Research Article | Open Access
A Novel Multicriteria Group Decision-Making Approach with Hesitant Picture Fuzzy Linguistic Information
In real-life group decision-making environment, different decision-makers (DMs) might be hesitant to provide their evaluation by more than two linguistic terms with positive, neutral, negative, and refusal information (e.g., attitude for support, neutral, oppose, and refusal) for criteria of alternative. In order to solve such kinds of decision-making problems, a novel definition of hesitant picture fuzzy linguistic sets (HPFLSs) is introduced, and the HPFLS-based methods are developed. Considering the operation laws appeared in previous papers are not well suitable for HPFLS operation, the novel operation laws of HPFLSs are developed. Then, two aggregation operators of HPFLSs are developed, including hesitant picture fuzzy linguistic weighted average operator (HPFLSWA) and hesitant picture fuzzy linguistic weighted geometric operator (HPFLSGA). Meanwhile, the related proofs are given in detail. Additionally, the comparison method of score and accuracy functions is provided to rank the alternative. Finally, a real-life case of teaching performance evaluation is used to verify the proposed methods. The same case-based comparisons are further conducted between the proposed method and previous methods. The results showed that the proposed method can well overcome the lack of operation rules appeared in previous multicriteria group decision-making (MCGDM) methods and demonstrate effectiveness.
In recent years, as one of the useful tools to solve MCGDM problems, the fuzzy linguistic set-based methods  have been a focus of research. Along with more and more complex decision-making environment, many scholars proposed different extensions of fuzzy linguistic term sets. In order to avoid the information loss for computing with words, Herrera and Martinez  put forward a 2-tuple fuzzy linguistic representation model. Then, some different 2-tuple fuzzy linguistic sets-based approaches are proposed to handle different decision-making problems [3–5]. Focusing on aggregating the criteria information, Wei  used 2-tuple linguistic information-based generalized weighted average and geometric operators, Wan  developed the 2-tuple linguistic hybrid aggregation operator, and Liu et al.  developed 2-tuple linguistic Heronian mean operators; these methods enormously enriched the decision-making theory. However, in some decision circumstances, interval number can better express the uncertainty of decision-maker information. Thus, uncertain linguistic set-based approaches [9–12] were proposed. Subsequently, to extend the 2-tuple fuzzy set with uncertain linguistic set, Zhang  defined several interval-valued 2-tuple linguistic aggregation operators, and Liu et al.  defined some new operation rules for 2-tuple linguistic term set and developed Bonferroni mean operators. All of the above approaches were illustrated by the practical application cases, and their effectiveness was demonstrated.
However, in some cases, the membership degree cannot be defined by one specific value; thus, Wang and Li  proposed intuitionistic fuzzy linguistic set. Then, intuitionistic fuzzy linguistic set-based approaches [16–19] were proposed. Due to the different demands of practical decision-making problems, the intuitionistic fuzzy linguistic set is extended by many scholars. With regard to preference relation, Zhang et al.  defined 2-tuple intuitionistic fuzzy linguistic preference relation. For the information aggregation operator, Liu et al.  defined the hesitant intuitionistic fuzzy linguistic weighted average (HIFLWA) operator, and Meng et al.  developed interval-valued intuitionistic uncertain linguistic hybrid Shapley operators. In addition, Tan et al.  put forward an extended single-valued neutrosophic projection-based qualitative flexible multicriteria decision-making method. All the mentioned approaches above can yield the effectiveness.
Considering that decision-maker may be hesitant to give the evaluation value using the linguistic term, hesitant fuzzy linguistic set  was defined. Then, hesitant fuzzy linguistic set-based methods [25–27] were proposed. In order to effectively address practical decision-making issues, many extensive approaches are conducted. Zhou et al.  extended hesitant linguistic set with evidence reasoning. Wang et al.  extended the hesitant linguistic set with interval number and defined the interval-valued hesitant linguistic set. Pang et al.  extended the hesitant fuzzy set with probability distribution and defined probability linguistic term sets. Lin et al.  extended the hesitant fuzzy linguistic term with probability distribution and interval number and proposed a probability uncertainty hesitant fuzzy linguistic set. All the above methods based on the extension of hesitant fuzzy linguistic set are verified by an illustration example and the effectiveness is demonstrated.
