1 Introduction

Tunnel collapse leaves the trapped people facing both physical and psychological challenges and also brings time constraints to the rescue work; trapped people will be physically and mentally injured, or even lose their lives, if the rescue effort is not complete in a timely fashion (Guo et al. 2022). Therefore, tunnel collapse rescue needs to be extremely efficient to save lives. In particular some special rescue resources must be transported to the disaster-affected point in a short period of time to open a life channel to trapped individuals before the subsequent rescue work can be carried out. This necessity means that preallocation of special emergency resources is especially meaningful for avoiding casualties.

The classic location problem of where best to position critical emergency resources requires reference to the P-center problem, the P-median problem, and the covering problem with corresponding models to solve the problem respectively (Boonmee et al. 2017). The coverage model (Toregas et al.1971; Church and ReVell 1974; Berman et al. 2010) can ensure that the configuration point provides services for all disaster-affected points within the coverage radius, so even if the location of the configuration point and disaster-affected point changes, the configuration point can still provide rescue services as long as the changed location is within its coverage area (Mukhopadhyay et al. 2017; Mrkela and Stanimirović 2022). Furthermore, the disaster-affected points of tunnel rescue are fixed for a short period, but also change when a new tunnel is added to the allocation matrix. This fluidity can emerge because tunnel collapse may occur at any tunnel under and after construction. With the coverage model, unless the new tunnel under construction is not covered by special rescue resources or has its own specific resource supply, there is no need for the original location scheme to be modified, effectively avoiding repetitive special resource adjustments. Hence this study used the coverage model to solve the proposed problem.

There is a large number of studies on the location selection of emergency facilities before disasters. Scholars apply the classic location model to the study of practical problems considering various factors and solve the proposed model with an appropriate algorithm (Rauchecker and Schryen 2019; Oksuz and Satoglu 2019; Shaw et al. 2022). Different factors were considered to optimize the location model of emergency facilities, like risk of disasters (Coutinho-Rodrigues et al. 2012; Akgün et al. 2015; Trivedi and Singh 2017), economic cost (Jalali et al. 2016; Karatas 2017; Xu et al. 2017; Zhao et al. 2017), timeliness of emergency material transportation (Bozorgi-Amiri and Khorsi 2016; Vahdani et al. 2018; Wang 2021), distance (Zhao et al. 2017; Ma et al. 2019; Xu et al. 2018), uncertainty of resource demand (Mestre et al. 2015; Bozorgi-Amiri and Khorsi 2016; Hajipour et al. 2022; Ren and Tan 2022), maximum coverage rate (Yushimito et al. 2012; Chanta et al. 2014; Zhao and Chen. 2015; Wang 2021), customer preference (Díaz et al. 2017; Atta et al. 2019; Küçükaydın and Aras 2020), weight of disaster-affected points (Zhang et al. 2013; Kılcı et al. 2015; Yenice and Samanlioglu 2020), and other factors such as victims’ suffering perception (Geng et al. 2021). Normally, the following four factors and their corresponding objectives are considered in the literature entirely or partly: economic cost (cost-effectiveness), timeliness of emergency material transportation (efficiency), distance (efficiency), and maximum coverage rate (efficiency). The consideration of other factors usually depends on specific disaster scenarios. To date, the existing location models with various factors focus more on natural hazard-related disasters, dam inundations, epidemics, and some proposed models for general disaster scenarios; models for tunnel collapse are sparse.

However, unlike these disasters, tunnel collapse has its unique characteristics. We summarize them into the following four points: (1) the number of potential disaster points is generally more than the special configuration points; (2) the weight of a potential disaster-affected point is related to its geological environment, natural conditions, and construction factors; (3) the oxygen content in the cavity formed by a tunnel collapse will decrease with time, which may kill all of the victims before help can arrive; and (4) the rescue might not be carried out at all due to a local lack of special rescue equipment, such as rescue rigs, that the rescue effort requires. The current preallocation of this kind of special resource cannot meet the needs of tunnel collapse management due to two fundamental factors—the high price of the equipment itself, and the need for professional personnel to operate during commissioning. The first and second characteristics require a modified coverage model for tunnel collapse rescue, and the latter two indicate that tunnel collapse rescue has strict time requirements for special emergency resources—untimely arrival of such equipment and supplies is likely to result in the death of all trapped people.

