Applications of complex picture fuzzy soft power aggregation operators in multi-attribute decision making.

The major theme of this analysis is to suggest a new theory in the form of complex picture fuzzy soft (CPFS) information and to initiate their major algebraic laws, score value, and accuracy values. The mathematical form of the CPFS set includes three main functions, called supporting, abstinence, and supporting against terms with a prominent characteristic that the sum of the triplet will lie in the unit interval. Further, in the consideration of the power aggregation operator using generalized t-norm and t-conorm and CPFS information, we diagnosed the mathematical concept of CPFS power averaging (CPFSPA), CPFS weighted power averaging (CPFSWPA), CPFS ordered weighted power averaging (CPFSOWPA), CPFS power geometric (CPFSPG), CPFS weighted power geometric (CPFSWPG), CPFS ordered weighted power geometric (CPFSOWPG). Moreover, the major results and their particular investigation of the invented approaches are also deliberated. Additionally, in the consideration of diagnosed operators using CPFS information, we illustrated a MADM ("multi-attribute decision-making") tool to find the best option from the family of decisions. Finally, we have shown the supremacy and feasibility of the diagnosed operators with the help of sensitive analysis and geometrical representations.

Introduction

Pattern recognition is a vital part of the decision-making strategy used as an application in the environment of engineering science, networking systems, and medical diagnoses. Pattern recognition is the computerized recognition of patterns and consistencies in information. It has been used in statistical information analysis, image analysis, signal procedure, bioinformatics, and machine learning. Pattern recognition has a lot of implementations in different areas, but various deficiencies were involved in the environment genuine life troubles in the consideration of crisp sets. To enhance the major theme of the research work, Atanassov[1] invented the major and well-known theory of intuitionistic fuzzy set (IFS), by generalizing the theory of fuzzy set (FS)[2]. The main IFS is very feasible and flexible because of its shape, the prominent condition of IFS is described: . The fundamental concept of IFS has massively improved than FS for handling awkward and unreliable information. Various applications are diagnosed in the circumstances of many fields. For sample, bipolar soft sets[3], geometric operators[4], generalized operators[5], simple operators[6], simple measures[7], measures using Hausdorff distance[8], similarity measures[9], Bonferroni operators[10], generalized Heronian operators[11], Heronian operators[12], interval-valued IFSs[13], cubic IFSs[14] and generalized cubic IFSs[15]. Cuong[16,17] diagnosed the main theory of picture FS (PFS) and its applications. The main PFS is very feasible and flexible because of its shape, the prominent condition of PFS is described: . The fundamental concept of PFS has massively improved over IFS and FS for handling awkward and unreliable information. Various applications are diagnosed in the circumstances of many fields of decision-making strategy[18-20] and operators[21-24].

FS has a huge number of implementations in the circumstances of medical diagnoses, pattern recognition, clustering analysis, and networking systems. But in various places, the theory of FS has also faced a lot of troubles. For example, if the owner of some well-known company wants to lunch a novel sort of various software in a market, they have given two sorts of information related to each software, called name and production date of the software. For handling the above-cited information, the novel theory of complex FS (CFS)[25] was diagnosed as a new strategy for managing genuine life ambiguities. Further, Alkouri and Salleh[26] introduced the strategy of complex IFS (CIFS) as a helper for scholars, those who faced complications in selecting the best option during the decision-making process. CIFS has its level for handling awkward and ambiguous information, certain people have considered the theory of CIFS for utilizing it in the circumstances of different fields, for instance, Ali et al.[27] diagnosed the complex intuitionistic fuzzy soft sets. Further, the concept of the relationships among any two CIFSs was invented by Jan et al.[28], the mathematical shape of complex interval-valued IFSs was diagnosed by Garg and Rani[29] and the theory of group utilized in the region of CIFSs was explored by Gulzar et al.[30]. Akram et al.[31] diagnosed the main theory of complex PFS (CPFS) and its applications. The main CPFS is very feasible and flexible because of its shape, the prominent condition of CPFS is described: and . The fundamental concept of CPFS has massively improved over CIFS and CFS for handling awkward and unreliable information. Various applications are diagnosed in the circumstances of many fields of decision-making strategy[32]. To explain the information in the above paragraph, we noticed that the prevailing information has the following major dilemmas:

How do we develop new and more effective ideas?

