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1987, Mathematical Modelling
The first axiom of the Analytic Hierarchy Process (AHP) states that the pairwise comparison judgments are reciprocal. Here we explore the relationships between the reciprocal property and preference relations with and without the axiom of transitivity and point out some important distinctions between utility theory and the AHP.
It is a fact that people make decisions and have been making decisions for a very long time. Contrary to what some of us who are interested in decision-making may like to believe, most people do not take seriously the existence of theories which purport to set their thinking and feeling right. They claim to know their own value system and what they want. They may wonder how anyone else can know well enough to tell them how best to organize their thinking in order to make better choices. Yet, research has shown that complex decisions are beyond the capacity of the brain to synthesize intuitively and efficiently. Since decision making is a natural characteristic of people, how do we describe what they do so that an ordinary mortal can understand what we are saying? We do not wish to legislate the method with which people should make decisions, but only to describe it even when it is prescribed by some method. In the process, we may learn things that can help people make better decisions. How?
Omega, 1995
The relationship between MAUT and AHP for three-level hierarchic structures is demonstrated based on a common framework interpreting multiattribute decision making techniques. A theorem showing the conditions under which two multiattribute decision making techniques result in a consistent preference structure is proved. It can be justified based on this theorem that the rank reversal problem in AHP resulting from the addition or deletion of alternatives is caused by multiplying inappropriate criteria priorities with local priorities for alternatives. A scaling method, AHW, is introduced synthesizing judgments from pairwise comparisons within and among criteria into value scales in MAUT.
Proceedings of the International Symposium on the Analytic Hierarchy Process, 1996
Arrow's Impossibility Theorem says that it is generally impossible to derive a rational group choice from ordinal comparisons made by the individual members. This paper demonstrates that, at appropriate consistency levels, and with the use of judgments on a cardinal scale, the Analytic Hierarchy Process negates Arrow's impossibility. Arrow's conditions are satisfied when aggregation is done at the judgment level when individual judgments are consistent, and at the priority level when they are near consistent.
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.
In this paper, we reexamine the axiomatic foundation of prioritarianism – a distributive ethical view originating from Derek Parfit (1991) that claims that “[b]enefiting people matters more the worse off these people are” (Parfit 1991: 19). In previous work, prioritarianism has been characterized by the following five axioms: Pigou-Dalton, Separability, Anonymity, Pareto, and Continuity. Among these axioms, many scholars have regarded Pigou-Dalton (along with Separability) as the key defining feature that distinguishes prioritarianism from other continuous welfarist views. We disagree: not because we think the Pigou-Dalton principle is incompatible with prioritarianism (it is), but because the Pigou-Dalton principle fails to distinguish prioritarianism from telic egalitarianism, which is what motivated Parfit to present prioritarianism as an alternative view of distributive ethics in the first place. Instead, we propose a new axiom, which we call “Priority,” which clearly expresses Parfit’s original prioritarian idea, as the main defining property of prioritarianism, and offer a new axiomatic characterization of prioritarianism in terms of this new axiom. We then analyze the precise logical relationships between Priority and the other axioms. Finally, we explore the important issue of measurability and interpersonal comparison of well-being in relation to prioritarianism. There have criticisms that the prioritarian social welfare function may not satisfy some information invariance property with respect to measurability and interpersonal comparability of well-being. It turns out that, compared to other social welfare orderings (such as utilitarianism, maximin, leximin, the general Gini ordering, etc.), prioritarianism may require a stronger well-being measure (viz., a translation-scale or a ratio-scale) than a cardinal measure with full interpersonal comparability to retain its normative and theoretical significance. From such observations, we specify the class of prioritarian social welfare functions free from this criticism.
Here we introduce the Analytic Hierarchy Process as a method of measurement with ratio scales and illustrate it with two examples. We then give the axioms and some of the central theoretical underpinnings of the theory. Finally, we discuss some of the ideas relating to this process and its ramifications. In this paper we give special emphasis to departure from consistency and its measurement and to the use of absolute and relative measurement, providing examples and justification for rank preservation and reversal in relative measurement.
ISAHP proceedings, 1996
The Analytic Hierarchy Process (AHP) is often used in group settings where group members engage in discussion and arrive at consensus judgments. Alternatively, each member of the group can make individual judgments. This paper considers alternative ways to aggregate individual judgments. We show why geometric, rather than arithmetic means be usedi We also consider that the method used to aggregate priorities or judgments depends on the nature of the group.
Proceedings of the International Symposium on the Analytic Hierarchy Process, 1999
The seven pillars of the AHP, some highlights of which are discussed in the paper, are: 1) ratio scales derived from reciprocal paired comparisons; 2) paired comparisons and the psychophysical origin of the fundamental scale used to make the comparisons; 3) conditions for sensitivity of the eigenvector to changes in judgments; 4) homogeneity and clustering to extend the scale from 1-9 to 1-Go; 5) additive synthesis of priorities, leading to a vector of multi-linear forms as applied within the decision structure of a hierarchy or the more general feedback network to reduce multi-dimensional measurements to a uthdimensional ratio scale; 6) allowing rank preservation (ideal mode) or allowing rank reversal (distributive mode); and 7) group decision making using a mathematically justifiable way for synthesizing individual judgments which allows the construction of a cardinal group decision compatible with the individual preferences.
