Post-Earthquake Scheduling of Rescuers: A Method Considering Multiple Disaster Areas and Rescuer Collaboration

Abstract

Reasonable and efficient scheduling of rescuers plays a crucial role in earthquake emergency relief, which can effectively reduce disaster losses and promote social stability and sustainable development. Due to the suddenness of disasters, the urgency of time, and the complexity of rescue efforts, scheduling of rescuers often involves multiple disaster areas, multiple departure areas, and diverse rescue tasks. However, most existing studies have paid little attention to the scheduling problem of rescuers considering multiple disaster areas, multiple departure areas, and multiple rescue tasks and have not comprehensively considered the collaboration of rescuers and task requirements. Thus, how to reasonably dispatch rescuers to disaster areas by considering the collaboration of rescuers and task needs is a noteworthy research problem. The objective of this paper is to propose a method considering multiple disaster areas and the collaboration of rescuers to solve the scheduling problem of rescuers after earthquakes. Firstly, the collaborative degrees of rescuers are calculated according to the collaborative performance among rescuers concerning collaborative feature indicators. Secondly, according to the performance of rescuers concerning professional skill evaluation indicators, the professional abilities of rescuers are calculated, and the comprehensive performance indicators for rescuers are obtained by aggregating the collaborative degrees and the professional abilities of rescuers. Thirdly, the time satisfaction degrees are calculated based on the times taken by rescuers from different departure areas to disaster areas and the time requirements of disaster areas. Then, the time satisfaction degrees and the comprehensive performance of rescuers are aggregated to obtain the comprehensive matching degrees. Furthermore, a rescuer scheduling model for earthquake emergency rescue is constructed to maximize the comprehensive matching degrees between rescuers and rescue tasks, and the optimal scheduling scheme is determined by solving the model. Finally, a case study and comparative analyses are presented to verify the rationality and feasibility of the proposed method. The results show that the proposed method can reasonably assign rescuers to quickly respond to the needs of rescue tasks in disaster areas, and is better than the other two methods in terms of rescue comprehensive capability evaluation. The proposed method can provide decision support for solving the post-earthquake scheduling problem of rescuers and help to improve the emergency response ability for large-scale geological disaster events.

1. Introduction

In recent years, large-scale earthquake disasters have occurred frequently around the world, claiming countless lives [1,2], causing serious damage to infrastructure, and inflicting huge economic losses [3]. For example, a devastating earthquake struck Wenchuan, China on 12 May 2008. It caused more than 80,000 deaths and missing people and the injury of 370,000 people, affected 10 counties (cities) in the severely affected areas, and caused economic losses of over USD 130 billion. On 25 April 2015, a magnitude 8.1 earthquake occurred in Nepal, affecting several countries, including Bangladesh, India, and China, resulting in nearly 9000 deaths and more than 20,000 injuries. Hundreds of thousands of buildings were damaged, and economic losses of approximately USD 6 billion were caused. As can be seen, massive earthquake disasters have a large impact range and affect many areas and people, causing serious damage to people’s production and livelihoods and even devastating consequences. After an earthquake, rapid and effective emergency relief operations become the most effective means to reduce earthquake-induced loss and casualties [4,5], which is conducive to promoting social stability and sustainable development. The main participants in earthquake emergency rescue operations are rescuers, and scheduling rescuers to carry out tasks in disaster areas is a central part of the earthquake response [6,7]. Hence, how to promptly and effectively dispatch rescuers plays an important role in post-earthquake emergency response and reducing casualties caused by earthquakes.

Due to the suddenness of earthquake disasters, the urgency of time, and the complexity of relief operations [8], the scheduling of rescuers often involves multiple disaster areas, multiple departure areas, and diverse rescue tasks. On the one hand, the rescue capabilities of various disaster areas are usually unable to meet the overwhelming rescue needs, and rescuers need to be dispatched from nearby areas to carry out rescue tasks. Different rescue tasks have different requirements for the professional abilities of rescuers. On the other hand, rescue tasks are difficult for individuals to complete independently and require the collaboration of multiple rescuers. However, rescuers have different rescue experiences and professional abilities, which increases the difficulty of setting up temporary rescue teams. In addition, the collapse of buildings has caused road congestion, making it difficult to accurately estimate the time for rescuers to travel from departure areas to disaster areas. Therefore, it is crucial to join the rescue task requirements, the collaboration and professional abilities of rescuers, and the rescue time to analyze the post-earthquake scheduling of rescuers.

Until now, studies on post-earthquake rescuer scheduling are still relatively scarce, and the relevant literature is limited. From the perspective of rescue participants, most of the existing methods for post-earthquake scheduling of rescuers can be divided into two categories, i.e., the dispatching of professional rescue forces funded by the government [9,10,11] and the dispatching of social forces (such as volunteers) [12,13,14]. The government-funded rescuers are usually the main emergency rescue forces, responsible for the majority of emergency rescue work, and the social forces are mainly composed of local enterprises, public organizations, and non-governmental organizations. These studies have made significant contributions to the dispatch of rescuers after earthquakes. However, existing studies have not fully investigated the post-earthquake rescuer scheduling problem considering multiple disaster areas and multiple departure areas, nor have they considered the collaboration of rescuers and the needs of rescue tasks. In order to comprehensively consider the various factors mentioned above for post-earthquake rescuer scheduling, the following three key issues need to be addressed: (1) how to accurately evaluate the professional abilities and collaboration degrees of rescuers by considering the needs of different rescue tasks, (2) how to depict the relationship between rescue time of rescuers and rescue effect to better reflect the timeliness of earthquake emergency rescue, and (3) how to construct a post-earthquake scheduling model of rescuers by comprehensively considering the rescue task requirements, the professional abilities and collaboration degrees of rescuers, the rescue time and other factors, and generate the optimal scheduling scheme.

Obviously, the research and analysis on the above issues can solve the scheduling problem of post-earthquake rescuers, provide valuable management insights for post-earthquake rescue work, and help to improve emergency response capabilities to cope with various emergency risks that arise in the process of unplanned urbanization.

The objective of this paper is to develop a method considering multiple disaster areas and the collaboration of rescuers to solve the scheduling problem of rescuers in the earthquake emergency response. Firstly, according to the collaborative performance among rescuers, the collaboration degrees of rescuers are calculated. Secondly, according to the performance of rescuers concerning professional skill indicators, the professional abilities of rescuers to complete rescue tasks are calculated, and the comprehensive performance of rescuers is obtained. Then, the time satisfaction degrees are obtained based on the times taken by the rescuers from the departure areas to the disaster areas and the time requirements of the disaster areas. Further, the time satisfaction degrees and the comprehensive performance of rescuers are aggregated to obtain the comprehensive matching degrees. On the basis of this, a rescuer scheduling model for earthquake emergency rescue is constructed with comprehensive matching degree maximization as the objectives, and the optimal scheduling scheme can be determined.

The remainder of this paper is organized as follows. Section 2 presents a literature review. In Section 3, the post-earthquake scheduling problem of rescuers is described and a scheduling method for rescuers is given. Section 4 presents a case study, Section 5 gives results analysis and comparative analyses, and Section 6 summarizes this research.

2. Literature Review

This study is related to two aspects, i.e., earthquake emergency management and post-earthquake scheduling of rescuers. A brief literature review with respect to each aspect is given below.

Earthquake emergency management activities include the preparation phase (period before the earthquake), the response phase (period during and shortly after an earthquake), and the recovery phase (period a long time after the earthquake) [15]. The tasks in the preparation phase include planning, training, monitoring and warning, developing earthquake emergency plans [16], etc. In the response phase, the main tasks include the dispatch of emergency supplies and rescue forces, earthquake emergency rescue, maintenance of social stability, emergency shelter and resettlement [17,18,19], etc. The tasks of the recovery phase involve post-earthquake disaster loss assessment, reconstruction, psychological intervention mechanisms, social conflict mitigation measures [20], etc. Scholars have conducted studies on earthquake emergency management from different perspectives to address the issues faced at different stages. The most prevalent topics include earthquake casualty analysis [21,22], rescue capability analysis [23,24], relief distribution [25,26] and routing [27,28,29], and facility location [30,31]. In contrast, manpower planning is another important topic in the context of earthquake emergency management, but it has received limited attention [32].