How do we develop a superior shape of aggregation operators, used for evaluating the collection of information?

How do we evaluate the beneficial preference from the collection of information?

Therefore, the main influence of this theory is to find the solution to the above dilemmas with the help of diagnosed power aggregation operators based on CPFS information.

Additionally, Molodtsov[33] introduced the theory of soft set (SS) by extending the theory of FS and because of their shape all scholars have employed it in different fields, for instance, the theory of fuzzy SS was invented by Maji et al.[34]. Further, the intuitionistic fuzzy SS was discovered by Maji et al.[35] and the interval-valued intuitionistic fuzzy SS was explored by Jiang et al.[36]. The generalized intuitionistic fuzzy SS was invented by Agarwal et al.[37]. Similarly, the theory of intuitionistic fuzzy SS and its application in decision-making was discovered by Jiang et al.[38]. The theory of power aggregation operators for IFS was discovered by Xu[39]. Further, Rani and Garg[40] diagnosed the power aggregation operators for CIFSs and CPFSs[32]. In various situations, the theory of CIFS information has failed because of many complications, for instance, if someone proved information in the shape of yes, abstinence, and no against the value of parameters, where the value of yes, abstinence, and no in the shape of a complex number, then the theory of complex intuitionistic fuzzy soft sets have been invalid. For evaluating the above-complicated situations, the theory of CPFSS and its operational laws are the parts of this manuscript. From the above-cited theory, we also clear that every theory has its limitations because of its structures. Similarly, all prevailing theories have a lot of benefits, keeping the benefits of the SSs and CPFSs, the main contribution of this analysis is to explore the well-known concept of CPFS setting and their laws. The major theme of this analysis is described with the help of various points:

To pioneer the theory of CPFS information and evaluated their major algebraic laws, score value, and accuracy values.

To present the theory of CPFSPA, CPFSWPA, CPFSOWPA, CPFSPG, CPFSWPG, and CPFSOWPG operators, and diagnosed their particular cases of the invented approaches.

To diagnose some real-life situations, we evaluate a MADM tool under the consideration of diagnosed operators to find the best option from the family of decisions.

To show the supremacy and feasibility of the diagnosed operators with the help of sensitive analysis and geometrical representations.

The major consequence of this organization is the shape: In section Preliminaries, we highlighted various principles of CIFSSs and their feasible and dominant algebraic laws. Additionally, we also recall the basic idea of PAOs and the generalized t-norm (TN) and t-conorm (TCN). In section “Complex picture fuzzy soft settings”, we diagnosed the mathematical concept of CPFSPA, CPFSWPA, CPFSOWPA, CPFSPG, CPFSWPG, and CPFSOWPG. Moreover, the major results and their particular investigation of the invented approaches are also deliberated. In section “Power aggregation operators under CPSF informationPower aggregation operators under CPSF information”, the major results and their particular investigation of the invented approaches are also deliberated. Additionally, in the consideration of diagnosed operators using CPFS information, we illustrated a MADM tool to find the best option from the family of decisions. Finally, we have shown the supremacy and feasibility of the diagnosed operators with the help of sensitive analysis and geometrical representations. Section “Conclusion” contains various concluding remarks.

Preliminaries

The fundamental theme of this section is to highlight various principles of CIFSSs and their feasible and dominant algebraic laws. Additionally, we also recall the basic idea of PAOs and the generalized TN and TCN. Where the mathematical term , proven universal sets and the truth, abstinence, and facility grades are demonstrated in the shape: and .

Definition 1

[27]In the presence of the universal set , the CIFSS is organized in the structure:

With and . Further, , represented the neutral grade. The mathematical organization , represented the CIFSNs.

