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 WP:SUBSTANTIATE. 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. 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, 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)
• 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 is sometimes used in designing highly specific procedures for particular situations, such as the rating of buildings by historic significance. 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.

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. 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.

Implementation

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

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.

Summary

The procedure can be summarized as:

1. The alternatives and the significant attributes are identified.
2. 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.
3. 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.
4. 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. 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.
5. 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.

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.

A simple AHP hierarchy. In practice, many Criteria have one or more layers of subcriteria. These are not shown in this simplified diagram. Also, to avoid clutter in AHP 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 criteria, subcriteria, etc. of which it is a child.

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:

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. 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 (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.

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, 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.

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. 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, showing LOCAL priorities. 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 (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:

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.

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:

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 the technical language of AHP, we will pairwise compare the alternatives with respect to their covering criteria.

Comparing Alternatives

The family can evaluate alternatives against their covering criteria in any order they choose. In this case, they choose the order of decreasing priority of the covering criteria. That means Purchase Price first.

Purchase price

The family has established a budget of $25,000 for buying the new car, but they are willing to consider alternatives whose price exceeds their budget. To refresh your mind, here are the six cars they are considering—in AHP terminology, the six alternatives—along with their purchase prices:

The Jones Family's alternatives, with purchase prices.

Knowing that they will have a lot of pairwise comparisons to make, the family prepared this worksheet to help them. It shows comparative information about the price and budget status of each pair of cars:

The Jones family's purchase price worksheet.

Now, what do they do?

First they might compare the purchase price of the Accord Sedan to that of the Accord Hybrid. If they stick purely to arithmetic, they could say that the Sedan is favored by 1.5, since the Hybrid's price is about 1.5 times that of the Sedan, and a lower price is better. They could follow that pattern through all 15 of the comparisons, and it would give a mathematically consistent set of comparisons.

But merely entering the numbers wouldn't take into account things like the $25,000 budget, or the value to the family of saving, say, $5,000 vs. $1,000 on a purchase. Things like that can be highly important in making decisions, and their importance can vary greatly with the situation and the people involved. Some families might never want to exceed their budget. Others might be willing to exceed it by a few dollars or a few percent, but very unwilling to go further. Still others might not care much if they spend double their budget on the car. Because the AHP allows decision makers to enter their judgments about the data, rather than just the data themselves, it can deal with all these situations and more.

Let's say that the Jones family is willing to exceed their budget by up to $1,000, but anything more is unacceptable. They "never say never," however—budget-busting cars will score as low as possible on purchase price, but won't be removed from the list of alternatives. And for cars priced under budget, a $1,000 difference in price doesn't matter much to the Joneses, but a $5,000 difference is strongly important, and a $10,000 difference is extreme. They might enter the following intensities into the AHP software (throughout this example, the judgments of decision makers are shaded in green):

Purchase price judgments entered by the Jones family, with the rationale for their choices. From the Fundamental Scale, 1 expresses that A and B are equally preferred, 3 that A is moderately preferred to B, 5 that A is strongly preferred, 7 that A is very strongly preferred, and 9 that A is extremely preferred to B. Intensities 2, 4, 6, and 8 express intermediate values.

You can follow the family's thinking by looking at the rationale for each judgment. Whenever a car that is under budget is compared with one that is over budget by more than $1,000, the former is extremely preferred. For cars under budget, a $1,000 less expensive car is slightly preferred, a $5,000 one is strongly preferred, and a $6,000 one is even more strongly preferred. When both cars are well over budget (comparison #3), they are equally preferred, which is to say they are equally undesirable. Because budget status and absolute price difference are enough to make each comparison, the ratio of prices never enters into the judgments.

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

Alternative	Local Priority	Global Priority
Element	0.362	0.089
Accord	0.242	0.060
CR-V	0.242	0.060
Odyssey	0.100	0.025
Pilot	0.027	0.007
Accord Hybrid	0.027	0.007
TOTAL	1.000	0.246

The local priorities show how much the purchase price of each model contributes to the subcriterion of Purchase Price. The global priorities show how much the purchase price of each model contributes to the overall goal of choosing the best car for the Jones family.

