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

Authors: Muhammad Riaz; Dragan Pamucar; Anam Habib; Nimra Jamil
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Bipolar fuzzy sets (BFSs) are effective tool for dealing with bipolarity and fuzziness. The sine trigonometric functions having two significant features, namely, periodicity and symmetry about the origin, are helping in decision analysis and information analysis. Taking the advantage of sine trigonometric functions and significance 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 modified 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 efficiency of suggested methodology.


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## Body

## 1. 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 insufficient for dealing with MCGDM problems. To deal with these matters, Zadeh [1] initiated the conception of fuzzy set (FS) and membership function. Later on, different researchers presented different 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) [10–12].The 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 flaw in common: they can only deal with one property and its not-property at a time. They 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 specific object. Some well-known contradictory features in decision analysis include effects and side effects, profit 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. The 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] applied IFDHWA and IFDHWG AO to evaluate enterprise financial performance. In 2021, Jana et al. [55] extended Dombi operations towards single-valued trapezoidal neutrosophic numbers (SVTrNNs). They also presented Dombi operation on SVTrNNs, and they proposed some new averaging and geometric averaging operators named as SVTrN Domi weighted averaging (SVTrNDWA) operator, SVTrN Dombi ordered weighted averaging (SVTrNDOWA) operator, SVTrN Dombi hybrid weighted averaging (SVTrNDHWA) operator, SVTrN Dombi weighted geometric (SVTrNDWGA) operator, SVTrN Dombi ordered weighted geometric (SVTrNDOWGA) operator, and SVTrN Dombi hybrid weighted geometric (SVTrNDHWGA) operator. In 2022, Ajay et al. [56] extended the STOLs for NSs and CNSs and defined the operational laws and their functionality. They also defined distance measures and ST-AOs. In 2022, Qiyas et al. [57] defined some reliable STOLs for SFNs and defined ST-OAs to deal real world problems.The superiority and inferiority ranking (SIR) technique is a generalization of the eminent PROMETHEE method. This 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).(2)
Averaging AOs are developed named as sine trigonometric bipolar fuzzy weighted averaging (ST-BFWA) operator, sine trigonometric bipolar fuzzy ordered weighted averaging (ST-BFOWA) operator, and sine trigonometric bipolar fuzzy hybrid weighted averaging (ST-BFHWA) operator.(3)
Geometric AOs are proposed including sine trigonometric bipolar fuzzy weighted geometric (ST-BFWG) operator, sine trigonometric bipolar fuzzy ordered weighted geometric (ST-BFOWG) operator, and sine trigonometric bipolar fuzzy hybrid weighted geometric (ST-BFHWG) operator.(4)
Certain aspects of proposed operators are also discussed, such as idempotency, boundedness, and monotonicity.(5)
A modified SIR method by using features of proposed operators is proposed to cope with MCGDM problems.(6)
A robust application of best medical tourism supply chain is presented by using a modified SIR technique involving sine trigonometric AOs.The layout of the remaining manuscript is as follows. In Section2, 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.


## 2. Preliminaries

This section includes some rudimentary abstractions related to BFSs. Throughout this manuscript, we considerY as universe of discourse.Definition 1.
(see [13]). A BFS B on Y can be described as(1)B=y,ℵB+y,ℵB−y:y∈Y,where ℵB+y∈0,1 denotes positive membership degree and ℵB−y∈−1,0 denotes negative membership degree of an element y∈Y. A bipolar fuzzy number (BFN) can be expressed as B=ℵB+,ℵB−.
In 2015, Gul proposed operational laws of BFNs in his M.Phil Thesis.Definition 2.
[62]. Let B1=ℵB1+,ℵB1− and B2=ℵB2+,ℵB2− be two BFNs and σ∈0,∞, then operational laws between them can be defined as(i)
B1⊕B2=ℵB1++ℵB2+−ℵB1+ℵB2+,−ℵB1−ℵB2−(ii)
B1⊗B2=ℵB1+ℵB2+,−−ℵB1−+−ℵB2−−ℵB1−ℵB2−(iii)
σB1=1−1−ℵB1+σ,−−ℵB1−σ(iv)
B1σ=ℵB1+σ,−1−1−−ℵB1−σ(v)
B1c=1−ℵB1+,−1−ℵB1−(vi)
B1≺¯B2 if ℵB1+≤ℵB2+ and ℵB1−≥ℵB2−(vii)
B1=B2 if B1≺¯B2 and B2≺¯B1Definition 3.
(see [20]). For a BFN B=ℵB+,ℵB−, score and accuracy functions can be expressed as(2)ScrB=1+ℵB++ℵB−2,(3)AcrB=ℵB+−ℵB−2.
The values of score and accuracy functions are used to compare two BFNs. For two BFNsB1 and B2,(i)
IfScrB1<ScrB2, then B1<B2(ii)
IfScrB1>ScrB2, then B1>B2(iii)
IfScrB1=ScrB2, then B1<B2 if AcrB1<AcrB2(iv)
IfScrB1=ScrB2, then B1>B2 if AcrB1>AcrB2(v)
IfScrB1=ScrB2, then B1=B2 if AcrB1=AcrB2Definition 4.
(see [21]). If B1 and B2 are two BFSs on Y=y1,y2,…,yn, then the normalized Hamming distance between them is calculated as(4)dB1,B2=12n∑i=1nℵB1+yi−ℵB2+yi+ℵB1−yi−ℵB2−yi.


