Integrated multi-criteria group decision-making model for supplier selection in an uncertain environment

Abstract The evaluation and selection of reliable suppliers are an extremely important operational activity for any supply chain management. It is a hard problem since supplier selection is typically a multi-criteria decision-making problem involving several conflicting qualitative and quantitative criteria on which the decision-maker’s knowledge is usually vague and imprecise. The selection process may be biased while the decision is made by single expert instead of multiple experts. This paper shows an integrated fuzzy approach to solve this problem. In this study, the proposed multi-criteria group decision-making model is developed by integrating intuitionistic fuzzy set (IFS), fuzzy analytic hierarchy process (fuzzy AHP), and fuzzy technique for order preference by similarity to ideal solution (fuzzy TOPSIS). The weight of the decision-maker is determined by using the intuitionistic fuzzy set, criteria weight is determined by the fuzzy AHP, and both weights are applied in the fuzzy TOPSIS method for ranking the suppliers. The efficiency and practicability of the presented approach are numerically demonstrated by solving a numerical example. Finally, sensitivity analysis is performed to test the robustness and stability of the introduced methodology. The result shows that the proposed model is more robust compared to other approaches.

ABOUT THE AUTHOR Md. Mohibul Islam completed his B.Sc. and M.Sc. Engineering program in the field of Industrial Engineering. He is an assistant professor at the Department of Industrial & Production Engineering of Rajshahi University of Engineering & Technology, Bangladesh. Currently, he is doing his Ph.D. research in Japan by attaining a Japanese government scholarship (MEXT) from 2019. Presently, he is a Ph.D. candidate in the graduate school of engineering at Nagoya Institute of Technology, Japan. His research interest is to develop an algorithm and mathematical model to solve industrial problems mainly related to the field of supply chain and logistics management. He is also interested to apply advanced fuzzy logic to solve decision-management problems. He is a member of the Japan Society of Mechanical Engineers and a member of the Institution of Engineers in Bangladesh. He is the owner of about 20 published articles in the field of Industrial Engineering. These articles have been published at different international conference proceedings and in different reputed international journals.

PUBLIC INTEREST STATEMENT
In this study, a new integrated multi-criteria group decision-making model is proposed for supplier selection under an uncertain environment.The intuitionistic Fuzzy Set, the Fuzzy AHP, and the Fuzzy TOPSIS methods have been integrated to develop the proposed model.The proposed approach is validated by benchmarking with other models such as the Fuzzy COPRAS, the Fuzzy EDAS, and the Fuzzy ELECTRE methods.The result shows that the proposed model is more robust and reliable than other existing models.Sensitivity analysis is performed to test the stability and highlight the robustness of the proposed model.This model can be applied to solve any decisionmanagement problem.

Introduction
A strong supply chain network is very important for any manufacturing and service organization to survive in the competitive business market. Different stakeholders such as suppliers, manufacturers, wholesalers, retailers, and customers are involved in this network. Supplier is the most important stakeholders in any high-tech industry since the smooth flow of productivity depends on their performance. Besides, approximately 80% of manufacturing costs are related to raw materials Zeydan et al. (2011). Hence, reliable supplier selection is a crucial function for any manufacturing organization. It plays a pivotal role in augmenting profitability by reducing operational costs. The evaluation and selection of desirable suppliers often involve several qualitative and quantitative criteria. The characteristics of several criteria are mutually conflicting. Hence, the supplier evaluation and selection processes are considered as a multi-criteria decision-making (MCDM) problem. There are many well-established MCDM models in the literature. These models have distinct mathematical characteristics and solution procedures. But they have two common objectives such as (a) deriving the relative weights of the considered criteria by evaluating one against the others, and (b) ranking of the candidate alternatives based on the accumulative score with respect to each criterion. To solve decision management problems by MCDM model mainly two types of weights such as objective and subjective weights are discussed in Liu et al. (2015). The objective approach ignores decision makers' opinions and is established on determining weights of criteria based on the information contained in the decision-making matrix using certain mathematical models. On the other hand, subjective approaches reflect the subjective thinking and intuition of the decision-makers. Uncertainty is involved in this process due to imprecise human judgment. Since the supplier rankings are sensitive to the weight of criteria, so especial attention must be paid to the model for determining the weights of criteria. From the literature review as presented in Section 2, it can be clearly noticed that different mathematical techniques have mainly been employed for two purposes (a) determination of weights to be assigned to various evaluation criteria, and (b) ranking of the competing suppliers. Gabus and Fontela (1972) developed the decision-making trial and evaluation laboratory (DEMATEL) method, which is used for estimating the objective weight of the criteria. Shemshadi et al. (2011) used the entropy method to determine the objective weight of the criteria. The best-worst method (BWM), stepwise weight assessment ratio (SWARA), full consistency method (FUCOM), and label-based weight assessment (LBWA) methods are mainly used to determine the subjective weight of the criteria. But these methods cannot deal with uncertainty. The fuzzy AHP and fuzzy SWARA can be used to determine the weight of the criteria considering uncertainty. The interrelationship between the criteria is very important and it should be taken into consideration during determining the weight of the criteria which is ignored by the fuzzy SWARA method. In this method, initially, criteria are ranked, then 2nd ranked criteria is compared with previous rank i.e., 1st ranked and in such a way it is continued to last rank. There is no scope to compare within each criterion. The hesitant fuzzy set (HFS), intuitionistic fuzzy set (IFS), neutrosophic fuzzy set (NFS) can also determine criteria weight considering uncertainty. Like fuzzy SWARA, these methods also avoid interrelationship between the criteria.

