Two short proofs regarding the logarithmic least squares optimality in Chen, K., : Bridging the g... more Two short proofs regarding the logarithmic least squares optimality in Chen, K., : Bridging the gap between missing and inconsistent values in eliciting preference from pairwise comparison matrices, Annals of Operations Research 235(1):155-175
Pairwise comparison matrices are frequently applied in multi-criteria decision making.
A weight... more Pairwise comparison matrices are frequently applied in multi-criteria decision making.
A weight vector is called efficient if no other weight vector is at least as good in approximating the elements of the pairwise comparison matrix, and strictly better in at least one position. A weight vector is weakly efficient if the pairwise ratios
cannot be improved in all non-diagonal positions. We show that the principal eigenvector is always weakly efficient, but numerical examples show that
it can be inefficient. The linear programs proposed test whether a given weight vector is (weakly) efficient, and in case of (strong) inefficiency, an efficient (strongly) dominating
weight vector is calculated. The proposed algorithms are implemented in Pairwise Comparison Matrix Calculator, available at pcmc.online.
Pairwise comparison matrices, a method for preference modelling and quantification in multi-attri... more Pairwise comparison matrices, a method for preference modelling and quantification in multi-attribute decision making and ranking problems, are naturally extended to the incomplete case, offering a wider range of applicability. The weighting problem is to find a weight vector that reflects the decision maker's preferences is as well as possible. The logarithmic least squares problem has a unique and simply computable solution. The spanning tree approach does not assume any metric in advance, instead it goes through all minimal sufficient subsets (spanning trees) of the set of pairwise comparisons, then the weight vectors are aggregated. It is shown that the geometric mean of weight vectors, calculated from all spanning tress, is the optimal solution of the well known logarithmic least squares problem, not only for complete, but for incomplete pairwise comparison matrices as well.
The queens graph Q m×n has the squares of the m × n chessboard as its vertices; two squares are a... more The queens graph Q m×n has the squares of the m × n chessboard as its vertices; two squares are adjacent if they are both in the same row, column, or diagonal of the board. A set D of squares of Q m×n is a dominating set for Q m×n if every square of Q m×n is either in D or adjacent to a square in D. The minimum size of a dominating set of Q m×n is the domination number, denoted by γ(Q m×n ).
A recent work of the authors on the analysis of pairwise comparison matrices that can be made con... more A recent work of the authors on the analysis of pairwise comparison matrices that can be made consistent by the modification of a few elements is continued and extended. Inconsistency indices are defined for indicating the overall quality of a pairwise comparison matrix. It is expected that serious contradictions in the matrix imply high inconsistency and vice versa. However, in the 35-year history of the applications of pairwise comparison matrices, only one of the indices, namely CR proposed by Saaty, has been associated to a general level of acceptance, by the well known ten percent rule. In the paper, we consider a wide class of inconsistency indices, including CR, CM proposed by Koczkodaj and Duszak and CI by Peláez and Lamata. Assume that a threshold of acceptable inconsistency is given (for CR it can be 0.1). The aim is to find the minimal number of matrix elements, the appropriate modification of which makes the matrix acceptable. On the other hand, given the maximal number of modifiable matrix elements, the aim is to find the minimal level of inconsistency that can be achieved. In both cases the solution is derived from a nonlinear mixed-integer optimization problem. Results are applicable in decision support systems that allow real time interaction with the decision maker in order to review pairwise comparison matrices.
Efficiency is a core concept of multi-objective optimization problems and multi-attribute decisio... more Efficiency is a core concept of multi-objective optimization problems and multi-attribute decision making. In the case of pairwise comparison matrices a weight vector is called efficient if the approximations of the elements of the pairwise comparison matrix made by the ratios of the weights cannot be improved in any position without making it worse in some other position. A pairwise comparison matrix is called double perturbed if it can be made consistent by altering two elements and their reciprocals. The most frequently used weighting method, the eigenvector method is analyzed in the paper, and it is shown that it produces an efficient weight vector for double perturbed pairwise comparison matrices.
The problem of a multiple player dice tournament is discussed and solved in the paper. A die has ... more The problem of a multiple player dice tournament is discussed and solved in the paper. A die has a finite number of faces with real numbers written on each. Finite dice sets are proposed which have the following property, defined by Sch¨utte for tournaments: for an arbitrary subset of k dice there is at least one die that beats each of the k with a probability greater than 1/2. It is shown that the proposed dice set realizes the Paley tournament, that is known to have the Schütte property (for a given k) if the number of vertices is large enough. The proof is based on Dirichlet’s theorem, stating that the sum of quadratic nonresidues is strictly larger than the sum of quadratic residues.
We confirm a conjecture of Littlewood: there exist seven infinite
circular cylinders of unit radi... more We confirm a conjecture of Littlewood: there exist seven infinite circular cylinders of unit radius which mutually touch each other.
Having a pairwise comparison matrix in a multi-attribute decision problem, two basic problems ari... more Having a pairwise comparison matrix in a multi-attribute decision problem, two basic problems arise: how to compute the weight vector, and, how to associate an inconsistency index to the matrix. Two key concepts of the Analytic Hierarchy Process, the eigenvector method and inconsistency index CR are discussed. (In)efficiency is a well-known property in multiple objective optimization. We introduce a restriction of it in the paper. Given a pairwise comparison matrix A = [a ij ]
Incomplete pairwise comparison matrix was introduced by Harker in 1987 for the case in which the ... more 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, eg, 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 (n–1)/2, an incomplete pairwise comparison is given, except for the last one where the matrix turns into complete.
