ΑΔΣ Advances in Decision Sciences

Purpose: As an extension of pythagorean fuzzy sets, the q‐rung orthopair fuzzy sets (q‐ROFS) is proposed by Yager in 2017. The q-ROFS offers a novel calculation form for the loss function and effectively deals with unclear information of multi-attribute decision-making (MADM) problems. The concept of q-rung orthopair fuzzy number (q-ROFN) is introduced to facilitate the use of q-ROFS in 2018. This study proposes a comprehensive q‐rung orthopair triangular fuzzy number (q-ROTFN) which is a special notation of q-ROFN, to cope with supplier selection problems. Design/methodology/approach: A new method is developed in this paper for supplier selection MADM problems in uncertain situations. The proposed technique utilizes experts' knowledge represented by q‐ROFN. It considers the selection of the most proper supplier taking into account flexibility, quality, price, supplier profile, and delivery criteria. Based on the advantages of q-ROFN, this article proposes an extended fuzzy TOPSIS method that does not require aggregation technology. Findings: To verify the proposed technique, a case study is conducted to evaluate and rank the alternative suppliers for an automotive company. As a result of the outcomes, it is shown that the proposed method is suitable for MADM problems. Originality/value: The main contributions of this paper are as follows: (i) Traditional TOPSIS method has been extended using the q-ROTFN to solve multi-attribute decision problems, (ii) It is shown that aggregation techniques are not needed for q-ROTFN based TOPSIS method, (iii) A novel expert weight calculation technique is proposed.


Introduction
Multi-attribute decision making (MADM) are widely known methods to make a decision using many conflicting criteria and attitudes. There are many well-known MADM methods like Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), The Analytic Hierarchy Process (AHP), Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE), etc. (Alibeigi et al., 2021). As observed from the literature, the results can vary when different methods are implemented for the same problem with the same data. While some techniques are related to calculating weights of decision-making criteria and alternatives such as AHP, others as TOPSIS are used to define the differences of alternatives according to their order preference concept. Due to the variety of methods, one of the important questions is to decide which method is suitable for a defined problem. The best way to determine the method is to understand the problem structure.
TOPSIS is one of the most popular methods introduced by Yoon and Hwang (1980) and is based on the distances of the decision points to the ideal solution. There are numerous applications of fuzzy TOPSIS to solve MADM problems (Tang et al., 2020;Jahanshahloo et al., 2006;Chen, 2000;Hwang et al., 1981). Compared to AHP, TOPSIS gives better results in a short calculation procedure. One of the other advantages of the TOPSIS method is calculation ease. However, the nature of decision-making problems has uncertainty which can be concluded with wrong results to solve using the traditional MADM methods. The fuzzy set theory also provides a better outcome for handling the vagueness of human thought, the primary input of MADM problems. Therefore, instead of using the classical decision-making methods, it is better to use the fuzzy extension.
In this study, our problem is the selection of appropriate alternative suppliers among a set of alternatives for supplier selection problems. TOPSIS is one of the most suitable traditional MADM methods compared to other methods. However, regarding all the aforementioned explanations and considering our problem structure, the fuzzy extension of TOPSIS can have a better solution.
The q-rung orthopair fuzzy set (q-ROFS), developed by Yager (2017), is a powerful set that effectively deals with uncertainty in real-life problems and covers complicated and hesitant fuzzy information. Therefore, this model has a more robust capability than other fuzzy sets, such as intuitionistic or pythagorean fuzzy sets to manage the uncertainty.  introduced the concept of q-rung orthopair fuzzy number (q-ROFN) to facilitate the use of q-ROFS.
Numerous researches apply the q-ROFS and q-ROFN methods to solve the MADM problems; these methods' procedure of decision information is too old and limited. Furthermore, the q-ROFS and q-ROFN are generally based on different aggregation operators. No matter what aggregation operators are used, it may distort decision information. Also, some decisionmakers might give the criteria evaluation value of a scheme too low or too high owing to individual bias.