Literature review
To survive in a competitive global market reliable and flexible supplier selection is an inevitable task for any manufacturing and service organization. Several tasks are involved in the supplier selection process such as identification of materials or products to be procured, assimilation of a list of potential suppliers, shortlisting of the key factors/criteria based on which the suppliers need to be evaluated, formation of a team of experts/decision-makers to extensively analyze and strategize this selection process, choosing of the most appropriate suppliers while disposing off the inefficient ones, and continuous performance evaluation of the finally selected suppliers Chakraborty et al. (2020). To achieve these goals, many MCDM models have been developed by different researchers over the decades. Many firms and researchers have applied single models as well as integrated models to make their decisions. In these models, certain/uncertain manufacturing environments have been considered. Table 1 provides a concise list of different approaches for solving supplier selection problems. Keršuliene et al. (2010) developed the SWARA method to determine the weights of the criteria using crisp numbers. A relatively new model known as the BWM is proposed by Rezaei (2015) to determine criteria weights. In this method, criteria are compared with respect to the best criteria and worst criteria, and relative ratings are given on criteria based on a crisp scale. This method is faster, but it cannot handle uncertainty and is not applicable in a group decision-making environment. The FUCOM method is developed by Pamuˇcar et al. (2018) to estimate the weights of the criteria using a crisp scale. In this method, criteria are initially ranked, and then compare other criteria with respect to the first rank. This method is applicable in a group decision-making environment, but it cannot work in a fuzzy environment. The Fuzzy pivot pairwise relative criteria importance assessment (PIPRECIA) method is developed by Stevi´c et al. (2018) to determine weights of the criteria. This model can determine fuzzy criteria weights. But the flexibility of this model is limited because it cannot work with other fuzzy scales. Besides, the 1-2 scale is used to assess the criteria but if 2 is used as a rating it generates an infeasible solution. The most recent model known as LBWA is proposed by Žižovi´c and Pamucar (2019) to determine the weight of the criteria. This model can handle a large number of criteria, but it cannot handle uncertainty and is not applicable in a group decision-making environment. It is noticed that weights of the criteria obtained from a particular method are used in another model for ranking the alternatives. Yazdani et al. (2019a) proposed an integrated model using the DEMATEL and EDAS methods for solving the green supplier selection problem in a construction company in Spain. They used the DEMATEL method to estimate the objective weights of the criteria and then used it in the EDAS method for ranking the suppliers. In their studies, they did not consider the uncertainty in the decision-making process. Stevi´c et al. (2020) proposed the MARCOS method to select a sustainable supplier in a health care industry in Bosnia. In their model, the weights of the criteria are determined using crisp ratings given by multiple experts. Then, these weights are used in the MARCOS method for ranking the alternatives. They did not show the interrelationship between the criteria. They also ignored the weight of the decision-makers in the evaluation process. Chakraborty et al. (2020) used the MARCOS method for ranking the alternative. This model works well in a crisp environment, but it cannot handle fuzzy data directly. Yazdani et al. (2019b) used the EDAS method with an extended SWARA method for ranking the alternative considering a crisp environment. They opined that their model can be extended into a fuzzy model for a more acceptable ranking. Rani et al. (2020) used the COPRAS method with a hesitant fuzzy SWARA method for ranking the suppliers. In this model, the weights of the criteria are determined using the hesitant fuzzy SWARA method, and then applied it to the COPRAS method for ranking the suppliers. The drawback of this research is that the interrelationships between the criteria were not taken into consideration which limits its application to some extent. Ataei et al. (2020) developed a new method called the ordinal priority approach (OPA) that can calculate the weights of experts and attributes as well as the rank of alternatives. According to their approach, initially, all experts, attributes, and alternatives are ranked using ordinal data. Then,a multi-objective non-linear mathematical model is formulated, and subsequently, this model is converted into a linear programming model to solve the problem. After solving this model, weights, and ranks of experts, attributes and alternatives are obtained. There are many advantages of this model compared with other MCDM models; however, it cannot handle multiple ranks during the decision-making process. Mahmoudi et al. (2021a) integrate the OPA model with grey systems theory to eliminate the multiple ranks problem from the OPA method. Subsequently, Mahmoudi et al. (2021b) proposed another gresilience supply chain model using a fuzzy OPA to solve the green supplier selection problem. In this model, linguistic variables are used to handle the uncertainty in the decision-making process. Since, gresilience is a new concept in supply chain management, so more theoretical and practical works need to be done for further refinement of this concept. Panchal et al. (2021) proposed a novel structure framework to identify the major risk factors of an ash handling unit of a coal-fired-based power plant. In their study, initially, they listed some possible failure causesand then rated them by fuzzy rating under the failure mode and effect analysis (FMEA) approach. Afterward, the enlisted causes are ranked using the fuzzy EDAS method and then verified by the fuzzy combinative distance-based assessment (CODAS) method. They mentioned that the accuracy of the result by the proposed method depends on the quality of the information/data provided by the experts. To achieve the accuracy of the analysis they used the fuzzy numbers to handle the uncertainty. Multiple experts evaluate the possible causes using fuzzy values, but they did not check the consistency of the judgment. They ignored the weight of the decision-makers. Besides, they did not consider the interrelationship between the discussed causes. Yazdani et al. (2020) proposed an integrated model using the DEMATEL, BWM, and modified EDAS methods. In their studies, initially, the best and worst criteria and their weights are determined by the DEMATEL method using objective values. Then, the best and worst criteria are used in the BWM method for determining the weights of the criteria. Finally, weights obtained from both methods are aggregated by simple average to obtain unique criteria weights. Subsequently, these weights are applied in a modified EDAS method for ranking the alternatives. The main advantage of this method is that the best and worst criteria can be identified by the DEMATEL method that is used in the BWM method. Without identifying the best and worst criteria, the BWM method cannot determine criteria weights. The outcome of the DEMATEL method is used as an input of the BWM method. This is the main benefit of their model. However, there are also limitations of their studies such as (a) the DEMATEL can be used in a group decision-making environment, whereas the BWM method is not applicable in a group decision-making environment, and (b) both methods cannot handle uncertainty and vagueness during the decision-making process. Therefore, extension is needed for handling uncertainty by this model. Yazdani et al. (2021a) proposed an integrated model for food supplier selection in Winery units in Spain. In their model, criteria weights are determined by the SWARA-D and LBWA-D methods separately using D-number, and then both weights are aggregated to get a unique criteria weight. Then, these weights are used in the MARCOS-D method for ranking the suppliers. In their studies, D-numbers are used to integrate the uncertainty in the decision-making process. Despite having several advantages like overcoming exclusiveness hypothesis and completeness constraint, authors opined that D-numbers having some associated limitations in terms of operations and use of combination rules to fulfill associative properties. To overcome operational problems, further research is required to improve combination rule algorithms like multiplicative and additive forms for flexible use of D-numbers. Yazdani et al. (2021b) proposed another integrated model to select sustainable suppliers in a dairy company in Iran. In their model, criteria weights are determined by the criteria importance through inter-criteria correlation (CRITIC) method using the interval valued neutrosophic numbers (IVNNs). Then, these weights are used in the combined compromise solution (CoCoSo) method for ranking the suppliers. Three membership degrees such as truth-membership (acceptance), falsitymembership (rejection), and indeterminacy-membership (uncertainty) are used for handling uncertainty in decision-making process. In their study, all decision makers are considered equally important and ignored their weights. This consideration is not realistic since the judgment capacity can be varied with respect to various aspects such as experiences, educational qualification, intellectual ability, working area, age, honesty, and so on. So, this model can be extended by integrating these factors in future. Kumari and Mishra (2020) used IFS to determine the weights of criteria considering uncertainty, then it applied into the COPRAS method for ranking the suppliers. Though IFS can handle fuzzy data but the COPRAS method cannot handle fuzzy data directly. That's why data de-fuzzification is needed for ranking the alternatives by the COPRAS method. From these literature reviews, it can be clearly noticed that different mathematical techniques have mainly been employed for two purposes such as (a) determination of weights to be assigned to various evaluation criteria, and (b) ranking of the competing suppliers using these weights. It is also noticed that the weights of the criteria can be determined using the crisp model as well as the fuzzy model. The BWM, DEMATEL, Entropy, SWARA, FUCOM, LBWA methods are mainly used to determine the weights of the criteria based on certain data. The fuzzy AHP, and fuzzy SWARA methods can be used to determine the weights of the criteria considering fuzzy data. The interrelationship between the criteria is very important, and it should be taken into consideration during estimating the weights of the criteria that are ignored by the fuzzy SWARA method. It is noticed that when fuzzy weights are being applied to other MCDM models for ranking the alternatives, in most of the cases the fuzzy criteria weights are being converted into crisp values. As a result, fuzzy data is not being used directly in the considered model during the analysis. It is also noticed that most of the previous studies did not consider the weights of the decision-makers. They assumed each decision expert is equally important, and their ratings are also equally valuable. This is not a realistic assumption. Because human judgment capability can be varied with respect to education, experiences, age, working area, honesty, intellectual ability etc. That's why uncertainty is involved in human judgment. To reduce this vagueness, dissimilar weights should be given on expert for evaluating their judgment. So, we are motivated to develop an integrated fuzzy model to handle the uncertainty properly and mitigate the identified problems. The objective of this study is to develop an integrated fuzzy model by integrating three fuzzy models such as intuitionistic fuzzy set (IFS), fuzzy analytic hierarchy process (fuzzy AHP), and fuzzy technique for order preference by similarity to ideal solution (fuzzy TOPSIS) for selecting the best supplier. Then, the model is verified with other MCDM models such as fuzzy EDAS, fuzzy COPRAS, and fuzzy ELECTRE methods to confirm its validity. Finally, sensitivity analysis is also performed to test the robustness and stability of the proposed model.

