International Journal of Machine Learning and Cybernetics https://doi.org/10.1007/s13042-018-0845-2

ORIGINAL ARTICLE

Multi-criteria decision-making method based on dominance degree and BWM with probabilistic hesitant fuzzy information Jian Li1   · Jian‑qiang Wang1 · Jun‑hua Hu1 Received: 5 August 2017 / Accepted: 7 June 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract In this paper, multi-criteria decision-making (MCDM) methods with probabilistic hesitant fuzzy information are proposed based on the dominance degree of probabilistic hesitant fuzzy elements (PHFEs) and best worst method (BWM). First, we discuss the probabilistic distribution function of PHFE and the dominance degree matrix between two PHFEs. The dominance degree matrix is constructed based on the probabilistic distribution function of PHFE, which can be characterized as a fuzzy complementary judgment matrix. Second, BWM is extended to fuzzy preference relations based on the constructed dominance degree matrix. Subsequently, an algorithm is designed for selecting the best and worst weight vectors, and then two models are developed based on additive consistency and multiplicative consistency of fuzzy preference relations to derive the criteria weights. In addition, an algorithm is presented to improve the consistency of the dominance degree matrix when a desired consistency level is not achieved. Finally, the selection of best investment company is provided as an example to demonstrate the feasibility and effectiveness of the proposed methods. Keywords  Probabilistic hesitant fuzzy element · Probabilistic distribution function · Dominance degree matrix · Best worst method · Fuzzy preference relations

1 Introduction Analytic hierarchy process (AHP), as a classic decisionmaking method, has been studied deeply by numerous scholars over the last decades [1–3]. In classical AHP, decisionmaking processes are based on the pairwise comparisons of real numbers. With rapid social, economic, and scientific development, fuzziness and uncertainty frequently occur in decision-making processes. In some cases, classical AHP is ineffective for solving fuzzy decision-making problems. A number of fuzzy theories have been applied to combine with classical AHP, such as fuzzy AHP [4, 5], intuitionistic fuzzy AHP [3, 6–8], hesitant fuzzy AHP [9–11], and linguistic AHP [12–14]. In fuzzy decision-making processes, these extended AHP methods provide more complex structures than that of the classical AHP method because they combine with qualitative and quantitative evaluation values. In addition, these methods have been applied to solve * Jian‑qiang Wang [email protected] 1



School of Business, Central South University, Changsha 410083, People’s Republic of China

numerous practical decision-making problems, such as the safety evaluation of coal mine [15], classification of urban emergency [16], evaluation of supply chains [17], and evaluation of intelligent green building policies [18]. Among the aforementioned extended AHP methods, hesitant fuzzy AHP represents the judgments of pairwise comparisons with a set of real numbers in the unit closed interval [0, 1]. In this way, hesitant fuzzy AHP is more convenient for describing hesitant and uncertain evaluations than fuzzy AHP. Consequently, hesitant fuzzy AHP receive increasing attention as a research topic [9–11, 19–22]. However, a defect of hesitant fuzzy element (HFE) was discussed, that is, the serious loss of information gradually emerged. In this study, we take the following example as an illustration. Suppose that seven experts evaluate alternatives with respect to the criteria on a scale of one (0.1, worst; 1, best). These experts provide evaluation values as follows: three experts provide the value 0.2; one expert provides two values 0.2 and 0.3; two experts provide the value 0.5; and one expert provides the value 0.7. According to Torra [23], the above evaluation information can be denoted as a HFE {0.2, 0.3, 0.5, 0.7} . Obviously, the HFE {0.2, 0.3, 0.5, 0.7} cannot fully represent the evaluation information from

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experts because the probabilities of each evaluation value are unequal. To remove this shortcoming, Zhu [24] introduced the concept of probabilistic hesitant fuzzy element (PHFE), which provides hesitant evaluation values corresponding with their probabilities. PHFE appears more accurate and descriptive than HFE and can be used to describe the example above. Then, the above evaluation information can be denoted using PHFE {0.2(1∕2), 0.3(1∕14), 0.5(2∕7), 0.7(1∕7)} . In this way, the probabilities are added to the evaluation values to cover the existing defect of HFE. Following the work of Zhu [24], the probabilistic hesitant fuzzy set (PHFS) receive increasing attention, and some significant studies are conducted based on PHFS [25–33]. For example, Xu and Zhou [27] constructed score and deviation functions, comparison laws, and basic operations of PHFS. Zhang et al. [29] observed some missing values in PHFE, and then proposed an improved PHFS. Li and Wang [32] introduced the Hausdorff distance between two PHFEs. Distance measures, scores, and deviation degrees for PHFEs were also discussed in [34]. With the development of probabilistic hesitant fuzzy theory, probabilistic hesitant fuzzy AHP was developed and applied in many fields [24, 35–38]. Zhu [24] introduced a stochastic method to improve the consensus degrees of probabilistic hesitant fuzzy preference relations (PHFPRs) and developed it to solve energy channel selection problems. Wu et al. [35] provided a novel consensus reaching method for group decision-making problems based on three levels (e.g., an alternative pair level, an alternative level, and a preference relations level). Wu and Xu [37] proposed a novel distance measure for probabilistic hesitant fuzzy preference elements (PHFPEs), and then, the consensus measures were investigated based on the proposed distance. Zhou and Xu [38] demonstrated the probability calculation method for PHFPEs based on expected consistency and obtained missing occurrence probabilities of the elements in PHFPEs. In addition, probabilistic hesitant fuzzy AHP was developed under linguistic environments [39, 40]. Several scholars have studied multi-criteria decisionmaking (MCDM) problems under probabilistic hesitant environment. However, some issues still exist and are discussed as follows. (1) Zhang et al. [29] proposed the normalization of PHFE. The normalization related to any two PHFEs should have the same length and arrangement. Moreover, the process heavily relies on the decision-makers’ subjectivity, that is, conservative decision-makers add minimum values in short PHFEs, while optimistic decisionmakers add maximum values. Therefore, this normalization process will generate various decision results. (2) Aggregation operator methods [27] assess information using PHFE. Xu and Zhou [27] provided several aggregation operators to integrate the evaluation information, and then the MCDM process was proposed based on the proposed aggregation

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operators. However, the randomness of the proposed method has received a little attention. (3) PHFPRs have been used in qualitative decision-making environments [24, 35–38]. These methods utilize expected consistency to integrate hesitant values with corresponding probabilities, and then, PHFPRs turn into fuzzy preference relations. However, the relationship between the hesitant values and corresponding probabilities has received a little attention. To overcome the aforementioned limitations, this paper focuses on presenting the dominance degrees of PHFEs and integrating the Best Worst Method (BWM) to develop a novel MCDM method. Rezaei [41, 42] introduced the BWM to obtain weight vectors. Subsequently, Mou et al. [43, 44] extended BWM to intuitionistic fuzzy environments. Moreover, Guo and Zhao [45] extended BWM to triangular fuzzy environments. In BWM, decision-makers should first identify the best and worst weight vectors and then make pairwise comparisons between these weight vectors. Finally, the optimal model is constructed to determine optimal weight vectors. This method has been widely used in various fields due to its simple logic [46–49]. Although many existing studies have explored BWM, only few studies have focused on 0.1–1 scale to perform the pairwise comparisons of decision-makers’ preferences. Given that the 0.1–1 scale has been applied to many fuzzy sets and fuzzy preference relations, it is necessary to focus on this scale. The main motivations and contributions of this study are summarized as follows: 1. To overcome the limited attention on the randomness of PHFEs, we discuss the probability distribution function of PHFEs and construct the dominance degree matrix between two PHFEs. The constructed dominance degree matrix can be characterized as a fuzzy complementary judgment matrix. 2. To combine the advantages of both dominance degree matrix and BWM simultaneously, a novel MCDM method with PHFE is developed. 3. In order to improve the ability of BWM by extending it to fuzzy preference relations. Two models are constructed to derive weight vectors based on the additive consistency and multiplicative consistency of fuzzy preference relations. This strategy can provide a more powerful decision-making tool. This paper is organized as follows. In Sect. 2, we review concepts related to dominance degree of the probability distribution function, HFS, and PHFS. In Sect. 3, we discuss the dominance degree between two PHFEs and then extend BWM-derived weight vectors to fuzzy preference relations. Two models are constructed based on the additive consistency and multiplicative consistency of fuzzy preference relations. In Sect. 4, a numerical example is applied to

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demonstrate the proposed MCDMs. Finally, we draw conclusions in Sect. 5.

