Linear Diophantine Fuzzy Einstein Aggregation Operators for Multi-Criteria Decision-Making Problems

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Introduction and Literature Review
e problem of vague and misleading information has become a major issue for decades. Aggregation of data is important for decision-making corporate, administrative, social, medical, technological, psychological, and artificial intelligence fields. Awareness of the alternative has traditionally been seen as a crisp number or linguistic number. However, due to its uncertainty, the data cannot easily be aggregated. Multicriteria decision-making (MCDM) is an approach where feasible objects are accessed by decision experts (DEs) under the multiple criterion.Traditionally, an object's evaluation is thought of as either a crisp number or a linguistic number. However, owing to the unpredictable existence of real-world problems, classical mathematics cannot solve these complex problems literally. In any reallife problem-solving technique, the complexity characterizes the behavior of an object, whose components interrelate in multiple ways and follow different logical rules, meaning there is no fixed rule to handle multiple challenges due to various uncertainties in real-life circumstances. Many scholars from all over the world have apparently studied MCDM management techniques extensively.
is effort resulted in a multitude of innovative solutions to complex real concerns. e frameworks for this objective are largely based on a summary of the issues at hand. To deal with uncertainties the researchers have been proposed various mathematical techniques. Zadeh [1] introduced fuzzy sets (FSs), Pawlak [2] developed rough sets, and Molodtsov [3] proposed soft sets. ese sets are independent generalizations of the crisp sets. Subsequently, the idea of intuitionistic fuzzy sets (IFSs) is presented by Atanassov [4,5] as an extension of FSs and the concept of PFSs is introduced by Yager [6,7] as a generalization of IFSs. A Pythagorean fuzzy number (PFN) developed by Zhang and Xu [8] is significantly superior than the intuitionistic fuzzy number (IFN). Yager [9] introduced the idea of generalized orthopair fuzzy sets which is also known as q-rung orthopair fuzzy set (q-ROFS). Ali [10] proposed two aspects of q-ROFS in terms of L-fuzzy sets and orbits. Ali and Shabir [11] investigated fuzzy soft sets and soft sets to discuss their logic connectives. Zhang [12] introduced bipolar fuzzy sets and relations. Smarandache [13] introduced the ideas of neutrosophy, neutrosophic sets, and neutrosophic probability.
Alcantud et al. [14][15][16] introduced a technique by utilizing N-soft set approach towards rough sets. ey presented some extensions of soft sets and fuzzy sets and their applications in MCDM under incomplete information. Karaaslan and Hunu [17] studied neutrosophic [18] soft sets and their applications in decision-making. Peng et al. [19,20] introduced some results on Pythagorean fuzzy information measures and their applications in MCDM. Naeem et al. [21] proposed some novel features of Pythagorean m-polar fuzzy sets with applications. Riaz and Tehrim [22] introduced the idea of bipolar fuzzy soft topology with decision-making application. Riaz et al. [23] presented the notion of soft multirough set topology and its applications to MCDM problems.
Aggregation operators for MCDM have been studied by numerous researchers. Garg [24] presented the idea of generalized Pythagorean fuzzy information aggregation using Einstein operators and its applications to decisionmaking. Garg [24] proposed applications of Einstein operations under PFS environment. Garg et al. [25] derived new generalized dice similarity measures for complex q-rung orthopair fuzzy sets and established certain properties of suggested information measures. Garg and Arora [26] studied Archimedean t-norm of the intuitionistic fuzzy soft set (IFSS) and developed new generalized Maclaurin symmetric mean aggregation operators for information aggregation. A novel complex q-rung orthopair fuzzy Bonferroni mean was developed by Liu et al. [27] for solving MCDM problems. A robust multiple attribute group decision-making (MAGDM) method was introduced by Liu and Wang [28] by using intuitionistic fuzzy Einstein interactive operations. Hesitant intuitionistic fuzzy linguistic aggregation operators were derived by Liu et al. [29] for MADM.
Akram et al. [30] presented an extension of Dombi's aggregation operators for decision-making under m-polar fuzzy information. An application towards hospital performance measurement was proposed by Yang et al. [31] using the idea of triangular single-valued neutrosophic data. Pythagorean fuzzy Einstein weighted geometric aggregation operator was studied by Rahman et al. [32] for solving MAGDM problems. New interaction power Bonferroni mean aggregation operators were developed by Wang and Li [33] for MADM of real-world problems. Riaz et al. [34][35][36][37] proposed q-rung orthopair fuzzy hybrid, Einstein, prioritized, and Einstein prioritized AOs. Einstein operators, studied by Wang and Liu [38], were based on uncertain intuitionistic fuzzy information.
