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A marijuana legalization model using benefits, opportunities, costs and risks (BOCR) analysis
Thomas Saaty Journal: International Journal of Strategic Decision Sciences Marijuana has been regarded as an evil in our society because of abuse and addiction. However, its legalization and use in several states have shown that from the standpoint of its benefits, opportunities, costs and risks (BOCR) the overall decision should be to legalize it in our greater society... Read More |
A measure of world influence
Thomas Saaty, Mohamad W. Khouja Journal: Journal of Peace Science It is not unusual to think of and write about power without adequate definition. Power is such a rich concept that even for special use it would carry more meaning than maybe intended. Power is closely identified with the ability to do something. Influence is the capacity to sway others to obtain... Read More |
A Model for the Control of Arms
Thomas Saaty Journal: Operations Research As the problems and ideas of arms control and disarmament have gained substantial momentum in recent years, the need has developed to look into the refinements of disarmament questions within the framework of national security. Here we have sought to initiate the development of a framework within... Read More |
A model of neural impulse firing and synthesis
Thomas Saaty, Luis Vargas Journal: Journal of Mathematical Psychology Neurons are decision makers that decide from instant to instant whether to fire or not to fire based on information received through neurotransmitter electric charge. In firing they accomplish two goals. First, they pass information to other neurons which in their turn make a decision to fire or ... Read More |
A natural way to make momentous decisions
Thomas Saaty Journal: Journal of scientific & industrial research A brief introduction to making decisions by the Analytic Hierarchy Process (AHP) is given. A simple illustration of how to decide on which of three people should get a heart transplant is provided. How to structure a decision problem and how to enter judgments semantically, transform them to pair... Read More |
A new approach to performance measurement the Analytic Hierarchy Process
Eren Erdener, Thomas Saaty Journal: Design Methods and Theories A new approach to performance measurement the Analytic Hierarchy Process Read More |
A new approach to the middle east conflict: the Analytic Hierarchy Process
Thomas Saaty, Luis Vargas We present an alternative process to address the Israeli Palestinian conflict. It does so in two ways that are different from past efforts. The first is by formally structuring the conflict and the second is the manner in which discussions are conducted and conclusions drawn. Read More |
A new macroeconomic forecasting and policy evaluation method using the Analytic Hierarchy Process
Thomas Saaty Journal: Mathematical Modelling The Analytic Hierarchy Process is used to show how forecasts can be made of the effects of monetarist, Keynesian and supply-side macroeconomic policies and to determine their impact on important variables such as unemployment, inflation and GNP growth. Read More |
A Nonlinear Programming Model in Optimum Communication Satellite Use
Thomas Saaty Journal: SIAM Review In this note we formulate the problem of allocating communication needs of different cities via relay-type satellites. Our model turns out to be a dynamic (over successive time periods) quadratic nonlinear program
requiring integer solutions. Such solutions have been recently studied by Kunzi a... Read More |
A note on the AHP and expected value theory
Thomas Saaty Journal: Socio-Economic Planning Sciences It is shown here that one cannot simply take columns of numbers, normalize them and add to obtain results corresponding to operations in the AHP. This is what traditional expected value theory using a single scale would lead one to do. Care needs to be exercised. What one must do is to interpret ... Read More |
A ratio scale metric and the compatibility of ratio scales: The possibility of arrow's impossibility theorem
Thomas Saaty Journal: Applied Mathematics Letters We develop a metric for ratio scales and explore the notion of compatibility of two sets of measurements of a set of objects or properties on a ratio scale. We briefly address Arrow's impossibility theorem. We maintain that it is not as impossible as claimed when, as in reality, a certain degree ... Read More |
A scaling method for priorities in hierarchical structures
Thomas Saaty Journal: Journal of mathematical psychology 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 ... Read More |
A structured scientific solution to the Israeli–Palestinian conflict: the Analytic Hierarchy Process approach
Thomas Saaty, Luis Vargas, H. J. Zoffer Journal: Decision Analytics While the Israeli–Palestinian conflict has raged for decades, in all of its ramifications there has never been a totally structured or scientific approach to the conflict with all of its details. The Analytic Hierarchy Process (AHP) approaches the problem along these lines. There are a plethora o... Read More |
A theory of analytical hierarchies applied to political candidacy
Thomas Saaty, J. P. Bennett Journal: Behavioral Science A theory of hierarchies is applied to structure complex decision problems about groups, organizations, societies or supranational systems in three steps. The problem is decomposed into a number of strata, each with several elements which may be people, variables, policies, and so on. We next anal... Read More |
About a hundred years of creativity in decision making
Thomas Saaty Journal: International Journal of the Analytic Hierarchy Process Can all the methods people invented to do decision making be right all the time? Maybe for making trivial decisions, but not in the case of those that involve dependence and feedback and need complete analysis of benefits, opportunities, costs and risks to get an overall justifiable outcome. Read More |
Absolute and relative measurement with the AHP. The most livable cities in the United States
Thomas Saaty Journal: Socio-Economic Planning Sciences In this paper it is shown that there are two types of measurement involved in the AHP, absolute and relative. The first requires a standard with which to compare elements, but mostly alternatives at the bottom of the hierarchy. The process leads to absolute preservation in the rank of the alterna... Read More |
Addressing with brevity criticism of the Analytic Hierarchy Process
Thomas Saaty, Luis Vargas, Rozann Whitaker Journal: International Journal of the Analytic Hierarchy Process The paper provides an overview that covers the main criticisms of the AHP and the authors replies to them. Because there have been many papers that reply to criticisms, the thrust here is to classify them and reply to them briefly in each category without giving lengthy repetitions of what is alr... Read More |
An Analytic Hierarchy Process model of group consensus
Qingxing Dong, Thomas Saaty Journal: Journal of Systems Science and Systems Engineering In group decision making, a certain degree of consensus is necessary to derive a meaningful and valid outcome. This paper proposes a consensus reaching model for a group by using the Analytic Hierarchy Process (AHP). It supports people to improve their group consensus level through an updating of... Read More |
An Analytic Network Process model for financial-crisis forecasting
Michael Niemira, Thomas Saaty Journal: International Journal of Forecasting We discuss and develop an imbalance-crisis turning point model to forecast the likelihood of a financial crisis based on an Analytic Network Process framework. The Analytic Network Process (ANP) is a general theory of relative measurement used to derive composite-priority-ratio scales from indivi... Read More |
An essay on how judgment and measurement are different in science and in decision making
Thomas Saaty Journal: International Journal of the Analytic Hierarchy Process In decision making the priority scales are derived objectively after subjective judgments are made, and they reflect the importance of the influences we considered. The process is the opposite of what is done in the physical sciences where the subjectivity of interpreting the final number comes a... Read More |
An essay on rank preservation and reversal
Thomas Saaty, Mujgan Sagir Ozdemir Journal: Mathematical and Computer Modelling Rank preservation and reversal, so fundamental in decision making, have been an unresolved issue in the field of economics and utility theory and came into focus when the Analytic Hierarchy Process was developed because it uses paired comparisons that inevitably make the priorities of the alterna... Read More |
An exposition of the AHP in reply to the paper "remarks on the Analytic Hierarchy Process"
Thomas Saaty Journal: Management science An exposition of the AHP in reply to the paper " remarks on the analytic hierarchy process" Read More |
An Indicator of One’s Life Satisfaction
Thomas Saaty, H. J. Zoffer, Lirong Wei Journal: European Journal of Pure and Applied Mathematics How satisfied is each of us with the fulfillment of his life? In this paper after a thorough search of the literature about satisfaction, 58 criteria and subcriteria related to satisfaction or fulfillment were identified and arranged in a hierarchic structure. A process of prioritization known as... Read More |
An innovative orders-of-magnitude approach to AHP-based mutli-criteria decision making: Prioritizing divergent intangible humane acts
Thomas Saaty, Jennifer Shang Journal: European Journal of Operational Research An innovative Analytic Hierarchy Process-based structure is developed to capture the relationship between various levels of activities contributed by people to society. Physical objects have widespread extension and degrees of importance that often differ by many orders of magnitude. Similarly, m... Read More |
An objective approach to faculty promotion and tenure by the Analytic Hierarchy Process
Thomas Saaty Journal: Research in Higher Education This paper proposes a novel evaluation approach for the selection of candidates for promotion and/or tenure. The decision problem is conceptualized as a hierarchy of factors and a mathematical procedure known as the Analytic Hierarchy Process (AHP) is used for successively “weighting” or prioriti... Read More |
Applications of analytical hierarchies
Thomas Saaty Journal: Mathematics and Computers in Simulation In a previous article which appeared in this journal I outlined some of the ideas involved in measuring
priorities in hierarchical systems. It was my hope to show that when dealing with complexity, a hierarchical structure with ratio scale measurement provides a natural expression of the mind to... Read More |
Applications of the Analytic Hierarchy Process to long range planning processes
James Emshoff, Thomas Saaty Journal: European Journal of Operational Research This paper considers the long range planning process from the point of view both of projecting forward likely or desired changes from the current position to define a possible future state, and of identifying desirable future states and working backwards from those to consider ways in which they ... Read More |
Architectural Design by the Analytic Hierarchy Process