Proposed research methodology
The aim of this study is to propose an integrated fuzzy model for ranking the suppliers considering uncertainty. To achieve this goal three different fuzzy models such as the IFS, fuzzy AHP, and fuzzy TOPSIS methods have been integrated. The IFS method is used to estimate the weights of the decision makers, the fuzzy AHP method is used to estimate the weights of the criteria, and finally the fuzzy TOPSIS method is used to rank the suppliers. The flow-diagram of this research methodology is illustrated in Figure 1, and its descriptions are described as follows: Step 1: Initially, a decision-making team is formed. Members of this team are selected from production, marketing, and quality control department. These members are selected by the top management of the company.
Step 2: The weight of each member is determined using intuitionistic fuzzy number by applying the logic of IFS theory.
Step 3: Decision-makers determine supplier's selection criteria through group discussion.
Step 4: Each member develops pair-wise comparison matrix individually for determining relative weights of the criteria.
Step 5: After developing comparison matrix consistency ratio is checked. If this ratio becomes greater than 10% then pair-wise comparison rating is modified in step 4, otherwise next step is followed.
Step 6: After checking consistency ratio, crisp ratings are expressed as triangular fuzzy rating using fuzzy scale.
Step 7: In this step, initially, fuzzy ratings are aggregated by multiplying the weights of the decision-makers, and aggregated decision matrix is formed. After that, fuzzy weights of the criteria are determined using the fuzzy geometric mean method. By this step the fuzzy AHP process is ended. Step 8: The fuzzy weights obtained from the fuzzy AHP method are subsequently used in the fuzzy TOPSIS method for ranking the alternatives.
Step 9: Obtained result is compared with other MCDM models such as, fuzzy COPRAS, fuzzy EDAS, and fuzzy ELECTRE method, and then sensitivity analysis is performed setting different conditions for evaluating model's performance.
Step 10: After evaluating model's performance, the best model as well as the best supplier is selected from this study.