Definition 2

[27] Considered any two , then

Definition 3

[27]Considered any two , the score value (SV) and accuracy value (AV) are simplified by:

Considered any two and , then

When then ;

When then ;

When then.

When then ;

When then ;

When then .

Definition 4

[32]Considered any collection of attributes , the PAO is simplified by:

The mathematical term , stated the support for and , particularized by: where , acknowledged the measure for and , with a technique that:

;

;

If then .

Definition 5

[15]A mapping is particularized TN when , justified the boundary, monotonicity, commutative, and associativity techniques. Where , diagnosed the TN and the general shape of Archimedean TN and TCN is diagnosed by: and , based on the continuous increasing (or decreasing) function with and .

Complex picture fuzzy soft settings

The major theme of this analysis is to suggest a new theory in the form of CPFS information and invented their major algebraic laws, score value, and accuracy values. The mathematical form of the CPFS set includes three main functions, called supporting, abstinence, and supporting against terms with a prominent characteristic that is the sum of the triplet will lie in the unit interval.

Definition 6

In the presence of the universal set , the CPFSS is organized in the structure:

With and . Further, , represented the neutral grade. The mathematical organization , represented the CPFSNs.

Definition 7

Considered any two , then

Definition 8

Considered any two , the score value (SV) and accuracy value (AV) are simplified by:

Considered any two and , then

When then ;

When then ;

When then.

When then ;

When then ;

When then .

Further, we try to construct the general shape of certain algebraic laws based on CPFSSs.

Definition 9

Considered any two with , then

Theorem 1

Considered any two with , then

;

;

;

;

;

.

Proof

We prove Eqs. (3) and (5), because Parts (1), (2), (4), and (6) are straightforward. Assume that , then

Hence, . Assume that , then

Hence, .

Power aggregation operators under CPSF information

The major theme of this analysis is to investigate the consideration of power aggregation operator using generalized t-norm and t-conorm and CPFS information, we diagnosed the mathematical concept of CPFSPA, CPFSWPA, CPFSOWPA, CPFSPG, CPFSWPG, CPFSOWPG. Moreover, the major results and their particular investigation of the invented approaches are also deliberated. Various important techniques called averaging, Einstein, and Hamacher operators are investigated using , and . Throughout this manuscript, the term , represented the CPFSNs with .

Definition 10

When , then the CPFSPA operator is simplified by: by where , and , and , simplified the support for and .

Theorem 2

Using the information in Eq. (21), we get

Assume , then using the information in Eq. (22), we get

Similarly, assume that , then we get

Hence Eq. (22) is held for . Additionally, we suppose that Eq. (22) is also held for and , then for , we have

Hence Eq. (22) is held for , therefore it is also true for all positive integer .

Property 1 (Idempotency)

When , then

When , then based on Eq. (22), we get

Property 2

When and be any two CPFSNs, then

When and be any two CPFSNs, then

Property 3

Prove that

Property 4

We know that , thenthen,

Assume that and , then

Assume , then then,

Important cases of the invented work using the information in Eq. (22) are described here.

Assume in Eq. (22), then

Stated the CPFS Archimedean weighted averaging (CPFSAWA) operator.

Assume in Eq. (22), then

Stated the CPFS Einstein weighted averaging (CPFSEWA) operator.

When in Eq. (22), then

Stated the CPFS Hamacher weighted averaging (CPFSHWA) operator.

Definition 11

When , the CPFSWPA operator is simplified by: by where , and , and , simplified the support for and , where and , expressed the weight vector with and .

Theorem 3

Considering the information in Eq. (30), we get

Omitted.

Definition 12

When , then CPFSOWPA operator is simplified by: by where , and , and , simplified the support for and , where and , expressed the weight vector with and with and .

Theorem 4

Considering the information in Eq. (32), we get

Omitted.

Definition 13

When , the CPFSPG operator is simplified by: by where , and , and , simplified the support for and .

