A hesitant group emergency decision making method based on prospect theory
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Abstract
Group emergency decisionmaking (GEDM) problems have drawn great attention in past few years due to its advantages of dealing with the emergency events (EEs) effectively. Due to the fact that EEs are usually featured by lack of information and time pressure, decision makers (DMs) are often bound rational and their psychological behaviors are very crucial to the GEDM process. However, DM’s psychological behaviors are neglected in current GEDM approaches. The assessments representing the individual wisdom provided by each expert are usually aggregated in the GEDM process. Nevertheless, the aggregation process always implies summarization of data that can result in loss of information. To overcome these limitations pointed out previously, this paper proposes a new GEDM method that considers the DM’s psychological behaviors in the decision process using prospect theory and replaces the aggregation process by a fusion method with hesitant fuzzy set, which keeps the experts’ information as much as possible. A case study is provided to illustrate the validity and feasibility of the proposed method.
Keywords
Group emergency decision making Hesitant fuzzy sets Prospect theoryIntroduction
Emergency event (EE) is defined as “events which suddenly take place, causing or having the possibility to cause intense death and injury, property loss, ecological damage and social hazards” [18], such as earthquakes, air crash, hurricanes, terrorist attacks, etc. When an EE occurs, it must be dealt with some measures to mitigate the losses of properties and lives, the process of selecting the measures is defined emergency decision making (EDM). EDM has received increasing attention and became a very active and important research field in recent years [12, 15, 20, 35, 36, 37] because it plays a crucial role in mitigating the losses of properties and lives caused by EE.
 1.
Losing information in aggregation process. Current GEDM studies [41, 42, 43], use aggregation process that may imply loss of useful information for the decision process from the very beginning. Therefore, an important challenge for GEDM is to keep as much information as possible about the group, avoiding such a lost.
 2.
DM’s psychological behaviors in selection process. Different studies [7, 8] have shown that the DM is bounded rational under risk and uncertainty and his/her psychological behavior plays an important role in GEDM processes. However, such an important issue has been neglected in current GEDM methods.
 1.
It considers different experts’ opinions as the group hesitancy and fuses them into HFSs.
 2.
At the same time, it takes into account the DM’s psychological behaviors using prospect theory (PT) in the selection process, because of its advantages of capturing human beings psychological behaviors under risk and uncertainty [16].
Preliminaries
This section provides a brief review of different concepts about HFS and PT that will be utilized in the proposed method and to make it understood easily.
Hesitant fuzzy sets
HFS was introduced by Torra [30] as an extension of fuzzy sets to model the hesitancy in quantitative contexts reviewed in depth [23, 25]. It is defined as below:
Definition 1
Summary of the related works on hesitant fuzzy set
Authors  Contributions  Year 

