A novel three-way group investment decision model under intuitionistic fuzzy multi-attribute group decision-making environment
Introduction
With the increasing complexity and uncertainty of investment decision-making problems, it is necessary to investigate a scientific and reasonable investment decision-making model to avoid decision risks and obtain high returns [28]. For a practical investment decision-making problem, if decision-makers (DMs) accept a bad alternative, it could lead to decision losses; in contrast, if DMs reject a good alternative, they may miss investment opportunities. In addition, the non-commitment decision is a newly added choice. In this case, it is hard for DMs to make a decision when they do not want to miss some opportunities. The non-commitment decision demands DMs to gather more information to make a choice. Therefore, some investment decision-making problems can be seen as three-way decision (3WD) problems [42]. Our work is to study a three-way group investment decision (3WGID) model under multi-attribute group decision-making (MAGDM) environment by taking advantage of the 3WD theory and intuitionistic fuzzy (IF) sets. In the following, we present some related researches of 3WD, MAGDM and IF sets briefly.
The theory of 3WD was introduced by Yao [42], which offers an appropriate approach for DMs to solve the uncertainty of decision-making problems. It can classify a universal set into three pairwise disjoint parts and give three different decision rules by designing appropriate strategies. This theory has been widely applied in many research fields, such as conflict analysis [7], formal concept analysis [10], [24], [43], attribute reduction [11], multi-label classification [50], pattern recognition [32], malware analysis [26], multi-attribute decision making (MADM) [3], [15], software development [36], etc. In general, there are two aspects to extend the model of 3WD: one is the loss function and the other is the conditional probability [17].
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For the loss function, it is difficult to measure the loss accurately under certain conditions. To solve this problem, there is a trend to reduce the accuracy of loss measurement by some fuzzy theories and methods [20]. For instance, Liang et al. [16] discussed a novel 3WD model by evaluating the loss function with an IF set. Sun et al. [33] utilized a linguistic term to express the loss function. Zhang et al. [47] considered a hesitant fuzzy linguistic term (HFLT) set as a new measurement format of the loss function. However, all the above-mentioned forms of loss functions do not take into account the discrepancy of the loss functions of different objects and do not give a computational method of them. Recently, Jia and Liu [6] proposed a novel 3WD model by utilizing evaluation values of attributes under a fuzzy MADM environment to educe the relative loss function. Similarly, Liu et al. [23] used evaluation values of attributes under an intuitionistic fuzzy MADM environment to measure the relative loss function. Liang et al. [15] and Lei et al. [8] also studied the relative loss functions induced from evaluation values of attributes with an interval type-2 fuzzy (IT2F) set and an HFLT set, respectively.
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For the conditional probability, one of the preconditions for determining the conditional probability is the decision attribute or the class label [17], [23]. To solve this problem, Yao and Zhou [45] computed the conditional probability based on the naive probabilistic independence assumption and Bayesian theory. Liu et al. [18] calculated the conditional probability by use of logistic regression. Nevertheless, we may face a universal situation in some practical decision problems where there are only conditional attributes but no decision attributes. For instance, in many MADM problems, there are only conditional attributes included but no decision attributes in information tables. To address this issue, Liang et al. [17] utilized the TOPSIS method to determine the conditional probability under Pythagorean fuzzy (PF) MADM environment. This provides an effective approach for us to obtain the conditional probability in MADM problems. Inspired by this viewpoint, we utilize intuitionistic fuzzy TOPSIS (IF-TOPSIS) method to estimate the conditional probability in intuitionistic fuzzy MAGDM (IF-MAGDM) problems.
MAGDM [8], [28] is a crucial research branch in the field of decision science by combining MADM [6], [23] with group decision-making (GDM) [14]. It usually offers a mechanism to fuse a group of DMs’ preferences to obtain an optimal alternative by using diverse decision-making models [8], [37], [48]. To date, MAGDM is widely utilized in many fields, such as emergency decisions [33], medical diagnosis [38], supplier selections [40] and so on.
Generally speaking, the main process of MAGDM is that a group of experts offer their opinions/preferences for alternatives on multiple attributes and try to obtain a suitable solution with the aid of diverse theoretical techniques and decision-making models [38], [48]. In addition, experts usually utilize qualitative and/or quantitative approaches to evaluate the values of alternatives aiming at multiple attributes, which depends on the characteristics of alternatives and their own knowledge. However, in many decision-making situations, it is difficult or even impossible to obtain accurate evaluation values because of the innate vagueness of human judgments and the continuous increase of complexity in practical problems [6], [15]. Fuzzy set theory [46], as an essential uncertainty technique, has been successfully applied to solve this uncertainty and vagueness in some MAGDM problems. For example, Gupta et al. [4] extended the TOPSIS method to MAGDM based on interval-valued IF (IVIF) sets. Liu and Li [22] constructed a novel MAGDM method by use of cloud distance operators with linguistic information.
