(PDF) A multiple attribute group decision making method based on two novel intuitionistic multiplicative distance measures

JID: INS ARTICLE IN PRESS [m3Gsc;June 11, 2018;10:4] Information Sciences 0 0 0 (2018) 1–18 Contents lists available at ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins A multiple attribute group decision making method based on two novel intuitionistic multiplicative distance measures Huchang Liao, Cheng Zhang∗, Li Luo Business School, Sichuan University, Chengdu 610064, China a r t i c l e i n f o a b s t r a c t Article history: Distance measure is used to measure the deviation degree between different arguments. Received 10 September 2017 Since the existing distance measures between intuitionistic multiplicative sets cannot rep- Revised 4 May 2018 resent the angle of two intuitionistic multiplicative sets, this paper introduces some new Accepted 7 May 2018 distance measures between intuitionistic multiplicative sets, which include the projection- Available online xxx based distance measure and the psychological distance measures. After that, a multiple Keywords: attribute group decision making method is proposed to handle the problem in which the Intuitionistic multiplicative set weights of experts and attributes are unknown or partially known. A case study concern- Distance measure ing the drug supplier selection in hospital management is provided to demonstrate the Psychological distance measure calculation process of the proposed method. Finally, some comparative analyses are given Projection-based distance measure to illustrate the efficiency and applicability of the proposed method. Multiple attribute group decision making © 2018 Elsevier Inc. All rights reserved. Intuitionistic multiplicative entropy 1. Introduction In multiple attribute decision making (MADM) problems, a finite set of alternatives are evaluated by decision-makers (DMs) over different attributes. However, ambiguity always exists due to the vague knowledge of DMs when assessing the alternatives [19]. Atanassov [3] proposed the intuitionistic fuzzy set (IFS) as a generalization of fuzzy set, which is character- ized by a membership degree, a non-membership degree and a hesitant degree. IFS is based on the traditional symmetric 0–1 scale that the grades are uniformly distributed around the middle number 0.5 [30]. However, in many cases, the grades are not symmetrically and uniformly distributed. This can be verified by the law of diminishing marginal utility in eco- nomics [29]. In this sense, the 0–1 scale is no longer applicable. Based on the 1/9–9 scale, Xia et al. [28,29] introduced the intuitionistic multiplicative set (IMS), in which the membership degree, non-membership degree and hesitant degree are asymmetrically distributed with respect to the 1/9–9 scale. The IMS has been investigated by many scholars over the past several years [11,12,18,21,28,34, 38–40]. Distance measure is not only the basic tool but also a significant component of some MADM methods [15]. It has been widely used in measuring the deviation degree between different objects [7]. Up to now, many scholars have paid atten- tion to this issue and obtained many results [11,12,17,18]. To describe the similarity degree under the intuitionistic mul- tiplicative environment, various distance measures have been put forward, such as the Manhattan distance measure, the weighted Manhattan distance measure [11], the Minkowski distance measure, the normalized Minkowski distance measure, the weighted Minkowski distance measure, the Euclidean distance measure and the weighted Euclidean distance measure ∗ Corresponding author. E-mail address:

(C. Zhang). https://doi.org/10.1016/j.ins.2018.05.023 0020-0255/© 2018 Elsevier Inc. All rights reserved. Please cite this article as: H. Liao et al., A multiple attribute group decision making method based on two novel intuition- istic multiplicative distance measures, Information Sciences (2018), https://doi.org/10.1016/j.ins.2018.05.023 JID: INS ARTICLE IN PRESS [m3Gsc;June 11, 2018;10:4] 2 H. Liao et al. / Information Sciences 000 (2018) 1–18 [12]. Recently, Liao et al. [18] proposed the Hausdorff distance measures, the hybrid distance measures, the weighted dis- tance measures, the ordered weighted distance measures, and the weighted distance measures in continuous forms for IMSs. Some interesting conclusions could be drawn by analyzing the previous literature. On the one hand, although the existing distance measures can measure the deviation degree between different IMSs, the included angle of these IMSs are not considered, and thus these distance measures could not fully depict the difference between two IMSs. It is observed that the projection-based distance measure can depict the degree that one object is close to another more exactly than traditional distance measures. The projection-based distance measure has been used in determining the weights of experts or attributes [31,35,36] and improving the grey relation method [37], and has been extended into intuitionistic trapezoidal fuzzy sets [37], simplified neutrosophic sets [33], multi-valued neutrosophic sets [10], and hesitant fuzzy linguistic term sets [26]. This paper intends to extend the projection-based distance measure into intuitionistic multiplicative environment and then use it to measure the deviation degree between different alternatives. On the other hand, DMs always have personal preferences on alternatives, and strong competition may exist among the alternatives with small differences. Kahneman [13] demonstrated that people prefer to minimize loss rather than maximize profit. In this sense, people are more sensitive to loss than gain. Thus, it is important to reflect the views of DMs from the point of psychology. The existing distance measures do not consider the preferential and competitive relationships between alternatives. The preference relationship between alternatives means the dominant or dominated status of alternatives [23]. Huber et al. [9] introduced the dominance vector and the indifference vector, and then used the distances along these vectors to represent the preferential relationship and the competitive relationship between alternatives, respectively. Con- sidering the subjective preferences on alternatives, Wedell [25] proposed a dimensional weighting model to illustrate the preferential relationship in distance measures. Berkowitsch et al. [4] proposed a generalized distance function which con- siders the differences between attributes and the preference relationships between alternatives. All these aforementioned methods were applied to solve the MADM problems with accurate information, but cannot measure the distance between IMSs. Motivated by the generalized distance function in Ref. [4], this paper proposes a psychological distance measure be- tween IMSs, which takes into account the psychological cognitions of DMs towards alternatives in terms of the preferential relationship and the competitive relationship between them. Combined with the projection-based distance measure and the psychological distance measure, in this paper, a new MAGDM method is introduced on the basis of the intuitionistic multiplicative entropy. It provides a new way to solve the MAGDM problems in which the weights of experts and attributes are unknown or partially known. Then, a case study concerning the drug supplier selection in hospital management is given to illustrate the effectiveness and practicability of the new approach. The contributions of this paper can be highlighted as follows: (1) The projection-based distance measure and the psychological distance measure are respectively proposed. The former can depict the deviation between IMSs more exactly than the traditional distance measures, while the latter can express imprecise or vague evaluation information more scientifically and effectively. In addition, the psychological distance mea- sure between IMSs can reduce the impact of personal preferences of DMs and the competitive relations between alter- natives on decision results. (2) Based on the proposed intuitionistic multiplicative entropy measure and the projection-based distance measure, a new method is introduced to determine the weights of attributes and experts for the MAGDM problems. (3) Based on the introduced distance measures, a method is proposed to solve the complex MAGDM problems in which the weights of experts and attributes are unknown or partially known. (4) The proposed intuitionistic multiplicative MAGDM method is applied to handle the problem of drug supplier selection. The obtained results may provide some reference for the hospitals in China. The rest of this paper is organized as follows: Section 2 briefly recalls the preliminaries related to IMSs. Section 3 pro- poses the projection measure, the normalized projection measure and the projection-based distance measures of IMSs, and then introduces a new distance measure called the psychological distance measure. In Section 4, the intuitionistic multi- plicative entropy is introduced and then the methods to determine the weights of experts and attributes are discussed, respectively. A new MAGDM method is proposed based on the above distance measures in Section 5. In Section 6, a prac- tical example concerning the evaluation of drug suppliers is given to describe the applicability and effectiveness of the proposed approach. The paper ends in Section 7. 