Torra [30]  Hesitant fuzzy set (HFS)  2010 
Bedregal et al. [3]  Typical hesitant fuzzy set (THFS)  2014 
Xia and Xu [40]  Hesitant fuzzy element (HFE)  2011 
Hesitant fuzzy weighted average operator  2011  
Hesitant fuzzy power average operator  2011  
Yu [45]  Hesitant fuzzy Choquet integral operator  2011 
Zhou [48]  Distance measures for hesitant fuzzy set  2012 
Wei [38]  Entropy measures for hesitant fuzzy set  2016 
Rodriguez et al. [25]  Hesitant fuzzy linguistic term set (HFLTS)  2014 
Cevik Onar [9]  Multicriteria decision making AHP method  2014 
Wei [39]  Multicriteria decision making VIKOR method  2014 
Xue [44]  Group decision making  2017 
Yu [46]  Personal evaluation  2013 
Aliahmadipour [2]  Clustering  2016 
Torra introduced in [30] the concept of envelop of a HFE and proved that is a intuitionistic fuzzy value (IFV) according to the following definition:
Definition 2
[30] Let h be a HFE, the IFV \(A_\mathrm{env}(h)\) is the envelop of h, in which \(A_\mathrm{env}(h) \)can be represented as \(\left( {h^{},1h^{+}} \right) \) being \(h^{}=\min \left\{ {\sigma \sigma \in h} \right\} \) and \(h^{+}=\max \left\{ {\sigma \sigma \in h} \right\} \).
Different operations and properties has been defined for HFSs [30] such operations together the managing of intuitionistic fuzzy sets and intervals [14, 30] allow us to interpret HFEs like an interval.
Many researchers have paid attention on HFSs because it is a useful approach to model experts’ hesitation. Therefore, different proposals have been introduced in the literature. Bedregal et al. [3] presented a special case of HFS named Typical Hesitant Fuzzy Set, that introduces some restrictions, because a HFS should be a finite and nonempty set. Many aggregation operators for HFSs have been defined such as, hesitant fuzzy weighted average [40], hesitant fuzzy power average [40], hesitant fuzzy choquet integral [45] and so on [23, 25]. Distance measures are widely used in different fields such as, machine learning and decision making, for this reason some of them have been extended to deal with HFS [48]. Some entropy measures have been also defined for HFS [38]. And there are many applications based on HFS such as, multicriteria decision making [9, 39], group decision making [44], evaluation [46], and clustering approaches [2].
Recently, Rodriguez et al. [27] proposed the concept of Hesitant Fuzzy Linguistic Term Set (HFLTS), which not only keeps the basis on the fuzzy linguistic approach [47], but also extends the idea of HFS to linguistic contexts [24]. It has drawn great attention since it has been applied to solve different decision problems [4, 19, 26].
For sake of clarity, we make a summary of the related works on hesitant fuzzy set, see the following Table 1.
Prospect theory
A hesitant group emergency decision making dealing with DM’s behaviors
This section introduces a novel hesitant GEDM method based on PT that aims at keeping experts’ information as much as possible during the decision process and taking into account the DM’s psychological behaviors during the selection process.
 1.
Definition framework;
 2.
Information fusion based on HFS;
 3.
Alternative selection based on PT.
Definition framework

\(A=\left\{ {a_1 ,\ldots ,a_i ,\ldots ,a_I } \right\} \): set of alternatives, where \(a_i \) denotes the ith alternative, \(i=1,2,\ldots ,I\).

\(C=\left\{ {c_1 ,\ldots ,c_j ,\ldots ,c_J } \right\} \): set of criteria, where\(c_j \)denotes the jth criterion, \(j=1,2,\ldots ,J\).

\(S=\left\{ {s_1 ,\ldots ,s_m ,\ldots ,s_M } \right\} \): set of emergency situations, where \(s_m \) denotes the mth emergency situation, \(m=1,2,\ldots ,M\).

\(W=\left( {w_{c_1 } ,\ldots ,w_{c_j } ,\ldots ,w_{c_J } } \right) \): vector of criteria weights, where \(w_{c_j } \)denotes the weight of the jth criterion, \(j=1,2,\ldots ,J\).

\(E=\left\{ {e_1 ,\ldots ,e_h ,\ldots ,e_H } \right\} \): set of experts, where\(e_h \)denotes the hth expert, \(h=1,2,\ldots ,H\).

\(C^{h}=\left\{ {c_{j }^h (a_i )} \right\} \): set of opinions provided by expert \(e_h \), where \(c_{j }^h (a_i )\in R\) denotes the preference over theith alternative regarding to thejth criterion, \(i=1,2,\ldots ,I\); \(h=1,2,\ldots ,H\); \(j=1,2,\ldots ,J\).

\(\bar{{C}}^{h}=\left\{ {\bar{{c}}_{j }^h (a_i )} \right\} \): denotes the normalization of \(C^{h}\), where \(\bar{{c}}_{j }^h (a_i )\in [0,1]i=1,2,\ldots ,I; h=1,2,\ldots ,H; j=1,2,\ldots ,J\).

\(h_M (a_i )=\left\{ {\bar{{c}}_1 (a_i ),\ldots ,\bar{{c}}_J (a_i )} \right\} \): denotes the HFS of experts’ preference, where \(\bar{{c}}_j (a_i)\) is the hesitant fuzzy element (HFE) and \(\bar{{c}}_j (a_i)=\left\{ \bar{{c}}_{j }^1 (a_i ),\ldots ,\bar{{c}}_{j }^h (a_i ),\ldots ,\bar{{c}}_{j }^H (a_i ) \right\} \), \(i=1,2,\ldots ,I\); \(h=1,2,\ldots ,H\); \(j=1,2,\ldots ,J\).

