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Article

Linear Diophantine Fuzzy Fairly Averaging Operator for Suitable Biomedical Material Selection

by
Hafiz Muhammad Athar Farid
1,
Rukhsana Kausar
1,
Muhammad Riaz
1,
Dragan Marinkovic
2,3,* and
Miomir Stankovic
4
1
Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan
2
Department of Structural Analysis, Technical University Berlin, 10623 Berlin, Germany
3
Faculty of Mechanical Engineering, University of Nis, 18000 Nis, Serbia
4
Mathematical Institute of the Serbian Academy of Sciences and Arts, 11001 Belgrade, Serbia
*
Author to whom correspondence should be addressed.
Axioms 2022, 11(12), 735; https://doi.org/10.3390/axioms11120735
Submission received: 21 November 2022 / Revised: 6 December 2022 / Accepted: 9 December 2022 / Published: 15 December 2022
(This article belongs to the Special Issue Multiple-Criteria Decision Making II)

Abstract

:
Nowadays, there is an ever-increasing diversity of materials available, each with its own set of features, capabilities, benefits, and drawbacks. There is no single definitive criteria for selecting the perfect biomedical material; designers and engineers must consider a vast array of distinct biomedical material selection qualities. The goal of this study is to establish fairly operational rules and aggregation operators (AOs) in a linear Diophantine fuzzy context. To achieve this goal, we devised innovative operational principles that make use of the notion of proportional distribution to provide an equitable or fair aggregate for linear Diophantine fuzzy numbers (LDFNs). Furthermore, a multi-criteria decision-making (MCDM) approach is built by combining recommended fairly AOs with evaluations from multiple decision-makers (DMs) and partial weight information under the linear Diophantine fuzzy paradigm. The weights of the criterion are determined using incomplete data with the help of a linear programming model. The enhanced technique might be used in the selection of compounds in a variety of applications, including biomedical programmes where the chemicals used in prostheses must have qualities similar to those of human tissues. The approach presented for the femoral component of the hip joint prosthesis may be used by orthopaedists and practitioners who will choose bio-materials. This is due to the fact that biomedical materials are employed in many sections of the human body for various functions.

1. Introduction

Numerous academic fields, including sociology, epistemology, intellectual technology, and machine learning, investigate how humans come to their conclusions and make decisions in response to the myriad of challenges that they confront on a daily basis. In general, several quantitative and analytical models are used in an effort to characterise these processes. The difficulty of making decisions is a challenge that develops during this procedure. The process of choosing one or more of the alternate forms of behaviour faced by individuals or an organisation in order to achieve a particular goal is referred to as “decision making”, and it is distinguished as the procedure of selecting one or more of the available options. According to research, while it is possible to get by with making many of your day-to-day judgments based just on gut instinct, this method is not sufficient for making significant and important choices on its own. MCDM refers to a group of analytic methods that analyse the benefits and drawbacks of potential options based on a number of different factors. Methods from MCDM are utilised to provide assistance for the decision-making process as well as to pick one or more alternatives from a group of alternatives featuring varying features based on criteria that are in conflict with one another or to rank these options. In different terms, while using MCDM approaches, decision-makers evaluate the various options based on a number of criteria in order to rank them according to the attributes that are most important to them.
The choice of materials is one of the most important parts of the process of designing, researching, and making products. The outcomes of the material procurement process have a direct impact on both the product’s quality and its budget [1]. When used in a particular product, the optimum material will allow the product to have the highest possible performance while also having the lowest possible cost. It is thus of utmost importance to determine how to choose the best material from among the available alternatives [2]. Because of developments in materials science and improvements in production processes, the types of materials that may be selected are increasing in number, and the product requirements that must be taken into account during the selection procedure are becoming more in depth. The difficulty of selecting the appropriate material is made more difficult by the extensive prerequisites as well as the extensive range of options. Therefore, it is of utmost importance to investigate suitable methods to handle the problem of material selection [3]. During the practise of selecting materials, several different product criteria, including productivity, affordability, stability, dependability, market dynamics, fashion, and so on, need to be taken into consideration in order to choose the best material [4]. These product needs can be thought of as selection criteria, and different materials can be assessed using the data on the factors [5].
A bio-material, which is defined as a material meant to come into contact with biological structures, can be used to develop, treat, or modify a human tissue, muscle, or physiological function. An implant is a device that is entirely or partly implanted under the epithelial surface. Implants are defined as devices that are made of one or more bio-materials. The body’s vastly different perimeter necessitates the use of biomedical materials. During the course of your daily activities, the bones in your body are subjected to a range of stressors. Similarly, when the body is in motion, orthopaedic materials are subjected to billions of loading cycles. Fatigue resistance and mechanical toughness are other important considerations. A number of natural and synthetic materials, referred to together as biomedical materials, can fulfil or assist the functions of live tissues in the human body. Because the human body is made up of proteins and oxygenated salt solutions, it is reasonable to expect that these materials will not bloat, deform, or corrode as a result of absorbing biological fluids. Under these conditions, some implant materials are accepted by the body, while others are rejected. Materials used in bio-medicine should not be corrosive, toxic, or carcinogenic; they should also have adequate mechanical strength; they should not cause reactions that are not naturally occurring in the body; and they should not decay. Biomedical materials are employed in orthopaedic applications such as joint prostheses and skeletal appropriate substitute materials. Other applications for these materials include frontal and reattachment surgery, dentistry, heart valves, artificial heart parts, catheters, backbone instrumentation, fixator material, metal parts, perforated screws, screw washers, fixator wires, nails, hip plates, angled plates, anatomical plates, and implantable devices. Metals are the preferred material for use in the biomedical sector due to their extended lifespan, malleability, and abrasion resistance. Metals, on the other hand, have limited bio-compatibility, exceedingly difficult corrosion in human fluids compared to bodily tissues, a high density, and the potential to elicit allergic tissue responses. Ceramics are dense, hard, brittle, and difficult to manufacture materials with low mechanical properties and excellent bio-compatibility. Furthermore, ceramics offer high bio-compatibility and corrosion resistance. Composite materials have been developed as a replacement for less attractive materials that were previously utilised. Polymers are utilised in general plastic surgery materials for the circulatory system and general plastic surgery operations, as opposed to metallic bio-materials and bio-ceramics used in orthopaedic and dental implants. Hip replacement surgery, commonly known as total hip arthroplasty, is the medical term for the procedure of replacing a damaged or worn out hip joint. The patient’s native joint will be replaced with an artificial joint during this treatment.
The information regarding criteria for alternative materials is typically hazy and imprecise. This is because human cognition is inherently cloudy, and the qualities of materials are not always completely clear. As a result, the criteria information is better suited for depiction by a fuzzy set (FS) [6], and the fuzzy MCDM approaches have been utilised in order to tackle a variety of material decision issues. Despite this, the fuzzy MCDM approach is still susceptible to the following drawbacks: the features of the material grow increasingly complicated and unclear, and the information needed to evaluate them cannot be adequately represented by FS. As a result, material attribute knowledge must be expressed using a more powerful quantitative tool because: (1) the consistency of the attribute knowledge expressed by FS is uncertain, and if the attribute knowledge is not precise, the final outcome of the material selection will be incorrect; and (2) the consistency of the attribute knowledge expressed by FS is uncertain.
Atanassov expanded FSs by developing the innovative concept of “intuitionistic fuzzy sets” (IFS) [7], which is described as having a “membership degree (MSD)” and a “non-membership degree (NMSD)” that is less than or equal to 1. When dealing with ambiguous, unclear, or insufficient data, one of the most powerful and effective techniques has been demonstrated to be the IFS hypothesis. This element of the IFS is important for a large number of professionals that work with real-world scenarios within the context of an IFS framework. In 2019, Riaz and Hashmi carried out an exhaustive evaluation of the constraints associated with MSDs and NMSDs in the structures of FS, IFS, “Pythagorean fuzzy set” (PFS) [8], and “q-rung orthopair fuzzy set” (q-ROFS) [9], and such limitations were defined. They came up with the “linear Diophantine fuzzy set” (LDFS) [10] as a solution to these problems by including IFS-specific reference parameters (RPs) into its design. They contend that the LDFS concept will eliminate the restrictions placed on the selection of features in exercise by the methodologies that are now in use for the various sets, and that this will make it possible to pick features with no limits. By utilising the arbitrary quality of the RPs, they were also able to prove that the universe of this set has a greater number of occurrences than the FS, IFS, PFS, and q-ROFS did. Table 1 shows a quick comparison of the proposed approach with current notions.
Despite the fact that much study has been done on material selection in the past, there is a requirement for a clear and comprehensive scientific technique or computational tool to aid user organisations in making the right material selection conclusion. A material selection procedure’s goal is to discover material selection qualities and acquire the best acceptable mix of material selection attributes in conjunction with actual demand. We utilised LDFS and fairly operations to construct fairly AOs for this purpose.
The remainder of the article is structured as follows: Section 2 conducts a literature study on the material selection process and AOs. Section 3 discusses some fundamental LDFS definitions. Section 4 introduces some fairly operations and verifies the corresponding theorems. Section 5 discusses certain LDF-fairly AOs and their properties. Section 6 elaborates on the MCDM material selection approach and discusses the computation procedure. The case study and decision-making procedure are proposed in Section 7. Section 8 is a summary of the whole study.

