# American Institute of Mathematical Sciences

June 2018, 8(2): 135-155. doi: 10.3934/naco.2018008

## Fuzzy target-environment networks and fuzzy-regression approaches

 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

+ 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

Fund Project: 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

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.

Citation: Erik Kropat, Gerhard Wilhelm Weber. Fuzzy target-environment networks and fuzzy-regression approaches. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 135-155. doi: 10.3934/naco.2018008
##### References:
 [1] U. Bronfenbrenner, The Ecology of Human Development: Experiments by Nature and Design, Cambridge, MA: Harvard University Press (ISBN 0-674-22457-4), 1979. [2] C. Carleos, F. Rodriguez, H. Lamelas and J. A. Baro, Simulating complex traits influenced by genes with fuzzy-valued effects in pedigreed populations, Bioinformatics, 19 (2003), 144-148. doi: 10.1093/bioinformatics/19.1.144. [3] S. Charfeddine, K. Zbidi and F. Mora-Camino, Fuzzy-regression analysis using trapezoidal fuzzy numbers, In Proceedings of the Joint 4th Conference of the European Society for Fuzzy Logic and Technology and the 11th Rencontres Francophones sur la Logique Floue et ses Applications(eds. E. Montseny, P. Sobrevilla), Barcelona, Spain, September 7-9, 2005. [4] Y.-H. O. Chang and B. M. Ayyub, Fuzz y-regression methods -a comparative assessment, Fuzzy Sets and Systems, 119 (2001), 187-203. doi: 10.1016/S0165-0114(99)00091-3. [5] P. Diamond, Fuzzy least squares source, Information Sciences, 46 (1988), 141-157. doi: 10.1016/0020-0255(88)90047-3. [6] J. Gebert, M. Lätsch, E. M. P. Quek and G. -W. Weber, Analyzing and optimizing genetic network structure via path-finding, Journal of Computational Technologies, 9 (2004), 3-12. [7] T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning, Springer, New York, 2009. doi: 10.1007/978-0-387-84858-7. [8] H. Ishibuchi and M. Nii, Fuzzy-regression using asymmetric fuzzy coefficients and fuzzified neural networks, Fuzzy Sets and Systems, 199 (2004), 273-290. doi: 10.1016/S0165-0114(98)00370-4. [9] T. Kubota, K. Miyake and T. Hirasawa, Epigenetic understanding of gene-environment interactions in psychiatric disorders: a new concept of clinical genetics Clinical Epigenetics, 4 (2012), 1. doi: 10.1201/b16680-14. [10] C. Kahraman, A. Beskese and F. Tunc Bozbura, Fuzzy-regression approaches and applications, In Fuzzy Applications in Industrial Engineering. Studies in Fuzziness and Soft Computing (ed. C. Kahraman), Volume 201. Springer Berlin, Heidelberg, (2006), 589-615. doi: 10.1007/3-540-33517-X_24. [11] G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic. Theory and Applications, Prentice Hall, New Jersey, 1995. [12] E. Kropat, Fuzzy-Unabhängigkeitssysteme und Fuzzy-Matroide, Wissenschaftlicher Verlag Berlin, Berlin, 2002. [13] E. Kropat, A. Özmen, G.-W. Weber, S. Meyer-Nieberg and Ö. Defterli, Fuzzy evolving networks and fuzzy-regression analysis for two-modal systems - prediction strategies for fuzzy target-environment networks, RAIRO-Operations Research, special issue on "Recent Advances in Operations Research in Computational Biology, Bioinformatics and Medicine", 50 (2012), 413-435. doi: 10.1051/ro/2015044. [14] E. Kropat, G. -W. Weber and B. Akteke-Öztürk, Eco-finance networks under uncertainty, in Proceedings of the International Conference on Engineering Optimization (ISBN 978857650156-5, CD)(eds. J. Herskovits, A. Canelas, H. Cortes and M. Aroztegui), EngOpt 2008, Rio de Janeiro, Brazil, 01-05 June 