Abstract
In this paper, we develop a multiattribute group decision-making (MAGDM) method to solve problems with interactive attributes under interval-valued q-rung orthopair fuzzy set (IVq-ROFS) environment. Firstly, the interval-valued q-rung orthopair fuzzy Choquet integral average (IVq-ROFCA) operator is proposed to aggregate interval-valued q-rung orthopair fuzzy information. Then, we investigate the interval-valued q-rung orthopair fuzzy Choquet integral geometric (IVq-ROFCG) operator and offer several related properties. More importantly, for handling problems with interdependence between attributes for IVq-ROFS, a MAGDM method is developed based on the IVq-ROFCA operator. Finally, an example of the warning management system for hypertension is given to illustrate the proposed method, and parameter analysis and comparison analysis further verify the feasibility and validity of the proposed method.
1. Introduction
As a powerful and practical approach, fuzzy set (FS) [1] has been deeply used in various areas, such as medical treatment, manufacturing, and education . With the increase in people’s awareness of complex and uncertain issues, fuzzy theories and methods have received great attention [2–5]. Since decision-makers need to deal with the possibilities of support, opposition, and neutrality in real life, Atanassov [6] proposed the intuitionistic fuzzy set (IFS), that is, the sum of the membership degree () and the nonmembership degree () satisfies . In response to the condition , Yager [7] investigated the Pythagorean fuzzy set (PFS) (). As a wider range compared with intuitionistic fuzzy, Verma and Merigo [8] defined a generalized hybrid trigonometric Pythagorean fuzzy similarity measure and developed the approach for Pythagorean fuzzy decision‐making; Bakioglu and Atahan [9] addressed the prioritization of risks involved with self-driving vehicles by the proposed new hybrid MCDM method based on AHP and TOPSIS VIKOR under Pythagorean fuzzy environment; Peng and Yang [10] defined interval-valued Pythagorean fuzzy sets (IVPFSs) and developed two interval-valued Pythagorean fuzzy aggregation operators to solve MAGDM problems. Gradually, for dealing with much more complicated problems, Yager [11] proposed the q-rung orthopair fuzzy set (q-ROFS) in 2016 and extended the fuzzy set to a wider range of applications based on the different values of q. Researchers have put forward numerous excellent results in recent years. Yager et al. [12, 13] further deduced the related properties and mathematical principles of q-ROFS. Liu and Wang proposed several weighted average operators of q-rung orthopair fuzzy numbers (q-ROFNs) [14]. Liu and Liu developed the q-rung orthopair fuzzy Bonferroni mean operator to deal with problems [15]. Xing et al. [16] developed the q-rung orthopair fuzzy point weighted aggregation operator and applied it to q-rung orthopair fuzzy decision-making. Garg investigated the trigonometric operation-based q-rung orthopair fuzzy aggregation operator for processing fuzzy information [17] and introduced a novel concept of the connection number‐based q‐rung orthopair fuzzy set (CN‐q‐ROFS) [18]. Hussain et al. [19] proposed q‐rung orthopair fuzzy soft weighted averaging, q‐rung orthopair fuzzy soft ordered weighted averaging, and q‐rung orthopair fuzzy soft hybrid averaging operators. Verma [20] defined two order‐α divergence measures for q-ROFS to quantify the information of discrimination that determine completely unknown or partially known attribute weights. Aydemir and Gunduz [21] presented neutrality average and neutrality geometric aggregation operators and further designed a general score function for q‐ROFS based on power aggregation operators. Liu et al. [22] constructed the normalized bidirectional projection model and generalized knowledge-based entropy measure under q-rung orthopair fuzzy environment. In particular, it is better to address strongly uncertain decision-making problems via IVq-ROFS, such that Garg [23] defined q-connection numbers (q-CNs) and gave some new q-exponential operation laws (q-EOLs) and operators over q-CNs for IVq-ROFS and presented the possibility of comparison between IVq-ROFSs [24]. Garg [25] introduced a novel concept of interval-valued q-rung orthopair fuzzy preference relations (IVq-ROFPRs) and then proposed additive consistent of IVq-ROFPR and programming model to derive the weights of alternatives.