\(E_{ij} =\left[ {E_{ij}^L ,E_{ij}^U } \right] \): be an interval value, where \(E_{ij} \) denotes the effective control scope [37] over the ith alternative with respect to the jth criterion.

\(R_j =\left[ {R_j^L ,R_j^U } \right] \): be an interval value, where \(R_j^L , R_j^U \) are preferences, and \(R_j \)denotes the RP provided by the DM with respect to the jth criterion.

\(\bar{{R}}_j =\left[ {\bar{{R}}_j^L ,\bar{{R}}_j^U } \right] \): denotes the normalization of \(R_j \), where \(\bar{{R}}_j \in [0,1]j=1,2,\ldots ,J\).
Information fusion based on HFS
As it was pointed out in the introduction, the aggregation always implies a summarization of original experts’ opinions that can imply loss of information from different points of view such as distribution, diversity of data, etc. This loss of information can either bias or lead to wrong decisions regardless the aggregation operator. To overcome such a limitation, the experts’ preferences in the GEDM problem are considered as the group hesitation about the alternatives and they will be fused by utilizing a HFS to keep as much information as possible.
 Step 1:
The experts involved in the GEDM problem provide the related information \(c_{j }^h (a_i )\) about the emergency alternative with regarding to different criteria through analyzing the emergency alternatives
 Step 2:Based on information \(c_{j }^h (a_i )\) provided by experts, the preference of experts \(\bar{{c}}_{j }^h (a_i )\) of effective control scope of the ith alternative with respect to the jth criterion can be calculated by Eq. (3):$$\begin{aligned} \bar{{c}}_{j }^h (a_i )=\frac{c_{j }^h (a_i )}{\max \left\{ {\mathop {\max }\limits _h c_{j }^h (a_i )} \right\} },\quad j=1,2,\ldots J \end{aligned}$$(3)
 Step 3:
From \(\bar{{c}}_{j }^h (a_i )\) calculated by Eq. (3), the HFEs \(\bar{{c}}_j (a_i )\) for the jth criterion with respect to the ith alternative and the HFS \(h_M (a_i )\) can be formed and managed according to their envelops as interval values.
 Step 4:Based on step 3, the lower bound \(E_{ij}^L \) and upper bound \(E_{ij}^U \) of the effective control scope \(E_{ij}\) can be calculated by Eqs. (4), (5) [31].$$\begin{aligned}&E_{ij}^L =\mathop {\min }\limits _h \left\{ {\bar{{c}}_{j }^1 (a_i ),\ldots ,\bar{{c}}_{j }^h (a_i ),\ldots ,\bar{{c}}_{j }^H (a_i )} \right\} \end{aligned}$$(4)The interval value \(E_{ij}\) is the result of fusion information that can avoid the loss of information and keep the experts’ opinions as much as possible. In order to facilitate the computations, the preferences \(R_j \) need to be transformed into \(\bar{{R}}_j \) by utilizing the Eqs. (6), (7):$$\begin{aligned} \nonumber \\&E_{ij}^U =\mathop {\max }\limits _h \left\{ {\bar{{c}}_{j }^1 (a_i ),\ldots ,\bar{{c}}_{j }^h (a_i ),\ldots ,\bar{{c}}_{j }^H (a_i )} \right\} \end{aligned}$$(5)$$\begin{aligned} \bar{{R}}_j^L =\frac{R_j^L }{\max \left\{ {\mathop {\max }\limits _h c_{j }^h (a_i )} \right\} } \end{aligned}$$(6)$$\begin{aligned} \bar{{R}}_j^U =\frac{R_j^U }{\max \left\{ {\mathop {\max }\limits _h c_{j }^h (a_i )} \right\} } \end{aligned}$$(7)
Alternative selection based on PT
Due to the fact that DM’s psychological behaviors play an important role in GEDM process, this proposal uses PT to address such an important issue, because of its advantages to capture the psychological behaviors.
Calculation of gains and losses
According to the RP \(\bar{{R}}_j \) and the effective control scope \(E_{ij} \) of emergency alternatives, gains and losses can be obtained.
Due to the fact that, we are dealing with interval values, before obtaining the gains and losses, the relationship between \(\bar{{R}}_j \) and \(E_{ij} \) should be determined. There are six possible cases of positional relationship between \(\bar{{R}}_j \) and \(E_{ij} \) as shown in Table 2.
To obtain the gains and losses with respect to each alternative, the following definition is provided.
Definition 3
Possible cases of positional relationship between \(\bar{{R}}_j \) and \(E_{ij} \)
Cases  Positional relationship between \(\bar{{R}}_j \) and \(E_{ij} \)  

