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Nowadays, wireless sensor networks (WSNs) are used to monitor and collect data in various environments. One of the main challenges in WSNs is the energy consumption due to the deployed sensor nodes in WSNs are energy-constrained. Clustering method is a solution for this problem and the cluster head (CH) selection process is a major part of the clustering method. This paper used the firefly algorithm (FA) and hesitant fuzzy to propose a new CH selection protocol. The proposed protocol uses three parameters of sensor nodes to calculate the score of each node to determine the best CHs. In order to describe the performance of the proposed protocol, three scenarios are simulated and evaluated. The simulation results show that the proposed protocol improves the energy saving and increases the network lifetime.

Inspired by the analysis of soft sets and fuzzy sets, in this paper, we study the relevance and interrelationship of the equivalence and congruence relation of lattice ordered fuzzy soft groups(l-FSGs). Moreover, we analyze some of the innate algebraic results accrued out of it theoretically. We construct l-FSG congruence class and l-FSG quotient set on a group X. We analyze the relationship between l-FSG congruence relation and l-FSG normal subgroups. In particular, we prove that in a group X, l-FSG normal subgroups form a modular lattice.

In this paper, we introduce the concept of fuzzy congruence relations on a pseudo BE-algebra and some of properties are investigated. We show that the set of all fuzzy congruence relations is a modular lattice and the quotient structure induced by fuzzy congruence relations is studied.

The notions of $$(\in ,\in \vee q)$$-cubic p- (a- and q-) ideals of BCI-algebras are introduced and some related properties are investigated. Several characterization for these generalized $$(\in ,\in \vee q)$$-cubic ideals are defined and relationship between $$(\in ,\in \vee q)$$-cubic p-ideals, $$(\in ,\in \vee q)$$-cubic q-deals and $$(\in ,\in \vee q)$$-cubic a-ideals of BCI-algebras are discussed.

As a generalization to hesitant fuzzy linguistic term set (HFLTS), double hierarchy hesitant fuzzy linguistic term set (DHHFLTS) is presented which circumvents the weakness of HFLTS in representing complex linguistic terms. DHHFLTS has two linguistic hierarchies with the second hierarchy supplementing the primary which enables decision-makers (DMs) to represent complex linguistic terms better. Motivated by the power of DHHFLTS, in this paper, a new decision framework is presented under DHHFLTS context. Initially, a new aggregation operator called double hierarchy hesitant fuzzy hybrid aggregation (DHHFHA) operator is proposed for sensible aggregation of DMs’ preference information. Further, weights of attributes are calculated by extending statistical variance (SV) method under DHHFLTS context. Objects are prioritized by extending the popular WASPAS (weighted aggregated sum product assessment) method to DHHFLTS context. The applicability and usefulness of the proposed framework are realized by demonstrating a risk management technique (RMT) selection problem for a construction project. Finally, the superiority and weakness of the proposed framework are discussed by comparison with other methods.

The aim of this study is to introduce fuzzy filters of Sheffer stroke Hilbert algebra. After defining fuzzy filters of Sheffer stroke Hilbert algebra, it is shown that a quotient structure of this algebra is described by its fuzzy filter. In addition to this, the level filter of a Sheffer stroke Hilbert algebra is determined by its fuzzy filter. Some fuzzy filters of a Sheffer stroke Hilbert algebra are defined by a homomorphism. Normal and maximal fuzzy filters of a Sheffer stroke Hilbert algebra and the relation between them are presented. By giving the Cartesian product of fuzzy filters of a Sheffer stroke Hilbert algebra, various properties are examined. © 2021 - IOS Press. All rights reserved., Ege Üniversitesi: 20772, The authors are thankful to the referees for a careful reading of the paper and for valuable comments and suggestions. This study is partially funded by Ege University Scientific Research Projects Directorate with the Project Number 20772.