Based on the advantages of q-ROFN, this study proposes an extended fuzzy TOPSIS method that does not require aggregation technology. The proposed method attempts to extend the traditional TOPSIS with q-ROTFN which is a special notation of q-ROFN. Another outstanding point of this paper a novel expert's weight calculation technique that can successfully cope with the expert's possible bias and solve experts' circumstances with significantly different opinions has been proposed.
The article's primary contributions are as follows: (i) to demonstrate how the proposed model, which is an extension of the traditional TOPSIS method based on q-ROTFN, can efficiently solve multi-attribute decision problems without the requiring aggregation operators, (ii) to introduce a new expert weight calculation technique, (iii) to conduct a case study in the automotive industry to show the applicability of the proposed method for assessing and ranking suppliers.
The second section presents a literature review about TOPSIS and q-rung orthopair fuzzy environment. The third section explains the theoretical fundamentals of q-ROFS and q-ROTFN. Moreover, the proposed model is shown in the third section. The case study results and sensitivity analysis are presented in the fourth and fifth sections. In the last section, the findings and conclusions are discussed.

Literature Review
Supply chain is the concept that defines all the logistics activities, starting from the required raw material supplied for producing a product to reaching the final consumer. All these processes include people, technologies, and resources involved in these activities. On the other hand, supply chain management defines all the managerial tasks of the material and information flow that occur during these processes. Supply chain management contains four core phases: planning, procurement, production, and distribution. Supplier selection appears in the procurement phase according to the conditions determined from the planning phase, followed by processes in producing and distributing the products from the supplied materials (David, 2016;Liang and Cao, 2019). Defining the proper supplier is a crucial and difficult problem for companies. The reason for this is that numerous criteria must be considered when selecting the most suitable supplier(s) to meet the expectations of the business. It is possible to see that there are abundant studies on supplier selection from the literature. One of the oldest studies was conducted by Youssef et al. (1996), in which existing methods have been grouped under five headings as categorical models, matrix models, cost-based models, multicriteria selection models and supplier profile evaluation model. These models can be reclassified as multi-attribute decision models, cost-based models, mathematical programming models, and statistical models for supplier selection problems. Researchers widely deal with supplier selection as a MADM problem involving conflicting and noncommensurable criteria, assuming that compromising is acceptable for conflict resolution.
TOPSIS, which was developed by Yoon and Hwang (1980), is one of the widely used techniques among MADM methodologies. It aims to evaluate the relative closeness of the alternatives to the ideal solution with the Euclidean distance approach. The basic layout of this technique can be described as the selected alternative should be geometrically the nearest distance from the positive ideal solution and the furthest from the negative ideal solution. Compared with other MADM methods like ELECTRE or VIKOR, the TOPSIS method has an easy calculation procedure; hence, it has been widely implemented in many different fields. However, exact numerical values for evaluation may be inadequate when real-life cases take into account. This situation comes from the uncertainty of human thoughts and judgments, especially preferences. Therefore, fuzzy TOPSIS proposed by Chen (2000) is a systematic approach that extends classical TOPSIS to solve decision-making problems under a fuzzy environment.
Fuzzy TOPSIS has linguistic expressions to reflect the judgments more realistic way. In this method, decision-makers verbally express their thoughts when evaluating alternatives using specified criteria. Then, these linguistic expressions are converted into fuzzy numbers. After that, the proximity coefficients of each alternative are calculated. Finally, calculated closeness coefficients are sorted according to their values, and the appropriate alternative is selected.
Fuzzy sets can be added to multi-criteria decision problems to get more robust results and cover real-life uncertainty (Ayyildiz, 2021). In this manner, the q-ROFS and q-ROFN are novel and effective tools for dealing with uncertainty in real-life problems and covering more complicated and uncertain fuzzy evaluation information. Moreover, representation of decision information effectively and reducing distortion put forward research for q-ROFS and q-ROFN.
