Emergency Material Scheduling Optimization Method Using Multi-Disaster Point Distribution Approach

Abstract

The outbreak of multiple disaster sites during the coronavirus disease 2019 (COVID-19) pandemic has presented challenges due to varying access time intensity, population density, and medical resources at each site. To address these issues, this study focuses on 13 districts and counties in Wuhan, China. The importance of each research area is analyzed using the improved PageRank and TOPSIS algorithms to determine the optimal site selection plan. Additionally, a particle swarm algorithm is used to construct an emergency material dispatching model that targets both distribution and site selection costs to solve the multi-distribution center dispatching problem. The results suggest that constructing 10 distribution centers can satisfy the demand for epidemic prevention and control in Wuhan city while saving costs associated with site selection and material distribution. Compared to the previous optimal solution, the distribution and site selection costs under the optimal solution decreased by 27.9% and 17.82%, respectively. This approach can serve as a basis for dispatching emergency materials during public health emergencies.

1. Introduction

With the rapid increase in urban populations and the development of society, various major public emergencies have continued to occur worldwide. Public events are now a major concern globally, and the emergency material security system is facing significant challenges. A failure to supply emergency materials promptly can lead to the rapid spread of events, posing a severe threat to public life, safety, and social and economic order. COVID-19 put significant pressure on healthcare systems worldwide in 2019 due to its rapid spread, the wide range of infections it can cause, and the difficulties in its prevention and control [1]. The late start of research in emergency management in China has led to the technology in the field of emergency material dispatching being relatively weak, which has resulted in the phenomena of insufficient reserves and untimely dispatching of emergency materials in response to public health emergencies [2]. Although the epidemic situation in China has gradually stabilized since September 2020, the continuous mutation of the novel coronavirus has increased its transmission capacity, leading to a multi-disaster outbreak and localized gathering in the region. The difficulty of tracing the source of the epidemic has also increased gradually. Under this new situation, delivering emergency supplies to a disaster area in the shortest possible time is a top priority of precise epidemic prevention work. This requires considering different situations, such as the intensity of access time, population density, and medical scale of each disaster site. Currently, numerous experts and scholars have conducted research on the optimization of material dispatching, leading to the establishment of a more mature system model. Zhang et al. [3] utilized the entropy weight method to determine the material tension of demand points and established a multi-objective emergency medical material dispatching model that considered different preconditions for the urgency of material demand points. To address the problem of protective supply shortages for different demand subjects, Song et al. [4] developed a reservoir site allocation model that minimized the weighted sum of unsatisfied demand at hospital material distribution points and maximized the fairness of distribution at material purchase concentration points. Hu et al. [5] factored in hospital priority level and degree of material satisfaction and adjusted the distribution scheme in real time based on changes in road condition information. Ni et al. [6] considered the disaster situation, geographic location, and material demand in the disaster area and constructed a clustering center of gravity method for the emergency logistics distribution center site selection model. Zhou et al. [7] introduced compromise fuzzy decision-making to optimize the overall site selection scheme with the objectives of minimizing transportation costs and maximizing their coverage of the population and medical resources. Li et al. [8] established an index evaluation system that utilized population density, hospital capacity, maximum travel time, and maximum travel distance as evaluation indexes for the location of a square cabin hospital. Wang et al. [9] constructed a nonlinear fractional distribution model that combined distribution center distributions and path information from the perspective of transportation time, cost, and reliability. They solved the model using NSGA-II and a differential evolutionary algorithm. Chang et al. [10] dynamically adjusted the distribution scheme of each supply point based on changes in demand at demand points to minimize the demand for resources and delivery time. They designed a multi-objective genetic algorithm based on a greedy algorithm to regulate the available resource allocation and generate feasible emergency coordination schedules for decision makers.

The early studies on emergency material dispatch structures focused mainly on two methods: dispatching emergency materials from one supply point to multiple demand points or from multiple supply points to one demand point. For instance, Knott [11] investigated the problem of transporting bulk food from one distribution center to multiple demand points, while Das [12] examined the transportation of emergency supplies from a single outlet point to multiple demand points. Similarly, Xue et al. [13] investigated the dispatching of emergency materials from one material distribution center to multiple affected points. However, due to the vast scale of disasters, the demand for emergency supplies significantly increases, and the number of supplies in a material distribution center or at a single relief point can hardly meet the needs of multiple disaster sites simultaneously. To address this issue, Wang et al. [14] established a dispatching model from multiple supply points to multiple demand points, considering the satisfaction level of the demand points in the initial stage after a disaster. Likewise, Liberatore et al. [15,16] studied emergency material dispatching methods based on the transfer of goods from multiple supply points to multiple demand points. Furthermore, Zhang et al. [17] proposed a scenario tree based on conditional probability to define the correlation between major and secondary disasters and constructed a multi-objective three-stage stochastic planning model that minimizes transportation time, costs, and unmet demands. Yang et al. [18], in the context of the Internet of Things, developed various homeland emergency material dispatching models that consider comprehensive factors, such as the demand of disaster areas, social reserve, road conditions, transportation mode, loading limits, the satisfaction rate of the disaster area, and road carrying capacity, with multiple rescue points and increased disaster areas. The objective of the model was to find the shortest rescue time. Regarding the stages of emergency material dispatching, previous studies did not change the emergency material supply hierarchy in different stages but rather focused on changes in supply and demand [19] or dispatching tasks [20,21].

