key: cord-1043142-ak14a1az authors: He, Shi-Fan; Wang, Ying-Ming; Pan, Xiao-Hong; Chin, Kwai-Sang title: A novel behavioral three-way decision model with application to the treatment of mild symptoms of COVID-19 date: 2022-05-25 journal: Appl Soft Comput DOI: 10.1016/j.asoc.2022.109055 sha: 68d9046ea5a81a9e4487027ea06ec80e50322bfe doc_id: 1043142 cord_uid: ak14a1az The Coronavirus Disease 2019 (COVID-19) has popularized since late December 2019. In present, it is still highly transmissible and has severe impact on the public health and global economy. Due to the lack of specific drug and the appearance of different variants, the selection of the antiviral therapy to treat the patients with mild symptom is of vital importance. Hence, in this paper, we propose a novel behavioral three-way decision (3WD) model and apply it to the medicine selection decision. First, a new relative utility function is constructed by considering the risk-aversion behavior and regret-aversion behavior of human beings. Second, based on the relative utility function, some new rules are defined to calculate the thresholds and conditional probabilities in 3WD and some corresponding theorems are explored and proved. Next, a new information fusion mechanism in the framework of evidential reasoning algorithm is developed. Then, the decision results are obtained based on the Bayesian decision procedure and the principle of maximum utility. Finally, an example with large-scale data set and an example about medicine selection for COVID-19 are provided to show the implementation process and effectiveness of the proposed method. Comparative analysis and sensitivity analysis are also performed to illustrate the superiority and the robustness of the current proposal. demonstrate the implementation and effectiveness of the proposed method. Comparative analysis and sensitivity analysis are also conducted in this section to show its superiority and robustness. Conclusions are drawn in Section 7. In this section, we will briefly review some studies related to medical decision based on MADM, 3WD and ER algorithm. Healthcare and medical industry play an important role in the living standard and well-being of people [12] . Making an accurate medical decision is a difficult task and normally needs to consider several attributes from different aspects [13, 14] . For example, when making a medical diagnosis, the doctor need to evaluate the disease from multiple aspects because diseases are normally accompanied by multiple symptoms [15] . Hence, the medical decision can be considered as a kind of MADM problems [13, 16] . For that, MADM methods have been applied in various fields of medical decision and healthcare [16] . Based on fuzzy integral and fuzzy measure, Dursun et al. [17] proposed a multi-attribute group decision making (MAGDM) method to deal with healthcare waste in Lstanbul, Turkey. Li and Wei [18] developed a large-scale group decision making method to conduct healthcare management, which considers the complexity of the management and the opinions of stakeholders. Considering experts may lack knowledge to handle critical diseases, Das and Kar [16] proposed an algorithm based on intuitionistic fuzzy soft set to explore a method that can reflect the opinions of all experts. To solve medical decision about acute inflammatory demyelinating disease, Chen et al. [19] defined an extended QUALIFLEX method to conduct MADM analysis. Tolga et al. [13] defined finite interval Type-2 (FIT2) Gaussian fuzzy numbers and extended TODIM method with FIT2 Gaussian fuzzy numbers to select J o u r n a l P r e -p r o o f Journal Pre-proof healthcare device. 3WD initially proposed by Yao [10, 20, 21] divides a whole into three regions, i.e., positive region (POS), negative region (NEG) and boundary region (BND), which can be interpreted as three decision actions, i.e., acceptance, rejection and non-commitment. The idea of 3WD is in line with people's cognition because they innovatively provide a deferment strategy [11] . When people have a full acknowledge about an event, they can make a quick rejection or acceptance judgments; but if they cannot make an immediate decision, they are usually willing to postpone the decision, that is, deferment [15] .The extensive studies of 3WD have led to the extension from narrow 3WD to wide 3WD. In narrow sense, 3WD was firstly introduced to interpret three types of classification rules in rough set theory [22] . Until now, narrow 3WD has developed many generalized models, such as three-way approximation models [23] , three-way analysis models [24] , three-way concept lattice models [25] and so on. These models are 3WD in various context, which have specific mathematical expressions [26] . In recent years, the wide 3WD has been studied in depth based on the common existed "three" phenomena in the fields of computer sciences, management, cognitive science and so on [27] . For wide 3WD, decision is viewed as computing, processing, analysis etc. [28] . It is thinking, problem solving and information processing in threes [22] . The wide 3WD changes the two-way consideration such as true/false, white/black, into threeway consideration like true/unsure/false, white/grey/black [28] , which is flexible and simple enough [22] . Hence, the wide 3WD has been a method that can be used in various research topics. 