# Routing algorithms as tools for integrating social distancing with emergency evacuation

### Dataset and challenge formulation

Motor vehicle Routing Complications (VRPs) in the context of pre-catastrophe evacuation are both of those critical and sophisticated54. Even though we simply cannot include all serious-world constraints, we can about formulate the problem of pre-catastrophe evacuation as the most examined variant of VRPs, the Capacitated Automobile Routing Difficulty (CVRP) with a rescue centre (usually named depot) and a given set of n properties denoted nodes or consumers, N = 1,2,…,n42,55.

Typically, there are homogeneous automobiles K with the same ability C, the place K = (), (C>). In this research, we work out whole evacuation time based mostly on just one unexpected emergency auto for just about every community (with all neighborhoods the similar dimension and currently being evacuated in parallel), so K is the very same car or truck for every single route in K in a community, and (|K |) is the range of routes needed to evacuate just about every community. The amount of individuals on the distinctive demands registry to be picked up (generally referred to as demand) from every single home is (d_j), in which < (d_j) (le) C, j (in) N. Let E = (e=i,j=j,i:i,j in 0,1,2,ldots ,n, i ne j) be the edge set55. An emergency vehicle moving from house i to j incurs travel cost (c_ij) for (i,j in E) or evacuation time in this study42,55. We define an integer decision variable (x_ij), where (x_ij)=1 if the emergency vehicle moves from house i to j while (x_ij)=0 if the emergency vehicle does not move from house i to j55,56,57.

The vehicle starts from the rescue center, goes to a subset of houses S (subseteq) N to pick up people, and finally returns to the rescue center before or until the vehicle capacity C is reached55. This process constitutes one route. The vehicle repeats this process until it collects every resident on the special needs registry, visiting each house only once (Fig. 1 and the other figures in Supplementary information)55. The goal is to evacuate every person on the evacuation registry within the minimum possible evacuation time. Results for a rescue operation consist of the routes utilized by this vehicle, evacuation time, the number of routes, and other detailed information about each route (see Supplementary information). The formulation of the CVRP can be expressed mathematically as follows55,56,57.

beginalignedMinimize sum _kin K sum _i=0^n sum _j=0, ine j^n c_ij x_ij^k, endaligned

(3)

beginalignedsum _kin K sum _i=0^n sum _j=1, ine j^n x_ij^k = 1, endaligned

(4)

beginalignedsum _jin N x_0j^k = 1, quad forall k in K, endaligned

(5)

beginalignedsum _i=0,i ne j^n x_ij^k = sum _i=0^n x_ji^k, quad j=0,1,ldots ,n, forall k in K, endaligned

(6)

beginalignedsum _i=0^n sum _j=1,i ne j^n d_j x_ij^k le C, quad forall k in K, endaligned

(7)

beginaligned- 1, quad forall S subseteq 1,2,ldots ,n, endaligned

(8)

beginalignedx_ij^k in 0,1, quad forall k in K, i,j in E endaligned

(9)

According to the parameters we defined earlier55,56,57, (3) is the objective function which minimizes the overall costs (total evacuation time) for the neighborhood. Constraint (4) makes sure that each house is visited only once. Constraint (5) indicates that the emergency vehicle can leave the rescue center only once for each route in K. Constraint (6) requires that the number of the emergency vehicles arriving at and leaving each house or the rescue center is the same (1 in this study). Constraint (7) ensures that the number of people picked up by the emergency vehicle in a single route is no more than the vehicle capacity C. (8) is called the subtour elimination constraint, which avoids any route that is disconnected from the rescue center55,56,58. Lastly, constraint (9) denotes that the integer decision variable should be either 1 (visited) or 0 (not visited).

To analyze the efficacy of pre-disaster evacuations using Deep Reinforcement Learning (DNN-based solution) and Sweep Algorithm (Non-DNN solution), we consider four datasets in this study.

#### Dataset selection

In our experiments, we used the A-n36-k5, A-n53-k7, A-n69-k9 datasets, commonly used in studies of capacitated vehicle routing problems50,51. Each dataset contains a unique set of coordinates of one depot and several nodes in a two-dimensional Cartesian coordinate system, plus the demand of each node and vehicle capacity. Since we are interested in exploring how size of neighborhood and social distancing policy could impact the total time and the number of routes in evacuation operations, we selected the first 20 locations in the A-n36-k5 dataset as our Dataset 1. We used all 35 locations in A-n36-k5 as our Dataset 2. Our Dataset 3 and 4 are A-n53-k7, A-n69-k9 datasets which contain 52 and 68 houses respectively. To accommodate the possibility that the depot is outside the neighborhood, we allowed for a series of fixed times (0, 0.5, 1.0, and 2.0 h) for transit to and from a central depot.

#### Dataset formation

In structuring the problem, we use the city of New Orleans, Louisiana, USA, as the reference case. New Orleans experienced massive losses and disruption in Hurricane Katrina (2005)59,60,61,62,63 and has implemented comprehensive plans to prepare for a future major hurricane. While our analytical framework is general, details like the average number of people in a household and the length of the time window for completing the evacuation are specific to New Orleans.

In New Orleans, the elderly, disabled, and those at high risk for severe illness from COVID-19 can sign up online or call 311 for the Special Needs Registry (SNR) before a mandatory evacuation order is announced43,44. The 72-h evacuation timeline announced by the City of New Orleans specifies a window of 42 h to collect evacuees from their houses and transport them to the Smoothie King Center, which serves as a transfer and processing center, and from there, evacuees will board a bus to state or federal shelters in other cities43. Finally, the city government will bring evacuees back to their homes or local shelters once it becomes safe to return to New Orleans43. As of February 23, 2021, the special needs registry for New Orleans included approximately 4000 individuals64.