2. Preliminaries For the simplicity of forthcoming presentation, this section reviews the concept, comparison method and aggregation operator of IMSs. The existing distance measures of IMSs and the strict intuitionistic fuzzy entropy are also reviewed. Xia et al. [29] introduced the concept of IMS as follows: Definition 1 [29]. Let X be fixed. An IMS in X is defined as: D = {< x, (ρD (x ), σD (x )) > | x ∈ X } (1) which assigns to each element x a membership degree ρD (x ) and a non-membership degree σD (x ), with the conditions: 1/9 ≤ ρD (x ), σD (x ) ≤ 9, ρD (x )σD (x ) ≤ 1, ∀x ∈ X. Please cite this article as: H. Liao et al., A multiple attribute group decision making method based on two novel intuition- istic multiplicative distance measures, Information Sciences (2018), https://doi.org/10.1016/j.ins.2018.05.023 JID: INS ARTICLE IN PRESS [m3Gsc;June 11, 2018;10:4] H. Liao et al. / Information Sciences 000 (2018) 1–18 3 For convenience, the pair (ρD (x ), σD (x )) is called an intuitionistic multiplicative number (IMN) and M is the set of all IMNs. For each IMN in X, τD (x ) = 1/(ρD (x )σD (x )) is interpreted as the uncertain or hesitant degree. Obviously, 1 ≤ τD (x ) ≤ 92 , ∀x ∈ X. An IMN can also be represented as (ρD (x ), σD (x ), τD (x )). In the following of this paper, we omit the third part τD (x ) for the simplicity of presentation. Definition 2 [29]. For an IMN α = (ρα , σα ), s(α ) = ρα /σ α is called the score function of α , and h(α ) = ρα σα is called the accuracy function of α . For two IMNs α1 and α2 , (1) If s(α1 ) > s(α2 ), then α1 > α2 ; (2) If s(α1 ) = s(α2 ), then (i) If h(α1 ) > h(α2 ), then α1 > α2 ; (ii) If h(α1 ) = h(α2 ), then α1 = α2 . Definition 3 [29]. Let α1 and α2 be two IMNs. The partial order is denoted as α1 ≥ p α2 if and only if ρα1 ≥ ρα2 and σα1 ≤ σα2 . Especially, α1 = α2 if and only if ρα1 = ρα2 and σα1 = σα1 . The top and bottom elements in M are 9 p = (9, 1/9) and 1/9 p = (1/9, 9), respectively. Definition 4 [28]. Let αi (i = 1, 2, ..., n) be a collection of IMNs, and w = (w1 , w2 , ..., wn )T be their weight vector with wi ∈ [0, 1] and ni=1 wi = 1. Then the intuitionistic multiplicative weighted averaging (IMWA) operator is defined as: ⎛ n n ⎞ ( 1 + 2 ρi ) i − 1 2 σi w i w n ⎜ i=1 i=1 ⎟ IMW Aw (α1 , α2 , ..., αn ) = (wi αi ) = ⎝ , n n ⎠ (2) 2 ( 2 + σi ) − σi i=1 wi wi i=1 i=1 Jiang et al. [12] defined the Minkowski distance measure shown as Eq. (3) and the weighted Minkowski distance measure shown as Eq. (4) between the IMSs A and B: λ λ λ 1/λ 1 n ρ ( x ) σ ( x ) τ ( x ) dgd (A, B ) = log9 A i + log9 A i + log9 A i , λ≥1 (3) 2λ+1 ρB ( x i ) σB ( x i ) τB ( x i ) i=1 λ λ λ 1 /λ 1 n ρ ( x ) σ ( x ) τ ( x ) wi log9 A i + log A i + log A i dgnd (A, B ) = , λ≥1 (4) 2λ+1 i=1 ρB ( x i ) 9 σ B ( x i ) 9 τ B ( x i ) Definition 5 [8]. Let X = {x1 , x2 , ..., xn } be fixed, and an IFS F in X be defined as F = {< x, (μ(x ), ν (x )) > | x ∈ X }. Then the strict intuitionistic fuzzy entropy of F is defined as: 1 n E (F ) = (δ (xi ) − σ δ (xi ) · (1 − 2 min (μ(xi ), ν (xi ) )) ) (5) n i=1 1−|μ(xi )−ν (xi )| where δ (xi ) = 1+|μ(xi )−ν (xi )| with the condition σ ∈ [0, 0.5]. 3. Some novel distance measures between IMSs This section proposes the projection-based distance measure and the psychological distance measure between IMSs. The former can broaden the application of the projection-based distance measure from real numbers to IMNs, while the latter can reflect the preferences of DMs more objectively and accurately than the existing distance measures between IMSs. 3.1. The projection-based distance measures between IMSs The projection-based distance measure between IMSs can measure the distance between two arguments more precisely than the existing distance measures by including the angle between these arguments. Definition 6. For an IMN α = (ρα , σα ) with 1/9 ≤ ρα ,σα ≤ 9, 0 < ρα σα ≤ 1, we define the module of α as 1 |α| = (1 + log9 ρα )2 + (1 + log9 σα )2 + (log9 τα )2 (6) 2 The inner product of two IMNs α1 and α2 is defined as: 1 α1 · α2 = [(1 + log9 ρα1 )(1 + log9 ρα2 ) + (1 + log9 σα1 )(1 + log9 σα2 ) + (log9 τα1 )(log9 τα2 )] (7) 4 Then, the included angle cosine of two IMNs α1 and α2 can be defined as: α1 · α2 Cos(α1 , α2 ) = |α1 ||α2 | Please cite this article as: H. Liao et al., A multiple attribute group decision making method based on two novel intuition- istic multiplicative distance measures, Information Sciences (2018), https://doi.org/10.1016/j.ins.2018.05.023 JID: INS ARTICLE IN PRESS [m3Gsc;June 11, 2018;10:4] 4 H. Liao et al. / Information Sciences 000 (2018) 1–18 [(1 + log9 ρα1 )(1 + log9 ρα2 ) + (1 + log9 σα1 )(1 + log9 σα2 ) + (log9 τα1 )(log9 τα2 )] = (8) (1 + log9 ρα1 )2 + (1 + log9 σα1 )2 + (log9 τα1 )2 · (1 + log9 ρα2 )2 + (1 + log9 σα2 )2 + (log9 τα2 )2 Definition 7. Let α1 = (ρα1 , σα1 ) and α2 = (ρα2 , σα2 ) be two IMNs. Then, α1 · α2 (1 + log9 ρα1 )(1 + log9 ρα2 ) + (1 + log9 σα1 )(1 + log9 σα2 ) + (log9 τα1 )(log9 τα2 ) Projα2 α1 = = (9) |α2 | 2 (1 + log ρα ) + (1 + log σα ) + (log τα ) 2 2 2 9 2 9 2 9 2 is called the projection of α1 on α2 . The larger the value of Projα2 α1 is, the higher the degree to which α1 approaches to α2 should be. Definition 8. Let A = (αi j )m×n and B = (βi j )m×n be two intuitionistic multiplicative matrices (IMMs), where αi j = (ρi j α , σi j α ) and βi j = (ρi j β , σi j β ) are IMNs. Then the inner product of A and B is defined as: 1 m n A·B= 1 + log9 ρi j α 1 + log9 ρi j β + 1 + log9 σi j α 1 + log9 σi j β + log9 τi j α log9 τi j β (10) 4 i=1 j=1 The modules of A and B are given, respectively, as: m n 2 2 2 |A| = 1 2 1 + log9 ρi j α + 1 + log9 σi j α + log9 τi j α i=1 j=1 n m 2 2 2 (11) |B| = 1 2 1 + log9 ρi j β + 1 + log9 σi j β + log9 τi j β i=1 j=1 The included angle cosine of the IMMs A and B is defined as: A·B Cos(A, B ) = |A||B| n m 1 + log9 ρi j α 1 + log9 ρi j β + 1 + log9 σi j α 1 + log9 σi j β + log9 τi j α log9 τi j β i=1 j=1 = m m n 2 2 2 n 2 2 2 1 + log9 ρi j α + 1 + log9 σi j α + log9 τi j α · 1 + log9 ρi j β + 1 + log9 σi j β + log9 τi j β i=1 j=1 i=1 j=1 (12) Property 1. The included angle cosine of A and B satisfies: (1) 0 ≤ Cos(A, B ) ≤ 1; (2) Cos(A, B ) = 1 if and only if A = B; (3) Cos(A, B ) = Cos(B, A ). Proof. (1) By Definition 8, we know that |A| = 0, |B| = 0 and A · B ≥ 0. Therefore, it is true that Cos(A, B ) ≥ 0. By the Cauchy- Schwarz inequality (m1 n1 + m1 n1 + · · · + mn nn )2 ≤ (m1 2 + m2 2 + · · · + mn 2 )(n1 2 + n2 2 + · · · + nn 2 ), 0 ≤ A · B ≤ |A||B| can be obtained. That is to say, Cos(A, B ) ≤ |AA||·BB| ≤ 1. Hence, 0 ≤ Cos(A, B ) ≤ 1. It is apparent that (2)-(3) in Property 1 are also true. The proof is omitted here. Definition 9. The projection of the IMM A over B can be defined as: n m 1 + log9 ρi j α 1 + log9 ρi j β + 1 + log9 σi j α 1 + log9 σi j β + log9 τi j α log9 τi j β A·B 1 i=1 j=1 ProjB A = = · m |B| 2 n 2 2 2 1 + log9 ρi j β + 1 + log9 σi j β + log9 τi j β i=1 j=1 (13) It should be noted that ProjB A = ProjA B. The greater the value of ProjB A is, the higher the degree to which the IMM A approaches to the IMM B should be. That is to say, the projection can be used to measure the deviation degree between two IMMs. Property 2. Let A, B, and C be three IMMs. Then, the projection measure satisfies: (1) 0 ≤ ProjB A ≤ |A|; (2) If A = B, then Pr ojB A = Pr ojA B = |A| = |B|; (3) If Pr ojC A ≤ Pr ojC B, then B is closer to C than A. Proof. Please cite this article as: H. Liao et al., A multiple attribute group decision making method based on two novel intuition- istic multiplicative distance measures, Information Sciences (2018), https://doi.org/10.1016/j.ins.2018.05.023 JID: INS ARTICLE IN PRESS [m3Gsc;June 11, 2018;10:4] H. Liao et al. / Information Sciences 000 (2018) 1–18 5 (1) By Property 1, |A| > 0, |B| > 0 and A · B ≥ 0. Since A · B = |A||B|Cos(A, B ), Pr ojB A = |A||B|Cos |B| (A,B ) = |A|Cos(A, B ). Therefore, 0 ≤ ProjB A ≤ |A|. (2) By Property 1, Cos(A, B ) = Cos(B, A ) = 1 when A = B. Moreover, |A| = |B|. Thus, Pr ojB A =Pr ojB A =|A| · Cos(A, B ) = |B| · Cos(B, A ) = Pr ojA B = |A| = |B|. (3) It is apparent that the