Innovative Bipolar Fuzzy Sine Trigonometric Aggregation Operators and SIR Method for Medical Tourism Supply Chain

Bipolar fuzzy sets (BFSs) are eective tool for dealing with bipolarity and fuzziness. e sine trigonometric functions having two signicant features, namely, periodicity and symmetry about the origin, are helping in decision analysis and information analysis. Taking the advantage of sine trigonometric functions and signicance of BFSs, innovative sine trigonometric operational laws (STOLs) are proposed. New aggregation operators (AOs) are developed based on proposed operational laws to aggregate bipolar fuzzy information. Certain characteristics of these operators are also discussed, such as boundedness, monotonicity, and idempotency. Moreover, a modied superiority and inferiority ranking (SIR) method is proposed to cope with multicriteria group decision-making (MCGDM) with bipolar fuzzy (BF) information. To exhibit the relevance and feasibility of this methodology, a robust application of best medical tourism supply chain is presented. Finally, a comprehensive comparative and sensitivity analysis is evaluated to validate the eciency of suggested methodology.


Introduction
Multicriteria group decision-making (MCGDM) is a process to seek an optimal alternative and ranking of feasible alternatives by a group of decision-experts under several stages and several criteria. However, this process is desperate with uncertainty due to data imprecision and vague perception. As a result, crisp theory is insu cient for dealing with MCGDM problems. To deal with these matters, Zadeh [1] initiated the conception of fuzzy set (FS) and membership function. Later on, di erent researchers presented di erent extensions of FSs including, intuitionistic fuzzy sets (IFSs) [2], Pythagorean fuzzy sets (PyFSs) [3,4], q-rung orthopair fuzzy sets (q-ROFSs) [5], hesitant fuzzy sets (HFSs) [6], neutrosophic sets (NSs) [7], single-valued NSs [8], picture fuzzy sets (PFSs) [9], and spherical fuzzy sets (SFSs) [10][11][12]. e fuzzy models are extremely useful in dealing with uncertain MCGDM problems, and they have been widely used by decision makers. Nevertheless, they all have one aw in common: they can only deal with one property and its not-property at a time. ey are unable to cope with any property's counter property. It is quite common in decision analysis to have to consider both the positive and negative aspects of a speci c object. Some well-known contradictory features in decision analysis include e ects and side e ects, pro t and loss, health and sickness, and so on. Zhang [13,14] propounded the abstraction of bipolar fuzzy sets (BFSs) which deal with both a property and its counter property. Lee [15] studied operations on bipolar-valued fuzzy sets. Tehrim and Riaz [16] introduced connection numbers of SPA theory for the decision support system by using the IVBF linguistic VIKOR method. Jana and Pal [17] proposed the BF-EDAS method for MCGDM problems. Liu et al. [18] suggested an integrated bipolar fuzzy SWARA-MABAC technique and utilized it for the safety risk and occupational health diagnosis. Jana et al. [19] introduced BF-Dombi AOs and Wei et al. [20] developed bipolar fuzzy Hamacher AOs.
Han et al. [21] proposed the TOPSIS method for YinYang bipolar fuzzy cognitive TOPSIS. Wei et al. [22] established MADM with IVBF information. Hamid et al. [23] initiated weighted aggregation operators for q-rung orthopair m-polar fuzzy set. Akram et al. [24] proposed the notion of complex fermatean fuzzy N-soft sets. AOs are crucial in information aggregation and are subject to a variety of operational laws. Based on algebraic operational laws, Xu [25] and Xu and Yager [26] propounded weighted averaging and geometric AOs for IFSs. Garg [27] introduced interactive operators for IFSs. Huang [28] proposed intuitionistic fuzzy Hamacher aggregation operators. Gou and Xu [29] suggested exponential operational laws (EOLs) for IFSs.
Li and Wei [30] proposed logarithmic operational laws (LOLs) for IFSs. Peng et al. [31] proposed EOLs for q-ROFSs. Similarly, the LOLs for PFSs [32] are also defined. Aside from the exponential and logarithmic functions, sine trigonometric function is another suitable choice for information fusion. e two main characteristics are periodicity and symmetry about the origin which aid in meeting the decision makers' expectations during object evaluation. Abdullah et al. [33] developed STOLs for PFSs. Kabani [34] studied Pakistan as a medical tourism destination. Muzaffar and Hussain [35] investigated medical tourism to discuss the challenge: are we ready to take the challenge. Zhang and Xu [36] proposed TOPSIS for PFSs and PFNs with MCDM.
Mahmood et al. [37] proposed an innovative MCDM method with spherical fuzzy soft rough (SFSR) average aggregation operators. Ihsan et al. [38] presented the MADM support model based on bijective hypersoft expert set. Karaaslan and Karamaz [39] introduced an innovative decision-making approach with HFPHFS. Alcantud [40] introduced the novel concepts of soft topologies and fuzzy soft topologies and investigated their relationships. Liu et al. [41] introduced the idea of mining temporal association rules based on temporal soft sets. Riaz et al. [42] introduced a novel TOPSIS approach based on cosine similarity measures and CBF-information. Zararsiz and Riaz [43] introduced the notion of bipolar fuzzy metric spaces with application. Riaz et al. [44] proposed distance and similarity measures for bipolar fuzzy soft sets with application to pharmaceutical logistics and supply chain management.
In 2021, Gergin et al. [45] modified the TOPSIS method to deal the supplier selection for automotive industry. Karamasa et al. [46] introduced the weighting factors which affect the logistics out-sourcing decision-making problem. Ali et al. [47] introduced Einstein geometric AO to deal complex IVPFS, and its novel principles and its operational laws are defined. Muhammad et al. [48] and Biswas et al. [49] propounded multicriteria decision-making techniques to deal real world problems. Milovanovic et al. [50] developed uncertainty modeling using intuitionistic fuzzy numbers.
In 2021, Garg [51] introduced some robust STOLs, its operational laws for PFSs, and AOs and algorithms to interpret MCDM. In 2021, Mahmood et al. [52] interpreted BCFHWA, BCFHOWA, BCFHHA, BCFHWG, BCFHOWG, and BCFHHG operators. Palanikumar et al. [53] proposed some new methods to solve MCDM based on PNSNIVS. A notion of PNSNIVWA, PNSNIVWG, GPNSNIVWA, and GPNSNIVWG is also discussed in the article. In 2021, Jana et al. [54]  is technique employs superiority and inferiority information to represent decision makers' behavior toward each criterion and to determine the degrees of domination and subordination of each alternative, from which superiority and inferiority flows are derived. It was introduced by Xu [58]. Chai and Liu [59] proposed the IF-SIR method to deal with MCGDM problems. Peng and Yang [60] extended the SIR technique to pythagorean fuzzy data. Zhu et al. [61] proposed the SIR approach for q-ROFSs.
Keeping in mind the importance of sine trigonometric function and SIR method, the aims and perks of this manuscript are as follows: (1) To address bipolarity and uncertainty, innovative sine trigonometric operational laws (STOLs) are proposed for bipolar fuzzy sets (BFSs). e layout of the remaining manuscript is as follows. In Section 2, some fundamental concepts about BFSs are reviewed. In Section 3, we define STOLs for BFSs and discuss their properties. In Sections 4 and 5, we introduce novel AOs based on BF-STOLs and explore their characteristics. Section 6 provides an extended version of the SIR technique for dealing with MCGDM problems using bipolar fuzzy data. A numerical illustration and a comparative analysis are also proffered to validate the efficaciousness of the propounded technique. Finally, in Section 7, there are some closing remarks.