Li et al. (2021) put forward the concept of demand time and applied the demand time service range to a hierarchical scheduling model. The demand time service range refers to the irregular rescue coverage area drawn with the disaster point as the center and the maximum resource transportation time under the condition of demand time minus respond time as the time impedance radius, which is also meaningful for the preallocation model. Combining the demand time service range with the preallocation of special emergency resources can fully meet the different needs of special resource preallocation in real life and effectively ensure an acceptable rescue time, which is of practical significance.

To apply demand time service range for the preallocation of special emergency resources, this study proposed a multi-objective preallocation model of special emergency resources that considers the weight of potential disaster-affected points and other normal factors. Furthermore, it conducted a national case research effort to provide a reliable reference for practical preallocation of rescue rigs. The primary contributions of this study include:

  • Consider the weight of potential disaster-affected points as an additional optimization factor and, taking the demand time as the time constraint, establish a multi-objective preallocation model of special emergency resources for tunnel collapse.

  • Take the rescue rig as the research object and carry out a case study of the major tunnel projects under construction in China, including 15 large tunnels, as potential disaster points and 18 alternative configuration points selected from 24 national emergency resource reserves. This study provides a theoretical reference for the pre-configuration of special rescue equipment like rescue rigs.

The rest of the article is organized as follows. Section 2 describes the multi-objective allocation model of special emergency resources. Section 3 uses a questionnaire survey and literature research to define the rescue demand time, and the response time is explained and assigned. Section 4 presents the solution algorithm, while Sect. 5 uses the preallocation model to make an allocation scheme of the rescue rig equipment that considers all the potential disaster-affected points in China. Finally, Section 6 describes the conclusions and future work.

2 Multi-Objective Allocation Model of Special Emergency Resources for Tunnel Collapse

Suppose there are M potential disaster points and N alternative configuration points of special emergency resource, J = {Ai|i = 1,2,..., m} is the set of potential disaster-affected points, and S = {Sj|j = 1,2,..., n} is the set of alternative configuration points. The basic assumptions of the model are as follows:

  1. (1)

    Because tunnel collapse accidents generally would not occur at the same time at two or more potential disaster-affected points, the resource scheduling process of each potential disaster-affected point does not affect others;

  2. (2)

    The tunnel collapse accident does not affect the road network and road traffic conditions basically. The road network damage factors are not included in the consideration of emergency resource layout and allocation; and

  3. (3)

    Each potential disaster-affected point needs access to at least one configuration point to provide services to ensure full coverage of rescue at potential disaster-affected points.

2.1 Multi-Objective Allocation Model

Considering that the aims of the allocation model are varied, we set three targets for the model: efficiency, multiple override, and cost-effectiveness. Also, the weight of potential disaster-affected points is introduced into the resource allocation model. Assuming that the total weight of all potential disaster-affected points equals 1, based on the collapse risk level of the tunnel under construction, the weight of each potential disaster-affected point is calculated:

$${\gamma }_{\text{i}}\text{ } = \frac{{L}_{\text{i}}}{\sum_{{\text{i}}= \text{1} }^{\text{m}}{L}_{\text{i}}}$$
(1)

where \({\gamma }_{\text{i}}\) is the weight of potential disaster-affected point i, and Li is the collapse risk level of potential disaster-affected point i. A more detailed solution process is shown in Sect. 4.1.