The majority of the previous studies focused on the material dispatching model in the initial stage of an outbreak without considering the impact of urban network topology on the epidemic’s spread or the problem of material dispatch and distribution in the case of multiple disaster site outbreaks under normalized epidemic prevention and control. To address these gaps, we utilized the improved PageRank algorithm and the improved TOPSIS algorithm to analyze the importance of the target area to be selected based on the topology of the emergency site network. Next, we developed a fusion siting optimization model that considered access time intensity, population density, and medical scale to determine the optimal location. Based on the site selection results, we constructed a multi-distribution center path optimization model that minimizes site selection costs and distribution costs. This model provides a reasonable material dispatching plan for each disaster site, assuming sufficient material resources.

2. Site Selection Area Node Importance Algorithm

2.1. TOPSIS Algorithm and Its Improved Algorithm

The TOPSIS algorithm, also known as the approximate ideal solution ranking method, is commonly used in the fields of node importance ranking and comprehensive economic efficiency evaluation [22]. It can make full use of the information from the original data, and its results can accurately reflect the gap between the evaluation schemes. In the site-selected regional nodes, due to the differences in population density, medical scale, and staffing in each region, the demand for emergency supplies in each region is not absolutely positively correlated, and there may be large differences in the number of patients in regional nodes with little difference in medical scale, so it is necessary to fully quantitatively consider the degree of influence of each index on the importance of the nodes. The entropy weighting method can be assigned according to the degree of difference of each indicator sign value to derive the corresponding weight of each indicator, and the indicators with large relative changes have larger weights. Combining the characteristics of TOPSIS and the entropy method, this paper uses the entropy method to improve the traditional TOPSIS algorithm by assigning weights to each indicator.

The algorithm’s steps are listed below.

Step 1: To eliminate the effect of different indicator magnitudes, the set $T$ of each indicator is normalized to obtain the set $T_i^{*}$ of each indicator after normalization. The calculation formula of each index of the $i$ regional node is:

$$t_{ij}^{*}=\frac{t_{ij}}{\sqrt{{{\displaystyle \sum }}_{i=1}^n{t}_{ij}^2}} \left(j=1,2,\cdots ,m\right)$$

(1)

Step 2: Based on the set of indicators processed by the standardized formula, the positive ideal solution is as follows:

$$T^{+}=\left(t_1^{+},t_2^{+},\cdots ,t_j^{+},\cdots ,t_m^{+}\right)=max\left\{t_{ij}\right\} \left(1\le j\le m,j=1,2,\cdots ,m\right)$$

(2)

Similarly, the negative ideal solution is as follows:

$$T^{-}=\left(t_1^{-},t_2^{-},\cdots ,t_j^{-},\cdots ,t_m^{-}\right)=max\left\{t_{ij}\right\} (1\le j\le m, j=1,2,\cdots ,m)$$

(3)

Step 3: The entropy weight method is used to assign weights to the indicators involved in site selection. The entropy value of each indicator is calculated by the following formula:

$$K_j=-\frac{{\displaystyle \sum _{i=1}^m{t}_{ij}\cdot \ln \left(t_{ij}^{*}\right)}}{\ln m} (j=1,2,\cdots ,m)$$

(4)

The formula for calculating the weights of each site selection indicator is as follows:

$$Z_j=\frac{1-H_j}{m-{\displaystyle \sum _{j=1}^m{H}_j}} (j=1, 2, \cdots ,m)$$

(5)

where $Q=\left(q_1, q_2,\cdots , q_i,\cdots ,q_m\right)$ is the set of each regional node; $T=\left(T_1,T_2,\cdots ,T_j,\cdots ,T_m\right)$ is the set of each indicator; $T_i=\left(t_{i1},t_{i2},\cdots ,t_{ij},t_{in}\right)$ is the set of indicators of node $q_i$; $T^{*}=\left(T_1^{*},T_2^{*},\cdots ,T_i^{*},\cdots ,T_m^{*}\right)$ is the set of indicators after normalization; and $T_i^{*}=\left(t_{i1}^{*},t_{i2}^{*},\cdots ,t_{ij}^{*},\cdots ,t_{in}^{*}\right)$ is the set of each indicator of regional node $i$ after normalization.

