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An intuitionistic fuzzy entropy approach for supplier selection

An intuitionistic fuzzy entropy approach for supplier selection Due to apparent flexibility of Intuitionistic Fuzzy Set (IFS) concepts in dealing with the imprecision or uncertainty, these are proving to be quite useful in many application areas for a more human consistent reasoning under imperfectly defined facts and imprecise knowledge. In this paper, we apply notions of entropy and intuitionistic fuzzy sets to present a new fuzzy decision-making approach called intuitionistic fuzzy entropy measure for selection and ranking the suppliers with respect to the attributes. An entropy-based model is formulated and applied to a real case study aiming to examine the rankings of suppliers. Furthermore, the weights for each alternative, with respect to the criteria, are calculated using intuitionistic fuzzy entropy measure. The supplier with the highest weight is selected as the best alternative. This proposed model helps the decision-makers in better understanding of the weight of each criterion without relying on the mere expertise. Keywords Multicriteria decision-making · Supplier selection · Intuitionistic fuzzy entropy · Intuitionistic fuzzy set Introduction recently, Bustince et al. [5] have focused on the history, defi- nition, and basic properties of fuzzy set types and relation- Decision-maker’s judgments, including preference informa- ships between the different types of fuzzy sets. tion, are usually stated in linguistic terms. There are many In last couple of years, many researchers also proposed approaches proposed for modeling the decision linguistic different functions for intuitionistic fuzzy sets (IFSs) and term sets. Zadeh [43–45] defined the linguistic variable as a applied them in various real-time applications. In 1983, IFS variable whose values are words or sentences in a natural or was introduced by Atanassov [1] as generalization of fuzzy artificial language in his three consecutive papers. Recently, sets. Basically intuitionistic fuzzy sets based models may Morente-Molinera et al. [23] provided a systematic review be adequate in situations when we face human testimonies, of the fuzzy linguistic modeling approaches developed over public opinions, etc. IFSs can be viewed as a generalization the last decade. The reviewed methods are classified into of fuzzy sets that may better model imperfect information six categories based on different approaches. In addition, which is present in any conscious decision-making (Atan- assov [2]). Intuitionistic fuzzy sets take into account both the degrees of membership and non-membership on the real unit * Mohamadtaghi Rahimi interval [0, 1] subject to the condition that their sum belongs Mohamadtaghi.rahimi@unbc.ca to the same interval. In recent years, several researchers Pranesh Kumar extended IFS based on various decision-making techniques. Pranesh.Kumar@unbc.ca For the first time, De Luca and Termini [11] integrated Behzad Moomivand the entropy concept (Shannon [15]) with fuzzy set theory Moomivand_stu@qom-iau.ac.ir (Zadeh [42]). The main purpose of entropy measures is Gholamhosein Yari to explain uncertainty degree. In recent years, numerous yari@iust.ac.ir studies have integrated the entropy with various fuzzy sets types, such as; Burillo and Bustince [4], Coban [9], Department of Mathematics and Statistics, University of Northern British Columbia, Prince George, BC, Canada Joshi and Kumar [21], Yari et  al. [38, 39], Szmidt and Kacprzyk [26], Farnoosh et al. [14], Ye [40], Hung and Department of Management, Qom Branch, Islamic Azad University, Qom, Iran Yang [18], Rahimi and Kumar [28], Rahimi et al. [29], Wei et al. [34], Zeng and Li [46], Szmidt and Kacprzyk Department of Mathematics, Iran University of Science and Technology, Tehran, Iran Vol.:(0123456789) 1 3 1870 Complex & Intelligent Systems (2021) 7:1869–1876 [27], Ye [41], and Zhang et al. [47]. Burillo and Bustince focused on to propose the new intuitionistic fuzzy entropy [4] have defined the interval-valued fuzzy sets and IFSs, measure for selection suppliers. and introduced the distance measure between IFSs using The paper is organized as following: “Literature review” the entropy measures. Joshi and Kumar [21] introduced the presents the literature review of entropy, IFSs, and applica- novel parametric (R, S)-norm intuitionistic fuzzy entropy tion of these methods in assessment of supplier selection. for solving problem of multiple-attribute decision-making In “Preliminaries”, we have provided some concepts and (MADM). Szmidt and Kacprzyk [26] have proposed the background about IFS, score function, and an Intuitionistic new non-probabilistic-type entropy measure for IFSs by Fuzzy Entropy measure. A new MCDM method is proposed considering IFSs and a ratio of distance between them. in “Proposed MCDM method and its application in selecting Ye [41] has introduced the fuzzy cross entropy based on the best supplier” which discusses our case study to show interval-valued intuitionistic fuzzy sets (IVIFSs) using the the validity of the proposed method. In “Conclusion”, we intuitionistic fuzzy (IF) cross entropy. Hung and Yang [18] conclude and state limitations and recommendations for have applied the probability concept for introducing the future studies. fuzzy entropy IFSs using two entropy measures for IFSs. Wei et al. [33] have introduced the entropy measure for IVIFSs by incorporating three kinds of entropy measures, Literature review and, finally, proposed the new entropy measure for IVIFSs. Some researchers have used the entropy and IFSs in In recent decades, several of previous studies used, inte- various application areas such as supplier and vendor grated, and introduced the entropy and IFS in numerous selection (Shahrokhi et al. [24], Wen et al. [35], Gerogi- application areas. Burillo and Bustince [4], defined the annis et al. [16], Xiao and Wei [37], Wang and Lv [31], interval-valued fuzzy sets and IFSs and introduced the dis- Krishankumar et al. [22], Song et al. [25], Guo et al. [17], tance measure between IFSs using the entropy technique. Chai et al. [6], Bali et al. [3], Wen et al. [35], and Xiao Wen et al. [35] used the IFS for selection of vendor based on and Wei [37]). Shahrokhi et al. [24] have proposed the some MADM approaches such as Simple Additive Weight- integrated approach based on IFS and linear programming ing (SAW), Weight Product Matrix (WPM), ELimination technique for selection of suppliers in a group decision- Et Choix Traduisant la REalité—Elimination (ELECTRE), making environment. Wen et  al. [35] have considered Order of Preference by Similarity to Ideal Solution (TOP- the IFS for selection of vendor based on some MADM SIS), and Lexicographic. Wang et al. [32] extended some approaches such as Simple Additive Weighting (SAW), operators including triangular intuitionistic fuzzy ordered Weight Product Matrix (WPM), ELimination Et Choix weighted averaging (TIFOWA), triangular intuitionis- Traduisant la REalité – Elimination (ELECTRE), Order tic fuzzy ordered weighted geometric (TIFOWG), hybrid of Preference by Similarity to Ideal Solution (TOPSIS), weighted averaging (IFHWA), triangular intuitionistic fuzzy and Lexicographic. Gerogiannis et  al. [16] have intro- generalized ordered weighted averaging (TIFGOWA), and duced the hybrid approach for assessment of biomass triangular intuitionistic fuzzy generalized hybrid weighted suppliers by integrating IFS, multi-periodic optimization averaging (TIFGHWA) based on TOPSIS and multi-objec- (MPO), and linear programming. Wang and Lv [31] have tive programming. Shahrokhi et al. [24] proposed the inte- investigated induced intuitionistic fuzzy Einstein hybrid grated approach based on IFS and linear programming tech- aggregation operator (I-IFEHA) for selection of supplier nique for selection of suppliers in a group decision-making in environment of group decision-making based on fuzzy environment. Joshi and Kumar [21] introduced the novel par- measures by introducing aggregation and Einstein opera- ametric (R, S)-norm intuitionistic fuzzy entropy for solving tor I-IFEHA. Krishankumar et al. [22] have introduced a problem of multiple-attribute decision-making (MADM). Jin novel approach for supplier selection using IVIF based et al. [20] proposed two new approaches for group decision- on statistical variance (SV) and ELECTRE methods. Wen making to derive the normalized intuitionistic fuzzy priority et al. [35] have used IFS for supplier selection in envi- weights from IFPRs based on multiplicative consistency and ronment of group decision-making. Xiao and Wei [37] the order consistency. Gerogiannis et al. [16] introduced the have presented a method to deal with the supplier selec- hybrid approach for assessment of biomass suppliers by inte- tion problem in supply chain management with interval- grating IFS, multi-periodic optimization (MPO), and linear valued intuitionistic fuzzy information. It may, however, programming. Wang et al. [32] proposed the new method be noted that although, these researchers have applied by integration OWA–TOPSIS and intuitionistic fuzzy set- and integrated entropy with IFSs in various application tings. Chen and Chang [7] proposed novel approach for areas, but there are gaps in application of these techniques fuzzy multiattribute decision-making based on three opera- in supplier selection. Therefore, in this paper, we have tors named IFWGA, IFOWGA, and IFHGA. Wang and Lv [31] investigated induced intuitionistic fuzzy Einstein hybrid 1 3 Complex & Intelligent Systems (2021) 7:1869–1876 1871 aggregation operator (I-IFEHA) which is investigated for Definition 1 (Atanassov [2]). An IFS over X is defined as selection of supplier in environment of group decision-mak- follows: ing based on fuzzy measures by introducing aggregation and � � A = ⟨x, 𝜇 (x), 𝛾 (x)⟩�x ∈ X ̃ ̃ A A Einstein operations for proposing the I-IFEHA. Szmidt and 𝜇 (x) ∶ X → [0,1], 𝛾 (x) ∶ X → [0,1], Kacprzyk [26] proposed the new entropy measure for IFSs in ̃ ̃ A A the non-probabilistic-type by interpreting of IFSs and a ratio where µ and γ, respectively, define the degree of mem- of distance between them. In 2007, Vlachos and Sergiadis bership and the degree of non-membership, and we have: [30] proposed the intuitionistic fuzzy divergence measure for 0 ≤ 𝜇 x + 𝛾 x ≤ 1 for every x ∈ X. ̃ ( ) ̃ ( ) A A the first time, and studied its application pattern recognition and medical diagnosis. Krishankumar et al. [22] introduced π = 1 − 𝜇 x − 𝛾 x denotes a measure of non-deter- ̃ ̃ ( ) ̃ ( ) A A A the novel approach for supplier selection using IVIF based minancy which is called the intuitionistic fuzzy (IF) index on statistical variance (SV) and ELECTRE methods. Ye [41] of the element x. Obviously, when 𝜇 = 0 , the set A is a introduced the fuzzy cross entropy based on interval-valued fuzzy set. If we denote the set of all the FSs on X by F(X), intuitionistic fuzzy sets (IVIFSs) using the intuitionistic ̃ ̃ the operations of IFSs are defined for every A, B ∈ F(X) as: fuzzy (IF) cross entropy. Furthermore, Wei and Ye [34] pro- � � posed an improved version of intuitionistic fuzzy divergence A = ⟨x, 𝛾 (x), 𝜇 (x)⟩�x ∈ X A A in Vlachos and Sergiadis [30] and developed a method for � � ̃ ̃ A ∧ B = ⟨x, μ (x) ∧μ (x), γ (x) ∨γ (x)⟩�x∈ X ̃ ̃ ̃ ̃ A B A B pattern recognition with intuitionistic fuzzy information. � � ̃ ̃ A ∨ B = ⟨x, 𝜇 (x) ∨ 𝜇 (x), 𝛾 (x) ∧ 𝛾 (x)⟩�x∈ X ̃ ̃ ̃ ̃ A A Wen et al. [35] used IFS for supplier selection in environ- B B � � ̃ ̃ A ⊗ B = ⟨x, 𝜇 (x) + 𝜇 (x) − 𝜇 (x)𝜇 (x), 𝛾 (x)𝛾 (x)⟩�x∈ X ment of group decision-making. Hung and Yang [18] used ̃ ̃ ̃ ̃ ̃ ̃ A A A B B B � � ̃ ̃ the probability concept for introducing the fuzzy entropy A ⊗ B = ⟨x, 𝜇 (x)𝜇 (x), 𝛾 (x) + 𝛾 (x) − 𝛾 (x)𝛾 (x)⟩�x∈ X ̃ ̃ ̃ ̃ ̃ ̃ A B A B A B � � � � � � 𝛼 𝛼 IFSs with two entropy measures for IFSs. Hung and Yang 𝛼 A = ⟨x,1 − 1 − 𝜇 (x) , 𝛾 (x) ⟩�x ∈ X ̃ ̃ A A [19] defined another divergence measure called ‘J-diver - � � � � � � 𝛼 𝛼 A = ⟨x, 𝜇 (x) ,1 − 1 − 𝛾 (x) ⟩�x ∈ X . ̃ ̃ A A gence’ for measuring the difference between two IFSs and then applied it to clustering analysis and pattern recognition. Burillo and Bustince [4] introduced the concept of entropy Definition 2 (Wu-Zhang [36]). Let A = a ̃ , a ̃ ,… , a ̃ be 1 2 n in intuitionistic fuzzy set theory, which allows us to measure an IFS and ã = 𝜇 , 𝛾 , i = 1,2,…,n, be intuitionistic fuzzy i i i the degree of intuitionism associated with an IFS. Vlachos values in A. Then, an Intuitionistic Fuzzy Entropy measure