Efficiency requires the minimization of the weighted distance from each potential disaster-affected point to the nearest configuration point (Farahani et al. 2010).

$$\min D = \sum\limits_{{i = 1}}^{m} ( \gamma _{i} \cdot \min (x_{j} x_{{ij}} d_{{ij}} ))$$
(2)

where xj is the decision variable for selecting the alternative configuration points. If the alternative configuration point is selected, xj equals 1, otherwise xj equals 0. xij is the coverage decision variable, if the demand time service range of the configuration point Sj covers the disaster-affected point Ai, xij equals 1, otherwise xij equals 0. dij is the shortest distance from the disaster-affected point Ai to the configuration point Sj.

Multiple override is to maximize the weighted coverage times of the disaster-affected points, that is, considering the risk weight coefficient of the disaster-affected points, so that the disaster-affected points can be provided with emergency rescue services from as many alternative configuration points as possible.

$$\max Z = \sum\limits_{{i = 1}}^{m} ( \gamma _{i} \cdot\sum\limits_{{j = 1}}^{n} {x_{j} } x_{{ij}} )$$
(3)

Cost-effectiveness means the minimum sum of the maintenance costs of the selected configuration points:

$$\min {\text{ }}C = \sum\limits_{{j = 1}}^{n} c x_{j}$$
(4)

where c is the fixed preallocation cost for each configuration point.

The objectives of the model consist of Eqs. 2–4. The maximization of rescue efficiency requires the selection of configuration points closer to the disaster-affected point, the maximization of coverage requires the selection of as many configuration points as possible, and the minimization of rescue cost requires the setting of as few configuration points as possible. There is a strong conflict between these three objectives. We believe that under the condition that all constraints in Eqs. 5–11 are satisfied, there needs to be a trade-off among the three objectives to achieve the system optimization.

2.2 Constraints

The constraints are shown in Eqs. 5–10:

$$d_{{ij}} = v\cdot t_{{ij}} ,{\text{if}}\,t_{{ij}} + a_{0} = X, x_{{ij}} = 1$$
(5)
$$x_{j} x_{{ij}} d_{{ij}} > 0,\forall j,{\text{ }}j = 1,2, \ldots ,n$$
(6)
$$\forall i,{\text{ }}\sum\limits_{{j = 1}}^{n} {x_{j} } x_{{ij}} \ge 1$$
(7)
$$\sum\limits_{{j = 1}}^{n} {cx_{j} } \le B$$
(8)
$$0 < \gamma _{i} < 1$$
(9)
$$x_{j} = 0{\text{ or }}1,{\text{ }}x_{{ij}} = 0{\text{ or }}1$$
(10)
$$S_{j} \in S,{\text{ }}A_{i} \in A$$
(11)

where v is the speed of recuse vehicle, tij is the shortest travel time from the disaster-affected point Ai to the configuration point Sj, a0 is rescue response time, X is the demand time. B is the budget of total preallocation cost. Equation 5 indicates that if the demand time service range of a certain configuration point covers the disaster-affected point, then the corresponding coverage decision variable is equal to 1. Equation 6 indicates that the distance from each potential disaster-affected point to the nearest configuration point is greater than 0, that is, the potential disaster-affected points cannot overlap with the configuration points. Equation 7 is a coverage constraint, ensuring that all disaster-affected points are covered at least once. Equation 8 is a cost constraint, which ensures that each scheme’s cost does not exceed the budget B. Equation 9 indicates the value range of the weight of each potential disaster-affected point. Equation 10 ensures that the variable takes the value 0 or 1. Equation 11 indicates the range of the disaster-affected points and the configuration points.

3 Concepts of Rescue Time

This section introduces three concepts about rescue time: demand time, response time, and demand time service range, as well as quantifies these concepts respectively.

3.1 Determination of Demand Time

From the perspective of the trapped persons, demand time refers to the critical time range in which relevant rescue personnel and vehicles must arrive at the accident site and carry out rescue operations within the specified time point after the collapse (Li et al. 2021). Therefore, demand time should meet the psychological, physiological, and life support needs of trapped persons as much as possible (Sun et al. 2010).

