The Analytic Hierarchy Process (AHP) is a problem solving framework. It is a systematic procedure for representing the elements of any problem. It organizes the basic rationality by breaking down a problem into its smaller constituent parts and then calls for only simple pairwise comparison judgments, to develop priorities in each hierarchy.
The AHP provides a comprehensive framework to cope with the intuitive, the rational, and the irrational in us at the same time It is a method we can use to integrate our perceptions and purposes into an overall synthesis. The AHP does not require that judgments be consistent or transitive. The degree of consistency (or inconsistency) of the judgment is revealed at the end of the AHP process.
When dealing with scientists, corporate managers, the academic community, lay people and others in solving problems or planning we have observed repeatedly that people provide subjective judgments based on feelings and intuition rather than on well worked out logical reasoning. Also when they reason together people tend to influence each other s thinking. Individual judgments are altered slightly to accommodate the group's logic and the group's interests. However, people have very short memories and if asked afterwards to support the group judgments, they instinctively go back to their individual judgments.
One also observes that people find it difficult to justify their judgments logically and to explain how strong these judgments are. As a result people make great compromise in their thinking to accommodate ideas and judgments.
Designing an analytic hierarchy - like the structuring of a problem by any other method necessitates substantial knowledge of the system in question. A very strong aspect of the AHP is that the knowledgeable individuals who supply judgments for the pairwise comparisons usually also play a prominent role in specifying the hierarchy.
Detailed Explanation and Simple Example
We are all fundamentally decision makers. Everything we do consciously or unconsciously is the result of some decision. The information we gather is to help us understand occurrences in order to develop good judgments to make decisions about these occurrences. Not all information is useful for improving our understanding and judgments. If we only make decisions intuitively, we are inclined to believe that all kinds of information are useful and the larger the quantity the better. But that is not true. There are numerous examples which show that too much information is as bad as little information. Knowing more does not guarantee that we understand better as illustrated by some author’s writing “Expert after expert missed the revolutionary significance of what Darwin had collected. Darwin, who knew less, somehow understood more.” To make a decision we need to know the problem, the need and purpose of the decision, the criteria of the decision, their sub-criteria, stakeholders and groups affected and the alternative actions to take. We then try to determine the best alternative, or in the case of resource allocation we need priorities for the alternatives to allocate their appropriate share of the resources.
Decision making, for which we gather most of our information, has become a mathematical science today (Figuera et al.2005). It formalizes the thinking we use so that what we have to do to make better decisions is transparent in all its aspects. We need to have some fundamental understanding of this most valuable process that nature endowed us with to make it possible for us to make choices that help us survive. Decision making involves many criteria and sub-criteria used to rank the alternatives of a decision. Not only does one need to create priorities for the alternatives with respect to the criteria or sub-criteria in terms of which they need to be evaluated, but also for the criteria in terms of a higher goal, or if they depend on the alternatives, then in terms of the alternatives themselves. The criteria may be intangible and have no measurements to serve as a guide to rank the alternatives and creating priorities for the criteria themselves in order to weight the priorities of the alternatives and add over all the criteria to obtain the desired overall ranks of the alternatives is a challenging task. How? We will cover some of the essentials of multi-criteria decision making here.
The measurement of intangible factors in decisions has for a long time defied human understanding. Number and measurement are the core of mathematics and mathematics is essential to science. So far mathematics has assumed that all things can be assigned numbers from minus infinity to plus infinity in some way and all mathematical modeling of reality has been described in this way by using axes and geometry. Naturally all this is predicated on the assumption that one has the essential factors and all these factors are measurable. But there are many more important factors that we do not know how to measure than there are ones that we have measurements for. Knowing how to measure such factors could conceivably lead to new and important theories that rely on many more factors for their explanations. After all, in an interdependent universe everything depends on everything else. Is this just a platitude or is there some truth behind it? If we knew how to measure intangibles, much wider room would be open to interpret everything in terms of many more factors than we have been able to do so far scientifically. One thing is clear, numerical measurement must be interpreted for meaning and usefulness according to its priority to serve our values in a particular decision. It does not have the same priority for all problems. Its importance is relative. Therefore, we need to learn about how to derive relative priorities in decision making.
Background
There are two possible ways to learn about anything - an object, a feeling or an idea. The first is to examine and study it in itself to the extent that it has various properties, synthesize the findings and draw conclusions from such observations about it. The second is to study that entity relative to other similar entities and relate it to them by making comparisons.
The cognitive psychologist Blumenthal (1977) wrote that "Absolute judgment is the identification of the magnitude of some simple stimulus...whereas comparative judgment is the identification of some relation between two stimuli both present to the observer. Absolute judgment involves the relation between a single stimulus and some information held in short-term memory, information about some former comparison stimuli or about some previously experienced measurement scale... To make the judgment, a person must compare an immediate impression with impression in memory of similar stimuli".