## 3. Sine Trigonometric Operational Laws for BFSs

In this section, we suggest sine trigonometric operational laws (STOLs) for BFNs and investigate some useful results.Definition 5.
LetB=y,ℵB+y,ℵB−y:y∈Y be a BFS on Y. A sine trigonometric operator on B can be defined as(5)sinB=y,sinπ2ℵB+y,sinπ21+ℵB−y−1:y∈Y.
Clearly,sinB is again a BFS on Y because sinπ/2ℵB+y∈0,1 and sinπ/21+ℵB−y−1∈−1,0 serve as positive and negative membership degrees, respectively, for every element y∈Y. The set sinB is called sine trigonometric-BFS (ST-BFS).Definition 6.
LetB=ℵB+,ℵB− be a BFN, then(6)sinB=sinπ2ℵB+,sinπ21+ℵB−−1,is called ST-BFN.Definition 7.
For two BFNsB1=ℵB1+,ℵB1− and B2=ℵB2+,ℵB2−, we propose STOLs as follows:(i)
sinB1⊕sinB2=1−1−sinπ/2ℵB1+1−sinπ/2ℵB1+,−sinπ/21+ℵB1−−1sinπ/21+ℵB2−−1(ii)
sinB1⊗sinB2=sinπ/2ℵB1+sinπ/2ℵB1+,−1−sinπ/21+ℵB1−sinπ/21+ℵB2−(iii)
σsinB1=1−1−sinπ/2ℵB1+σ,−−sinπ/21+ℵB1−−1σ; σ>0(iv)
sinB1σ=sinπ/2ℵB1+σ,−1−sinπ/21+ℵB1−σ; σ>0Theorem 1.
LetB1=ℵB1+,ℵB1− and B2=ℵB2+,ℵB2− be two BFNs and σ>0,σ1>0,σ2>0 be three real numbers, then(i)
σsinB1⊕sinB2=σsinB1⊕σsinB2(ii)
sinB1⊗sinB2σ=sinB1σ⊗sinB2σ(iii)
σ1sinB1⊕σ2sinB1=σ1+σ2sinB1(iv)
sinB1σ1⊗sinB1σ2=sinB1σ1+σ2Proof.
We substantiate (i) and (iv), and others can be substantiated similarly.(i)
Forσ>0,(7)σsinB1⊕sinB2=1−1−sinπ2ℵB1+σ1−sinπ2ℵB2+σ,−−sinπ21+ℵB1−−1σ−sinπ21+ℵB2−−1σ=1−1−sinπ2ℵB1+σ,−−sinπ21+ℵB1−−1σ⊕1−1−sinπ2ℵB2+σ,−−sinπ21+ℵB2−−1σ=σsinB1⊕σsinB2.(iv
) Forσ1,σ2>0,(8)sinB1σ1⊗sinB1σ2=sinπ2ℵB1+σ1,−1−sinπ21+ℵB1−σ1⊗sinπ2ℵB1+σ2,−1−sinπ21+ℵB1−σ2=sinπ2ℵB1+σ1sinπ2ℵB1+σ2,−1−sinπ21+ℵB1−σ1sinπ21+ℵB1−σ2=sinπ2ℵB1+σ1+σ2,−1−sinπ21+ℵB1−σ1+σ2=sinB1σ1+σ2.Definition 8.
LetB=ℵB+,ℵB− be a BFN and sinB be the corresponding ST-BFN, then(9)sinBc=1−sinπ2ℵB+,−sinπ21+ℵB−,is called complement of sinB.Theorem 2.
LetB1=ℵB1+,ℵB1− and B2=ℵB2+,ℵB2− be two BFNs and σ>0, then(i
)σsinB1c=sinB1σc(ii)
sinB1cσ=σsinB1c(iii)
sinB1⊕sinB2c=sinB1c⊗sinB2c(iv)
sinB1⊗sinB2c=sinB1c⊕sinB2cProof.
We substantiate (i) and (iv), and others can be substantiated similarly.(i)(10)sinB1c=1−sinπ2ℵB1+,−sinπ21+ℵB1−,σsinB1c=1−sinπ2ℵB1+σ,−−−sinπ21+ℵB1−σ.Now,(11)sinB1σ=sinπ2ℵB1+σ,−1−sinπ21+ℵB1−σ,sinB1σc=1−sinπ2ℵB1+σ,−sinπ21+ℵB1−σ=σsinB1c.(iv)(12)sinB1⊗sinB2c=1−sinπ2ℵB1+sinπ2ℵB2+,−sinπ21+ℵB1−sinπ21+ℵB2−,sinB1c⊕sinB2c=1−sinπ2ℵB1+,−sinπ21+ℵB1−⊕1−sinπ2ℵB2+,−sinπ21+ℵB2−=1−sinπ2ℵB1+sinπ2ℵB2+,−−sinπ21+ℵB1−−sinπ21+ℵB2−=sinB1⊗sinB2c.Theorem 3.
. LetB1 and B2 be two BFNs with B1≺¯B2, i.e., ℵB1+≤ℵB2+ and ℵB1−≥ℵB2−, then sinB1≺¯sinB2.Proof.
Since sine is an increasing function on the interval0,π/2 so for ℵB1+≤ℵB2+, we have sinπ/2ℵB1+≤sinπ/2ℵB2+. Likewise, for ℵB1−≥ℵB2−, we obtain 1+ℵB1−≥1+ℵB2−. This implicates that sinπ/21+ℵB1−≥sinπ/21+ℵB2− which further implicates that sinπ/21+ℵB1−−1≥sinπ/21+ℵB2−−1. Hence, by Definition 2 (part (vi)), we have sinB1=sinπ/2ℵB1+,sinπ/21+ℵB1−−1≺¯sinπ/2ℵB2+,sinπ/21+ℵB2−−1=sinB2.


## 4. Bipolar Fuzzy Sine Trigonometric Averaging Aggregation Operators

In this section, some new averaging AOs have been proposed on the basis of STOLs of BFNs. These aggregation operators include (i) ST-BFWA operator, (ii) ST-BFOWA operator, and (iii) ST-BFHWA operator.