To determine the demand time, we conducted a questionnaire survey and literature research. A total of 92 valid questionnaires of rescuers and victims were collected, the statistical results of the questionnaire show that 41.6% of the rescue participants arrive at the scene of the accident within 2 h after the collapse accident, 91.7% of the rescue participants arrive within 6 h (including 2 h), and the remaining rescue participants need a longer time to arrive at the scene. Of the 92 valid questionnaires, 37 were collected from the trapped people, and the statistical results of the questionnaires show that 78.4% of the trapped people had psychological reactions such as tension, panic, at a loss, blank mind, and slow response within 6 h; 83.8% of the trapped people had extreme hunger, thirst, cold, and other physiological needs within 12 h, and 18.9% had other physical discomforts (cold, cough, fever, and other diseases); without the support of external relief materials, 67.6% of the trapped people believe that a trapped person can survive for 24 h, that is the time period within which the possibility of survival is the greatest.

Combined with the literature, the arrival time of local rescue forces, which include fire brigades, ambulances, rescue teams, and other rescue workers and equipment, of 26 tunnel collapse accidents is statistically analyzed. The survey results show that 26.9% of the local rescue forces arrived within 0–3 hours, 61.5% within 3–6 hours, and 11.6% required more than 6 hours. In the vast majority of tunnel collapse accidents, the local rescue forces have arrived at the site for rescue within 6 hours. The National Comprehensive Disaster Prevention and Reduction Plan (2016–2020) issued by the general office of the State Council (State Council of the people’s Republic of China 2017) requires that the basic life support of the affected persons can be effectively ensured within 12 h when a natural hazard-related disaster occurs.

In a nutshell, it is determined that 6 h, 12 h, and 24 h are the psychological demand time point, physiological demand time point, and life demand time point respectively. Because the rescue rig is a necessary machine to open the life channel when other rescue materials generally have arrived and are waiting for transportation through the life channel within 6 hours and that the required time should meet all demands of the trapped people as far as possible, we take 6 h as the preallocated constraint time of the rescue rig, which means the rescue rig should arrive at the disaster-affected point within 6 h.

3.2 Response Time

Rescue response time a0 refers to the sum of the delay time of information transmission after the collapse and the preparation time before the departure of emergency vehicles, personnel, and materials (Li et al. 2021). The rescue response time of 26 tunnel collapse accidents was investigated, and the results show that a accident rescue response time within 2 h accounted for 88.2% of recorded incidents. Combined with the actual collapse accident investigation, considering different types of rescue vehicles, and referring to the emergency medical rescue time (Liu et al. 2021; David and Luc 2022), the response time of rescue vehicles, heavy-duty and out of gauge vehicles, and ordinary vehicles are determined to be 0.5 h, 1.5 h, and 2 h respectively. The rescue rig is disassembled first and then transported by heavy-duty vehicles, so the response time is 1.5 h.

3.3 Method for Determining Demand Time Service Range

In this study, the irregular rescue coverage area is drawn with the disaster-affected point as the center and the maximum resource transportation time under the condition of demand time minus respond time as the time impedance radius, which is called the demand time service range. According to the characteristics of highway tunnel collapse accidents and relevant highway network data, we use the ArcGIS software to draw the demand time service range, with a powerful network analysis ability that can realize the functions of path analysis, service area analysis, and nearest facility point analysis.

Taking the ArcGIS 10.4 software as the platform and vector road network map as the data basis, we selected the appropriate geographic coordinate system and projection, established a new geographic database, created a new feature dataset, and imported the vector road network in ESRI Shapefile (shp) format. After topological processing, we verified and modified topology errors, and finally created the road network dataset.

4 Solve Method

This section designs the solution algorithm, which includes four parts: tunnel collapse risk assessment, potential disaster-affected points weight solution: shortest distance solution; coverage decision variable solution; and the optimal special emergency resource preallocation solution by non-dominated sorting genetic algorithm II (NSGA-II) and the technique for order of preference by similarity to ideal solution (TOPSIS).

