Multi-attribute decision-making using Archimedean aggregation operator in T-spherical fuzzy environment

ABSTRACT


Introduction
Uncertainty always exists in almost all types of information that are based on human options.To reduce this uncertainty (Zadeh 1965), announced the notion of the fuzzy set (FS) with membership degree (MD).In addition, Atanassov (1989) offered the thought of the intuitionistic fuzzy (IF) set (IFS) with MD and nonmember ship degree (NMD) always lies between the range of [0,1].IFS provides greater freedom to study imprecise and ambiguous information.Furthermore, Yager (2013), provided the awareness of the Pythagorean fuzzy set (PyFS) their sum of the square of its NMD and MD lies between [0, 1].The generalization of the PyFS set is provided by Yager (2016), in the form of taking q th power of MD and NMD is called a q-rung ortho pair fuzzy set (q-ROFS).
Human opinion has also a certain degree of refusal and abstinence.This indicates that the previous structures of the generalized form of FS are unable to solve these types of confusing problems.To cover these types of problems, a Picture fuzzy set (PFS) with the involvement of abstinence degree (AD) along with MD and NMD firstly defined by Cuong (2015).The range of PFS also lies between the interval [0, 1].Later on, Mahmood et al. (2019), presented that the generalized form of PFS is said to be spherical FS (SFS), and after some modification, by taking the q th power of SFS he introduced a new idea called the T-spherical FS (TSFS).
In FS theory MADM approach is a trending technique nowadays for the aggregation of information.We use the MADM approach for the selection of the most suitable option from the given list of multiple options by under consideration of the multiple standards at the same time.In FS theory number of AOs based on TN and TCNs was defined by several mathematicians.For example, Munir et al. (2020), examined the Einstein interactive AOs of TSFS.Liu and Wang (2018) proposed the concepts of geometric and averaging AOs for orthopair FS and also examined their application to MADM problems.Wei (2010), presented the IF trapezoidal fuzzy operators, and Jana et al. (2019) proposed the concept of Dombi AOs in IFS theory as well as in PyFS theory.The q-ROFS Aczel-Alsina AOs defined by Khan et al. (2022).Ullah, Mahmood, and Garg (2020) provided the thought of Hamacher AOs in the TSF environment while Hamacher AOs in the IFSs system based on entropy measurement developed by Garg (2019).In FS theory IF hybrid geometric and arithmetic, AOs were proposed by Ye (2017).The idea of Maclaurin symmetric AOs proposed by Ullah (2021).The IF cubic AOs developed and their applicability to the MADM problem were discussed by Kaur and Garg (2018), and Liu et al. (2019) discussed power Muirhead Maen AOs for TSFS and their application to the MADM problems.In addition to that, some applications of TSFS theory by utilizing several types of operators and measures are summaries by Ullah (2021).
In the theory of probabilistic fuzzy metric space Menger (1942), firstly proposed the concept of the triangular norm.Over time, many TN and their corresponding TCNs and their operations are discussed in the FS theory.For example, Strict ATCN and ATN were discussed by Nguyen, Kreinovich, and Wojciechowski (1998).IF integral-based ATCN and ATN were presented by Lei et al. (2016).Continuous integral-based ATN and ATCNs were introduced by Ai et al. (2020).The idea of maximal discrete ATN and ATCNs was presented by (Bejines and Navara 2022).The interactive ATN and ATCN were presented by Wang and Garg (2021).
Almost all the AOs are developed on the bases of operational laws of algebra like the product, sum, scalar multiplication, and power operation.However, ATN and ATCN is the generalized form of many kinds of FS theoretic operations.ATN and ATCN provide better understanding and preciseness in results than other present TN and TCN in the FS system.Because in ATN and ATCN have obtained all AOs by changing the value of the function and it covers a wide class of TN and TCNs like algebraic, Frank, Hamacher, Einstein, and many other classes of TN and TCNs.So, AOs based on ATN and ATCN are more significant than other existing AOs.
For medical science problems, the MADM approach is a useful technique, such as the smart medical devices selection for diagnosing the problem in the human body based on IF Choquet integral by Büyüközkan and Göçer (2019).The selection of LASER surgical instruments for surgery by using the MADM technique based on neutrosophic FS and fuzzy TOPSIS method by Farooq and Saqlain (2021).Pre-operative surgical tool ordering by utilizing the MADM approach by Miller et al. (2008), and risk evaluation in the selection of the prioritization of the medical devices by Jamshidi et al. (2015).Pamucar et al. (2022) delivered the concept of supplier selection of healthcare instruments during the covid-19 pandemic situation.An efficient surgery instruments supply selection methodology to hospital pharmacy by using fuzzy the MADM approach by Manivel and Ranganathan (2019).By using the MADM algorithm to measure the effect of air quality on the surgical instruments in operation theater by Colella et al. (2022).Rahman and Lee (2013), by using fuzzy logic studied the assessment of the disturbance of surgical instruments during surgery.Tian and Juan proposed the method of selection of the best surgical instruments through manipulation and perception.By using intervalvalued IF model evaluation of surgical instruments risk during organ transplant by Salimian et al. (2022).
A surgical instrument is a tool or apparatus used during an operation or surgery to carry out particular tasks or achieve desired results such as modification of the biological tissues.Many surgical instruments have been invented by several manufacturing companies.It is very difficult to decide for the hospital management departments for giving the tender to the surgical company which offers the best quality instruments at reasonable price rates.In this critical situation, we have proposed an ATSFWA and ATSFWG AOs model for dealing with this kind of problem.In this article, we have considered the alternatives such as nature of the material, purity of the material, accuracy in the functionality, designed by computer numerical control (CNC) machines or by hand for the selection, and using the proposed ATSFWA and ATSFWG AOs algorithm for the aggregating the fuzzy information.
The article offers the following information: Section 2, discussed some fundamental definitions for a better understanding of the article.In section 3. we introduce operational laws for the aggregation of TSF information.Section 4, proposed new AOs like ATSFWA, Archimedean TSF ordered weighted averaging (ATSFOWA), and Archimedean TSF hybrid weighted averaging (ATSFOWA).ATSFWG, Archimedean TSF ordered weighted geometric (ATSFOWG), Archimedean TSF hybrid weighted geometric (ATSFOWG).Section 5 also discussed some desirable axioms.In section 6, develop the algorithm for explaining the MADM problematic issue.On the behalf of our proposed AOs, solve the MADM problem in section 7. Compare our aggregation outcomes with other existing AOs in section 8. Section 9, discussed the advantages of this article.Section 10, provided the comprehensive conclusion and future research interest mentioned.