and Sergiadis [30] proposed another measure of intuitionis- is formulated in the following way: tic fuzzy entropy and revealed an intuitive and mathemati- cal connection between the notions of entropy for fuzzy set 𝜇 𝛾 −1 i i + 𝛾 Ln 𝜀 ã = 𝜋 − (Ln2) 𝜇 Ln . i i i i and intuitionistic fuzzy set. Wei et al. [33], introduced the 𝜇 + 𝛾 𝜇 + 𝛾 i i i i entropy measure for IVIFSs by incorporating three kinds (1) of entropy measures, and finally, proposed the new entropy This measure satisfies the four axioms in Szmidt and measure for IVIFSs. Zhang and Jiang [48] defined a measure Kacprzyk [24] for IF value entropy measure. of intuitionistic fuzzy entropy for intuitionistic fuzzy sets by generalizing of the De et al. [10], logarithmic fuzzy entropy. Definition 3 (Chen and Tan [ 8]). Let ã = 𝜇 , 𝛾 , i i i Xiao and Wei [37] presented a method to deal with the sup- i = 1,2,…,n, be intuitionistic fuzzy values, and then, the plier selection problem in supply chain management with score of a ̃ is: interval-valued intuitionistic fuzzy information. Although, previous mentioned papers have investigated the important S ã = 𝜇 − 𝛾 , i = 1,2,… , n. (2) i i i role of entropy and IFS in assessment of supplier selection, but there is gap in literature regarding to these issues, how- ever; this study based on current literature, attempted to review these issues comprehensively. Proposed MCDM method and its application in selecting the best supplier We consider that one of the largest companies in Iran would Preliminaries like to select the best supplier firm to provide the materials in production line. In this context, we propose a new method Some basic definitions of IFS, Intuitionistic Fuzzy Entropy based on Intuitionistic Fuzzy Entropy to identify the best measure, and the score function are reviewed for the sake supplier. First, we have to recognize the main criteria which of completeness. 1 3 1872 Complex & Intelligent Systems (2021) 7:1869–1876 can influence our decision. After the criteria selection, the Table 1 Supplier performance response data next step is how to choose the best supplier. Performance C C C C C 1 2 3 4 5 Supplier 1 1.2 M VH L 1 Criteria selection Supplier 2 1.5 VH H H 3 Supplier 3 1.3 M L M 6 Using Dickson’s [13] 23 criteria in supplier selection and Supplier 4 1.7 H VH H 2 the addition of one local criterion which is pay off time (an Supplier 5 1.3 H M H 3 important factor in Iran’s business market), a questionnaire containing 24 questions was constructed. This question- naire was sent to 30 firm’s managers and firm’s sale man- agers. In each question, the importance of one criterion is qualitative, we convert them on the quantitative scale by the five-point Likert scale as it is shown in Table  2. That is; in evaluated. The applicant would choose among: “very low”, “low”, “medium”, “high”, and “very high”. All responses the qualitative questions, the criteria with respect to each supplier are given a number among 1, 3, 5, 7 and 9 (Fig. 2). were converted to the five-point Likert scale. Then, using SPSS, we have compared the means of the criteria points at To convert the values into the Intuitionistic Fuzzy Values, we have extended the method introduced in Deng-Chan [12] 95% confidence level. As follows, five criteria were selected as the most important ones such as; price, quality, deliver, as follows: To get the degrees of membership, non-membership, and technical capability, and pay off factors. intuitionistic fuzzy index, we have calculated the distance of each value with the lowest value as the value of membership, The selection model the distance of each value with the highest value as the value of non-membership, and the distance of each value with the In the presented selection model (shown in Fig. 1), we have tabulated the information of five suppliers in Table1 with average of others as the value of intuitionistic fuzzy index. Then, these calculated values are, respectively, divided by respect to the above criteria. Note that when the values are their total sum. If the criterion is a kind of cost, the degrees of membership and non-membership are replaced with each Criteria other. For example, the intuitionistic fuzzy degrees for the Selection two first values of C1 are calculated here: �1.2−1.2� ⎧ = 0  = 0 �1.2−1.2�+�1.2−1.7�+�1.2−1.4� Gathering the Supplier Performance Response Data with ⎪ �1.2−1.7� = 0.714  = 0.714 ⎨ 11 respect to each criterion �1.2−1.2�+�1.2−1.7�+�1.2−1.4� �1.2−1.4� = 0.286  = 0.286 ⎩ 11 �1.2−1.2�+�1.2−1.7�+�1.2−1.4� �1.5−1.2� ⎧ = 0.50  = 0.333 �1.5−1.2�+�1.5−1.7�+�1.5−1.4� Calculation of Degrees of Membership, non-Membership �1.5−1.7� = 0.333  = 0.50 and intuitionistic fuzzy index ⎨ 21 �1.5−1.2�+�1.5−1.7�+�1.5−1.4� �1.5−1.4� = 0.167  = 0.167. �1.5−1.2�+�1.5−1.7�+�1.5−1.4� Calculation of intuitionistic fuzzy entropy for each supplier From the first column of Table  2, we now that Supplier with respect to each criterion 1 has the worst performance in delivery, where Supplier 4 has the best. With this proposed method, in Table  3, we see that Suppliers 4 and 1 have the highest and the lowest Determination of the weight for each supplier with respect to each criterion Table 2 Supplier performance response data on five-point Likert scale Supplier scoring with respect to each criterion Performance C C C C C 1 2 4 9 24 Supplier 1 1.2 5 9 3 1 Supplier 2 1.5 9 7 7 3 Supplier Supplier 3 1.3 5 3 5 6 Ranking Supplier 4 1.7 7 9 7 2 Supplier 5 1.3 7 5 7 3 Fig. 1 The flowchart of the criteria with respect to the suppliers 1 3 Complex & Intelligent Systems (2021) 7:1869–1876 1873 degrees of membership, respectively, where the Suppliers 3 value is added to 1 and shown as normalized value (similarly and 5 also the same degrees, since they are doing the same for the column). The reason of adding 1 is because of the in this criterion. opposing behavior of the number less and more than one. For intuitionistic fuzzy index in Table 3, we see that Sup- For example, in the vertical group, since the biggest value plier 4 has the highest value because of having the farthest is 3.9055, the first normalized value becomes: distance from the average. That is, there is a better confi- dence for the suppliers having the value close to the average (0.9832 − 3.9055) + 1 = 3.9223. and it results to a lower intuitionistic