Preliminaries
is section includes some rudimentary abstractions related to BFSs. roughout this manuscript, we consider Y as universe of discourse.
Definition 1 (see [13]). A BFS B on Y can be described as
〉 be two BFNs and σ ∈ (0, ∞), then operational laws between them can be defined as Definition 3 (see [20]). For a BFN B � 〈ℵ + B , ℵ − B 〉, score and accuracy functions can be expressed as e values of score and accuracy functions are used to compare two BFNs. For two BFNs B 1 and B 2 , Definition 4 (see [21]). If B 1 and B 2 are two BFSs on Y � y 1 , y 2 , . . . , y n , then the normalized Hamming distance between them is calculated as

Sine Trigonometric Operational Laws for BFSs
In this section, we suggest sine trigonometric operational laws (STOLs) for BFNs and investigate some useful results.
, 0] serve as positive and negative membership degrees, respectively, for every element y ∈ Y . e set sin B is called sine trigonometric-BFS (ST-BFS).
is called ST-BFN.
is called complement of sin B.
Proof. We substantiate (i) and (iv), and others can be substantiated similarly.

Mathematical Problems in Engineering
Now, Proof. Since sine is an increasing function on the interval

Mathematical Problems in Engineering
Proof. To prove the theorem, we employ mathematical induction on n. For n � 2, we have is shows that our assertion is correct for n � 2. Assume that the result holds true for n � k, i.e., Now, for n � k + 1, we have Now, . . , n, be another collection of BFNs such that B i ≺ B * i , ∀i � 1, 2, . . . , n, then

Mathematical Problems in Engineering
Proof (i) Let B i � B ∀i � 1, 2, . . . , n. en, by using (13), we have Due to the monotonicity of sine function, we get Similarly, Since (iii) It is similar to the preceding proof, so we exclude it.

by utilizing ST-BFOWA operator is still a BFN and is given by
Proof. Straightforward.

by utilizing ST-BFHWA operator is still a BFN and is given by
Proof. Straightforward.