3WD offers new opportunities for studying MADM problems [29] . It provides not only a reasonable semantic interpretation for decision results, but also a powerful and scientific tool to address MADM problems [30] . Hence, the fusion of 3WD with MADM has become a hot research topic. Jia and Liu [31] applied 3WD to MADM by using attribute values to express loss functions and preliminarily manifested the correlation between them. Huang et al. [32] explored a new 3WD method to MADM, in which they provided a new calculation method of loss function and conditional probabilities. Zhu et al. [33] defined a new 3WD method based on the regret theory, which includes optimistic, neutral and pessimistic strategies. Liang et al. [34] proposed a behavioral 3WD model based on the prospect theory under interval type-2 fuzzy environment and applied it to solve MADM problems. Considering several departments or agents may be involved in the decision-making process, Sun et al. [35] proposed a 3WD method to handle multiple attribute group decision making (MAGDM) problems with linguistic information. Wang et al. [36] took the hesitancy of decision makers into consideration and developed a three-way MADM method under hesitant fuzzy environments. In real world, there exist many MADM problems with incomplete information. To deal with these problems, Zhan et al. [37] proposed a novel 3WD-based MADM model based on utility theory. Dempster-Shafer (D-S) theory is easy to understand and can comprehensively process the uncertain and inaccurate information [38] . However, D-S theory has some drawbacks in dealing with conflicting evidences [39] . To tackle this issue, Yang and Xu [39] developed the ER algorithm based on the combination rule of D-S theory. Till now, the ER algorithm has been testified as a good tool to aggregate the information and has many successful applications in aggregating the uncertain and imprecise information in MADM problems. Xue et al. [40] introduced the ER algorithm to multi-scale hesitant fuzzy linguistic environment to combine the attributes in the hazard assessment of landslide dames. To reasonably assess the renewable energy projects, Liang et al. [41] and Pan et al. [42] ( ) x NEG X ∈ , respectively. The loss function in different states is given by a 3 2 × matrix, as shown in Table 1 . Pr X x is the conditional probability of an object belonging to X . According to the minimum-risk decision rules derived from the Bayesian decision procedure, the decision rules can be determined as: where α , β and γ are the decision thresholds, which are determined by: The rules (P1)-(N1) can be simplified as follows: Prospect theory was developed by Kahneman and Tversky [46] on the basis of utility theory. It can forecast the actual decision behavior of decision maker under risk [47] . Prospect theory consists of two phases: the editing phase and the evaluation phase [46] . In the editing phase, the outcomes of alternatives are coded as gains or losses relative to the reference point. In the evaluation phase, the prospect values are calculated by a value function and the alternative with the highest prospect value is chosen. The prospect value function is defined as follows: where p and q are coefficients of risk attitude, 0 , 1 p q ≤ ≤ ; x ∆ represents the deviation between the existing value and reference point, which denotes gain ( 0 x ∆ ≥ ) or loss ( 0 x ∆ < ); and θ is the risk aversion coefficient, 1 θ > . In [48] , Kahneman and Tversky found that setting 0.88 p q = = and 2.25 θ = will make the results keep consistent with empirical data. J o u r n a l P r e -p r o o f Journal Pre-proof Shadowed set coined by Pedrycz [49, 50] shows its superiority in charactering fuzzy information. The construction of shadowed set is based on balancing the uncertainty, which is also called as uncertainty relocation. It maps the membership grade of object in universe to a set [ ] { } 0,1, 0,1 , which is defined as: Definition 1 [49, 50] . Let U be a given universe of discourse, a shadowed set S can be represented as follows: where 0, 1 and [ ] 0,1 respectively mean full exclusion, full belongingness and uncertainty, which respectively correspond to the exclusion, the core and the shadowed area in the shadowed set. The exclusion area of shadowed set consists of ( ) composes the core of shadowed set; and the shadowed area is the regions of U where ( ) [ ] According to the definition of shadowed set, Landowski [51] defined the shadowed number ( ) , , , S x x x x = is shown in Fig.2 . , , , S x x x x = Li et al. [52] proposed a data-driven method to construct shadowed sets used to model