Theorem 5

Considering the information Eq. (34), we determine

Omitted.

Property 5

When , then

Omitted.

Property 6

When and be any two CIFSNs, then

Omitted.

Property 7

Prove that

Omitted.

Property 8

When , then

Omitted.

Important cases of the invented work using the information in Eq. (35) are described here.

Assume in Eq. (35), then

Stated the CPFS Archimedean weighted geometric (CPFSAWG) operator.

Assume in Eq. (35), then

Stated the CPFS Einstein weighted geometric (CPFSEWG) operator.

Assume in Eq. (35), then

Stated the CPFS Hamacher weighted geometric (CPFSHWG) operator.

Definition 14

When , the CPFSWPG operator is simplified by: by where , and , and , stated the support for and , where and , expressed the weight vector with and .

Theorem 6

Considering the information in Eq. (33), we get

Omitted.

When , the CPFSOWPG operator is simplified by: by where , and , and , simplified the support for and , where and , expressed the weight vector with and with and .

Theorem 7

Considering the information in Eq. (35), we get

Omitted.

Application (“MADM Processes using Proposed Operators”)

In the consideration of diagnosed operators using CPFS information, we illustrated a MADM tool to find the best option from the family of decisions.

Strategic decision-making processes

For managing awkward and problematic information that occurred in genuine life dilemmas, the MADM tool plays an important role in the circumstance of FS theory. Here, we have discussed a procedure for resolving the above issues, we choose alternatives and attributes whose representations are described: and , with weight vector and , provided for attributes and parameters. Then, we give a CPFSNs , with and for each alternative in the shape of a matrix. To handle the above scenario, we described various stages for resolving the above issues.

Stage 1: Deliberated information in the form of a closed matrix that contains CPFSNs for each alternative and their attributes, the mathematical term , stated the matrix.

Stage 2: Summarized the , stated the support for the intellectuals, we have where .

Stage 3: Considering the weighted averaging/geometric operators, we initiate , stated the support for the intellectuals, we have where .

Stage 4: Considering the CPFSPA and CPFSPG operators, we deliberated the values from the matrices into exact values in the availability of TN and TCN .

Stage 5: Averaged the SV of the deliberated values.

Stage 6: Ranking all results and diagnosed the best optimal.

Illustrated example

The existing shortage of energy in various constructing countries over a decade is causing the economic growth of the country. Between the distinct energy supplies, electrical energy is the main energy supplier which is valuable in the market by the distinct area of the economy. For various years, Pakistan is suffering from various electricity issues. The intensification of the electric shortage has become a main political dilemma in Pakistan which is causing the strong construction of the economy. The problem of electricity in Pakistan does not only affect the people and the economy but also affects many other dilemmas, because of some brain-dead employees and unsuccessful planning and policy. Therefore, the management of the government of Pakistan must give special attention to these issues and try to increase the quality of the electricity demand. For this, we describe the various new source of energy that will be increased the level of electricity in the future. Here, we suggest some causes of generating energy in the form of alternatives, such that : Hydropower, Solar Energy; Fuel; and Coal. Assume four experts in form of attributes: : Environmental, Economic; Technical and : Social-Political with parameters, described in the form: : Client Facilities, : Bandwidth, : Package, : Total Cost, and Internet Speed. Here, we choose the values and , stated the weighty vector for . To handle the above scenario, we described various stages for resolving the above issues.

Stage 1: Deliberated information in the form of a closed matrix that contains CPFSNs for each alternative and their attributes in the form of Tables 1, 2, 3, and 4, the mathematical term , stated the matrix.

