*3.1. Definition Framework*

Due to the information about EE usually being inadequate or incomplete, especially in the early stage in a real-world situation, and related emergency situations become more and more complicated with the dynamic evolution of EE across time, it is hard for DM to describe the EE using just one type of information; thus, for convenience, different types of information will be used to describe the situation of EE and emergency response alternatives [15,16]. Thus, in our proposal, both interval and numerical values are employed, in which the interval values are used to estimate the damages or losses caused by EE and numerical values are used to describe the cost of alternatives.

The following notations that will be used in our proposal are defined below:


the *m*-th criterion. The weighting vector is usually provided by the DM satisfying ∑ *<sup>m</sup>*=1 *wXm* = 1,

$$w\_{\mathcal{X}\_m} \in [0,1]\_{\prime} \\ \underset{\cdot}{m} = 1,2,\dots,M.$$


#### *3.2. Calculation of Gains and Losses*

When an EE occurs, it may have different possible emergency situations. The DM needs to collect related information about possible situations and losses to make a decision. According to the collected

information, DM forms the corresponding RP, *Rθ<sup>m</sup>*, of the *m*-th criterion *Xm* in the *θ*-th situation *S*2*θ*. Gains and losses can be determined on the basis of the RPs *Rθm* and the pre-defined effective control scope *Eδm* of different alternatives.

Because both the RPs and the pre-defined effective control scopes are expressed in the form of interval values, the relationship between the interval values *Rθm* and *Eδm* should be analyzed before determining the gains and losses. To simplify, the relationship between *Rθm* and *Eδm* and the computation formulas for obtaining gains and losses taken from Wang et al. [17] will be utilized in our proposal.

The positional relationship between *Rθm* and *Eδm* is summarized in Table 2. Tables 3 and 4 provide the computation formulas of gains and losses for all possible relationships between *Rθm* and *Eδ<sup>m</sup>*, in which Tables 3 and 4 are for cost criteria and benefit criteria, respectively.

Based on the computation formulas of gain and loss provided in Tables 3 and 4, the gain matrix *GMθ* and the loss matrix *LMθ* can then be formed. Afterwards, the overall prospect values can be calculated by the value function on the basis of the gain and loss matrix *GMθ*, *LMθ*.


**Table 2.** Positional relationship between interval values *Rθm* and *Eδm* [17].

**Table 3.** Computation formulas of gain and loss for cost criteria [17].



**Table 4.** Computation formulas of gain and loss for benefit criteria [17].

#### *3.3. Computation of Overall Prospect Values*

Assume that the gain matrix of the *θ*-th situation is denoted by *GMθ* = (*<sup>G</sup>θδm*)*<sup>δ</sup>*<sup>×</sup>*m*, and similarly, the loss matrix and value matrix of the *θ*-th situation are denoted by *LMθ* = (*<sup>L</sup>θδm*)*<sup>δ</sup>*×*m* and *VMθ* = (*<sup>v</sup>θδm*)*<sup>δ</sup>*<sup>×</sup>*m*, respectively.

$$w\_{\theta\delta m} = G\_{\theta\delta m}{}^{\mu} + [-\lambda(-L\_{\theta\delta m})^{\beta}], \ \delta = 1, 2, \dots, k\_1; \ \theta = 1, 2, \dots, k\_2; \ m = 1, 2, \dots, M \tag{2}$$

where *vθδm* means the value with respect to the alternative *S*1*δ*, concerning criterion *Xm*, in the situation *S*2*θ*. According to [30], the parameters *α*, *β* and *λ* can employ different values. In this proposal, the following ones will be employed, i.e., *α* = *β* = 0.88, *λ* = 2.25. According to PT, Equation (2) is usually utilized to measure the degree of gains and losses, in which different feelings of DM towards gains and losses are reflected by using prospect values; the greater *vθδ<sup>m</sup>*, the more DM satisfies, which denotes that the DM satisfies his/her decisions; otherwise, he/she regrets or feels depressed about his/her decisions. In this way, the DM's psychological behavior can be described clearly and comprehensively.