fuzzy index. Using Eq. (1) from Definition 2, the Intuitionistic fuzzy Now, multiplying the normalized values of each row by entropy measures are easily calculated and presented in the normalized values of each column represents the coef- Table 4. ficient of each criterion with respect to each supplier. For example, the coefficient of C1 with respect supplier 1 is Normalization: The entropy measures uncertainty and it 3.9223 × 1.1978 = 4.6981 (the difference between 4.6981 indicates that more is its value, more is uncertainty. Then, and 4.6983 is because of rounding two numbers 3.9223 and calculating the sum of each row in Table 4, the distance of 1.1978 which are not rounded in calculations). All the coef- each summed value of each row with the largest summed ficients are shown in Table  5. The sum of the coefficients Fig. 2 The diagram of the crite- ria with respect to the suppliers Table 3 The intuitionistic fuzzy values Performance C C C C C 1 2 4 9 24 Supplier 1 (0,0.714,0.286) (0,0.714,0.286) (0.714,0,0.286) (0,0.588,0.412) (0,0.714,0.286) Supplier 2 (0.5,0.333,0.167) (0.625,0,0.375) (0.625,0.312,0.063) (0.769,0,0.231) (0.4,0.6,0) Supplier 3 (0.167,0.666,0.167) (0,0.714,0.286) (0,0.625,0.376) (0.417,0.417,0.166) (0.625,0,0.375) Supplier 4 (0.625,0,0.375) (0.455,0.455,0.09) (0.714,0,0.286) (0.769,0,0.231) (0.167,0.666,0.167) Supplier 5 (0.167,0.666,0.167) (0.455,0.455,0.09) (0.263,0.526,0.211) (0.769,0,0.231) (0.4,0,0.6) Table 4 The intuitionistic fuzzy Performance C C C C C Sum Normalized 1 2 4 9 24 entropy measures Supplier 1 0.2857 0.2857 0.2857 0.4118 0.2857 0.9832 3.9223 Supplier 2 0.9758 0.3750 0.9234 0.2308 0.9710 3.4759 1.4296 Supplier 3 0.7683 0.2857 0.3750 1.0000 0.3750 2.5183 2.3872 Supplier 4 0.3750 1.0000 0.2857 0.2308 0.7683 2.6598 2.2457 Supplier 5 0.7683 1.0000 0.9355 0.2308 0.9710 3.9055 1.0000 Sum 3.1731 2.3750 2.5196 2.1042 3.3709 Normalized 1.1978 1.9959 1.8513 2.2667 1.0000 1 3 1874 Complex & Intelligent Systems (2021) 7:1869–1876 Table 5 Total weights of each criterion method, we have considered the problem of selecting the best supplier firm to provide the materials in production Performance C C C C C 1 2 4 9 24 line of a large company in Iran. For economic considera- Supplier 1 4.6983 7.8285 7.2614 8.8907 3.9224 tions, every company wants to use a method of decision- Supplier 2 1.7124 2.8532 2.6465 3.2404 1.4296 making to select the best supplier. Obviously, criterion Supplier 3 2.8595 4.7647 4.4195 5.4111 2.3873 based only on expertise is infeasible some time. By the Supplier 4 2.6900 4.4822 4.1575 5.0904 2.2458 use of intuitionistic fuzzy entropy, we have attained a new Supplier 5 1.1978 1.9959 1.8512 2.2666 1.0000 method to provide a standard measurement to select the Total 13.1579 21.9245 2.5196 24.8992 10.9850 best supplier. In literature, earlier researchers have dem- Weight 0.1441 0.2401 1.8513 0.2727 0.1203 onstrated that expertise had a strong effect in the selec- tion of best supplier especially in determining the range of the weight. However, in our proposed, novelty lies in corresponding to each criterion shows the total coefficient of the fact that a standard method is applied for determina- each criterion. At the end, dividing each coefficient by sum tion of the weight and wherein the expertise effect on the of the coefficients determines its weight. Table  5 presents the decision-making has been reduced, thus, making the pro- sum of the coefficients of each criterion and the total weight posed method more applicable. of each criterion. In continuation for the future work, we are going to Finally, multiplying the weight of each criterion by the construct the matrix of the optimal weights based on the score of each intuitionistic fuzzy value which is calculated intuitionistic fuzzy entropy values for decision-makers in Eq. 2 shows the importance of each criterion with respect with respect to the attributes of the alternatives. Then, to each supplier. For example, the importance of criterion 1 based on this matrix of weights, and some operators such with respect to supplier 1, since its score is 0.714, is equal as weighted averaging operator and the score function, the to 0.714 × 0.1441 = 0.1029. In Table 6, the sum of degree of rank of the suppliers will be denoted by the scores which importance of each supplier shows their rankings. they gain. As a hint for other authors, the method provided From Table 6, it is noted that the values of total rank of in this paper can also be used in portfolio optimization criterion for the suppliers are − 0.0859, 0.4508, − 0.1634, when the calculated weights can represent the share of each 0.2185, and 0.1991, respectively. Thus, the selection prefer- stock. ences of suppliers may be stated as: Supplier 2 >> Supplier 4 >> Supplier 5 >> Supplier 1 >> Supplier 3, Open Access This article is licensed under a Creative Commons Attri- indicating that Supplier 2 is the best. bution 4.0 International License, which permits use, sharing, adapta- tion, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes Conclusion were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated In this investigation, we have introduced a new entropy- otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not based model which extends the notion of intuitionistic permitted by statutory regulation or exceeds the permitted use, you will fuzzy sets. To show the applicability of the proposed need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://cr eativ ecommons. or g/licen ses/ b y/4.0/ . Table 6 Total rank of the Performance C C C C C Total Rank- 1 2 4 9 24 criteria ing order Supplier 1 0.1029 − 0.1714 0.1590 − 0.1603 − 0.0859 − 0.0859 4 Supplier 2 − 0.0241 0.1501 0.1392 0.2097 − 0.0241 0.4508 1 Supplier 3 0.0721 − 0.1714 − 0.1392 0.0000 0.0752 − 0.1634 5 Supplier 4 − 0.0901 0.0000 0.1590 0.2097 − 0.0602 0.2185 2 Supplier 5 0.0721 0.0000 − 0.0586 0.2097 0.0241 0.1991 3 1 3 Complex & Intelligent Systems (2021) 7:1869–1876 1875 22. 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An intuitionistic fuzzy entropy approach for supplier selection

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Springer Journals
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Copyright © The Author(s) 2021
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2199-4536