Using judgments has been considered to be a questionable practice when objectivity is the norm. But a little reflection shows that even when numbers are obtained from a standard scale and they are considered objective, their interpretation is always, I repeat, always, subjective. We need to validate the idea that we can use judgments to derive tangible values to provide greater credence for using judgments when intangibles are involved.
The Analytic Hierarchy Process
To make a decision in an organized way to generate priorities we need decompose the decision into the following steps.
1. Define the problem and determine the kind of knowledge sought.
2. Structure the decision hierarchy from the top with the goal of the decision, then the objectives from a broad perspective, through the intermediate levels (criteria on which subsequent elements depend) to the lowest level (which usually is a set of the alternatives).
3. Construct a set of pairwise comparison matrices. Each element in an upper level is used to compare the elements in the level immediately below with respect to it.
4. Use the priorities obtained from the comparisons to weight the priorities in the level immediately below. Do this for every element. Then for each element in the level below add its weighted values and obtain its overall or global priority. Continue this process of weighting and adding until the final priorities of the alternatives in the bottom most level are obtained.
To make comparisons, we need a scale of numbers that indicates how many times more important or dominant one element is over another element with respect to the criterion or property with respect to which they are compared. Table 1 exhibits the scale. Table 2 exhibits an example in which the scale is used to compare the relative consumption of drinks in the United States. One compares a drink indicated on the left with another indicated at the top and answers the question: How many times more, or how strongly more is that drink consumed in the US than the one at the top? One then enters the number from the scale that is appropriate for the judgment: for example enter 9 in the (coffee, wine) position meaning that coffee consumption is 9 times wine consumption. It is automatic that 1/9 is what one needs to use in the (wine, coffee) position. Note that water is consumed more than coffee, so one enters 2 in the (water, coffee) position, and ½ in the (coffee, water) position. One always enters the whole number in its appropriate position and automatically enters its reciprocal in the transpose position.
The priorities, (obtained in exact form by raising the matrix to large powers and summing each row and dividing each by the total sum of all the rows, or approximately by adding each row of the matrix and dividing by their total) are shown at the bottom of the table along with the true values expressed in relative form by dividing the consumption of each drink (volume) by the sum of the consumption of all drinks. The information about actual consumption was obtained from the US Statistical Abstracts. We see the answers are very close and pair-wise comparison judgments of someone who knows can lead to very accurate results of drink consumption.

An Example of a Simple DecisionThe following is a simple decision examined by someone to determine what kind of job would be best for her after getting her PhD: either to work at two kinds of companies or to teach at two kinds of schools. The Goal is to determine the kind of job for which she is best suited as spelled out by the criteria. Because of space limitations we will not define them in detail here. For more detail see Saaty, (1994 and 2000).

There are 12 pairwise comparison matrices in all: One for the criteria with respect to the goal, which we show here in Table 3, two for the sub-criteria the first of which for the sub-criteria under flexibility: location, time, and work, that we show in Table 4 and one for the sub-criteria under opportunity that we do not show here. Then there are nine comparison matrices for the four alternatives with respect to all the “covering criteria”, the lowest level criteria or sub-criteria connected to the alternatives. The 9 covering criteria are: flexibility of location, time and work, entrepreneurial company, possibility for salary increases and a top-level position, job security, reputation and salary. The first six are sub-criteria in the second level and the last three are criteria from the first level. We only show one of these 9 matrices comparing the alternatives with respect to potential increase in salary in Table 5.
The criteria listed on the left are one by one compared with each criterion listed on top as to which one is more important with respect to the goal of selecting a best job. The sub-criteria on the left are then compared with the sub-criteria on top as to their importance with respect to flexibility. The alternatives on the left are then compared with those on top with respect to relative preference for potential increase in salary. The sub-criteria priorities are weighted by the priority of their parent criterion flexibility (.036) to obtain their global priority.
The priorities for each matrix are obtained as they were from the matrix of comparisons for the drinks in the US. In Table 6 the rankings of the alternatives are shown against the nine covering criteria (only one of the matrices leading to the rankings was shown, in Table 5). We need to multiply each ranking by the priority of its criterion or sub-criterion and add the resulting weights for each alternative to get its final priority. We call this part of the process, synthesis. It is shown in Table 6. Because Table 6 is horizontally long, it is divided into two pieces where the lower piece follows to the right of the upper piece.
The overall priorities for the alternative jobs, shown on the far right of the lower piece of Table 6, are the sums across each row for the alternatives. Note that they sum to 1. These priorities may also be expressed in the ideal form by dividing each priority by the largest one, .333 for International Company, as shown in Table 7. The effect is to make this alternative the ideal one with the others getting their proportionate value. One may then interpret the results to mean that a State University job is about 78% as good as one with an International Company and so on.