Step 4: The formula for calculating the distance of each solution from the positive ideal solution and the negative ideal solution is given below.

Define the distance of the evaluation object $i$ ($i$ = 1, 2,$\cdots$, n) from the maximum value $D_i^{+}=\sqrt{{\displaystyle \sum _{j=1}^m{Z}_j{\left(t_j^{+}-t_{ij}^{*}\right)}^2}}$.

Define the distance of the evaluation object $i$ ($i$ = 1, 2,$\cdots$, n) from the minimum value $D_i^{-}=\sqrt{{\displaystyle \sum _{j=1}^m{Z}_j{\left(t_j^{-}-t_{ij}^{*}\right)}^2}}$.

Calculate the combined sensitivity value. The $S_i$ value is less than 1. The closer the $S_i$ value is to 1, the stronger the sensitivity is:

$$S_i=\frac{D_i^{-}}{D_i^{+}+D_i^{-}} (i=1,2,\cdots ,n)$$

(6)

2.2. PageRank Algorithm and Its Improved Algorithm

The PageRank algorithm is the classical node importance analysis method [23]. In this study, the characteristics of the PageRank algorithm were used to model the regional nodes in the site network as web pages and the connectivity lines between regional nodes as links. The regional node with more connected lines has a higher access degree; the longer the length of the connecting lines between neighboring regional nodes, i.e., the greater the proportion of the access time weight, the better the conditions for fast material transportation, and the higher the access degree. Since the vehicle transportation of emergency supplies is continuous on roads, the regional nodes connected to the regional nodes with higher access degrees also have higher access degrees. Therefore, the PR value of the PageRank algorithm is an important indicator suitable for the importance of regional nodes in the emergency material site network.

The algorithm’s steps are listed below.

Step 1: Initialize the PR value of each region node, denoted as $T_1=\left(P{R}_{11},P{R}_{21},\cdots ,P{R}_{i1},\cdots ,P{R}_{m1}\right)$, where $P{R}_{i1}$ is the initial PR value of the region node $i$.

Step 2: Using the topology of the network, the PR value distribution of each regional node after the $k$-th iteration is calculated as $T_k=\left(P{R}_{1k},P{R}_{2k},\cdots ,P{R}_{ik},\cdots ,P{R}_{\mathrm{mk}}\right)$, and the PR value is calculated as follows:

$$P{R}_{ik}=\frac{1-q}{m}+q{\displaystyle \sum _{j\in N_i}\frac{P{R}_j^k}{d_i}}$$

(7)

Since the traditional PageRank algorithm is for unweighted networks, in the emergency distribution center site node network, the PR value calculation formula is improved using the characteristics of weighted networks. The PR value calculation formula of the weighted regional node $i$ is:

$$P{R}_i=\frac{1-q}{m}+q{\displaystyle \sum _{j\in N_i}\frac{P{R}_j}{T_{ij}{d}_i}}$$

(8)

where q is the resistance coefficient, taken as 0.85, until the PR value distribution of each regional node is stabilized.

2.3. Importance Fusion Algorithm for Siting Regional Nodes

Although the traditional TOPSIS algorithm considers multi-feature factors, it does not consider the differences in the influence of different feature objects on the study objectives. Setting the weights of different features quantitatively helps to improve the algorithm’s performance. The fusion of two algorithms applicable to networks with different topological characteristics is an important innovation in emergency logistics siting algorithms.

Access time intensity, population density, and medical scale are selected as indicators affecting the ranking of the importance of regional nodes in the emergency site network. The PageRank algorithm itself focuses on the network node connection structure for network node importance analysis and is suitable for analyzing the physicality relationship of regional node connections in the emergency site network. The TOPSIS algorithm is a method for ranking according to the proximity of limited evaluation objects and idealized targets. It is suitable for multi-objective decision analysis, including the social relationship analysis of the population density and medical scale of each regional node in the emergency site network. A regional node importance fusion algorithm based on PageRank-TOPSIS is proposed by combining the characteristics of the emergency site network and the features of the two algorithms. Then, the weighted regression coefficients $m,n$ are set to linearly fuse the regional node importance PR values of the PageRank algorithm and the regional node importance $s$ of the TOPSIS algorithm.

The PR-TOPSIS algorithm’s steps are as below.

Step 1: Based on the topology of the emergency site network, conduct field trips to count the access time intensity attribute values of each regional node, as well as the population density intensity and medical scale intensity attribute values of each regional node.