2008. [15] E. Kropat, G. -W. Weber, S. Z. Alparslan-Gök and A. Özmen, Inverse problems in complex multi-modal regulatory networks based on uncertain clustered data, In Modeling, Optimization, Dynamics and Bioeconomy (eds. A. Pinto and D. Zilberman), Springer International Publishing, (2014), 437-451. doi: 10.1007/978-3-319-04849-9_25. [16] E. Kropat, G. -W. Weber and S. Belen, Dynamical gene-environment networks under ellipsoidal uncertainty -set-theoretic regression analysis based on ellipsoidal OR, In Dynamics, Games and Science I, Springer Proceedings in Mathematics (eds. M. M. Peixoto, A. A. Pinto, D. A. Rand):, Vol. 1, Springer-Verlag Berlin-Heidelberg, (2011), 545-571. doi: 10.1007/978-3-642-11456-4_35. [17] E. Kropat, G. -W. Weber and C. S. Pedamallu, Regulatory networks under ellipsoidal uncertainty -Data analysis and prediction by optimization theory and dynamical systems, In Data Mining: Foundations and Intelligent Paradigms (eds. MD. E. Holmes, L. S. Jain): Volume 2: Statistical, Bayesian, Time Series and other Theoretical Aspects, Springer-Verlag Berlin, (2012), 27-56. doi: 10.1007/978-3-642-23241-1_3. [18] E. Kropat, G.-W. Weber and J.-J. Rückmann, Regression analysis for clusters in gene-environment networks based on ellipsoidal calculus and optimization, Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications & Algorithms, 17 (2010), 639-657. [19] D. S. Malik and J. N. Mordeson, Fuzzy Discrete Structures, Physica-Verlag, Heidelberg, 2000. doi: 10.1007/978-3-7908-1838-3. [20] M. Mazandarani and M. Najariyan, Differentiability of type-2 fuzzy number-valued functions, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 710-725. doi: 10.1016/j.cnsns.2013.07.002. [21] M. J. Meaney, Epigenetics and the biology of gene × environment interactions, In Gene-Environment Transactions in Developmental Psychopathology: The Role in Intervention Research (eds. P. H. Tolan and B. L. Leventhal), Springer International Publishing, Cham, (2017), 59-94. [22] G. Peters, Fuzzy linear regression with fuzzy intervals, Fuzzy Sets and Systems, 63 (1994), 45-55. doi: 10.1016/0165-0114(94)90144-9. [23] D. T. Redden and W. H. Woodall, Properties of certain fuzzy linear regression methods, Fuzzy Sets and Systems, 64 (1994), 361-375. doi: 10.1016/0165-0114(94)90159-7. [24] L. -K. Pries, S. Gülöksüz and G. Kenis, DNA methylation in schizophrenia, In Neuroepigenomics in Aging and Disease (ed. R. Delgado-Morales), Springer International Publishing, Cham, (2017), 211-236. doi: 10.1007/978-3-319-53889-1_12. [25] D. A. Savic and W. Pedrycz, Evaluation of fuzzy linear regression models, Fuzzy Sets and Systems, 39 (1991), 51-63. doi: 10.1016/0165-0114(91)90065-X. [26] M. Sakawa and H. Yano, Multiobjective fuzzy linear regression analysis for fuzzy input-output data, Fuzzy Sets and Systems, 47 (1992), 173-181. doi: 10.1016/0020-0255(92)90069-K. [27] H. Tanaka, K. Uejima and K. Asai, Linear regression analysis with fuzzy model, IEEE, Systems, Trans. Systems Man Cybernet., SMC-2, (1982), 903-907. [28] H. Tanaka, I. Hayashi and J. Watada, Possibilistic linear regression analysis for fuzzy data, European Journal of Operational Research, 40 (1989), 389-396. doi: 10.1016/0377-2217(89)90431-1. [29] H. Tanaka and H. Ishibuchi, Possibilistic regression analysis based on linear programming, In Fuzzy-Regression Analysis (eds. J. Kacprzyk, M. Fedrizzi). Physica-Verlag, Heidelberg, (1992), 47-60. [30] H. Tanaka, H. Ishibuchi and S. G. Hwang, Fuzzy model of the number of staff in local government by fuzzy-regression analysis with similarity relations, J. Jpn. Indust. Management Assoc., 41 (1990), 99-104. [31] H. Tanaka, S. Uejima and K. Asai, Linear regression analysis with fuzzy model, IEEE Trans. Systems Man and Cybernetics, 12 (1982), 903-907. doi: 10.1109/TSMC.1982.4308923. [32] M. Taştan, Analysis and prediction of gene expression patterns by dynamical systems, and by a combinatorial algorithm, Sc Thesis, Institute of Applied Mathematics, Middle East Technical University (METU), Ankara, Turkey, 2005. [33] Ö. Uǧur and G.