Fuzzy operations
Let, Ã and B two triangular fuzzy numbers, then the fuzzy values and its algebraic operations are expressed as follows: Two triangular fuzzy numbers are expressed by Eqs. (1) and (2). The reciprocal value of triangular fuzzy number is expressed by Eq.

Intuitionistic fuzzy set
Intuitionistic fuzzy set (IFS) introduced by Atanassov (1986) is an extension of the classical fuzzy set, which is a suitable way to deal with vagueness. IFS contains three elements known as membership degree, non-membership degree and hesitation degree. If the degree of one element becomes higher, accordingly other two elements will be lower. In real-life, three uncertain situations may have occurred when a person evaluates a thing. The person may think the evaluated thing is good, bad or he/she may be in hesitation. When the top management evaluates of a particular decision expert compare to other experts, the management team may face the above mentioned three situations during giving rating. These situations can be handled by IFS nicely. That's why, in this study, IFS is used to determine the weights of the decision-makers. The mathematical formulation of IFS is described as follows: Intuitionistic fuzzy set A in a finite set X can be written as: A third parameter of IFS is π A x ð Þ, known as the intuitionistic hesitation degree of whether x belongs to A or not.
It is obviously seen that for every x 2 X : Þ for all elements of the universe, the ordinary fuzzy set concept is recovered Shu et al. (2006).
� be an intuitionistic fuzzy number for rating of k th decision-maker. Then the weight of k th decision-maker (λ k ) can be obtained using following formula Boran et al. (2009).
where, μ k ; ν k ; andπ k are the membership degree, non-membership degree, and hesitation degree, respectively of k th decision-makers given by the management.
where, λ k is the weight of k th decision-maker. k denotes total number of decision makers.

