Analytic Hierarchy Process

The Analytic Hierarchy Process (AHP) is a technique for dealing with complex decisions where a number of competing factors demand consideration. It is especially suited to problems with high stakes, involving human perceptions and judgments, whose resolutions have long-term repercussions. AHP provides a comprehensive and rational framework for structuring a problem, for representing its elements and quantifying them, for relating those elements to overall goals, and for evaluating alternative solutions.

While it can be used by individuals working on straightforward problems, AHP is most commonly applied where teams of people are working on highly complex situations, particularly where some of the elements are difficult to quantify. Computer software is available to assist in the application of the process.

Developed in the 1970s by mathematician Thomas L. Saaty, AHP has been applied worldwide, in a wide variety of decision situations, in fields such as government, business, industry, healthcare, quality, and education.

Uses and applications of the AHP

The applications of AHP to complex decision situations number in the thousands, typically where problems are important and complex. Many such applications are never reported to the outside world, 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)
• Quantifying the overall quality of software systems (Microsoft Corporation)
• Selecting university faculty (Bloomsburg University of Pennsylvania)
• Deciding where to locate offshore manufacturing plants (University of Cambridge)
• Assessing risk in operating cross-country petroleum pipelines (American Society of Civil Engineers)
• Deciding how best to manage U.S. watersheds (U.S. Department of Agriculture)

AHP 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.

The process 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. AHP was introduced in China in 1982, and its application 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.

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. AHP is considered an important subject in many institutions of higher learning, including schools of engineering and graduate schools of business. AHP is also an important subject in the quality field, and is taught in many specialized courses including Six Sigma and QFD.

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.

Description of the process

The introductory material for this section is Under Construction
AHP is a method that breaks complexity into manageable pieces, works with each piece, then collectively evaluates all the pieces. The steps are...

Though the AHP can be used by an individual, our discussion will assume it's used by a group.

We say "decision problem," but there might be a more general term.

Make it clear that this is a simplified description, and that it is based on previously published information, especially Decision Making for Leaders, but also including lots and lots of other books and articles.

Hello again, Lou....

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 build the hierarchy, they increase their understanding of the problem, of its context, and of each other's thoughts and feelings about both.

Hierarchies defined

The hierarchy of authority aboard a commercial airliner. Each person is subordinate to a single other person.

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 a single other element. 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. In the airliner example shown to the right, the hierarchical system is used for assigning responsibilities, exercising leadership, and facilitating communication.

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 understanding of whatever 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.

A simple AHP hierarchy. In practice, many Criteria have one or more layers of subcriteria. These are not shown in this simplified diagram. To avoid clutter in these diagrams, the lines between the Alternatives and Criteria are often omitted or reduced in number. Regardless of any such changes to the diagram, in the actual hierarchy each Criterion is connected to every Alternative.

The specifics 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.

An elementary 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 criteria to be considered might be cost, safety, style, and capacity. The cost criterion might be subdivided into purchase price, fuel costs, maintenance costs, and resale value. Capacity might be subdivided into cargo capacity and passenger capacity. The alternatives for the family, which for personal reasons always buys Hondas, might be the Accord Sedan, Accord Hybrid Sedan, Pilot SUV, CR-V SUV, Element SUV, and Odyssey Minivan.

The Jones' hierarchy could be diagrammed like this:

File:AHPHierarchy2.png

AHP hierarchy for the Jones family car 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 directly above it.

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 5,000 miles a year, etc.). It would definitely be different for a 25-year-old playboy who has millions to spend on cars, knows he will never have a wreck, and is intensely interested in speed, handling, and the numerous aspects of style.

Additional information. More information about hierarchies, including further reading and some complex real-world examples, can be found HERE.

Establish Priorities

Once the hierarchy has been constructed, the participants use AHP to establish priorities for all its elements. 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. The following sections describe a simple, straightforward case. (Rozann: pairwise comparisons using judgment)

Priorities defined

NOTE: These are rough notes to get the ideas straight. Better wording will follow.

Priorities are numbers associated with the elements or nodes of the hierarchy. By definition, the priority of the Goal is 1.000. Priorities for the Criteria in the level immediately below the Goal can vary in magnitude, but will always add up to 1.000. The priorities for the subcriteria of any node at that level can also vary but will add up to 1.000, as will those for the subcriteria of any node at that level, and so on.

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.

AHP hierarchy for the Jones family auto decision, with some associated priorities.

If you understand what has been said so far, you will see that if we added 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 an AHP hierarchy, the priorities indicate the relative weights given to the items in a given group of elements. Depending on the problem at hand, "weight" can refer to importance, or preference, or likelihood, or whatever factor is being considered.

If all the priorities in a group are equal, each member 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.

In the example above, all priorities in a given group are equal, indicating that no item in that group has more or less weight than any other. This is the situation before we begin to evaluate the relative importance of the items. (For example before we assess whether price or safety is more important to the Joneses in buying a car.)

In this situation, the priorities are called default priorities. As the analytic hierarchy process continues, the priorities will change to reflect our judgments about the various items in each group.

As we enter judgments, the priorities will change but will still add to one for each group. For example, if we judge passenger capacity to be more important than cargo capacity, the priority for the former will increase and the latter will decrease.

More specifically, if we judge passenger capacity to be three times as important as cargo capacity, passenger capacity's new priority will be .750, and cargo capacity's will be .250 (because .750 = 3 x .250, and .750 + .250 = 1.000. Don't worry—the software takes care of keeping track of all this).

Make judgments

Under Construction
Establish priorities for the elements of the hierarchy. Includes inputting data. Be careful about data. There are scales involved. Some are meaningful to us and the problem, some are not. This includes the consistency check step (repeated at every level)

Derive priorities

Under Construction
This is done by "AHP Magic Math." It turns your pairwise comparisons of items at one level into weights for the items at that level. Maybe we are "developing" priorities, or "calculating" them, or ????

Synthesize priorities throughout the structure

Under Construction
More "AHP Magic Math" It takes the priorities for the various items at the various levels and turns them into one big integrated set of priorities for the whole hierarchy.

Check sensitivity

Under Construction
Take a look at what would happen if your judgments changed.

Using AHP with groups

Under Construction

When you get to this part, go through the whole article and harmonize "user", "participants", etc.

Underlying philosophy

Under Construction

There is a lot of good material on this in the books. If it can be put here in a suitable way, this maybe should be the first section of the article, or at least somewhere way up at the top.

Criticisms and drawbacks

Under Construction

See also

Multi Criteria Decision Making