the multidimensional aspects of quality and quality improvement

The applications of AHP to complex decision situations have numbered in the thousands,[4] and have produced extensive results in problems involving planning, resource allocation, priority setting, and selection among alternatives.[2] Other areas have included forecasting, total quality management, business process re-engineering, quality function deployment, and the Balanced Scorecard.[3] Many AHP applications are never reported to the world at large, because they take place at high levels of large organizations where security and privacy considerations prohibit their disclosure. But some uses of AHP are discussed in the literature. Recently these have included:

AHP is sometimes used in designing highly specific procedures for particular situations, such as the rating of buildings by historic significance.[10] 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, then to justify its budget to lawmakers.[11]

Education and scholarly research

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[12] and graduate schools of business.[13] 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.[14][15][16]

Nearly a hundred Chinese universities offer courses in AHP, and many doctoral students choose AHP as the subject of their research and dissertations. Over 900 papers have been published on the subject in China, and there is at least one Chinese scholarly journal devoted exclusively to AHP.[17]

The International Symposium on the Analytic Hierarchy Process (ISAHP) holds biennial meetings of academics and practitioners interested in the field. At its 2007 meeting in Valparaiso, Chile, over 90 papers were presented from 19 countries, including the U.S., Germany, Japan, Chile, Malaysia, and Nepal. Topics covered ranged from Establishing Payment Standards for Surgical Specialists, to Strategic Technology Roadmapping, to Infrastructure Reconstruction in Devastated Countries.[18]

Full text copies of all ISAHP papers from 2001-2007 can be found here.

Using the Analytic Hierarchy Process

A typical device for entering judgments in an AHP group decision making session

As 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:

  1. Model the problem as a hierarchy containing the decision goal, the alternatives for reaching it, and the criteria for evaluating the alternatives.
  2. 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 real-estate purchases, the investors might say they prefer location over price and price over timing.
  3. 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.
  4. Check the consistency of the judgments.
  5. Come to a final decision based on the results of this process.[19]

These steps are more fully described below.

Model the problem as a hierarchy

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.[19]

Hierarchies defined

A hierarchy is a 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. 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.[19]

AHP hierarchies explained

An AHP hierarchy is a structured means of modeling the problem 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.

The hierarchy can be visualized as a diagram like the one below, with the goal at the top, the alternatives at the bottom, and the criteria in the middle. There are useful terms for describing the parts of such diagrams: Each box is called a node. The boxes descending from any node are called its children. The node from which a child node descends is called its parent. Groups of related children are called comparison groups. The parents of an Alternative, which are often from different comparison groups, are called its covering criteria.

Applying these definitions to the diagram, the four Criteria are children of the Goal, and the Goal is the parent of each of the four Criteria. Each Alternative is a child of its four covering criteria. There are two comparison groups: a group of four Criteria and a group of three Alternatives.

A simple AHP hierarchy. To avoid clutter in AHP diagrams, the lines connecting alternatives and their covering criteria are often omitted or reduced in number. Regardless of any such simplifications in the diagram, in the actual hierarchy each alternative is connected to every one of its covering criteria.

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 process. Published descriptions of AHP applications often include diagrams and descriptions of their hierarchies. These have been collected and reprinted in at least one book.[20] You can see some more complex AHP hierarchies HERE.

As the AHP proceeds through its other steps, the hierarchy 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.[19]

Establish priorities

Once the hierarchy has been constructed, the participants use AHP to establish priorities for all its nodes. In doing so, information is elicited from the participants and processed mathematically. This section explains priorities, shows how they are established, and provides a simple example.

Priorities defined and explained

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 absolute numbers 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.

Simple AHP hierarchy with associated default priorities.

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 you understand what has been said so far, you will see that 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.

A more complex AHP hierarchy, with local and global default priorities. In the interest of clarity, the decision alternatives do not appear in the diagram.

The local priorities, shown in gray, represent the relative weights of the nodes within a group of siblings with respect to their parent. You can easily see that 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.

A practical example

In an AHP hierarchy for the practical case of buying a vehicle, the goal might be to choose the best car for the Jones family. The family might decide to consider cost, safety, style, and capacity as the criteria for making their decision. They might subdivide the cost criterion into purchase price, fuel costs, maintenance costs, and resale value. They might separate Capacity into cargo capacity and passenger capacity. The family, which for personal reasons always buys Hondas, might decide to consider as alternatives the Accord Sedan, Accord Hybrid Sedan, Pilot SUV, CR-V SUV, Element SUV, and Odyssey Minivan.

