Research on the Distribution of Emergency Supplies during the COVID-19 Epidemic

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Introduction
Nowadays, the mutated strain of COVID-19, Omicron, is outbreaking and spreading in many provinces and cities in China, such as Shanghai, Guangzhou, and Beijing, respectively. It spreads fast, and it is so insidious and penetrating that it makes the outbreak rapid and large scale. Not only does it seriously afect the normal social order, but it also increases the pressure on medical resources and causes huge casualties and economic losses. According to national and provincial health statistics, as of November 28, 2022, China has a total of 9,036,539 cases of cumulative national diagnosis and cumulative deaths of 30,166 cases. Te situation is unprecedentedly serious. Te virus is transmitted by a wide range of routes, including respiratory droplet transmission, close contact transmission, and aerosol transmission. Te closed prevention and control make the distribution channel of emergency supplies obstructed and inefcient, such as drugs, protective clothing, masks, nucleic acid testing reagents, and so on. It not only cannot ensure the daily treatment of infected people but also afects the selfprotection of ordinary citizens and healthcare workers, which seriously endangers the health and safety of the human masses and also has a signifcant impact on social and economic development [1]. Terefore, as the epidemic continues to recur, it is imperative to achieve efcient distribution of emergency supplies to facilitate emergency treatment and normalized prevention and control during the epidemic.
In recent years, there have been numerous studies on the distribution of emergency supplies, and the research objects include not only natural disasters including earthquakes [2], typhoons [3], and foods [4], but also cover SARS, infuenza A (H1N1), African swine fever epidemics, and the COVID-19 epidemics [5][6][7][8]. Te long duration and high frequency of the COVID-19 pandemic have put forward higher requirements for the safety, efectiveness, stability, and sustainability of emergency supplies distribution in modern society [9]. Bi et al. [10] propose a reward-based trafc control mechanism that generates cooperative behavior by issuing credit money and thus optimizes resource allocation. Liu et al. [11] incorporated new criteria such as dynamism, synergy, and policy into the evaluation system from the perspective of the COVID-19 pandemic to comprehensively evaluate the state of the transportation infrastructure in the COVID-19 epidemic. At this stage, the research results related to emergency materials are mainly focused on the following two aspects. One is the dispatching of emergency supplies after emergencies, which mainly includes the distribution of supplies [7] and the supply order to each demand point [2] to realize the efcient distribution of emergency supplies and the optimal selection of distribution paths; Liu and Song [12] established a discrete-time mixed integer linear programming model to optimize the emergency blood supply system in the disaster relief process. Secondly, the location of the emergency supplies warehouse is selected to achieve efcient distribution of emergency supplies by determining the optimal location of the warehouse, e.g., Cong and Yu [3] analyzed the characteristics of the regional emergency supplies reserve location problem after a disaster, and Peng et al. [13] considered the situation where an emergency event occurs that prevents the normal distribution of supplies. And the safety and environmental friendliness [14] of the material transportation means used is also a perspective worth studying.
In addition, the emergency material distribution problem can be extended into the following fve aspects according to the focus of the research: (1) focus on emergency material transportation tools, such as "truck + drone" joint dispatch [6] and helicopter dispatch [15]; (2) optimization models focusing on multiple objectives, with the main objectives including minimizing rescue cost and rescue time [16] and minimizing total delay time and total system loss [17], for example, Wang et al. [18] proposed a data-driven multiobjective optimization method, Safaei et al. [19] established a dual-objective two-stage optimization model by considering the infuence of rescue material distribution locations and the risk level of diferent suppliers; (3) focus on multiparty cooperation and coordination, such as Xie et al. [20] and Zhang et al. [21] considered the government and proposed new response strategies for the emergency material security problem; (4) focused on dividing the emergency system into levels [8], phases [22], or cycles [23] for detailed research, such as Zhang and Miao [4] integrated and optimized two phases of predisaster preparation and postdisaster response, while Fu et al. [5] divided the system into two subsystems of epidemic spread and material security for research; and (5) focusing on the impact brought by service level, for example, Liu et al. [1] defned an innovative class of emergency service level functions, and Wang et al. [24] considered that the afected people would have pain perception due to the lack of supplies and services, and constructed a pain function by designing a numerical rating scale (NRS) to portray the pain perception cost of the victims and included it in the decision of total emergency cost.
Te emergence of the COVID-19 epidemic has raised a series of urgent questions, and many scholars have conducted in-depth studies on this issue. For example, Kamran et al. [25] focused on the supply chain network of vaccines and combined the variable neighborhood search (VNS) and whale optimization algorithm (WOA) to design a novel metaheuristic algorithm to optimize the decision making of vaccines. Goodarzian et al. [26] frst proposed a responsivegreen-cold vaccine supply chain network during the COVID-19 pandemic and used the modifed gray wolf optimization (MGWO) algorithm to fnd the Pareto solution and near-optimal solution. Shirazi et al. [27] optimized the plasma supply chain network in case of COVID-19 outbreak. All of these literatures provide the theoretical basis for the distribution problem of emergency supplies of this paper. Diferent from them, the problem studied in this paper focuses on the need to hire a third-party feet to deliver supplies to the demand point, and no longer need to return to the distribution center after the distribution is completed. And this paper combines the brain storm optimization algorithm with the genetic algorithm and the large neighborhood search algorithm, which can substantially improve the quality of the solution and thus optimize the emergency material distribution system.
Te emergency material distribution problem proposed in this paper is essentially an open vehicle path problem with load limit and time window limit, which is an NP-hard problem from the theoretical point of view. It is difcult to obtain its optimal solution in a short period, while the actual emergency decision often has a high requirement for timeliness [1]. Related research results in recent years are as follows: Yin and Zhang [28] constructed three hybrid bat algorithms to solve multiobjective vehicle path problems with hard time windows; Wu et al. [29] improved the ant colony algorithm based on brainstorming to improve the solution of vehicle path problems with soft time windows; the literature [30][31][32] conducted in-depth studies on open vehicle path problems under diferent conditions. Brandao [33] and Ruiz et al. [34] considered the case of multiple warehouses and demand detachability, respectively; on this basis, the iterative local search algorithm for jet chains proposed by Brandao [35] and the variable neighborhood search algorithm designed by Chen et al. [36] both improved the quality of OVRPTW solutions.
In summary, based on the unpredictability of unexpected epidemics, this paper establishes a mixed integer linear programming mathematical model satisfying the load constraint, the demand constraint, and the time window constraint from a practical point of view. We use the brain storm optimization algorithm for the solution of the model and introduce the crossover operation of the genetic algorithm and the destruction operator and repair operator of the large neighborhood search algorithm in the brain storm optimization algorithm to enhance the diversity of solutions 2 Complexity while making the solutions develop in the optimal direction. By solving two sets of examples, the solution results of 14 algorithms are compared and analyzed, and the experiments fnd that (i) the brain storm optimization algorithm proposed in this paper can produce shorter total distribution routes when solving CVRPTW. Compared with the other eight algorithms, it produces the shortest average path length and better solution quality and has signifcant algorithmic advantages; (ii) in comparing the efect of seven algorithms for solving COVRPTW, the best solutions obtained by most of the cases using the brain storm optimization algorithm are better than the best solutions in the known literature. Meanwhile, the use of the brain storm optimization algorithm can signifcantly reduce the length of the distribution route and lower the distribution cost, which proves the excellent quality of the BSOA solution; (iii) the solution efect of the two sets of cases is comprehensive. Te BSOA algorithm in this paper has excellent convergence and stability, and the solution quality is superior, which can verify the efectiveness and stability of the algorithm solution from multiple dimensions and help to improve the distribution efciency of emergency supplies. Terefore, it is of great criticality to help the emergency reception and regular prevention and control during the COVID-19 epidemic.