Bipolar Fuzzy Sine Trigonometric Geometric Aggregation Operators
In this section, we propose geometric aggregation operators including (i) ST-BFWG operator, (ii) ST-BFOWG operator, and (iii) ST-BFHWG operator.

ST-BFWG Operator
Definition 12. For n BFNs B i , a ST-BFWG operator is explicated as

then their cumulative value obtained by utilizing ST-BFWG operator is expressed as
Mathematical Problems in Engineering 9 Now, e properties mentioned in eorem 5, namely, idempotency, monotonicity, and boundedness, also apply to the ST-BFWG operator.

by utilizing ST-BFOWG operator is expressed as
Proof. It is obvious.
Idempotency, monotonicity, and boundedness are all satisfied by the ST-BFOWG operator.  � (c 1 , c 2 , . . . , c n ) with c i > 0 and n i�1 c i � 1 can be described as where

by utilizing ST-BFHWG operator is expressed as
Proof. Straightforward.

Bipolar Fuzzy SIR Method
An MCGDM problem is made up of a finite number of alternatives, a set of criteria, and a set of decision makers. To solve an MCGDM problem, the most apposite alternative must be chosen among those available. Let  A � a 1 , a 2 , . . . , a m be a set of alternatives and C � c 1 , c 2 , . . . , c n be a set of criteria. Suppose that the set of decision makers is E � e 1 , e 2 , . . . , e l and their weight vector is ϑ � ϑ 1 , ϑ 2 , . . . , ϑ l where all the weights are BFNs. Construct the individual decision matrices H k � (h k ij ) m×n in which h k ij denotes the evaluation information of the alternative a i w.r.t the criterion c j provided by the decision maker e k in the form of BFNs. Assume that φ � (φ k j ) l×n is the criterion weight matrix in which φ k j is the weight of criterion c j assigned by the decision maker e k in the form of BFNs. In this section, we set up the BF-SIR technique to address this MCGDM problem. e steps in this technique are outlined as follows: Step 1. Compute the relative propinquity coefficient of each ϑ k , k � 1, 2, . . . , l, by the equation Step 2. Normalize δ k , k � 1, 2, . . . , l, by the equation In this way, we get a normalized vector ζ � ζ 1 , ζ 2 , . . . , ζ l of relative propinquity coefficients.
Step 3. Acquire the accumulated bipolar fuzzy decision matrix and the criterion weight vector by utilizing ST-BFWA operator as follows: Step 4. Obtain the relative efficiency function f ij as follows: where h � 〈min

Mathematical Problems in Engineering
Step 5. Compute the preference intensity PI j (a i , a t ) (i, t � 1, 2, . . . , m, i ≠ t) which provides the degree of preference of alternative a i over alternative a t w.r.t the criterion c j and it can be defined as follows: where λ j is a threshold function given by Step 6. Construct the superiority matrix S � (S ij ) m×n and inferiority matrix I � (I ij ) m×n by utilizing the following equations: Step 7. Calculate the superiority flow (S-flow) and inferiority flow (I-flow) as follows: Step 8. Compute the score functions of λ > (a i ) and λ < (a i ), i � 1, 2, . . . , m, by using (2).

Case Study.
e process of seeking medical treatment supply chain from a foreign country is known as medical tourism. In the past, patients from underdeveloped parts of the world used to travel to Europe and America for diagnostics and treatment. However, in recent years, this trend has flipped as medical tourism, in which individuals from developed countries travel to developing countries for medical treatment.
ere are a variety of reasons why people from developed countries prefer less developed countries. e low cost of treatment is the main factor. Healthcare prices are dependent on a country's per capita gross domestic product (GDP), which serves as a procurator for income levels. e low cost of offshore medical care is indebted to low medicolegal and administrative costs. Second, people seek medical guidance from outside the country for the procedures for which health insurance is not provided, such as cosmetic surgery, fertility therapy, dental reconstruction, gender reassignment surgeries, and so on. Patients in countries where access to healthcare is regulated by the government, such as Canada and the United Kingdom, desire to avoid the delays that come with extensive waiting lists. Another factor could be the lack of availability of a certain operation in their home country, such as stem cell therapy, which may be inaccessible or limited in developed countries but widely available in emerging markets. Some patients believe that their privacy will be better protected in a remote location. Another motive for offshore treatment is the recreational aspect. As a result of these factors, medical tourism is expanding globally. Medical tourism was worth 54.4 billion US dollars in 2020, and by 2027, it was expected to be worth more than 200 billion US dollars (https://www.statista.com/ statistics/1084720/medical-tourism-market-size-worldwide/). Figure 1 depicts the gradual expansion of the medical tourism industry from 2016 to 2020, with projections for 2027. e medical tourism market in Asia-Pacific has a lot of room for expansion. Due to economic development, this region is expected to see rapid market expansion. Singapore, Japan, India, ailand, and the Philippines are among the most popular medical tourism destinations. Singapore and India are well-known for their cardiac and orthopaedic surgery. ailand is well-known for its dental procedures and gender reassignment surgeries. Japan has one of the best oncology treatment facilities in the world. e Philippines is famous for its cosmetic surgery, dentistry, and fertility treatment. e Medical Tourism Index (MTI) evaluates a country's suitability as a medical tourism destination by taking into account its overall environment, healthcare costs, tourist attractions, and the standard of medical amenities and services. e higher the MTI, the better the destination. Medical tourism is seen as an unexplored sector in Pakistan that might be transformed into a lucrative potential if the government addresses some critical issues such as security, brain drain, and service quality. According to Pakistani medical professionals, Pakistan has "huge potential" to become a competitive medical tourism hub in Asia. In what follows, we will use the BF-SIR method to determine the best medical tourism destination in Pakistan.

