471 research outputs found
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
An inconsistency control system based on incomplete pairwise comparison matrices
Incomplete pairwise comparison matrix was introduced by Harker in 1987 for the case in which the decision maker does not fill in the whole matrix completely due to, e.g., time limitations. However, incomplete matrices occur in a natural way even if the decision maker provides a completely filled in matrix in the end. In each step of the total n(nâ1)/2, an incomplete pairwise comparison is given, except for the last one where the matrix turns into complete. Recent results on incomplete matrices make it possible to estimate inconsistency indices CR and CM by the computation of tight lower bounds in each step of the filling in process. Additional information on ordinal inconsistency is also provided. Results can be applied in any decision support system based on pairwise comparison matrices. The decision maker gets an immediate feedback in case of mistypes, possibly causing a high level of inconsistency
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
An LP-based inconsistency monitoring of pairwise comparison matrices
A distance-based inconsistency indicator, defined by the third author for the consistency-driven pairwise comparisons method, is extended to the incomplete case. The corresponding optimization problem is transformed into an equivalent linear programming problem. The results can be applied in the process of filling in the matrix as the decision maker gets automatic feedback. As soon as a serious error occurs among the matrix elements, even due to a misprint, a significant increase in the inconsistency index is reported. The high inconsistency may be alarmed not only at the end of the process of filling in the matrix but also during the completion process. Numerical examples are also provided
On pairwise comparison matrices that can be made consistent by the modification of a few elements
Pairwise comparison matrices are often used in Multi-attribute Decision Making forweighting the attributes or for the evaluation of the alternatives with respect to a criteria. Matrices provided by the decision makers are rarely consistent and it is important to index the degree of inconsistency. In the paper, the minimal number of matrix elements by the modification of which the pairwise comparison matrix can be made consistent is examined. From practical point of view, the modification of 1, 2, or, for larger matrices, 3 elements seems to be relevant. These cases are characterized by using the graph representation of the matrices. Empirical examples illustrate that pairwise comparison matrices that can be made consistent by the modification of a few elements are present in the applications
A simplified implementation of the least squares solution for pairwise comparisons matrices
This is a follow up to "Solution of the least squares method problem of pairwise comparisons matrix" by BozĂłki published by this journal in 2008. Familiarity with this paper is essential and assumed. For lower inconsistency and decreased accuracy, our proposed solutions run in seconds instead of days. As such, they may be useful for researchers willing to use the least squares method (LSM) instead of the geometric means (GM) method
On Saaty's and Koczkodaj's inconsistencies of pairwise comparison matrices
The aim of the paper is to obtain some theoretical and numerical properties of Saatyâs and Koczkodajâs inconsistencies of pairwise comparison matrices (PRM). In the case of 3 Ă 3 PRM, a differentiable one-to-one correspondence is given between Saatyâs inconsistency ratio and Koczkodajâs inconsistency index based on the elements of PRM. In order to make a comparison of Saatyâs and Koczkodajâs inconsistencies for 4 Ă 4 pairwise comparison matrices, the average value of the maximal eigenvalues of randomly generated n Ă n PRM is formulated, the elements aij (i < j) of which were randomly chosen from the ratio scale ... ...
with equal probability 1/(2M â 1) and a ji is defined as 1/a ij . By statistical analysis, the empirical distributions of the maximal eigenvalues of the PRM depending on the dimension number are obtained. As the dimension number increases, the shape of distributions gets similar to that of the normal ones. Finally, the inconsistency of asymmetry is dealt with, showing a different type of inconsistency
SĂșlyok meghatĂĄrozĂĄsa pĂĄros összehasonlĂtĂĄs mĂĄtrixok legkisebb nĂ©gyzetes közelĂtĂ©se alapjĂĄn
A pĂĄros összehasonlĂtĂĄsok mĂłdszere a többszempontĂș döntĂ©si feladatok megoldĂĄsĂĄnak egy lehetsĂ©ges eszköze mind a szempontsĂșlyok meghatĂĄrozĂĄsĂĄban, mind az alternatĂvĂĄk Ă©rtĂ©kelĂ©sĂ©ben. A szempontokat pĂĄronkĂ©nt összehasonlĂtva, fontossĂĄgaiknak a döntĂ©shozĂł ĂĄltal megĂtĂ©lt arĂĄnyait mĂĄtrixba rendezve a feladat a sĂșlyvektor meghatĂĄrozĂĄsa Ășgy, hogy annak komponensei valamilyen Ă©rtelemben jĂłl illeszkedjenek a döntĂ©shozĂł ĂĄltal megadott Ă©rtĂ©kekhez. A pĂĄros összehasonlĂtĂĄs mĂĄtrixbĂłl a sĂșlyok kiszĂĄmĂtĂĄsĂĄra leggyakrabban hasznĂĄlt sajĂĄtvektor mĂłdszer (Analytic Hierarchy Process) mellett szĂĄmos tĂĄvolsĂĄgminimalizĂĄlĂł mĂłdszer is lĂ©tezik. Ezek egyike a legkisebb nĂ©gyzetek mĂłdszere, melynek megoldĂĄsa nemlineĂĄris, nemkonvex fĂŒggvĂ©ny feltĂ©teles optimalizĂĄlĂĄsĂĄt jelenti. A cikkben olyan mĂłdszereket mutatunk be a pĂĄros összehasonlĂtĂĄs mĂĄtrixok legkisebb nĂ©gyzetes becslĂ©sĂ©re, amelyek a cĂ©lfĂŒggvĂ©ny összes lokĂĄlis Ă©s globĂĄlis minimumhelyĂ©nek meghatĂĄrozĂĄsĂĄra alkalmasak
A method for solving LSM problems of small size in the AHP
The Analytic Hierarchy Process (AHP) is one of the most popular methods used in Multi-Attribute Decision Making. It provides with ratio-scale measurements of the prioirities of elements on the various leveles of a hierarchy. These priorities are obtained through the pairwise comparisons of elements on one level with reference to each element on the immediate higher level. The Eigenvector Method (EM) and some distance minimizing methods such as the Least Squares Method (LSM), Logarithmic Least Squares Method (LLSM), Weighted Least Squares Method (WLSM) and Chi Squares Method (X2M) are of the tools for computing the priorities of the alternatives. This paper studies a method for generating all the solutions of the LSM problems for 3 Ă 3 matrices. We observe non-uniqueness and rank reversals by presenting numerical results
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