projection measure satisfies (3). The proof is omitted here. Inspired by Yue and Jia [36], the normalized projection of one IMM over the other can be defined as follows: Definition 10. Let A = (αi j )m×n and B = (βi j )m×n be two IMMs, where αi j = (ρi j α , σi j α ) and βi j = (ρi j β , σi j β ). Then, the normalized projection of A over B is defined as: ProjB A/|B| ProjB A NProjB A = = (14) ProjB A/|B| + |1 − ProjB A/|B|| ProjB A + ||B| − ProjB A| The projection-based distance measure between two IMMs A and B can be introduced to depict the deviation degree between two IMMs: dProj (A, B ) = NProjS∗ A − NProjS∗ B (15) where S∗ is any given IMM satisfying = (si j )m×n with si j being an IMN. S∗ Combining Eqs. (14) and (15), the projection-based distance measure between two IMMs A and B can also be represented as: ProjS∗ A ProjS∗ B dProj (A, B ) = − (16) ProjS∗ A + ||S∗ | − ProjS∗ A| ProjS∗ B + ||S∗ | − ProjS∗ B| Based on Property 2, the normalized projection measure and the projection-based distance measure satisfy the following properties: Property 3. Let A, B and C be three IMMs. Then, we have (1) 0 ≤ NProjB A ≤ 1; (2) If A = B, then NProjB A = 1; (3) If NProjC A ≤ NProjC B, then B is closer to C than A. Proof. ≥ 0. Furthermore, 0 ≤ ProjB A ≤ ProjB A + ||B| − ProjB A|. It ProjB A (1) By Property 2, ProjB A ≥ 0. Therefore, NProjB A = ProjB A+||B|−ProjB A| ProjB A is true that NProjB A = ProjB A+||B|−ProjB A| ≤ 1. Hence, 0 ≤ NProjB A ≤ 1. (2) When A = B, Pr ojB A = Pr ojA B = |A| = |B|. Thus, NProjB A = 1. (3) It is apparent that (3) holds and thus the proof is omitted here. Property 4. The projection-based distance between two IMMs A and B satisfies: (1) −1 ≤ dProj (A, B ) ≤ 1; (2) If A = B, then dProj (A, B ) = 0; (3) dProj (A, B ) = −dProj (B, A ). Proof. (1) By Property 3, 0 ≤ NProjS∗ A ≤ 1 and 0 ≤ NProjS∗ B ≤ 1. Thus, −1 ≤ dProj (A, B ) ≤ 1. (2) By Property 3, NProjS∗ A = NProjS∗ B when A = B. Thus, dProj (A, B ) = NProjS∗ A−NProjS∗ B = 0. (3) According to Eq. (15), dProj (A, B ) = NProjS∗ A − NProjS∗ B = −(NProjS∗ B − NProjS∗ A ) = −dProj (A, B ). This completes the proof of Property 4. 3.2. The psychological distance measures between IMNs/IMSs In this section, the psychological distance measures between IMNs and that between IMSs are proposed, respectively. 3.2.1. The psychological distance measure between IMNs In MADM process, DMs prefer to evaluate alternatives in pair and then make a choice according to the overall pair- wise evaluations, rather than evaluate each alternative independently. The DMs need to calculate the correlative distances between alternatives first, based on which, the pairwise perceived similarities between alternatives can be derived. The greater the perceived similarity is, the more similar the two alternatives are. In this sense, the evaluation result of one alter- native will affect that of the other. The perceived similarity measure is a decreasing function with respect to the correlative distance measure. In the three dimension coordinate space shown as Fig. 1, let the points A, B and C represent three options expressed in IMNs. In analogous to the MADM process, we can take the membership degree ρ , the non-membership degree σ and the Please cite this article as: H. Liao et al., A multiple attribute group decision making method based on two novel intuition- istic multiplicative distance measures, Information Sciences (2018), https://doi.org/10.1016/j.ins.2018.05.023 JID: INS ARTICLE IN PRESS [m3Gsc;June 11, 2018;10:4] 6 H. Liao et al. / Information Sciences 000 (2018) 1–18 Fig. 1. The Euclidean distance (a) and the psychological distance (b) between IMNs A, B and C. hesitant degree τ as three evaluation attributes that are used to evaluate the options A, B and C. Comparing the IMNs A, B and C according to Definition 3, the membership degree of C is greater than that of A and the non-membership degree of C is greater than that of A as well, denoted as ρC > ρA , σC > σA . Thus, it is hard for DMs to determine which one is better. The bigger membership degree is preferred if we choose C. On the contrary, the smaller non-membership degree is preferred if we choose A. Both the IMNs A and C are acceptable. The loss in membership degree of C is compensated by the loss in non-membership degree of A. The decision result is determined by the subjective importance of membership degree and non-membership degree. Thus, we define AC as the indifference vector in this three dimension coordinate space. The indifference vector contains information about the exchange ratios between attributes, which indicate how many units of the other attribute should give up in order to obtain one unit of an attribute. In analogous, we have ρA > ρB , σB > σA and ρC > ρB , σB > σC , and then the partial orders A> p B and C > p B can be derived. Obviously, comparing to option B, DMs are more willing to choose option A and option C, because option B has not only the smallest membership degree but also the biggest non-membership degree. In other words, option B is dominated by option A or C. We define BC as the dominance vector in this three dimension coordinate space. It should be noted that the number of indifference vectors is determined by the number of attributes, but the number of dominance vectors is always one in sense that one option is either better or worse than the other option. Fig. 1(a) shows the Euclidean distance while Fig. 1(b) shows the psychological distance. Although the Euclidean distances AC and BC in Fig. 1(a) are the same, the distances AC and BC illustrated in Fig. 1(b) are different. Two reasons can be given to account for this phenomenon: one is that DMs give different importance for the membership degree and non-membership degree; the other is that there are preferential relations between options. Take the psychological distance between A and B as an example. First of all, two classes of distances along the indiffer- ence direction and the dominance direction, respectively, can be calculated, and we mark them as indifference line (dashed line in Fig. 1) and dominance line (continuous line in Fig. 1). To project the distance between two IMNs onto the indif- ference vector and the dominance vector, we introduce a basic transformation matrix to map the initial distance vector between IMNs to the direction axis of the indifference vector and the dominance vector. Multiplying the inverse of the ba- sic transformation matrix by the initial distance vector distinitial (A, B ), the transformed distance vector disttrans (A, B ) can be calculated. The first n − 1 entries of disttrans (A, B ) illustrate the distances in unit of each indifference vector, while the last entry of disttrans (A, B ) illustrates the distance in unit of the dominance vector. Huber et al. [9] pointed out that the distance in dominance direction should be given more attention than that along the indifference direction. That is to say, more weight should be given in this direction to express the preferential relational of DMs. Motivated by this, a diagonal matrix Aw , in which only the last entry of diagonal is assigned and the other values of diagonal are equal to 1, is introduced. The parameter wd (wd > 1), which means the weight along the dominance direction, makes the distance in the dominance direction obtain more weight but that in the indifference direction remain the same. If wd = 1, the weights given in these two directions are the same. Then the Euclidean distance is obtained and the dominance effect would not be considered. Let α = (ρα , σα ) and β = (ρβ , σβ ) be two IMNs. As mentioned above, the membership degree and non-membership degree should assign different weights. Let the weight vector of the three parts ρ , σ and τ be w = (wρ , wσ , wτ )T , where 0 ≤ wρ ≤ 1, 0 ≤ wσ ≤ 1, 0 ≤ wτ ≤ 1, wρ + wσ + wτ = 1. Based on the above analysis, the steps to calculate the psychological distance between IMNs can be described as follows and illustrated in Algorithm 1. Step 1. Take the parameters ρ , σ and τ of an IMN as three attributes. To avoid the distortions resulting from the different types and value ranges of attributes, a normalization process needs to be done by Eqs. (17) and (18). 1 1 ( τα − 1 ) 9 − 1 α = ( ρα , σ α , τ α ) = ρα , , + 9 (17) σα 9 81 − 1 Please cite this article as: H. Liao et al., A multiple attribute group decision making method based on two novel intuition- istic multiplicative distance measures, Information Sciences (2018), https://doi.org/10.1016/j.ins.2018.05.023 JID: INS ARTICLE IN PRESS [m3Gsc;June 11, 2018;10:4] H. Liao et al. / Information Sciences 000 (2018) 1–18 7 Algorithm 1 (Calculate the psychological distance between IMNs). Input: two IMNs α and β . Output: the psychological distance d psy (α , β ) between α and β . 