Abstract
Under the framework of decision matrix, this chapter focuses on GDM problems whose decision information takes the form of EHFLTSs. A general framework of multiple groups decision-making (MGDM) will be introduced at first for the problems with complex structure of groups. Then some specific MGDM approaches will be introduced based on information fusion of EHFLTSs.
Under the framework of decision matrix, this chapter focuses on GDM problems whose decision information takes the form of EHFLTSs. A general framework of multiple groups decision-making (MGDM) will be introduced at first for the problems with complex structure of groups. Then some specific MGDM approaches will be introduced based on information fusion of EHFLTSs. Especially, an admissible order-based GDM approach will be presented based on the admissible order defined in Sect. 2.3.
1 A Framework of Multiple Groups Decision-Making
In complex GDM problems, the scale of a group would be big or huge, and the structure of a group would be very complex. The existing contributions mainly focus on the sizes of groups, but pay little attention on the structures of groups. This section treats a group as several sub-groups based on its inner structure where members of each sub-groups own similar or the same knowledge and expertise. The GDM problems are thus called MGDM problems [19, 21]. The necessity of considering structures of groups could be summarized as follows:
In real applications, GDM is usually taken into account instead of individual decision-making although the derived decisions might be not always better than those derived by individual decision-making. This is caused by several reasons. Firstly, one expert can not afford the whole task of assessments due to the complexities of the problems. Then a group is necessary, in which each expert only needs to complete partial work. Secondly, groups own some notable advantages. It can be expected that a group takes advantage of the diverse strengths and expertise of its members, and reaches the superior solutions than the individuals. Michaelsen et al. [13] demonstrated that groups outperform their most proficient group members 97% of the time. Daily and Steiner [3] also showed that groups can achieve a higher number of brainstormed ideas.
However, there are also some disadvantages involved with GDM. It is apparent that groups are generally slower to arrive at decisions than individuals. Groupthink, sometimes, occurs when the members in the group feel pressure to confront what seems to be the dominant view of the group. Group polarization is another potential disadvantage of GDM. In addition, the members may perform the tasks quite differently. For example, individuals tend to take more risks [7] and act more selfishly [8] when they make decisions in a group.
In order to overcome the limitations of GDM, Wang and Xu [21] presented a novel decision-making framework, i.e., MGDM, to serve as a generation of traditional GDM. MGDM refers to as making decisions over available alternatives by a decision organization that is characterized by several groups. There are generally two categories of groups: heterogeneous groups and homogeneous groups. The experts in a heterogeneous group are adept at distinct fields of disciplines, and have diverse cultural backgrounds. Each expert is good at evaluating alternatives with respect to a subset of criteria. Heterogeneous group is adapted for the problems whose evaluations are interdisciplinary. The experts in a homogeneous group are from adjacent (or the same) disciplines. They have the same or similar cultural backgrounds. Each expert can conduct the same evaluation task in isolation. This kind of group is used to overcome the potential disadvantages of individuals. In the novel framework of [21], as can be seen in Part 1 of Fig. 3.3, the experts in each group are homogeneous while the experts in different groups are heterogeneous. Each group deals with partial evaluations according to its disciplines and backgrounds. The individuals of a group work in isolation if possible. The organization acts collectively to complete the whole work. Obviously, if there is only one group, the organization is reduced to a homogeneous group; while if there is only one member in each homogeneous group, the organization is reduced to a heterogeneous group.
1.1 Mathematical Description of MGDM
In the MGDM problems, several groups of experts act collectively to select the most relevant alternative(s) among the available ones. Formally, this problem can be described as follows [21]:
A decision organization is formed by r groups (denoted by \(\{G_l |l=1,2,\ldots , r \}\)) of experts. The t homogeneous experts in \(G_l\) are denoted by \(E = \{ e_{lk} | k = 1,2,\ldots ,t, l =1,2,\ldots ,r \}\). The experts of different groups are heterogeneous. The relative weights of experts within a group are indifferent. The organization is authorized to evaluate a set of m alternatives \(A = \{ a_1, a_2, \ldots , a_m \}\) in terms of n criteria \(C = \{ c_1, c_2, \ldots , c_n\}\). The weight vector of criteria is \(\mathbf {w} = (w_1, w_2, \ldots , w_n)\), where \(\sum \nolimits _{j=1}^{n}w_j = 1\), \(w_j \in [0,1] (j = 1,2,\ldots ,n)\). The group \(G_l\) evaluates a subset of C in isolation, denoted by \(SC_l\), such that \(SC_l \ne \emptyset (l = 1,2,\ldots ,r)\), \(\cup _{l=1}^{r} SC_l = C\). The performance of \(a_i\), provided by \(e_{lk}\), with respect to the criterion \(c_j\), is represented by a function \(V: A \times E \times C \rightarrow S\), where S represents the range of performance values in specific problems and could be [0, 1], the LTS \(S^{(\tau )}\) or \(2^{S^{(\tau )}}\). Thus, the performance is denoted by \(V(a_i, c_j, e_{lk} )\). The aim of the MGDM problem is to synthesize the evaluation values of each alternative and then reach a final decision.
1.2 Process of MGDM
In this section, we mainly focus on the solutions of three specific MGDM problems based on different scenarios. Let’s begin with a simple example. Suppose that a company is going to select and import the most valuable product from some alternatives. Three main criteria are the production cost, marketing cost and after-sales service cost. The manager authorizes three relative departments, i.e., the producing department, the marketing department and after-sales service department, to evaluate the product. If one criterion is only evaluated by a single department (for instance, the production costs of alternatives are only focused by the producing department), we call this case the 1-to-n scenario (because more than one criterion may be assessed by the same department). Moreover, distinct departments may pay their attention on the same criterion. For example, the production cost influences the work of all departments. Thus, they will all express their opinions about the costs of alternatives. We call this case the m-to-n scenarios. We focus on the solution of these scenarios in this section.
(1) The process for the 1-to-n scenario
As shown in Fig. 3.1, we have \(SC_{l_1} \cap SC_{l_2} = \emptyset (\forall l_1, l_2 = 1,2,\ldots , r)\) in this scenario. This is close to the case of traditional GDM problems. The process of this scenario is as follows [21]:
Step 1: Forming the decision matrix \(D = (V(a_i,c_j))_{m \times n}\). The performance of \(a_i\) with respect to \(c_j\), denoted by \(V(a_i,c_j)\), is synthesized by the opinions of all experts who contribute the corresponding piece of opinion, i.e.,
where \(\cup \) means the consideration of all the opinions.
Step 2: The choice of the aggregation operator Agg. Associated with \(\mathbf {w}\), we obtain the overall performance of each alternative \(V(a_i)\) by one aggregation operator:
Step 3: The choice of the best alternative(s).
(2) The processes for the m-to-n scenario
Generally, as shown in Fig. 3.2, a criterion may be evaluated by more than one group. Formally, there may exist \(l_1, l_2 \in \{1,2,\ldots ,r \}\) such that \(SC_{l_1} \cap SC_{l_2} \ne \emptyset \). Given \(c_j\), the groups which participate in evaluating \(c_j\) are denoted by \(G^{(j)}\). Apparently, \(G^{(j)} \subseteq G\) and \(\cup _{j = 1,2,\ldots ,n} G^{(j)} = G\). The experts in \(G^{(j)}\) are denoted by \(e_k^{(j)} (m = 1,2,\ldots , \#G^{(j)})\), where \( \#G^{(j)}\) is the number of experts in \( G^{(j)}\). In the following, we discuss two distinct cases of this scenario and present the corresponding processes.
Firstly, we assume that the weights of groups in \( G^{(j)}\) are equal. In other words, they have the same confidence level when evaluating \(c_j\). The corresponding decision-making process is [21]:
Step 1: Forming the decision matrix \(D = (V(a_i,c_j))_{m \times n}\). For \(c_j\), \(V(a_i,c_j)\) is derived by
where \(\cup \) means the consideration of all the opinions.
Step 2: See Step 2 of the process for the 1-to-n scenario.
Step 3: See Step 3 of the process for the 1-to-n scenario.