### 4.1. ST-BFWA Operator

Definition 9.
. LetBi, i=1,2,…,n, be a compendium of BFNs and φ=φ1,φ2,…,φn be the weights of Bi, i=1,2,…,n, with φi>0 and ∑i=1nφi=1. Then, ST-BFWA operator is described as(13)ST−BFWAB1,B2,…,Bn=φ1sinB1⊕φ2sinB2⊕⋯⊕φnsinBn.Theorem 4.
. LetBi=ℵBi+,ℵBi− be n BFNs, then their cumulative value acquired by using (13) is again a BFN and is given by(14)ST−BFWAB1,B2,…,Bn=1−∏i=1n1−sinπ2ℵBi+φi,−∏i=1n−sinπ21+ℵBi−−1φi.Proof.
To prove the theorem, we employ mathematical induction onn. For n=2, we have(15)ST−BFWAB1,B2=φ1sinB1⊕φ2sinB2=1−1−sinπ2ℵB1+φ1,−−sinπ21+ℵB1−−1φ1⊕1−1−sinπ2ℵB2+φ2,−−sinπ21+ℵB2−−1φ2=1−∏i=121−sinπ2ℵBi+φi,−∏i=12−sinπ21+ℵBi−−1φi.
This shows that our assertion is correct forn=2. Assume that the result holds true for n=k, i.e.,(16)ST−BFWAB1,B2,…,Bk=φ1sinB1⊕φ2sinB2⊕⋯⊕φksinBk=1−∏i=1k1−sinπ2ℵBi+φi,−∏i=1k−sinπ21+ℵBi−−1φi.
Now, forn=k+1, we have(17)ST−BFWAB1,B2,…,Bk+1=φ1sinB1⊕φ2sinB2⊕⋯⊕φksinBk⊕φk+1sinBk+1=1−∏i=1k1−sinπ2ℵB1+φi,−∏i=1k−sinπ21+ℵB1−−1φi⊕1−1−sinπ2ℵBk+1+φk+1,−−sinπ21+ℵBk+1−−1φk+1=1−∏i=1k+11−sinπ2ℵBi+φi,−∏i=1k+1−sinπ21+ℵBi−−1φi.
Hence, the result holds∀n.Example 1.
. LetB1=0.41,−0.39, B2=0.66,−0.21, B3=0.59,−0.46, and B4=0.72,−0.56 be four BFNs and φ=0.23,0.31,0.27,0.19 be the corresponding weight vector, then(18)∏i=141−sinπ2ℵBi+φi=1−sinπ2ℵB1+φ1×1−sinπ2ℵB2+φ2×1−sinπ2ℵB3+φ3×1−sinπ2ℵB4+φ4=1−sinπ20.410.23×1−sinπ20.660.31×1−sinπ20.590.27×1−sinπ20.720.19=0.1821,∏i=14−sinπ21+ℵBi−−1φi=−sinπ21+ℵB1−−1φ1×−sinπ21+ℵB2−−1φ2×−sinπ21+ℵB3−−1φ3×−sinπ21+ℵB4−−1φ4=−sinπ21−0.39−10.23×−sinπ21−0.21−10.31×−sinπ21−0.46−10.27×−sinπ21−0.56−10.19=0.1550.
Now,(19)ST−BFWAB1,B2,B3,B4=1−∏i=141−sinπ2ℵBi+φi,−∏i=14−sinπ21+ℵBi−−1φi=1−0.1821,−0.1550=0.8179,−0.1550.Theorem 5.
. LetBi=ℵBi+,ℵBi−, i=1,2,…,n, be a compendium of BFNs and φ=φ1,φ2,…,φn be the weight vector with φi>0 and ∑i=1nφi=1, then ST-BFWA operator holds the properties listed as follows:(i
)Idempotency. If all BFNs are equal, i.e., Bi=B=ℵB+,ℵB−, then(20)ST−BFWAB1,B2,…,Bn=sinB.(ii)
Monotonicity. Let Bi∗=ℵBi∗+,ℵBi∗−, i=1,2,…,n, be another collection of BFNs such that Bi≺¯Bi∗, ∀i=1,2,…,n, then(21)ST−BFWAB1,B2,…,Bn≺¯ST−BFWAB1∗,B2∗,…,Bn∗.(iii)
Boundedness. Let B¯=miniℵBi+,maxiℵBi− and B¯=maxiℵBi+,miniℵBi−, then(22)sinB¯≺¯ST−BFWAB1,B2,…,Bn≺¯sinB¯.Proof.
(i)
LetBi=B∀i=1,2,…,n. Then, by using (13), we have(23)ST−BFWAB1,B2,…,Bn=1−∏i=1n1−sinπ2ℵBi+φi−∏i=1n−sinπ21+ℵBi−−1φi=1−1−sinπ2ℵB+∑i=1nφi−−sinπ21+ℵB−−1∑i=1nφi=1−1−sinπ2ℵB+−−sinπ21+ℵB−−1=sinπ2ℵB+,sinπ21+ℵB−−1=sinB.(ii)
SinceBi≺¯Bi∗, this implies that ℵBi+≤ℵBi∗+ and ℵBi−≥ℵBi∗−, ∀i=1,2,…,n. Suppose that ST−BFWAB1,B2,…,Bn=ℵ˜+,ℵ˜− and ST−BFWAB1∗,B2∗,…,Bn∗=ℵ˜∗+,ℵ˜∗−. Due to the monotonicity of sine function, we get(24)sinπ2ℵBi+≤sinπ2ℵBi∗+,⟹1−sinπ2ℵBi+≥1−sinπ2ℵBi∗+,⟹1−sinπ2ℵBi+φi≥1−sinπ2ℵBi∗+φi,⟹∏i=1n1−sinπ2ℵBi+φi≥∏i=1n1−sinπ2ℵBi∗+φi,⟹ℵ˜+=1−∏i=1n1−sinπ2ℵBi+φi≤1−∏i=1n1−sinπ2ℵBi∗+φi=ℵ˜∗+.Similarly,(25)sinπ21+ℵBi−≥sinπ21+ℵBi∗−,⟹sinπ21+ℵBi−−1≥sinπ21+ℵBi∗−−1,⟹−sinπ21+ℵBi−−1φi≤−sinπ21+ℵBi∗−−1φi,⟹ℵ˜−=−∏i=1n−sinπ21+ℵBi−−1φi≥−∏i=1n−sinπ21+ℵBi∗−−1φi=ℵ˜∗−.Sinceℵ˜+≤ℵ˜∗+ and ℵ˜−≥ℵ˜∗−, we have(26)ST−BFWAB1,B2,…,Bn⪯¯ST−BFWAB1∗,B2∗,…,Bn∗.(iii)
It is similar to the preceding proof, so we exclude it.


### 4.2. ST-BFOWA Operator

Definition 10.
LetBi, i=1,2,…,n, be a compendium of BFNs, then ST-BFOWA operator is explicated as(27)ST−BFOWAB1,B2,…,Bn=φ1sinBη1⊕φ2sinBη2⊕⋯⊕φnsinBηn,where η1,η2,…,ηn is an arrangement of 1,2,…,n with the constraint that Bηi−1≥Bηi∀i=2,3,…,n. It is noteworthy that the weights φi with φi>0 and ∑i=1nφi=1 are associated with the ordered positions of BFNs Bi.Theorem 6.
The cumulative value ofn BFNs Bi=ℵBi+,ℵBi− acquired by utilizing ST-BFOWA operator is still a BFN and is given by(28)ST−BFOWAB1,B2,…,Bn=1−∏i=1n1−sinπ2ℵBηi+φi,−∏i=1n−sinπ21+ℵBηi−−1φi.Proof.
Straightforward.Theorem 7.
LetBi=ℵBi+,ℵBi−, i=1,2,…,n, be a compendium of BFNs and φ=φ1,φ2,…,φn be the weight vector with φi>0 and ∑i=1nφi=1, then ST-BFOWA operator satisfies the following properties:(i)
Idempotency. If Bi=B=ℵB+,ℵB−, ∀i=1,2,…,n, then(29)ST−BFOWAB1,B2,…,Bn=sinB.(ii)
Monotonicity. Let Bi∗=ℵBi∗+,ℵBi∗−, i=1,2,…,n, be another collection of BFNs such that Bi≺¯Bi∗, ∀i=1,2,…,n, then(30)ST−BFOWAB1,B2,…,Bn⪯¯ST−BFOWAB1∗,B2∗,…,Bn∗.(iii)
Boundedness. If fi¯=miniℵBi+,maxiℵBi− and B¯=maxiℵBi+,miniℵBi−, then(31)sinB¯¯≺¯ST−BFOWAB1,B2,…,Bn⪯¯sinB¯.Proof.
It is obvious.