Problem Description.
After the outbreak of the COVID-19 epidemic, the demand for medical supplies in hospitals, communities, and isolation sites increased dramatically. But due to the highly contagious nature of the virus, closed management was adopted everywhere. And volunteers transporting supplies had to wear protective clothing with passes to distribute supplies within a specifed time frame according to a specifc route, which is essentially a typical vehicle routing problem (VRP). It can be described as the existing multiple loading or unloading points, need to organize the appropriate transportation routes in an orderly manner according to the actual situation, so that one or more vehicles carrying goods pass through them in a certain order, and under certain constraints (such as the number of goods demanded by the customer point, the capacity of the vehicle, and the mileage that the vehicle can travel). To achieve the corresponding goals, there can be one or more goals, such as the shortest distance traveled by the vehicle, the smallest transportation cost, and the least time spent. Te schematic diagram of the service model is shown in Figure 1.
Due to the unpredictability of the epidemic, the distribution center may need a third party to provide sufcient vehicles to distribute emergency supplies. All vehicles will depart from the distribution center and return to the thirdparty organization after the distribution is completed without returning to the distribution center. Tis problem belongs to the category of open vehicle routing problem (OVRP), and the specifc service model is shown in Figure 2. Terefore, the emergency material distribution problem at the material demand point can be simply summarized as constraint open vehicle routing problem with time windows (COVRPTW), which is described as the existing multiple demand points, and a demand point can only be served by one vehicle. Te vehicle departs from the distribution center and delivers several demand points along a specifc route within a specifed time window, without returning to the distribution center afterwards. And we need to design a distribution plan that minimizes the sum of the total distance traveled by all vehicles.

