Yager aggregation operators based on complex interval-valued q-rung orthopair fuzzy information and their application in decision making

As a more massive feasible and prominent tool than the complex interval-valued Pythagorean fuzzy (CIVPF) set and complex interval-valued intuitionistic fuzzy (CIVIF) set, the complex interval-valued q-rung orthopair fuzzy (CIVQROF) set has been usually used to represent ambiguity and vagueness for real-life decision-making problems. In this paper, we firstly proposed some distance measures, Yager operational laws, and their comparison method. Further, we developed CIVQROF Yager weighted averaging (CIVQROFYWA), CIVQROF Yager ordered weighted averaging (CIVQROFYOWA), CIVQROF Yager weighted geometric (CIVQROFYWG), CIVQROF Yager ordered weighted geometric (CIVQROFYOWG) operators with CIVQROF information, and some certain well-known and feasible properties and outstanding results are explored in detail. Moreover, we proposed a new and valuable technique for handling multi-attribute decision-making problems with CIVQROF information. Lastly, a practical evaluation regarding the high blood pressure diseases of the patient is evaluated to illustrate the feasibility and worth of the proposed approaches.


Introduction
Multi-attribute decision-making (MADM) has the wide applications in real-life decision-making problems. For an MADM problem, we firstly express the evaluation information for each alternative under each attribute. However because of ambiguity and complexity of decision environment [1], it is very difficult to give the decision information by real number, fuzzy set (FS) [2] and intuitionistic FS (IFS) [3] are very valid tools to express the uncertain information, and they have a lot of utilization in different scenarios. The IFS is more reliable than the FS because IFS can express two different types of functions at the same time, called supporting degree and supporting-against degree, and their sum should be formulated in the unit interval. Further, some questions have been raised related to the range of the degrees in IFS because in many situations experts have faced a lot of dilemmas related to information that has been given by intervals. In order to solve these dilemmas, Atanassov [4] proposed interval-valued IFS (IVIFS) which described two functions by the interval numbers, called supporting degree and supporting-against degree, and their sum of the upper parts should be restricted to the unit interval. Further, IFS and IVIFS have been used in different fields [5][6][7][8], many modifications and extensions have been done, such as Pythagorean FS (PFS) [9] interval-valued PFS (IVPFS) [10], decisionmaking [11][12][13][14][15], q-rung orthopair FS (QROFS) [16] and interval-valued QROFS (IVQROFS) [17]. For an MADM problem, the second work is to aggregate all attribute values for each alternative, and get the comprehensive values for all alternative, then ranking them. So the aggregation operators are important tools for this goals, many scholars developed different types of operators [18], power Maclaurin symmetric mean operators [19], and neutrality aggregation operators for QROFS in [20].
The previous introduced FSs have a lot of advantages, but they have also a lot of limitations because they are not able to deal with the two-dimension information in a singleton set. In our daily life, in many places, experts have faced two-dimension information, for instance, if a person "A" wants to organize different sort of software in a market and the employees of the enterprises can give information regarding each software, firstly name of the software and secondly production data of the software. The first information represented the amplitude term, and the second term expressed the phase term of the information. Obviously, the existing IFS, PFS, and QROFS cannot deal with this decision situation. For this, the complex IFS (CIFS) [21] was proposed, which is the generalization of complex FS (CFS) [22]. Further, some questions have been raised related to the range of the degrees in CIFS, because in many situations experts have faced a lot of dilemmas related to information that has been given in the shape of intervals. To process this situation, Garg and Rani [23] proposed the complex IVIFS (CIVIFS) which is described by two functions in the interval, called supporting degree and supporting-against degree, and their sum (separately for real and unreal parts) of the upper parts should be formulated in the unit interval. Further, the CIFS and CIVIFS have been used in different fields [23][24][25][26], many modifications and extensions have been done, such as complex PFS (CPFS) [27], complex IVPFS (CIVPFS) [28], complex QROFS (CQROFS) [29,30] and complex IVQROFS (CIVQROFS) [31]. Some extensions of the existing operators for the different complex FSs are done [32], such as logarithmic aggregation operators for IFS using tnorm and t-conorm [33], logarithmic operational laws and logarithmic aggregation operators for PFS [14] and the logarithmic and exponential aggregation operators for CIFS [34]. The collection of Yager t-norms is an important operation, derived in the early 1980s by Yager, which is described as (0 ≤ θ ≤ +∞) The Yager t-norm T Y θ is nilpotent ⇐⇒ 0 < θ < +∞, when θ 1, it becomes Lukasiewicz t-norm. Where the additive generator of T Y θ for 0 < θ < +∞ is f Y θ (x) (1 − x) θ . In order to fully use the advantages of the Yager t-norm and t-conorm, Shahzadi et al. [35] proposed the Yager weighted aggregation operators for PFSs, Akram et al. [36] proposed the Yager aggregation operators for CPFSs, Garg et al. [37] developed the Yager aggregation operators for Fermatean FSs, and Akram and Shahzadi [38] developed the Yager aggregation operators for QROFSs.
