Papers
This section presents a collection of AHP/ANP papers. Whether you are a student, academic and/or professional in the field of decision making, you can help us expand this collection by sharing your papers on decision making with the Analytic Hierarchy Process (AHP) and the Analytic Network Process (ANP).
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Paper results for keyword: EigenvectorNeurons the decision makers, Part II: The firings of many neurons and their density; the neural network its connections and field of firings
Thomas SaatyJournal: Neural Networks This paper is concerned with the firing of many neurons and the synthesis of these firings to develop functions and their transforms which relate chemical and electrical phenomena to the physical world. The density of such functions in the most general spaces that we encounter allows us to use li...
Read MoreContinuous pairwise comparisons
Thomas SaatyJournal: Fundamenta InformaticaeOne often assumes that comparisons are discrete and carried out in a matrix of numbers. However, our eyes and other senses perform comparisons in a continuous way by making many simultaneously. Here the mathematics of pairwise comparisons is generalized to the continuous case. It is more likely t...
Read MoreThe Analytic Hierarchy Process without the theory of oskar perron
Thomas SaatyJournal: International Journal of the Analytic Hierarchy ProcessIt is known and has been mathematically proven that the principal eigenvector is necessary for deriving priorities from judgments in the Analytic Hierarchy Process (AHP). According to the work of Oskar Perron, the principal eigenvector can be obtained as the limiting power of a positive matrix. I...
Read MoreOn the measurement of intengibles. A principal eigenvector approach to relative measurement derived from paired comparisons
Thomas SaatyJournal: Notices of the American Mathematical SocietyNearly all of us have been brought up to believe that clear-headed logical thinking is our only sure way to face and solve problems. But experience suggests that logical thinking is not natural to us. Indeed, we have to practice, and for a long time, before we can do it well. Since complex proble...
Read MoreFuzzy judgments and fuzzy sets
Thomas Saaty, Liem TranJournal: International Journal of Strategic Decision Sciences Using fuzzy set theory has become attractive to many people. However, the many references cited here and in other works, little thought is given to why numbers should be made fuzzy before plunging into the necessary simulations to crank out numbers without giving reason or proof that it works to ...
Read MoreDispersion of group judgments
Thomas Saaty, Luis VargasJournal: Mathematical and Computer ModellingTo achieve a decision with which the group is satisfied, the group members must accept the judgments, and ultimately the priorities. This requires that (a) the judgments be homogeneous, and (b) the priorities of the individual group members be compatible with the group priorities. There are three...
Read MorePriority as dominance in derived measurement: Invariance of the principal eigenvector
Mujgan Sagir Ozdemir, Thomas SaatyRanking is a process of prioritization. Priorities, as measurement rather than pure guessing, can be derived from paired comparison judgments that generalize on ratios of actual measurements. Paired comparisons involve the selection of the smaller of the two objects being compared as the unit and...
Read MoreDeriving the AHP 1-9 scale from first principles
Thomas SaatyJournal: Proceedings of the Sixth International Symposium on the Analytic Hierarchy ProcessWe demonstrate how the integers 1 to 9 used in the Fundamental Scale of the AHP to represent pairwise comparison judgments can be derived from stimulus-response theory. The conditions required for the stability of the eigenvector of priorities, known from the mathematics literature, are briefly m...
Read MoreRanking by eigenvector versus other methods in the Analytic Hierarchy Process
Thomas Saaty, G HuJournal: Applied Mathematics LettersCounter-examples are given to show that in decision making, different methods of deriving priority vectors may be close for every single pairwise comparison matrix, yet they can lead to different overall rankings. When the judgments are inconsistent, their transitivity affects the final outcome...
Read MoreEigenvector and logarithmic least squares
Thomas SaatyJournal: European journal of operational researchThe eigenvector method deals with two questions simultaneously, closeness and order. Together, they belong to the field topology of order. The metric idea of closeness is inadequate to judge what is a good approximation to data involving order relations. There is usually a condition which relates...
Read MoreComparison of eigenvalue, logarithmic least squares and least squares methods in estimating ratios
Thomas Saaty, Luis VargasJournal: Mathematical modellingThree methods—the eigenvalue, logarithmic least squares, and least squares methods—used to derive estimates of ratio scales from a positive reciprocal matrix are analyzed. The criteria for comparison are the measurement of consistency, dual solutions, and rank preservation. It is shown that the e...
Read MoreMathematical modeling of dynamic decisions; priorities and hierarchies with time dependence
Thomas SaatyJournal: Mathematics and computers in SimulationThe question often arises in regard to the use of Analytic Hierarchy Process: What would one do if the judgments where to change? A simple answer to that problem is that one should solve the new problem. But this is not what people usually have in mind. Presumably what they would like is a parame...
Read MoreThe forward and backward processes of conflict analysis
Joyce M. Alexander, Thomas SaatyJournal: Behavioral ScienceIn this study, the forward process of planning is applied to conflict resolution, by structuring a conflict according to the levels of a conceptual hierarchy. The parties to the conflict form the first level, their objectives are at the second level, and possible solutions are at the third and fi...
