Abstract
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.
Highlights
Being one of the most important field of Operational Research, Linear Programming (LP) concerns to models that optimize linear objective function and constraints on the decision variables
Efficient algorithms has been developed for LP, among which we point out the interior point [12], and the simplex algorithm [3]
One of the important issues of LP is that the modeling needs well-defined and precise data, which in general involves high information costs. This is a task almost impossible to accomplish in many cases, due to risk or uncertainty in some data [9]. This problem of the classical LP can be contoured with the use of fuzzy numbers
Summary
Being one of the most important field of Operational Research, Linear Programming (LP) concerns to models that optimize linear objective function and constraints on the decision variables. 420 FUZZY LINEAR PROGRAMMING: OPTIMIZATION OF AN ELECTRIC CIRCUIT MODEL of a LP as fuzzy numbers, a constraint violation is allowed, and the degree of satisfaction of a constraint is defined as the membership function of the constraint [9]. A resistance of 48 ohms can have ± 10% tolerance, due to temperature, time of use, among other reasons [10] This gradualness motivated this study to use the fuzzy linear programming approach for a problem of a voltage divider. For the numerical simulation for all cases it is use the classical interior point method [12], using the linear programming algorithm linprog of the software Matlab R , and an algorithm that reproduces the representation theorem.
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