4.1 Tunnel Risk Assessment

The possibility and risk degree of a collapse accident to a tunnel project under construction is called collapse accident risk. According to the Analysis of Safety Risk Assessment System and Guide for Highway Bridge and Tunnel Construction (Ministry of Transport of the People’s Republic of China, Engineering Quality Supervision Bureau 2011) and the Code for Investigation of Geotechnical Engineering (GB 50021–2001) (Ministry of Construction of the People’s Republic of China 2009), the collapse risk assessment of tunnels under construction is carried out from three aspects: geological environment, natural conditions, and construction factors, and a total of nine risk factors are selected. Based on the literature (Kim et al. 2021; Xu et al. 2021; Meng et al. 2022), all evaluation indicators are introduced into the same scoring standard system, the scores of different indicators are specified as 0–4, and each risk evaluation indicator of tunnel under construction is scored. Our collapse risk evaluation indicators and scoring standards of tunnels under construction (Ou et al. 2021; Li et al. 2017) are shown in Table 1.

Table 1 Risk evaluation indicators and scoring standards of tunnels under construction
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To demonstrate the importance of the evaluation indicators, the construction status of the tunnel under construction is first scored according to Table 1, and the initial score of each indicator is obtained. Then the engineering weightFootnote 1 of the tunnel under construction for the index ω(0)i is calculated according to the index weight ωi given in Table 1. The calculation formula is as follows:

$$\omega (0)_{i} = \frac{\omega _{i} \cdot s(0)_{i} }{\sum\nolimits_{{k = 1}}^{n} {\omega _{i} } \cdot s(0)_{k} }$$
(12)

where ω(0)i is the engineering weight of item i; ωi is the weights of item i; s(0)i is the initial evaluation score of item i; i is the serial number of the evaluation item; k is the serial number of the evaluation indicator (k = 1,2,…,9).

The sum of the weighted mean value of each evaluation indicator score was used as the calculated score S:

$$S = \sum \omega (0)_{i} \cdot s(0)_{i}$$
(13)

Finally, we determine the risk level of collapse according to the interval of S based on the risk level standard shown in Table 2 (Tang et al. 2018).

Table 2 Standard for collapse risk grade
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4.2 Solution Algorithm

  1. (1)

    Solve γi: The risk assessment of the tunnel under construction is carried out according to nine evaluation indicators, and then the collapse risk level of the potential disaster-affected point can be obtained. According to the risk level of potential disaster points, take 1 as the sum of weights, and calculate the weight coefficient of each potential disaster point γi.

  2. (2)

    Calculate dij: Using the ArcGIS software as the platform, the road length data are directly obtained from the road attributes of the vector map, then use the shortest path tool in the network analysis window to solve the shortest distance dij between any potential disaster point and the alternative configuration point.

  3. (3)

    Determine xij: Calculate minimum distance travel time tij with Eq. 5 and dij, if the sum of tij and response time a0 ≤ X, then the demand time service range of the configuration point Sj covers the disaster-affected point Ai, that is, xij equals 1; if the sum of tij and response time a0 > X, then it is considered that the disaster-affected point Ai is beyond rescue jurisdiction of Sj, that is, xij equals 0.

  4. (4)

    Solve decision variable xi by combining the NSGA-II and TOPSIS: The special emergency resource preallocation model built in this study is a multi-objective linear integer planning problem. Multi-objective evolutionary algorithms (MOEAs) like NSGA-II were frequently used as the solving algorithm in the literature, as they are able to generate population of solutions and provide the whole set of Pareto optimal solutions in single run MOEA applications to different multi-objective facility location problems (Lücken et al. 2014; Falcón-Cardona et al. 2020). The solution procedure of the NSGA-II algorithm is shown in Fig. 1.

Fig. 1
figure 1

Flowchart of solving NSGA-II algorithm

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On the basis of the Pareto frontier of multi-objective optimization that is obtained by the NSGA-II algorithm, the program uses the technique for order preference by similarity to an ideal solution (TOPSIS) (Tang et al. 2018) to select and output a set of optimal solutions.

