Linear Diophantine Fuzzy Relations and Their Algebraic Properties with Decision Making
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 o...
| Main Authors: | , , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-05-01
|
| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/13/6/945 |
| _version_ | 1797532536032722944 |
|---|---|
| author | Saba Ayub Muhammad Shabir Muhammad Riaz Muhammad Aslam Ronnason Chinram |
| author_facet | Saba Ayub Muhammad Shabir Muhammad Riaz Muhammad Aslam Ronnason Chinram |
| author_sort | Saba Ayub |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-10T11:01:36Z |
| format | Article |
| id | doaj.art-8b9956a4ebd84d03a40b9e5169d133c6 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T11:01:36Z |
| publishDate | 2021-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-8b9956a4ebd84d03a40b9e5169d133c62023-11-21T21:29:02ZengMDPI AGSymmetry2073-89942021-05-0113694510.3390/sym13060945Linear Diophantine Fuzzy Relations and Their Algebraic Properties with Decision MakingSaba Ayub0Muhammad Shabir1Muhammad Riaz2Muhammad Aslam3Ronnason Chinram4Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, PakistanDepartment of Mathematics, Quaid-i-Azam University, Islamabad 45320, PakistanDepartment of Mathematics, University of the Punjab, Lahore 54590, PakistanDepartment of Mathematics, College of Sciences, King Khalid University, Abha 61413, Saudi ArabiaAlgebra and Applications Research Unit, Division of Computational Science, Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla 90110, ThailandBinary 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.https://www.mdpi.com/2073-8994/13/6/945linear Diophantine fuzzy setslinear Diophantine fuzzy relationsequivalence linear Diophantine fuzzy relationssymmetrydecision-making |
| spellingShingle | Saba Ayub Muhammad Shabir Muhammad Riaz Muhammad Aslam Ronnason Chinram Linear Diophantine Fuzzy Relations and Their Algebraic Properties with Decision Making Symmetry linear Diophantine fuzzy sets linear Diophantine fuzzy relations equivalence linear Diophantine fuzzy relations symmetry decision-making |
| title | Linear Diophantine Fuzzy Relations and Their Algebraic Properties with Decision Making |
| title_full | Linear Diophantine Fuzzy Relations and Their Algebraic Properties with Decision Making |
| title_fullStr | Linear Diophantine Fuzzy Relations and Their Algebraic Properties with Decision Making |
| title_full_unstemmed | Linear Diophantine Fuzzy Relations and Their Algebraic Properties with Decision Making |
| title_short | Linear Diophantine Fuzzy Relations and Their Algebraic Properties with Decision Making |
| title_sort | linear diophantine fuzzy relations and their algebraic properties with decision making |
| topic | linear Diophantine fuzzy sets linear Diophantine fuzzy relations equivalence linear Diophantine fuzzy relations symmetry decision-making |
| url | https://www.mdpi.com/2073-8994/13/6/945 |
| work_keys_str_mv | AT sabaayub lineardiophantinefuzzyrelationsandtheiralgebraicpropertieswithdecisionmaking AT muhammadshabir lineardiophantinefuzzyrelationsandtheiralgebraicpropertieswithdecisionmaking AT muhammadriaz lineardiophantinefuzzyrelationsandtheiralgebraicpropertieswithdecisionmaking AT muhammadaslam lineardiophantinefuzzyrelationsandtheiralgebraicpropertieswithdecisionmaking AT ronnasonchinram lineardiophantinefuzzyrelationsandtheiralgebraicpropertieswithdecisionmaking |