This paper presents a novel time series forecasting method based on fuzzy cognitive maps (FCMs) induced by intuitionistic fuzzy set integrated with fuzzy C-means clustering and Particle Swarm Optimization (PSO). As FCM has inherent characteristics of scalability and adaptability, it is used in developed model and includes non-determinacy by using induced fuzzy set that are constructed from intuitionistic fuzzy sets (IFSs). The proposed approach aims to increase the precision in forecasting outputs by leveraging the strengths of each individual component in the presence of uncertainty and non-determinacy. FCMs are utilized to capture the underlying causal relationships in time series data, providing a dynamic representation of system behaviors. Fuzzy C-means clustering is used to create IFSs, enabling more effective handling of uncertainty and imprecision in the data. PSO in the proposed forecasting model is used to fine-tune the parameters of FCM, optimizing the learning process and improving forecasting performance. The proposed hybrid model of time series forecasting is evaluated on time series datasets of market price of State Bank of India and enrolments of the University of Alabama to demonstrate its superior execution compared to existing methods of fuzzy time series forecasting.

Though fuzzy cognitive maps have gained significant popularity in time series forecasting in recent years, more sophisticated methods are required for precise forecasting of non-linear and complex time series data in the environment of uncertainty and hesitation. In this research paper, we integrate hesitant fuzzy cognitive map (HFCM) and complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) and propose a novel method for time series forecasting. CEEMDAN offers better approach than empirical mode decomposition (EMD) and ensemble empirical mode decomposition (EEMD) in handling non-stationary datasets and breaks original time series data into intrinsic mode functions and residues. Proposed forecasting method uses an enhanced cognitive map with two-steps hesitancy in the weights of relationship between the nodes. Two-steps hesitancy in the weights of relationship between the nodes is implemented in a novel way by using Fermi function. Proposed forecasting method uses differential evolution to learn the weights of HFCM to avoid premature convergence and maintain population diversity. Proposed forecasting method is applied to eight diversified time series data of Sunspot, S&P 500, Lake Erie levels, CO2 at Mauna Loa, MCRF, RUT, Milk production, and MG Chaos. Minimum and maximum root mean square error 0.008 and 18.96 are observed in forecasting of time series data of MG Chaos and Milk production. Minimum and maximum mean absolute deviation of 0.006 and 18.521 are also observed in the same time series data of MG Chaos and Milk production. Acceptable range of evaluation parameter (delta(r) < 1 ) and performance parameter (PP > 0) confirm accurate and unbiased forecasted outputs computed using proposed CEEMDAN and HFCM-based time series forecasting method.

A fuzzy set lacks to determine the hesitancy of an element in terms of belongingness to a set, the same problem arises in forecasting time series data by the fuzzy set when there is the availability of multiple fuzzification methods to fuzzify the time series data to remove hesitancy in the system. In the present study, the HFS has been applied in time series forecasting and a HFTSF method is proposed by introducing essential concepts of weighted hesitant fuzzy Cartesian product, HFR, HFLRs, HFLGs and hesitant fuzzy defuzzification method. The basic steps followed in the mechanism of the proposed method are the construction of HFS by a partition of the UOD into intervals of equal and unequal length, distribution of weights based on the length of the interval, fuzzification of the data by using triangular membership function for equal and unequal intervals, construction of HFLRs and HFLGs, relation matrix obtained by weighted hesitant fuzzy Cartesian product, computation of hesitant fuzzy row vectors by max-min composition operation and finally hesitant defuzzification of the data. The proposed method is implemented over the enrollment data of the University of Alabama and the share price of SBI at Bombay stock exchange, India. Performance test, validity test, and statistical test are also examined on the forecasted value by the proposed method and well-known existing methods to examine the superiority of the proposed method. This article presents a novel prediction model that authentically captures methodological hesitancy which were absent in prior HFS based forecasting models due to reliant on aggregation operator that transform HFS into conventional fuzzy set. By directly formulating HFLRs through a weighted Cartesian product, our framework eliminates information loss. The model delivers a triple advantage: it maintain complete information integrity, guaranteeing interpretablity through a transparent calculus, and demonstrating superior accuracy and robustness against existing benchmarks in hesitant environment.

Selecting the most suitable warehouse site is a complex task involving multiple conflicting criteria, such as infrastructure, market, cost, and characteristics. These factors’ inherent ambiguity and uncertainty demand a structured and efficient decision-making method. Polytopic Fuzzy Sets (POFs) are well-suited to model such uncertain information. This study introduces an advanced Weighted Aggregated Sum Product Assessment (WASPAS) method based on POFs, utilizing the Weighted Average (WA) and Weighted Geometric (WG) to improve decision-making in warehouse site selection. A two-stage decision-making framework is then proposed, where criteria weights are determined through POFWA, and final rankings are computed using both the Weighted Sum Method (WSM) based on POFWA and the Weighted Product Method (WPM) based on POFWG. The effectiveness of this method is demonstrated through a case study on the selection of a warehouse site. Comparative analysis confirms the method’s reliability and superiority over existing approaches.