Emergency manpower planning, that is, assigning and dispatching rescuers, is a key element in the process of earthquake disaster management. Although the research on rescuers in earthquake emergency management has slightly increased, there is still relatively little research on post-earthquake rescuer scheduling. Song et al. [33] established an optimization model for the assignment problem of earthquake emergency rescue teams, with maximum satisfaction of emergency rescue time and the highest competencies of emergency rescue teams as objectives. Zheng et al. [34] developed an efficient multi-objective biogeographic optimization algorithm for the fuzzy multi-objective optimization problem of rescue task scheduling, which takes into account the efficiency of task scheduling and the operational risk of rescue teams. Hooshangi et al. [11] used queuing theory to simulate urban search and rescue in earthquake environments by considering the number of damaged and injured buildings, as well as the service time after task allocation, so as to determine an appropriate number of rescue teams and improve survival rates. Xu et al. [35] used a local search optimization algorithm based on discrete teaching-learning to solve the allocation problem of emergency rescue forces, so as to minimize the average completion time of all rescue teams. The above study considered various rescue tasks, but did not simultaneously consider multiple departure areas and disaster areas, which is not in line with the actual situation of earthquake emergency rescue. For this, Fei and Wang [36] used the Dempster–Shafer theory to develop a rescuer scheduling model that considers the actual needs of rescue tasks, and verified the advantages of the model in emergency multi-task group decision-making through an empirical analysis using an earthquake emergency rescue as an example. Chu and Zhong [10] proposed a decision-making method to support the allocation of medical staff after earthquakes and established an optimization model based on the utility principle and the random transfer probability of triage levels. Zhou et al. [37] presented a joint programming model for the dispatch of emergency rescue teams and road reconstruction under earthquake disaster to maximize the survival probability of trapped people and the rescue effect. Although these studies considered multiple departure areas and disaster areas, they did not consider the situation of different rescue tasks in disaster areas due to different levels of disaster. Chen et al. [38] developed a model for earthquake disaster rescue based on complex adaptive systems theory. The model not only considers the complexity and diversity of tasks but also takes into account both timeliness and efficiency. However, the aforementioned research did not consider collaboration and cooperation among rescuers. Collaboration among personnel plays an important role in improving rescue effects. La et al. [39] aimed at the deployment of geological disaster rescuers, conducted a synergy analysis of rescue forces and predicted the required rescuers based on rescue tasks, and constructed a rescuer deployment model with the goal of optimal deployment effect by considering the urgency of rescue tasks.

Table 1. Summary of the main studies on post-earthquake rescuer scheduling.

Publication	Considering Multiple Disaster Areas	Considering Multiple Departure Areas	Consider Diverse Rescue Tasks	Considering Rescuer Collaboration
Zheng et al. [34]	No	No	Yes	No
Chu and Zhong [10]	Yes	Yes	No	No
Zhou et al. [37]	Yes	Yes	No	No
Song et al. [33]	No	Yes	Yes	No
Chen et al. [38]	Yes	Yes	Yes	No
Fei and Wang [36]	Yes	Yes	No	No
Hooshangi et al. [11]	No	Yes	Yes	No
Xu et al. [35]	Yes	No	Yes	No
La et al. [39]	No	No	Yes	Yes
This study	Yes	Yes	Yes	Yes

presents a summary of the main studies on post-earthquake rescuer scheduling.

Through the above studies’ review, we can find that existing studies provide some methods and ideas that can be used as a reference in this paper, such as the consideration of rescuer capability and rescue time [26,33,36] and the construction of a rescuer dispatch model [10,38]. However, existing research still has some limitations. Relevant research has not fully investigated the post-earthquake rescuer scheduling problem considering multiple departure areas and multiple disaster areas. In fact, large-scale earthquakes have a large sphere of influence, and the affected areas are scattered, so rescuers need to be dispatched from nearby areas to different disaster areas. Usually, the time taken for each departure area to reach different disaster areas to carry out rescue tasks varies, and timeliness is the key in earthquake emergency rescue work and also an important factor affecting rescue effectiveness. In addition, in terms of evaluating the rescue capabilities of rescuers, existing research often only considers the professional abilities and rarely mentions cooperation between rescuers. In the actual process of earthquake emergency rescue, a single rescue worker only has limited capabilities and resources; it is difficult to complete the rescue work independently. Only close cooperation and collaboration between rescuers can effectively complete complex rescue tasks and create a good collaborative effect between rescuers. Therefore, in the dispatch of rescuers after the earthquake, it is necessary to match the professional abilities of rescuers with the needs of rescue tasks as much as possible and consider rescue time, as well as consider the collaborative performance of rescue personnel, so as to improve rescue efficiency.

3. Method for Post-Earthquake Scheduling of Rescuers

In this section, the dispatch problem of earthquake emergency rescuers and related symbols are first explained. Then, the calculation methods of the collaboration degrees of rescuers, the professional abilities of rescuers, the comprehensive performance of rescuers, and time satisfaction degrees are introduced, respectively. Finally, a scheduling model for rescuers is established.

3.1. Problem Description

After a large-scale earthquake disaster, multiple towns or villages often require external rescue at the same time. At this time, each nearby rescue area (i.e., departure area) needs to immediately assign rescuers to multiple disaster areas to carry out rescue missions. The scheduling problem of rescuers after earthquakes is shown in Figure 1. Assuming there are $l$ disaster areas that need to be rescued, in which the rescue tasks required for each disaster area are $M_g^{}$, the number of rescuers required for each rescue task in each disaster area is $N_g^l$, there are $n$ departure points, and the number of rescuers that can be dispatched at each departure area is $a_i$. Usually, rescuers have different levels of professional skills, and different disaster-stricken areas have different requirements for rescue time considering that rescue tasks require collaborative cooperation among rescuers to be effectively completed. In view of this, how to reasonably allocate rescuers from different departure areas to disaster areas to complete corresponding tasks by comprehensively considering factors such as the collaboration of rescuers, professional abilities of rescuers, and rescue time in order to achieve the best effect of rescue task completion is the concerned problem of this paper. Here, it is assumed that each rescuer can only be dispatched to one disaster area to perform one rescue task at most, and the number of rescuers meets the needs of rescue tasks in each disaster area.

To clearly represent the scheduling problem of rescuers after earthquakes, the symbols involved are defined, as shown in

Table 2. The symbols used in this paper and the corresponding explanations.