2 Preliminaries This section briefly reviews some definitions related to the dominance degrees of probability distribution function, HFS, and PHFS.

2.1 Dominance degrees of probability distribution function The probability distribution function is an important concept in probability theory. In the following, the probability distribution function of discrete random variables is described. Definition 1 [50]. Let X be a discrete random variable, and x be the possible value of X  , then p(x) is the probability distribution function of X if and only if 1. The probability of possible value x takes a specific value p(x) , that is, P(X = x) = p(x). 2. p(x) is non-negative for all possible values. 3. The sum of p(x) over all possible values is 1, that is, ∑+∞ p(x) = 1. x=−∞ Based on the probability distribution function of two discrete random variables X1 and X2 , the probabilities of x1 > x2 , x1 = x2 and x1 < x2 can be obtained as follows. Definition 2 [51]. Let X1 and X2 be two independent discrete random( variables, ) (x1 and ) x2 be the outcomes of X1 and X2 , and p1 x1 and p2 x2 be the probability distribution functions of X1 and X2 . Then, the probabilities of x1 > x2 , x1 = x2 , and x1 < x2 are obtained from the following formulas: +∞ ∑ ( ) p x1 > x2 =

x1 ∑

+∞ ∑ ( ) ( ) ( ) ( ) p1 x1 p2 x2 − p1 x1 p2 x1 ,

x1 =−∞ x2 =−∞

x1 =−∞

+∞ ∑ ( ) ( ) ( ) p x1 = x2 = p1 x1 p2 x1 ,

+∞ ∑ ( ( ) ( )) D p1 x1 ≳ p2 x2 =

x1 ∑

( ) ( ) p1 x1 p2 x2

x1 =−∞ x2 =−∞

− 0.5

+∞ ∑

(1)

( ) ( ) p1 x1 p2 x1 ,

x1 =−∞

( ) ( ) and the dominance degree of p2 x2 over p1 x1 is defined as +∞ +∞ ∑ ( ) ( ) ∑ ( ( ) ( )) D p2 x2 ≳ p1 x1 = p1 x1 p2 x2 x1 =−∞ x2 =x1

− 0.5

+∞ ∑

(2)

( ) ( ) p1 x1 p2 x1 .

x1 =−∞

( ( ) ( )) D p x ≳ p x2 and The dominance degrees of 1 1 2 ( ( ) ( )) D p2 x2 ≳ p1 x1 have the following relationship, which has been proven in [51]. ( ( )

( ))

( ( )

( ))

Property 1 Let D p1 x1 ≳ p2 x2 and D p2 x2 ≳ p1 x1 ( ) ( ) ( ) be the dominance degrees of p1 x1 over p2 x2 and p2 x2 ( ) ( ( ) ( )) over p1 x1  , respectively. Then, D p1 x1 ≳ p2 x2 + ( ( ) ( )) D p2 x2 ≳ p1 x1 = 1.

2.2 Hesitant fuzzy set and probabilistic hesitant fuzzy set In this subsection, the concepts of HFS and PHFS are introduced, and their difference is described. Definition 4 [23]. Let X be a reference set, A HFS on X is a function hA that returns a subset of values in [0, 1]. To be easily understood, Xia and Xu [52] presented HFS by a mathematical symbol as follows: { } A = < x, hA (x) > |x ∈ X ,

x1 =−∞

+∞ +∞ +∞ ∑ ∑ ( ) ( ) ∑ ( ) ( ) ( ) p x1 < x2 = p1 x1 p2 x2 − p1 x1 p2 x1 . x1 =−∞ x2 =x1

Definition 3 [51]. Let X1 and X2 be two independent(discrete ) random distributions p1 x1 and � � ( ) variables with probability∑ +∞ p x = 1 and p2 x2  , respectively, where x1 =−∞ 1 1 � � ( ) ∑+∞ p x = 1 . The dominance degree of p1 x1 over x2 =−∞ 2 2 ( ) p2 x2 is defined as

x1 =−∞

Based on the above formulas, ( ) Liu et al. ( )[51] proposed ( ) the dominance degrees of p x over p x and p x2 over 1 1 2 2 2 ( ) p1 x1 as follows.

where the function hA (x) is a set of different values in [0, 1], representing the possible membership degrees of element x to A . For convenience, hA (x) is called HFE. To collect the decision-makers’ preferences effectively, Zhu [24] introduced the concept of PHFS as follows: Definition 5 [24]. Let X be a reference set. Then, PHFS on X is expressed by a mathematical symbol:

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{ ( ) } AP = < x, hx px > |x ∈ X , where the function hx is a set of different values in [0, 1], which is described by the corresponding probabilistic distribution px . And hx denotes the possible membership degrees ( ) of the element to A . For convenience, hx px is called PHFE and indicated as ( ) { ( ) } hx px = 𝛾i pi |i = 1, 2, … , #h , where pi is the probability of the possible value 𝛾i satisfy( ) ∑#h ing i=1 pi = 1 , and #h is the number of all PHFE 𝛾i pi in ( ) hx px . If the probabilities of the possible values in a PHFE are ignored, then the values are considered with the same probability. In this case, PHFE turns into HFE. The out-degree of node is applied to determine the best and worst weight vectors in the following section. We restate its concept as follows. Definition 6 [53]. The out-degree of node i is denoted as  , which is the number of all arcs whose arrow tails are Dout i the node i .

where xi , i=1, 2, ⋯ , #h is the discrete random variables denoting all possible values in a PHFE. Probability distribution function p(x) is an alternative to the representation of PHFE. In probability distribution function p(x) , condition values are determined from the possible values ( 𝛾i , and ) the probability of possible value is pi , that is, P X = 𝛾i = pi , (i=1, 2, ⋯ , #h). Given that the possible values in PHFEs belong to [0, 1], the probabilities ( ) of x1(> x)2 , x1 = x2 , and x1 < x2 relate to PHFEs hx1 px1 and hx2 px2 can be determined based on Definition 2.

( ) { ( ) } Definition 8 Let hx1 px1 = 𝛾i pi | i=1, 2, ⋯ , #h1 and ( ) { ( ) } hx2 px2 = 𝛾j pj | j=1, 2, ⋯ , #h2 be two PHFEs, and ( ) ( ) p1 x1 and p2 x2 be the probability distribution functions ( ) ( ) � � ∑1 of hx1 px1 and hx2 px2  , respectively, where x =0 p1 x1 = 1 1 � � ∑1 and x =0 p2 x2 = 1 . Let x1 and x2 be the outcomes of 2 ( ) ( ) PHFEs hx1 px1 and hx2 px2  , respectively. Then, the probability of x1 > x2 is determined as x1 1 1 ∑ ( ( ) ( ) ∑ ( ) ( ) ) ∑ p1 x1 p2 x2 − p1 x1 p2 x1 , p x1 > x2 = x1 =0 x2 =0

3 Dominance degrees of probabilistic hesitant fuzzy elements and best worst method

and the probability of x1 = x2 is determined as

In this section, the dominance degrees of PHFEs are defined. Next, the fuzzy preference relations are obtained based on the dominance degrees matrix. Finally, BWM-derived weight vectors are extended to fuzzy preference relations.

3.1 Dominance degrees of probabilistic hesitant fuzzy elements

( ) { ( ) } Definition 7 Let hx px = 𝛾i pi |i=1, 2, … , #h be a PHFE, and its probability distribution function p(x) can be obtained as follows:

13

1 ( ) ∑ ( ) ( ) p x1 = x2 = p1 x1 p2 x1 ,

(3)

(4)

(5)

x1 =0

and the probability of x1 < x2 is determined as 1 1 1 ∑ ( ) ∑ ( ) ( ) ∑ ( ) ( ) p x1 < x2 = p1 x1 p2 x2 − p1 x1 p2 x1 . x1 =0 x2 =x1

For every PHFE with two parameters 𝛾i and pi . 𝛾i is the condition values, and pi is the probability of 𝛾i . In this case, each PHFE can be regarded as a discrete random variable. In the following, the probability distribution function and dominance degrees related to PHFE are introduced.