linguistic terms. First, interval data pre-processing is conducted on the collected interval data. After bad data processing, outlier processing, tolerance limit processing and reasonable interval processing, ineffective data will be deleted, and the data remained will be used in phase 2, i.e., construction of shadowed sets based on interval data. As word means different things for different people, this difference can be considered by shadowed sets. Inspired by their innovative work, we also tried to construct the shadowed sets corresponding to seven-level linguistic terms in our previous research [53] as shown in Fig.3 . For more details, please kindly refer to [52, 53] . The ER algorithm is developed based on the D-S theory proposed by Dempster and Shafer, which is well suited to address the imprecise and uncertain information. To better understand ER algorithm, we first introduce some primary concepts about traditional D-S theory and ER algorithm. where Φ is an empty set, A is any subset of H , 2 H is the power set of H and consists of all the subsets of H including empty set and universal set.  constitutes the interval that supports A . The difference between the belief and the plausibility of set A describes the ignorance of the assessment for the set A . Definition 3 [54, 55] . The core of the D-S theory of evidence is the Dempster's combination rule. It provides a way to fuse evidence with different sources, which is defined as follows: where A and B are both focal elements, is named as the normalization factor. In classical 3WD, the loss function is fixed for alternatives belonging to the same state and taking the same action. That cannot effectively distinguish each alternative. To conquer this drawback, Jia and Liu [31] defined relative loss functions with respect to multiple attributes. Based on their work, many scholars have studied the relative utility functions considering the risk attitude of decision maker based on the prospect theory, such as [34, 56] . This makes a meaningful extension for 3WD theory. But some researchers think that feeling such as regret and rejoice is also a fact of life and it is irrational to ignore them [57] . Hence, in this paper, we are going to simultaneously take the risk-aversion and the regretaversion behavior into account by combining the prospect theory and the regret-rejoice function. The definition of the prospect theory is introduced in subsection 3.2 and the regret-rejoice function is defined as follows: where δ ( i N x s = . Then, the relative regret function and the relative rejoice function can be respectively determined by the formulas in Table 2 and Table 3 . After obtaining the relative regret function and the relative rejoice function, the regret or rejoice perceived by the decision maker is calculated by Table 4 . The regret-rejoice function is derived from regret theory, which can effectively describe the psychological behavior of decision maker. That is, the decision maker will rejoice when the gain of selected alternative is more than others and will regret if the loss of the selected alternative is more than others [60] . However, as stated in prospect theory, the decision maker is more sensitive to losses than to equal gains [47] . In other word, the decision maker is loss aversion. This characteristic is largely ignored by the regret-rejoice function including in regret theory. For that, this paper combines the regret-rejoice function with prospect theory to describe the psychological behavior of decision maker more comprehensively. In this way, both the regret aversion and loss aversion of decision maker can be well reflected. After obtaining the utility function, we can determine the expected utilities of taking different actions, which are calculated as follows: J o u r n a l P r e -p r o o f Inspired by the Bayesian minimum-risk decision rules and the idea of maximizing expected utility, Lei et al. [56] proposed decision rules that maximize the expected utility. In this paper, we also adopt these decision rules, therefore, the three decision rules are derived as follows: as follows: Similarly, Obviously, 0 Then, we prove li as follows: For the three actions, namely acceptance p a , delay B a and rejection N a , the utility of accepting the right alternative exceeds those of delaying and rejecting the right alternative. The utility of accepting the right alternative is the highest and the utility of rejecting the right alternative is the least. In similar ways, the utility of rejecting the wrong alternative exceeds those of delaying and accepting the wrong alternative. The utility of rejecting the wrong alternative is the highest and the utility of accepting the wrong alternative is the least. Thus, li In this section, we will discuss the determination of thresholds. Based on the fact that Then, Therefore, the thresholds are determined. It is known that ( ) According to prospect theory, the decision maker is more sensitive to the losses than to the equal