Step 2: Macroscopically, an analysis of node importance is performed according to the improved TOPSIS algorithm based on the population density intensity and medical scale intensity attribute values of the site selection area nodes. Then, a comprehensive sensitivity analysis is performed for each regional node.

Step 3: Microscopically, a node importance analysis is performed according to the improved PageRank algorithm on the access time intensity attributes of each regional node to obtain the PR value of each regional node.

Step 4: The PR value and the result of the sensitivity analysis $s$ of each regional node are processed by linear regression to obtain the improved regional node importance G. The improved node importance $G=ns+mPR$ of the node $i$. Here, $s$ and PR are normalized values, and $n+m=1$, where $n$ and $m$ are incomplete quantitative parameters and can study the influence of different weighted regression coefficients on a regional node’s importance.

3. Distribution Model Construction

3.1. Problem Description

After a sudden health event, there are multiple demand and supply points in any road network, and multiple emergency distribution centers organize and deploy vehicles to find distribution paths in multiple single-loop distribution paths. Most of the existing studies discuss the single distribution center problem; however, this is significantly different from the actual situation. Therefore, this study studies a dispatching method with multiple distribution centers, as shown in Figure 1. In this study, we specifically analyzed a typical case of Wuhan to ensure that the distribution vehicles are sufficient to meet the demands of each demand point.

3.2. Model Construction

Due to the irregularity, suddenness, and outbreak contagiousness of epidemic events, the dispatch of medical emergency supplies has greater requirements for the timeliness and accuracy of material transportation. To meet the requirements of this problem, the following assumptions are made for this problem.

• We assume that the locations of the material distribution center and emergency material demand points are known;
• We assume that all emergency vehicles are of the same type;
• We assume that the demand point is satisfied by only one vehicle;
• We assume that the vehicles that have completed the distribution of emergency supplies need to return to the distribution center;
• We assume that the demand at all demand points is less than the maximum load of the vehicle.

In summary, the present dispatch optimization problem can be described by $C_0$ and $C_1$ to indicate the fixed cost of vehicle start-up and the driving cost per unit distance of vehicle operation. Assuming that the number of distribution centers is $M$ and $N$ is the total hub point, the number of locations to be distributed is $N-M$, and the set of point nodes is $n=\left\{1, 2, \cdots , M, M+1, M+2, \cdots , N\right\}$. $q_i\left(i\in \left\{1, 2, 3, L, N\right\}\right)$ to indicate the demand quantity of each required distribution location. $Q$ is the maximum emergency material distribution quantity of each distribution vehicle. The number of distribution vehicles in the distribution center is $K$. $m$ indicates the set of vehicles $\left(m=1, 2, 3, \cdots , K\right)$. $D$ indicates the maximum running distance of the distribution vehicle. We denote the distance between location $i$ and location $j$ by $d_{ij}, d_{ij}=\sqrt{{(x_i-x_j)}^2+{(y_i-y_j)}^2}$. $x_{ij}^k$ indicates whether the distribution process of the distribution vehicle is continuous between location $i$ and $j$. If $i$ and $j$ are continuous, the value of $x_{ij}^k$ is 1, and vice versa, the value of $x_{ij}^k$ is 0. The scheduling optimization model is shown in

Table 1. Scheduling optimization model parameters.

Formula		Indexes
${\mathrm{d}}_{ij}=\sqrt{{(x_i-x_j)}^2+{(y_i-y_j)}^2}$	(9)	The distance between location $i$ and location j.
$\min [C_{0}K+C_1{\displaystyle \sum _{k=1}^K{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N(d_{ij}\times x_{ij}^k)}}}]$	(10)	Minimizes the total cost of distribution by controlling distribution vehicles.
${\displaystyle \sum _{k=1}^K{\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=M+1}^N{x}_{ij}^k}}}={\displaystyle \sum _{k=1}^K{\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=M+1}^N{x}_{ji}^k}}}=K$	(11)	Ensures that each distribution vehicle to be distributed will depart from the distribution center and return to the distribution center after completing the distribution task.
${\displaystyle \sum _{i=M+1}^N{x}_{ij}^k}={\displaystyle \sum _{i=M+1}^N{x}_{ji}^k}$	(12)	Ensures that the flow of distribution demand points is balanced and the number of distribution vehicles arriving at the demand point is the same as the number of distribution vehicles leaving the demand point.
${\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N{x}_{ij}^k{q}_j}}\le Q,(k\in m)$	(13)	Ensures that the distribution volume of emergency supplies for each distribution vehicle does not exceed the maximum distribution volume.
${\displaystyle \sum _{k=1}^K{\displaystyle \sum _{i=1}^N{x}_{ij}^k}}=1,(j=M+1,\dots ,\mathsf{N})$	(14)	Ensures that demand is met at each demand point and that demand is met by one and only one distribution vehicle.
${\displaystyle \sum _{i=1}^N{\displaystyle \sum _{\mathrm{j}=1}^N{x}_{ij}^k}}{d}_{ij}\le D,(k\in m)$	(15)	Ensures that the total distribution distance of distribution vehicles does not exceed the maximum distribution distance.
$K\ge \left[\frac{{\displaystyle \sum _{j=M+1}^N{q}_j}}{Q}\right],(K\in N\ast )$	(16)	Ensures that the number of distribution vehicles is sufficient to meet the needs of each demand point.

