Comparative Ranking Preferences Decision Analysis through a Novel Fuzzy TOPSIS Technique for Vehicle Selection

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Introduction
Te automotive industry is a signifcant contributor to India's GDP.Between 2010 and 2022, there was a substantial increase in passenger vehicle sales.Today, engine performance is not the only factor when purchasing a vehicle.Consumers look for additional distinguishing features to help them make informed decisions and compare various brands.Customer preferences have become increasingly intricate and dynamic due to the availability of diverse information sources.Terefore, the automotive sector must consider customer preferences to ensure its sustainability.Achieving customer satisfaction is crucial for its prosperity.According to existing research, the purchase decision process consists of fve stages: problem identifcation, information acquisition, alternative assessment, purchase selection, and post-purchase behaviour.Tis study focuses on the third phase of the decision-making process, which involves examining options through the mathematical technique.
Te evaluation stage of alternatives involves choosing the best among the available alternatives, which can be a complex process.Tere are many characteristics and options to consider, and it is not easy to articulate these precisely in an uncertain environment.Mathematical methods often assume that model parameters accurately describe the characteristics of the real-world decisionmaking problem.However, it needs to ensure a perfect method to deal with uncertain data through a suitable mathematical model.Also, the future state of a system may be unknown and uncertain.In such cases, fuzzy set theory can provide a better approach.It can help to better understand the criteria and alternatives in uncertain or imprecise situations by assigning membership values to quantities.In 1965, Zadeh [1] introduced the concept of fuzzy sets.Tese sets use a fuzzy number to represent ambiguity, imprecision, and haziness.Te fuzzy-based model is suitable for making decisions in complex scenarios where the data is imprecise or vague.Fuzzy sets ofer a deep investigation of imprecise stochastic uncertainty based on a strict mathematical framework.Choosing an alternative involves conficting criteria, which makes it an MCDM problem.When dealing with fuzziness and conficting criteria, a particular situation becomes a fuzzy MCDM problem.Many researchers have used MCDM ( [2][3][4][5]) techniques and multi-attribute group decisionmaking (MAGDM) ( [6,7]to solve real-life decision-making problems.In day-to-day reality, people often consider numerous variables implicitly and may be content with making judgments based solely on intuition [8]. Although there are several approaches for solving MCDM and multi-attribute decision-making (MADM) problems, TOPSIS [2] is the most efective and traditional method for solving MCDM, MCGDM, and MAGDM in real-life application problems.Te traditional TOPSIS approach has been expanded into the fuzzy TOPSIS [9] method, which is used to solve MCDM problems in which the criteria or alternatives are placed in an uncertain environment.Several authors extended their research in FTOPSIS, such as Chen [9] extended TOPSIS under fuzziness, interval-valued fuzzy number ( [10,11]), Boran et al. [12] extended Chen's methods [10,11] to intuitionistic fuzzy number, and Park et al. [13] extended Boran et al. [12] methods to interval-valued intuitionistic fuzzy number.Moreover, Chen and Lee [14] developed an interval type-2 [15] and its extension proposed by Dymova et al. [16], which is the same approach utilised by Deveci et al. [5].Roszkowska and Kacprzak [17] extended the aforementioned fuzzy TOPSIS methods based on ordered fuzzy numbers.Lourenzutti and Krohling [6] explored a generalised TOPSIS technique for group decision-making with the heterogeneous input.Abootalebi et al. [18] proposed a MAGDM using modifed TOPSIS techniques with interval information to overcome the shortcomings of the aforementioned fuzzy TOPSIS method.Furthermore, Salih et al. [19] surveyed the developments in fuzzy TOPSIS on FMCDM between 2007 and 2017.Tis extended fuzzy TOPSIS approach deals with real-world application problems in a variety of felds, such as river valley water quality management [20], aircraft ( [21,22]), supplier selection ( [23,24]), project risk [25], supply chain management in food industries [26], automobile industry [27,28], and transition supply chain [29] via diferent decision-making methods.Te techniques mentioned above ofer a range of benefts.However, using only numerical values may not always be sufcient to represent real-world scenarios accurately.When faced with subjective human judgements, such as when there are competing criteria, it needs to consider the situation carefully.Under these circumstances, decision-makers ought to take vague or imprecise information into account.Instead of using exact numbers, a more sensible strategy may include using GIT2TrFNs for the proposed MCDM problem.It would mean considering the potential use of GIT2TrFNs to evaluate the ratings of attributes under consideration.Te primary objective of this study is to present a novel FTOPSIS method that integrates interval data to tackle the issues associated with MCDM.Two types of proposed ranking methods evaluate FTOPSIS: defuzzifcation and comparison of preference relations.Defuzzifcation involves generating a crisp value from the aggregated output of a fuzzy set, which removes the inherent uncertainty.On the other hand, a fuzzy pairwise comparison is difcult and time-consuming, but it preserves the inherent uncertainty.Tis work ofers a defuzzifcation of GIT2TrFNs for the MCDM-FTOPSIS technique.
Section 1 of, this study uses a literature review to illustrate the importance of MCDM issues and the development of FTOPSIS.Section 2 explains the essential concepts and preliminaries of GIT2TrFNs.Te proposed novel FTOPSIS method for determining the ranking preference of MCDM problems is explained in Section 3. Te proposed ranking system is applied numerically in Section 4. Finally, comparison studies of the proposed method and the conclusion are discussed in Sections 5 and 6.Tis study conducts a comprehensive review of existing models and decisionmaking methodologies based on the FTOPSIS method under various fuzzy numbers and environments from 2000 to 2022.Table 1 shows a wide literature survey on FTOPSIS.
To support the recognition of the relevance of the selection of the proposed study, Figure 1 shows the signifcant number of publications that fall into diferent fuzzy categories (TFN, TrFN, IVFN, and IT2FNs) and reveals that the fuzzy TOPSIS is the method that has been utilized the most frequently for performing the decision-making of various applications.