Fuzzy AHP method
The original AHP method is proposed by Saaty (1980). Subsequently, this method is extended by Buckley (1985) to determine fuzzy weight considering uncertainty. The steps involved in this method are described as follows: Step 1: Each decision-maker develop a pair-wise comparison matrix on the basis of given criteria using fuzzy scale. For example, if criteria C 1 is preferred to criteria C 2 , then a particular cell (1, 2) in the decision matrix fuzzy value is given as (l, m, u), and in the opposite cell (2, 1), the corresponding reciprocal value is given as 1 u ; 1 m ; 1 l À � . In such way, a complete pair-wise decision matrix is formed by each expert as shown in Eq. (15). Where, x k ij indicates the k th decision-maker's preference of i th criteria over j th criterion. This notation represents triangular fuzzy number as follows: , where x l k ij denotes lower bound, x m k ij denotes middle bound, and x u k ij denotes upper bound rating given by k th decision-maker.
Step 2: In this step, consistency property of the developed matrix Dk is checked to ensure the consistency of judgment using Eqs. (16)-(18). where, w is the eigen vector, and λ max is the largest eigen value of matrix Dk developed by k th expert, CI is the consistency index that measure the inconsistency within the pair-wise comparison matrix, n is the number of criteria, CR is the consistency ratio which is used to measure the degree of CI, and RI is the random consistency index, its value is related to the dimension of the matrix as listed in Table 2. If CR < 0.10, the inconsistency degree of the comparison matrix Dk is considered acceptable, otherwise, the comparison matrix needs to be adjusted.
CR ¼ CI RI Step 3: If multiple decision-makers are involved in the decision-making team, then the preferences are aggregated using Eq. (19), where, λ k is the weight of the k th decision-maker, and where x l k ij denotes lower bound, x m k ij denotes middle bound, and x u k ij denotes upper bound rating given by k th decision-maker. Step 4: According to aggregated preferences, the pair-wise aggregated comparison matrix is formed as shown in Eq. (20).
Step 5: The geometric mean of the fuzzy values of each criterion is calculated using Eq. (21) Step 6: Finally, fuzzy weights of the criterion i, is determined by multiply each r i with this reverse vector as shown Eq. (22). Then, normalized weight is estimated using Eqs. (23) and (24).

Fuzzy TOPSIS method
The fuzzy TOPSIS method is used to solve MCDM problem considering uncertainty. This method can handle fuzzy data directly during the analysis. The procedure involved in this method is explained as follows: Let, the number of decision-makers, k = 1, 2, . . . K, alternative A i i ¼ 1; 2; . . . ; m ð Þ, criteria C j j ¼ 1; 2; . . . ; n ð Þ, and p k ij is the fuzzy rating of the i th alternative with respect to j th criterion given by the k th decision maker. The notation p k ij represents triangular fuzzy number as follows: p k ij = (p l k ij , p m k ij , p u k ij ), where p l k ij denotes lower bound, p m k ij denotes middle bound, and p u k ij denotes upper bound rating given by k th decision maker. The steps involved in this methods are described as follows: Step 1: Each decision-maker evaluates every alternative with respect to every criteria, and accordingly, a decision matrix is formed as shown in Eq. (25) Step 2: If multiple decision-makers are involved for evaluating the alternatives, then the ratings are aggregated using Eq. (26), where, λ k is the weight of the k th decision-maker, and p l k ij denotes lower bound, p m k ij denotes middle bound, and p u k ij denotes upper bound ratings given by k th decision-maker.
Step 3: According to the aggregated rating, the combined decision matrix is formed as presented by Eq. (27) Q ¼q Step 4: In this step, the combined matrix Q is normalized. For doing normalization, the selection criteria are classified into two ways such as Benefit Criteria (BC) and Non-Benefit Criteria (NBC). The criteria with an expected higher value is known as the Benefit Criteria, whereas the opposite is considered as the Non-Benefit Criteria. After this classification, the combined matrix Q is normalized using Eqs. (29)-(32), and the normalized decision matrix R is formed as shown in Eq. (28).
Step 5: In this step, the weighted normalized decision matrix Ṽ is formed as shown in Eq. (33). For doing this, each normalized element r ij is multiplied by fuzzy criteria weight w j as shown in Eq. (34).
where;ṽ ij ¼r ij �w j Step 6: Defined the Fuzzy Positive Ideal Solution (FPIS, A þ ) and Fuzzy Negative Ideal Solution (FNIS, A À ) as follows: For benefit criteria, a higher value is better, so the higher value is A þ and the lower value is A À , whereas for the non-benefit criteria a lower value is better, so the lower value is A þ and the higher value is A À .
Step 7: The distance of each alternative from FPIS or A þ , and from FNIS or A À can be calculated using Eq. (37).

dx;z ð Þ ¼
ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ð Þ represents the distance between two fuzzy numbers.
Step 8: In this step, the closeness coefficient, CC i , is determined. For doing this, initially, distances of each alternative from FPIS, and from FNIS with respect to each criteria are added using Eqs. (38) and (39), respectively. Then, closeness coefficient, CC i is estimated using Eq. (40).
Step 9: Finally, the alternatives are ranked according to the value of closeness coefficient (CC i ) in decreasing order. The best alternative is close to the FPIS, and farthest from the FNIS.