-W. Weber, Optimization and dynamics of gene-environment networks with intervals, J. Ind. Manag. Optim., 3 (2007), 357-379. doi: 10.3934/jimo.2007.3.357. [34] G. -W. Weber, Charakterisierung struktureller Stabilität in der nichtlinearen Optimierung, in Aachener Beiträge zur Mathematik 5 (eds. H. H. Bock, H. T. Jongen, and W. Plesken), Augustinus publishing house (now: Mainz publishing house) Aachen, 1992. [35] G. -W. Weber, Generalized Semi-Infinite Optimization and Related Topics, Heldermann Publishing House, Research and Exposition in Mathematics 29, Lemgo, 2003. [36] G.-W. Weber, S. Z. Alparslan-Gök and N. Dikmen, Environmental and life sciences: gene-environment networks -optimization, games and control -a survey on recent achievements, Journal of Organisational Transformation and Social Change, 5 (2008), 197-233. doi: 10.1386/jots.5.3.197_1. [37] G.-W. Weber, S. Z. Alparslan-Gök and B. Söyler, A new mathematical approach in environmental and life sciences: gene-environment networks and their dynamics, Environmental Modeling & Assessment, 14 (2007), 267-288. doi: 10.1007/s10666-007-9137-z. [38] G.-W. Weber, E. Kropat, B. Akteke-Öztürk and Z.-K. Görgülü, Survey on OR and mathematical methods applied on gene-environment networks, Central European Journal of Operations Research (CEJOR), 17 (2009), 315-341. doi: 10.1007/s10100-009-0092-4. [39] G.-W. Weber, E. Kropat, A. Tezel and S. Belen, Optimization applied on regulatory and eco-finance networks -survey and new developments, Pacific Journal of Optimization, 6 (2010), 319-340. [40] G.-W. Weber, S. Özögür-Akyüz and E. Kropat, A review on data mining and continuous optimization applications in computational biology and medicine, Embryo Today, Birth Defects Research (Part C), 87 (2009), 165-181. doi: 10.1002/bdrc.20151. [41] G.-W. Weber, A. Tezel, P. Taylan, A. Soyler and M. Çetin, Mathematical contributions to dynamics and optimization of gene-environment networks, Optimization, 57 (2008), 353-377. doi: 10.1080/02331930701780037. [42] G.-W. Weber, P. Taylan, S.-Z. Alparslan-Gök, S. Özöğür and B. Akteke-Öztürk, Optimization of gene-environment networks in the presence of errors and uncertainty with Chebychev approximation, TOP, the Operational Research journal of SEIO (Spanish Statistics and Operations Research Society), 16 (2008), 284-318. doi: 10.1007/s11750-008-0052-5. [43] G.-W. Weber, Ö. Uğur, P. Taylan and A. Tezel, On optimization, dynamics and uncertainty: a tutorial for gene-environment networks, Discrete Applied Mathematics, 157 (2009), 2494-2513. doi: 10.1016/j.dam.2008.06.030. [44] F. B. Yılmaz, H. Öktem and G. -W. Weber, Mathematical modeling and approximation of gene expression patterns and gene networks, In Operations Research Proceedings 2004: Selected Papers of the Annual International Conference of the German Operations Research Society (GOR) (eds. F. Fleuren, D. den Hertog and P. Kort). Jointly Organized with the Netherlands Society for Operations Research (NGB) Tilburg, September 1-3, 2004, Springer, Berlin, Heidelberg, (2005), 280-287. [45] H.-J. Zimmermann, Fuzzy Set Theory - and Its Applications, Springer Netherlands, 2001. doi: 10.1007/978-94-010-0646-0.