However, attributes are not independent in decision-making, and there are more mutual influences and correlations that can be appropriately solved by the Choquet integral [26]. The Choquet integral comes with decision-making problems with independent attributes in handy [27–29], which had been conducted in an in-depth research study Tan and Chen defined the intuitionistic of the fuzzy measure based on the Choquet integral [30] and investigated the induced Choquet ordered averaging operator for aggregating expert evaluation [31]. Tan developed the generalized interval-valued intuitionistic fuzzy geometric aggregation (GIIFGA) operator and combined TOPSIS to deal with MAGDM problems [32]. Xu [33] investigated the intuitionistic fuzzy Choquet integral average operator and geometric operator, which are applied to address MAGDM problems with IFS. It is significant to mention that Grabisch [34] presented a synthesis on the application of fuzzy integrals and extended the application of fuzzy integrals in various fields. Considering the advantages of fuzzy integrals that simulate the interaction between standards in a flexible way, scholars have conducted a lot of studies on fuzzy integrals (including Choquet integral) group decision-making (GDM) methods to effectively solve decision problems. Wu et al. [35] extended some operational properties of intuitionistic fuzzy values (IFVs) and then studied the aggregation properties of the intuitionistic fuzzy-valued Choquet integral (IFCI) and the intuitionistic fuzzy-valued conjugate Choquet integral (IFCCI). Xing et al. [36] developed the Choquet integral based on q-rung orthopair fuzzy environment and proposed q-rung orthopair fuzzy decision-making methods, Keikha et al. [37] combined the Choquet integral and the TOPSIS method to process fuzzy information, and Teng and Liu [38] developed the generalized Shapley probabilistic linguistic Choquet average (GS-PLCA) operator and investigated a method to solve large group decision-making (LGDM) issues. The in-depth study of the fusion application of the Choquet integral and GDM methods is of practical significance for solving complex and uncertain problems.
Although the aforementioned studies brilliantly handle complex decision-making problems, they cannot be applied to the decision-making problem where attributes are dependent under the interval-valued q-rung orthopair fuzzy environment. Accordingly, this paper develops two interval-valued q-rung orthopair fuzzy Choquet integral operators for aggregating fuzzy information. Subsequently, a MAGDM method is constructed by employing the proposed IVq-ROFCA operator, where interaction attributes of alternatives among the MAGDM problem are taken into account. Finally, compared with existing methods, the practicability and superiority of the proposed method are demonstrated.
The remainder of this paper is constructed as follows: Section 2 reviews the concept of the IVq-ROFS and Choquet integral operator, Section 3 proposes the IVq-ROFCA operator and the IVq-ROFCG operator and extends several weighted and ordered operators, Section 4 introduces a MAGDM method based on the IVq-ROFCA operator, Section 5 gives an illustrated case and then provides parameter analysis and comparison analysis, and Section 6 concludes this paper.
2. Preliminaries
In this section, we make a brief review of the IVq-ROFS and Choquet integral.
Definition 1 (see [11]). Let be a fixed set, is a q-ROFS, where , and the following equation holds:where . For all , is the degree of membership, is the degree of nonmembership, and the degree of indeterminacy is shown in as follows:
Definition 2 (see [39]). Given a fixed set , IVq-ROFS on is defined asThe membership is represented by interval values and the nonmembership is , . The indeterminacy degree of is shown as follows:In particular, IVq-ROFS extends the application of interval-valued fuzzy sets in decision-making problems. When q = 1, IVq-ROFS would reduce to the interval-valued intuitionistic fuzzy set (IVIFS); when q = 2, IVq-ROFS would transform to IVPFS.
Definition 3 (see [39]). Let and be two interval-valued q-rung orthopair fuzzy numbers (IVq-ROFNs), . Equations (5)–(8) hold
Definition 4 (see [39]). For the IVq-ROFN , the score function is defined as
Definition 5 (see [39]). For the IVq-ROFN , the accuracy function is defined as
Definition 6 (see [39]). For two IVq-ROFNs and , the comparison method is defined as follows:(1)If , then (2)If , then (3)If , and if , then ; if , then ; if , then
Definition 7 (see [26]). Let be a universe of discourse, be a positive real-valued function, and be the fuzzy measure on . Then, the discrete Choquet integral of on fuzzy measure is defined aswhere is a permutation of that satisfies .
3. Several Interval-Valued q-Rung Orthopair Fuzzy Choquet Integral Operators
In this section, we investigate IVq-ROFCA and IVq-ROFCG operators, discuss their properties, and extend their weighted operators.
3.1. IVq-ROFCA
Definition 8. Let be the fuzzy measure on the nonempty finite set () and are IVq-ROFNs. The IVq-ROFCA operator is defined aswhere indicates the Choquet integral, and denotes a permutation of that satisfies The IVq-ROFCA operator aggregates information according to the fuzzy measures between attributes, and it is easy to find that the aggregation results obtained are still IVq-ROFNs.
