Thomas L. Saaty

The Analytic Hierarchy Process

This chapter provides an overview of Analytic Hierarchy Process (AHP), which is a systematic procedure for representing the elements of any problem hierarchically. It organizes the basic rationality by breaking down a problem into its smaller constituent parts and then guides decision makers through a series of pair-wise comparison judgments to express the relative strength or intensity of impact of the elements in the hierarchy. These judgments are then translated to numbers. The AHP includes procedures and principles used to synthesize the many judgments to derive priorities among criteria and subsequently for alternative solutions. It is useful to note that the numbers thus obtained are ratio scale estimates and correspond to so-called hard numbers. Problem solving is a process of setting priorities in steps. One step decides on the most important elements of a problem, another on how best to repair, replace, test, and evaluate the elements, and another on how to implement the solution and measure performance.

How to make a decision: The analytic hierarchy process

Abstract This paper serves as an introduction to the Analytic Hierarchy Process — A multicriteria decision making approach in which factors are arranged in a hierarchic structure. The principles and the philosophy of the theory are summarized giving general background information of the type of measurement utilized, its properties and applications.

A Scaling Method for Priorities in Hierarchical Structures

Abstract The purpose of this paper is to investigate a method of scaling ratios using the principal eigenvector of a positive pairwise comparison matrix. Consistency of the matrix data is defined and measured by an expression involving the average of the nonprincipal eigenvalues. We show that λmax = n is a necessary and sufficient condition for consistency. We also show that twice this measure is the variance in judgmental errors. A scale of numbers from 1 to 9 is introduced together with a discussion of how it compares with other scales. To illustrate the theory, it is then applied to some examples for which the answer is known, offering the opportunity for validating the approach. The discussion is then extended to multiple criterion decision making by formally introducing the notion of a hierarchy, investigating some properties of hierarchies, and applying the eigenvalue approach to scaling complex problems structured hierarchically to obtain a unidimensional composite vector for scaling the elements falling in any single level of the hierarchy. A brief discussion is also included regarding how the hierarchy serves as a useful tool for decomposing a large-scale problem, in order to make measurement possible despite the now-classical observation that the mind is limited to 7 ± 2 factors for simultaneous comparison.

Decision making with the analytic hierarchy process

Decisions involve many intangibles that need to be traded off. To do that, they have to be measured along side tangibles whose measurements must also be evaluated as to, how well, they serve the objectives of the decision maker. The Analytic Hierarchy Process (AHP) is a theory of measurement through pairwise comparisons and relies on the judgements of experts to derive priority scales. It is these scales that measure intangibles in relative terms. The comparisons are made using a scale of absolute judgements that represents, how much more, one element dominates another with respect to a given attribute. The judgements may be inconsistent, and how to measure inconsistency and improve the judgements, when possible to obtain better consistency is a concern of the AHP. The derived priority scales are synthesised by multiplying them by the priority of their parent nodes and adding for all such nodes. An illustration is included.

Models, methods, concepts & applications of the analytic hierarchy process

1. How to Make a Decision. 2. The Seven Pillars of the AHP. 3. Architectural Design. 4. Designing a Mousetrap. 5. Designing the Best Catamaran. 6. The Selection of a Bridge. 7. Measuring Dependence Between Activities: Input Output Application to the Sudan. 8. Technological Choice in Less Developed Countries. 9. Market Attractiveness of Developing Countries. 10. An AHP Based Approach to the Design and Evaluation of a Marketing Driven Business and Corporate Strategy. 11. New Product Pricing Strategy. 12. Incorporating Expert Judgment in Economic Forecasts - the Case of the U.S. Economy in 1992. 13. A New Macroeconomic Forecasting and Policy Evaluation Method. 14. Forecasting the Future of the Soviet Union. 15. Abortion and the States: How Will the Supreme Court Rule on the Upcoming Pennsylvania Abortion Issue. 16. The Benefits and Costs of Authorizing Riverboat Gambling. 17. The Case of the Spotted Owl vs. the Logging Industry. 18. Selection of Recycling Goal Most Likely to Succeed. 19. To Drill or Not to Drill: A Synthesis of Expert Judgments. 20. Modeling the Graduate Business School Admissions Process. 21. Infertility Decision Making. 22. The Decision by the US Congress on Chinas Trade Status: A Multicriteria Analysis. 23. Deciding Between Angioplasty and Coronary Artery Bypass Surgery. Index.