Thomas Saaty, Miguel H. Beltran Journal: Design Methods and Theories The Analytic Hierarchy Process (AHP), which is explained and applied in another article in the same journal, is used to illustrate the architectural design of a house for a family of three. The plan of the house was develop to satisfy the family's needs by considering the size of the lot, the siz... Read More |
Automatic decision-making: Neural firing and response
Thomas Saaty Journal: Journal of Systems Science and Systems Engineering In contrast with conscious decision-making, there are numerous subconscious decisions that we make without thinking about them. Some are biological and are made by different parts of our body to keep it alive and functioning normally. Others are a result of repetition and training that we can the... Read More |
Axiomatic foundation of the Analytic Hierarchy Process
Thomas Saaty Journal: Management science This paper contains an axiomatic treatment of the Analytic Hierarchy Process (AHP). The set of axioms corresponding to hierarchic structures are a special case of axioms for priority setting in systems with feedback which allow for a wide class of dependencies. The axioms highlight: (1) the recip... Read More |
Axioms of the Analytic Hierarchy Process (AHP) and its Generalization to Dependence and Feedback: The Analytic Network Process (ANP)
Thomas Saaty, Konrad Kułakowski Journal: eprint arXiv:1605.05777 The AHP/ANP are multicriteria decision-making theories that deal with both hierarchic structures when the criteria are independent of the alternatives and with networks when there is any dependence within and between elements of the decision. Both of them have been repeatedly used in practice by ... Read More |
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Basic theory of the Analytic Hierarchy Process: How to make a decision
Thomas Saaty Journal: Revista de la Real Academia de Ciencias Exactas Fisicas y Naturales An assumption arising from the practice of science and engineering since the middle ages is that because nature is physical, we should be able to relate all measurement to physical dimensions. But that is not true. Human thinking and feeling exist in the physical world but they are not matter or ... Read More |
Better world through better decision making
Thomas Saaty Journal: Symp. Anal. Hierarchy Process To change this world from its primitive habit of using words and language to arrive at decisions by making tradeoffs which require the use of numbers, we need to work hard to promote this way of thinking to teach the whole world and its politicians how to measure the intangibles involved in the d... Read More |
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Center for conflict resolution
Thomas Saaty Journal: Bulletin of the Atomic Scientists There are many organizations in the world today that are devoted to studying arms control and conflict resolution. None is really able to provide direct assistance to conflicting parties. what is needed is an international center for Conflict Resolution, located in a neutral territory, that will ... Read More |
Choosing the best city of the future
Thomas Saaty, Mujgan Sagir Ozdemir Journal: Journal of Urban Management This paper describes various possibilities of the cities of futures considering various constraints and demand of society, environment and geography. The need for future cities arises because of the rapid growth in population and thereby causing a decline in the living standards. In the United St... Read More |
Coefficient perturbation of a constrained extremum
Thomas Saaty Journal: Operations Research A schedule of allocating labor (in a shipping operation) whose available amount is a function of time, to different tasks, in order to minimize the total cost, is given. The problem is cast in linear-programming form in which all the coefficients are parameterized. An illustration is given follow... Read More |
Comparison of eigenvalue, logarithmic least squares and least squares methods in estimating ratios
Thomas Saaty, Luis Vargas Journal: Mathematical modelling Three 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 More |
Conflict resolution and the falkland islands invasions
Thomas Saaty Journal: Interfaces The analytic hierarchy process (AHP) is compared with game and metagame theories for analyzing conflicts. The AHP helps to structure the problem, generate and synthesize judgments, and carry out a benefit/cost analysis of various policies. The Falkland Islands crisis is examined using the judgmen... Read More |
Continuous pairwise comparisons
Thomas Saaty Journal: Fundamenta Informaticae One 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 More |
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Decision making with the Analytic Hierarchy Process
Thomas Saaty Journal: International journal of services sciences 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 thro... Read More |
Decision making with the Analytic Hierarchy Process
Thomas Saaty Journal: Scientia Iranica In this paper, an introduction is given to decision-making theory, the Analytic Hierarchy Process. In the AHP, a decision hierarchy is constructed with a goal, criteria and alternatives. The criteria are pairwise compared for their importance with respect to the goal to derive a scale of relative... Read More |
Decision Making with The Analytic Network Process (ANP) and Its Super-Decisions
Thomas Saaty Journal: Software The National Missile Defence (NMD) Example There are three phases in the creation of the structure of a complex decision, 1) the subjective personal, group or corporate values, 2) the interface- merits of the decision between the first (the subjective), and the third (the objective) phases: Benefits, opportunities, costs and risks and the... Read More |
Decision making, new information, ranking and structure
Thomas Saaty Journal: Mathematical Modelling The rank of a set of alternatives can change if a new criterion is introduced into the set of criteria, but it can also change if the importances of the criteria depend on the number of alternatives and on the strength of their ranking. As a result a new alternative may change the relative order ... Read More |
Decision making, scaling, and number crunching
Thomas Saaty Journal: Decision Sciences The purpose of this note is to comment on the artificiality of using known scales of measurement (or conveniently improvised numbers) to make decisions. In particular, the pitfalls of trying to please the decision maker and of using normalization as a number crunching tool are discussed. Finally,... Read More |
Decision making—the Analytic Hierarchy and Network Processes (AHP/ANP)
Thomas Saaty Journal: Journal of systems science and systems engineering This is the first part of an introduction to multicriteria decision making using the Analytic Hierarchy Process (AHP) and its generalization, the Analytic Network Process (ANP). The discussion involves individual and group decisions both with the independence of the criteria from the alternatives... Read More |
Decision-making with the AHP: Why is the principal eigenvector necessary
Thomas Saaty Journal: European journal of operational research 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... Read More |
Decisions, structure, and natural law
Thomas Saaty Journal: International Journal of the Analytic Hierarchy Process The object here is to show that our thinking processes and our physical forms and those of all things that exist, are a result of response in nature to influences as stimuli, brought about by natural occurrences. The ideas are developed through a generalization of the role judgment plays in decis... Read More |
Dependence and independence: From linear hierarchies to nonlinear networks
Thomas Saaty, Masahiro Takizawa Journal: European journal of operational research Most decisions need to be free from assumptions of independence to be faithful to the complex problems in which they arise. This paper illustrates how to generate priorities for decisions involving general types of dependence of criteria on alternatives, criteria on criteria and alternatives on a... Read More |
Deriving the AHP 1-9 scale from first principles
Thomas Saaty Journal: Proceedings of the Sixth International Symposium on the Analytic Hierarchy Process We 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 More |
Diagnosis with dependent symptoms: Bayes theorem and the Analytic Hierarchy Process
Thomas Saaty, Luis Vargas Journal: Operations Research Judgments are needed in medical diagnosis to determine what tests to perform given certain symptoms. For many diseases, what information to gather on symptoms and what combination of symptoms lead to a given disease are not well known. Even when the number of symptoms is small, the required numbe... Read More |
Dirac distributions and threshold firing in neural networks
Thomas Saaty, Luis Vargas Journal: Applied Mathematics Letters We extend our earlier work on positive reciprocal kernels of Fredholm integral operators [9] to study firings and their synthesis in neural networks. First, we show that, in general, neural response in
both spontaneous and non-spontaneous firing give rise to generalized functions of the Dirac t... Read More |
Discrete search problem in pattern recognition
Thomas Saaty Journal: IEEE Transactions on Information Theory Suppose we have number of sensors identified by indexes 1 to N each related to a source that is either active or inactive. The set of sensors is sampled in parallel at regular intervals of wine and a record is made of the output of only one of the sensors, randomly chosen from the set if more tha... Read More |
Dispersion of group judgments
Thomas Saaty, Luis Vargas Journal: Mathematical and Computer Modelling To 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 More |
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Economic forecasting with tangible and intangible criteria
Thomas Saaty Journal: Economic Horizons This paper provides a summary of a mathematical theory about the use of expert judgments in paired comparisons and how to derive priorities from them particularly when intangible factors are involved. An example to validate the process when applied to tangibles is given along with a simple decisi... Read More |
Eigenvector and logarithmic least squares
Thomas Saaty Journal: European journal of operational research The 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 More |
Estimating technological coefficients by the Analytic Hierarchy Process
Thomas Saaty, Luis Vargas Journal: Socio-Economic Planning Sciences We give a summary and an example of a new, systems oriented, method for estimating the input-output coefficients of a given economy. Our approach is based on pairwise comparisons among the sectors of the economy ranking them according to their priority on a ratio scale. What we obtain corresponds... Read More |
Evaluating and Optimizing Technological Innovation Efficiency of Industrial Enterprises Based on Both Data and Judgments
Wei Gu, Thomas Saaty, Lirong Wei Journal: International Journal of Information Technology & Decision Making Technological innovation as one of the most important competitive strategies for companies has attracted the attentions of companies and governments. In this paper, we present an evaluation method based on data and judgments to rank the technological innovation capability and technological innova... Read More |