Numerical experiment
In this study, the proposed model is applied to a real automobile factory to evaluate their potential suppliers in order to select the best supplier. In this experiment, three decision-makers (DM 1 , DM 2 , DM 3 ) are involved to evaluate five suppliers (S1, S2, S3, S4, S5) with respect to five criteria (C1, C2, C3, C4, C5). The criteria are defined by the decision-makers as follows: Quality (C1): related to quality of conformance, quality management, and after-sale service quality.
Price ( In this study, procedure for the selection of supplier contains the following steps: Step 1 (Determination of weights of decision-makers by applying the IFS method): In this step, the weight of each decision-maker is estimated using intuitionistic fuzzy numbers (IFNs). For doing this, the management used linguistic terms as shown in Table 3 as comparative rating for the experts. By using these ratings decision-makers are evaluated as shown in Table 4. According to these linguistic ratings and it's corresponding IFNs are then used in Eq. (13) to determine the weight of the decision-makers. For example, linguistic terms "Very important" is given to DM 1 by the management as comparative rating. According to this rating, the membership degree μ k¼1 = 0.90, and non-membership degree ν k¼1 = 0.10. Then, the hesitation degree π k¼1 = 0.0 is obtained using Eq. (11). After that, using these values in Eq. (13), the weight of the DM 1 is obtained 0.406. Similarly, weight of DM 2 and DM 3 are also determined according to their ratings as presented in Table 4. These weights are used subsequently in the fuzzy AHP method, and in the fuzzy TOPSIS method.   Fairly preferable (FP) 3, 5, 7 Extremely preferable (EP) 5, 7, 9 Absolutely preferable (AP) 7, 9, 9 Islam & Arakawa, Cogent Engineering ( Step 2 (Determination of weights of criteria by applying the fuzzy AHP method): In this step, the weights of the criteria are estimated. For doing this, the fuzzy AHP method is used. The linguistic    term and their triangular fuzzy numbers are used for rating the importance of the criteria as shown in Table 5. By using this rating, the decision-makers developed pair-wise comparison matrix individually as shown in Table 6-8. Then, consistency ratio (CR) of the judgment of each decisionmaker is checked. For example, by using data from Table 6, λ max = 5.35 is estimated using Eq. (16). Then, using equation Eq. (17), consistency index (CI) = 0.09 is determined. Where, the number of criteria, n = 5 is used. Random index (RI) = 1.12 is obtained from Table 2. Finally, CR = 0.08 is obtained using Eq. (18). The obtained CR (0.08) is less than the allowed CR (0.10). This confirms that the pair-wise comparison matrix developed by DM 1 is consistent. Similarly, consistency ratio of the comparison matrix developed by DM 2 and DM 3 is also checked as shown in Table 9. From this table, pair-wise comparison matrices developed by three decision-makers are consistent. Now, using Eq. (19) the comparison matrices are aggregated. For example, pair-wise comparison rating between C1 and C5 are given (1,3,5), (3,5,7) and (1,3,5) by DM 1 , DM 2 and DM 3 , respectively. Now, using the weight of each decision-maker we got the aggregated rating of this cell is as follows: (      in Figure 2. The fuzzy criteria weights obtained in this step is used in the fuzzy TOPSIS method for ranking the suppliers. Step 3 (Suppliers' ranking by applying the fuzzy TOPSIS method): Fuzzy criteria weights as illustrated in Figure 2 are used in the fuzzy TOPSIS method for ranking the suppliers. For evaluating suppliers a linguistic scale, and its corresponding triangular fuzzy numbers are used as presented in Table 12. Suppliers are evaluated by three decision-makers with respect to each criteria using this linguistic scale, as shown in Tables 13-15. The linguistic scale is subsequently converted into triangular fuzzy number, and then, the aggregated fuzzy decision matrix is formed. For example, Good (G), Good (G), and High (H) ratings are given to supplier S1 with respect to C1 by DM 1 , DM 2 , and DM 3 , respectively, and its corresponding triangular fuzzy ratings are (2.50, 5.0, 7.5), (2.50, 5.0, 7.5), and (5.0, 7.5, 10.0). For this cell, the aggregated fuzzy ratings are calculated using Eq. (26) as follows: q 11 = (2.50*0.406 +2.50*0.356 +5.0*0.238, 5.0*0.406 +5.0*0.356 +7.5*0.238, 7.5*0.406 +7.5*0.356 +10.0*0.238) = (3.1, 5.6, 8.1). Similarly, aggregated ratings for remaining cells for every supplier with respect to every criteria are also determined, and finally, aggregated fuzzy decision matrix is developed as shown in Table 16. Then, this aggregated decision matrix is normalized. For example, C1 is a benefit criteria, so, using Eqs. (29)  (32), the rating of this particular criteria is normalized as follows: a À 1 ¼ min 1 3:5; 7:5; 5:0; 6:5; 4:4 ð Þ ¼ 3:5, and r 11 ¼ 3:5 8:5 ; 3:5 6:0 ; 3:5 3:5 À � = (0.41, 0.58, 1.0). Similarly, other benefit and non benefit criteria are also normalized, and then normalized decision matrix is formed as shown in Table 17. Then, the weighted normalized decision matrix is formed. For doing this, normalized ratings are multiplied by the fuzzy weight of the criteria. For example, normalized values of S1 with respect to criteria C1 are 0.31, 0.56, 0.81, and fuzzy weights of C1 are 0.14, 0.32,   0.73, then the weighted normalized values are determined using Eq. (34) as follows: ṽ 11 ¼r 11 �w 1 = (0.31*0.14, 0.56*0.32, 0.81*0.73) = (0.04, 0.18, 0.59). Similarly, weighted normalized ratings of other suppliers with respect to other criteria are also determined, and then, weighted normalized decision matrix is formed as shown in Table 18. From this matrix, fuzzy positive ideal solutions (FPIS) or A þ , and fuzzy negative ideal solutions (FNIS) or A À are identified using equation Eq. (35) and (36). For example, C1 is a benefit criteria, and we got the FPIS or A þ = (0.11, 0.32, 0.73) and FNIS or A À = (0.04, 0.17, 0.57). Similarly, the values of FPIS and FNIS for others criteria are also determined as shown in Table 18. Then the distance of each alternative from FPIS and from FNIS are calculated. For example, the distance of S1 from FPIS and FNIS with respect to C1 criteria is determined using Eq. (37) as follows:  Table 19. Then, the distances are added using Eqs. (38) and (39). For example, distances of S1 from FPIS and from FNIS with respect to five criteria are 0.12, 0.26, 0.12, 0.02, 0.04 and 0.01, 0.00, 0.00, 0.00 and 0.00, respectively, and their summation are d þ 1 = 0.56, and d À 1 = 0.01. Finally, the closeness coefficient of the suppliers is determined using Eq. (40). For example, the closeness coefficient of S1 is obtained, CC 1 = 0:01 0:56þ0:01 = 0.02. Similarly, closeness coefficient of others suppliers are also determined as shown in Table 19. Then, suppliers are ranked in decreasing order with respect to the value of closeness coefficient, and the order is S2 > S3 > S4 > S5 > S1 as shown in Table 19 in the last column. According to this rank the best supplier is S2.

