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.
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