The CIVQROFS can express the two-dimensional information, and it has been widely used in decision making fields, for example, if someone decided to buy a new and well-brand car considering two main factors such as the name of the car and the model of the car. Where the name and model of the car expressed the two-dimension information such as amplitude and phase terms, and the traditional IVQROF cannot deal with this problems. In addition, because Yager t-norms are more general than the other operations, and the aggregation operators based on Yager t-norms are also more general than some existing operators. Now there are no researches about the aggregation operators based on Yager t-norms for CIVQROFS, so it is necessary to extend the aggregation operators based on Yager t-norms to CIVQROFS, and then use them to solve the MADM problems.
The motivation for this study is provided as: 1. Some scholars extended Yager aggregation operators for PFSs, Fermatean FSs, and QROFSs, however, there are still a lot of limitations, the PFSs, Fermatean FS, and QROFSs failed to express two-dimensional information. Therefore, we propose CIVQROFS to express the complex two-dimensional information. 2. The CIVQROF set is a very valuable and feasible modification of the CIVPF set and CIVIF set, which can express the wider scope than the CIVPF set and CIVIF set because the CIVPF set and CIVIF can only meet 0 ≤ Δ +2 3. Some scholars developed some aggregation operators based on IVIFSs, IVPFSs, QROFSs, CIVIFS, and CIVPFSs, however, because Yager t-norms are more general than the other operations, and the aggregation operators based on Yager t-norms are also more general than some existing operators, so it is necessary to extend the aggregation operators based on Yager t-norms to CIVQROFS. 4. The MADM method is an important tool to solve some decision problems, especially for some decision-making problems based on CIVQROF information, so it is necessary to develop the MADM method based on the Yager aggregation operators of CIVQROFS.
The real part of Interval-valued non-membership grade The real part of Interval-valued refusal grade The imaginary part of the Interval-valued refusal grade The major contributions of this study are given as follows.
1. We propose some distance measures, Yager operational laws, and their important results. 2. We develop CIVQROFYWA, CIVQROFYOWA, CIVQROFYWG, and CIVQROFYOWG operators, and discuss some certain well-known and feasible properties and outstanding results in detail. 3. We propose a new and valuable technique for handling multi-attribute decision-making (MADM) problems with CIVQROF information. 4. We do a comparative analysis to show the advantages of the proposed operators by a practical evaluation regarding the high blood pressure diseases of the patient.
To achieve these goals, the main construction of this paper is given as follows. The CIVQROFSs and Yager tnorm and t-conorm are introduced in Sect. 2. In Sect. 3, we develop some distance measures and Yager operational laws. In Sect. 4, we develop the CIVQROFYWA, CIVQROFY-OWA, CIVQROFYWG, and CIVQROFYOWG operators, and some certain well-known and feasible properties and outstanding results are analyzed in detail. In Sect. 5, we propose multi-attribute decision-making methods under the CIVQROF circumstances for evaluating high blood pressure threat management, and the advantages of the proposed methods are discussed by the comparative analysis. The final remarks of this paper are explained in Sect. 6.

Preliminaries
In this section, we give some existing concepts and theories such as CIVQROFSs and their operational laws, and Yager t-conorm and t-conorm. The symbol X is used for fixed sets and the information Δ μ ∅ represented the supporting and supporting-against. All symbols which are used in this paper are explained in Table 1.

and the refusal grade is
is called score value.
is called accuracy value.
Using Eqs. (2) and (3), we have , then the operational laws are defined as follows:

Operational laws and distance measures for CIVQROFNs
In this section, we propose some new operational laws, distance measures for CIVQROFNs.
Similarly, we can get the same result for the imaginary part, i.e., 0 ≤ Δ − 1, 2, then the distance measure of CIVQROF (CIVQROFDM) is defined by: Theorem 2 Equation (12) holds the following properties:

Definition 5 Let
then the compromise distance of CIVQROF (CIVQROFCD) is defined by: Were, d μ ∅−1 , 1 is the distance between μ ∅−1 and the positive ideal solution (PIS) and, then different score values are defined below:
We only prove the Eq. (26), such that

Yager aggregation operators for CIVQROFNs
In this part, we develop the CIVQROFYWA, CIVQROFY-OWA, CIVQROFYWG, and CIVQROFYOWG operators, and explore some well-known and feasible properties in detail.

