In this paper, we introduce a new type of fuzzy preference structure, called Pythagorean fuzzy preference relations (PFPRs), to describe uncertain evaluation information in group decision‐making process. Moreover, it allows decision makers to offer effectively handle uncertain information more flexibly than intuitionistic fuzzy preference relations when one compares two alternatives in the process of decision making. Using PFPRs, we propose an approach for group decision making based on group recommendations and consistency matrices. First, the proposed approach constructs the collective consistency matrix, the weight collective preference relations (PRs), and the group collective PR. Then, it construct a consensus relation for each expert and determinate the group consensus degree for all experts. If the group consensus degree is smaller than a predefined threshold value, then it identify the consensus values in each consensus relation which are smaller than the group consensus degree and updates the Pythagorean fuzzy preference values corresponding to the identified consensus values. The above process is continued, until the group consensus degree is larger than or equal to the predefined threshold value. Finally, based on the group collective PR, we calculate the row arithmetic mathematical average values and with respect to that values the various methods are applied for ranking the preference order of the alternatives. Numerical example are provided to illustrate the proposed approach.
Decision-theoretic rough sets (DTRSs), which provide a classical model of three-way decisions (3WDs), play an important role in risk decision-making problems. The risk is associated with the loss function of DTRSs, which is evaluated by the decision makers. As a new extension of fuzzy sets, Pythagorean fuzzy sets can handle uncertain information more flexibly than intuitionistic fuzzy sets in the process of decision making and it gives a new measure for the determination of loss functions of DTRSs.More specifically, we take into account the loss functions of DTRSs with Pythagorean fuzzy numbers and propose a Pythagorean fuzzy decision-theoretic rough set (PFDTRS) model. Some properties of the expected losses are carefully investigated. Then we further design three approaches for deriving 3WDs with the PFDTRS model. The group decision making (GDM) based on the PFDTRS model is also discussed. It provides a novel interpretation for the determination of loss functions. With the aid of the Pythagorean fuzz weighted averaging operator, we aggregate the loss functions, as suggested by the all experts, which support a coherent way of designing information granules in the presence of numerics. An algorithm for 3WDs in GDM based on the PFDTRS model is designed. Then, an example is presented to elaborate on 3WDs with the PFDTRS model. K E Y W O R D Sdecision-theoretic rough sets, group decision making, loss function, Pythagorean fuzzy sets, three-way decisions 818
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.
Cloud computing technologies have been prospering in recent years and have opened an avenue for a wide variety of forms of adaptable data sharing. Taking advantage of these state-of-the-art innovations, the cloud storage data owner must, however, use a suitable identity-based cryptographic mechanism to ensure the safety prerequisites while sharing data to large numbers of cloud data users with fuzzy identities. As a successful way to guarantee secure fuzzy sharing of cloud data, the identity-based cryptographic technology still faces an effectiveness problem under multireceiver configurations. The chaos theory is considered a reasonable strategy for reducing computational complexity while meeting the cryptographic protocol's security needs. In an identity-based cryptographic protocol, public keys for individual clients are distributed, allowing the clients to separately select their own network identities or names as their public keys. In fact, in a public-key cryptographic protocol, it is for the best that the confirmation of the public key is done in a safe, private manner, because this way the load of storage on the server's side can be considerably relieved. The objective of this paper is to outline and examine a conversion process that can transfer cryptosystems using Chebyshev's chaotic maps over the Galois field to a subtree-based protocol in the cloud computing setting for fuzzy user data sharing, as opposed to reconcocting a different structure. Furthermore, in the design of our conversion process, no adjustment of the original cryptosystem based on chaotic maps is needed. K E Y W O R D Schaotic hash function, Chebyshev chaotic maps, fuzzy user, public key cryptography, subtree | INTRODUCTIONCloud storage has introduced new ways to store, retrieve, and share digital data, making the concept of sharing ondemand information a reality. Cloud storage service suppliers have been merging into cloud information centers these days, and nominal costs are charged for the service of data sharing. Officers in companies and other organizations can easily acquire precious data with the assistance of cloud storage or share the latest information anywhere with other
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