Secondly, we assume that the weights of groups in \( G^{(j)}\) cannot be ignored. Generally, because of the distinguished professional area, the groups in \( G^{(j)}\) may have different confidence levels when evaluating \(c_j\). The \(L^{(j)}\) groups in \( G^{(j)}\) are denoted by \( G_l^{(j)}\), \(l = 1,2,\ldots ,L^{(j)}\). For simplicity, the group \( G_l^{(j)}\) is advised to provide a confidence level associated with the evaluation of \( G_l^{(j)}\), denoted by \(cl_l^{(j)} (\in [0,1])\). The decision-making process is [21]:
Step 1: Synthesizing evaluations within each group. Similar to the above cases, we consider all the opinions in each group by the operator \(\cup \), and then the performance of \(a_i\) with respect to \(c_j\) provided by \(G_l\), denoted by \(V(a_i, c_j, G_l)\), can be derived.
Step 2: Forming the decision matrix \(D = (V(a_i,c_j))_{m \times n}\). The weights of groups in \( G^{(j)}\) can be derived by \({\lambda } = (\lambda _1, \lambda _2, \ldots , \lambda _{L^{(j)}})\), where \(\lambda _l = cl_l^{(j)} / \sum \nolimits _{l=1} ^{L^{(j)}} cl_l^{(j)}\). Utilize the aggregation operator Agg1 to synthesize the opinions of the \(L^{(j)}\) groups:
Step 3: The choice of the aggregation operator Agg2. See Step 2 of the process for the 1-to-n scenario.
Step 4: See Step 3 of the process for the 1-to-n scenario.
Obviously, we only present the general framework of the decision-making processes. In application, the operator \(\cup \) and the aggregation operators should be specified based on the forms of performance values and the preferences of decision makers. If the performance values take the form of EHFLTSs, the implementation of these processes will be developed in the next section.
2 A MGDM Approach Based on Information Fusion
2.1 Some Aggregation Operators of EHFLTSs
Information fusion is a key technique for GDM. Similar to Torra [17], we have the following extension principle:
Definition 3.1
([19]) Let \(\varTheta \) be a mapping \(\varTheta : (\bar{S}^{(g)})^n \rightarrow \bar{S}^{(g)}\), where \(\bar{S}^{(g)} = \{ s_\alpha | \alpha \in [-g, g] \}\), \(H = \{ h_S^1, h_S^2, \ldots , h_S^n \}\) is a set of n EHFLTSs defined on the reference set X based on \(\bar{S}^{(g)}\). The extension of \(\varTheta \) on H is defined for each \(x \in X\) by:
According to Definition 3.1, the existing aggregation operators of VLTs can be extended to the setting of EHFLTSs. In the rest of this section, two sets of EHFLTSs are denoted by \(H = \{ h_S^j \}\) and \({\dot{H}} = \{ \dot{h}_S^j \}\), \(j = 1,2,\ldots ,n\). The number of linguistic terms in \(h_S^j\) and \(\dot{h}_S^j\) are denoted by \(\# h_S^j\) and \(\# \dot{h}_S^j\). Linguistic terms in \(h_S^j\) and \(\dot{h}_S^j\) are denoted by \(s_{\alpha _j}\) and \(s_{\dot{\alpha }_j}\), respectively. Furthermore, the \(i_j\) linguistic terms in \(h_S^j\) and \(\dot{h}_S^j\) are specified by \(s_{\alpha _{ji_j}}\) and \(s_{\dot{\alpha }_{ji_j}}\) if necessary. We will introduce two classes of aggregation operators according to the form of weighting vector in the sequel.
2.1.1 Aggregation Operators with Linguistic Weights
Given a set of n EHFLTs are denoted by \(\{h_S^j \} (j = 1,2,\ldots ,n)\), their corresponding weights take the form of linguistic terms from \(\bar{S}^{(g)}\). Herrera and Herrera-Viedma [5] provided the linguistic weighted disjunction (LWD) operators in the setting values and the weights of objects are represented by simple linguistic terms. As the extension of the LWD operator, we have the following operator:
Definition 3.2
([19]) Let \(\{h_S^j \} (j = 1,2,\ldots ,n)\) be n EHFLTSs, \(\omega = (\omega _1, \omega _2, \ldots , \omega _n)\) be the weighting vector such that \(\omega _j \in \bar{S}^{(g)} (j = 1,2, \ldots , n)\). A mapping \(EHFLWD: (\bar{S}^{(g)})^n \rightarrow \bar{S}^{(g)}\) is called an extended hesitant fuzzy linguistic weighted disjunction (EHFLWD) operator of dimension n if
If \(\omega = (s_g, \ldots , s_g)\), where \(s_g\) is the largest term in \(\bar{S}^{(g)}\), then \(\omega _j \wedge h_S^j = h_S^j\), we have
However, \(\mathop \vee \limits _{j = 1}^n h_S^j \ne \mathop {\max \nolimits } \limits _{j} \{ h_S^j \}\).
The OWA operator [25] provides an aggregation strategy to lie between the max and min operators because of its re-ordering step. In linguistic setting, Yager [26] presented an ordinal form of the OWA operator. Motivated by which, we introduce the OWA operator in the extended hesitant fuzzy linguistic setting.
Definition 3.3
([19]) Let \(\{h_S^j \} (j = 1,2,\ldots ,n)\) be n EHFLTSs. An extended hesitant fuzzy ordinal OWA (EHFOOWA) operator of dimension n is a mapping \(EHFOOWA: (\bar{S}^{(g)})^n \rightarrow \bar{S}^{(g)}\) such that
where \(\mathbf {w} = (w_1, w_2, \ldots , w_n)\) is the associated weighting vector, \(w_j \in \bar{S}^{(g)} (j = 1,2, \ldots , n)\), \(h_S^{\sigma (j)}\)is the j-th largest element of \(\{h_S^j \}\).
If \(\mathbf {w} = (s_g, s_{-g}, \ldots , s_{-g})\), then \(w_1 \wedge h_S^{\sigma (1)} = h_S^{\sigma (1)}\), \(w_j \wedge h_S^{\sigma (j)} = w_j (\forall j = 2, 3, \ldots , n)\), we have
The EHFOOWA operator is reduced to the extended hesitant fuzzy linguistic max (EHFLM1) operator. Similarly, if \(\mathbf {w} = (s_{-g}, \ldots , s_{-g}, s_g)\), \( EHFOOWA (h_S^1, h_S^2, \ldots , h_S^n) = \mathop {\min } \limits _{j} \{ h_S^j \}\), The EHFOOWA operator is reduced to the extended hesitant fuzzy linguistic min (EHFLM2) operator in this case.
Consider that the LWD operator weights only the values themselves, while the ordinal OWA operator weights the re-ordered positions of the values only, Xu [23] proposed an ordinal hybrid aggregation (OHA) operator to reflect the importance degrees of both the linguistic arguments and their ordered positions. Based on the same idea, the follow aggregation operator can be defined:
Definition 3.4
([19]) Let \(\{h_S^j \} (j = 1,2,\ldots ,n)\) be n EHFLTSs. An extended hesitant fuzzy ordinal hybrid aggregation (EHFOHA) operator of dimension n is a mapping \(EHFOHA: (\bar{S}^{(g)})^n \rightarrow \bar{S}^{(g)}\) such that
where \(\mathbf {w} = (w_1, w_2, \ldots , w_n)\) is the associated weighting vector, \(w_j \in \bar{S}^{(g)} (j = 1,2, \ldots , n)\), \(h_S^{\dot{\sigma }(j)}\) is the j-th largest element in \(\{\bar{h}_S^j = \omega _j \wedge h_S^j \}\), \(\omega = (\omega _1, \omega _2, \ldots , \omega _n)\) is the weighting vector of \(\{h_S^j \}\), \(\omega _j \in \bar{S}^{(g)} (j = 1,2, \ldots , n)\).
Especially, if \(\mathbf {w} = (s_g, s_g, \ldots , s_g)\), then \(\forall h_S^{\dot{\sigma }(j)}\), \(w_j \wedge h_S^{\dot{\sigma } (j)} = h_S^{\dot{\sigma } (j)}\),
thus the EHFOHA operator is reduced to the EHFLWD operator. If \(\omega = (s_g, s_g, \ldots , s_g)\), then \(\omega _j \wedge h_S^j = h_S^j (\forall h_S^j)\), thus the EHFOHA operator is reduced to the EHFOOWA operator.