Xu et al. [39][40][41][42] proposed various aggregation operators using intuitionistic fuzzy sets and hesitant fuzzy sets for MCDM. Ye [43][44][45] introduced [46] weighted aggregation operators for MCDM with interval-valued hesitant fuzzy sets, single-valued neutrosophic sets, and simplified neutrosophic sets. Liu et al. [47] introduced linguistic intuitionistic cubic fuzzy AOs, Abdullah et al. [48] presented a novel concept of sine trigonometric picture fuzzy AOs, and Naeem et al. [49] initiated the concept of similarity measures for fractional orthotriple fuzzy sets and their applications towards MCDM. Ashraf et al. [50] initiated the novel concept of spherical fuzzy sets. e idea of linear Diophantine fuzzy set (LDFS) was defined by Riaz and Hashmi [51] as a new concept for modeling uncertainties in MADM problems. A LDFS is a strong model to relax the limitations of MD and NMD due to existence of reference/control parameters. Riaz et al. [52] extended LDFS to the idea of soft rough LDFS sets with application to sustainable material handling equipment. Riaz et al. [53] introduced spherical linear Diophantine fuzzy sets with modeling uncertainties in MCDM. e rest of this article is as follows. In Section 2, the basic concept of LDFS with its operational laws is quickly analyzed. In Section 3, some LDFS-based t-norm, t-conorm, and Einstein operational laws are defined. In Section 4, several linear Diophantine fuzzy Einstein AOs are developed. In Section 5, a robust MCDM approach with proposed operators is shown by a practical example to demonstrate the effectiveness of LDFSs for a country's national health administration (NHA) to create fully developed postacute care (PAC) model network to improve the health recovery of patients suffering from cerebrovascular diseases (CVDs). Finally, we conclude the results of this manuscript in Section 6.
to choose optimal or convincing alternatives. erefore, this article intends to tackle these difficulties with LDFS-based aggregation operators designated as the LDFEWA operator, LDFEOWA operator, LDFEWG operator, and LDFEOWG operator.
ese operators not only can derive a ranking knowledge but also can have a prominent influence in identifying the optimal choice. e inspiration for the proposed work is addressed in every part of this article. is paper has multiple objectives as follows: (1) e existing models IFSs, PFSs, and q-ROFSs have their strict limitations for truthness/membership grades (MGs) and falsity/nonmembership grades (NMGs). LDFS is an innovative flexible approach to relax these limitations. e decision makers (DMs) can choose these grades in [0, 1] without any restriction. Additionally, the reference or control parameters are used as a weight vector such that the sum of reference parameters is less than unity. ese parameters classify the physical sense of the objects and help to deal with the uncertain information about the objects under consideration.
(2) Einstein AOs based on t-norm and t-conorm are utilized to assemble the information data into a particular point. ese operators remove the inconsistency and irrationality of the operational laws and provide us a broad range for the decisionmaking patterns.
(3) e third and vital intention is to assemble a powerful association of the proposed model with the MCDM obstacles. We develop four novel operators to determine MCDM difficulties under the effect of control parameterizations of the proposed model. A useful example related to a country's national health administration (NHA) to create a fully developed postacute care (PAC) model network for the health recovery of patients suffering from cerebrovascular diseases (CVDs) is exhibited to demonstrate the practicability and efficacy of the intended approach. e set of control parameters represents an essential role in decision-making models. ey accommodate us to improve the valuation area of satisfaction and dissatisfaction functions and parameterize the model, which gives us a variety of taking alternatives under different physical situations. e deficiency in IFS, PFS, and q-ROFS is that they have no parameterizations.
is novel idea enhances the existing methodologies.
Many researchers [9,10,29] worked on the novel idea of q-ROFSs which is the annex of IFSs and PFSs with the ρ D (η ⌣ ) represent MG and NMG corresponding to alternative η ⌣ , respectively. For very large values of q, the valuation space of q-ROFS approaches to that of LDFS, but there is deficiency in q-ROFS, and it cannot deal with reference parameters, which is a key factor of LDFS. For small values of q, the valuation space of q-ROFS is smaller than that of LDFS, but this affects MCDM. For the input data in MCDM problems if σ D (η ⌣ ) and ρ D (η ⌣ ) equal to 1, then q-ROFS fails to deal with this situation because 1 + 1 > 1. However, with the suitable choice of reference parameters, we can easily deal these types of values in LDFSs (i.e., (1)(0.6) + (1)(0.3) < 1; the choice of numeric values of reference parameters is according to the situation and decision-making problem with the condition that there sum is less than 1). From all the discussion, it is clear that our proposed idea is more suitable and superior to others and contains a variety of reference parameters. We can use this approach in various applications of engineering, medical, artificial intelligence, and MADM methods. In Table 1, we can see the comparison between the proposed approach with the existing concepts. Figure 1 shows the graphical comparison of LDFNs with some existing fuzzy numbers.

