Fundamenta Informaticae - Volume 144, issue 3-4

Authors: Janicki, Ryszard | Kułakowski, Konrad

Article Type: Other

DOI: 10.3233/FI-2016-1329

Citation: Fundamenta Informaticae, vol. 144, no. 3-4, pp. i-iii, 2016

Authors: Saaty, Thomas L.

Article Type: Research Article

Abstract: 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 that in evolution, little creatures had and now still have a “feeling” about the environment in which they find themselves by sensing many things at once. Through evolution, sensing has been transformed to thinking and to discrete comparisons. This paper presents some material from other works written about continuous comparisons.

Keywords: Pairwise comparisons, continuous comparisons, systems of equations, Fredholm’s equation of the second kind, eigenvectors and eigenfunctions

DOI: 10.3233/FI-2016-1330

Citation: Fundamenta Informaticae, vol. 144, no. 3-4, pp. 213-221, 2016

Authors: Sandrasagra, Barbara | Soltys, Michael

Article Type: Research Article

Abstract: We propose a research program for developing a formal framework for ranking procedures based on the Pairwise Comparisons method. We are interested in the case where relatively few items are to be ranked with a complex procedure and according to a large number of criteria. A typical example of this scenario is a competition where several companies bid for a contract, and where the selection of the winner is done with multiple criteria according to an intricate selection procedure. Several other applications are suggested.

DOI: 10.3233/FI-2016-1331

Citation: Fundamenta Informaticae, vol. 144, no. 3-4, pp. 223-240, 2016

Authors: Schumann, Andrew | Woleński, Jan

Article Type: Research Article

Abstract: This paper examines two main possibilities of pairwise comparisons analysis: first, pairwise comparisons within a lattice, in this case these comparisons can be measurable by numbers; second, comparisons beyond any lattice, in this case these comparisons cannot be measurable in principle. We show that the first approach to pairwise comparisons analysis is based on the conventional square of opposition and its generalization, but the second approach is based on unconventional squares of opposition. Furthermore, the first approach corresponds to lateral inhibition in transmission signals and the second approach corresponds to lateral activation in transmission signals.

Keywords: Logical relations, logical square, logical hexagon, logical octagon, lateral inhibition, lateral activation, pairwise comparison

DOI: 10.3233/FI-2016-1332

Citation: Fundamenta Informaticae, vol. 144, no. 3-4, pp. 241-254, 2016

Authors: Pérez, Joaquín | Mokotoff, Ethel

Article Type: Research Article

Abstract: This paper shows an example of strong rank reversal in group decision making. Decision makers have preferences expressed through a reciprocal paired comparison matrix. Every one of them applies the eigenvector priority function to her paired comparison matrix to obtain her individual priority vector and then a group priority vector is computed by any of the following two procedures: a) Averaging the already computed individual priority vectors, and b) Averaging the entries of the comparison matrices to obtain a group comparison matrix, and applying to it the eigenvector priority function. Strong rank reversal means that there is one alternative that …has the highest priority for every decision maker, and consequently the highest priority in the averaged priority vector obtained by procedure (a), but loses such highest priority when procedure (b) is applied. Show more

Keywords: Rank Reversal, Paired Comparison Matrix, Reciprocal matrix, AHP

DOI: 10.3233/FI-2016-1333

Citation: Fundamenta Informaticae, vol. 144, no. 3-4, pp. 255-261, 2016

Authors: Choo, Eng U. | Wedley, William C. | Wijnmalen, Diederik J.D.

Article Type: Research Article

Abstract: A ratio pairwise comparison matrix estimates another matrix of true ratios between objects. From the pairwise comparison matrix, various methods are used to derive a priority vector and associated consistent matrix that also estimates the matrix of true ratios. The distance from the consistent matrix and the true matrix measures the accuracy of a method. The geometric mean is shown to be the only method with error below a basic threshold while being invariant to any reordering and rescaling of columns. Besides being simple to calculate, the geometric mean has excellent performance and many desirable properties.

Keywords: Ratio scale preferences, pairwise ratio comparisons, relevant ratio information

DOI: 10.3233/FI-2016-1334

Citation: Fundamenta Informaticae, vol. 144, no. 3-4, pp. 263-278, 2016

Authors: Ábele-Nagy, Kristóf | Bozóki, Sándor

Article Type: Research Article

Abstract: Efficiency, the basic concept of multi-objective optimization is investigated for the class of pairwise comparison matrices. A weight vector is called efficient if no alternative weight vector exists such that every pairwise ratio of the latter’s components is at least as close to the corresponding element of the pairwise comparison matrix as the one of the former’s components is, and the latter’s approximation is strictly better in at least one position. A pairwise comparison matrix is called simple perturbed if it differs from a consistent pairwise comparison matrix in one element and its reciprocal. One of the classical weighting methods, …the eigenvector method is analyzed. It is shown in the paper that the principal right eigenvector of a simple perturbed pairwise comparison matrix is efficient. An open problem is exposed: the search for a necessary and sufficient condition of that the principal right eigenvector is efficient. Show more

