A Method with Feedback for Aggregation of Group Incomplete Pair-Wise Comparisons

A METHOD WITH FEEDBACK FOR AGGREGATION OF GROUP INCOMPLETE PAIR-WISE COMPARISONS Authors: Vitaliy Tsyganok* (ORCID 0000-0002-0821-4877), Sergii Kadenko*, Oleh Andriichuk*, Pavlo Roik* *Institute for information recording of the National academy of sciences of Ukraine 2, Shpak str. 03113, Kyiv, Ukraine ABSTRACT A method for aggregation of expert estimates in small groups is proposed. The method is based on combinatorial approach to decomposition of pair-wise comparison matrices and to processing of expert data. It also uses the basic principles of AHP/ANP approaches, such as building of criteria hierarchy to decompose and describe the problem, and evaluation of objects by means of pair-wise comparisons. It allows to derive priorities based on group incomplete pair-wise comparisons and to organize feedback with experts in order to achieve sufficient agreement of their estimates. Double entropy inter-rater index is suggested for usage as agreement measure. Every expert is given an opportunity to use the scale, in which the degree of detail (number of points/grades) most adequately reflects this expert’s competence in the issue under consideration, for every single pair comparison. The method takes all conceptual levels of individual expert competence (subject domain, specific problem, individual pair-wise comparison matrix, separate pair-wise comparison) into consideration. The method is intended to be used in the process of strategic planning in weakly-structured subject domains. Keywords: group decision making; incomplete pair-wise comparisons; combinatorial method; feedback with experts; scales with different numbers of grades. 1. Introduction During the last several decades expert estimation proved to be the most effective (and sometimes the only) reliable approach, allowing people to make competent and informed decisions in weakly structured subject domains. Weakly-structured domains are domains, influenced by multiple inter-related factors, or criteria, both tangible and intangible. Not all of these factors can be described by quantitave indicators. The only way to describe decision variants according to intangible factors (criteria) is to ask experts from the target subject domain to evaluate them. Aggregation of data, obtained from multiple experts to facilitate decision-making in a given domain, described by multiple criteria, calls for particular techniques. These techniques were developed and improved for decades, resulting in emergence of several areas of decision science, such as group decision-making, multi-criteria decision aid (MCDA), multi-criteria decision-making (MCDM), and others. Group decision-making represents one of the most important components of decision science, as, in order to improve the reliability of the information obtained from experts it is preferable to use a group of experts, rather than just an individual expert, no matter how competent he or she is (as stated, among others, by (Saaty & Peniwati, 2007)). Beside that, obtaining of data from a group of experts allows the decision-maker to utilize the redundancy of information, which is a powerful and universal property. Various approaches have been utilized to facilitate group decision-making in different areas. Most recent academic efforts in the area of group decision-making were covered, among others, in the following papers: (He, He, & Huang, 2017), (Khaleie & Fasanghari 2012), (Lu, Zhang, & Ruan, 2008), (Mohammadi & Makui 2016), (Pérez, Wikström, Mezei et al., 2013), (Xu & Chen, 2008), (Zhang, Ma, Liu et al. 2012). The best way to evaluate decision variants, that cannot be numerically described, is to compare them among themselves. This suggestion, made and empirically proved by Tom Saaty and his followers (Saaty, 2008), provided the basis for a whole family of AHP/ANP expert evaluation methods, that is widely used by decision-makers around the world. Pair-wise comparisons still remain in the focus of academic attention. As of now, pair-wise comparison-based approaches are used in combination with neural networks, fuzzy logic, group theory, and even genetic algorithms. Some recent publications on pair-wise comparison-based approaches include (Dong et al. 2015), (Fedrizzi & Brunelli, 2010), (Krejci, 2015), (Zhang & Chen, 2016), (Cavallo & D’Appuzzo, 2012), (Tsyganok, Kadenko, & Andriichuk, 2015). Aggregation of pair-wise comparison matrices (PCM) is an important problem, for which multiple solution methods were suggested, including geometric mean, arithmetic mean, logarithmic least squares, and others. Well-known approaches to aggregation of individual expert estimates are described by (Forman & Peniwati, 1998). Comparative analysis of different PCM aggregation methods was attempted by many authors, including, for example, (Choo & Wedley, 2004), (Tsyganok, 2010), (Lundi, Siraj, & Greco, 2016). We feel that publications, dedicated to PCM aggregation methods themselves, and to th

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