.

3.3. Model Solving

3.3.1. Objective Function

To control the minimum total cost of distribution vehicles, the objective function is the sum of the fixed cost of starting the distribution vehicle and the increased cost per kilometer. When the total cost is the minimum value, it is the optimal solution for this individual, called the individual extreme value, as shown in Equation (12). From these optimal solutions, we aim to find a global value called the global optimal solution. We thus compare it with the historical global optimum and update it.

$$\min [C_{0}K+C_1{\displaystyle \sum _{k=1}^K{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N(d_{ij}\times x_{ij}^k)}}}]$$

(17)

3.3.2. Position and Speed Update

The formula for the globally optimal position and the formula for the velocity magnitude to find the globally optimal position are shown in Equations (18) and (19).

$$v_{id}\left(t+1\right)=\omega v_{id}\left(t\right)+c_1\ast random\left(0,1\right)\ast \left(p_{id}-x_{id}\left(t\right)\right)+c_2\ast random\left(0,1\right)\ast \left(p_{gd}-x_{id}\left(t\right)\right)$$

(18)

$$x_i\left(t+1\right)=x_i\left(t\right)+v_i\left(t\right)$$

(19)

In Equations (18) and (19), $v_{id}\left(t\right) and x_{id}\left(t\right)$, respectively, denote the velocity and position of the individual at time $t$. $\omega$ is the inertia factor, and its value is non-negative. When the value of $\omega$ is large, the global seeking ability and the local seeking ability are strong. When the value of $\omega$ is small, the global seeking ability is weak, but the local seeking ability is strong. By adjusting the size of $\omega$, the global and local search performance can be adjusted. $c_1$ and $c_2$ are called acceleration constants; $c_1$ is the individual learning factor of each particle, and $c_2$ is the social learning factor of each particle, generally taken as $c_1=c_2\in \left[0,4\right]$. $random\left(0,1\right)$ denotes the random number on the interval [0,1]; $p_{id}$ is the $d$-th dimension of the individual extreme value of the i-th variable, and $p_{gd}$ is the $d$-th dimension of the global optimal solution.

4. Instance Verification

4.1. Data Declaration

In late 2019, COVID-19 first appeared in the public view, followed by the first large-scale pandemic infection outbreak in Wuhan, Hubei, China, in early 2020, which had a huge impact on society and the Chinese economy. Wuhan, a central city in central China, is located in the eastern part of the Jianghan Plain and the middle reaches of the Yangtze River, with the Yangtze River and its largest tributary, the Han River, intersecting in the city. It has 13 administrative districts under it, namely Jiangan District, Jianghan District, Qiaokou District, Hanyang District, Wuchang District, Qingshan District, Hongshan District, Caidian District, Jiangxia District, Huangpi District, Xinzhou District, Dongxihu District, and Hannan District, with a land area of 8569.15 km² in the city, as Figure 2 shows. In terms of health care, Wuhan has 67 tertiary hospitals and 31 health prevention and control institutions. Wuhan is a typically successful case of fighting the pandemic successfully in 2020.

The number of residents in each district of Wuhan in 2020 is shown in

Table 2. Statistics of Wuhan districts in 2020.

	Area/Square Kilometer	Number of Residents Population/People	Weight/%	Fever Clinic
Hongshan District	509	2,500,000	20.28	63
Jiangxia District	2014.5	1,280,000	10.39	12
Huangpi District	2261	1,151,644	9.34	21
Wuchang District	82.2	1,102,200	8.94	45
Jiangan District	64.24	965,260	7.83	54
Caidian District	1100.81	890,618	7.23	54
Xinzhou District	1409.7	860,377	6.98	18
Dongxihu District	495.3	845,782	6.86	8
Hanyang District	108	837,263	6.79	24
Qiaokou District	41.9	666,661	5.41	21
Jianghan District	33.49	647,932	5.26	18
Qingshan District	61.1	431,818	3.50	6
Hannan District	287	145,103	1.18	4