Symbols	Explanations
$O_i$	The $i$th departure area for earthquake emergency rescue, $i=1,2,\dots ,n$.
$a_i$	The number of earthquake emergency rescuers that can be dispatched at the departure area $O_i$, $i=1,2,\dots ,n$.
$P_{ij}$	The $j$th earthquake emergency rescuer at the departure area $O_i$, $i=1,2,\dots ,n$, $j=1,2,\dots ,a_i$.
$D_l$	The $l$th disaster area in earthquake emergency rescue, $l=1,2,\dots ,I$.
$M=\{M_1,M_2,\dots ,M_m\}$	The set of rescue tasks in earthquake emergency rescue, where $M_g$ denotes the $g$th rescue task, such as health and epidemic prevention task and on-site rescue task in earthquake emergency rescue, $g=1,2,\dots ,m$.
$\lambda { }_l$	The set of rescue tasks that need to be performed in the disaster area $D_l$. Usually, due to the different conditions at each disaster area, the rescue tasks to be performed differ, $\lambda { }_l^{}\subseteq M$. Additionally, let $G_l$ be the subscript set of $\lambda { }_l^{}$, $G_1^{}\cup G_2^{}\cup \cdots \cup G_I^{}=\{1,2,\dots ,m\}$, $l=1,2,\dots ,I$, $G_l\ne \varnothing$.
$N_g^l$	The number of rescuers required for the rescue task $M_g$ in the disaster area $D_l$. Here, assuming a sufficient number of rescuers are available for dispatch, i.e., ${\displaystyle \sum _{i=1}^n{a}_i}\ge {\displaystyle \sum _{l=1}^I{\displaystyle \sum _{g\in G_l}^{}{N}_g^l}}$, $l=1,2,\dots ,I$, $g=1,2,\dots ,m$.
$C_p$	The $p$th collaborative feature indicator required to complete rescue tasks, i.e., the indicator used to assess the collaborative relationship between rescuers, $p=1,2,\dots ,\mu$.
$c_{ijuz}^p$	The collaborative performance of rescuers $P_{ij}$ and $P_{uz}$ concerning the collaborative feature indicator $C_p$. Usually, $c_{ijuz}^p$ is obtained based on the rescuer’s historical rescue records of collaboration and historical exercise records of collaboration, and the performance of the rescuer concerning the collaborative feature indicator is expressed by the score level, $i,u=1,2,\dots ,n$, $i\ne u$, $j,z=1,2,\dots ,{\alpha }_i$, $j\ne z$, $p=1,2,\dots ,\mu$.
$w_p^g$	The weight of the collaborative feature indicator $C_p$ concerning the task $M_g$, which satisfies $0\le w_p^g\le 1$, ${\displaystyle \sum _{p=1}^{\mu }{w}_p^g}=1$, $g=1,2,\dots ,m$.
$E=\{E_1,E_2,\dots ,E_q\}$	The set of professional skill evaluation indicators for rescuers required to complete rescue tasks, where $E_k$ denotes the $k$th professional skill evaluation indicator, $k=1,2,\dots ,q$.
$b_{ij}^k$	The performance of the rescuer $P_{ij}$ concerning the professional skill evaluation indicator $E_k$. Usually, $b_{ij}^k$ is obtained based on the historical rescue records, historical exercise records, and the qualifications of the rescuer, and $b_{ij}^k$ is expressed by the score level, $i=1,2,\dots ,n$, $j=1,2,\dots ,a_i$, $k=1,2,\dots ,q$.
${\mathsf{\Gamma }}_g$	The set of professional skill evaluation indicators required to complete the rescue task $M_g$. Due to the different nature and content of rescue tasks, the professional skills required for different rescue tasks also vary, where ${\mathsf{\Gamma }}_g\subseteq E$, ${\mathsf{\Gamma }}_1\cup {\mathsf{\Gamma }}_2\cup \cdots \cup {\mathsf{\Gamma }}_m=E$. Additionally, let $H_{}^g$ be the subscript set of ${\mathsf{\Gamma }}_g$, where $H_{}^1\cup H_{}^2\cup \cdots \cup H_{}^m=\{1,2,\dots ,q\}$, $H_{}^g\ne \varnothing$, $g=1,2,\dots ,m$.
${\epsilon }_k^g$	The weight of the professional skill evaluation indicator $E_k$ concerning the task $M_g$, which satisfies $0\le {\epsilon }_k^g\le 1$, ${\displaystyle \sum _{k\in H^g}{\epsilon }_k^g}=1$, $g=1,2,\dots ,m$, $k=1,2,\dots ,q$.
$t_i^l$	The time from the departure area $O_i$ to the disaster area $D_l$ for rescuers.
$T_l^o$	The optimal rescue time limit for the disaster area $D_l$.
$T_l^e$	The effective rescue time limit for the disaster area $D_l$.
$T_l^f$	The failure rescue time limit for the disaster area $D_l$.

.

The problem to be solved in this paper is how to construct an optimization scheduling model for post-earthquake rescuers, considering $N_g^l$ (i.e., the number of rescuers required for each rescue task in each disaster area), the collaborative performance $c_{ijuz}^p$, the performance of professional skill $b_{ij}^k$, and the time $t_i^l$.

3.2. Collaboration Degrees of Rescuers

In the process of earthquake emergency rescue, rescuers often face complex and diverse rescue tasks, and the rapid and effective completion of rescue tasks relies on close cooperation and efficient collaboration among rescuers [40]. Only the collaboration among rescuers can give full play to the synergy effect among rescuers [41], so the collaboration of rescuers is crucial for earthquake emergency rescue scheduling. Therefore, according to the collaborative evaluation information between rescuers, the degrees of collaboration of rescuers are calculated in this paper.

To calculate the collaboration degrees of rescuers, the collaborative performance evaluation value $c_{ijuz}^p$ of the collaborative feature indicator $C_p$ between the rescuers is first standardized to obtain the standardized evaluation value ${\overline{c}}_{ijuz}^p$. Let $C^a$ and $C^b$ be the set of benefit-based collaborative feature indicators and the set of cost-based collaborative feature indicators, respectively, $C^a\cup C^b=C$, $C^a\cap C^b=\varnothing$. Let $C{O}^a$ and $C{O}^b$ be the set of subscripts for collaborative feature indicators in the sets $C^a$ and $C^b$, respectively, $C{O}^a\cup C{O}^b=\{1,2,\dots ,\mu \}$, $C{O}^a\cap C{O}^b=\varnothing$.

If $C_p$ is a benefit-based collaborative feature indicator, i.e., $C_p\in C^a$, $p\in C{O}^a$, then ${\overline{c}}_{ijuz}^p$ can be calculated by Equation (1)

$${\overline{c}}_{ijuz}^p=\frac{c_{ijuz}^p-c_{\min }^p}{c_{\max }^p-c_{\min }^p}$$

(1)

If $C_p$ is a cost-based collaborative feature indicator, i.e., $C_p\in C^b$, $p\in C{O}^b$, then ${\overline{c}}_{ijuz}^p$ can be obtained as follows:

$${\overline{c}}_{ijuz}^p=\frac{c_{\max }^p-c_{ijuz}^p}{c_{\max }^p-c_{\min }^p}$$

(2)

where $c_{\min }^p=\min \left\{c_{ijuz}^p\right|i,u=1,2,\dots ,n;j,z=1,2,\dots ,{\alpha }_i,i\ne u,j\ne z\}$, and $c_{\max }^p=\max \left\{c_{ijuz}^p\right|i,u=1,$$2,\dots ,n;j,z=1,2,\dots ,{\alpha }_i,i\ne u,j\ne z\}$, $p=1,2,\dots ,\mu$.

Then, the collaborative performance ${\delta }_{ij}^p$ of rescuer $P_{ij}$ can be obtained using the following equation:

$$\begin{gathered} {\delta }_{ij}^p=\frac{1}{{\displaystyle \sum _{i=1}^n{a}_i-1}}[{\displaystyle \sum _{u=1}^n{\displaystyle \sum _{z=1}^{a_n}{\overline{c}}_{ijuz}^p}}], \\ i=1,2,\dots ,n, j=1,2,\dots ,{\alpha }_i, p=1,2,\dots ,\mu \end{gathered}$$

(3)

Based on ${\delta }_{ij}^p$ and the weight $w_p^g$ of the collaborative feature indicator $C_p$ for the rescue task $M_g$, the degree of collaboration of the rescuer, $R_{ij}^g$, can be calculated, i.e.,

$$R_{ij}^g={\displaystyle \sum _{p=1}^{\mu }{w}_p^g{\delta }_{ij}^p}, i=1,2,\dots ,n, j=1,2,\dots ,{\alpha }_i, g=1,2,\dots ,m$$

(4)

3.3. Professional Abilities of Rescuers to Complete Rescue Tasks

In earthquake emergency rescue, different disaster areas have different situations, and there are differences in the rescue tasks that need to be completed. And the professional abilities required for rescue tasks are different, so different levels of the professional abilities of rescuers need to be considered [42].

