Analytic Hierarchy Process

Analytic Hierarchy Process

The Analytic Hierarchy Process (AHP) is a structured technique for helping people deal with complex decisions. Rather than prescribing a "correct" decision, the AHP helps people to determine one. Based on mathematics and human 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 problem, for representing and quantifying its elements, for relating those elements to overall goals, and for evaluating alternative solutions. It is used throughout 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 applying 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, comparing them to one another in pairs. 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.

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 derived for each of the decision alternatives. Since these numbers represent the alternatives' relative ability to achieve the decision goal, they allow a straightforward consideration of the various courses of action.

Uses and applications

While it can be used by individuals working on straightforward decisions, 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.cite book | last = Bhushan | first = Navneet | authorlink = | coauthors = Kanwal Rai | title = Strategic Decision Making: Applying the Analytic Hierarchy Process | publisher = Springer-Verlag | date = January, 2004| location = London | pages = | url = | doi = | id = | isbn = 1-8523375-6-7 ] It has unique advantages where 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.

The applications of AHP to complex decision situations have numbered in the thousands, Citation| first=J.E. | last=de Steiguer| coauthors=Jennifer Duberstein, Vicente Lopes| contribution=The Analytic Hierarchy Process as a Means for Integrated Watershed Management| title=First Interagency Conference on Research on the Watersheds| editor-first=Kenneth G.| editor-last=Renard| coeditors=et al| publisher=U.S. Department of Agriculture, Agricultural Research Service| place=Benson, Arizona| pages=736–740| date=October, 2003| year=| id= | contribution-url=| format=| accessdate= ] and have produced extensive results in problems involving planning, resource allocation, priority setting, and selection among alternatives. Many such 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:

:*Deciding how best to reduce the impact of global climate change ("Fondazione Eni Enrico Mattei") [ Citation| first=M. | last=Berrittella| coauthors=A. Certa, M. Enea, P. Zito| contribution=An Analytic Hierarchy Process for the Evaluation of Transport Policies to Reduce Climate Change Impacts| title=Fondazione Eni Enrico Mattei (Milano)| editor-first=| editor-last=| coeditors=| publisher=| place=| pages=| date=January, 2007| year=| id= | contribution-url=| format=| accessdate= ] :*Quantifying the overall quality of software systems ("Microsoft Corporation") cite journal|title=Test Run: The Analytic Hierarchy Process|journal=MSDN Magazine|date=June, 2005|first=James|last=McCaffrey|coauthors=|volume=|issue=|pages=|id= |url=|format=|accessdate=2007-08-21 ] :*Selecting university faculty ("Bloomsburg University of Pennsylvania") [ cite journal|title=Improving the Faculty Selection Process in Higher Education: A Case for the Analytic Hierarchy Process|journal=IR Applications|date=August, 2005|first=John R.|last=Grandzol|coauthors=|volume=6|issue=|pages=|id= |url=|format=|accessdate=2007-08-21 ] :*Deciding where to locate offshore manufacturing plants ("University of Cambridge") [ Citation| first=Walailak | last=Atthirawong| coauthors=Bart McCarthy| contribution=An Application of the Analytical Hierarchy Process to International Location Decision-Making| title=Proceedings of The 7th Annual Cambridge International Manufacturing Symposium: Restructuring Global Manufacturing| editor-first=Mike| editor-last=Gregory| coeditors=Yongjiang Shi| publisher=University of Cambridge| place=Cambridge, England| pages=1–18| date=September, 2002| year=2002| id= | contribution-url=| format=| accessdate= ] :*Assessing risk in operating cross-country petroleum pipelines ("American Society of Civil Engineers") [ cite journal|title=Analytic Hierarchy Process Analyzes Risk of Operating Cross-Country Petroleum Pipelines in India|journal=Natural Hazards Review|date=November, 2003|first=Prasanta Kumar|last=Dey|coauthors=|volume=4|issue=4|pages=213–221|id= |url=|format=|accessdate=2007-08-20|doi=10.1061/(ASCE)1527-6988(2003)4:4(213) ] :*Deciding how best to manage U.S. watersheds ("U.S. Department of Agriculture")