show all references

##### References:
 [1] U. Bronfenbrenner, The Ecology of Human Development: Experiments by Nature and Design, Cambridge, MA: Harvard University Press (ISBN 0-674-22457-4), 1979. [2] C. Carleos, F. Rodriguez, H. Lamelas and J. A. Baro, Simulating complex traits influenced by genes with fuzzy-valued effects in pedigreed populations, Bioinformatics, 19 (2003), 144-148. doi: 10.1093/bioinformatics/19.1.144. [3] S. Charfeddine, K. Zbidi and F. Mora-Camino, Fuzzy-regression analysis using trapezoidal fuzzy numbers, In Proceedings of the Joint 4th Conference of the European Society for Fuzzy Logic and Technology and the 11th Rencontres Francophones sur la Logique Floue et ses Applications(eds. E. Montseny, P. Sobrevilla), Barcelona, Spain, September 7-9, 2005. [4] Y.-H. O. Chang and B. M. Ayyub, Fuzz y-regression methods -a comparative assessment, Fuzzy Sets and Systems, 119 (2001), 187-203. doi: 10.1016/S0165-0114(99)00091-3. [5] P. Diamond, Fuzzy least squares source, Information Sciences, 46 (1988), 141-157. doi: 10.1016/0020-0255(88)90047-3. [6] J. Gebert, M. Lätsch, E. M. P. Quek and G. -W. Weber, Analyzing and optimizing genetic network structure via path-finding, Journal of Computational Technologies, 9 (2004), 3-12. [7] T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning, Springer, New York, 2009. doi: 10.1007/978-0-387-84858-7. [8] H. Ishibuchi and M. Nii, Fuzzy-regression using asymmetric fuzzy coefficients and fuzzified neural networks, Fuzzy Sets and Systems, 199 (2004), 273-290. doi: 10.1016/S0165-0114(98)00370-4. [9] T. Kubota, K. Miyake and T. Hirasawa, Epigenetic understanding of gene-environment interactions in psychiatric disorders: a new concept of clinical genetics Clinical Epigenetics, 4 (2012), 1. doi: 10.1201/b16680-14. [10] C. Kahraman, A. Beskese and F. Tunc Bozbura, Fuzzy-regression approaches and applications, In Fuzzy Applications in Industrial Engineering. Studies in Fuzziness and Soft Computing (ed. C. Kahraman), Volume 201. Springer Berlin, Heidelberg, (2006), 589-615. doi: 10.1007/3-540-33517-X_24. [11] G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic. Theory and Applications, Prentice Hall, New Jersey, 1995. [12] E. Kropat, Fuzzy-Unabhängigkeitssysteme und Fuzzy-Matroide, Wissenschaftlicher Verlag Berlin, Berlin, 2002. [13] E. Kropat, A. Özmen, G.-W. Weber, S. Meyer-Nieberg and Ö. Defterli, Fuzzy evolving networks and fuzzy-regression analysis for two-modal systems - prediction strategies for fuzzy target-environment networks, RAIRO-Operations Research, special issue on "Recent Advances in Operations Research in Computational Biology, Bioinformatics and Medicine", 50 (2012), 413-435. doi: 10.1051/ro/2015044. [14] E. Kropat, G. -W. Weber and B. Akteke-Öztürk, Eco-finance networks under uncertainty, in Proceedings of the International Conference on Engineering Optimization (ISBN 978857650156-5, CD)(eds. J. Herskovits, A. Canelas, H. Cortes and M. Aroztegui), EngOpt 2008, Rio de Janeiro, Brazil, 01-05 June 2008. [15] E. Kropat, G. -W. Weber, S. Z. Alparslan-Gök and A. Özmen, Inverse problems in complex multi-modal regulatory networks based on uncertain clustered data, In Modeling, Optimization, Dynamics and Bioeconomy (eds. A. Pinto and D. Zilberman), Springer International Publishing, (2014), 437-451. doi: 10.1007/978-3-319-04849-9_25. [16] E. Kropat, G. -W. Weber and S. Belen, Dynamical gene-environment networks under ellipsoidal uncertainty -set-theoretic regression analysis based on ellipsoidal OR, In Dynamics, Games and Science I, Springer Proceedings in Mathematics (eds. M. M. Peixoto, A. A. Pinto, D. A. Rand):, Vol. 1, Springer-Verlag Berlin-Heidelberg, (2011), 545-571. doi: 10.1007/978-3-642-11456-4_35. [17] E. Kropat, G. -W. Weber and C. S. Pedamallu, Regulatory networks under ellipsoidal uncertainty -Data analysis and prediction by optimization theory and dynamical systems, In Data Mining: Foundations and Intelligent Paradigms (eds. MD. E. Holmes, L. S. Jain): Volume 2: Statistical, Bayesian, Time Series and other Theoretical Aspects, Springer-Verlag Berlin, (2012), 27-56. doi: 10.1007/978-3-642-23241-1_3. [18] E. Kropat, G.