Preliminaries
Tis section provides a brief overview of the fundamental concepts related to interval type-2 fuzzy numbers (IT2TrFNs), GIT2TrFNs, and the process of defuzzifcation for GIT2TrFNs.
)) be a IT2TrFN with an upper membership function μ U (x) and a lower membership function μ U (x) [49] (where u 1 , u 2 , u 3 , u 4 and u 1 , u 2 , u 3 , u 4 are the lower and upper trapezoidal elements, respectively, and ω and ω Table 1: Literature survey on decision-making approaches using TOPSIS under fuzziness.
where 0 ≤ μ  U (x, u) ≤ 1 (Figure 2), J x is the primary membership of x, and μ  U (x, u) is the secondary grade or secondary membership for x ∈ X, u ∈ J x .
Note: If the elements of  U are continuous, then it represents by where J denotes union over all admissible x and u.Furthermore,  is replaced by  for the discrete universe of discourse.In (3), when μ and it has type-1 interval set membership function.

The Proposed Novel FTOPSIS Method
Te TOPSIS approach for analysing real-valued data was initially proposed by Hwang and Yoon [54].Subsequently, Chen [30] expanded upon the approach by using T1FSs to account for the inherent uncertainty in the fuzzy environment.Chen et al. [55] enhanced the approach by IT2FSs.Tere has been a notable focus in current academic research on the topic of fuzzy TOPSIS, but comparatively less attention has been given to the study of IT2FSs [16].Te TOPSIS approach has been employed in this study.Te authors Rashid et al. [56] have expanded the original PIS and NIS, which were initially designed for IVFSs, to encompass IT2FSs.Te extended vertex approach is employed to ascertain the disparity between viable alternatives and optimal solutions.Its design idea has been changed to include a fuzzy notion with the concept of fuzziness in weight vector of attributes.Let us consider an MCDM problem which is composed of "m" alternatives P i for i � 1, . .., m, and "n" criteria C j for j � 1, . .., n.Te decision matrix D � [d ij ] m×n is formed with all the attributes and the alternatives.Te weight vector of attributes is W � [w 1 , w 2 , . . ., w n ] T and  n j�1 w j � 1, 0 ≤ w j ≤ 1.
Steps for the FTOPSIS process are as follows: (1) Create the decision matrix and assign a weight to each criterion.Let X � [x ij ] m×n be a decision matrix and the weight to each criterion is assigned through W � [w 1 , w 2 , . . ., w n ] T which is known as a weight vector (2) Compute defuzzifed matrix where (4) Compute the normalized weighted decision matrix (5) Obtain the ideal solutions, both positive and negative.
Te positive ideal solution (PIS) P + has the following form: Te negative ideal solution (NIS) P − has the following form: where I b denote the beneft criteria (more is better) and J c denote the cost criteria (less is better); i � 1, . . ., m, j � 1, . . ., n. (6) Calculate the distance measures d i + and d i − of the alternatives far from PIS and NIS.Te most utilized conventional n-dimensional Euclidean distance is applied for this purpose.Journal of Engineering (7) Compute the relative closeness coefcient (RCC) to the ideal alternatives.
where 0 ≤ RCC i ≤ 1, i � 1, 2, . .., m. (8) Rank the alternatives based on RCC to the ideal alternatives.On the basis of RCC i rank, the alternatives are in the descending order.
Te concise nature of the fuzzy method of TOPSIS described can engage the reader efectively.Figure 4 showcases the proposed fowchart for TOPSIS approaches, utilizing a fuzzy-based approach.

Linguistic Terms for the Proposed FTOPSIS Method.