2. Literature Review

This section discusses some relevant research on the MCDM approach for the selection of materials and various AOs.

2.1. Literature Review on MCDM Method for Material Selection

The material selection technique focuses most of its research on the MCDM approach, and it has been used a lot to solve a wide range of material selection problems. Shanian and Savadogo [11,12] initially used the traditional MCDM techniques “Elimination and Choice Expressing REality” (ELECTRE) and “Technique for Order Preference by Similarity to an Ideal Solution)” (TOPSIS) to the problem of material selection. Using these approaches, they selected the optimal materials for certain products. Liu et al. [13] introduced a unique hybrid MCDM approach based on an enhanced version of “VIsekriterijumska Optimizacija I Kompromisno Resenje” (VIKOR) to handle the issue of interrelated multiple factors and industry standard in material selection.
The interval-valued VIKOR method was presented by Jahan and Edwards [14] as a solution to the target-dependent materials selection difficulties. Peng and Xiao [15] developed a unique MCDM approach that combines the “Preference Ranking Organisation Methods for Enrichment Evaluations ” (PROMETHEE) and “analytic network process” (ANP) to determine the optimal material for a given application. Rao and Davim [16] used TOPSIS and the “analytical hierarchy process” (AHP) to handle the problem of selecting materials for a given technological design. Moreover, Rao [4] offered a systematic procedure for the selection of materials by combining the VIKOR and AHP. Combining AHP, grey correlation, and TOPSIS, Tian et al. [17] developed a hybrid MCDM strategy to address the issues involved with the selection of ecologically sustainable ornamental materials. Zhang et al. [18] developed a hybrid MCDM approach by combining the tactical decision and assessment laboratories, grey relational analysis, ANP, and TOPSIS to determine the most effective environmentally friendly material. Jahan et al. [19] created an aggregation process in an effort to address the problem of material selection. The inputs for this approach are the material ranking orders generated by various MCDM techniques, and the outputs are the aggregate rankings.
When there is a degree of unpredictability in the knowledge about the requirements for materials, fuzzy MCDM is the method that is most suited for finding solutions to material selection issues. Steel alloys materials selection difficulties were the focus of the presentation of Wang and Chang [20] on the fuzzy MCDM technique, which proposes that the ideal material may be identified through the aggregate and ordering of fuzzy numbers. Rathod and Kanzaria [21] came up with the idea for an assessment model that is based on fuzzy AHP and TOPSIS so that they could choose a suitable phase transition material. An incorporated fuzzy MCDM technique was given by Sa et al. [22] to address the issue of the green materials segment. This method integrates the fuzzy AHP and fuzzy TOPSIS approaches. Girubha and Vinodh [23] came up with the idea of using the fuzzy VIKOR approach to choose the best material for the electricity instrument cluster. Khabbaz et al. [24] came up with a technique that is a simplified form of fuzzy logic. This approach is able to readily cope with the qualitative character of materials and the fuzzy space that corresponds to it, which allows for improved material selection. Material selection was one of the applications Yang and Ju [25] used for their newly constructed fuzzy variable, which they named uncertain member language variable. An evaluation framework was presented by Roy et al. [26] for the purpose of solving the material selection problem. Farid and Riaz [27] introduced Einstein interactive AOs related to a neutrosophic set with its use to material selection in engineering design. Zhang et al. [28] proposed the MCDM approach for material selection based on group generalised AOs related to PFSs.
In terms of implementation possibilities, fuzzy MCDM is superior when it comes to material selection. When it comes to material selection, expanded variants of FS such as LDFS are utilised rather infrequently. In the same vein, the issue of determining whether or not criteria information has been accurately represented is seldom ever researched. As a result, the purpose of this study is to attempt to address these gaps.