Result comparison
In this study, the ranking order generated by the proposed model is compared with other models such as the fuzzy COPRAS, fuzzy EDAS, and fuzzy ELECTRE method. For doing comparison, the same data are applied in these methods sequentially for ranking the suppliers. The results are shown in Table 20. Initially, proposed ranking order S2 > S3 > S4 > S5 > S1 is verified by the Model-2 whose order is S2 > S3 > S1 > S5 > S4. From this order, it is found that suppliers, S2, S3, and S5 are ranked same as the 1 st , 2 nd , and 4 th by both models, whereas suppliers, S1 and S4 rankings are differents. Ranking variations are presented by these two models. Again, the proposed ranking order is verified by the Model-3 whose order is S3 > S2 > S1 > S5 > S4. From this order, it is found that only S5 occupied 4 th position which is same as the proposed Model-1. From Model-1 and Model-2, it is found that S2 is the best supplier which is ranked as the 1 st , whereas S3 is the best supplier by Model-3. For more justification about the best supplier, the proposed ranking order is again verified by Model-4. From Table 20, it is seen that ranking order generated by the Model-4 is same as the Model-1. This result shows that the ranking order generated by the proposed model is more acceptable, and this model is validated. After comparing the proposed model with other three models, it is noticed that ranking order variations are presented in these models. Under this condition, these models are further tested setting two conditions to evaluate their performance for ranking the suppliers.

