Bipolar complex fuzzy credibility aggregation operators and their application in decision making problem

: A bipolar complex fuzzy credibility set (BCFCS) is a new approach in computational intelligence and decision-making under uncertainty.


Introduction
In today's culture, scientific and technological advances have resulted in scientific and technological discoveries that have decreased the complications in our daily lives. Yet, despite scientific progress that has made life easier, some concerns, such as decision making (DM), remain complicated. DM, particularly multi-criteria group decision making (MCGDM), has been widely adopted in a variety of sectors where traditional methods have failed in recent years. As in real life, information is usually applications to MADM. Limited number of researchers produced numerous applications, such as the Hamacher aggregation information examined by Mahmood et al. [26]. Also, Wei et al. [32] defined a bipolar fuzzy Hamacher aggregation operators in multiple attribute decision making. Additionally, Mahmood and Rehman [27] proposed the core theory of Dombi operators based on BCFS. Jana et al. [19] proposed bipolar fuzzy Dombi prioritized aggregation operators in MADM.
The main motivation for this analysis is explained below: (1) To define new operations for bipolar complex fuzzy credibility numbers (BCFCNs), and to investigate properties these numbers. A BCFCN is superior that a bipolar complex fuzzy number as it carries more comprehensive and reasonable information. Bipolar complex fuzzy credibility numbers is the extension of bipolar complex fuzzy numbers to deal with two-sided contrasting features, which can describe the information with a bipolar complex fuzzy number and an credibility number simultaneously.
(2) A secondary objective of this paper is to introduce some fundamental operations on BCFCNs, their key properties, and related significant results. Suggested operations are very helpful to strengthen BCFCS theory.
(3) Since aggregation operators for bipolar complex fuzzy credibility numbers (BCFCNs) have not been established so far, motivated by the above discussion, this paper presents novel averaging and geometric aggregation operators under bipolar complex fuzzy credibility information are proposed.
(4) An algorithm for new MCDM technique is developed based on proposed aggregation operators using bipolar complex fuzzy credibility information. Proposed technique is also demonstrated by a numerical illustration.
(5) To demonstrate the validity and capability of the proposed technique, conduct a comparative examination of the developed operators with various current theories.
The framework of this study is as follows: Section 2 includes certain prevalent ideas such as BCFCS, aggregation operator, and their operational laws. In Section 3, we employed the theory of averaging/geometric aggregation operators to diagnose the well-known operators, such as bipolar complex fuzzy credibility weighted average (BCFCWA), bipolar complex fuzzy credibility ordered weighted average (BCFCOWA), bipolar complex fuzzy credibility hybrid average (BCFCHA), bipolar complex fuzzy credibility weighted geometric (BCFCWG), bipolar complex fuzzy credibility ordered weighted geometric (BCFCOWG), and bipolar complex fuzzy credibility hybrid geometric (BCFCHG) operators, as well as analyses their strategic features and related outcomes. Section 4 proposes an algorithm for multiple criteria group decision making utilizing stated operators. Then, a numerical example of a case study of Hospital selection is discussed. In Section 5, we compared the described operators to existing methodologies to demonstrate the validity and capabilities of the proposed approach. Finally, write the study's conclusion.

Bipolar complex fuzzy credibility average operator
Here, we are going to define some aggregation operators like, BCFCWA, BCFCOWA, and BCFCHA operators.

Bipolar complex fuzzy credibility weighted average operator
.., n) be a set of BCFCNs with the weights Φ = (Φ 1 , ..., Φ n ) T , such as n i=1 Φ i = 1 and 0 ≤ Φ i ≤ 1. Then, the BCFCWA operator is obtain as utilizing Definition 3.1, aggregated value for BCFCWA operator is shown in Theorem 3.2.
.., n) be the set of BCFCNs with weights Φ = (Φ 1 , ..., Φ n ) T , such as n =1 Φ = 1 and 0 ≤ Φ ≤ 1. Then, BCFCWA operator is obtained as Proof. To prove this theorem, we used mathematical induction principle. As we know that Let Eq (3.2) is true for n = 2. Then, The result hold for n = 2. Now, let Eq (3.2) is true for n = τ. Then, we get Next, let Eq (3.2) is true for n = τ + 1, which show that Eq (3.2) true for n = τ + 1. Hence, the given result is hold for n ≥ 1. The BCFCWA operator satisfied the following properties.
are the maximum and minimum BCFCNs. Then, Proof. We studied two cases (for real and imagined components) separately for MG and credibility degree.
(1) For membership degree, we have Similarly, we can prove for ib + K , and As n =1 Φ = 1. Then Similarly, we can prove for ib − K i . (2) For credibility degree, we have Similarly, we can prove for id + K . Additionally, As n =1 Φ = 1. Then Similarly, we have for id − K .
Proof. Proof is follow from Theorem 3.2. The BCFCOWA operator satisfied the following properties.
Proof. Proof is follow from Theorem 3.2.