Case 1  \(E_{ij}^U <\bar{{R}}_j^L \)  
Case 2  \(\bar{{R}}_j^U <E_{ij}^L \)  
Case 3  \(E_{ij}^L<\bar{{R}}_j^L \le E_{ij}^U <\bar{{R}}_j^U \)  
Case 4  \(\bar{{R}}_j^L<E_{ij}^L \le \bar{{R}}_j^U <E_{ij}^U \)  
Case 5  \(E_{ij}^L<\bar{{R}}_j^L<\bar{{R}}_j^U <E_{ij}^U \)  
Case 6  \(\bar{{R}}_j^L \le E_{ij}^L <E_{ij}^U \le \bar{{R}}_j^U \)  
Gains and losses for all possible cases (cost criteria)
Cases  Gain \(G_{ij} \)  Loss \(L_{ij} \)  

Case 1  \(E_{ij}^U <\bar{{R}}_j^L \)  \(\bar{{R}}_j^L 0.5(E_{ij}^L +E_{ij}^U )\)  0 
Case 2  \(\bar{{R}}_j^U <E_{ij}^L \)  0  \(\bar{{R}}_j^U 0.5(E_{ij}^L +E_{ij}^U)\) 
Case 3  \(E_{ij}^L<\bar{{R}}_j^L \le E_{ij}^U <\bar{{R}}_j^U \)  \(0.5(\bar{{R}}_j^L E_{ij}^L )\)  0 
Case 4  \(\bar{{R}}_j^L<E_{ij}^L \le \bar{{R}}_j^U <\bar{{E}}_{ij}^U\)  0  \(0.5(\bar{{R}}_j^U E_{ij}^U )\) 
Case 5  \(E_{ij}^L<\bar{{R}}_j^L<\bar{{R}}_j^U <E_{ij}^U \)  \(0.5(\bar{{R}}_j^L E_{ij}^L )\)  \(0.5(\bar{{R}}_j^U E_{ij}^U )\) 
Case 6  \(\bar{{R}}_j^L \le E_{ij}^L <E_{ij}^U \le \bar{{R}}_j^U \)  0  0 
From Table 2, the calculation of gains and losses is discussed. In general, the criteria can be classified into two types: benefit and cost [21]. A benefit criterion means the higher the better while a cost criterion the higher the worse. Note that for cost criteria, if \(E_{ij}^U <\bar{{R}}_j^L \), the expert feels gains, and if \(E_{ij}^L >\bar{{R}}_j^U \), the expert feels losses. The following discussion is for cost criterion only.
Gains and losses for all possible cases (benefit criteria)
Cases  Gain \(G_{ij} \)  Loss \(L_{ij} \)  