⎧p1 x1 = 𝛾1 , ⎪ ⎪p2 x2 = 𝛾2 , p(x) = ⎨ ⎪ ⋯ #h ⎪p ⎩ #h x = 𝛾#h .

x1 =0

x1 =0

(6) ( ) h p = 0.6(0.3), 0.7(0.5)} Example 1 Let and {0.4(0.2), x1 x1 ( ) hx2 px2 = {0.3(0.1), 0.4(0.3), 0.5(0.4), 0.6(0.2)} be two PHFEs. According ( to ) Definition ( 7, ) the probability distribution functions of hx1 px1 and hx2 px2 can be obtained as follows:

⎧0.2 � � ⎪ p1 x1 = ⎨0.3 ⎪ ⎩0.5

⎧0.1 ⎪ � � ⎪0.3 p2 x2 = ⎨ ⎪0.4 ⎪0.2 ⎩

x11 = 0.4; x12 = 0.6; x13 = 0.7.

x21 = 0.3; x22 = 0.4; x23 = 0.5; x24 = 0.6.

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According to Definition 8, the probabilities of x1 > x2 , x1 = x2 , and x1 < x2 can be calculated as follows: (

)

p x1 > x2 = 0.2 × (0.1 + 0.3) + 0.3 × (0.1 + 0.3 + 0.4 + 0.2) + 0.5 × (0.1 + 0.3 + 0.4 + 0.2)

−0.2 × 0.3 − 0.3 × 0.2 = 0.76 ( ) p x1 = x2 = 0.2 × 0.3 + 0.3 × 0.2 = 0.12 ( ) p x1 < x2 = 0.2 × (0.3 + 0.4 + 0.2) + 0.3 × 0.2 − 0.2 × 0.3 − 0.3 × 0.2 = 0.12 To combine(Definition 3 with ) ( )Definition ( 8, )the dominance ( ) degrees of hx1 px1 over hx2 px2 and hx2 px2 over hx1 px1 are given as follows. ( ) { ( ) } Definition 9 Let hx1 px1 = 𝛾i pi | i=1, 2, ⋯ , #h1 and ( ) { ( ) } hx2 px2 = 𝛾j pj | j=1, 2, ⋯ , #h2 be two PHFEs, and ( ) ( ) p1 x1 and p2 x2 be the probability distribution functions ( ) ( ) of PHFEs hx1 px1 and hx2 px2  , respectively, where � � � � ∑1 ∑1 p x = 1 and x =0 p2 x2 = 1 . The dominance x1 =0 1 1 2 ( ) ( ) degree of hx1 px1 over hx2 px2 is then defined as ( ( ) ( )) ( ) ( ) D hx1 px1 ≳ hx2 px2 = p x1 > x2 + 0.5p x1 = x2 ,

(7) ( ) ( ) and the dominance degree of hx2 px2 over hx1 px1 is defined as

(

( ) ( )) ( ) ( ) D hx2 px2 ≳ hx1 px1 = p x1 < x2 + 0.5p x1 = x2 .

(8) ( ( ) ( )) of D hx1 px1 ≳ hx2 px2 and (The( dominance ) ( degrees )) D hx2 px2 ≳ hx1 px1 have the following relationship, which is similar to Property 1.

( ( ) ( )) ( ( ) Property 2 Let D hx1 px1 ≳ hx2 px2 and D hx2 px2 ≳ ( ) ( ) ( )) hx1 px1 be the dominance degrees of hx1 px1 over hx2 px2 ( ) ( ) and hx2 px2 over hx1 px1  , respectively. Then, we have ( ( ) ( )) ( ( ) ( )) D hx1 px1 ≳ hx2 px2 + D hx2 px2 ≳ hx1 px1 = 1. The proof of Property 2 can be easily obtained from that of Property 1, and is excluded here. Example 2  Based on ( the ) data given ( )in Example 1, the dominance degree of hx1 px1 over hx2 px2 is calculated as follows:

( ( ) ( )) ( ) ( ) D hx1 px1 ≳ hx2 px2 = p x1 > x2 + 0.5p x1 = x2 = 0.76 + 0.5 × 0.12 = 0.82; ( ) ( ) and the dominance degree of hx2 px2 over hx1 px1 is calculated as follows:

( ( ) ( )) ( ) ( ) D hx2 px2 ≳ hx1 px1 = p x1 < x2 + 0.5p x1 = x2 = 0.12 + 0.5 × 0.12 = 0.18. ( ( ) ( )) ( ( ) O v i o u s l y , D hx1 px1 ≳ hx2 px2 + D hx2 px2 ≳ ( b )) hx1 px1 = 0.82 + 0.18 = 1.

3.2 Best worst method In BWM, the best and worst weight vectors should be identified first. Then, the pairwise comparisons between the best to other weight vectors and other weight vectors to the worst are conducted. Finally, the optimal model is constructed to determine the optimal weight vectors. In this subsection, BWM-derived weight vectors are extended to fuzzy preference relations. First, an algorithm is designed for selecting the best and worst weight vectors. Second, the consistency of fuzzy preference relations is checked according to additive consistency and multiplicative consistency. Finally, an algorithm is designed to improve the consistency of the fuzzy preference relations when a desired consistency level is not achieved. The dominance degree matrix can be obtained with respect to Definition 9, and according to Property 2, the dominance degree matrix is a fuzzy preference relation. In the following subsection, the weight vectors of dominance degree matrix can be derived according to BWM. The process is shown in Fig. 1. 3.2.1 Algorithm for selecting the best and worst weight vectors Motivated by the algorithm proposed by Mou et al. [44], a similar algorithm is designed to select the best and worst weight vectors of the dominance degree matrix in the following. Algorithm 1 Step [ ]1 Establish the dominance degree matrix D = Dij n×n using Eqs. (7) and (8). Step 2 Calculate the out-degrees of all weight vectors:

Dout = i

∑n

v , j=1 ij

(i = 1, 2, … , n)

where the value of vij can be calculated using the following binary relation: { 1, if Dij ≥ 0.5 vij = 0, if Dij < 0.5 of the weight vectors. Step 3 Output all out-degrees Dout i Step 4 The best and the worst weights vector can be determined according to the values of Dout as i

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Set up the dominance degree matrix

Calculate the out-degree values

Determine the best and worst weight vectors Additive consistency

Feedback

Multiplicative consistency

Determine the optimal solution

Determine the optimal solution

Determine the deviation variable

Determine the deviation variable

Determine the consistency ratio

Determine the consistency ratio

No

Acceptable consistency? Yes

Determine the weight variable and the acceptable consistency matrix

Fig. 1  Process of derived weight vectors by best and worst method

{ { } , wB = max } Dout , Dout , … , Dout   , and wW = min Dout n 1 1 2 out Dout , … , D . n 2

If fuzzy preference relation is an additive consistency, then the element 𝛾best,j and the weights wbest and wj are relative, and there is

3.2.2 Check the consistency of dominance degree matrix Tanino [54] introduced two types of consistency (additive consistency and multiplicative consistency) for fuzzy preference relations based on the additive and multiplicative transitivity of pairwise comparisons. Motivated by the idea of BWM [41], BWM-derived weight vectors are extended to fuzzy preference relations related to additive consistency and multiplicative consistency. 1. Additive consistency fuzzy preference relations

( ) Definition 10 Let A = 𝛾ij n×n be fuzzy preference relations. Then, A is an additive consistency if { 𝛾best,i + 𝛾ij = 𝛾best,j + 0.5 𝛾ij + 𝛾j,worst = 𝛾i,worst + 0.5

13

,

(best ≤ i ≤ j ≤ worst), (9)

𝛾best,j =

) n − 1( wbest − wj + 0.5, 2

(10)

where n is the order of fuzzy preference relations. Similarly, the relationship between the element 𝛾j,worst and the weights wj and wworst is given as

𝛾j,worst =

) n − 1( wj − wworst + 0.5. 2

(11)

If fuzzy preference relations are inconsistent, then Eqs. (10) and (11) do not hold. Motivated by the idea in [41, 44], the priority weight vectors can be obtained by minimizing the ) | ( | w − w + 0.5 − 𝛾 maximum absolute differences | n−1 best j best,j || | 2 ) | ( | and | n−1 wj − wworst + 0.5 − 𝛾j,worst | . Thus, the following | 2 | mathematical model can be constructed: Model 1 min 𝜍

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s.t.

In the following, the BWM derive weight vector is extended to fuzzy preference relations with multiplicative consistency. Similar results can be obtained according to the additive consistency of fuzzy preference relations discussion above.

