Sanjay Kumar

Hesitant intuitionistic fuzzy sets (HIFSs) are more useful than hesitant fuzzy sets (HFSs) and intuitionistic fuzzy sets (IFSs) to model the uncertainty with non-determinism caused by the availability of multiple membership and non-membership grades. In this research paper, we propose the HIFS-based intuitionistic fuzzy time series (IFTS) forecasting method. The proposed method uses adaptive radius clustering for optimal partitioning and incorporates uncertainty with non-determinism in the process of fuzzification of time series data through HIFS. Intuitionistic fuzzy logical relations (IFLRs) used in proposed forecasting method are based on IFSs obtained by aggregating elements of HIFS. Probabilistic λ-cutting algorithm that groups IFSs according to their similarity measure is used in proposed forecasting method to establish IFLRs. Proposed IFTS forecasting method uses a simple computational rule to forecast the outputs and make the process of forecasting simple. Practicability and utility of the suggested forecasting method is shown by implementing it on time series data of the share price of State Bank of India (SBI) at the Bombay Stock Exchange (BSE), India, and the Taiwan Stock Exchange (TAIEX) data. Reduced error measures and valid statistical measure confirm superiority of the proposed method in forecasting of financial time series data.

Computational methods for time series forecasting have always an edge over conventional methods of forecasting due to their easy implementation and prominent characteristics of coping with large amount of time series data. Many computational methods for fuzzy time series (FTS) forecasting have been developed in past using fuzzy set, intuitionistic fuzzy set (IFS), and hesitant fuzzy set (HFS) for incorporating uncertainty, non-determinism, and hesitation in time series forecasting. Since probabilistic fuzzy set (PFS) incorporates both probabilistic and non-probabilistic uncertainties simultaneously, we have proposed PFS and particle swarm optimization (PSO) based computational method for FTS forecasting. First, we have developed a PFS based computational method for FTS forecasting and then it is integrated with PSO to enhance the accuracy in forecasted outputs. Unlike other PSO based for FTS forecasting method, PSO is used to optimize both number of partitions and length of intervals. Three diversified time series data of enrolments of the University of Alabama, market price of State Bank of India (SBI) share at Bombay stock exchange (BSE) India, and death cases due to COVID-19 in India are used to compare the performance of PFS based computational method of FTS forecasting before and after its integration with PSO in terms of root mean square error (RMSE). After integration of PFS based computational method with PSO, accuracy in the forecasted outputs is increased significantly and its performance is found better than many other existing FTS forecasting methods. Goodness of the proposed FTS forecasting method is also tested using tracking signal and Willmott index.

In this paper, limiting behaviour of intuitionistic fuzzy parameterized soft set (intuitionistic FP-soft set) based decision making method is shown by a counter example. Then we define picture fuzzy parameterized soft set (picture FP-soft set). This notion can be used to express fuzzy information in more general and effective way. In particular, some basic operations such as union, intersection, complement and subset of picture FP-soft sets are developed. We also propose a conversion method to construct picture fuzzy set from intuitionistic fuzzy set. Furthermore, reduced PF-soft set and reduced intuitionistic fuzzy set are proposed. Based on these definition decision making method based on picture FP-soft set is developed. Finally, using proposed construction method we convert intuitionistic fuzzy information of counter example in picture fuzzy information and then apply picture FP-soft set based decision making method leading a more clear decision of problem which shows efficiency of proposed set and decision making method.

Fuzzy Time Series (FTS) techniques are gaining popularity among researchers. Fuzzification, fuzzy relation determination, and defuzzification are the three step