Safety

Comparing the alternatives on the basis of Safety is much less objective than comparing them on Purchase Price. Purchase prices are measured in dollars and can be determined to the penny. People can easily agree on the meaning of a $20,360 purchase price, and can rationally compare it to all the other prices, using methods and calculations that are understood and accepted by all.

But "safety" eludes our efforts even to define it in an objective way. Not only that, but the objective measurements of safety are limited and not readily comparable from car to car.

The government conducts objective crash tests, but they are incomplete measures of the "safety" of a given car. Also, the crash tests only compare the members of a single class of cars, such as Midsize Cars or Minivans. Is a midsize car with 100% 5-star safety ratings equally as safe as a minivan with the same ratings? It's not exactly clear. And when evaluating minivans that have 5-star ratings in all categories but one, who can say if the one with four stars for "Frontal Impact, Driver's Side" is safer than the one whose four stars are in "Side Impact, Rear Occupant?" There's really no way to tell.

In spite of these difficulties, the AHP provides a rational way to evaluate the relative safety of different cars.

Let's assume that the Jones family has researched the Safety of the six Hondas they are considering. They will have found that all of them are among the safest cars on the road. All six are "Top Safety Picks" of the IIHS safety standards organization. All of them do very well in the crash testing programs of the National Highway Traffic Safety Administration. But there are differences between them, and the family wants to factor the differences into their decision. "Your car can never be too safe."

The worksheet below includes the data that the family has decided to evaluate. They believe that a heavier car is a safer car, so they've documented the curb weights of their alternatives. They have investigated the results of government crash tests, and they've summarized the results on the worksheet:

Safety data to be evaluated by the Jones family. Curb Weight is from manufacturer's literature. Crash Ratings are from NHTSA and are given for various classes of vehicle. For each class, they include Frontal Impact, Driver's Side; Frontal Impact, Passenger's Side; Side Impact, Front Occupant; Side Impact, Rear Occupant; and Rollover Rating. Ratings are from one to five stars, with five stars being the best.

The family will consider everything in the worksheet as they compare their alternatives. They are not safety experts, but they can apply their life experience to making decisions about the safety ratings. They all feel safer when driving a car that is significantly heavier than another one. One family member has seen two gruesome rollover accidents, and is terrified of a vehicle rolling over with her inside. She insists that the family car has the highest possible Rollover Rating.

Here are the weights that the Jones family enters for the alternatives regarding Safety (throughout this example, orange shading is used for judgments where A is favored; yellow shading is used for B):

Safety judgments entered by the Jones family, with the intensity and rationale for each. From the Fundamental Scale, 1 expresses that A and B are equally preferred, 3 that the better of the pair is moderately preferred to the worse, 5 that the better is strongly preferred, 7 that the better is very strongly preferred, and 9 that the better is extremely preferred to the worse. Intensities 2, 4, 6, and 8 express intermediate values.

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

Alternative	Local Priority	Global Priority
Odyssey	0.424	0.100
Accord	0.215	0.051
Accord Hybrid	0.215	0.051
Pilot	0.083	0.020
CR-V	0.038	0.009
Element	0.025	0.006
TOTAL	1.000	0.237

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:

Passenger Capacity of the Jones family's alternatives.

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:

Passenger Capacity judgments entered by the Jones family. The rationale is that four passengers is barely enough, five is perfect for their needs, and eight is just a little bit better than five.

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

Alternative	Local Priority	Global Priority
Pilot	0.273	0.049
Odyssey	0.273	0.049
Accord	0.136	0.025
Accord Hybrid	0.136	0.025
CR-V	0.136	0.025
Element	0.046	0.008
TOTAL	1.000	0.181

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

Fuel costs

After careful consideration, the Jones family believes that no matter which car they buy, they will drive it the same number of miles per year. In other words, there is nothing about any of the alternatives, including the price of fuel or the car's fuel consumption per mile, that would cause it to be driven more or fewer miles than any other alternative. They also believe that the government MPG rating is an accurate basis on which to compare the fuel consumption of the cars. Here is a worksheet showing the government MPG ratings of the Jones family alternatives:

Fuel consumption figures for the Jones family's alternatives.