2.2. Literature Review on Fuzzy AOs

AOs are capable of processing and making use of data effectively. Different AOs will be produced as a result of different sets of rules or by focusing on various components of the available information. Jana et al. [29] proposed a new MCDM dynamic approach with the help of some complex AOs. Feng et al. [30] introduced some notions about IFSs. Ashraf et al. [31] developed some sine trigonometric AOs for a single valued neutrosophic. Liu et al. [32] introduced “power Maclaurin symmetric mean” AOs for q-ROFNs with applications to MCDM. Xing et al. [33] presented the concept of point weighted AOs for q-ROFNs. Liu and Wang [34] gave the idea of “Archimedean Bonferroni AOs” for q-ROFNs. Liu et al. [35] developed a heterogeneous relationship among the criteria for q-ROFNs. Mahmood and Ali [36] proposed complex q-ROF Hamacher AOs for MCDM. Hussain et al. [37] proposed AOs for hesitant q-ROFSs, and Jana et al. [38] initiated the concept of AOs for the MCDM method using a bipolar fuzzy soft set. Saha et al. [39] gave the concept of fairly AOs for q-ROFSs.
Liu and Shi [40] presented linguistic Heronian mean AOs, Lu and Ye [41] proposed some exponential AOs and Li et al. [42] gave generalised Einstein AOs for SVNNs. Saha et al. [43], Wei and Zhang, and Wei and Wei [44] gave some different AOs related to different extensions of FSs. Alcantud [45] presented some extensive results related to soft sets. Karaaslan and Ozlu [46] developed some work related to dual type-2 hesitant FSs. Senapati et al. [47] proposed Aczel–Alsina geometric AOs for interval-valued IFSs. Wang and Zhang [48] introduced MCDM based on rough sets and fuzzy measures. Gergin et al. [49] proposed some extensive MCDM approch for supplier selection. Narang et al. [50] introduced the decision-making approach based on Heronian mean AOs. Karamaşa et al. [51] gave the idea of neutrosophic operational sciences techniques. Some brilliant work related to some extension of fuzzy sets can be seen in [52,53,54].
While dominating AOs intervene to settle MCDM difficulties within the LDF framework, they seldom investigate the impartiality of their peers when addressing with MSD and NMSD. For instance, the values derived by AOs that previously exist in pieces of literature cannot be distinguished when a DM supervises all MSD and NMSD for a comparable work. This indicates that the ultimate conclusion is unquestionably biased. Thus, new procedures are necessary to handle MSD and NMSD properly and to ensure that LDFN operates fairly or neutrally. Using the notion of proportionate distributing rules for all functions, we develop two neutral or fair procedures in order to attain genuine pleasure while assessing MSD and NMSD.
As a result of this issue, the key aims of the article are as follows:
  • LDFNs are very good at tackling challenges with a two-degree complexity scale. LDFSs are employed to develop new AOs.
  • Construct several distinct fairly or neutral procedures for handling the MSD and NMSG using the interaction coefficient in an acceptable manner.
  • We proposed two new AOs, namely, the “linear Diophantine fuzzy fairly weighted averaging (LDFFWA) operator” and the “linear Diophantine fuzzy fairly ordered weighted averaging (LDFFOWA) operator”.
  • Several innovative concepts linked with freshly generated AOs for data fusion are demonstrated by a sufficient number of illustrative cases. The suggested operators provide information that is more general, trustworthy, and accurate than previous methods.
  • Using the provided AOs, a novel MCDM approach for modelling uncertainty in material selection is developed.

