Күрделі шешімдерді талдау және ұйымдастырудың аналитикалық иерархиялық әдісі

Structured technique for organizing and analyzing complex decisions

In the theory of decision making, the analytic hierarchy process (AHP), also analytical hierarchy process, is a structured technique for organizing and analyzing complex decisions, based on mathematics and psychology. It was developed by Thomas L. Saaty in the 1970s; Saaty partnered with Ernest Forman to develop Expert Choice software in 1983, and AHP has been extensively studied and refined since then. It represents an accurate approach to quantifying the weights of decision criteria. Individual experts’ experiences are utilized to estimate the relative magnitudes of factors through pair wise comparisons. Each of the respondents compares the relative importance of each pair of items using a specially designed questionnaire. The relative importance of the criteria can be determined with the help of the AHP by comparing the criteria and, if applicable, the sub criteria in pairs by experts or decision makers. On this basis, the best alternative can be found.

Though using the analytic hierarchy process requires no specialized academic training, it is considered an important subject in many institutions of higher learning, including schools of engineering and graduate schools of business. It is a particularly important subject in the quality field, and is taught in many specialized courses including Six Sigma, Lean Six Sigma, and QFD. The International Symposium on the Analytic Hierarchy Process (ISAHP) holds biennial meetings of academics and practitioners interested in the field. A wide range of topics is covered. Those in 2005 ranged from "Establishing Payment Standards for Surgical Specialists", to "Strategic Technology Roadmapping", to "Infrastructure Reconstruction in Devastated Countries". At the 2007 meeting in Valparaíso, Chile, 90 papers were presented from 19 countries, including the US, Germany, Japan, Chile, Malaysia, and Nepal. A similar number of papers were presented at the 2009 symposium in Pittsburgh, Pennsylvania, when 28 countries were represented. Subjects of the papers included Economic Stabilization in Latvia, Portfolio Selection in the Banking Sector, Wildfire Management to Help Mitigate Global Warming, and Rural Microprojects in Nepal.

A hierarchy is a stratified 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. Though the concept of hierarchy is easily grasped intuitively, it can also be described mathematically. 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. More complex hierarchies can be found on a special talk page for this article. 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 decision making process. Constructing a hierarchy typically involves significant discussion, research, and discovery by those involved. Even after its initial construction, it 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.

This section explains priorities, shows how they are established, and provides a simple example.

Priorities are numbers associated with the nodes of an AHP hierarchy. They represent the relative weights of the nodes in any group. Like probabilities, priorities are absolute numbers between zero and one, without units or dimensions. A node with priority .200 has twice the weight in reaching the goal as one with priority .100, ten times the weight of one with priority .020, and so forth. Depending on the problem at hand, "weight" can refer to importance, or preference, or likelihood, or whatever factor is being considered by the decision makers. Priorities are distributed over a hierarchy according to its architecture, and their values depend on the information entered by users of the process. Priorities of the Goal, the Criteria, and the Alternatives are intimately related, but need to be considered separately. By definition, the priority of the Goal is 1.000. The priorities of the alternatives always add up to 1.000. Things can become complicated with multiple levels of Criteria, but if there is only one level, their priorities also add to 1.000. All this is illustrated by the priorities in the example below. Observe that the priorities on each level of the example—the goal, the criteria, and the alternatives—all add up to 1.000. The priorities shown are those that exist before any information has been entered about weights of the criteria or alternatives, so the priorities within each level are all equal. They are called the hierarchy's default priorities. If a fifth Criterion were added to this hierarchy, the default priority for each Criterion would be .200. If there were only two Alternatives, each would have a default priority of .500. Two additional concepts apply when a hierarchy has more than one level of criteria: local priorities and global priorities. Consider the hierarchy shown below, which has several Subcriteria under each Criterion. The local priorities, shown in gray, represent the relative weights of the nodes within a group of siblings with respect to their parent. The local priorities of each group of Criteria and their sibling Subcriteria add up to 1.000. The global priorities, shown in black, are obtained by multiplying the local priorities of the siblings by their parent's global priority. The global priorities for all the subcriteria in the level add up to 1.000. The rule is this: Within a hierarchy, the global priorities of child nodes always add up to the global priority of their parent. Within a group of children, the local priorities add up to 1.000. So far, we have looked only at default priorities. As the Analytical Hierarchy Process moves forward, the priorities will change from their default values as the decision makers input information about the importance of the various nodes. They do this by making a series of pairwise comparisons.

Experienced practitioners know that the best way to understand the AHP is to work through cases and examples. Two detailed case studies, specifically designed as in depth teaching examples, are provided as appendices to this article:

Simple step by step example with four Criteria and three Alternatives: Choosing a leader for an organization. More complex step by step example with ten Criteria/Subcriteria and six Alternatives: Buying a family car and Machinery Selection Example. Some of the books on AHP contain practical examples of its use, though they are not typically intended to be step by step learning aids.

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 — that "rank reversal" must not occur. 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 some situations in which rank reversal can reasonably be expected. The original formulation of AHP allowed rank reversals. In 1993, Forman introduced a second AHP synthesis mode, called the ideal synthesis mode, to address choice situations in which the addition or removal of an 'irrelevant' alternative should not and will not cause a change in the ranks of existing alternatives. 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 AHP are extensively discussed in a 2001 paper in Operations Research, The latter 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 AHP. A new form of rank reversal of AHP was found in 2014 in which AHP produces rank order reversal when eliminating irrelevant data, this is data that do not differentiate alternatives. There are different types of rank reversals. Also, other methods besides the AHP may exhibit such rank reversals. More discussion on rank reversals with the AHP and other MCDM methods is provided in the rank reversals in decision making page.

Within a comparison matrix one may replace a judgement with a less favorable judgment and then check to see if the indication of the new priority becomes less favorable than the original priority. In the context of tournament matrices, it has been proven by Oskar Perron that the principal right eigenvector method is not monotonic. This behaviour can also be demonstrated for reciprocal n x n matrices, where n > 3. Alternative approaches are discussed elsewhere.