INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL Online ISSN 1841-9844, ISSN-L 1841-9836, Volume: 17, Issue: 5, Month: October, Year: 2022 Article Number: 4468, https://doi.org/10.15837/ijccc.2022.5.4468 CCC Publications Supplier Selection Model Based on D Numbers and Transformation Function Leihui Xiong, Xiaoyan Su, Hong Qian Leihui Xiong School of Automation Engineering, Shanghai University of Electric Power Shanghai 200090, China xionglh@mail.shiep.edu.cn Xiaoyan Su* School of Automation Engineering, Shanghai University of Electric Power Shanghai 200090, China *Corresponding author: suxiaoyan@shiep.edu.cn. Hong Qian School of Automation Engineering, Shanghai University of Electric Power Shanghai 200090, China qianhong.sh@163.com Abstract Selecting reasonable suppliers can effectively improve the efficiency of enterprise supply chain management. Among them, expert evaluation is an important part of supplier selection problem, but the uncertainty, fuzziness and incompleteness of expert opinions make supplier selection prob- lem difficult to solve. In order to systematically and effectively solve the uncertainty, ambiguity and incompleteness in supplier selection problem, this paper presents a new supplier selection method based on D numbers and transformation function. First, fuzzy preference relation is generated based on the decision matrix of pairwise comparisons given by experts. D numbers which can effectively deal with uncertain information extend fuzzy preference relation (D matrix). Second, the D matrix is converted into a crisp matrix form based on the integration representation of D numbers according to different situations whether or not the information in D matrix is complete. Third, the crisp matrix is converted into judgement matrix by using the transformation functions. Finally, analytic hierarchy process (AHP) method is applied based on the judgment matrix to give a priority weights for decision making. Three numerical examples and application of the supplier selection are used to show the feasibility and effectiveness of the proposed method. Keywords: D numbers, transformation functions, analytic hierarchy process, fuzzy preference relation. https://doi.org/10.15837/ijccc.2022.5.4468 2 1 Introduction Multi-attribute decision making is an important part of decision theory and modern decision sci- ence [1]. It has been widely used in investment decision making [2], project evaluation[3], scheme selection[4], factory site selection [5, 6], comprehensive evaluation of economic benefits [7], etc. There- fore, it is of great practical and theoretical significance to study effective and practical multi-attribute decision making methods. As the market environment becomes more and more complex and the competition among enter- prises becomes more and more fierce, the supplier selection in multi-attribute decision making problem attracts more and more attention [8]. Supplier selection and evaluation is an important activity for enterprises to determine their own product suppliers, and it is also the premise to optimize the sup- plier management system. Enterprise supplier selection evaluation has a set of strict process steps. The enterprise selects suppliers mainly from four aspects: enterprise performance, business capability, quality system and enterprise environment [9]. Different index elements can be obtained by refining these four aspects, and then the evaluation of each index can be established by analyzing these index elements. Finally, AHP and other methods are used to make decisions [10]. Traditional supplier selection process can solve this problem to a certain extent, but with the deepening of the research on supplier selection, scholars find that: due to the complexity of practical problems and the limitation of decision-makers’ knowledge and experience, there are many uncertain- ties and incomplete semantic information in decision makers’ determination of attribute values and expression of preference information, which leads to a large amount of uncertain information to be dealt with in supplier selection [11, 12]. In order to improve the supplier selection method, some scholars use fuzzy number theory, grey theory, D-S evidence theory to deal with uncertain information in supplier selection problem. Rashidi et al.[13] applied fuzzy data envelopment analysis (DEA), and the technique for order of preference by similarity to ideal solution (TOPSIS) to the selection of logistics service providers, and compared and analyzed the two models to demonstrate the effectiveness of these two methods. Chen et al. [14] combined TOPSIS, decision making trial and evaluation laboratory (DEMATEL) and rough fuzzy method in dealing with supplier selection problem. This method uses fuzzy number to represent internal and external uncertainties, and DEMATEL method to reflect the interaction between attributes. Xing et al. [15] proposed a new supplier selection model based on interval two-type trapezoidal fuzzy partial order and Choquet integral, which can effectively deal with the interaction between criteria and consider consensus. In general, although fuzzy number theory can deal with subjective opinions effectively, fuzzy number itself is better at dealing with fuzzy information rather than other uncertain information. Hekmat et al. [16] used grey principal component analysis (PCA) and DEA to deal with supplier selection in uncertain environment. This method can deal with uncertain information as well as insufficient data information and related situations. Babak et al.