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2198-6053
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10.1007/s40747-020-00224-6
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Abstract

Due to apparent flexibility of Intuitionistic Fuzzy Set (IFS) concepts in dealing with the imprecision or uncertainty, these are proving to be quite useful in many application areas for a more human consistent reasoning under imperfectly defined facts and imprecise knowledge. In this paper, we apply notions of entropy and intuitionistic fuzzy sets to present a new fuzzy decision-making approach called intuitionistic fuzzy entropy measure for selection and ranking the suppliers with respect to the attributes. An entropy-based model is formulated and applied to a real case study aiming to examine the rankings of suppliers. Furthermore, the weights for each alternative, with respect to the criteria, are calculated using intuitionistic fuzzy entropy measure. The supplier with the highest weight is selected as the best alternative. This proposed model helps the decision-makers in better understanding of the weight of each criterion without relying on the mere expertise. Keywords Multicriteria decision-making · Supplier selection · Intuitionistic fuzzy entropy · Intuitionistic fuzzy set Introduction recently, Bustince et al. [5] have focused on the history, defi- nition, and basic properties of fuzzy set types and relation- Decision-maker’s judgments, including preference informa- ships between the different types of fuzzy sets. tion, are usually stated in linguistic terms. There are many In last couple of years, many researchers also proposed approaches proposed for modeling the decision linguistic different functions for intuitionistic fuzzy sets (IFSs) and term sets. Zadeh [43–45] defined the linguistic variable as a applied them in various real-time applications. In 1983, IFS variable whose values are words or sentences in a natural or was introduced by Atanassov [1] as generalization of fuzzy artificial language in his three consecutive papers. Recently, sets. Basically intuitionistic fuzzy sets based models may Morente-Molinera et al. [23] provided a systematic review be adequate in situations when we face human testimonies, of the fuzzy linguistic modeling approaches developed over public opinions, etc. IFSs can be viewed as a generalization the last decade. The reviewed methods are classified into of fuzzy sets that may better model imperfect information six categories based on different approaches. In addition, which is present in any conscious decision-making (Atan- assov [2]). Intuitionistic fuzzy sets take into account both the degrees of membership and non-membership on the real unit * Mohamadtaghi Rahimi interval [0, 1] subject to the condition that their sum belongs Mohamadtaghi.rahimi@unbc.ca to the same interval. In recent years, several researchers Pranesh Kumar extended IFS based on various decision-making techniques. Pranesh.Kumar@unbc.ca For the first time, De Luca and Termini [11] integrated Behzad Moomivand the entropy concept (Shannon [15]) with fuzzy set theory Moomivand_stu@qom-iau.ac.ir (Zadeh [42]). The main purpose of entropy measures is Gholamhosein Yari to explain uncertainty degree. In recent years, numerous yari@iust.ac.ir studies have integrated the entropy with various fuzzy sets types, such as; Burillo and Bustince [4], Coban [9], Department of Mathematics and Statistics, University of Northern British Columbia, Prince George, BC, Canada Joshi and Kumar [21], Yari et  al. [38, 39], Szmidt and Kacprzyk [26], Farnoosh et al. [14], Ye [40], Hung and Department of Management, Qom Branch, Islamic Azad University, Qom, Iran Yang [18], Rahimi and Kumar [28], Rahimi et al. [29], Wei et al. [34], Zeng and Li [46], Szmidt and Kacprzyk Department of Mathematics, Iran University of Science and Technology, Tehran, Iran Vol.:(0123456789) 1 3 1870 Complex & Intelligent Systems (2021) 7:1869–1876 [27], Ye [41], and Zhang et al. [47]. Burillo and Bustince focused on to propose the new intuitionistic fuzzy entropy [4] have defined the interval-valued fuzzy sets and IFSs, measure for selection suppliers. and introduced the distance measure between IFSs using The paper is organized as following: “Literature review” the entropy measures. Joshi and Kumar [21] introduced the presents the literature review of entropy, IFSs, and applica- novel parametric (R, S)-norm intuitionistic fuzzy entropy tion of these methods in assessment of supplier selection. for solving problem of multiple-attribute decision-making In “Preliminaries”, we have provided some concepts and (MADM). Szmidt and Kacprzyk [26] have proposed the background about IFS, score function, and an Intuitionistic new non-probabilistic-type entropy measure for IFSs by Fuzzy Entropy measure. A new MCDM method is proposed considering IFSs and a ratio of distance between them. in “Proposed MCDM method and its application in selecting Ye [41] has introduced the fuzzy cross entropy based on the best supplier” which discusses our case study to show interval-valued intuitionistic fuzzy sets (IVIFSs) using the the validity of the proposed method. In “Conclusion”, we intuitionistic fuzzy (IF) cross entropy. Hung and Yang [18] conclude and state limitations and recommendations for have applied the probability concept for introducing the future studies. fuzzy entropy IFSs using two entropy measures for IFSs. Wei et al. [33] have introduced the entropy measure for IVIFSs by incorporating three kinds of entropy measures, Literature review and, finally, proposed the new entropy measure for IVIFSs. Some researchers have used the entropy and IFSs in In recent decades, several of previous studies used, inte- various application areas such as supplier and vendor grated, and introduced the entropy and IFS in numerous selection (Shahrokhi et al. [24], Wen et al. [35], Gerogi- application areas. Burillo and Bustince [4], defined the annis et al. [16], Xiao and Wei [37], Wang and Lv [31], interval-valued fuzzy sets and IFSs and introduced the dis- Krishankumar et al. [22], Song et al. [25], Guo et al. [17], tance measure between IFSs using the entropy technique. Chai et al. [6], Bali et al. [3], Wen et al. [35], and Xiao Wen et al. [35] used the IFS for selection