Machine Learning and Computational Intelligence in Supply ChainsView this Special Issue
Innovative Bipolar Fuzzy Sine Trigonometric Aggregation Operators and SIR Method for Medical Tourism Supply Chain
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.
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  initiated the conception of fuzzy set (FS) and membership function. Later on, different researchers presented different extensions of FSs including, intuitionistic fuzzy sets (IFSs) , Pythagorean fuzzy sets (PyFSs) [3, 4], q-rung orthopair fuzzy sets (q-ROFSs) , hesitant fuzzy sets (HFSs) , neutrosophic sets (NSs) , single-valued NSs , picture fuzzy sets (PFSs) , 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  studied operations on bipolar-valued fuzzy sets. Tehrim and Riaz  introduced connection numbers of SPA theory for the decision support system by using the IVBF linguistic VIKOR method. Jana and Pal  proposed the BF-EDAS method for MCGDM problems. Liu et al.  suggested an integrated bipolar fuzzy SWARA-MABAC technique and utilized it for the safety risk and occupational health diagnosis. Jana et al.  introduced BF-Dombi AOs and Wei et al.  developed bipolar fuzzy Hamacher AOs.
Han et al.  proposed the TOPSIS method for YinYang bipolar fuzzy cognitive TOPSIS. Wei et al.  established MADM with IVBF information. Hamid et al.  initiated weighted aggregation operators for q-rung orthopair m-polar fuzzy set. Akram et al.  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  and Xu and Yager  propounded weighted averaging and geometric AOs for IFSs. Garg  introduced interactive operators for IFSs. Huang  proposed intuitionistic fuzzy Hamacher aggregation operators. Gou and Xu  suggested exponential operational laws (EOLs) for IFSs.
Li and Wei  proposed logarithmic operational laws (LOLs) for IFSs. Peng et al.  proposed EOLs for q-ROFSs. Similarly, the LOLs for PFSs  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.  developed STOLs for PFSs. Kabani  studied Pakistan as a medical tourism destination. Muzaffar and Hussain  investigated medical tourism to discuss the challenge: are we ready to take the challenge. Zhang and Xu  proposed TOPSIS for PFSs and PFNs with MCDM.
Mahmood et al.  proposed an innovative MCDM method with spherical fuzzy soft rough (SFSR) average aggregation operators. Ihsan et al.  presented the MADM support model based on bijective hypersoft expert set. Karaaslan and Karamaz  introduced an innovative decision-making approach with HFPHFS. Alcantud  introduced the novel concepts of soft topologies and fuzzy soft topologies and investigated their relationships. Liu et al.  introduced the idea of mining temporal association rules based on temporal soft sets. Riaz et al.  introduced a novel TOPSIS approach based on cosine similarity measures and CBF-information. Zararsiz and Riaz  introduced the notion of bipolar fuzzy metric spaces with application. Riaz et al.  proposed distance and similarity measures for bipolar fuzzy soft sets with application to pharmaceutical logistics and supply chain management.
In 2021, Gergin et al.  modified the TOPSIS method to deal the supplier selection for automotive industry. Karamasa et al.  introduced the weighting factors which affect the logistics out-sourcing decision-making problem. Ali et al.  introduced Einstein geometric AO to deal complex IVPFS, and its novel principles and its operational laws are defined. Muhammad et al.  and Biswas et al.  propounded multicriteria decision-making techniques to deal real world problems. Milovanovic et al.  developed uncertainty modeling using intuitionistic fuzzy numbers.
In 2021, Garg  introduced some robust STOLs, its operational laws for PFSs, and AOs and algorithms to interpret MCDM. In 2021, Mahmood et al.  interpreted BCFHWA, BCFHOWA, BCFHHA, BCFHWG, BCFHOWG, and BCFHHG operators. Palanikumar et al.  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.  applied IFDHWA and IFDHWG AO to evaluate enterprise financial performance. In 2021, Jana et al.  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.  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.  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 . Chai and Liu  proposed the IF-SIR method to deal with MCGDM problems. Peng and Yang  extended the SIR technique to pythagorean fuzzy data. Zhu et al.  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 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.
This section includes some rudimentary abstractions related to BFSs. Throughout this manuscript, we consider as universe of discourse.
Definition 1. (see ). A BFS on can be described aswhere denotes positive membership degree and denotes negative membership degree of an element . A bipolar fuzzy number (BFN) can be expressed as .
In 2015, Gul proposed operational laws of BFNs in his M.Phil Thesis.
Definition 2. . Let and be two BFNs and , then operational laws between them can be defined as(i)(ii)(iii)(iv)(v)(vi) if and (vii) if and
Definition 3. (see ). For a BFN , score and accuracy functions can be expressed asThe values of score and accuracy functions are used to compare two BFNs. For two BFNs and ,(i)If , then (ii)If , then (iii)If , then if (iv)If , then if (v)If , then if
Definition 4. (see ). If and are two BFSs on , then the normalized Hamming distance between them is calculated as
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. Let be a BFS on . A sine trigonometric operator on can be defined asClearly, is again a BFS on because and serve as positive and negative membership degrees, respectively, for every element . The set is called sine trigonometric-BFS (ST-BFS).
Definition 6. Let be a BFN, thenis called ST-BFN.
Definition 7. For two BFNs and , we propose STOLs as follows:(i)(ii)(iii); (iv);
Theorem 1. Let and be two BFNs and be three real numbers, then(i)(ii)(iii)(iv)
Proof. We substantiate (i) and (iv), and others can be substantiated similarly.(i)For , (iv) For ,
Definition 8. Let be a BFN and be the corresponding ST-BFN, thenis called complement of .
Theorem 2. Let and be two BFNs and , then(i)(ii)(iii)(iv)
Proof. We substantiate (i) and (iv), and others can be substantiated similarly. (i) Now, (iv)
Theorem 3. . Let and be two BFNs with , i.e., and , then .
Proof. Since sine is an increasing function on the interval so for , we have . Likewise, for , we obtain . This implicates that which further implicates that . Hence, by Definition 2 (part (vi)), we have .
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. . Let , , be a compendium of BFNs and be the weights of , , with and . Then, ST-BFWA operator is described as
Theorem 4. . Let be BFNs, then their cumulative value acquired by using (13) is again a BFN and is given by
Proof. To prove the theorem, we employ mathematical induction on . For , we haveThis shows that our assertion is correct for . Assume that the result holds true for , i.e.,Now, for , we haveHence, the result holds .
Example 1. . Let , , , and be four BFNs and be the corresponding weight vector, thenNow,
Theorem 5. . Let ,