[17] ranked suppliers based on TOPSIS and grey theory, and applied this method to the selection of battery suppliers for electric vehicles. In a word, grey theory can express the incompleteness and uncertainty of information to a certain extent, but it is mainly used for fuzzy prediction and cannot express the uncertain information intuitively. Sureeyatanapas et al. [18] solve the supplier selection problem based on D-S evidence theory and TOPSIS method. In this method, D-S evidence theory is used to construct decision matrix, and then TOPSIS method is used for further fusion. Zhang et al. [19] combined D-S evidence theory and analytic network process (ANP), and this model effectively solved the problem of supplier selection in an uncertain environment and considered the interaction between indicators. Fei et al. [20] solved the supplier selection problem based on D-S evidence theory and an elimination and choice translating reality (ELECTRE), which can better analyze the priority relationship between suppliers while dealing with uncertain information. In general, D-S evidence theory [22] [23] can better process and express uncertain information, so it is widely used in this field. However, D-S evidence theory cannot deal with the incomplete identification framework and the correlation of basic elements. To sum up, the above methods have certain defects in dealing with uncertain information in supplier selection. Therefore, the method applying D numbers to multi-attribute decision making is proposed [21]. D numbers can effectively deal with the uncertainty and fuzziness of information. At https://doi.org/10.15837/ijccc.2022.5.4468 3 Table 1: Comparison of methods to deal with uncertain information in supplier selection problem Method Feature 1 Feature 2 Feature 3 DEA-TOPSIS method [13] √ × × TOPSIS, DEMATEL and rough fuzzy method [14] √ × × Interval Type-2 trapezoidal fuzzy numbers method [15] √ × × Grey PCA-DEA method [16] √ × × TOPSIS and grey theory method [17] √ × × D-S evidence theory and TOPSIS method [18] √ √ × D-S evidence theory and ELECTRE method [19] √ √ × D-AHP method [39] √ √ √ D-ANP method [41] √ √ √ The proposed method √ √ √ the same time, D numbers extends D-S evidence theory to a certain extent , which can deal with the situation where the identification framework is incomplete and the basic elements are not independent from each other. Considering the advantages of D numbers[24, 25, 26, 27, 28, 29, 30, 31] in dealing with uncertain information, D numbers is widely used in various fields [32, 33, 34, 35, 36, 37, 38]. Deng et al. [39, 40] solved the supplier selection problem based on D numbers and analytic hierarchy process (AHP). On the basis of Deng, Fei et al. [41] selected the optimal supplier based on D numbers and network analysis method (ANP). However, the applications of these methods may get counter-intuitive results under some situations. This paper presents a new supplier selection method based on D numbers and transformation function. Compared with the method of Deng et al. [39], the method in this paper can deal with the situations where the expert opinions are quite uncertain and the decision-making objects are similar. In addition, in the process of consistency evaluation, the method in this paper is more consistent with the actual situation and has a good theoretical basis. (Table 1 shows the comparison of methods for dealing with uncertain information in supplier selection, in which Feature 1 indicates that uncertain information is expressed to a certain extent, Feature 2 indicates that uncertain information can be expressed intuitively, and Feature 3 indicates that it can handle the situation where the identification framework is incomplete and the basic elements are not independent.) The rest of this paper is organized as follows. In section 2, the preliminaries on AHP, D numbers and transformation functions are briefly introduced. In section 3, a new method to process uncertain information in supplier selection problem is proposed and some examples which can be handled by the method in this paper but cannot be handled by Deng et al.’s method [39] are given. In section 4, an application of the supplier selection is illustrated to show the rationality of this new method. In section 5, conclusions. 2 Preliminaries 2.1 Analytic hierarchy process (AHP) [45] The steps for the AHP are shown below: Firstly, the hierarchical structure of the problem is constructed, and then the pairwise comparison matrix is constructed based on certain criteria. Definition 1:When the numbers of elements is n, the pairwise comparison matrix is defined as follow. W =   w11 · · ·w1n w21 ... ... wn1 · · ·wnn   (1) Where wij measures the relationship that i is more important than j. Thirdly, we calculate the eigenvector corresponding to the largest eigenvalue of matrix W . Finally, the matrix W ’s consistency https://doi.org/10.15837/ijccc.2022.5.4468 4 Table 2: The value of RI Dimension 1 2 3 4 5 6 7 8 9 10 RI 0 0 0.52 0.89 1.12 1.26 1.36 1.41 1.46 1.49 is verified by CR which is defined as follow: CI = λmax −n n− 1 (2) CR = CI RI (3) The value of random consistency index (RI) is shown in Table 2. We can accept the consistency of the pairwise comparison matrix, when CR is less than 0.1. Finally, normalizing the eigenvector to obtain the final weights. 