1. Derive the normalized IMNs α and β by Eqs. (17) and (18). 2. Calculate iv j by Eqs. (19) and (20), and then determine dv. 3. Derive distinitial (α , β ) and E, and then calculate disttrans (α , β ) by Eqs. (21) and (22). 4. Define a diagonal matrix Aw , and then calculate d psy (α , β ) by Eq. (23). 1 1 τβ − 1 9 − 19 β = ρβ , σ β , τ β = ρβ , , + (18) σβ 9 81 − 1 Step 2. Calculate the indifference vector iv j by Eq. (19). The dominance vector dv which is orthogonal to all indifference vectors in multi-attribute coordinate space, denoted as vdommance ⊥vindi f f erence , can be derived by Eq. (20). The indiffer- ence vector reveals the exchange information between ρ , σ and τ in the psychological distance measure. For example, iv j represents that the DM obtains wσ /wρ (or wτ /wρ ) units of non-membership attribute (or hesitance attribute) for giving up one unit of membership attribute. −wσ /wρ −wτ /wρ iv1 = w ρ /w ρ , iv2 = 0 (19) 0 wρ /wρ T d v = wρ /wρ , wσ /wρ , wτ /wρ (20) Step 3. Define a distance along the dominance direction and the indifference direction. In multi-attribute coordinate space, the line connecting two IMNs is called the initial distance vector distinitial (α , β ). It is represented by the indifference vector and the dominance vector in the form of the transformed distance vector disttrans (α , β ). To obtain iv disttrans (α , β ), we define the basis transformation matrix E = ( iv j , ddvv ), where iv j and dv are the Euclidean j length of iv j and dv, respectively. Accordingly, each column of E indicates the corresponding standardized vector. The standardization is used to facilitate the subsequent comparisons. Step 4. The initial distance vector distinitial (α , β ) contains three distance values between α and β , which can be calcu- lated by Eq. (21) (here we takes the Manhattan distance as an example). disttrans (α , β ) can be derived by Eq. (22), where the first two entries represent the related changes on indifference vectors and the last entry represents the related change on the dominance vector. In other words, disttrans (α , β ) can indirectly reflect how many unit values we need to move along the indifference vector and the dominance vector, respectively. T distinitial (α , β ) = 1 log9 ρα , log9 σα , log9 τα (21) 2 ρβ σβ τβ iv dv disttrans (α , β ) = E −1 distinitial (α , β ), E = , (22) iv dv Step 5. Since the distance along the dominant direction is more important than the indifference direction, we introduce 1 0 0 a diagonal matrix Aw = 0 1 0 , where only the last entry of diagonal is assigned and other values of diagonal 0 0 wd are equal to 1. Finally, the psychological distance between α and β are calculated by Eq. (23). 1 d psy α , β = · dist T trans (α , β ) · Aw · disttrans (α , β ), ωd > 1 (23) 3wd where dist T trans (α , β ) is the transposition of disttrans (α , β ). In addition, 3wd is a balancing coefficient to make the psycho- logical distance meet the conditions of distance measures and wd can be assigned any real number greater than 1. Step 6. End. Example 1. Let α = (1/6, 4, 3/2) and β = (7, 1/8, 8/7) be two IMNs. According to Eqs. (17) and (18), we transfer them to α = (1/6, 1/4, 1/6), β = (7, 8, 8/63). The weight vector of the three parameters is w = (1/3, 1/3, 1/3 )T and Aw = Please cite this article as: H. Liao et al., A multiple attribute group decision making method based on two novel intuition- istic multiplicative distance measures, Information Sciences (2018), https://doi.org/10.1016/j.ins.2018.05.023 JID: INS ARTICLE IN PRESS [m3Gsc;June 11, 2018;10:4] 8 H. Liao et al. / Information Sciences 000 (2018) 1–18 Algorithm 2 (Calculate the psychological distance between IMSs). Input: IMSs A and B . Output: The psychological distance d psy (A , B ). 1. Derive the normalized IMSs A(xi ) and B(xi ) by Eqs. (24) and (25). 2. Calculate iv j and dv by Eqs. (26) and (27). 3. Calculate E by Eq. (28). 4. Calculate distinitial (A(xi ), B(xi )) and disttrans (A(xi ), B(xi )) by Eqs. (29) and (30). 5. Define a diagonal matrix Aω and then calculate d psy (A , B ) by Eq. (31). 1 1 . Based on Eqs. (19)–(23), the psychological distance between α and β can be calculated as follows: 5 ⎛ √ √ √ ⎞ −1 − 1 1 −1/√ 2 −1/ 2 1/√3 0.8505 iv = 10 , dv = 1 , E3×3 = ⎝ 1/ 2 0√ 1/√3⎠, distinitial (α , β ) = 0.7887 , 01 1 0 1/ 2 1/ 3 0.0619 disttrans (α , β ) = (0.3134, −0.7144, 0.9821 )T , d psy (α , β ) = 0.6017. It should be noted that the transformed distance vector disttrans (α , β ) reveals the changes between α and β , which represents that to reach IMN α from IMN β , we need to move by 0.3134 and 0.7144 units along the first and second indifference vectors, respectively, and 0.9821 units along the dominance vector. 3.2.2. The psychological distance measures between IMSs This section further investigates the psychological distance measure between IMSs to broaden its applications in MADM under the intuitionistic multiplicative environment. Let X = {x1 , x2 , ..., xn } be a finite set, A = {(ρA (xi ), σA (xi ))|xi ∈ X } and B = {(ρB (xi ), σB (xi ))|xi ∈ X } be two IMSs in X associated with the weight vector ω = (ω1 , ω2 , ...ωn )T , where 0 ≤ ωi ≤ 1 and ni=1 ωi = 1. The other parameters are given as above. The steps to calculate the psychological distance between IMSs can be summarized as follows and illustrated in Algorithm 2: Step 1. Check the attributes to ensure that all of them are in the same type. Transfer the cost type attributes to benefit type attributes by the complement operator: (ρ (x j ), σ (x j ))c = (σ (x j ), ρ (x j )). The standardized IMSs A and B are derived by Eqs. (24) and (25). ⎧ ⎨ ρA (xi ), 1 , 1 + (τA (xi )−1 )(9− 9 ) , for benefit attributes 1 σA (xi ) 9 81−1 A = ( ρA ( x i ) , σ A ( x i ) , τ A ( x i ) ) = (24) ⎩ σA (xi ), 1 , 1 + (τA (xi )−1 )(9− 9 ) , for cost attributes 1 ρA ( x i ) 9 81−1 ⎧ ⎨ ρB (xi ), 1 , 1 + (τB (xi )−1 )(9− 9 ) , for benefit attributes 1 σB (xi ) 9 81−1 B = ( ρB ( x i ) , σ B ( x i ) , τ B ( x i ) ) = (25) ⎩ σB (xi ), 1 , 1 + (τB (xi )−1 )(9− 9 ) , for cost attributes 1 ρB ( x i ) 9 81−1 Step 2. Calculate the indifference vectors iv j and the dominance vector dv of the IMSs A and B according to Eqs. (26) and (27). ω ω1 T j+1 iv j = − , 0, · · · , 0, , 0, · · · , 0 , where iv j, j+1 = 1, j = 1, 2, ..., n − 1 (26) ω1 ω1 dv = (ω1 /ω1 , ω2 /ω1 , ..., ωn /ω1 )T (27) Step 3. Calculate the basic transformation matrix E by Eq. (28). ⎛ ⎞ −ω2 / ω12 + ω22 −ω3 / ω12 + ω32 ··· −ωn / ω12 + ωn2 ω1 /ω12 + ω22 + · · · + ωn2 ⎜ ω1 / ω 2 + ω 2 ··· ω2 /ω12 + ω22 + · · · + ωn2⎟ ⎜ 1 2 0 0 ⎟ ⎜ 0 ω1 / ω12 + ω32 ··· 0 ω3 / ω12 + ω22 + · · · + ωn2⎟ En×n =⎜ ⎟ (28) ⎜ .. .. .. .. .. ⎟ ⎝ . . . ⎠ . . 0 0 ··· ω1 / ω12 + ωn2 ωn / ω12 + ω22 + · · · + ωn2 Step 4. Calculate the initial distance vector distinitial (A(xi ), B(xi )) and the transformed distance vector disttrans (A(xi ), B(xi )) by Eqs. (29) and (30), where distinitial (A(xi ), B(xi )) is made up of the psychological distance values between any two Please cite this article as: H. Liao et al., A multiple attribute group decision making method based on two novel intuition- istic multiplicative distance measures, Information Sciences (2018), https://doi.org/10.1016/j.ins.2018.05.023 JID: INS ARTICLE IN PRESS [m3Gsc;June 11, 2018;10:4] H. Liao et al. / Information Sciences 000 (2018) 1–18 9 IMNs calculated by Algorithm 1. ⎛ ⎞ d psy A (x1 ), B (x1 ) ⎜d psy A (x2 ), B (x2 ) ⎟ ⎜ ⎟ distinitial (A(xi ), B(xi ) ) = ⎜ .. ⎟ (29) ⎝ . ⎠ d psy A (xn ), B (xn ) disttrans (A(xi ), B(xi ) ) = E −1 distinitial (A(xi ), B(xi ) ) (30) ⎛ ⎞ 1 ⎜ 1 ⎟ ⎜ ⎟ Step 5. Define the diagonal matrix Aω = ⎜ ⎜ .. . ⎟, and then the psychological distance between IMSs A ⎟ ⎝ 1 ⎠ ωd and B can be calculated by Eq. (31). 1 d psy A , B = · dist T trans (A(xi ), B(xi ) ) · Aw · disttrans (A(xi ), B(xi ) ) (31) n · ωd Step 6. End. Example 2. Let A and B be two IMSs in X = {x1 , x2 , x3 , x4 } and A = {(1/6, 4, 3/2), (2, 1/6, 3), (6, 1/8, 4/3), (3, 1/5, 5/3)}, B = {(6, 1/9, 3/2), (7, 1/9, 9/7), (5, 1/6, 6/5), (1/7, 2, 7/2)}. The weight vector of them is w = (3/5, 1/4, 1/7, 1/140)T . As- 1 sume that the three parts of the IMNs share the same weight, i.e., w = (1/3, 1/3, 1/3) . AwN = T 1 and AwS = 5 ⎛ ⎞ 1 ⎜ 1 ⎟ ⎝ 1 ⎠, where AwN and AwS represent the diagonal matrices associated to the psychological distance be- 10 tween IMNs and IMSs, respectively. Similar to Example 1, the psychological distances between IMNs in A and B can be calculate as: d psy (A (x1 ), B (x1 )) = 0.5872, d psy (A (x2 ), B (x2 )) = 0.1934, d psy (A (x3 ), B (x3 )) = 0.0449, d psy (A (x4 ), B (x4 )) = 0.4746. Then, the psychological distance between A and B can be calculated by Eqs. (24)–(31) and we obtain ⎛ ⎞ ⎛ ⎞ −5/12 −5/21 −1/84 1 ⎜ 1 0 0 ⎟ ⎜5/12⎟ iv = ⎝ 0 ⎠ , dv = ⎝ 5/21⎠ , 0 1 ⎛0 0 1 1/84 ⎞ ⎛ ⎞ −5/13 −293/1265 −168/14113 897/995 0.5872 ⎜ 12/13 0 0 299/796 ⎟ ⎜0.1934 ⎟ E4×4 = ⎝ , distinitial (A(xi ), B(xi ) ) = ⎝ 299/1393⎠ 0.0449⎠ , 0 787/809 0 0 0 14112/14113 79/7361 0.4746 T disttrans (A(xi ), B(xi ) ) = E −1 distinitial (A(xi ), B(xi ) ) = −0.0415 −0.0899 0.4680 0.6167 d psy (A , B ) = 1 4×10 · dist T trans (A, B ) · AwS · disttrans (A, B ) = 0.3666. 