Preliminaries
In this part, we recall certain rudiments of LDFSs, some of its operations, and score functions. roughout the study, we utilize Q as a universal set.
Definition 1 (see [51]). A linear Diophantine fuzzy set (LDFS) D in Q is defined as 1] are the membership grade (MG), the nonmembership grade (NMG), and the corresponding reference parameters, respectively, which satisfy the basic conditions: for all η ⌣ ∈ Q. e LDFS is called the absolute LDFS in Q. e LDFS is called the null LDFS in Q.
e reference or control parameters are valuable for modeling or analyzing a critical system. We can characterize different systems by altering the dynamic representation of these parameters. In addition, ) is called the indeterminacy degree and its corresponding reference parameter of η ⌣ to D.

Journal of Mathematics
Example 1 (reimbursement cases for medical treatment). Medical expenses must be primarily to relieve or check a physical or mental sickness, illness, or disorder. ey do not include expenses that are for cosmetic purposes or are merely beneficial to general health. Usually, the reimbursement is organized directly between the establishments involved, so you need not to pay for the treatment. However, in many cases, the treatment involves highly qualified doctors at well-established hospitals. So, people take their treatment by self-payment and tend to reimbursement cases. ese cases will be sent to national health authority (NHA) for their approval and releasing expenses. However, the decision experts (DEs) will decide the amount of expenses to be paid. e DEs also provide particular list of medicine that are, and are not, eligible for reimbursement. Since medicines are compounds or chemicals products that are used to cure, ease symptoms, prevent disease, or help in the treatment of illnesses, so it is critical to decide which medicine is really used, or not used, for treatment and which is used for other purposes. Let be the list some lifesaving medicine. Two or more drugs can be combined in an appropriate ratio during preparation of medicines to get a high impact of such medicines. If the reference or control parameter is considered as σ Λ � "Membership grades or favorable in health recovery," ρ Λ � "Non − membershi pgrades or not favorable in health recovery," α Λ � "Less side effect safter surgeries," β Λ � "More side effect safter surgeries," then the LDFS is given in Table 2.

Grade of dissatisfaction Grade of dissatisfaction
Grade of satisfaction Grade of satisfaction q-power sum of satisfaction and dissatisfaction grades approaches to one Sum of product of satisfaction and dissatisfaction grades with their corresponding reference parameters equal to one Square sum of satisfaction and dissatisfaction grades equal to one.
Square sum of satisfaction and dissatisfaction grades equal to one.
Sum of satisfaction and dissatisfaction grades equal to one.