Experiments on rank preservation and reversal in relative measurement
Thomas Saaty, Luis Vargas Journal: Mathematical and computer modelling We show through simulation that three methods of scaling, distributive (uniqueness is important), ideal (uniqueness is not important), and utility (use of interval scales for the ideal) modes, yield the same ranking of alternatives with surprisingly high frequency, except for the case of copies o... Read More |
Expert System for Ice Hockey Game Prediction: Data Mining with Human Judgment
Wei Gu, Thomas Saaty, Rozann Whitaker Journal: International Journal of Information Technology & Decision Making This paper describes an expert system to predict National Hockey League (NHL) game outcome. A new method based on both data and judgments is used to estimate the hockey game performance. There are many facts and judgments that could influence an outcome. We employed the support vector machine to ... Read More |
Exploring optimization through hierarchies and ratio scales
Thomas Saaty Journal: Socio-Economic Planning Sciences This paper explores the concept of optimization by solely using the AHP and compares outcomes with those obtained in traditional optimization theory without and with constraints. The difference is essentially in the absence of the traditional black box involving complex manipulations in algebra o... Read More |
Exploring the interface between hierarchies, multiple objectives and fuzzy sets
Thomas Saaty Journal: Fuzzy sets and systems The complexity of experience acquired through our senses and as interpreted by our mind, is fuzzy and must remain so as long as the meaning of things change as they are embedded in larger or different contexts to relate them to new ideas and new experiences. Here we give a method for measuring th... Read More |
Extending the measurement of tangibles to intangibles
Thomas Saaty, Mujgan Sagir Ozdemir Journal: International Journal of Information Technology & Decision Making Tangibles have measurements generally on ratio scales with arbitrary units that are always interpreted by using judgments as to what particular purpose the measurements serve. How two measurements on a ratio scale are related with respect to dominance leads to forming their ratio which is a dimen... Read More |
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Financial and intangible factors in fleet lease or buy decision
Luis Vargas, Thomas Saaty Journal: Industrial Marketing Management We show that there is no single best solution in the leasing versus company ownership problem for the fleet administration industry as a whole. Because of the different styles and traditions of firms and because of the emphasis on intangibles beyond economics, a unifying framework of reference in... Read More |
Five Papers by Conny Palm
Thomas Saaty Journal: Operations Research Information from The Royal Board of Swedish Tele-Communications, edited by E. MALMGREN, contains five papers by Conny Palm prefaced by a short article, "Research on Telephone Traffic Carried by Full Availability Groups." It would be desirable to say more regarding these useful papers, but... Read More |
Five Ways to Combine Tangibles with Intangibles
Thomas Saaty Journal: International Journal of the Analytic Hierarchy Process This paper presents five different ways to establish weights for the criteria that govern making comparisons. Four of these can be done in the context of the AHP, but the fifth and most reliable one is obtained by using the ANP. Read More |
Forecasting foreign exchange rates: an expert judgement approach
Andrew Blair, Robert Nachtmann, Josephine Olson, Thomas Saaty Studies have indicated that forecasts by market experts can be more accurate than time series forecasts. This article describes a process for structuring an expert foreign exchange forecast using Saaty's Analytic Hierarchy Process (AHP). The specific example developed is a forecast of the yen/dol... Read More |
Forecasting the resurgence of the US economy in 2001: an expert judgment approach
Andrew Blair, Gershon Mandelker, Thomas Saaty, Rozann Whitaker Journal: Socio-Economic Planning Sciences This paper describes a forecast of the date for the resumption of growth of the US economy in 2001. It uses an expert judgment approach within the framework of decision theory, the Analytic Hierarchy Process, as well as its generalization to dependence and feedback, the Analytic Network Process. Read More |
Forecasting the resurgence of the US economy in 2010: An expert judgment approach
Andrew Blair, Gershon Mandelker, Thomas Saaty, Rozann Whitaker Journal: Socio-Economic Planning Sciences This paper describes a forecast, performed in December 2008, of the time of the recovery of the U.S. economy from the contraction that began in December 2007. As in two earlier papers, the forecast uses an expert judgment approach, the Analytic Hierarchy Process (AHP), within the framework of dec... Read More |
Foresight—Insight—Hindsight
James C. Frauenthal, Thomas Saaty Journal: Two-Year College Mathematics Journal Many mathematical models which attempt to describe problems stated in words appear trivial in hindsight. This is the case because the crucial insight needed to formulate the model is divulged by the structure of the model. It would be desirable to be able to teach this critical, inductive step in... Read More |
Free trade discussions between Canada and the United States
Thomas Saaty, William Wedley Journal: Canadian Journal of Administrative Sciences/Revue Canadienne des Sciences de l'Administration The free trade negotiations between Canada and the United States are a type of conflict where each side considers not only its own gains and losses but also Us perceptions of the gains and losses of the other side. The assessment involves intangibles and tradeoffs which can be structured and anal... Read More |
Fundamentals of the Analytic Network Process—Dependence and feedback in decision-making with a single network
Thomas Saaty Journal: Journal of Systems science and Systems engineering The Analytic Network Process (ANP) is a multicriteria theory of measurement used to derive relative priority scales of absolute numbers from individual judgments (or from actual measurements normalized to a relative form) that also belong to a fundamental scale of absolute numbers. These judgment... Read More |
Fundamentals of the Analytic Network Process—multiple networks with benefits, costs, opportunities and risks
Thomas Saaty Journal: Journal of systems science and systems engineering The general theory of the ANP enables one to deal with the benefits, opportunities, costs, and risks (the BOCR merits) of a decision, by introducing the notion of negative priorities for C and R along with the rating (not comparison) of the top priority alternative synthesized for each of the fou... Read More |
Fuzzy judgments and fuzzy sets
Thomas Saaty, Liem Tran Journal: 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 More |
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Group decision making and the AHP
Thomas Saaty Journal: The analytic hierarchy process This paper focuses on the application of the Analytic Hierarchy Process in a group setting. In particular, we present observations and suggestions that are intended to help in the planning and execution of a group decision-making effort in which AΗΡ plays a major role. Read More |
Group decision making using the Analytic Hierarchy Process
Indrani Basak, Thomas Saaty Journal: Mathematical and computer modelling Various methods of consensus of preference rankings of individuals are described in applying group decision making. Here, we discuss two different approaches, one deterministic and the other stochastic. The first is applicable to situations where the group is small, and the second applies to opin... Read More |
Group decision-making: Head-count versus intensity of preference
Thomas Saaty, Jennifer Shang Journal: Socio-Economic Planning Sciences This paper puts forth a framework for reshaping the group decision-making process. The proposed framework extends from the usual one-issue-at-a-time decision-making to one that involves several related issues simultaneously. Weaknesses of the traditional majority voting mechanism are first identi... Read More |
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Hierarchical analysis of behavior in competition: Prediction in chess
Thomas Saaty, Luis Vargas Journal: Behavioral science In this paper we use the analytic hierarchy process to combine technical and behavioral characteristics of chess players and predict the outcome of a championship match. The method also applies to decision making in living systems at the level of the group. Our approach to prediction deals with t... Read More |
Hierarchies, reciprocal matrices, and ratio scales
Thomas Saaty Journal: Discrete and System Models We are interested in the problem of finding a scale which reflects the relative intensity of a property shared by n objects. The objects may be n stones and the properties may be their weights. What is needed is a theory that would enable us to conduct measurement which produces not only known re... Read More |
High-level nuclear waste management: analysis of options
Thomas Saaty, H Gholamnezhad Journal: Environment and Planning B: Planning and Design A new approach is introduced for evaluating strategies for the safe disposal of high-level nuclear waste. Five strategies are considered, and by means of the effective method of ‘analytic hierarchy process’ these strategies are prioritized with respect to a set of tangible and intangible criteria... Read More |
High‐level decisions: A lesson from the Iran hostage rescue operation
Thomas Saaty, Luis Vargas, Amos Barzilay We use the analytic hierarchy process to analyze the role of subjective factors in decision making as illustrated in the Iran rescue operation. Essentially, we show that a decision maker and that decision maker's advisors may differ in their estimates on whether an action should or should not... Read More |
Higher education in the United States (1985–2000): Scenario construction using a hierarchical framework with eigenvector weighting
Thomas Saaty, Paul C. Rogers Journal: Socio-Economic Planning Sciences The object of this paper is to illustrate an application of hierarchies and eigenvalues developed by the first author to the area of planning in higher education. We start with a brief discussion of the role of scenario construction in planning and follow it with a discussion of the role of hiera... Read More |
Highlights and critical points in the theory and application of the Analytic Hierarchy Process
Thomas Saaty Journal: European Journal of operational research 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, Judgm... Read More |
Hoover s Problem
Ke-Yuan Chen, Thomas Saaty Journal: Mathematics Magazine J. Edgar Hoover, the former FBI Chief, is reported to have had a great aversion to left turns arising from th time his chauffeured car was hit while attempting a lft-hand turn. Thereafter, particularly when the old lawman visited a new city, the driver had to work out the route in advance to avoi... Read More |
How many judges should there be in a group?
