With the rapid development of metaverse technology, many industries are exploring how to achieve efficient decision-making in virtual environments. However, decision- making issues in the metaverse often involve multiple interests, dynamic environments, and uncertain factors. Especially when faced with multiple selections, decision-makers need to process subjective evaluation information that includes hesitation, fuzziness, temporal dependence, and bi-directional correlation. Then, how to mathematically model the above complex information and achieve effective classification decisions is an interesting and crucial issue. To do this, we first define two new fuzzy sets in the uni-directional and bi-directional hesitant fuzzy environments, namely the uni-directional hesitant fuzzy set and the bi-directional hesitant fuzzy set. Next, a bi-directional hesitant fuzzy-long short-term memory-transformer model is proposed to deal with the defined information and address the above issues. Then, a parallel probability optimization algorithm is designed to derive the optimal results and parallel occurrence probabilities of an alternative. Finally, the proposed methods are verified by a metaverse application example, which provides a new idea for the further development of related technologies.

Recently, the uncrewed aerial vehicle (UAV) autonomous navigation has been widely used in logistics transportation and environmental monitoring. However, the interaction of subjective evaluation information has been the critical challenge for UAVs and its operations. Also, the correlation among different attributes of the evaluation information is unclear, and the interaction between early and late attributes is often overlooked. Then, how to describe this interactive evaluation information and realize classification decisions has become a key issue. To do it, this paper introduces the hesitant fuzzy set (HFS) and develop the temporal and interactive HFS. Further, we propose a self-attention interactive hesitant fuzzy-long short-term memory network to processes and recognizes the defined information. Then, a multi-probability classification algorithm is given to show the calculation process. At last, its feasibility and applicability are verified by an example of UAV autonomous navigation, which provides a new direction for the development of related UAV technologies.

Power and energy balancing technology is at the core of smart grids. Expert subjective judgments play a critical role in the balancing control process. However, due to the subjectivity, ambiguity, and large-scale nature of the assessment data, traditional models struggle to effectively model and analyze such information. To address the large-scale nature of the data, we introduced long short-term memory (LSTM) to develop the LSTM-Transformer model. Considering the uncertainty and ambiguity of the data, we further constructed the hesitation-fuzzy-based LSTM-Transformer (HF-LSTM-Transformer) model. This is a novel deep learning method capable of handling complex subjective and fuzzy information in power balance control decision making. By incorporating the hesitant fuzzy set, the HF-LSTM-Transformer enhances the LSTM-Transformer’s ability to process uncertainty. In addition, we designed the forward and backward propagation processes, parameter update procedures, and an optimization algorithm to enhance the classification prediction accuracy of the new model. Finally, we provided two smart grid examples from both comparative and application perspectives to fully demonstrate the robustness and accuracy of the classification results obtained by the proposed method.

As the deep learning algorithm, the long short-term memory (LSTM) network is an emerging and hot tool to address classification issues. In the classification process, to provide more reasonable suggestions, the subjective evaluation and qualitative selection given by decision makers and experts are important and cannot be ignored. Thus, this paper proposes the hesitant fuzzy LSTM (HF-LSTM) network model to simultaneously address these two issues. To do this, we first define the series-connection hesitant fuzzy set (SHFS) and the classified SHFS to show the internal logic and the sequential order among the qualitative data. They could be more suitable presentation tools to present subjective evaluation information than other fuzzy sets. Concerning the new characters and properties of these new fuzzy sets, the HF-LSTM network is further constructed; also, the generalized and dispersed properties are proven mathematically. Then, the HF-LSTM network model is suitable for these new fuzzy environments and can provide classification results so that subjective evaluation and qualitative selection can be achieved in the whole classification process. Further, this paper derives the global optimization algorithm of the HF-LSTM network which can make the final classification and improve the credibility of the classification results. Finally, we apply the proposed methods to intelligent building selection and the results present the feasibility of the HF-LSTM network and its global optimization.

Subjective evaluation is a commonly used method in the real recognition process. Generally, two fuzziness can be found in evaluation information, namely what values should be given to fully describe the information and how to distinguish different values. The recently developed probabilistic hesitant fuzzy set could perfectly address these issues. In this article, we propose a dual-fuzzy convolutional neural network (DF-CNN) by fusing the hot neural network algorithm into the probabilistic hesitant fuzzy environment and then using it in a practical handwritten image recognition process. For this new DF-CNN, we provide the whole calculation process including the forward propagation, backward propagation, and parameter updating calculations. Also, the optimization algorithm of the DF-CNN is given to derive its optimal results. Finally, we apply the DF-CNN and its optimization algorithm to deal with a real issue, namely the handwritten numeral image recognition. The calculation process and the comparison fully demonstrate the feasibility and effectiveness of the proposed new model and algorithm.

Smart agriculture can enhance agricultural production efficiency, improve the ecological environment, and realize the sustainable development of agriculture. Many countries and companies are working hard to develop or introduce smart agricultural solutions. Because of the shackles of traditional agricultural management methods and fierce competition with a variety of different solutions, it is a difficult task for enterprises to select and implement smart agricultural solutions smoothly. Hence, enterprises must assess alternative solutions and select a feasible solution in advance. This study drew a novel assessment and selection for smart agriculture solutions using an information error-based Pythagorean fuzzy cloud algorithm. First, an evaluation index system built on smart agriculture solutions was constructed from four aspects. Then, a new concept of Pythagorean fuzzy clouds was defined to express the evaluation information for each indicator. Simultaneously, the Pythagorean fuzzy cloud weighted Bonferroni mean (PFCWBM) operator was developed to aggregate the assessment information of multiple indicators. Next, an assessment and selection decision framework for smart agriculture solutions based on the PFCWBM operator was presented. In addition, an example was given to illustrate the effectiveness of the proposed algorithm. Finally, a discussion was conducted to verify the superiority of our approach. The results showed that our algorithm can characterize and evaluate complex information and has high sensitivity and environmental adaptability.