### 4.3. ST-BFHWA Operator

Definition 11.
LetBi, i=1,2,…,n, be a compendium of BFNs and φ=φ1,φ2,…,φn be the weight vector of Bi with φi>0 and ∑i=1nφi=1. A ST-BFHWA operator with associated weight vector γ=γ1,γ2,…,γn with γi>0 and ∑i=1nγi=1 can be described as(32)ST−BFHWAB1,B2,…,Bn=γ1sinB˙η1⊕γ2sinB˙η2⊕⋯⊕γnsinB˙ηn,where B˙i=nφiBi and η1,η2,…,ηn is an arrangement of 1,2,…,n with the stipulation that B˙ηi−1≥B˙ηi∀i=2,3,…,n.Theorem 8.
The cumulative value ofn BFNs Bi=ℵBi+,ℵBi− acquired by utilizing ST-BFHWA operator is still a BFN and is given by(33)ST−BFHWAB1,B2,…,Bn=1−∏i=1n1−sinπ2ℵB˙ηi+γi,−∏i=1n−sinπ21+ℵB˙ηi−−1γi.Proof.
Straightforward.


## 4.1. ST-BFWA Operator

Definition 9.
. LetBi, i=1,2,…,n, be a compendium of BFNs and φ=φ1,φ2,…,φn be the weights of Bi, i=1,2,…,n, with φi>0 and ∑i=1nφi=1. Then, ST-BFWA operator is described as(13)ST−BFWAB1,B2,…,Bn=φ1sinB1⊕φ2sinB2⊕⋯⊕φnsinBn.Theorem 4.
. LetBi=ℵBi+,ℵBi− be n BFNs, then their cumulative value acquired by using (13) is again a BFN and is given by(14)ST−BFWAB1,B2,…,Bn=1−∏i=1n1−sinπ2ℵBi+φi,−∏i=1n−sinπ21+ℵBi−−1φi.Proof.
To prove the theorem, we employ mathematical induction onn. For n=2, we have(15)ST−BFWAB1,B2=φ1sinB1⊕φ2sinB2=1−1−sinπ2ℵB1+φ1,−−sinπ21+ℵB1−−1φ1⊕1−1−sinπ2ℵB2+φ2,−−sinπ21+ℵB2−−1φ2=1−∏i=121−sinπ2ℵBi+φi,−∏i=12−sinπ21+ℵBi−−1φi.
This shows that our assertion is correct forn=2. Assume that the result holds true for n=k, i.e.,(16)ST−BFWAB1,B2,…,Bk=φ1sinB1⊕φ2sinB2⊕⋯⊕φksinBk=1−∏i=1k1−sinπ2ℵBi+φi,−∏i=1k−sinπ21+ℵBi−−1φi.
Now, forn=k+1, we have(17)ST−BFWAB1,B2,…,Bk+1=φ1sinB1⊕φ2sinB2⊕⋯⊕φksinBk⊕φk+1sinBk+1=1−∏i=1k1−sinπ2ℵB1+φi,−∏i=1k−sinπ21+ℵB1−−1φi⊕1−1−sinπ2ℵBk+1+φk+1,−−sinπ21+ℵBk+1−−1φk+1=1−∏i=1k+11−sinπ2ℵBi+φi,−∏i=1k+1−sinπ21+ℵBi−−1φi.
Hence, the result holds∀n.Example 1.
. LetB1=0.41,−0.39, B2=0.66,−0.21, B3=0.59,−0.46, and B4=0.72,−0.56 be four BFNs and φ=0.23,0.31,0.27,0.19 be the corresponding weight vector, then(18)∏i=141−sinπ2ℵBi+φi=1−sinπ2ℵB1+φ1×1−sinπ2ℵB2+φ2×1−sinπ2ℵB3+φ3×1−sinπ2ℵB4+φ4=1−sinπ20.410.23×1−sinπ20.660.31×1−sinπ20.590.27×1−sinπ20.720.19=0.1821,∏i=14−sinπ21+ℵBi−−1φi=−sinπ21+ℵB1−−1φ1×−sinπ21+ℵB2−−1φ2×−sinπ21+ℵB3−−1φ3×−sinπ21+ℵB4−−1φ4=−sinπ21−0.39−10.23×−sinπ21−0.21−10.31×−sinπ21−0.46−10.27×−sinπ21−0.56−10.19=0.1550.
Now,(19)ST−BFWAB1,B2,B3,B4=1−∏i=141−sinπ2ℵBi+φi,−∏i=14−sinπ21+ℵBi−−1φi=1−0.1821,−0.1550=0.8179,−0.1550.Theorem 5.
. LetBi=ℵBi+,ℵBi−, i=1,2,…,n, be a compendium of BFNs and φ=φ1,φ2,…,φn be the weight vector with φi>0 and ∑i=1nφi=1, then ST-BFWA operator holds the properties listed as follows:(i
)Idempotency. If all BFNs are equal, i.e., Bi=B=ℵB+,ℵB−, then(20)ST−BFWAB1,B2,…,Bn=sinB.(ii)
Monotonicity. Let Bi∗=ℵBi∗+,ℵBi∗−, i=1,2,…,n, be another collection of BFNs such that Bi≺¯Bi∗, ∀i=1,2,…,n, then(21)ST−BFWAB1,B2,…,Bn≺¯ST−BFWAB1∗,B2∗,…,Bn∗.(iii)
Boundedness. Let B¯=miniℵBi+,maxiℵBi− and B¯=maxiℵBi+,miniℵBi−, then(22)sinB¯≺¯ST−BFWAB1,B2,…,Bn≺¯sinB¯.Proof.
(i)
LetBi=B∀i=1,2,…,n. Then, by using (13), we have(23)ST−BFWAB1,B2,…,Bn=1−∏i=1n1−sinπ2ℵBi+φi−∏i=1n−sinπ21+ℵBi−−1φi=1−1−sinπ2ℵB+∑i=1nφi−−sinπ21+ℵB−−1∑i=1nφi=1−1−sinπ2ℵB+−−sinπ21+ℵB−−1=sinπ2ℵB+,sinπ21+ℵB−−1=sinB.(ii)
SinceBi≺¯Bi∗, this implies that ℵBi+≤ℵBi∗+ and ℵBi−≥ℵBi∗−, ∀i=1,2,…,n. Suppose that ST−BFWAB1,B2,…,Bn=ℵ˜+,ℵ˜− and ST−BFWAB1∗,B2∗,…,Bn∗=ℵ˜∗+,ℵ˜∗−. Due to the monotonicity of sine function, we get(24)sinπ2ℵBi+≤sinπ2ℵBi∗+,⟹1−sinπ2ℵBi+≥1−sinπ2ℵBi∗+,⟹1−sinπ2ℵBi+φi≥1−sinπ2ℵBi∗+φi,⟹∏i=1n1−sinπ2ℵBi+φi≥∏i=1n1−sinπ2ℵBi∗+φi,⟹ℵ˜+=1−∏i=1n1−sinπ2ℵBi+φi≤1−∏i=1n1−sinπ2ℵBi∗+φi=ℵ˜∗+.Similarly,(25)sinπ21+ℵBi−≥sinπ21+ℵBi∗−,⟹sinπ21+ℵBi−−1≥sinπ21+ℵBi∗−−1,⟹−sinπ21+ℵBi−−1φi≤−sinπ21+ℵBi∗−−1φi,⟹ℵ˜−=−∏i=1n−sinπ21+ℵBi−−1φi≥−∏i=1n−sinπ21+ℵBi∗−−1φi=ℵ˜∗−.Sinceℵ˜+≤ℵ˜∗+ and ℵ˜−≥ℵ˜∗−, we have(26)ST−BFWAB1,B2,…,Bn⪯¯ST−BFWAB1∗,B2∗,…,Bn∗.(iii)
It is similar to the preceding proof, so we exclude it.


