The approximation of fuzzy numbers by parametric functions has been the object on many papers since the start of fuzzy sets in the 1960s by Zadeh. In this paper we suggest the use of piecewise monotonic Hermite splines to approximate and represent a fuzzy number (or interval) and to derive a procedure to control the absolute error associated to the arithmetic operations between fuzzy numbers, in order to reduce the distance between the true result of the operation and its approximation. Several computational experiments are given to show the performance of the suggested procedure. An extensive list of techniques for monotonic splines or piecewise polynomial approximation that preserve monotonicity have considered both local and global approximations. As we are interested in maintaining the simplicity of computations, we suggest and briefly describe the adoption of a local cubic Hermite approximation that is guarantied to be monotonic and can be obtained without solving sets of linear equations or constrained minimization problems as is generally required for global curve fitting. The preposed approximation is the basis to obtain simple parametric representations of the fuzzy numbers or intervals, by the use of piecewise monotonic functions of different forms.
Approximate Fuzzy Arithmetic Operations Using Monotonic Splines
STEFANINI, LUCIANO
2005
Abstract
The approximation of fuzzy numbers by parametric functions has been the object on many papers since the start of fuzzy sets in the 1960s by Zadeh. In this paper we suggest the use of piecewise monotonic Hermite splines to approximate and represent a fuzzy number (or interval) and to derive a procedure to control the absolute error associated to the arithmetic operations between fuzzy numbers, in order to reduce the distance between the true result of the operation and its approximation. Several computational experiments are given to show the performance of the suggested procedure. An extensive list of techniques for monotonic splines or piecewise polynomial approximation that preserve monotonicity have considered both local and global approximations. As we are interested in maintaining the simplicity of computations, we suggest and briefly describe the adoption of a local cubic Hermite approximation that is guarantied to be monotonic and can be obtained without solving sets of linear equations or constrained minimization problems as is generally required for global curve fitting. The preposed approximation is the basis to obtain simple parametric representations of the fuzzy numbers or intervals, by the use of piecewise monotonic functions of different forms.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.