The Jones' hierarchy could be diagrammed as shown below:

AHP hierarchy for the Jones family car buying decision. The Goal is green, the Criteria and Subcriteria are yellow, and the Alternatives are pink. All the alternatives (six different models of Hondas) are shown below the lowest level of each criterion. Later in the process, each alternative (each model) will be rated with respect to the criterion or subcriterion directly above it.
Alternatives for the Jones family car buying decision. To save space in the diagrams, we have represented them as stacks of papers.

As they build their hierarchy, the Joneses should investigate the values or measurements of the different elements that make it up. If there are published safety ratings, for example, or manufacturer's specs for cargo capacity, they should be gathered as part of the process. This information will be needed later, when the criteria and alternatives are evaluated.

Note that the measurements for some criteria, such as purchase price, can be stated with absolute certainty. Others, such as resale value, must be estimated, so must be stated with somewhat less confidence. Still others, such as style, are really in the eye of the beholder and are hard to state quantitatively at all. The AHP can accommodate all these types of criteria, even when they are present in a single problem.

Also note that the structure of the vehicle-buying hierarchy might be different for other families (ones who don't limit themselves to Hondas, or who care nothing about style, or who drive less than 5,000 miles (8,000 km) a year, etc.). It would definitely be different for a 25-year-old playboy who doesn't care how much his cars cost, knows he will never wreck one, and is intensely interested in speed, handling, and the numerous aspects of style.[19]

Pairwise comparisons

To incorporate their judgments about the various elements in the hierarchy, decision makers compare the elements two by two. How they are compared will be shown later on. Right now, let's see which items are compared. Our example will begin with the four Criteria in the second row of the hierarchy, though we could begin elsewhere if we wanted to. The Criteria will be compared as to how important they are to the decision makers, with respect to the Goal.

Each pair of items in this row will be compared; there are a total of six pairs (Cost/Safety, Cost/Style, Cost/Capacity, Safety/Style, Safety/Capacity, and Style/Capacity). You can use the diagram below to see these pairs more clearly.

AHP hierarchy for the Jones family car buying decision. Can you see the six possible comparisons in the Criteria row?
Alternatives for the Jones family car buying decision. These six models are pairwise compared with each of the covering criteria. First do them with respect to Purchase Price, then with respect to Fuel Costs, and so on until the end, with Passenger Capacity.

In the next row, there is a group of four subcriteria under the Cost criterion, and a group of two subcriteria under the Capacity criterion.

In the Cost subgroup, each pair of subcriteria will be compared regarding their importance with respect to the Cost criterion. (As always, their importance is judged by the decision makers.) Once again, there are six pairs to compare (Purchase Price/Fuel Costs, Purchase Price/Maintenance Costs, Purchase Price/Resale Value, Fuel Costs/Maintenance Costs, Fuel Costs/Resale Value, and Maintenance Costs/Resale Value).

In the Capacity subgroup, there is only one pair of subcriteria. They are compared as to how important they are with respect to the Capacity criterion.

Things change a bit when we get to the Alternatives row. Here, the cars in each group of alternatives are compared pair-by-pair with respect to the covering criterion of the group, which is the node directly above them in the hierarchy. What we are doing here is evaluating the models under consideration with respect to Purchase Price, then with respect to Fuel Costs, then Maintenance Costs, Resale Value, Safety, Style, Cargo Capacity, and Passenger Capacity. Because there are six cars in the group of alternatives, there will be fifteen comparisons for each of the eight covering criteria.

When the pairwise comparisons are as numerous as those in our example, specialized AHP software can help in making them quickly and efficiently. We will assume that the Jones family has access to such software, and that it allows the opinions of various family members to be combined into an overall opinion for the group.

The family's first pairwise comparison is Cost vs. Safety. They need to decide which of these is more important in choosing the best car for them all. This can be a difficult decision. On the one hand, "You can't put a price on safety. Nothing is more important than the life of a family member." But on the other hand, the family has a limited amount of money to spend, no member has ever had a major accident, and Hondas are known as very safe cars. In spite of the difficulty in comparing money to potential injury or death, the Jones family needs to determine its judgment about Cost vs. Safety in the car they are about to buy. They have to say which criterion is more important to them in reaching their goal, and how much more important it is (to them) than the other one. In making this judgment, they should remember that since the AHP is a flexible process, they can change their judgment later on.

You can imagine that there might be heated family discussion about Cost vs. Safety. It is the nature of the AHP to promote focused discussions about difficult aspects of the decisions to which it is applied. Such discussions encourage the communication of differences, which in turn encourages cooperation, compromise, and agreement among the members of the group.