Generally, studies about q-ROFN focus on theoretical research, especially on aggregation operators. Peng et al. (2018) presented a new score function of q-ROFN to solve the failure problems when comparing two q-ROFN. They defined a new exponential operational law about q-ROFN, in which the bases are positive real numbers, and the exponents are q-ROFN. The notion of interval-valued q-rung orthopair fuzzy sets (IVq-ROFS) that allows decisionmakers to present their satisfactory and unsatisfactory ratings about a given set of alternatives based on a range value is presented by Joshi et al. (2018). They provided the aggregation of IVq-ROFS based on essential operations of the proposed model, such as negation, union, and intersection. In another study, to quantify the loss function of decision-theoretic rough sets, Liang and Cao (2019) utilized q-ROFSs and constructed a new three-way decisions model by TOPSIS with projection-based distance measures. To aggregate different thoughts based on the power average (PA) operator and power geometric (PG) operator, the authors designed four operators: (i) q-rung orthopair fuzzy power average (q-ROFPA), (ii) q-rung orthopair fuzzy power weighted average (q-ROFPWA), (iii) q-rung orthopair fuzzy power geometric (q-ROFPG), (iv) q-rung orthopair fuzzy power weighted geometric (q-ROFPWG).  introduced four multi-criteria selection model operators to use q-ROFI, including q-ROFMSM, the q-ROFWMSM, q-ROFDMSM and q-ROFWDMSM operators. A new notion for q-rung orthopair regular fuzzy set is developed by . Yang and Pang (2019) introduced some q-rung orthopair fuzzy partitioned Bonferroni mean operators. After that Liu D. et al. (2019) presented two novel q-rung orthopair fuzzy extended Bonferroni mean (q-ROFEBM) operators and their weighted form (q-ROFEWEBM) to select a location.
The main advantage of the proposed operators is that they can aggregate input arguments with heterogeneous relationships more intuitively and effectively. Liu and Wang (2020) proposed two new operators: the q-rung orthopair fuzzy weighted averaging operator (q-ROFWAO) and the q-rung orthopair fuzzy weighted geometric operator (q-ROFWGO). Another study suggested advanced weighted generalized Maclaurin symmetric mean (q-ROFWGMSM) and the weighted generalized geometric Maclaurin symmetric mean (q-ROFWGGMSM) operator (Liu and Wang, 2020).
As a result of current research on q-ROFS and q-ROFN, it can be mentioned that most of the studies focused on the forms of various collecting operators. Associated with deeper research, these aggregation operators become more complex. However, q-ROFN is a comprehensive and effective information expression tool that is able to deal with uncertain knowledge and adjust the representation of information based on different decision-makers and decision scenarios.
Apart from theoretical research, there have been some q-ROFS and q-ROFN empirical studies with real-world applications in recent years, as this method is well suited for complex decision-making situations in a variety of fields, such as stock investment. (Tang et al., 2020), e-commerce (Liang and Cao, 2019), energy source selection (Krishankumar et al., 2021), education (Hussain, 2019), partner selection , construction project selection , investment selection (Liu, P. et al., 2019), location selection (Liu, D. et al., 2019), etc. Some investment types, like stock investment, are characterized by high risk and immense profit. Therefore, the selection method should avoid investment risks and obtain increased returns. Tang et al. (2020) combined q-ROFS with decision-theoretic rough sets considering these critical advantages. Thanks to the q-ROFS, a new generalized form of Pythagorean fuzzy sets (PFS) and intuitionistic fuzzy sets (IFS) show uncertain information more extensively and flexibly. Thanks to the reasonableness and effectiveness of the q-ROFS, it has also been implemented for other risky investment selections.  introduced the q-rung orthopair hesitant fuzzy set (q-ROHFS) TOPSIS method to select energy projects. Another study for selecting renewable energy sources from a range of sources based on sustainability characteristics has been conducted using the q-ROF information to minimize vagueness and subjective randomness by providing a flexible and generalized preference style (Krishankumar et al., 2021). Hussain et al. (2019) presented the q-rung orthopair fuzzy TOPSIS (q-ROF-TOPSIS) methodology for the MADM problem in the education field thanks to the core advantage of the q-ROF information, which cope with complexities and uncertainties efficiently. The developed model depends on the cover-based q-rung orthopair fuzzy sets (Cq-ROFRS). Wang et al. (2019) introduce the q-rung orthopair fuzzy MABAC (multi-attributive border approximation area comparison) model on account of the traditional MABAC model to compute the distance between each alternative.
