1. Introduction
In real-world practices, we always face tasks and activities in which it is necessary to use decision-making processes. Generally, decision making is a cognitive process based on different mental and reasoning processes that lead to the choice of a suitable alternative from a set of possible alternatives in a decision situation [
1,
2,
3,
4,
5]. Because of the inherent complexity and uncertainty of the decision situation or the existence of multiple and conflicting objectives, decision-making problems are complex and difficult; particularly in the era of big data, decision making becomes more complicated because the huge amounts of decision information and alternatives are continuously growing. Many new decision-making methods, such as granular computing techniques [
1,
6,
7,
8,
9,
10], have been proposed for expressing complex or uncertain information in decision-making processes and solving decision-making problems [
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23].
Nowadays, decision-making methods with hesitant fuzzy linguistic term sets (HFLTSs) are a focus point in linguistic decision making (LDM). In many qualitative decision environments, experts think of several possible linguistic values or richer expressions than a single term for an indicator, alternative, variable, and so forth. Accordingly, Rodríguez et al. [
24] proposed the concept of HFLTSs to overcome the drawback of existing fuzzy linguistic approaches: the elicitation of single and very simple terms to encompass and express the qualitative information. Formally, by taking into account the idea of hesitant fuzzy sets [
25] and using extended context-free grammars of a linguistic variable [
26], HFLTSs allow us to use different and great flexible forms to represent decision makers’ knowledge and preferences in LDM. To make a multi-criteria linguistic decision with HFLTSs, Rodríguez et al. developed the “min_upper” and “max_lower” operators to combine HFLTSs and obtain a linguistic interval for each alternative; then the linguistic intervals are used to build a preference relation between alternatives, and a nondominance choice degree is applied to obtain a solution set of alternatives for the decision problem. The use of the min_upper and max_lower operators produced the first method to deal with hesitant fuzzy linguistic information; since then, many researchers have paid attention to linguistic decision making with HFLTSs, such as in [
27], where Lee and Chen proposed likelihood-based comparison relations of HFLTSs and several hesitant fuzzy linguistic aggregation operators to overcome the drawbacks of the methods in [
24,
28]. In [
29], Liu and Rodríguez proposed a fuzzy envelope of HFLTSs for linguistic decision making with HFLTSs. In [
30], Montserrat-Adell et al. provided a lattice structure of the set of HFLTSs by means of the operations intersection and connected union, and presented two distances between hesitant fuzzy linguistic sets in the lattice structure, which can be used in linguistic decision making with HFLTSs. In [
31], Rodríguez et al. presented a group decision-making model based on HFLTSs. In [
28], Wei et al. defined new negation, max-union and min-intersection closed operations for HFLTSs; then they proposed a hesitant fuzzy linguistic weighting averaging operator and a hesitant fuzzy linguistic ordered weighting averaging operator to deal with multi-criteria decision-making problems with HFLTSs. Up to now, operations and extensions of HFLTSs [
32,
33,
34,
35,
36,
37,
38], hesitant fuzzy linguistic measures and aggregation operators [
39,
40,
41,
42,
43], and HFLTSs in decision making [
44,
45,
46,
47] have been widely studied.
In the existing decision-making methods, despite the existence of different decision-making processes in the literature that are composed of different phases, the TOPSIS method proposed in [
5] is a useful, important and widely studied multiple-attribute group decision-making method; formally, the TOPSIS method originates from the concept that the selected alternative should have the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. Its decision-making process can be expressed in the following five steps [
48]: (1) The normalization of the decision matrix; (2) the construction of the weighted normalized decision matrix; (3) the determination of positive and negative ideal solutions; (4) the calculation of separation measures and relative closeness; (5) the ranking of alternatives. Since then, many extended TOPSIS methods have been applied to different multiple-attribute decision making scenarios [
49,
50,
51,
52,
53,
54,
55]; for example, Chen [
56] proposed an extended TOPSIS method for multiple-attribute decision making by considering triangular fuzzy numbers and defining the crisp Euclidean distance between two fuzzy numbers. Similarly, Ashtiani et al. [
57] extended the TOPSIS method to solve a multiple-attribute decision-making problem with interval-valued fuzzy sets. He and Gong [
58] provided a natural generalization of the TOPSIS method to solve a multiple-attribute decision-making problem with intuitionistic fuzzy sets. Liu et al. [
59] developed a new TOPSIS method for decision-making problems with interval-valued intuitionistic fuzzy data. Yue [
60] presented a method for solving decision-making problems with an interval number and extended his method to intuitionistic fuzzy sets. In [
61], Liang et al. proposed an extended TOPSIS method with linguistic neutrosophic numbers to evaluate investment risks of metallic mines. In [
62], Sałabun proposed a new method to estimate the mean error of TOPSIS with the use of a fuzzy reference model.