Let's say that the family decides that in this case, Cost is moderately more important to them than Safety. The software requires them to express this judgment by entering a number. They can use this table to determine it; in this case they would enter a 3 in favor of Cost:

Continuing our example, let's say they make the following judgments about all the comparisons of criteria, entering them into the software as numbers gotten from the table: as stated, Cost is moderately important (3) over Safety; also, Cost is very strongly important (7) over Style, and is moderately important (3) over Capacity. Safety is extremely more important (9) than Style, and of equal importance (1) to Capacity. Capacity is very strongly important (7) over Style.

We could show those judgments in a table like this:

Judgments made by the Jones family and entered into the AHP software. Throughout this article, the family's judgments are shown with a green background. The orange and yellow colors show which alternative predominates.

The AHP software uses mathematical calculations to convert these judgments to priorities for each of the four criteria. The details of the calculations are beyond the scope of this article, but are readily available elsewhere.[2][21][22][23] The software also calculates a consistency ratio that expresses the internal consistency of the judgments that have been entered.

In this case the judgments showed acceptable consistency, and the software used the family's inputs to assign these new priorities to the criteria:

AHP hierarchy for the Jones family car buying decision. The Criteria have been pairwise compared, and their new priorities are shown. (Before the comparisons, each Criterion had a default priority of 0.250.) Note that the priorities still total 1.000, and the priorities for the subcriteria have not changed. The family has said that Cost is quite important to them, Style is relatively unimportant, and Safety and Capacity are each roughly half as important as Cost, with Safety having a slight edge over Capacity. Family members can look at these priorities and see how they feel about them. If they are uncomfortable about something, they can redo their pairwise comparisons.

You can duplicate this analysis at this online demonstration site; use the Line by Line Method by clicking its button, and don't forget to enter a negative number if the Criterion on the left is less important than the one on the right. If you are having trouble, click here for help. IMPORTANT: The demo site is designed for convenience, not accuracy. The priorities it returns may differ significantly from those returned by rigorous AHP calculations. Nevertheless, it is useful in showing the mechanics of the pairwise comparison process. Once you are comfortable with the demo, you can experiment by entering your own judgments for the criteria in question. If your judgments are different from those of the Jones family, your priorities will possibly be quite different from theirs.[24]

It is widely recognized that the best way to understand the AHP is to work through one or more detailed examples. Continuing this example is beyond the scope of an encyclopedia article, but a completely worked-out version is available here.

Criticisms

The AHP is now 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.[3] While the general consensus is that it is both technically valid and practically useful, the method does have its critics.[6]

In the early 1990s a series of debates between critics and proponents of AHP was published in Management Science[25][26][27][28] and The Journal of the Operational Research Society.[29][30][31] These debates seem to have been settled in favor of AHP:

  • An in-depth paper discussing and rebutting the academic criticisms of AHP was published in Operations Research in 2001.[3]
  • A 2008 Management Science paper reviewing 15 years of progress in all areas of Multicriteria Decision Making showed that AHP publications have far outnumbered those in any other area, characterizing their growth as "enormous."[32]
  • Also in 2008, the major international society for operations research and the management sciences formally recognized AHP's broad impact on its fields.[33]

Occasional criticisms still appear. A 1997 paper examined possible flaws in the verbal (vs. numerical) scale often used in AHP pairwise comparisons.[34] Another from the same year claimed that innocuous changes to the AHP model can introduce order where no order exists.[35] A 2006 paper found that the addition of criteria for which all alternatives perform equally can alter the priorities of alternatives.[36]

Most of the criticisms involve a phenomenon called rank reversal, discussed in the following section.

Rank reversal

Decision making involves ranking alternatives in terms of criteria or attributes of those alternatives. It is an axiom of some decision theories (including AHP's) 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.

The validity of this assumption is disputed by some. Adding new alternatives can change the rank of the old ones. These rank reversals do not occur often, but the possibility of their occurrence has substantial logical implications about the methodology used to make decisions, the underlying assumptions of various decision theories, etc.

The 2000 U.S. presidential election is an example of a decision that can be understood as involving rank reversal. Ralph Nader was an 'irrelevant' alternative, in that he was dominated by both the Democrat and Republican candidates. However, since he attracted more votes from those who would have voted Democrat rather than Republican, his presence caused the ranks to reverse. Put another way, if Nader were not in the race, it is widely accepted that Al Gore would have won.

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 both situations in which rank reversal is not reasonable as well as situations where they are to be expected. 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 the ideal alternative are extensively discussed in the previously-mentioned Operations Research paper[3] as well as a chapter entitled Rank Preservation and Reversal, in the current basic book on AHP.[21] 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 the AHP.