A complex Pythagorean fuzzy set, an extension of Pythagorean fuzzy set, is a powerful tool to handle two dimension phenomenon. Dombi operators with operational parameters have outstanding flexibility. This article presents certain aggregation operators under complex Pythagorean fuzzy environment, including complex Pythagorean Dombi fuzzy weighted arithmetic averaging (CPDFWAA) operator, complex Pythagorean Dombi fuzzy weighted geometric averaging (CPDFWGA) operator, complex Pythagorean Dombi fuzzy ordered weighted arithmetic averaging (CPDFOWAA) operator and complex Pythagorean Dombi fuzzy ordered weighted geometric averaging (CPDFOWGA) operator. Moreover, this paper explores some fundamental properties of these operators with appropriate elaboration. A decision‐making numerical example related to the selection of bank to purchase loan is given to demonstrate the significance of our proposed approach. Finally, a comparative analysis with existing operators is given to demonstrate the peculiarity of our proposed operators.

This paper introduces and applies the notion of fuzzy hyper EQ-subalgebras as a generalization of the notation of fuzzy EQ-subalgebras and constructs extendable fuzzy EQ-subalgebras via these concepts. Moreover, by using the fuzzy hyper EQ-filters, it defines a congruence relation on hyper EQ-algebras that under some conditions is strongly regular and the quotient of any hyper EQ-algebra via this relation is an EQ-algebra.

Supply chain management (SCM) is an attractive area for research which has seen tremendous growth in the past decades. From the literature we observe that, supplier outsourcing (SO) is a highly explored research field in SCM which lacks significant scientific contribution. The major concern in SO is the decision makers’ (DMs) viewpoint which are often vague and imprecise. To better handle such imprecision, in this paper, we propose a new two-stage decision-making framework called TSDMF, which uses hesitant fuzzy information as input. In the first stage, the DMs’ preferences are aggregated using a newly proposed simple hesitant fuzzy-weighted geometry operator, which uses hesitant fuzzy weights for better understanding the importance of each DM. Following this, in the second stage, criteria weights are estimated using newly proposed hesitant fuzzy statistical variance method and finally, a new ranking method called three-way hesitant fuzzy VIKOR (TWHFV) is proposed by extending the VIKOR ranking method to hesitant fuzzy environment. This ranking method uses three categories viz., cost, benefit and neutral along with Euclid distance for its formulation. The practicality of the proposed TSDMF is verified by demonstrating a supplier outsourcing example in an automobile factory. The robustness of TWHFV is realized by using sensitivity analysis and other strengths of TSDMF are discussed by comparison with another framework.

In the online published article, the third author’s “A. Borumand Saeid” affiliation is updated as following “Dept. of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran”.

Nowadays, the most important aspects of wireless sensor networks (WSNs) are to make optimal use of and direct the limited energy of sensor nodes towards the desired application and prolong the network lifetime for that application. Although a few studies exist that have addressed these special goals, they have been mostly focused on the process of selecting cluster heads (CHs) and forwarders. No studies have been conducted so far on the selection of fuzzy input parameters in clustering and routing processes as well as the application-based parameter selection process. Generally, a fixed number of parameters have always been selected by designers. Hence, the shuffled frog leaping algorithm (SFLA) was employed in this paper to propose a technique for selecting fuzzy input parameters in a fuzzy multi-hop clustering protocol named the PS-SFLA. This technique includes three main phases, introduced in three versions for the sake of stepwise evaluation. Based on the literature review, the most frequent and diverse parameters were extracted and formulated in the first version. The proposed technique used the SFLA in the second version to select the appropriate parameters fitting the application specifics and scenario and determine the coefficients of parameters simultaneously so that they could be used as the inputs of the fuzzy inference system. It was also utilized in the final version for the automated, accurate, application-based tuning of fuzzy rules before the network was set up. By design, different versions of the PS-SFLA act as the starting points of the next version in addition to the fact that they can be evaluated separately. The PS-SFLA was compared with the LEACH, ASLPR, SIF, ERA, and FSFLA in different scenarios and two applications from the perspective of the alive nodes, the number of packets received by the BS, lifetime, and other factors. According to the simulation results, the PS-SFLA outperformed all of the other methods greatly in all scenarios and applications and PS-SFLA increases the lifetime due to the appropriate selection of fuzzy input parameters based on application and purpose.