Symbol Description
(1) Set COVRPTW can be defned in a directed graph G � (V, A), where V � 0, 1, 2, · · ·n { } denotes the set of all points, 0 is the distribution center, N � V\ 0 { } is the set of demand points, n is the number of demand points, and A denotes the set of arcs. From the assumption, a reasonable distribution route must start from node 0 and need not return to node 0. In addition, K denotes the set of distribution vehicles (kϵK). ∆ + (i) denotes the set of all arcs that start from node i and ∆ − (j) denotes the set of all arcs that return to node j.
Te parameter design is shown in Table 1. x k ij � 1, vehicle k travels from node i to node j, 0, otherwise. (1)

Mathematical Model Construction.
Before the model is constructed, there are the following requirements for the time window constraints. Te demand point allows the vehicle to arrive before the left time window, but it needs to wait until the left time window time point to start the service. However, the demand point will not accept the service for the vehicle that arrives after the right time window, that is, the vehicle can only serve the demand point if it arrives before the right time window. Te mathematical model of the emergency material distribution problem is established as follows: Objective function: Subject to k∈K jϵ∆ + (i) where (2) is the objective function, which indicates that the sum of all distances traveled by the vehicles is minimized, and (3) to (11) are all constraints.
(3) means that each demand point can only be assigned to one distribution route, (4) means that each vehicle departs from the distribution center to the demand point, (5) means that the vehicles are balanced in and out, which means that each demand point must have a vehicle drive in to have a vehicle drive out, (6) means that the travel time of the vehicle from node i to node j is the ratio of the distance between node i and node j and the travel speed of the vehicle ratio, (7) indicates that the travel time of vehicle k is continuous, (8) indicates that the start time of service of vehicle k to demand point i must be between the left time window and the right time window of demand point i, (9) indicates that the departure of the vehicle from the distribution center must be after the left time window of the distribution center, (10) indicates that the initial load of vehicle k at the distribution center must be no greater than the maximum load capacity, and (11) indicates that no vehicle returns to the distribution center.

Algorithm Design
In this paper, the brain storm optimization algorithm (BSOA) is improved by introducing the crossover operation of the genetic algorithm and the destruction operator and repair operator of the large neighborhood search algorithm, which not only enhances the diversity of the solution, but also makes the solution develop in the optimal direction. Tus, the optimal distribution scheme is fltered out and the efciency of emergency material distribution is improved.

Introduction to BSOA.
BSOA is inspired by the brainstorming method. Te brainstorming method refers to organizing a number of people for a meeting, and anyone who has an idea can express it. Te others are not allowed to deny or criticize it. After the meeting, all the ideas mentioned are summarized and organized, so as to come up with more new ideas to solve the problem. BSOA is a new population intelligence optimization algorithm proposed by Shi [37], which is similar to genetic algorithm. Te principles of both are similar, but the updated solutions are diferent. Te fow of its solution problem is shown in Figure 3.

