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The geometric-arithmetic mean inequality is undoubtedly the most important one in the area of information aggregation. Recently, some q-rung orthopair fuzzy aggregation operators were proposed based on the Hamacher operations. In this paper, we give a detailed theoretical and practical analysis of the developed Hamacher arithmetic and geometric operators for q-rung orthopair fuzzy values. First, we investigate the monotonicity of these Hamacher aggregation operators on q-rung orthopair fuzzy values with respect to the parameter within Hamacher operations. Then, we discuss the limiting cases of these q-rung orthopair fuzzy Hamacher aggregation operators as the parameter therein approaches to zero or infinity and give a new characterization of the boundedness of these aggregation operators. Subsequently, we establish the geometric-arithmetic mean inequality for q-rung orthopair fuzzy information based on Hamacher operations. Finally, we present a decision making method by use of these aggregation operators and apply it to the problem of enterprise resource planning system selection. [ABSTRACT FROM AUTHOR]

As an anode material for sodium ion batteries (SIBs), carbon materials have attracted people's interest because of their abundant resources, good structural stability and low cost. Among most carbon precursors, pitch is viewed as a promising one because of a higher carbon content, good oxidation reversibility and low cost. However, the pitch-based carbon obtained with direct pyrolysis of pitch displays a high degree of graphitization and small layer spacing, which is unfavorable for the storage of sodium ions. In recent years, with the aid of the development of the nanoengineering process, the storage of sodium ions with pitch-based carbon has been drastically improved. This review article summarizes the recent progress of pitch nanoengineering to obtain the carbon anode for high-performance SIBs, including porous structure adjustment, heteroatom doping, co-carbonization and pre-oxidation. In addition, the merits and demerits of a variety of nanoengineering processes are discussed, and future research directions of pitch-based carbon are prospected. [ABSTRACT FROM AUTHOR]

Aggregation of q-rung orthopair fuzzy information serves as an important branch of the q-rung orthopair fuzzy set theory, where operations on q-rung orthopair fuzzy values (q-ROFVs) play a crucial role. Recently, aggregation operators on q-ROFVs were established by employing the Einstein operations rather than the algebraic operations. In this paper, we give a further investigation on operations and aggregation operators for q-ROFVs based on the Einstein operational laws. We present the operational principles of Einstein operations over q-ROFVs and compare them with those built on the algebraic operations. The properties of the q-rung orthopair fuzzy Einstein weighted averaging (q-ROFEWA) operator and q-rung orthopair fuzzy Einstein weighted geometric (q-ROFEWG) operator are investigated in detail, such as idempotency, monotonicity, boundedness, shift-invariance and homogeneity. Then, the developed operators are applied to multiattribute decision making problems under the q-rung orthopair fuzzy environment. Finally, an example for selecting the design scheme for a blockchain-based agricultural product traceability system is presented to illustrate the feasibility and effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

Granular computing is a relatively new platform for constructing, describing and processing information or knowledge. For crisp information granulation, the universe is decomposed into granules by binary relations on the universe, say, preorder, tolerance and equivalence relations. A knowledge structure is composed of all information granules induced by a relation that corresponds to the granulation. This paper establishes a novel theoretical framework for the measurement of information granularity of knowledge structures. First, two new relations between knowledge structures are introduced through the use of their respective Boolean relation matrices, where the granular equality relation is defined based on an orthogonal transformation with the transformation matrix being a permutation matrix, and the granularly finer relation is presented by combining the classical finer relation and the orthogonal transformation. Then, it is demonstrated that the simplified knowledge structure base with the granularly finer relation is a partially ordered set, which can be represented by a Hasse diagram. Subsequently, an axiomatic definition of information granularity is proposed to satisfy the constraints regarding these two relations. Moreover, a general form of the information granularity is given, and some existing measures are proved to be its special cases. Finally, as an application of the proposed measure, the attribute significance measure is developed based on the information granularity. [ABSTRACT FROM AUTHOR]

Dombi operations which include the Dombi product and Dombi sum are special cases of t-norms and t-conorms besides the algebraic operations. Recently, operations and aggregation operators for q-rung orthopair fuzzy values (q-ROFVs) based on Dombi operations were proposed. In this paper, we further discuss some additional issues relating to Dombi operations and Dombi aggregation operators of q-ROFVs. First, we give a reasonable explanation for the definition of the Dombi scalar multiplication and Dombi exponentiation which are constructed respectively by the Dombi sum and Dombi product over q-ROFVs, and then investigate the fundamental properties of these operations. Subsequently, the shift-invariance and homogeneity properties of the q-rung orthopair fuzzy Dombi weighted averaging/geometric operators are analyzed. And the boundedness of aforementioned aggregation operators are precisely characterized with respect to the parameter in Dombi operations. Finally, a method for multiattribute decision making is proposed by utilizing the developed operators under the q-rung orthopair fuzzy environment and an example of the selection of investment companies is given to illustrate the detailed decision making process. [ABSTRACT FROM AUTHOR]