2.2 D numbers [39] D numbers is an extension of Dempster-Shafer theory, which can deal with the uncertainty infor- mation. D numbers have wider application than Dempster-Shafer theory, because D numbers do not have constrain of integral and independent assumption. Definition 2:Let Ω be a finite nonempty set, D number is a mapping D: 2Ω → [0, 1] , which is given as follows: ∑ B⊆Ω D (B) ≤ 1 and D (φ) = 0 (4) where φ is an empty set and B is a subset of Ω. If ∑ B⊆Ω D (B) = 1 the information is assumed to be complete. If ∑ B⊆Ω D (B) < 1 , which means the information is incomplete. There is an example to show a D numbers. Suppose a system’s security needs to be accessed by 10 experts. The "medium" security level is supported by 4 experts. The "high" security level is supported by 3 experts. The remaining experts do not generalize a conclusion because of their limitations in knowledge. We can express this situation using D numbers as follow. D ({high}) = 0.3,D ({medium}) = 0.4 The sum of high and medium level less than one. Because the information is incomplete. Fuzzy numbers and D-S theory cannot describe this well. Definition 3: Let D = ({b1,v1} ,{b2,v2} , · · · ,{bi,vi} , · · ·{bn,vn}) be a D number, the integration representation of D numbers can be calculated as follow: I (D) = n∑ i=1 bivi (5) 2.3 Transformation functions [42] Definition 4: Suppose that we have a set of alternatives, X = {x1, · · · ,xn}, and associated with it a reciprocal multiplicative preference relation A = (aij ) with aij ∈ [1/9, 9]. Then, the corresponding reciprocal fuzzy preference relation, P = (pij ) with pij ∈ [0, 1] , associated with A is given as follows: pij = g (aij ) = 1 2 × (1 + log9aij ) (6) With such a transformation function g we can relate the research issues obtained for both kinds of preference relations. https://doi.org/10.15837/ijccc.2022.5.4468 5 D numbers extended fuzzy preference relation (D matrix) Crisp matrix Judgement matrix Weights of alternatives The maximum eigenvector I(RD) D numbers extended fuzzy preference relation (D matrix) Crisp matrix Probability matrix Triangular matrix Weights of alternatives Triangulated crisp matrix I(RD) step1 step2 step3 step4 Ranking of alternatives Local information T p R T c R step1 step2 step3 P R C R D R D R C R tr R (a) Deng et al.'s method [54] (b) The proposed method input input output output output Figure 1: Flow chart of Deng et al.’s method and the proposed method in deriving priority weights of alternatives https://doi.org/10.15837/ijccc.2022.5.4468 6 3 Proposed method The main flow chart of the proposed method compared with Deng et al.’s method [39] are shown in Figure 1. More details are illustrated as follows. 3.1 Introduction of Deng et al.’s method [39] After D numbers extended fuzzy preference relation (D matrix) is constructed, we can gain the crisp matrix by the integration representation of D numbers. After that the main problem is to gain the probability matrix. Two different processes are given in Deng et al.’s method for dealing with complete information and incomplete information to gain the probability matrix. Assume that cij is the element in the i-th row and j-th column of crisp matrix. (1) D matrix with complete information When cij + cji = 1.0, We believe that cij has complete information. (i) When the cij is greater than 0.5, we replace cij with 1. (ii) When the cij is lesser than 0.5, we replace cij with 0. (2) D matrix with incomplete information When cij + cji < 1.0, We believe that cij has incomplete information. (i) If cij ≥ 0.5 or cji ≥ 0.5 , the larger one is replaced by 1 and the smaller one is replaced by 0. (ii) If cij < 0.5,cji < 0.5, we can use Equation (7) and Equation (8) to get elements of the probability matrix. C ′ ij = 1 − (0.5 −Cij ) 1 − (Cij + Cji) (7) C ′ ji = 1 − (0.5 −Cji) 1 − (Cij + Cji) (8) where Cij means that the row i and column j of the crisp matrix. C ′ ij means that the row i and column j of the probability matrix. And then using triangularization method to convert the probability matrix into the triangular matrix which is used to rank the elements. This method sums up the elements in each row, records the number of rows corresponding to the largest item as i, removes the i-th row and the j-th column of the matrix, and loops until the matrix is empty. The order of deletion is sort of elements. The triangular matrix can be obtained by using this sort. Finally, the interval of weights are calculated by introducing the variable λ which reflects the information of the pairwise comparison. 3.2 An improved D-AHP method 3.2.1 D matrix D matrix is an extension of fuzzy preference relation. Tanino et al. [43] propose the fuzzy preference relation to deal with ambiguity in expert language. For the multiplicative preference relation, the diagonal elements are the inverse of each other, i.e. aij ×aji = 1. While the fuzzy preference relation is subjected to an additive reciprocal, i.e. rij + rji = 1. Suppose A = {A1,A2, ...,An} is a set of alternatives. A fuzzy set A×A, which is calculated by a membership function (µR : A×A → [0, 1]), represent a fuzzy preference relation R as follows. R = A1 A2 ... An A1 A2 · · · An  r11 r12 · · · r1n r21 r22 · · · r2n ... ... ... ... rn1 rn2 · · · rnn   (9) where rij = µR (Ai,Aj ) ,∀i,j ∈ {1, 2, . . . ,n}. rij represents the degree of preference for alterna- tive Ai over alternative Aj. The five scenarios for rij are as follows: (1)When rij = µR (Ai,Aj ) = µR (Ai,Aj ) = 0, it means that Aj is absolutely preferred to Ai; (2)When rij = µR (Ai,Aj ) ∈ (0, 0.5), it means that Aj is preferred to Ai to some degree; (3)When rij = µR (Ai,Aj ) = 0.5, it means that https://doi.org/10.15837/ijccc.2022.5.4468 7 indifference between Ai and Aj; (4)When rij = µR (Ai,Aj ) ∈ (0.5, 1), it means that Ai is preferred to Aj to some degree; (5)When rij = µR (Ai,Aj ) = 1, it means that Ai is absolutely preferred to Aj. At the same time, R must satisfy three constraints: (1) rij ≥ 0; (2) rij + rji = 1,∀i,j ∈{1, 2, . . . ,n}; (3) rii = 0.5,∀i,j ∈{1, 2, . . . ,n}. With this method, linguistic