Einstein Operational Laws for LDFNs
Within this section, the t-norm and t-conorm are given as an outline, and we shall provide examples of these concepts. Some of the characteristics of Einstein's operations are given to LDFNs.
For the very first time, triangular norms have indeed been introduced in the form of probabilistic metric spaces, as we are using them currently. It also plays an essential role in statistics, decision-making, and cooperative games. Most parameterized t-norm groups are renowned for their functional equation solutions. T-norms have been used for fuzzy set theory at the intersection of two fuzzy sets. T-conorms are used for modeling disjunction or union. Triangular norms and conorms are operations that generalize logical conjuncture and logical disjuncture to fuzzy logic. ese are basic descriptions of the conjunction and disjunction in mathematical fuzzy logic semantics which are used in the MCDM to integrate the prerequisite. e t-norm is expressed by the binary operation ∐ that meets the required constraints at the interval [0, 1]: (iv) ∐(Γ, 1) � 1 (neutral element 1) and ∐(0, 0) � 0 A t-conorm is a binary operation ⋓ that meets specific constraints at the interval [0, 1]: > 0 be any real number; then, Einstein operations for LDFNs is defined as follows: > 0 be any real number; then, Proof. Here, we prove (ii), (iii), (iv), and (viii), remaining are similar to the following. Similarly, Now, suppose ,

Journal of Mathematics
Hence, we prove 8 Journal of Mathematics , On right-hand sides, Journal of Mathematics 9 Hence, we prove . erefore, Journal of Mathematics is is a trivial case. Hence, we omit the proof. □ is is a trivial case. Hence, we omit the proof. □

Linear Diophantine Fuzzy Einstein Aggregation (LDFEA) Operators
In this segment, we propose certain fresh Einstein operators for LDFNs, namely, LDFEWA operator, LDFEOWA operator, LDFEWG operator, and LDFEOWG operator with some of their properties.
is called LDFEWA operator and can be written as Proof. By using the process of Mathematical induction, we prove this result. For Λ � 2, As we know that both ℶ 1 . E Λ 1 and ℶ 2 . E Λ 2 are LDFNs and ℶ 1 . en, We proved that Λ � 2. Presume that the result for Λ � k is correct, and we have obtained

Journal of Mathematics 13
Now, we are going to prove n � k + 1: Result holds for Λ � k + 1. In this way, we have completed the proof. □ Theorem 5. Let Λ Λ � ( σ Λ , ρ Λ , α Λ , β Λ ) be a collection of LDFNs. en, Proof. We can easily proof this theorem by definition, so we omit the proof.

□
Proof. We can easily proof this theorem by definition, so we omit the proof. where
Presume that the result for Λ � k is correct, and we have obtained Now, we are going to prove n � k + 1: Result holds for Λ � k + 1. In this way, we complete proof. 1, 2, 3, . . . , n} be a collection of LDFNs. en, 1, 2, . . . , n) for all Λ, then Proof. We can easily proof this theorem by definition, so we omit the proof.

MCDM to Hospital-Based PAC-CVD
In this portion, the first case study regarding the selection of hospitals in the PAC-CVD program is presented. We set some attributes related to the selected list of alternatives for the appropriate decision. en, we establish a novel algorithm based on LDFEWA and LDFEWG operators to deal with this MCDM problem. A numerical example is shown to verify the consistency, versatility, and superiority of our suggested aggregation operators and algorithms.

