Multi-attribute decision making with hesitant fuzzy information based on least common multiple principle and reference ideal method

Computers & Industrial Engineering 137 (2019) 106021

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Multi-attribute decision making with hesitant fuzzy information based on least common multiple principle and reference ideal method

T

Donghai Liua, , Lizhen Wangb ⁎

a b

Department of Mathematics, Hunan University of Science and Technology, China School of Insurance, Central University of Finance and Economics, China

ARTICLE INFO

ABSTRACT

Keywords: Hesitance degree Reference ideal method Distance measure TOPSIS The least common multiple expansion principle

The TOPSIS method is a technique for establishing order preference by similarity to the ideal solution, however, the traditional TOPSIS method cannot deal with the situation where the ideal solution vary between the minimum value and the maximum value. Considering the reference ideal can be any set or a point between the minimum value and maximum value, we introduce a reference ideal-TOPSIS method under the hesitant fuzzy decision environment in this paper. Firstly, we propose some new distance measures between hesitant fuzzy sets (HFSs), which take the hesitance degree of hesitant fuzzy element and the least common multiple expansion (LCME) principle into consideration. Then the reference ideal-TOPSIS method is developed by combing the proposed distance measures and reference ideal theory. Furthermore, we apply the proposed reference idealTOPSIS method to illustrate its practical use and make sensitivity analysis of distance measure parameters to observe the stability of the decision making results. A comparison analysis with the existing methods is also given to verify its rationality and effectiveness.

1. Introduction Multi-attribute group decision making problems are a very important research topic in complex economics, engineering, management, there are many literatures on multi-attribute decision making theory (Dong, Zhou, & Martínez, 2019; Mo & Deng, 2018; Liu, Zhang, Wu, & Dong, 2019; Liu, Liu, & Chen, 2019; Wu, Li, Chiclana, & Yager, 2019; Wu, Sun, Fujita, & Chiclana, 2019; Wu, Chang, Cao, & Liang, 2019; Zhou, Liu, Yang, Chen, & Wu, 2019). Due to the complexity of the decision making environment and the difference of human cognitive ability, there are some uncertainties in decision making problems. In order to deal with the uncertainty information well, Zadeh (1965) provided a fuzzy set E = { x i , µE (x i ) x i X } for coping with the imprecise judgements mathematically, where 0 µE (x i ) 1 represents the membership degree of x i X to the set E. One of its characteristics is that the membership degree of an element can be represented by a single numerical value in [0,1]. However, in some practical decision problems, it is difficult to use a numerical value to describe the membership degree of an element to the set. For example, two experts evaluate the possibility of making a profit from an investment plan, one expert thinks the possibility of making a profit is 0.5, the other expert thinks the possibility of making a profit is 0.8. The evaluation information cannot be represented by the fuzzy set at this time. Thus ⁎

Torra and Narukawa (2009) proposed the concept of hesitant fuzzy set (HFS), which contains the information about all membership degrees of x i X to the set. So the evaluation of the investment plan can be represented by a hesitant fuzzy set H = { xi , 0.5, 0.8 x i X } , where 0.5 and 0.8 represent its possible membership degree of x i X . Since the HFS has been put forward, it has attracted many scholars’ attention (Bedregal et al., 2016; Liu, Chen, & Peng, 2018; Meng & Chen, 2015; Na, Xu, & Xia, 2013; Qian, Wang, & Feng, 2013; Torra, 2010; Xu & Xia, 2011a, 2011b). Distance measure is also an important topic in fuzzy set theory, it is widely used in the field of decision making, such as engineering and manufacturing system (Han, Deng, Han, & Hou, 2011), medical diagnosis (Liu, Chen, & Peng, 2018), supplier evaluation (Gitinavard, Ghaderi, & Pishvaee, 2017) etc. The distance measure can be used to compare the alternatives with “the best one” or “the worst one”. The commonly used distance measures of HFSs are the Hamming distance measure, the Euclidean distance measure and the generalized distance measure (Bian, Zheng, Yin, & Deng, 2018; Dong, Zhang, Li, Hu, & Deng, 2019; Farhadinia, 2013, 2014; Garg & Arora, 2017; Liao, Xu, & Zeng, 2014; Liao & Xu, 2015; Liu, Chen, & Peng, 2017, 2019; Peng, Gao, & Gao, 2013; Xu & Xia, 2011a). For example, Xu and Xia (2011a) proposed a variety of ordered weighted distance measures between HFSs based on the well known Hamming distance measure, Euclidean

Corresponding author. E-mail address: [email protected] (D. Liu).

https://doi.org/10.1016/j.cie.2019.106021 Received 10 March 2019; Received in revised form 7 August 2019; Accepted 17 August 2019 Available online 24 August 2019 0360-8352/ © 2019 Elsevier Ltd. All rights reserved.

Computers & Industrial Engineering 137 (2019) 106021

D. Liu and L. Wang

distance measure and their generalizations, some numerical examples are also provided to illustrate the feasibility of these distance measures. Liao et al. (2014) proposed a family of distance measures between two hesitant fuzzy linguistic term sets(HFLTSs) and applied them to multiattribute decision making problems. Wang, Wu, Wang, Zhang, and Chen (2016) introduced the Hausdorff distance between hesitant fuzzy linguistic numbers based on the linguistic scale function. In fact, the hesitancy degree of hesitant fuzzy set is an important feature, which describes the decision maker’s hesitation in decision making process. Recently, some scholars have begun to pay attention to the hesitancy degree of hesitant fuzzy set. For example, Li, Zeng, and Li (2015) proposed some new distance and similarity measures between hesitant fuzzy sets based on their hesitancy degrees. Liu et al. (2018) proposed some distance measures between hesitant fuzzy linguistic term sets which include the hesitance degree of hesitant fuzzy element and linguistic scale function together. Because the numbers of hesitant fuzzy elements(HFEs) is different, when we calculate the distance measure between HFSs, the number of their elements should be equal, the currently used method is to extend the set of fewer elements until the number of elements in two HFSs is equal. For example, the shorter HFS is extended by adding the minimum value, the maximum value or any value between them in it. The added element is determined by the decision maker, which may weaken the computation objectivity and result in loss of information. To overcome this shortcoming, Wu, Zhou, Chen, and Tao (2018) introduced the least common multiple expansion (LCME) principle to extend the hesitant fuzzy set, and they got the conclusion that hesitant fuzzy decision information remains unchanged by utilizing the LCME principle. That is to say, HFS can keep the quality of information or avoid the loss of information based on the LCME principle. Furthermore, Zhang, Chen, Li, and Huang (2018) proposed some distance measures between HFSs based on the LCME principle and applied them to pattern recognition. On the other hand, the technique for order preference by similarity to ideal solution(TOPSIS) is one of the most popular methods for multiattribute decision making problems. The TOPSIS method is to obtain the best alternatives that are nearest to the positive ideal solution(PIS) and furthest to the negative ideal solution(NIS). The most important of the method is to search for the ideal solution in the maximum value or the minimum value. However, in real world, the ideal solution is not always the maximum value or the minimum value, it may be an interval between the minimum value and the maximum value. For instance, the best suitable pH value for the human health, the most suitable temperature for plants to growth, etc. Thus Cables, Lamata, and Verdegay (2016) proposed the reference ideal method(RIM) to deal with this situation, where the reference ideal solution is some value between the minimum value and the maximum value. Motivated by this, we intend to develop the RIM to hesitant fuzzy decision environment. Firstly, we propose some new distance measures between HFSs which take the hesitance degree of HFE and the LCME principle into consideration, then we develop the reference ideal-TOPSIS method to the proposed distance measures in the paper. In general, the contributions of the paper is not only to provide a new angle to construct the hesitant fuzzy distance measures, but also to develop the RIM-TOPSIS method which the traditional TOPSIS method cannot solve the problem. The rest of the paper is organized as follows. In Section 2, some basic concepts to be used are defined. In Section 3, we first propose some new distance measures between HFSs based on the hesitance degree of HFE and the LCME principle, then we develop the reference ideal-TOPSIS method to the proposed distance measures in hesitant fuzzy decision environment. In Section 4, we give an application of the proposed reference ideal-TOPSIS method to illustrate its practical use and make sensitivity analysis of distance measure parameters on decision results. Furthermore, a comparison analysis with existing methods is given to verify the reasonability and effectiveness of the proposed method. The paper ends with the conclusions in Section 5.