Thomas Saaty, Mujgan Sagir Ozdemir Journal: Annals of Data Science This paper briefly examines the question of how many judges are needed to obtain valid and consistent judgments when using the analytic hierarchy process. It turns out that if a judge is experienced and well versed in an area, he can be sufficient to provide the judgments instead of diluting his ... Read More |
How to Handle Dependence with the Analytic Hierarchy Process
Thomas Saaty Journal: Mathematical Modelling Hierarchic and network systems are discussed as basic frameworks of unstructured problems modeled by the Analytic Hierarchy Process. A hierarchy represents a linear chain of interactions, whereas a network allows for feedback in the form of cycles and loops. A theory is provided for the prioritie... Read More |
How to make a decision: the Analytic Hierarchy Process
Thomas Saaty Journal: Interfaces People make three general types of judgments to express importance, preference, or likelihood and use them to choose the best among alternatives in the presence of environmental, social, political, and other influences. They base these judgments on knowledge in memory or from analyzing benefits, ... Read More |
How to make a decision: the Analytic Hierarchy Process
Thomas Saaty Journal: European journal of operational research 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 measuremen... Read More |
How to prioritize inventions
Thomas Saaty, Elena Rokou Journal: World Patent Information In this paper we consider many intangible criteria that influence the priorities and ranks of inventions as they contribute to human welfare, by using the Analytic Hierarchy Process. We apply the ideas to evaluate a number of modern inventions along with a few old ones. Both the categories of inv... Read More |
Hypermatrix of the Brain
Thomas Saaty Journal: Artificial Neural Nets and Genetic Algorithms Decision-making, a natural and fundamental process of the brain, involves the use of pairwise comparisons. They are represented by a matrix whose entries belong to a fundamental scale, and from which an eigenvector of priorities that belongs to a ratio scale is derived. A simple decision is repre... Read More |
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Impact of disarmament nuclear package reductions
Thomas Saaty Journal: Quantitative Assessment in Arms Control The traditional view of conflict is that it is a struggle between people who are in the right and people who are in the wrong. If one wished to be more religious about it, our imprinting from childhood seems to say that conflict is a struggle between good and evil. A more realistic view of confli... Read More |
Implementing neural firing: towards a new technology
Thomas Saaty, Luis Vargas Journal: Mathematical and Computer modelling Following earlier work on neural firing and synthesis, we use the expressions we derived elsewhere in the literature for that purpose, to show how we can represent images and sounds with the aid of CLOS (Common Lisp Object System) computer program of the LISP computer language. Read More |
Inconsistency and rank preservation
Thomas Saaty, Luis Vargas Journal: Journal of Mathematical Psychology Conditions for rank preservation in a positive reciprocal matrix that is inconsistent are provided. Three methods of deriving ratio estimates are examined: the eigenvalue, the logarithmic least squares, and the least squares methods. It is shown that only the principal eigenvector directly deals ... Read More |
International center for conflict resolution
Thomas Saaty Journal: Defense Analysis Everyone agrees that international conflicts carry the seed of worldwide destruction, thus requiring an effective approach to their resolution. The major problem of international conflict resolution today, however, is how to avoid war rather than how to attain peace. Still, the continuance of a g... Read More |
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Knapsack allocation of multiple resources in benefit-cost analysis by way of the Analytic Hierarchy Process
James Bennett, Thomas Saaty Journal: Mathematical and computer modelling Benefit-cost analysis has traditionally required monetary measures of benefits and costs of candidate projects. This paper demonstrates that priorities of both tangible and intangible attributes can be used in a benefit-cost setting to address the long standing problem of allocating multiple reso... Read More |
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Letter to the Editor—A Conjecture Concerning the Smallest Bound on the Iterations in Linear Programming
Thomas Saaty Journal: Operations Research In solving linear programming problems accurate estimates of the number of iterations needed to reach the optimum are important to have. It has been mentioned in the literature that computing experience indicates this number of iterations to be of the order of twice the number of constraints. We ... Read More |
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Making and validating complex decisions with the AHP/ANP
Thomas Saaty Journal: Journal of Systems Science and Systems Engineering Several examples that serve to validate the AHP/ANP with matrices hierarchies and networks are given in this paper. They are then followed by a discussion of the real numbers and how they are generated without the need for an absolute zero, and how they define an absolute scale of measurement tha... Read More |
Making decisions in hierarchic and network systems
Thomas Saaty, Mariya Sodenkamp Journal: International Journal of Applied Decision Sciences This paper summarises a mathematical theory of the measurement of both tangible and intangible factors, the Analytic Hierarchy Process (AHP) and its generalisation to dependence and feedback, the Analytic Network Process (ANP) and illustrates their application to making complex multicriteria deci... Read More |
Marketing applications of the Analytic Hierarchy Process
Yoram Wind, Thomas Saaty Journal: Management science Several marketing applications of the Analytic Hierarchy Process (AHP) are reviewed. The paper starts with a brief description of this process, which was developed by Thomas Saaty in 1971, including an eight-point outline of how to apply it. The thrust of the paper is a discussion of a number of ... Read More |
Mathematical modeling of dynamic decisions; priorities and hierarchies with time dependence
Thomas Saaty Journal: Mathematics and computers in Simulation The 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 More |
Measuring the fuzziness of sets
Thomas Saaty Journal: Journal of Cybernetics Set theory begins to be useful when there is some natural criterion for defining belonging to a set. Sets of objects without properties are uninteresting. Elements are assigned to sets because they share properties or conform to a rule. A set of elements is said to be fuzzy when we allow some ele... Read More |
Modeling the graduate business school admissions process
Thomas Saaty, James France, Kathleen Riehle Valentine Journal: Socio-economic planning sciences Each year graduate business admissions committees must carefully evaluate prospective students. Committees must judge the performance of applicants with respect to a wide range of quantitative and qualitative criteria in order to select the best students. The overall process is complex and time c... Read More |
Modeling unstructured decision problems—the theory of analytical hierarchies
Thomas Saaty Journal: Mathematics and computers in simulation Quantitative modeling of unstructured decision problems with social implications is new and challenging and has pressing needs. A new approach to scaling using largest eigenvalues and reciprocal matrices and the effect of inconsistent judgment are introduced and relevant theory discussed. In this... Read More |
Multi-decisions decision-making: In addition to wheeling and dealing, our national political bodies need a formal approach for prioritization
Thomas Saaty Journal: Mathematical and Computer Modelling Individuals, corporations and governments constantly face the extremely complex problem of ordering and prioritizing their numerous decisions according to urgency and importance. They need to sequence expenditures and allocate scarce resources to optimize the returns on their investments over tim... Read More |
Multicriterion decisions in systems with feedback
Thomas Saaty Journal: Mathematical Modelling in Science and Technology The process of setting priorities in systems with a network structure is more complex than it is in a hierarchy which is essentially linear. The composition of priorities in a hierarchy is a special case of composition in a system. Simple examples of composition in hierarchies with cycles between... Read More |
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Negative priorities in the Analytic Hierarchy Process
Thomas Saaty, Mujgan Sagir Ozdemir In decision-making, there are often criteria that are opposite in direction to other criteria as in benefits (B) versus costs (C), and in opportunities (O) versus risks (R), and sometimes need to be distinguished by using negative numbers. In making paired comparisons of alternatives with respect... Read More |
Negotiating the Israeli–Palestinian controversy from a new perspective
Thomas Saaty, H. J. Zoffer Journal: Information Technology & Decision Making In most long-lasting conflicts, each party's grievances increase while the concessions they are willing to make decline in number, quality, and perceived value. Both parties lose sight of what they are willing to settle for, generally exaggerate their own needs, and minimize the needs of the ... Read More |
Neural synthesis of firing decisions in the brain
Thomas Saaty Journal: Istanbul University Journal of the School of Business, The brain generally miniaturizes its perceptions into what may be regarded as a model of what happens outside. We experience the world according to the capacity of our nervous system to register the stimuli we receive. In order to understand and control the environment there needs to be proportio... Read More |
Neurons the decision makers, Part I: The firing function of a single neuron
Thomas Saaty Journal: Neural Networks This paper is concerned with understanding synthesis of electric signals in the neural system based on making pairwise comparisons. Fundamentally, every person and every animal are born with the talent to compare stimuli from things that share properties in space or over time. Comparisons always ... 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 Saaty Journal: 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 More |
New light on the theorem of perron
Thomas Saaty Journal: Trabajos de Estadistica Y de Investigacion Operativa We prove that the principal eigenvector of a positive matrix represents the relative dominance of its rows of ranking of alternatives in a decision represented by the rows of a pairwise comparison matrix. Read More |
