Research Article
A Simple Method to Improve the Multiplicative Consistency of an Interval Fuzzy Preference Relation
Hyonil Oh*
,
Whonchol Hwang
Issue:
Volume 11, Issue 3, September 2025
Pages:
90-101
Received:
10 March 2025
Accepted:
3 April 2025
Published:
4 August 2025
Abstract: In this paper, we propose a method that improves the multiplicative consistency and minimizes indeterminacy (the sum of widths of all interval membership degrees) by using only off-diagonal elements of an interval fuzzy preference relation (IFPR). In addition, this method keeps the initial information as much as possible. To do so, we formulate a concept of the multiplicative consistency that satisfies the additive reciprocity between the related preferences of the IFPR and is invariant under any permutation of objects. Next, the equations which are equivalent to the multiplicative consistency for the IFPR and uses only off-diagonal elements are derived. Based on these equations, the linear models to judge the multiplicative consistency of the IFPR and calculate multiplicatively consistent IFPR minimizing indeterminacy by using only off-diagonal elements are constructed. Based on linear models, we construct an algorithm that calculates the acceptable consistent IFPR keeping the initial information as much as possible and prove that a consistency index of algorithm converges to zero. The proposed method can reduce a large amount of calculations and is correct in judging and improving the multiplicative consistency for the IFPR in comparison with previous results because it uses only off-diagonal elements of the initial IFPR. In addition, a numerical example is provided to show the feasibility and efficiency of the proposed method.
Abstract: In this paper, we propose a method that improves the multiplicative consistency and minimizes indeterminacy (the sum of widths of all interval membership degrees) by using only off-diagonal elements of an interval fuzzy preference relation (IFPR). In addition, this method keeps the initial information as much as possible. To do so, we formulate a c...
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Research Article
Generated Fuzzy Quasi-ideals in Ternary Semigroups
Ravi Srivastava*
Issue:
Volume 11, Issue 3, September 2025
Pages:
102-108
Received:
8 June 2025
Accepted:
1 July 2025
Published:
5 August 2025
Abstract: Here in this paper, we provide characterizations of fuzzy quasi-ideal in terms of level and strong level subsets. Along with it, we provide expression for the generated fuzzy quasi-ideal generated by level subsets and strong level subsets of the given fuzzy set in explicit manner. We also establish the existence for a generated fuzzy quasi-ideal in ternary semigroup by showing that the non-empty intersection of an arbitrary family of fuzzy quasi-ideals is again a fuzzy quasi-ideal. This paper introduces and explores the concept of generated fuzzy quasi-ideals in ternary semigroups, extending classical algebraic notions into the fuzzy domain. A fuzzy quasi-ideal is defined as a fuzzy set that satisfies specific conditions analogous to those of crisp quasi-ideals under the ternary operation. A foundational result established in this work is that the non-empty intersection of any family of fuzzy quasi-ideals in a ternary semigroup remains a fuzzy quasi-ideal, reinforcing the internal consistency of the structure. Furthermore, we explore key properties of these generated fuzzy quasi-ideals, including their relationships with level sets and strong level subsets. The central focus of the paper is on how fuzzy quasi-ideals can be generated by arbitrary fuzzy sets within a ternary semigroup. We establish methods for constructing the smallest fuzzy quasi-ideal containing a given fuzzy set, along with expressions for this generated structure in terms of the quasi-ideals generated by its level and strong level subsets. Through constructive proofs, we demonstrate the existence by showing the non-empty intersection of an arbitrary family of fuzzy quasi-ideals in a ternary semigroup is itself a fuzzy quasi-ideal and uniqueness of such generated fuzzy quasi-ideals. The findings contribute to a deeper understanding of the internal structure of ternary semigroups and provide a foundational framework for further research in fuzzy algebraic systems. In summary, this work establishes a comprehensive framework for generated fuzzy quasi-ideals in ternary semigroups, revealing their structural properties, generation mechanisms, and theoretical importance. These results contribute meaningfully to the study of fuzzy algebraic systems and open new avenues for further research in fuzzy ternary algebra.
Abstract: Here in this paper, we provide characterizations of fuzzy quasi-ideal in terms of level and strong level subsets. Along with it, we provide expression for the generated fuzzy quasi-ideal generated by level subsets and strong level subsets of the given fuzzy set in explicit manner. We also establish the existence for a generated fuzzy quasi-ideal in...
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