## 4.2. ST-BFOWA Operator

Definition 10.
LetBi, i=1,2,…,n, be a compendium of BFNs, then ST-BFOWA operator is explicated as(27)ST−BFOWAB1,B2,…,Bn=φ1sinBη1⊕φ2sinBη2⊕⋯⊕φnsinBηn,where η1,η2,…,ηn is an arrangement of 1,2,…,n with the constraint that Bηi−1≥Bηi∀i=2,3,…,n. It is noteworthy that the weights φi with φi>0 and ∑i=1nφi=1 are associated with the ordered positions of BFNs Bi.Theorem 6.
The cumulative value ofn BFNs Bi=ℵBi+,ℵBi− acquired by utilizing ST-BFOWA operator is still a BFN and is given by(28)ST−BFOWAB1,B2,…,Bn=1−∏i=1n1−sinπ2ℵBηi+φi,−∏i=1n−sinπ21+ℵBηi−−1φi.Proof.
Straightforward.Theorem 7.
LetBi=ℵBi+,ℵBi−, i=1,2,…,n, be a compendium of BFNs and φ=φ1,φ2,…,φn be the weight vector with φi>0 and ∑i=1nφi=1, then ST-BFOWA operator satisfies the following properties:(i)
Idempotency. If Bi=B=ℵB+,ℵB−, ∀i=1,2,…,n, then(29)ST−BFOWAB1,B2,…,Bn=sinB.(ii)
Monotonicity. Let Bi∗=ℵBi∗+,ℵBi∗−, i=1,2,…,n, be another collection of BFNs such that Bi≺¯Bi∗, ∀i=1,2,…,n, then(30)ST−BFOWAB1,B2,…,Bn⪯¯ST−BFOWAB1∗,B2∗,…,Bn∗.(iii)
Boundedness. If fi¯=miniℵBi+,maxiℵBi− and B¯=maxiℵBi+,miniℵBi−, then(31)sinB¯¯≺¯ST−BFOWAB1,B2,…,Bn⪯¯sinB¯.Proof.
It is obvious.


## 4.3. ST-BFHWA Operator

Definition 11.
LetBi, i=1,2,…,n, be a compendium of BFNs and φ=φ1,φ2,…,φn be the weight vector of Bi with φi>0 and ∑i=1nφi=1. A ST-BFHWA operator with associated weight vector γ=γ1,γ2,…,γn with γi>0 and ∑i=1nγi=1 can be described as(32)ST−BFHWAB1,B2,…,Bn=γ1sinB˙η1⊕γ2sinB˙η2⊕⋯⊕γnsinB˙ηn,where B˙i=nφiBi and η1,η2,…,ηn is an arrangement of 1,2,…,n with the stipulation that B˙ηi−1≥B˙ηi∀i=2,3,…,n.Theorem 8.
The cumulative value ofn BFNs Bi=ℵBi+,ℵBi− acquired by utilizing ST-BFHWA operator is still a BFN and is given by(33)ST−BFHWAB1,B2,…,Bn=1−∏i=1n1−sinπ2ℵB˙ηi+γi,−∏i=1n−sinπ21+ℵB˙ηi−−1γi.Proof.
Straightforward.


## 5. 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.


### 5.1. ST-BFWG Operator

Definition 12.
Forn BFNs Bi, a ST-BFWG operator is explicated as(34)ST−BFWGB1,B2,…,Bn=sinB1φ1⊗sinB2φ2⊗⋯⊗sinBnφn.Theorem 9.
LetBi=ℵBi+,ℵBi− be n BFNs, then their cumulative value obtained by utilizing ST-BFWG operator is expressed as(35)ST−BFWGB1,B2,…,Bn=∏i=1nsinπ2ℵBi+φi,−1−∏i=1nsinπ21+ℵBi−φi.Proof.
It is obvious.Example 2.
LetB1=0.41,−0.39, B2=0.66,−0.21, B3=0.59,−0.46, and B4=0.72,−0.56 be four BFNs and φ=0.23,0.31,0.27,0.19 be the corresponding weight vector, then(36)∏i=14sinπ2ℵBi+φi=sinπ2ℵB1+φ1×sinπ2ℵB2+φ2×sinπ2ℵB3+φ3×sinπ2ℵB4+φ4=sinπ20.410.23×sinπ20.660.31×sinπ20.590.27×sinπ20.720.19=0.7841∏i=14sinπ21+ℵBi−φi=sinπ21+ℵB1−φ1×sinπ21+ℵB2−φ2×sinπ21+ℵB3−φ3×sinπ21+ℵB4−φ4=sinπ21−0.390.23×sinπ21−0.210.31×sinπ21−0.460.27×sinπ21−0.560.19=0.7973.
Now,(37)ST−BFWGB1,B2,B3,B4=∏i=14sinπ2ℵBi+φi,−1−∏i=14sinπ21+ℵBi−φi=0.7841,−1−0.7973=0.7841,−0.2027.
The properties mentioned in Theorem5, namely, idempotency, monotonicity, and boundedness, also apply to the ST-BFWG operator.