of vendor based on and Wei [37]). Shahrokhi et al. [24] have proposed the some MADM approaches such as Simple Additive Weight- integrated approach based on IFS and linear programming ing (SAW), Weight Product Matrix (WPM), ELimination technique for selection of suppliers in a group decision- Et Choix Traduisant la REalité—Elimination (ELECTRE), making environment. Wen et  al. [35] have considered Order of Preference by Similarity to Ideal Solution (TOP- the IFS for selection of vendor based on some MADM SIS), and Lexicographic. Wang et al. [32] extended some approaches such as Simple Additive Weighting (SAW), operators including triangular intuitionistic fuzzy ordered Weight Product Matrix (WPM), ELimination Et Choix weighted averaging (TIFOWA), triangular intuitionis- Traduisant la REalité – Elimination (ELECTRE), Order tic fuzzy ordered weighted geometric (TIFOWG), hybrid of Preference by Similarity to Ideal Solution (TOPSIS), weighted averaging (IFHWA), triangular intuitionistic fuzzy and Lexicographic. Gerogiannis et  al. [16] have intro- generalized ordered weighted averaging (TIFGOWA), and duced the hybrid approach for assessment of biomass triangular intuitionistic fuzzy generalized hybrid weighted suppliers by integrating IFS, multi-periodic optimization averaging (TIFGHWA) based on TOPSIS and multi-objec- (MPO), and linear programming. Wang and Lv [31] have tive programming. Shahrokhi et al. [24] proposed the inte- investigated induced intuitionistic fuzzy Einstein hybrid grated approach based on IFS and linear programming tech- aggregation operator (I-IFEHA) for selection of supplier nique for selection of suppliers in a group decision-making in environment of group decision-making based on fuzzy environment. Joshi and Kumar [21] introduced the novel par- measures by introducing aggregation and Einstein opera- ametric (R, S)-norm intuitionistic fuzzy entropy for solving tor I-IFEHA. Krishankumar et al. [22] have introduced a problem of multiple-attribute decision-making (MADM). Jin novel approach for supplier selection using IVIF based et al. [20] proposed two new approaches for group decision- on statistical variance (SV) and ELECTRE methods. Wen making to derive the normalized intuitionistic fuzzy priority et al. [35] have used IFS for supplier selection in envi- weights from IFPRs based on multiplicative consistency and ronment of group decision-making. Xiao and Wei [37] the order consistency. Gerogiannis et al. [16] introduced the have presented a method to deal with the supplier selec- hybrid approach for assessment of biomass suppliers by inte- tion problem in supply chain management with interval- grating IFS, multi-periodic optimization (MPO), and linear valued intuitionistic fuzzy information. It may, however, programming. Wang et al. [32] proposed the new method be noted that although, these researchers have applied by integration OWA–TOPSIS and intuitionistic fuzzy set- and integrated entropy with IFSs in various application tings. Chen and Chang [7] proposed novel approach for areas, but there are gaps in application of these techniques fuzzy multiattribute decision-making based on three opera- in supplier selection. Therefore, in this paper, we have tors named IFWGA, IFOWGA, and IFHGA. Wang and Lv [31] investigated induced intuitionistic fuzzy Einstein hybrid 1 3 Complex & Intelligent Systems (2021) 7:1869–1876 1871 aggregation operator (I-IFEHA) which is investigated for Definition 1 (Atanassov [2]). An IFS over X is defined as selection of supplier in environment of group decision-mak- follows: ing based on fuzzy measures by introducing aggregation and � � A = ⟨x, 𝜇 (x), 𝛾 (x)⟩�x ∈ X ̃ ̃ A A Einstein operations for proposing the I-IFEHA. Szmidt and 𝜇 (x) ∶ X → [0,1], 𝛾 (x) ∶ X → [0,1], Kacprzyk [26] proposed the new entropy measure for IFSs in ̃ ̃ A A the non-probabilistic-type by interpreting of IFSs and a ratio where µ and γ, respectively, define the degree of mem- of distance between them. In 2007, Vlachos and Sergiadis bership and the degree of non-membership, and we have: [30] proposed the intuitionistic fuzzy divergence measure for 0 ≤ 𝜇 x + 𝛾 x ≤ 1 for every x ∈ X. ̃ ( ) ̃ ( ) A A the first time, and studied its application pattern recognition and medical diagnosis. Krishankumar et al. [22] introduced π = 1 − 𝜇 x − 𝛾 x denotes a measure of non-deter- ̃ ̃ ( ) ̃ ( ) A A A the novel approach for supplier selection using IVIF based minancy which is called the intuitionistic fuzzy (IF) index on statistical variance (SV) and ELECTRE methods. Ye [41] of the element x. Obviously, when 𝜇 = 0 , the set A is a introduced the fuzzy cross entropy based on interval-valued fuzzy set. If we denote the set of all the FSs on X by F(X), intuitionistic fuzzy sets (IVIFSs) using the intuitionistic ̃ ̃ the operations of IFSs are defined for every A, B ∈ F(X) as: fuzzy (IF) cross entropy. Furthermore, Wei and Ye [34] pro- � � posed an improved version of intuitionistic fuzzy divergence A = ⟨x, 𝛾 (x), 𝜇 (x)⟩�x ∈ X A A in Vlachos and Sergiadis [30] and developed a method for � � ̃ ̃ A ∧ B = ⟨x, μ (x) ∧μ (x), γ (x) ∨γ (x)⟩�x∈ X ̃ ̃ ̃ ̃ A B A B pattern recognition with intuitionistic fuzzy information. � � ̃ ̃ A ∨ B = ⟨x, 𝜇 (x) ∨ 𝜇 (x), 𝛾 (x) ∧ 𝛾 (x)⟩�x∈ X ̃ ̃ ̃ ̃ A A Wen et al. [35] used IFS for supplier selection in environ- B B � � ̃ ̃ A ⊗ B = ⟨x, 𝜇 (x) + 𝜇 (x) − 𝜇 (x)𝜇 (x), 𝛾 (x)𝛾 (x)⟩�x∈ X ment of group decision-making. Hung and Yang [18] used ̃ ̃ ̃ ̃ ̃ ̃ A A A B B B � � ̃ ̃ the probability concept for introducing the fuzzy entropy A ⊗ B = ⟨x, 𝜇 (x)𝜇 (x), 𝛾 (x) + 𝛾 (x) − 𝛾 (x)𝛾 (x)⟩�x∈ X ̃ ̃ ̃ ̃ ̃ ̃ A B A B A B � � � � � � 𝛼 𝛼 IFSs with two entropy measures for IFSs. Hung and Yang 𝛼 A = ⟨x,1 − 1 − 𝜇 (x) , 𝛾 (x) ⟩�x ∈ X ̃ ̃ A A [19] defined another divergence measure called ‘J-diver - � � � � � � 𝛼 𝛼 A = ⟨x, 𝜇 (x) ,1 − 1 − 𝛾 (x) ⟩�x ∈ X . ̃ ̃ A A gence’ for measuring the difference between two IFSs and then applied it to clustering analysis and pattern recognition. Burillo and Bustince [4] introduced the concept of entropy Definition 2 (Wu-Zhang [36]). Let A = a ̃ , a ̃ ,… , a ̃ be 1 2 n in intuitionistic fuzzy set theory, which allows us to measure an IFS and ã = 𝜇 , 𝛾 , i = 1,2,…,n, be intuitionistic fuzzy i i i the degree of intuitionism associated with an IFS. Vlachos values in A. Then, an Intuitionistic