2. Preliminaries In this section, we introduce some related basic knowledge about fuzzy set, HFS, the hesitance degree of HFS and some existing distance measures between HFSs, which may be used in other sections. Throughout the paper, let X = {x1, x2 , …, xn} be a discrete and finite discourse set. 2.1. Fuzzy set Definition 1 (Zadeh, 1965). Let X = {x1, x2 , …, xn} be a fixed set, a fuzzy set E on X can be defined as:

E = { x i , µE (x i ) x i

X },

where µE (xi ) is a mapping from X to the closed interval [0, 1] and µE (xi ) is the degree of membership of x i X . 2.2. Hesitant fuzzy set Definition 2 (Torra and Narukawa, 2009). Let X = {x1, x2 , …, xn} be a fixed set, a HFS HE on X can be defined as:

HE = { xi , hE (x i ) x i

X },

where hE (x i ) is a set of some values between 0 and 1, denoting the possible membership degree of x i X to HE . For convenience, we call hE (x i ) a hesitant fuzzy element. Definition 3 (Li et al., 2015). Let HE = { xi , hE (x i ) x i X } be a HFS on X, if l (hE (x i )) represents the number of elements in hE (x i ) , then the hesitance degree of hE (x i ) can be defined as

u hE (x i ) = 1

1 . l (hE (xi ))

Example 1. For a fixed x i in X, let h E1 (xi ) = {0.5, 0.6, 0.8, 0.9, } and h E2 (x i ) = {0.6, 0.8} be two hesitant fuzzy elements, then 1 1 1 3 u (h E1 (x i )) = 1 4 = 4 and u (h E2 (xi )) = 1 2 = 2 , the hesitance degree of h E1 (xi ) is greater than that of h E2 (x i ) . 2.3. Multi-fuzzy set Definition 4 (Sebastian and Ramakrishnan, 2011). Let X = {x1, x2 , …, xn} be a fixed set, N the set of all natural numbers and {Li : i N } a family of complete lattices, the multi-fuzzy set E can be defined as follows:

E = { x i , µ1 (x i ), µ 2 (x i ), …, µj (x i ), where µj (x i )

Li , for i

xi

X },

N.

2.4. Existing distance measures between hesitant fuzzy sets Distance measure is a basic tool to describe the discrepancy between the alternatives, the most widely used distance measures between hesitant fuzzy sets are given as follows: Definition 5 (Xu and Xia, 2011a). Let HE1 = { x i , h E1 (xi ) x i X } and HE2 = { x i , h E2 (x i) xi X } be two hesitant fuzzy sets on X = {x1, x2 , …, xn} , the normalized fuzzy Hamming distance between HE1 and HE2 can be defined as:

dH HE1, HE2 =

1 n

n i=1

1 l (hE (x i ))

l (hE (xi))

h Ej1 (x i )

h Ej2 (x i ) .

j=1

The normalized fuzzy Euclidean distance between HE1 and HE2 can be defined as: 2

Computers & Industrial Engineering 137 (2019) 106021

D. Liu and L. Wang

dE HE1, HE2

1 = n

n i=1

1 l (hE (xi ))

l (hE (xi))

h Ej1

j=1

() xi

h Ej2

()

2

where

1 2

xi

. l (Hj )

h1j (x i ), …, h1j (x i ) , hj2 (x i ), …, hj2 (xi ) , …, hj

h Hj (xi ) = The normalized fuzzy generalized distance between E1 and E2 can be defined as

dG HE1, HE2

1 = n

n i=1

1 l (hE (x i ))

l (hE (xi))

N h1j (xi)

, hj

is

(x i )

N hm j (xi )

1

h Ej1

j=1

(x ) i

h Ej2

(x ) i

,

, N hm j (x i )

where is a positive real number, h Ej1 (x i ) and h Ej2 (x i ) are the jth values h E1 (xi ) h E2 (x i ) , in and respectively, and l (hE (x i )) = max{l (h E1 (x i )), l (h E2 (x i))} .

the number of hm j (x i )(1 by

Definition 6 (Xu and Xia, 2011a). Let HE1 = { x i , h E1 (xi ) x i X } and HE2 = { x i , h E2 (x i) xi X } be two hesitant fuzzy sets on X = {x1, x2 , …, xn} , the distance measure d (HE1, HE2 ) satisfies the following properties:

l (Hj )

l (Hj )) in h Hj (xi ) , which is determined

m

(j = 1, 2, …, k ) .

H1 = {0.2, 0.2, 0.3, 0.3, 0.5, 0.5},

In multi-attribute decision making problems, the decision makers evaluates the alternatives according to different attributes and selects the optimal one based on the distance measure of the alternative and the ideal solution. But in some situations, the ideal solution may not be the maximum or minimum value, but be an interval or a point between them, some of traditional decision-making methods are no longer be used. Motivated by this, we propose a reference ideal method with hesitant fuzzy information to deal with the problem in this section.

X = {x1, x2 , …, xn} Definition 8. Let be a fixed set, H2 = {h21 (xi ), h22 H1 = {h11 (x i ), h12 (xi ), …, h1l (H1) (xi ) x i X } and (x i ), …, h2l (H2) (x i ) xi X } be any two HFSs on X, if the weight of n 1) , then the element x i is i , satisfying with i = 1 i = 1(0 i weighted Hamming distance measure dWH (H1, H2) based on the hesitance degree of HFS and LCME principle can be defined as:

3.1. Distance measures for hesitant fuzzy sets based on the hesitancy degree and least common multiple expansion principle

dWH H1, H2

It is known that the distance measure is an important tool for describing the discrepancy between the alternatives. If we calculate the distance measure between HFSs with different numbers of elements, the existing method of extending the shorter HFEs by adding the maximum value, minimum value or any value until they have an equal number of elements, which may cause the initial information loss and distortion. In order to overcome this disadvantage, we propose some new distance measures between HFSs based on the hesitancy degree and LCME principle in this subsection. Firstly, we give the concept of multiple hesitant fuzzy set based on the LCME principle as follows.

