Approximation Of Fuzzy Numbers By Step-type Fuzzy Numbers Under Weighted Metric

With the development of science and technology and modern mathematics,fuzzy set theory is a mathematical tool used to deal with problems(phenomena,objects,events)with uncertain attributes on the boundary of its own concepts.In fuzzy set theory,the membership function of fuzzy set can be used to characterize fuzzy set itself.As a special fuzzy set in the real number field,fuzzy numbers play an important role in characterizing fuzzy phenomena with uncertain or inaccurate digital information.With the increasing complexity of information system science issues,the structure of fuzzy numbers and their membership functions used to describe uncertain or inaccurate numerical information is often quite complex.In order to facilitate calculation and maintain as much information as possible about the original problem to be dealt with,the best nearest approximation among special fuzzy numbers with relatively simple structures can be used to replace this fuzzy number.Therefore,exploring the method of approximating general fuzzy numbers with relatively simple structures is meaningful both in theory and application.In the dissertation,the approximation of multi-order step type fuzzy numbers under the weighted metric is discussed,and the concrete approximation methods(give the concrete calculation formulas of the best approximation)are established.Compared with the approximation of multi-order step type fuzzy numbers proposed by the predecessors,the best approximation obtained in the dissertation should be closer to the original approximation of fuzzy numbers.That is to say,for the approximated fuzzy numbers with more complex structure,the best approximation obtained in this dissertation should be more reasonable and objective than the best approximation given by the predecessors.The reasons are shown in two aspects:1.Here the weighted metric is adopted in the dissertation,while the former used a relatively simple unweighted metric(from the structure of the two distances,we can see that the weighted metric can more objectively and reasonably describe the difference between two fuzzy numbers than the unweighted metric);2.The previous work was to determine the best approximation position of general fuzzy number when the level value of the order(ladder)of the multi-order step type fuzzy number is known.The work of this article is to determine the best approximation position when the level value of the order(ladder)of the step type fuzzy number is unknown.That is,the work of the dissertation is not only to optimize the left and right positions of the best approximation of the step type fuzzy number,but also to optimize its up and down positions.The specific contents of each chapter are as follows:1.The first chapter outlines the background and development of the fuzzy set theory,the development of fuzzy numbers,and the research status,purpose and significance of the fuzzy number approximation.2.The second chapter introduces the relevant concepts of the fuzzy sets and fuzzy numbers and the definitions of two distances,which pave the way for the next chapter.3.The third chapter firstly introduces the required knowledge of the step type fuzzy number approximation method,such as the definition of m-n-step type fuzzy number and related theorems.Secondly,based on the weighted Euclidean metric,the definitions of the best approximation of two step type fuzzy numbers satisfying different conditions are given,and the comparison results are obtained.Finally,in the form of theorems,the methods of approximating general fuzzy numbers with multi-order step type fuzzy numbers are given,and the concrete calculation methods of the best approximation of m-n-step type fuzzy numbers of general fuzzy numbers are obtained.4.The fourth chapter gives the specific calculation methods of approximation of several special forms of multi-order step type fuzzy numbers,and gives the relevant examples to find the best approximation in order to facilitate the use of the results obtained in the previous chapter,and also to illustrate the feasibility and rationality of the obtained results,5.The fifth chapter makes a summary and outlook of the article.