Solution
Strategy. BSOA solves the emergency supplies problem by performing eight key steps, as shown below.

Encoding and Decoding.
Te coding method used is to refect both the distribution center and the demand point in the individual. Suppose there are four demand points, numbered 1, 2, 3, and 4, and the distribution center allows at most two vehicles for distribution. At this time, the distribution center can be represented by the numbers 5 and 6 and inserted into the arrangement of the four demand points. Tus, the individual is decoded into fve scenarios of allocation.
(1) 5 and 6 are inserted separately into the interior of the demand point arrangement, meaning that the individual is represented as 125364. At this time, the distribution centers 5 and 6 split 1234 into three distribution routes. Represent it as follows, where 0 indicates the distribution center: Route 1: 0 ⟶ 1 ⟶ 2 ⟶ 0 Route 2: 0 ⟶ 3 ⟶ 0 Route 3: 0 ⟶ 4 ⟶ 0 (2) 5 and 6 are inserted together into the interior of the demand point arrangement, meaning that the individual is represented as 125634. At this time, distribution centers 5 and 6 split 1234 into two distribution routes. Represent it as follows, where 0 indicates the distribution center: 5 and 6 are inserted together at the head of the demand point arrangement, meaning that the individual is represented as 561234. At this time, the distribution centers 5 and 6 will split 1234 into one distribution route. Represent it as follows, where 0 indicates the distribution center: Route: 0 ⟶ 1 ⟶ 2 ⟶ 3 ⟶ 4 ⟶ 0 (4) 5 and 6 are inserted together at the end of the demand point arrangement, meaning that the individual is represented as 123456. At this time, the distribution centers 5 and 6 will split 1234 into one distribution route. Represent it as follows, where 0 indicates the distribution center: Route: 0 ⟶ 1 ⟶ 2 ⟶ 3 ⟶ 4 ⟶ 0 (5) 5 and 6 are inserted into the head and end of the demand point arrangement, respectively, that is, the individual is expressed as 512346. At this time, the distribution center 5 and 6 will split 1234 into one distribution route. Represent it as follows, where 0 indicates the distribution center: In summary, if the number of demand points is N and the distribution center allows at most K vehicles for distribution, then the individuals in the problem of emergency material distribution using BSOA are expressed as a random arrangement of 1 ∼ (N + K − 1). And the abovementioned fve cases should be considered in detail in the process of decoding the individuals into distribution schemes without omission.

Objective Function.
Te decoding method cannot guarantee that each distribution route satisfes the load weight constraint of the vehicle and the time window constraint of the node. Terefore, the way of adding penalties to the distribution routes that violate the constraints can be used to achieve the abovementioned constraints. Te total cost of the distribution scheme is calculated as follows: In the formula, s denotes the distribution scheme, f(s) denotes the total cost of the current distribution scheme, c(s) denotes the total distance traveled by the vehicles in the current distribution scheme, g(s) denotes the sum of the load constraints violated by all the routes of the current distribution scheme, w(s) denotes the sum of the time window constraints violated by all the routes of the current distribution scheme, α is the penalty factor for violating the load constraints, and β is the penalty factor for violating the time window constraint, see above for the meanings of other parameters.
Te smaller the distribution scheme's total cost, the smaller the value of the objective function, that is, it indicates the better quality of the current individual.

Population Initialization.
A random initialization is used to construct the initial population of BSOA. Assuming that the number of populations is NIND, the number of demand points is N, and the distribution center can allow at most K vehicles to deliver for the demand points, then any individual in the initial population is a random arrangement of 1 ∼ (N + K − 1).