Intellectual Capital Assets (ICAs) refer to the intangible resources that are critical to the creation of organizational value out of human, customer, innovation and process capital. High impact ICAs should be identified and prioritized for investment in by organizations as they strive to increase their value. This research aims at assessing and ranking ICAs by using Analytic Network Process (ANP) model for capturing the interactions between the assets. The method used to gather the pairwise data aimed at supervising CEOs, scientific supervisors, shareholders, employees and customers in order to determine the relative value each ICA contributes to value creation. A mathematical model is employed to translate qualitative assessments into quantitative results with regard to Consistency Index (CI) and Consistency Ratio (CR) to check the reliability of the judgments. Cost benefit analysis shows the ICAs which yield the greatest return on investment. Our findings suggest that knowledge sharing and innovation are the most dominant ICA that positively and significantly affects value creation. These assets involve moderate investment but have huge returns, while other assets such as employee training and customer relations also have greater potential but involve high investment. The results offer practical guidance to organizations in terms of how to effectively prioritize ICAs to support their achievement of strategic objectives in creating innovative value.

Polytopic fuzzy sets are useful in multi-criteria group decision-making (MCGDM) to capture uncertainty when it occurs in such a way that the sum of squares of positive-membership, neutral-membership and negative-memberships exceeds by 1. This research paper introduces interval-valued polytopic fuzzy set and integrates it with linguistic variable to define linguistic interval-valued polytopic fuzzy set (LIVPOFS) for capturing the uncertainty more efficiently using qualitative assessment. We define basic operations, score function, and accuracy function for LIVPOFS. Using prominent characteristic of cloud model to transform qualitative insights into quantitative data, we propose a LIVPOFS and a trapezium cloud-based technique for order of preference by similarity to ideal solution (TOPSIS) method for MCGDM problems. Proposed MCGDM developed in this study allows user feedback to be translated into quantitative data more efficiently, enhancing decision-making accuracy in qualitative MCGDM. Recognizing the essential role of taxi services in transportation networks, we have applied the proposed MCGDM method to rank taxi services of Ola, Uber, Mega, and Meru in the Delhi region of India. The qualitative phase in implementation of proposed method begins with a structured questionnaire designed to collect user opinions on different taxi services. To streamline computations, users are divided into groups, and their responses are collected via Google Forms. The effectiveness of the proposed LIVPOFS-based trapezium cloud TOPSIS in ranking taxi services is further validated through comparisons with existing MCGDM methods.

Ranking of the states for business climate is often done by survey analysis and hence prejudiced by untrustworthy and instable responses of the participants. In the present study, we integrate spherical fuzzy set (SFS) and linear assignment model (LAM) with analytic hierarchy process (AHP) to develop a multi criteria group decision making (MCGDM) method to rank Indian states for their business climate and potential index. The proposed MCGDM method is developed using state investment potential index (SIPI) published by the National Council of Applied Economic Research (NCAER), New Delhi to analyse the 'ease of doing business' aspects of various states based on six parameters. We take these six parameters, sub parameters as criteria and sub criteria in proposed MCGDM method and find their global weight using AHP method. Advantages of the proposed method are that it uses SFSs which are more efficient than other extensions of fuzzy sets in handling non-determinacy, and applies LAM to avoid subjectivity in the preferences of decision makers. We also do the sensitivity analysis of the ranking by changing the weights of the criteria and sub criteria to confirm the robustness of the proposed MCGDM method.

Intuitionistic fuzzy cognitive maps (IFCMs) have been proven effective in modeling and forecasting stationary time series with hesitancy and uncertainty. However, challenges persist in dealing when in time varying non stationary time series influenced by dynamic statistical features and availability of multiple intuitionistic fuzzy numbers (IFNs) for node in cognitive map. Hesitant intuitionistic fuzzy set (HIFS) provides an efficient tool that allows associating multiple IFNs time series data. In this study, we present the notion of hesitant intuitionistic fuzzy cognitive map (HIFCM) and propose a HIFCM based fuzzy time series forecasting approach. Weights associated with membership and non-membership grades of IFN are optimized by using particle swarm optimization. With a view to reveal the applicability and utility of the suggested forecasting technique, it is applied to time series data of enrolments of the University of Alabama. The model's performance is gauged in accordance to Root mean square error (RMSE), and a low value shows the model's superiority over other previously existing methods.

Inter-dependency among the decision criteria and difficulty of handling the information in form of “yes,” “abstain,” “no,” and “refusal” are two important issues to be addressed in multicriteria group decision-making (MCGDM) problems in the environment of uncertainty. Recently, spherical fuzzy set (SFS) has gained the attention of the researchers for MCGDM problems due to their capability of handling decision makers’ preferences in the form “yes,” “abstain,” “no,” and “refusal” more efficiently than picture fuzzy set. Choquet integral operator has an edge over traditional aggregation operator in the modeling of interaction among the preferences of decision makers and decision criteria and therefore, we define Choquet integral operator for SFS. The main objective of this study is to define the spherical fuzzy Choquet integral (SFCI) operator and to extend the VIsekriterijumska optimizacija I Kmpromisno Resenje (VIKOR) for spherical fuzzy environment. In this study we define spherical fuzzy Choquet average, spherical fuzzy Choquet geometric and spherical fuzzy Choquet integral distance (SFCID) operators and propose SFCI-based VIKOR method for MCGDM problems. Proposed SFCI-based VIKOR method is applied on warehouse selection problems and results are compared with existing MCGDM methods.