2.1.2 Aggregation Operators with Numerical Weights
Except for linguistic weights, numerical weights are often used in application as well. In this case, several aggregation operators are developed in linguistic setting, such as the linguistic weighted averaging (LWA) operator [24], the linguistic OWA operator [24], the linguistic hybrid aggregation (LHA) operator [23], the induced linguistic OWA operator [24] and so on. Based on the extension principle and some existing linguistic aggregation operator, some new aggregation operators can be developed as follows:
Definition 3.5
([19]) Let \(\{h_S^j \} (j = 1,2,\ldots ,n)\) be n EHFLTSs, their weighting vector is \(\omega = (\omega _1, \omega _2, \ldots , \omega _n)\), where \(\omega _j \in [0,1] (j = 1,2, \ldots , n)\) and \(\sum \nolimits _{j=1}^{n} \omega _j =1\). A mapping \(EHFLWA: (\bar{S}^{(g)})^n \rightarrow \bar{S}^{(g)}\) is called an extended hesitant fuzzy linguistic weighted averaging (EHFLWA) operator of dimension n if
where \(\dot{\bar{\alpha }} = \sum \nolimits _{j=1}^{n} \omega s_{\alpha _j}\).
The EHFLWA operator extends both the weighted averaging (WA) operator and the LWA operator. In fact, if \(\omega = (1/n, 1/n, \ldots , 1/n)\), then the EHFLWA operator is reduced to the extended hesitant fuzzy linguistic averaging (EHFLA) operator:
Definition 3.6
([19]) Let \(\{h_S^j \} (j = 1,2,\ldots ,n)\) be n EHFLTSs. An EHFLOWA operator of dimension n is a mapping \(EHFLOWA: (\bar{S}^{(g)})^n \rightarrow \bar{S}^{(g)}\)such that
where \(\bar{\beta } = \sum \nolimits _{j=1}^n w_j \beta _j\), \(h_S^{\sigma (j)}\) is the j-th biggest element of \(\{h_S^j \}\), \(\mathbf {w} = (w_1, w_2, \ldots , w_n)\) is the associated weighting vector such that \(w_j \in [0,1] (j = 1,2, \ldots , n)\) and \(\sum \nolimits _{j=1}^{n} w_j =1\).
If \(\mathbf {w} = (1,0,\ldots ,0)\), then \(w_1 h_S^{\sigma (1)} = h_S^{\sigma (1)} \) and \(w_j h_S^{\sigma (j)} = s_0~(\forall j=2,3, \ldots ,n)\), thus
The EHFLOWA operator is reduced to the EHFLM1 operator. Similarly, if \(\mathbf {w} = (0, \ldots , 0,1)\), then the EHFLOWA operator is reduced to the EHFLM2 operator.
According to Definitions 3.5 and 3.6, it is clear that the EHFLWA operator weights the input EHFLTs, while the EHFLOWA operator weights the ordered position instead. The weights represent distinct aspects of inputs in these two operators. The following operator could overcome this drawback:
Definition 3.7
([19]) Let \(\{h_S^j \} (j = 1,2,\ldots ,n)\) be n EHFLTSs, An extended hesitant fuzzy linguistic hybrid aggregation (EHFLHA) operator of dimension n is a mapping \(EHFLHA: (\bar{S}^{(g)})^n \rightarrow \bar{S}^{(g)}\) such that
where \(\dot{\bar{\beta }} = \sum \nolimits _{j=1}^n w_j \dot{\beta }_j\), \(h_S^{\dot{\sigma }(j)}\) is the j-th biggest element of \(\{ \bar{h}_S^j = n \omega _j h_S^j \}\), \(\omega = (\omega _1, \omega _1, \ldots , \omega _n)\) is the weighting vector of \(\{ h_S^j \}\) such that \(\omega _j \in [0,1] (j \!=\! 1,2, \ldots , n)\) and \(\sum \nolimits _{j=1}^{n} \omega _j =1\), n is the balancing coefficient, \(\mathbf {w} = (w_1, w_2, \ldots , w_n)\) is the associated weighting vector such that \(w_j \in [0,1] (j = 1,2, \ldots , n)\) and \(\sum \nolimits _{j=1}^{n} w_j =1\).
If \(\mathbf {w} = (1/n, 1/n, \ldots , 1/n)\), according to Theorems 2.15 and 2.16,
then the EHFLHA operator is reduced to the EHFLWA operator. Similarly, if \(\omega = (1/n, 1/n, \ldots , 1/n)\), then \(\bar{h}_S^j = h_S^j\), the EHFLHA operator is reduced to the EHFLOWA operator.
2.2 Properties of the Aggregation Operators
We will discuss some properties of the presented aggregation operators in this subsection. Because of the operation \(\cup \), most of the operators do not possess excellent mathematical properties, such as monotonicity, idempotency, commutativity and boundary. But, luckily, we will see some operators own properties like these four.
Theorem 3.8
([19]) Let \(\{h_S^j \}\) and \(\{ \dot{h}_S^j \}\) (\(j = 1,2,\ldots ,n\)) be two sets of EHFLTSs. If \(\exists i \in \{ 1,2,\ldots , n \}\) such that \(\#h_S^i = \# \dot{h}_S^i = N_i\) and \(\forall s_{\alpha _{ij_i}} \in h_S^1, s_{\dot{\alpha }_{ij_i}} \in \dot{h}_S^i\), \(s_{\alpha _{ij_i}} \le s_{\dot{\alpha }_{ij_i}}\), and \(\forall j \ne i\), \(h_S^i = \dot{h}_S^i\), then we have
Proof
Since \(s_{\alpha _{ij_i}} \le s_{\dot{\alpha }_{ij_i}}\), we have \(\forall \omega _i \in S^{(g)}\), \(\omega _i \wedge s_{\alpha _{ij_i}} \le \omega _i \wedge s_{\dot{\alpha }_{ij_i}}\), then
which means \(\omega _i \wedge h_S^i \le \omega _i \wedge \dot{h}_S^i\), thus \(EHFLWD(h_S^1, h_S^2, \ldots , h_S^n) = (\omega _1 \wedge h_S^1) \vee \cdots \vee (\omega _i \wedge h_S^i) \vee \cdots \vee (\omega _n \wedge h_S^n) \!\le \! (\omega _1 \wedge h_S^1) \vee \cdots \vee (\omega _i \wedge \dot{h}_S^i) \vee \cdots \vee (\omega _n \wedge h_S^n) = (\omega _1 \wedge \dot{h}_S^1) \vee \cdots \vee (\omega _i \wedge \dot{h}_S^i) \vee \cdots \vee (\omega _n \wedge \dot{h}_S^n) = EHFLWD(\dot{h}_S^1, \dot{h}_S^2, \ldots , \dot{h}_S^n)\). \(\square \)
Theorem 3.9
([19] Quasi-Boundary) Let \(\{h_S^j \}(j = 1,2,\ldots ,n)\) be n EHFLTSs. Then
where \(s_{-L} = \mathop {\min }\limits _j \{ \min \{ \omega _j ,\mathop {\min }\limits _{s_{\alpha _j} \in h_S^j} \{ s_{\alpha _j } \} \} \} \), \(s_L = \mathop {\max \nolimits }\limits _j \{ \max \{ \omega _j ,\mathop {\max }\limits _{s_{\alpha _j} \in h_S^j} \{ s_{\alpha _j } \} \} \} \).
