WASPAS method and Aczel-Alsina aggregation operators for managing complex interval-valued intuitionistic fuzzy information and their applications in decision-making

Aczel-Alsina t-norm and t-conorm are a valuable and feasible technique to manage ambiguous and inconsistent information because of their dominant characteristics of broad parameter values. The main theme of this analysis is to explore Aczel-Alsina operational laws in the presence of the complex interval-valued intuitionistic fuzzy (CIVIF) set theory. Furthermore, we derive the theory of aggregation frameworks based on Aczel-Alsina operational laws for managing the theory of CIVIF information. The CIVIF Aczel-Alsina weighted averaging (CIVIFAAWA), CIVIF Aczel-Alsina ordered weighted averaging (CIVIFAAOWA), CIVIF Aczel-Alsina hybrid averaging (CIVIFAAHA), CIVIF Aczel-Alsina weighted geometric (CIVIFAAWG), CIVIF Aczel-Alsina ordered weighted geometric (CIVIFAAOWG) and CIVIF Aczel-Alsina hybrid geometric (CIVIFAAHG) operators are proposed, and their well-known properties and particular cases are also detailly derived. Further, we derive the theory of the WASPAS method for CIVIF information and evaluate their positive and negative aspects. Additionally, we demonstrate the multi-attribute decision-making (MADM) strategy under the invented works. Finally, we express the supremacy and dominancy of the invented methods with the help of sensitive analysis and geometrical shown of the explored works.


INTRODUCTION
MADM strategies aims to identify the best of a few relative other options or positioning choices as per their importance as far as the assessed objective. The techniques are utilized for choosing the most acceptable other option/arrangement, because there is no such option for which all rules' esteems are awesome. MADM strategy is the sub-part of the decision-making technique that has been used in the region of discrete fields. However, it is massively difficult to apply the MADM technique to the phenomena of fuzzy sets rather than crisp sets. To achieve this idea in the real scenario, Zadeh (1965) explored the fuzzy set (FS), which only depends on the supporting grade (SG) M C C ∈ [0,1]. In facilitating that sort of situation, FS suffers from an obvious deficiency for not describing the data in the shape of yes or no, not addressing expert opinion, namely, non-SG (NSG). To conquer this imperfection, Atanassov (1986) proposed the methodology of intuitionistic FS (IFS) with an SG and NSG. The well-known prominent of IFS is as followed: 0 ≤ M R + N R ≤ 1. Interval-valued (IV) data is mostly utilized to depict the ambiguity and problematic occurrences, like the difference in temperature, the vacillation of stock cost, and the scope of circulatory strain. Besides, the IV information might be gotten from various areas or sources. For this, the IV intuitionistic FS (IVIFS), was stated by Atanassov & Gargov (1989) Based on above discussions, we have obtained the result that the prevailing theories neglect to manage two-domination data in the shape of SG and NSG, and simultaneously neglect to survive with inconsistent and fluctuational at a provided phase of time. However, the data got from ''medical research'' such that the biometric and facial acknowledgment data set consistently changes with the entry of the time. Along these lines, Ramot et al. (2002) extended the scope of SG from a genuine subset to the unit circle of the complicated plane and henceforth established the principle of complex FS (CFS). The mathematical structure of SG in the circumstances of CFS is of the form Since CFS restricts only up to SG and does not take into account NSG, Alkouri & Salleh (2012) produced the principle of complex IFS (CIFS) in the shape Recently, Garg & Rani (2019a) studied the form of CIFS in the interval environment and proposed the mathematical structure of CIVIFS. Because of its strong ability in dealing with uncertain information, CIVIFS has been promoted in many ways, but the results in Aczel-Alsina operational laws still need to be enriched.

MOTIVATION AND MAIN CONTRIBUTION
CIVIFS theory is the modified version of the FS, IFS, IVIFS, CFS, and CIFS because of their valuable and dominant structure. Further, the theory of Aczel-Alsina is also very famous and reliable because it is the generation of the algebraic t-norm and t-conorm. Moreover, discovering the theory of aggregation operators in the presence of Aczel-Alsina information for managing CIVIF values is a very challenging task for new fuzzy scholars, because up to date no one can derive the theory of Aczel-Alsina aggregation operators for CIVIF values. Furthermore, deriving the theory of the WASPAS technique (Zavadskas et al., 2012) is also a very awkward and challenging task for fuzzy researchers. Keeping the benefits of the above prevailing operators, the major contribution of this analysis is illustrated below: (1) To initiate the Aczel-Alsina operational laws and their related results.
(2) To invent the principle of CIVIFAAWA, CIVIFAAOWA, CIVIFAAHA, CIVIFAAWG, CIVIFAAOWG, and CIVIFAAHG operators, and illustrated their well-known properties and results. (3) To derive the theory of the WASPAS method for CIVIFSs.
(4) To demonstrate the MADM strategy under the invented works.
(5) To express the supremacy and dominancy of the invented works with the help of sensitive analysis and geometrical shown of the explored works. Presentation of our analysis is implemented in the shape: Section 2 covers all the prevailing methodologies. In Section 3, we initiate the Aczel-Alsina operational laws and Lower bound of real part in neutral grade upper bound of imaginary part in neutral grade their related results. Section 4 produces the principle of CIVIFAAWA, CIVIFAAOWA, CIVIFAAHA, CIVIFAAWG, CIVIFAAOWG, and CIVIFAAHG operators, and illustrates their well-known properties. In Section 5, we derive the WASPAS method for CIVIFSs. In Section 6, we demonstrate the effectiveness of the MADM strategy under the invented works. The conclusion of this study is illustrated in Section 7. Before starting the proposed work, all variables and indexes used in this study are defined in Table 1.