In [
63], Beg and Rashid firstly proposed the TOPSIS method for HFLTSs, in which, the one decision matrix
X is calculated by aggregating the opinions of decision makers; the HFLTS positive- and negative-ideal solutions are obtained by the minimization of the minimal and maximal assessments of cost criteria and the maximization of the minimal and maximal assessments of benefit criteria; then the positive-ideal separation matrix (negative-ideal separation matrix) is constructed by distances between
X and the positive-ideal (negative-ideal) solution, which can be used to obtain the relative closeness of each alternative and rank all the alternatives. In this paper, we develop a new hesitant fuzzy linguistic TOPSIS method for group multi-criteria linguistic decision making, in which, we use the preference degree to define a pseudo-distance between two HFLTSs, and we present the positive and negative information of each criterion provided by each decision maker. Making use of the weighted 2-tuple linguistic aggregation operator, we aggregate the positive and negative information provided by
m decision makers to obtain the HFLTS positive- and negative-ideal solutions. Finally, we utilize the pseudo-distance to calculate distances between the assessments of the decision maker and the HFLTS positive- and negative-ideal solutions, and provide a new relative closeness degree of each alternative to rank all the alternatives. The rest of this paper is structured as follows: In
Section 2, we briefly review basic concepts and operations of HFLTSs and Beg and Rashid’s TOPSIS method. In
Section 3, we define the pseudo-distance between two HFLTSs and analyze its properties. We provide the positive and negative information of each criterion and aggregate these to obtain the HFLTS positive- and negative-ideal solutions. Accordingly, we propose the new hesitant fuzzy linguistic TOPSIS method for group multi-criteria linguistic decision making and design an algorithm to carry out hesitant fuzzy linguistic decision making. In
Section 4, we utilize an example to illustrate the practicality of the new hesitant fuzzy linguistic TOPSIS method and compare the method with Rodriguez’s method [
24], Beg and Rashid’s method [
63] and Liao’s method [
45]. We conclude the paper in
Section 5.
2. Preliminaries
In this section, we briefly review concepts and operators of HFLTSs and the TOPSIS method for HFLTSs, and we present the two important hesitant fuzzy linguistic decision-making methods, that is, Rodriguez’s method and Beg and Rashid’s method.
Definition 1. [24] Let S be a linguistic term set, , a HFLTS, , is an ordered finite subset of the consecutive linguistic terms of S. The basic operations on HFLTSs are as follows [
24]:
Lower bound: , and ;
Upper bound: , and ;
Complement: and ;
Union: or ;
Intersection: and ;
Envelope: .
Rodríguez et al. [
24] proposed the min_upper and max_lower operators to obtain the core information of hesitant fuzzy linguistic assessments of each alternative; then preference degrees [
64] are used to deal with multi-criteria linguistic decision making with HFLTSs. Formally, the min_upper and max_lower operators are as follows: Let
be a set of alternatives,
be a set of criteria,
be a linguistic term set, and
be a set of HFLTSs. The min_upper operator consists of the following two steps:
Apply the upper bound
for each HFLTS that is associated with each alternative:
Obtain the minimum linguistic term for each alternative:
The max_lower operator consists of the following two steps:
Apply the lower bound
for each HFLTS that is associated with each alternative:
Obtain the maximum linguistic term for each alternative:
Let
and
; then the core information of hesitant fuzzy linguistic assessments of alternative
is
On the basis of the core information of each alternative and the preference degrees [
64] between two sets of core information, the nondominance degree NDD
of each alternative can be calculated, and the best alternatives are the set of nondominated alternatives
.
Example 1. [24] Let , and nothing (), very low (), low (), medium (), high (), very high (), perfect (). Assessments provided by the decision maker are shown in Table 1. The min_upper operator and the max_lower operator are used to obtain the core information of each alternative, such as for alternative , , , , , , and . Similarly, and .
On the basis of , and , preference degrees [64] between them can be calculated; for example, for and , their preference degree is and the binary preference relation between the three alternatives isThen the nondominance degree of can be calculated, that is, and . Such as , similarly, and ; is selected. Beg and Rashid proposed an alternative hesitant fuzzy linguistic group decision method, that is, the TOPSIS method for HFLTSs [
63]. In the TOPSIS method, the main concepts are a distance between two HFLTSs and the HFLTS positive- and negative-ideal solutions, which can be formalized as follows: Let
and
be the two HFLTSs on
,
and
; then the distance between
and
is
Let
be
m HFLTS decision matrices provided by
m decision makers; then the one decision matrix formed by aggregating the opinions of
m decision makers is
, where
and
Let
and
be collections of benefit and cost criteria, respectively. The HFLTS positive-ideal (negative-ideal) solution
is defined as follows:
where
is the
ith considered alternative;
is the
jth criterion used for evaluating the alternatives;
;
; and
or
have the form
. On the basis of the distance between two HFLTSs and the HFLTS positive- and negative-ideal solutions, the positive-ideal separation matrix
and negative-ideal separation matrix
between
X and the positive- and negative-ideal solutions can be calculated as follows:
Accordingly, the relative closeness (
) of each alternative to the ideal solution is as follows:
where
and
. Ranking alternatives are carried out by using the following rule: the greater the value of
, the better the alternative
.