The Analytic Hierarchy Process (AHP) and its generalization to dependence and feedback, the Analytic Network Process (ANP), are theories of relative measurement of intangible criteria. With this approach to relative measurement, a scale of priorities is derived from pairwise comparison measurements only after the elements to be measured are known. The ability to do pairwise comparisons is our biological heritage and we need it to cope with a world where everything is relative and constantly changing. In traditional measurement one has a scale that one applies to measure any element that comes along that has the property the scale is for, and elements are measured one by one, not by comparing them with each other. In the AHP paired comparisons are made with judgments using numerical values taken from the AHP absolute fundamental scale of 1-9. A scale of relative values is derived from all these paired comparisons and it also belongs to an absolute scale that is invariant under the identity transformation like the system of real numbers. The AHP/ANP is useful for making multicriteria decisions involving benefits, opportunities, costs and risks. The ideas are developed in stages and illustrated with examples of real life decisions. The subject is transparent and despite some mathematics, it is easy to understand why it is done the way it is along the lines discussed here.
The analytic hierarchy process (AHP) is a decision-making procedure widely used in management for establishing priorities in multicriteria decision problems. Underlying the AHP is the theory of ratio-scale measures developed in psychophysics since the middle of the last century. It is, however, well known that classical ratio-scaling approaches have several problems. We reconsider the AHP in the light of the modern theory of measurement based on the so-called separable representations recently axiomatized in mathematical psychology. We provide various theoretical and empirical results on the extent to which the AHP can be considered a reliable decision-making procedure in terms of the modern theory of subjective measurement.
ISAHP proceedings, 1994
Expert Systems with Applications, 2013
Representing and reasoning over different forms of preferences is of crucial importance to many different fields, especially where numerical comparisons need to be made between critical options. Focusing on the well-known Analytical Hierarchical Process (AHP) method, we propose a two-layered framework for addressing different kinds of conditional preferences which include partial information over preferences and preferences of a lexicographic kind. The proposed formal two-layered framework, called CS-AHP, provides the means for representing and reasoning over conditional preferences. The framework can also effectively order decision outcomes based on conditional preferences in a way that is consistent with well-formed preferences. Finally, the framework provides an estimation of the potential number of violations and inconsistencies within the preferences. We provide and report extensive performance analysis for the proposed framework from three different perspectives, namely time-complexity, simulated decision making scenarios, and handling cyclic and partially defined preferences.
Mathematical and Computer Modelling, 1993
Here we study the problem of determining the ranking of the alternatives that one should infer when decision makers use interval judgments rather than point estimates in the Analytic Hierarchy Process. How many rankings could one infer from the matrix of interval judgments, and which is the most likely to be selected?
Management Science, 1990
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International Journal of Operational Research, 2009
We provide an explicit derivation of a new aggregation rule that naturally incorporates the relative strength of normalisation with respect to different criteria. The proposed aggregation procedure avoids unwanted rank reversals in the additive Analytic Hierarchy Process. In the case of absolute measure, the strength of normalisation is manifested by the priority vector's norm that can be interpreted as the unit of measure. However, when we have only relative judgements, norms are unknown and thus the decision maker has to estimate the relative strength of normalisation. Simultaneously, the new aggregation mechanism allows acceptable rank reversals if imperfections of human behaviour are recognised.
2008
Arguments have been provided against the use of the eigenvector as the operator that derives priorities. A highlight of the arguments is that the eigenvector solution does not always respect the condition of ordinal preference (COP) based on the decision maker's judgments. While this condition may be reasonable when dealing with measurable concepts (such as distance or time) that lead to consistent matrices, it is questionable whether it is to be expected in all situations, particularly when the information provided by the decision maker is not fully consistent. The judgments that lead to inconsistency may also contain valuable information that must be considered in the priority assessment process as well. By the other hand, the analytic hierarchy process (AHP) use the eigenvector operator to derive the priorities that represent the cardinal decision maker preferences from a pairwise comparison matrix, which do not always respect the COP condition. The AHP and still deeper the ANP (the mathematical generalization of AHP) start from concepts of ordinal metric of dominance and system theory, which is well supported by graph theory and ordinal topology with the Cesaro sum as its fundamental pillar to build metric of dominance. These mathematic concepts has no relation with COP preservation moreover, this two way of thinking are in a course of collision since the second (COP) inhibit the first (Cesaro sum). Systems theory claims that the whole is more than its standalone components, and that internal relationships provide additional information as well. Given that the pairwise comparison matrix is an interrelated system and not just a collection of standalone judgments, we plan to show that the eigenvector, because it is a systemic operator, is the most suitable to represent and capture the behavior of the whole system and its emerging properties.
IJSRD, 2013
This article presents a review of the applications of Analytic Hierarchy Process (AHP). AHP is a multiple criteria decision-making tool that has been used in almost all the applications related with decision-making. Decisions involve many intangibles that need to be traded off. 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 also included.
1994
Some authors have proposed that the Analytic Hierarchy Process (AHP) axiom of independence be relaxed to accommodate observations drawn from: a) examples of pairwise comparisons of alternatives in clusters in single criterion AHP problems, and b) examples of problems in which the criteria have the same underlying measurement, and both the achievement of the goal and the alternatives are measured objectively. We show that the illustrations given are actually single criterion problems according to the AHP. Thus the AHP axiom of independence is inapplicable in both situations and therefore not violated. We also consider the consequence of failure to distinguish between a criterion as an attribute of alternatives and a cluster of alternatives, the two being different in the hierarchic structure. Finally, we discuss transformable problems, which look like multicriteria problems but are actually single criterion problems and how failure to recognize this fact may lead to incorrect syntheses and false conclusions.
Mathematical Modelling, 1987
Theories that attempt to represent decision makers' preferences are usually based on the axiom of transitivity. On the other hand, Saaty's Eigenvalue Method of deriving ratio scales from pairwise comparisons is based on the assumption that the numerical preferences satisfy the reciprocal property. We show that this property cannot be derived from the usual set of axioms used to obtain order preserving value functions, but that they in turn are consequences of the reciprocal property.
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