Clustering
Operation. K-means clustering is used to cluster all the individuals in the population, and the clustering object is the objective function value of the individuals. Suppose the number of populations is NIND and the number of clusters is K, the steps are as follows: STEP 1: From the NIND individuals, K individuals are randomly selected as the initial clustering centers. STEP 2: Te diference between the objective function values of NIND individuals and the objective function values of K initial clustering centers is calculated one by one, and its absolute value is taken, and the individuals are grouped with the clustering center corresponding to the smallest absolute value. STEP 3: Te mean value of the objective function value of all individuals in each cluster is found separately, and then compared one by one. Te individual whose objective function value is the closest to the mean value is the new cluster center of the cluster. STEP 4: Determine whether the preset maximum number of iterations is reached. If yes, the termination condition is reached, the loop is terminated and the clustering result is output; if not, STEP 2 is returned.

Replacement Operation.
Tat is, with a certain probability, randomly generated individuals are used to replace randomly selected clustering centers. Te replacement operation can increase the diversity of populations in the subsequent search process.

Update Operation.
Te update of population is the core of BSOA. But the update of individual position of population by BSOA is usually applicable to continuous optimization problems, which cannot be directly applied to the emergency material distribution problem proposed in 6 Complexity this paper. To better realize the update of individual positions, this paper introduces the crossover operation in the genetic algorithm while combining the characteristics of the emergency material distribution problem, which makes the update of individuals more efective. Te update operation involves the swap operation and crossover operation of individuals, and the two operations are explained as follows: Te swap operation is for one individual, which means that two positions are randomly selected from the current individual and the elements in these two positions are swapped, as shown in Figure 4. Suppose there is the following individual Figure 4(a), positions a and b are randomly selected. If a � 2 and b � 7 (see Figure 4(b)), it will be swapped and the elements in other positions remain unchanged. Ten, the exchanged individual is Figure 4(c).
Te crossover operation is for two individuals, which means that two crossover positions are randomly selected frst, and then the crossover fragment between the crossover positions of two individuals is moved to the head of the other individual, respectively. Finally, the duplicate elements that appear afterwards in each individual are deleted. As shown in Figure 5, suppose there are two individuals as follows (see Figure 5(a)), randomly select two crossover positions a and b, if a � 2 and b � 7, then get the crossover fragment (see Figure 5(b)), move the intersection fragment of individual 2 to the head of individual 1, move the intersection fragment of individual 2 to the head of individual 1 (see Figure 5(c)), and get two new individuals after deleting the duplicate elements that appear afterwards (see Figure 5(d)).
To perform an update operation, frst defne the parameters, constants, and variables associated with it as follows: rand: indicates a random number between 0 and 1 p − one: denotes the probability of selecting a cluster and takes a value between 0 and 1 p − two: denotes the probability of selecting two clusters, taking values between 0 and 1 and p − two � 1 − p − one p − one − center: denotes the probability of selecting a cluster center in a cluster and takes a value between 0 and 1 p − two − center: denotes the probability of selecting the cluster center in two clusters and takes a value between 0 and 1 X select 1: denotes the selected individual 1 X select 2: denotes the selected individual 2 Assuming that the number of populations is NIND, the number of clusters is k, and k ≥ 2 is satisfed. Ten, the specifc fow of the update operation for the individual i in the population is shown in Figure 6.