Table 1

Information is given for Scholars 1.

	e_1	e_2	e_3	e_4	e_5
L_1	0.4e^i2π0.3,0.3e^i2π0.2,0.1e^i2π0.1	0.41e^i2π0.31,0.31e^i2π0.21,0.11e^i2π0.11	0.42e^i2π0.32,0.32e^i2π0.22,0.12e^i2π0.12	0.43e^i2π0.33,0.33e^i2π0.23,0.13e^i2π0.13	0.44e^i2π0.34,0.34e^i2π0.24,0.14e^i2π0.14
L_2	0.3e^i2π0.2,0.3e^i2π0.1,0.1e^i2π0.3	0.31e^i2π0.21,0.31e^i2π0.11,0.11e^i2π0.31	0.32e^i2π0.22,0.32e^i2π0.12,0.12e^i2π0.32	0.33e^i2π0.23,0.33e^i2π0.13,0.13e^i2π0.33	0.34e^i2π0.24,0.34e^i2π0.14,0.14e^i2π0.34
L_3	0.3e^i2π0.3,0.1e^i2π0.1,0.2e^i2π0.3	0.31e^i2π0.31,0.11e^i2π0.11,0.21e^i2π0.31	0.32e^i2π0.32,0.12e^i2π0.12,0.22e^i2π0.32	0.33e^i2π0.33,0.13e^i2π0.13,0.23e^i2π0.33	0.34e^i2π0.34,0.14e^i2π0.14,0.24e^i2π0.34
L_4	0.5e^i2π0.2,0.1e^i2π0.4,0.2e^i2π0.1	0.51e^i2π0.21,0.11e^i2π0.41,0.21e^i2π0.11	0.52e^i2π0.22,0.12e^i2π0.42,0.22e^i2π0.12	0.53e^i2π0.23,0.13e^i2π0.43,0.23e^i2π0.13	0.54e^i2π0.24,0.14e^i2π0.44,0.24e^i2π0.14
L_5	0.6e^i2π0.2,0.1e^i2π0.1,0.1e^i2π0.1	0.61e^i2π0.21,0.11e^i2π0.11,0.11e^i2π0.11	0.62e^i2π0.22,0.12e^i2π0.12,0.12e^i2π0.12	0.63e^i2π0.23,0.13e^i2π0.13,0.13e^i2π0.13	0.64e^i2π0.24,0.14e^i2π0.14,0.14e^i2π0.14

Table 2

Information is given for Scholars 2.

	e_1	e_2	e_3	e_4	e_5
L_1	0.5e^i2π0.2,0.1e^i2π0.4,0.2e^i2π0.1	0.51e^i2π0.21,0.11e^i2π0.41,0.21e^i2π0.11	0.52e^i2π0.22,0.12e^i2π0.42,0.22e^i2π0.12	0.53e^i2π0.23,0.13e^i2π0.43,0.23e^i2π0.13	0.54e^i2π0.24,0.14e^i2π0.44,0.24e^i2π0.14
L_2	0.6e^i2π0.2,0.1e^i2π0.1,0.1e^i2π0.1	0.61e^i2π0.21,0.11e^i2π0.11,0.11e^i2π0.11	0.62e^i2π0.22,0.12e^i2π0.12,0.12e^i2π0.12	0.63e^i2π0.23,0.13e^i2π0.13,0.13e^i2π0.13	0.64e^i2π0.24,0.14e^i2π0.14,0.14e^i2π0.14
L_3	0.4e^i2π0.3,0.3e^i2π0.2,0.1e^i2π0.1	0.41e^i2π0.31,0.31e^i2π0.21,0.11e^i2π0.11	0.42e^i2π0.32,0.32e^i2π0.22,0.12e^i2π0.12	0.43e^i2π0.33,0.33e^i2π0.23,0.13e^i2π0.13	0.44e^i2π0.34,0.34e^i2π0.24,0.14e^i2π0.14
L_4	0.3e^i2π0.2,0.3e^i2π0.1,0.1e^i2π0.3	0.31e^i2π0.21,0.31e^i2π0.11,0.11e^i2π0.31	0.32e^i2π0.22,0.32e^i2π0.12,0.12e^i2π0.32	0.33e^i2π0.23,0.33e^i2π0.13,0.13e^i2π0.33	0.34e^i2π0.24,0.34e^i2π0.14,0.14e^i2π0.34
L_5	0.3e^i2π0.3,0.1e^i2π0.1,0.2e^i2π0.3	0.31e^i2π0.31,0.11e^i2π0.11,0.21e^i2π0.31	0.32e^i2π0.32,0.12e^i2π0.12,0.22e^i2π0.32	0.33e^i2π0.33,0.13e^i2π0.13,0.23e^i2π0.33	0.34e^i2π0.34,0.14e^i2π0.14,0.24e^i2π0.34