Proof
\(\forall s_{\alpha _1} \in h_S^1, s_{\alpha _2} \in h_S^2, \ldots , s_{\alpha _n} \in h_S^n\),
Thus \(s_{-L} \le E( \cup _{s_{\alpha _1} \in h_S^1, \ldots , s_{\alpha _n} \in h_S^n} \{ \mathop {\max }\limits _j \{ \min \{ \omega _j, s_{\alpha _j} \} \} \} ) \le s_L\), where E() is the expected term defined in Definition 2.18. According to Definition 2.20, \(s_{-L} \le EHFLWD(h_S^1, h_S^2, \ldots , h_S^n) \le s_L\). \(\square \)
Similarly, using the same approach, we can easily prove the following theorem:
Theorem 3.10
([19] Quasi-Boundary) Let \(\{h_S^j \}(j = 1,2,\ldots ,n)\) be n EHFLTSs. Then
where \(s_{-L} = \mathop {\min }\limits _j \{ \min \{ w _j ,\mathop {\min }\limits _{s_{\alpha _j} \in h_S^j} \{ s_{\alpha _j } \} \} \} \), \(s_L = \mathop {\max \nolimits }\limits _j \{ \max \{ w _j ,\mathop {\max }\limits _{s_{\alpha _j} \in h_S^j} \{ s_{\alpha _j } \} \} \} \).
Theorem 3.11
([19] Commutativity) Let \(\{h_S^j \}(j = 1,2,\ldots ,n)\) be n EHFLTSs. Then we have
(1) \(EHFOOWA(h_S^1, h_S^2, \ldots , h_S^n) = EHFOOWA(\dot{h}_S^1, \dot{h}_S^2, \ldots , \dot{h}_S^n)\);
(2) \(EHFLOWA(h_S^1, h_S^2, \ldots , h_S^n) = EHFLOWA(\dot{h}_S^1, \dot{h}_S^2, \ldots , \dot{h}_S^n)\);
where \(\dot{h}_S^1, \dot{h}_S^2, \ldots , \dot{h}_S^n\) is any permutation of \(h_S^1, h_S^2, \ldots , h_S^n\).
Proof
(1) Since \(\dot{h}_S^1, \dot{h}_S^2, \ldots , \dot{h}_S^n\) is any permutation of \(h_S^1, h_S^2, \ldots , h_S^n\), then \(h_S^{\sigma (j)} = \dot{h}_S^{\sigma (j)} (\forall j = 1,2,\ldots ,n)\), and thus \(EHFOOWA(h_S^1, h_S^2, \ldots , h_S^n) = \mathop {\vee }\limits _{j=1}^n (w_j \wedge h_S^{\sigma (j)}) = \mathop {\vee }\limits _{j=1}^n (w_j \wedge \dot{h}_S^{\sigma (j)}) = EHFOOWA(\dot{h}_S^1, \dot{h}_S^2, \ldots , \dot{h}_S^n)\).
(2) The proof is similar to (1). \(\square \)
Theorem 3.12
([19] Quasi-Idempotency) Let \(h_S\) be an EHFLTS, \(\cong \) be the equivalent relation defined in Definition 2.20, then
Proof
Assume \(\# h_S = N\), \(h_S = \{ s_{\beta _1}, s_{\beta _2}, \ldots , s_{\alpha _N} \}\), then
Firstly, we prove \(E(EHFLWA(h_S, h_S, \ldots , h_S)) = E(h_S)\). We only prove the case of \(n=2\), other cases (\(n \ge 2\)) could be proven by mathematical induction. Since \(n=2\),
When \(\alpha _1\) and \(\alpha _2\) traverse \(\beta _1, \beta _2,\ldots , \beta _N\), Eq. (3.8) includes \(N^2\) possible values whose sum is \(N\omega _1 \sum \nolimits _{j=1}^N \beta _j + N \omega _2 \sum \nolimits _{j=1}^N \beta _j = N \sum \nolimits _{j=1}^N \beta _j\). Therefore,
Secondly, we prove \( D(EHFLWA(h_S,h_S, \ldots ,h_S) ) = D(h_S)\), where the function D is the hesitation degree defined in Definition 2.19. The max and min linguistic terms of \(h_S\) are denoted by \(s_{\alpha ^{+}}\) and \(s_{\alpha ^{-}}\), thus \(D(h_S) = (\alpha ^{+} - \alpha ^{-}) / (2g+1)\). According to the basic operations of EHFLTSs and the definition of the EHFLWA operator, the max virtual linguistic term in \(EHFLWA(h_S,h_S, \ldots ,h_S) \) is derived if \(s_{\alpha _1} = \cdots = s_{\alpha _N} = s_{\alpha ^{+}}\), thus \(\mathop {\oplus }\limits _{j=1}^N \omega _j s_{\alpha _j} = s_{\alpha ^{+}}\). Similarly, the min virtual linguistic term in \(EHFLWA(h_S, h_S, \ldots ,h_S) \) is \(s_{\alpha ^{-}}\), thus \( D(EHFLWA(h_S,h_S, \ldots ,h_S) ) = D(h_S)\).
According to Definition 2.20, \(EHFLWA(h_S, h_S, \ldots , h_S) \cong h_S\). \(\square \)
Theorem 3.13
([19] Quasi-Boundary) Let \(\{h_S^j \}(j = 1,2,\ldots ,n)\) be n EHFLTSs. Then
where \(s_{-L} = \mathop {\min }\limits _j \{ \mathop {\min }\limits _{s_{\alpha _j} \in h_S^j} \{ s_{\alpha _j } \} \} \), \(s_L = \mathop {\max \nolimits }\limits _j \{ \mathop {\max }\limits _{s_{\alpha _j} \in h_S^j} \{ s_{\alpha _j } \} \} \).
Proof
Since \(\forall i = 1,2,\ldots ,n\),
Then \(s_{-L} \le \mathop {\oplus } \limits _{i=1}^n \omega _i s_{\alpha _i} \le s_L\), which leads to
Thus \(s_{-L} \le EHFLWA(h_S^1, h_S^2, \ldots , h_S^n) \le s_L\). \(\square \)
2.3 Implementation of the MGDM Processes
This section will implement the MGDM processes presented in Sect. 3.1 in the setting of EHFLTSs. The problem description can be found in Sect. 3.1.1. Figure 3.3 illustrates the three parts of the proposed approach [19].
Part 1. Structure of groups in the organization. Experts in each group are homogeneous while experts in different groups are heterogeneous. Each group deals with a part of evaluations according to its knowledge and speciality. Individuals of a group work in isolation if possible. The whole organization works collectively to complete the entire evaluations.
Part 2. Evaluation and transformation. Given a linguistic term set \(S^{(g)}\), HFLTSs can be directly used by the experts to elicit several linguistic values for a linguistic variable when they hesitate among several values. However, such elements are not similar to human beings’ way of thinking and reasoning. Therefore, Rogríguez et al. [16] defined a context-free grammar to generate linguistic expressions that are more similar to human beings’ expressions. Then the linguistic expressions provided by experts are transformed into HFLTSs by using a transformation function. According to the way of individual thinking in fuzzy uncertain circumstance and the proposed construction axiom, in this model, individual evaluations are represented by linguistic expressions similar to human beings’ way of thinking and reasoning and then transformed to HFLTSs.
Part 3. Synthesis for decision-making. In this phase, two tools are used for synthesis. The union operation is used to transform HFLTSs to a generalized case, EHFLTSs. Aggregation operators are used to synthesize opinions represented by EHFLTSs. The most difference between union and aggregation operator is that all the original information are kept when the former is used while averaging value is obtained by some means when an aggregation operator is used. Thus this model can reduce the use of aggregation operators and eliminate loss of information.
Suppose that the decision organization G is divided into r groups \(G= \{G_l | l = 1,2,\ldots ,r \}\) whose weights are denoted by \(\omega ^{(G)} = (\omega _1^{(G)}, \omega _2^{(G)}, \ldots , \omega _r^{(G)})\). The weights of criteria \(C = \{c_j |j=1,2,\ldots ,n \}\) are denoted by \(\omega ^{(C)} = (\omega _1^{(C)}, \omega _2^{(C)}, \ldots , \omega _n^{(C)})\). Group \(G_l\) is authorized to evaluate the set of alternatives \(A = \{a_1, a_2, \ldots , a_m \}\) with respect to a subset of criteria \(SC_l\). In this section, we implement the MGDM processes in Sect. 3.1.2 by two different scenarios with different types of weights [19]. For simplicity, we suppose that every group is authorized to evaluate alternatives with respect to the whole set of criteria.