Theorem 1. For IVq-ROFNs , is the fuzzy measure on the nonempty finite set , and the IVq-ROFCA operator can be expressed as
Proof. Based on Definition 3, we prove equation (13) by mathematical induction.
If n = 2,If n = k,If n = k + 1, the results of IVq-ROFCA are as follows:Equation (13) holds, and Theorem 1 holds.
Example 1. Suppose there are three suppliers to be chosen. The core competitiveness of suppliers can be evaluated by three criteria : denotes the level of technological innovation, denotes the ability of circulation control, and denotes the capability of management. The decision matrix is generated by evaluation of experts, in which the IVq-ROFN evaluation values are represented in Table 1. Now, it needs to evaluate the core competitiveness of according to the decision matrix , which provides the reference for the manufacturing enterprise to select the best solution. The fuzzy measure between attributes are set as follows:Let q = 3, all the IVq-ROFNs in satisfy , and the results are obtained as follows:It is easy to find , , and are IVq-ROFNs. According to equation (9), can be derived, which indicates that the supplier is better than and .
Theorem 2. Suppose are IVq-ROFNs, is the fuzzy measure on a nonempty finite set , and then, the following four properties of the IVq-ROFCA operator hold(1)Idempotency: for an IVq-ROFN , if , then (2)Boundedness: if and , then (3)Commutativity: suppose is a permutation of , then (4)Monotonicity: if are IVq-ROFNs and , then The proof of Theorem 2 is given as follows.
Proof. (1)Because all elements in the are equal and ,(2)According to equation (9), it can be seen that For any , it satisfies Namely, Similarly, sorted out(3)According to Definition 8, it can be easily derived that(4)Based on and Definition 8, it can be found thatThus,Similarly,According to Definition 6, it is obviously discovered that .
For some special relationships that exist between evaluation values in decision-making, we then develop the following operators to facilitate calculation.
Definition 9. Let be the fuzzy measure on the nonempty finite set () and be IVq-ROFNs.(1)If for all , , and it is independent for any elements in that means . The IVq-ROFCA operator transforms to the interval-valued q-rung orthopair fuzzy weighted Choquet average (IVq-ROFWCA) operator, which is expressed as(2)If for all , we have , where and . Furthermore, the IVq-ROFCA operator reduces to the interval-valued q-rung orthopair fuzzy order Choquet average (IVq-ROFOCA) operator that is represented as(3)If for all , where Qis a basic unit-interval monotonic function, satisfies monotonicity in [0,1] and follows properties: . For , where and , the IVq-ROFCA operator changes to the interval-valued q-rung orthopair fuzzy order weighted Choquet average (IVq-ROFOWCA) operator that is proposed asEspecially, if , then the IVq-ROFOWCA operator reduces to the IVq-ROFOCA operator. In addition, we develop the IVq-ROFCG operator in Section 3.2.
3.2. IVq-ROFCG
Definition 10. Let be the fuzzy measure on the nonempty finite set () and be IVq-ROFNs. Then, the IVq-ROFCG operator is defined aswhere indicates the Choquet integral, is a permutation of , which satisfies
Theorem 3. If is the fuzzy measure on the nonempty finite set and are IVq-ROFNs, . Then, the IVq-ROFCG operator is represented by
The proof of Theorem 3 is similar to Theorem 1, and the proof is omitted.
Theorem 4. If is a set of IVq-ROFNs, is the fuzzy measure on the nonempty finite set , then the following properties of the IVq-ROFC operator hold:(1)Idempotency: for an IVq-ROFN , if , then (2)Boundedness: if and , then (3)Commutativity: suppose that is a permutation of , then (4)Monotonicity: if is a set of IVq-ROFNs and , then
The proof of Theorem 4 is similar to Theorem 2, and the proof is omitted.
Definition 11. Let be the fuzzy measure on the nonempty finite set () and be n IVq-ROFNs. Then,(1)If for all , , and it is independent for any elements in that means . Then, the IVq-ROFCG operator transforms to the interval-valued q-rung orthopair fuzzy weighted Choquet geometric (IVq-ROFWCG) operator that is represented by(2)If for all , we have , where and , the IVq-ROFCG operator reduces to the interval-valued q-rung orthopair fuzzy order Choquet geometric (IVq-ROFOCG) operator expressed by(3)If for all , where Q is a basic unit-interval monotonic function, satisfies monotonicity in [0,1] and follows properties: for . Then, we let , where and . Then, the IVq-ROFCG operator changes to the interval-valued q-rung orthopair fuzzy order weighted Choquet geometric (IVq-ROFOWCG) operator that is shown asEspecially, if , it is obvious to find that the IVq-ROFOWCG operator reduces to the IVq-ROFOCG operator.