In solving the MCDM problem within fuzzy contexts, a novel FTOPSIS approach proves to be highly efective.By representing the criteria weights and scores as linguistic variables, this method employs GIT2TrFN to assign values to these variables, expressed in linguistic terms.Tis approach is valuable when grappling with real-world situations that are too intricate or ambiguous to be satisfactorily conveyed via quantitative expressions.Table 2 shows the linguistic terms in the form of GIT2TrFNs as an example of the proposed application for vehicle selection.

Numerical Example
Tis section explains the proposed TOPSIS method by solving a numerical example problem.Buying or investing in a vehicle (alternative) is often considered a luxury, especially for middle-class households.On the other hand, lower-income groups tend to compare diferent vehicle models and brands based on their needs and budgets.People consider various criteria, such as the type, size, capacity, engine power, fuel efciency, safety features, and life-cycle cost.Tese criteria are numerous, and people search for the best alternatives to fulfl their needs.Tis situation falls under the MCDM problem, and it encompasses ambiguous information, which is crucial to articulate with expertise and understanding rather than rigid limitations.Te following example focuses on selecting the best alternative when faced with competing criteria such as cost, comfort, service, maintenance, and other factors.Tese are some of the essential factors to consider when buying a vehicle.Interestingly, the least expensive, most comfortable, and safest alternative is preferable.
Tis example shows how to use the proposed MCDM model to choose a vehicle.Assume a customer wishes to purchase a vehicle, and the person has to pick out the best one among the fve alternatives (vehicle) with specifc criteria.Let P 1 , P 2 , P 3 , P 4 , and P 5 denote the variety of vehicles as alternatives (P i ) available.Tese vehicles are distinct in their own way and the stakeholder (decision-maker) is to select the best alternative available with the set C = {style (C 1 ), reliability (C 2 ), fuel-economy (C 3 ), cost (C 4 )} of certain criterion (C j ).Te alternatives are to be ranked according to the given criteria.Since the criteria are not defned in sharp boundaries, it is appropriate to represent them using GIT2TrFN.From the expert opinion, each of these criteria is given some weights, and this is represented by the weight vector W = (0.25, 0.25, 0.25, 0.25) T .Figure 5 shows the procedures for computing critical features for the MCDM, which will assist in selecting a vehicle.2. Te essence of this decision matrix is to portray the diferent alternatives that are displayed against attributes that consumers may consider when buying a vehicle.In Table 3, P i ' s represents the variety of vehicles available to consumers.Tey would choose their vehicle from amongst these alternatives, of course, subject to the diferent criteria which are represented by GIT2TrFNs.However, these fuzzy quantities are transformed back into crisp numbers for further calculations.Tis is represented in Table 3.Also, since all the criteria may not be of equal importance, some weights are assigned to these criteria, and with these weights in place, a weighted decision matrix is calculated as shown in Table 4. Te rest of the steps that follow in TOPSIS are shown in Tables 3-8.
4.1.8.Ranking Preference.Te ranking preference given here is the justifcation of P 2 ≺ P 4 ≺ P 1 ≺ P 5 ≺ P 3 based on RCC's values.Te alternative P 3 has a higher value of the relative closeness among the other alternatives.As a result, based on the abovementioned evaluation criteria, the alternative P 3 is the best automotive vehicle among the fve alternatives.