They believe, therefore, that the fuel cost of any alternative vs. any other depends exclusively on the MPG ratings of the two cars. So the pairwise judgments they enter for any two cars will be inversely proportional to their MPG ratings. In other words, if car A has exactly twice the MPG rating of car B, the Fuel Cost for car B will be exactly twice that of car A. This table shows the judgments they will enter for all the comparisons:

Fuel Cost judgments entered by the Jones family. Judgments are inversely proportional to the MPG ratings of the cars being compared.

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

Alternative	Local Priority	Global Priority
Accord Hybrid	0.212	0.027
Accord	0.188	0.024
CR-V	0.160	0.020
Odyssey	0.156	0.020
Element	0.151	0.019
Pilot	0.133	0.017
TOTAL	1.000	0.127

The local priorities show how much the fuel cost of each model contributes to the subcriterion of Fuel Costs. The global priorities show how much the fuel cost of each model contributes to the overall goal of choosing the best car for the Jones family.

Resale Value

When the family researched Resale Value, they learned that lending institutions keep statistics on the market value of different models after various time periods. These estimated "residual values" are used for leasing, and are typically based on a limit of 12,000 miles (19,000 km) driven per year. Actual residual values depend on the condition of the car, and can vary with market conditions.

The Joneses are going to buy their car, not lease it, and they expect to drive it more than 12,000 miles per year, but they agree among themselves that the leasing figures are a good basis on which to compare the alternatives under consideration. Their bank gave them this table showing the residual value of each alternative after four years and 48,000 miles (77,000 km):

Four-year residual values. A value of 0.52 means that this model is worth 52% of its original price.

As they look at the table of residual values, they see that the residual value of a CR-V is 25% higher than that of a Pilot (0.55 is 125% of 0.44). They reason that such a greatly higher residual value is an indication of a better or more desirable car, so they want to place a premium on cars with relatively high residual value. After some thought and discussion, they decide that, when comparing residual values, they want to look at the higher one as a percentage of the lower, and assign their intensities on that basis. Where one model has a residual value that is less than 105% of another, they consider the residual values as equal for all practical purposes. Where one model has a residual value that is 125% of the residual value of another, they consider the former model as quite strongly more important, desirable, valuable, etc., as indicated by its much higher resale value. With a bit more thought and discussion, they decide to make their judgments on this basis:

The Jones rationale for comparing residual values. It is based on the higher residual value as a percentage of the lower one.

They realize that not every family would do it this way, but this way seems best for them. This table shows the judgments they will enter for their Resale Value comparisons:

Resale Value comparisons entered by the Jones family. They are based on the rationale shown above.

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

Alternative	Local Priority	Global Priority
CR-V	0.415	0.034
Accord	0.225	0.018
Element	0.105	0.009
Odyssey	0.105	0.009
Accord Hybrid	0.095	0.008
Pilot	0.055	0.004
TOTAL	1.000	0.081

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 60,000 miles (97,000 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:

Jones family worksheet for Maintenance Costs. Plus signs indicate good maintenance history; the more plus signs, the lower the 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:

Maintenance Cost judgments entered by the Jones family. From the Fundamental Scale, 1 means that A and B are equally preferred, 3 that A is moderately preferred, and 5 that A is strongly preferred. Intermediate numbers express intermediate preferences.

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

Alternative	Local Priority	Global Priority
Accord	0.357	0.018
Accord Hybrid	0.312	0.016
CR-V	0.100	0.005
Element	0.089	0.004
Pilot	0.084	0.004
Odyssey	0.058	0.003
TOTAL	1.000	0.050

The local priorities show how much the projected maintenance cost of each model contributes to the subcriterion of Maintenance Costs. The global priorities show how much the maintenance cost of each model contributes to the overall goal of choosing the best car for the Jones family.