To calculate the professional abilities of rescuers, the performance $b_{ij}^k$ of the rescuer $P_{ij}$ concerning the professional skill evaluation indicator $E_k$ is first standardized. Let $E^a$ and $E^b$ be the set of benefit-based professional skill evaluation indicators and the set of cost-based professional skill evaluation indicators, respectively, $E^a\cup E^b=E$, $E^a\cap E^b=\varnothing$. Let $E{O}^a$ and $E{O}^b$ be the set of subscripts for professional skill evaluation indicators in the sets $E^a$ and $E^b$, respectively, $E{O}^a\cup E{O}^b=\{1,2,\dots ,q\}$, $E{O}^a\cap E{O}^b=\varnothing$.

If $E_k$ is a benefit-based professional skill evaluation indicator, i.e., $E_k\in E^a$, $k\in E{O}^a$, then the standardized professional skill evaluation value $S_{ij}^k$ can be calculated, i.e.,

$$S_{ij}^k=\frac{b_{ij}^k-b_{\min }^k}{b_{\max }^k-b_{\min }^k}$$

(5)

If $E_k$ is a cost-based professional skill evaluation indicator, i.e., $E_k\in E^b$, $k\in E{O}^b$, then $S_{ij}^k$ can be obtained as follows:

$$S_{ij}^k=\frac{b_{\max }^k-b_{ij}^k}{b_{\max }^k-b_{\min }^k}$$

(6)

where $b_{\min }^k=\min \left\{b_{ij}^k\right|i=1,2,\dots ,n;j=1,2,\dots ,a_i\}$, and $b_{\max }^k=\max \left\{b_{ij}^k\right|i=1,2,\dots ,$$n;j=1,2,\dots ,a_i\}$, $k=1,2,\dots ,q$.

According to $S_{ij}^k$ and the weight ${\epsilon }_k^g$ of the professional skill evaluation indicator $E_k$ for the rescue task $M_g$, the professional ability of the rescuer, $S{C}_{ij}^g$, can be obtained using the following equation:

$$S{C}_{ij}^g={\displaystyle \sum _{k\in H^g}{S}_{ij}^k{\epsilon }_k^g}, i=1,2,\dots ,n, j=1,2,\dots ,{\alpha }_i, g=1,2,\dots ,m$$

(7)

3.4. Comprehensive Performance of Rescuers

The comprehensive performance $C{P}_{ij}^g$ of the rescuer can be calculated based on the collaboration degree of the rescuer and the professional ability of the rescuer. $C{P}_{ij}^g$ can be obtained as follows:

$$\begin{gathered} C{P}_{ij}^g=\theta \cdot R_{ij}^g+(1-\theta )S{C}_{ij}^g, i=1,2,\dots ,n, j=1,2,\dots ,{\alpha }_i, \\ g=1,2,\dots ,m \end{gathered}$$

(8)

where $\theta \in [0,1]$. $\theta$ is the coefficient which is usually given by earthquake emergency decision makers based on actual situations. The larger $\theta$ is, the greater the priority given to the collaboration degrees of rescuers in the earthquake emergency rescue dispatch decisions; otherwise, the greater priority is given to the professional abilities of rescuers.

3.5. Time Satisfaction Degrees

In earthquake emergency rescue, the time from the departure area to the disaster area directly affects the effectiveness of the rescue [43,44]. The timely rescue will maximize the effectiveness of the rescue; otherwise, the rescue will be meaningless. As there are different situations in different disaster areas, the requirements of the disaster areas for rescue time also vary [45]. Time satisfaction degrees can be determined based on the actual travel times of rescuers and the requirements of different disaster areas for rescue time. To effectively measure the relationship between travel time and the effectiveness of earthquake emergency rescue, the following emergency rescue time function was constructed for rescuers from the departure area $O_i$ to reach the disaster area $D_l$ [46,47]:

$$\psi (t_i^l)=\left\{\begin{array}{ll}1,\hfill & t_i^l\le T_l^o\hfill \\ (1-z^l){(\frac{T_l^e-t_i^l}{T_l^e-T_l^o})}^{l_1}+z^l,\hfill & T_l^o<t_i^l\le T_l^e,l_1>1\hfill \\ z^l{(\frac{T_l^f-t_i^l}{T_l^f-T_l^e})}^{l_2},\hfill & T_l^e<t_i^l\le T_l^f,0<l_2<1\hfill \\ 0,\hfill & t_i^l>T_l^f\hfill \end{array}\right.$$

(9)

In Equation (9), $z^l$ represents time satisfaction degree of the rescuer arriving at the disaster area $O_i$ at time $T_l^e$, and the values of $T_l^o$, $T_l^e$, $T_l^f$ and $z^l$ should be provided by the emergency decision maker based on the actual situation of the earthquake response, $i=1,2,\dots ,n$, $l=1,2,\dots ,I$.

The time satisfaction degree function is shown in Figure 2. In Figure 2, if the rescuer is able to reach the disaster area $D_l$ before the optimal rescue time limit $T_l^o$, the time satisfaction degree is 1. If the rescuer reaches the disaster area $D_l$ between the optimal rescue time limit $T_l^o$ and the effective rescue time limit $T_l^e$, time satisfaction degree shows a slight downward trend with the increase in $t_i^l$. If the rescuer reaches the disaster area $D_l$ between the effective rescue time limit $T_l^e$ and the failure rescue time limit $T_l^f$, time satisfaction degree tends to decrease significantly with increasing $t_i^l$. If the rescuer reaches the disaster area $D_l$ after the failure rescue time limit $T_l^f$, the rescue is completely ineffective and the time satisfaction is 0.

In the process of earthquake emergency rescue, it is difficult to accurately estimate the time $t_i^l$ of rescuers from the departure area $O_i$ to the disaster area $D_l$ due to road congestion, bridge damage, and other situations. In this paper, it is assumed that $t_i^l$ obeys a uniform distribution in the interval $[t_i^{lL},t_i^{lU}]$, $0\le t_i^{lL}\le t_i^{lU}$, and the expected value $T{S}_i^l$ of time satisfaction degree within $[t_i^{lL},t_i^{lU}]$ can be used to represent time satisfaction degree. $T{S}_i^l$ can be obtained as follows:

$$T{S}_i^l=\frac{{\displaystyle {\int }_{t_i^{lL}}^{t_i^{lU}}\psi (t_i^l)d{t}_i^l}}{t_i^{lU}-t_i^{lL}}$$

(10)

The values of $t_i^{lL}$ and $t_i^{lU}$ may have multiple size relationships with the key turning points $T_l^o$, $T_l^e$ and $T_l^f$ in the time satisfaction degree function. Here, referring to the processing method of [40], the estimation of time $t_i^l$ should be as accurate as possible, assuming that the estimation interval $[t_i^{lL},t_i^{lU}]$ contains at most one key turning point. Thus, there may be six different situations for the key turning points in the time satisfaction degree function, and the calculation formula for the time satisfaction degree $T{S}_i^l$ can be obtained as follows:

$$T{S}_i^l=\left\{\begin{array}{ll}1,& t_i^{lL}\le t_i^{lU}\le T_l^o\\ 1+\frac{(z^l-1)(t_i^{lU}-T_l^o{)}^2}{2(t_i^{lU}-t_i^{lL})(T_l^e-T_l^o)},& t_i^{lL}\le T_l^o<t_i^{lU}\le T_l^e\\ 1+\frac{(z^l-1)(t_i^{lU}+t_i^{lL}-2T_l^o)}{2(t_i^{lU}-t_i^{lL})},& T_l^o<t_i^{lL}\le t_i^{lU}\le T_l^e\\ \frac{z^l(t_i^{lU}-T_l^e)(T_l^e+t_i^{lU}-2T_l^f)}{2(t_i^{lU}-t_i^{lL})(T_l^e-T_l^f)}+\frac{z^l(T_l^e-t_i^{lL})(t_i^{lL}+T_l^e-2T_l^o)+(T_l^e-t_i^{lL}{)}^2}{2(t_i^{lU}-t_i^{lL})(T_l^e-T_l^o)},& T_l^o<t_i^{lL}\le T_l^e<t_i^{lU}\le T_l^f\\ \frac{z^l(t_i^{lU}+t_i^{lL}-2T_l^f)}{2(T_l^e-T_l^f)},& T_l^e<t_i^{lL}\le t_i^{lU}\le T_l^f\\ 0,& t_i^{lU}>T_l^f\end{array}\right.$$