-W. Weber and J.-J. Rückmann, Regression analysis for clusters in gene-environment networks based on ellipsoidal calculus and optimization, Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications & Algorithms, 17 (2010), 639-657. [19] D. S. Malik and J. N. Mordeson, Fuzzy Discrete Structures, Physica-Verlag, Heidelberg, 2000. doi: 10.1007/978-3-7908-1838-3. [20] M. Mazandarani and M. Najariyan, Differentiability of type-2 fuzzy number-valued functions, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 710-725. doi: 10.1016/j.cnsns.2013.07.002. [21] M. J. Meaney, Epigenetics and the biology of gene × environment interactions, In Gene-Environment Transactions in Developmental Psychopathology: The Role in Intervention Research (eds. P. H. Tolan and B. L. Leventhal), Springer International Publishing, Cham, (2017), 59-94. [22] G. Peters, Fuzzy linear regression with fuzzy intervals, Fuzzy Sets and Systems, 63 (1994), 45-55. doi: 10.1016/0165-0114(94)90144-9. [23] D. T. Redden and W. H. Woodall, Properties of certain fuzzy linear regression methods, Fuzzy Sets and Systems, 64 (1994), 361-375. doi: 10.1016/0165-0114(94)90159-7. [24] L. -K. Pries, S. Gülöksüz and G. Kenis, DNA methylation in schizophrenia, In Neuroepigenomics in Aging and Disease (ed. R. Delgado-Morales), Springer International Publishing, Cham, (2017), 211-236. doi: 10.1007/978-3-319-53889-1_12. [25] D. A. Savic and W. Pedrycz, Evaluation of fuzzy linear regression models, Fuzzy Sets and Systems, 39 (1991), 51-63. doi: 10.1016/0165-0114(91)90065-X. [26] M. Sakawa and H. Yano, Multiobjective fuzzy linear regression analysis for fuzzy input-output data, Fuzzy Sets and Systems, 47 (1992), 173-181. doi: 10.1016/0020-0255(92)90069-K. [27] H. Tanaka, K. Uejima and K. Asai, Linear regression analysis with fuzzy model, IEEE, Systems, Trans. Systems Man Cybernet., SMC-2, (1982), 903-907. [28] H. Tanaka, I. Hayashi and J. Watada, Possibilistic linear regression analysis for fuzzy data, European Journal of Operational Research, 40 (1989), 389-396. doi: 10.1016/0377-2217(89)90431-1. [29] H. Tanaka and H. Ishibuchi, Possibilistic regression analysis based on linear programming, In Fuzzy-Regression Analysis (eds. J. Kacprzyk, M. Fedrizzi). Physica-Verlag, Heidelberg, (1992), 47-60. [30] H. Tanaka, H. Ishibuchi and S. G. Hwang, Fuzzy model of the number of staff in local government by fuzzy-regression analysis with similarity relations, J. Jpn. Indust. Management Assoc., 41 (1990), 99-104. [31] H. Tanaka, S. Uejima and K. Asai, Linear regression analysis with fuzzy model, IEEE Trans. Systems Man and Cybernetics, 12 (1982), 903-907. doi: 10.1109/TSMC.1982.4308923. [32] M. Taştan, Analysis and prediction of gene expression patterns by dynamical systems, and by a combinatorial algorithm, Sc Thesis, Institute of Applied Mathematics, Middle East Technical University (METU), Ankara, Turkey, 2005. [33] Ö. Uǧur and G.