Case 1  \(E_{ij}^U <\bar{{R}}_j^L \)  0  \(0.5(E_{ij}^L +E_{ij}^U )\bar{{R}}_j^L \) 
Case 2  \(\bar{{R}}_j^U <E_{ij}^L \)  \(0.5(E_{ij}^L +E_{ij}^U )\bar{{R}}_j^U \)  0 
Case 3  \(E_{ij}^L<\bar{{R}}_j^L \le E_{ij}^U <\bar{{R}}_j^U \)  0  \(0.5(E_{ij}^L \bar{{R}}_j^L )\) 
Case 4  \(\bar{{R}}_j^L<E_{ij}^L \le \bar{{R}}_j^U <E_{ij}^U \)  \(0.5(E_{ij}^U \bar{{R}}_j^U )\)  0 
Case 5  \(E_{ij}^L<\bar{{R}}_j^L<\bar{{R}}_j^U <E_{ij}^U \)  \(0.5(E_{ij}^U \bar{{R}}_j^U )\)  \(0.5(E_{ij}^L \bar{{R}}_j^L )\) 
Case 6  \(\bar{{R}}_j^L \le E_{ij}^L <E_{ij}^U \le \bar{{R}}_j^U \)  0  0 
Furthermore, based on Tables 3 and 4, the gain and loss matrix \(\hbox {GM}\) and \(\hbox {LM}\) can be constructed, which are used to calculate prospect values using value function.
Calculation of overall prospect values
Case study and comparison
Case study
To illustrate the validity and feasibility of the proposed method, this section presents an adapted real case about a barrier lake emergency caused by a huge earthquake that occurred in southwestern China.

\(c_1 \): The cost of alternatives (10,000 RMB which is the acronym of “renminbi”, the official currency of the People’s Republic of China).

\(c_2 \): The number of casualties.
 \(c_3 \): Property loss (10,000 RMB). The emergency alternatives are described as follows:

\(a_1 \): Evacuate people from the most dangerous upstream and downstream areas of the barrier lake to safe areas, and inform people in potentially dangerous areas to prepare for evacuation. At the same time, combine repeated small batch quantities of artificial blasting and excavation of drain grooves to meet the requirements of the discharged barrier lake floods;

\(a_2 \): Based on \(a_1 \), increase the joint scheduling of the reservoir and hydropower station in the upstream and downstream areas to reduce the pressure of the barrier lake;

\(a_3 \): Based on \(a_2 \), mobilize large, heavy machinery and implement largescale blasting to reduce the water level of the barrier lake as much as possible to lower the risk of dam break;

\(a_4 \): Based on \(a_3 \), increase the joint scheduling of the reservoir and hydropower station in the upstream and downstream areas. Meanwhile, mobilize large, heavy machinery and implement largescale blasting to reduce the water level of the barrier lake as much as possible to lower the risk of dam break. Analyzing by professional experts, the barrier lake might be evolve into four possible emergency situations in 72 h, the emergency situations are as follows:

\(s_1\): The dam body of the barrier lake will not break;

\(s_2\): 1 / 3 of the dam body of the barrier lake will break;

\(s_3\): 1 / 2 of the dam body of the barrier lake will break;

\(s_4\): The entire dam body of the barrier lake will break. Assume that three experts are invited to participate in the decision process to help DM makes a final decision. First, they are asked to define the effective control scope of the four emergency alternatives mentioned above. Through analyzing these emergency alternatives, the preferences \(c_{j }^h (a_i )\) and \(\bar{{c}}_{j }^h (a_i )\) of the effective control scopes for alternatives are given (see Table 5), where \(\bar{{c}}_{j }^h (a_i )\) is calculated by Eq. (3).

\(c_{j}^h (a_i )\) and \(\bar{{c}}_{j }^h (a_i )\) of effective control scope for alternatives
Alternatives  Experts  Criteria (weights)  

\(c_1\) (0.3)  \(c_2\) (0.4)  \(c_3\) (0.3)  
\(c_1^h (a_i )\)  \(\bar{{c}}_1^h (a_i )\)  \(c_2^h (a_i )\)  \(\bar{{c}}_2^h (a_i )\)  \(c_3^h (a_i )\)  \(\bar{{c}}_3^h (a_i )\)  
\(e_1\)  250  0.42  5000  0.59  3500  0.64  
\(a_1\)  \(e_2\)  280  0.47  5000  0.59  3000  0.55 
\(e_3\)  300  0.50  4000  0.47  4000  0.73  
\(e_1\)  300  0.50  6500  0.76  4000  0.73  
\(a_2\)  \(e_2 \)  300  0.50  5500  0.65  4000  0.73 
\(e_3\)  350  0.58  5000  0.59  4500  0.82  
\(e_1\)  400  0.67  7000  0.82  4500  0.82  
\(a_3 \)  \(e_2\)  350  0.58  7500  0.88  4500  0.82 
\(e_3\)  400  0.67  6500  0.76  5000  0.91  
\(e_1\)  600  1.00  8000  0.94  5000  0.91  
\(a_4\)  \(e_2\)  500  0.83  8500  1.00  5300  0.96 
\(e_3\)  550  0.92  7500  0.88  5500  1.00 
The HFEs \(\bar{{c}}_j (a_i)\) and the effective control scope \(E_{ij} \) for the alternatives
Alternatives  Criteria  