3. The Proposed TOPSIS for HFLTSs
In this section, we develop a new hesitant fuzzy linguistic TOPSIS method for linguistic decision-making problems. Compared with Beg and Rashid’s TOPSIS method, there are three different aspects: (1) We use the preference degree to define a pseudo-distance between two HFLTSs; (2) We present the positive and negative information of each criterion provided by each decision maker; considering weights of decision makers, we aggregate the positive and negative information provided by all decision makers to obtain the HFLTS positive- and negative-ideal solutions, respectively; (3) We propose a new relative closeness degree to rank alternatives. All of these are elaborated on in the following subsections.
3.1. A Pseudo-Distance between Two HFLTSs
The preference degree between two HFLTSs has been studied by many researchers [
24,
27,
28,
65]; generally, we let
and
be the two HFLTSs on
,
and
. Then the preference degree
between
and
is as follows:
For example, let
nothing
, very low
, low
, medium
, high
, very high
, perfect
. For HFLTSs
and
,
and
; then
Formally, the preference degree between two HFLTSs has the following properties: (1) ; (2) ; (3) if , then ; (4) if , then .
Definition 2. Let and be any two HFLTSs on and the HFLTS on S be the reference set; then we define the following: Intuitively, is the difference of preference degrees between two HFLTSs ( and ) and the the reference set . According to Equations (7) and (8), we have the following property:
Proposition 1. Let , and be HFLTSs on and the HFLTS on S be the reference set; then
- 1.
;
- 2.
;
- 3.
.
Proof. According to Equation (8),
and
is apparent. Here, we prove Proposition 1. as follows:
That is,
holds. ☐
Proposition 1 means that
is the pseudo-distance between HFLTSs
and
on
. In fact, we let
,
and
. According to Equation (8), if
, then
is apparent. If
, we have
, and according to Equation (7), we have
If
,
and
, then
; this means that
if and only if
does not always hold. As a special case, in Equation (8), we consider the condition
, that is, the pseudo-distance between HFLTS
and the reference set
. According to property 2 of the preference degree,
; thus
, that is,
, and hence the pseudo-distance between
and the reference set
is reduced as follows:
Accordingly, we can develop an ordering of HFLTSs on on the basis of the reference set; that is, for any two HFLTSs and on and the reference set , if and only if . Intuitively, the order on HFLTSs means that the closer the HFLTS is to the reference set , the bigger is. According to Equation (9), it can be easily proved that the order on HFLTSs is a pre-order, that is, satisfies the following:
The reflexive property: .
Transitivity: if and , then .
We note that if is the set of all HFLTSs on , then is a pre-order set.
Example 2. Let nothing , very low , low , medium , high , very high , perfect . For the HFLTSs and , and and , suppose the reference set and . According to Equation (9), we have and , that is, as a result of .
3.2. The HFLTS Positive- and Negative-Ideal Solutions
A group multi-criteria hesitant fuzzy linguistic decision-making problem is described as follows:
m decision makers
are asked to assess
n alternatives
with respect to
r criteria
by using HFLTSs on
; formally, decision maker
provides the decision matrix to express his or her assessments, that is,
where
means that decision maker
assesses alternative
with respect to criterion
by using the HFLTS
on
. On the basis of the decision matrix
, we provide the following definitions.
Definition 3. In the decision matrix , the positive information of each provided by decision maker is The negative information of provided by decision maker is Example 3. Let be a set of three alternatives, be a set of criteria defined for each alternative and nothing (), very low (), low (), medium (), high (), very high (), perfect ()} be the linguistic term set. The assessments provided by decision maker are shown in Table 2. For the criterion , we have , and ; hence , , and ; that is, the positive information and negative information of provided by decision maker are and , respectively.
We can notice from Example 3 that the positive information of
is the optimistic information according to assessments of all alternatives provided by decision maker
; the negative information of
is the pessimistic information according to assessments of all alternatives provided by decision maker
. Compared with Beg and Rashid’s method [
63], Equations (1) and (2) are aimed at aggregating the opinions of
m decision makers; the result is the one decision matrix
. However, Equations (11) and (12) are used to aggregate the opinions of
n alternatives provided by decision maker
with respect to the criterion
; the results are the optimistic information vector
and the pessimistic information vector
provided by decision maker
with respect to the criteria.