1 = 2

n i

u (H1)

u (H2) +

i=1

(x i )

h2j (x i )

1 2 L (Hm )m =1

L (Hm )2m = 1

h1j

× j =1

.

The weighted Euclidean distance measure dWE (H1, H2) based on the hesitance degree of HFS and LCME principle can be defined as: dWE H1, H2 =

1 2

2

n i

u (H1)

u (H2)

+

i=1

1 L (Hm )2m = 1

×

L (Hm)2m = 1

2

h1j (xi)

1 2

h2j (xi)

.

j =1

The weighted generalize distance measure dWG (H1, H2) based on the hesitant degree of HFS and LCME principle can be defined as:

Definition 7. Let H = {H1, H2, …, Hk } be k HFSs on X = {x1, x2 , …, xn} , for any x i X , the possible membership degree of x i X to the set l (H ) Hj (1 j k ) can be given as Hj = {h1j (x i ), hj2 (x i ), …, hj j (x i )} , where m the value of hj (x i)(1 m l (Hj )) belongs to [0, 1], l (Hj ) represents the number of elements in Hj . If L (Hj ) kj = 1 is the least common multiple (LCM) of l (H1), l (H2), …, l (Hk ) , the multiple hesitant fuzzy set can be defined as:

X },

H2 = {0.4, 0.4, 0.4, 0.6, 0.6, 0.6}.

According to Theorems 2–5 in Wu, Lin, Zhou, Chen, and Chen (2019), it is already known that the important decision information of HFS remains unchanged when we extend the shorter hesitant fuzzy element based on the LCME principle, so it is reasonable to define the distance measure between HFSs using the LCME principle. In the following, we propose some new distance measures between HFSs based on the hesitance degree of HFS and the LCME principle.

3. Reference ideal method with hesitant fuzzy information in multi-attribute decision making problems

Hj = { x i , h Hj (x i ) x i

L (Hj )kj = 1

Example 2. Let H1 and H2 be two HFSs on X = {x1, x2 , …, xn} , for a fixed x i X , H1 = x i , h1 (x i ) = {0.2, 0.3, 0.5} , H2 = x i , h2 (x i ) = {0.4, 0.6} , the numbers of elements in H1 and H2 are 3 and 2, respectively, the LCM of 3 and 2 is 6, then the multiple hesitant fuzzy set can be defined as H = {H1, H2} , where

(1) 0 d (HE1, HE2 ) 1; (2) d (HE1, HE2 ) = 0 if and only if HE1 = HE2 ; (3) d (HE1, HE2 ) = d (HE2, HE1) .

H = {H1, H2, …, Hk },

N h2j (xi)

l (Hj )

(x i ),

dWG H1, H2 =

1 2

n i i=1

u (H1)

u (H2)

+

1 2 L (Hm )m =1

×

L (Hm )2m = 1

1

h1j (xi)

h2j (xi)

,

j= 1

> 0, u (Hm) is the corresponding hesitance degree of where Hm (m = 1, 2), h1j (x i ) and h2j (x i ) are the jth values in H1 and H2 , respectively. L (Hm)2m = 1 is the LCM of l (H1) and l (H2) , where l (H1) and l (H2) represent the number of elements in H1 and H2 , respectively.

(j = 1, 2, …, k ),

3

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D. Liu and L. Wang

Example 3. Let X = {x1, x2}, H1 = { x i , h1 (x i) x i X } = { x1, {0.2, 0.3, 0.5} , x2 , {0.7, 0.8} } and H2 = { x i , h2 (x i ) x i X } = { x1, {0.4, 0.6} , x2, {0.5, 0.6} } be two HFSs on X, the weight vector of (x1, x2) is = ( 1, 2) = (0.6, 0.4) . We know the hesitance degree 2 1 1 1 u (h1 (x1)) = 3 , u (h1 (x2)) = 2 , u (h2 (x1)) = 2 and u (h2 (x2 )) = 2 . For , the LCM of l (h1 (x1)) and l (h2 (x1)) is 6; for x2 X , the LCM of l (h1 (x2 )) and l (h2 (x2 )) is 2, then the multiple hesitant fuzzy set based on the LCME H1 = {{0.2, 0.2, 0.3, 0.3, 0.5, 0.5}, principle can be given as {0.7, 0.8}}, H2 = {{0.4, 0.4, 0.4, 0.6, 0.6, 0.6}, {0.5, 0.6}} , so we have dWH (H1, H2) = 0.39 and dWE (H1, H2) = 0.1623. Remark 1. The distance measure dWG H1, H2 =

1 2

n i=1

i

u (H1)

u (H2)

(x i ), …, h2l (H2) (x i ) xi X } be any two HFSs on X, if the weight of n 1, 1) , for 0 element x i is i , satisfying with i = 1 i = 1(0 i the preference weighted Hamming distance measure dPWH (H1, H2) based on the hesitance degree of HFS and LCME principle can be defined as: n

u (H1)

i

1 × 2 L (Hm )m =1

u (H2) +

i=1

h1j (xi )

h 2j (xi )

L (Hm )2m = 1 j =1

.

1

+

1 l (xi)

×

l (xi ) j =1

h1j (xi )

h 2j (x i )

The preference weighted Euclidean distance measure dPWE (H1, H2) based on the hesitance degree of HFS and LCME principle can be defined as:

, where (l (xi ) = max{l (h1 (x i )), l (h2 (x i ))}) , which defined in Li et al. (2015) is affected by the subjective factors of the decision makers, and it may cause the decision information distortion. Let us reconsider the Example 3, the number of elements in Hi (i = 1, 2) are not equal, if the shorter one is extended by adding the minimum value, H2 can be extended as H2 = { x1, {0.4, 0.4, 0.6} , x2 , {0.5, 0.6} , using the distance measure defined in Li et al. (2015), we have dWH (H1, H2) = 0.13; if the shorter one is extended by adding the maximum value, H2 can be extended as H2 = { x1, {0.4, 0.6, 0.6} , x2 , {0.5, 0.6} . Similarly, the same distance measure is used for calculation, we have dWH (H1, H2) = 0.15, obviously they are not equal. That is to say, the calculation results based on the same distance measure are different, which may cause the decision information distortion in decision process. However, the distance measures defined in Definition 8 overcomes this disadvantage. Theorem 1. Let H1 = {h11 (x i ), h12 (xi ), …, h1l (H1) (xi ) x i

dPWE H1, H2 =

dPWG H1, H2 =

u (H1)

i

u (H2)

i=1

1 + × L (Hm )2m = 1

2

L (Hm)2m = 1

h1j (xi)

j=1

1 2

h2j (xi)

.

