Determining of Level Sets for a Fuzzy Surface Using Gröbner Basis

Determining of Level Sets for a Fuzzy Surface Using Gröbner Basis

Hamed Farahani, Sajjad Rahmany, Abdolali Basiri
Copyright: © 2015 |Pages: 14
DOI: 10.4018/IJFSA.2015040101
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In this paper, a manner to determine the level sets of a fuzzy surface using the benefits of Gröbner basis is presented. Fuzzy surfaces are constructed from incomplete datasets or from data that contain uncertainty which has not statistical nature. The authors firstly define the concept of level sets for the fuzzy surfaces. Then, employing Gröbner bases benefits a criterion is proposed for when the level sets of the fuzzy surface are nonempty sets. Moreover, a new algorithm is designed to determine the level sets. The big advantage of the proposed method lies in the fact that it attains all members of the level sets of the fuzzy surface at a time. Finally, some applied numerical examples are illustrated to demonstrate the proficiency of the given approach.
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1. Introduction

Fuzzy surfaces play a major role in various areas of sciences such as geosciences (Burrough and McDonnell 1998, Caha et al. 2012, Janoska and Dvorsky 2012, Lodwick 2008, Waelder 2007), medical sciences (Dubey and Nagpal 2011). In these areas, the information used for modelling is often incomplete. For example points used to the model surface are only samples from the set of all points on the surface and their precision can be unknown. From such inputs it is not possible to create precise and absolutely certain surface. It is necessary to include the uncertainty, that has origin in both input data and interpolation process, in the surface. Since none of the mentioned types of uncertainty is of statistical nature it is convenient to conceptualize the uncertainty by an alternative method i.e. fuzzy set theory. Fuzzy set theory and fuzzy logic dispose of methods and tools for modelling surfaces based on biased input data. Especially fuzzy numbers are suited for modelling surfaces with uncertainty. Triangular and trapezoidal fuzzy numbers are the most common ones that are used in applications because of their easy implementation.

In this work, a novel approach is presented dealing with fuzzy surfaces of arbitrary degree. Firstly, consider an arbitrary fuzzy fixed constant is considered. Then, with equalization the fuzzy fixed constant and the fuzzy surface (with IJFSA.2015040101.m01 real independent variables) a multivariate fuzzy polynomial equation will be obtained. The set of all ordered IJFSA.2015040101.m02-tuples in IJFSA.2015040101.m03 that satisfy this multivariate fuzzy polynomial equation is called the level set for the fuzzy surface. Our aim in this work is to determine the level sets for the fuzzy surfaces using Gröbner basis. The presented idea in this paper is to compute the IJFSA.2015040101.m04-cuts of the multivariate fuzzy polynomial equation and obtain its parametric form with respect to IJFSA.2015040101.m05 and the sign of variables. Then, the parametric form of the fuzzy polynomial equation with respect to IJFSA.2015040101.m06 is collected and polynomials as IJFSA.2015040101.m07 are obtained, where IJFSA.2015040101.m08. We compute a Gröbner basis IJFSA.2015040101.m09 for the ideal generated by IJFSA.2015040101.m10 with respect to the lexicographical order. The basis has an upper triangular structure. Then the system can be solved using the forward substitution. We will show that the set of solutions of the Gröbner basis system and the level set of the fuzzy surface are equal. Also, a criterion is proposed based on Gröbner basis for when the level sets of the fuzzy surfaces are empty sets.

The structure of this paper is organized as follows. Section 2 is divided to two subsections. Some required definitions and results about fuzzy numbers is given in the first subsection. The required concepts and results about the Gröbner basis are presented in the second subsection. Section 3 is divided to two subsections. Some required definitions and results about fuzzy surfaces is given in the first subsection. In the second subsection the main idea for determining the level sets for the fuzzy surfaced is presented. Furthermore, an algorithm is proposed to determine the level sets. Some examples are given to illustrate the algorithm in Section 4. Finally, conclusions are presented in Section 5.

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