Local Search Operation.
In the population Population obtained by the update operation, a local search operation is performed with the individuals whose objective function values are in the top 60%, so that better individuals are obtained and the population as a whole move in a better direction.
Te destruction operator and repair operator in large neighborhood search (LNS) algorithm are introduced in the local search operation. Te destruction operator is mainly used to remove a group of customers from the current solution, and the repair operator is mainly used to reinsert the removed demand points into the corrupted solution. Te specifc operations are as follows: (1) Destruction Operator. Te destruction operator is removed according to the correlation between the demand points, and the correlation between the two nodes is calculated as In the formula, d ij ′ is the value after normalizing d ij , d ij ′ � (d ij /max d ij ), which is between [0, 1]. d ij is the Euclidean distance between node i and node j. V ij indicates whether node i is on the same route as node j. If yes, it is 0, otherwise it is 1.
Te larger R(i, j) indicates the greater correlation between node i and node j. On this basis, assuming that the number of demand points is N, the number of demand points to be removed is L, and the random element is D: the specifc operation steps of the destruction operator are as follows: STEP 1: A demand point i is randomly selected from N demand points, then the set of removed demand points is R � [i] and the set of unremoved demand points is U � [the remaining N − 1 demand points]. STEP 2: Compare the number of demand points in the two sets of R and L. If the number of demand points of R is less than or equal to the number of demand points of L, go to STEP 3; otherwise, go to STEP 5. STEP 3: Choose a random demand point r from R and calculate the correlation between all demand points in U and the demand point R. Te demand points in U are sorted in order of decreasing relevance, and the result is denoted as S. Ten, the next removed demand point Complexity next is calculated according to the formula rand D × |U| .(rand ∈ (0, 1), |U| denotes the number of demand points in the set U, and ⌈⌉ indicates rounding up.) STEP 4: Add the demand point next to R, remove the demand point next to U, and go to STEP 2. STEP 5: Remove all demand points in R from the current solution, output the set of removed demand points R, and the destroyed solution S destroy .
(2) Repair Operations. To facilitate the repair operation, two concepts of "insertion cost" and "regret value" are introduced in this paper.
Te solution obtained after the destruction operation is S destroy . Under the condition that the load constraint and the time window constraint are not violated, a demand point in R is inserted back to a certain position in S destroy , and the diference between the total distance traveled by the solution and the total distance traveled by the solution before the insertion of S destroy is the "insertion cost" of inserting the demand point into this position.
Under the condition that the constraint is not violated, assume that when inserting demand point i in R into S destroy , there are lr insertion positions to choose from and lr "insertion costs" will also appear. All "insertion costs" will be sorted in order from the smallest to the largest; the result is up delta. Ten, the "regret value" of inserting the demand point lr back into S destroy is the "insertion cost" in the second place after sorting minus the "insertion cost" in the frst place, which is the diference between the second smallest value and the smallest value of the "insertion cost." It can be expressed in mathematical notation as up delta(2) − up delta (1).
Te specifc steps of the repair operation are as follows: STEP 1: Initialize the repaired solution S repair , S repair � S destroy . STEP 2: If R is not empty, go to STEP 3; otherwise, go to STEP 6. STEP 3: Calculate the number of demand points nr in the current R. Calculate the "regret value" regret that interpolates each demand point in R back to S repair , regret is a matrix of nr rows and 1 column. STEP 4: Find the maximum "regret value" in the regret corresponding to the serial number max index. Ten, determine the demand point rd � R(max index) that will be inserted back, and fnally insert rd back to the position with the smallest "insertion cost" in S repair . STEP 5: Update R(max inde x) � [], return to STEP 2. STEP 6: End of repair, output the repaired solution S repair .

Merge Operation.
Suppose the population after the update operation is Population, the number of population is NIND and the local search operation is performed on the individuals whose objective function values are in the top 60%. Te local population obtained is ofspring. Te merge operation is used to merge Population and ofspring to form a new population, Population, where the population size is determined and some individuals are removed to ensure that the population size remained consistent with Population. Moreover, the local operation makes the individuals develop in a better direction, and the quality of the obtained individuals will not be worse than the original population. So, all the individuals in the ofspring should be retained, and only the individuals with the top 40% of the objective function value should be retained in the original Population.
In summary, the specifc fow of using BSOA to solve the emergency material distribution problem is shown in Figure 7.