Definition 8:
Let , then the CIVQROFYWA operator is defined by: Proof: If 2, then Thus, Eq. (31) holds for 2. Further, assume that it also holds for k, such that Then, we will prove that it is also held for k + 1, we have Hence, Eq. (31) holds for all possible value of "positive".

Definition 9: Let
, then the CIVQROFYOWA operator is defined by: 1) and the term μ W v− represents the position weight vector with

Definition 10: Let
, then the CIVQROFYWG operator is defined by:

Multi-attribute decision-making methods
In this section, we propose a new and valuable MADM method with unknown weights and CIVQROF information. For this, suppose C ∅−1 , C ∅−2 , . . . , C ∅− m is the family of alternatives and Further, R r i j m×n represents the decision matrix where Based on the proposed operators, we give the decision-making steps as follows. High risk of illness Orange Not susceptible Blue C C Q−5 Normal Green Step 1: because there are two types of information, called benefit type and cost type: Case A: if the information is cost type, then normalize it by Case B: if the information is benefit type, then not needed to be normalized.
Step 2: compute the aggregated values by using the CIVQROFYWA and CIVQROFYWG operators.
Step 3: get the score values of the aggregated values.
Step 4: rank alternatives based on the score values and get the better optimal.

Illustrated example
With the advancement of data innovation, it was applied to hypertension counteraction, and what's more, treatment for the local area of the executives. Albeit the current administration framework fulfills the premise of clinical administration, proficiency should be moved along. To foster a local area hypertension the board framework, to give more precise checking and forecast of hypertension powerless populaces, the forecast model that the framework needs to work in the wake of utilizing large information handling actually should be investigated and decided by chiefs. Occupants' defenselessness to hypertension is separated into five levels: standard, not powerless, in danger of illness, high gamble of illness, and exceptionally high gamble of sickness lately. Hypertension the executive's framework utilizes green, blue, yellow, orange, and red are demonstrated, and they are isolated into five choices C ∅−1 , C ∅−2 , C ∅−3 , C ∅−4 , C ∅−5 , each tone and the comparing risk level are displayed in Table 2. For each plan, the admonition level is connected with many variables of hypertensive patients. Taking into account that the gathered information data is actual assessment data, the specialists judge whether inhabitants are defenseless against risk principally relies upon current pulse estimation esteem, hereditary history, The five signs of changing pattern of different circulatory strain estimations, age, furthermore, related risk factors are addressed by μ AT −1 , μ AT −2 , μ AT −3 and μ AT −4 . All pointers are proficiency pointers.
To proceed with the above decision making problems, we give the following decision steps.
Step 1: Get the decision matrix in CIVQROFNs (see Table 3), because the information is benefit type, then not needed to be normalized.
Step 2: Compute the aggregated values by using the CIVQROFYWA and CIVQROFYWG operators, given in Table 4.
Step 3: Get the score values of the above-aggregated information shown in Table 5.
Step 4: Rank the alternatives based on the score values and get the better optimal, shown in Table 6.
Noticed that the CIVQROFYWA operator is give the best optimal C C Q−5 and the by using the CIVQROFYWG operator, we also obtained the same best decision as a C C Q−5 . Further, we remove the imaginary parts from the information given in Table 3, and we obtained the data shown in Table 7.
To show the advantages of the proposed information, we need to consider some existing types of information. For this, we consider the value ℯ 0 1, then by this rule, we convert the information in Table 7 to the information shown in Table 8.
Further, we compute the aggregated values by the CIVQROFYWA and CIVQROFYWG operators, given in Table 9.
Then calculate the score values of the above-aggregated information in Table 9, and get the results shown in Table 10.
Finally, rank the alternatives based on the score values and get the better one, shown in Table 11.
Noticed that the better alternative based on the CIVQRO-FYWA operator is C C Q−4 and there is the same result C C Q−4 produced by the CIVQROFYWG operator.
Further, in order to show the advantages of the proposed method, we do the comparative analysis of the proposed operators with some existing operators.