Sensitivity Analysis.
In this sensitivity analysis, this study examines the efect of diferent scenarios on purchasing a vehicle.Te decision-maker evaluated fve alternatives based on various criteria.Table 9 shows the performance of the fuzzy TOPSIS method, and alternatives are ranked equally for diferent membership values.As shown in Figure 6, the proposed FTOPSIS method is less afected by changes in the membership values of GIT2TrFN but more afected by changes in the weights of the criteria.If a particular criterion adds more weight, it will alter the order of the alternatives.Moreover, the FTOPSIS technique can accurately distinguish between diferent variations.Tis capability of the FTOPSIS method can prove particularly advantageous for decisionmakers when dealing with highly subjective criteria and complex judgements.

Comparison of Ranking Preferences with Existing Methods
Te FTOPSIS approaches were evaluated and compared to existing methods across various TOPSIS outcomes.Several researchers have investigated the FTOPSIS methodology in diverse practical contexts inside fuzzy settings, employing triangular fuzzy numbers to explore the inherent fuzziness.
To conduct a comparative analysis between the proposed technique and current methods, we have found scholarly works that have employed the FTOPSIS method and those that have focused on TrFNs and IT2FNs in Table 1.For a quick view of the real-world application, Table 10 shows the results of the existing approaches relating to the RCC for ranking preference.Chu [57] proposed FTOPSIS by developing the membership function of two TrFNs.In addition to this, there is a drawback associated with arranging the generalised and interval-valued trapezoidal fuzzy numbers [59] in inconsistent order.On the other hand, Ashtiani et al. [60] tackled the FMADA problem using interval-valued triangular fuzzy numbers.Te decision-making method proposed by [60] utilised the lower and upper limits of interval-valued triangular fuzzy numbers to compute the relative closeness coefcient using the TOPSIS method.However, this approach must only consider the holistic nature of interval-valued triangular fuzzy numbers.Dymova et al. [16] used an IT2FV α-cut representation for the type-2 interval fuzzy extension of the FTOPSIS method to overcome the limitations and drawbacks of earlier techniques.Many uncertainties cannot be dealt with using a type-1 interval fuzzy set.Celik et al. [61] reviewed many articles based on fuzzy sets of type-2 intervals to identify uncertainty in solving MCDM problems, despite shortcomings in addressing the generalised interval type-2 fuzzy sets in solving MCDM problems.To address the defciency mentioned earlier, Ilieva [43] proposed a modifed TOPSIS based on IT2FNs with drawbacks on the defuzzifcation of GIT2TrFN.To overcome all the abovementioned limitations and disadvantages, Meniz [62] compared defuzzications ([53, 63, 64]) and contributed the new idea of fuzzy metric for obtaining the optimum solution to the MCDM problems.Te extensions of the fuzzy TOPSIS approach, created by various authors, possess certain limits and drawbacks.However, it is important to note that these extensions are particularly advantageous in tackling problems related to MCDM/MADM, MCGDM, and other comparable issues.Tis study introduces GIT2TrFN as a proposed method for extending FTOPSIS, along with its corresponding defuzzifcation technique.Table 10 presents a comparison between existing methods and the proposed novel FTOPSIS outcomes using diferent methodologies.