) | |n − 1( | | | 2 wbest − wj + 0.5 − 𝛾best,j | ≤ 𝜍 | | ) | |n − 1( | | | 2 wj − wworst + 0.5 − 𝛾j,worst | ≤ 𝜍 | | ∑n wj = 1

( ) Definition 11 Let A = 𝛾ij n×n be fuzzy preference relations. Then, A is multiplicative consistency if

j=1

𝜍 ≥ 0, wj ≥ 0, j = 1, 2, ⋯ , n

{

𝜍A∗ and optimal weight vectors ( Optimal solution ) w1A , w2A , ⋯ , wnA can be obtained by solving Model 1. If the optimal solution 𝜍A∗ is sufficiently small, then the consistency of the fuzzy preference relations is acceptable. To measure the consistency of fuzzy preference relations, the consistency ratio is discussed in the following. Motivated by the idea in [41, 44], the maximum consistency degree of fuzzy preference relations can be calculated according to the following equalities: {

𝛾ij 𝛾j,worst 𝛾worst,i = 𝛾ji 𝛾worst,j 𝛾i,worst

𝛾best,j =

0.1 ≤ 𝛾best,worst ≤ 0.5

𝛾best,j − 𝛿A + 𝛾j,worst − 𝛿A − 0.5 = 𝛾best,worst + 𝛿A ,

0.5 ≤ 𝛾best,worst ≤ 1

(12)

and the relationship between the element 𝛾j,worst and the weights wj and wworst is described as

0.1 ≤ 𝛾best,worst ≤ 0.5

𝛾best,worst − 𝛿A + 𝛾best,worst − 𝛿A − 0.5 = 𝛾best,worst + 𝛿A ,

0.5 ≤ 𝛾best,worst ≤ 1

After solving Eq. (13), the deviation variable 𝛿A can be obtained as follows:

𝛾j,worst =

𝛿A

wj wj + wworst

(18)

.

| w | | | best s.t. | − 𝛾best,j | ≤ 𝜍 | wbest + wj | | | | | wj | | − 𝛾j,worst | ≤ 𝜍 | | | wj + wworst | | ∑n wj = 1

(15)

.

(13)

.

If fuzzy preference relations are inconsistent, then Eqs. (17) and (18) do not hold. Priority weight vectors can be obtained by minimizing the maximum absolute differences | wbest | | wj | | | | | | wbest +wj − 𝛾best,j | and | wj +wworst − 𝛾j,worst | . Thus, the following | | | | mathematical model can be constructed: Model 2 min 𝜍

(14)

Placing the values of 𝛾best,worst into the Eq. (14), the consistency index 𝛿A can be obtained. And some consistency index values are shown in Table 1. Therefore, the consistency ratio can be calculated as follows:

CRA =

(17)

,

𝛾best,worst + 𝛿A + 𝛾best,worst + 𝛿A − 0.5 = 𝛾best,worst − 𝛿A ,

𝜍A∗

(best ≤ i ≤ j ≤ worst).

wbest , wbest + wj

𝛾best,j + 𝛿A + 𝛾j,worst + 𝛿A − 0.5 = 𝛾best,worst − 𝛿A ,

⎧ 0.5 − 𝛾best,worst , 0.1 ≤ 𝛾 best,worst ≤ 0.5 ⎪ 3 . 𝛿A = ⎨ ⎪ 𝛾best,worst − 0.5 , 0.5 ≤ 𝛾 best,worst ≤ 1 ⎩ 3

⋅

(16) If fuzzy preference relations with multiplicative consistency, then the element 𝛾best,j and the weights wbest and wj are relative, and there is

where 𝛿A is a deviation variable. When 𝛾best,j = 𝛾j,worst = 𝛾best,worst , Eq. (12) can be converted into

{

𝛾best,i 𝛾ij 𝛾j,best = 𝛾i,best 𝛾ji 𝛾best,j

2. Multiplicative consistency of fuzzy preference relations

j=1

𝜍 ≥ 0, wj ≥ 0, j = 1, 2, … , n Table 1  Consistency index for additive fuzzy BWM

𝛾best,worst

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

𝛿A

0.13

0.1

0.07

0.03

0

0.03

0.07

0.1

0.13

0.17

13



International Journal of Machine Learning and Cybernetics

The smaller the consistency ratio, the better the consistency of the fuzzy preference relations is. If CRA , CRM ≤ 0.1 , then the optimal weight vectors obtained from Models 1 and 2 are reliable. Otherwise, the consistency will be improved in the following section.

∗ 𝜍M and optimal weight vectors ( The optimal solution ) w1M , w2M , ⋯ , wnM can be obtained by solving Model 2. In the following, the consistency ratio can be obtained similar to fuzzy preference relations with additive consistency. The maximum consistency degree of fuzzy preference relations can be calculated from the following equalities:

{(

)( )( ) ( )( )( ) 𝛾best,j + 𝛿M 𝛾j,worst + 𝛿M 𝛾worst,best + 𝛿M = 𝛾j,best − 𝛿M 𝛾worst,j − 𝛿M 𝛾best,worst − 𝛿M , ( )( )( ) ( )( )( ) 𝛾best,j − 𝛿M 𝛾j,worst − 𝛿M 𝛾worst,best − 𝛿M = 𝛾j,best + 𝛿M 𝛾worst,j + 𝛿M 𝛾best,worst + 𝛿M ,

0.1 ≤ 𝛾best,worst ≤ 0.5 0.5 ≤ 𝛾best,worst ≤ 1

, (19)

where 𝛿M is a deviation variable. When 𝛾best,j = 𝛾j,worst = 𝛾best,worst  , 𝛾j,best = 𝛾worst,j = 𝛾worst,best  , then Eq.  (19) can be transformed as: {( )( )( ) ( )( )( ) 𝛾best,worst + 𝛿M 𝛾best,worst + 𝛿M 𝛾worst,best + 𝛿M = 𝛾worst,best − 𝛿M 𝛾worst,best − 𝛿M 𝛾best,worst − 𝛿M , ( )( )( ) ( )( )( ) 𝛾best,worst − 𝛿M 𝛾best,worst − 𝛿M 𝛾worst,best − 𝛿M = 𝛾worst,best + 𝛿M 𝛾worst,best + 𝛿M 𝛾best,worst + 𝛿M ,

0.1 ≤ 𝛾best,worst ≤ 0.5 0.5 ≤ 𝛾best,worst ≤ 1

.

(20)

After simplifying the above equations, Eq. (20) can be changed into:

{(

)2 ( ) ( )2 ( ) 𝛾best,worst + 𝛿M 1 − 𝛾best,worst + 𝛿M = 1 − 𝛾best,worst − 𝛿M 𝛾best,worst − 𝛿M , ( )2 ( ) ( )2 ( ) 𝛾best,worst − 𝛿M 1 − 𝛾best,worst − 𝛿M = 1 − 𝛾best,worst + 𝛿M 𝛾best,worst + 𝛿M ,

Placing the values of 𝛾best,worst into the above equations, the consistency index 𝛿M can be obtained. And some consistency index values are shown in Table 2. Therefore, the consistency ratio can be calculated as follows:

CRM =

∗ 𝜍M

𝛿M

When a desired consistency level is not achieved, we should modify some elements in the dominance degree matrix so that a predefined accepted consistency level can be satisfied. The key problems include which elements should be revised and how to revise the elements. To solve these problems, an algorithm is designed to improve the consistency of the dominance degree matrix in the following. Only the condition of fuzzy preference relations with additive consistency is discussed, and that of fuzzy preference relations with multiplicative consistency can be determined similarly.

In order to derive the weights of dominance degree matrix relative to the two consistency definitions, mathematical modeling and the consistency of the fuzzy preference relations are used to construct two models above. The additive consistency of fuzzy preference relations is based on the additive transitivity 𝛾ij + 𝛾jk = 𝛾ik + 0.5 , whereas the multiplicative consistency of fuzzy preference relations is based on the multiplicative transitivity 𝛾ij 𝛾jk 𝛾ki = 𝛾ji 𝛾kj 𝛾ik . Decisionmakers can effectively select definitions and models according to their preferences [43]. However, a disadvantage exists in Model 2 because its nonlinear programming model leads to complexity in computing. Whereas, a simple linear programming model exists in Model 1.

Table 2  Consistency index for multiplicative fuzzy BWM

13

(21)

.

0.5 ≤ 𝛾best,worst ≤ 1

3.2.3 Improve the consistency of the dominance degree matrix

(22)

.

0.1 ≤ 𝛾best,worst ≤ 0.5

Algorithm 2 Step 1 Determine the value of 𝛾best,worst and adjust its value from 𝛾best,worst to 𝛾best,worst + 𝜀. Step 2 Identify the best and worst weight vectors according to Algorithm 1. Step 3 Calculate the optimal solution 𝜍A∗ according to Model 1.