(1) Weights take the form of linguistic terms. Suppose \(\omega _l^{(G)} \in S^{(g)}\), \(\omega _j^{(C)} \in S^{(g)}\), \(l = 1,2,\ldots , r\), \(j = 1,2,\ldots ,n\). The process of the second scenario in Sect. 3.1.2 can be specified by the following Approach 1.
Step 1: Union within each group. The evaluation information of \(a_i\), with respect to the criterion \(c_j\), provided by the group \(G_l\), denoted by \(h_S(a_i, c_j, G_l)\), is derived by:
where \(\#G_l\) is the number of experts in \(G_l\), \(l = 1,2,\ldots , r\), \(i=1,2,\ldots ,m\), \(j=1,2,\ldots ,n\).
Step 2: The collective overall preference values of \(a_i\) with respect to the criterion \(c_j\), denoted by \(h_S(a_i,c_j)\), are obtained by the EHFOHA operator:
where \(i=1,2,\ldots ,m\), \(j=1,2,\ldots ,n\), \(\mathbf {w} = (w_1, w_2, \ldots , w_r)\) is the associated weighting vector of the EHFOHA operator, \(w_l \in S^{(g)}\), \(l=1,2,\ldots ,r\).
Step 3: The overall preference values of \(a_i\), denoted by \(h_S(a_i)\), are obtained by utilizing the EHFLWD operator:
where \(i = 1,2,\ldots ,m\).
Step 4: Utilize the overall preference values \(h_S(a_i)\) to rank the alternatives.
(2) Weights take the form of real numbers. Let \(\omega _l^{(G)} \in [0,1]\), \(l = 1,2,\ldots , r\), \(\sum \nolimits _{l=1}^r \omega _l^{(G)} = 1\), \(\omega _j^{(C)} \in [0,1]\), \(j = 1,2,\ldots ,n\), \(\sum \nolimits _{j=1}^n \omega _l^{(C)} = 1\). Then the process of the second scenario in Sect. 3.1.2 can be specified by the following Approach 2.
Step 1: See Step 1 of Approach 1.
Step 2: The collective overall preference values of \(a_i\) with respect to the criterion \(c_j\), denoted by \(h_S(a_i,c_j)\), are obtained by the EHFLHA operator:
where \(i=1,2,\ldots ,m\), \(j=1,2,\ldots ,n\), \(\mathbf {w} = (w_1, w_2, \ldots , w_r)\) is the associated weighting vector of the EHFLHA operator such that \(w_l \in [0,1]\), \(l=1,2,\ldots ,r\), \(\sum \nolimits _{l=1}^n w_l = 1\).
Step 3: The overall preference values of \(a_i\), denoted by \(h_S(a_i)\), are obtained utilizing the EHFLWA operator:
where \(i = 1,2,\ldots ,m\).
Step 4: See Step 4 of Approach 1.
We assumed that the weighting vectors are completely known in the above scenarios. In fact, if the weighting vector of criteria is absolutely unknown, then we can use the corresponding OWA operator instead of the weighted averaging operator. That is, in Step 3 of the two scenarios, we can replace the EHFLWD operator and the EHFLWA operator with the EHFOOWA operator and the EHFLOWA operator respectively. Further, the associated weighting vector of the EHFLOWA operator can be determined by the normal distribution based method or others [22].
2.4 Applications
In this section, we apply the proposed linguistic MGDM approaches in a practical example of evaluating university faculty for tenure and promotion [1, 23]. The criteria used in some universities are \(c_1\): teaching, \(c_2\): research, and \(c_3\): service (whose weighting vector is \(\omega ^{(C)} = (0.14, 0.26, 0.6)\)). Five alternatives (faculty candidates), \(\{ a_i | i=1,2,3,4,5 \}\), are to be evaluated using the LTS \(S^{(3)} = \{s_{-3}, \ldots , s_0, \ldots , s_3 \}\) by two groups of experts (whose weighting vector is \(\omega ^{(G)} = (0.6, 0.4)\)). After elicitation, the experts’ evaluation information is listed in Table 3.1. As the weighting vectors take the form of real numbers, we utilize Approach 2 to meet a decision [19].
Step 1: The evaluation information of the two groups is derived by the union of HFLTSs provided by the experts of each group. The results are listed in Table 3.2.
Step 2: The performances of alternatives with respect to the set of criteria can be derived by using the EHFLHA operator with \(\mathbf {w} = (0.5,0.5)\), which are shown in Table 3.3.
Step 3: The overall performances of alternatives are derived by the EHFLWA operator. The resultant expected terms are:
Step 4: Based on Definition 2.20, the ranking of 5 alternatives is:
Thus \(a_2\) is the best alternative.
2.5 Comparative Analysis
As an alternative solution of the above problem, the LA operator, the LWA operator and LHA operator are used for comparison. Without the idea of hesitation, we cannot deal with several possible values at the same time. Thus if experts have hesitancy among several possible linguistic terms, a pre-aggregation step has to be done, and then an averaging value is computed by some means. Following the advice of Xu [23], the problem can be processed by some steps.
Step 1: If an expert has hesitancy among several possible linguistic terms, the LA operator is used to obtain the corresponding averaging values. For example, the resultant decision matrix provided by \(e_{11}\) is transformed into Table 3.4.
Step 2: The evaluations within each group are synthesized by the LA operator as well as the relative weights of experts within a group are indifferent. For example, the resultant decision matrix of \(G_1\) is shown in Table 3.5.
Step 3: We aggregate the results of Step 2 by the LHA operator to obtain the final decision matrix. The weighting vector and the associated weighting vector are the same as those of Sect. 3.2.4. The result is presented in Table 3.6.
Step 4: The overall performances of alternatives are:
Step 5: The alternatives can be ranked as: \(a_3 \prec a_5 \prec a_1 \prec a_4 \prec a_2\).
Based on the procedures of the comparable processes, we discuss their differences by the following aspects:
(1) Number of times of using aggregation operators. As the idea of HFS is used, the introduced model and process need less aggregation. Compared to the existing process, the pre-aggregation step in expert level is eliminated. Further, aggregation within each group is conducted by the operation \(\cup \) rather than an aggregation operator.
(2) Possible values versus averaging values. Because of less aggregation operator is used, all possible values, rather than only averaging values, are maintained for consideration. As in Tables 3.1 and 3.4, the introduced process uses HFLTSs to represent experts’ opinions, while the existing process has to synthesize all possible linguistic terms to an averaging value. The same phenomenon happens in each level of aggregation. Therefore, the introduced process handles all possible values along with the procedure of aggregation, it is more meaningful than considering just averaging values as there is no loss of information.
(3) Final decisions. The two methods are different but agree on the first choice \(a_2\), which validates that the introduced process is reasonable and it is useful to consider all possible values. We can also see that the priorities of five alternatives are distinct. There is a rank reversal between \(a_4\) and \(a_5\). The introduced process uses all possible values for synthesis and needs less aggregation, as analyzed above, thus the final decision would be more rational.
3 A Two-Phase GDM Approach Based on Admissible Orders
We shall focus on the use of admissible orders developed in Sect. 2.3 in GDM in this section. In order that, the extended hesitant fuzzy linguistic OWA (EHFLOWA) operator based on admissible orders is introduced at first.
3.1 Defining the EHFLOWA Operator Based on Admissible Orders
Zhang and Wu [27] proposed the EHFLOWA operator by defining the order of EHFLTSs with the expected term. Moreover, Wang [19] defined an improved version according to the partial order \(\preceq _W\) defined in Eq. (2.25). However, it is not sufficient if some input elements are not comparable with respect to these partial orders. This section introduces a new definition of the EHFLOWA operator. The linguistic OWA operator for CWW is as follows:
Definition 3.14
([24]) Let \(\{s_{\alpha _1}, s_{\alpha _2}, \ldots , s_{\alpha _m} \}\) be m linguistic terms, \(\mathbf {w} = (w_1,w_2,\ldots ,w_m) \in [0,1]^m\) be the associated vector such that \(\sum \nolimits _{i=1}^m w_i = 1\). A linguistic OWA operator of dimension m associated with \(\mathbf {w}\) is a mapping \(LOWA_{\mathbf {w}}: (S^{(g)})^m \rightarrow S^{(g)}\) such that
where \(s_{\alpha _{\sigma (i)}}\) is the i-th largest of \(\{s_{\alpha _1}, s_{\alpha _2}, \ldots , s_{\alpha _m} \}\).