(11)

3.6. Construction and Solution of Rescuer Scheduling Model

In the scheduling of earthquake emergency rescue, it is necessary to consider both the comprehensive performance of rescuers and time satisfaction degrees. Hence, it is necessary to aggregate the comprehensive performance of rescuers and time satisfaction degrees and calculate the comprehensive matching degrees between rescuers and rescue tasks. In actual earthquake emergency rescues, when either the comprehensive performance of rescuers and time satisfaction degrees are low, the comprehensive matching degrees should also be small [36]. Considering this reality, this paper uses the ordered weighted geometric mean operator to aggregate these two factors. The ordered weighted geometric mean operator is a common information aggregation method proposed by Xu and Da [48], which sorts the aggregation parameters in descending order and assigns weights based on the position of the parameters. It has the characteristic of “emphasizing equilibrium” and can reduce the influence of parameters with large deviations in the aggregation results. The comprehensive matching degree $C{M}_{ij}^{lg}$ between rescuers and rescue tasks can be calculated, i.e.,

$$C{M}_{ij}^{lg}=\left\{\begin{array}{cc}{(C{P}_{ij}^g)}^{{\eta }_1}\cdot {(T{S}_i^l)}^{{\eta }_2},& C{P}_{ij}^g>T{S}_i^l\\ {(T{S}_i^l)}^{{\eta }_1}\cdot {(C{P}_{ij}^g)}^{{\eta }_2},& C{P}_{ij}^g\le T{S}_i^l\end{array}\right.$$

(12)

where ${\eta }_1$ and ${\eta }_2$ represent the importance of the comprehensive performance of rescuers and the importance of time satisfaction degrees, respectively. And ${\eta }_1+{\eta }_2=1$, ${\eta }_1={0.5}^{(1-\pi )/\pi }$, ${\eta }_2=1-{0.5}^{(1-\pi )/\pi }$, where $\pi$ represents the attitude of the earthquake emergency decision makers. The larger $\pi$ is, the more optimistic the emergency decision maker will be, $\pi \in [0,1]$, $i=1,2,\dots ,n$, $j=1,2,\dots ,{\alpha }_i$, $l=1,2,\dots ,I$, $g\in G_l$.

Based on the comprehensive matching degree $C{M}_{ij}^{lg}$, a rescuer scheduling model for earthquake emergency rescue can be further established. Here, a binary variable $x_{ij}^{lg}$ is introduced. If the rescuer $P_{ij}$ is assigned to the disaster area $D_l$ to perform the rescue task $M_g$, then $x_{ij}^{lg}=1$; otherwise, $x_{ij}^{lg}=0$. In order to maximize the comprehensive matching degrees in the scheduling results of rescuers, the objective function Equation (13a) is set. Equation (13a) represents maximizing the sum of the comprehensive matching degrees between rescuers and rescue tasks. To satisfy the assumption that there is a one-to-one relationship between the rescuer and the rescue task in the disaster area, the constraint condition (13b) is provided. It indicates that each rescuer can only be dispatched to one disaster area to perform one rescue task. To ensure that the number of rescuers dispatched to the disaster area to carry out the rescue task is equal to the number of rescuers required for the rescue task in the disaster area, the constraint condition (13c) is provided. To sum up, the post-earthquake rescuer scheduling model is modeled as follows:

$$\max Z={\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^{{\alpha }_i}{\displaystyle \sum _{l=1}^I{\displaystyle \sum _{g\in G_l}^{}C{M}_{ij}^{lg}{x}_{ij}^{lg}}}}}$$

(13a)

$$s.t.{\displaystyle \sum _{l=1}^I{\displaystyle \sum _{g\in G_l}^{}{x}_{ij}^{lg}}\le 1,}i=1,2,\dots ,n;j=1,2,\dots ,a_i$$

(13b)

$${\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^{{\alpha }_i}{x}_{ij}^{lg}=N_g^l,}}l=1,2,\dots ,I;g\in G_l$$

(13c)

$$x_{ij}^{lg}=0\hspace{0.33em}or\hspace{0.33em}1,\hspace{1em}i=1,2,\dots ,n\hspace{0.33em};j=1,2,\dots ,{\alpha }_i;$$

(13d)

Since the Model (13) is a 0–1 integer programming model, Lingo 11.0, Cplex 9.0 and other software can be used to solve it in practical operation.

In summary, the specific steps for solving the optimization scheduling model for post-earthquake rescuers are as follows:

Step 1. Calculate the collaboration degree $R_{ij}^g$ of the rescuer for the rescue task based on Equations (1)–(4).

Step 2. Calculate the professional ability $S{C}_{ij}^g$ of the rescuer to complete the rescue task according to Equations (5)–(7).

Step 3. Calculate the comprehensive performance $C{P}_{ij}^g$ of the rescuer according to Equation (8).

Step 4. Calculate the time satisfaction degree $T{S}_i^l$ of the rescuer from the departure area to the disaster area according to Equations (9)–(11).

Step 5. Calculate the comprehensive matching degree $C{M}_{ij}^{lg}$ between the rescuer and the rescue task according to Equation (12).

Step 6. Construct a scheduling Model (13) and solve it to determine the optimal scheduling scheme.

4. Case Study

In this section, the use of the proposed scheduling method is illustrated by a case study in the background of emergency medical rescue during the earthquake disaster in Ludian County, Yunnan Province, China.

On 3 August 2014, a magnitude 6.5 earthquake disaster struck Ludian County, Zhaotong City, Yunnan Province, China, with over 1300 aftershocks. Tens of thousands of people were affected, thousands of people were injured or killed, and many towns in the county were affected. Among them, the severely affected areas include Longtoushan Town ($D_1$), Lehong Town ($D_2$), and Huodehong Town ($D_3$). After the earthquake, local relevant departments quickly launched the emergency plan and organized personnel to carry out search and rescue work. To actively respond to the call for earthquake relief, the volunteer emergency rescue organization of Yunnan Province dispatched twelve medical personnel around Ludian County to complete rescue tasks in the disaster areas. They come from four departure areas, i.e., Yaoshan Town ($O_1$), Yiche Town ($O_2$), Yongfeng Town ($O_3$), and Chongxi Town ($O_4$). The numbers of medical personnel in $O_1$, $O_2$, $O_3$, and $O_4$ are $a_1=4$, $a_2=3$, $a_3=3$, and $a_4=2$, respectively. These medical personnel face three rescue tasks: (1) first aid at search and rescue sites ($M_1$), i.e., medical staff directly enter the disaster site to provide on-site rescue and emergency treatment for wounds; (2) medical treatment at temporary settlements ($M_2$), i.e., medical staff provide medical treatment for the disaster victims rescued from the scene, including performing various surgical tasks; and (3) health and epidemic prevention ($M_3$), i.e., medical staff carry out environmental disinfection and sterilization, epidemic prevention, food hygiene assurance, health education and other work in the disaster area. Due to the different actual situations in different disaster areas, the rescue tasks required to be completed also vary. There are two rescue tasks that need to be completed in the disaster area $D_1$, i.e., $M_1$ and $M_2$. The numbers of medical personnel required for the rescue tasks ($M_1$ and $M_2$) in the disaster area $D_1$ are $N_1^1=2$ and $N_2^1=1$. Three rescue tasks need to be completed in the disaster area $D_2$, i.e., $M_1$, $M_2$, and $M_3$. The numbers of medical personnel required for the rescue tasks ($M_1$, $M_2$, and $M_3$) in the disaster area $D_2$ are $N_1^2=2$, $N_2^2=2$, and $N_3^2=2$. Two rescue tasks need to be completed in the disaster area $D_3$, i.e., $M_2$ and $M_3$. The numbers of medical personnel required for the rescue tasks ($M_2$ and $M_3$) in the disaster area $D_3$ are $N_2^3=2$ and $N_3^3=1$. The positions of departure areas and disaster areas are shown in Figure 3.