Although these existing linguistic set-based approaches can demonstrate the effectiveness for solving the practical MCGDM issues, under some decision-making situations, they cannot be applied effectively. For example, one listed company carried on portfolio optimization, and there are four plans for the voting by the stakeholder. Some stakeholders may give “vote for,” some stakeholders may give “vote against,” some stakeholders may give “abstain,” and the rest of the stakeholders may refuse to vote. For such kind of decision-making problems, it is required to propose a new method.
Picture fuzzy set (PFS) was firstly proposed by Cường , which is the extension of the intuitionistic fuzzy set. After then, in order to solve the MCGDM voting problems, several picture fuzzy set-based MCGDM methods have been proposed [33–38]. With regard to measurement, Singh et al. developed the approach with picture fuzzy correlation coefficients , Wei et al.  introduced a novel method to solve MCGDM voting problems using cross-entropy of the picture fuzzy set. As for aggregation operators, Wei et al.  used the picture 2-tuple linguistic-based operators, including Bonferroni mean operator, weighted averaging operator, ordered weighted averaging operator, and hybrid averaging operator . In addition, Nie et al.  put forward the MCGDM voting method based on 2-tuple linguistic picture preference relation and applied to the selection of voting the proxy advisor firm by a stakeholder. Nevertheless, the proposed picture fuzzy set-based methods are enriched and used to well solve the MCGDM issues. In the practical cases, such as the students rank their teacher’s teaching performance, it is common for students to provide the assessment value with positive, indeterminacy, negative, or refusal attitude information. For example, “I cannot give the teaching performance with the score “outstanding excellent,” but it is ok for the “excellent”” or “I refuse to give any evaluation value.” For such kind of practical MCGDM problems, they cannot be appropriately solved by the existing fuzzy linguistic set-based MCGDM approaches. Thus, it is necessary to extend the existing fuzzy linguistic set to HPFLSs.
HPFLSs can elaborate the advantages of both hesitant linguistic set and picture fuzzy set, which are more suitable for the practical hesitant case of the decision-makers with positive indeterminacy, negative, and refusal information. For example, ten students are required to give the score for teaching quality (one of the criteria of teaching performance) for mathematics. Three students refuse to give any evaluation value, seven students oppose to give the teaching performance with the score “outstanding excellent,” but they agree for the “excellent.” Thus, if the evaluation term set is , then the evaluation score can be collected as with HPFLSs.
In the following, the contribution of this paper is presented:(1)In view of the defects of operation rules appeared in the methods of linguistic term set (LTS) and picture fuzzy linguistic set methods, the new operation rules based on sigmoid and equivalent transformation functions are proposed to overcome the existing limitation.(2)The new definition of hesitant picture fuzzy linguistic sets (HPFLSs) is introduced, it is more flexible to describe the two more possible linguistic terms for voting decision-making than picture fuzzy linguistic sets.(3)In order to solve MCGDM voting problems under the completely unknown criteria weight environment, the max-min deviation model for HPFLSs is constructed, and then the criteria weighted is computed by using Lagrange functions.(4)In order to develop an effective approach with HPFLSs to solve practical decision-making problems, the average weighted operator of HPFLS (HPFLSWA) and hesitant picture fuzzy linguistic weighted geometric operator (HPFLSWG) are developed in this paper. Meanwhile, the comparison methods for HPFLSs are introduced. Then, based on the proposed operators and comparison approaches, an effective multicriteria group decision-making (MCGDM) voting method under HPFLSs is developed.