Abstract One of the most important steps in solving Multi-Attribute Decision Making (MADM) proble... more Abstract One of the most important steps in solving Multi-Attribute Decision Making (MADM) problems is to derive the weights of importance of the attributes. The decision maker is requested to compare the importance for each pair of attributes. The result expressed in numbers is written in a pairwise comparison matrix. The aim is to determine a weight vector w=(w1, w2,, wn), which reflects the preferences of the decision maker, in the positive orthant of the n-dimensional Euclidean space.
Pairwise comparison (PC) matrices are used in multi-attribute decision problems (MADM) in order t... more Pairwise comparison (PC) matrices are used in multi-attribute decision problems (MADM) in order to express the preferences of the decision maker. Our research focused on testing various characteristics of PC matrices. In a controlled experiment with university students (N = 227) we have obtained 454 PC matrices. The cases have been divided into 18 subgroups according to the key factors to be analyzed. Our team conducted experiments with matrices of different size given from different types of MADM problems. Additionally, the matrix elements have been obtained by different questioning procedures differing in the order of the questions. Results are organized to answer five research questions. Three of them are directly connected to the inconsistency of a PC matrix. Various types of inconsistency indices have been applied. We have found that the type of the problem and the size of the matrix had impact on the inconsistency of the PC matrix. However, we have not found any impact of the questioning order. Incomplete PC matrices played an important role in our research. The decision makers behavioral consistency was as well analyzed in case of incomplete matrices using indicators measuring the deviation from the final order of alternatives and from the final score vector.
A distance-based inconsistency indicator, defined by the third author for the consistency-driven ... more 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.
Central European Journal of Operations …, Jan 1, 2011
Pairwise comparison matrices are often used in Multi-attribute Decision Making for weighting the ... more Pairwise comparison matrices are often used in Multi-attribute Decision Making for weighting 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.
An important variant of a key problem for multi-attribute decision making is considered. We study... more An important variant of a key problem for multi-attribute decision making is considered. We study the extension of the pairwise comparison matrix to the case when only partial information is available: for some pairs no comparison is given. It is natural to define the inconsistency of a partially filled matrix as the inconsistency of its best, completely filled completion. We study here the uniqueness problem of the best completion for two weighting methods, the Eigenvector Method and the Logarithmic Least Squares Method. In both settings we obtain the same simple graph theoretic characterization of the uniqueness. The optimal completion will be unique if and only if the graph associated with the partially defined matrix is connected. Some numerical experiences are discussed at the end of the paper.
An extension of the pairwise comparison matrix is considered when some comparisons are missing. A... more An extension of the pairwise comparison matrix is considered when some comparisons are missing. A generalization of the eigenvector method for the incomplete case is introduced and discussed as well as the Logarithmic Least Squares Method. The uniqueness problem regarding both weighting methods is studied through the graph representation of pairwise comparison matrices. It is shown that the optimal completion/solution is unique if and only if the graph associated with the incomplete pairwise comparison matrix is connected. An algorithm is proposed for solving the eigenvalue minimization problem related to the generalization of the eigenvector method in the incomplete case. Numerical examples are presented for illustration of the methods discussed in the paper. Keywords -Multi-attribute decision making, incomplete pairwise comparison matrix, eigenvalue optimization 1 Research was supported in part by OTKA grants K 60480, K 77420, NK 63066, NK 72845, K77476 Least Squares Method [4,3] are two well-known and often cited weighting methods.
The aim of the paper is to obtain some theoretical and numerical properties of Saaty's and Koczko... more The aim of the paper is to obtain some theoretical and numerical properties of Saaty's and Koczkodaj 's inconsistencies of pairwise comparison matrices (PRM ). In the case of 3 × 3 PRM, a differentiable one-to-one correspondence is given between Saaty's inconsistency ratio and Koczkodaj 's inconsistency index based on the elements of PRM. In order to make a comparison of Saaty's and Koczkodaj 's inconsistencies for 4 × 4 pairwise comparison matrices, the average value of the maximal eigenvalues of randomly generated n × n PRM is formulated, the elements a ij (i < j) of which were randomly chosen from the ratio scale
Central European Journal of Operations Research, Jan 1, 2008
The aim of the paper is to present a new global optimization method for determining all the optim... more 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.
Ezúton szeretnék köszönetet mondani témavezetőmnek, dr. Rapcsák Tamásnak, amiért a szakmai támoga... more Ezúton szeretnék köszönetet mondani témavezetőmnek, dr. Rapcsák Tamásnak, amiért a szakmai támogatáson túl idejét nem kímélve rendkívül sokat tett mind e dolgozat, mind publikációim színvonalának növeléséért. Hálás vagyok azért, hogy az MTA SZTAKI Operációkutatás és Döntési Rendszerek Laboratórium és Osztály vezetőjeként az elméleti kutatások mellett gyakorlati problémák megoldásába is bevont.
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Papers by Sándor Bozóki
A weight vector is called efficient if no other weight vector is at least as good in approximating the elements of the pairwise comparison matrix, and strictly better in at least one position. A weight vector is weakly efficient if the pairwise ratios
cannot be improved in all non-diagonal positions. We show that the principal eigenvector is always weakly efficient, but numerical examples show that
it can be inefficient. The linear programs proposed test whether a given weight vector is (weakly) efficient, and in case of (strong) inefficiency, an efficient (strongly) dominating
weight vector is calculated. The proposed algorithms are implemented in Pairwise Comparison Matrix Calculator, available at pcmc.online.
circular cylinders of unit radius which mutually touch each other.