Simulation Experiments and Results
To test the performance of BSOA in solving the emergency material distribution problem prototyped by COVRPTW more comprehensively, this paper uses two sets of   In this paper, BSOA is implemented programmatically using MATLAB R2016a software in the Win11 environment with Intel(R) Core(TM) i5-8250U CPU @ 1.60 GHz (8.00 GB RAM). Te algorithm parameters are set as follows: penalty function coefcient alpha = 10 for violating the load constraint, penalty function coefcient delta = 100 for violating the time window constraint, maximum number of iterations MAXGEN = 1000, number of populations NIND = 100, and the algorithm is run randomly 10 times.    Table 2. Te maximum load capacity of the vehicle is 8 t and the speed v is 40 km/h and is kept constant. Randomly run 10 times, the resulting average path length is 1005.01 km, the optimal distribution path length is 1004.32 km, the optimal number of vehicles is 6, and the specifc distribution routes are 0-12-19-6-13-0, 0-4-15-9-0, 0-18-11-3-0, 0-10-20-17-2-0, 0-7-5-14-0, and 0-1-16-8-0. Te roadmap of the optimal distribution scheme is shown in Figure 8, and the convergence diagram of the algorithm in the optimization process is shown in Figure 9.
Te results obtained from BSOA were multidimensionally analyzed and compared with the simple genetic algorithm, simulated annealing algorithm based on or-opt, hybrid genetic algorithm proposed by Jian et al. [38], the basic bat algorithm, elite genetic hybrid bat algorithm, multipoint recombination elite hybrid bat algorithm, singlepoint recombination elite hybrid bat algorithm proposed by 10 Complexity Yin and Zhang [28], and a total of nine algorithms proposed by Chen et al. [36], as shown in Table 3. From the comparison results of the nine optimization algorithms in Table 3, it can be seen that in solving the same distribution problem, the BSOA proposed in this paper and the variable neighborhood search algorithm proposed by Chen et al. [36] yield the shortest optimal path length and use the least number of vehicles, which is better compared to the other seven optimization algorithms for solving the problem. Meanwhile, the average optimal path length solved by BSOA is the shortest and is superior to the variable neighborhood search algorithm in terms of superiority in solution quality.

Example 2 Verifcation and Analysis.
Te standard set of 56 cases created by Solomon [39] for VRPTW is used in example 2, which can be classifed into six types, including R1, C1, RC1, R2, C2, and RC2, according to the demand point time window width and location distribution. Tis basically covers all the current situation of demand point distribution, which is in line with the actual situation and is very practical. Te characteristics of each type are shown in Table 4.
Since the standard set of arithmetic cases is written for VRPTW, and the prototype of the emergency material distribution problem proposed in this paper is COVRPTW, the distance traveled by the vehicle after serving the last demand point is not considered when performing the algorithm validation, that is, the vehicle does not need to return to the distribution center. In this paper, the BSOA is compared with the unifed variable neighborhood search (UVNS), unifed hybrid genetic search (UHGS), iterated local search algorithm (ILSA) proposed by Brandao [35], the    Complexity evolutionary algorithm (EA), and greedy randomized multistart trajectory local search algorithm (GRMSTS) by Repoussis et al. [40], and the solution results of the variable neighborhood search algorithm proposed by Chen et al. [36] as well as the currently known best known solutions (BKS) for solving OVRPTW. Tus, the validity of the BSOA solution is verifed.
Te detailed results are shown in Tables 5 and 6, where TD denotes the total distance traveled and GAP denotes the gap between the optimal solution found by the algorithm and the currently known best solution. Te calculation equation is as follows: GAP � the optimal solution obtained by the algorithm − BKS BKS × 100%.
According to the simulation results in Table 5, it can be seen that when the BSOA proposed in this paper solves 29 cases with wide time windows such as R1, C1, and RC1 in Solomon's standard algorithm, the BSOA solutions of 19 cases are better than the currently known best solutions, and 2 cases are the same as the currently known best solutions, and the optimization is better for the demand points whose locations have the characteristics of random distribution. Compared with the other seven algorithms, the BSOA solution yields the smallest travel cost, with an average savings of 3.62%.
According to the simulation results in Table 6, it can be seen that BSOA solves 27 cases with narrow time windows such as R2, C2, and RC2, and the BSOA solutions of 19 cases are better than the currently known best solutions, and the optimization is better for the demand points whose locations have the characteristics of random distribution. Compared with the other seven algorithms, the BSOA solution yields the smallest travel cost, with an average savings of 12.93%.
Terefore, based on the detailed results in Tables 5 and 6, as well as the analysis and comparison, it can be concluded that BSOA has excellent solution quality and solution     performance, especially for clusters of demand points with random distribution characteristics of geographic locations, which is in line with the realistic situation. Te feasibility and efectiveness of using BSOA for solving COVRPTW are further verifed from multiple dimensions.