For convenience, a set of m EHFLTSs, normalized by the technique of Sect. 2.3, is denoted by \(H = \{ h_S^i | i=1,2,\ldots ,m \}\), where \(h_S^i = \{s_{\alpha _j}^i | j = 1,2,\ldots , n \} \in \mathcal{E} ^n (S)\). Given the admissible order \(\preceq \) defined in Definition 2.24, the EHFLOWA operator can be generalized as follows [20]:
Definition 3.15
([24]) Let \(H = \{ h_S^i | i=1,2,\ldots ,m \}\) be m EHFLTSs, \(\mathbf {w} = (w_1,w_2,\ldots ,w_m) \in [0,1]^m\) be the associated vector such that \(\sum \nolimits _{i=1}^m w_i = 1\), and \(\preceq \) be an admissible order on \(\mathcal{E} ^n (S)\). An EHFLOWA operator of dimension m associated with \(\preceq \) and \(\mathbf {w}\) is a mapping \(EHFLOWA_{\mathbf {w}}^{\preceq }: (\mathcal{E} ^n (S))^m \rightarrow \mathcal{E} ^n (S)\) such that
where \(h_S^{\sigma (i)}\) is the i-th largest of H with respect to the order \(\preceq \).
Based on Definition 2.22, Eq. (3.10) can be rewritten as [20]:
where \(s_{\alpha _j}^{\sigma (i)}\) is the j-th linguistic term in \(h_S^{\sigma (i)}\), \(i=1,2,\ldots ,m\), \(j=1,2,\ldots ,n\).
Example 3.16
Given the LTS \(S^{(3)} = \{s_{-3}, \ldots , s_0, \ldots , s_3 \}\), let \(h_S^1 = \{s_1, s_3 \}\) and \(h_S^2 = \{s_{-1}, s_0, s_2, s_3 \}\), \(\mathbf {w} = (0.4, 0.6)\). If we let \(\xi = 0.5\), then \(h_S^1\) is normalized to \(h_S^1 = \{s_1, s_2, s_2, s_3 \}\). Using the order \(\preceq _{D_n}\) defined in Eq. (2.37), we have \(h_S^2 \preceq _{D_n} h_S^1\), and \( EHFLOWA_{\mathbf {w}}^{\preceq } (h_S^1,h_S^2) = \{0.4 s_1 \oplus 0.6 s_{-1}, 0.4 s_2 \oplus 0.6 s_0, 0.4 s_2 \oplus s_2, 0,4 s_3 \oplus 0.6 s_3 \} = \{ s_{-0.2}, s_{0.8}, s_2, s_3 \} \). Furthermore, let \(h_S^3 = \{s_0, s_2, s_2, s_4 \}\), then \(h_S^2 \preceq _{D_n} h_S^3\), \(EHFLOWA_{\mathbf {w}}^{\preceq } (h_S^3,h_S^2) = \{s_{-0.6}, s_{0.8}, s_2, s_{3.4} \}\). Thus we have
but \(h_S^3 \preceq _{D_n} h_S^1\).
Example 3.16 demonstrates that the EHFLOWA operator may be not increasingly monotonic with respect to the admissible order \(\preceq \). According to Sect. 3.2.2, the EHFLOWA operator based on the partial order \(\preceq _W\) is not monotonic as well. Fortunately, if the admissible order is generated by the mapping \(K_\lambda \) defined in Eq. (2.41), we have the following theorem [20]:
Theorem 3.17
([24]) Let \(\preceq \) be an admissible order on \(\mathcal{E} ^n (S)\) generated by the n-tuple \((K_\lambda , f_2, \ldots , f_n)\), \(EHFLOWA_{\mathbf {w}}^{\preceq }\) be the EHFLOWA operator defined by Eq. (3.11). Then
Proof
According to the definition of \(K_\lambda \), it is obvious that \(h_S^{\sigma (1)} \preceq h_S^{\sigma (2)} \preceq \cdots \preceq h_S^{\sigma (m)}\) if \(K_\lambda (h_S^{\sigma (1)}) \le K_\lambda (h_S^{\sigma (2)}) \le \cdots \le K_\lambda (h_S^{\sigma (m)})\). Thus,
We can see that Eq. (3.12) is irrelevant to \((f_2, \ldots , f_n)\). Then the increasing monotonicity of the EHFLOWA operator can be clarified [20]:
Theorem 3.18
([24]) Let \(\preceq \) be an admissible order on \(\mathcal{E} ^n (S)\) generated by \((K_{\lambda _1},K_{\lambda _2}, \ldots , K_{\lambda _n})\). Then the EHFLOWA operator \(EHFLOWA_{\mathbf {w}}^{\preceq }\) is an aggregation function on \(\mathcal{E} ^n (S)\) with respect to \(\preceq \).
Proof
Suppose that the inputs of the EHFLOWA operator are \(H = \{ h_S^i | i=1,2,\ldots ,m \}\), where \(S^{(g)} = \{s_{-g}, \ldots , s_0, \ldots , s_g \}\).
(1) Since the top and the bottom of \((\mathcal{E} ^n (S), \preceq )\) are \(\{s_g\}\) and \(\{s_{-g}\}\), it is obvious that \(EHFLOWA_{\mathbf {w}}^{\preceq }( \{s_g\}, \{s_g\}, \ldots , \{s_g\}) = \{s_g\}\), \(EHFLOWA_{\mathbf {w}}^{\preceq } (\{s_{-g}\}, \{s_{-g}\}, \ldots , \{s_{-g}\} ) = \{s_{-g}\}\).
(2) To show the increasing monotonicity of \(EHFLOWA_{\mathbf {w}}^{\preceq }\), we increase any EHFLTS \(h_S^i\) to \(\dot{h}_S^i\). Because \(h_S^i \preceq \dot{h}_S^i\), \(K_{\lambda _j} (h_S^i) \le K_{\lambda _j} (\dot{h}_S^i) (\forall j = 1,2,\ldots ,n)\). According to Theorem 3.17,
If this inequality is strict, then the result follows. If it reduces to equality, it must be caused by \(K_{\lambda _1} (h_S^i) = K_{\lambda _1} (\dot{h}_S^i)\). Then there exists \(j \in \{ 2,3,\ldots ,n \}\) such that \(K_{\lambda _1} (h_S^i) < K_{\lambda _1} (\dot{h}_S^i)\). We discuss the problem in two situations:
(i) If the increase from \(h_S^i\) to \(\dot{h}_S^i\) does not change the ordinal relation of the m inputs, then
(ii) Otherwise, there exists an EHFLTS \(h_S^k\) such that
and \(K_{\lambda _j}(h_S^i) < K_{\lambda _j}(h_S^k)\). Moreover,
only depends on \(h_S^i\), \(\dot{h}_S^i\) and \(h_S^k\). Simply speaking, if there is only one \(h_S^k\), then the ordered positions of \(h_S^i\) and \(h_S^k\) in \(EHFLOWA_{\mathbf {w}}^{\preceq } (h_S^1,\ldots , h_S^i, \ldots , h_S^m)\) are replaced by \(h_S^k\) and \(\dot{h}_S^i\), respectively, in \(EHFLOWA_{\mathbf {w}}^{\preceq } (h_S^1,\ldots ,\dot{h}_S^i, \ldots , h_S^m)\). Due to the linearity of \(K_{\lambda _{j}}\), we have \(\delta < 0\).
According to (i) and (ii), the increasing monotonicity of the EHFLOWA operator is demonstrated. Then, the result follows immediately. \(\square \)
Based on Theorem 3.18, it is clear that the EHFLOWA operators \(EHFLOWA_{\mathbf {w}}^{\preceq _{Lex1}}\) and \(EHFLOWA_{\mathbf {w}}^{\preceq _{Lex2}}\) are aggregation functions.