3. Certain Fundamental Concepts

This portion of the article will provide various essential notions connected to LDFSs, and it will do so in the context of the universal set X.
Definition 1
([10]). A “linear Diophantine fuzzy set (LDFS)” ℜ in X is defined as
= ϖ , ξ ( ϖ ) , ρ ( ϖ ) , η ( ϖ ) , β ( ϖ ) : ϖ X ,
where ξ ( ϖ ) , ρ ( ϖ ) , η ( ϖ ) , β ( ϖ ) [ 0 , 1 ] are the MSD, the NMSD and the corresponding reference parameters (RPs), respectively. Moreover,
0 ϖ ( ϖ ) + β ( ϖ ) 1 ,
and
0 η ( ϖ ) ξ ( ϖ ) + β ( ϖ ) ρ ( ϖ ) 1
for all ϖ X . The LDFS
X = { ( ϖ , 1 , 0 , 1 , 0 ) : ϖ X }
is known as the “absolute LDFS” in X. The LDFS
ϕ = { ( ϖ , 0 , 1 , 0 , 1 ) : ϖ X }
is known as the “null LDFS” in X.
Specific components can be modeled or classified using the RPs. We can categorise distinct systems by changing the physical significance of the RPs. In addition, η ( ϖ ) π ( ϖ ) = 1 ( η ( ϖ ) ξ ( ϖ ) + β ( ϖ ) ρ ( ϖ ) ) is called the “indeterminacy degree” and its corresponding reference parameter of ϖ to .
It is clear that our proposed approach is more acceptable and superior to others, and it incorporates a diverse set of reference variables. This technique may be used for a wide range of technological, medical, intelligent systems, and MADM applications.
Definition 2
([10]). A “linear Diophantine fuzzy number (LDFN)” is a tuple
α γ = ( ξ α γ , ρ α γ , η α γ , β α γ ) satisfying the following conditions:
(1)
0 ξ α γ , ρ α γ , η α γ , β α γ 1 ;
(2)
0 η α γ + β α γ 1 ;
(3)
0 η α γ ξ α γ + β α γ ρ α γ 1 .
Definition 3
([10]). Let α γ = ( ξ α γ , ρ α γ , η α γ , β α γ ) be a LDFN; then, the “score function” Υ ˘ ( α γ ) can be defined by the mapping Υ ˘ α γ ) : L D F N ( X ) [ 1 , 1 ] and given by
Υ ˘ ( α γ ) = 1 2 [ ( ξ α γ ρ α γ ) + ( η α γ β α γ ) ]
where L D F N ( X ) is a collection of LDFNs on X.
Definition 4
([10]). Let α γ = ( ξ α γ , ρ α γ , η α γ , β α γ ) be a LDFN; then, the “accuracy function” can be defined by the mapping Θ : L D F N ( X ) [ 0 , 1 ] and given as
Θ ( α γ ) = 1 2 ξ α γ + ρ α γ 2 + ( η α γ + β α γ )
Definition 5
([10]). Let α γ 1 and α γ 2 be two LDFNs; then, by using the score function and accuracy function, we have:
(i) 
If Υ ˘ ( α γ 1 ) < Υ ˘ ( α γ 2 ) then α γ 1 < α γ 2 ,
(ii) 
If Υ ˘ ( α γ 2 ) < Υ ˘ α γ 1 ) then α γ 2 < α γ 1 ,
(iii) 
If Υ ˘ ( α γ 2 ) = Υ ˘ α γ 1 ) then,
(a) 
If Θ ( α γ 1 ) < Θ ( α γ 2 ) then α γ 1 < α γ 2 ,
(b) 
If Θ ( α γ 2 ) < Θ ( α γ 1 ) then α γ 2 < α γ 1 ,
(c) 
If Θ ( α γ 1 ) = Θ ( α γ 2 ) then α γ 1 = α γ 2 .
Definition 6
([10]). Let α γ 1 = ( ξ 1 , ρ 1 , η 1 , β 1 ) be a LDFN and X > 0 . Then
  • α γ 1 c = ( ρ 1 , ξ 1 , β 1 , η 1 ) ;
  • X α γ 1 = 1 ( 1 ξ 1 ) X , ρ 1 X , 1 ( 1 η 1 ) X , β 1 X ;
  • α γ 1 X = ξ 1 X , 1 ( 1 ρ 1 ) X , η 1 X , 1 ( 1 β 1 ) X .
Definition 7
([10]). Let α γ i = ( ξ i , ρ i , η i , β i ) be two LDFNs with i = 1 , 2 . Then
  • α γ 1 α γ 2 ξ 1 ξ 2 , ρ 2 ρ 1 , η 1 η 2 , β 2 β 1 ;
  • α γ 1 = α γ 2 ξ 1 = ξ 2 , ρ 1 = ρ 2 , η 1 = η 2 , β 1 = β 2 ;
  • α γ 1 α γ 2 = ξ 1 + ξ 2 ξ 1 ξ 2 , ρ 1 ρ 2 , η 1 + η 2 η 1 η 2 , β 1 β 2 ;
  • α γ 1 α γ 2 = ξ 1 ξ 2 , ρ 1 + ρ 2 ρ 1 ρ 2 , η 1 η 2 , β 1 + β 2 β 1 β 2 .
Definition 8
([10]). Let α γ i = ( ξ i , ρ i , η i , β i ) be the assemblage of LDFNs with i Δ . Then
  • i Δ α γ i = sup i Δ ξ i , inf i Δ ρ i , sup i Δ η i , inf i Δ β i ;
  • i Δ α γ i = inf i Δ ξ i , sup i Δ ρ i , inf i Δ η i , sup i Δ β i .
Example 1.
Consider α γ 1 = ( 0.810 , 0.470 , 0.520 , 0.390 ) and α γ 2 = ( 0.910 , 0.360 , 0.640 , 0.270 ) be two LDFNs. Then, it is clear that α γ 1 α γ 2 . One can verify that
  • α γ 1 c = ( 0.470 , 0.810 , 0.390 , 0.520 ) ;
  • α γ 1 α γ 2 = ( 0.910 , 0.360 , 0.640 , 0.270 ) = α γ 2 ;
  • α γ 1 α γ 2 = ( 0.810 , 0.470 , 0.520 , 0.390 ) = α γ 1 ;
  • α γ 1 α γ 2 = ( 0.9829 , 0.1692 , 0.8272 , 0.1053 ) ;
  • α γ 1 α γ 2 = ( 0.7371 , 0.6608 , 0.3328 , 0.5547 ) .
In addition, if X = 0.1 , then we have
  • X α γ 1 = ( 0.1530 , 0.9272 , 0.0707 , 0.9101 ) ;
  • α γ 1 X = ( 0.9791 , 0.0615 , 0.9366 , 0.0482 ) .
Definition 9.
Let α γ 1 = ξ 1 , ρ 1 , η 1 , β 1 and α γ 2 = ξ 2 , ρ 2 , η 2 , β 2 be two LDFNs and , 1 , 2 > 0 be the real numbers, then we have,
  • α γ 1 α γ 2 = α γ 2 α γ 1 ;
  • α γ 1 α γ 2 = α γ 2 α γ 1 ;
  • α γ 1 α γ 2 = α γ 1 α γ 2 ;
  • α γ 1 α γ 2 = α γ 1 α γ 2 ;
  • 1 + 2 α γ 1 = 1 α γ 1 2 α γ 2 ;
  • α γ 1 1 + 2 = α γ 1 1 α γ 2 2 .
If ξ α γ 1 = ρ α γ 1 and ξ α γ 2 = ρ α γ 2 then we obtain ξ α γ 1 α γ 2 ρ α γ 1 α γ 2 , ξ α γ 1 α γ 2 ρ α γ 1 α γ 2 , ξ α γ 1 ρ α γ 1 , ξ α γ 1 ρ α γ 1 . Thus, none of the operations α γ 1 α γ 2 , α γ 1 α γ 2 , α γ 1 , α γ 1 are located to be impartial or honest, indeed. So, at the very start, our attention has to be in the direction of expanding a few fairly operations among IFNs.