-W. Weber, Optimization and dynamics of gene-environment networks with intervals, J. Ind. Manag. Optim., 3 (2007), 357-379. doi: 10.3934/jimo.2007.3.357. [34] G. -W. Weber, Charakterisierung struktureller Stabilität in der nichtlinearen Optimierung, in Aachener Beiträge zur Mathematik 5 (eds. H. H. Bock, H. T. Jongen, and W. Plesken), Augustinus publishing house (now: Mainz publishing house) Aachen, 1992. [35] G. -W. Weber, Generalized Semi-Infinite Optimization and Related Topics, Heldermann Publishing House, Research and Exposition in Mathematics 29, Lemgo, 2003. [36] G.-W. Weber, S. Z. Alparslan-Gök and N. Dikmen, Environmental and life sciences: gene-environment networks -optimization, games and control -a survey on recent achievements, Journal of Organisational Transformation and Social Change, 5 (2008), 197-233. doi: 10.1386/jots.5.3.197_1. [37] G.-W. Weber, S. Z. Alparslan-Gök and B. Söyler, A new mathematical approach in environmental and life sciences: gene-environment networks and their dynamics, Environmental Modeling & Assessment, 14 (2007), 267-288. doi: 10.1007/s10666-007-9137-z. [38] G.-W. Weber, E. Kropat, B. Akteke-Öztürk and Z.-K. Görgülü, Survey on OR and mathematical methods applied on gene-environment networks, Central European Journal of Operations Research (CEJOR), 17 (2009), 315-341. doi: 10.1007/s10100-009-0092-4. [39] G.-W. Weber, E. Kropat, A. Tezel and S. Belen, Optimization applied on regulatory and eco-finance networks -survey and new developments, Pacific Journal of Optimization, 6 (2010), 319-340. [40] G.-W. Weber, S. Özögür-Akyüz and E. Kropat, A review on data mining and continuous optimization applications in computational biology and medicine, Embryo Today, Birth Defects Research (Part C), 87 (2009), 165-181. doi: 10.1002/bdrc.20151. [41] G.-W. Weber, A. Tezel, P. Taylan, A. Soyler and M. Çetin, Mathematical contributions to dynamics and optimization of gene-environment networks, Optimization, 57 (2008), 353-377. doi: 10.1080/02331930701780037. [42] G.-W. Weber, P. Taylan, S.-Z. Alparslan-Gök, S. Özöğür and B. Akteke-Öztürk, Optimization of gene-environment networks in the presence of errors and uncertainty with Chebychev approximation, TOP, the Operational Research journal of SEIO (Spanish Statistics and Operations Research Society), 16 (2008), 284-318. doi: 10.1007/s11750-008-0052-5. [43] G.-W. Weber, Ö. Uğur, P. Taylan and A. Tezel, On optimization, dynamics and uncertainty: a tutorial for gene-environment networks, Discrete Applied Mathematics, 157 (2009), 2494-2513. doi: 10.1016/j.dam.2008.06.030. [44] F. B. Yılmaz, H. Öktem and G. -W. Weber, Mathematical modeling and approximation of gene expression patterns and gene networks, In Operations Research Proceedings 2004: Selected Papers of the Annual International Conference of the German Operations Research Society (GOR) (eds. F. Fleuren, D. den Hertog and P. Kort). Jointly Organized with the Netherlands Society for Operations Research (NGB) Tilburg, September 1-3, 2004, Springer, Berlin, Heidelberg, (2005), 280-287. [45] H.-J. Zimmermann, Fuzzy Set Theory - and Its Applications, Springer Netherlands, 2001. doi: 10.1007/978-94-010-0646-0.
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 $Z'_{i0}$
The symmetric triangular fuzzy coefficient $A_{jr} = (A^C_{jr}, A^W_{jr})$
The $\alpha$-cut of the fuzzy model $\mathcal{F}_j$
The asymmetric triangular fuzzy coefficient $A_{jr} = (A^L_{jr}, A^C_{jr}, A^U_{jr})$
The asymmetric triangular fuzzy coefficient $A_{jr} = (A^L_{jr}, A^M_{jr}, A^N_{jr}, A^U_{jr})$
The fuzzy inclusion relation with fuzzy model $\mathcal{F}_j$ and fuzzy output $\widehat{X}^{(\kappa+1)}_j$
The fuzzy inclusion relation with asymmetric trapezoidal fuzzy model $\mathcal{F}_j$ and asymmetric triangular fuzzy output $\widehat{X}_j$
fuzzy-regression algorithms for target-environment data
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