The rest of this paper is organized as follows: several definitions and operation laws related to PFS and TLSs are reviewed in Section 2. In Section 3, the definition and novel basic operation rules of HPFLSs are introduced, and the HPFLSWA and HPFLSWG operators are developed and min-max deviation method, which is used to gain the criteria weight for completely unknown, is conducted; moreover, the ordering comparison methods for HPFLSs are presented. Section 4 provides a voting method of MCGDM based on the HPFLSWA and HPFLSWG operators, and its steps are described in detail. In Section 5, the real teaching performance evolution from Hubei University of Automotive Technology is used to illustrate the effectiveness of the proposed method, and according to the decision-maker preference, how to select the approximate value of parameter is discussed. Conclusions are drawn in Section 6.
In this section, the definitions of PFS, 2-tuple fuzzy linguistic sets (2TFLSs), picture 2-tuple fuzzy linguistic sets (P2TFLSs), and 2-tuple linguistic picture fuzzy sets (2TLPFSs) are presented, and the operation rules of P2TFLSs are reviewed to lay the groundwork for later analysis.
2.1. Definitions of PFS, 2TFLSs, P2TFLSs, and 2TLPFSs
Definition 1 (see ). Let be a universe space, and a PFS is defined aswhere is called the degree of positive membership of in , is called the degree of neutral membership of in , is called the degree of negative membership of in , and , , and satisfy the condition . Moreover, can be called the degree of refusal.
Let be a linguistic term set, and symbolic method aggregation linguistic information obtains a value , and if , then an approximation function is used to express the index result of .
Definition 2 (see ). Let be the aggregation result of the indices of a set of labels assessed in a linguistic term set , i.e., the results of symbolic aggregation operation, and be the cardinality of . Let and be two values such that and ; then, is called the symbolic translation.
Based on the PFS and Definition 2, Wei et al.  defined the picture 2-tuple fuzzy linguistic set.
Definition 3. A picture 2-tuple fuzzy linguistic set in is defined aswhere , , and with the condition , , , and , the numbers , , and represent the positive membership degree, neutral membership degree, negative membership degree of the element to linguistic variable , respectively. Then, could be called the refusal membership degree of the element to linguistic variable .
For convenience to collect the data in the selection of the proxy advisor firm problems, Nie et al.  defined the concept of 2-tuple linguistic picture fuzzy sets (2TLPFSs).
Definition 4. Let be a linguistic set and be a linguistic 2-tuple fuzzy set. Suppose that , , and , if , then is called 2TLPFSs, and , , and could be called the positive membership degree, neutral membership degree, and negative membership degree of , respectively.
2.2. Operation Rules of Linguistic Terms Sets (LTSs) and 2TLPFSs
Motivated by the aggregation function of t-norm, the operations of LTSs and 2TLPFSs are constructed, as follows.
Definition 5 (see ). Let and be two 2TLPFSs, and the operation rules of 2TLPFSs can be defined as follows:(1)(2)(3)(4)However, in some typical circumstances, the results obtained by the above operation rules can be unreasonable, and the examples are shown as follows.
Example 1. Suppose that two alternatives are evaluated under three criteria, the evaluation values are represented in the form of 2TLPFS as , , , and , , . The criteria weight vector is . Obviously, the aggregated result of alternative is superior than that of . However, in accordance with the above operation rules, the results are and , which indicate that the aggregated result of alternative is superior than that of . Thus, this result cannot be accepted.
Example 2. Assume that experts give three criteria score by 2TLPFS for two alternatives that , , , and , , . The criteria weight vector is , and it is easy to see that should be superior than . However, according to the operation rules of Definition 5, the results are and , which indicate that the alternative is equal to alternative . So, the result is not reasonable.
Definition 6 (see ). Let be a LTS, and be two linguistic terms, and be two equivalent transformation function of linguistic terms, and be a real number; then,(1)(2)(3)(4)Although the above operation rules can overcome some defects existed in the previous paper, it still can yield the unreasonable results under some typical situations. The example is similar to the above Example 2; thus, it is omitted here.