4. Fairly Operations on LDFNs

In this section, we develop some fairly operations between LDFNs and study their primary properties.
Definition 10.
Consider α γ 1 = ξ α γ 1 , ρ α γ 1 , η α γ 1 , β α γ 1 and α γ 2 = ξ α γ 2 , ρ α γ 2 , η α γ 2 , β α γ 2 are the two LDFNs and > 0 . Then, we define.
α γ 1 ˜ α γ 2 = 1 2 ξ α γ 1 ξ α γ 2 ξ α γ 1 ξ α γ 2 + ρ α γ 1 ρ α γ 2 × 1 + 2 ξ α γ 1 ρ α γ 1 2 ξ α γ 2 ρ α γ 2 , 1 2 ρ α γ 1 ρ α γ 2 ξ α γ 1 ξ α γ 2 + ρ α γ 1 ρ α γ 2 × 1 + 2 ξ α γ 1 ρ α γ 1 2 ξ α γ 2 ρ α γ 2 , η α γ 1 η α γ 2 η α γ 1 η α γ 2 + β α γ 1 β α γ 2 × 1 1 η α γ 1 β α γ 1 1 η α γ 2 β α γ 2 , β α γ 1 β α γ 2 η α γ 1 η α γ 2 + β α γ 1 β α γ 2 × 1 1 η α γ 1 β α γ 1 1 η α γ 2 β α γ 2
α γ 1
= 1 2 ξ α γ 1 ξ α γ 1 + ρ α γ 1 × 1 + 2 ξ α γ 1 ρ α γ 1 , 1 2 ρ α γ 1 ξ α γ 1 + ρ α γ 1 × 1 + 2 ξ α γ 1 ρ α γ 1 , η α γ 1 η α γ 1 + β α γ 1 × 1 1 η α γ 1 β α γ 1 , β α γ 1 η α γ 1 + β α γ 1 × 1 1 η α γ 1 β α γ 1
   It can be easily verified that α γ 1 ˜ α γ 2 , α γ 1 are the LDFNs.
Theorem 1.
Consider α γ 1 = ξ α γ 1 , ρ α γ 1 , η α γ 1 , β α γ 1 and α γ 2 = ξ α γ 2 , ρ α γ 2 , η α γ 2 , β α γ 2 are the two LDFNs. If ξ α γ 1 = ρ α γ 1 , ξ α γ 2 = ρ α γ 2 , η α γ 1 = β α γ 1 and η α γ 2 = β α γ 2 then we have
(i) 
ξ α γ 1 ˜ α γ 2 = ρ α γ 1 ˜ α γ 2 , and  η α γ 1 ˜ α γ 2 = β α γ 1 ˜ α γ 2 ,
(ii) 
ξ α γ 1 = ρ α γ 1 , and  η α γ 1 = β α γ 1
Proof.
(i)  As given ξ α γ 1 = ρ α γ 1 and ξ α γ 2 = ρ α γ 2
ξ α γ 1 ˜ α γ 2 ρ α γ 1 ˜ α γ 2 = 1 2 ξ α γ 1 ξ α γ 2 ξ α γ 1 ξ α γ 2 + ρ α γ 1 ρ α γ 2 × 1 + 2 ξ α γ 1 ρ α γ 1 2 ξ α γ 2 ρ α γ 2 1 2 ρ α γ 1 ρ α γ 2 ξ α γ 1 ξ α γ 2 + ρ α γ 1 ρ α γ 2 × 1 + 2 ξ α γ 1 ρ α γ 1 2 ξ α γ 2 ρ α γ 2 = 1
Consequently, ξ α γ 1 ˜ α γ 2 = ρ α γ 1 ˜ α γ 2 . If ξ α γ 1 = ρ α γ 1 and ξ α γ 2 = ρ α γ 2 .
As given, η α γ 1 = β α γ 1 and η α γ 2 = β α γ 2
η α γ 1 ˜ α γ 2 β α γ 1 ˜ α γ 2 = η α γ 1 η α γ 2 η α γ 1 η α γ 2 + β α γ 1 β α γ 2 × 1 1 η α γ 1 β α γ 1 1 η α γ 2 β α γ 2 β α γ 1 β α γ 2 η α γ 1 η α γ 2 + β α γ 1 β α γ 2 × 1 1 η α γ 1 β α γ 1 1 η α γ 2 β α γ 2 = 1
Consequently, η α γ 1 ˜ α γ 2 = β α γ 1 ˜ α γ 2 . If η α γ 1 = β α γ 1 and η α γ 2 = β α γ 2 .
(ii)  As given ξ α γ 1 = ρ α γ 1 and ξ α γ 2 = ρ α γ 2
ξ α γ 1 ρ α γ 1 = 1 2 ξ α γ 1 ξ α γ 1 + ρ α γ 1 × 1 + 2 ξ α γ 1 ρ α γ 1 1 2 ρ α γ 1 ξ α γ 1 + ρ α γ 1 × 1 + 2 ξ α γ 1 ρ α γ 1 = 1
Consequently, ξ α γ 1 = ρ α γ 1 . If ξ α γ 1 = ρ α γ 1 and ξ α γ 2 = ρ α γ 2 .
As given, η α γ 1 = β α γ 1 and η α γ 2 = β α γ 2
η α γ 1 β α γ 1 = η α γ 1 η α γ 1 + β α γ 1 × 1 1 η α γ 1 β α γ 1 β α γ 1 η α γ 1 + β α γ 1 × 1 1 η α γ 1 β α γ 1 = 1
Consequently, η α γ 1 = β α γ 1 . If η α γ 1 = β α γ 1 and η α γ 2 = β α γ 2 . □
The above theorem shows that the operations α γ 1 ˜ α γ 2 , α γ 1 show the fairly or neutral nature to the DMs, when MSG, NMSG and RPs are equal initially. This is why we call the operations ˜ , * fairly operations.
Theorem 2.
Consider α γ 1 = ξ α γ 1 , ρ α γ 1 , η α γ 1 , β α γ 1 and α γ 2 = ξ α γ 2 , ρ α γ 2 , η α γ 2 , β α γ 2 are the LDFNs and , 1 and 2 are any three real numbers, then we have
(i)   α γ 1 ˜ α γ 2 = α γ 2 ˜ α γ 1
(ii)  α γ 1 ˜ α γ 2 = α γ 1 ˜ α γ 2
(iii)  1 + 2 α γ 1 = 1 α γ 1 ˜ 2 α γ 1
Proof.
(i)   This one is trivial.
(ii)  α γ 1 ˜ α γ 2