AHP is sometimes used in designing highly specific procedures for particular situations, such as the rating of buildings by historic significance. [ cite journal|title=HIST 1.0; Decision Support Software for Rating Buildings by Historic Significance|journal=National Institute of Standards and Technology, NISTIR 5683|date=October, 1995|first=Barbara C.|last=Lippert|coauthors=Stephen F. Weber|volume=|issue=|pages=|id= |url=|format=|accessdate=2007-08-20 ] 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. [ Citation| first=Charles D. | last=Larson| coauthors=Ernest H. Forman| contribution=Application of the Analytic Hierarchy Process to Select Project Scope for Videologging and Pavement Condition Data Collection| title=86th Annual Meeting Compendium of Papers CD-ROM| editor-first=| editor-last=| coeditors=| publisher=Transportation Research Board of the National Academies| place=| pages=| date=January, 2007| year=| id= | contribution-url=| format=| accessdate= ]

AHP is widely used in countries around the world. At a recent international conference on AHP, 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". [ cite web|url= |title=Participant Names and Papers, ISAHP 2005, Honolulu, Hawaii |accessdate=2007-08-22 |date=July, 2005 ] AHP was introduced in China in 1982, and its use in that country has expanded greatly since then—its methods are highly compatible with the traditional Chinese decision making framework, and it has been used for many decisions in the fields of economics, energy, management, environment, traffic, agriculture, industry, and the military. Citation| first=Hongkai | last=Sun| coauthors=| contribution=AHP in China| title=Proceedings of the 8th International Symposium on the Analytic Hierarchy Process| editor-first=Jason| editor-last=Levy| coeditors=| publisher=| place=Honolulu, Hawaii| pages=| date=July, 2005| year=2005| id= | contribution-url=| format=| accessdate= ]

Though using AHP requires no specialized academic training, the subject is widely taught at the university level—one AHP software provider lists over a hundred colleges and universities among its clients. [ cite web|url= |title=List of Expert Choice education clients |accessdate=2007-08-23 ] AHP is considered an important subject in many institutions of higher learning, including schools of engineering [ cite journal|title=Using the Analytic Hierarchy Process in Engineering Education|journal=International Journal of Engineering Education|date=1998|first=P.R.|last=Drake|coauthors=|volume=14|issue=3|pages=191–196|id= |url=|format=|accessdate=2007-08-20 ] and graduate schools of business. [ cite journal|title=Exercises for Teaching the Analytic Hierarchy Process|journal=INFORMS Transactions on Education|date=January, 2004|first=Lawrence|last=Bodin|coauthors=Saul I. Gass|volume=4|issue=2|id= |url=|format=dead link|date=July 2008 – [ Scholar search] |accessdate=2007-08-20 ] AHP is also an important subject in the quality field, and is taught in many specialized courses including Six Sigma, Lean Six Sigma, and QFD. [ cite journal|title=Analytical Hierarchy Process (AHP) – Getting Oriented||date=January, 2005|first=David L.|last=Hallowell|coauthors=|volume=|issue=|pages=|id= |url=|format=|accessdate=2007-08-21 ] [ cite journal|title=Analytic Hierarchy Process (AHP)|journal=QFD Institute|date=|first=|last=|coauthors=|volume=|issue=|pages=|id= |url=|format=|accessdate=2007-08-21 ] [ cite journal|title=Analytical Hierarchy Process: Overview||date=|first=|last=|coauthors=|volume=|issue=|pages=|id= |url=|format=|accessdate=2007-08-21 ]

In China, nearly a hundred schools 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 that country, and there is at least one Chinese scholarly journal devoted exclusively to AHP.


As can be seen in the examples that follow, 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. [ cite web|url= |title=Decision Lens web site |accessdate=2008-07-26 ] [ cite web|url= |title=Expert Choice web site |accessdate=2008-07-26 ]


The procedure can be summarized as:

# The alternatives and the significant attributes are identified.
# For each attribute, and each pair of alternatives, the decision makers specify their preference (for example, whether the location of alternative "A" is preferred to that of "B") in the form of a fraction between 1/9 and 9.
# Decision makers similarly indicate the relative significance of the attributes. For example, if the alternatives are comparing potential real-estate purchases, the investors might say they prefer location over price and price over timing.
# Each matrix of preferences is evaluated by using eigenvalues to check the consistency of the responses. This produces a "consistency coefficient" where a value of "1" means all preferences are internally consistent.Fact|date=February 2008 This value would be lower, however, if a decision maker said X is preferred to Y, Y to Z but Z is preferred to X (such a position is internally inconsistent). It is this step that causes many users to believe that AHP is theoretically well founded.Fact|date=August 2007
# A score is calculated for each alternative.