### 5.2. ST-BFOWG Operator

Definition 13.
A ST-BFOWG operator is defined as(38)ST−BFOWGB1,B2,…,Bn=sinBη1φ1⊗sinBη2φ2⊗⋯⊗sinBηnφn,where η1,η2,…,ηn is an arrangement of 1,2,…,n such that Bηi−1≥Bηi∀i=2,3,…,n.Theorem 10.
The cumulative value ofn BFNs Bi=ℵBi+,ℵBi− acquired by utilizing ST-BFOWG operator is expressed as(39)ST−BFOWGB1,B2,…,Bn=∏i=1nsinπ2ℵBηi+φi,−1−∏i=1nsinπ21+ℵBηi−φi.Proof.
It is obvious.
Idempotency, monotonicity, and boundedness are all satisfied by the ST-BFOWG operator.


### 5.3. ST-BFHWG Operator

Definition 14.
A ST-BFHWG operator with associated weight vectorγ=γ1,γ2,…,γn with γi>0 and ∑i=1nγi=1 can be described as(40)ST−BFHWGB1,B2,…,Bn=sinB˙η1γ1⊗sinB˙η2γ2⊗⋯⊗sinB˙ηnγn.where B˙i=Binφi and η1,η2,…,ηn is an arrangement of 1,2,…,n such that B˙ηi−1≥B˙ηi∀i=2,3,…,n.Theorem 11.
The cumulative value ofn BFNs Bi=ℵBi+,ℵBi− obtained by utilizing ST-BFHWG operator is expressed as(41)ST−BFHWGB1,B2,…,Bn=∏i=1nsinπ2ℵB˙ηi+γi,−1−∏i=1nsinπ21+ℵB˙ηi−γi.Proof.
Straightforward.


## 5.1. ST-BFWG Operator

Definition 12.
Forn BFNs Bi, a ST-BFWG operator is explicated as(34)ST−BFWGB1,B2,…,Bn=sinB1φ1⊗sinB2φ2⊗⋯⊗sinBnφn.Theorem 9.
LetBi=ℵBi+,ℵBi− be n BFNs, then their cumulative value obtained by utilizing ST-BFWG operator is expressed as(35)ST−BFWGB1,B2,…,Bn=∏i=1nsinπ2ℵBi+φi,−1−∏i=1nsinπ21+ℵBi−φi.Proof.
It is obvious.Example 2.
LetB1=0.41,−0.39, B2=0.66,−0.21, B3=0.59,−0.46, and B4=0.72,−0.56 be four BFNs and φ=0.23,0.31,0.27,0.19 be the corresponding weight vector, then(36)∏i=14sinπ2ℵBi+φi=sinπ2ℵB1+φ1×sinπ2ℵB2+φ2×sinπ2ℵB3+φ3×sinπ2ℵB4+φ4=sinπ20.410.23×sinπ20.660.31×sinπ20.590.27×sinπ20.720.19=0.7841∏i=14sinπ21+ℵBi−φi=sinπ21+ℵB1−φ1×sinπ21+ℵB2−φ2×sinπ21+ℵB3−φ3×sinπ21+ℵB4−φ4=sinπ21−0.390.23×sinπ21−0.210.31×sinπ21−0.460.27×sinπ21−0.560.19=0.7973.
Now,(37)ST−BFWGB1,B2,B3,B4=∏i=14sinπ2ℵBi+φi,−1−∏i=14sinπ21+ℵBi−φi=0.7841,−1−0.7973=0.7841,−0.2027.
The properties mentioned in Theorem5, namely, idempotency, monotonicity, and boundedness, also apply to the ST-BFWG operator.


## 5.2. ST-BFOWG Operator

Definition 13.
A ST-BFOWG operator is defined as(38)ST−BFOWGB1,B2,…,Bn=sinBη1φ1⊗sinBη2φ2⊗⋯⊗sinBηnφn,where η1,η2,…,ηn is an arrangement of 1,2,…,n such that Bηi−1≥Bηi∀i=2,3,…,n.Theorem 10.
The cumulative value ofn BFNs Bi=ℵBi+,ℵBi− acquired by utilizing ST-BFOWG operator is expressed as(39)ST−BFOWGB1,B2,…,Bn=∏i=1nsinπ2ℵBηi+φi,−1−∏i=1nsinπ21+ℵBηi−φi.Proof.
It is obvious.
Idempotency, monotonicity, and boundedness are all satisfied by the ST-BFOWG operator.


## 5.3. ST-BFHWG Operator

Definition 14.
A ST-BFHWG operator with associated weight vectorγ=γ1,γ2,…,γn with γi>0 and ∑i=1nγi=1 can be described as(40)ST−BFHWGB1,B2,…,Bn=sinB˙η1γ1⊗sinB˙η2γ2⊗⋯⊗sinB˙ηnγn.where B˙i=Binφi and η1,η2,…,ηn is an arrangement of 1,2,…,n such that B˙ηi−1≥B˙ηi∀i=2,3,…,n.Theorem 11.
The cumulative value ofn BFNs Bi=ℵBi+,ℵBi− obtained by utilizing ST-BFHWG operator is expressed as(41)ST−BFHWGB1,B2,…,Bn=∏i=1nsinπ2ℵB˙ηi+γi,−1−∏i=1nsinπ21+ℵB˙ηi−γi.Proof.
Straightforward.