= 1 2 ξ α γ 1 ξ α γ 2 ξ α γ 1 ξ α γ 2 + ρ α γ 1 ρ α γ 2 ξ α γ 1 ξ α γ 2 ξ α γ 1 ξ α γ 2 + ρ α γ 1 ρ α γ 2 + ρ α γ 1 ρ α γ 2 ξ α γ 1 ξ α γ 2 + ρ α γ 1 ρ α γ 2 × 1 + 1 + 1 + 2 ξ α γ 1 ρ α γ 1 2 ξ α γ 2 ρ α γ 2 , 1 2 ρ α γ 1 ρ α γ 2 ξ α γ 1 ξ α γ 2 + ρ α γ 1 ρ α γ 2 ξ α γ 1 ξ α γ 2 ξ α γ 1 ξ α γ 2 + ρ α γ 1 ρ α γ 2 + ρ α γ 1 ρ α γ 2 ξ α γ 1 ξ α γ 2 + ρ α γ 1 ρ α γ 2 × 1 + 1 + 1 + 2 ξ α γ 1 ρ α γ 1 2 ξ α γ 2 ρ α γ 2 , η α γ 1 η α γ 2 η α γ 1 η α γ 2 + β α γ 1 β α γ 2 η α γ 1 η α γ 2 η α γ 1 η α γ 2 + β α γ 1 β α γ 2 + β α γ 1 β α γ 2 η α γ 1 η α γ 2 + β α γ 1 β α γ 2 × 1 1 1 1 η α γ 1 β α γ 1 1 η α γ 2 β α γ 2 , β α γ 1 β α γ 2 η α γ 1 η α γ 2 + β α γ 1 β α γ 2 η α γ 1 η α γ 2 η α γ 1 η α γ 2 + β α γ 1 β α γ 2 + β α γ 1 β α γ 2 η α γ 1 η α γ 2 + β α γ 1 β α γ 2 × 1 1 1 1 η α γ 1 β α γ 1 1 η α γ 2 β α γ 2
= 1 2 ξ α γ 1 ξ α γ 2 ξ α γ 1 ξ α γ 2 + ρ α γ 1 ρ α γ 2 1 + 2 ξ α γ 1 ρ α γ 1 2 ξ α γ 2 ρ α γ 2 , 1 2 ρ α γ 1 ρ α γ 2 ξ α γ 1 ξ α γ 2 + ρ α γ 1 ρ α γ 2 1 + 2 ξ α γ 1 ρ α γ 1 2 ξ α γ 2 ρ α γ 2 , η α γ 1 η α γ 2 η α γ 1 η α γ 2 + β α γ 1 β α γ 2 1 1 η α γ 1 β α γ 1 1 η α γ 2 β α γ 2 , β α γ 1 β α γ 2 η α γ 1 η α γ 2 + β α γ 1 β α γ 2 1 1 η α γ 1 β α γ 1 1 η α γ 2 β α γ 2
= 1 2 ξ α γ 1 ξ α γ 1 + ρ α γ 1 × ξ α γ 2 ξ α γ 2 + ρ α γ 2 ξ α γ 1 ξ α γ 1 + ρ α γ 1 × ξ α γ 2 ξ α γ 2 + ρ α γ 2 + + ρ α γ 1 ξ α γ 1 + ρ α γ 1 + × ρ α γ 2 ξ α γ 2 + ρ α γ 2 + × 1 + 1 + 1 + 2 ξ α γ 1 ρ α γ 1 × 1 + 1 + 2 ξ α γ 2 ρ α γ 2 , 1 2 ρ α γ 1 ξ α γ 1 + ρ α γ 1 × ρ α γ 2 ξ α γ 2 + ρ α γ 2 ξ α γ 1 ξ α γ 1 + ρ α γ 1 × ξ α γ 2 ξ α γ 2 ρ α γ 2 + ρ α γ 1 ξ α γ 1 + ρ α γ 1 × ρ α γ 2 ξ α γ 2 + ρ α γ 2 × 1 + 1 + 1 + 2 ξ α γ 1 ρ α γ 1 × 1 + 1 + 2 ξ α γ 2 ρ α γ 2 , η α γ 1 η α γ 1 + β α γ 1 × η α γ 2 η α γ 2 + β α γ 2 η α γ 1 η α γ 1 + β α γ 1 × η α γ 2 η α γ 2 + β α γ 2 + + β α γ 1 η α γ 1 + β α γ 1 + × β α γ 2 η α γ 2 + β α γ 2 + × 1 1 1 1 η α γ 1 β α γ 1 × 1 1 1 η α γ 2 β α γ 2 , β α γ 1 η α γ 1 + β α γ 1 × β α γ 2 η α γ 2 + β α γ 2 η α γ 1 η α γ 1 + β α γ 1 × η α γ 2 η α γ 2 β α γ 2 + β α γ 1 η α γ 1 + β α γ 1 × β α γ 2 η α γ 2 + β α γ 2 × 1 1 1 1 η α γ 1 β α γ 1 × 1 1 1 η α γ 2 β α γ 2
= 1 2 ξ α γ 1 ξ α γ 2 ξ α γ 1 ξ α γ 2 + ρ α γ 1 ρ α γ 2 1 + 2 ξ α γ 1 ρ α γ 1 2 ξ α γ 2 ρ α γ 2 , 1 2 ρ α γ 1 ρ α γ 2 ξ α γ 1 ξ α γ 2 + ρ α γ 1 ρ α γ 2 1 + 2 ξ α γ 1 ρ α γ 1 2 ξ α γ 2 ρ α γ 2 , η α γ 1 η α γ 2 η α γ 1 η α γ 2 + β α γ 1 β α γ 2 1 1 η α γ 1 β α γ 1 1 η α γ 2 β α γ 2 , β α γ 1 β α γ 2 η α γ 1 η α γ 2 + β α γ 1 β α γ 2 1 1 η α γ 1 β α γ 1 1 η α γ 2 β α γ 2
Hence, α γ 1 ˜ α γ 2 = α γ 1 ˜ α γ 2 .
(iii)  1 α γ 1 ˜ 2 α γ 1
= 1 2 ξ α γ 1 1 ξ α γ 1 1 + ρ α γ 1 + × 1 + 2 ξ α γ 1 ρ α γ 1 1 , 1 2 ρ α γ 1 1 ξ α γ 1 1 + ρ α γ 1 + × 1 + 2 ξ α γ 1 ρ α γ 1 1 , η α γ 1 1 η α γ 1 1 + β α γ 1 + × 1 1 η α γ 1 β α γ 1 1 , β α γ 1 1 η α γ 1 1 + β α γ 1 + × 1 1 η α γ 1 β α γ 1 1 ˜ 1 2 ξ α γ 1 2 ξ α γ 1 2 + ρ α γ 1 + × 1 + 2 ξ α γ 1 ρ α γ 1 2 , 1 2 ρ α γ 1 2 ξ α γ 1 2 + ρ α γ 1 + × 1 + 2 ξ α γ 1 ρ α γ 1 2 , η α γ 1 2 η α γ 1 2 + β α γ 1 + × 1 1 η α γ 1 β α γ 1 2 , β α γ 1 2 η α γ 1 2 + β α γ 1 + × 1 1 η α γ 1 β α γ 1 2 = 1 2 ξ α γ 1 1 + 2 ξ α γ 1 1 + 2 + ρ α γ 1 + × 1 + 2 ξ α γ 1 ρ α γ 1 1 + 2 , 1 2 ρ α γ 1 1 + 2 ξ α γ 1 1 + 2 + ρ α γ 1 + × 1 + 2 ξ α γ 1 ρ α γ 1 1 + 2 , η α γ 1 1 + 2 η α γ 1 1 + 2 + β α γ 1 + × 1 1 η α γ 1 β α γ 1 1 + 2 , β α γ 1 1 + 2 η α γ 1 1 + 2 + β α γ 1 + × 1 1 η α γ 1 β α γ 1 1 + 2
and 1 + 2 α γ 1
= 1 2 ξ α γ 1 1 + 2 ξ α γ 1 1 + 2 + ρ α γ 1 + × 1 + 2 ξ α γ 1 ρ α γ 1 1 + 2 , 1 2 ρ α γ 1 1 + 2 ξ α γ 1 1 + 2 + ρ α γ 1 + × 1 + 2 ξ α γ 1 ρ α γ 1 1 + 2 , η α γ 1 1 + 2 η α γ 1 1 + 2 + β α γ 1 + × 1 1 η α γ 1 β α γ 1 1 + 2 , β α γ 1 1 + 2 η α γ 1 1 + 2 + β α γ 1 + × 1 1 η α γ 1 β α γ 1 1 + 2
Hence, 1 + 2 α γ 1 = 1 α γ 1 ˜ 2 α γ 1    □