3. Hesitant Picture Fuzzy Linguistic Sets (HPFLSs)
In this section, a new concept of HPFLSs is introduced firstly. Subsequently, motivated by the operation rules of the linguistic term set  and picture 2-tuple linguistic term sets , to avoid the existing limitations pointed out in Examples 1 and 2, the novel operation laws of HPFLSs are proposed. Next, two aggregation operators of HPFLSs are introduced, and the related properties are discussed.
3.1. Hesitant Picture Fuzzy Linguistic Sets (HPFLSs)
Definition 7. Let be an LTS, a HPFLS is defined aswhere , , , and . , , and represent the positive membership, indeterminacy membership, and negative membership of the linguistic term , respectively; is the refusal membership of the linguistic term . And its complementary set is .
3.2. Novel Operation Rules of HPFLSs
In this subsection, two equivalent transformation functions and are defined. Then, the novel operation laws of HPFLSs are introduced based on the defined functions.
Definition 8. Let be an LTS; then, the linguistic term can be equivalently transformed to according to the function , and can be equivalently transformed to the linguistic term according to the function as follows:where , , and are the adjustment parameters described in Definition 9.
Definition 9. For any , can be equivalently transformed to according to the function , and can be equivalently transformed to according to the function as follows:where , by default, .
Definition 10. Let and be two HPFLSs, , and , be the equivalent transformation functions, and ; then, the operation rules can be defined below:(1)(2)(3)(4)where , , , and .
Based on the above operation rules in Definition 10, we have concluded the following Theorem 1.
Theorem 1. Let Hs1, Hs2, Hs3 be three HPFLS, and λ1 > 0, λ2 > 0, λ3 > 0, then, the following properties are true.(1)(2)(3)(4)(5)(6)(7)(8)
Example 3. Let be an LTS, and be two alternatives, be the criteria set, and be the weight vector. The corresponding evaluation values are presented in Table 1.
According to Definitions 8–10, we have , , and . By using the operation rules (1) and (3) in Definition 10, the aggregation results of is obtained as , and that of is . The aggregation results show that . Similarly, by using the operation rules (2) and (4) in Definition 10, we obtain the aggregation results for as and the aggregation results for as . The aggregation results show that as well. Intuitively, should be superior than according to the evaluation values shown in Table 1. Thus, the above results state that our proposed operation rule can effectively overcome the defects as pointed out in Examples 1 and 2. The detailed computing process of the aggregation result is omitted here.
3.3. Aggregation Operators of HPFLSs
In this subsection, two aggregation operators of HPFLSs are presented and can be utilized to solve the real-life decision-making problems.
Definition 11. Let be HPFLSs, and their weight vector is . Then, the hesitant picture fuzzy linguistic weighted average operator (HPFLWA) can be defined as follows:
Definition 12. Let be HPFLSs, and their weight vector is . Then, the hesitant picture fuzzy linguistic weighted geometric operator (HPFLWG) is referred to asBased on Definition 11 and the operation rules described in Definition 10, Theorem 2 can be gained.
Theorem 2. For HPFLSs, , ; and their weight vector is , with and , and the aggregation results can be obtained by the HPFLWA operator as
Proof. (1)When , the following result can be gained: then(2)Assume that equation (8) holds for any ; thus, we have(3)When , the following results are acquired:So, the equation holds for any . The proof is completed now.
Based on Definition 12 and the operation rules of Definition 10, Theorem 3 can be obtained.
Theorem 3. For HPFLSs, , ; their weight vector is ; for any , and , and the aggregation results can be calculated by the HPFLWG operator:
Proof. (1)When , in accordance with the operation rules in Definition 10, the following processing is presented:Assume that the equation holds for any :(3)When , the following results are acquired:Thus, the equation holds for any , and the proof is completely down.
3.4. Criteria Weight Determination
In the real-life decision-making circumstance, usually, the criteria weight is completely unknown due to the time pressure. Thus, the min-max deviation method is used to identify the criteria weight. Obviously, the criteria with the max-deviation should be assigned with the largest weight; the criteria with the min-deviation should be given the smallest weight. In this subsection, according to the deviation method constructed by the hamming distance, the deviation of the alternative