Analytic Hierarchy Process

The Analytic Hierarchy Process (AHP) is a structured technique for helping people deal with complex decisions. 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.

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, roughly estimated or carefully measured, 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 their judgments about the elements' relative meaning and importance, or they can use concrete data about the elements. 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 the judgments to numerical values that can be processed, evaluated 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.

At the end of the process, numerical priorities are derived for each of the decision alternatives. It is then a simple matter to pick the best alternative, or to rank them in order of relative preference.

Uses and applications

While it can be used by individuals working on straightforward decisions, 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. 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. Computer software is available to assist in the application of the process.

The applications of AHP to complex decision situations have numbered in the thousands, and have produced extensive results in problems involving alternative selection, planning, resource allocation, and priority setting. 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 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.

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, Lean 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.

Steps in using the process

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.

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 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. In the airliner example shown to the right, 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. Inside the tower is another hierarchy, with the top level motherboard and its subordinate hard drive(s), CD-ROM drive, and sometimes a floppy drive or DVD writer.

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 simplifications in the diagram, in the actual hierarchy each Alternative is connected to every one of its covering criteria—the lowest-level criterion or subcriterion that appears immediately above it.

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 like this:

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 directly above it.

File:AHPHierarchy2a.png

Alternatives for the Jones family car buying decision. To save space, we have represented them as stacks of papers.

Some simple terms apply to the boxes in the diagrams above. Each of the boxes 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, the four Criteria are children of the Goal. The Goal is the parent of each of the four Criteria. Cargo Capacity and Passenger Capacity are the children of their parent, the Capacity criterion.

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 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.

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 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.

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

If you understand what has been said so far, you will see that if we were to add a "Handling" criterion to this hierarchy, making 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. As the analytic hierarchy process continues, the default priorities will change 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. (Don't worry—the AHP software does all this arithmetic automatically.)

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 them 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?

File:AHPHierarchy2a.png

Alternatives for the Jones family car buying decision. These six models will be pairwise compared with respect to each of the covering criteria—the items in the nodes immediately above each group of alternatives. In the example, these are Purchase Price, Fuel Costs, Maintenance Costs, Resale Value, Safety, Style, Cargo Capacity, and 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 a total of 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 will be 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 each of 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.

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 made these 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.

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. 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.

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:

AHP hierarchy for the Jones family car buying decision. The items in each group of Subcriteria have been pairwise compared, and their resulting priorities are shown. The priorities in each group total 1.000. These are called local priorities.

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 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:

AHP hierarchy for the Jones family car buying decision. Showing global priorities. The priority of the Goal is 1.000, as always. The global priorities shown in red for the Criteria and Subcriteria also add up to 1.000. Each item with a red global priority contributes that amount to the priority of the Goal.

Now we know, according to the judgments entered by the family, the priority assigned to each of the factors that apply to their decision:

Factor	Priority
Purchase Price	0.246
Safety	0.237
Passenger Capacity	0.181
Fuel Costs	0.127
Resale Value	0.081
Maintenance Costs	0.050
Style	0.042
Cargo Capacity	0.036

The next step is to evaluate each of the cars with respect to these factors. In AHP lingo, we will pairwise compare the alternatives with respect to their covering criteria.

Criticisms

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. Their criticisms have included: A) since there is no theoretical basis for constructing hierarchies, AHP users can construct different hierarchies for identical decision situations, possibly producing different solutions, B) AHP rankings are claimed to be arbitrary because they are based on subjective opinions using a ratio scale, a situation which can produce "rank reversal," C) there are said to be flaws in the methods of combining individual weights into composite weights, and D) the process has no sound underlying statistical theory.