gains [47, 61] . Hence, we can prove 0 Table 5 . In 3WD, the estimation and evaluation of the conditional probability is a crucial problem. Liang et al. [62] demonstrated that the positive ideal solution (PIS) and negative ideal solution (NIS) in TOPSIS method correspond to the two decision states in 3WD, and the conditional probability can be calculated by means of the relative closeness degree. Inspired by [62] , this paper implies the TOPSIS method to elaborate the determination of conditional probability. Normally speaking, the maximum of the attribute will be selected as PIS and the minimum of the attribute will be as NIS. Hence, the PIS and NIS of alternatives l a are first confirmed by: Then, the relative closeness of alternative l a is calculated as follows. 3WD can not only classify all alternatives into POS, BND or NEG regions, but also can provide a complete ranking order. This is important because in some case, it may be hard to make final decision only based on the classification [30] . Besides, the ranking of alternatives can assist decision maker in selecting an optimal alternative and allocating limited resources [63] . Hence, the ranking regulation is necessary. In this paper, the ranking regulations consist of two phases. First, the priority principle is determined. To be specific, the alternatives in If there is an assumption that ( )( ) ( )( ) , then the thresholds meet 0 1 β γ α < < < < . The rules (P4)-(N4) can be simplified as follows: Second, the ranking of alternatives in the same regions follows the principle of maximizing utility. Utility is the main factor deciding whether an alternative should be chosen or not, which can also be used to explain the semantics of three rules in 3WD [63] . The overall utility value can be calculated by the following function. The higher the utility value is, the better the alternative will be. In this section, we present a complete decision process based on the proposed 3WD method. In this decision process, we first construct the utility function based on the prospect theory and regret-rejoice function. Then, we design a mechanism to employ the ER algorithm to combine multiple attributes information. Finally, the 3WD rules are developed to obtain the decision results. Table 6 . , the information is characterized by linguistic terms, which cannot be computed directly. In our previous study [53] , we proposed a method that employs the shadowed sets to model linguistic terms and proposed a distance measure model based on shadowed sets to measure the relationships between linguistic terms, which is defined as follows: Journal Pre-proof For more details, please refer to [53] . Step 2. Calculate the thresholds li α and li β by Eqs. (18) and (20). Step 3. Construct decision matrix with distributed assessment Step 4. Obtain the basic probability masses n ζ by the ER algorithm. Then, we can obtain the combined decision information Therefore, Wang et al. [64] proposed an analytical ER algorithm, which is equivalence to the recursive ER algorithm and makes the ER algorithm more flexible in aggregating attributes. First, the basic probability masses are obtained by combining the relative weights and the degree of In this section, two numerical examples are presented. The first one is emulation study with a largescale data set, which aims at manifesting the feasibility and applicability of the proposed method. The other one is from the case study about COVID-19 drug selection in Mishra et al' s research [8] , which is J o u r n a l P r e -p r o o f Journal Pre-proof an application of the proposed method and contributes to illustrate the implementation process of the proposed method. Then, a comparative analysis is conducted to show its superiority and a sensitivity analysis is designed to test the robustness of the proposed method. In this section, a numerical example with large-scale data set is provided to illustrate the feasibility and applicability of the proposed method. The numerical example used in this section is randomly generated, which consists of 300 alternatives denotes as { } . As stated in subsection 3.3, a data-driven method is employed to obtain the shadowed sets corresponding to linguistic terms in our previous study. According to this method, the seven-level linguistic terms and their corresponding shadowed numbers are shown in Table 7 . For more details, please refer to [53] . 8.72,9.31,9.59,9. 