𝛾best,worst

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

𝛿M

0.0633

0.0746

0.0599

0.0325

0

0.0325

0.0599

0.0746

0.0633

0

International Journal of Machine Learning and Cybernetics

Step 4 Calculate the consistency ratio CRA using Eq. (15). If CRA ≤ 0.1 , then go to Step 6; else, go to the next step. Step 5 Return to Step 1. If the improvement of CRA is not obvious, then find out the maximum value of 𝛾best,j , (j ≠ worst) , and adjust its value from 𝛾best,j to 𝛾best,j + 𝜀 . If the improvement of CRA is not obvious, determine the value of minimum 𝛾j,worst , (j ≠ best) and adjust its value from 𝛾j,worst to 𝛾j,worst − 𝜀 . Subsequently, return to Step 2. ∗ Step (6 Output the optimal ) solution 𝜍A , the optimal weight vectors w1A , w2A , ⋯ , wnA  , and the acceptable consistency matrix D′. Remark 1  Adjusted values 𝛾best,worst , 𝛾best,j , or 𝛾j,worst should be in the closed unit interval [0, 1]. Moreover, the sensitivity of the improvement CRA can be determined by a threshold. Remark 2  The adjusted quantity 𝜀 is any small real number greater than 0. Similar to the automatic iterative algorithm proposed by Wu and Xu [55], we can set it as different real numbers, such as 0.1 and 0.05. Remark 3  The principle for this adjustment is increasing ( ) the values related to the best to other weight vectors 𝛾best,j and reducing the ( ) values related to the other weight vectors to worst 𝛾j,worst .

4 Steps of proposed methods and illustrative example In this section, decision-making problems will be adopted to demonstrate how to apply the proposed methods. For MCDM problems with PHFE information, there { } are m alternatives, denoted by A = A1 , A2 , … , Am  . Each{alternative is } assessed under n criteria denoted by C = c1 , c2 , … , cn  . All criteria are independent with each other, and the criteria weights { } are completely unknown. Moreover, D = d1 , d2 , … , dP is a set of decision-makers (assuming the weights of the decision-makers are equal). The evaluation values of alternative Ai with respect to criterion are denoted by ( c)k provided { ( ) by the decision makers } hxk pxk = 𝛾i pi , i = 1, 2, … , #h  , which is a PHFE.

4.1 Steps of the proposed method In this section, the dominance degree is integrated with BWM to develop a novel MCDM method to solve best potential industry selection problems. We only describe the steps of method based on the fuzzy preference relations with additive consistency. Similar steps can be obtained when the fuzzy preference relations with multiplicative consistency are applied.

The steps can be summarized in the following and shown in Fig. 2. Step 1 Normalize the decision-making matrix. Benefit and cost criteria are generally included in MCDM problems. According to the method proposed by Xu and Hu [56], cost type criteria and benefit type criteria have the following relation: { 𝛼ij , for benef it type criteria cj 𝛽ij = , ( )c 𝛼ij , for cost type criteria cj (23)

(i = 1, 2, … , m; j = 1, 2, … , n), ( )c where 𝛼ij is the complement of 𝛼ij , and the complement of PHFE is obtained as follows [27]: {( ( )c )( )} hx px = ∪𝛾i (pi )∈hx (px ) 1 − 𝛾i pi . Step 2 Calculate the overall values of the evaluation information. The overall values of the evaluation information can be obtained according to the following formula [37]: } ( ) { ( )| hox pox = hoi poi |i = 1, 2, ⋯ , #hox , (24) | where hoi is the element that appeared in 𝛾i , and the corresponding probabilities poi are calculated as

poi =

1 ∑m (k) p . k=1 i p

Step 3 Construct dominance degree matrices. [ ] Dominance degree matrices Dk = Dijk n×n , (k = 1, 2, ⋯ , m) can be established using Eqs. (7) and (8). Step 4 Obtain optimal weight vectors and acceptable consistency matrices. According to the process described in Sect. 3.2, overall weight vectors can be obtained as follows: ( ) w1A , w2A , … , wnA =

(

) m m m 1∑ 1∑ 1∑ w1Ak , w2Ak , … , wnAk . m k=1 m k=1 m k=1

(25) In addition, acceptable consistency matrices can be [ ] � �  , (k = 1, 2, … , m). obtained as Dk = Dijk n×n

Step 5 Calculate the ranking index. The ranking index can be determined as follows:

𝜇k =

m n ∑ ∑ i=1 j=1

�

Dijk wiA , (k = 1, 2, ⋯ , m),

(26)

Step 6 Determine the best alternative. The best alternative can be derived according to the maximal value of 𝜇k , (k = 1, 2, … , m).

13



International Journal of Machine Learning and Cybernetics Start

Expert 1

Expert 2

Expert p

Provide the PHFE

Provide the PHFE

Provide the PHFE

Construct the overall PHFE Construct the dominance Matrix D1 Determine the weight vector

Construct the dominance Matrix D2

Determine the weight vector

Obtain the acceptable consistency Matrix

Construct the dominance Matrix Dm

Determine the weight vector

Obtain the acceptable consistency Matrix

Obtain the acceptable consistency Matrix

Determine the overall weight vector

Calculate ranking index Alternative End

Fig. 2  Framework of MCDMs proposed in this paper

4.2 Illustrative example In this section, an investment and selection problem [29, 57] is provided to present the application of the proposed methods. An overseas investment company wants to invest the most potential industry in China. The following four alternatives to be considered: (1) automobile industry A1 , (2) food industry A2 , (3) clothing industry A3 , and (4) computer industry A4 . When making a decision, three criteria are investigated: (1) profit c1 , (2) growth c2 , and (3) environment c3 . Criteria weights are independent with each other and completely unknown. Suppose that three overseas experts are invited to

participate in a decision-making analysis. To get more decision information, each overseas expert surveyed ten experts in China, suppose the ten experts they surveyed were not exactly similar. Three overseas experts provided their preference evaluations for alternatives in the form of PHFEs, as shown in Tables 3, 4 and 5, respectively (the data comes from [29]). Take the evaluation values {0.7(0.7), 0.8(0.3)} from Overseas Expert 1 for example, evaluation information are obtained from the ten experts relative to car industry A1 with respect to environment c3 . Seven of them give a value 0.7, whereas three of them provide a value 0.8. Thus, the probability of the vale 0.7 is 0.7, and the probability of the

Table 3  Evaluation information given by Overseas Expert 1

c1 c2 c3

13

A1

A2

A3

A4

{0.6(0.25), 0.7(0.5), 0.8(0.25)} {0.5(0.5), 0.6(0.5)} {0.7(0.7), 0.8(0.3)}

{0.5(0.6), 0.6(0.4)} {0.4(0.3), 0.5(0.3), 0.6(0.4)} {0.3(0.2), 0.4(0.4), 0.5(0.4)}

{0.6(0.3), 0.7(0.3), 0.8(0.4)} {0.7(0.5), 0.8(0.5)} {0.6(0.3), 0.7(0.7)}

{0.3(0.4), 0.4(0.6)} {0.6(0.8), 0.7(0.2)} {0.5(0.4), 0.6(0.6)}

International Journal of Machine Learning and Cybernetics

vale 0.8 is 0.3. Other entries in Tables 3, 4 and 5 can be similarly explained. The procedures for obtaining the optimal alternative by using the proposed methods are described in the following. 1. If the dominance degree matrices are additive consistency Step 1 Normalize the decision-making matrices. Given that all criteria are benefit types, the normalization is not needed. Step 2 Calculate the overall values of the evaluation information. The overall values of the evaluation information can be obtained according to Eq. (24), as shown in Table 6. Step 3 Construct the dominance degree matrices. The dominance degree matrices of each criterion under all alternatives can be constructed using Eqs. (7) and (8) as follows:

⎛0.5000 0.3390 0.3187⎞ ⎜ ⎟ D4 = ⎜0.6610 0.5000 0.4759⎟ ⎜0.6813 0.5241 0.5000⎟ ⎝ ⎠ Step 4 Obtain the optimal weight vectors and acceptable consistency matrices. The process for obtaining the optimal weight vectors and acceptable consistency matrices is described as follows: According to Algorithm 1, the best weight vector w1 and the worst weight vector w2 of dominance degree matrix D1 can be obtained. If D1 is additive consistency, then we can construct the following model to derive the weight vectors: Model 3 min 𝜍

s.t.