values can be a way of generating preference information. But the fuzzy preference relation still has some drawbacks. Firstly, there will be inconsistencies in the fuzzy preference. How to measure the consistency of the fuzzy preference is still an open question. Secondly, the fuzzy preference can’t use in some situations. For example, there are two schemes A1,A2 and 10 experts give their idea about them. Situation 1: 8 experts give the conclusion that r12 = 0.7. The other 2 experts give the conclusion that r12 = 0.6. Situation 2: 6 experts give the conclusion that r12 = 0.8. Because of the lack of relevant knowledge, the other 4 experts did not give suggestions. In both cases, the use of the fuzzy preference alone does not adequately represent the information. To solve the above shortcomings, this paper combine the fuzzy preference relation with D number and replaces the rij in R with the D number, then the resulting new matrix is called D matrix. We can transform the fuzzy preference relation R (see Equation (9)) into D matrix RD (see Equation (10)). RD = A1 A2 ... An A1 A2 · · · An  D11 D12 · · · D1n D21 D22 · · · D2n ... ... ... ... Dn1 Dn2 · · · Dnn   (10) where RD satisfies the following constraints: (1)Dij = {( b ij 1 ,v ij 1 ) , ( b ij 2 ,v ij 2 ) , . . . , ( bijm,v ij m )} ,∀i,j ∈{1, 2, . . . ,n} (2)Dji = {( 1 − bij1 ,v ij 1 ) , ( 1 − bij2 ,v ij 2 ) , . . . , ( 1 − bijm,vijm )} ,∀i,j ∈{1, 2, . . . ,n} (3)bijk ∈ [0, 1],∀k ∈{1, 2, . . . ,m} Viewed from another perspective, the fuzzy preference is a special case of D matrix. When m = 1,vij1 = 1,∀i,j ∈{1, 2, . . . ,n}, b ij 1 is the same thing as rij. Situation 1 and Situation 2 can be expressed in terms of D numbers as follows. Situation 1: RD1 = A1 A2 A1 A2[ {(0.5, 1.0)} {(0.7, 0.8) , (0.6, 0.2)} {(0.3, 0.8) , (0.4, 0.2)} {(0.5, 1.0)} ] (11) Situation 2: RD2 = A1 A2 A1 A2[ {(0.5, 1.0)} {(0.8, 0.6)} {(0.2, 0.6)} {(0.5, 1.0)} ] (12) 3.2.2 Priority weights of alternatives based on D matrix In the previous section we defined the D matrix and explained the advantages of the D matrix. The generation of priority weight is the key step of the MADM and we can calculate priority weight by fuzzy preference relation. But how to get it by matrix D. The proposed method will discuss the problem in two cases. Case 1: D matrix with complete information Assume all the experts participate in the evalua- tion, and we get a D matrix with complete information on all the elements. That is to say, in Equation (10) vij1 + v ij 2 + . . . + vijm = 1,∀i,j ∈ {1, 2, . . . ,n}. For example, there is a D matrix with complete https://doi.org/10.15837/ijccc.2022.5.4468 8 information. (see Equation (13)). Take the following steps to get the priority weights. RD = A1 A2 A3 A1 A2 A3  {(0.5, 1.0)} {(0.2, 1.0)} {(0.7, 0.8) , (0.8, 0.2)}{(0.8, 1.0)} {(0.5, 1.0)} {(0.9, 1.0)} {(0.3, 0.8) , (0.2, 0.2)} {(0.1, 1.0)} {(0.5, 1.0)}   (13) Step1. Use the integration representation of D numbers (see Definition 3) to calculate the crisp matrix. For example, we can calculate D13 = {(0.7, 0.8), (0.8, 0.2)} which is the element in Equation (13), according to Equation (5). I (D13) = 0.7 × 0.8 + 0.8 × 0.2 = 0.72 And then, we can get the crisp matrix Rc. Rc = I (RD) = A1 A2 A3 A1 A2 A3   0.5000 0.2000 0.72000.8000 0.5000 0.9000 0.2800 0.1000 0.5000   (14) Step 2. Convert a crisp matrix to a judgement matrix. For instance, we can calculate the element Rc(1, 3) = 0.72 of the crisp matrix Equation (14), according to Equation (6). Rtr(1, 3)=9−1+2×0.72 = 2.6295 Consequently, a judgement matrix Rtr is derived as follow. Rtr = A1 A2 A3 A1 A2 A3   1.0000 0.2676 2.62953.7372 1.0000 5.7995 0.3803 0.1724 1.0000   (15) Step 3. Calculate the maximum eigenvector and eigenvalue of the judgement matrix. The eigenvalues and eigenvectors can express the information of the matrix to the greatest extent. According to the above steps, the judgement matrix we get is similar to the paired comparison matrix in AHP, both of which are multiplicative preference matrices. The elements in the judgement matrix are on an interval of [1/9, 9] , which is same as the pairwise comparison matrix in AHP method. So we use the same treatment as AHP for the judgement matrix. According to formula Rtrx = λmaxx and after normalization, we can get the final weights of alternatives A1, A2 and A3 in D matrix shown in Equation (13). ω1 : ω2 : ω3 = 0.2180 : 0.6832 : 0.0988 The ranking of alternatives is A2 � A1 � A3 (The ’�’ sign means ’better than’). Similarly, consistency indicators CR can be obtained. In this example, CR = 0.0155 < 0.1, so the consistency of the D matrix in Equation (13) is acceptable. Case 2: D matrix with incomplete information Assume part of experts participate in the evaluation, and we get a D matrix with incomplete information on some of the elements. That is to say, in Equation (10), vij1 + v ij 2 + . . . + vijm < 1,∀i,j ∈ {1, 2, . . . ,n}. For example, there is a D matrix with incomplete information. (see Equation (16)). Take the following steps to get the priority weights. RD = A1 A2 A3 A1 A2 A3  {(0.5, 1.0)} {(0.2, 1.0)} {(0.7, 0.8)}{(0.8, 1.0)} {(0.5, 1.0)} {(0.9, 1.0)} {(0.3, 0.8)} {(0.1, 1.0)} {(0.5, 1.0)}   (16) https://doi.org/10.15837/ijccc.2022.5.4468 9 Step1. In this case, elements with complete information and incomplete information is processed separately. (1) Dij in D matrix with complete information If m∑ k=1 v ij k = 1 , then the integration representation of D numbers is used to calculate the row i and column j element in the crisp matrix. (2) Dij in D matrix with incomplete information If m∑ k=1 v ij k < 1 , a new processing method is proposed as follow. I ′ (Dij ) = n∑ m=1 bmvm + 0.5 × (1 − n∑ m=1 vm) (17) where I′ (Dij ) is the row i and column j element in the crisp matrix. When the pros and cons of two events are not clear, people are more likely to assume that they are similar. In fuzzy preference relation, 0.5 is used to express this situation when alternatives are assumed to be equal with respect to a certain criterion. Thus, this method applies 0.5 to represent the preference assignment under the condition of incomplete information. For example, in Equation (16), D13 = {(0.7, 0.8)} and D31 = {(0.3, 0.8)} are elements with incom- plete information. According to Equation (17) I ′ (D13) = 0.7 × 0.8 + 0.5 × (1 − 0.8) = 0.66 I ′ (D31) = 0.3 × 0.8 + 0.5 × (1 − 0.8) = 0.34 After dealing with the above two cases, then we can get the results of the crisp matrix Rc. Rc = I (RD) = A1 A2 A3 A1 A2 A3   0.5000 0.2000 0.66000.8000 0.5000 0.9000 0.3400 0.1000 0.5000   (18) Step 2. Convert a crisp matrix to a judgement matrix. Just like Case 1, for instance, we can calculate the element Rc13 = 0.66 of the crisp matrix Equation (18), according to Equation (6). Rtr13=9−1+2×0.66 = 2.0200 Consequently, a judgement matrix Rtr is derived as follow. Rtr = A1 A2 A3 A1 A2 A3   1.0000 0.2676 2.02003.7372 1.0000 5.7995 0.4950 0.1724 1.0000   (19) Step 3. Calculate the maximum eigenvector and eigenvalue of the judgement matrix. The treatment is the same as Case 1. According to formula Rtrx = λmaxx and after normalization, we can get the final weights of alternatives A1, A2 and A3 in D matrix shown in Equation (16). ω1 : ω2 : ω3 = 0.2015 : 0.6896 : 0.1089 The ranking of alternatives is A2 � A1 � A3 (The ’�’ sign means ’better than’). Similarly, consistency indicators CR can be obtained. As CR = 0.0038 < 0.1, the consistency of the D matrix of Equation (16) is acceptable. https://doi.org/10.15837/ijccc.2022.5.4468 10 3.2.3 Summary of Case 1 and Case 2 Case 1 and Case 2 represent the cases of D matrix with complete information and D matrix with incomplete information, respectively. Here we express the pairwise comparison opinions of experts in the form of a D matrix, that is, our input is a D matrix. It can be concluded from the above that D-matrix is an extension of fuzzy preference relation, which can deal with uncertain and incomplete information more effectively. The process can be concluded as follows (shown in Figure 1). Firstly, convert D matrix to a crisp matrix according to two different situations: when the information is complete, the integration representation of D numbers is directly used; when the information is in- complete, a new method is proposed. Secondly, convert the crisp matrix to the judgement matrix based on the transformation functions to get a good preparation for using AHP. Finally, AHP is used to deal with the judgement matrix. The eigenvector corresponding to the largest eigenvalue of the judgement matrix is calculated to get the weights. The CR is also used to measure the consistency. 4 Numerical examples In this part, three examples are illustrated to show effectiveness and superiority of the proposed method compared with the Deng et al.’s method [39]. Assume that the following three examples are expert opinions in different situations. Here we directly use the D matrix to express this information. Deng et.al’s method and the proposed method are applied to get the priority weights of alternatives, respectively. 4.1 Example 1 RD = A1 A2 A3 A1 A2 A3  {(0.5000, 1)} {(0.4990, 1)} {(0.5010, 1)}{(0.5010, 1)} {(0.5000, 1)} {(0.4990, 1)} {(0.4990, 1)} {(0.5010, 1)} {(0.5000, 1)}   (20) In the first example, the elements of the D matrix are complete information. The pairwise com- parison values are all close to 0.5, indicating that A1,A2, and A3 are approximately equally important. 4.1.1 Deng et al.’s method Firstly, dealing with the D matrix by the integration representation of D numbers. Then, the crisp matrix Rc is obtained in Equation (21). Rc = I (RD) = A1 A2 A3 A1 A2 A3   0.5000 0.4990 0.50100.5010 0.5000 0.4990 0.4990 0.5010 0.5000   (21) Secondly, construct the probability matrix. The elements in the D numbers are all information com- plete. So there are only two choices of 0 or 1 of the elements in the probability matrix Rp . The probability matrix can be obtained as follow in Equation (22). Rp = A1 A2 A3 A1 A2 A3   0.0000 0.0000 1.00001.0000 0.0000 0.0000 0.0000 1.0000 0.0000   (22) Finally, the triangularization method is used to rank the alternatives. However, based on the proba- bility matrix the sum of each row is equal, the subsequent steps cannot be carried out. https://doi.org/10.15837/ijccc.2022.5.4468 11 4.1.2 The proposed method Step 1 The information of the D matrix is all complete. So the process just like the Deng’s method. The crisp matrix is same as the Equation (21). Step 2 According to Equation (6), the judgement matrix can be obtained. For instance, Rtr13=9−1+2×I(D13) = 0.9956. The judgement matrix is as follow in Equation (23). Rtr = A1 A2 A3 A1 A2 A3   1.0000 0.9956 1.00441.0044 1.0000 0.9956 0.9956 1.0044 1.0000   (23) Step 3 According to the Rtrx = λx, the largest eigenvalue of the judgement matrix is calculated. Normalized the eigenvector of the largest eigenvalue and then we can get the final weights. The final weights of the three alternatives are 0.3333 0.3333 0.3333 and CR < 0.1. 4.1.3 Discussion of Example 1 In this example, the alternatives are quite similar, and it can be seen that the D number matrix has good consistency. The correct result can be obtained by the proposed method, and CR is less than 0.1. However, Deng et al.’s method cannot be applied since the sum of each row in the probability matrix Rp is equal, so that the following interval cannot be processed, and the consistency index cannot be obtained. 