In this article, we propose a novel intuitionistic fuzzy decision making approach from the perspective of envelopment analysis. This method helps make a decision by calculating the intuitionistic fuzzy efficiencies of all the alternatives. The calculated efficiency values are relative indexes instead of stock indexes such as the aggregated and measured values. A prominent advantage is that this method can distinguish the efficient and inefficient alternatives and further improve the inefficient ones by modifying their membership and nonmembership degrees. Consequently, we can optimize the inefficient alternatives in the future, which cannot be achieved by other similar methods. To do this, we construct two intuitionistic fuzzy envelopment analysis (IFEA) models based on the membership and nonmembership degrees, respectively, which are different from the data envelopment analysis (DEA) and the fuzzy DEA. Their dual forms are derived so that these models can be transformed into the linear programming forms. With respect to the attributes' differences, two preference IFEA models are developed by integrating subjective preference relations. Moreover, we provide the membership improvement formulas and summarize the specific decision making and membership improvement processes for practical applications. Finally, an example of ranking and improving the six listed companies is presented to demonstrate these approaches.

Qualitative portfolio selection approach is a suitable technique for obtaining an optimal portfolio when quantitative data are unavailable and traditional portfolio models are ineffective. However, few studies focus on this issue. This study addresses the lack of research by defining the score-hesitation trade-off rule and introducing the intuitionistic fuzzy set (IFS), based on which an intuitionistic fuzzy portfolio selection (IFPS) model is proposed. The IFS is introduced because of its comprehensive consideration of preference and nonpreference, and is used to represent qualitatively evaluated information from investors and experts. Furthermore, an intuitionistic fuzzy investment scenario is established and a trisection approach is designed to distinguish three types of risk investors, based on which three corresponding IFPS models are constructed. After this, a portfolio selection process under the intuitionistic fuzzy environment is provided, and a simple example is given to show the application of the process. In addition, the investment opportunities and efficient frontier of the IFPS model are investigated to demonstrate the effectiveness of the proposed portfolio selection model. Finally, an example of calculating optimal investment ratios and selecting an optimal portfolio for four newly listed stocks in China is provided to demonstrate the feasibility and practicability of the proposed approaches.

On the basis of the hesitant fuzzy membership, this study proposes the extended intuitionistic fuzzy set (EIFS) and the extended intuitionistic fuzzy number (EIFN) to synthesize the characters of the intuitionistic fuzzy set and the hesitant fuzzy set. We further develop two simplified and applied EIFSs, namely the credible EIFS (C-EIFS) and the possible EIFS (P-EIFS), to comprehensively mine the hesitant fuzzy membership information and to avoid the logical difficulty of simultaneously providing the membership and non-membership in each EIFS or EIFN. Then we investigate the foundations of C-EIFS and P-EIFS, including their expressions, operations, functions, differences, and selection rules. The corresponding aggregation operators are also proposed, and the calculation and relationships of these operators are proven. The prominent properties of C-EIFNs and P-EIFNs are focused on the boundary and average values, respectively; that is, the C-EIFN tends to aggregate the extreme information, whereas the P-EIFN prefers aggregating complete information. Therefore, applying them to decision making with risk preference is suitable, and two risk preference investment cases are provided to demonstrate the applications of these concepts and approaches.

As a generalized fuzzy number, the hesitant fuzzy element (HFE) has received increasing attention. However, it is noted that the occurring probabilities of the elements in the HFE are equal, which is obviously problematic; thus, the preference relations on the HFEs can be inaccurate. To address this issue, this paper proposes the probabilistic hesitant fuzzy preference relations (PHFPRs) based on the probabilistic HFE (PHFE). It seems difficult to provide accurate probabilities that describe the occurring possibilities of the elements in the PHFPRs. This paper further demonstrates the probability calculation method for the PHFPEs based on a proposed variable, which is called the expected consistency. Moreover, the expected consistency index and the judgment principle are designed to evaluate the degree to which the PHFPRs are consistent. For the inconsistent PHFPRs, this paper presents an iterative optimization algorithm to improve their expected consistency by optimizing some elements in the PHFPRs. When the iteration terminates, the consistent PHFPRs and the priorities of the alternatives are identified. Finally, an example that selects a Ph.D. candidate is presented, and the results demonstrate the feasibility and effectiveness of using the PHFPRs and the consistency methods.

Numerical preference relations constitute a useful decision-making technique and can be classified into two types on the basis of the 0.1-0.9 and 1-9 scales. However, these numerical scales cannot fully describe some desired properties, such as asymmetry, consistency, variability, and diminishing utility, in accordance with the preference relations. To address this issue, we first develop a generalized sigmoid function and define a continuous preference set, on the basis of which we propose the generalized sigmoid scale. Then, we introduce the optimal discrete fitting and risk preference selection approaches to solve the risk appetite parameters in the generalized sigmoid scale. Using these new methods, we further propose the asymmetric fuzzy preference relation (AFPR), examine its additive transitivity property and some weak transitivity properties, and design an approximate consistency test. The corresponding five-step modeling process under an asymmetric fuzzy preference environment is constructed, which can be applied in decision making involving different risk appetites. Finally, two examples are used to demonstrate the properties and advantages of these new methods. The first example is simple and shows the differences between the traditional fuzzy preference relations and the AFPRs, while the second is a practical case and is used to illustrate the feasibility and reasonability of the AFPRs.