Fuzzy Linear Programming: Optimization of an Electric Circuit Model
DOI:
https://doi.org/10.5540/tema.2017.018.03.419Keywords:
Electrical circuit model, Linear programming, Fuzzy linear programming, Fuzzy sets of type-1, Fuzzy sets of type-2Abstract
A problem of a voltage division circuit is modeled in order to determine the values of the resistors, centered in away that the impedance of the resistance voltage divider is minimal. This problem is equivalent to maximizing the admittance, associated to the resistance, which is defined as the quotient of the electric current and its voltage, measured in Siemens. Three cases are analyzed for the components of the linear programming: real numbers, fuzzy numbers of type-1, and fuzzy set of type-2. The first case is considered in order to validate the other two cases. The optimal solution in the fuzzy linear programming of type-1 is obtained through a total linear order defuzzification function, defined in the trapezoidal fuzzy numbers subspace of fuzzy numbers vector space, which allows to solve the corresponding linear programming problem with real components. A discussion upon the parameter for the linear defuzzification is establish to determined the best representative of the parameters family. The α−levels representation theorem is the method to obtain the optimal solution of type-2. For each α−level is solved a fuzzy linear programming problem of type-1, using the previous methodology. Numerical simulations illustrate the results in the three cases.References
J. M. Adamo, Fuzzy decision trees, Fuzzy Sets and Systems, 4, (1980), 207-219.
R. E. Bellman, L. A. Zadeh, Decision-making in a fuzzy environment, Management Science, 17, (1973), 149-156.
G. B. Dantzig, "Origins of the simplex method" , Technical Report SOL 87-5, Department of Operations Research, Stanford University, Stanford, CA, 1987.
D. Dubois, H. Prade, Operations on fuzzy numbers, International Journal of Systems Science, 9, (1978), 613-626.
N. N. Karnik, J. M. Mendel, Q. Liang, Type-2 fuzzy logic systems, IEEE Trans. on Fuzzy Systems, 7, (1999), 643-658.
J. M. Mendel, M. R. Rajatia, P. Sussner, On clarifying some definitions and notations used for type-2 fuzzy sets as well as some recommended changes, Information Sciences, 340-341, (2016), 337-345.
W. Pedryz, F. Gomide, "An Introduction to Fuzzy Sets: Analysis and Design", MIT Press, 1998.
H. Prade, R.R. Yager, D. Dubois, editors, "Readings in Fuzzy Sets for Intelligent Systems", Morgan Kaufmann Publishers, 1993.
N. Sahinidis, Optimization under uncertainty: state-of-the-art and opportunities, Computers and Chemical Engineering, 28, (2004), 971-983.
J. J. Salazar González, Optimización matemática: Ejemplos y aplicaciones, Technical report Universidad de La Laguna, España,
https://imarrero.webs.ull.es/sctm03.v2/, 2003.
J. L. Verdegay, "Fuzzy Mathematical Programming", in: Fuzzy Information and Decision Processes, pp. 231-237, M.M. Gupta and E. Sánchez Editors, 1982.
J. Nocedal, S. J. Wright, "Numerical Optimization", Springer-Verlag, 2006.
Downloads
Additional Files
Published
How to Cite
Issue
Section
License
Authors who publish in this journal agree to the following terms:
Authors retain copyright and grant the journal the right of first publication, with the work simultaneously licensed under the Creative Commons Attribution License that allows the sharing of the work with acknowledgment of authorship and initial publication in this journal.
Authors are authorized to assume additional contracts separately, for non-exclusive distribution of the version of the work published in this journal (eg, publish in an institutional repository or as a book chapter), with acknowledgment of authorship and initial publication in this journal.
Authors are allowed and encouraged to publish and distribute their work online (eg, in institutional repositories or on their personal page) at any point before or during the editorial process, as this can generate productive changes as well as increase impact and the citation of the published work (See The effect of open access).
This is an open access journal which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the
author. This is in accordance with the BOAI definition of open access
Intellectual Property
All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License under attribution BY.