In practice, the associated weights may take the form of linguistic terms as well, like the case of Definition 3.3. Apparently, it is easy to extend Definition 3.15 to this kind of weighting vectors. We omit the discussion in this section.
3.2 The Two-Phase GDM Approach
The mathematical description of the focused GDM problem is: t experts in the set \(E=\{e_1, e_2, \ldots , e_t\}\) are authorized to evaluate the set of alternatives \(A=\{a_1, a_2, \ldots , a_m \}\) with respect to the set of criteria \(C = \{ c_1, c_2, \ldots , c_n \}\). Our aim is to figure out the most desirable alternative from A according to the experts’ opinions. It is common that the experts prefer to express their evaluations by linguistic terms in qualitative setting. Moreover, the experts may have hesitancy on several terms. Thus, we assume that, generally, the experts express their opinions by the form of EHFLTSs based on the predefined LTS \(S^{(g)} = \{s_{-g}, \ldots , s_0, \ldots , s_g \}\). The performance of \(a_i\) with respect to \(c_j\), provided by \(e_k\), is an EHFLTS denoted by \(h_S(a_i, c_j, e_k)\), \(i=1,2,\ldots ,m\), \(j=1,2,\ldots ,n\), \(k = 1,2,\ldots ,t\). Then the linguistic information of \(e_k\) can be represented by a matrix \(MH_k = (h_S(a_i, c_j, e_k))_{m \times n}\), \(k = 1,2,\ldots ,t\).
The two-phase GDM approach is divided into two phases [20]: the aggregation phase and the exploitation phase. The former obtains the overall evaluation of each alternative by the EHFLOWA operator and ranks alternatives according to the selected parameters. If different rankings are derived by distinct parameters, then the latter synthesizes all possible rankings by the social choice theory to obtain a final ranking.
Phase 1. Aggregating through the EHFLOWA operator
In the aggregation phase, the following aggregation process is presented [20].
Step 1: Determine the risk preference and choose an admissible order \(\preceq \). The risk preference has to be determined at first so that original linguistic information can be normalized and an admissible order can be selected. It should be noted that obtaining the risk preference is very complicated because both the decision maker’s preferences and the practical problem should be considered simultaneously. For example, if the decision maker is optimistic, then \(\xi \) can be close to 1 and the order \(\preceq _{Lex2}\) can be used; if he/she is pessimistic, then \(\xi \) should be close to 0 and the order \(\preceq _{Lex1}\) is more logical; if he/she is more or less risk-neutral, then \(\xi \) should be close to 0.5 and the order \(\preceq _{D_n}\) is more suitable.
Step 2: Determine the weighting vector \(\mathbf {w}\). Suppose that the weights of experts and criteria are unknown or indifferent, or the decision maker prefers to conduct aggregations only according to the provided information, then the EHFLOWA operator is very suitable. In this case, some existing methods for determining OWA weights can be used, such as the quantifiers proposed in [25] and the method based on normal distribution [22].
Step 3: Aggregation. Firstly, the set of experts’ opinions are aggregated to form the group’s decision matrix by the EHFLOWA operator. The resultant matrix is denoted by \(MH = (h_S(a_i, c_j))_{m \times n}\), where \(h_S(a_i, c_j)\) is the averaging performance of \(a_i\) with respect to \(c_j\), \(i=1,2,\ldots ,m\), \(j = 1,2,\ldots ,n\). Secondly, the overall performance of \(a_i\), denoted by \(h_S(a_i)\), is obtained by aggregating the i-th row of MH using the EHFLOWA operator, \(i=1,2,\ldots ,m\).
Step 4: Ranking. The solution is the alternative which is superior to others with respect to the order \(\preceq \).
Comparing with the methods proposed by Rodríguez et al. [16], we use the EHFLOWA operator to fuse information provided by the experts based on the admissible order \(\preceq \). The order used here is a total order rather than partial order. Moreover, according to Theorem 3.18, the operator we used is an aggregation function which is of increasing monotonicity. While the advantage of Rodríguez et al. [16] is that it uses all possible terms directly in the aggregation of the criteria level.
Phase 2. Exploiting by the social choice theory
The result of the aggregation algorithm is highly restricted to \(\xi , \mathbf {w}\) and \(\preceq \). If this information is completely known in the application, we use the above process straightly and the final decision can be reached. However, some of them, such as the order which reflects the risk preferences of the decision makers, are very complex to determine. If we hesitate about some weighting strategies or some appropriate orders, we can run the aggregation process with each combination. Also, if all combinations return the same result, then we can choose the best alternative easily. Generally, if we obtain different rankings of alternatives, an exploiting algorithm based on the social choice theory is introduced as follows [20]: Suppose that a collection of rankings are derived by running the aggregation process.
Step 1: Construct the defeat relation \(DR= (dr_{ij})_{m \times m}\), in which \(dr_{ij}\) represents the percentage of defeat of \(a_i\) by \(a_j\), \(i,j = 1,2,\ldots ,m\).
Step 2: Run the MinLexMax Algorithm 1, with DR.
Step 3: Select the winning alternative according to the output of Step 2.
The MinLexMax algorithm outputs a sole winning alternative if a tiny condition is satisfied.
Theorem 3.19
([20]) The MinLexMax algorithm returns a sole winning alternative if and only if there does not exist two equal row vectors (after rearrangements in ascending order) in \(DR^{(2)}\).
Proof
According to the MinLexMax algorithm, the largest value of each column is removed from the matrix in each loop. If the algorithm stops with \(k \ge m\), then there exist at least two rows in \(DR^{(2)}\) such that:
(1) In each loop the same value is removed;
(2) The only entries of each of the two rows in the (\(m-1\))-th loop are equal.
Thus, after rearrangements in ascending order, the two row vectors are equal. \(\square \)
Using the social choice theory, the alternative defeated by the smallest percentages of other alternatives can be figured out. The algorithm enables each possible ranking serves as a voter. All information of the rankings is used to form the defeat relation. The algorithm is very easy to use in real application. We will apply this algorithm in a real GDM problem to clarify its reasonableness.
3.3 Application in Evaluations of Energy Technologies
Nowadays, a sustainable energy system is crucial for any countries. The implementation of new and innovative energy technologies is very important for governments [4]. The function of governments is to appraise and select energy technologies and then support the outstanding ones with funding and other incentives for private sector efforts. However, the process of evaluations is really complex because a series of uncertainties and implications might be encountered, such as: (1) the analysis has to face several uncertainties such as fossil fuel price, environmental regulations, market structure, technological, and demand and supply uncertainty [18]; (2) sustainability is inherently vague and complex, and the implications of sustainable developments as a policy objective is difficult to be defined or measured [14]; (3) the necessary information for the evaluations of technologies with respect to their sustainability may be unquantifiable due to its nature or even unavailable since the cost of its computation is too high [4].
The government formed a working organization with 4 experts from the relevant energy “actors”. To assess the technologies’ impact on the environmental, social, economical and technological dimension of sustainable development, a number of criteria have been selected and shown in Table 3.7. Furthermore, the organization looked systematically into the longer-term future, sought for the technologies which have not been used in any energy sector or have been applied at the initial stage, but are likely to uphold the sustainable development by the four dimensions in Table 3.7. At last, the technologies listed in Table 3.8 have been pre-selected as alternatives. We solve the problem by the approach proposed in the above subsection.
It is natural that most of the criteria are qualitative and cannot be quantified. When expressing the evaluation results, it is more convenient for the experts to use linguistic expressions instead of crisp numbers. Moreover, because of the high level of uncertainties mentioned above, the experts may feel not very confident to use a certain term to represent their opinions. Assume that the experts agree with evaluating alternatives with the LTS \(S^{(4)} = \{ s_{-4}, \ldots , s_0, \ldots , s_4 \}\) in Fig. 2.1. Whenever they can not express their opinions by a certain term, EHFLTSs or HFLTSs can be used. Tables 3.9, 3.10, 3.11 and 3.12 show the linguistic information provided by 4 experts.