Comparative analysis
In this analysis, we compare the proposed operators with various existing operators to show the advantages of the proposed method. For this, we use the following existing operators: aggregation operators based on generalized improved score function using interval-valued IFSs [7], aggregation operators based on improved accuracy function using interval-valued PFSs [10], aggregation operators based on interval-valued QROFSs [17], aggregation operators for   Table 5 The score values of the aggregated information CIVQROFYWA operator CIVQROFYWG operator 0.8141 0.9632 Table 6 The ranking results Method Ranking results  Table 8 The complex interval-valued QROFSs from Table 7 μ AT −1 μ AT −2  Table 9 The aggregated values based on CIVIFSs [23], Einstein geometric aggregation operators for CIVPFSs [28], aggregation operators for CIVQROFSs [31]. Then based on the information in Table 3, the comparative analysis is given in Table 12.

CIVQROFYWA operator
Results in Table 11 show that some operators can give their ranking results and some operators failed. Some detailed reasons are discussed below: 1. The operators from Ref. [7] are based on IVIFS and we know that the IVIFS is the special case of the proposed CIVQROF information, and they are not able to process the CIVQROF types of information. 2. The operators from Garg [10] are based on IVPFS, and they have failed to process the CIVQROF information, because they are the special cases of the Yager aggregation operators based on CIVQROF information. Table 12 The comparative result based on the information in Table 3 Methods CIVQROFYWA operator 0.7195, 0.7966, 0.4395, 0.6821, 0.8141 CIVQROFYWG operator 0.8749, 0.9285, 0.7291, 0.9208, 0.9632 Table 10 The score values based on Table 9 CIVQROFYWA operator CIVQROFYWG operator Table 11 The ranking values based on Table 10 Method Ranking values 3. The operators from Joshi et al. [17] are based on IVQROFS, and they have failed to process the CIVQROF information, because they are the special cases of the Yager aggregation operators based on CIVQROF information. 4. The operators from Garg and Rani [23] are based on CIVIFS, and they have failed to process the CIVQROF information, because they are the special cases of the Yager aggregation operators based on CIVQROF information. 5. The operators from Ali et al. [28] are based on CIVPFS, and they have failed to process the CIVQROF information, because they are the special cases of the Yager aggregation operators based on CIVQROF information.
Because the operators from Refs. [7,10,17,23,28] are based on IVIFSs, IVPFSs, IVQROFSs, CIVIFSs, and CIVPFSs, and they are the special cases of the proposed operators, and they cannot process the decision-making problems with CIVQROF information. But the operator in Ref. [31] was developed based on CIVQROFSs and the considered information in Table 3 is also CIVQROFNs, so the operators in Ref. [31] and in this paper can process this decision problems. Further, we use the information in Table 8, the comparative analysis is shown in Table 13.
Results in Table 13 show that some operators can give their ranking results and some operators failed. Some detailed reasons are discussed below: 1. The operators from Ref. [7] is based on IVIFS and we know that the IVIFS is the special case of the CIVQROF information, so the operators in Ref. [7] are not able to process the CIVQROF information. 2. The operators from Garg [10] are based on IVPFS, and they have failed to process the CIVQROF information, because they are the special cases of the Yager aggregation operators based on CIVQROF information. 3. The operators from Garg and Rani [23] are based on CIVIFS, and they have failed to process the CIVQROF information, because they are the special cases of the Yager aggregation operators based on CIVQROF information. 4. The operators from Ali et al. [28] are based on CIVPFS, and they have failed to process the CIVQROF information, because they are the special case of the Yager aggregation operators based on CIVQROF information. Table 13 The comparative analysis based on the information in Table 8 Methods

Conclusion
The contributions of this paper are shown as follows.
1. Propose some distance measures, Yager operational laws, and their important results. 2. Develop CIVQROFYWA, CIVQROFYOWA, CIVQRO-FYWG, and CIVQROFYOWG operators, and discuss some certain well-known and feasible properties and outstanding results in detail. 3. Propose a new and valuable technique for handling multiattribute decision-making (MADM) problems with CIVQROF information. 4. Show the advantages of the proposed operators by comparative Analysis by a practical evaluation regarding the high blood pressure diseases of the patient.

Limitation of the proposed work
Yager aggregation operators for CIVQROF information are proposed in this paper, but in many cases, the CIVQROF set has a lot of limitations because of its structure and conditions, if an expert gives information by yes, no, abstinence, and refusal, then they cannot be processed by Yager aggregation operators based on CIVQROF information. For this, we need to utilize Yager aggregation operators based on complex interval-valued picture fuzzy sets, complex interval-valued spherical fuzzy sets, and complex interval-valued T-spherical fuzzy sets.
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