4. Methods to determine the weights of experts and attributes This section introduces some methods to calculate the weights of experts and attributes. These methods are based on the projection-based distance measure and the intuitionistic multiplicative entropy, respectively. 4.1. Calculate the experts’ weights based on the projection-based distance measure For a MAGDM problem, let X = {X1 , X2 , ..., Xm } be a set of alternatives, C = {C1 , C2 , ..., Cn } be a set of attributes and F = {F1 , F2 , ..., Ft } be a set of DMs (or experts) whose weight vector is g = (g(1) , g(2) , · · · , g(t ) )T . Suppose that the evaluation values for all alternatives with respect to the attributes are expressed by the IMNs (ρi j , σi j ), i = 1, 2, ..., m, j = 1, 2, ..., n, where ρi j and σi j are the degrees of membership and non-membership of alternative Xi on attribute C j with 1/9 ≤ ρi j ,σi j ≤ 9, Please cite this article as: H. Liao et al., A multiple attribute group decision making method based on two novel intuition- istic multiplicative distance measures, Information Sciences (2018), https://doi.org/10.1016/j.ins.2018.05.023 JID: INS ARTICLE IN PRESS [m3Gsc;June 11, 2018;10:4] 10 H. Liao et al. / Information Sciences 000 (2018) 1–18 (k ) 0 < ρi j σi j ≤ 1. An intuitionistic multiplicative decision matrix R(k ) = (ri j )m×n is constructed as ⎛ C1 C2 ··· Cn ⎞ X1 (ρ11 , σ11 ) (ρ12 , σ12 ) ··· ( ρ1 n , σ 1 n ) R ( k ) = ri j (k ) = X2 ⎜ (ρ21 , σ21 ) (ρ22 , σ22 ) ··· (ρ2n , σ2n ) ⎟, k = 1, 2, ..., t m×n .. ⎜ ⎝ .. .. .. ⎟ ⎠ . . . ··· . Xm (ρm1 , σm1 ) ( ρm 2 , σ m 2 ) ··· (ρmn , σmn ) To facilitate the calculations, first of all, the attribute values need to be standardized. Usually, the attributes can be divided into beneficial and cost attributes. This paper unifies attributes by transforming cost attributes into benefi- (k ) cial attributes. In this way, R(k ) = (ri j )m×n can be transformed into the intuitionistic multiplicative decision matrices (k ) D ( k ) = ( di j )m×n , where (k) ρi j (k) , σi j (k) , for benefit attribute C j , k = 1, 2, ..., t (k ) (k ) (k ) (k ) di j = ρi j , σ i j = (32) σi j , ρi j , for cost attribute C j (∗ ) Take the mean values di j = (ρi j (∗ ) , σi j (∗ ) ) as the ideal decision values of all evaluated attribute values, where ρi j (∗ ) = 1 t (k ) σ (∗ ) = 1 t (k ) , τ (∗ ) = 1 t (k ) . The projection-based distance measure can be used to calculate the t ρ k=1 i j , ij t σ k=1 i j ij t k=1 τi j ∗ deviation between each assessment and di j . The calculation formulas are shown as follows: n m 1 + log9 ρi j S 1 + log9 ρi j Dk + 1 + log9 σi j S 1 + log9 σi j Dk + log9 τi j S log9 τi j Dk ∗ ∗ ∗ 1 i=1 j=1 ProjS∗ Dk = · m 2 (33) 2 n 2 2 1 + log9 ρi j S∗ + 1 + log9 σi j S∗ + log9 τi j S∗ i=1 j=1 ProjS∗ Dk ProjS∗ D∗ dProj (Dk , D ) = ∗ − (34) ProjS∗ Dk + ||S∗ | − ProjS∗ Dk | ProjS∗ D∗ + ||S∗ | − ProjS∗ D∗ | ⎛ ⎞ 9, 1/9 · · · 9, 1/9 S∗ = ⎝ .. .. . . ⎠ ··· (35) 9, 1/9 ··· 9, 1/9 (k ) (∗ ) It is reasonable that the smaller the deviation between di j and di j is, the bigger the experts’ weights should be assigned. To avoid big differences among experts, the weights of experts can be calculated by 0.8 1 1 1 1 g(k ) = + 0.2 × 1 dPr o j (Dk , D ) · ∗ + + + ... + , t dPr o j (D1 , D∗ ) dPr o j (D2 , D∗ ) dPr o j (D3 , D∗ ) dPr o j (Dt , D∗ ) k = 1, 2, ..., t (36) Using the IMWA operator, the aggregated decision matrix D = (di j )m×n can be derived, where t (1 ) (2 ) (t ) g(k ) di j ( ) , i = 1, 2, ..., m k di j = IMW Ag di j , di j , ..., di j = j = 1, 2, ..., n, (37) k=1 4.2. Calculate the weights of attributes based on the entropy of IMS In MADM problems, the attribute weights are usually determined by DMs directly based on their subjective judgments. To obtain objective weights, this section introduces two methods to derive the weights of attributes in the cases that the weight information is completely unknown or partly known. 4.2.1. Method to derive the weights of attributes when the weight information is completely unknown Entropy of a fuzzy set describes the fuzziness degree of the set. Fuzzy entropy has been widely used to determine the weights of attributes. For the evaluation values given by DMs, the greater the degree of variation is, the more information can be derived, and thus the attributes should be assigned more weights. For the intuitionistic fuzzy entropy, some axiomatic definitions and formulas have been proposed, but some of them ignore the effect caused by the hesitancy degrees of IFSs. Fan et al. [8] proposed a new axiomatic definition for intuitionistic fuzzy entropy and gave a general formula of the strict intuitionistic fuzzy entropy. We find that this definition can effectively overcome the previous defects and is more scientific and practical than others. Motivated by the intuitionistic fuzzy entropy measure [8,24,27], this subsection introduces the intuitionistic multiplicative entropy. Please cite this article as: H. Liao et al., A multiple attribute group decision making method based on two novel intuition- istic multiplicative distance measures, Information Sciences (2018), https://doi.org/10.1016/j.ins.2018.05.023 JID: INS ARTICLE IN PRESS [m3Gsc;June 11, 2018;10:4] H. Liao et al. / Information Sciences 000 (2018) 1–18 11 Algorithm 3 (MAGDM with psychological distance). Input: DMs’ evaluation values with IMNs. Output: The ranking of alternatives. 1. Construct D(k ) through Eq. (32). 2. Derive each expert’s weight g(k ) according to Eq. (36) and then calculate D through Eq. (37). 3. Calculate each attribute’s λ j based on Model 2. (∗ ) (− ) 4. Determine the IMPIS di j and the IMNIS di j . 5. Calculate the relative closeness coefficient f (Xi ) of each alternative using Eq. (40). Definition 11. Let X = {x1 , x2 , ..., xm } be a given universal set and A = {x, ρA (x ), σA (x ) |x ∈ X } be an IMS. Then, an intuition- istic multiplicative entropy can be defined as: 1 m E (A ) = (δ (xi ) + ξ δ (xi ) · min {log9 ρ (xi ), log9 σ (xi )} ) (38) m i=1 ρ (x ) ρ (x ) where δ (xi ) = (2 − |log9 σ (xi ) | )/(2 + |log9 σ (xi ) |) and ξ ∈ [0, 0.5]. i i In decision making process, sometimes the weights of attributes are completely unknown. According to Eq. (38), the in- tuitionistic multiplicative entropy of each value in the aggregated decision matrix D = (di j )m×n can be calculated. Motivated by Ye [32], the weight of attribute C j can be derived as: 1 m 1 − Ej λj = , where E j = E d j , j = 1, 2, ..., n (39) n m n− Ej i=1 j=1 4.2.2. Method to derive the weights of attributes when the weight information is partly known It is also common that there are some constraints for the weights of attributes. Let H be the set of known weight infor- mation. To address the issue with partly known attribute weight information, Wu and Zhang [27] proposed a programming model on the basis of the minimum principle to determine the attribute weights. A similar approach was proposed by Wang and Wang [24]. In this section, combined with the transformation mechanism between IMS and IFS [11], we use the intuitionistic multiplicative entropy to derive the weights of attributes via a programming model shown below: Model 1 n n min E (Xi ) = λ j E d j = λ j · [δ (xi ) + ξ δ (xi ) · min {log9 ρ (xi ), log9 σ (xi )}] j=1 j=1 n s.t. λ j = 1, λ ∈ H j=1 ρ (x ) ρ (x ) where δ (xi ) = (2 − |log9 σ (xi ) | )/(2 + |log9 σ (xi ) |), and ξ ∈ [0, 0.5]. i i Since the alternatives are treated equally, Model 1 can be simplified to Model 2: Model 2 m m n min E (Xi ) = λ j · [δ (xi ) + ξ δ (xi ) · min {log9 ρ (xi ), log9 σ (xi )}] i=1 i=1 j=1 n s.t. λ j = 1, λ ∈ H j=1 Solving Model 2, the optimal weight vector for attributes can be determined. 