Style

The family decided that Style is important to them, but how can they determine the "style" of each of the six alternatives? "Style" is a pretty subjective concept—it can truly be said that "style is in the eye of the beholder." Yet through the method of pairwise comparison, the AHP gives the Jones family a way to evaluate the "style" of the cars they are considering.

Honda's web site provides photos of each of the alternatives. It also has videos, commercials, rotatable 360° views, color chips, and more, all available to help family members evaluate the Style of each car. The family can compare their alternatives two-by-two on Style, using the tools on the web site to help them make their judgments. They did just that, and here is the record of their judgments:

Style judgments entered by the Jones family. The family looked at each pair of cars as shown in detail on a web site, decided which of the two they preferred, then entered the intensity of their preference according to the Fundamental Scale.

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

Alternative	Local Priority	Global Priority
Accord	0.346	0.015
Accord Hybrid	0.346	0.015
CR-V	0.160	0.007
Odyssey	0.078	0.003
Pilot	0.045	0.002
Element	0.025	0.001
TOTAL	1.000	0.042

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

Cargo Capacity

The Cargo Capacity of each alternative, measured in cubic feet, is listed in the manufacturer's specifications for each vehicle. The Joneses don't really know how it is calculated, but they trust that it's a good indication of how much cargo can be packed into a vehicle. This worksheet shows the cargo capacities of the Jones' alternatives:

Cargo Capacity of the Jones' alternatives.

Cargo capacities for the alternatives vary from 14 to 148 cubic feet (4.2 m³). If they wanted to, the Jones family could enter these capacities directly into the AHP software. But that would mean that, when considering Cargo Capacity, a car with 148 cu ft (4.2 m³). of it would be over ten times as desirable as one with only 14. Given the car's use as a family vehicle, that doesn't seem quite right. So the family looks at the available capacities and determines that a 14 cu ft (0.40 m³). trunk is perfectly fine for their needs, that something about five times larger is slightly better, and that something about ten times larger is moderately so. These judgments correspond to values of 1, 2, and 3 on the AHP's Fundamental Scale.

Here are the judgments they would enter into the AHP software:

Cargo Capacity judgments entered by the Jones family. The rationale is that 14 cubic feet (0.40 m³) is totally satisfactory, five times that much (70 cu. ft.) is slightly better, and ten times that much (140 cu. ft.) is moderately better than 14.

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

Alternative	Local Priority	Global Priority
Odyssey	0.310	0.011
Pilot	0.170	0.006
CR-V	0.170	0.006
Element	0.170	0.006
Accord	0.090	0.003
Accord Hybrid	0.090	0.003
TOTAL	1.000	0.036

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:

Global priorities for the Jones family car buying decision. Note that the priorities for each group of children total that of their parent.

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.

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. In the early 1990s a series of debates between critics and proponents of AHP was published in Management Science and The Journal of the Operational Research Society. Occasional criticisms still appear today.

Rank reversal

Most criticisms of AHP have involved a phenomenon called "rank reversal." Many people hear about rank reversal and assume that there is some sort of proven principle about it that needs to be upheld in making decisions. That unwarranted assumption has led to much misunderstanding of AHP and other decision making techniques. In actuality, rank reversal is a complex matter about which there are many conflicting ideas and opinions. This section offers a simplified explanation of the situation.

Decision making involves ranking alternatives in terms of criteria or attributes of those alternatives. It is an axiom of some decision theories that when new alternatives are added to a decision problem, the ranking of the old alternatives must not change. But in the real world, 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.

A simple example will demonstrate the phenomenon of rank reversal:

Consider a pretty girl in a small town. She's having a party next week, and she wants to buy a dress that will impress her guests. She visits the town's only dress store and goes to the rack of party dresses. There are five such dresses, and after long consideration she ranks them by desirability as follows:

Rank	Style	Color	Price
1	Style A	Blue	$109
2	Style A	Green	$109
3	Style B	Red	$119
4	Style C	Yellow	$99
5	Style D	Off-White	$149

Now imagine that she enters the back room and sees the store's entire inventory of dresses. The dresses she has looked at in Styles B, C, and D are the only ones of their kind, but there are four more Style A dresses in green and eight more Style A dresses in blue. In the language of decision science, these dresses are copies of the existing alternatives. In our one-store small town scenario, there's a reasonable chance that one or more party guests would buy and wear one of the copies.