The two basic steps in the process are to model the problem as a hierarchy, then to establish priorities for its elements. These 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.cite book | last = Saaty | first = Thomas L. | authorlink = | coauthors = | title = Decision Making for Leaders: The Analytic Hierarchy Process for Decisions in a Complex World | publisher = RWS Publications | date = 1999-05-01 | location = Pittsburgh, Pennsylvania | pages = | url = | doi = | id = | isbn = 0-9620317-8-X (This book is the primary source for the sections in which it is cited.) ]

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.

AHP hierarchies explained

An AHP hierarchy is a structured means of describing 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. In most cases the criteria are 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 filling up the middle. In 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". Applying these definitions to the diagram below, the five Criteria are children of the Goal, and the Goal is the parent of each of the five Criteria. Each Alternative is the child of each of the Criteria, and each Criterion is the parent of three Alternatives.

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.

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.

A simple example

In an AHP hierarchy for the simple 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:

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. Information about the Jones' alternatives, including color photos, can be found [ HERE] .

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 convert|5000|mi|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.

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 activity is somewhat complex, and the participants have many options on the road to completing it. This and the following sections describe a simple, straightforward example of establishing priorities.

As our first step, we will define priorities and show how they interact.

Priorities defined

"Priorities" are numbers associated with the nodes of the hierarchy. By definition, the priority of the Goal is 1.000. The priorities of the Criteria (which are the children of the Goal) can vary in magnitude, but will always add up to 1.000. The priorities of the children of any Criterion can also vary but will always add up to 1.000, as will those of their own children, and so on down the hierarchy.

This illustration shows some priorities for the Jones car buying hierarchy. We'll say more about them in a moment. For now, just observe that the priorities of the children of each parent node add up to 1.000, and that there are three such groups of children in the illustration.

If you understand what has been said so far, you will see that if we were to add a "Handling" criterion to this hierarchy, giving it five Criteria instead of four, the priority for each would be .200. You will also know that if the Safety criterion had three children, each of them would have a priority of .333.

In our example as it stands, the priorities within every group of child nodes are equal. In this situation, the priorities are called "default priorities." Throughout this article, default priorities will be shown in gray. As the analytic hierarchy process continues, the priorities will change from their default values to reflect our judgments about the various items in each group.

As you may have guessed by now, the priorities indicate the relative weights given to the items in a given group of nodes. Depending on the problem at hand, "weight" can refer to importance, or preference, or likelihood, or whatever factor is being considered by the participants.

If all the priorities in a group of nodes are equal, each member of the group has equal weight. If one of the priorities is two times another, or three, (or whatever), that member has two, or three, (or whatever) times the weight of the other one. For example, if we judge cargo capacity to be three times as important as passenger capacity, cargo capacity's new priority will be .750, and passenger capacity's priority will be .250, because .750 = 3 × .250, and .750 + .250 = 1.000.

AHP priorities have another important feature. The priority of any child node represents its contribution to the priority of its parent. In the diagram above, Cost, Safety, Style and Capacity each contribute .250 of the 1.000 priority of the Goal. Cargo capacity and passenger capacity each contribute half of the priority belonging to the Capacity criterion. Working through the arithmetic, Passenger Capacity contributes .500 × .250 = .125 of the 1.000 priority of the Goal.

As we move ahead through the Analytical Hierarchy Process, the priorities will change but will still add to 1.000 for each group of child nodes.

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.

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:

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.cite book | last = Saaty | first = Thomas L. | authorlink = Thomas L. Saaty | coauthors = | title = Fundamentals of Decision Making and Priority Theory | publisher = RWS Publications | date = 2001 | location = Pittsburgh, Pennsylvania | pages = | url = | doi = | id = | isbn = 0-9620317-6-3 ] cite web|url=|accessdate=2008-03-02 |last=Trick |first=Michael A. |date=1996-11-23 |title=Analytic Hierarchy Process |work=Class Notes |publisher=Carnegie Mellon University Tepper School of Business ] cite book | last = Meixner | first = Oliver | authorlink = | coauthors = Reiner Haas | title = Computergestützte Entscheidungs-findung: Expert Choice und AHP – innovative Werkzeuge zur Lösung komplexer Probleme | language=German| publisher = Redline Wirtschaft bei Ueberreuter | date = 2002 | location = Frankfurt/Wien | pages = | url = | doi = | id = | isbn = 3-8323-0909-8 ] 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:

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. [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. ]

Look again at the above diagram and note that the Subcriteria still show their default priorities. This is because the decision makers haven't entered any judgments about them. So next on the family's agenda is to pairwise compare the four Subcriteria under Cost, then the two Subcriteria under Capacity. They will compare them following the same pattern as they did for the Criteria.