PRELIMINARIES
Here, we utilized the weighted sum model (WSM) and the weighted product model (WPM) to review the concept of the WASPAS method (Zavadskas et al., 2012). Moreover, the extended WASPAS method was derived from Zavadskas et al. (2013). Some valuable and effective steps of the WASPAS method are listed below: Step 1: The input data of the technique is represented in the form of a matrix of alternatives and attributes, which is based on the data received from the expert.
Step 2: Normalize the decision matrix in the presence of the information in Eq. (1): where B represented the benefit types of data and C stated the cost type of criteria.
Step 3: Compute WSM and WPM of each alternative: Step 4: Calculate the score value by using the theory of WSM and WPM information referring to the following way: There exist some special cases: Step 5: Deriving the best preference by the score value in Step 4. Next, the algebraic theories of some prevailing principles like CIVIFSs, the concept of Aczel-Alsina t-norm and t-conorm will be discussed. Of note, the notation X U , stated for universal sets.
Definition 1: (Garg & Rani, 2019a) The mathematical structure of CIVIFS C C is shown in the shape of: and indicate the TD and FD states the neutral grade, and denotes the complex interval-valued intuitionistic fuzzy number (CIVIFN). Definition 2: (Garg & Rani, 2019a) Suppose there are two CIVIFNs    ,j = 1,2, then: Definition 3: (Garg & Rani, 2019b) By taking any two CIVIFNs    , then the score value (SV) and accuracy value (AV) are determined by the following formulas: Definition 4: (Garg & Rani, 2019a) By taking any two CIVIFNs states the Aczel-Alsina TN, its expression is listed as follows: Definition 8: (Aczél & Alsina, 1982) states the Aczel-Alsina TCN, the detailed expression is shown as follows:

ACZEL-ALSINA OPERATIONAL LAWS FOR CIVIFSS
This section mainly introduces the Aczel-Alsina operational laws for CIVIFS that keeps the benefits of the IV information and explores these elementary properties.

ACZEL-ALSINA AGGREGATION OPERATORS FOR CIVIFS
This section proposes a group of aggregation operators by utilizing the Aczel-Alsina operational laws for CIVIFSs such that CIVIFAAWA, CIVIFAAOWA, CIVIFAAHA, CIVIFAAWG, CIVIFAAOWG, and CIVIFAAHG operators, and illustrates their wellknown properties.

Definition 11: For CIVIFNs
then the CIVIFAAWA operator is interpreted as: then by using Eq. (20), we elaborate Furthermore, we derive the theory of idempotency, boundedness, and monotonicity for the information in Eq. (21).
if C C j = C, then the detailed expression is shown as: Proof: Property 2: For CIVIFNs , by using inequality, we have Similarly, for the upper part, we have Therefore, we obtained

Definition 13: For CIVIFNs
then the CIVIFAAHA operator is invented by:

WASPAS METHOD FOR CIVIFSS
The main theme of this section is to illustrate the WASPAS method for CIVIFSs and verify the validity of the proposed method with the help of some numerical examples. Some valuable and effective steps of the WASPAS method are listed below: Step 1: The input data of the technique is represented in the form of a matrix of alternatives and attributes, which is based on the data received from the expert.
Step 2: Further, we normalize the information in decision matrix by using the below theory: where the data in Eq. (48) is used for benefit types of data, such as Where the data in Eq. (49) is used for cost types of data, such as Step 3: Utilizing CIVIFAAWA and CIVIFAAWG operators to obtain the WSM and WPM of each alternative: Step 4: Compute the score value according to WSM and WPM, the detailed formula is listed as follows.
Step 5: Rank the alternatives and derive the best one referring to the score value S i in Step 4. Further, we justify the above-mentioned method by some practical examples. Example 1: To verify the WASPAS technique under the consideration of some CIVIF information, we applied it for practical CIVIF decision matrix to obtain the best alternative. Four alternatives: S 1 , S 2 , S 3 , S 4 ; and four criteria C 1 , C 2 , C 3 , C 4 , and C IJ indicates the assessment information of S I (I = 1,2, 3, 4) under the criterion C J (J = 1,2, 3, 4). Some valuable and effective steps of the WASPAS method are listed below: Step 1: The input data of the technique is represented in the form of a matrix of alternatives and attributes, which is based on the data received from the expert.
Step 5: Identify the ranking information for evaluating or deriving the best preference.
From the above analysis, we obtain the best preferences as S 2 .

