Fuzzy target-environment networks and fuzzy-regression approaches

Erik Kropat; Gerhard Wilhelm Weber

doi:10.3934/naco.2018008

Article Contents

2018, Volume 8, Issue 2: 135-155. Doi: 10.3934/naco.2018008

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Fuzzy target-environment networks and fuzzy-regression approaches

Erik Kropat^1, , and
Gerhard Wilhelm Weber^2,+,

1.
University of the Bundeswehr Munich, Faculty of Informatics, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany
2.
Poznan University of Technology, Faculty of Engineering Management, Chair of Marketing and Economic Engineering, uI. Strzelecka 11, 60-965 Poznan, Poland

* Corresponding author: Erik Kropat
* Corresponding author: Erik Kropat

⁺ Honorary positions: Faculty of Economics, Business and Law, University of Siegen, Germany; School of Science, Information Technology and Engineering, University of Ballarat, Australia; Center for Research & Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Portugal; University of North Sumatra, Medan, Indonesia

Received: December 2016

Revised: December 2017

Published: May 2018

The reviewing process of this paper was handled by Editors A. (Nima) Mirzazadeh, Kharazmi University, Tehran, Iran, and Gerhard-Wilhelm Weber, Middle East Technical University, Ankara, Turkey. This paper was for the occasion of the 12th International Conference on Industrial Engineering (ICIE 2016), which was held in Tehran, Iran during 25-26 January, 2016

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Abstract

In systems sciences, the role of the environment is considered as a key factor for a deeper understanding of interconnected complex systems. The framework of target-environment networks allows for an investigation of regulatory systems under various kinds of uncertainty. Parameter-dependent models are applied to predict the future states of the system with respect to uncertain observations. In particular, fuzzy possibilistic regression models have been introduced that are based on crisp measurements. In this study, the concept of fuzzy target-environment networks is further extended towards fuzzy-regression models with fuzzy data sets. Regression models for various shapes of fuzzy coefficients and fuzzy model outputs are presented.

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Mathematics Subject Classification: Primary: 62J86, 05C72, 62A86.

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Figure 1. The fuzzy target-environment network (cf. [13]). The nodes are the targets and environmental items. The branches are weighted by the fuzzy coefficients of the linear fuzzy models $\mathcal{F}_j$ and $\mathcal{G}_i$. An additional node "0" is introduced which corresponds to the intercepts $Z_{j0}$ and $Z'_{i0}$

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Figure 2. The symmetric triangular fuzzy coefficient $A_{jr} = (A^C_{jr}, A^W_{jr})$

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Figure 3. The $\alpha$-cut of the fuzzy model $\mathcal{F}_j$

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Figure 4. The asymmetric triangular fuzzy coefficient $A_{jr} = (A^L_{jr}, A^C_{jr}, A^U_{jr})$

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Figure 5. The asymmetric triangular fuzzy coefficient $A_{jr} = (A^L_{jr}, A^M_{jr}, A^N_{jr}, A^U_{jr})$

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Figure 6. The fuzzy inclusion relation with fuzzy model $\mathcal{F}_j$ and fuzzy output $\widehat{X}^{(\kappa+1)}_j$

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Figure 7. The fuzzy inclusion relation with asymmetric trapezoidal fuzzy model $\mathcal{F}_j$ and asymmetric triangular fuzzy output $\widehat{X}_j$

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Table 1. fuzzy-regression algorithms for target-environment data

Data	Coefficients	Fuzzy Model Output	Objective Function	Algorithm
crisp	symmetric, triangular	symmetric, triangular	minimize total spread of fuzzy coefficients	(CFR1)
crisp	symmetric, triangular	symmetric, triangular	minimize total spread of fuzzy model output	(CFR2)
crisp	symmetric, triangular	symmetric, triangular	minimize total spread of fuzzy model output with membership grades	(CFR3)
crisp	asymmetric, triangular	asymmetric, triangular	minimize total spread of fuzzy model output	(CFR4)
crisp	asymmetric, trapezoidal	asymmetric, trapezoidal	minimize sum of total spread and inner spread of fuzzy model output	(CFR5)
crisp	asymmetric, trapezoidal	asymmetric, trapezoidal	minimize sum of total spread and inner spread of fuzzy model output with membership grades	(CFR6)

symmetric, triangular	symmetric, triangular	symmetric, triangular	minimize total spread of fuzzy coefficients	(FFR1)
symmetric, triangular	symmetric, triangular	symmetric, triangular	minimize total spread of fuzzy model output	(FFR2)
symmetric, triangular	symmetric, triangular	symmetric, triangular	minimize total spread of fuzzy model output with membership grades	(FFR3)
asymmetric, triangular	asymmetric, trapezoidal	asymmetric, trapezoidal	minimize sum of total spread and inner spread of fuzzy model output	(FFR4)

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