Case Study for Hospital-Based PAC-CVD.
A country's national health administration (NHA) is deciding to create a fully developed postacute care (PAC) model system to improve the quality of recovery of patients with cerebrovascular diseases (CVDs). Cerebrovascular accident (CVA) is usually known as stroke. Stroke results in weakness of limbs, one-sided weakness, and whole body weakness means paralysis. It may cause paralysis of tongue and inability to speak even due either blockage of vessels supplying specific area of brain (thrombosis of vessels) in which case it is called infarctive stroke, and in other case, it may be due to rupture of vessels supplying specific area of the brain in which case it is called haemorrhagic stroke, weakness, or paralysis and depends on type of vessel damaged that causes brain tissue damage, and ultimately, the part of the body that is controlled by that area of the brain is paralyzed. ere are some other types such as stenosis which is due to narrowing of blood vessels supplying the brain and results in decreased blood supply to the brain causing different effects on the body depending on vessel, and it may include carotid stenosis and vertebral stenosis. Also, vascular malformations may be congenital (present at time of birth) and acquired (that are caused by different diseases at any stage of life). e management flowchart of PAC-CVD program is given in Figure 2. Improving the efficiency of hospital-based PAC in emergency stroke treatment, assessment, and selection of proper medical care facilities is crucial in the program. However, the selection of pilot hospitals is an extremely confused and MCDM problem, due to the complexity of NHA in the country. To illustrate the workability and applicability of the suggested operators in the medical and health care domains, an interpretive problem concerning the assortment of pilot hospitals in the PAC-CVD program is implemented.
Participants considered for PAC-CVD must include a network of health care organizations at different levels, directed by a primary institution. A pilot hospital must accommodate restoration services (including physical therapy, occupational therapy, and speech therapy) and develop a PAC team of doctors of PAC-CVD qualifications including neurologists, neurosurgeons, internists, and family physicians. e risk factors for this problem are given in Figure 3. It is important to take serious decision about this problem. e death rate chart per 100, 000 population due to CVD is given in Figure 4. e criteria needed for pilot hospital recruitment are set out in Table 3.
A clear and sufficient budget was given for the PAC-CVD program. Five function-related groups (FRGs) were designed for reasonably practicable patient care passageways. Groups are shown in Table 4.

Proposed Methodology.
We discuss MCDM concern with the use of proposed operators. Take into account the set of alternatives Λ � Λ 1 , Λ 2 , . . . , Λ m with m elements, and . n is the set of criterions with n elements. Decision makers have their specific opinion matrix D � (P ij ) m×n , where P ij is given for the alternatives Λ i ∈ Λ with respect to the criteria O .. j ∈ O .. by the decision maker in the form of LDFNs. You can also address the MCDM concern in terms of a LDFNs decision matrix provided by D � (P ij ) m×n � ( σ ij , ρ ij , α ij , β ij ) m×n . e proposed operators will apply to the MCDM which will include the following points.
Step 1: obtain the judgement matrix D � (P ij ) m×n in the format of LDFNs from the decision maker (Table 5).
Step 2: the parameters in the judgement matrix are characterized by two categories, such as the cost form criteria (τ c ) and the benefit form criteria (τ b ). If all parameters are from the same form, normalization is not needed, so there are various forms of parameters in the MCDM; in this case, the D matrix has been converted into a normalization matrix using the normalization formula Y(k ij ) � ( σ ij , ρ ij , α ij , β ij ): where P c ij shows the compliment of P ij .

(54)
Step 4: determine the score value through all combined alternatives evaluations.
Step 5: alternative options are sorted first by score function and the best alternative can be selected. e pictorial view of the algorithm is shown in Figure 5.

Numerical Example.
All participating hospitals had the right to financial rewards, surcharges for outpatient diagnosis and treatment, and requests for reimbursement of other types of medical expenses. e suggested approach was used to assist NHA in selecting the best set of hospitals. Here, we use LDFN, Λ Λ � ( ρ Λ , σ Λ , β Λ , α Λ 〉): Λ � 1, 2, 3, . . . , n}, in this approach if we consider the control parameters as α � highly effect on patients and β � not highly effect on patients.
Stroke patients in neurology or neurosurgery wards who met the criteria of the PAC-CVD program PAC setting in one regional hospital (n = 169) 1. Case manager, physiatrist (medical director), physical therapist, occupational therapist, speech therapist, nutritionist, pharmacist, or social worker provide PAC service 2. Assessment of function and quality of life at admission, every 3 weeks and before discharge (n = 1) Death during hospitalization due to cardiac event (n = 129) Return to home and community (n = 11) Referral to long-term care facilities (n = 28) Referral back to acute wards due to infection or recurrent-stroke   1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 Year   Table 3, is the finite set of criterions.
e decision maker provides a matrix of their own opinion D � (P ij ) 5×6 given in Table 6.
Step 2: we get the normalized decision matrix given in Table 7 by taking complement of cost-type criterions given in Table 3, where O .. 6 � operational cost is the cost-type criterion.
Step 3: evaluate combined evaluations of alternatives utilizing the LDFEWA operator: For i � 1, For i � 5, Table 4: Function-related groups.