When made aware of these new alternatives, our fashion-conscious girl might rank her choices in a different order. Considering her great embarrassment if a guest were to wear the same dress that she did, she might rank her choices like this:

Rank	Old Rank	Style	Color	Price
1	3	Style B	Red	$119
2	4	Style C	Yellow	$99
3	5	Style D	Off-White	$149
4	2	Style A	Green	$109
5	1	Style A	Blue	$109

Notice that the rankings of the two Style A dresses have reversed (since there are more copies of the blue dress than of the green one). Not only that, but Style A has gone from the most preferred style to the least preferred. Rank reversal has occurred. Axioms of decision theories have been violated. Scholars and researchers can cry "foul," or impugn the method by which the girl has made her choice, but there is no denying that in the world of our example, ranks have been reversed. There is no doubt that the reversal is due to the introduction of additional alternatives that are no different than the existing ones.

The above example, is but one example of rank reveral. Rank reversal can also occur when additional alternatives are added/removed that are not copies of the original alternatives. Another example of rank reveral is the 2004 U.S. Presidential elections (as well as earlier elections). Ralph Nader was an 'irrelevant' alternative, in that he was dominated by both the Democratic and Republican candidates. However, since he attracted more votes from those who would have voted Democratic rather than Republican, his presence caused the ranks to reverse. Put another way, if Nader were not in the race, it is generally 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 Operations Research as well as a chapter entitled Rank Preservation and Reversal, in the current basic book on AHP. It 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. An online copy of the chapter is available HERE.

Other criticisms

Arrow's Impossibility Theorem

Arrow's Impossibility Theorem can be used to explain the limitations of the analytic hierarchy process. . Arrow's Impossibility Theorem demonstrates that no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a certain set of reasonable criteria with three or more discrete options.

Looked at in another light, using AHP's ratio scale measurement, the paradox of Arrow's Imposssibility Theorem is no longer a paradox.

The use of arbitrary scales

AHP is based on pairwise comparisons where the relative importance of different attributes can be elicited using a verbal comparisons (referred to as the fundamental scale by Saaty), numerical comparisons, or graphical comparisons. The verbal comparisons of 'equal', 'moderate', 'strong', 'very strong', 'extereme' and intermediate intensisties, such as between moderate and strong, are given a value on a scale of 1 to 9 or the inverse (1/9th to 1). 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.". Whatever verbal scale is used, the measures are ordinal at best and should not be used in arithmetic operations. However, if there is enough variety and redundancy in the judgments then it has been demonstrated in numerous experiments that the principal right hand eigenvector of the matrix of comparisons produces priorities that are close to actual ratio scale priorites. When this is not the case, pairwise numerical or graphical judgments can be made instead.

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."

The Schenkerman article is based on an example where rectangular fields of different lengths and widths were evaluated. Schenkerman's conclusion is based on his eroneous assumption that length and width were of equal importance. This is not the case because, in his example, the measurements were such that the total of the lengths and widths were not equal. Had he used the actual lengths and widths of the shapes to specify the priorities of the length versus width, the results would have been what he had expected them to be.

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." 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." This criticism is a variant of the rank reversal controversy. If the 'ideal' synthesis mode of AHP is used, there are no such alterations.

Responses by AHP proponents

Proponents argue that these concerns are expressed by those not aware of actual practice of AHP, in particular, the ideal synthesis mode .

They assert that the AHP process is theoretically sound. It works well in practice and is extremely popular among decision-makers in the private and public sectors.

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 almost twenty years ago. 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.

See also

• Decision making
• 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.
• http://www.geocities.com/thor4bp/ 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
• Ergo decision support system from TEC, free trial download