سنتز بهترین بردارهای اولویتی محلی فردی در تصمیم گیری تحلیل سلسله مراتبی گروهی

An assessment of the individual judgments and AHP-produced priority vectors for involved decision-makers indicates that the individual consistencies of decision makers may vary significantly, thus making the final group decision less reliable. In this paper, an approach is proposed as to how to combine decision makers’ local priority vectors in AHP synthesis and reduce so-called group inconsistency. Instead of aggregating individual judgments (AIJ), or aggregating individually derived final priorities (AIP), we propose to perform an AHP synthesis of the best local priority vectors taken from the most consistent decision makers. The approach and related algorithm we label as MGPS after the key terms ‘multicriteria group prioritization synthesis.’ The concept is analogous to the one proposed by Srdjevic [1] for individual AHP applications where the best local priority vectors are selected based on the consistency performance of several of the most popular prioritization methods. Here, decision makers are combined instead of prioritization methods, and group context is fully implemented. After completing an evaluation of the decision makers inconsistencies in each node of the hierarchy, the selected best local priority vectors are synthesized in a standard manner, and the final solution is declared to be an AHP-group decision. Two numerical examples indicate that the developed approach and algorithm generate the final priorities of alternatives with the lowest overall inconsistency (in the multicriteria sense).

Group decision making problems arise from many real-world situations in many fields such as human spaceflight mission planning, water management, and selection of advanced technology (Choudhury et al. [2], Srdjevic [3], Tavana and Hatami-Marbini [4]). In recent times, most of these problems are attacked and successfully solved with the application of the analytic hierarchy process (AHP), developed by Saaty [5]. In fact, the so-called AHP-group application means that the standard AHP-individual application is extended to provide for certain types of aggregations, consensus procedures, etc. Rich, pertinent literature in this regard is referenced appropriately in the remaining part of this text. Three mainstream theories and applications comprise AHP. The first theory applies to the preference relations - linguistic, numerical, and fuzzy (e.g. Saaty [5], Chiclana et al. [6], Fu and Yang [7], Herrera and Martínez [8], Herrera et al. [9]) - that are widely used in individual and group decision making. The second one applies to the prioritization methods for extracting cardinal information (weights of the decision elements, usually the criteria and alternatives) from so-called judgment matrices (i.e. multiplicative preference relations) at each node of a hierarchy. The third mainstream theory applies to consensus building, aggregation methods, and measuring (in)consistency in group decision making. This paper falls within this third theory and focuses on the AHP-group synthesis of the best local priority vectors identified by multicriteria evaluation of inconsistency contained in individual vectors obtained for participating members of a group. To our best knowledge, the approach we propose is novel and the related computational procedure is labeled as MGPS algorithm, after the key terms ‘multicriteria group prioritization synthesis.’ The AHP offers a variety of options in supporting the decision making processes, including group contexts. Being a soft computing technique as well as a multicriteria analysis method, the proposed approach is aimed to contribute to the soft computing community by introducing an objective method of synthesizing locally computed priority vectors for all involved individuals in a group. In general, computations within AHP are inherently simple. However, an issue of (in)consistency in decision makers in modern times presents the challenge of employing evolution strategies, genetic algorithms, particle swarm optimization and other soft computing techniques and heuristics in deriving priorities from inconsistent (or low consistency) matrices. In a way, a new paradigm aims to provide more efficient solving of the spectrum of decision making problems where individual judgments of decision makers play a leading role. The MGPS algorithm we propose is inspired by an idea for making objective the process of prioritizing alternatives vs. a global goal. We developed it to be straightforward and easy to program and implement in real life applications. Being fully programmed in FORTRAN, on a proof-of-concept level it was successfully used in many test applications as well as in two described examples presented in the second part of this paper. Most of the research papers dominantly report theoretical works and application results concentrating on just one ratio-scale matrix at a time. Much fewer research reports relate to the whole hierarchy where the original AHP philosophy belongs to and the concept of AHP synthesis is fully implemented. This paper relates to the latter case only; that is, we consider a complete AHP hierarchy, not just one local matrix. In addition, the presented work is aimed to cover several topics related to AHP application in group contexts where decision makers demonstrate different consistencies. The core of our approach is to perform AHP synthesis of the best local priority vectors taken from the most consistent decision makers. Without losing generality, if we consider a three level hierarchy (goal-criteria-alternatives), the number of matrices is 1 + na, where na is the number of alternatives. If there are K decision makers then there are K(1 + na) matrices, and in every node of a hierarchy there is a set of K matrices with their priority vectors. Some vectors are more consistent than others, and it is possible to identify the best one if certain criteria are adopted for assessing their quality regarding consistency. On the other hand, consistency measures applicable to AHP can be divided into two groups: (a) general (e.g. Euclidean Distance or Minimum Violation), and (b) specific (e.g. prioritization method related, such as CR for eigenvector method or GCI for logarithmic least square method). If consistency measures are used as criteria for assessing the quality of priority vectors derived from decision makers at a given node of hierarchy, then the best (optimal in multicriteria sense) priority vector can be identified and propagated to the final AHP synthesis. Obviously, the final synthesis is performed with the best vectors and therefore the final (group) priorities of alternatives are objectively the best possible. The paper is organized as follows: in Section 2, we briefly present reviewed related research, basic preliminary knowledge of AHP, including its most commonly used prioritization method known as the eigenvector (EV) method. In Section 3, we develop an approach to AHP-group synthesis following an idea presented in Srdjevic [1], and instead of AIJ or AIP aggregations, we propose to combine the local priorities derived from group members based on a multicriteria evaluation of their demonstrated consistencies. In Sections 4 and 5 two illustrative examples from real-life AHP-group applications are provided, and the results of MGPS application are discussed. Section 6 presents concluding remarks and an agenda for future research in the subject area