\(c_1 \)  \(c_2 \)  \(c_3 \)  
\(\bar{{c}}_1 (a_i )\)  \(E_{i1} \)  \(\bar{{c}}_2 (a_i )\)  \(E_{i2} \)  \(\bar{{c}}_3 (a_i )\)  \(E_{i3} \)  
\(a_1 \)  \(\left\langle {\bar{{c}}_1 (a_1 ),\left\{ {0.42, 0.47, 0.50} \right\} } \right\rangle \)  [0.42, 0.50]  \(\left\langle {\bar{{c}}_2 (a_1 ),\left\{ {0.59, 0.59, 0.47} \right\} } \right\rangle \)  [0.47, 0.59]  \(\left\langle {\bar{{c}}_3 (a_1 ),\left\{ {0.64, 0.55, 0.73} \right\} } \right\rangle \)  [0.55, 0.73] 
\(a_2 \)  \(\left\langle {\bar{{c}}_1 (a_2 ),\left\{ {0.50, 0.50, 0.58} \right\} } \right\rangle \)  [0.40, 0.58]  \(\left\langle {\bar{{c}}_2 (a_2 ),\left\{ {0.76, 0.65, 0.59} \right\} } \right\rangle \)  [0.59, 0.76]  \(\left\langle {\bar{{c}}_3 (a_2 ),\left\{ {0.73, 0.73, 0.82} \right\} } \right\rangle \)  [0.73, 0.82] 
\(a_3 \)  \(\left\langle {\bar{{c}}_1 (a_3 ),\left\{ {0.67, 0.58, 0.67} \right\} } \right\rangle \)  [0.58, 0.67]  \(\left\langle {\bar{{c}}_2 (a_3 ),\left\{ {0.82, 0.88, 0.76} \right\} } \right\rangle \)  [0.76, 0.88]  \(\left\langle {\bar{{c}}_3 (a_3 ),\left\{ {0.82, 0.82, 0.91} \right\} } \right\rangle \)  [0.82, 0.91] 
\(a_4 \)  \(\left\langle {\bar{{c}}_1 (a_4 ),\left\{ {1.00, 0.83, 0.92} \right\} } \right\rangle \)  [0.83, 1.00]  \(\left\langle {\bar{{c}}_2 (a_4 ),\left\{ {0.94, 1.00, 0.88} \right\} } \right\rangle \)  [0.88, 1.00]  \(\left\langle {\bar{{c}}_4 (a_4 ),\left\{ {0.91, 0.96, 1.00} \right\} } \right\rangle \)  [0.91, 1.00] 
Based on the data \(\bar{{c}}_{j }^h (a_i )\) in Table 5, the HFEs \(\bar{{c}}_j (a_i )\) can be obtained and the effective control scope for the alternatives with respect to different criteria can be calculated by Eqs. (4), (5). Table 6 shows the results.
According to the four possible emergency situations of the barrier lake, the DM provided the RP according to his/her professional knowledge and experience by using interval values. The \(R_j \) and \(\bar{{R}}_j \) are shown in Table 7, where \(\bar{{R}}_j \) are obtained by Eqs. (6), (7).
The RP \(R_j \) and \(\bar{{R}}_j \)
RP  \(c_1 \)  \(c_2 \)  \(c_3 \) 

\(R_j \)  [300, 350]  [6000, 7000]  [2000, 3000] 
\(\bar{{R}}_j \)  [0.5, 0.58]  [0.71, 0.82]  [0.36, 0.55] 
Overall prospect values and the ranking of alternatives
Alternatives  \(a_1\)  \(a_2 \)  \(a_3 \)  \(a_4\) 

