A Simple Method to Improve the Multiplicative Consistency of an Interval Fuzzy Preference Relation

Hyonil Oh; Whonchol Hwang

doi:doi:10.11648/j.ijmfs.20251103.11

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A Simple Method to Improve the Multiplicative Consistency of an Interval Fuzzy Preference Relation

Hyonil Oh^*

, Whonchol Hwang

Published in International Journal of Management and Fuzzy Systems (Volume 11, Issue 3)

Received: 10 March 2025 Accepted: 3 April 2025 Published: 4 August 2025

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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.

Published in	International Journal of Management and Fuzzy Systems (Volume 11, Issue 3)
DOI	10.11648/j.ijmfs.20251103.11
Page(s)	90-101
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Interval Fuzzy Preference Relation (IFPR), Multiplicative Consistency, Off-diagonal Elements, Additive Reciprocity, Permutation of Objects

References

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[9]	Mishra, L. N., Raiz, M., Rathour, L. Mishra, V. N. Tauberian theorems for weighted means of double sequences in intuitionistic fuzzy normed spaces. Yugoslav Journal of Operations Research. 2022, 32(3), 377-388.
[10]	Mo, J., Huang, H. L. Continuous probability-interval valued fuzzy preference relations and its application in group decision making. Iranian Journal of Fuzzy Systems. 2021, 18(4), 185-200.
[11]	Liu, F., Huang, M. J., Huang, C. X., Pedrycz, W. Measuring consistency of interval-valued preference relations: Comments and Comparison. Operational Research. 2022, 22, 371-399.
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APA Style

Oh, H., Hwang, W. (2025). A Simple Method to Improve the Multiplicative Consistency of an Interval Fuzzy Preference Relation. International Journal of Management and Fuzzy Systems, 11(3), 90-101. https://doi.org/10.11648/j.ijmfs.20251103.11

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ACS Style

Oh, H.; Hwang, W. A Simple Method to Improve the Multiplicative Consistency of an Interval Fuzzy Preference Relation. Int. J. Manag. Fuzzy Syst. 2025, 11(3), 90-101. doi: 10.11648/j.ijmfs.20251103.11

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AMA Style

Oh H, Hwang W. A Simple Method to Improve the Multiplicative Consistency of an Interval Fuzzy Preference Relation. Int J Manag Fuzzy Syst. 2025;11(3):90-101. doi: 10.11648/j.ijmfs.20251103.11

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@article{10.11648/j.ijmfs.20251103.11,
  author = {Hyonil Oh and Whonchol Hwang},
  title = {A Simple Method to Improve the Multiplicative Consistency of an Interval Fuzzy Preference Relation
},
  journal = {International Journal of Management and Fuzzy Systems},
  volume = {11},
  number = {3},
  pages = {90-101},
  doi = {10.11648/j.ijmfs.20251103.11},
  url = {https://doi.org/10.11648/j.ijmfs.20251103.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijmfs.20251103.11},
  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.},
 year = {2025}
}

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TY  - JOUR
T1  - A Simple Method to Improve the Multiplicative Consistency of an Interval Fuzzy Preference Relation

AU  - Hyonil Oh
AU  - Whonchol Hwang
Y1  - 2025/08/04
PY  - 2025
N1  - https://doi.org/10.11648/j.ijmfs.20251103.11
DO  - 10.11648/j.ijmfs.20251103.11
T2  - International Journal of Management and Fuzzy Systems
JF  - International Journal of Management and Fuzzy Systems
JO  - International Journal of Management and Fuzzy Systems
SP  - 90
EP  - 101
PB  - Science Publishing Group
SN  - 2575-4947
UR  - https://doi.org/10.11648/j.ijmfs.20251103.11
AB  - 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.
VL  - 11
IS  - 3
ER  -

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Author Information

Hyonil Oh

Institute of Mathematics, State Academy of Sciences, Pyongyang, DPR Korea

Contact Email

http://orcid.org/0000-0002-5624-6240
Whonchol Hwang

Department of Mathematics, University of Science, Pyongyang, DPR Korea

http://orcid.org/0009-0005-0072-832X

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Plain Text BibTeX RIS

APA Style

Oh, H., Hwang, W. (2025). A Simple Method to Improve the Multiplicative Consistency of an Interval Fuzzy Preference Relation. International Journal of Management and Fuzzy Systems, 11(3), 90-101. https://doi.org/10.11648/j.ijmfs.20251103.11

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ACS Style

Oh, H.; Hwang, W. A Simple Method to Improve the Multiplicative Consistency of an Interval Fuzzy Preference Relation. Int. J. Manag. Fuzzy Syst. 2025, 11(3), 90-101. doi: 10.11648/j.ijmfs.20251103.11

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AMA Style

Oh H, Hwang W. A Simple Method to Improve the Multiplicative Consistency of an Interval Fuzzy Preference Relation. Int J Manag Fuzzy Syst. 2025;11(3):90-101. doi: 10.11648/j.ijmfs.20251103.11

Copy | Download

@article{10.11648/j.ijmfs.20251103.11,
  author = {Hyonil Oh and Whonchol Hwang},
  title = {A Simple Method to Improve the Multiplicative Consistency of an Interval Fuzzy Preference Relation
},
  journal = {International Journal of Management and Fuzzy Systems},
  volume = {11},
  number = {3},
  pages = {90-101},
  doi = {10.11648/j.ijmfs.20251103.11},
  url = {https://doi.org/10.11648/j.ijmfs.20251103.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijmfs.20251103.11},
  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.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - A Simple Method to Improve the Multiplicative Consistency of an Interval Fuzzy Preference Relation

AU  - Hyonil Oh
AU  - Whonchol Hwang
Y1  - 2025/08/04
PY  - 2025
N1  - https://doi.org/10.11648/j.ijmfs.20251103.11
DO  - 10.11648/j.ijmfs.20251103.11
T2  - International Journal of Management and Fuzzy Systems
JF  - International Journal of Management and Fuzzy Systems
JO  - International Journal of Management and Fuzzy Systems
SP  - 90
EP  - 101
PB  - Science Publishing Group
SN  - 2575-4947
UR  - https://doi.org/10.11648/j.ijmfs.20251103.11
AB  - 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.
VL  - 11
IS  - 3
ER  -

Copy | Download