According to the requirements of rescue tasks, earthquake emergency managers developed professional skill evaluation indicators for medical personnel, i.e., nursing rescue capability ($E_1$), emergency rescue experience ($E_2$), level of operation of medical instruments ($E_3$), epidemic prevention knowledge and experience ($E_4$). The indicators of collaborative feature were determined: communication and coordination ability ($C_1$), teamwork spirit ($C_2$), goal consistency ($C_3$), and level of mutual trust ($C_4$). Emergency managers evaluated the performance of medical staff concerning different indicators based on a score of 1 to 5. The higher the score is, the better the performance of medical staff concerning the indicator will be. The professional skill evaluation information and collaborative evaluation information of medical staff are shown in

Table 3. Professional skill evaluation information of medical staff.

$P_{ij}$	$E_1$	$E_2$	$E_3$	$E_4$
$P_{11}$	4	3	4	4
$P_{12}$	4	1	3	2
$P_{13}$	3	4	5	3
$P_{14}$	4	5	5	3
$P_{21}$	2	4	3	4
$P_{22}$	4	4	2	3
$P_{23}$	3	3	4	2
$P_{31}$	5	3	3	5
$P_{32}$	2	1	4	3
$P_{33}$	3	4	5	5
$P_{41}$	4	3	3	4
$P_{42}$	2	5	5	4

and

Table 4. Collaborative evaluation information of medical staff.

	$C_1$
$P_{ij}$	$P_{11}$	$P_{12}$	$P_{13}$	$P_{14}$	$P_{21}$	$P_{22}$	$P_{23}$	$P_{31}$	$P_{32}$	$P_{33}$	$P_{41}$	$P_{42}$
$P_{11}$	-	5	2	4	2	5	3	4	2	3	4	3
$P_{12}$	-	-	4	5	3	2	5	1	2	5	2	4
$P_{13}$	-	-	-	3	4	4	3	5	5	4	5	1
$P_{14}$	-	-	-	-	5	4	2	4	2	3	4	2
$P_{21}$	-	-	-	-	-	3	2	5	3	3	5	4
$P_{22}$	-	-	-	-	-	-	1	4	4	3	2	2
$P_{23}$	-	-	-	-	-	-	-	3	2	5	4	4
$P_{31}$	-	-	-	-	-	-	-	-	4	3	3	4
$P_{32}$	-	-	-	-	-	-	-	-	-	1	3	3
$P_{33}$	-	-	-	-	-	-	-	-	-	-	5	2
$P_{41}$	-	-	-	-	-	-	-	-	-	-	-	1
$P_{42}$	-	-	-	-	-	-	-	-	-	-	-	-
	$C_2$
$P_{ij}$	$P_{11}$	$P_{12}$	$P_{13}$	$P_{14}$	$P_{21}$	$P_{22}$	$P_{23}$	$P_{31}$	$P_{32}$	$P_{33}$	$P_{41}$	$P_{42}$
$P_{11}$	-	4	3	1	3	2	2	4	3	4	3	4
$P_{12}$	-	-	5	3	5	2	4	2	4	3	2	2
$P_{13}$	-	-	-	5	3	2	3	4	5	2	5	5
$P_{14}$	-	-	-	-	3	2	3	4	4	2	4	2
$P_{21}$	-	-	-	-	-	1	4	3	2	5	3	5
$P_{22}$	-	-	-	-	-	-	2	3	1	4	5	3
$P_{23}$	-	-	-	-	-	-	-	5	3	2	3	1
$P_{31}$	-	-	-	-	-	-	-	-	4	4	5	4
$P_{32}$	-	-	-	-	-	-	-	-	-	2	5	3
$P_{33}$	-	-	-	-	-	-	-	-	-	-	2	2
$P_{41}$	-	-	-	-	-	-	-	-	-	-	-	4
$P_{42}$	-	-	-	-	-	-	-	-	-	-	-	-
	$C_3$
$P_{ij}$	$P_{11}$	$P_{12}$	$P_{13}$	$P_{14}$	$P_{21}$	$P_{22}$	$P_{23}$	$P_{31}$	$P_{32}$	$P_{33}$	$P_{41}$	$P_{42}$
$P_{11}$	-	3	3	1	4	3	3	4	4	3	2	4
$P_{12}$	-	-	5	3	4	2	3	4	3	2	5	3
$P_{13}$	-	-	-	4	2	2	2	2	1	2	4	3
$P_{14}$	-	-	-	-	3	2	4	3	2	3	2	1
$P_{21}$	-	-	-	-	-	5	3	5	5	3	3	2
$P_{22}$	-	-	-	-	-	-	2	3	4	4	2	5
$P_{23}$	-	-	-	-	-	-	-	2	2	5	3	2
$P_{31}$	-	-	-	-	-	-	-	-	3	4	2	4
$P_{32}$	-	-	-	-	-	-	-	-	-	5	5	1
$P_{33}$	-	-	-	-	-	-	-	-	-	-	4	4
$P_{41}$	-	-	-	-	-	-	-	-	-	-	-	2
$P_{42}$	-	-	-	-	-	-	-	-	-	-	-	-
	$C_4$
$P_{ij}$	$P_{11}$	$P_{12}$	$P_{13}$	$P_{14}$	$P_{21}$	$P_{22}$	$P_{23}$	$P_{31}$	$P_{32}$	$P_{33}$	$P_{41}$	$P_{42}$
$P_{11}$	-	2	4	3	3	2	1	2	4	3	4	2
$P_{12}$	-	-	4	3	4	3	3	4	2	1	2	4
$P_{13}$	-	-	-	2	5	3	2	3	3	3	5	3
$P_{14}$	-	-	-	-	4	5	4	2	4	4	3	4
$P_{21}$	-	-	-	-	-	3	2	5	2	3	3	3
$P_{22}$	-	-	-	-	-	-	5	3	4	2	1	4
$P_{23}$	-	-	-	-	-	-	-	1	5	4	2	5
$P_{31}$	-	-	-	-	-	-	-	-	5	3	4	1
$P_{32}$	-	-	-	-	-	-	-	-	-	3	4	2
$P_{33}$	-	-	-	-	-	-	-	-	-	-	5	4
$P_{41}$	-	-	-	-	-	-	-	-	-	-	-	3
$P_{42}$	-	-	-	-	-	-	-	-	-	-	-	-

, respectively. The weights of professional skill evaluation indicators for rescue tasks and the weights of collaborative feature indicators for rescue tasks are determined by earthquake emergency managers, as shown in

Table 5. The weights of professional skill evaluation indicators for rescue tasks.

$M_g$	$E_1$	$E_2$	$E_3$	$E_4$
$M_1$	0.3	0.5	0.2	-
$M_2$	0.5	-	0.4	0.1
$M_3$	0.2	-	0.2	0.6

and

Table 6. The weights of collaborative feature indicators for rescue tasks.

$M_g$	$C_1$	$C_2$	$C_3$	$C_4$
$M_1$	0.3	0.2	0.3	0.2
$M_2$	0.2	0.2	0.3	0.3
$M_3$	0.4	0.2	0.2	0.2

, respectively. The time for medical personnel from each departure area to arrive in the disaster area and the requirements for rescue time for each disaster area are shown in

Table 7. Time for medical staff from each departure place to arrive at each disaster area (unit: hour).

$O_i$	$D_1$	$D_2$	$D_3$
$O_1$	[1, 3]	[2, 3]	[3, 4]
$O_2$	[3, 4]	[5, 6]	[2, 3]
$O_3$	[2, 3]	[1, 2]	[1, 2]
$O_4$	[1, 3]	[2, 3]	[2, 3]

and

Table 8. Requirements for rescue time at each disaster area (unit: hour).

$D_l$	$T_l^o$	$T_g^e$	$T_g^f$	$z^l$
$D_1$	1	3	4	0.7
$D_2$	1	3	5	0.6
$D_3$	2	4	5	0.6

.

