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Fuzzy target-environment networks and fuzzy-regression approaches

  • * 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

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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  • 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.

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

    Citation:

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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 $Z'_{i0}$

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

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

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

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

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

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

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

    Data Coefficients Fuzzy Model Output Objective Function Algorithm
    crispsymmetric, triangularsymmetric, triangularminimize total spread of fuzzy coefficients(CFR1)
    crispsymmetric, triangularsymmetric, triangularminimize total spread of fuzzy model output(CFR2)
    crispsymmetric, triangularsymmetric, triangularminimize 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, triangularsymmetric, triangularsymmetric, triangularminimize total spread of fuzzy coefficients(FFR1)
    symmetric, triangularsymmetric, triangularsymmetric, triangularminimize total spread of fuzzy model output(FFR2)
    symmetric, triangularsymmetric, triangularsymmetric, triangularminimize total spread of fuzzy model output with membership grades(FFR3)
    asymmetric, triangularasymmetric, trapezoidalasymmetric, trapezoidalminimize sum of total spread and inner spread of fuzzy model output(FFR4)
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