, and the population distribution is shown in Figure 3a. Among the 13 districts in Wuhan, the districts with a resident population of more than 1 million are Hongshan District, Jiangxia District, Huangpi District, and Wuchang District. There are 2 districts with a population of less than 500,000, namely Qingshan District and Hannan District. The total resident population of the top four districts accounted for 48.96% of the city’s total resident population, nearly half of the total population. The distribution of medical resources needs to be considered under the premise of emergency public health. According to public documents, the list of medical institutions in Wuhan, and the number of practitioners at the end of 2019, the number of fever outpatient clinics in each Wuhan district during the epidemic emergency is shown in Table 2, and the clinics’ distribution is shown in Figure 3b. Combined with the population distribution, there is a clearly uneven distribution of medical resources. The area of Jiangxia District and Huangpi District is more than 2000 square kilometers, and the population is more than 1 million, but the number of fever clinics combined is only 33, which is nearly half of the number of fever clinics in Hongshan District.

Access time intensity, population density, and medical scale were selected as indicators affecting the importance ranking of regional nodes in the emergency site selection network. Each district in Wuhan identified its government site as the material distribution center in its administrative area, i.e., the above-mentioned demand center and the candidate sites for the emergency material distribution center were also considered in the same location. According to the actual road network, only county roads and above are considered for emergency logistics channels. All nodes and sections in the road network have been numbered, and the drawn road network topology is shown in Figure 4. The figure contains 62 important network nodes in Wuhan, where red circles with numbers represent the administrative district center, white circles with numbers represent interchanges, intersections, or overpasses, and the number in the circle is the network node number. The green numbered boxes represent high-speed roads and expressways, orange numbered boxes represent an elevated road, white numbered boxes represent the main city road, and the numbers in the boxes represent the traffic time of the road section. The basic data consist of the Wuhan map and statistical yearbook.

4.2. Site Selection Result of Distribution Center

The 13 districts under the Wuhan management jurisdiction selected in this study suffered from different degrees of impact during COVID-19, and the number of patients was more prominent. At present, epidemic prevention and control have entered normalization. We analyzed the network topology model of the selected districts with the PageRank algorithm and analyzed the road traffic conditions, population density and the number of medical institutions with the TOPSIS algorithm. Then, we combined our method with the fusion algorithm to derive the importance ranking of each district to determine the location of emergency medical supplies to respond to possible major public health emergencies. Based on the network topology of 13 districts in Wuhan, a total of 62 nodes, the PR value of each district was obtained using the PageRank algorithm and Python programming, such that the higher the PR value, the higher the importance of each district. The results of the sensitivity analysis are shown in

Table 3. Comprehensive sensitivity analysis results.

ID	Location	${\mathit{D}}^{+}$	${\mathit{D}}^{-}$	S
1	Zongguan, Guozikou	0.0766	0.6080	0.8881
2	Mengjiapu	0.0832	0.5671	0.8721
3	Baofeng Road, Qintai Avenue	0.1354	0.4393	0.7644
4	Qintai Road, Hangkong Road	0.1467	0.4454	0.7522
5	Evergreen Road, Gushaoshu Road	0.2568	0.4360	0.6293
6	Fuxing Road, Huaihai Road, Fanhu	0.2500	0.4087	0.6205
7	Han Shi Road, Heping Road, Chenjiaji	0.2519	0.4111	0.6201
8	Qiaokou District	0.2753	0.3782	0.5787
9	Wuchang District	0.2805	0.3222	0.5346
10	Jianghan District	0.3679	0.4112	0.5278

according to the TOPSIS algorithm’s ranking of the advantages and disadvantages. Combined sensitivity values of less than 0.5 that were assigned through our sensitivity analysis and according to the importance of nodes were discarded. A total of 10 distribution centers were selected from 62 nodes; that is, the construction of 10 distribution centers can meet the demand for epidemic prevention and control in Wuhan.

The PR value of each regional node and the results of the sensitivity analysis were linearly regressed to obtain improved regional node importance, resulting in a specific score ranking, as shown in

Table 4. Algorithm results.

ID	Location	S	PR	S-PR
1	Zongguan, Guozikou	0.8881	0.0675	0.6419
2	Mengjiapu	0.8721	0.0634	0.6294
3	Qintai Road, Hangkong Road	0.7522	0.0594	0.5443
4	Baofeng Road, Qintai Avenue	0.7644	0.0375	0.5433
5	Evergreen Road, Gushaoshu Road	0.6293	0.0301	0.4495
6	Han Shi Road, Heping Road, Chenjiaji	0.6201	0.0320	0.4436
7	Fuxing Road, Huaihai Road, Fanhu	0.6205	0.0300	0.4433
8	Qiaokou District	0.5787	0.0310	0.4143
9	Jianghan District	0.5278	0.0399	0.3814
10	Wuchang District	0.5346	0.0202	0.3802

, and further saving site selection, construction, and materials. The corresponding site selection results are shown in Figure 5. In this study, we selected the top three distribution centers with the highest scores, which are Zongguan Guozkou, Mengjiapu, and Qintai Road/Hangkong Road.