T-Spherical Fuzzy Set
In this segment, we discussed a few fundamental concepts related to TSFS, TN, and TCN also their operational laws.These key concepts will make our article easy to understand for the reader.Definition 1: (Mahmood et al., 2019), Consider the TSFS on  is  = {〈,   (),   (),   ()〉} and   ,   and   are represents the MD, AD and NMG of  ∈  respectively lies between the range [0, 1] and 0 ≤ (   ,    ,    ) ≤ 1 for  ∈ ℤ.Here √1 − (  (),   (),   ()) is the refusal grade.So, this triplet (   ,    ,    ) is called TSF number (TSFN).Remark 1 In the light of the above definition 1.We have proved that TSFS is comprehensive form of FS as compare to the other existing fuzzy structures.That is, i.
When we take  = 2 in the above structure then TSFS becomes SFS.ii.
When we take  = 1 in the above structure then TSFS becomes PFS.iii.
When we take  = 2 and  = 0 in the above structure then TSFS becomes PyFS.iv.
When we take  = 1 and  = 0 in the above structure then TSFS becomes IFS.v.
And similarly, a decreasing generator  for ATN is expressed as Klement and Mesiar (2005), presented some TN and TCN for such following functions which are given below, such as:

3.1Archimedean TCN and TN on TSFNs
In the FS theory, ATCN and ATN play a vital role in the aggregation of fuzzy data and in solving MADM problems (Garg and Arora, 2021).In this segment, on the behalf of ATCN and ATN, we define some basic operational rules for the TSF environment as follows: Definition 12: Let three TSFNs such as  = (, , ),  1 = ( 1 ,  1 ,  1 ) and  2 = ( 2 ,  2 ,  2 ) and ℷ > 0, now on the bases of TSFNs we defined some new operations for ATCN and ATN as follows: Reports in Mechanical Engineering, Vol. 4, No. 1, 2023: 18 -38   22 • (Algebraic) (Nguyen, Walker, and Walker, 2018), When () = − log  then operational laws are defined in the Definition 4. Are found.

𝑟
), then the operational laws are defined as follows: ) , ℷ > 0 These are Einstein's operational laws on TSFs.
) ,  > 0, ℷ > 0, the following operational laws are defined as These are the Hamacher operational laws on TSFs.
) ,  > 0, and ℷ > 0, then the following operational laws are defined as

