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

The Analytic Hierarchy Process

A Scaling Method for Priorities in Hierarchical Structures

Fuzzy Set Theory - And Its Applications, Third Edition is a textbook for courses in fuzzy set theory. It can also be used as an introduction to the subject. The character of a textbook is balanced with the dynamic nature of the research in the field by including many useful references to develop a deeper understanding among interested readers. The book updates the research agenda (which has witnessed profound and startling advances since its inception some 30 years ago) with chapters on possibility theory, fuzzy logic and approximate reasoning, expert systems, fuzzy control, fuzzy data analysis, decision making and fuzzy set models in operations research. All chapters have been updated. Exercises are included.

https://www.springer.com/productFlyer_978-0-7923-7435-0.pdf?SGWID=0-0-1297-33300503-0

Fuzzy Set Theory - and Its Applications

I have developed "tennis elbow" from lugging this book around the past four weeks, but it is worth the pain, the effort, and the aspirin. It is also worth the (relatively speaking) bargain price. Including appendixes, this book contains 894 pages of text. The entire panorama of the neural sciences is surveyed and examined, and it is comprehensive in its scope, from genomes to social behaviors. The editors explicitly state that the book is designed as "an introductory text for students of biology, behavior, and medicine," but it is hard to imagine any audience, interested in any fragment of neuroscience at any level of sophistication, that would not enjoy this book. The editors have done a masterful job of weaving together the biologic, the behavioral, and the clinical sciences into a single tapestry in which everyone from the molecular biologist to the practicing psychiatrist can find and appreciate his or

Principles of Neural Science

How to make a decision: The analytic hierarchy process

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 decline. But the total volume of oil consumption will still rise, and it will remain the largest single source of energy for the next two decades.

Oil Prices: 1985 and 1990

Neurons the decision makers, Part I: The firing function of a single neuron

where the aij, b1, c; are constants. In general, a set of linear inequalities defines a convex set in affine space; n-dimensional affine space A. is the set of points (cl, c2, * *, c.) where cl, C2, . . *, cn are real numbers. A set is convex in affine space if it contains the segment joining any two of its points. The convex set defined by (2) is the intersection of half spaces. It has vertices, edges, etc. Geometrically, a solution to the above problem is a point of the convex set defined by (2) which also maximizes (minimizes) (1). This point is generally a vertex of the boundary. For simplicity assume that (1) and (2) are normalized. Then this is the vertex (xi, x, * , x?) for which (1) assumes its maximum (or minimum) distance from the origin. Briefly, the "simplex process" is one of several iterative procedures used in solving this problem. The iterations of this process consist of translating the hyperplane corresponding to (1) in a parallel direction each time evaluating its distance from the origin at the intersected vertex of the convex set defined by the inequalities. The process is repeated in such a way that the translated hyperplane in each step yields improved results leading to the optimal distance. The simplex process also involves a useful criterion which eliminates some of the vertices of the convex set as possible trial points [3]. The number of iterations involved is decided for each problem separately. Unfortunately, so far, this number is only known to be dominated by Cn+m,n and as will be seen below this is an unsatisfactory estimate of the number of vertices of the convex sets of this problem. One would like an upper bound to the number of trial vertices knowing the number of inequalities and variables involved (the number of variables determines n, the dimension of the space). Hence the upper bound we wish to study here is of the form: (number of vertices)n< (a function of the number of inequalities) , n being the dimension of the space. Since each inequality defines at most a half-space with a hyperplane of dimension (n-1) as boundary, it suffices to study this problem in general in the form: (number of vertices)n< (a function of the number of (n-1) dimensional hyperplanes),. A rough upper bound which does not always take into consideration the polyhedral [1] property of the problem is available. If we denote by Fi(i = 0, 1, ... , n -1) the number of ith dimensional faces of a convex polyhedron in n-dimensional affine space then we have,

http://www.jstor.org/stable/pdfplus/2307037.pdf

The Number of Vertices of a Polyhedron

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 same, regardless of the kind of medal involved. Perhaps one reason it is done this way is because there has not been a scientific way to assign appropriate weights to each type of medal. This paper explores use of the measurement theory, the Analytic Hierarchy Process (AHP), to quantify the values of gold, silver and bronze medals and use these values to compute the total value of the medals won by the leading countries in order to determine which country may be considered the winner of the 21st Winter Olympics held February 12–28, 2010, in Vancouver, Canada. Copyright © 2010 John Wiley & Sons, Ltd.

Who Won the Winter 2010 Olympics?: A Quest into Priorities and Rankings

In model building using the AHP, sensitivity analysis is a crucial step in determining if the solution is implementable and robust. For example, Zhong and Gu (2010) developed an AHP model to assess black-start schemes for fast restoration of a power system.

Thomas L. Saaty

Papers

Oil Prices: 1985 and 1990

Neurons the decision makers, Part I: The firing function of a single neuron

The Number of Vertices of a Polyhedron

Who Won the Winter 2010 Olympics?: A Quest into Priorities and Rankings

Sensitivity Analysis in the Analytic Hierarchy Process