⎛ 0.5000 0.7585 0.5963⎞ ⎛ 0.5000 0.5650 0.3295⎞ ⎜ ⎜ ⎟ ⎟ D1 = ⎜ 0.2415 0.5000 0.3345⎟ , D2 = ⎜ 0.4350 0.5000 0.4705⎟ , ⎜ ⎜ ⎟ ⎟ ⎜ 0.4037 0.6655 0.5000⎟ ⎜ 0.6705 0.5295 0.5000⎟ ⎝ ⎝ ⎠ ⎠

⎛ 0.5000 0.7118 0.6019⎞ ⎜ ⎟ D3 = ⎜ 0.2882 0.5000 0.5283⎟ , ⎜ ⎟ ⎜ 0.3981 0.4717 0.5000⎟ ⎝ ⎠

) | |( | w1 − w2 + 0.5 − 0.7585| ≤ 𝜍 | | ) |( | | w1 − w3 + 0.5 − 0.5963| ≤ 𝜍 | | ) | |( | w3 − w2 + 0.5 − 0.6655| ≤ 𝜍 | | w1 + w2 + w3 = 1 𝜍 ≥ 0, wj ≥ 0, j = 1, 2, 3.

and

We obtain the optimal solution 𝜍A∗ = 0.0128 and the optimal weight vector w1A = (0.4516, 0.1920, 0.3564) by solving Model 3.

Table 4  Evaluation information given by Overseas Expert 2

c1 c2 c3

A1

A2

A3

A4

{0.5(0.5), 0.7(0.5)} {0.5(0.4), 0.6(0.6)} {0.5(0.3), 0.6(0.4), 0.7(0.3)}

{0.7(0.7), 0.8(0.2), 0.9(0.1)} {0.6(0.7), 0.7(0.3)} {0.7(0.6), 0.8(0.4)}

{0.7(0.4), 0.8(0.6)} {0.5(0.7), 0.6(0.2), 0.8(0.1)} {0.5(0.5), 0.7(0.5)}

{0.5(0.3), 0.6(0.4), 0.7(0.3)} {0.6(0.5), 0.7(0.5)} {0.6(0.5), 0.7(0.2), 0.8(0.3)}

Table 5  Evaluation information given by Overseas Expert 3

c1 c2 c3

A1

A2

A3

A4

{0.6(0.3), 0.7(0.5), 0.8(0.2)} {0.6(0.7), 0.7(0.3)} {0.5(0.3), 0.6(0.7)}

{0.6(0.5), 0.7(0.5)} {0.5(0.3), 0.6(0.4), 0.7(0.3)} {0.5(0.5), 0.7(0.5)}

{0.6(0.2), 0.7(0.8)} {0.5(0.4), 0.6(0.3), 0.7(0.3)} {0.7(0.4), 0.8(0.6)}

{0.6(0.4), 0.7(0.6)} {0.6(0.8), 0.8(0.2)} {0.6(0.2), 0.7(0.8)}

Table 6  Overall evaluation information A1

A2

A3

A4

c1

} { 0.5(0.17), 0.6(0.18)

} { 0.5(0.2), 0.6(0.3), 0.7(0.4)

} { 0.6(0.17), 0.7(0.5)

} { 0.3(0.13), 0.4(0.2), 0.5(0.1)

c2

0.7(0.5), 0.8(0.15) } { 0.5(0.3), 0.6(0.6)

0.8(0.07), 0.9(0.03) } { 0.4(0.1), 0.5(0.2)

0.8(0.33) } { 0.5(0.37), 0.6(0.17)

0.6(0.27), 0.7(0.3) } { 0.6(0.7), 0.7(0.23)

c3

0.7(0.1) } { 0.5(0.2), 0.6(0.37)

0.6(0.5), 0.7(0.2) } { 0.3(0.07), 0.4(0.13), 0.5(0.3)

0.7(0.27), 0.8(0.2) } { 0.5(0.17), 0.6(0.1)

0.8(0.07) } { 0.5(0.13), 0.6(0.43)

0.7(0.53), 0.8(0.2)

0.7(0.34), 0.8(0.1)

0.7(0.33), 0.8(0.1)

0.7(0.37), 0.8(0.13)

13



International Journal of Machine Learning and Cybernetics

The deviation variable is 𝛿A = 0.0862 , and the consistency ratio is CRA = 0.0128<0.1 . The optimal weight vector obtained from Model 3 is reliable. Thus, the acceptable � consistency matrix is D1 = D1. Similarly, the optimal weight vectors of dominance degree matrix D2 , D3 , and D4 can be obtained as follows: w2A = (0.2982, 0.2352, 0.4667) , w3A = (0.4879, 0.2722, 0.2399) , and w4A = (0.2192, 0.3790, 0.4018). ′ ′ In addition, the acceptable consistency matrices D2 , D3 , ′ and D4 are as follows: ⎛ 0.5000 0.5650 0.3295⎞ ⎛ 0.5000 0.7118 0.7519⎞ ⎜ ⎜ ⎟ ⎟ � D2 = ⎜ 0.4350 0.5000 0.2705⎟ , D3 = ⎜ 0.2882 0.5000 0.5283⎟ , ⎜ ⎜ ⎟ ⎟ ⎜ 0.6705 0.7295 0.5000⎟ ⎜ 0.2481 0.4717 0.5000⎟ ⎝ ⎝ ⎠ ⎠ �

and D4 = D4. According to Eq. (25), the overall weight vectors can be obtained as follows: �

wA = (0.3627, 0.2696, 0.3662) Step 5 Calculate the ranking index. ′ If Dk is an additive consistency, then the ranking index can be determined as follows: 𝜇1 = 1.5375 , 𝜇2 = 1.5266 , 𝜇3 = 1.5139 , and 𝜇4 = 1.4904. Step 6 Determine the best alternative. Given that 𝜇1 > 𝜇2 > 𝜇3 > 𝜇4 and the best alternative is A1 , then the car industry is the most potential industry in China. 2. If the dominance degree matrices are multiplicative consistency If Dk , (k = 1, ⋯ , 4) is multiplicative consistency fuzzy preference relation, the procedures for obtaining the optimal alternative are described in the following: Steps 1′–3′ The same as those described in Steps 1–3. Step 4′ Obtain the optimal weight vectors and the acceptable consistency matrices. The process of obtaining the optimal weight vectors and acceptable consistency matrices is described as follows: According to Algorithm 1, the best weight vector w1 and the worst weight vector w2 of dominance degree matrix D1 can be obtained. If D1 is a multiplicative consistency, then we can construct the following model: Model 4 min 𝜍

| | w1 − 0.7585|| ≤ 𝜍 s.t. || | | w1 + w2 | | w1 | | | w + w − 0.5963| ≤ 𝜍 | | 1 3 | | w3 | | | w + w − 0.6655| ≤ 𝜍 | | 3 2 w1 + w2 + w3 = 1 𝜍 ≥ 0, wj ≥ 0, j = 1, 2, 3

13

∗ = 0.0047 and the optiWe obtain the optimal solution 𝜍M mal weight vectors w1M = (0.5024, 0.1641, 0.3335) by solving Model 4. The deviation variable is 𝛿M = 0.0707 , and the consistency ratio is CRM = 0.0665<0.1 . The optimal weight vector obtained from Model 6 is reliable. Thus, the acceptable �� consistency matrix is D1 = D1. Similarly, the optimal weight vectors of dominance degree matrices D2 , D3 , and D4 can be obtained as follows: w2M = (0.2622, 0.2007, 0.5371) , w3M = (0.5768, 0.2263, 0.1969) , and w4M = (0.1965, 0.3826, 0.4208). ′′ ′′ In addition, the acceptable consistency matrices D2 , D3 , ′′ and D4 are as follows:

⎛ 0.5000 0.5650 0.3295⎞ ⎛ 0.5000 0.7118 0.7519⎞ ⎜ ⎜ ⎟ ⎟ �� D2 = ⎜ 0.4350 0.5000 0.2705⎟ , D3 = ⎜ 0.2882 0.5000 0.5283⎟ , ⎜ ⎜ ⎟ ⎟ ⎜ 0.6705 0.7295 0.5000⎟ ⎜ 0.2481 0.4717 0.5000⎟ ⎝ ⎝ ⎠ ⎠ ��

and D4 = D4. Similar to Eq. (25), the overall weight vectors can be obtained as follows: ��

wM = (0.3845, 0.2434, 0.3721) Step 5′ Calculate the ranking index. ′′ If Dk is multiplicative consistency, then the ranking index can be determined as follows: 𝜈1 = 1.5590 , 𝜈2 = 1.5366 , 𝜈3 = 1.5294 , and 𝜈4 = 1.4781. Step 6′ Determine the best alternative. Given that 𝜈1 > 𝜈2 > 𝜈3 > 𝜈4 and best alternative is A1 , then the car industry is the most potential in China.