Portfolio optimization has always been a topic of wide interest for investors. They always want to maximize their return for a given level of risk or minimize the risk for a given level of return. Modern Portfolio Theory (MPT) helps investors in portfolio selection but doesn’t consider the uncertainty and complexity associated with the real market. Thus, to deal with the uncertainty of the real market, we use fuzzy logic in portfolio selection. In this paper, we have found the results with Statistical method (using Lagrange’s multipliers method) and then by using Fuzzy logic toolbox of MATLAB (Triangular membership function and Gaussian membership function). The results obtained by both the methods are then compared. This study also examines the testing data sets.

Recently, hesitant fuzzy sets (HFSs) have been used extensively in time series forecasting. HFSs have inherent characteristics of addressing problem of non-stochastic hesitancy that is developed as a result of the availability of numerous techniques for fuzzification of time series data. In the present study, we have developed an HFS-based computational method for weighted fuzzy time series (WFTS) forecasting. The proposed method addresses the three main issues of appropriate partitioning of the universe of discourse (UOD) into unequal-length intervals, inclusion of hesitancy during fuzzification of time series data, recurrence, and weighting of fuzzy logical relation (FLR) in fuzzy time series forecasting. The proposed method uses a non-parametric clustering approach of adaptive radius clustering for accurate partitioning of UOD and HFS for inclusion of hesitancy in time series during process of fuzzification. The recurrence and weighting of FLRs are handled using Markov weights, which are then subsequently optimized by utilizing the popular swarm intelligence technique of grey wolf optimization. A simple computational method is provided that incorporates the optimized weights, thus simplifying the forecasting process. The proposed WFTS forecasting method is implemented in the Python programming language to forecast benchmark time series data of the University of Alabama and financial time series data of Taiwan stock exchange (TAIEX), market price of State Bank of India (SBI) at Bombay Stock Exchange (BSE), India. The model's performance is measured by means of root-mean-square error (RMSE), and its reduced amount demonstrates the model's outperformance in forecasting of three diversified time series data taken in the study.

In this paper, we mention the limitation of the intuitionistic fuzzy parameterized soft set (IFPSS) in multi-criteria decision-making (MCDM) problems. We define a spherical fuzzy parameterized soft set (SFPSS) to overcome the limitation of IFPSS and develop a method to construct SFPSS from IFS. A simple MCDM method is also proposed in this study of SFPSS.

Abstract 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.

Abstract 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.

Sentiment analysis (SA) is an important part of psychology that helps to predict the attitude or personality traits of a human. In the present study extension of traditional fuzzy sets namely hesitant fuzzy sets (HFS) along with hesitant fuzzy aggregation operators are used to analyze Twitter data to predict sentiment parameters using fuzzy rule-based approach. Tools such as TextBlob and NaiveBayesClassifier of Natural Language Processing toolkit (NLTK) were used to perform initial Sentiment Analysis of Twitter data in Google Collaboratory. The results so obtained are used as inputs in the proposed model. Validation of the proposed model is done in terms of RMSE by comparing results with the traditional TSK model for equal and unequal interval lengths of fuzzy sets.

Probabilistic fuzzy set (PFS) is an ideal tool to touch uncertainties due to randomness (probabilistic) and fuzziness (non-probabilistic) in a single framework. In the present study, we divide time series data into clusters and propose a novel weighted fuzzy time series (FTS) forecasting method using PFS. Proposed method models non-probabilistic uncertainty due to imprecision and linguistic representation of time series data and probabilistic uncertainties in assigning membership grades to time series datum along with occurrence of recurrence of fuzzy logical relations. In proposed forecasting method, probabilities to membership grades are assigned using Gaussian probability distribution function (PDF). Time series data of SBI share price are forecasted using proposed forecasting method in order to show its suitability and applicability. Root mean square error and average forecasting error are used as performance indicator to confirm the outperformance of proposed weighted fuzzy time series forecasting method based on PFS.

A spherical linguistic fuzzy set is the addition of linguistic variables and a spherical fuzzy set. It includes both qualitative and quantitative estimates which can more effectively depict decision-maker’s actual choices and uncertainty. In this paper, we have mentioned the limitation of the existing score and accuracy function for spherical linguistic fuzzy numbers and proposed an improved score function and accuracy function using linguistic scale function to compare spherical linguistic fuzzy numbers. We have also proposed a spherical linguistic fuzzy weighted aggregation operator $$(\check{\text{S}}\text{LFWA})$$ and a spherical linguistic fuzzy ordered weighted aggregation operator $$(\check{\text{S}}\rm{LFOWA})$$ . The proposed score, accuracy functions, and aggregation operator are applied in a numerical application.