As a result of the aforementioned studies, q-ROFS or q-ROFN are efficient tools to solve a multi-variable problem, and the results are satisfactory. Table 1 presents a summary of these studies, including the using methodologies. Moreover, Table 1 consists of alternatives for each reviewed research, a list of criteria, the number of decision-makers, and the q value used.
The literature review showed that almost all the research on q-ROFN agrees that this method is an efficient and influential instrument for gathering different decision-makers evaluations under uncertain and complicated conditions. In the other word, q-ROFN is capable of encompassing more complex and hesitant fuzzy evaluation information. However, our q-ROFN research is generally based on complex aggregation operators. Even if aggregation operators are a powerful tool for solving decision problems, while it has been applied to these new types of fuzzy sets with defective operational rules, the result has significant information distortion frequently. To solve this problem, Liu P. et al. (2019) proposed a novel decisionmaker weight calculation model applying the TOPSIS method with q-ROFN. The outstanding point of this calculation, it hasn't included complex aggregation operators.
In this study, an extension of TOPSIS using q-ROTFN has been used to solve a supplier selection problem in the automotive sector. The reasons why the q-ROTFN based TOPSIS method is used in the state of traditional fuzzy TOPSIS can be listed as follows: (i) the form of decision information used by fuzzy TOPSIS methods is too restricted and an old procedure, (ii) traditional fuzzy TOPSIS is not capable of dealing with existing complicated decision situations, (iii) the aggregation methods used in the traditional fuzzy TOPSIS method distort decision information, (iv) each decision-makers' weight sometimes cannot be determined by fuzzy TOPSIS directly and accurately (v) the defect of relative closeness calculation of the TOPSIS method which is nearest ideal solution to positive ideal solution is not necessarily the farthest from the negative ideal solution, which makes the evaluation result inaccurate. From the application perspective to the best of our knowledge, to solve group decision problems, even supplier selection problems, q-ROTFN-based TOPSIS method has never been used. Zadeh (1965) introduced the concept of fuzzy sets in 1965 as an extension of classical sets. Fuzzy sets use a similar form of probability to express a person's degree of satisfaction with a particular entity, which is referred to as the membership degree (u). Further, Atanassov (1986) extended the fuzzy sets by adding non-membership degree (v) to express one's degree of dissatisfaction with a particular entity. This novel formula of information has been called intuitionistic fuzzy sets (IFSs) and specified that it must satisfy the constraints of u ϵ [0, 1], v ϵ [0, 1], and 0 ≤ u + v ≤ 1. IFS also includes a hesitation degree (π), which is an indicator of an entity's level of uncertainty, and must satisfy π = 1u -v. Xu (2017) refers to the single element of IFS as an intuitionistic fuzzy number (IFN). However, due to the uncertainty and conflict inherent in people's subjective cognition, expert evaluation values do not always fully satisfy the constraints of 0 ≤ u + v ≤ 1. To solve this problem, Yager (2014) proposed pythagorean fuzzy sets (PFSs) that extend 0 ≤ u + v ≤ 1 to 0 ≤ u 2 + v 2 ≤ 1. A single element of PFS is called pythagorean fuzzy number (PFN). While PFN broadens the representation scope of decision-making information, its capacity to express vague and hesitant information remains limited as people's vague consciousness and degree of hesitation increase. Subsequently, Yager (2017) proposed q-rung orthopair fuzzy sets (q-ROFS), which provide a novel calculation form of the loss function. As shown in Eq. (1), q-ROFSs can be expressed by the sum of q th power of the membership degree and q th power of the non-membership degree.  introduced the concept of q-rung orthopair fuzzy number (q-ROFN) to facilitate the use of q-ROFS. 0 ≤ u q + v q ≤ 1, where q≥1
is represented by hesitancy degree or indeterminacy degree ̃ ,for each member of y Y.