The Choquet integral is proven quite reasonable as an integral form with respect to monotone measures, where the credibility measure is a specific case with self-duality. The main objective of this paper is to propose the Choquet integral of measurable functions on the credibility space, which bridges the gap between the Choquet integral and credibility theory. First, the Choquet integrals for nonnegative functions with respect to the credibility measure are introduced, and their properties are investigated such as the monotonicity and translatability. Then, the symmetric Choquet integrals and translatable Choquet integrals of any measurable functions are developed through the use of the Choquet integrals of nonnegative functions. Finally, Choquet integrals on finite sets based on the credibility measure are presented to simplify the calculation procedures. [ABSTRACT FROM AUTHOR]

Weighted power means with weights and exponents serving as their parameters are generalizations of arithmetic means. Taking into account decision makers' flexibility in decision making, each attribute value is usually expressed by a q‐rung orthopair fuzzy value (q‐ROFV, q≥1), where the former indicates the support for membership, the latter support against membership, and the sum of their qth powers is bounded by one. In this paper, we propose the weighted power means of q‐rung orthopair fuzzy values to enrich and flourish aggregations on q‐ROFVs. First, the q‐rung orthopair fuzzy weighted power mean operator is presented, and its boundedness is precisely characterized in terms of the power exponent. Then, the q‐rung orthopair fuzzy ordered weighted power mean operator is introduced, and some of its fundamental properties are investigated in detail. Finally, a novel multiattribute decision making method is explored based on developed operators under the q‐rung orthopair fuzzy environment. A numerical example is given to illustrate the feasibility and validity of the proposed approach, and it is shown that the power exponent is an index suggesting the degree of the optimism of decision makers. [ABSTRACT FROM AUTHOR]

Pan-integral is an important kind of nonlinear integral, and credibility measure is a monotone measure with self-duality. In this paper, we define the pan-integral on credibility space, a new type of integral construction, whose basic properties are discussed in detail. The relationship between the pan-integral and H-fuzzy integral on a special pan-credibility space is established as well. Moreover, some monotone convergence theorems for a sequence of nonnegative measurable functions are investigated. [ABSTRACT FROM AUTHOR]

The four fundamental operations of arithmetic for real (and complex) numbers are well known to everybody and quite often used in our daily life. And they have been extended to classical and generalized fuzzy environments with the demand of practical applications. In this paper, we present the arithmetic operations, including addition, subtraction, multiplication, and division operations, over q‐rung orthopair membership grades, where subtraction and division operations are both defined in two different ways. One is by solving the equation involving addition or multiplication operations, whereas the other is by determining the infimum or supremum of solutions of the corresponding inequality. Not all of q‐rung orthopairs can be performed by the former method but by the latter method, and it is proved that the former is a special case of the latter. Moreover, the elementary properties of arithmetic operations as well as mixed operations are extensively investigated. Finally, these arithmetic operations are pointwise defined on q‐rung orthopair fuzzy sets in which the membership degree of each element is a q‐rung orthopair. [ABSTRACT FROM AUTHOR]

Generalized orthopair fuzzy sets are extensions of ordinary fuzzy sets by relaxing restrictions on the degrees of support for and support against. Correlation analysis is to measure the statistical relationships between two samples or variables. In this paper, we propose a function measuring the interrelation of two q‐rung orthopair fuzzy sets, whose range is the unit interval [0,1]. First, the correlation and correlation coefficient of q‐rung orthopair membership grades are presented, and their basic properties are investigated. Second, these concepts are extended to q‐rung orthopair fuzzy sets on discrete universes. Then, we discuss their applications in cluster analysis under generalized orthopair fuzzy environments. And, a real‐world problem involving the evaluation of companies is used to illustrate the detailed processes of the clustering algorithm. Finally, we introduce the correlation and correlation coefficient of q‐rung orthopair fuzzy sets on both bounded and unbounded continuous universes and provide some numerical examples to substantiate such arguments. [ABSTRACT FROM AUTHOR]

Abstract: The generalized orthopair fuzzy set inherits the virtues of intuitionistic fuzzy set and Pythagorean fuzzy set in relaxing the restriction on the support for and support against. The very lax requirement provides decision makers great freedom in expressing their beliefs about membership grades, which makes generalized orthopair fuzzy sets having a wide scope of application in practice. In this paper, we present the Minkowski‐type distance measures, including Hamming, Euclidean, and Chebyshev distances, for q‐rung orthopair fuzzy sets. First, we introduce the Minkowski‐type distances of q‐rung orthopair membership grades, based on which we can rank orthopairs. Second, we propose several distances over q‐rung orthopair fuzzy sets on a finite discrete universe and subsequently discuss their applications to multiattribute decision‐making problems. Then we extend these results to a continuous universe, both bounded and unbounded cases are considered. Some illustrative examples are employed to substantiate the conceptual arguments. [ABSTRACT FROM AUTHOR]