NINA's decision: How to make better decisions and resolve conflicts-an essay for the layman
Thomas Saaty, H. J. Zoffer Journal: International Journal of the Analytic Hierarchy Process Decision making is a central activity of all people, usually done so automatically that we do not even realize that we are doing it every moment of every day of every year for all our lives. This silent and inarticulate approach worked well when humanity was fragmented and individuals and groups ... Read More |
Nuclear balance and the parity index: The role of intangibles in decisions
Ami Arbel, Thomas Saaty, Luis Vargas Journal: IEEE Transactions on Systems, Man, and Cybernetics An approach to understanding and assessing issues concerning arms control is presented. Arms control has two main objectives: (1) decreasing the possibility of a nuclear exchange; and (2) limiting the scope and intensity of such an exchange. A framework is presented for dealing with those issues ... Read More |
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Oil prices: 1985 and 1990
Thomas Saaty, Luis Vargas Journal: The Logic of Priorities Today oil is the world’s major energy resource. It accounts for about 54 percent of the world’s total energy consumption. Because of conservation and the development of alternative resources in industrialized countries, the share of oil in the world’s total energy consumption is expected to decli... Read More |
On Mathmatical Structures in Some Problems of Politics
Thomas Saaty Journal: The Mathematics Teacher In recent years there has been a remarkable surge of effort to develop quantitative foundations for the solution of political problems, both to increase understanding of the underlying structure and to improve predictions. Read More |
On nonlinear optimization in integers
Thomas Saaty Journal: Naval Research Logistics Quarterly The purpose of this paper is to give some elementary theorems on optimization in integers of nonlinear expressions subject to simple equality constraints. Other than the complicated algorithms of integer linear-programming, there is a dirth of examples and a
tradition to follow which might help ... Read More |
On polynomials and crossing numbers of complete graphs
Thomas Saaty Journal: Journal of Combinatorial Theory A long-standing, unsolved problem is that of finding the minimum number of crossings of the edges in a complete graph when embedded on a surface of genus zero. Read More |
On the invalidity of fuzzifying numerical judgments in the Analytic Hierarchy Process
Thomas Saaty, Liem Tran Journal: Mathematical and Computer Modelling Fuzzy set theory has serious difficulties in producing valid answers in decision-making by fuzzifying judgments. No theorems are available about its workability when it is applied indiscriminately as a number crunching approach to numerical measurements that represent judgments. When judgments ar... Read More |
On the measurement of intengibles. A principal eigenvector approach to relative measurement derived from paired comparisons
Thomas Saaty Journal: Notices of the American Mathematical Society Nearly 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 More |
On the relativity of relative measures–accommodating both rank preservation and rank reversals in the AHP
Ido Millet, Thomas Saaty The analytic hierarchy process (AHP) has been criticized for allowing the introduction of new alternatives to cause rank reversals among existing alternatives. While in many cases this is a perfectly valid phenomenon there are also many cases where rank should be preserved. To address both contin... Read More |
Operational gaming for energy policy analysis
Thomas Saaty, Fred Ma, Peter Blair Journal: Energy Policy A new approach to energy policy analysis is developed by structuring the interactive dynamic processes in energy environment systems Current energy system modelling efforts do not effectively account for multiple interests and the interaction of policy markers. This paper presents a methodology f... Read More |
Operations Research: Some Contributions to Mathematics: Applied mathematics gets a new surge of life from techniques of operations research
Thomas Saaty Journal: Science The discussion of fuzzy sets indicates that set theory should be extended to make it more suitable for the development of algebraic structures with wider applications. Stochastic optimization, a synthesis of the three areas of continuum mathematics, is a rapidly growing field particularly in the ... Read More |
Optimization and the geometry of numbers: packing and covering
Thomas Saaty, Joyce M. Alexander Journal: SIAM Review There are a number of useful applications gradually arising out of the geometry of numbers. In this paper we give a brief survey of the fields of packing and covering, stating some results, and giving illustrations of methods of proof and approaches to the subject. We then mention and discuss a v... Read More |
Optimum positions for m airports
Thomas Saaty There is a set of n factories, which are located throughout the United States. The wares produced by these factories are delivered to local airports, where they are flown to one of two overseas airports, called distribution centers. The centers distribute the wares to a number of neighboring area... Read More |
Origin of Neural Firing and Synthesis in Making Comparisons
Thomas Saaty, Luis Vargas Journal: European Journal of Pure and Applied Mathematics The nervous system uses its own kind of mathematical function patterns for both external and internal realities. The conscious part of the nervous system is there to respond to what happens outside by regulating externally received information signals from the senses and the skin and muscles of t... Read More |
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Parametric objective function (part 1)
Saul Guss, Thomas Saaty Journal: Journal of the Operations Research Society of America In the linear programming problem where there is a linear function to be optimized (called the objective function), it is desirable to study the behaviour of solutions when the (cost) coefficients in the objective function are parametrized. The problem is then to find the set of xj (j = 1, 2, …, ... Read More |
Parametric objective function (part 2)—generalization
Saul Guss, Thomas Saaty Journal: Journal of the Operations Research Society of America In Part 1 (Saaty, T. L., S. I. Gass. 1954. Parametric objective function (Part 1). J. Opns. Res. Soc. Am.2 316.), the cost function in the general linear programming problem was parametrized with one parameter and the problem of generating solutions completely studied. In Part 2, a generalization... Read More |
Physics as a decision theory
Thomas Saaty Journal: European Journal of Operational Research We show that there is a close mathematical relationship between physics and the Analytic Hierarchy Process. We argue that numerical scales used in physics must be interpreted in terms which the scientist understands through experience and through theories advocated by experts in the field. In phy... Read More |
Portfolio selection through hierarchies
Thomas Saaty, Paul C. Rogers, Ricardo Pell Journal: The journal of portfolio management The method we will discuss addresses the huge task of ranking and choosing from among a large set of stocks in order to devide how much of our recources should be allocated to each of the stocks chosen. Read More |
Principles for Implementing a Potential Solution to the Middle East Conflict
Thomas Saaty, H. J. Zoffer Journal: Notices of the AMS The purpose of this paper is to show how the Analytic Hierarchy Process (AHP) applies mathematics to deal with the most complex problems of the world in a comprehensive and holistic way—in this case the Israeli-Palestinian conflict. This process has been helpful in finding solutions to some of th... Read More |
Principles of the Analytic Hierarchy Process
Thomas Saaty Journal: Expert judgment and expert systems Cognitive psychologists have classified thinking into two types. This division has gone by many names. Aristotle referred to it as active versus passive reason [35]; Freud [10] as secondary versus primary process thinking; and Hobbes [14] as thought with or without “designe.” More recently the di... Read More |
Priority as dominance in derived measurement: Invariance of the principal eigenvector
Mujgan Sagir Ozdemir, Thomas Saaty Ranking 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 More |
Priority setting in complex problems
Thomas Saaty Journal: IEEE Transactions on Engineering Management There are three principles which one can recognize in problem solving. They are the principles of decomposition, comparative judgments, and synthesis of priorities. The Analytic Hierarchy Process (AHP) provides a comprehensive framework to cope with the intuitive, the rational, and the irrational... Read More |
Procedures for synthesizing ratio judgements
J. Aczel, Thomas Saaty Journal: Journal of mathematical Psychology 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. Read More |
Projecting average family size in rural India by the Analytic Hierarchy Process
Thomas Saaty, Molly Wong Journal: Journal of Mathematical Sociology In this study, cultural, economic as well as certain crucial demographic factors are considered as the determinants for projecting the average family size in rural India. We use the Analytic Hierarchy Process to analyze influences of the factors which enter implicitly in a rural couple's deci... Read More |
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Questions and answers: Valentina Ferretti interviews Tom Saaty
Valentina Ferretti, Thomas Saaty Journal: International Journal of the Analytic Hierarchy Process The following interview aims to stimulate the discussion about the key characteristics of Multicriteria Analysis techniques. In particular, the interview will work through the conceptual lens of decision processes taking place in the field of urban and territorial transformations. Read More |
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Rank According to Perron: A New Insight
Thomas Saaty Journal: Mathematics Magazine Rank According to Perron: A New Insight Read More |
Rank from comparisons and from ratings in the Analytic Hierarchy/Network Processes
Thomas Saaty Journal: European Journal of Operational Research Rank preservation and reversal are important subjects in multi-criteria decision-making particularly if a theory uses only one of two ways of creating priorities: rating alternatives one at a time with respect to an ideal or standard, or comparing them in pairs. It is known that our minds can do ... Read More |
Rank Generation, Preservation, and Reversal in the Analytic Hierarchy Decision Process