Conclusion
Te outbreak of the COVID-19 epidemic has put forward higher requirements on the personnel arrangement and route design of emergency material distribution. Based on this background, in this paper, the open vehicle routing problem with load restrictions and time window restrictions is explored in depth for the specifc situation that the distribution center needs to be provided with vehicles by a third party, coupled with the requirements of demand points on demand quantity and time window. Assuming that the vehicle does not need to return to the distribution center after the distribution is completed, a COVRPTW mixed integer linear programming model with the minimization of travel distance as the objective function and the coexistence of multiple constraints is constructed, and a brain storm optimization algorithm is designed to solve it. In this paper, the validity of BSOA is demonstrated using two sets of examples, that is, the CVRPTW and COVRPTW are solved separately using BSOA, and the solution results are compared and analyzed with the existing literature and the best known solutions available. Te experimental results show that (i) the brain storm optimization algorithm proposed in this paper can produce shorter total distribution routes when solving CVRPTW compared with the other eight algorithms, all of which are capable of fnding good quality solutions under certain specifc conditions. But as for the BSOA, it produces the shortest average path length and better solution quality and has signifcant algorithmic advantages; (ii) in comparing the efect of seven algorithms for solving COVRPTW, the best solutions obtained by most of the cases using the brain storm optimization algorithm are better than the best solutions in the known literature. Meanwhile, the use of the brain storm optimization algorithm can signifcantly reduce the length of the distribution route and lower the distribution cost, which proves the excellent quality of the BSOA solution; (iii) the solution efect of the two sets of cases is comprehensive, the BSOA algorithm in this paper has excellent convergence and stability, and the solution quality is superior, which can verify the efectiveness and stability of the algorithm solution from multiple dimensions and help to improve the distribution efciency of emergency supplies. Terefore, it is of great criticality to help the emergency reception and regular prevention and control during the COVID-19 epidemic. Future research on the distribution of emergency supplies can further consider the impact of trafc congestion, real-time road conditions, and vehicle types on route optimization, make reasonable assumptions, establish mathematical models, and design algorithms to solve them to obtain a distribution solution that is more in line with the actual situation, with the shortest distribution distance and the lowest cost.
When designing algorithms in the future, the focus can be on more novel and advanced algorithms. For example, in some recently published literature on responding to outbreaks of the COVID-19 epidemic, the customized multiobjective hybrid metaheuristic solution algorithm [41] can substantially improve supply efciency, and nondominated sorting genetic algorithm II and multiobjective particle swarm optimization [42] perform efectively in ambulance route planning. In addition, advanced algorithms in other felds such as online learning, scheduling, multiobjective optimization, transportation, medicine, data classifcation, and others can provide theoretical guidance for subsequent algorithm design. For example, the learning-based algorithm [43] for optimizing many objective problems, the adaptive polyploid memetic algorithm [44] that performs well in truck scheduling, and the island-based metaheuristic algorithm [45] as well as the memetic algorithm with a deterministic parameter control [46] to that work great for berth scheduling at marine container terminals. In the brain storm optimization algorithm proposed in this paper, K-means clustering ensures the quality of the initial population, and crossover, destruction, and repair operations enhance the diversity of the solution, but the solution time is long and there are still limitations. Future research can combine the brain storm optimization algorithm with novel advanced algorithms to design an efcient and general algorithm to solve the most scheduling problems.
With the rapid development of the logistics industry, the open vehicle path problem study is applicable to an increasingly wide range of felds, such as courier services, fresh food transportation, waste furniture recycling, and other types of transport logistics industry are involved. It is of great practical signifcance to conduct an in-depth study on it. And, it is also crucial to explore the design and optimization of efcient solution algorithms in an efort to achieve efcient distribution of goods, which can best meet the existing needs of the society. It not only provides guidance for the staf and managers of the logistics industry in actual operation, helping them to obtain the optimal distribution solution with the minimum cost and maximize efciency. And at a deeper level, it can also significantly improve the efciency of logistics distribution and promote the rapid development of the logistics industry.

Data Availability
Te data that support the fndings of this study are available on request from the corresponding author, Qing Yang. Email: yangqing923@126.com.