89 Due to the restriction of space, we only present the decision results as shown in Fig.5 . The detail implementation process of the proposed method will be given in numerical example 2. From Fig.5, we can not only know the ranking order, that is, for all 300 alternatives, the 200rd alternative ranks the first and the 176th alternative ranks the last, but also know the classification results. With the help of 3WD rules, all alternatives are divided into three parts, i.e., POS, BND and NEG. For alternatives in NEG regions, they shouldn't be chosen in any case; for alternatives in BND regions, selecting it or not needs more information and consideration; for alternatives in POS regions, they can be chosen and the order of selecting them is further determined by expected utilities. Results of all alternatives in numerical example 1. To show the effectiveness and the superiority of proposed method in tackling drug selection problem, we further applied the proposed method to solve the COVID-19 drug selection problem. This numerical example is from the case study in Mishra et al.' s research [8] . Table 8 . Table 8 . The evaluation matrix in the form of linguistic terms Employing the proposed method to solve this problem mainly includes the following steps: Step 1. Determine the utility function ( ) li v x according to Table 4 . The results are shown in Table 9 . Step 2. Calculate the thresholds li α and li β by Eqs. (18) and (20) . The results are shown in Table 10 . Table 11 . Step 4. Obtain the basic probability masses n ζ according to Eqs. (28)- (41) . The results are shown in Table 12 . Step 5. Calculate the thresholds l α and l β . The results are shown in Table 13 . Table 13 . The thresholds l α and l β . Step 7. Classify all alternatives based on the decision rules (P5) -(N5) and rank all alternatives in the same region according to the expected utilities. Then, we can obtain a a a a a     . The decision results are different even paradox. This is common when dealing with a new disease. In this case, the involvement of new experts and the discussion between group is advised to reach consensus. (3) When making the final group decision, the classification should be priority to the ranking. To be specific, the alternative medicines in the positive region will be first considered. If all experts agree with the medicines in the positive region, then the decision results will be generated based on the ranking order. In order to show the effectiveness and superiority of the proposed method, we respectively compare it with the hesitant fuzzy group decision-making method proposed by Mishra et al. [8] , 3WD under MADM proposed by Jia and Liu [31] , 3WD based on regret theory proposed by Huang and Zhan [65] and 3WD based on prospect theory proposed by Liang et al. [34] . Except for Mishra et al.' s method, other methods didn't involve group decision. To make a fair comparison, we employ the aggregated matrix in [8] (as shown in Table 14 ) to conduct the comparative analysis and the results are summarized in Table 15 . [8] was also developed for the drug selection to treat the mild symptoms of COVID-19. Compared with other 3WD-based methods, their method directly output the ranking order of alternatives. That means that the decision maker only relies on the ranking order to make the decision regardless of whether the alternative is good or not. This will increase the risk in medical decision and even lead to a dangerous situation. For example, when evaluating the performance of medicines, five alternatives may all belong to negative region, namely, 1 2 3 4 5 , , , , NEG a a a a a = . But when J o u r n a l P r e -p r o o f Journal Pre-proof ranking them by two-way decision methods, a ranking order will be obtained such as a a a a a     . According to the result of two-way decision, 4 a will be chosen. However, even though 4 a is the first option, it is still of bad quality because it is in the negative region. Choosing it will inevitably increase the decision risk. In medical decision, this might be a threat to the patients' life. In this case, 3WD is a sensible tool. It can not only provide a complete ranking order of alternatives, but also provide their classification. This can largely decrease the risk in medical decision. (2) The consideration of psychological behavior. Mishra et al.' s method [8] as well as Jia and Liu' s method [31] didn't consider the psychological behavior of decision makers. However, human is bounded rational, which was initially proposed by Simon [66] and has been approved by many scholars. That means in the decision-making process, the decision results are inevitably affected by the psychological behavior of decision maker. Huang and Zhan [65] proposed a 3WD method based on regret theory to reflect the regret-aversion behavior of decision maker. From Table 16 , we can notice that the results obtained by Huang and Zhan' s method is different from our method. The reason can be summarized as: on the one hand, they didn't consider the risk-aversion behavior of decision maker; on the other hand, their calculation of conditional probabilities is based on the cardinality of class, which ignore the degree of difference. Liang et al. [34] proposed a 3WD method based on the prospect theory to reflect the risk-aversion behavior of decision maker. The different outcomes between Liang et al.' s method and Jia and Liu' s method can also reflect the great influence of psychological behavior on decision results. Besides, from Table 16 , we can observe that the result