## 6. 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. LetA=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=ϑ1,ϑ2,…,ϑl where all the weights are BFNs. Construct the individual decision matrices ℍk=hijkm×n in which hijk 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=φjkl×n is the criterion weight matrix in which φjk 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. The steps in this technique are outlined as follows:Step 1. Compute the relative propinquity coefficient of each ϑk, k=1,2,…,l, by the equation(42)δk=dϑk,ϑ¯dϑk,ϑ¯+dϑk,ϑ¯.whereϑ¯=minkℵϑk+,maxkℵϑk− and ϑ¯=maxkℵϑk+,minkℵϑk−. It is evident that if ϑk⟶ϑ¯, then δk⟶0, and if ϑk⟶ϑ¯, then δk⟶1.Step 2. Normalize δk, k=1,2,…,l, by the equation(43)ζk=δk∑k=1lδk.In this way, we get a normalized vectorζ=ζ1,ζ2,…,ζl 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:(44)h˜ij=ST−BFWAζkhij1,hij2,…,hijl=1−∏k=1l1−sinπ2ℵhijk+ζk,−∏k=1l−sinπ21+ℵhijk−−1ζk.(45)φ˜j=ST−BFWAζkφj1,φj2,…,φjl=1−∏k=1l1−sinπ2ℵφjk+ζk,−∏k=1l−sinπ21+ℵφjk−−1ζk.Step 4. Obtain the relative efficiency function fij as follows:(46)fij=dh˜ij,h˜¯dh˜ij,h˜¯+dh˜ij,h˜¯,whereh˜¯=miniℵh˜ij+,maxiℵh˜ij− and h˜¯=maxiℵh˜ij+,miniℵh˜ij−. It follows that if h˜ij⟶h˜¯, then fij⟶0, and if h˜ij⟶h˜¯, then fij⟶1.Step 5. Compute the preference intensity PIja^i,a^ti,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:(47)PIja^i,a^t=λjfij−ftj,whereλj is a threshold function given by(48)λjx=0.01,x>0,0,x≤0.Step 6. Construct the superiority matrix S=Sijm×n and inferiority matrix I=Iijm×n by utilizing the following equations:(49)S−indexSij=∑t=1mPIja^i,a^t=∑t=1mλjfij−ftj,(50)I−indexIij=∑t=1mPIja^t,a^i=∑t=1mλjftj−fij.Step 7. Calculate the superiority flow (S-flow) and inferiority flow (I-flow) as follows:(51)λ>a^i=ST−BFWASijφ˜1,φ˜2,…,φ˜n=1−∏j=1n1−sinπ2ℵφ˜j+Sij,−∏j=1n−sinπ21+ℵφ˜j−−1Sij.(52)λ<a^i=ST−BFWAIijφ˜1,φ˜2,…,φ˜n=1−∏j=1n1−sinπ2ℵφ˜j+Iij,−∏j=1n−sinπ21+ℵφ˜j−−1Iij.Step 8. Compute the score functions of λ>a^i and λ<a^i, i=1,2,…,m, by using (2).Step 9. Apply the superiority ranking laws (SR-laws) and inferiority ranking laws (IR-laws) as follows:SR-Law 1. If λ>a^i≻λ>a^t and λ<a^i≺λ<a^t, then a^i≻a^tSR-Law 2. If λ>a^i≻λ>a^t and λ<a^i=λ<a^t, then a^i≻a^tSR-Law 3. If λ>a^i=λ>a^t and λ<a^i≺λ<a^t, then a^i≻a^tIR-Law 1. If λ>a^i≺λ>a^t and λ<a^i≻λ<a^t, then a^i≺a^tIR-Law 2. If λ>a^i≺λ>a^t and λ<a^i=λ<a^t, then a^i≺a^tIR-Law 3. If λ>a^i=λ>a^t and λ<a^i≻λ<a^t, then a^i≺a^tStep 10. Integrate the SR-laws with the IR-laws to determine the optimal alternative.


### 6.1. Case Study

The 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. There are a variety of reasons why people from developed countries prefer less developed countries. The 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. The 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.Figure 1
Medical tourism market size.The 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, Thailand, and the Philippines are among the most popular medical tourism destinations. Singapore and India are well-known for their cardiac and orthopaedic surgery. Thailand is well-known for its dental procedures and gender reassignment surgeries. Japan has one of the best oncology treatment facilities in the world. The Philippines is famous for its cosmetic surgery, dentistry, and fertility treatment. The 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. The higher the MTI, the better the destination. Figure2 depicts the medical tourism index for the aforementioned Asian countries (https://www.medicaltourism.com/mti/2020-2021/region/asia).Figure 2
Medical tourism index (2020-2021).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.


### 6.2. 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 makerse^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. The weights of criteria c^j given by the decision makers e^k are given in Table 3. The decision makers evaluate each alternative a^i w.r.t each criterion c^j and give their assessment via BFNs. Three decision matrices are given in Tables 4–6.Step 1. The relative propinquity coefficients δk (k=1,2,3) are computed using (42) as follows:(53)δ=0.12,0.72,0.52.Step 2. The normalized vector is obtained using (43)as follows:(54)ζ=0.0882,0.5294,0.3824.Step 3. The 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:(55)φ˜1=0.9357,−0.1744,φ˜2=0.9423,−0.0799,φ˜3=0.9183,−0.0632.Step 4. The relative efficiency function is calculated using (46)as follows:(56)fij=0.34270.22340.46930.43200.42410.596310.88340.48200.74630.28070.0246.Step 5, 6. The superiority and inferiority matrices are constructed using (49) and (50) as follows:(57)S=000.010.010.020.030.030.030.020.020.010,I=0.030.030.020.020.010000.010.010.020.03.Step 7, 8. The 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:(58)a^3≻a^2≻a^4≻a^1.Applying IR-laws to Table8 yields the following ranking order:(59)a^3≻a^2≻a^4≻a^1.Step 10. According to both SR and IR-laws, a^3 is the best alternative.Table 1
Bipolar fuzzy weights of decision makers.
Decision makersWeightse^1ϑ1=0.79,−0.28e^2ϑ2=0.85,−0.37e^3ϑ3=0.92,−0.25Table 2
Criteria for the selection of the best medical tourism destination.
CriteriaDescription(i) Service qualityc^1This includes modern equipment, qualified staff, and variety of medical treatments.(ii) Securityc^2This includes life and fiscal security of the tourists.(iii) Infrastructure facilitiesc^3This includes transportation and maintenance of hospitals and equipment.Table 3
Bipolar fuzzy weights of criteria.
c^1c^2c^3e^10.73,−0.260.65,−0.360.81,−0.29e^20.82,−0.380.76,−0.190.78,−0.27e^30.69,−0.420.83,−0.360.65,−0.17Table 4
BF decision matrixℍ1.
c^1c^2c^3a^10.82,−0.210.92,−0.230.78,−0.26a^20.76,−0.190.52,−0.410.66,−0.24a^30.86,−0.170.87,−0.180.79,−0.34a^40.67,−0.310.42,−0.380.76,−0.12Table 5
BF decision matrixℍ2.
c^1c^2c^3a^10.69,−0.330.82,−0.190.89,−0.17a^20.82,−0.210.66,−0.290.77,−0.32a^30.91,−0.360.79,−0.260.87,−0.29a^40.79,−0.290.56,−0.210.82,−0.26Table 6
BF decision matrixℍ3.
c^1c^2c^3a^10.76,−0.220.88,−0.130.96,−0.41a^20.89,−0.160.62,−0.240.69,−0.56a^30.73,−0.290.92,−0.260.71,−0.31a^40.81,−0.320.63,−0.460.56,−0.21Table 7
Accumulated BF decision matrix.
c^1c^2c^3a^10.9128,−0.08950.9747,−0.03420.9922,−0.0740a^20.9713,−0.04310.8396,−0.09380.9135,−0.1775a^30.9751,−0.11620.9762,−0.07710.9585,−0.1102a^40.9459,−0.11110.7886,−0.10740.9183,−0.0611Table 8
The BF-SIR flows.
Alternativesλ>a^iScrλ>a^iλ<a^iScrλ<a^ia^10.0469,−0.94820.04940.3425,−0.70450.319a^20.2641,−0.74880.25760.1483,−0.89200.1282a^30.3425,−0.70450.3190.0469,−0.94820.0494a^40.1483,−0.89200.12820.2641,−0.74880.2576