For convenience, ̃ ( ) ̃ ( ) is called q-ROFN by , which can be denoted by ̃ ̃ ̃ . Assume ̃ and ̃ are two q-ROFN. Some operations rules between these two q-ROFN are listed below.

q-Rung Orthopair Triangular Fuzzy Number (q-ROTFN)
The q-rung orthopair triangular fuzzy number (q-ROTFN) is a special notation of orthopair fuzzy set. For universal set Y, a q-rung orthopair triangular fuzzy set (q-ROTFS) ̃ in Y is defined by -+ be two q-ROTFNs, then the normalized Hamming Distance d ( ̃ ̃) between the q-ROTFN ̃ and ̃ is calculated using Eq. (8).

The Proposed Model
The q-ROFN is a very efficient and influential instrument for expressing information of different decision-makers. Generally, q-ROFN is based on aggregation operators. Assume the decision matrix given by decision-maker DM k is = [ ] mxn, and ƞ= (ƞ 1 , ƞ 2 , …., ƞ p ) is the degree of decision-maker importance in this field and ϵ [0,1]. w = (w 1 , w 2 , …, w n ) is criteria weight vectors and satisfies the constraints w j ϵ [0,1], ∑ . With this information, the goal of the MADM problem is to choose the best alternative from a set of alternatives.
Step 1: Construct the normalized decision matrix. There are two types of criteria in a decision-making environment: benefit type and cost type. While the former implies that the bigger is the better, the latter means that the smaller is the better. To eliminate the impact of different criteria types, criteria should be standardized. Step2: Construct the attribute weighted decision matrix of each decision-maker. This matrix is shown by = and calculated using Eq. (9).
Step 3: Compute support degree between attribute decision matrix and attribute decision matrix using Eq. (10) and represent it as Sup ( ).
Step 4: For each calculate total support degree T ( ) by using Eq. (11).
Step 5: Expert evaluation data frequently appear to be of extreme value due to the influence of personal prejudice. When the expert's weights are equal, the conventional method eliminates the highest and lowest scores, but this approach loses decision information. This study uses expert assessment rationality to preserve complete information and effectively deal with unreasonable excess data. The rationality degree of evaluation means that the assessment information provided by the experts for each alternative is close to the assessment information provided by other experts. Calculate the rationality degree of expert evaluation using Eq. (12).
Step 6: Calculate the comprehensive index of experts by combining the rationality degree of the evaluation and the importance level of the expert. The formulation of the comprehensive index is shown in Eq. (13).
Step 8: Construct the weighted decision matrix ̈ of expert by using Eq. (15).
⟩ where w k signifies the weight of each k expert. (15) Step 9: Convert each final weighted decision matrix of expert ̈ into the alternative decision matrix . corresponds to the ̈ in step 8.
Step 10: Using Eq. (16) and Eq. (17), calculate A+ (positive ideal decision matrix) and A-(negative ideal decision matrix). should contain the best evaluation data in all alternative decision matrix . ,( where should contain the worst evaluation data in all alternative decision matrix Step 12: Compute the relative closeness of the alternative decision matrix to the ideal decision matrix. Relative closeness is calculated by Eq. (20). - Step 13: Determine the priority of the alternatives. The optimal alternative is determined using the ranking rule that the larger the , the better the alternative.

Application Case in the Automotive Industry
One of the foremost manufacturers of spare parts for automobiles needs to determine a supplier of joint components used in various spare parts. A1, A2, A3 express three potential suppliers (i=1,2,3). The alternatives are evaluated against five decision criteria (j=1,2,3,4,5), the most important ones for the automotive industry. The criteria are defined by the decisionmakers using a survey among Turkey's leading automotive companies as flexibility (C1), quality (C2), price (C3), supplier profile (C4), and delivery (C5). The assessment of the potential suppliers in each criteria is based on linguistic decisions provided by the decisionmakers D1, D2, D3 (k=1,2,3), who are responsible for purchasing at a different company level. According to these three decision makers' professional backgrounds and specialists, the importance index vector is determined as η = (0.8, 0.65, 0.95) T . The assessment data provided by each decision-maker for the criteria of each alternative is symbolized by; where i=1,2,3; j=1,2,3,4,5; k=1,2,3.