4. A MAGDM Method Based on the IVq-ROFCA Operator
In this section, we develop a MAGDM method based on the IVq-ROFCA operator. For a MAGDM problem, is the set of alternatives, is the set of interaction between attributes, and is the fuzzy measure of attributes, indicates the set of experts, and is the fuzzy measure of experts. For each expert, a decision matrix of different alternatives is , where are IVq-ROFNs that satisfy . For the k th expert , the decision matrix is expressed as , denotes the kth expert's evaluation of alternative with the attribute . Because of the differences between physical dimensions of attributes, the decision matrix needs to be standardized. Correspondingly, indicates the benefit type and indicates the cost type, and the standardization processing is shown in equation (37) and the complement operation of fuzzy numbers is shown in equation (38):
The steps of the MAGDM method based on the IVq-ROFCA operator are as follows:(1)According to the evaluation matrix given by experts, the appropriate value of q is chosen. When the amount of data is small, the value of q can be determined by observation. If the amount of data is large, q can be determined by the traversal method utilizing .(2)Using equations (37) and (38), the decision matrix is converted into the standardize matrix .(3) is aggregated into employing the IVq-ROFCA operator, and the collection matrix is derived by where represents the IVq-ROFNs in the decision matrix given by the t-th expert.(4)Then, the collective value of is derived by equation (40), which aggregates the fuzzy information corresponding to attributes:(5)The score value and the accuracy value of are obtained according to equations (9) and (10), which determine the ranking of alternatives.
5. Case of the Warning Management System for Hypertension
5.1. Decision Result by the Proposed MAGDM Method
In order to improve the management efficiency of doctors, we plan to develop a daily follow-up warning management system for hypertension. The warning degrees are divided into five levels that are represented in five colors denoted by . indicates that the resident shall seek treatment immediately, showing red; indicates the resident needs treatment in time, showing orange; indicates the resident should be treated for the timely follow-up to promote management, showing yellow; indicates that blood pressure management may be required in the future, showing blue; and indicates that the blood pressure of the resident is normal, and the color is green. The warning levels are related to numerous factors of hypertension patients, among which the blood pressure measurements, related diseases, related risk factors, and follow-up time intervals are represented as the attributes denoted by , , , . From a managerial perspective, , , and are benefit types, and is the cost type. With different importance, the fuzzy measures of attributes are listed in Table 2.
Without taking medicine, a 60-year-old resident has a measured value and related factors as follows: systolic blood pressure is 153 mmHg, diastolic blood pressure is 98 mmHg, there is no relevant disease record, obesity and family genetic history, and the doctor had no follow-up record of him. Three experts are invited to evaluate the warning degree of this resident using IVq-ROFNs, and the fuzzy measures of them are , , . Tables 3–5 are the evaluation values of the three experts after standardization. Subsequently, the decision result is determined utilizing the proposed MAGDM method.
We set q = 3 via observation, and then, the collective matrix R is obtained by equation (39) which is presented in Table 6.
The comprehensive attribute values of each alternative are obtained through equation (40), and the results are obtained as follows:
The score of each alternative is determined by equation (9):
The ranking result of alternatives’ scores is derived as follows:
Therefore, the ranking of alternatives is , and is the optimal selection, which means the doctor is required for timely follow-up to promote management of blood pressure of this patient who receives yellow warning.
5.2. Parameter Analysis
For further studying the influence of q, we analyze different values of q in this subsection, which are 2, 3, 4, and 5 in this case, and the results are listed in Table 7 and Figure 1.
It can be seen from Table 7 that the score rankings are always , and the early-warning results are all yellow warnings, which are consistent with the results obtained in Table 7. Hence, the decision result does not change due to the variation in the value of q. However, with the increase in the q value, it can be discovered that scores of the alternatives are declining completely but show different trends of downward. Considering the practical application, there is not much practical significance for IVq-ROFNs with the large value of q, which often does not exceed 5. Therefore, the value of q will not have an impact on the final decision result.