Decision making for leaders

Decision Making for Leaders is an introduction to Saatys analytic hierarchy process (AHP) aimed at an audience of leaders in business, industry, and government. As such, the book can be viewed as a popularized version of Saatys more technical works on AHP (for example, see [1]).

Decision-making with the AHP: Why is the principal eigenvector necessary

In this paper it is shown that the principal eigenvector is a necessary representation of the priorities derived from a positive reciprocal pairwise comparison judgment matrix A=(aij) when A is a small perturbation of a consistent matrix. When providing numerical judgments, an individual attempts to estimate sequentially an underlying ratio scale and its equivalent consistent matrix of ratios. Near consistent matrices are essential because when dealing with intangibles, human judgment is of necessity inconsistent, and if with new information one is able to improve inconsistency to near consistency, then that could improve the validity of the priorities of a decision. In addition, judgment is much more sensitive and responsive to large rather than to small perturbations, and hence once near consistency is attained, it becomes uncertain which coefficients should be perturbed by small amounts to transform a near consistent matrix to a consistent one. If such perturbations were forced, they could be arbitrary and thus distort the validity of the derived priority vector in representing the underlying decision.

Highlights and critical points in the theory and application of the Analytic Hierarchy Process

Abstract This paper provides a detailed discussion with references on the fundamentals of the Analytic Hierarchy Process and in particular of relative measurement. The points discussed are grouped under the following categories: Structure in the AHP — Hierarchies and Networks, Scales of Measurement, Judgments, Consistency and the Eigenvector, Synthesis and Normative vs. Descriptive. The paper also includes a discussion of rank and a number of citations of rank reversals attributed to a variety of factors ranging from intransitivity to procedure invariance, that are thought to be unexplained by Utility Theory with its underlying principle to always preserve rank. It is shown that when there is synergy due to the number of elements the AHP can be used to both preserve rank when it is desired to preserve it and allow it to reverse when it should reverse.

Relative Measurement and Its Generalization in Decision Making: Why Pairwise Comparisons are Central in Mathematics for the Measurement of Intangible Factors The Analytic Hierarchy/Network Process

According to the great mathematician Henri Lebesgue, making direct comparisons of objects with regard to a property is a fundamental mathematical process for deriving measurements. Measuring objects by using a known scale first then comparing the measurements works well for properties for which scales of measurement exist. The theme of this paper is that direct comparisons are necessary to establish measurements for intangible properties that have no scales of measurement. In that case the value derived for each element depends on what other elements it is compared with. We show how relative scales can be derived by making pairwise comparisons using numerical judgments from an absolute scale of numbers. Such measurements, when used to represent comparisons can be related and combined to define a cardinal scale of absolute numbers that is stronger than a ratio scale. They are necessary to use when intangible factors need to be added and multiplied among themselves and with tangible factors. To derive and synthesize relative scales systematically, the factors are arranged in a hierarchic or a network structure and measured according to the criteria represented within these structures. The process of making comparisons to derive scales of measurement is illustrated in two types of practical real life decisions, the Iran nuclear show-down with the West in this decade and building a Disney park in Hong Kong in 2005. It is then generalized to the case of making a continuum of comparisons by using Fredholm’s equation of the second kind whose solution gives rise to a functional equation. The Fourier transform of the solution of this equation in the complex domain is a sum of Dirac distributions demonstrating that proportionate response to stimuli is a process of firing and synthesis of firings as neurons in the brain do. The Fourier transform of the solution of the equation in the real domain leads to nearly inverse square responses to natural influences. Various generalizations and critiques of the approach are included.

Procedures for Synthesizing Ratio Judgements

Requirements which seem reasonable for functions synthesizing judgements (quantities or their ratios), in particular separability, associativity or bisymmetry, cancellativity, consensus, reciprocal or homogeneity properties are investigated and all functions satisfying them are determined.