T-Spherical Fuzzy Archimedean Weighted Averaging AOs
This part, of the article, develops the ATSFWA, ATSFOWA, and ATSFHWA operators and discussed their fundamental characteristics in detail.Definition 13: Let   ( = 1, 2, … , ) ∈  be the set of TSFNs, and . Then a mapping of an ATSFWA AOs is:   → , and it can be defined as Some fundamental axioms of the ATSFWA AOs are discussed below as follows: Theorem 1 Consider   = (, , ), ( = 1, 2, … , ) be the family of ATSFNs, and results after aggregation ATSFWA AOs is also TSFN and it can be defined as )) (5) And )) Thus, the statement is true for  = 2. Now consider the statement is true for  = .( 1 ,  2 , … ,   ) )) Now taking  =  + 1, then we have )) ) Hence, the statement is true for  =  + 1.
Definitions 13 and 14 make it clear that the ATSFWA and ATSFOWA operators aggregate TSFNs by only weighting them and by ordering their weighting, respectively.As a result, weights demonstrate the many aspects of both ATSFWA and ATSFOWA operators.This shortcoming is not covered by any of the operators.To solve the problem, we define the ATSFHWA operator as follows.Where  ̈() =   and  is any balancing coefficient and ( ̈(1) ,  ̈(2) , … ,  ̈() ) is the permutation of  for weights TSFNs.
Almost all characteristics of the ATSFHWA operator are similar to ATSFWA that we discussed in detail in Theorem 1, 2, 3, and 4.However, as stated in the following Theorem 8, the ATSFHWA tool is a stronger model of the ATSFOWA operator.Theorem 9 The ATSFHWA operator has special cases known as the ATSFWA and ATSFOWA operators.Proof: Similarly, we can also explain that ATSFOWA is a unique case of ATSFHWA.

T-Spherical Fuzzy Archimedean Geometric Averaging AOs
In this part, we proposed ATSFWG, ATSFOWG, and ATSFHWG operators based on Archimedean operational laws and discuss their fundamental characteristics.Definition 16 Consider   (1, 2, … , ) be the family of TSFNs, and . Then ATCN and ATN are based on the TSF geometric (TSFWG) operator with mapping   →  * .
Definition 18 Consider   = (ρ, φ, τ) be any set of TSFNs and  i ′ be the WV.Then TSFHWG operator of dimension  is mapping ATSFHWG:   →  * can be written as Where  ̈() =   and  is any balancing coefficient and ( ̈(1) ,  ̈(2) , … ,  ̈() ) is the permutation  with the weight of TSFNs.Almost all characteristics of the ATSFHWG operator are similar to ATSFWG which we discussed in detail in Theorem 1, 2, 3, and 4.However, as stated in the following Theorem 8, the ATSFHWG operator is a stronger model of the ATSFOWG operator.Theorem 14 The ATSFHWG operator has special cases known as the ATSFWG and ATSFOWG operators.Proof: Similarly, we can also explain that ATSFOWG is a unique case of ATSFHWG.Remark The following changes can be observed in the weighted geometric operator as given below: • (Algebraic) If () = − log , then ATSFWG operator can be reduced into the TSF weighted averaging TSFWG operator, which can be defined as given below: • (Einstein) If () = − log((2 − )/), then ATSFEWA operator reduces in the TSFEWA operator, which can be defined as given below • (Hamacher) If () = − log (( + (1 − ))/) ,  > 0, then the ATSFWA operator reduces into the TSFHWA operator defined as If we take  is 1 and 2 in this equation ( 19), then the TSFHWA operator turns into the TSFWA and TSFEWA operators, respectively.) log  ) (20)

Propose algorithm for solving MADM problem
In this section, under the TSF environment, we solve the MADM problem by utilizing the proposed ATSFWA and ATSFWG operators.For this, we are taking  = { 1 ,  2 , … ,   } be the family of alternatives,  = { 1 ,  2 , … ,   } be the family of attributes, and and . Consider the TSF decision matrix can be represented as  = (  ) × .Then the proposed ATSFWA and ATSFWG operators are applied to solve the MADM problem for the TSF system.The proposed algorithm is explained by using the following steps: Step 1. Firstly, aggregate the TSFNs in the decision matrix   , for each alternative , by utilizing the ATSFWA and ATSFWG operators as follows: ( )) Step 2. Calculation of the SV of the aggregated findings by using Definition 2.
Step 3. Evaluate the ranking of alternatives by utilizing Definition 3.

Numerical Example
In this section, by using the ATSFWA and ATSFWG operators solve the real-world problem.The explanation of the challenging problem is given below: The health of people has considerable attention in modern society.So, it is necessary that in case of surgery doctors use neat and sterilized instruments.For successful surgery, no doubt the experience of a doctor is very important, but surgical instruments play vital in surgery.Using surgical instruments, doctors can cut through soft tissue, remove bone, dissect and isolate lesions, and eliminate or remove aberrant structures.So, multiple surgical companies offer newly invented and the latest instruments manufactured from machines.So, it is a challenging issue in a time, when many companies offer their best production quality and claim that the functionality of their instruments is very precise and accurate during the surgery.We construct a numerical example for the illustration of our proposed work.Suppose that, a list of four ℸ  ̃= (1, 2, … , 4) surgical instruments manufacturing companies.To choose the best company, by the consideration of four attributes   ̃= (1, 2, … , 4) in our mind, as follows: i.  1 ̃ is the nature of the material (stainless steel magnetic or nonmagnetic).
ii.  2 ̃ purity of the material.
iii.  3 ̃ is the accuracy in the functionality.
iv.  4 ̃ is designed by computer numerical control (CNC) machines or by hand.The WV distributed by the experts is given as (0.25, 0.25, 0.14, 0.36)  .By using the proposed AOs, the decision-maker evaluate the data of four surgical instruments manufacturing companies under the consideration of four attributes   ̃= (1, 2, … , 4).
Step 1. Collection of fuzzy data by anonymous decision-makers.Step 3. The aggregation outcomes of the ATSFWA and ATSFWG operators is provided in Table 3. Step 4. The ordering of the score values can be seen in Table 4.