4.2 Example 2 RD = A1 A2 A3 A1 A2 A3  {(0.5, 1.0)} {(0.1, 0.5)} {(0.0, 0.4)}{(0.9, 0.5)} {(0.5, 1.0)} {(0.4, 0.2)} {(1.0, 0.4)} {(0.6, 0.2)} {(0.5, 1.0)}   (24) In the second example, from the first column of the D-matrix, the pairwise comparisons of A2 and A1, A3 and A1, are both greater than 0.5. And the pairwise comparison of A3 and A1 is larger than the pairwise comparison of A2 and A1. This shows that A1 is the worst, A3 and A2 are both superior to A1, and A3 is superior to A1 to a greater extent than A2. Meanwhile, the value of the pairwise comparison between A3 and A2 is greater than 0.5, and the value of the pairwise comparison between A2 and A3 is less than 0.5, which further indicates that A3 is superior to A2. So given the D matrix, the order from best to worst should be A3, A2, A1. 4.2.1 Deng et al.’s method Firstly, same as the first step of Example 1. The Equation (5) is used to obtain the crisp matrix Rc as follows. Rc = I (RD) = A1 A2 A3 A1 A2 A3   0.5000 0.0050 1.00000.4500 0.5000 0.0800 0.4000 0.1200 0.5000   (25) Secondly, in this D matrix, some elements are information complete and other elements are information incomplete. The probability matrix(Rp) can be obtained in Equation (26). Rp = A1 A2 A3 A1 A2 A3   0.0000 0.1000 0.17000.9000 0.0000 0.4750 0.8300 0.5250 0.0000   (26) Finally, the sum of the first row of the Rp matrix is 0.27, the sum of the second row is 1.375, and the sum of the third row is 1.355, so A2 is ranked first, A3 is second, and A1 is third. https://doi.org/10.15837/ijccc.2022.5.4468 12 4.2.2 The proposed method Step 1 Because of incomplete information, processing is divided into two categories. If the in- formation is complete, the operation shown as Equation (5) is implemented. If the information is incomplete, the Equation (17) is used. The crisp matrix is derived. Rc = I (RD) = A1 A2 A3 A1 A2 A3   0.5000 0.3000 0.30000.7000 0.5000 0.4800 0.7000 0.5200 0.5000   (27) Step 2 The operator of the crisp matrix is shown in Equation (6). We can get the judgement matrix as is shown in Equation (28). Rtr = A1 A2 A3 A1 A2 A3   1.0000 0.4152 0.41522.4082 1.0000 0.9159 2.4082 1.0918 1.0000   (28) Step 3 Same as the third step of the method proposed in this article in example 1. The weights can be obtained. The CR can be calculate by Equation (3). Finally, the weights of three alternatives A1, A2, A3 are: 0.1718 0.3989 0.4292, and CR = 0.0013 < 0.1. Sort the weight of three alternatives from the largest to the smallest, and the ranking from the best to the worst can be A3, A2, A1. 4.2.3 Discussion of Example 2 In this example, as discussed in the first paragraph in Section 4.2, the reasonable ranking of the alternatives should be A3 � A2 � A1. The weights of the alternatives by the proposed method are 0.1718 0.3989 0.4292, which is consistent with the conclusion made in Section 4.2. However, using Deng et al.’s method, the ranking of the alternatives is A2 � A3 � A1, which is inconsistent with the actual situation. The reason for this result is that Deng et al’s method directly uses the distance ratio to divide the reliability of the unknown part into the information we know from the beginning, which lacks solid physical foundation. 4.3 Example 3 RD = A1 A2 A3 A4 A1 A2 A3 A4  {(0.5, 1.0)} {(0.1, 1.0)} {(0.6, 0.8)} {(0.3, 0.6)} {(0.9, 1.0)} {(0.5, 1.0)} {(0.8, 1.0)} {(0.6, 1.0)} {(0.4, 0.8)} {(0.2, 1.0)} {(0.5, 1.0)} {(0.9, 1.0)} {(0.7, 0.6)} {(0.4, 1.0)} {(0.1, 1.0)} {(0.5, 1.0)}   (29) In the third example, the value of the pairwise comparison of A4 and A1 is greater than the value of the pairwise comparison of A3 and A1 and both are greater than 0.5, indicating that A4 is superior to A3. However, the pairwise comparison value of A3 and A4 is greater than 0.5, indicating that A3 is superior to A4. That is to say, D matrix has obvious inconsistency. 4.3.1 Deng et al.’s method The processing method is the same as the Deng et al.’s method in Example 2. We can get the crisp matrix and the probability matrix as are shown in Equation (30) and Equation (31). Further I.D. can be obtained, according to the following calculation. I.D. = 0.1 + 0.8 4 (4 − 1) /2 = 0.15 https://doi.org/10.15837/ijccc.2022.5.4468 13 Rc = A1 A2 A3 A4 A1 A2 A3 A4  0.5000 0.1000 0.4800 0.1800 0.9000 0.5000 0.8000 0.6000 0.3200 0.2000 0.5000 0.9000 0.4200 0.4000 0.1000 0.5000   (30) Rc = A1 A2 A3 A4 A1 A2 A3 A4  0.0000 0.0000 0.9000 0.2000 1.0000 0.0000 1.0000 1.0000 0.1000 0.0000 0.0000 1.0000 0.8000 0.0000 0.0000 0.0000   (31) 4.3.2 The proposed method The processing method is the same as the method proposed in this article in Example 2. We can get the crisp matrix and the judgement matrix in Equation (32) and Equation (33). We can get CR according to the Equation (3). Rc = A1 A2 A3 A4 A1 A2 A3 A4  0.5000 0.1000 0.5800 0.6000 0.9000 0.5000 0.8000 0.6000 0.4200 0.2000 0.5000 0.9000 0.4000 0.4000 0.1000 0.5000   (32) Rtr = A1 A2 A3 A4 A1 A2 A3 A4  1.0000 0.1724 1.4228 0.5902 5.7995 1.0000 3.7372 1.5519 0.7036 0.2676 1.0000 5.7995 1.6944 0.6444 0.1724 1.0000   (33) Finally, the weights of alternatives A1, A2, A3, A4 are: 0.1300 0.4746 0.2549 0.1406 and CR = 0.3952 > 0.1. 4.3.3 Discussion of Example 3 From the first column of the D number matrix given in Example 3, it can be seen that A3 is worse than A1 with a greater probability, and A4 is better than A1 with a greater probability. However, A3 can be obtained A4 is less than A3 with a particularly large probability, which is obviously in conflict. In other words, the elements (evaluations given by experts) of this D number matrix is inconsistent. Using the proposed method, CR is close to 0.4 and far greater than 0.1, which correctly illustrates this point. However, using Deng et al.’s method, the consistency index is 0.15 which shows that the D number matrix is consistent. This is obviously inappropriate. 