This article focuses on a group decision-making (GDM) approach based on the multiplicative consistency of linguistic preference relation (LPR) with experts' self-confidence and self-doubting (SC&SD) levels. To give their preferences, the experts use their knowledge of the experience according to their degree of SC&SD levels. First, we propose the concepts of multiplicative consistent LPR-SC&SD using the experts' general minimum self-confidence level and the maximum self-doubting level. We suggest then a consensus-building iterative process, that is, a consensus reaching process (CRP) algorithm to achieve multiplicative consistency of LPR-SC&SD according to identification and adjustment rules. A theorem is given for convergence of the CRP algorithm. In a GDM problem, social network analysis is studied for the experts to obtain their weight according to the degree of SC&SD. When we achieve the acceptable preferences of all the experts using the CRP algorithm of the multiplicative consistent LPR-SC&SD, then we aggregate all the preferences by the experts' weights. The aggregation of all preferences is also an LPR-SC&SD, known as the weight collective LPR-SC&SD. Finally, a case-by-case example and several comparative analyses are done with the current GDM processes to demonstrate the viability and applicability of the proposed GDM system.

In this paper, we introduce a new type of fuzzy set, called Pythagorean linguistic sets (PLSs), to address the preferred and nonpreferred degrees of linguistic variables. Moreover, it allows decision makers to offer effectively handle uncertain information more flexible than intuitionistic linguistic sets (ILSs) when one compares two alternatives in the process of decision making. Some of the fundamental operational laws, score, accuracy, and aggregation operators are defined, and their properties are investigated. Preference relation (PR) is a useful and efficient tool for decision making that only requires the decision makers to compare two alternatives at one time. Taking the advantages of PLSs and PRs, this paper also introduces Pythagorean linguistic preference relations (PLPRs) and studies their application. We propose an approach for group decision making using group recommendations based on consistency matrices and feedback mechanism. First, the proposed method constructs the collective consistency matrix, the weight collective PRs, and the group collective PRs. Then, it constructs a consensus relation for each expert and determines the group consensus degree (GCD) for all experts. If the GCD is smaller than a predefined threshold value, then a feedback mechanism is activated to update the PLPRs. Finally, after the GCD is greater than or equal to the predefined threshold value, we calculate the arithmetic mathematical average values of the updated group collective PR to select the most appropriate alternative.

Fuzzy planar graph is a very important subclass of fuzzy graph. In this paper, two types of edges are mentioned for fuzzy graphs: effective edges and considerable edges. In addition, a comparative study between Kuratowski's graphs and fuzzy planar graph is made. A new concept of a strong fuzzy planar graph is introduced. Some related results are established. These results have certain applications in subway tunnels, routes, oil/gas pipelines representation, etc. It is also shown that an image can be represented by a fuzzy planar graph, and contraction of such an image can be made with the help of a fuzzy planar graph.