Due to *vθδm* not usually having the same units, a normalization process for removing the effect of units is needed. The normalized value matrix *VMθ* = (*<sup>v</sup>θδm*)*<sup>δ</sup>*×*m* can be obtained by using:

$$\overline{v}\_{\theta\delta m} = \frac{v\_{\theta\delta m}}{v\_{\theta\delta}^\*}, \delta = 1, 2, \dots, k\_1; \; \theta = 1, 2, \dots, k\_2; \; m = 1, 2, \dots, M \tag{3}$$

where *<sup>v</sup>θδ*<sup>∗</sup> = max *m*∈*M* |*<sup>v</sup>θδm*|.

On the basis of the normalized value matrix *VMθ* and the weighting vector *WXm* provided by DM, the overall prospect values of alternative *S*1*δ* can be calculated by using the following equation,

$$O\_{\theta\delta} = \sum\_{m=1}^{M} \overline{v}\_{\theta\delta m} w\_{X\_m \prime} \; \delta = 1, 2, \dots, k\_1; \; \theta = 1, 2, \dots, k\_2; \; m = 1, 2, \dots, M \tag{4}$$

#### *3.4. Selecting Optimal Alternative Based on Payoffs*

In this section, the payoffs of EE and DM will be determined on the basis of the overall prospect values, *Oθδ*, obtained above. Then, according to the payoffs of EE and DM, the optimal alternative can be selected as the proper response regarding different emergency situations.

#### 3.4.1. Determining the Payoffs of the Players

Due to the fact that the game between EE and DM is a zero-sum game and EE is unconscious of the benefits or costs that it will ge<sup>t</sup> or lose, just determining the payoffs of the DM is adequate for the emergency response.

Because *Oθδ* is a comprehensive value that reflects the DM's psychological behavior, it is regarded as the part of the payoffs of DM. Since each alternative has its own cost, it is more reasonable to consider the prospect values of per unit cost rather than the overall prospect values. The payoffs of DM are determined as follows:

$$P\_1(\mathcal{S}\_1) = f(\mathcal{O}\_{\theta\delta}, \mathcal{C}\_{\delta}) = \frac{O\_{\theta\delta}}{\mathcal{C}\_{\delta}}, \ \delta = 1, 2, \cdots \cdot k\_1; \ \theta = 1, 2, \cdots \cdot k\_2 \tag{5}$$

Then, the payoffs of EE can be obtained as:

$$P\_2(S\_2) = -P\_1(S\_1) \tag{6}$$

From Equations (5) and (6), the selection process of the optimal alternative can be determined in the coming subsection.

3.4.2. Selection of the Optimal Alternative with Respect to Each Emergency Situation

As mentioned previously, the game between EE and DM is a zero-sum game, and EE is a special player, which has no consciousness about the real world, so it is adequate to determine the optimal strategy of the DM.

The equation for selecting the optimal strategy of DM with respect to each possible emergency situation goes as follows:

$$P\_1(S\_{2\theta}, S\_{1\delta}^\*) = \max\_{\delta \in k\_1} P\_1(S\_{2\theta}, S\_{1\delta}), \ \theta = 1, 2, \dots, k\_2 \tag{7}$$

The vector strategy (*<sup>S</sup>*2*θ*, *S*∗ 1*δ*) means if the EE has taken *S*2*θ* as its strategy, the best response for the DM is the strategy *S*∗ 1*δ*. In other words, the strategy *S*∗ 1*δ* will be the optimal strategy of DM to deal with the emergency situation *S*2*θ*.

For a clear understanding, the procedures of the new proposed method are summarized as the following steps:


#### **4. Case Study and Comparison**

## *4.1. Case Study*

This part will provide a case study on a typhoon emergency event to demonstrate the validity and rationality of the proposed method.

In summer, it is quite common for coastal cities to suffer from different kinds of losses (lives, property, environment, etc.) caused by typhoons. In order to take effective measures to reduce the losses caused by typhoon as much as possible in the real world, this section takes typhoon landfall as an application background to demonstrate the validity and rationality of our proposal. Suppose that a typhoon is approaching and will possibly make landfall at one city located on the southeast coast of China. When it makes landfall, it might cause various losses, such as lives, properties, environment damages, etc. Thus, the following criteria are concerned in this case study:

*c*1: The number of casualties. *c*2: Property losses (in 1000\$).

*c*3: The negative effects on the environment on a scale of 0–100 (0: no negative effect; 100: serious negative effect).