Nowadays, companies are paying more attention to supply chain management in order to achieve success in the competitions and keep the customers satisfied. Therefore, supplier selection is of prime importance in a supply chain which has a key role in determining the success or failure of a business. In this paper, fuzzy multi-objective linear programming model with group decision making is presented. Firstly, linguistic variables in the form of interval-valued intuitionistic fuzzy numbers and group decisions are defined. By using TOPSIS method, the weight of each criterion (objective) and weight of each constraint are determined. Then, with defining membership functions of the objectives and membership functions of suppliers’ constraints and using the achieved weights from the previous phase, the supplier selection problem will be converted to fuzzy multi-objective linear programming model which determines the appropriate ordering quantities for each supplier. Finally, two numerical examples for supplier selection are used to demonstrate the practicality and effectiveness of the proposed approach.

This paper introduces the notion of fuzzy hyper EQ-subalgebras as generalization of the notation of fuzzy EQ-subalgebras and constructs extendable fuzzy EQ-subalgebras via these concepts. Moreover, we investigated some properties in fuzzy EQ-subalgebras, fuzzy hyper EQ-subalgebras by their α-level sets and connect them via positive fundamental relation C(σ∗).

Since Wireless Sensor Networks (WSNs) are composed of a set of sensor nodes that limit resource constraints such as energy constraints, energy consumption in WSNs is one of the challenges of these networks. One of the solutions to reduce energy consumption in WSNs is to use clustering. In clustering, cluster members send their data to their Cluster Head (CH), and the CH after collecting the data, sends them to the Base Station (BS). In clustering, choosing CHs is very important; so many methods have proposed to choose the CH. In this study, a hesitant fuzzy method with three input parameters namely, remaining energy, distance to the BS, distance to the center of cluster is proposed for efficient cluster head selection in WSNs. We define different scenarios and simulate them, then investigate the results of simulation.

Fuzzy models are becoming useful because of their aim of reducing the differences between the traditional numerical models used in engineering and sciences and the symbolic models used in expert systems. A vague graph is a generalized structure of a fuzzy graph that gives more precision, flexibility, and compatibility to a system when compared with systems that are designed using fuzzy graphs. In this paper, we introduce various types of vague bridges, vague cut-vertices, vague cycles and vague trees in vague graphs, and investigate some of their interesting properties. Most of these various types are defined in terms of levels. We also describe comparison of these types.

This paper focuses on the investigation of fuzzy filters of BL-algebras, an important and popular generic logical algebra. By studying the equivalent conditions of fuzzy fantastic filter and fuzzy normal filter of BL-algebras, the relation between these two filters are revealed. Two open problems of BL-algebras, i.e.,”Under what suitable condition a normal filter becomes a fantastic filter? ” and ” Under what suitable condition extension property for normal filter holds? (Extension property for a normal filter)”, are solved accordingly.

In this paper, we introduce the notion of fuzzy filters(ideals) in CI-algebras. We study fuzzy congruence relation in detail and construct quotient of \((X/\bar{\theta}; {\circ}, 1/\bar{\theta})\) via a fuzzy congruence relation on CI-algebra X.

In this note, by using the concept of Bipolar-valued fuzzy set, the notion of bipolar-valued fuzzy BM-algebra is introduced. Moreover, the notions of (strong) negative s-cut (strong) positive t-cut are introduced and the relationship between these notions and crisp sub-algebras are studied.

In this paper, we introduce the notion of intuitionistic (T,S)-fuzzy subalgebras in CI-algebras and study their fundamental properties. We get a fuzzy subalgebra from an intuitionistic (T,S)-fuzzy subalgebra. Also the notion of intuitionistic (T,S)-fuzzy (closed) filters of CI-algebras is introduced. We investigate the relationship between intuitionistic (T,S)-fuzzy subalgebras and intuitionistic (T,S)-fuzzy (closed) filters of CI-algebras.

In this article, we introduced the notions fuzzy soft subalgebras and fuzzy soft ideal of subtraction algebras. We showed that fuzzy soft image and fuzzy soft inverse image of fuzzy soft ideal are both fuzzy soft ideal under certain conditions. We have studied their related properties in details.

