ElsevierBenítez López, J.; Izquierdo Sebastián, J.; Pérez García, R.; Ramos Martinez, E. (2014). A simple formula to find the closest consistent matrix to a reciprocal matrix. Applied Mathematical Modelling. 38(15-16):3968-3974. doi:10.1016Modelling. 38(15-16):3968-3974. doi:10. /j.apm.2014 A simple formula to nd the closest consistent matrix to a reciprocal matrix J. Benítez * , J. Izquierdo † , R. Pérez-García † , E. Ramos-Martínez † Abstract: Achieving consistency in pair-wise comparisons between decision elements given by experts or stakeholders is of paramount importance in decisionmaking based on the AHP methodology. Several alternatives to improve consistency have been proposed in the literature. The linearization method (Benítez et al., Achieving matrix consistency in AHP through linearization, Applied Mathematical Modelling 35 (2011) 4449-4457), derives a consistent matrix based on an original matrix of comparisons through a suitable orthogonal projection expressed in terms of a Fourier-like expansion. We propose a formula that provides in a very simple manner the consistent matrix closest to a reciprocal (inconsistent) matrix. In addition, this formula is computationally efficient since it only uses sums to perform the calculations. A corollary of the main result shows that the normalized vector of the vector, whose components are the geometric means of the rows of a comparison matrix, gives the priority vector only for consistent matrices.
The survival and regrowth of microorganisms in drinking water distribution systems (DWDSs) can be affected not only by biological aspects but also by the interaction of various other factors. Some of these factors have been found to be clearly related to biofilm development in DWDSs. However, the complexity of the microenvironment under study and the biofilm growth characteristics have so far led the various methodologies applied to produce ambiguous or not easily comparable, and thus not very useful, results. In this study we compile the information currently available on biofilm ecology in DWDSs and apply various machine learning algorithms based on naïve Bayesian networks. In addition, as a step forward, we also use ensemble methods. These methods have been widely adopted to improve the robustness and the overall prediction accuracy of single models through their accumulative experience on the performance of multiple applications in learning systems. We claim that they also reduce the high uncertainty associated with the process of biofilm development in DWDSs. Specifically, we propose alternatives for the base naïve Bayesian model to outperform its individual results while maintaining its simplicity. These alternatives include augmentation of the arcs in the graph and bagging initial approaches. Then, both ideas are combined by a hybrid algorithm that produces explanatory decision trees. As a result, it is possible to achieve a deeper understanding of the consequences that the interaction of the relevant hydraulic and physical factors of DWDSs has for biofilm development.
In many real-world applications we have at our disposal a limited number of inputs in a theoretical database with full information, and another part of experimental data with incomplete knowledge for some of their features. These are cases that can be addressed by a label propagation process. It is a widely studied approach that may acquire complexity if new constraints in the new unlabeled data that should be taken into account are found. This is the case of the membership to a group or community in graphs. The proposal is to add the Laplacian matrix as well as another different similarity measures (may be not found in the original database) in the label propagation. A kernel embedding process together with a simple label propagation algorithm will be the main tools to achieve this approach by the use of all types of available information. In order to test the functionality of this new proposal, this work introduces an experimental study of biofilm development in drinking water pipes. Then, a label propagation through pipes belonging to a complete water supply network is approached. These pipes have their own properties depending on their network location and environmental co-variables. As a result, the proposal is a suitable and efficient way to deal with practical data, based on previous theoretical studies by the constrained label propagation process introduced.
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