Φ1′, Φ1″, f1(Φ1′), and f1(Φ1″); Φ2′, Φ2″, f2(Φ2′), and f2(Φ2″); and Φ3′, Φ3″, f3(Φ3′), and f3(Φ3″) are the optimal solution, the worst solution, the optimal value, and the worst value of smallest weighted distance; largest weighted coverage times; and minimum total preallocation cost respectively. The relative distance Ra, ra from the positive, negative ideal points, and non-inferior scheme Φa is calculated according to the TOPSIS, the calculation equations are Eqs. 13 and 14.

$$R_{a} = \sqrt {\beta _{1} \left( {\frac{{f_{1} \left( {\Phi _{a} } \right) - f_{1} \left( {\Phi ^{\prime}_{1} } \right)}}{{f_{1} \left( {\Phi ^{\prime}_{1} } \right)}}} \right)^{2} + \beta _{2} \left( {\frac{{f_{2} \left( {\Phi _{a} } \right) - f_{2} \left( {\Phi ^{\prime}_{2} } \right)}}{{f_{2} \left( {\Phi ^{\prime}_{2} } \right)}}} \right)^{2} + \beta _{3} \left( {\frac{{f_{3} \left( {\Phi _{a} } \right) - f_{3} \left( {\Phi ^{\prime}_{3} } \right)}}{{f_{3} \left( {\Phi ^{\prime}_{3} } \right)}}} \right)^{2} }$$
(14)
$$r_{a} = \sqrt {\beta _{1} \left( {\frac{{f_{1} \left( {\Phi _{a} } \right) - f_{1} \left( {\Phi ^{\prime\prime}_{1} } \right)}}{{f_{1} \left( {\Phi ^{\prime\prime}_{1} } \right)}}} \right)^{2} + \beta _{2} \left( {\frac{{f_{2} \left( {\Phi _{a} } \right) - f_{2} \left( {\Phi ^{\prime\prime}_{2} } \right)}}{{f_{2} \left( {\Phi ^{\prime\prime}_{2} } \right)}}} \right)^{2} + \beta _{3} \left( {\frac{{f_{3} \left( {\Phi _{a} } \right) - f_{3} \left( {\Phi ^{\prime\prime}_{3} } \right)}}{{f_{3} \left( {\Phi ^{\prime\prime}_{3} } \right)}}} \right)^{2} }$$
(15)

where β1, β2, β3 are the relevant weights of associated weighted distance, weighted coverage times, and total maintenance cost respectively, β1 + β2 + β3 = 1.

Non-inferior scheme Φa relative proximity to the ideal point εa is:

$$\varepsilon _{a} = \frac{{r_{a} }}{{R_{a} + r_{a} }}$$
(16)

where εa is a non-inferior scheme Φa relative proximity to the ideal point.

From the above formula, the value range of εa is [0,1], according to the size of εa we sort the schemes in the Pareto optimal solution set. The greater the value of relative proximity, the better the scheme, and so we can obtain a set of optimal solutions. When the relative proximity to the ideal solution ε of the optimal scheme is smaller than 0.5, we will re-select the optimal scheme and repeat the calculation until ε > 0.5. To sum up, the Pareto frontier of multi-objective optimization is obtained by applying NSGA-II. On this basis, we applied TOPSIS to select and output the optimal scheme.

5 National Preallocation of Rescue Rigs

For the collapse accident of highway tunnel under construction, we conducted the location and configuration analysis of special emergency resources for potential disaster-affected points. Based on the data of the national highway vector network and the ArcGIS software, we selected GCS_WGS_1984 as the geographic coordinate system, and WGS_1984_Web_Mercator_Auxiliary_Sphere as the projected coordinate system, created a geographic database and feature dataset, and imported a vector map shp file.