See also

References

  1. ^ Saaty, Thomas L. (2008-06). "Relative Measurement and its Generalization in Decision Making: Why Pairwise Comparisons are Central in Mathematics for the Measurement of Intangible Factors - The Analytic Hierarchy/Network Process" (PDF). RACSAM (Review of the Royal Spanish Academy of Sciences, Series A, Mathematics). 102 (2): 251–318. Retrieved 2008-12-22. {{cite journal}}: Check date values in: |date= (help); Cite has empty unknown parameter: |coauthors= (help)
  2. ^ a b c Bhushan, Navneet (2004). Strategic Decision Making: Applying the Analytic Hierarchy Process. London: Springer-Verlag. ISBN 1-8523375-6-7. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |month= ignored (help)
  3. ^ a b c d e Forman, Ernest H. (2001-07). "The analytical hierarchy process—an exposition". Operations Research. 49 (4): 469–487. doi:10.1287/opre.49.4.469.11231. {{cite journal}}: |access-date= requires |url= (help); Check date values in: |date= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
  4. ^ a b de Steiguer, J.E. (October 2003), "The Analytic Hierarchy Process as a Means for Integrated Watershed Management" (PDF), in Renard, Kenneth G. (ed.), First Interagency Conference on Research on the Watersheds, Benson, Arizona: U.S. Department of Agriculture, Agricultural Research Service, pp. 736–740 {{citation}}: Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |coeditors= ignored (help)
  5. ^ Berrittella, M. (January, 2007), "An Analytic Hierarchy Process for the Evaluation of Transport Policies to Reduce Climate Change Impacts" (PDF), Fondazione Eni Enrico Mattei (Milano) {{citation}}: Check date values in: |date= (help); Cite has empty unknown parameter: |coeditors= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
  6. ^ a b McCaffrey, James (2005). "Test Run: The Analytic Hierarchy Process". MSDN Magazine. Retrieved 2007-08-21. {{cite journal}}: Cite has empty unknown parameter: |coauthors= (help); Unknown parameter |month= ignored (help)
  7. ^ Grandzol, John R. (2005). "Improving the Faculty Selection Process in Higher Education: A Case for the Analytic Hierarchy Process" (PDF). IR Applications. 6. Retrieved 2007-08-21. {{cite journal}}: Cite has empty unknown parameter: |coauthors= (help); Unknown parameter |month= ignored (help)
  8. ^ Atthirawong, Walailak (September, 2002), "An Application of the Analytical Hierarchy Process to International Location Decision-Making" (PDF), in Gregory, Mike (ed.), Proceedings of The 7th Annual Cambridge International Manufacturing Symposium: Restructuring Global Manufacturing, Cambridge, England: University of Cambridge, pp. 1–18 {{citation}}: Check date values in: |date= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |coeditors= ignored (help)
  9. ^ Dey, Prasanta Kumar (2003). "Analytic Hierarchy Process Analyzes Risk of Operating Cross-Country Petroleum Pipelines in India". Natural Hazards Review. 4 (4): 213–221. doi:10.1061/(ASCE)1527-6988(2003)4:4(213). Retrieved 2007-08-20. {{cite journal}}: Cite has empty unknown parameter: |coauthors= (help); Unknown parameter |month= ignored (help)
  10. ^ Lippert, Barbara C. (1995). "HIST 1.0; Decision Support Software for Rating Buildings by Historic Significance" (PDF). National Institute of Standards and Technology, NISTIR 5683. Retrieved 2007-08-20. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |month= ignored (help)
  11. ^ Larson, Charles D. (January, 2007), "Application of the Analytic Hierarchy Process to Select Project Scope for Videologging and Pavement Condition Data Collection", 86th Annual Meeting Compendium of Papers CD-ROM, Transportation Research Board of the National Academies {{citation}}: Check date values in: |date= (help); Cite has empty unknown parameter: |coeditors= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
  12. ^ Drake, P.R. (1998). "Using the Analytic Hierarchy Process in Engineering Education" (PDF). International Journal of Engineering Education. 14 (3): 191–196. Retrieved 2007-08-20. {{cite journal}}: Cite has empty unknown parameter: |coauthors= (help)
  13. ^ Bodin, Lawrence (January, 2004). "Exercises for Teaching the Analytic Hierarchy Process" ([dead link] – Scholar search). INFORMS Transactions on Education. 4 (2). Retrieved 2009-03-11. {{cite journal}}: Check date values in: |date= (help); External link in |format= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
  14. ^ Hallowell, David L. (2005). "Analytical Hierarchy Process (AHP) – Getting Oriented". ISixSigma.com. Retrieved 2007-08-21. {{cite journal}}: Cite has empty unknown parameter: |coauthors= (help); Unknown parameter |month= ignored (help)