Thomas Saaty Journal: Decision sciences Decision making has the objective of finding the best alternative or set of alternatives by considering a number of goals, objectives, criteria, competitors, and other important factors. The analytic hierarchy process is a decision aid used to assist a decision maker in sorting out the complexity... Read More |
Ranking by eigenvector versus other methods in the Analytic Hierarchy Process
Thomas Saaty, G Hu Journal: Applied Mathematics Letters Counter-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 More |
Ranking countries more reliably in the summer olympics
Thomas Saaty, Mujgan Sagir Ozdemir Journal: International Journal of the Analytic Hierarchy Process In this paper we consider the many intangible criteria that influence the outcome of the Summer Olympics by using the Analytic Network Process, and apply the ideas to evaluate the medals won and the country scores in the 2012 London Olympics. Both the categories of games and the events in each ga... Read More |
Ratio Scales are Critical for Modeling Neural Synthesis in the Brain
Thomas Saaty Journal: Artificial Neural Nets and Genetic Algorithms The brain generally miniaturizes its perceptions into what may be regarded as a model of what happens outside. We experience the world according to the capacity of our nervous system to register the stimuli we receive. In order to understand and control the environment there needs to be proportio... Read More |
Ratio scales derived from perturbations of consistent judgments
Thomas Saaty Journal: Behaviormetrika We derive ratio scales from paired comparison judgments in a reciprocal matrix A. When the judgments are consistent, we have a principal eigenvalue structure which is preserved when A is perturbed. Mathematical conditions are given on the size of the perturbations to produce a good approximation ... Read More |
Rationing energy to industries: priorities and input-output dependence
Thomas Saaty, Luis Vargas Journal: The Logic of Priorities A short time ago it was unthinkable and deemed an academic exercise to speak of rationing. People thought that there could be no crippling energy crisis because our energy czars and planners would presumably take our needs into their projections. Today the situation looks very different. Witness ... Read More |
Reflections and projections on creativity in Operations Research and Management Science: a pressing need for a shift in paradigm
Thomas Saaty Journal: Operations Research This paper is an outgrowth of a talk given at a plenary session of the national meeting of ORSA/TIMS (now INFORMS) in May of 1996. Rather than speculate on what might be the possible domain of OR/MS within a systems framework, the paper gives a sketch of what basic ideas there are today that can ... Read More |
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
Thomas Saaty Journal: RACSAM-Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 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 s... Read More |
Remarks on the four color problem; the Kempe catastrophe
Thomas Saaty Journal: Mathematics Magazine The original error discovered by Heawood in Kempe's attempt to prove the four-color conjecture is often encountered by many of those who follow the inductive argument approach to the problem. Perhaps the nature of the difficulty is not well appreciated. An old Chinese proverb urges that to know t... Read More |
Representation of visual response to neural firing
Thomas Saaty, Luis Vargas Journal: Mathematical and computer modelling In this note, we give examples of the use of linear combinations of impulse response functions to represent visual images. We also derive expressions for estimating the parameters of these functions using the Fourier transform of stimuli. Read More |
Resolution of retributive conflicts
Thomas Saaty Journal: IFAC Proceedings Volumes In prolonged confrontations, known here as retributive, the parties not only exact their own demands but also want to incur a high cost on the opponent. Success in resolving a conflict is measured by the gain: each party's benefits and the costs to the opponent compared with the perceived gain fo... Read More |
Response to Holder's comments on the Analytic Hierarchy Process
Thomas Saaty Journal: Journal of the Operational Research Society Response to Holder's comments on the analytic hierarchy process Read More |
Response to holder's comments on the Analytic Hierarchy Process: response to the response to the response
Thomas Saaty Journal: Journal of the Operational Research Society It would have been fitting and proper were Holder, in his call for more validation of the analytic hierarchy process (AHP), to have validated his own vague suggestions for a better optics experiment with chairs; a power scale to fulfil his desire for non-linearity, and its credibility in applicat... Read More |
Resume of useful formulas in queuing theory
Thomas Saaty Journal: Operations Research This paper is intended as a convenient summary of some results in queuing, which in the author's opinion would be of value to investigators applying the theory to operational problems. The paper is neither intended as an elementary introduction, nor is it an exposition of the techniques used ... Read More |
Risk: Its Priority and Probability, The Analytic Hierarchy Process
Thomas Saaty Journal: Risk Analysis Risk estimation involves priorities and probabilities which are themselves a form of priority of natural alternatives. This paper provides illustrations of how one can deal with risk and uncertainty using the Analytic Hierarchy Process, a new approach to measurement by ratio scales. The paper als... Read More |
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Scaling the membership function
Thomas Saaty Journal: European Journal of Operational Research To scale the membership function, a method is needed that can assess the importance of elements
with respect to a property or criterion that is common among them. It is conceivable that membership should be determined in terms of several criteria tangible and intangible. The approach needed is a... Read More |
Scenarios and priorities in transport planning: Application to the Sudan
Thomas Saaty Journal: Transportation Research This paper describes two related aspects of a transportation planning study recently completed for the Sudan—the use of scenarios and the prioritization of those scenarios and the associated transport projects. In the use of scenarios, considerable care was taken in the specification of different... Read More |
Seven is the magic number in nature
Thomas Saaty Journal: Proceedings of the American Philosophical Society Where there is structure, the parts of the structure must function together with a degree of consistency and purpose. Specifically, I am thinking of dynamic systems in which there are action and reaction among the parts and their functions and also friction and resistance. Natural systems, such a... Read More |
Seven more years of queues. A lament and a bibliography
Thomas Saaty Journal: Naval research logistics quarterly Everyone today knows that a queue is a waiting line. If one also takes the trouble to examine the literature, which now is nearing 2000 references on the subject, he might get the idea that all those contributing to the understanding of congestion phenomena are interested in doing something about... Read More |
Should the UK have Brexited the European Union?
Lirong Wei, Thomas Saaty Journal: International Journal of the Analytic Hierarchy Process This paper is an analysis of Brexit, and asked the question, Should the UK have Brexited the European Union? We use the Analytic Hierarchy Process (AHP) to model the decision based on Benefit-Opportunities-Costs and Risks (BOCR). The AHP structure considers various factors that may be taken into... Read More |
Some mathematical concepts of the Analytic Hierarchy Process
Thomas Saaty Journal: Behaviormetrika In this paper we give a skeletal outline of the foundations of the Analytic Hierarchy Process and some of its highlights. We also show that the principal eigenvector solution is essential for deriving the scale of priorities from the fundamental scale of judgments given in the matrix. Read More |
Some mathematical topics in the Analytic Hierarchy Process
Thomas Saaty Journal: Mathematical models for decision support Decisions involving several criteria and alternatives need a correct procedure to combine weights to make tradeoffs and select the best alternatives. There are two parts to the problem, both of which need proof. The first is how to generate judgments and associate numbers with them, and the secon... Read More |
Some stochastic processes with absorbing barriers
Thomas Saaty Journal: Journal of the Royal Statistical Society: Series B Models for growth distributions of biological organisms are investigated in this paper. They are specializations of the birth–death equations with two absorbing barriers. The solutions we seek are the distributions of the time to grow from an initial dose to an infective dose. Equations for these... Read More |
Speculating on the future of Mathematics
Thomas Saaty Journal: Applied Mathematics Letters Modern mathematics continues to lag the real world by not offering new theories and concepts. Despite great excitement within the field about all sorts of progress in the subject, the availability and application of mathematics to new political and social structures is primitive and its relevance... Read More |
Stability analysis of the forward‐backward process: Northern Ireland case study
Joyce M. Alexander, Thomas Saaty Journal: Behavioral Science In this study, the forward and backward processes of planning are applied to conflict resolution in a way similar to that in which the forward process was applied in a previous paper by the authors . By repeated analysis of the policies of the parties to the conflict and possible reactions to the... Read More |
Stimulus-response with reciprocal kernels: The rise and fall of sensation
Thomas Saaty, Luis Vargas Journal: Journal of Mathematical Psychology The paper presents a theory for constructing response scales based on the reciprocal property of paired comparisons of stimuli from the same sensory continuum. Reciprocal paired comparisons define the pair estimator function K(s, t), the kernel of a Fredholm integral equation of the second kind. ... Read More |
Synthesis of complex criteria decision making: a case towards a consensus agreement for a Middle East conflict resolution
H. J. Zoffer, Asma Bahurmoz, Mohammed K. Hamid, Marcel Minutolo Journal: Group Decision and Negotiation The title of this paper reflects both the process and the outcome of the current undertaking. Frustrated with the current state of the Middle East but encouraged by earlier attempts at modeling complex problems, the authors participated in a panel discussion assembled to address the conflict and ... Read More |
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Technological choice in the less developed countries: An analytic hierarchy approach