4.3 Comparative analysis and discussion To validate the feasibility of the proposed methods, a comparative study is conducted with existing methods based on the same illustrative example. In the following, the integrated maximizing score deviation method with aggregation operators introduced in Xu and Zhou [27] is applied to the illustrative example. The overall PHFE score values can be calculated as shown in Table 7. Criteria weights can be obtained through the maximizing score deviation method as follows: w1 = 0.3403 , w2 = 0.3230 , and w3 = 0.3367. Then, using the hesitant probabilistic fuzzy weighted averaging (operator )(HPFWA), the( alternative ) evaluation is s HPFWA1 = 0.1036  , s HPFWA2 = 0.0224  , ( ( ) ) s HPFWA3 = 0.2394 , and s HPFWA4 = 0.1991 . Thus, the final ranking is A3 ≻ A4 ≻ A1 ≻ A2 , and A3 is the optimal alternative. While using the hesitant probabilistic fuzzy weighted geometric (operator )(HPFWG), the ( alternative ) evaluation is s HPFWG = 0.0998  , s HPFWG 1 2 = 0.0201 , ( ( ) ) s HPFWG3 = 0.2321 , and s HPFWG4 = 0.1974 . Thus,

International Journal of Machine Learning and Cybernetics Table 7  Overall PHFE score values

c1 c2 c3

A1

A2

A3

A4

0.6630 0.5800 0.6330

0.6430 0.5800 0.5860

0.7160 0.6360 0.6760

0.5410 0.6370 0.6410

the final ranking is A3 ≻ A4 ≻ A1 ≻ A2 , and A3 is the optimal alternative. The results obtained by Xu and Zhou’s methods [27] and the proposed methods are shown in Table 8. The ranking results and the best alternative derived from the HPFWA and HPFWG operators are same, that is, A3 ≻ A4 ≻ A1 ≻ A2 , and the best alternative is A3 . Furthermore, the ranking results and the best alternative derived from additive consistency and multiplicative consistency are also same, that is, A1 ≻ A2 ≻ A3 ≻ A4 , and the best alternative is A1 . The same ranking result and the best alternative described above have the following explanation. Xu and Zhou’s methods [27] are based on aggregation operators, whereas the proposed methods are based on consistency. The ranking results and the best alternative are obtained based on similar models and ranking indexes. In addition, the ranking results and the best alternative derived from Xu and Zhou’s methods [27] and the proposed methods are different. The ranking results derived from Xu and Zhou’s methods [27] are A3 ≻ A4 ≻ A1 ≻ A2 , and the best alternative is A3 , whereas the ranking results derived from the proposed methods are A1 ≻ A2 ≻ A3 ≻ A4 , and the best alternative is A1 . The ranking difference can be explained as follows. First, the proposed methods utilize the dominance degree of PHFEs and BWM to derive the criteria weights, whereas Xu and Zhou’s methods [27] utilize the maximizing score deviation method to obtain the criteria weights. The maximizing score deviation method presented in [27] translate PHFEs into real numbers, and then the criteria weights can be obtained by calculating the Euclidean distance between any two real numbers. This method does not fully take advantage of probabilistic hesitant fuzzy information and it may cause the loss of information. However, the proposed methods effectively utilize the dominance degree of PHFEs, it can then avoid the loss of information. Second, the proposed methods use the acceptable consistency

Table 8  Ranking results of different methods

matrices and the criteria weights to obtain the ranking index, while Xu and Zhou’s methods [27] employ the hesitant probabilistic fuzzy aggregation operators and the criteria weights. The acceptable consistency matrices presented in this study can ensure the consensus among all experts, whereas methods presented in [27] pay a little feedback to experts. In this case, only the acceptable information can be used in the proposed methods, whereas Xu and Zhou’s methods [27] use all the evaluation information provided by the experts, including unreasonable information. Therefore, the ranking results obtained from the proposed methods are more reasonable. Furthermore, the overall evaluation information obtained from Xu and Zhou’s methods [27] is complex and tedious. Take a21 in Table  6 for example, we find that the integration result obtained using Eq. (24) is {0.5(0.3), 0.6(0.6), 0.7(0.1)} . However, when we use the HPFWA operator to integrate the evaluation values, the result is {0.5358(0.14), 0.5691(0.35), 0.5783(0.06), 0.6085(0.15), 0.6(0.21), 0.6366(0.09)} . If we use the HPFWG operator, then the result is {0.5313(0.14), 0.5593(0.21), 0.5646(0.2), 0.5944(0.15), 0.6(0.21), 0.6316(0.09)} . Obviously, the burden of computation can be significantly decreased when using Eq. (24) to integrate the evaluation values. According to the comparative analysis, the methods proposed in this paper have the following advantages over other methods. 1. The calculations required for the proposed methods are relatively straightforward. The burden of computation can be significantly decreased using Eq. (24). 2. Dominance degree matrices are constructed from the probability distribution function of PHFEs, and it includes good properties, such as complementary. Moreover, the dominance degree of PHFEs is integrated with BWM to derive the weight vectors. The weights derived from the proposed methods are reliable. 3. BWM is provided to derive the weight vectors from the dominance degree matrix based on the additive consistency or multiplicative consistency of fuzzy preference relations. By constructing the optimal model to determine the optimal weight vectors and the acceptable consistency matrices, the proposed methods easily meet

Methods

Xu and Zhou’s methods [27] Proposed methods

Ranking

HPFWA operator HPFW G operator Additive consistency Multiplicative consistency

A3 A3 A1 A1

≻ A4 ≻ A4 ≻ A2 ≻ A2

The best alternative ≻ A1 ≻ A1 ≻ A3 ≻ A3

≻ A2 ≻ A2 ≻ A4 ≻ A4

A3 A3 A1 A1

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consensus among decision-makers. However, methods based on aggregation operators pay a little feedback to decision makers. 4. BWM with fuzzy preference relations can be used to derive weights. In addition, it can also be combined with other MCDMs to solve practical problems.

5 Conclusion To add the probabilities to the values in HFS, PHFS is then introduced, and it is valuable for decision-makers to present their preferences for MCDM problems. In this paper, we first discussed the probabilistic distribution function of PHFE. Second, the dominance degree matrix between two PHFEs was constructed. Third, we developed two optimal models to derive weight vectors of the dominance degree matrix based on the additive consistency and multiplicative consistency of fuzzy preference relations. Finally, we provided an investment company selection problem to demonstrate the usefulness of the proposed methods. This study made several significant contributions to MCDM domains. First, the dominance degree proposed in this paper did not require the same length of PHFEs or the arrangement of their possible values. The results obtained using the proposed methods are objective. Second, we developed two optimal models to derive the weight vectors of the dominance degree matrix based on the additive consistency and multiplicative consistency of fuzzy preference relations, and the proposed methods easily meet consensus. Finally, BWM was extended to fuzzy preference relations, which can be used to derive weights and be combined with other MCDMs to solve practical problems. Acknowledgements  The authors thank the editors and anonymous reviewers for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (No. 71571193) and the Fundamental Research Funds for the Central Universities of Central South University (Nos. 2018zzts095).