Based on the method proposed in this paper, the solution process of medical staff scheduling problem in earthquake emergency medical rescue is given below.

Firstly, the collaboration degree $R_{ij}^g$ of the medical staff $P_{ij}$ concerning the rescue task $M_g$ was calculated based on Equations (1)–(3), as shown in

Table 9. The collaboration degree of medical staff.

$P_{ij}$	$M_1$	$M_2$	$M_3$
$P_{11}$	0.5205	0.5045	0.5273
$P_{12}$	0.5705	0.5568	0.5727
$P_{13}$	0.5864	0.5795	0.6091
$P_{14}$	0.5227	0.5227	0.5455
$P_{21}$	0.6182	0.6136	0.6182
$P_{22}$	0.4955	0.4977	0.4955
$P_{23}$	0.4932	0.4932	0.5000
$P_{31}$	0.6091	0.5932	0.6182
$P_{32}$	0.5364	0.5523	0.5273
$P_{33}$	0.5727	0.5682	0.5682
$P_{41}$	0.5909	0.5864	0.6000
$P_{42}$	0.4841	0.4955	0.4818

.

Secondly, the professional ability $S{C}_{ij}^g$ of the medical staff $P_{ij}$ to complete the rescue task $M_g$ was calculated based on Equations (4) and (5), as shown in

Table 10. Professional abilities of medical staff to complete rescue tasks.

$P_{ij}$	$M_1$	$M_2$	$M_3$
$P_{11}$	0.5833	0.6667	0.6667
$P_{12}$	0.2667	0.4667	0.2000
$P_{13}$	0.6750	0.6000	0.4667
$P_{14}$	0.9000	0.7667	0.5333
$P_{21}$	0.4417	0.2000	0.4667
$P_{22}$	0.5750	0.3667	0.3333
$P_{23}$	0.4833	0.4333	0.2000
$P_{31}$	0.6167	0.7333	0.8667
$P_{32}$	0.1333	0.3000	0.3333
$P_{33}$	0.6750	0.6667	0.8667
$P_{41}$	0.5167	0.5333	0.6000
$P_{42}$	0.7000	0.4667	0.6000

.

Then, the comprehensive performance $C{P}_{ij}^g$ of the medical staff $P_{ij}$ was calculated based on Equation (6). Here, it is assumed that the collaboration degree $R_{ij}^g$ and the professional ability $S{C}_{ij}^g$ are equally important in earthquake emergency rescue scheduling, i.e., $\theta =0.5$. The calculation results of the comprehensive performance are shown in

Table 11. Comprehensive performance of medical staff.

$P_{ij}$	$M_1$	$M_2$	$M_3$
$P_{11}$	0.5519	0.5856	0.5970
$P_{12}$	0.4186	0.5117	0.3864
$P_{13}$	0.6307	0.5898	0.5379
$P_{14}$	0.7114	0.6447	0.5394
$P_{21}$	0.5299	0.4068	0.5424
$P_{22}$	0.5352	0.4322	0.4144
$P_{23}$	0.4883	0.4633	0.3500
$P_{31}$	0.6129	0.6633	0.7424
$P_{32}$	0.3348	0.4261	0.4303
$P_{33}$	0.6239	0.6174	0.7174
$P_{41}$	0.5538	0.5598	0.6000
$P_{42}$	0.5920	0.4811	0.5409

.

Further, the time satisfaction degree $T{S}_i^l$ was calculated according to Equations (7)–(9). Here, assume the parameter values in the time satisfaction function are $l_1=2$ and $l_2=0.5$. The calculation results of the time satisfaction degree are shown in

Table 12. Time satisfaction degrees of medical staff from each departure area to each disaster area.

$O_i$	$D_1$	$D_2$	$D_3$
$O_1$	0.8500	0.7000	0.7000
$O_2$	0.3500	0.0000	0.9000
$O_3$	0.7750	0.9000	1.0000
$O_4$	0.8500	0.7000	0.9000

.

According to Equation (10), the comprehensive matching degree $C{M}_{ij}^{lg}$ between medical staff and rescue task was calculated. Here, it is assumed that $\pi =0.5$. And the calculation results of the comprehensive matching degree are shown in

Table 13. Comprehensive matching degrees between medical staff and rescue tasks.

$P_{ij}$	$D_1$		$D_2$			$D_3$
	$M_1$	$M_2$	$M_1$	$M_2$	$M_3$	$M_2$	$M_3$
$P_{11}$	0.6849	0.7055	0.6216	0.6403	0.6464	0.6403	0.6464
$P_{12}$	0.5965	0.6595	0.5413	0.5985	0.5201	0.5985	0.5201
$P_{13}$	0.7322	0.7080	0.6644	0.6425	0.6136	0.6425	0.6136
$P_{14}$	0.7776	0.7403	0.7057	0.6718	0.6145	0.6718	0.6145
$P_{21}$	0.4307	0.3773	0.0000	0.0000	0.0000	0.6051	0.6987
$P_{22}$	0.4328	0.3889	0.0000	0.0000	0.0000	0.6237	0.6107
$P_{23}$	0.4134	0.4027	0.0000	0.0000	0.0000	0.6457	0.5612
$P_{31}$	0.6892	0.7170	0.7427	0.7726	0.8174	0.8144	0.8616
$P_{32}$	0.5094	0.5747	0.5490	0.6193	0.6223	0.6528	0.6560
$P_{33}$	0.6953	0.6917	0.7493	0.7454	0.8035	0.7858	0.8470
$P_{41}$	0.6861	0.6898	0.6226	0.6260	0.6481	0.7098	0.7348
$P_{42}$	0.7094	0.6395	0.6438	0.5803	0.6153	0.6580	0.6977

.

Finally, Lingo software was used to solve the scheduling Model (13), and the obtained medical personnel scheduling results are shown in Figure 4. In Figure 4, the blue, red, and purple shapes represent the rescue tasks to be completed in disaster areas $D_1$, $D_2$, and $D_3$, respectively. And the shapes of the connecting line are the rescue tasks that the medical personnel are dispatched to perform at the corresponding disaster areas, i.e., the optimal scheduling scheme. The final scheduling scheme is shown in Figure 5. Specifically, the medical staff $P_{11}$ from the departure area $O_1$ was dispatched to the disaster area $D_1$ to carry out the rescue task $M_2$; the medical staff $P_{12}$ from the departure area $O_1$ and $P_{32}$ from the departure area $O_3$ were dispatched to the disaster area $D_2$ to carry out the rescue task $M_2$; the medical staff $P_{13}$ and $P_{14}$ from the departure area $O_1$ were dispatched to the disaster area $D_1$ to carry out the rescue task $M_1$; the medical staff $P_{21}$ from the departure area $O_2$ was dispatched to the disaster area $D_3$ to carry out the rescue task $M_3$; the medical staff $P_{22}$ and $P_{23}$ from the departure area $O_2$ were dispatched to the disaster area $D_3$ to carry out the rescue task $M_2$; the medical staff $P_{31}$ and $P_{33}$ from the departure area $O_3$ were dispatched to the disaster area $D_2$ to carry out the rescue task $M_3$; and the medical staff $P_{41}$ and $P_{42}$ from the departure area $O_4$ were dispatched to the disaster area $D_2$ to carry out the rescue task $M_1$.

5. Result Analysis and Comparative Analyses

To verify the feasibility and rationality of the method proposed in this paper, result analysis and comparative analyses were conducted in this section.