Numerical Illustration.
Suppose that ministry of health of a Pakistan needs to assess some true potential of medical tourism supply chain. For this purpose, the ministry hires three decision makers e 1 , e 2 , and e 3 and assigns them weights which are given in Table 1. Let A � a 1 , a 2 , a 3 , a 4 be the set of alternatives where a 1 � Islamabad, a 2 � Karachi, a 3 � Lahore, and a 4 � Peshawar. Table 2 lists the criteria for determining the best alternative. e weights of criteria c j given by the decision makers e k are given in Table 3. e decision makers evaluate each alternative a i w.r.t each criterion c j and give their assessment via BFNs. ree decision matrices are given in Tables 4-6.
e relative propinquity coefficients δ k (k � 1, 2, 3) are computed using (42) as follows: Step 2. e normalized vector is obtained using (43)as follows: Step 3. e accumulated bipolar fuzzy decision matrix is acquired using (44), which is given in Table 7. Equation (45) is used to determine accumulated weights of criteria, which are as follows: Step 4. e relative efficiency function is calculated using (46)as follows: Step 5, 6. e superiority and inferiority matrices are constructed using (49) and (50) Step 7, 8. e S-flow and I-flow are computed using (51) and (52), which are given in Table 8.
Step 9. Applying SR-laws to Table 8 yields the following ranking order: Applying IR-laws to Table 8 yields the following ranking order: Step 10. According to both SR and IR-laws, a 3 is the best alternative.    is includes transportation and maintenance of hospitals and equipment.

Comparative and Sensitivity Analysis.
In this section, we compare our suggested BF-SIR technique to some existing approaches in order to evaluate its validity. Table 9 summarizes the comparative study of various techniques. It can be seen from Table 9 that our suggested approach is compatible with the existing techniques.

Conclusion
In daily life, we encounter many situations where we must deal with uncertainty as well as bipolarity when making a decision. Taking this into consideration, the bipolar fuzzy set (BFS) is a sophisticated model that can handle bipolarity and fuzziness at the same time. e main contributions of this manuscript are listed as follows: (1) Since the sine trigonometric function is periodic and symmetric about the origin, it can accommodate the decision expert's choices during object appraisal. erefore, we proposed sine trigonometric operational laws (STOLs) for BFSs. We explored some of their properties as well.
(2) Based on BF-STOLs, we suggested the following averaging AOs: bipolar fuzzy sine trigonometric weighted averaging (BF-STWA) operator; bipolar fuzzy sine trigonometric ordered weighted averaging (BF-STOWA) operator; and bipolar fuzzy sine trigonometric hybrid weighted averaging (BF-STHWA) operator. (3) Based on BF-STOLs, we suggested the following geometric AOs: bipolar fuzzy sine trigonometric weighted geometric (BF-STWG) operator; bipolar fuzzy sine trigonometric ordered weighted geometric (BF-STOWG) operator; and bipolar fuzzy sine trigonometric hybrid weighted geometric (BF-STHWG) operator. (4) We investigated some important characteristics of these operators, such as idempotency, monotonicity, and boundedness. (5) We established an extended superiority and inferiority ranking (SIR) method to handle MCGDM problems in a bipolar fuzzy environment. We applied this technique to the selection of the best medical tourism supply chain. (6) We compared our suggested model with some existing ones to exhibit its validity and efficiency.
In the future, we will develop bipolar fuzzy sine trigonometric power aggregation operators, bipolar fuzzy sine trigonometric Hamy mean operators, bipolar fuzzy sine trigonometric Bonferroni mean operators, bipolar fuzzy sine trigonometric prioritized operators, and bipolar fuzzy sine trigonometric Dombi operators.

Data Availability
No data were used in this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest. Table 9: Comparative analysis of the suggested and existing methodologies.