References 1. Ivanco M, Hou G, Michaeli J (2017) Sensitivity analysis method to address user disparities in the analytic hierarchy process. Expert Syst Appl 90:111–126 2. Ahn BS (2017) The analytic hierarchy process with interval preference statements. Omega 67:177–185 3. Ren PJ, Xu ZS, Liao HC (2016) Intuitionistic multiplicative analytic hierarchy process in group decision making. Comput Ind Eng 101:513–524 4. Zhü KY (2014) Fuzzy analytic hierarchy process: fallacy of the popular methods. Eur J Oper Res 236:209–217 5. Kahraman C, Öztayşi B, Uçal Sarı İ, Turanoğlu E (2014) Fuzzy analytic hierarchy process with interval type-2 fuzzy sets. Knowl Based Syst 59:48–57

13

International Journal of Machine Learning and Cybernetics 6. Liao HC, Xu ZS (2015) Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process. Appl Soft Comput 35:812–826 7. Otay İ, Oztaysi B, Onar SC, Kahraman C (2017) Multi-expert performance evaluation of healthcare institutions using an integrated intuitionistic fuzzy AHP&DEA methodology. Knowl Based Syst 133:90–106 8. Cao YX, Zhou H, Wang JQ (2018) An approach to interval-valued intuitionistic stochastic multi-criteria decision-making using set pair analysis. Int J Mach Learn Cybernet 9:629–640 9. Zhu B, Xu ZS (2014) Analytic hierarchy process-hesitant group decision making. Eur J Oper Res 239:794–801 10. Zhu B, Xu ZS, Zhang R, Hong M (2016) Hesitant analytic hierarchy process. Eur J Oper Res 250:602–614 11. Zhou W, Xu ZS, Asymmetric hesitant fuzzy sigmoid preference relations in the analytic hierarchy process. Inf Sci 358–359:191– 207 (2016) 12. Santos LFdOM, Osiro L, Lima RHP (2017) A model based on 2-tuple fuzzy linguistic representation and analytic hierarchy process for supplier segmentation using qualitative and quantitative criteria. Expert Syst Appl 79:53–64 13. Franco CA (2014) On the analytic hierarchy process and decision support based on fuzzy-linguistic preference structures. Knowl Based Syst 70:203–211 14. Peng HG, Wang JQ, Cheng PF (2018) A linguistic intuitionistic multi-criteria decision-making method based on the Frank Heronian mean operator and its application in evaluating coal mine safety. Int J Mach Learn Cybernet 9:1053–1068 15. Abdullah L, Najib L (2014) A new type-2 fuzzy set of linguistic variables for the fuzzy analytic hierarchy process. Expert Syst Appl 41:3297–3305 16. He YQ, Du SP (2016) Classification of urban emergency based on fuzzy analytic hierarchy process. Proced Eng 137:630–638 17. Mangla SK, Govindan K, Luthra S (2017) Prioritizing the barriers to achieve sustainable consumption and production trends in supply chains using fuzzy analytical hierarchy process. J Clean Prod 151:509–525 18. Kuo CFJ, Lin CH, Hsu MW, Li MH (2017) Evaluation of intelligent green building policies in Taiwan—using fuzzy analytic hierarchical process and fuzzy transformation matrix. Energy Build 139:146–159 19. Zhou W, Xu ZS, Chen MH (2015) Preference relations based on hesitant-intuitionistic fuzzy information and their application in group decision making. Comput Ind Eng 87:163–175 20. García-Lapresta JL, Pérez-Román D (2016) Consensus-based clustering under hesitant qualitative assessments. Fuzzy Sets Syst 292:261–273 21. Wang JQ, Peng JJ, Zhang HY, Chen XH (2017) Outranking approach for multi-criteria decision-making problems with hesitant interval-valued fuzzy sets. Soft Comput. https​://doi. org/10.1007/s1304​2-13016​-10589​-13049​ 22. Wang J, Wang JQ, Tian ZP, Zhao DY (2018) A multi-hesitant fuzzy linguistic multicriteria decision-making approach for logistics outsourcing with incomplete weight information. Int Trans Oper Res 25:831–856 23. Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25:529–539 24. Zhu B (2014) Decision method for research and application based on preference relation. Southeast University, Nanjing 25. Yu SM, Wang J, Wang JQ, Li L (2018) A multi-criteria decisionmaking model for hotel selection with linguistic distribution assessments. Appl Soft Comput 67:739–753 26. Zhou H, Wang JQ, Zhang HY (2017) Stochastic multicriteria decision-making approach based on SMAA-ELECTRE with extended gray numbers. Int Trans Oper Res. https​: //doi.org/10.1111/ itor.12380​

International Journal of Machine Learning and Cybernetics 27. Xu ZS, Zhou W (2017) Consensus building with a group of decision makers under the hesitant probabilistic fuzzy environment. Fuzzy Optim Decis Mak 16:481–503 28. Peng HG, Zhang HY, Wang JQ (2018) Probability multi-valued neutrosophic sets and its application in multi-criteria group decision-making problems. Neural Comput Appl 30:363–383 29. Zhang S, Xu ZS, He Y (2017) Operations and integrations of probabilistic hesitant fuzzy information in decision making. Inf Fusion 38:1–11 30. Wang JQ, Cao YX, Zhang HY (2017) Multi-criteria decisionmaking method based on distance measure and choquet integral for linguistic Z-numbers. Cognit Comput 9:827–842 31. Zhou W, Xu ZS (2017) Expected hesitant VaR for tail decision making under probabilistic hesitant fuzzy environment. Appl Soft Comput 60:297–311 32. Li J, Wang JQ (2017) An extended QUALIFLEX method under probability hesitant fuzzy environment for selecting green suppliers. Int J Fuzzy Syst 19:1866–1879 33. Li J, Wang JQ (2017) Multi-criteria outranking methods with hesitant probabilistic fuzzy sets. Cognit Comput 9:611–625 34. Ding J, Xu ZS, Zhao N (2017) An interactive approach to probabilistic hesitant fuzzy multi-attribute group decision making with incomplete weight information. J Intell Fuzzy Syst 32:2523–2536 35. Wu ZB, Jin BM, Xu JP (2018) Local feedback strategy for consensus building with probability-hesitant fuzzy preference relations. Appl Soft Comput 67:691–705 36. Zhou W, Xu ZS (2017) Group consistency and group decision making under uncertain probabilistic hesitant fuzzy preference environment. Inf Sci 414:276–288 37. Wu ZB, Xu JP (2018) A consensus model for large-scale group decision making with hesitant fuzzy information and changeable clusters. Inf Fusion 41:217–231 38. Zhou W, Xu ZS (2018) Probability calculation and element optimization of probabilistic hesitant fuzzy preference relations based on expected consistency. IEEE Trans Fuzzy Syst 26:1367–1378 39. Zhang YX, Xu ZS, Wang H, Liao HC (2016) Consistency-based risk assessment with probabilistic linguistic preference relation. Appl Soft Comput 49:817–833 40. Zhang YX, Xu ZS, Liao HC (2018) A consensus process for group decision making with probabilistic linguistic preference relations. Inf Sci 414:260–275 41. Rezaei J (2015) Best-worst multi-criteria decision-making method. Omega 53:49–57 42. Rezaei J (2016) Best-worst multi-criteria decision-making method: some properties and a linear model. Omega 64:126–130

43. Mou Q, Xu ZS, Liao HC (2017) A graph based group decision making approach with intuitionistic fuzzy preference relations. Comput Ind Eng 110:138–150 44. Mou Q, Xu ZS, Liao HC (2016) An intuitionistic fuzzy multiplicative best-worst method for multi-criteria group decision making. Inf Sci 374:224–239 45. Guo S, Zhao H (2017) Fuzzy best-worst multi-criteria decisionmaking method and its applications. Knowl Based Syst 121:23–31 46. Gupta H, Barua MK (2016) Identifying enablers of technological innovation for Indian MSMEs using best–worst multi criteria decision making method. Technol Forecast Soc Chang 107:69–79 47. Ren JZ, Liang HW, Chan FTS (2017) Urban sewage sludge, sustainability, and transition for Eco-City: multi-criteria sustainability assessment of technologies based on best-worst method. Technol Forecast Soc Chang 116:29–39 48. Rezaei J, Nispeling T, Sarkis J, Tavasszy L (2016) A supplier selection life cycle approach integrating traditional and environmental criteria using the best worst method. J Clean Prod 135:577–588 49. Rezaei J, Wang J, Tavasszy L (2015) Linking supplier development to supplier segmentation using best worst method. Expert Syst Appl 42:9152–9164 50. Liu BD (2005) A course in uncertainty theory. Tsinghua University Press, Beijing 51. Liu Y, Fan ZP, Zhang Y (2011) A method for stochastic multiple criteria decision making based on dominance degrees. Inf Sci 181:4139–4153 52. Xia MM, Xu ZS (2011) Hesitant fuzzy information aggregation in decision making. Int J Approx Reason 52:395–407 53. Kay E (1977) Graph theory with applications. J Oper Res Soc 28:237–238 54. Tanino T (1984) Fuzzy preference orderings in group decision making. Fuzzy Sets Syst 12:117–131 55. Wu ZB, Xu JP (2012) A concise consensus support model for group decision making with reciprocal preference relations based on deviation measures. Fuzzy Sets Syst 206:58–73 56. Xu ZS, Hu H (2010) Projection models for intuitionistic fuzzy multiple attribute decision making. Int J Inf Technol Decis Mak 9:267–280 57. Wei GW, Zhao XF, Lin R (2013) Some hesitant interval-valued fuzzy aggregation operators and their applications to multiple attribute decision making. Knowl Based Syst 46:43–53 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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