Abstract

Background: It has been a matter of discussion in years even after using an FST. Because of being a single valued membership, a fuzzy set can't express desired information. Further extended as HFSs which allows all possible membership degree lying between [0,1] is used widely where hesitancy occurs in taking preference over matters in Decision Making.

Objective: The aim in this paper is to create a diversified portfolio where the return is maximum and the risk is minimal.

Methods: Decision making methods like Fuzzy Soft Set, Mean Potentially Approach, and Soft Hesitant Fuzzy Rough Set which are based on Fuzzy Soft Set theory for construct the optimal portfolio. And non-fuzzy set method is applied for the optimal portfolio. It is found that a Soft hesitant fuzzy rough set is best as compare to other methods. Then, the ratio of optimal portfolio is obtained with the help of firefly optimization.

Results: Soft Hesitant Fuzzy Rough Set has better outcomes on the basis of Performance Measure of these methods. With the help of Soft Hesitant Fuzzy Rough Set, diversified portfolio is also constructed. After constructing the optimal portfolio, Firefly algorithm is applied for obtain the proportions of seven assets. It is clearly showing the firmness of the ranked portfolio having maximum return and minimum risk on comparing without rank portfolio.

Conclusion: Firefly algorithm is applied for optimization to the proportion of optimal assets of seven assets. The main result is, return and dividend is better and risk is less when compared to without ranking method. It is clear that the optimal portfolio with the ranking method is better than without ranking method.

Keywords: Portfolio optimization, fuzzy soft set, hesitant fuzzy set, hesitant soft set, rough set, firefly optimization.

Graphical Abstract