  15. ^ "Analytic Hierarchy Process (AHP)". QFD Institute. Retrieved 2007-08-21. {{cite journal}}: Cite has empty unknown parameter: |coauthors= (help)
  16. ^ "Analytical Hierarchy Process: Overview". TheQualityPortal.com. Retrieved 2007-08-21. {{cite journal}}: Cite has empty unknown parameter: |coauthors= (help)
  17. ^ Sun, Hongkai (July, 2005), "AHP in China" (PDF), in Levy, Jason (ed.), Proceedings of the 8th International Symposium on the Analytic Hierarchy Process, Honolulu, Hawaii {{citation}}: Check date values in: |date= (help); Cite has empty unknown parameters: |coeditors= and |coauthors= (help)
  18. ^ "Participant Names and Papers, ISAHP 2005, Honolulu, Hawaii". 2005. Retrieved 2007-08-22. {{cite web}}: Unknown parameter |month= ignored (help)
  19. ^ a b c d e Saaty, Thomas L. (1999-05-01). Decision Making for Leaders: The Analytic Hierarchy Process for Decisions in a Complex World. Pittsburgh, Pennsylvania: RWS Publications. ISBN 0-9620317-8-X. {{cite book}}: Cite has empty unknown parameter: |coauthors= (help) (This book is the primary source for the sections in which it is cited.)
  20. ^ Saaty, Thomas L. (1992). The Hierarchon: A Dictionary of Hierarchies. Pittsburgh, Pennsylvania: RWS Publications. ISBN 0-9620317-5-5. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help) 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.
  21. ^ a b Saaty, Thomas L. (2001). Fundamentals of Decision Making and Priority Theory. Pittsburgh, Pennsylvania: RWS Publications. ISBN 0-9620317-6-3. {{cite book}}: Cite has empty unknown parameter: |coauthors= (help)
  22. ^ Trick, Michael A. (1996-11-23). "Analytic Hierarchy Process". Class Notes. Carnegie Mellon University Tepper School of Business. Retrieved 2008-03-02.
  23. ^ Meixner, Oliver (2002). Computergestützte Entscheidungs-findung: Expert Choice und AHP – innovative Werkzeuge zur Lösung komplexer Probleme (in German). Frankfurt/Wien: Redline Wirtschaft bei Ueberreuter. ISBN 3-8323-0909-8. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  24. ^ Note that the demo site expresses priorities in percentages rather than decimal fractions as we do. It also uses different numbers to represent the verbal descriptions presented here. It's only a demo, but you should use our numbers, not theirs, and you should convert the percentages to decimal fractions. IMPORTANT: The demo site is designed for convenience, not accuracy. The priorities it returns may be significantly different from those returned by rigorous AHP calculations.
  25. ^ Dyer, J. S. (1990): Remarks on the Analytic Hierarchy Process. In: Management Science, 36 (3), S. 249-258.
  26. ^ M. V. Mikhalevic "Remarks on the Dyer-Saaty controversy" Cybernetics and Systems Analysis, Volume 30, Number 1 / January, 1994
  27. ^ 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
  28. ^ Dyer, J.S. (1990b), "A clarification of ‘Remarks on the analytic hierarchy process’", Management Science, Vol. 36 No.3, pp.274-5.
  29. ^ Holder, R.D., Some Comment on the Analytic Hierarchy Process, Journal of the Operational Research Society, 1990, 41, 11 1073-1076.
  30. ^ 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
  31. ^ 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
  32. ^ Wallenius, Jyrki (July, 2008). "Multiple Criteria Decision Making, Multiattribute Utility Theory: Recent Accomplishments and What Lies Ahead". Management Science. 54 (7): 1339–1340. doi:10.1287/mnsc.1070.0838. Retrieved 2009-03-13. {{cite journal}}: Check date values in: |date= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
  33. ^ The Institute for Operations Research and the Management Sciences (INFORMS) is an international society for practitioners in the fields of operations research and management science. They are the publishers of Management Science, one of the journals in which the original critical debates were published. Their Impact Prize is awarded every two years to recognize contributions that have had a broad impact on the fields of operations research and the management sciences; emphasis is placed on the breadth of the impact of an idea or body of research. In 2008, Thomas L. Saaty received the INFORMS Impact Prize for his development of the Analytic Hierarchy Process.
  34. ^ 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
  35. ^ Stan Schenkerman "Inducement of nonexistent order by the analytic hierarchy process", Decision Sciences, Spring 1997
  36. ^ 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
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