The emergency alternatives are described as follows:

Regarding the coming typhoon, the following alternatives can be carried out:

*S*11: Broadcast and send short messages to remind citizens regarding the coming typhoon and sugges<sup>t</sup> that citizens prepare food, water, medicine and other daily necessities in advance; furthermore, local governmen<sup>t</sup> organizes related departments to check the evacuation solutions and paths to ensure the citizens' safety as much as possible;

*S*12: Based on *S*11, inform schools and plants to check the safety issues; classes and work can be stopped if necessary. Meanwhile, employees in ocean transport, fishermen and mariculture are required to come back to or go closer to harbors to take shelter from the typhoon. In addition, check the stability of high-altitude facilities and dangerous buildings.

*S*13: Based on *S*12, telecom operators and power supply departments strengthen their checking and maintenance to ensure all different lines of communication and power supply are open. Meanwhile, check the urban drainage pipelines to avoid urban waterlogging.

*S*14: Based on *S*13, vindicate public security in preventing criminal issues from occurring; meanwhile, hospitals prepare enough ambulances and staff to ensure that injured citizens can be rescued and treated immediately. Furthermore, the reservoirs and hydropower stations near the city should make reasonable schedules to avoid floods.

*Cδ* is the cost of the *δ*-th alternative (in 1000\$). The criteria weights of each criterion are provided by DM in this case study. The pre-defined effective control scope *Eδ<sup>m</sup>*, the cost *Cδ* and related weights *wXm* are given in Table 5.


**Table 5.** The *Eδ<sup>m</sup>*, *Cδ* and *wXm* of the typhoon emergency.

Analyzing by the weather forecast and historical data, there are four possible situations of a typhoon in the coming 72 h, as follows:

*S*21: The typhoon will not make landfall at the city, and it just brings light rain and wind;

*S*22: The typhoon will make landfall at part of the area of the city and bring moderate rain and gales;

*S*23: The typhoon will make landfall over the entire city and bring rainstorms and strong wind;

*S*24: The typhoon will have a front landfall over the entire city and bring downpours and blustery weather;

The reference points *Rθm* regarding the four possible emergency situations provided by DM are shown in Table 6.


**Table 6.** Reference points (RPs) regarding the four emergency situations.

According to the information shown in Tables 5 and 6, the positional relationship between *Rθm* and *Eδm* in Table 2 and the equations provided in Tables 3 and 4, the gain and loss matrix *GMθ*, *LMθ* can be obtained as follows,


Based on *GMθ* and *LMθ*, the value matrix *VMθ* and its normalized form *VMθ* can be obtained according to Equations (2) and (3), respectively, i.e.,

$$\begin{aligned} VM\_1 &= \begin{bmatrix} -2.25 & -70.35 & 7.59\\ 6.91 & 455.67 & 19.95\\ 9.88 & 568.45 & 25.69 \end{bmatrix}, VM\_2 = \begin{bmatrix} -2.25 & -129.47 & 2.24\\ 4.12 & 378.35 & 13.96\\ 7.25 & 493.64 & 19.95 \end{bmatrix},\\ VM\_3 &= \begin{bmatrix} -140.2 & -340.45 & -5.04\\ -5.92 & 105.90 & 2.24\\ 3.01 & 378.35 & 13.96\\ 3.01 & 378.35 & 13.96 \end{bmatrix}, VM\_4 = \begin{bmatrix} -2.29 & -53.67 & -12.0\\ -14.02 & 57.54 & -5.04\\ -4.14 & 132.9 & 2.24\\ 208.79 & 75.9\\ 0.699 & 0.816 & 0.763 \end{bmatrix}, \text{and} VM\_5 &= \begin{bmatrix} -0.310 & -0.2623 & 0.112\\ 0.1379 & 0.4805 & 0.303\\ 0.668 & 0.7664 & 0.699\\ 0.685 & 0.764 & 0.699\\ 1 & 1 & 1 \end{bmatrix},\\ VM\_5 &= \begin{bmatrix} -1 & -0.8998 & -0.3610\\ 0.0713 & 0.6818 & 0.504\\ 0.2147 & 1 & 1 \end{bmatrix}, \begin{bmatrix} -1 & -1 & -1\\ -0.180 & 0.2472 & 0.1312\\ -0.180 & 0.599 & 0.1444\\ 0 & 0.599 & 0.4444 \end{bmatrix}. \end{aligned}$$

According to Equation (4), the overall prospect values *Oθδ* of the *θ*-th alternatives in *δ*-th emergency situation are calculated and shown in Table 7.