5. Fairly Aggregation Operators for LDFNs

This section discusses fairly AOs for LDFNs and their characteristics.

5.1. LDFFWA Operator

Definition 11.
Let α γ h = ξ h , ρ h , η h , β h be the accumulation of LDFNs, and LDFFWA: F n F , be a n dimension mapping. If
L D F F W A ( α γ 1 , α γ 2 , α γ e ) = ϑ γ 1 α γ 1 ˜ ϑ γ 2 α γ 2 ˜ , ˜ ϑ γ e α γ e
then the mapping LDFFWA is called “linear Diophantine fuzzy fairly weighted averaging (LDFFWA) operator”, here, ϑ γ i is the weight vector (WV) of α γ i with ϑ γ i > 0 and i = 1 e ϑ γ i = 1 .
Moreover, as demonstrated in the theorem similarly below, we can consider LDFFWA using fairly operational laws.
Theorem 3.
Let α γ h = ξ h , ρ h , η h , β h be the accumulation of LDFNs; we can also find LDFFWA by
L D F F W A ( α γ 1 , α γ 2 , , α γ e )
= 1 2 i = 1 e ξ i ϑ γ i i = 1 e ξ i ϑ γ i + i = 1 e ρ i ϑ γ i × 1 + i = 1 e 2 ξ i ρ i ϑ γ i , 1 2 i = 1 e ρ i ϑ γ i i = 1 e ξ i ϑ γ i + i = 1 e ρ i ϑ γ i × 1 + i = 1 e 2 ξ i ρ i ϑ γ i , i = 1 e η i ϑ γ i i = 1 e η i ϑ γ i + i = 1 e β i ϑ γ i × 1 i = 1 e 1 η i β i ϑ γ i , i = 1 e β i ϑ γ i i = 1 e η i ϑ γ i + i = 1 e β i ϑ γ i × 1 i = 1 e 1 η i β i ϑ γ i
where ϑ γ i is the WV of α γ i with ϑ γ i > 0 and i = 1 e ϑ γ i = 1 .
Proof.
This proof will begin with mathematical induction.
For e = 1 , we have α γ 1 = ξ 1 , ρ 1 and ϑ γ = 1 .    L D F F W A ( α γ 1 ) = ϑ γ 1 α γ 1
= 1 2 ξ 1 ϑ γ 1 ξ 1 ϑ γ 1 + ρ 1 1 ϑ γ × 1 + 2 ξ 1 ρ 1 ϑ γ 1 , 1 2 ρ 1 ϑ γ 1 ξ 1 ϑ γ 1 + ρ 1 ϑ γ 1 × 1 + 2 ξ 1 ρ 1 ϑ γ 1 , η 1 ϑ γ 1 η 1 ϑ γ 1 + β 1 1 ϑ γ × 1 1 η 1 β 1 ϑ γ 1 , β 1 ϑ γ 1 η 1 ϑ γ 1 + β 1 ϑ γ 1 × 1 1 η 1 β 1 ϑ γ 1
Even as theorem holds true for e = 1 , we now anticipate it to hold proper for e = g , i.e.,
LDFFWA ( α γ 1 , α γ 2 , , α γ g ) = 1 2 i = 1 g ξ i ϑ γ i i = 1 g ξ i ϑ γ i + i = 1 g ρ i ϑ γ i × 1 + i = 1 g 2 ξ i ρ i ϑ γ i , 1 2 i = 1 g ρ i ϑ γ i i = 1 g ξ i ϑ γ i + i = 1 g ρ i ϑ γ i × 1 + i = 1 g 2 ξ i ρ i ϑ γ i , i = 1 g η i ϑ γ i i = 1 g η i ϑ γ i + i = 1 g β i ϑ γ i × 1 i = 1 g 1 η i β i ϑ γ i , i = 1 g β i ϑ γ i i = 1 g η i ϑ γ i + i = 1 g β i ϑ γ i × 1 i = 1 g 1 η i β i ϑ γ i
We will prove for e = g + 1 .
LDFFWA ( α γ 1 , α γ 2 , , α γ g + 1 )  =  LDFFWA ( α γ 1 , α γ 2 , , α γ g ) ˜ ( ϑ γ g + 1 α γ g + 1 )
= 1 2 i = 1 g ξ i ϑ γ i i = 1 g ξ i ϑ γ i + i = 1 g ρ i ϑ γ i × 1 + i = 1 g 2 ξ i ρ i ϑ γ i , 1 2 i = 1 g ρ i ϑ γ i i = 1 g ξ i ϑ γ i + i = 1 g ρ i ϑ γ i × 1 + i = 1 g 2 ξ i ρ i ϑ γ i , i = 1 g η i ϑ γ i i = 1 g η i ϑ γ i + i = 1 g β i ϑ γ i × 1 i = 1 g 1 η i β i ϑ γ i , i = 1 g β i ϑ γ i i = 1 g η i ϑ γ i + i = 1 g β i ϑ γ i × 1 i = 1 g 1 η i β i ϑ γ i ˜ 1 2 ξ α γ g + 1 ϑ γ g + 1 ξ α γ g + 1 ϑ γ g + 1 + ρ α γ g + 1 ϑ γ g + 1 × 1 + 2 ξ α γ g + 1 ρ α γ g + 1 ϑ γ g + 1 , 1 2 ρ α γ g + 1 ϑ γ g + 1 ξ α γ g + 1 ϑ γ g + 1 + ρ α γ g + 1 ϑ γ g + 1 × 1 + 2 ξ α γ g + 1 ρ α γ g + 1 ϑ γ g + 1 , η α γ g + 1 ϑ γ g + 1 η α γ g + 1 ϑ γ g + 1 + β α γ g + 1 ϑ γ g + 1 × 1 1 η α γ g + 1 β α γ g + 1 ϑ γ g + 1 , β α γ g + 1 ϑ γ g + 1 η α γ g + 1 ϑ γ g + 1 + β α γ g + 1 ϑ γ g + 1 × 1 1 η α γ g + 1 β α γ g + 1 ϑ γ g + 1