Application in MADM
The significant commitment of this examination is to apply MADM method under CIVIFS for deciding the optimal scheme from the group of complex interval-valued intuitionistic fuzzy data. To determine the best one, we expounded a dynamic interaction. There are m alternative C C = C C 1 ,C C 2 ,...,C C m and n criteria looking like

Procedure of decision-making
To achieve the acquirement of the best one, we built the dynamic calculation looking like the accompanying stages: Stage 1: Construct the CIVIF decision matrix utilizing the CIVIF evaluation information.
Stage 2: Normalize the CIVIF decision matrix. The specific conversion process is shown below when dealing with beneficial data and cost data: Stage 4: Using Eq. (6) to derive the score information.
Stage 5: Evaluate the ranking information in the availability of score information.

Represented example
The significant finding of this investigation is to break down the explained administrators in the conditions of the MADM methodology. For this, we examined some pragmatic information to decide the practicality and probability of the introduced works.

Clarification of the problem
Permit us to ponder a creation association that expects to enroll a publicizing director for an unfilled post. Here, we consider five competitors C C j ,j = 1,2,3,4,5, allocated for extra appraisals, such as: C C 1 : Oral presentation capacity; C C 2 : History; C C 3 : Overall tendency; and C C 4 : confidence. For this, we consider weight vectors such as 0.4,0.3,0.2,0.1.
The five specialists C C j ,j = 1,2,3,4,5 are to oversee vagueness under CIVIF information by utilizing dynamic strategies.

Method under CIVIFAAWA and CIVIFAAWG operators
Determine the useful individual from the gathering of people (Five up-and-comers) by utilizing the MADM procedure under CIVIFAAWA and CIVIFAAWG operators. For obtaining the ideal one, we developed the dynamic calculation looking like the accompanying stages: Stage 1: Construct the CIVIF decision matrix. The specific data covering the cost and beneficial sorts are listed as Table 2.
Stage 2: Normalize the decision matrix referring to the subsequent conversion process.   Stage 4: Here, we compute the score values of the aggregated information in Stage 3, see Table 4.
Stage 5: Obtain the ranking information based on the score values, the detailed result is stated in Table 5.

Influence of parameter
Here, we discuss the stability and influence of the derived operators based on the different values of parameters ψ. Therefore, by using the information in Table 2 and various parameter values, we obtain the subsequent consequence listed in Table 6. We have gotten the consistent advantageous ideal C C 5 based on diverse operators by utilizing the particular upsides of the boundary. This result shows that our calculation model has a good stability.

Notes.
'' ×'' denotes it is unsuitble to calculate the score values.

Comparative analysis
Here, our main theme to evaluate the comparison between proposed method with few existing analyses to show the stability and effectiveness of the proposed method. For this, we use various existing operators such as Aggregation operators (AOs Table 7 for the data in Table 2. Under the various kinds of operators, we have gotten a completely consistent optimal judgment C C 5 by utilizing the particular upsides of the boundary. The best optimal is C C 5 according to the theory which was proposed by Garg & Rani (2019a) and Garg & Rani (2019b) based on CIVIFSs and proposed operators. Further, the derived theory of Aggregation operators (AOs) (Xu, 2007), geometric AOs (Xu & Yager, 2006), information AOs (Wang & Liu, 2012), Einstein geometric AOs (Wang & Liu, 2012), Hamacher AOs (Huang, 2014), Dombi AOs (Seikh & Mandal, 2021) under the IFSs have been failed, due to various limitations, because these operators or information ware proposed based on FSs, IFSs, IVIFSs, CFSs, and CIFSs which are the particular cases of the proposed information and hence they are not able to evaluate our suggestion information (CIVIF values).
Therefore, it can be inferred that the presented information and MADM model are very valuable and dominant for handling awkward information.

CONCLUSION
In this manuscript, we combined four main theories such as CIVIF information, Aczel-Alsina operational laws, averaging/geometric aggregation operators, and the WASPAS technique. Furthermore, the theory of CIVIF information is the modified version of the FSs, IFSs, CFSs, CIFSs, and IVIFSs, because these are the special cases of the invented theory. Further, we derived the theory of aggregation operators based on Aczel-Alsina