Table 3

Information is given for Scholars 3.

	e_1	e_2	e_3	e_4	e_5
L_1	0.3e^i2π0.2,0.3e^i2π0.1,0.1e^i2π0.3	0.31e^i2π0.21,0.31e^i2π0.11,0.11e^i2π0.31	0.32e^i2π0.22,0.32e^i2π0.12,0.12e^i2π0.32	0.33e^i2π0.23,0.33e^i2π0.13,0.13e^i2π0.33	0.34e^i2π0.24,0.34e^i2π0.14,0.14e^i2π0.34
L_2	0.5e^i2π0.2,0.1e^i2π0.4,0.2e^i2π0.1	0.51e^i2π0.21,0.11e^i2π0.41,0.21e^i2π0.11	0.52e^i2π0.22,0.12e^i2π0.42,0.22e^i2π0.12	0.53e^i2π0.23,0.13e^i2π0.43,0.23e^i2π0.13	0.54e^i2π0.24,0.14e^i2π0.44,0.24e^i2π0.14
L_3	0.6e^i2π0.2,0.1e^i2π0.1,0.1e^i2π0.1	0.61e^i2π0.21,0.11e^i2π0.11,0.11e^i2π0.11	0.62e^i2π0.22,0.12e^i2π0.12,0.12e^i2π0.12	0.63e^i2π0.23,0.13e^i2π0.13,0.13e^i2π0.13	0.64e^i2π0.24,0.14e^i2π0.14,0.14e^i2π0.14
L_4	0.4e^i2π0.3,0.3e^i2π0.2,0.1e^i2π0.1	0.41e^i2π0.31,0.31e^i2π0.21,0.11e^i2π0.11	0.42e^i2π0.32,0.32e^i2π0.22,0.12e^i2π0.12	0.43e^i2π0.33,0.33e^i2π0.23,0.13e^i2π0.13	0.44e^i2π0.34,0.34e^i2π0.24,0.14e^i2π0.14
L_5	0.3e^i2π0.2,0.3e^i2π0.1,0.1e^i2π0.3	0.31e^i2π0.21,0.31e^i2π0.11,0.11e^i2π0.31	0.32e^i2π0.22,0.32e^i2π0.12,0.12e^i2π0.32	0.33e^i2π0.23,0.33e^i2π0.13,0.13e^i2π0.33	0.34e^i2π0.24,0.34e^i2π0.14,0.14e^i2π0.34

Table 4

Information is given for Scholars 4.