We could imagine the result of their comparisons yielding the priorities shown here: [Their comparisons under "Cost" were Purchase Price 2 over Fuel Cost, 5 over Maintenance Cost, and 3 over Resale Value; Fuel Cost 2 over Maintenance Cost and 2 over Resale Value; Maintenance Cost -2 vs. Resale Value. Their comparisons under "Capacity" were Cargo Capacity -5 vs. Passenger Capacity.]

At this point, all the comparisons for Criteria and Subcriteria have been made, and the AHP software has derived the local priorities for each group at each level. One more step can be made here. We know how much the priority of each Criterion contributes to the priority of the Goal. Since we also know how much the priority of each Subcriterion contributes to the priority of its parent, we (and the AHP software) can calculate the "global priority" of each Subcriterion. That will show us the priority of each Subcriterion with respect to the Goal. The global priorities throughout the hierarchy will add up to 1.000, like this:

Based on the judgments entered by the family, the AHP has derived the priorities for the factors against which each of the six cars will be compared. They are shown, from highest to lowest, in the table below. Notice that Cost and Capacity will not be evaluated directly, but that each of their Subcriteria will be evaluated on its own:

The local priorities show how much the safety of each model contributes to the Criterion of Safety. The global priorities show how much the Safety of each model contributes to the overall goal of choosing the best car for the Jones family.

Passenger capacity

This characteristic is easy to evaluate. The alternatives can carry either four or five or eight passengers. Here are the figures:

The family has decided that four is barely enough, five is perfect for their needs, and eight is just a little bit better than five. Here are their judgments:

When the judgments shown above are entered, the AHP software returns the following priorities for the six alternatives with respect to Passenger Capacity:

The local priorities show how much the resale value of each model contributes to the Subcriterion of Resale Value. The global priorities show how much the resale value of each model contributes to the overall goal of choosing the best car for the Jones family.

Maintenance Costs

The Jones family researched maintenance costs for the cars under consideration, but they didn't find any hard figures. The closest they got was Consumer Reports magazine, which publishes 17 separate maintenance ratings for every car on the market. Their Hondas ranked very well, with all ratings "Much Better Than Average," except for a few on the Pilot and Odyssey. The Pilot got "Better Than Average" for its audio system and the user rating, and "Average" for body integrity. The Odyssey got "Better Than Average" for body hardware and power equipment, and "Average" for brakes, body integrity, and user rating.

The Joneses also asked their favorite mechanic to evaluate the maintenance costs for their six cars. Using tire prices and mileage estimates, he came up with figures for tire costs over convert|60000|mi|km of driving. He didn't have figures for brake costs, but he said they'd be about twice as much for the SUVs and minivans as they would for the sedans. He also cautioned them that the battery in the Accord Hybrid was an expensive repair item, and that the engine placement on the Odyssey made it a more expensive car to work on.

The family created this worksheet to keep track of all their information about maintenance costs:

Even though every column on the worksheet contains a different type of information, the Joneses can use it to make reasonable, rational judgments about Maintenance Costs. Here are the judgments they will enter:

When the judgments shown above are entered, the AHP software returns the following priorities for the six alternatives with respect to Maintenance Costs:

The local priorities show how much the cargo capacity of each model contributes to the subcriterion of Cargo Capacity. The global priorities show how much the cargo capacity of each model contributes to the overall goal of choosing the best car for the Jones family.

Make the decision

In the end, the AHP software arranges and totals the global priorities for each of the alternatives. Their grand total is 1.000, which is identical to the priority of the goal. Each alternative has a global priority corresponding to its "fit" to all the family's judgments about all those aspects of Cost, Safety, Style and Capacity. Here is a summary of the global priorities of the alternatives:

The Odyssey Minivan, with a global priority of 0.220, is the alternative that contributes the most to the goal of choosing the best car for the Jones family. The Accord Sedan is a close second, with a priority of 0.213. The other models have considerably less priority than those two. In descending order, they are CR-V SUV, Accord Hybrid, Element SUV, and Pilot SUV.

The Analytic Hierarchy Process has shown the Joneses that the Odyssey Minivan best satisfies all their criteria and judgments, followed closely by the Accord Sedan. The other alternatives fall significantly short of meeting their criteria. The family's next step is up to them. They might just go out and buy an Odyssey, or they might use the AHP or other means to refine their decision between the Odyssey and the Accord Sedan.