obtained by Liang et al.' s method is different from that produced by our method. The reason is that Liang et al.' s method does not take into account the regret-aversion of decision maker. However, the regret feeling is also a fact of life and it is irrational to ignore it [57] . Hence, the proposed method considers the risk-aversion behavior and regret-aversion behavior at the same time and produces a more reliable result. In this section, sensitivity analysis is conducted to explore the robustness of the proposed method. The parameter involved in this method includes: the parameter used to calculate the value of adopting noncommitment η ( 0 0.5 η < ≤ ), the regret aversion coefficient δ ( 0 δ > ), the coefficients of risk attitude p and q ( 0 , 1 p q ≤ Fig.6 The influence of parameters η , δ , p , q and θ From Fig.6 , it can be noticed that: (1) No matter how the parameters values change, it always obeys the priority principle. That is, the alternatives in POS are superior to the alternatives in BND and alternatives in BND are better than those in NEG . (2) The classification and ranking order vary with different value of η , δ , p , q and θ . From the definition of prospect theory and regret-rejoice function, we already know that different risk and regret attitudes lead to different results. Hence, in a specific decision-making problem, their values should be determined according to the specific decisionmaking situation and decision maker. (3) Although the classification and ranking order may have some fluctuation, the variation of the decision results is almost stable. That is, alternative 4 a will rank first in all case even though 4 a is not always in the positive region. For example, for 0.18 η ≤ and 0.185 p ≤ , all alternatives will be in the boundary region. The ranking results will be further based on their expected utilities and alternative 4 a will still be the first option. In other words, the proposed method maintains its validity when selecting the best alternative. The epidemic of COVID-19 has led to unprecedented societal influence, especially for the public health and the global economy. The scientists from all over the world are trying their best to control this epidemic. In this paper, a new method with respect to the treatment of mild symptoms of COVID-19 is proposed, which helps to select the most desirable therapy. The method developed in this paper is based on behavioral 3WD model, which can help the managers and the doctors in making sensible judgements, reducing the decision risks and working on practical applications. From the results of numerical example and comparative analysis, we can conclude that different from other therapy selection methods, the proposed method not only can provide the ranking order of alternatives, but also classifies them into the positive region, boundary region and negative region, which can decrease the decision risks involved in the medical decision. Medical decision is related with the life of patient. When facing some unknown or unfamiliar diseases, the ranking of alternative medicines may not be as the same importance as their classification. For medicines in negative region, it shouldn't be chosen even though it ranks the first. Besides, the sensitivity analysis also shows the robustness of the proposed method. The medicine selection method provided in this paper provides a new perspective for the therapy selection of COVID-19, which can be used in other or future possible medical problems. Modeling the lockdown relaxation protocols of the Philippine government in response to the COVID-19 pandemic: An intuitionistic fuzzy DEMATEL analysis Prediction of COVID-19 confirmed, death, and cured cases in India using random forest model Predictions of COVID-19 Infection Severity Based on Coassociations between the SNPs of Co-morbid Diseases and COVID-19 through Machine Learning of Genetic Data Identification of dominant risk factor involved in spread of COVID-19 using hesitant fuzzy MCDM methodology JCS: An Explainable COVID-19 Diagnosis System by Joint Classification and Segmentation, IEEE Transactions on Image Processing Diagnosis Assistant for identifying affected cases globally using MCDM, Materials Today: Proceedings, (2021) Comparative Evaluation of the Treatment of COVID-19 with Multicriteria Decision-Making Techniques An extended fuzzy decisionmaking framework using hesitant fuzzy sets for the drug selection to treat the mild symptoms of Coronavirus Disease Z-uncertain probabilistic linguistic variables and its application in emergency decision making for treatment of COVID-19 patients Three-way decisions with probabilistic rough sets Failure mode and effect analysis: A three-way decision approach An integrated fuzzy clustering cooperative game data envelopment analysis model with application in hospital efficiency Finite-interval-valued Type-2 Gaussian fuzzy numbers applied to fuzzy TODIM in a healthcare problem Application of fuzzy sets theory to evaluate the