Linear Diophantine Fuzzy Relations and Their Algebraic Properties with Decision Making

Abstract

Binary relations are most important in various fields of pure and applied sciences. The concept of linear Diophantine fuzzy sets (LDFSs) proposed by Riaz and Hashmi is a novel mathematical approach to model vagueness and uncertainty in decision-making problems. In LDFS theory, the use of reference or control parameters corresponding to membership and non-membership grades makes it most accommodating towards modeling uncertainties in real-life problems. The main purpose of this paper is to establish a robust fusion of binary relations and LDFSs, and to introduce the concept of linear Diophantine fuzzy relation (LDF-relation) by making the use of reference parameters corresponding to the membership and non-membership fuzzy relations. The novel concept of LDF-relation is more flexible to discuss the symmetry between two or more objects that is superior to the prevailing notion of intuitionistic fuzzy relation (IF-relation). Certain basic operations are defined to investigate some significant results which are very useful in solving real-life problems. Based on these operations and their related results, it is analyzed that the collection of all LDF-relations gives rise to some algebraic structures such as semi-group, semi-ring and hemi-ring. Furthermore, the notion of score function of LDF-relations is introduced to analyze the symmetry of the optimal decision and ranking of feasible alternatives. Additionally, a new algorithm for modeling uncertainty in decision-making problems is proposed based on LDFSs and LDF-relations. A practical application of proposed decision-making approach is illustrated by a numerical example. Proposed LDF-relations, their operations, and related results may serve as a foundation for computational intelligence and modeling uncertainties in decision-making problems.

Linear Diophantine Fuzzy Set Theory Applied to BCK/BCI-Algebras

“…In this section, we present some basic results and examples related to linear Diophantine fuzzy sets [3,4] and to BCK/BCI-algebras [23].…”

“…To eliminate these restrictions by using reference parameters, Riaz and Hashmi [3] in 2019 found a new extension of fuzzy sets and called it linear Diophantine fuzzy sets (LDFS). Using the corresponding reference parameters to the membership and non-membership fuzzy relations, S. Ayub et al [4] established a robust fusion of LDFSs and binary relations and introduced linear Diophantine fuzzy relations.…”

Linear Diophantine Fuzzy Rough Sets: A New Rough Set Approach with Decision Making

“…In 1983, Atanassov [21] proposed the idea of intuitionistic fuzzy relation (IF relation) by promulgating the constraint that the sum of association and disassociation grades should not be greater than 1. Recently, Ayub et al [22], proposed a beautiful extension of the IF relation, named linear Diophantine fuzzy relation (LDF relation), with a robust application in decision-making, by the influence of the novel concepts of LDFSs. Hashmi et al [23] suggested the conceptualization of m-polar neutrosophic topology with applications to MADM.…”

Rough Approximation Operators on a Complete Orthomodular Lattice

“…In 1936, Birkhoff and von Neumann [20] considered the orthomodular lattice as quantum logic for studying the algebraic structure of quantum mechanics. There are many extensions of Pawlak's rough set, such as fuzzy rough sets [21,22], covering based rough sets [23][24][25], probabilistic rough sets [26,27], soft rough sets [28][29][30][31], Diophantine fuzzy rough sets [32][33][34][35], multi-granulation rough sets [36][37][38], hesitant fuzzy rough set [39] and so on. However, as far as I know, there is only a little literature addressing the rough sets and quantum logics together.…”