Vasudevan Ramanujam, Thomas Saaty Journal: Technological Forecasting and Social Change A novel and practical approach is proposed for the assessment and selection of imported technology by the less developed countries. Technological choice is conceptualized as a multiobjective, multicriterion problem wherein subjective judgments and political processes play key roles. A new plannin... Read More |
Terrorism, Patterns for Negotiations: Three Case Studies Through Hierarchies and Holarchies: a Study for the Arms Control and Disarmament Agency
James P. Bennett, Thomas Saaty Journal: Wharton School, University of Pennsylvania AS PART OF A STUDY FOR THE ARMS CONTROL AND DISARMAMENT AGENCY THE USE OF ANALYTIC HIERARCHIES AND HOLARCHIES WERE FOUND TO BE A USEFUL GUIDE FOR NEGOTIATORS CONFRONTING TERRORISTS HOLDING HOSTAGES. WHEN THIS FINDING WAS APPLIED TO THE MUNICH MASSACRE OF 1972, THE OBJECTIVES AND INFLUENCES OF THE... Read More |
That is not the Analytic Hierarchy Process: what the AHP is and what it is not
Thomas Saaty Journal: Journal of Multi‐Criteria Decision Analysis There are two basic principles that are totally missed by Salo and Hamalaninen in their attempt to reformulate the AHP, which is based on ratio scales, to fit multiattribute value theory, which is based on interval scales. Read More |
The allocation of intangible resources: the Analytic Hierarchy Process and linear programming
Thomas Saaty, Luis Vargas, Klaus Dellmann Journal: Socio-Economic Planning Sciences An intangible is an attribute that has no scale of measurement. Intangibles such as effort and skill arise in conjunction with resource allocation but are not usually included directly in a mathematical model because of the absence of a unit of measurement. However, intangibles can be quantified ... Read More |
The analytic hierarchy and analytic network measurement processes: applications to decisions under risk
Thomas Saaty Journal: European Journal of Pure and Applied Mathematics Mathematics applications largely depend on scientific practice. In science measurement depends on the use of scales, most frequently ratio scales. A ratio scale there is applied to measure various physical attributes and assumes a zero and an arbitrary unit used uniformly throughout an applicatio... Read More |
The Analytic Hierarchy Process and human resource allocation: Half the story
Thomas Saaty, Kirti Peniwati, Jennifer Shang Journal: Mathematical and Computer Modelling The Analytic Hierarchy Process (AHP) provides a way to rank the alternatives of a problem by deriving priorities. A question that occurs in practice is: what is the best combination of alternatives that has the largest sum of priorities and satisfies given constraints? This leads one to consider ... Read More |
The Analytic Hierarchy Process applied to complexity
Thomas Saaty, Nina Begicevic Journal: International Journal of Economics and Business Research Even in science, there is always the human element of using our own assumptions in logic, our feelings, our values and numerous criteria we must deal with. In addition, we often have to include the measurements of tangibles that always need agreed upon subjective judgement to interpret what exact... Read More |
The Analytic Hierarchy Process in conflict management
Thomas Saaty Journal: International Journal of Conflict Management The Analytic Hierarchy Process (AHP) is a theory of measurement. When applied in decision‐making, it assists one to describe the general decision operation by decomposing a complex problem into a multi‐level hierarchic structure of objectives, criteria, subcriteria and alternatives. The AHP provi... Read More |
The Analytic Hierarchy Process without the theory of oskar perron
Thomas Saaty Journal: International Journal of the Analytic Hierarchy Process It 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 More |
The Analytic Hierarchy Process: A new approach to deal with fuzziness in architecture
Thomas Saaty Journal: Architectural Science Review A degree of fuzziness accompanies complex architectural problems. The concept of fuzziness can be compared to the hierarchy of the nervous system. Three types of fuzziness can be isolated: instantaneous, ongoing and long term. In this paper, the need for the development of a conceptually simple f... Read More |
The Analytic Hierarchy Process: how to measure intangibles in a meaningful way side by side with tangibles
Thomas Saaty Journal: International Symposium on QFD, ISQFD Read More |
The Analytic Hierarchy Process: Some observations on the paper by Apostolou and Hassell
Thomas Saaty Journal: Journal of Accounting Literature Apostolou and Hassell in their overview of the Analytic Hierarchy Process (AHP) and its applications to accounting included a section on some theoretical questions about the AHP that differ considerably from my own understanding as the creator of that theory. There are now more than 1,200 papers ... Read More |
The Analytic Hierarchy Process: wash criteria should not be ignored
Thomas Saaty, Luis Vargas Journal: International Journal of Management and Decision Making In this short paper it is shown that care is required to perform the appropriate normalisation needed to derive the right set of priorities from paired comparisons when criteria are added or deleted in a decision problem. This is particularly true for criteria with respect to which the alternativ... Read More |
The computational algorithm for the parametric objective function
Saul Guss, Thomas Saaty Journal: Naval research logistics quarterly If a linear programming problem involves two objective functions, it is desirable to learn all solutions depending on the relative weight attached to the two functions. This paper presents details of an algorithm which finds these solutions systematically. Read More |
The Conflict in South Africa
Thomas Saaty, Luis Vargas Journal: The Logic of Priorities The domestic strife in South Africa between ruling whites and subordinated blacks threatens to become a major conflict of violent dimensions in the African continent. This strife is primarily a consequence of racial policies promulgated by the white apartheid regime. Donald Woods (1978) describes... Read More |
The Conflict in South Africa
Thomas Saaty, Luis Vargas Journal: The Logic of Priorities The domestic strife in South Africa between ruling whites and subordinated blacks threatens to become a major conflict of violent dimensions in the African continent. This strife is primarily a consequence of racial policies promulgated by the white apartheid regime. Donald Woods (1978) describes... Read More |
The conflict in South Africa: directed or chaotic
David Tarbell, Thomas Saaty Journal: Journal of Peace Science The domestic strife in South Africa between ruling whites and subordinated blacks threatens to become a major conflict of violent dimensions in the African continent. This strife is primarily a consequence of racial policies promulgated by the white, apartheid regime. Donald Woods (1978) describe... Read More |
The decision by the US congress on China’s trade status: a multicriteria analysis
Thomas Saaty, Luis Vargas Journal: Models, Methods, Concepts & Applications of the Analytic Hierarchy Process In this paper, we used a decision making tool, the Analytic Hierarchy Process (AHP), to analyze a decision to select a trade status for China that is in the best interest of the United States before that decision came before Congress for a vote. Read More |
The Eigenvector in lay language
Thomas Saaty Journal: International Journal of the Analytic Hierarchy Process Decision making depends on identifying a structure of criteria and alternatives of a decision. It also depends on experience and judgments to select the best alternative. In the Analytic Hierarchy Process (AHP) for decision making the criteria and alternatives are prioritized by forming matrices ... Read More |
The forward and backward processes of conflict analysis
Joyce M. Alexander, Thomas Saaty Journal: Behavioral Science In 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 More |
The future of operations research in the government
Thomas Saaty Journal: Interfaces The crucial thrust of this report develops some ideas about how operations research and management science may increase their effective participation in high level governmental decision making. The opportunity confronting operations research to participate actively in charting the future of our s... Read More |
The general case of dependence in hierarchic decision theory
Thomas Saaty Journal: Toward Interactive and Intelligent Decision Support Systems A hierarchy is a simple structure used to represent the simplest type of functional (contextual or semantic) dependence of one level or component of a system on another in a sequential manner. It is also a convenient way to decompose a complex problem in search of causeeffect explanations in step... Read More |
The legitimacy of rank reversal
Thomas Saaty, Luis Vargas Journal: Omega WE ARE GLAD to be finally'answering Belton and Gear's concerns expressed in their article in Omega [l]. There are two points they make in their letter. The first has to do with rank reversal, and the second with how to ask the question by minimizing fuzziness. Our response is organized into four ... Read More |
The minimum number of intersections in complete graphs
Thomas Saaty Journal: Proceedings of the National Academy of Sciences of the United States of America Zarankiewicz has developed results concerning the minimum number of edge intersections, when drawn in a plane, of the bipartite graph consisting of two sets of vertices with one edge joining each element of one set to each element of the other set. When each set has three vertices, we have one of... Read More |
The Modern Science of Multicriteria Decision Making and Its Practical Applications: The AHP/ANP Approach
Thomas Saaty Journal: Operations Research This paper presents a summary of the discrete mathematical part of my work, the Analytic Hierarchy Process (AHP) and its generalization to dependence and feedback, the Analytic Network Process (ANP), for measuring tangible and intangible factors, particularly as applied to decision making. The fa... Read More |
The Need for Adding Judgment in Baysian Prediction
Thomas Saaty, Lei Zhang Journal: International Journal of Information Technology & Decision Making There are many examples of successful application of the famous Bayes’ theorem to forecast the outcome of causes or influences of events. However, there are also many examples of failures of just using probabilities and statistics without also using judgments in each particular application. This ... Read More |
The number of vertices of a polyhedron
Thomas Saaty Journal: The American Mathematical Monthly The number of vertices of a polyhedron Read More |
The Peruvian Hostage Crisis of 1996–1997: What Should the Government Do?