Bipolar complex fuzzy credibility geometric operator
Here, we are going to defined some aggregation operators like as, BCFCWG, BCFCOWG, and BCFCHG operators.
The BCFCWG operator satisfied the following properties.

Proof. Proof follows from Theorem 3.2.
The BCFCOWG operator satisfied the following properties.
The largest permutation value from the set of BCFCNs is represented by K * σ() = K nϑ , and n stands for the balancing coefficient.
Proof. Proof is follow from Theorem 3.2.

An approach for MCGDM based on bipolar complex fuzzy credibility information
In this section, we construct an approach to tackle the MCGDM problem using the proposed bipolar complex intuitionistic fuzzy set. Assume thatÉ = É 1 , ...,É n is the collection of n criteria and ℘ = {℘ 1 , ..., ℘ m } is the set of m alternatives for a MCGDM problem. Let the weights for the criterionÉ as Φ = (Φ 1 , ..., Φ n ) T , such as n =1 Φ and 0 ≤ Φ. The following are the key steps: Step 1: Create a decision matrix using the assessment data collected in accordance with the criteriá E for qualified experts for each alternative ℘; 11 12 . . 1n 21 22 . . . .
Step 2: Evaluate the aggregate information given by experts with the help of BCFCWA operators.
Step 3: Evaluate the aggregate information using BCFCOWA operators.
Step 4: Evaluate the score value of the information obtained with the help of operators.
Step 5: Give alternatives a ranking based on the score value.

Example
As per predictions, chronic diseases (CDs) can cause one-third of all deaths in the world and are one of the main causes of death and disability in the world. There are numerous diseases related with CDs, like diabetes, hypertension heart disease and cardiovascular diseases (CVDs), some of them present higher risks than others (in particular, CVD). It is one of the main causes of disability and presents threats to population vitality. Because it causes diseases like hypertension, arrhythmia, stroke, and heart attacks, as well as deaths, CVD is a global crisis. The diagnosis, monitoring, and treatment of CVD are necessary since it is a life-threatening disease. However, there are also other problems that can make diagnosis and treatment more difficult, like the lack of qualified cardiologists or patients who live in remote areas far from hospitals. Modern technologies, such as the Internet of Things (IoT), are utilized to monitor patients with CD in order to address these issues. IoT has set the stage for a variety of uses, and it has played a remarkable role in telemedicine in the health care field and in managing patients who are located elsewhere.
In this real life example we have to select the best hospital using our proposed work. There are four alternatives (Hospitals) and six criteria with the weights are Φ = (0.20, 0.30, 0.15, 0.25, 0.10) T , which is discussed as follows: (1) Surgical Doctors (SD): In this type of criteria, we have discussed a special groups of doctors, which have the ability to treat all the surgical patient.   Tables 1-3.
Step 1: The total information given by experts for each alternative ℘ i under the criteriaÉ in Tables 1-3.
Step 2: Using the BCFCWA operators and expert-provided data against their weights. Table 1. BCF evaluation information given by expert one. Table 2. BCF evaluation information given by expert two. Table 3. BCF evaluation information given by expert three.
Step 3: Using the BCFCOWA operator and the aggregated value of Step 4: Analyze the score value of the data we collected using various operators.   Step 5: Utilize the ranking values shown in Table 6 to determine which choice is the best.

Comparative study
The comparative study of determined techniques was discussed in this section, along with certain common operators based on accepted concept like as, BCFSs.
The method described in [13,19,[25][26][27] contains bipolar fuzzy set details, but the given model cannot be solved using this method. Reviewing Table 5 reveals that the methods now in use lack basic information and are unable to solve or rank the case that has been provided. Compared to other methods already in use, the strategy suggested in this study is more capable and dependable. The main analysis of the identified and proposed hypotheses is presented in Table 7.

Conclusions
We define a number of operations, the scoring function, and the accuracy function for BCFCS in this paper. We also established several aggregation operators based on BCFC operational laws, such as BCFCWA, BCFCOWA, BCFCHA, BCFCWG, BCFCOWG, and BCFCHG operators. We explored the essential properties of the aforementioned operators' specific situations, such as idempotency, boundedness, and monotonous. Next, utilizing these operators, we solved the bipolar complex fuzzy MCGDM problem. To validate the interpreted techniques, we provided a numerical example of selecting fire extinguishers. Finally, we compared our findings to those of existing operators to establish the usefulness and applicability of our method.
In the future, we will use our proposed operators in different domains, like as, complex Pythagorean fuzzy set, complex picture fuzzy set, complex Spherical fuzzy set, and complex fractional orthotriple fuzzy set.