\(\mathrm{OPV}_i \)  \(\)0.1278  0.0150  0. 0912  \(\)0.04823 
Ranking  4  2  1  3 
The aggregated information of the effective control scope \(\tilde{E}_{ij} \)
Alternatives  Experts (weights)  Criteria (weights)  

\(c_1\) (0.3)  \(c_2\) (0.4)  \(c_3\) (0.3)  
\(\bar{{c}}_1^h (a_i )\)  \(\tilde{E}_{i1} \)  \(\bar{{c}}_2^h (a_i )\)  \(\tilde{E}_{i2} \)  \(\bar{{c}}_3^h (a_i )\)  \(\tilde{E}_{i3} \)  
\(e_1 \)(1/3)  0.42  0.59  0.64  
\(a_1 \)  \(e_2 \)(1/3)  0.47  0.4607  0.59  0.5494  0.55  0.6364 
\(e_3 \)(1/3)  0.50  0.47  0.73  
\(e_1 \)(1/3)  0.50  0.76  0.73  
\(a_2 \)  \(e_2 \)(1/3)  0.50  0.5275  0.65  0.6676  0.73  0.7573 
\(e_3 \)(1/3)  0.58  0.59  0.82  
\(e_1 \)(1/3)  0.67  0.82  0.82  
\(a_3 \)  \(e_2 \)(1/3)  0.58  0.6392  0.88  0.8235  0.82  0.8482 
\(e_3 \)(1/3)  0.67  0.76  0.91  
\(e_1 \)(1/3)  1.00  0.94  0.91  
\(a_4 \)  \(e_2 \)(1/3)  0.83  0.9175  1.00  0.9412  0.96  0.9571 
\(e_3 \)(1/3)  0.92  0.88  1.00 
Comparison
To illustrate the validity and feasibility of the proposed method, a comparison between the new information fusion process using HFSs and the aggregation process using weighted average method is performed.
The weighted average method is widely used to aggregate the experts’ opinion in the aggregation process of the group decision making problem. In the weighted average method, the weight is assigned to each expert. In this paper, we assume that three experts’ opinions are equally important, i.e., (1 / 3, 1 / 3, 1 / 3). Let \(\tilde{E}_{ij} \) be the aggregated information of the effective control scope. In order to make a validity comparison between the results of the two different methods, the \(\bar{{c}}_1^h (a_i )\) will be utilized to generate the effective control scope \(\tilde{E}_{ij} \), where \(\tilde{E}_{ij} =\frac{1}{3}\sum _{h=1}^3 {\bar{{c}}_{j }^h (a_i )} \). The results are shown in Table 9.
Overall prospect values and the ranking of alternatives
Alternatives  Our proposal  Weighted average method  

\(\mathrm{OPV}_i\)  Ranking  \(\widetilde{\mathrm{OPV}_i }\)  Ranking  
\(a_1 \)  \(\)0.1278  4  \(\)0.1096  4 
\(a_2 \)  0.0150  2  0.0319  2 
\(a_3 \)  0.0912  1  0.0573  1 
\(a_4 \)  \(\)0.0482  3  \(\)0.0480  3 
As it can be seen that the ranking of alternatives obtained by different methods is the same, it verifies the validity and feasibility of our proposal.
The value obtained for the best alternative based on our proposal is greater than the value obtained based on the weighted average method because the proposal considers all the information provided by experts avoiding the loss of information.
Conclusion
Current GEDM approaches aggregate experts’ individual assessments that may incur in loss of information that bias the decision process. Therefore, to take all the experts’ opinions into account and also, DM’s psychological behaviors during the GEDM process, this paper has introduced a new GEDM that considers DM’s psychological behaviors using PT and the aggregation process is replaced by a fusion process using HFSs. Eventually, a case study and a comparison with the weighted average method about a barrier lake EE that happened in real world is provided to illustrate the validity and feasibility of the proposed method.
Notes
Acknowledgements
This paper was partially supported by the Spanish National research project TIN201566524P, Spanish Ministry of Economy and Finance Postdoctoral Fellow (IJCI201523715), ERDF, the National Natural Science Foundation of China (Project No. 71371053) and the Young Doctoral Dissertation Project of Social Science Planning Project of Fujian Province (Project No. FJ2016C202).
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