5. An algorithm to solve intuitionistic multiplicative MADM problems In the above section, the weights λ j and gk with respect to the evaluation attributes and experts have been calculated, respectively. Based on that, a MAGDM method using the psychological distance measure developed in Section 3 is proposed. The steps are as follows and the corresponding algorithm is shown in Algorithm 3. (k ) Step 1. Construct the initial decision matrices R(k ) = (ri j )m×n , k = 1, 2, ..., t, according to the evaluation opinions of t (k ) experts, based on which, the intuitionistic multiplicative decision matrix D(k ) = (di j )m×n can be derived according to Eq. (32). Step 2. Calculate each expert’s weight g(k ) according to Eq. (36), and then calculate the weighted decision matrix D = (di j )m×n using the IMWA operator shown as Eq. (37). Step 3. Calculate the attribute weights λ j based on Model 2. Please cite this article as: H. Liao et al., A multiple attribute group decision making method based on two novel intuition- istic multiplicative distance measures, Information Sciences (2018), https://doi.org/10.1016/j.ins.2018.05.023 JID: INS ARTICLE IN PRESS [m3Gsc;June 11, 2018;10:4] 12 H. Liao et al. / Information Sciences 000 (2018) 1–18 Step 4. Determine the intuitionistic multiplicative positive-ideal solution (IMPIS) and the intuitionistic multiplicative (∗ ) (− ) negative-ideal solution (IMNIS), which are expressed as di j = (ρi j (∗ ) , σi j (∗ ) ) and di j = (ρi j (− ) , σi j (− ) ), respectively. Based on the proposed psychological distance measure, the relative closeness coefficients f (Xi ) of each alternative can be determined using Eq. (40). (k ) (k ) (k ) f (Xi ) = d psy di j , di j ( ) / d psy di j , di j ( ) + d psy di j , di j ( ) − − ∗ (40) Step 5. Rank the alternatives according to f (Xi ) in descending order. The bigger f (Xi ) is, the better the alternative should be. Step 6. End. The projection-based distance measure and the psychological distance measure are foundation parts of Algorithm 3. In Algorithm 3, the intuitionistic multiplicative decision matrices R(k ) , k = 1, 2, ..., t, are constructed first and then the nor- malized decision matrices D(k ) , k = 1, 2, ..., t, are produced. Afterwards, the experts’ weights are obtained based on the projection-based distance measure, and then the collective decision matric D can be obtained. Furthermore, the attribute weights are derived based on the intuitionistic multiplicative entropy. Finally, the alternatives are ranked according to the relative closeness coefficients. 6. Case study: Drug supplier selection In this section, a case study concerning drug supplier selection is applied to illustrate the applicability of the proposed algorithm for MAGDM. 6.1. Case introduction In the medical and healthcare system reform in China, how to strengthen the hospital management and enhance the core competitiveness of hospitals becomes an important issue. Drug supply, medical equipment and medical consumables, providing support for hospital medical treatment, are deemed as the important parts of hospital management. In the course of drug purchase, we need to choose the best drug supplier first. Supplier management is an important activity of hospital drug administration. Evaluating drug suppliers promotes the formation with survival of the fittest concerning the drug qual- ity and service level. In addition, only in this way can the selection principles of equity and justness be reflected. In the past, people used to employ empirical or habitual methods to determine the drug suppliers, but now, numerous group decision methods are utilized in drugs procurement department. Usually, the hospital pharmacy management committee (or Drug procurement group) establishes drug suppliers by means of collective discussion and purchase plans. Even though it can avoid some disadvantages compared to individual decision making, there exists blindness and artificiality in the process of drug supplier selection. Consequently, how to evaluate the drug suppliers and select the best one turns out to be an urgent problem. As we know, supplier selection is a complex MADM problem which consists of qualitative and quantitative attributes [22]. The drug supplier selection problem has attracted many researchers [1,2,6,14,15]. Most of them focused on the purchase of raw materials for pharmaceutical companies, and they mainly used the traditional methods, such as Analytic Hierarchy Pro- cess (AHP) [2,15] and Analytic Network Process (ANP) [14], to evaluate the suppliers. Asamoah et al. [2] identified quality, price and reliability/capacity as the attributes and then used the AHP method to select the raw material suppliers in a phar- maceutical manufacturing firm in Ghana. Kirytopoulos et al. [14] applied the ANP method to evaluate the optimal suppliers for medicinal materials based on the understanding of the pharmaceutical industry’s supply chain process. However, these approaches turn out to be less effective in dealing with the imprecise assessments [6]. Hence, Datta et al. [6] used the fuzzy AHP method to derive the weights of attributes and then combined the TOPSIS method with the generalized trape- zoidal fuzzy numbers to propose an integrated fuzzy MAGDM approach to solve the supplier selection problem. Alinezad et al. [1] used the fuzzy AHP method to determine the weights of evaluation criteria, and then used the Quality Function Deployment (QFD) to solve the supplier selection problem in pharmaceutical enterprises. In China, many approaches have been used to solve the drug supplier selection problems, such as the AHP method [38] and the multi-level gray evaluation method [5]. In addition, many hospitals tried to build the drug supplier evaluation and scoring system. After consulting the experts and staffs in the hospital pharmacy management committee and referring to the above literature, we can derive the following attributes: - C1 (Supplier reputation) Including GSP certification, management quality, credit level and economic strength; - C2 (Drug price) It is related to the economic efficiency of hospitals. Those factors are generally expressed as discount rate, plus rate and difference rate, while the total discount rate is more intuitive; - C3 (Quantity of drug) The actual quantity supplied by drug suppliers; - C4 (Arrival time) The time from the order to warehouse; - C5 (Mode of payment) Fund payment time due to delaying the appropriation can increase the utilization rate and turnover rate of the hospital funds; - C6 (Ability to solve shortages) The ability to solve the problem about scarce drugs. Please cite this article as: H. Liao et al., A multiple attribute group decision making method based on two novel intuition- istic multiplicative distance measures, Information Sciences (2018), https://doi.org/10.1016/j.ins.2018.05.023 JID: INS ARTICLE IN PRESS [m3Gsc;June 11, 2018;10:4] H. Liao et al. / Information Sciences 000 (2018) 1–18 13 Table 1 The intuitionistic multiplicative judgment matrix for E1 . 1 C1 C2 C3 C4 C5 C6 X1 (6,1/8,4/3) (3,1/5,5/3) (4,1/7,7/4) (1/3,2,3/2) (5,1/6,6/5) (6,1/7,7/6) X2 (1,1/4,4) (8,1/9,9/8) (3,1/5,5/3) (1/7,2,7/2) (3,1/4,4/3) (7,1/8,8/7) X3 (3,1/6,2) (4,1/5,5/4) (1,1/8,8) (1/5,5,1) (7,1/8,8/7) (7,1/8,8/7) X4 (7,1/8,8/7) (3,1/6,2) (5,1/6,6/5) (1/8,6,4/3) (6,1/7,7/6) (4,1/7,7/4) X5 (4,1/6,3/2) (7,1/8,8/7) (7,1/8,8/7) (1/6,4,3/2) (5,1/9,9/5) (3,1/7,7/3) Table 2 The intuitionistic multiplicative judgment matrix for E2 . 2 C1 C2 C3 C4 C5 C6 X1 (7,1/8,8/7) (5,1/6,6/5) (4,1/6,3/2) (1/4,4,1) (6,1/8,4/3) (6,1/8,4/3) X2 (2,1/4,2) (7,1/8,8/7) (7/2,1/4,8/7) (1/8,6,4/3) (3,1/6,2) (6,1/7,7/6) X3 (10/3,1/6,9/5) (9/2,1/6,4/3) (4,1/6,3/2) (1/6,5,6/5) (7,1/8,8/7) (6,1/8,4/3) X4 (6,1/8,4/3) (5,1/7,7/5) (7,1/8,8/7) (1/7,5,7/5) (6,1/8,4/3) (9/2,1/7,14/9) X5 (4,1/7,7/4) (13/2,1/7,14/13) (5,1/6,6/5) (1/6,5,6/5) (6,1/7,7/6) (7,1/8,8/7) Table 3 The intuitionistic multiplicative judgment matrix for E3 . 3 C1 C2 C3 C4 C5 C6 X1 (5,1/8,8/5) (4,1/5,5/4) (5,1/8,8/5) (1/5,3,5/3) (7,1/8,8/7) (6,1/8,4/3) X2 (1,1/5,5) (8,1/8,1) (3,1/7,7/3) (1/7,6,7/6) (7/2,1/5,10/7) (4,1/6,3/2) X3 (7/2,1/7,2) (5,1/7,7/5) (2,1/6,3) (1/4,2,2) (8,1/8,1) (13/2,1/8,16/13) X4 (7,1/8,8/7) (3,1/7,7/3) (5,1/7,7/5) (1/6,5,6/5) (13/2,1/7,14/13) (5,1/6,6/5) X5 (6,1/7,7/6) (7,19/144,144/133) (6,1/8,4/3) (1/7,6,7/6) (6,1/9,3/2) (4,1/5,5/4) Table 4 The intuitionistic multiplicative averaging decision matrix. C1 C2 C3 C4 C5 C6 X1 (6,1/8,428/315) (4,17/90,247/180) (13/3,73/504,97/60) (47/180,3,25/18) (6,5/36,386/315) (6,11/84,23/18) X2 (4/3,7/30,11/3) (23/3,13/108,61/56) (19/6,83/420,12/7) (23/168,14/3,2) (19/6,37/180,100/63) (17/3,73/504,80/63) X3 (59/18,10/63,29/15) (9/2,107/630,239/180) (7/3,11/72,25/6) (37/180,4,7/5) (22/3,1/8,23/21) (13/2,1/8,1012/819) X4 (20/3,1/8,76/63) (11/3,19/126,86/45) (17/3,73/504,131/105) (73/504,16/3,59/45) (37/6,23/168,31/26) (9/2,19/126,811/540) X5 (14/3,19/126,53/36) (41/6,403/3024,797/724) (6,5/36,386/315) (10/63,5,58/45) (17/3,23/189,67/45) (14/3,131/840,397/252) To better monitor the procurement channels and select the best drug supplier with good service quality and high man- agement levels, for the H hospital in China, every quarter there is a quantitative evaluation for supply companies. By an- alyzing the values of the evaluation indices, we can derive the results as the attributes for screening drug suppliers. The working mode of drug suppliers has been highly valued and received comments from other hospitals. Now, the H hospi- tal invites three experts from the hospital pharmacy management committee (or drug purchasing group) to provide their assessments in IMSs to select the best one from five pharmaceutical enterprises. The evaluation values are listed in Tables 1–3. The weights of experts and attributes are unknown. 