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Journal of Mathematics   (60) Step 4: the scoring values pertaining to any k i are (61) Step 5: rank the value obtained in Step 4 using the score function:
e decision maker provides a matrix of their own opinion D � (P ij ) 5×6 given in Table 8.
Step 2: we get the normalized decision matrix given in Table 9 by taking compliment of cost type criterions given in Table 3, where O .. 6 � operational cost is the cost type criterion.
Step 3: determine cumulative assessments of the alternatives by using the LDFEWG operator: k i � LDFEWG k i1 , k i2 , . . . , k in .

Comparison Analysis.
In this section, we compare proposed AOs with some existing AOs. e uniqueness of our proposed operators is that they both yield the same result. We equate our results by solving the information data with some preexisting operators and arriving at the same optimal decision. is demonstrates the robustness and validity of our proposed models. Because of their reference parameterizations, the presented techniques on LDFNs are more effective and superior to some existing theories. e beauty of this structure is that it creates independence between membership and nonmembership grades and creates categorization criteria due to parameterizations. e comparison of presented aggregation operators with some existing operators is given in Table 10.

Conclusion
Multi-criteria decision-making (MCDM) is an important real decision problem, and its most basic and most important research direction is how to express these uncertain information. e IFSs, PFSs, and q-ROFSs are all a good way to deal with fuzzy information. However, LDFSs are more general; their outstanding ability is to relax the strict constraints of IFS, PFS, and q-ROFS by considering reference/ control parameters. MCDM is an important branch of operation research. e procedures extended for this assignment essentially depend on the nature of problem under judgement. Our daily-life circumstances are unpredictable, imprecise, and obscure. e existing structures were assembled below the hypothesis that decision makers (DMs) consider certain constraints while evaluating multiple alternatives and attributes. However, this kind of state becomes difficult for DMs so that they will assign MGs and NMGs under various restrictions. In order to relax these restrictions, LDFS is a new approach towards uncertainty and decision-making problems which incorporate pair of reference or control parameters against MGs and NMGs. We have utilized LDFSs for evaluating the integrity of DMS knowledge in the fundamental framework and to eliminate any deformation in the decision analysis. e influential privilege of computing the control parameters into the examination is to decrease the feasibility of errors that are created by the theoretical knowledge depending on fundamental evaluations of MGs and NMGs. Additionally, we have developed several aggregation operators named as LDFEWA operator, LDFEOWA operator, LDFEWG operator, and LDFEOWG operator. Many interesting characteristics of the proposed operators are studied and their illustration is well proved. A brief discussion regarding LDFS-based t-norm and t-conorm is expressed. A practical application of the proposed method for MCDM to the national health administration (NHA) for the creation of a fully developed postacute care (PAC) model network is given. Comparative studies with some existing approaches demonstrate the feasibility and reliability of the proposed operators. e superiority of the proposed work has been justified with the validity test. us, we conclude that the proposed approach presents a better and easier way to solve the uncertainties of real-life problems. e proposed work exhibits a broad scope of potential applications. For further research, recognizing the perfection of novel LDFSs, it may be extended to any other aggregation operators, such as prioritized AOs, power mean AOs, Dombi's AOs, Bonferroni mean AOs, and Heronian mean AOs. We hope that our research results will be successful for researchers working in the fields of information aggregation, statistical techniques, intelligent systems, machine learning, neural networks, and psychiatric disorder.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare no conflicts of interest.