5.1. Result Analysis

According to the results of the scheduling model, the scheduling scheme of medical staff can be determined. In the scheme, medical personnel $P_{13}$ and $P_{14}$ were assigned to perform the task $M_1$ in the disaster area $D_1$. From Table 12, it can be seen that $P_{13}$ and $P_{14}$ departed from $O_1$ and arrived at disaster area $D_1$ in a relatively short time, with the highest time satisfaction degree. From Table 11 and Table 13, it can be seen that the comprehensive performance of $P_{13}$ and $P_{14}$ are more in line with the needs of $M_1$, and the comprehensive matching degrees of $P_{13}$ and $P_{14}$ are relatively high, so this plan is reasonable. Then, medical personnel $P_{11}$ was assigned to perform task $M_2$ of disaster area $D_1$. The reason for choosing this plan is high time satisfaction degree, the professional ability of $P_{11}$ meets the needs of $M_2$, and the comprehensive matching degree is high. Next, the medical staff of the departure area $O_4$ were dispatched to perform task $M_1$ in $D_2$. Due to the relatively short time from $O_4$ to $D_2$, high time satisfaction degrees, high collaboration degree of $P_{41}$, and the professional ability of $P_{42}$ is more in line with the needs of $M_1$, so this plan is also reasonable. The reason for assigning medical personnel $P_{12}$ and $P_{32}$ to perform task $M_2$ in disaster area $D_2$ is that $P_{12}$ and $P_{32}$ have high time satisfaction degrees and collaboration degrees. The reason for assigning medical personnel $P_{31}$ and $P_{33}$ to perform task $M_3$ in disaster area $D_2$ is that $P_{31}$ and $P_{33}$ have the highest time satisfaction, high comprehensive performance, and the highest comprehensive matching degrees. The reason for assigning medical personnel $P_{22}$ and $P_{23}$ to perform task $M_2$ in $D_3$ is that high time satisfaction. Further, medical personnel $P_{21}$ was assigned to perform task $M_3$ in disaster area $D_3$ due to high time satisfaction degree and collaboration degree of $P_{21}$. Therefore, this plan meets the actual earthquake emergency needs. To sum up, it can be found that the goal of scheduling results is to maximize the comprehensive matching degree of comprehensive performance and time satisfaction degree. In other words, the scheduling scheme is consistent with the core idea of the proposed method. Therefore, the scheduling scheme of medical staff is consistent with the actual situation and is reasonable.

5.2. Comparative Analyses

To more comprehensively analyze the advantages of the proposed method in this paper, comparative analyses need to be conducted. The method proposed in this paper was compared with the method proposed in the literature [45], because the two methods are used to solve similar problems. The data of this study were applied to the method in the literature [45], and the scheduling results obtained are shown in

Table 14. Scheduling results of medical staff obtained by different methods.

Methods	$D_1$		$D_2$			$D_3$
	$M_1$	$M_2$	$M_1$	$M_2$	$M_3$	$M_2$	$M_3$
The method proposed by Li et al. [45]	$P_{14}$$P_{42}$	$P_{11}$	$P_{13}$$P_{41}$	$P_{12}$$P_{32}$	$P_{31}$$P_{33}$	$P_{22}$$P_{23}$	$P_{21}$
The method proposed by Fei and Wang [36]	$P_{21}$$P_{22}$	$P_{12}$	$P_{14}$$P_{42}$	$P_{11}$$P_{13}$	$P_{32}$$P_{41}$	$P_{23}$$P_{31}$	$P_{33}$
The method proposed in this paper	$P_{13}$$P_{14}$	$P_{11}$	$P_{41}$$P_{42}$	$P_{12}$$P_{32}$	$P_{31}$$P_{33}$	$P_{22}$$P_{23}$	$P_{21}$

.

From Table 14, it can be seen that the scheduling results obtained by the two methods are basically consistent. However, the scheduling results of a few medical staff are different, such as $P_{13}$ and $P_{42}$, and the main reasons for the differences included two aspects. On the one hand, in terms of rescue capabilities, Li et al. [45] only considered the evaluation values of professional skill evaluation indicators of medical staff. This study not only considers the professional abilities of medical staff, but also considers the collaborative performance of medical staff, which can make the scheduling results more comprehensive. On the other hand, in the calculation of the comprehensive matching degree, Li et al. [45] used the geometric mean operator for the calculation, while this paper used the ordered weighted geometric mean operator considering the attitude characteristic of the emergency decision maker in the calculation process, which is more consistent with the facts.

In addition, the optimization method for rescuer assignments provided by Fei and Wang [36] is introduced for comparison. Similarly, based on the background and data of this study, the scheduling results obtained by using the method in the literature [36] are shown in Table 14. It can be found that the scheduling results obtained by the method in the literature [36] differ from those in this paper. The main reason is that the literature [36] only considers the professional abilities of rescuers and does not consider the collaboration between rescuers. Meanwhile, although the literature [36] considers multiple disaster areas, it does not consider the different rescue tasks faced by different disaster areas due to different disaster situations. However, this paper not only considers multiple disaster areas and diverse rescue tasks but also considers the collaboration of rescuers, which is more in line with the reality of earthquake rescue. The results show the effectiveness of the proposed method to a large extent.

6. Conclusions

In this paper, a rescuer scheduling method considering multiple disaster areas and the collaboration of rescuers is proposed for the earthquake emergency rescuer scheduling problem. In the method, the collaborative performance between rescuers, the professional abilities of rescuers to complete rescue tasks, the arrival time of the rescuer from each departure area to the disaster area, and the different needs of rescue time in each disaster area are considered. By solving the constructed scheduling model for earthquake emergency rescue, the optimal scheduling result can be obtained. The contributions of the proposed method are as follows:

(1) This study considers the collaborative performance of rescuers, compensating for the limitations of traditional scheduling methods that only consider the professional abilities of rescuers. It can help to improve the rationality of post-earthquake rescuer scheduling results and have certain guiding significance for the arrangement of earthquake emergency rescue work.

(2) Some key factors that need to be grasped in real earthquake emergency scheduling are concerned in this study, such as multiple departure areas, multiple disaster areas, multiple tasks, the difference in the professional abilities of rescuers, the collaboration of rescuers, the uncertainty of travel time, and the urgency of rescue time. This is more in line with the actual situation of earthquake emergency rescue and can provide decision support for earthquake emergency rescue dispatch. In addition, the method proposed in this paper has a clear logic, a simple solution, and a certain practical application value.

The research outcomes have certain reference value for earthquake emergency management. Meanwhile, the proposed method has strong practical significance for guiding post-earthquake emergency rescue work and improving the effectiveness of rescue scheduling plans. Firstly, dispatching rescuers reasonably based on the needs of different tasks is beneficial for fully utilizing rescue resources and improving rescue effectiveness. Due to the different requirements of time and rescuer for different rescue tasks, the effectiveness of assigning different rescuers to the same disaster area to carry out the same rescue task also varies. Therefore, in the process of personnel deployment, we should strive to make the best use of the talents of the people, make the best use of the personnel, and reasonably dispatch rescuers based on the needs of rescue tasks. Secondly, in earthquake emergency rescue, different rescuers need to collaborate and cooperate with each other to achieve the expected results. In fact, the same rescue task usually requires multiple rescuers, and the requirements for rescuers vary among rescue tasks. Therefore, it is necessary to consider the collaboration and professional ability of rescuers during the dispatch process, so as to improve post-earthquake rescue performance. Thirdly, establish a decision support system for the dispatch of rescuers in the field of earthquake emergency response. In the system, promptly update the qualifications of rescuers, historical rescue records, and road data. The proposed method can be used as a systematic program to obtain a reasonable scheduling scheme, thereby providing support for the scheduling decisions of rescuers.

Although this study has achieved certain results, further research is needed on the issue of the emergency rescue scheduling, such as considering the coordination among professional rescuers and volunteers, dynamic adjustment of rescue tasks and other emergency rescue scheduling issues in complex situations. In terms of practical application of the proposed method, the emergency rescuer information database needs to be established, and the qualifications of rescuers, historical rescue records of rescuers, and road data need to be updated in a timely manner in the information database. Then, the evaluation index system and evaluation model of the rescuers’ ability and coordination degree need to be constructed by literature retrieval, expert questionnaire survey, and other methods, and the optimization model for rescuer scheduling needs to be constructed by considering the comprehensive ability of the rescuer and rescue time. To improve the efficiency of emergency response, the decision support system for earthquake emergency rescuer scheduling needs to be developed for implementation in emergency drills or actual disasters so as to further verify the practicality of the proposed method.