Conclusion
Tis study investigates the novel FTOPSIS technique for selecting the best vehicle from an individual's perspective.Using GIT2TrFN instead of traditional fuzzy sets simplifes and allows more ambiguity inputs in MCDM problems, improving the strategy's resilience and intelligence.Furthermore, the vehicle selection MCDM problem demonstrates the novel FTOPSIS technique's high efectiveness when using GIT2TrFN to address real-world uncertain scenarios.Tis study helps to determine the existence of contradictory criteria when we use fuzzy linguistic terms using GIT2TrFN rather than exact numbers.Te presented sensitivity analysis gives more information about the reasonable study of the proposed method.Figures 7 and 8 provide more details about the proposed FTOPSIS and the existing methods.Te performance of the PIS and NIS of the proposed alternatives is nearly the same as that of the Liu [38] methods (Figure 7).In Figure 8, the proposed FTOPSIS method shows that the RCC of alternative performance is neutral compared to the existing method.Te proposed FTOPSIS technique evaluates alternatives in scenarios, including contradictory criteria .Also, it gives better ranking results for the numerical example of the vehicle selection problem.As a result, the decision-maker may choose a vehicle P 3 as the best option among the fve alternatives.Te practical scenario involved the application of the proposed methodology, where an automotive company chooses the most suitable vehicle for its manufacturing development.
Te proposed decision process holds considerable managerial signifcance, as it ofers potential assistance to top-level management in efectively managing proposed projects by optimizing resource allocations and improving productivity.Given our inclination towards employing more realistic mathematical abstractions to depict human decision-making, it is evident that further investigation on this specialized subject is warranted.
6.1.Limitations.Tis study is unsuitable for determining the weights of the best and worst alternatives, as it focuses solely on the ideal and anti-ideal options through the FTOPSIS methodology.
6.2.Future Recommendations.Tis study holds promise for future growth and deals with many real-life decision-making problems in exploring the wide range of mathematical frameworks using extended fuzzy numbers, Fermatean fuzzy numbers, q-rung fuzzy sets, and neutrosophic fuzzy numbers.

Figure 4 :
Figure 4: Flow diagram for the proposed weighted decision matrix-based TOPSIS under fuzziness.

Figure 5 :
Figure 5: Computational steps of numerical solution of vehicle selection.

Figure 7 :
Figure 7: Performance of PIS and NIS with existing methods.

Figure 8 :
Figure 8: RCC performance of the proposed method with existing methods.

Table 3
of the decision matrix D = [x ij ] m×n (i = 1, ..., 5; j = 1, ..., 4) is formed using the linguistic terms presented in Table2.Te diferent criteria that consumers consider are represented by fuzzy numbers of type-2 interval values.Tey are assigned membership values based on subjective judgements which are illustrated in Table

Table 2 :
Linguistic terms in the form of general interval type-2 fuzzy sets.

Table 4 :
Te weighted normalized decision matrix.

Table 6 :
Normalized decision matrix.Table5gives defuzzifed values to the decision matrix D in the form of D f � [x ij ] m×n (see Defnition 4).Defuzzifcation is ensured to convert the fuzzy values to crisp numbers, and the next step is normalising the achieved defuzzifed values.

Table 8 :
RCC to the ideal solutions.

Table 9 :
Performance of TOPSIS under various linguistic values and distinct membership values of GIT2TrFN

Table 10 :
Comparison of ranking preferences of diferent methods.