1 2

n

u (H1)

i

u (H2)

i=1

+

1

2 L (Hm)m =1

1 × 2 L (Hm)m =1

j

h1 (xi)

j =1

j

h2 (xi)

> 0, u (Hm) is the corresponding hesitance degree of where Hm (m = 1, 2), h1j (x i ) and h2j (x i ) are the jth values in H1 and H2 , respectively. L (Hm)2m = 1 is the LCM of l (H1) and l (H2) , where l (H1) and l (H2) represent the number of elements in H1 and H2 , respectively.

= 0 , which means the hesitancy degree is not Remark 3. If considered in the preference distance measures, then the distance measures dPWH , dPWE and dPWG are reduced to the metric distance measures, which are proposed based on the LCME principle.

0 dWG (H1, H2) 1; dWG (H1, H2) = dWG (H2, H1) ; dWG (H1, H2) = 0 if and only if H1 = H2 . hk (x ) = {hk1 (x ), hk2 (x ), …, hkL (Hm) (x )} , h1j (xi ) h 2j For if j (x i ) h3 (xi )(j = 1, 2, …, L (Hm)) and (H1) l (H2 ) l (H3) , then dWG (H1, H2) dWG (H1, H3) and dWG (H2, H3) dWG (H1, H3) .

Theorem 2. The distance measures dPWH , dPWE and dPWG also satisfy the properties (1)-(4) in Theorem 1. Proof. The proof is obvious, we omit it here. □

Proof. The properties (1), (2) and (3) are obvious, we only give the proof of property 4. Since l (H1) l (H2 ) l (H3) , we have u (H1) u (H2) u (H3) , then u (H1) u (H2) u (H1) u (H3) is obtained. On the other hand, for the L (Hm), h1j (x i ) h 2j (x i ) h3j (xi ) , same then h1j (x i ) h 2j j j (x i ) h1 (x i ) h3 (x i ) , so dWG (H1, H2) dWG (H1, H3) is obtained. Similarly, we can obtain dWG (H2, H3) dWG (H1, H3). □

In some decision making problems, if the universe of discourse X and the weights of elements are continuous, the distance measures defined above are no longer suitable for these situations, thus we define the continuous distance measures for hesitant fuzzy sets based on the hesitancy degree and LCME principle as follows: Definition 10. Let X = [a, b] be a continuous universe of discourse, 1 2 l (H ) 1 2 l (H ) H1 = {h1 (x ), h1 (x ), …, h1 1 } and H2 = {h2 (x ), h 2 (x ), …, h2 2 } be any two HFSs over the element x(x X ) , if the weight of x is (x ) , b (x ) 1) , then the continuous satisfying with a (x ) dx = 1(0 weighted Hamming distance measure dCWH (H1, H2) based on the hesitance degree of HFS and LCME principle can be defined as:

1

Remark 2. If the weight 1 = 2 = = n = n , the weighted Hamming distance measure dWH , the weighted Euclidean distance dWE and the weighted generalize distance measure dWG are reduced to the normalized Hamming distance measure dNH , the normalized Euclidean distance dNE and the normalized generalize distance measure dNG , respectively. The distance measures defined in Definition 8 include both the hesitancy degrees and the membership values. But in the decision making process, decision makers may have different preferences for the hesitancy degree and membership values, if we assume that (0 1) is a preference coefficient of the hesitation degree and 1 is a preference coefficient of the membership degree, then the distance measures between HFSs with preference can be given as follows: a

2

n

(3.1)

X }, H2 = and

xi X } H3 = {h31 (x i ), h32 (x i ), three HFSs on X = {x1, x2 , …, xn} , then the weighted generalize distance measure dWG satisfies the following properties:

X = {x1, x2 , …, xn} Definition 9. Let be H1 = {h11 (x i ), h12 (xi ), …, h1l (H1) (xi ) x i X } and

1 2

The preference weighted generalize distance measure dPWG (H1, H2) based on the hesitant degree of HFS and LCME principle can be defined as:

{h 21 (x i ), h 22 (x i ), …, h 2l (H2) (xi ) …, h3l (H3) (x i ) x i X } be

(1) (2) (3) (4)

1 2

dPWH H1, H2 =

1

dCWH (H1, H2) = 2 ( +

b a

b a

(x ) u (H1)

(x ) L (H

1

2 m )m = 1

u (H2) dx ×

L (Hm )2m = 1

j

h 1 (x )

j

h2 (x ) dx .

j=1

The continuous weighted Euclidean distance measure dCWE (H1, H2) based on the hesitance degree of HFS and LCME principle can be defined as:

fixed set, H2 = {h21 (xi ), h22 4

,

Computers & Industrial Engineering 137 (2019) 106021

D. Liu and L. Wang

b 1 ( a 2

dCWE H1, H2 =

+

b a

(x )

u (H2) 2 dx

(x ) u (H1)

1 2 L (Hm )m =1

×

j h 1 (x )

j h 2 (x )

dx

.

+b

j=1

The continuous weighted generalize distance measure dCWG (H1, H2) based on the hesitant degree of HFS and LCME principle can be defined as: b 1 ( a 2

dCWG H1, H2 =

+

b a

(x ) u (H1)

1 2 m )m = 1

(x ) L (H

×

u (H2) dx

+b

L (Hm )2m = 1

j

h 1 (x )

j

h2 (x ) dx

3

0.6x 2

a

1 2

(

1 a

b 1 a L (Hm )2m = 1

b a

1 b

a

u (H2) 2 dx

u (H1)

×

j h1 (x )

j h 2 (x )

1 2

3

.

u (H1)

b 1 a L (Hm )2m = 1

×

u (H2) dx 1

L (Hm)2m = 1

j h1 (x )

j h 2 (x )

dx

,

j=1

Definition 11. Let X = [a, b] be a continuous universe of discourse, 1 2 l (H ) 1 2 l (H ) H1 = {h1 (x ), h1 (x ), …, h1 1 (x )} and H2 = {h2 (x ), h 2 (x ), …, h2 2 (x )} be any two HFSs over the element x(x X ) , if the weight of x is (x ) , b (x ) 1) , then the continuous satisfying with a (x ) dx = 1(0 preference weighted Hamming distance measure dCPWH (H1, H2) based on the hesitance degree of HFS and LCME principle can be defined as: b a

1

dCPWH (H1, H2) = 2 (

(x ) u (H1) b a

+ 1 L (Hm)2m = 1

×

j

(x )

u (H2) dx 1

2 L (Hm )m =1

j

h1 (x )

h 2 (x ) dx .

j=1

2 3 1

1

dx

j=1

x dx + 4

1

3 1

1 × 3

0.2x 2

1 +

0.4x 2

The continuous preference weighted Euclidean distance measure dCPWE (H1, H2) based on the hesitance degree of HFS and LCME principle can be defined as:

1 +

x dx = 0.45. 4

dCPWE H1, H2 =

1 2(

b a

(x ) u (H1)

u (H2) 2 dx

dCWH H2, H =

1 2

2

L (Hm)2m = 1

where is a positive real number. Similar to the preference discrete distance measures, we propose the continuous weighted distance measures of HFSs with different preferences for the hesitance degree of HFSs and their membership values, which are also defined based on LCME principle.