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The blue dots in the graph represents the aggregated values of the ATSFWA operator while the red dots show the aggregated results of the ATSFWG operators.
Step 5.The ranking ordering of the aggregated results by using the proposed AOs is given in Table 4.
It is observed in Table 4. by using the ATSFWA operator ℸ 1 ̃ is the best option from the list of options, while using the ATSGWG operator ℸ 3 ̃ is the best option.It depends upon the experts whether they choose ATSFWA or ATSFWG operator for the aggregation of the data.
To demonstrate the significance of ATN and ATCN on TSFS theory.Discuss the following important observations by changing the generating function.Also discussed their SF values and rank order in Table 5

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A small briefing of aggregated findings of proposed AOs with other existing AOs in Table 5. is represented gematrically in Figure 2.For more clarity, the outcomes of Table 5 are discussed in Figure 2. In Table 5. we deeply analyze our aggregated findings with Munir et al. (2020), Ullah et al. (2020), Ullah et al. (2020a) and Mahmood et al. (2021a) by applying the AOs described in these references in our proposed example.Since the proposed work of Munir et al. (2020), Ullah et al. (2020), Ullah et al. (2020a) and Mahmood et al. (2021), depends upon the Algebraic sum and product operational laws while our proposed AOs depend upon the changeability of generator.Due to this fact, we believe that our proposed AOs are more reliable than other present AOs.

Advantages
ATN and ATCN are very valuable and feasible to evaluate the family of information into a singleton set, because it is the general form of all AOs such as averaging/geometric, Einstein, Hamacher, and frank AOs to compute with the help of algebraic, Einstein, Frank and Hamacher TN and TCN.With the help of different values of function TN and TCN, we can easily obtain these all operators from our one proposed operator, called Archimedean AOs.To enhance the quality and worth of the proposed idea, we describe some special cases of the proposed operators by putting some different values of a function ATN and ATCN, such as: ) ,  > 1, then the ATSFFWA and ATSFFWG operator turn into the TSFFWA, and the TSFFWG operator is defined by Mahnaz et al. (Mahnaz et al. 2022).

Conclusion
For selecting the best preferences, decision-making is valuable and critical technique.The highly notable key points of the analysis are discussed below: Firstly, discussed the Archimedean operational laws for TSFS and justify them with the help of a numerical example.Diagnosed the theory of ATSFWA, ATSFWG, ATSFOWA, ATSFOWG, ATSFHWA, and ATSFHWG.Some axioms ("Boundedness, Monotonicity, and Idempotency") and the findings of the To demonstrate the MADM approach on the bases of given TSF information and also discussed the comparative study with other existing prevailing AOs.In this article, geometrical despeciation of proposed information has also discussed the purpose of better understanding.
The following are upcoming aspects of the work: We aim to apply the proposed technique in complex TSFs power AOs (Khan et al., 2022a), and intervalvalued TSF frank AOs (Hussain et al., 2022).

)
decision-making using Archimedean aggregation operator in T-spherical …(M.R. Khan) These are the Frank operational laws in TSFs Einstein) If () = − log((2 − )/), then ATSFEWA operator reduces in the TSFEWA operator, which can be defined as given below Reports in Mechanical Engineering ISSN: 2683-5894  Multi-attribute decision-making using Archimedean aggregation operator in T-spherical …(M.R. Khan)

Figure 2 .
Figure 2. The above graph represents the geometrical view of the comparative analysis, whereas the lines in the graph depicted the score value of the AOs.The aggregation findings given in row 1 are from Mahmood et al. (2021), the aggregation findings given in row 2 are from Ullah et al. (2020a), the aggregation findings given in row 3 are from Ullah et al. (2020), while the aggregation findings are given in row 4 are from Munir et al. (2020).

Table 1 .
T-SF decision matrixThe aggregated values by utilizing the ATSFWA and ATSFWG operators are represented in Table2.

Table 2 .
Shows Aggregation outcomes

Table 3 .
Shows the score value of aggregated data Multi-attribute decision-making using Archimedean aggregation operator in T-spherical …(M.R.Khan)

Table 4 .
Ranking of score function