Although the Analytic Hierarchy Process has been the subject of many research papers and the general consensus is that the technique is both technically valid and practically useful, there are critics of the method. In the early 1990s a series of debates between critics and proponents of AHP was published in Management Science [Dyer, J. S. (1990): Remarks on the Analytic Hierarchy Process. In: Management Science, 36 (3), S. 249-258.] [M. V. Mikhalevic "Remarks on the Dyer-Saaty controversy" Cybernetics and Systems Analysis, Volume 30, Number 1 / January, 1994 ] [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] [Dyer, J.S. (1990b), "A clarification of ‘Remarks on the analytic hierarchy process’", Management Science, Vol. 36 No.3, pp.274-5.] and The Journal of the Operations Research Society. [Holder, R.D., Some Comment on the Analytic Hierarchy Process, Journal of the Operational Research Society, 1990, 41, 11 1073-1076. ] [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] [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] [Dyer, J. S. (1990): Remarks on the Analytic Hierarchy Process. In: Management Science, 36 (3), S. 249] Those debates, augmented by additional criticisms, continue even today.

Many of the criticisms are discussed at length in a chapter entitled "Rank Preservation and Reversal", in the current basic book on AHP.cite book | last = Saaty | first = Thomas L. | authorlink = Thomas L. Saaty | coauthors = | title = Fundamentals of Decision Making and Priority Theory | publisher = RWS Publications | date = 2001 | location = Pittsburgh, Pennsylvania | pages = | url = | doi = | id = | isbn = 0-9620317-6-3 ]

The use of arbitrary scales

AHP is based on pairwise comparisons where the relative importance of different attributes are given a value on a scale of 1 to 9 or the inverse (1/9th to 1). These values are in practice assigned by verbal elicitation of decision makers. For example, if a person says attribute A is "moderately more important" than attribute B, A is said to have a relative weight of 3 times that of B while being "extremely more important" will give A a weight of 9 times that of B. While this scale is commonly used in AHP, it is arbitrary and alternative scales have been proposed. One study found "...that the perceived meaning of the verbal expressions varies from one subject to the next and also depends on the set of elements involved in the comparison." [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] However, the researchers felt the problem was correctable in that the scales could be based on empirical evidence of AHP user perceptions.

Inducement of nonexistent order

Another problem is the inducement of "nonexistent order" by innocuous changes even without the addition or deletion of suboptimal alternatives. AHP critic Stan Schenkerman writes in Decision Sciences:"An apparently unreported problem facing decision makers who use AHP is described [in this paper] . It is demonstrated that conventional AHP and some of its variants (the ideal mode, and the pairwise aggregated approach, PAHAP) can induce ordering even when no order exists. It is also shown that all three approaches can induce different orderings and that the orderings are sensitive to innocuous changes. Thus, even absent addition or deletion of alternatives, the decision maker relying on AHP or these variants can be seriously misled." [Stan Schenkerman "Inducement of nonexistent order by the analytic hierarchy process", Decision Sciences, Spring 1997]

Rank Change Due to Addition of Indifferent Criteria

A recently published criticism finds a flaw in "...another feature of AHP which may be, and in many application contexts will indeed be, an even stronger shortcoming of the method." [ 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] It consists in the fact that the addition of indifferent criteria (for which all alternatives perform equally) causes a significant alteration of the aggregated priorities of alternatives, with important consequences. The authors conclude that, as a result of this error "..almost all applications of AHP are potentially flawed."

Responses by AHP proponents

Proponents argue that in spite of these concerns, the process works well in practice and is extremely popular among decision-makers in the private and public sectors.

ee also

*Multi-Criteria Decision Analysis
*Thomas L. Saaty


External links

* [ An illustrated guide (pdf)] - Dr. Oliver Meixner university of Wien - "Analytic Hierarchy Process", a very easy to understand summary of the mathematical theory
* [ Decision Lens Official Site of AHP Software from Saatys, Founders of AHP]
* [ For over 25 years, Expert Choice is the Offical Site of AHP software developed by Ernest Forman]
* [ Analytic Hierarchy Process (AHP) Tutorial] - Dr. Kardi Teknomo AHP Tutorial using MS Excel.
* An AHP Application In Greek/English Language (documentation only in Greeks)
* [ AHPproject - Free Web-Based Decision Support Tool]
* [ Decision Duck - A free AHP, Decision Support Tool]
* [ AHP in IT Options Analysis]
* [ 123 AHP - A free web-based AHP application, easy to use]
* [ Ergo decision support system from TEC, free trial download]

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