,

j=1

dCWH H1, H

1

a

1

Example 4 (Alternative selection). Energy is an indispensable factor in x social development. For X = [1, 3], the attribute weight is (x ) = 4 , suppose there are two alternatives H1 and H2 to be invested, the possible evaluations for the alternatives with respect to the attribute can be 0.3x 0.4x 0.2x 0.4x 0.6x represented as HFSs H1 = { 2 , 2 , 2 } and H2 = { 2 , 2 } , we can select the best alternative by calculating the distance between the alternative and the ideal solution.Here we take dCWH as an example to calculate the distance between the alternative and the ideal solution, suppose that the ideal solution H = {1} , which can be seen as a special 2 1 HFS. It is easy to know u (H1) = 3 and u (H2) = 2 . Using the formula dCWH , we have

1 2

b a

1 b

1

dCNG H1, H2 =

> 0, u (Hm) is the corresponding hesitance degree of where j j Hm (m = 1, 2), h1 (x ) and h2 (x ) are the jth values in H1 and H2 , 2 respectively. L (Hm)m = 1 is the LCM of l (H1) and l (H2) , where l (H1) and l (H2) represent the number of elements in H1 and H2 , respectively.

=

(

1 2

2

L (Hm )2m = 1

1 2

dCNE H1, H2 =

1 2

x dx + 4

1

3 1

1 × 2

0.3x 2

1 +

0.4x 2

1

+ 1

x dx 4

= 0.56.

Remark 4. For x [a, b], if H1 and H2 are two HFSs, (x ) = b a , then the continuous weighted Hamming distance measure dCWH (H1, H2) , the continuous weighted Euclidean distance measure dCWE (H1, H2) and the continuous weighted generalize distance measure dCWG (H1, H2) are reduced to the continuous normalized Hamming distance measure dCNH (H1, H2) , the continuous normalized Euclidean distance measure dCNE (H1, H2) and the continuous normalized generalize distance measure dCNG (H1, H2) , respectively, which are given as follows: 1

+b

1 2

(

1 a

1 b

a

b a

u (H1)

b 1 a L (Hm )2m = 1

×

dCPWG H1, H2 =

+ 1

j

h1 (x )

1 2 L (Hm)m =1

×

j

h1 (x )

j

h2 (x ) dx

.

j =1

b a

1 ( 2

(x )

b a

(x ) u (H1)

1 2 L (Hm)m =1

×

L (Hm )2m = 1

u (H2) dx

1 j h1 (x )

j h 2 (x )

dx

,

j =1

1, u (Hm) is the corresponding hesitance degree of where > 0, 0 j j Hm (m = 1, 2), h1 (x ) and h2 (x ) are the jth values in H1 and H2 , 2 respectively. L (Hm)m = 1 is the LCM of l (H1) and l (H2) , where l (H1) and l (H2) represent the number of elements in H1 and H2 , respectively.

u (H2) dx L (Hm)2m = 1

(x )

1 2

2

The continuous preference weighted generalize distance measure dCPWG (H1, H2) based on the hesitant degree of HFS and the LCME principle can be defined as:

Because dCWH (H2, H ) > d CWH (H1, H ) , that is to say the alternative H2 is closer to the ideal solution H than the alternative H1, so the best alternative is H1.

dCNH H1, H2 =

b a

L (Hm )2m = 1

Remark 5. It is easy to know that the distance measures defined in Definition 10 and Definition 11 satisfy the properties (1)–(3) in Theorem 1.

j

h 2 (x ) dx .

j=1

5

Computers & Industrial Engineering 137 (2019) 106021

D. Liu and L. Wang

3.2. The reference ideal-TOPSIS method with proposed distance measures

between Hik and the reference ideal interval [Bk , Bk+], from which we can find the maximum distance and the minimum distance between the alternative Hik and the reference ideal interval [Bk , Bk+], so the reference positive ideal solution and the reference negative ideal solution are obtained, respectively, which can be given as follows:

It is already known that the reference ideal solution may not be the minimum value or maximum value in practical decision making problems, it may be an interval or a point between them, so the traditional TOPSIS method cannot deal with this situation. In this subsection, we develop the reference ideal-TOPSIS method to the proposed distance measures under the hesitant fuzzy decision making environment.

+

3.2.1. Reference ideal method normalization According to the reference ideal method in Cables et al. (2016), we should normalize the values of decision matrix. There have some concepts that should be identified, such as:

(d (hikl , [Bk , Bk+]))

=

+

Step 5 Calculate the relative closeness coefficient

where

d hikl , Bk , Bk+

=

min{

Bk ,

hikl

0,

Bk+

},

hikl hikl

0.4 , 0.2

0.6 ) + min( 0.3

Energy is a very important factor in socio-economic development of societies. Assume that there are five energy projects Hi (i = 1, 2, 3, 4, 5) to be invested, the decision makers evaluate the five possible projects anonymously from the following four attributes C1: technological; C2 : environmental; C3 : social political; C4 : economic. The weight vector of the attributes is = ( 1, 2 , 3 , 4 )T =(0.15, 0.3, 0.2, 0.35)T . It is reasonable to allow the evaluation values repeated many times or appear only once, the corresponding evaluations are represented as hesitant fuzzy decision matrix H = (Hik )5 × 4 , which are given in Table 1. where Hik (i = 1, 2, 3, 4, 5;k = 1, 2, 3, 4) are hesitant fuzzy evaluation values of the alternative Hi with respect to criteria Ck . The reference ideal intervals of attributes are given in Table 2. In the following subsection, we develop the reference-TOPSIS method to the proposed distance measures to find the best energy project.

0.4 ,

0.6 ) + 0 = 0.3.

In the following, we present the whole algorithm of reference idealTOPSIS method to the proposed distance measures. 3.2.2. Algorithm to apply the reference ideal method Step 1 Define the work context, the following aspects are defined, including: The range of decision evaluation rk . The reference ideal interval Bk = [Bk , Bk+] (it can be any set between the minimum value and the maximum value). The weight k of the criterion, which is considered under the criterion Ck . Step 2 For the evaluation of the alternative Hi under the criterion Ck , the hesitant fuzzy decision matrix H = (Hik )m × n is given as follows:

• • •

H=

H11 H21

H12 H22

H13 H23

H1n H2n

Hm1 Hm2 Hm3

Hmn

, where

4.1. Case description

Example 5. Let H1 = {0.2, 0.3, 0.5} be a HFS and the reference ideal interval is [0.4, 0.6], if the distance parameter = 1, then the hesitant fuzzy reference distance measure

0.3

Di

Di + Di+

In this section, we present a numerical example (adapted from Li, Zeng, & Zhao, 2015; Xu & Xia, 2011a) to illustrate the feasibility of the proposed reference-TOPSIS method in hesitant fuzzy decision making problems. A comparison analysis with other existing methods is given to verify its reasonability and effectiveness.