Most of the literature on emergency material dispatching considers a single emergency demand point, which is to be supplied by multiple material supply points. This study discusses the more practical problem of dispatching multiple stockpile points to meet the needs of multiple emergency demand points and considers the problem of uncertainty in delivery time due to road damage and other reasons. In this study, taking Wuhan in Hubei Province as an example, 62 points were selected according to its epidemic and specific facilities distribution map, including 3 distribution centers close to the central location. Due to the fact that each district in Wuhan City determines its government site as the material distribution center, i.e., demand point, within its administrative area, a total of 16 distribution locations are obtained based on the algorithm results. The dispatch data include the coordinates of 3 distribution centers and 16 demand points; the distribution centers can use up to three distribution vehicles to distribute the emergency supplies, and the detailed data are shown in

Table 5. Location coordinates and demand.

Location Number	Location	Category	x Coordinate/km	y Coordinate/km	Demand
28	Zongguan Guozikou	Distribution center	31.9382	25.852	Self-assigned
27	Mengjiapu	Distribution center	30.5068	24.8792	Self-assigned
3	Qintai Road Hangkong Road	Distribution center	34.8582	27.168	Self-assigned
30	Qiaokou District	Point of demand	27.9201	27.7378	4
32	Evergreen Road, Gushaoshu Road	Point of demand	33.7235	30.3409	12
19	Han Shi Road, Heping Road, Chenjiaji	Point of demand	39.1837	34.1248	14
31	Fuxing Road, Huaihai Road, Fanhu	Point of demand	32.4535	26.4242	2
15	Jianghan District	Point of demand	35.1444	28.341	2
1	Wuchang District	Point of demand	37.1197	28.7416	2
14	Baofeng Road, Qintai Avenue	Point of demand	32.6825	25.7375	5
60	Huangpo District	Point of demand	36.0192	46.5166	4
62	Xinzhou District	Point of demand	59.0629	49.6175	2
38	Jiangxia District	Point of demand	40.9474	15.2786	3
42	Hannan District	Point of demand	29.8872	8.5011	3
46	Caidian District	Point of demand	26.5164	22.0616	4
50	Dongxihu District	Point of demand	22.4886	24.1741	5
18	Qingshan District	Point of demand	40.7163	32.2689	6
10	Hongshan District	Point of demand	44.5342	25.3942	11
26	Hanyang District	Point of demand	32.9401	24.9365	7

.

The specific parameters of the algorithm are taken as shown in

Table 6. Particle swarm algorithm parameter setting.

Parameter	Number of Particles	Maximum Number of Iterations	${\mathit{c}}_1$	${\mathit{c}}_2$	$\mathit{\omega }$
Parameter values	30	100	0.4	0.4	0.2

.

4.3. Path Scheme

The vehicle path problem for multiple distribution centers is an extension of the classical VRP problem. It studies multiple parking lots serving several customers at the same time, and each customer has a certain cargo demand. What is to be determined is which distribution center serves the customer point and optimizes the vehicle path during service. Most of the current solutions to this problem are split into two stages. Firstly, the customer points are assigned to distribution centers, which are transformed into a single distribution center problem. Secondly, the paths are optimized at each distribution center. In this study, the particle swarm algorithm is used to solve the scheduling optimization model, and an optimal solution, which is the optimal path scheme, is obtained after the algorithm iterates to the best solution, as shown in

Table 7. Optimal path solution.

Distribution Center	Distribution Route
Zongguan Guozikou	28→26→42→46→30→28
Mengjiapu	27→31→14→27
Qintai Road, Hangkong Road	3→15→32→1→18→19→60→62→10→38→3

. According to the logistics industry guidelines, the general distribution path of the distribution center that is within its distribution range and does not need to pass through other areas shall not exceed 5 km. The iterative process and the scheduling line are shown in Figure 6 and Figure 7.