healthcare and medical problems: A review of three decades of research with recent developments A novel three-way decision approach under hesitant fuzzy information Group decision making in medical system: An intuitionistic fuzzy soft set approach A fuzzy multi-criteria group decision making framework for evaluating health-care waste disposal alternatives A large scale group decision making approach in healthcare service based on subgroup weighting model and hesitant fuzzy linguistic information The extended QUALIFLEX method for multiple criteria decision analysis based on interval type-2 fuzzy sets and applications to medical decision making Constructing shadowed sets and three-way approximations of fuzzy sets The superiority of three-way decisions in probabilistic rough set models Extending characteristic relations on an incomplete data set by the three-way decision theory Constructing shadowed sets and three-way approximations of fuzzy sets Resilience Analysis of Critical Infrastructures: A Cognitive Approach Based on Granular Computing Three-way fuzzy concept lattice representation using neutrosophic set A three-way decision approach with risk strategies in hesitant fuzzy decision information systems Strategy selection under entropy measures in movement-based three-way decision Three-way decision and granular computing Complex network analysis of three-way decision researches A three-way decision method based on fuzzy rough set models under incomplete environments A novel three-way decision model under multiple-criteria environment A three-way decision method with pre-order relations A regret theory-based three-way decision approach with three strategies Heterogeneous multi-attribute nonadditivity fusion for behavioral threeway decisions in interval type-2 fuzzy environment Three-way decisions approach to multiple attribute group decision making with linguistic information-based decision-theoretic rough fuzzy set Three-way multi-attribute decision making under hesitant fuzzy environments A novel three-way decision model based on utility theory in incomplete fuzzy decision systems An interval evidential reasoning-based dynamic performance evaluation method for complex systems On the evidential reasoning algorithm for multiple attribute decision analysis under uncertainty Hazard assessment of landslide dams using the evidential reasoning algorithm with multi-scale hesitant fuzzy linguistic information Multi-granular linguistic distribution evidential reasoning method for renewable energy project risk assessment, Information Fusion The evidential reasoning approach for renewable energy resources evaluation under interval type-2 fuzzy uncertainty Approach for multi-attribute decision making based on novel intuitionistic fuzzy entropy and evidential reasoning A general evidential reasoning algorithm for multi-attribute decision analysis under interval uncertainty Utilizing the Evidential Reasoning approach to determine a suitable wireless sensor network orientation for asset integrity monitoring of an offshore gas turbine driven generator Prospect theory: An analysis of decision under risk Consensus building in multiperson decision making with heterogeneous preference representation structures: A perspective based on prospect theory Advances in Prospect Theory: Cumulative Representation of Uncertainty From fuzzy sets to shadowed sets: Interpretation and computing Interpretation of clusters in the framework of shadowed sets Shadowed numbers and their standard and multidimensional arithmetic Interval data driven construction of shadowed sets with application to linguistic word modelling A shadowed set-based TODIM method and its application to large-scale group decision making The evidential reasoning approach for MADA under both probabilistic and fuzzy uncertainties The evidential reasoning approach for multiple attribute decision analysis using interval belief degrees Multigranulation behavioral three-way group decisions under hesitant fuzzy linguistic environment Regret theory: State dominance and expected utility Algorithms for interval-valued fuzzy soft sets in stochastic multi-criteria decision making based on regret theory and prospect theory with combined weight Regret in decision making under uncertainty Grey stochastic multi-criteria decision-making based on regret theory and TOPSIS Loss Aversion Under Prospect Theory: A Parameter-Free Measurement Method for three-way decisions using ideal TOPSIS solutions at Pythagorean fuzzy information A novel three-way group investment decision model under intuitionistic fuzzy multi-attribute group decision-making environment Environmental impact assessment using the evidential reasoning approach TWD-R: A three-way decision approach based on regret theory in multi-scale decision information systems Effects of increased productivity upon the ratio of urban to rural population This research was supported by the National Natural Science Foundation of China (Grant no. 61773123).