4.4 Discussion and comparison Deng et al.’s method proposed an interesting method, i.e., the AHP method extended by D numbers preference relation, which can deal with the complex decision problem under uncertain environment [39]. However, some situations cannot be well handled in Deng et al.’s method. Three examples in this section shows the certain situations. From these examples, some conclusions can be made. (1) The proposed method can handle situations that cannot be handled by Deng et al.’s method, as is shown in example 1. (2) For the processing of incomplete information, this article considers that the incomplete information is because some of the information is unknown, and the unknown part should be expressed as the same importance of the two alternatives to be pairwise compared, that is, they cannot be distinguished. In the proposed method, 0.5 is multiplied by the degree of uncertainty. This method of expression is more in line with thinking habits and has a clearer physical meaning. Deng et https://doi.org/10.15837/ijccc.2022.5.4468 14 Global supplier selection Cost(C1) Qualitiy(C2) Service performance(C3) Supplier s profile(C4) Risk factor(C5) Supplier S1 Supplier S2 Supplier S3 A1 Level 1: Overall Objective (O) Level 2: Criteria (C) Level 3: Attributes (A) Level 4: Decision Alternatives (S) A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19 Figure 2: The hierarchical structure for the supplier selection al.’s method redistributes information according to distance which may not work out well under certain situations. Meanwhile, information is artificially added during the redistribution process. Example 2 shows this point. (3) In the evaluation of consistency of D matrix, the consistency ration CR in AHP can be directly adopted in the proposed method since a bridge (the transformation functions) is built between the crisp matrix of D matrix and the judgement matrix, which has a relatively sufficient theoretical basis. The consistency evaluation process in Deng et al.’s method may draw intuitive results. In example 3, the consistency indicator obtained is unreasonable and shows this point. (4)As can be seen in Figure 1, the proposed method is more simple and convenient to use. In summary, the proposed method has the merit of dealing with uncertain information, and can also improve the flexibility, efficiency and performance compared with Deng et al.’s method. 5 Case study In this section, the problem of selecting the best supplier [44] is adopted to illustrate the use and effectiveness of the proposed method. The hierarchical structure of the problem is shown in Figure 2 [44]. As shown in the figure, there are four levels. The overall objective is placed at the first level. The second level is the criteria. The third level is the attributes. The final level is the decision alternatives. The second level is consisted of five components. It contains the product’s cost(C1), product’s quality(C2), service performance (C3), supplier’s profile(C4) and factor of the risk(C5). Each item of the third level corresponds to each item of the second level. The define of attribute is as follow. A1: Product’s total price; A2: Prices of delivery; A3: Tariff and custom duties; A4: Rejection rate of the product; A5: Increased lead time; A6: Quality assessment; A7: Remedy for quality problems; A8: Delivery schedule; A9: Technological and support; A10: Response to changes; A11: Ease of com- munication; A12: Financial status; A13: Customer base; A14: Performance history; A15: Production facility and capacity; A16: Geographical location; A17: Political stability; A18: Political stability; A19: Terrorism. More details are given in [44]. For this problem, we can construct the D matrix at each level based on expert evaluation. For example, We can construct D matrix between the overall objective and criteria in Table.3 [39]. Ac- cording to the method proposed in the paper, we can get the priority weights of criteria. WDeng and WP roposed on the last two columns of Table 3 respectively represent the relative weights obtained by Deng et al.’s method and the relative weights obtained by the proposed method. It can be seen that the weights derived by Deng et al’s method and the proposed method are consistent. In a similar way, we can get the D matrix of the attributes corresponding to the criteria, and then calculate the priority weights of the attributes corresponding to the criteria as shown from column 2 to column 6 of Table 4. By integrating the weights of the criteria level and attributes level, we can get the contribution ratio and ranking of attributes relative to the overall objective. The results are shown on column 7 of Table 4, and you can see that the most important attribute is A1. It can be https://doi.org/10.15837/ijccc.2022.5.4468 15 0 0.05 0.1 0.15 0.2 0.25 0.3 A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19 W e ig h ts Attributes Deng et al.'s method Proposed method Figure 3: Compare Deng et al.’s method with the proposed method to produce the weights of attributes with respect to the overall objective Table 3: D numbers preference relation of criteria with respect to the overall objective. O C1 C2 C3 C4 C5 WDeng WP roposed C1 {(0.5,1.0)} {(0.6,1.0)} {(0.65,0.6)} {(0.85,1.0)} {(0.9,1.0)} 0.3650 0.3786 C2 {(0.4,1.0)} {(0.5,1.0)} {(0.8,1.0)} {(0.7,1.0)} {(0.7,1.0)} 