[Bk , Bk+], [Bk , Bk+].

According to the maximum distance and the minimum distance between the alternative and the reference ideal interval, we can find the reference negative ideal solution(R-NIS) and the reference positive ideal solution(R-PIS), respectively.

d (H1, [0.4, 0.6]) = min( 0.2

=

4. Illustrative case

(3.2)

hikl

i

0 < i < 1, i = 1, 2, …, m . Step 6 Rank the alternatives Hi , the greater value of i , the better alternative Hi will be. The flow diagram of the proposed method is depicted in Fig. 1.

,

l=1

+

where Hi = max{d (Hi1, [B1 , B1+]), d (Hi2, [B2 , B2+]), …, d (Hin, [Bn , Bn+])}, (i = 1, 2, …, m) ; Hi = min{d (Hi1, [B1 , B1+]), d (Hi2, [B2 , B2+]), …, d (Hin , [Bn , Bn+])}, (i = 1, 2, …, m) . Step 4 Applying the proposed distance measures in (3.1) to calculate the separation of the alternatives Hi (i = 1, 2, …, m) between the + reference ideal solution H and H , which can be denoted as + + Di = dPWG (Hi, H ) and Di = dPWG (Hi, H ) , respectively.

1

d Hik , Bk , Bk+

+

H = {H1 , H2 , …, Hn },

(1) For the evaluation about alternative Hi under the attribute Ck , it is easy to know the range of hesitant fuzzy evaluation values Hik = {hikl l = 1, 2 ,l (Hik )} belong to [0, 1]. (2) Assume the reference ideal interval of each attribute Ck is [Bk , Bk+], which represents the maximum importance or relevance in a given range. The reference ideal may changes between the extreme ranges. We can give the distance measure between Hik and the reference ideal interval [Bk , Bk+], which is represented by the following function: l (Hik )

+

H = {H1 , H2 , …, Hn };

4.2. Calculation process of the proposed method The calculation process of the reference-TOPSIS method is organized in the following way. Under the hesitant fuzzy environment, it is already known the range of all attributes is included in [0, 1], the reference ideal intervals of the attributes Ck (k = 1, 2, 3, 4) are given in Table 2 and the weight vector of attributes is = ( 1, 2 , 3 , 4 )T =(0.15, 0.3, 0.2, 0.35)T . In order to obtain the reference positive ideal solution(R-PIS) and the reference negative ideal solution(R-NIS), we first calculate the distances between the evaluation values and the reference ideal intervals by formula (3.2), for convenience, let = 2 , the results are obtained in Table 3. According to Table 3, we can give the reference positive ideal solutions and the reference negative ideal solutions in Table 4.

,

where Hij (i = 1, 2, …, m;j = 1, 2, …, n) are HFSs. Step 3 Applying the formula (3.2) to calculate the distance measure 6

Computers & Industrial Engineering 137 (2019) 106021

D. Liu and L. Wang

Furthermore, we can apply formula to calculate the distances between alternatives and R-PIS, the distances between alternatives and RNIS. Firstly, the hesitance degree of Hik (i = 1, 2, 3, 4, 5;k = 1, 2, 3, 4) are given in Table 5. We can also obtain the LCM L (H1k ) k4 = 1 = 12, L (H2k )k4 = 1 = 60 , L (H3k )k4= 1 = 6, L (H4k )k4 = 1 = 12 and L (H5k ) k4= 1 = 60 . Then the hesitant fuzzy decision matrix H = (Hik )5 × 4 can be extended to the multiple hesitant fuzzy decision matrix H = (Hik )5 × 4 based on the LCME principle. For example, the element H11 = {0.5, 0.4, 0.3} can be extended to {0.5, 0.5, 0.5, 0.5, 0.4, 0.4, 0.4, 0.4, 0.3, 0.3, 0.3, 0.3} . Furthermore, we + apply the distance measure dPWG to calculate Di+ (Hi , H ) and Di (Hi , H )(i = 1, 2, 3, 4, 5) , respectively. If we assume the preference parameter = 0.5 and the distance parameter = 2 , the corresponding results are obtained in Table 6. The relative closeness coefficients of the alternatives are obtained in Table 7. From Table 7, we can conclude that the ranking order of the energy projects is given by H3 H2 H4 H1 H5.

distance parameter attains a certain value, the change of the preference parameter no longer affects the ranking results of the projects. In other words, the results obtained by the proposed method are relatively stable. Furthermore, we consider the influence of distance parameter on the ranking results. For a given preference parameter , we adjust the different distance parameter = 1, 2, 3, 6, 10, 15, 20 to observe the stability of the decision results, the ranking results obtained by dPWG are given in Tables 15–19. As can be (3.1)seen from the computation results in Tables 15–19, the best energy project is always H3 as the preference parameter changes in different distance measures. Furthermore, when the distance parameter takes a smaller value, we should pay more attention to the divergence between the hesitance degrees, because the change of the parameter of the hesitance degree is more sensitive to the decision results at this time. On the other hand, as the value of distance parameter increases, the influence of hesitance degree on the decision results gets smaller and smaller. In other words, the ranking results mainly depend on the divergence between the membership values, we should pay more attention to the divergence between the membership values than the hesitance degree at this time.

4.3. Comparison analysis 4.3.1. Sensitivity analysis of the parameters In this paper, the proposed distance measure dPWG involves the distance parameter and the preference parameter , they are determined by the decision makers or the decision environment. In order to understand the impact of these parameters on the decision results, we need to make sensitivity analysis of them. Firstly, for a given distance parameter , if the preference parameter of the hesitance degree is 0.1, 0.2, 0.4, 0.6, 0.8, then the preference parameter of the membership degree takes the value 1 = 0.9, 0.8, 0.6, 0.4, 0.2 , respectively, the decision results obtained by dPWG are given in Tables 8–14. As can be seen from these tables, we can conclude that the optimal project is always H3 . When the preference parameter changes, the ranking results are slightly different in some cases. However, if the

4.3.2. Comparison analysis with existing method In order to illustrate the feasibility and effectiveness of the reference-TOPSIS method to the proposed distance measures, we use the existing distance measures proposed in Xu and Xia (2011a) and Li et al. (2015) to calculate the same numerical examples for comparative analysis. The best energy project obtained in Xu and Xia (2011a) and Li et al. (2015) are also H3 , the results obtained by them are same as the proposed method in this paper, which illustrates the feasibility of the reference-TOPSIS method of the proposed distance measures. In the following, we give some advantages of reference-TOPSIS method of the proposed distance measures: (1) The proposed distance measures in this paper include the hesitance degree of HFSs, which can describe the

Problem Set of alternatives

Experts discussion

Feedback to experts

Preferences

Hesitant fuzzy decision matrix

Reference ideal solution

Distance measure (based on LCME and hesitance degree)

Relative closeness coefficient

Selection procress Fig. 1. The proposed method based on LCMP and RIM.

