Background: This paper generalizes the results of Embedding problem of Fuzzy Number Space and its extension into a Fuzzy Banach Space C(Ω) × C(Ω), where C(Ω) is the set of all real-valued continuous functions on an open set Ω.
Objectives: The main idea behind our approach consists of taking advantage of interplays between fuzzy normed spaces and normed spaces in a way to get an equivalent stochastic program. This helps avoiding pitfalls due to severe oversimplification of the reality.
Method: The embedding theorem shows that the set of all fuzzy numbers can be embedded into a Fuzzy Banach space. Inspired by this embedding theorem, we propose a solution concept of fuzzy optimization problem which is obtained by applying the embedding function to the original fuzzy optimization problem.
Results: The proposed method is used to extend the classical Mean-Variance portfolio selection model into Mean Variance-Skewness model in fuzzy environment under the criteria on short and long term returns, liquidity and dividends.
Conclusion: A fuzzy optimization problem can be transformed into a multiobjective optimization problem which can be solved by using interactive fuzzy decision making procedure. Investor preferences determine the optimal multiobjective solution according to alternative scenarios.
The purpose of this paper is to show that the category of normed spaces can be embedded in the category of Menger probabilistic normed spaces, and that C(Ω) is probabilistic normable, whereas it is not normable in the classical case, when Ω is an open subset of R n. So, the spectrum of the category of Menger probabilistic normed spaces is broader than the category of classical normed spaces. Therefore, it can be a meaningful replacement in some model of security markets. As our model is suitably generalized, that fact can help us to adapt and improve within natural problems of finance, especially, in portfolio optimization of insurance.
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