In 1986, Atanassov [1] extended fuzzy sets to introduce the concept of IF sets. IF sets are characterized by a membership degree (MD), a non-membership degree (N-MD) and a hesitation degree (HD). In a fuzzy set [46], the MD of an element x is only a value between 0 and 1 and the N-MD is directly. This MD combines the evidences of support and opposition for x. In fact, the information of an element belonging to a fuzzy concept may be insufficient and incomplete. Therefore, a single number can not account for the lack of knowledge. In particular, in some realistic situations, the N-MD of an element in a fuzzy set is not certainly equal to 1 minus the MD. In other words, there may be some degrees of hesitation. For instance, in a voting campaign, there is usually “abstention”, in addition to “support” and “opposition” [41]. Indeed, abstention implies the hesitation or indeterminacy of the voter to the candidate. Nevertheless, a fuzzy set only includes the MD, but ignores the hesitation and the uncertainty often involved in decision-making processes. Besides, there is no method to incorporate the lack of knowledge of the MD in a fuzzy set. In IF sets, the MD and the N-MD are two independent evaluation indicators, which can be regarded as two fuzzy sets. The only connection between the MD and the N-MD is that the N-MD should be less than or equal to 1 minus the MD. Meanwhile, the HD of IF sets reflects the uncertainty of DMs’ cognition and the complexity of the decision-making environment. Compared with fuzzy sets represented only by an MD, using IF sets to depict evaluation information can better describe the characteristics of affirmation, negation and hesitation in human cognitive performance. In recent years, IF sets have been successfully applied in decision-making problems under uncertainty. For instance, Boran et al. [2] utilized an IF set to depict the uncertain information in supplier selection problems. Park et al. [29] explored the uncertain information in robot selection problems by using an IF set to characterize evaluation information. Huang et al. [5] used an IF set to express vague information of the security audit in information systems.
Based on the above statements, we intend to discuss a 3WGID model by using the 3WD theory and IF sets under the MAGDM environment. The motivations of our work can be summarized as follows:
(1) The theories of 3WD and IF sets are two influential mathematical theories to characterize vague and uncertain concepts, which are all suitable for solving uncertain decision-making problems. There are connections and differences between these two theories. Specifically speaking, both of them can describe uncertain concepts in decision-making problems. At the same time, they have the limitations of subjective and objective descriptions. We can combine the advantages of these two theories. Therefore, it is necessary to conduct in-depth research by combining the 3WD theory and IF sets.
(2) The traditional MAGDM models only determine an optimal alternative based on the evaluation values on multiple attributes. However, they often ignore the necessity of further testing to obtain the final decision results in the practical decision-making processes. This makes the final decision results too harsh because the result is a two-way decision (2WD) [6], [23]. Therefore, we need to design a new model for MAGDM. The intuitive approach is to add uncertainty to the final decision results for further investigation. This idea is consistent with the theory of 3WD. Therefore, utilizing 3WD to address and explain some real MAGDM problems is more reasonable.
(3) Although scholars have studied some 3WD models under the uncertain environment [13], [14], [16], [33], they did not give an objective approach to calculate the loss functions and did not take into account the discrepancy of the loss functions of different objects. Therefore, in light of the results in [6], [8], [15], [23], we plan to discuss a data-driven approach to compute the relative cost and revenue functions of alternatives by using the evaluation values of attributes. Furthermore, considering that there are only conditional attributes but no decision attributes in MAGDM problems [17], we plan to utilize the IF-TOPSIS method to estimate the conditional probability.
(4) According to our knowledge, most 3WD models abide by the minimum-cost principle. In fact, when we face an investment decision-making problem in real life, profit or utility is the main factor to decide whether we should invest in an alternative. Liu et al. [19] designed a profit-based three-way investment decision (3WID) model by use of the revenue and cost functions. Recently, Li and Huang [9] proposed a novel profit-based 3WID model with a hesitant fuzzy (HF) set. In addition, for a practical investment decision-making problem, an investor needs to consider the characteristics and differences of alternatives from multiple attributes. However, the models in [9], [19] do not consider the differences of the evaluation values of different alternatives under multiple attributes. Therefore, it is appropriate to discuss a 3WID model under a multi-attribute environment. Furthermore, in consideration of the effectiveness of GDM [21], [33], [39] in many real-life situations, we plan to study a 3WGID model in the IF-MAGDM context.
The structure of this paper is arranged as follows. In Section 2, we briefly review several preliminary definitions and concepts of 3WD based on decision-theoretic rough sets (DTRS), IF sets and IF-MAGDM. In Section 3, we explore a novel 3WGID model under the IF-MAGDM environment. In Section 4, we investigate an IF-MAGDM method with the help of the proposed 3WGID model. In Section 5, a real-life case study with respect to a coalfield investment is presented to demonstrate the effectiveness and feasibility of the established IF-MAGDM method. In the last section, we conclude our paper and outline several study issues in the future.
Section snippets
Preliminaries
In this section, for the convenience of the following statements, we present some basic definitions and concepts.
The construction of 3WGID under the IF-MAGDM environment
The theories of 3WD and IF sets are two influential mathematical theories to characterize vague and uncertain concepts, which are all suitable for solving uncertain decision-making problems. As a novel decision-making theory and methodology, 3WD can classify a collection of alternatives into three pairwise disjoint parts and give three decision actions (acceptance, rejection and non-commitment). IF sets can describe human’s inaccurate cognitions in terms of affirmation, negation and hesitation.
An IF-MAGDM method with the aid of the 3WGID model
In what follows, we plan to develop an IF-MAGDM method with the aid of the 3WGID model established in Section 3. In addition to forming the classification rule of each alternative discussed in Section 3, we further study the ranking of all alternatives.
An illustrative example
In this section, we intend to demonstrate the decision-making process and the validity of the new constructed IF-MAGDM method by an illustrative example. In addition, a sensitivity analysis and a comparative analysis are carried out to illustrate the effectiveness of the proposed IF-MAGDM method.
Conclusions
In conclusion, this paper introduces a 3WGID model based on the theory of 3WD in the IF-MAGDM context. It can be applied to cope with the uncertainty of some investment decision-making problems. The research results strengthen the understanding of the 3WD model and extend its applications in profit-based investment decision-making problems. Compared with some existing MAGDM models and methods, the main innovations of this paper can be summarized as follows. (1) Different from the existing
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to express their sincere thanks to the Editors and anonymous reviewers for their most valuable comments and suggestions in improving this paper greatly. The work described in this paper was supported by grants from the National Natural Science Foundation of China (Grant Nos. 11971365 and 11571010).
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