7

Computers & Industrial Engineering 137 (2019) 106021

D. Liu and L. Wang

Table 1 The hesitant fuzzy decision matrix H . C1 {0.5, {0.5, {0.7, {0.8, {0.9,

H1 H2 H3 H4 H5

C3

C2 0.4, 0.3} 0.3} 0.6} 0.7, 0.4, 0.3} 0.7, 0.6, 0.3, 0.1}

{0.9, {0.9, {0.9, {0.7, {0.8,

0.8, 0.7, 0.1} 0.7, 0.6, 0.5, 0.2} 0.6} 0.4, 0.2} 0.7, 0.6, 0.4}

Table 2 The reference ideal intervals of attributes Ck .

The reference ideal intervals

{0.5, {0.8, {0.7, {0.8, {0.9,

0.4, 0.2} 0.6, 0.5, 0.1} 0.5, 0.3} 0.1} 0.8, 0.7}

{0.9, {0.7, {0.6, {0.9, {0.9,

0.6, 0.5, 0.3} 0.4, 0.3} 0.4} 0.8, 0.6} 0.7, 0.6, 0.3}

Table 5 The hesitance degree of hesitant fuzzy information HiK .

C1

C2

C3

C4

[0.5, 0.7]

[0.6, 0.8]

[0.4, 0.6]

[0.6, 0.7]

H1 H2

Table 3 Distances between the hesitant evaluation values and the reference ideal intervals.

H1 H2 H3 H4 H5

C4

C1

C2

C3

C4

0.2236 0.2 0 0.2449 0.4899

0.5099 0.3606 0.1 0.4472 0.2

0.2 0.3606 0.1414 0.3606 0.3742

0.3742 0.3606 0.2 0.2236 0.3606

H3 H4

H5

C1

C2

C3

C4

2 3 1 2 1 2 3 4 4 5

3 4 4 5 1 2 2 3 3 4

2 3 3 4 2 3 1 2 2 3

3 4 2 3 1 2 2 3 3 4

Table 6 The distances between the alternatives and the R-PIS/ R-NIS.

hesitant fuzzy information in a more concise way and avoid information loss and distortion in decision process. (2) The most existing distance measures between hesitant fuzzy sets are calculated by extending the set with fewer element until they have the same number of elements, which mainly depends on the individual interest of the decision maker. In other words, the extending process weakens the objectivity of the calculation, but the distance measures based on the LCME principle can avoid weakening the objectivity of the computation results. (3) It is already known that “the positive ideal solution” is not always the fuzzy number 1 and “the negative ideal solution” is not the fuzzy number 0, but an interval between 0 and 1, so the traditional TOPSIS method may lead to unreasonable results in hesitant fuzzy decision making problems. The reference-TOPSIS method of the proposed distance measures can be seen as the extension of the traditional TOPSIS method, which can deal with the decision information more effective and can be applied in a wider range of situations.

Di+ Di

H1

H2

H3

0.1487

0.1508

0

0.1114

0.1294

H4

H5

0.1668

0.1618

0.1619

0.1333

0.1021

Table 7 The relative closeness coe?cients of the alternatives.

i

H1

H2

H3

H4

H5

0.4283

0.4618

1

0.4442

0.3867

Table 8 The closeness coe?cients obtained by dPWG with 1

= 0.1 = 0.2 = 0.4 = 0.6 = 0.8

5. Conclusions In some practical decision making problems, the ideal solutions cannot be the minimum value or the maximum value, the existing traditional decision making methods cannot deal with these problems. Considering the characteristics of the reference ideal can be any set or a point between the minimum value and the maximum value, we develop the reference ideal-TOPSIS method to the proposed new distance measures in multi-attribute decision making problems. The main contributions of the paper are given as follows: (1) Some new distance measures between HFSs are proposed, which take the hesitance degree of hesitant fuzzy element and the LCME principle into consideration. The proposed distance measure keeps the quality of decision information and avoids the decision information loss. (2) The reference ideal-

0.4 0.3739 0.3163 0.2517 0.1794

2

3

0.4413 0.4271 0.3981 0.3687 0.3381

1 1 1 1 1

4

0.4178 0.4123 0.4002 0.3858 0.3682

= 0.1 = 0.2 = 0.4 = 0.6 = 0.8

0.4966 0.4825 0.4490 0.4042 0.3370

2

3

0.5096 0.4986 0.4749 0.4479 0.4161

1 1 1 1 1

4

0.4649 0.4610 0.4509 0.4361 0.4110

Ranking results

5

0.3008 0.2747 0.2181 0.1543 0.0819

Table 9 The closeness coefficients obtained by dPWG with 1

= 1.

H3 H3 H3 H3 H3

H2 H2 H4 H4 H4

H4 H4 H2 H2 H2

H1 H1 H1 H1 H1

H5 H5 H5 H5 H5

H4 H4 H1 H1 H1

H5 H5 H5 H5 H5

= 2.

5

0.4614 0.4459 0.4092 0.3599 0.2829

Ranking results

H3 H3 H3 H3 H3

H2 H2 H2 H2 H2

H1 H1 H4 H4 H4

Table 4 The R-PIS and R-NIS.