5.1 Study Area and Initial Configuration Point Selection

Considering the construction scale and regional impact, the current tunnel projects under construction nationwide are investigated, and a total of 15 large tunnels under construction (are determined. The specific conditions of the selected tunnels under construction are shown in Table 3, and the layout of the tunnels under construction is shown in Fig. 2.

Table 3 Specific conditions of selected highway tunnels under construction
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Fig. 2
figure 2

Layout of selected tunnels under construction. See Table 3 for the name and other information of the 15 tunnels.

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5.2 Preparation before Configuration

To prove the feasibility and superiority of the proposed model, we set up two cases. One is preallocated according to the above method, and the other is not preallocated and calculated according to the original reserve location of rescue rigs shown in Table 4 and Fig. 3a. The 12 rescue rigs are reserved in the big construction companies. Due to the high capital cost, low utilization rate, and time-consuming and laborious work required for the new resource preallocation points, the site selection and preallocation are considered in combination with the existing 24 national emergency resource reserves. These 24 points need to be screened according to two principles (Lu and Hou 2010; Guan et al. 2021): (1) we will select one or two closed reserves according to the number of tunnels under construction around; and (2) the reserve closer to the tunnels under construction or has big construction companies is with priority. After preliminary screening, 18 emergency reserves are selected as ideal alternative points for emergency resource preallocation for tunnel collapse, assuming that each point has a rescue rig. Table 5 shows the alternative configuration points of special emergency resource after the screening, and Fig. 3b displays the layout of the alternative points.

Table 4 Original points of rig reserve
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Fig. 3
figure 3

Layout of the original rescue rig reserve points (a); and layout of alternative configuration points (b). See Tables 4 and 5 for site names.

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Table 5 Alternative configuration points of rigs
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5.3 Preallocation of Rescue Rigs Nationwide

The model solution is divided into four parts, including tunnel collapse risk assessment, potential disaster-affected points weight solution, shortest distance solution, and coverage decision variable solution and the optimal special emergency resource preallocation solution by NSGA-II and TOPSIS. According to Table 1, the comprehensive risk assessment of each tunnel under construction is carried out with the help of news reports, online material, and the relevant literature. After initial scoring, engineering standard weight calculation, and total score evaluation, the collapse risk grade of each tunnel under construction is obtained. The weight of potential disaster points is determined according to Eq. 1 and tunnel risk level. The greater the weight, the higher the risk level. The weight γi of each potential disaster-affected point Ai is shown in Table 6.

Table 6 Weight of potential disaster-affected points
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By using the new origin-destination (OD) cost matrix in the network analysis toolbox of the ArcGIS software, we solve both the shortest distance dij from each alternative point and each original rescue rig reserve point to the potential disaster-affected point. Figure 4 shows the solution process of the generated OD cost matrix (the starting point position is shown in the form of a straight line, but not the straight-line distance), and Tables 7 and 8 show the sorted shortest distance results respectively.

Fig. 4
figure 4

Origin-destination (OD) solution process between original reserve points and potential disaster-affected points (a); and OD solution process between alternative points and potential disaster-affected points (b)

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Table 7 The shortest distance from original reserve points to potential disaster-affected points
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Table 8 The shortest distance from alternative points to potential disaster-affected points
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Next, we solve the coverage decision variables xij both from the alternative points and the original rescue rig reserve points to the potential disaster-affected points. When the alternative configuration point covers the potential disaster-affected point (that is, the alternative configuration point can provide rescue services for the potential disaster-affected point), the decision variable xij = 1; When the alternative configuration point does not cover the potential disaster-affected point, the decision variable xij = 0. After that, we gain the weighted distance and weighted coverage times of the not preallocated case through calculation with the potential disaster-affected point weight and the shortest distance from original reserve points to potential disaster-affected points.