### 6.3. Comparative and Sensitivity Analysis

In this section, we compare our suggested BF-SIR technique to some existing approaches in order to evaluate its validity. Table9 summarizes the comparative study of various techniques. It can be seen from Table 9 that our suggested approach is compatible with the existing techniques.Table 9
Comparative analysis of the suggested and existing methodologies.
MethodsRanking of alternativesOptimal alternativeAlgorithm (Jana and Pal [17])a^3≻a^2≻a^1≻a^4a^3Algorithm (Wei et al. [20])a^3≻a^2≻a^4≻a^1a^3Algorithm (Hamid et al. [23])a^3≻a^4≻a^2≻a^1a^3Algorithm (Akram et al. [24])a^3≻a^2≻a^4≻a^1a^3Algorithm (Peng and Yang [60])a^3≻a^2≻a^4≻a^1a^3Algorithm (Zhang and Xu [36])a^3≻a^1≻a^2≻a^4a^3Algorithm (Proposed)a^3≻a^2≻a^4≻a^1a^3


## 6.1. Case Study

The 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. There are a variety of reasons why people from developed countries prefer less developed countries. The 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. The 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.Figure 1
Medical tourism market size.The 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, Thailand, and the Philippines are among the most popular medical tourism destinations. Singapore and India are well-known for their cardiac and orthopaedic surgery. Thailand is well-known for its dental procedures and gender reassignment surgeries. Japan has one of the best oncology treatment facilities in the world. The Philippines is famous for its cosmetic surgery, dentistry, and fertility treatment. The 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. The higher the MTI, the better the destination. Figure2 depicts the medical tourism index for the aforementioned Asian countries (https://www.medicaltourism.com/mti/2020-2021/region/asia).Figure 2
Medical tourism index (2020-2021).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.


## 6.2. 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 makerse^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. The weights of criteria c^j given by the decision makers e^k are given in Table 3. The decision makers evaluate each alternative a^i w.r.t each criterion c^j and give their assessment via BFNs. Three decision matrices are given in Tables 4–6.Step 1. The relative propinquity coefficients δk (k=1,2,3) are computed using (42) as follows:(53)δ=0.12,0.72,0.52.Step 2. The normalized vector is obtained using (43)as follows:(54)ζ=0.0882,0.5294,0.3824.Step 3. The 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:(55)φ˜1=0.9357,−0.1744,φ˜2=0.9423,−0.0799,φ˜3=0.9183,−0.0632.Step 4. The relative efficiency function is calculated using (46)as follows:(56)fij=0.34270.22340.46930.43200.42410.596310.88340.48200.74630.28070.0246.Step 5, 6. The superiority and inferiority matrices are constructed using (49) and (50) as follows:(57)S=000.010.010.020.030.030.030.020.020.010,I=0.030.030.020.020.010000.010.010.020.03.Step 7, 8. The 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:(58)a^3≻a^2≻a^4≻a^1.Applying IR-laws to Table8 yields the following ranking order:(59)a^3≻a^2≻a^4≻a^1.Step 10. According to both SR and IR-laws, a^3 is the best alternative.Table 1
Bipolar fuzzy weights of decision makers.
Decision makersWeightse^1ϑ1=0.79,−0.28e^2ϑ2=0.85,−0.37e^3ϑ3=0.92,−0.25Table 2
Criteria for the selection of the best medical tourism destination.
CriteriaDescription(i) Service qualityc^1This includes modern equipment, qualified staff, and variety of medical treatments.(ii) Securityc^2This includes life and fiscal security of the tourists.(iii) Infrastructure facilitiesc^3This includes transportation and maintenance of hospitals and equipment.Table 3
Bipolar fuzzy weights of criteria.
c^1c^2c^3e^10.73,−0.260.65,−0.360.81,−0.29e^20.82,−0.380.76,−0.190.78,−0.27e^30.69,−0.420.83,−0.360.65,−0.17Table 4
BF decision matrixℍ1.
c^1c^2c^3a^10.82,−0.210.92,−0.230.78,−0.26a^20.76,−0.190.52,−0.410.66,−0.24a^30.86,−0.170.87,−0.180.79,−0.34a^40.67,−0.310.42,−0.380.76,−0.12Table 5
BF decision matrixℍ2.
c^1c^2c^3a^10.69,−0.330.82,−0.190.89,−0.17a^20.82,−0.210.66,−0.290.77,−0.32a^30.91,−0.360.79,−0.260.87,−0.29a^40.79,−0.290.56,−0.210.82,−0.26Table 6
BF decision matrixℍ3.
c^1c^2c^3a^10.76,−0.220.88,−0.130.96,−0.41a^20.89,−0.160.62,−0.240.69,−0.56a^30.73,−0.290.92,−0.260.71,−0.31a^40.81,−0.320.63,−0.460.56,−0.21Table 7
Accumulated BF decision matrix.
c^1c^2c^3a^10.9128,−0.08950.9747,−0.03420.9922,−0.0740a^20.9713,−0.04310.8396,−0.09380.9135,−0.1775a^30.9751,−0.11620.9762,−0.07710.9585,−0.1102a^40.9459,−0.11110.7886,−0.10740.9183,−0.0611Table 8
The BF-SIR flows.
Alternativesλ>a^iScrλ>a^iλ<a^iScrλ<a^ia^10.0469,−0.94820.04940.3425,−0.70450.319a^20.2641,−0.74880.25760.1483,−0.89200.1282a^30.3425,−0.70450.3190.0469,−0.94820.0494a^40.1483,−0.89200.12820.2641,−0.74880.2576


## 6.3. Comparative and Sensitivity Analysis

In this section, we compare our suggested BF-SIR technique to some existing approaches in order to evaluate its validity. Table9 summarizes the comparative study of various techniques. It can be seen from Table 9 that our suggested approach is compatible with the existing techniques.Table 9
Comparative analysis of the suggested and existing methodologies.
MethodsRanking of alternativesOptimal alternativeAlgorithm (Jana and Pal [17])a^3≻a^2≻a^1≻a^4a^3Algorithm (Wei et al. [20])a^3≻a^2≻a^4≻a^1a^3Algorithm (Hamid et al. [23])a^3≻a^4≻a^2≻a^1a^3Algorithm (Akram et al. [24])a^3≻a^2≻a^4≻a^1a^3Algorithm (Peng and Yang [60])a^3≻a^2≻a^4≻a^1a^3Algorithm (Zhang and Xu [36])a^3≻a^1≻a^2≻a^4a^3Algorithm (Proposed)a^3≻a^2≻a^4≻a^1a^3


## 7. 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. The 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. Therefore, 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.



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*Source: 4182740-2022-06-20.xml*