**Table 7.** The overall prospect values *Oθδ* of the *θ*-th alternatives in the *δ*-th emergency situation.


Based on Equations (5) and (6) and the results of *Oθδ* shown in Table 7, the payoff matrix of EE and DM is provided in Table 8.


**Table 8.** The payoff matrix of EE and DM.

Then, based on Table 8 and Equation (7), the best strategy of DM for different possible situations can be obtained as follows:

If the EE has selected the strategy *S*21, the best strategy of DM is the one with the biggest payoff value, which can be obtained by using Equation (7):

$$\begin{aligned} P\_1(S\_{21}, S\_{1\delta}^\*) &= \max\_{\delta \in k\_1} P\_1(S\_{21}, S\_{1\delta}) \\ &= \max\_{\delta} \left\{ P\_1(S\_{21}, S\_{11}), P\_1(S\_{21}, S\_{12}), P\_1(S\_{21}, S\_{13}), P\_1(S\_{21}, S\_{14}) \right\} \\ &= \max\_{\delta} \left\{ -0.0071, 0.0136, 0.0106, 0.0077 \right\} \\ &= 0.0136 \\ \text{where } P\_1(S\_{21}, S\_{21}) &= P\_1(S\_{21}, S\_{11}) + 1 = 1 = 1 = 1 = 1 \text{ and } 0.1 = 1, 1 = 1, \dots \text{ to } 1. \end{aligned}$$

That is *<sup>P</sup>*1(*<sup>S</sup>*21, *<sup>S</sup>*<sup>∗</sup>1*δ*) = *<sup>P</sup>*1(*<sup>S</sup>*21, *<sup>S</sup>*12), which means if the EE has selected the strategy *S*21, the best strategy for DM is *S*12.

Similarly, the best strategies of DM regarding different possible situations are the ones with the biggest payoffs, which are underlined and bolded in Table 9.


**Table 9.** Best strategies of DM.

The four optimal solutions with respect to each emergency situation are (*<sup>S</sup>*21, *<sup>S</sup>*12), (*<sup>S</sup>*22, *<sup>S</sup>*12), (*<sup>S</sup>*23, *<sup>S</sup>*13) and (*<sup>S</sup>*24, *<sup>S</sup>*14), which means if the EE has selected *S*21, the best strategy of DM is to select *S*12; if the EE has selected *S*22, the best strategy of DM is to select *S*12; if the EE has selected *S*23, the best strategy of DM is to select *S*13; the EE has selected *S*24, the best strategy of DM is to select *S*14.

#### *4.2. Comparison with Other Methods*

In order to demonstrate the superiority and novelty of our proposal, a comparison with other methods will be conducted. Because there are no existing approaches that are based on PT and GT simultaneously, thus, some characteristics have been studied to highlight the superiority of our proposal; see Table 10.

**Table 10.** Comparison with other emergency decision making (EDM) methods.


According to Table 10, it can be seen clearly that our proposal considers not only the DM's psychological behavior, but also the coping with the different emergency situations. The proposed EDM method is closer to the real-world situations than other EDM methods.

#### **5. Conclusions and Future Works**

A new EDM method based on GT and PT is proposed in this paper aiming at overcoming the limitations in previous EDM approaches. Due to the inadequate and incomplete information about EEs, interval values are employed in our proposal to estimate the possible losses caused by different situations. DM's psychological behavior and coping with different emergency situations have been considered simultaneously, which is the significant difference between our proposal and the existing EDM approaches. An example about a typhoon and related comparison with existing EDM approaches have been conducted to demonstrate the novelty and rationality of our proposal. It is hoped that our proposed method can be applied to solve real-word problems in the near future.

The research in the near future should consider the different types of information in the game process, such as linguistic information, hesitant fuzzy linguistic information, and so on, which are common information types in the real world when DM hesitates in his/her assessments.

**Author Contributions:** All authors have contributed equally to this paper.

**Funding:** This work was partly supported by the National Natural Science Foundation of China (Project Nos. 61773123, 71701050, 71801050), the Spanish National Research Project (Project No. TIN2015-66524-P) and the Spanish Ministry of Economy and Finance Postdoctoral Fellow (IJCI-2015-23715)

**Acknowledgments:** The authors would like to thank the anonymous reviewers for their valuable time in reviewing our paper and for their helpful suggestions to improve our paper.

**Conflicts of Interest:** The authors declare no conflict of interest.
