Podrobný výpis o publikaci

We first recall the classical notion of a cooperative game with transferable utility, and also the classical solution concept of nucleolus (Schmeidler, 1969) of the TU-game. ¶ We then turn our attention to pairwise comparison methods, which are often used in multiple-criteria decision-making, e.g., in the Analytic Hierarchy Process (AHP) in particular: Given several objects, such as alternatives, the decision-maker (DM) evaluates them pairwise with respect to some criterion. The relative importance of the two elements in a pair is usually rated by a positive real number with multiplicative interpretation; that is, the number indicates how many times one element is better (or more important) than the other in the pair. By considering all pairs of the objects in this way, a pairwise comparison matrix (PCM) is obtained. The purpose is to find the priority vector of the given PCM; that is, numerical weights of the objects so that the ratio of two weights is close to the pairwise comparison value given by the DM when assessing the relative importance of the respective two elements. Saaty’s Eigenvector Method (EVM) and the Geometric Mean Method (GMM) are prominently used to find the priority vector of the PCM. ¶ In this paper, we allow the PCM entries to be elements of a divisible alo-group (Abelian linearly ordered group), cf. the “general unified framework for pairwise comparison matrices in multicriteria methods” by Cavallo and D’Apuzzo (2009), which includes the aforementioned positive real numbers as a special case. Then, while Saaty’s EVM cannot be used in this setting due to its intrinsic properties, the GMM can easily be adapted to find the priority vector of the PCM with entries from a divisible alo-group (Cavallo & D’Apuzzo, 2012) and, to our best knowledge, is the only currently known method that can be used in this setting. ¶ Inspired by the aforementioned concept of nucleolus from cooperative game theory (Schmeidler, 1969), we propose a new nucleolus-like method to compute the priority vector of a pairwise comparison matrix with entries from any divisible alo-group. The method utilizes the theory of linear programming in abstract spaces (Bartl, 2007). ¶ References ¶ Bartl, D. (2007). Farkas’ Lemma, other theorems of the alternative, and linear programming in infinite-dimensional spaces: a purely linear-algebraic approach. Linear and Multilinear Algebra, 55(4), 327–353. https://doi.org/10.1080/03081080600967820 ¶ Cavallo, B., & D’Apuzzo, L. (2009). A general unified framework for pairwise comparison matrices in multicriterial methods. International Journal of Intelligent Systems, 24(4), 377–398. https://doi.org/10.1002/int.20329 ¶ Schmeidler, D. (1969). The Nucleolus of a Characteristic Function Game. SIAM Journal on Applied Mathematics, 17(6), 1163–1170. https://doi.org/10.1137/0117107