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.
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 aswhere are the MSD, the NMSD and the corresponding reference parameters (RPs), respectively. Moreover,andfor all . The LDFSis known as the “absolute LDFS” in X. The LDFSis 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, is called the “indeterminacy degree” and its corresponding reference parameter of to ℜ.
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 satisfying the following conditions:
- (1)
;
- (2)
;
- (3)
.
Definition 3 ([
10]).
Let be a LDFN; then, the “score function” can be defined by the mapping and given bywhere is a collection of LDFNs on X. Definition 4 ([
10]).
Let be a LDFN; then, the “accuracy function” can be defined by the mapping and given as Definition 5 ([
10]).
Let and be two LDFNs; then, by using the score function and accuracy function, we have:- (i)
If then ,
- (ii)
If then ,
- (iii)
If then,
- (a)
If then ,
- (b)
If then ,
- (c)
If then .
Definition 6 ([
10]).
Let be a LDFN and . Then;
;
.
Definition 7 ([
10]).
Let be two LDFNs with . Then;
;
;
.
Definition 8 ([
10]).
Let be the assemblage of LDFNs with . Then;
.
Example 1. Consider and be two LDFNs. Then, it is clear that . One can verify that
;
;
;
;
.
In addition, if , then we have
;
.
Definition 9. Let and be two LDFNs and be the real numbers, then we have,
;
;
;
;
;
.
If and then we obtain . Thus, none of the operations 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 and are the two LDFNs and . Then, we define.
It can be easily verified that are the LDFNs.
Theorem 1. Consider and are the two LDFNs. If , , and then we have
- (i)
, and ,
- (ii)
, and
Proof.
(i) As given
and
Consequently, . If and .
As given,
and
Consequently, . If and .
(ii) As given
and
Consequently, . If and .
As given,
and
Consequently, . If and . □
The above theorem shows that the operations 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 and are the LDFNs and and are any three real numbers, then we have
(i)
(ii)
(iii)
Proof.
(i) This one is trivial.
(ii)
Hence, .
(iii)
and
Hence, □