In a group multi-criteria hesitant fuzzy linguistic decision-making problem, for each criterion
, we denote the positive and negative information of
as
and
provided by decision maker
. We suppose weights
of
m decision makers; then we can use the weighted 2-tuple linguistic aggregation operator [
66,
67] to obtain the positive and negative information of
provided by
m decision makers, that is,
where
,
,
and
, and
is the usual round operation. For example, let the positive information of
be
,
and
, which are provided by three decision makers with weights
in a group multi-criteria hesitant fuzzy linguistic decision-making problem; then the positive information of
provided by the three decision makers is
Definition 4. In a group multi-criteria hesitant fuzzy linguistic decision-making problem, and are called the HFLTS positive-ideal solution (HPIS) and the HFLTS negative-ideal solution (HNIS), where and () are decided by Equations (13) and (14), respectively.
3.3. The New Hesitant Fuzzy Linguistic TOPSIS Method
On the basis of
Section 3.1 and
Section 3.2, we propose a hesitant fuzzy linguistic TOPSIS method that involves the following steps:
Step 1: Let m decision makers be asked to assess n alternatives with respect to r criteria by using HFLTSs on ; decision maker with weight provides the decision matrix to express his or her assessments, where and .
Step 2: For each decision matrix
, making use of Equations (11) and (12), we obtain the positive information
and the negative information
of
. Then we utilize weight
and Equations (13) and (14) to calculate the positive and negative information
and
of each
provided by
m decision makers; we can obtain the HFLTS positive- and negative-ideal solutions as follows:
Step 3: We calculate the one decision matrix
D by aggregating assessments of decision makers; that is, we use weights
and the weighted 2-tuple linguistic aggregation operator to aggregate
m decision matrices
:
where
and
for every
and
, respectively, and
is the usual round operation.
Step 4: On the basis of Equation (9) and the HFLTS positive- and negative-ideal solutions of Equation (15), we calculate the positive-ideal separation matrix
and the negative-ideal separation matrix
between assessments of decision makers and the HFLTS positive- and negative-ideal solutions, that is,
where
is the pseudo-distance between
and the reference set
, and
is the pseudo-distance between
and the reference set
.
If we consider weights
of
r criteria such that
and
, then the positive-ideal separation matrix
and the negative-ideal separation matrix
have the following forms:
Step 5: The ranking of alternatives in the original TOPSIS method is based on “the shortest distance from the positive-ideal solution and the farthest from the negative-ideal solution”; formally, this is also fulfilled by the relative closeness degree of each alternative in the existing TOPSIS methods. In the paper, on the basis of
and
, we provide the following relative closeness degree
of each alternative:
Formally, the relative closeness degree of each alternative is in . More importantly, is a monotone function in its components; that is, is increasing for and decreasing for . This is coincidental with “the shortest distance from the positive-ideal solution and the farthest from the negative-ideal solution”.
Step 6: Rank all the alternatives according to the relative closeness degree . The greater the value , the better the alternative ; that is, for any , if and only if .
On the basis of the above-mentioned six steps, we provide the following algorithm to implement the new hesitant fuzzy linguistic TOPSIS method to solve hesitant fuzzy linguistic group multi-criteria decision-making problems.
5. Conclusions
Motivated by the TOPSIS method in decision making, in this paper, we have developed a new hesitant fuzzy linguistic TOPSIS method for group multi-criteria hesitant fuzzy linguistic decision making. In the proposed method, we presented the positive and negative information of provided by decision maker to express the optimistic and pessimistic information of all alternatives provided by each decision maker. Making use of the weighted 2-tuple linguistic aggregation operator, we aggregated the positive and negative information of provided by m decision makers to obtain the HFLTS positive- and negative-ideal solutions. We defined the pseudo-distance between two HFLTSs and used this to measure the distance between assessments of the decision maker and the HFLTS positive- and negative-ideal solutions. On the basis of the obtained positive- and negative-ideal separation matrices, we proposed a new relative closeness degree of each alternative, which could be used to rank all the alternatives; intuitively, the greater the value of the relative closeness degree, the better the alternative. We utilized an example to illustrate the performance, usefulness and effectiveness of the new hesitant fuzzy linguistic TOPSIS method, and compared it with the symbolic aggregation-based method, the HFL-TOPSIS method and the HFL-VIKOR method.
It seems that the pseudo-distance between two HFLTSs and the relative closeness degree of the alternative are useful and alternative tools in hesitant fuzzy linguistic decision making. We will use the two concepts in the other decision making method and consider the proposed hesitant fuzzy linguistic TOPSIS to carry out hesitant fuzzy linguistic decision making with huge amounts of decision information and alternatives in the future works.