0.3150 0.2929 C3 {(0.35,0.6)} {(0.2,1.0)} {(0.5,1.0)} {(0.65,1.0)} {(0.6,1.0)} 0.1650 0.1520 C4 {(0.15,1.0)} {(0.3,1.0)} {(0.35,1.0)} {(0.5,1.0)} {(0.55,1.0)} 0.0900 0.0917 C5 {(0.1,1.0)} {(0.3,1.0)} {(0.4,1.0)} {(0.45,1.0)} {(0.5,1.0)} 0.0650 0.0849 seen from Figure 3 that the weights of attributes with respect to the overall objective obtained by Deng et al.’s method and proposed method is not significantly different. In the same way, we can construct the D matrix of suppliers corresponding to attributes. From column 3 to column 5 of Table 5 shows the priority weight of suppliers with respect to each attribute. The final weight of suppliers with respect to the overall objective is obtained at the bottom of Table 5. Figure 4 shows the final result of using Deng et al.’s method and the proposed method to deal with the problem, it can be seen that the suppliers generated by the two methods have the same ranking: S1 � S3 � S2. The result is consistent with the expert’s final assessment. Furthermore, the results obtained by the method in this paper are more accurate, and the difference of priority weights between two suppliers is larger, which indicates that the division of different suppliers is more obvious. The reason for this may be concluded that the proposed method deal with the uncertain information more reasonably, especially with the incomplete information cases. 6 Conclusion In this study, a new supplier selection method based on D numbers and transformation function is proposed. Firstly, D numbers theory is used to express the uncertain information in preference relation and form a D matrix; Secondly, the D matrix is converted into a crisp matrix form according to different situations whether or not the information in D matrix is complete. Thirdly, the crisp matrix are converted into judgement matrix by using the transformation functions. Finally, AHP is used to select the best supplier. The proposed method has the following advantages: (1) Due to the characteristics of D numbers itself, the method in this paper is more advantageous in the expression of uncertain information. (2) Be able to deal with a variety of situations, including situations with large uncertainties in expert opinions or situations with similar decision-making objects. (3) The consistency evaluation index is consistent with the actual situation and has a good theoretical basis. In the future research, the integration rules of D numbers should be further improved to make full use of incomplete information rather than simple discount processing. In addition, the method in this https://doi.org/10.15837/ijccc.2022.5.4468 16 Table 4: The weights and ranking of attributes with respect to the overall objective based on proposed method. Criteria C1 C2 C3 C4 C5 Weight Ranking0.3786 0.2929 0.1520 0.0917 0.0849 Attributes A1 0.6330 0.2397 1 A2 0.2330 0.0882 3 A3 0.1340 0.0507 8 A4 0.4620 0.1353 2 A5 0.1870 0.0548 7 A6 0.2630 0.0770 4 A7 0.0880 0.0258 13 A8 0.4610 0.0701 5 A9 0.2010 0.0306 11 A10 0.2360 0.0359 10 A11 0.1020 0.0155 14 A12 0.6180 0.0567 6 A13 0.0970 0.0089 18 A14 0.1670 0.0153 15 A15 0.1180 0.0108 17 A16 0.1700 0.0144 16 A17 0.4400 0.0374 9 A18 0.3200 0.0272 12 A19 0.0700 0.0059 19 Table 5: The priority weights and ranking of suppliers with respect to the overall objective based on the proposed method. Attributes Suppliers Ai S1 S2 S3 A1 0.2397 0.4670 0.1660 0.3670 A2 0.1353 0.4270 0.1960 0.3770 A3 0.0882 0.6170 0.1670 0.2160 A4 0.0770 0.5340 0.2330 0.2330 A5 0.0701 0.5410 0.1180 0.3410 A6 0.0567 0.2830 0.5330 0.1840 A7 0.0548 0.5930 0.1640 0.2430 A8 0.0507 0.4170 0.1160 0.4670 A9 0.0374 0.6170 0.0660 0.3170 A10 0.0359 0.4330 0.4330 0.1340 A11 0.0306 0.6730 0.0140 0.3130 A12 0.0272 0.3330 0.3330 0.3340 A13 0.0258 0.6000 0.2000 0.2000 A14 0.0155 0.3170 0.5170 0.1660 A15 0.0153 0.3240 0.0230 0.6530 A16 0.0144 0.6000 0.3000 0.1000 A17 0.0108 0.4330 0.2340 0.3330 A18 0.0089 0.2930 0.5130 0.1940 A19 0.0059 0.5870 0.1060 0.3070 Proposed method 0.4858 0.2054 0.3089 Supplier’s ranking 1 3 2 https://doi.org/10.15837/ijccc.2022.5.4468 17 0.0 0.1 0.2 0.3 0.4 0.5 S1 S2 S3 P ri o ri ty w e ig h t Supplier Proposed method Deng et al.'s method Figure 4: Compare Deng et al.’s method with the proposed method to produce the final priority weights paper did not consider the interaction between indicators, so ANP and other methods were considered for improvement. 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This is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial 4.0 International License. Journal’s webpage: http://univagora.ro/jour/index.php/ijccc/ This journal is a member of, and subscribes to the principles of, the Committee on Publication Ethics (COPE). https://publicationethics.org/members/international-journal-computers-communications-and-control Cite this paper as: Xiong, L.; Su, X.; Qian, H. (2022). Supplier Selection Model Based on D Numbers and Trans- formation Function, International Journal of Computers Communications & Control, 17(5), 4468, 2022. https://doi.org/10.15837/ijccc.2022.5.4468 Introduction Preliminaries Analytic hierarchy process (AHP) saaty1982analytic D numbers Deng2014 Transformation functions herrera2004some Proposed method Introduction of Deng et al.'s method Deng2014 An improved D-AHP method D matrix Priority weights of alternatives based on D matrix Summary of Case 1 and Case 2 Numerical examples Example 1 Deng et al.'s method The proposed method Discussion of Example 1 Example 2 Deng et al.'s method The proposed method Discussion of Example 2 Example 3 Deng et al.'s method The proposed method Discussion of Example 3 Discussion and comparison Case study Conclusion