6.2. Method application Algorithm 3 is used to select the drug suppliers below. Step 1. Using Eq. (33), the intuitionistic multiplicative averaging decision matrix (Table 4) can be established based on the initial matrices given by the experts. Step 2. Derive the weights of experts by Eq. (37). Here the weight vector of the three experts is g(k ) = (0.2826, 0.2976, 0.4198)T . Then, using Eq. (38), the weighted decision matrix D can be calculated and shown in Table 5. Step 3. The intuitionistic multiplicative entropy E j of the weighted decision matrix D can be calculated as E j = (0.1664, 0.1017, 0.1369, 0.1632, 0.0845, 0.0868)T . Let λ = 0.1. Based on Model 2, the attribute weight vector can be calculated as λ j = (0.1585, 0.1708, 0.1641, 0.1591, 0.1740,0.1736 )T . Step 4. Determine the IMPIS and the IMNIS, which are shown in Table 6. Then, calculate the relative closeness coefficients f (Xi ). It should be noted that C1 , C3 , C5 and C6 are benefit attributes while C2 and C6 are cost attributes. Thus, all the cost attributes should be transformed to the benefit attributes and then the standardized decision matrix is obtained, shown in Table 7. Please cite this article as: H. Liao et al., A multiple attribute group decision making method based on two novel intuition- istic multiplicative distance measures, Information Sciences (2018), https://doi.org/10.1016/j.ins.2018.05.023 JID: INS ARTICLE IN PRESS [m3Gsc;June 11, 2018;10:4] 14 H. Liao et al. / Information Sciences 000 (2018) 1–18 Table 5 The weighted intuitionistic multiplicative decision matrix. C1 C2 C3 C4 C5 C6 X1 (5.8821,0.1237) (3.9807,0.1876) (4.4344,0.1399) (0.2520,2.8374) (6.1412,0.1342) (6.0603,0.1285) X2 (1.2532,0.2255) (7.7732,0.1195) (3.1675,0.1834) (0.1381,4.0085) (3.2278,0.1998) (5.3569,0.1451) X3 (3.3301,0.1546) (4.5916,0.1628) (2.0878,0.1519) (0.2110,3.1842) (7.4821,0.1237) (6.5505,0.1237) X4 (6.7577,0.1237) (3.5321,0.1477) (5.5841,0.1419) (0.1481,5.2320) (6.2676,0.1359) (4.5916,0.1508) X5 (4.7929,0.1477) (6.9196,0.1316) (6.0031,0.1346) (0.1572,4.9917) (5.7540,0.1184) (4.4140,0.1562) Table 6 The IM-PIS and the IM-NIS for decision information. C1 C2 C3 IM-PIS (6.7577,0.1237,1.1963) (3.5321,0.1876,1.5902) (6.0031,0.1346,1.2376) IM-NIS (1.2532,0.2255,3.5386) (7.7732,0.1195,1.0765) (2.0878,0.1834,2.6116) C4 C5 C6 IM-PIS (0.2502,2.8374,1.3986) (7.4821,0.1184,1.1288) (4.4140,0.1562,1.4504) IM-NIS (0.1381,5.2320,1.3840) (3.2278,0.1998,1.5506) (6.5505,0.1237,1.2341) Table 7 The standardized intuitionistic multiplicative decision matrix. C1 C2 C3 X1 (5.8821,8.0871,0.1528) (0.1876,0.2512,0.1488) (4.4344,7.1493,0.1791) X2 (1.2532,4.4344,0.3932) (0.1195,0.1286,0.1196) (3.1675,5.4528,0.1913) X3 (3.3301,6.4665,0.2158) (0.1628,0.2178,0.1486) (2.0878,6.5838,0.3504) X4 (6.7577,8.0871,0.1330) (0.1477,0.2831,0.2130) (5.5841,7.0455,0.1402) X5 (4.7929,6.7696,0.1569) (0.1361,0.1445,0.1220) (6.0031,7.4282,0.1375) C4 C5 C6 X1 (0.2520,0.3524,0.1554) (6.1412,7.4525,0.1348) (0.1285,0.1650,0.1427) X2 (0.1381,0.2495,0.2008) (3.2278,5.0050,0.1723) (0.1451,0.1867,0.1430) X3 (0.2110,0.3141,0.1653) (7.4821,8.0871,0.1201) (0.1237,0.1527,0.1372) X4 (0.1481,0.1911,0.1434) (6.2676,7.3609,0.1305) (0.1508,0.2178,0.1605) X5 (0.1572,0.2003,0.1416) (5.7540,8.4490,0.1632) (0.1562,0.2266,0.1611) Table 8 The standardized IMPIS and IMNIS for decision matrix. C1 C2 C3 IM-PIS (6.7577,8.0871,0.1329) (0.1876,0.2831,0.1767) (6.0031,7.4282,0.1375) IM-NIS (1.2532,4.4344,0.3932) (0.1195,0.1286,0.1196) (2.0878,5.4528,0.2902) C4 C5 C6 IM-PIS (0.2520,0.3524,0.1554) (7.4821,8.4490,0.1254) (0.1562,0.2266,0.1612) IM-NIS (0.1381,0.1911,0.1538) (3.2278,5.0050,0.1723) (0.1237,0.1527,0.1371) 1 Assume that the three parts of IMN share the same weight and define Aw = 1 and Aω = 10 ⎛ ⎞ 1 ⎜ 1 ⎟ ⎜ ⎟ ⎜ 1 ⎟ ⎜ ⎟. The normalized psychological distance between IMPIS and each alternative can be calculated ⎜ 1 ⎟ ⎝ 1 ⎠ 10 Please cite this article as: H. Liao et al., A multiple attribute group decision making method based on two novel intuition- istic multiplicative distance measures, Information Sciences (2018), https://doi.org/10.1016/j.ins.2018.05.023 JID: INS ARTICLE IN PRESS [m3Gsc;June 11, 2018;10:4] H. Liao et al. / Information Sciences 000 (2018) 1–18 15 Table 9 The psychological distances between alternatives and IMPIS/IMNIS. X1 X2 X3 X4 X5 The psychological distance between IMNs (IMPIS) 0.0220 0.2575 0.1084 0 0.0524 0.0226 0.1248 0.0440 0.0334 0.1068 0.0471 0.0974 0.1644 0.0111 0 0 0.0917 0.0271 0.0956 0.0880 0.0302 0.1284 0.0067 0.0273 0.0415 0.0488 0.0296 0.0607 0.0062 0 X1 X2 X3 X4 X5 The psychological distance between IMNs (IMNIS) 0.2360 0 0.1492 0.2575 0.2054 0.1032 0 0.0809 0.1207 0.0181 0.1151 0.0658 0.0291 0.1511 0.1619 0.0965 0.0411 0.0775 0.0111 0.0199 0.0982 0 0.1282 0.1012 0.0902 0.0120 0.0313 0 0.0546 0.0607 Table 10 The relative closeness coefficient of each alternative. Alternatives Si + Si − fi X1 0.0300 0.1087 0.7837 X2 0.1202 0.0254 0.1745 X3 0.0715 0.0783 0.5227 X4 0.0314 0.1163 0.7874 X5 0.0507 0.0941 0.6499 according to Eqs. (26)–(31) and the results are shown as below. ⎛ √ √ √ ⎞ −1/√ 2 −1/ 2 1/√3 E3×3 = ⎝ 1/ 2 0√ 1/√3⎠ 0 1/ 2 1/ 3 ⎛ ⎞ −1499/2045 −515/716 −1368/1931 −706/955 −1717/2325 841/2168 ⎜ 987/1451 0 0 0 0 232/555 ⎟ ⎜ 0 751/1081 0 0 0 347/864 ⎟ E6×6 =⎜⎜ ⎟ 0 0 367/520 0 0 1137/2920 ⎟ ⎝ ⎠ 0 0 0 965/1433 0 313/735 0 0 0 0 296/439 3461/8146 (∗ ) Finally, the psychological distance between each alternative Xi and the IMPIS di j , and the psychological distance be- (− ) tween each alternative Xi and the IMNIS di j can be calculated, shown in Table 9. Step 5. Calculate the relative closeness coefficient of each alternative according to Eq. (40). The results are shown in Table 10. Rank the alternatives according to fi in descending order and the ranking result is X4 X1 X5 X3 X2 . Hence, the best drug supplier is X4 . In practice, the alternative X4 is determined as the optimal supplier for the H hospital. Meanwhile, the managers also consider that there are little difference between the alternatives X1 and X4 . In the past, the managers used to select the alternative with long-term partnership. To determine the optimal drug supplier, both the preferences of the managers over the drug suppliers and the competition relations between the drug suppliers should be considered simultaneously. In this paper, both the weights of experts and attributes are determined objectively by the proposed method and the final rank- ing of suppliers is obtained according to the relative closeness coefficient values of all suppliers. Based on the calculation results, the difference between the relative closeness coefficients of X1 and X4 is small, namely 0.7874 and 0.7837. Both the alternatives X1 and X4 can be determined as the final drug supplier. That is in line with the actual situation. 6.3. Comparative analyses Algorithm 3 is used to obtain the optimal decision results in MAGDM. The psychological distance measures in Algorithm 3 can not only express the preference information of DMs on attributes but also measure the deviations be- tween different alternatives. To validate the proposed method, this section provides some comparative analyses from both numerical and theoretical points of view. Please cite this article as: H. Liao et al., A multiple attribute group decision making method based on two novel intuition- istic multiplicative distance measures, Information Sciences (2018), https://doi.org/10.1016/j.ins.2018.05.023 JID: INS ARTICLE IN PRESS [m3Gsc;June 11, 2018;10:4] 16 H. Liao et al. / Information Sciences 000 (2018) 1–18 Table 11 Results of the IM-TOPSIS method with different distance measures and parameters. The Minkowski distance The generalized weighted The generalized hybrid weighted distance Hausdorff distance λ=1 λ=2 λ=3 λ=1 λ=2 λ=3 λ=1 λ=2 λ=3 X1 0.7907 0.7879 0.7932 0.7865 0.7914 0.7985 0.7888 0.7898 0.7963 X2 0.1565 0.2131 0.2237 0.1573 0.1941 0.2010 0.1570 0.2026 0.2109 X3 0.5332 0.5107 0.4925 0.5362 0.5226 0.5097 0.5347 0.5171 0.5025 X4 0.8043 0.7571 0.7366 0.8002 0.7661 0.7489 0.8021 0.7621 0.7436 X5 0.6584 0.6492 0.6537 0.6613 0.6522 0.6553 0.6600 0.6509 0.6547 Ranking lists 41,532 14,532 14,532 41,532 14,532 14,532 41,532 14,532 14,532 6.3.1. Numerical comparison with the IM-TOPSIS and IM-VIKOR methods Firstly, the IM-TOPSIS method [18] is used to solve the drug supplier selection problem. The calculation results are shown in Table 11. Then, the IM-VIKOR method [18] is applied to solve this problem and the ranking result is obtained as follows: if λ = 0.2, then X1 X4 X5 X3 X2 ; if λ = 0.5, 0.8, then X4 X1 X5 X3 X2 . The compromise solutions are X1 and X4 in each case. Either the IM-TOPSIS method or the IM-VIKOR method can yield the same result that the best drug supplier is X1 or X4 . From Table 11, these ranking results are quite similar using different distance measures, including the Minkowski distance measure, the generalized weighted Hausdorff distance measure and the generalized hybrid weighted distance measure. But as the parameter λ changes, the ranking result changes slightly: when λ = 1, the ranking result is X4 X1 X5 X3 X2 ; when λ =2 or 3, the ranking result is X1 X4 X5 X3 X2 . All of these results show that X1 or X4 is the optimal decision result for the hospital. In addition, the compromise solution is X1 and X4 by the IM-VIKOR method no matter the DMs choose the largest group utility or the minimum individual regret value. As can be seen, the proposed psychological distance measure is more stable than the above-mentioned methods and thus can be widely used in MAGDM. 