Keywords: pairwise comparison matrix, efficiency, Pareto optimality, eigenvector

DOI: 10.3233/FI-2016-1335

Citation: Fundamenta Informaticae, vol. 144, no. 3-4, pp. 279-289, 2016

Authors: Koczkodaj, Waldemar W. | Mikhailov, Ludmil | Redlarski, Grzegorz | Soltys, Michael | Szybowski, Jacek | Tamazian, Gaik | Wajch, Eliza | Yuen, Kevin Kam Fung

Article Type: Research Article

Abstract: This study has been inspired by numerous requests for clarification from researchers who often confuse Saaty’s Analytic Hierarchy Process (AHP) with the pairwise comparisons (PC) method, taking AHP as the only representation of PC. This study should be regarded as an interpretation and clarification of past investigations of PC. In addition, this article is a reflection on general PC research at a higher level of abstraction: the philosophy of science. It delves into the foundations and implications of pairwise comparisons. Some results of this study are based on a recently published work by Koczkodaj and Szwarc. Finally, open problems have …also been reported for future research. Show more

Keywords: pairwise comparisons, inconsistency analysis, ratio scale, approximation

DOI: 10.3233/FI-2016-1336

Citation: Fundamenta Informaticae, vol. 144, no. 3-4, pp. 291-307, 2016

Authors: Csató, László | Rónyai, Lajos

Article Type: Research Article

Abstract: A special class of preferences, given by a directed acyclic graph, is considered. They are represented by incomplete pairwise comparison matrices as only partial information is available: for some pairs no comparison is given in the graph. A weighting method satisfies the linear order preservation property if it always results in a ranking such that an alternative directly preferred to another does not have a lower rank. We study whether two procedures, the Eigenvector Method and the Logarithmic Least Squares Method meet this axiom. Both weighting methods break linear order preservation, moreover, the ranking according to the Eigenvector Method depends …on the incomplete pairwise comparison representation chosen. Show more

Keywords: Directed acyclic graph, incomplete pairwise comparison matrix, Eigenvector Method, Logarithmic Least Squares Method

DOI: 10.3233/FI-2016-1337

Citation: Fundamenta Informaticae, vol. 144, no. 3-4, pp. 309-320, 2016

Authors: Brunelli, Matteo

Article Type: Research Article

Abstract: This paper recalls the definition of consistency for pairwise comparison matrices and briefly presents the concept of inconsistency index in connection to other aspects of the theory of pairwise comparisons. By commenting on a recent contribution by Koczkodaj and Szwarc, it will be shown that the discussion on inconsistency indices is far from being over, and the ground is still fertile for debates.

Keywords: Pairwise comparisons, valued preference relations, consistency, inconsistency index, analytic hierarchy process

DOI: 10.3233/FI-2016-1338

Citation: Fundamenta Informaticae, vol. 144, no. 3-4, pp. 321-332, 2016

Authors: Hou, Fujun

Article Type: Research Article

Abstract: Pairwise comparisons (PC) method is an efficient technique and hierarchical analysis is a popular means coping with complex decision problems. Based on two proposed theorems, this paper shows that the PC-based hierarchical decision models stem from the weighted average methods (including the arithmetic form and the geometric form). Some issues (including the rank reversal, the criterion for acceptable consistency and the method for deriving priorities, etc) associated with the current PC-based hierarchical models (including the AHP, the multiplicative AHP and the FPR-AHP) are investigated. Another PC-based hierarchical decision model, which is different from the Saaty’s AHP, is introduced for applications …by virtue of its desirable traits (such as the rank preservation, the isomorphic correspondence, etc). Show more

Keywords: hierarchical decision model, weighted average method, pairwise comparisons matrix (PCM), fuzzy preference relations (FPR), isomorphic correspondence