4.4. Analysis of Results

In this study, according to the Wuhan government bidding documents in the past two years, medical supply reserves are calculated based on 90 days for key supplies and 30 days for non-key supplies. The storage area of medical supplies required per 10,000 people is about 15–17 square meters, and the cost of equipment, such as warehouse electromechanics, is about CNY 2 million/project. Therefore, we ran our calculations based on 16 square meters per 10,000 people and an equipment cost of CNY 2 million per centralized distribution center. The total cost under non-optimized conditions is CNY 153,069,200, and the total cost under the optimized solution is CNY 126,662,600, which is 17.28% lower than the non-optimized conditions and reduces the distribution path. After the iterative solution of the scheduling optimization model is found by the particle swarm algorithm, the optimal solution is found without violating the maximum distribution volume constraint of each distribution vehicle. At present, diesel fuel in Wuhan is CNY 8.33/L, and the distribution personnel with the average working hour price of Wuhan is CNY 29.5/h. Thus, the distribution cost is CNY 7399.6/day before optimization and CNY 5781.13/day after optimization. Therefore, the goal of reasonable and efficient optimization of the distribution scheme is achieved. The comparison graph before and after optimization is shown in

Table 8. Comparison before and after optimization.

Index	Distribution Costs (CNY/Day)	Number of Sites (pcs)	Site Selection Cost (Million CNY)
Before optimization	7399.6	13	15,306.92
After optimization	5781.13	10	12,662.26

.

5. Conclusions and Prospect

5.1. Conclusions

(1) The process of selecting and evaluating a site for an urban emergency medical material distribution center under a public health event situation requires consideration of its unique characteristics, making it different from general site selection problems. To address this, we combined access time intensity, population density, and medical scale as importance-ranking indexes affecting regional nodes. We then used the improved web ranking algorithm and the improved TOPSIS algorithm to analyze the importance of the target area to be selected and determine the initial site selection plan. The results can serve as a valuable reference for the site selection of urban emergency medical supply distribution centers.

(2) By constructing 10 distribution centers, comprehensive coverage of emergency demand points and important high-risk demand points in the region can be achieved, effectively meeting the epidemic prevention and control needs of Wuhan while reducing construction and distribution costs. Compared with the previous solution, the optimal solution resulted in a 27.9% decrease in distribution costs and a 17.82% decrease in site selection costs. The findings of this study have implications for analyzing emergency material dispatching problems in general.

5.2. Theoretical Significance

(1) The effective and efficient management of emergency supplies is a crucial aspect of emergency relief efforts, which involve the prompt collection and distribution of supplies in the aftermath of an incident. The stockpiling and dispatching of emergency supplies is a significant element in the emergency relief field and is a challenge faced by all rescue agencies and organizations. In both developed and developing countries, it is essential to ensure the swift collection and dispatch of relief materials after a disaster to avoid wasting social resources and manpower. The management of infectious disease emergency logistics plays a critical role in ensuring the timely and organized supply and distribution of various medical supplies after an epidemic outbreak, thereby contributing to effective infectious disease emergency management and epidemic prevention and control.

(2) By addressing the challenges encountered in the current emergency response field, we can advance the theory and practice of disaster management. In particular, developing optimal decision models for emergency material dispatching under non-decisional information system environments can enhance the construction of intelligent decision systems for emergency management. These efforts can significantly improve the management level in emergency response and promote progress in disaster management theory.

5.3. Practical Significance

(1) It is conducive for the government to quickly develop a reasonable dispatching plan. Following a disaster, the primary objective is to promptly design a rescue material allocation plan based on available information to facilitate agile dispatching and an efficient response. This plan should not only consider time constraints but also fairness and effectiveness. The scientific and rational allocation of resources can decrease rescue time, maximize material utility, and diminish casualties and property damage. The investigation of material dispatching based on demand urgency is valuable in ensuring fairness and effectiveness in the distribution process. It provides government departments with scientifically based distribution solutions.

(2) During infectious disease emergencies, the shortage and delayed dispatch of medical supplies can heighten the risk of epidemic spread. Furthermore, the distribution of supplies may be inequitable, and response times may vary, leading to conflicts among epidemic areas. In this study, the dispatching model prioritizes the intensity of access time, population density, and medical scale as significant factors influencing regional nodes. The dispatching priority parameter indicates the level of urgency for emergency supplies in infected areas. This approach guarantees the timeliness and fairness of medical supply distribution, enabling a rapid, equitable, and organized dispatch of emergency medical supplies. Consequently, it improves the efficiency of infectious disease control.

5.4. Prospect

In this study, we have only considered the case of fixed distribution centers. However, temporary centralized distribution points can be established as per actual requirements, with the quantity and location of supplies adjusted accordingly. Hence, the dynamic distribution model of urban emergency medical supplies, considering the location of temporary centralized distribution centers, can be further improved in the future. Moreover, in the investigation of emergency material dispatching, the material demand urgency of all emergency material demand points suffering from different degrees of damage is assumed to be identical, and the demand level of emergency material demand points is not classified. In actual rescue operations, it is essential to enhance the supervision of emergency materials for enterprises’ reserves, social donations, or assault production. Transparent management of emergency materials is necessary to ensure the effectiveness of emergency material dispatching.