	e_1	e_2	e_3	e_4	e_5
L_1	0.6e^i2π0.2,0.1e^i2π0.1,0.1e^i2π0.1	0.61e^i2π0.21,0.11e^i2π0.11,0.11e^i2π0.11	0.62e^i2π0.22,0.12e^i2π0.12,0.12e^i2π0.12	0.63e^i2π0.23,0.13e^i2π0.13,0.13e^i2π0.13	0.64e^i2π0.24,0.14e^i2π0.14,0.14e^i2π0.14
L_2	0.4e^i2π0.3,0.3e^i2π0.2,0.1e^i2π0.1	0.41e^i2π0.31,0.31e^i2π0.21,0.11e^i2π0.11	0.42e^i2π0.32,0.32e^i2π0.22,0.12e^i2π0.12	0.43e^i2π0.33,0.33e^i2π0.23,0.13e^i2π0.13	0.44e^i2π0.34,0.34e^i2π0.24,0.14e^i2π0.14
L_3	0.3e^i2π0.2,0.3e^i2π0.1,0.1e^i2π0.3	0.31e^i2π0.21,0.31e^i2π0.11,0.11e^i2π0.31	0.32e^i2π0.22,0.32e^i2π0.12,0.12e^i2π0.32	0.33e^i2π0.23,0.33e^i2π0.13,0.13e^i2π0.33	0.34e^i2π0.24,0.34e^i2π0.14,0.14e^i2π0.34
L_4	0.5e^i2π0.2,0.1e^i2π0.4,0.2e^i2π0.1	0.51e^i2π0.21,0.11e^i2π0.41,0.21e^i2π0.11	0.52e^i2π0.22,0.12e^i2π0.42,0.22e^i2π0.12	0.53e^i2π0.23,0.13e^i2π0.43,0.23e^i2π0.13	0.54e^i2π0.24,0.14e^i2π0.44,0.24e^i2π0.14
L_5	0.6e^i2π0.2,0.1e^i2π0.1,0.1e^i2π0.1	0.61e^i2π0.21,0.11e^i2π0.11,0.11e^i2π0.11	0.62e^i2π0.22,0.12e^i2π0.12,0.12e^i2π0.12	0.63e^i2π0.23,0.13e^i2π0.13,0.13e^i2π0.13	0.64e^i2π0.24,0.14e^i2π0.14,0.14e^i2π0.14

Stage 2: Summarized the , stated the support for the intellectuals, we have

Stage 3: Considering the weighted averaging/geometric operators, we initiate , stated the support for the intellectuals, we have

Stage 4: Considering the CPFSPA and CPFSPG operators, we deliberated the values from the matrices into exact values in the availability of TN and TCN , we have

Stage 5: Averaged the SV of the deliberated values, such that

Stage 6: Ranking all results and diagnosed the best optimal, such that

Hence, we obtained two different sorts of ranking results in the shape of and , using CPFSPA and CPFSPG operators. To further enhance the standard of the invented approaches, we discuss the sensitive analysis of diagnosed work with various suggested approaches.

Sensitive analysis

The main idea of this analysis is to prove the invented work is more beneficial and realistic than the existing operators with the help of some comparison. Many scholars have diagnosed this procedure as demonstrating the proposed approaches are more utilized than the existing operators. For this, we consider various prevailing theories are tried to compare them with our diagnosed operators. The information related to existing theories is described in Refs.[32,40-44].

Information given in Ref.[40] contained the power aggregation operators for CIFSs, the invested power aggregation operators based on CPFS information are more able to resolve intuitionistic, picture, complex intuitionistic, and complex picture fuzzy information. But one deficiency that occurred in the existing operator is that it contained two grades in the shape of truth and falsity in the form of polar coordinates, which is not suitable because the proposed work contained information in the shape of truth, abstinence, falsity grades with parameters. Therefore, the prevailing operators in Ref.[40] failed.

Information given in Ref.[32] contained the power aggregation operators for CPFSs, the invested power aggregation operators based on CPFS information are more able to resolve intuitionistic, picture, complex intuitionistic, and complex picture fuzzy information. But one deficiency that occurred in the existing operator is that it contained three grades in the shape of truth, abstinence, and falsity in the form of polar coordinates, which is not suitable because the proposed work contained information in the shape of truth, abstinence, falsity grades with parameters. Therefore, the prevailing operators in Ref.[32] failed.

Information given in Ref.[41] contained the aggregation operators for PFSSs, the invested power aggregation operators based on CPFS information are more able to resolve intuitionistic, picture, complex intuitionistic, and complex picture fuzzy information. But one deficiency that occurred in the existing operator is that it contained three grades in the shape of truth, abstinence, and falsity with parameters in the form of one dimension, which is not suitable because the proposed work contained information in the shape of truth, abstinence, falsity grades with parameters in the shape of polar coordinates. Therefore, the prevailing operators in Ref.[41] failed. The comparative analysis is described in Table 5.

Table 5

Represented the sensitivity analysis.

