7,764 research outputs found
Methods of tropical optimization in rating alternatives based on pairwise comparisons
We apply methods of tropical optimization to handle problems of rating
alternatives on the basis of the log-Chebyshev approximation of pairwise
comparison matrices. We derive a direct solution in a closed form, and
investigate the obtained solution when it is not unique. Provided the
approximation problem yields a set of score vectors, rather than a unique (up
to a constant factor) one, we find those vectors in the set, which least and
most differentiate between the alternatives with the highest and lowest scores,
and thus can be representative of the entire solution.Comment: 9 pages, presented at the Annual Intern. Conf. of the German
Operations Research Society (GOR), Helmut Schmidt University Hamburg,
Germany, August 30 - September 2, 201
Solution of the Least Squares Method problem of pairwise comparison matrices
The aim of the paper is to present a new global optimization
method for determining all the optima of the Least Squares Method (LSM) problem of pairwise comparison matrices. Such matrices are used, e.g., in the Analytic Hierarchy Process (AHP). Unlike some other distance minimizing methods, LSM is usually hard to solve because of the corresponding nonlinear and non-convex objective function. It is found that the optimization problem can be reduced to solve a system of polynomial equations. Homotopy method is applied which is an efficient technique for solving nonlinear systems. The paper ends by two numerical example having multiple global and local minima
Solving the Least Squares Method problem in the AHP for 3 X 3 and 4 X 4 matrices
The Analytic Hierarchy Process (AHP) is one of the most popular methods used in Multi-Attribute Decision Making. The Eigenvector Method (EM) and some distance minimizing methods such as the Least Squares Method (LSM) are of the possible tools for computing the priorities of the alternatives. A method for generating all the solutions of the LSM problem for 3 Ă 3 and 4 Ă 4 matrices is discussed in the paper. Our algorithms are based on the theory of resultants
The Use of the AHP in Civil Engineering Projects
Most engineering, economic, social and institutional decisions are made with explicit notions of optimal behavior and implicit human motivations. In such a process, manipulation of both tangible and intangible data and satisfaction of multiple criteria are essential to the success of decision-making. In this paper an approach to multiple-criteria decision making known as the analytic hierarchy process (AHP) is presented. Some mathematical details of the procedure are briefly discussed. The application of the method to a real life civil engineering project for the selection of an appropriate bridge design is also presented.multi-criteria decision making, analytic hierarchy process, bridge design
Rating alternatives from pairwise comparisons by solving tropical optimization problems
We consider problems of rating alternatives based on their pairwise
comparison under various assumptions, including constraints on the final scores
of alternatives. The problems are formulated in the framework of tropical
mathematics to approximate pairwise comparison matrices by reciprocal matrices
of unit rank, and written in a common form for both multiplicative and additive
comparison scales. To solve the unconstrained and constrained approximation
problems, we apply recent results in tropical optimization, which provide new
complete direct solutions given in a compact vector form. These solutions
extend known results and involve less computational effort. As an illustration,
numerical examples of rating alternatives are presented.Comment: 16 pages. arXiv admin note: substantial text overlap with
arXiv:1503.0400
Budapest Bridges Benchmarking
This paper is concerned with the comparison of different scaling methods which are applied to a complex bridge evaluation problem. It is shown that both tangible and intangible data and satisfaction of multiple criteria are essential to the success of such projects. Some new inconsistency measures for the matrices emerging in the decision making process are also used. A detailed numerical analysis of the results is presented.
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