R - P IS R - N IS

C1

C2

C3

C4

{0.7, 0.6} {0.9, 0.7, 0.6, 0.3, 0.1}

{0.9, 0.6} {0.9, 0.8, 0.7, 0.1}

{0.7, 0.5, 0.3} {0.9, 0.8, 0.7}

{0.6, 0.4} {0.9, 0.6, 0.5, 0.3}

8

Computers & Industrial Engineering 137 (2019) 106021

D. Liu and L. Wang

Table 10 The closeness coefficients obtained by dPWG with 1

= 0.1 = 0.2 = 0.4 = 0.6 = 0.8

0.5147 0.5076 0.4896 0.4630 0.4156

2

0.5407 0.5327 0.5145 0.4921 0.4630

3

1 1 1 1 1

4

0.4933 0.4913 0.4859 0.4764 0.4561

1

= 0.1 = 0.2 = 0.4 = 0.6 = 0.8

0.5111 0.5105 0.5084 0.5047 0.4950

2

0.5802 0.5770 0.5689 0.5569 0.5357

3

1 1 1 1 1

4

0.5413 0.5411 0.5406 0.5397 0.5367

0.5300 0.5213 0.4990 0.4673 0.4122

= 0.1 = 0.2 = 0.4 = 0.6 = 0.8

0.5026 0.5025 0.5024 0.5023 0.5017

2

0.5930 0.5921 0.5899 0.5859 0.5768

3

1 1 1 1 1

4

0.5654 0.5653 0.5653 0.5653 0.5651

1

= 0.1 = 0.2 = 0.4 = 0.6 = 0.8

0.4994 0.4994 0.4994 0.4994 0.4994

2

0.5967 0.5964 0.5960 0.5953 0.5930

3

1 1 1 1 1

4

0.5747 0.5748 0.5748 0.5747 0.5747

1

= 0.1 = 0.2 = 0.4 = 0.6 = 0.8

0.4989 0.4989 0.4988 0.4989 0.4988

2

0.5977 0.5977 0.5976 0.5975 0.5970

3

1 1 1 1 1

4

0.5779 0.5779 0.5779 0.5779 0.5778

1

= = = = = = =

1 2 3 6 10 15 20

0.4 0.4966 0.5147 0.5111 0.5026 0.4994 0.4989

2

0.4413 0.5091 0.5407 0.5802 0.5930 0.5967 0.5977

3

1 1 1 1 1 1 1

4

0.4178 0.4649 0.4933 0.5413 0.5654 0.5747 0.5779

H1 H1 H1 H5 H1

= = = = = = =

H4 H4 H4 H1 H5

H3 H3 H3 H3 H3

H5 H5 H5 H5 H5

H2 H2 H2 H2 H4

H3 H3 H3 H3 H3

H5 H5 H5 H5 H5

H2 H2 H2 H2 H2

H4 H4 H4 H4 H2

H1 H1 H1 H1 H1

H4 H4 H4 H4 H4

H1 H1 H1 H1 H1

H4 H4 H4 H4 H4

H1 H1 H1 H1 H1

= = = = = = =

H3 H3 H3 H3 H3

H2 H2 H2 H2 H5

H5 H5 H5 H5 H2

= = = = = = =

H3 H3 H3 H3 H3

H2 H2 H2 H2 H2

H5 H5 H5 H5 H5

H4 H4 H4 H4 H4

0.3008 0.4614 0.5300 0.5883 0.5966 0.5950 0.5927

= = = = = = =

H1 H1 H1 H1 H1

0.4123 0.4610 0.4913 0.5411 0.5653 0.5748 0.5779

0.3981 0.4749 0.5145 0.5689 0.5899 0.5960 0.5976

3

1 1 1 1 1 1 1

4

0.4002 0.4509 0.4859 0.5406 0.5653 0.5748 0.5779

1 2 3 6 10 15 20

0.2517 0.4042 0.4630 0.5047 0.5023 0.4994 0.4988

2

0.3687 0.4479 0.4921 0.5569 0.5859 0.5953 0.5975

3

1 1 1 1 1 1 1

4

0.3858 0.4361 0.4764 0.5397 0.5653 0.5747 0.5779

1 2 3 6 10 15 20

0.1794 0.3370 0.4156 0.4950 0.5017 0.4994 0.4988

2

0.3381 0.4161 0.4630 0.5357 0.5768 0.5930 0.5970

3

1 1 1 1 1 1 1

4

0.3682 0.4110 0.4561 0.5367 0.5651 0.5747 0.5778

= 0.2. Ranking results

5

0.2747 0.4459 0.5213 0.5868 0.5965 0.5949 0.5927

H3 H3 H3 H3 H3 H3 H3

H2 H2 H2 H5 H5 H2 H2

H4 H1 H5 H2 H2 H5 H5

H1 H4 H1 H4 H4 H4 H4

H5 H5 H4 H1 H1 H1 H1

H1 H1 H1 H4 H4 H4 H4

H5 H5 H4 H1 H1 H1 H1

H1 H1 H5 H4 H4 H4 H4

H5 H5 H1 H1 H1 H1 H1

H1 H1 H1 H2 H4 H4 H4

H5 H5 H5 H1 H1 H1 H1

= 0.4. Ranking results

5

0.2181 0.4092 0.4990 0.5823 0.5961 0.5949 0.5927

H3 H3 H3 H3 H3 H3 H3

H4 H2 H2 H5 H5 H2 H2

H2 H4 H5 H2 H2 H5 H5

= 0.6. Ranking results

5

0.1543 0.3599 0.4673 0.5747 0.5953 0.5949 0.5927

H3 H3 H3 H3 H3 H3 H3

H4 H2 H2 H5 H5 H2 H2

H2 H4 H4 H2 H2 H5 H5

= 0.8.

5

0.0819 0.2829 0.4122 0.5576 0.5931 0.5948 0.5927

Ranking results

H3 H3 H3 H3 H3 H3 H3

H4 H2 H2 H5 H5 H5 H2

H2 H4 H4 H4 H2 H2 H5

will be extended to the hesitant fuzzy linguistic decision environment and applied to other related decision fields. Data availability The data used to support the findings of this study are included within the article.

Ranking results

H3 H3 H3 H3 H3 H3 H3

0.3163 0.4490 0.4896 0.5084 0.5024 0.4994 0.4988

1

= 0.1.

5

1 1 1 1 1 1 1

4

Table 19 The closeness coefficients obtained by dPWG with

Ranking results

0.5927 0.5927 0.5927 0.5927 0.5927

1 2 3 6 10 15 20

2

1

= 20.

5

0.4271 0.4986 0.5327 0.5770 0.5921 0.5964 0.5977

3

Table 18 The closeness coefficients obtained by dPWG with

Ranking results

0.5950 0.5949 0.5949 0.5949 0.5948

0.3739 0.4825 0.5076 0.5105 0.5025 0.4994 0.4989

1

= 15.

5

1 2 3 6 10 15 20

2

Table 17 The closeness coefficients obtained by dPWG with

Ranking results

0.5966 0.5965 0.5961 0.5953 0.5931

Table 15 The closeness coefficients obtained by dPWG with

H5 H5 H5 H4 H4

= 10.

5

Table 14 The closeness coefficients obtained by dPWG with

H2 H2 H2 H2 H2

Ranking results

0.5883 0.5868 0.5823 0.5747 0.5576

Table 13 The closeness coefficients obtained by dPWG with

H3 H3 H3 H3 H3

1

= 6.

5

Table 12 The closeness coefficients obtained by dPWG with 1

Ranking results

5

Table 11 The closeness coefficients obtained by dPWG with

Table 16 The closeness coefficients obtained by dPWG with

= 3.

H2 H2 H2 H5 H5 H2 H2

H4 H1 H5 H2 H2 H5 H5

H1 H4 H1 H4 H4 H4 H4

H5 H5 H4 H1 H1 H1 H1

Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgement

TOPSIS method is developed to the proposed distance measures, which can solve the problems that the traditional TOPSIS method cannot solve in some situation. In future research, the developed distance measures

This research is fully supported by a grant by Social Science Foundation of China (15BTJ028). We thank the editor and anonymous reviewers for their helpful comments on an earlier draft of this paper. 9

Computers & Industrial Engineering 137 (2019) 106021

D. Liu and L. Wang

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