6.3.2. Theoretical comparison with other methods Besides the IM-TOPSIS and the IM-VIKOR methods, there are some other methods for MAGDM problems with IMNs. (1) The IMAHP method and IMGAHP method Ren et al. [21] introduced the intuitionistic multiplicative analytic hierarchy process (IMAHP), which applies the intuition- istic multiplicative preference relation (IMPR) to calculate the attribute weights and then uses some aggregation operators to yield the ranking of alternatives. In this method, the IMPR of attributes and the decision matrix of alternatives with re- spect to attributes need to be constructed first. As usual, the consistency of the IMPR given by DMs should be checked and the inconsistent IMPR should be repaired till it with acceptable consistency before determining the ranking of alternatives. However, this iterative process is time-consuming and may be complex if DMs are under the pressure of time. In addition, Ren et al. [21] assigned the weights of DMs directly by a priori, which may lead the final ranking result unreasonable as the result is effected by the subjectivity of DMs. Moreover, the concept of the IMPR they defined was questionable [38] and the priority-generation method was not based on the consistency of IMPRs. To solve the drawbacks in Ren et al. [21], Zhang and Pedrycz [38] developed the intuitionistic multiplicative group an- alytic hierarchy process (IMGAHP) method, which contains four main steps, including the expressing, decomposition, and pairwise comparison stage, the consistency checking and improving state, the information aggregation stage, and the rank- ing exploitation stage. Compare with the IMGAHP method, our method mainly contains the weight determination stage and the ranking exploitation stage. It is apparent that the IMPRs of attributes are not needed in our method, and the cumber- some iteration processes for consistency checking and repairing are avoided as well. In this sense, the calculation complexity of our method is lower than the IMGAHP method. In addition, the projection-based distance measure and psychological dis- tance measure are applied in our method, which can represent DMs’ cognition comprehensively. (1) Aggregation operator-based methods Different aggregation operators were proposed to fuse the intuitionistic multiplicative information. Based on these oper- ators, some MAGDM methods were proposed. For example, Qian and Liu [20] introduced a MAGDM approach, which uses the IMWA and IMWG operators to aggregate experts’ evaluation information and then utilizes the comparison laws to rank the alternatives. Xia et al. [29] used the extended intuitionistic multiplicative power average operator and the intuitionis- tic multiplicative Choquet ordered average operator to aggregate the decision information of DMs, and then obtained the ranking results according to the score function and accuracy function of IMNs. Similarly, Yu and Fang [34] also proposed a MAGDM method based on the proposed aggregation operators of IMNs. As we can see, in this type of methods, the aggregation operators and the comparison laws are the core parts. As it is known to all, the decision information may be lost in the aggregation process. At the same time, both the weights of attributes and experts were determined by the DMs subjectively, which may lead the final results not reasonable. However, Please cite this article as: H. Liao et al., A multiple attribute group decision making method based on two novel intuition- istic multiplicative distance measures, Information Sciences (2018), https://doi.org/10.1016/j.ins.2018.05.023 JID: INS ARTICLE IN PRESS [m3Gsc;June 11, 2018;10:4] H. Liao et al. / Information Sciences 000 (2018) 1–18 17 Table 12 Methods for group decision making with IMNs. Paper Decision information Weights (experts/attributes) Methods Xia et al. [29] IMPR Experts’ weights were given by DMs Aggregation operators Comparison laws Liao et al. [18] Decision matrix with IMNs All weights were derived from DMs IM-TOPSIS IM-VIKOR Qian and Niu [20] Decision matrix with IMNs Experts’ weights were given by DMs Aggregation operators Yu and Fang [34] Comparison laws Ren et al. [21] IMPR Experts’ weights were given by DMs IMAHP Zhang and Pedrycz [38] IMPRs All weights are derived from IMPRs IMGAHP This paper Decision matrix with IMNs All weights were derived from DMs A MAGDM method based on psychological distance and projection distance our method not only operates the decision information given by DMs reasonably but also considers the weight information of attributes and experts. This paper uses the proposed projection-based method and the intuitionistic multiplicative entropy to calculate the weights of experts and attributes based on the evaluation information. The projection-based distance measure can measure the deviation between IMSs from two aspects, including the distance and angle, while the psychological distance measure can measure the deviation between IMSs by fully considering the DMs’ psychological preference over alternatives and the competitive relations among alternatives. The weights generated by the distance-based methods is more reasonable. It can not only avoid the subjectivity of DMs but also express the DMs’ preference over alternatives. Besides the method in Ref. [38], no other method considered the case that the weights of experts and attributes are unknown or partly unknown si- multaneously, but our proposed method can tackle this issue perfectly. Furthermore, the MAGDM method proposed in this paper sorts the alternatives according to the relative coefficients of them. There is no aggregation of decision information in our method, except the weighted decision matrix as auxiliary decision tools. The main differences between our method and the above-mentioned methods are shown in Table 12. 7. Conclusion The IMS, as a new form of information representation, has been widely used in MAGDM problems. As a basic component of MAGDM method, several distance measures have been proposed in the intuitionistic multiplicative environment. However, we found that there are some problems in the existing distance measures. Thus, this paper defined the projection-based distance measure and the psychological distance measure of IMSs. Then, a new intuitionistic multiplicative entropy was introduced, which can be used to derive the weights of experts and attributes that are completely unknown or partly known. Based on these concepts, a new MAGDM method was proposed and then employed to the drug supplier selection problem for hospital management. Comparative analyses between our method with other existing methods were given to show the efficiency and practicability of the proposed method. In the future, the MAGDM method can be implemented to other practical decision-making problems. Meanwhile, it can be extended to other decision environments, such as interval-valued intuitionistic multiplicative number and hesitant fuzzy linguistic term set [16]. We will try to investigate some new methods by combining our method with other well-known MADM methods, such as the PROMETHEE and ELECTRE. Acknowledgments The authors would like to thank the editors and anonymous reviewers for their insightful and constructive com- mendations that have led to an improved version of this paper. The work was supported by the National Natu- ral Science Foundation of China (71501135, 71771156, 71532007, 71131006), the China Postdoctoral Science Foundation (2016T90863, 2016M602698), and the Scientific Research Foundation for Excellent Young Scholars at Sichuan University (No. 2016SCU04A23). References [1] A. Alinezad, A. Seif, N. 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