Thomas Saaty, Enrique Mu Journal: Socio-Economic Planning Sciences How should the Peruvian Government resolve the Japanese embassy takeover hostage crisis? As of this writing, MRTA operatives continue to hold 72 hostages and demand the release of their jailed comrades among other concessions. We have used the Analytic Hierarchy Process decision methodology to an... Read More |
The Possibility of Group Choice: Pairwise Comparisons and Merging Functions
Thomas Saaty, Luis Vargas Journal: Social Choice and Welfare Preferences in Arrow’s conditions are ordinal. Here we show that when intensity of preference represented by reciprocal pairwise comparisons is considered, it is always possible to construct an Arrowian social welfare function using a two-stage social choice process. In stage 1, the individual pa... Read More |
The possibility of group choice: pairwise comparisons and merging functions
Thomas Saaty, Luis Vargas Journal: Social Choice and Welfare Preferences in Arrow’s conditions are ordinal. Here we show that when intensity of preference represented by reciprocal pairwise comparisons is considered, it is always possible to construct an Arrowian social welfare function using a two-stage social choice process. In stage 1, the individual pa... Read More |
The possibility of group welfare functions
Thomas Saaty, Luis Vargas Journal: International Journal of Information Technology & Decision Making This paper gives a brief overview of the well-known impossibility-possibility theorem in constructing a social welfare function from individual functions. The Analytic Hierarchy Process uses a fundamental scale of absolute numbers to represent judgments about dominance in paired comparisons. It i... Read More |
The rationality of punishment—measuring the severity of crimes: an AHP-based orders-of-magnitude approach
Thomas Saaty, Mujgan Sagir Ozdemir, Jennifer Shang We propose an innovative AHP-based model to assess the severity of the harms a criminal commits to society in a comprehensive and coherent way. Different from the traditional approach of structuring alternatives into one level, we organize the alternatives into multiple levels of that hierarchy. ... Read More |
The scope of human values and human activities in decision making
Thomas Saaty, Nina Begicevic Journal: Applied Soft Computing During and at the end of Olympic games, we are always given the number of gold, silver and bronze medals won by each country and often the total number won as an indicator of the surmised winner. The groups that report the medal count in this manner indicate that they believe all medals are the s... Read More |
The serious omission of comparisons in aristotle
Thomas Saaty Journal: International Journal of the Analytic Hierarchy Process Comparisons are the bedrock of all human thinking no matter where it is done on this earth in every culture. We compare things with one another and with themselves through memory to know if they are what we are looking for. We need memory and time to do that regardless of how brief and spontaneou... Read More |
The seven pillars of the Analytic Hierarchy Process
Thomas Saaty Journal: Multiple Criteria Decision Making in the New Millennium 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 eigen... Read More |
The Sudan transport study
Thomas Saaty Journal: Interfaces The Sudan, population 18.2 million, in area the largest country in Africa is traversed by the White and Blue Niles meeting at Khartoum and blessed with rain. It has the potential to be a breedbasket for Africa and the Middle East. Of an estimated 200 million agricultural acres, only 17 million ar... Read More |
The Three Laws of Thought, Plus One: The Law of Comparisons
Thomas Saaty Journal: Axioms The rules of logic are nearly 2500 years old and date back to Plato and Aristotle who set down the three laws of thought: identity, non-contradiction, and excluded middle. The use of language and logic has been adequate for us to develop mathematics, prove theorems, and create scientific knowledg... Read More |
The unknown in decision making: What to do about it
Mujgan Sagir Ozdemir, Thomas Saaty Journal: European journal of operational research The unknown or “other” that affects our lives is what we usually very much want to know about to cope with uncertainty. We often suspect that it affects us with partial and indefinite evidence that it exists but we only have uncertain feelings about it. Even when we do not know what it is we woul... Read More |
The US-OPEC energy conflict the payoff matrix by the Analytic Hierarchy Process
Thomas Saaty Journal: International Journal of Game Theory In this paper we study the U.S.-OPEC oil conflict and compute the payoff matrix for each of the two players using the Analytic Hierarchy Process by first evaluating the strategies of each player according to their intrinsic merits and then according to their relative strengths when considered aga... Read More |
Theory of measurement of impacts and interactions in systems
Thomas Saaty Journal: Systems Methodology in Social Science Research What is being frequently tested in society today is not our engineering skill or our ability to bring technology to bear on life’s problems, but our sense of priorities and our capacity to make trade-offs between various factors that have an impact on life. The trade-offs cannot all be measured i... Read More |
There is no mathematical validity for using fuzzy number crunching in the Analytic Hierarchy Process
Thomas Saaty Journal: Journal of Systems Science and Systems Engineering Fuzzy logic has difficulty producing valid answers in decision-making. Absent are theorems to prove that it works to produce results already known that are being estimated with judgments by transforming such judgments numerically. The numerical representation of judgments in the AHP is already fu... Read More |
Thirteen colorful variations on Guthrie
Thomas Saaty Journal: The American Mathematical Monthly After careful analysis of information regarding the origins of the four-color conjecture, Kenneth O. May concludes that:
It was not the culmination of a series of individual efforts that flashes across the mind of Francis Guthrie while coloring a map of England... his brother communicated the co... Read More |
Time dependent decision-making dynamic priorities in the AHP/ANP: Generalizing from points to functions and from real to complex variables
Thomas Saaty Journal: Mathematical and Computer Modelling Because good decisions depend on the conditions of the future, and because conditions vary over time, to make a good decision requires judgments of what is more likely or more preferred over different time periods. There are at least three ways to deal with dynamic decisions. One is to include in... Read More |
Time-dependent solution of the many-server Poisson queue
Thomas Saaty Journal: Operations Research In this paper we obtain the Laplace transform of the transient probabilities of the ordered queuing problem, with Poisson inputs, multiple channels, and exponential service times. Explicit expressions are derived for the two-channel case and known equilibrium conditions are shown to hold. The pro... Read More |
Transport planning with multiple criteria: the Analytic Hierarchy Process applications and progress review
Thomas Saaty Journal: Journal of advanced transportation Five examples of applications of the Analytic Hierarchy Process (AHP) are made to illustrate the different uses of this ratio scale multicriteria decision method in transportation. They include a commuter route selection hierarchy, a best mix of routes to Pittsburgh's new International Airport, a... Read More |
Two theorems on the minimum number of intersections for complete graphs
Thomas Saaty Journal: Journal of Combinatorial Theory After summarizing results of work on a conjecture regarding the minimum number of intersections of a complete graph drawn in the plane, the paper gives two theorems: one regarding a realization scheme for the conjectured quantity, and the other regarding a proof that essentially, the result would... Read More |
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Uncertainty and rank order in the Analytic Hierarchy Process
Thomas Saaty, Luis Vargas Journal: European Journal of Operational Research The Analytic Hierarchy Process uses paired comparisons to derive a scale of relative importance for alternatives. We investigate the effect of uncertainty in judgment on the stability of the rank order of alternatives. The uncertainty experienced by decision makers in making comparisons is measur... Read More |
Using the analysis of options technique to analyze a community conflict
HENRY BAIN, NIGEL HOWARD Journal: Journal of Conflict Resolution Perhaps one of the greatest needs in dealing with social and political conflict is to develop methods that spell out and analyze in detail who the participants in a conflict are, what their options are, and how they are expected to act given their own set of preferences. In general we need to sor... Read More |
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What is relative measurement? The ratio scale phantom
Thomas Saaty Journal: Mathematical and Computer Modelling Traditionally, measurement scales with a unit are assumed to be available to measure and rank alternatives. However, such scales are scarce and it is often desired to rank alternatives on many tangible and intangible criteria. How? Relative measurement is a method for deriving ratio scales from p... Read More |
When is a decision-making method trustworthy? Criteria for evaluating multi-criteria decision-making methods
Thomas Saaty, Daji Ergu Decision makers often face complicated decision problems with intangible and conflicting criteria. Numerous multi-criteria decision-making (MCDM) methods have been proposed to handle the measurement of the priorities of conflicting tangible/intangible criteria and in turn use them to choose the b... Read More |
Who won the 2008 Olympics? A multicriteria decision of measuring intangibles
Thomas Saaty Journal: Journal of Systems Science and Systems Engineering It often happens that at the end of Olympics games the medals won by more than one country are many and close in total number as in the 2008 games where China won 100 medals with many gold ones and the United States won 110 medals but with a lesser number of gold medals. The question is: Although... Read More |
Why the magic number seven plus or minus two
Thomas Saaty, Mujgan Sagir Ozdemir Journal: Mathematical and computer modelling In 1956, Miller conjectured that there is an upper limit on our capacity to process information on simultaneously interacting elements with reliable accuracy and with validity. This limit is seven plus or minus two elements. He noted that the number 7 occurs in many aspects of life, from the seve... Read More |
Words from the AHP Creator
Thomas Saaty Journal: International Journal of the Analytic Hierarchy Process Let me take this occasion to wish this journal and its contributors my very best. It is a joyful occasion to have our own journal for decision making with the AHP/ANP and my message to you follows. Read More |