We considered four datasets, each consisting of a depot (rescue center) and several nodes (houses) randomly distributed within a flat 2D grid world. The distance metric is Manhattan distance, as in an urban area laid out in blocks (see Supplementary Fig. S29 online). Each dataset is in the shape of a square (about 3 km (times) 3 km)65. The land area of New Orleans is (438.8,mathrmkm^2) ((169.42,mathrmmile^2))66 so the size of each dataset represents about 2% of the area of the city.

To enhance the efficiency of computation, we normalize the neighborhood as a 1 (times) 1 box, and transform the normalized distance back to actual distance to get total evacuation time.

To estimate the number of people in each house, we used the average household size (2014-2018), which is 2.44 persons per household in New Orleans, based on the American Community Survey (ACS) of U.S. Census Bureau66, assuming a normal distribution with mean=2.44 and standard deviation=0.5. If the household size is not an integer, it is rounded to the nearest integer. So, the household size ranges from 1-4 people. For simplicity, we assume that, if one person in a house is on the special needs registry, then all of the people in that house are on the registry as well. Supplementary Table S1 contains the summary of the datasets used in our study.

We ran the sweep algorithm and the pre-trained deep reinforcement learning model on a GPU NVIDIA Tesla K80, which enabled us to complete all experiments within one second with a batch size of 25642. For our conceptual model of a pre-planned evacuation map, calculation time is not relevant, but the possibility of quick calculation is consistent with real-time adjustments. In particular, a timely and scalable solution is of vital importance in real-world disaster response and evacuation route planning.

### Emergency vehicle and social distancing

We chose the average speed of 8 km/h (equivalent to 5 mph) for our simulation, based on the idea that emergency vehicles usually move slowly during evacuation operations, especially when the elderly and people with disabilities need extra time to get on and off the vehicles. To inv
estigate how social distancing in an emergency vehicle impacts the total time and the number of routes in disaster evacuation operation, we consider a nominal evacuation vehicle as a bus that seats up to 64 people. Social distancing could decrease that to 32, 16, 8, 4, or 2 people per emergency vehicle. The analysis can also be applied to a nominal capacity of 32, 16, 8, or 4, with an increasingly restricted range of social distancing. To our knowledge, this study is the first research which incorporates FEMA’s official guidelines53 into the investigation of the impacts of social distancing on emergency evacuation. In light of FEMA’s guidance on Recommended Evacuee Queuing and Boarding Process53, with social distancing protocols, only up to 25-28 passengers are allowed in a 56-passenger motor coach, which is the most widely available motor coach. In other words, the vehicle capacity decreases to half of its original value when social distancing is applied. All of the specifics like the size of the neighborhoods, vehicle speed, and vehicle capacity are reasonable, chosen to illustrate the general issues.

### Evacuees and evacuation planning

This is a model for a pre-planned evacuation, with a map of locations and the number of people per location known in advance. Such a map could be based on a registry of advance requests for evacuation assistance, or it could be developed in parallel with the forecast leading to an evacuation. With this concept, we know in advance the spatial distribution of the demand in the capacitated vehicle routing problem. In addition, we considered the rescue center to have unlimited accommodation capacity. Consistent with the Guide of City-Assisted Evacuation (CAE) for Hurricanes in New Orleans, people transported from their houses to the rescue center will be moved from there to be treated by appropriate emergency health care services at other disaster relief centers, such as state or federal long-term shelters43,67.

### Algorithm design

In this section, we described the general design of the DNN-based and non-DNN algorithms in this study.

#### Non-DNN solution–Sweep Algorithm

The sweep algorithm is a computationally efficient non-DNN solution typically used in real-world evacuations, business logistics, and supply chain management68,69. It starts with an arbitrary line from the depot (the rescue center). The order of houses to be visited by an emergency vehicle is determined by sweeping this line counter-clock wise and adding houses one by one when the line intersects these houses. In addition, the emergency vehicle must return to the rescue center when it reaches its passenger limit. After sweeping the line for 360 degrees, the evacuation operation is complete.

#### DNN-based solution–Deep Reinforcement Learning

In this study, we selected deep reinforcement learning as the DNN-based solution. Deep reinforcement learning is good at searching for optimal solutions in a relatively short period of time and is well-known for its capacity of adaptively resolving similarly complex problems, such as the Game of Go, robotics, and computer games. In particular, we used the Attention Model (AM)42 because this algorithm outperformed several common baseline algorithms and models for various routing problems, including the Capacitated Vehicle Routing Problem (CVRP). The Attention Model integrates the REINFORCE algorithms70 with greedy rollout baseline to the attention-based Transformer model71 and the variant of Graph Attention Networks (GATs)72 whose Convolutional Neural Networks (CNNs) with masked self-attention layers analyze graph-structured data efficiently. For all of our experiments of deep reinforcement learning, we adapted the algorithm designs in42 to our scenarios of evacuation with social distancing.

### Evaluation of DNN-based and non-DNN solutions

According to the 72-h hurricane evacuation timeline announced by New Orleans43, there will be only 42 h to pick up evacuees (Fig. 2). To evaluate the performance of DNN-based and non-DNN solutions, we used 24 and 42 h as the thresholds of categorizing the results into three levels of performance: satisfactory (< 24 h), borderline (24–42 h), and not allowed (> 42 h) (Fig. 2).

### Disclaimer

This disclaimer informs audience that this study should really only provide as their possess references alternatively than formal suggestions for any jurisdictions. The situations of simulations done in this review are only illustrations for the reason of academic analysis. Any steps, which includes but not confined to decisions, guidelines, and reports, taken dependent on any section of this study is the sole legal responsibility of readers, not authors in this review.

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