DOI: 10.3233/FI-2016-1339

Citation: Fundamenta Informaticae, vol. 144, no. 3-4, pp. 333-348, 2016

Authors: Huang, Han-Chen | Yang, Xiaojun

Article Type: Research Article

Abstract: The analytic hierarchy process (AHP) is the most popular extension to the pairwise comparisons method which is based on the observation that it is much easier to rank several objects when restricted to two objects at one time. As the pairwise comparisons are subjective, the use of linguistic expressions rather than numerical values is straightforward and friendlier due to the uncertainties that are inherent in human judgments. In this paper, to handle the uncertainty and hesitancy in practical decisionmaking situations, we represent pairwise comparisons in AHP using hesitant cloud linguistic term sets (HCLTSs) which are proposed based on hesitant fuzzy …linguistic term sets (HFLTSs) and normal cloud models. Then, the synthetic cloud model aggregation algorithm is proposed to transform the HCLTS pairwise comparison matrix into the positive reciprocal synthetic cloud matrix. A prioritization method using the geometric mean technique is adopted, and the ranking method based on comparing of the parameters of normal cloud models is proposed. Thus, we extend the traditional AHP method in hesitant and uncertain environment, and we call it HCLTS-AHP method. The comparative linguistic expressions of preferences become more flexible and richer and are more similar to human beings’ cognitive models. Furthermore, the synthetic cloud model is consistent with objectivity and the calculations are easy to implement. An illustrated example is applied to the ranking of four alternatives to show the usefulness of the proposed HCLTS-AHP method. Show more

Keywords: Pairwise comparison, Analytic hierarchy process (AHP), Normal cloud model, Hesitant cloud linguistic term set, Synthetic cloud model aggregation algorithm

DOI: 10.3233/FI-2016-1340

Citation: Fundamenta Informaticae, vol. 144, no. 3-4, pp. 349-362, 2016

Authors: Park, Jin Han | Kim, Ji Yu | Kwun, Young Chel

Article Type: Research Article

Abstract: The geometric Bonferroni mean (GBM) is an important aggregation technique which reflects the correlations of aggregated arguments. Based on the GBM, in this paper, we develop the optimized weighted geometric Bonferroni mean (OWGBM) and the generalized optimized weighted geometric Bonferroni mean (GOWGBM), whose characteristics are to reflect the preference and interrelationship of the aggregated arguments. Furthermore, we develop the intuitionistic fuzzy optimized weighted geometric Bonferroni mean (IFOWGBM) and the generalized intuitionistic fuzzy optimized weighted geometric Bonferroni mean (GIFOWGBM), and study their desirable properties such as idempotency, commutativity, monotonicity and boundedness. Finally, based on the IFOWGBM and GIFOWGBM, we present an …approach to multi-criteria decision making and illustrate it with a practical example. Show more

Keywords: Intuitionistic fuzzy optimized weighted geometric Bonferroni mean (IFOWGBM), generalized intuitionistic fuzzy optimized weighted geometric Bonferroni mean (GIFOWGBM), multi-criteria decision making

DOI: 10.3233/FI-2016-1341

Citation: Fundamenta Informaticae, vol. 144, no. 3-4, pp. 363-381, 2016

Authors: Blagojevic, Bosko | Srdjevic, Bojan | Srdjevic, Zorica | Zoranovic, Tihomir

Article Type: Research Article

Abstract: In AHP group decision making it is desirable that decision makers achieve the highest degree of consensus concerning the group priority vector at both levels, local and the final. Based on this philosophy, we have developed a method to derive local group priority vector respecting the three group consistency measures: geometric cardinal consensus index (GCCI ), group minimum violation coefficient (GMV ) and ordinal consensus measure (OCM ). Consistency of individual decisions against the group decision serves as an input to determine the weights of decision makers participating in the group and generate the group local priority vector. Proposed method …rewards cooperation between members of the group and raises chances for their consensus. Show more

Keywords: AHP group decision making, weights of decision makers, group consistency measures

DOI: 10.3233/FI-2016-1342

Citation: Fundamenta Informaticae, vol. 144, no. 3-4, pp. 383-395, 2016

Authors: Farkas, András

Article Type: Research Article

Abstract: We discuss the development and use of a recursive rank-one residue iteration (triple R-I) to balancing pairwise comparison matrices (PCMs). This class of positive matrices is in the center of interest of a widely used multi-criteria decision making method called analytic hierarchy process (AHP). To find a series of the ‘best’ transitive matrix approximations to the original PCM the Newton-Kantorovich (N-K) method is employed for the solution to the formulated nonlinear problem. Applying a useful choice for the update in the iteration, we show that the matrix balancing problem can be transformed to minimizing the Frobenius norm. Convergence proofs for …this scaling algorithm are given. A comprehensive numerical example is included to illustrate the useful features to measuring and reducing perturbation errors and inconsistency of a PCM as a result of the respondents’ judgments on the pairwise comparisons. Show more

Keywords: numerical mathematics, matrix balancing, diagonal similarity scaling, pairwise comparison matrix

DOI: 10.3233/FI-2016-1343

Citation: Fundamenta Informaticae, vol. 144, no. 3-4, pp. 397-417, 2016

Article Type: Other

Citation: Fundamenta Informaticae, vol. 144, no. 3-4, pp. 419-420, 2016