= 1 2 i = 1 g ξ i ϑ γ i × ξ g + 1 ϑ γ g + 1 i = 1 g ξ i ϑ γ i × ξ g + 1 ϑ γ g + 1 + i = 1 g ρ i ϑ γ i × ρ g + 1 ϑ γ g + 1 × 1 + i = 1 g 2 ξ i ρ i ϑ γ i × 2 ξ g + 1 ρ g + 1 ϑ γ g + 1 , 1 2 i = 1 g ρ i ϑ γ i × ρ g + 1 ϑ γ g + 1 i = 1 g ξ i ϑ γ i × ξ g + 1 ϑ γ g + 1 + i = 1 g ρ i ϑ γ i × ρ g + 1 ϑ γ g + 1 × 1 + i = 1 g 2 ξ i ρ i ϑ γ i × 2 ξ g + 1 ρ g + 1 ϑ γ g + 1 , i = 1 g η i ϑ γ i × η g + 1 ϑ γ g + 1 i = 1 g η i ϑ γ i × η g + 1 ϑ γ g + 1 + i = 1 g β i ϑ γ i × β g + 1 ϑ γ g + 1 × 1 i = 1 g 1 η i β i ϑ γ i × 1 η g + 1 β g + 1 ϑ γ g + 1 , i = 1 g β i ϑ γ i × β g + 1 ϑ γ g + 1 i = 1 g η i ϑ γ i × η g + 1 ϑ γ g + 1 + i = 1 g β i ϑ γ i × β g + 1 ϑ γ g + 1 × 1 i = 1 g 1 η i β i ϑ γ i × 1 η g + 1 β g + 1 ϑ γ g + 1 = 1 2 i = 1 g + 1 ξ i ϑ γ i i = 1 g + 1 ξ i ϑ γ i + i = 1 g + 1 ρ i ϑ γ i × 1 + i = 1 g + 1 2 ξ i ρ i ϑ γ i , 1 2 i = 1 g + 1 ρ i ϑ γ i i = 1 g + 1 ξ i ϑ γ i + i = 1 g + 1 ρ i ϑ γ i × 1 + i = 1 g + 1 2 ξ i ρ i ϑ γ i , i = 1 g + 1 η i ϑ γ i i = 1 g + 1 η i ϑ γ i + i = 1 g + 1 β i ϑ γ i × 1 i = 1 g + 1 1 η i β i ϑ γ i , i = 1 g + 1 β i ϑ γ i i = 1 g + 1 η i ϑ γ i + i = 1 g + 1 β i ϑ γ i × 1 i = 1 g + 1 1 η i β i ϑ γ i
Consequently, the statement holds genuine for e = g + 1 as well. Therefore, the belief is authentic for every e using the principle of induction.    □
Example 2.
Assume α γ 1 = 0.86 , 0.76 , 0.45 , 0.25 , α γ 2 = 0.67 , 0.98 , 0.53 , 0.29 and α γ 3 = 0.58 , 0.68 , 0.65 , 0.15 are three LDFNs with WV ϑ γ = ( 0.30 , 0.45 , 0.25 ) , then
1 2 i = 1 3 ξ i ϑ γ i i = 1 3 ξ i ϑ γ i + i = 1 3 ρ i ϑ γ i × 1 + i = 1 3 2 ξ i ρ i ϑ γ i = 0.327101 1 2 i = 1 3 ρ i ϑ γ i i = 1 3 ξ i ϑ γ i + i = 1 3 ρ i ϑ γ i × 1 + i = 1 3 2 ξ i ρ i ϑ γ i = 0.389193 i = 1 3 η i ϑ γ i i = 1 3 η i ϑ γ i + i = 1 3 β i ϑ γ i × 1 i = 1 3 1 η i β i ϑ γ i = 0.543735 i = 1 3 β i ϑ γ i i = 1 3 η i ϑ γ i + i = 1 3 β i ϑ γ i × 1 i = 1 3 1 η i β i ϑ γ i = 0.240855
L D F F W A ( α γ 1 , α γ 2 , α γ 3 )
= 1 2 i = 1 3 ξ i ϑ γ i i = 1 3 ξ i ϑ γ i + i = 1 3 ρ i ϑ γ i × 1 + i = 1 3 2 ξ i ρ i ϑ γ i , 1 2 i = 1 3 ρ i ϑ γ i i = 1 3 ξ i ϑ γ i + i = 1 3 ρ i ϑ γ i × 1 + i = 1 3 2 ξ i ρ i ϑ γ i , i = 1 3 η i ϑ γ i i = 1 3 η i ϑ γ i + i = 1 3 β i ϑ γ i × 1 i = 1 3 1 η i β i ϑ γ i , i = 1 3 β i ϑ γ i i = 1 3 η i ϑ γ i + i = 1 3 β i ϑ γ i × 1 i = 1 3 1 η i β i ϑ γ i
= 0.327101 , 0.389193 , 0.543735 , 0.240855
The proposed AO meets a number of special prerequisites, which are described in the following theorems.
Theorem 4.
Let α γ i = ξ i , ρ i , η i , β i be the accumulation of LDFNs and α γ = ( ξ , ρ , η , β ) be the LDFN such that, α γ i = α γ i . Then
L D F F W A ( α γ 1 , α γ 2 , , α γ e ) = α γ
Proof. 
Given that α γ i = α γ i , by this, ξ i = ξ , ρ i = ρ , η i = η and β i = β i
LDFFWA ( α γ 1 , α γ 2 , , α γ e )
= 1 2 i = 1 e ξ i ϑ γ i i = 1 e ξ i ϑ γ i + i = 1 e ρ i ϑ γ i × 1 + i = 1 e 2 ξ i ρ i ϑ γ i , 1 2 i = 1 e ρ i ϑ γ i i = 1 e ξ i ϑ γ i + i = 1 e ρ i ϑ γ i × 1 + i = 1 e 2 ξ i ρ i ϑ γ i , i = 1 e η i ϑ γ i i = 1 e η i ϑ γ i + i = 1 e β i ϑ γ i × 1 i = 1 e 1 η i β i ϑ γ i , i = 1 e β i ϑ γ i i = 1 e η i ϑ γ i + i = 1 e β i ϑ γ i × 1 i = 1 e 1 η i β i ϑ γ i = 1 2 i = 1 e ξ ϑ γ i i = 1 e ξ ϑ γ i + i = 1 e ρ ϑ γ i × 1 + i = 1 e 2 ξ ρ ϑ γ i , 1 2 i = 1 e ρ ϑ γ i i = 1 e ξ ϑ γ i + i = 1 e ρ ϑ γ i × 1 + i = 1 e 2 ξ ρ ϑ γ i , i = 1 e η ϑ γ i i = 1 e η ϑ γ i + i = 1 e β ϑ γ i × 1 i = 1 e 1 η β ϑ γ i , i = 1 e β ϑ γ i i = 1 e η ϑ γ i + i = 1 e β ϑ γ i × 1 i = 1 e 1 η β ϑ γ i = 1 2 ξ i = 1 e ϑ γ i ξ i = 1 e ϑ γ i + ρ i = 1 e ϑ γ i × 1 + 2 ξ ρ i = 1 e ϑ γ i , 1 2 ρ i = 1 e ϑ γ i ξ i = 1 e ϑ γ i + ρ i = 1 e ϑ γ i × 1 + 2 ξ ρ i = 1 e ϑ γ i , η i = 1 e ϑ γ i η i = 1 e ϑ γ i + β i = 1 e ϑ γ i × 1 1 η β i = 1 e ϑ γ i , β i = 1 e ϑ γ i η i = 1 e ϑ γ i + β i = 1 e ϑ γ i × 1 1 η β i = 1 e ϑ γ i = ξ , ρ , η , β = α γ
   □
Theorem 5.
Let α γ i = ξ i , ρ i , η i , β i be the accumulation of LDFNs. Then, for
L D F F W A ( α γ 1 , α γ 2 , , α γ e ) = ξ