Fuzzy Entropy measure and Sergiadis [30] proposed another measure of intuitionis- is formulated in the following way: tic fuzzy entropy and revealed an intuitive and mathemati- cal connection between the notions of entropy for fuzzy set 𝜇 𝛾 −1 i i + 𝛾 Ln 𝜀 ã = 𝜋 − (Ln2) 𝜇 Ln . i i i i and intuitionistic fuzzy set. Wei et al. [33], introduced the 𝜇 + 𝛾 𝜇 + 𝛾 i i i i entropy measure for IVIFSs by incorporating three kinds (1) of entropy measures, and finally, proposed the new entropy This measure satisfies the four axioms in Szmidt and measure for IVIFSs. Zhang and Jiang [48] defined a measure Kacprzyk [24] for IF value entropy measure. of intuitionistic fuzzy entropy for intuitionistic fuzzy sets by generalizing of the De et al. [10], logarithmic fuzzy entropy. Definition 3 (Chen and Tan [ 8]). Let ã = 𝜇 , 𝛾 , i i i Xiao and Wei [37] presented a method to deal with the sup- i = 1,2,…,n, be intuitionistic fuzzy values, and then, the plier selection problem in supply chain management with score of a ̃ is: interval-valued intuitionistic fuzzy information. Although, previous mentioned papers have investigated the important S ã = 𝜇 − 𝛾 , i = 1,2,… , n. (2) i i i role of entropy and IFS in assessment of supplier selection, but there is gap in literature regarding to these issues, how- ever; this study based on current literature, attempted to review these issues comprehensively. Proposed MCDM method and its application in selecting the best supplier We consider that one of the largest companies in Iran would Preliminaries like to select the best supplier firm to provide the materials in production line. In this context, we propose a new method Some basic definitions of IFS, Intuitionistic Fuzzy Entropy based on Intuitionistic Fuzzy Entropy to identify the best measure, and the score function are reviewed for the sake supplier. First, we have to recognize the main criteria which of completeness. 1 3 1872 Complex & Intelligent Systems (2021) 7:1869–1876 can influence our decision. After the criteria selection, the Table 1 Supplier performance response data next step is how to choose the best supplier. Performance C C C C C 1 2 3 4 5 Supplier 1 1.2 M VH L 1 Criteria selection Supplier 2 1.5 VH H H 3 Supplier 3 1.3 M L M 6 Using Dickson’s [13] 23 criteria in supplier selection and Supplier 4 1.7 H VH H 2 the addition of one local criterion which is pay off time (an Supplier 5 1.3 H M H 3 important factor in Iran’s business market), a questionnaire containing 24 questions was constructed. This question- naire was sent to 30 firm’s managers and firm’s sale man- agers. In each question, the importance of one criterion is qualitative, we convert them on the quantitative scale by the five-point Likert scale as it is shown in Table  2. That is; in evaluated. The applicant would choose among: “very low”, “low”, “medium”, “high”, and “very high”. All responses the qualitative questions, the criteria with respect to each supplier are given a number among 1, 3, 5, 7 and 9 (Fig. 2). were converted to the five-point Likert scale. Then, using SPSS, we have compared the means of the criteria points at To convert the values into the Intuitionistic Fuzzy Values, we have extended the method introduced in Deng-Chan [12] 95% confidence level. As follows, five criteria were selected as the most important ones such as; price, quality, deliver, as follows: To get the degrees of membership, non-membership, and technical capability, and pay off factors. intuitionistic fuzzy index, we have calculated the distance of each value with the lowest value as the value of membership, The selection model the distance of each value with the highest value as the value of non-membership, and the distance of each value with the In the presented selection model (shown in Fig. 1), we have tabulated the information of five suppliers in Table1 with average of others as the value of intuitionistic fuzzy index. Then, these calculated values are, respectively, divided by respect to the above criteria. Note that when the values are their total sum. If the criterion is a kind of cost, the degrees of membership and non-membership are replaced with each Criteria other. For example, the intuitionistic fuzzy degrees for the Selection two first values of C1 are calculated here: �1.2−1.2� ⎧ = 0  = 0 �1.2−1.2�+�1.2−1.7�+�1.2−1.4� Gathering the Supplier Performance Response Data with ⎪ �1.2−1.7� = 0.714  = 0.714 ⎨ 11 respect to each criterion �1.2−1.2�+�1.2−1.7�+�1.2−1.4� �1.2−1.4� = 0.286  = 0.286 ⎩ 11 �1.2−1.2�+�1.2−1.7�+�1.2−1.4� �1.5−1.2� ⎧ = 0.50  = 0.333 �1.5−1.2�+�1.5−1.7�+�1.5−1.4� Calculation of Degrees of Membership, non-Membership �1.5−1.7� = 0.333  = 0.50 and intuitionistic fuzzy index ⎨ 21 �1.5−1.2�+�1.5−1.7�+�1.5−1.4� �1.5−1.4� = 0.167  = 0.167. �1.5−1.2�+�1.5−1.7�+�1.5−1.4� Calculation of intuitionistic fuzzy entropy for each supplier From the first column of Table  2, we now that Supplier with respect to each criterion 1 has the worst performance in delivery, where Supplier 4 has the best. With this proposed method, in Table  3, we see that Suppliers 4 and 1 have the highest and the lowest Determination of the weight for each supplier with respect to each criterion Table 2 Supplier performance response data on five-point Likert scale Supplier scoring with respect to each criterion Performance C C C C C 1 2 4 9 24 Supplier 1 1.2 5 9 3 1 Supplier 2 1.5 9 7 7 3 Supplier Supplier 3 1.3 5 3 5 6 Ranking Supplier 4 1.7 7 9 7 2 Supplier 5 1.3 7 5 7 3 Fig. 1 The flowchart of the criteria with respect to the suppliers 1 3 Complex & Intelligent Systems (2021) 7:1869–1876 1873 degrees of membership, respectively, where the Suppliers 3 value is added to 1 and shown as normalized value (similarly and 5 also the same degrees, since they are doing the same for the column). The reason of adding 1 is because of the in this criterion. opposing behavior of the number less and more than one. For intuitionistic fuzzy index in Table 3, we see that Sup- For example, in the vertical group, since the biggest value plier 4 has the highest value because of having the farthest is 3.9055, the first normalized value becomes: distance from the average. That is, there is a better confi- dence for the suppliers having the value close to the average (0.9832 − 3.9055) + 1 = 3.9223. and it