# The Solution Equivalence to General Models for the RIM Quantifier Problem

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**.**A fuzzy subset Q on the real line is called a RIM quantifier if $Q\left(0\right)=0,Q\left(1\right)=1$ and $Q\left(x\right)\ge Q\left(y\right)$ for $x>y$.

**Definition**

**2.**

## 3. The General Model for the Minimax RIM Quantifier Problem

**∗ The general model for the minimax RIM quantifier problem.**

**Theorem**

**1.**

**Example**

**1.**

## 4. The General Model for the Minimum RIM Quantifier Problem

**∗ The general model for the minimum RIM quantifier problem.**

**Theorem**

**2**

**.**There is a unique optimal solution for (3), and the optimal solution has the form

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

## 5. Numerical Example

## 6. Conclusions

## Funding

## Conflicts of Interest

## References

- Amin, G.R.; Emrouznejad, A. An extended minimax disparity to determine the OWA operator weights. Comput. Ind. Eng.
**2006**, 50, 312–316. [Google Scholar] [CrossRef] - Amin, G.R. Notes on priperties of the OWA weights determination model. Comput. Ind. Eng.
**2007**, 52, 533–538. [Google Scholar] [CrossRef] - Emrouznejad, A.; Amin, G.R. Improving minimax disparity model to determine the OWA operator weights. Inf. Sci.
**2010**, 180, 1477–1485. [Google Scholar] [CrossRef] - Filev, D.; Yager, R.R. On the issue of obtaining OWA operator weights. Fuzzy Sets Syst.
**1988**, 94, 157–169. [Google Scholar] [CrossRef] - Fullér, R.; Majlender, P. An analytic approach for obtaining maximal entropy OWA operators weights. Fuzzy Sets Syst.
**2001**, 124, 53–57. [Google Scholar] [CrossRef] - Hagan, M.O. Aggregating template or rule antecedents in real-time expert systems with fuzzy set logic. In Proceedings of the 22nd Annual IEEE Asilomar Conference on Signals, Systems, Computers, Pacific Grove, CA, USA, 31 October–2 November 1988; pp. 681–689. [Google Scholar]
- Hong, D.H. A note on solution equivalence to general models for RIM quantifier problems. Fuzzy Sets Syst.
**2018**, 332, 25–28. [Google Scholar] [CrossRef] - Hong, D.H. On proving the extended minimax disparity OWA problem. Fuzzy Sets Syst.
**2011**, 168, 35–46. [Google Scholar] [CrossRef] - Liu, X.; Lou, H. On the equivalence of some approaches to the OWA operator and RIM quantifier determination. Fuzzy Sets Syst.
**2007**, 159, 1673–1688. [Google Scholar] [CrossRef] - Wang, Y.M.; Parkan, C. A minimax disparity approach obtaining OWA operator weights. Inf. Sci.
**2005**, 175, 20–29. [Google Scholar] [CrossRef] - Yager, R.R. Ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Trans. Syst. Man Cybern.
**1988**, 18, 183–190. [Google Scholar] [CrossRef] - Yager, R.R. OWA aggregation over a continuous interval argument with application to decision making. IEEE Trans. Syst. Man Cybern. Part B
**2004**, 34, 1952–1963. [Google Scholar] [CrossRef] - Yager, R.R. Families of OWA operators. Fuzzy Sets Syst.
**1993**, 59, 125–148. [Google Scholar] [CrossRef] - Yager, R.R.; Filev, D. Induced ordered weighted averaging operators. IEEE Trans. Syst. Man Cybern. Part B
**1999**, 29, 141–150. [Google Scholar] [CrossRef] [PubMed] - Liu, X. On the maximum entropy parameterized interval approximation of fuzzy numbers. Fuzzy Sets Syst.
**2006**, 157, 869–878. [Google Scholar] [CrossRef] - Liu, X.; Da, Q. On the properties of regular increasing monotone (RIM) quantifiers with maximum entropy. Int. J. Gen. Syst.
**2008**, 37, 167–179. [Google Scholar] [CrossRef] - Hong, D.H. The relationship between the minimum variance and minimax disparity RIM quantifier problems. Fuzzy Sets Syst.
**2011**, 181, 50–57. [Google Scholar] [CrossRef] - Hong, D.H. The relationship between the maximum entropy and minimax ratio RIM quantifier problems. Fuzzy Sets Syst.
**2012**, 202, 110–117. [Google Scholar] [CrossRef] - Liu, X. A general model of parameterized OWA aggregation with given orness level. Int. J. Approx. Reason.
**2008**, 48, 598–627. [Google Scholar] [CrossRef] [Green Version] - Fullér, R.; Majlender, P. On obtaining minimal variability OWA operator weights. Fuzzy Sets Syst.
**2003**, 136, 203–215. [Google Scholar] [CrossRef] [Green Version] - Sang, X.; Liu, X. An analytic approach to obtain the least square deviation OWA operater weights. Fuzzy Sets Syst.
**2014**, 240, 103–116. [Google Scholar] [CrossRef] - Wang, Y.M.; Luo, Y.; Liu, X. Two new models for determining OWA operater weights. Comput. Ind. Eng.
**2007**, 52, 203–209. [Google Scholar] [CrossRef] - Wheeden, R.L.; Zygmund, A. Measure and Integral: An Introduction to Real Analysis; Marcel Dekker, Inc.: New York, NY, USA, 1977. [Google Scholar]
- Rustagi, J.S. Variational Methods in Statistics; Academic Press: New York, NY, USA, 1976. [Google Scholar]

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hong, D.H.
The Solution Equivalence to General Models for the RIM Quantifier Problem. *Symmetry* **2019**, *11*, 455.
https://doi.org/10.3390/sym11040455

**AMA Style**

Hong DH.
The Solution Equivalence to General Models for the RIM Quantifier Problem. *Symmetry*. 2019; 11(4):455.
https://doi.org/10.3390/sym11040455

**Chicago/Turabian Style**

Hong, Dug Hun.
2019. "The Solution Equivalence to General Models for the RIM Quantifier Problem" *Symmetry* 11, no. 4: 455.
https://doi.org/10.3390/sym11040455