Read MoreA Systemic Rebuttal to the Criticism of Using the Eigenvector for Priority Assessment in Decision Making
Claudio Garuti, Valerio Salomon, Isabel SpencerJournal: Computacion y SistemasArguments 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 conditio...
Read More
Neurons the decision makers, Part II: The firings of many neurons and their density; the neural network its connections and field of firings
Thomas SaatyJournal: Neural Networks This paper is concerned with the firing of many neurons and the synthesis of these firings to develop functions and their transforms which relate chemical and electrical phenomena to the physical world. The density of such functions in the most general spaces that we encounter allows us to use li...
Read MoreContinuous pairwise comparisons
Thomas SaatyJournal: Fundamenta InformaticaeOne often assumes that comparisons are discrete and carried out in a matrix of numbers. However, our eyes and other senses perform comparisons in a continuous way by making many simultaneously. Here the mathematics of pairwise comparisons is generalized to the continuous case. It is more likely t...
Read MoreThe Analytic Hierarchy Process without the theory of oskar perron
Thomas SaatyJournal: International Journal of the Analytic Hierarchy ProcessIt is known and has been mathematically proven that the principal eigenvector is necessary for deriving priorities from judgments in the Analytic Hierarchy Process (AHP). According to the work of Oskar Perron, the principal eigenvector can be obtained as the limiting power of a positive matrix. I...
Read MoreOn the measurement of intengibles. A principal eigenvector approach to relative measurement derived from paired comparisons
Thomas SaatyJournal: Notices of the American Mathematical SocietyNearly all of us have been brought up to believe that clear-headed logical thinking is our only sure way to face and solve problems. But experience suggests that logical thinking is not natural to us. Indeed, we have to practice, and for a long time, before we can do it well. Since complex proble...
Read MoreFuzzy judgments and fuzzy sets
Thomas Saaty, Liem TranJournal: International Journal of Strategic Decision Sciences Using fuzzy set theory has become attractive to many people. However, the many references cited here and in other works, little thought is given to why numbers should be made fuzzy before plunging into the necessary simulations to crank out numbers without giving reason or proof that it works to ...
Read MoreDispersion of group judgments
Thomas Saaty, Luis VargasJournal: Mathematical and Computer ModellingTo achieve a decision with which the group is satisfied, the group members must accept the judgments, and ultimately the priorities. This requires that (a) the judgments be homogeneous, and (b) the priorities of the individual group members be compatible with the group priorities. There are three...
Read MorePriority as dominance in derived measurement: Invariance of the principal eigenvector
Mujgan Sagir Ozdemir, Thomas SaatyRanking is a process of prioritization. Priorities, as measurement rather than pure guessing, can be derived from paired comparison judgments that generalize on ratios of actual measurements. Paired comparisons involve the selection of the smaller of the two objects being compared as the unit and...
Read MoreDeriving the AHP 1-9 scale from first principles
Thomas SaatyJournal: Proceedings of the Sixth International Symposium on the Analytic Hierarchy ProcessWe demonstrate how the integers 1 to 9 used in the Fundamental Scale of the AHP to represent pairwise comparison judgments can be derived from stimulus-response theory. The conditions required for the stability of the eigenvector of priorities, known from the mathematics literature, are briefly m...
Read MoreRanking by eigenvector versus other methods in the Analytic Hierarchy Process
Thomas Saaty, G HuJournal: Applied Mathematics LettersCounter-examples are given to show that in decision making, different methods of deriving priority vectors may be close for every single pairwise comparison matrix, yet they can lead to different overall rankings. When the judgments are inconsistent, their transitivity affects the final outcome...
Read MoreEigenvector and logarithmic least squares
Thomas SaatyJournal: European journal of operational researchThe eigenvector method deals with two questions simultaneously, closeness and order. Together, they belong to the field topology of order. The metric idea of closeness is inadequate to judge what is a good approximation to data involving order relations. There is usually a condition which relates...
Read MoreComparison of eigenvalue, logarithmic least squares and least squares methods in estimating ratios
Thomas Saaty, Luis VargasJournal: Mathematical modellingThree methods—the eigenvalue, logarithmic least squares, and least squares methods—used to derive estimates of ratio scales from a positive reciprocal matrix are analyzed. The criteria for comparison are the measurement of consistency, dual solutions, and rank preservation. It is shown that the e...
Read MoreMathematical modeling of dynamic decisions; priorities and hierarchies with time dependence
Thomas SaatyJournal: Mathematics and computers in SimulationThe question often arises in regard to the use of Analytic Hierarchy Process: What would one do if the judgments where to change? A simple answer to that problem is that one should solve the new problem. But this is not what people usually have in mind. Presumably what they would like is a parame...
Read MoreThe forward and backward processes of conflict analysis
Joyce M. Alexander, Thomas SaatyJournal: Behavioral ScienceIn this study, the forward process of planning is applied to conflict resolution, by structuring a conflict according to the levels of a conceptual hierarchy. The parties to the conflict form the first level, their objectives are at the second level, and possible solutions are at the third and fi...
Read MoreA Systemic Rebuttal to the Criticism of Using the Eigenvector for Priority Assessment in Decision Making
Claudio Garuti, Valerio Salomon, Isabel SpencerJournal: Computacion y SistemasArguments 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 conditio...
Read More