In this paper, we present a novel approach to AHP-group applications based on the synthesis of individually optimal local priority vectors obtained by participating decision makers. The approach is called the MGPS (group-related) algorithm and is analogous to the MPS (prioritization method-related) algorithm developed in the authors’ earlier work. In every node of a given hierarchy, decision makers perform judgments independently from each other and create multiplicative preference relations, known also as judgment or pairwise comparison matrices. After a prioritization method is applied to derive weights from these matrices for all decision makers (members of a group), a multicriteria analysis is performed to identify the best priority vector across the group. The decision matrix for multicriteria analysis consists of individually computed priority vectors as alternatives, and a criteria set represented by consistency measures that are applicable when rating the performance of individual priority vectors. The identified best node-dependent (local) priority vectors are used in the standard final AHP synthesis. This way, the final result is obtained in an objective way, avoiding the application of any consensus model or associating weights to the decision makers that can be subjected to various manipulations that can be both positive and negative. Our approach is based on use of the eigenvector (EV) method for local prioritization, and we selected three relevant consistency measures that go with the EV method for multicriteria analysis of individual priority vectors obtained in all nodes of a hierarchy. Multicriteria analysis is performed locally for one node at a time. Regarding the adopted consistency measures, recall that the consistency ratio (CR) applies to the eigenvector method only, while Euclidean distance (ED) and minimum violation (MV) are general consistency measures applicable to whichever prioritization method is used. Future research agenda will include the other two well-known consistency measures: (a) geometric consistency index (GCI) applicable if the logarithmic least squares (LLS) method is used for prioritization, or (b) fuzzy consistency measure (μ) that applies to the fuzzy preference programming (FPP) method. Note that these two measures cannot be used when EV prioritization is performed, as is the case in our approach; when prioritizations are performed with either method, LLS or FPP, in such cases CR should be replaced accordingly with GCI or μ, both computed as global values for the whole hierarchy; the other two measures, ED and MV, remain eligible evaluation criteria in both cases. The issue of consistency measures is a hot topic in sophisticated research environments. Further research will probably rely on recent works such as Wang et al. [35] and the presented intuitionistic fuzzy AHP (IF-AHP) approach which synthesizes the eigenvectors of an intuitionistic fuzzy comparison matrix and at the same time handles consistency and satisfactory consistency defined after some basal knowledge is introduced. IF-AHP has been shown successful in combination with the extend analysis method and modified fuzzy logarithmic least squares method. Moreover, in Fallahi et al. [36] there is an interesting application of ant colony optimization (ACO) which is used to obtain optimal solutions satisfying some path planning criteria. Fuzzy AHP is then employed in the decision making judgments due to their inherent vagueness and uncertainty. Finally, a bi-criteria evolution strategy has been used in Srdjevic and Srdjevic [37] to identify priority vectors from a given matrix. There are other works in the subject area and discussion on the tocic can be considered as still open. In general, various consistency measures can be used as criteria when evaluating local priority vectors derived by participating decision makers. Some of the measures can be highly correlated (e.g. ED and CR), and this issue deserves additional research. Also, different multicriteria methods can be used for evaluating local vectors, such as the additive weighting method (used in MGPS) and the product weighting method, or, in the case of many decision makers, the ideal-point methods TOPSIS or CP. In all cases, it is necessary to allocate weights to selected consistency measures (criteria), and this is subject to an analyst's decision. If the number of decision makers is high (say, more than ten), the entropy method can be additionally employed to assess the decision matrix, recognize the objective importance of the consistency measures, and in turn generate the weights for multicriteria analysis. Finally, in the presented approach, we commented on the results when a single consistency measure (criterion) is used for the evaluation of local vectors. This is also a possible direction for future research and a challenge to combine our approach with goal programming optimization aimed at identifying target (benchmark) points in group syntheses based on consensus models. The algorithm itself is independent of a prioritization method, but the same method should be used at all nodes and for all decision makers. In fact, this is no limitation of the MGPS algorithm. Either ‘classical’ or ‘evolution based’ prioritization method is applicable uniquely over the whole hierarchy. Related consistency measures should also be used uniquely in regards to which prioritization method is used (e.g. CR with EV, and GCI with LLS). General consistency measures such as Euclidean distance or a rank reversal indicator are directly applicable with the MGPS algorithm regardless of the prioritization method used. The next step in development could be letting the algorithm identify at each node of a hierarchy the best vector possible by computing priority vectors of all individual matrices with all approved prioritization methods and identify the one with best consistency indicators.