The q-ROTFN method is applied to this case to show the applicability of q-ROTFN for supplier selection problems. One of the most critical parameters for q-ROFN TOPSIS is determining of q value. It is observed that the most applied value for this parameter is 3 in the literature (see Table 1). The main reason for using this parameter as 3 in our study is that it has developed better results with good performance. Moreover, the other parameter values have been implemented to see the differences in our problem solution.
In this study, three experts who are responsible for the company's purchasing evaluate the criteria's weight and alternatives' rating. It is worth to say the experience years of these experts in this field changes between 3-10 years. This long-term experience not only gives a chance for a proper evaluation of alternatives but also provides a different perspective assessment. Based on Chen's (2000) triangular fuzzy numbers (TFN), the evaluations are presented in Tables 2 and 3, respectively.  (5, 7, 9) Good (G) (7, 9, 10) Very good (VG) (9, 10, 10) Tables 4, 5, and 6 present the linguistic judgments of the weights of the criteria and the ratings of the alternatives for each three decision-makers.  The decision-making procedure for the extended fuzzy TOPSIS method based on q-ROTFNs is detailed below.
The initial step is the creation of a normalized decision matrix. For analysis, the linguistic variables must be converted to triangular fuzzy numbers. After the linguistic terms have been converted to fuzzy numbers, a procedure for normalization can be performed. In the second step, the weighted decision matrix for each decision-maker is calculated. As an example, the normalized and weighted decision matrices for are presented in Tables 7 and 8, respectively. The following step concerns the determination of the support degree. The support degrees calculated by Eq. (10) are listed in Tables 9, 10, and 11. Following the computation of the support degree, the total support degree is calculated using Eq. (11) and is presented in Tables 12, 13, and 14. For convenience, this study replaced T( ) with matrix . The next step is calculating the rationality degree (Step 5). Using Eq. (12); , and are calculated as 0.7732, 0.7631, and 0.7059, respectively.
Experts' comprehensive indexes and weights are calculated in the following two steps. Consistent with the professional background and specialist of the three decision-makers, the importance level of experts is determined as (0.8, 0.65, 0.95) T , and parameter is set to 0.5. The comprehensive index of each expert is found as 0.78569, 0.70653, and 0.82795. The weight of each expert is found 0.33889, 0.304398, and 0.356712, respectively.
Then final weighted decision matrix of each decision-maker is computed by multiplying the experts' weight and support degree. The final weighted decision matrix for each decisionmaker is presented in Tables 15, 16, and 17. q is small (q = 3), the RC is small for alternatives A1 and A2; when q is increased, the RC becomes more significant for these two alternatives. By contrast, changes in the q value have a negligible effect on the RC of alternative A3. As shown in Table 25, when the q value increases, the RC of alternatives increases by different ratios. As a result, the ranking results may vary as the q values change. When q is increased from three to five, the ranking of the alternatives changes from A2>A3>A1 to A2>A1>A3, but the best alternatives remain the same. As q increases, the optimal choice changes from A2 to A1. When q=7 or q=9, the ranking results remain constant, but the alternatives' relative closeness values change.

Conclusion and Future Studies
This paper develops a new approach as an extension of the traditional TOPSIS method using the q-rung orthopair fuzzy tools to solve MADM problems. The technique utilizes experts' evaluations has been gathered using q-ROTFN which is a special notation of q-ROFN. Additionally, a novel expert weight calculation procedure which is an effective approach to deal with the expert's potential bias and different opinions, is proposed. The outstanding point of this calculation, it hasn't included complex aggregation operators. Last but not least, the result of the automotive industry case study illustrates that the proposed method is suitable for supplier selection problems.
This method is just a new extension of TOPSIS used q-ROTFN, and there are rare studies using this method. Therefore, different MADM problems from various fields can be conducted. Moreover, a sensitivity analysis is set to show the results according to the changing q value, which is the most important parameter of this method.