Single variable differential calculus under q ‐rung orthopair fuzzy environment: Limit, derivative, chain rules, and its application

Abstract

The q‐rung orthopair fuzzy set ( q‐ROFS) that the sum of the qth power of the membership degree and the qth power of the nonmembership degree is restricted to one is a generalization of fuzzy set (FS). Recently, many researchers have given a series of aggregation operators to fuse q‐rung orthopair fuzzy discrete information. Subsequently, although some scholars have also focused on studying q‐rung orthopair fuzzy continuous information and give its continuity, derivative, differential, and integral, those studies are only considered from the perspective of multivariable fuzzy functions. Thus, the main aim of the paper is to study the q‐rung orthopair fuzzy continuous single variable information. In this paper, we first define the concept of q‐rung orthopair single variable fuzzy function ( q‐ROSVFF) to describe the fuzzy continuous information, and give its domain to make sure that this kind of function is meaningful. Afterward, we propose the limits, continuities, and infinitesimal of q‐ROSVFFs, and offer the relationship between the limit of q‐ROSVFF and that of q‐ROSVFF infinitesimal. On the basis of the definition of derivative in mathematical analysis, we define the subtraction and division derivatives and basic operational rules, and offer the simpler proofs for the derivatives of q‐ROSVFFs. What is more, we propose the subtraction and division differential invariances, and give the approximate calculation formulas of q‐ROSVFFs when the value of independent variable is changed small enough. In the real situation, fundamental functions cannot be used to express more complicated functions, thus we define the compound q‐ROSVFFs and give their chain rules of subtraction and division derivatives. Finally, we use numerical examples by simulation to verify the feasibility and veracity of the approximate calculation on q‐ROSVFFs.

Fuzzy Decision Making Method Based on Cocoso With Critic for Financial Risk Evaluation

“…Peng and Dai (2019) explored the q-rung orthopair fuzzy decision making method based on CODAS (combinative distance-based assessment) and multiparametric similarity measure, and successfully applied them in assessing the classroom teaching quality. -Calculus: Ye et al (2019) defined the notion of q-rung orthopair single variable fuzzy function (q-ROSVFF) for depicting the fuzzy continuous information. Gao et al (2019) explored derivatives, continuities and differentials of q-ROSVFF.…”

Multi Criteria Group Decision Making Based On Q-Rung Orthopair Trapezoidal Hesitant Dombi Fuzzy Aggregation Operators

“…In the …eld of qROFSs conventional wide of GDM throughout the historical centuries. The qROFSs for MCGDM concerning investigation themes, such association coe¢ cient of qROFSs [18], qROFS of distance actions [19], qROFS of similarity measures [20], qROFSs used to real life application of GDM issues [21,22], operational rules of qROFNs [23][24][25], and qROFNs of aggregation operator [26][27][28][29][30][31][32][33][34][35][36][37][38].…”

A Multi-Attribute Decision-Making Algorithm Using Q-Rung Orthopair Power Bonferroni Mean Operator and Its Application

“…Nevertheless, with the increasing complexity of the decision-making environment, how to improve accuracy in information expression of decision attributes and reliability of decision-making results are the key concerns in MADM theory. Where, the tools used by many scholars to handle the above problems include MADM methods based on intuitionistic fuzzy set (IFS) [9][10][11], Pythagorean fuzzy set (PFS) [12][13][14], picture fuzzy set (PtFS) [15][16][17] and q-rung orthopair fuzzy set (q-ROFS) [18][19][20][21][22][23]. Of which, q-ROFS is a powerful tool to solve issues of information representation and information aggregation.…”