Methods	Score values	Ranking values
Rani and Garg[40]	Notjustified	Notjustified
Liu et al.[32]	Notjustified	Notjustified
Khan et al.[41]	Notjustified	Notjustified
Garg and Arora[42]	Notjustified	Notjustified
Zulqarnain et al.[43]	Notjustified	Notjustified
Wang et al.[44]	Notjustified	Notjustified
CIFSPA operator	L_1=0.0522,L_2=0.027,L_3=0.0024, L_4=0.0217,L_5=0.0454	L_1≥L_5≥L_2≥L_4≥L_3
CIFSPG operator	L_1=0.0247,L_2=0.0437,L_3=0.0628, L_4=0.0819,L_5=0.1011	L_5≥L_4≥L_3≥L_2≥L_1

The theory diagnosed by Garg and Arora[42], is called generalized Maclaurin symmetric information based on IFSSs. Noticed that the theory presented by Garg and Arora[42] based on IFSSs is the special case of the diagnosed CPFS information and because of this reason, the theory of Garg and Arora[42] is not able to use for evaluating information discussed in section “Illustrated example”. Therefore, the theory of diagnosed information is massively powerful, and dominant compared to the prevailing theory discussed by Garg and Arora[42].

Robust aggregation operators diagnosed in Ref.[43] contained the robust aggregation operators for intuitionistic hypersoft sets (IHSS), the invested robust aggregation operators based on IHSS information are more able to resolve intuitionistic information. But one deficiency that occurred in the existing operator is that it contained three grades in the shape of truth and falsity in the form of one-dimension information, which is not suitable because the proposed work contained information in the shape of truth, abstinence, falsity grades with parameters. Therefore, the prevailing operators in Ref.[43] failed.

The theory diagnosed by Wang et al.[44], called Hamy means information based on Pythagorean uncertain linguistic information. Noticed that the theory presented by Wang et al.[44] based on Pythagorean uncertain linguistic information is the special case of the diagnosed CPFS information and because of this reason, the theory of Wang et al.[44] is not able to use for evaluating information discussed in section “Illustrated example”. Therefore, the theory of diagnosed information is massively powerful, and dominant compared to the prevailing theory discussed by Wang et al.[42].

From the above-cited information, we obtained the final result in the shape of and . Therefore, the invented operators based on CPFSS are very effective and dominant as compared to prevailing work[32,40,41].

Conclusion

The major theme of this analysis is diagnosed below:

We invented the new theory in the form of CPFS information and invented their major algebraic laws, score value, and accuracy values. The mathematical form of the CPFS set includes three main functions, called supporting, abstinence, and supporting against terms with a prominent characteristic that is the sum of the triplet will lie in the unit interval.

In the consideration of the power aggregation operator using generalized t-norm and t-conorm and CPFS information, we diagnosed the mathematical concept of CPFSPA, CPFSWPA, CPFSOWPA, CPFSPG, CPFSWPG, CPFSOWPG.

The major results and their particular investigation of the invented approaches are also deliberated with the help of t-norm and t-conorm , and .

In the consideration of diagnosed operators using CPFS information, we illustrated a MADM tool to find the best option from the family of decisions.

Finally, we showed the supremacy and feasibility of the diagnosed operators with the help of sensitive analysis and geometrical representations.

Limitations of the proposed approaches

No doubt, the theory of CPFS information has a lot of benefits, but in some cases, the theory of CPFS information, if someone provided such type of information whose sum is exceeded from the unit interval, then for managing with such sort of situation, we need to propose the theory of complex spherical fuzzy soft sets and complex T-spherical fuzzy soft sets.

Future work

In the upcoming times, we will try to modify the principle of complex q-rung orthopair fuzzy sets[45], complex spherical fuzzy sets[46,47], T-spherical fuzzy sets[48], and decision-making[49-57] to enhance the excellence and capacity of the exploration performs.

Ethics declaration statement

The authors state that this is their original work, and it is neither submitted nor under consideration in any other journal simultaneously.