Sensitivity analysis and discussion
The robustness and ranking stability of the proposed model along with other three models are tested by setting two experimental conditions as follows: Condition-1: Changing the weights of decision-makers and criteria: In this condition, suppliers' ranking order is observed by changing decision-makers weights, and criteria wights. For changing the weight of the decision-makers Eq. (41) is used Yazdani et al. (2021a).
where, $ nβ represents the adjusted value of the decision-maker, $ nα represents reduced value of the best decision-maker, w β represents the original value of the decision-maker, and w n represents original value of the best decision-maker. Weights of the decision-makers are adjusted with respect to the most influential weight. In this study, the most influential weight is 0.4060 (DM 1 ). This weight is reduced gradually from 1% to 20%, and accordingly, weights of other two decisionmakers are adjusted using Eq.     Table 21, and the effect of changing the most influential decision-maker's weight on other decision-makers weights are shown in Figure 3. From this figure, it is observed that weights of DM 2 , and DM 3 have been increased slightly with decreasing the weight of DM 1 . After that, these adjusted decision-makers' weights are applied in Eq. (19), and accordingly, fuzzy criteria weights are determined. Then, using Eqs. (23) and (24), the normalized criteria weights are estimated as given in Table 22. The effect of changing decision-makers' weights on criteria weights are shown in Figure 4. From this figure, it is found that minor changes have been occurred in criteria weights. Subsequently, in Model-1, these adjusted weights of both decision-makers, and criteria are applied in Eqs. (26) and (34), respectively, and finally, the suppliers' scores are determined as given in Table 23. The effect of changing the weights of the decision-makers, and criteria on suppliers' scores is shown in Figure 5. From this figure, it is found that the scores of the suppliers have not been changed significantly, and hence, the ranking order of the suppliers is remained constant as     illustrated in Figure 6. Similarly, these adjusted weights of both decision-makers, and criteria are used in Model-2, Model-3, and Model-4, consecutively, and its ranking order are presented in Figures 7, Figure 8 and 9, respectively. From these results, it is observed that steady rankings were generated by the Model-1, Model-3, and Model-4 under this changing condition. On the other hand, Model-2 was sensitive under this condition, and its ranking order has been changed. From these results, it is found that Model-1, Model-3, and Model-4 were more stable than Model-2 for ranking the suppliers in Condition-1.
Condition-2: New suppliers consideration: In this condition, five additional suppliers (S6, S7, S8, S9, S10) have been considered to evaluate jointly with previous five suppliers (S1, S2, S3, S4, S5). Rating of these additional suppliers are newly set up as shown in Table 24. Using these ratings, Model-1, along with other three models were executed again for ranking the suppliers. From Table 24, it is noticed that S5 and S6 were equally rated, accordingly, their ranked also equally expected which is achieved by all models as shown in Table 25. Again, it is noticed that rating of S7  Random rating is given by all decision-makers S10 Random rating is given by all decision-makers was the most superior than other ten suppliers, whereas rating of S8 was the least superior. Accordingly, 1 st ranking was expected for S7, and last ranking for S8. This is achieved by only Model-1, and Model-4 as shown in Table 25. Again, it is observed that rating of S9 and S10 were given randomly, and their ratings were lower than S7, and greater than S8, accordingly, their rankings were expected as lower than S7 but greater than S8. This is also achieved by Model-1, and Model-4. From Table 25, it is seen that S10 is ranked as the 1 st position by Model-2 and Model-3 which is inconsistent according to rating. From this sensitivity analysis, it is observed that Model-2 and Model-3 could not generate consistent ranking properly under this changing environment compare to Model-1 and Model-4. It can be said that Model-1 and Model-4 are more robust than Model-2, and Model-3. However, in Model-4, many interactive steps are involved to execute the solution. For this reason, in changing environment, Model-4 is inconvenient to generate results quickly. On the contrary, few interactive steps are involved in Model-1, and its computational process is faster. Hence, it is capable to generate results quickly. Therefore, Model-1 is better than Model-4. According to sensitivity analysis on the basis of Condition-1 and Condition-2, we conclude that proposed model is more robust and more superior than other three models, and the best supplier is S2 when 5 suppliers are evaluated, and S7 when 10 suppliers are evaluated.

Conclusion
In this study, an integrated fuzzy approach is proposed for solving supplier selection problem considering imprecise ratings. Three fuzzy approaches such as an intuitionistic fuzzy set (IFS), the fuzzy AHP, and the fuzzy TOPSIS methods have been integrated to solve suppliers selection problem. The IFS is used to determine the weights of the decision-makers, and the fuzzy AHP is used to determine the weights of the criteria, and finally, the fuzzy TOPSIS method is used to select the best supplier. The advantages of this model are as follows: (a) Pair-wise interrelationship between criteria is considered to determine criteria weight. (b) Inconsistency of each decisionmaker judgment is also checked to reduce the chances of errors to determine criteria weight. (c) Fuzzy data is used directly with out any defuzzification. (d) Multiple experts and their weights have been taken into consideration in decision-making process. In this study, the best supplier S2 is selected by the proposed model. The proposed model is also verified with other three fuzzy MCDM models such as the fuzzy COPRAS, fuzzy EDAS, and fuzzy ELECTRE methods. Sensitivity analysis is also perfomred to evaluate its performance for ranking suppliers under changing environment. From this analysis, it is found that the proposed model is more robust and strengthen than other three models. This model can also be applied to solve other areas of decision management problems such as manufacturing methods selection, maintenance strategy selection, project