For the preallocated case, take the potential disaster-affected point weight, the shortest distance from alternative points to potential disaster-affected points, and the coverage decision variables as the initial data, and use the MATLAB software to program and calculate on the servers of Intel i7, CPU 2.3 GHz and ram 8 GB. Set the population number of the algorithm to 200, the number of iterations to 100, the variation rate to 0.1, and the crossover rate to 0.8, the demand time as 6 hours, and the budget B as RMB 600,000 yuan. The optimal special emergency resource preallocation scheme is obtained, and the 6 hours demand time service range coverage of the two cases is shown in Fig. 5 respectively. It is not hard to see that the original reserve points are unreasonable and cannot cover all the potential disaster-affected points within 6 hours, while the optimal preallocation scheme is just the opposite.

Fig. 5
figure 5

The 6 hours demand time service range coverage of not preallocated cases (a); and the 6 hour demand time service range coverage of preallocated cases (b)

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Table 9 shows the comparison of the calculation results of the two cases. The explanation for the immense numeric value of weighted distance in the not preallocated case is that when a disaster-affected point does not become covered in any demand time service range of reserve points, the shortest distance of the disaster-affected point is infinity. But in order for the program to be able to calculate normally instead of outputting an infinite value, we set the shortest distance of such a disaster-affected point to 100,000 km. In the not preallocated case, there are four points uncovered among the 15 potential disaster-affected points, and this leads to an immense numeric value of weighted distance. Comparing the results of the two cases, when the rescue rigs are preallocated by the proposed model, the number of rescue rigs needed is reduced by 8.3%, which means a significant savings in the cost of purchasing equipment and preallocation, considering that the equipment itself is expensive. The number of covered potential disaster-affected points is increased by 36.36%, which means that more potential disaster-affected points can be rescued timely and more people can be protected from dying without life support. The weighted coverage times are increased from 0.853 to 1.383, which means that all of the potential disaster-affected points are covered at least once and indicates a substantial increase in equality. Finally, the weighted distance is significantly reduced by 99%, which means that the preallocation model can improve the rescue efficiency markedly. Meanwhile, the proposed model can give the corresponding optimal pre-configuration scheme according to the different demand times, and it provides a theoretical reference for the preallocation of special rescue equipment.

Table 9 Calculation results of the two cases under 6 hours demand time
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6 Conclusion

This study took the rescue rigs as the research object and established a multi-objective preallocation model of special emergency resources. The unique nature of tunnel collapse decides that rescue work cannot be carried out without some special huge heavy machines, such as rescue rigs existing only in small number, which indicates the practical significance of reasonable preallocation of such special emergency resource to improve the rescue efficiency. The demand time service range is considered as one of the constraints in the model, which ensures that the transportation time of special emergency resources is sufficiently limited. No matter how the optimization objectives and their weights in the model are adjusted, the special emergency resource will arrive at an accident site and carry out rescue work within an acceptable time. The additional factor—the weight of potential disaster-affected points included in the pre-configured model—can fully make use of the special characteristics of tunnel collapse and combine the distribution of disaster sites or service areas for location optimization.

Further, a case study of preallocation of main tunnel projects under construction in China is presented with two cases of whether to preallocate rescue rigs. These uncovered potential disaster-affected points in the not preallocated case can be rescued in time after preallocation, and the optimal scheme provided by the proposed model can cover all the potential disaster-affected points at least once, which means that equality was greatly improved. Meanwhile, compared with the not preallocated case, the number of rescue rigs needed is reduced and the weighted distance significantly declines, which means that the preallocation model can improve the rescue efficiency markedly and save the configuration cost to some degree.

This research can be improved from the following aspects. Conducting an additional questionnaire survey that includes the survivors and rescuers of the four levels of the tunnel collapse accidents would further refine the demand time. Moreover, the real highway and traffic conditions needed to take into consideration the cost function of the national road network, so the cost can vary realistically. When more complex factors are taken into account, the optimal scheme selected by TOPSIS will also change. Comparing the rescue efficiency and rescue cost under the two cases in the case study on these bases can further verify the effectiveness of the special emergency resource preallocation model. It can provide a more reliable basis for the implementation and application of a special emergency resource preallocation scheme.