In this paper, generalization of the fuzzy BCK/BCI-algebras of type (α, β)* are introduced. Then we state and prove some theorems that determine the relationship between these notions and subalgebras of BCK/BCI-algebras. Characterizations of (α, β)*-fuzzy subalgebras in a BCK/BCI-algebras are given.

In this paper the notion of fuzzy BE-algebras are introduced. We state and prove some theorems in fuzzy BE-algebras and level subalgebras. Some characterizations of fuzzy subalgebras of BE-algebras are given. Using fuzzy subalgebras, characterizations of Artinian and Noetherian BE-algebras are also established. Finally the notion of fuzzy topological BE-algebras is introduced. We state and prove the Foster’s results on homomorphic images and inverse images in fuzzy topological BE-algebras.

In this paper the notion of fuzzy BM-algebra and fuzzy topological BM-algebras are introduced. We stated and proved some theorem in fuzzy BM-algebras and level subalgebras. Finally fuzzy topological BM-algebras are studied.

By using the concept of fuzzy points, we generalize the notion of hyper BCK-algebra and the notions of fuzzy point hyper BCK-(sub) algebras, fuzzy point (weak, strong) hyper BCK-ideals, quasi hyper BCK-(sub) algebras and quasi (weak, strong) hyper BCK-ideals are introduced. The relationship between these notions are stated and proved. Finally, we study the condition QH on quasi hyper BCK-algebras.

In this paper, we introduce the notion of fuzzy n-folds (P, implicative and fantastic) ideals in BCH-algebras which is a natural generalization of notion of n-folds (P, implicative and fantastic) ideals in BCH-algebras and we stated and proved some theorems which determines the relationship between these notions. Finally we give some computational Algorithms for these notions.

By two relations belonging to and quasi-coincidence (q) between fuzzy points and fuzzy sets, we define the concept of (±, ±)-fuzzy subalgebras where ±, ± are any two of $${\{\in, q, \in\vee q, \in\wedge q\}}$$ with $${\alpha\neq \in\wedge q}$$. We state and prove some theorems in (±, ±)-fuzzy B-algebras.

In this paper, we introduce the notion of hesitant fuzzy (implicative) filters and get some results on BE-algebras and show that every hesitant fuzzy implicative filter is a hesitant fuzzy filter but not the converse.

In this paper, we introduce the notion of hesitant fuzzy (implicative) filters and get some results on BE- algebras and show that every hesitant fuzzy implicative filter is a hesitant fuzzy filter but not the converse. Finally, we state and prove the relationship between hesitant fuzzy (implicative) filters and γ-inclusive sets.

In this paper we introduced notion of fuzzy filters and fuzzy ideals on CI–algebras. We give the construction of quotient (Xθ;o,1θ) via a fuzzy congruence relation of CI–algebra X.

In this note, by using the concept of Bipolar-valued fuzzy set, the notion of bipolar-valued fuzzy BCK/BCI-algebra is introduced. Moreover, the notions of (strong) negative s-cut, (strong) positive t-cut are introduced and the relationship between these notions and crisp subalgebras are studied.

In this note, by using the concept of vague sets, the notions of vague BCK/BCI-algebra is introduced. And the notions of α-cut and vague-cut is introduced and the relationship between these notions and crisp subalgebras are studied.

By using the concept of fuzzy points, we generalize the notion of hyper BCK-algebra and the notions of fuzzy point hyper BCK-(sub) algebras, fuzzy point (weak, strong) hyper BCK-ideals, quasi hyper BCK-(sub) algebras and quasi (weak, strong) hyper BCK-ideals are introduced. The relationship between these notions are stated and proved.

In this paper the notion of fuzzy QS-subalgebra and fuzzy topological QS-algebras are introduced. We state and prove some theorem in fuzzy QS-subalgebras and level subalgebras. Finally the Foster’s results on homomorphic images and inverse images in fuzzy topological QS-algebras are studied.

In this paper, we introduce the concept of bipolar-valued fuzzification of ordered AG-groupoids and discuss some structural properties of bipolarvalued fuzzy two-sided ideals of an intra-regular ordered AG-groupoid.