Phase 1. Suppose that the decision maker hesitates about choosing the values of parameters of the aggregation process, we run it by the combination of several reasonable values. Specifically, \(\xi = 0, 0.5, 1\), \(\preceq = \preceq _{Lex1}, \preceq _{Lex2}, \preceq _{D_n} \), \(\mathbf {w}\) is implemented by the fuzzy quantifiers: “at least half”, “as many as possible” and “most” [6]. Then all the possible rankings of alternatives are summarized in Table 3.13. Note that the values of parameters used here is only for the purpose of illustration.
As can be seen in Table 3.13, there are 12 distinct rankings among 27 combinations of values of parameters. The only thing we are sure is that \(a_5\) is the one with the worst performance. The alternatives \(a_1\), \(a_3\) and \(a_6\) win the first place for 9, 15 and 3 times, respectively. Thus, it is necessary to derive a more admissible ranking by the proposed exploitation algorithm.
Phase 2. The defeat relation of 10 alternatives is derived as follows:
The maximum defeats of alternatives are
Then the minimum of maximum defeats is 0.33, which corresponds to the alternative \(a_3\). Therefore, it reaches the conclusion that \(a_3\) (Natural Gas Combined Cycle) is the wining technology.
3.4 Comparisons and Further Discussions
In the approach proposed by Zhang and Wu [27], the EHFLTSs are aggregated at the group level and the criteria level, respectively. Then, the alternatives are ranked by comparing the scores of overall performances. Besides, the approach in Sect. 3.2 aggregated EHFLTSs at the group level using the union operation rather than any other aggregation operators so that all possible linguistic terms provided by the experts can be considered directly in the second aggregation. The selected order is \(\preceq _W\).
Furthermore, as can be seen in Tables 3.9, 3.10, 3.11 and 3.12, all provided linguistic information is HFLTSs. Thus, the proposed approach can be compared with the existing methods proposed for this special setting. Liu and Rodríguez [11] developed a MCDM method based on the fuzzy envelopes of HFLTSs and the classical TOPSIS method. The fuzzy envelope of a HFLTS is represented by a trapezoidal fuzzy numbers. A kind of Minkowski distance is utilized to measure the distance of an alternative to the fuzzy positive and negative ideal solutions. To enable their method in GDM, we use the trapezoidal fuzzy OWA operator proposed in [10] to aggregate 4 matrices of the above example. In this sense, the approach proposed by Riera et al. [15] based on DNFs is also interesting for comparison. As the degree of one alternative satisfying each criterion is unknown, each evaluation is transformed into a DFN by fixing the degree to 1. For instance, the chain corresponding to the used LTS is \(L_n = \{0,1,\ldots ,8 \}\) and the subjective evaluation \(\{s_1,s_2,s_3\}\) is transformed into \(\{ 1 /5, 1/6, 1/7\}\). In addition, Tables 3.9, 3.10, 3.11 and 3.12 should be aggregated to form the final decision table because DFNs can only used for expressing individual opinions (the support of a DFN is consecutive natural numbers). Then, the aggregation among criteria can reach the overall performances of the alternative. In this comparison, the extension of the kernel aggregation function [12] is used and the parameter is fixed to 4.
Another similar approach based on membership functions can be found in [2], in which, a HFLTS is transformed into a trapezoidal fuzzy number and then reduces to an interval associated with the risk preferences of the decision maker. The order of intervals is defined by a likelihood-based method. The operators they used are based on the min and max operators.
Some features of the proposed approach and five existing approaches are summarized in Table 3.14. Notice that the approach of Liu and Rodríguez [11] orders the alternatives by distances to ideal solutions rather than comparing HFLTSs. Chen and Hong’s approach [2] takes use of a partial order of intervals. Riera et al. [15] compared DFNs by constructing a dominance relation which does not define a total order either. Due to the basic operation of EHFLTSs used in Zhang and Wu’s approach [27] and the approach in Sect. 3.2, the increasing monotonicity of their selected operators is not satisfied. However, the EHFLOWA operator used in the proposed approach is increasingly monotonic if the total order satisfies Theorem 3.18. Although the total order of DFNs has not been developed, the aggregation function on DFNs can be monotonic with respect to the predefined partial orders [9].
The rankings of six approaches are listed in Table 3.15. Notice that in the computational process, we use the OWA strategy associated with the quantifier “most” whenever an operator is needed (except for Chen and Hong’s approach [2] and Riera’s approach [15]). We set \(\xi = 0.5\) and \(\preceq = \preceq _{Lex1}\) in the proposed approach and consider the optimistic risk preference in Chen and Hong’s approach [2].
We discuss the comparable approaches in the following aspects:
(1) Concerning the orders. Total orders are essential to rank a set of values. If partial orders are considered, two values, which are not equal, may be unable to distinguish. As shown in Table 3.15, \(a_{10} = a_4\) is derived by \(E(a_{10}) = E(a_4)\). In fact, \(a_{10}\) and \(a_4\) do not coincide but are equivalent with respect to the partial order defined by expected terms. If the degree of hesitancy is further considered, we have \(a_{10} \preceq _W a_4\) according to the partial order \(\preceq _W\). However, this would not happen if total orders, such as total orders introduced in this book, are taken into account.
(2) Concerning risk preferences of the experts. Generally, modeling risk preferences are indispensable and inevitable for decision-making. In Zhang and Wu’s approach [27] and the approach of Sect. 3.2, this is reflected by the weighting strategies of aggregation operators. In Chen and Hong’s approach [2], the trapezoidal membership functions are converted into numerical intervals with the help of risk preferences. In Riera’s approach [15], the risk preference is represented by the parameter of the aggregation function. While in the approach of this section, risk preferences are reflected by the parameter \(\xi \) and the total orders. It is common that risk preferences can affect the final decision-making but are very complex to be determined accurately. Thus this approach is a solution to consider all possible situations to reach a sounder decision.
(3) Concerning the EHFLOWA operators. The EHFLOWA operator was used in Zhang and Wu’s approach [27], the approach of Sect. 3.2 and the approach of this section. However, there are some differences: (i) The operator in the two existing approaches is based on partial orders rather than total orders in the approach of this section. (ii) They utilize the basic operation defined in Definition 2.13. But the resultant operator is not increasingly monotonic. We use a special form of the operator as shown in Eq. (2.22). According to the main idea of hesitant fuzzy sets, we may ignore some possible values. But the monotonicity of the EHFLOWA operator can be satisfied by appropriate definition of admissible orders.
(4) EHFLTSs (HFLTSs) vs. DFNs. All the six approaches can solve the problems whose decision information takes the form of HFLTSs. Whereas Liu and Rodríguez’s approach [11], Chen and Hong’s approach [2] and Riera’s approach [15] can not deal with linguistic information with the form of EHFLTSs. But this fact is not an evidence to show that EHFLTSs are more advanced than HFLTSs and DFNs. We can only draw the following conclusions: (i) HFLTSs and DFNs are more rational to express individual evaluations than EHFLTSs; (ii) EHFLTSs and DFNs are more convenient for computation. EHFLTSs enable us to conduct computation by each possible term involved in them. If the computation is not conducted according to each involved term, then HFLTSs would be reduced to the uncertain linguistic terms. (iii) DFNs are more general and flexible linguistic representational model because they can include more information than EHFLTSs. However, it leads to additional work for the experts to provide this kind of information. If the membership degrees are available, then DFNs are the best choice; otherwise, EHFLTSs (or HFLTSs) can be used instead.
4 Conclusions
This chapter has studied GDM problems in the setting of EHFLTSs, based on the framework of decision matrices. Different from traditional GDM approaches, we have started with the analysis of inner structures of groups, and introduced the processes of MGDM. The processes have been implemented in the setting of EHFLTSs. The involved aggregation operators possess very good properties. Besides, a new OWA operator has been introduced based on the admissible order on the set of EHFLTSs. The operator has been applied in a GDM approach. We have discussed how subjective preferences of decision makers affect final decisions. The presented approach based on social choice theory could provide admissible decisions if different rankings are derived by using different values of parameters.
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Wang, H., Xu, Z. (2019). Group Decision-Making Based on EHFLTSs Under the Framework of Decision Matrix. In: Theory and Approaches of Group Decision Making with Uncertain Linguistic Expressions. Uncertainty and Operations Research. Springer, Singapore. https://doi.org/10.1007/978-981-13-3735-2_3
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