5.3. Comparison Analysis
In this subsection, the proposed method is used to solve the case in [32], which is compared with the method given in [32, 40]. There are 5 alternatives with 4 attributes , , , and for 3 experts , , and to make a decision, and the evaluation matrices , , and of interval-valued intuitionistic fuzzy numbers (IVIFNs) given are listed in Tables 8–10. Besides, the fuzzy measures of experts are , , and , and the fuzzy measures of attributes are shown in Table 11.
If is the fuzzy measure on the nonempty finite set (), for an IVIFS , the GIIFGA operator and the interval-valued intuitionistic fuzzy Einstein geometric Choquet integral (IVIFEGC) operator are defined as follows:where is a permutation of that satisfies
In this case, we set q = 1, IVq-ROFNs are equal to IVIFNs, which are applied to the comparison with existing methods. The obtained results of different methods are shown in Figure 2 and Table 12.
In this case, compared with the method proposed in [40], the same best and worst solutions are obtained by the method proposed in this paper, and obvious differences in the scores of alternatives are discovered. However, there are differences between and , and the small deviation is derived between them. From the method proposed in [32], the decision result is completely different from the other two methods, and the closeness of each alternative has a few differences. We analyze the reason as follows.
In this case, the membership degrees of IVIFNs are generally greater than nonmembership degrees. From the method proposed in [32], multiple decision matrices are aggregated by CITOPSIS, which reduces the difference in the influence of the membership and nonmembership of each alternative on the results. Thus, the deviations of alternatives are not obvious. Contrarily, the method proposed in this paper employs IVq-ROFCA to process fuzzy information, which further highlights the degree of expert support for the alternatives. The collective matrices of GIIFGA and IVq-ROFCA of Tables 8–10 are listed in Tables 13 and 14, respectively. According to equation (9), the scores of these two matrices are presented in Figure 3.
From Figure 3, the abscissa in the figure represents the alternatives with different attributes and the ordinate indicates their scores. It is easy to find there are only few differences between membership and nonmembership, except for the evaluation of with . Correspondingly, the span of the interval in membership with a wide range can be found in the experts’ evaluation of with from Tables 8–10, such as [0.2, 0.7]. Considering the above, there are differences in the selection of between the method proposed in this paper and the proposed method in [32]. On the other hand, it reflects that more advantages of the method proposed in this paper will be discovered, while the degree of support of experts for the alternatives is significantly higher than that of the opposition.
Above all, the proposed method employs the Choquet integral to aggregate interval-valued q-rung orthopair fuzzy information in this paper, which is a powerful tool to deal with MAGDM problems with dependent attributes. Moreover, the proposed method will highlight the support of experts for the alternatives and better reflect the superiority of the optimal alternative. Nevertheless, the proposed operators and the MAGDM method are used to handle decision problems for IVq-ROFNs, which are not able to be applied to different fuzzy environments. Besides, as a great impact on decision results remains to be studied, the degree of consensus among experts needs to be further studied in the proposed method.
6. Conclusion
The Choquet integral is an efficient tool to solve decision problems with interaction between attributes. For addressing complex MAGDM problems under interval-valued q-rung orthopair fuzzy information, we develop the IVq-ROFCA operator and the IVq-ROFCG operator and discuss some properties of them, including idempotency, commutativity, monotonicity, and boundedness. Particularly, we further design Choquet integral weighted and ordered operators for IVq-ROFS. Subsequently, a novel method is devised to process fuzzy information employing the IVq-ROFCA operator. Finally, a case of early-warning in the daily management of hypertension is given to illustrate the proposed method, and the results obtained are consistent with the actual situation provided by medical experts. The feasibility and effectiveness of the proposed method are proved by sensitivity analysis and comparative analysis additionally. Ulteriorly, our future research will focus on consensus models of IVq-ROFS, for instance, the MAGDM method with the Choquet integral based on consistency and consensus of experts. Moreover, considering the broad development prospect of fuzzy theory in the era of big data [41], the integration of the decision-making method, machine learning and big data are also one of our research directions in the future.
Data Availability
All data used to support the findings of the study are included within the article.
Disclosure
The innovation of this paper has been preprinted in arXiv: “Interval-Valued q-Rung Orthopair Fuzzy Choquet Integral Operators and Its Application in Group Decision Making” [42] by the authors of this paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported in part by the Department of Education of Jiangxi Province of China under Grant GJJ180270, National Natural Science Foundation of China grant no. 61866014, the Department of Shenzhen Local Science and Technology Development under Grant 2021Szvup052, and Jiangxi Provincial Natural Science Foundation (20202BAB202006).