results to a lower intuitionistic fuzzy index. Using Eq. (1) from Definition 2, the Intuitionistic fuzzy Now, multiplying the normalized values of each row by entropy measures are easily calculated and presented in the normalized values of each column represents the coef- Table 4. ficient of each criterion with respect to each supplier. For example, the coefficient of C1 with respect supplier 1 is Normalization: The entropy measures uncertainty and it 3.9223 × 1.1978 = 4.6981 (the difference between 4.6981 indicates that more is its value, more is uncertainty. Then, and 4.6983 is because of rounding two numbers 3.9223 and calculating the sum of each row in Table 4, the distance of 1.1978 which are not rounded in calculations). All the coef- each summed value of each row with the largest summed ficients are shown in Table  5. The sum of the coefficients Fig. 2 The diagram of the crite- ria with respect to the suppliers Table 3 The intuitionistic fuzzy values Performance C C C C C 1 2 4 9 24 Supplier 1 (0,0.714,0.286) (0,0.714,0.286) (0.714,0,0.286) (0,0.588,0.412) (0,0.714,0.286) Supplier 2 (0.5,0.333,0.167) (0.625,0,0.375) (0.625,0.312,0.063) (0.769,0,0.231) (0.4,0.6,0) Supplier 3 (0.167,0.666,0.167) (0,0.714,0.286) (0,0.625,0.376) (0.417,0.417,0.166) (0.625,0,0.375) Supplier 4 (0.625,0,0.375) (0.455,0.455,0.09) (0.714,0,0.286) (0.769,0,0.231) (0.167,0.666,0.167) Supplier 5 (0.167,0.666,0.167) (0.455,0.455,0.09) (0.263,0.526,0.211) (0.769,0,0.231) (0.4,0,0.6) Table 4 The intuitionistic fuzzy Performance C C C C C Sum Normalized 1 2 4 9 24 entropy measures Supplier 1 0.2857 0.2857 0.2857 0.4118 0.2857 0.9832 3.9223 Supplier 2 0.9758 0.3750 0.9234 0.2308 0.9710 3.4759 1.4296 Supplier 3 0.7683 0.2857 0.3750 1.0000 0.3750 2.5183 2.3872 Supplier 4 0.3750 1.0000 0.2857 0.2308 0.7683 2.6598 2.2457 Supplier 5 0.7683 1.0000 0.9355 0.2308 0.9710 3.9055 1.0000 Sum 3.1731 2.3750 2.5196 2.1042 3.3709 Normalized 1.1978 1.9959 1.8513 2.2667 1.0000 1 3 1874 Complex & Intelligent Systems (2021) 7:1869–1876 Table 5 Total weights of each criterion method, we have considered the problem of selecting the best supplier firm to provide the materials in production Performance C C C C C 1 2 4 9 24 line of a large company in Iran. For economic considera- Supplier 1 4.6983 7.8285 7.2614 8.8907 3.9224 tions, every company wants to use a method of decision- Supplier 2 1.7124 2.8532 2.6465 3.2404 1.4296 making to select the best supplier. Obviously, criterion Supplier 3 2.8595 4.7647 4.4195 5.4111 2.3873 based only on expertise is infeasible some time. By the Supplier 4 2.6900 4.4822 4.1575 5.0904 2.2458 use of intuitionistic fuzzy entropy, we have attained a new Supplier 5 1.1978 1.9959 1.8512 2.2666 1.0000 method to provide a standard measurement to select the Total 13.1579 21.9245 2.5196 24.8992 10.9850 best supplier. In literature, earlier researchers have dem- Weight 0.1441 0.2401 1.8513 0.2727 0.1203 onstrated that expertise had a strong effect in the selec- tion of best supplier especially in determining the range of the weight. However, in our proposed, novelty lies in corresponding to each criterion shows the total coefficient of the fact that a standard method is applied for determina- each criterion. At the end, dividing each coefficient by sum tion of the weight and wherein the expertise effect on the of the coefficients determines its weight. Table  5 presents the decision-making has been reduced, thus, making the pro- sum of the coefficients of each criterion and the total weight posed method more applicable. of each criterion. In continuation for the future work, we are going to Finally, multiplying the weight of each criterion by the construct the matrix of the optimal weights based on the score of each intuitionistic fuzzy value which is calculated intuitionistic fuzzy entropy values for decision-makers in Eq. 2 shows the importance of each criterion with respect with respect to the attributes of the alternatives. Then, to each supplier. For example, the importance of criterion 1 based on this matrix of weights, and some operators such with respect to supplier 1, since its score is 0.714, is equal as weighted averaging operator and the score function, the to 0.714 × 0.1441 = 0.1029. In Table 6, the sum of degree of rank of the suppliers will be denoted by the scores which importance of each supplier shows their rankings. they gain. As a hint for other authors, the method provided From Table 6, it is noted that the values of total rank of in this paper can also be used in portfolio optimization criterion for the suppliers are − 0.0859, 0.4508, − 0.1634, when the calculated weights can represent the share of each 0.2185, and 0.1991, respectively. Thus, the selection prefer- stock. ences of suppliers may be stated as: Supplier 2 >> Supplier 4 >> Supplier 5 >> Supplier 1 >> Supplier 3, Open Access This article is licensed under a Creative Commons Attri- indicating that Supplier 2 is the best. bution 4.0 International License, which permits use, sharing, adapta- tion, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes Conclusion were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated In this investigation, we have introduced a new entropy- otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not based model which extends the notion of intuitionistic permitted by statutory regulation or exceeds the permitted use, you will fuzzy sets. To show the applicability of the proposed need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://cr eativ ecommons. or g/licen ses/ b y/4.0/ . Table 6 Total rank of the Performance C C C C C Total Rank- 1 2 4 9 24 criteria ing order Supplier 1 0.1029 − 0.1714 0.1590 − 0.1603 − 0.0859 − 0.0859 4 Supplier 2 − 0.0241 0.1501 0.1392 0.2097 − 0.0241 0.4508 1 Supplier 3 0.0721 − 0.1714 − 0.1392 0.0000 0.0752 − 0.1634 5 Supplier 4 − 0.0901 0.0000 0.1590 0.2097 − 0.0602 0.2185 2 Supplier 5 0.0721 0.0000 − 0.0586 0.2097 0.0241 0.1991 3 1 3 Complex & Intelligent Systems (2021) 7:1869–1876 1875 22. 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Journal

Complex & Intelligent SystemsSpringer Journals

Published: May 5, 2021

Keywords: Multicriteria decision-making; Supplier selection; Intuitionistic fuzzy entropy; Intuitionistic fuzzy set

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