Routing algorithms as tools for integrating social distancing with emergency evacuation

One of the lessons from the COVID-19 pandemic is the importance of social distancing, even in challenging circumstances such as pre-hurricane evacuation. To explore the implications of integrating social distancing with evacuation operations, we describe this evacuation process as a Capacitated Vehicle Routing Problem (CVRP) and solve it using a DNN (Deep Neural Network)-based solution (Deep Reinforcement Learning) and a non-DNN solution (Sweep Algorithm). A central question is whether Deep Reinforcement Learning provides sufficient extra routing efficiency to accommodate increased social distancing in a time-constrained evacuation operation. We found that, in comparison to the Sweep Algorithm, Deep Reinforcement Learning can provide decision-makers with more efficient routing. However, the evacuation time saved by Deep Reinforcement Learning does not come close to compensating for the extra time required for social distancing, and its advantage disappears as the emergency vehicle capacity approaches the number of people per household.


Results
Trade-offs in size of neighborhood and social distancing. In this section, we analyzed the tradeoffs in size of neighborhood and social distancing by simulating the CVRP process of an emergency vehicle repeatedly starting from a rescue center to collect residents house by house until its vehicle capacity is met. In this study, we used the locations of nodes (or "houses") and a depot (a rescue center) in the standard CVRP benchmarking datasets published by 50,51 . We generated the demand of each node (household size or the number of people in each house to be picked up by the rescue vehicle) using the average household size in New Orleans (see "Methods"). The sizes of neighborhood in the four datasets we used in this study were 20, 35, 52, and 68 houses on the special needs registry. Social distancing limited the number of passengers allowed in one rescue vehicle, with capacities of 64,32,16,8,4, and 2 passengers per rescue vehicle. One can think of this range of vehicle capacities as representing the span from no social distancing to very strict social distancing in a large bus. But it can also be a way to explore modest distancing (2-to 4-fold capacity changes) in vehicles ranging from a large bus to a small van or sedan.
After the emergency vehicle picked up every resident on the special needs registry in the neighborhood, we summed the total time and the number of routes as the outputs (see Supplementary information for more information about the outputs). Figure 1 is an example of the output.
New Orleans plans to pick up residents starting 54 h before the storm reaches the coast and collect the last residents by 30 h before the storm reaches the coast, for a 24-h window (Fig. 2) 43    To accommodate the transit time required to travel from each neighborhood to the centrally located Smoothie King Center and back, we added 0, 0.5, 1, or 2 h to each evacuation route. In a realistic city layout, transit times would differ among neighborhoods, but they should be similar for each route within a neighborhood. With zero transit time, the de facto assumption is that the Smoothie King Center is in each neighborhood. With additional fixed transit time per route, the black squares in Fig. 1 and other figures in Supplementary information represent the closest freeway on-ramp or intersection with a major thoroughfare, marking the transition from collecting people within a neighborhood to transporting people to the Smoothie King Center.
Size of neighborhood. With any level of capacity per rescue vehicle (social distancing), the total time for evacuations increased as the size of neighborhood (number of people on the special needs registry) increased (Fig. 3). The total time of both DNN-based and non-DNN solutions rose more steeply with stronger social distancing across all sizes of neighborhood. For example, with 32 people in an emergency vehicle instead of 64, increasing the neighborhood size from 20 houses to 68 (a factor of 3.4), increased the total time by 1.89 h and 4.79 h for DNN-based and non-DNN solutions respectively. However, with more stringent social distancing (2 people per vehicle), increasing the neighborhood size from 20 houses to 68 increased the total time by 23.37 h and 22.18 h for DNN-based and non-DNN solutions respectively. The average total time across six vehicle capacities (2,4,8,16,32, and 64 people per emergency vehicle) and two algorithms (deep reinforcement learning and sweep algorithm) rises with the neighborhood size, and the change is linearly proportional to the change in neighborhood size (Fig. 4). For example, with 16 people per emergency vehicle, increasing the neighborhood size from 20 houses to 68 (a factor of 3.4), increased the total time by 2.37-fold and 2.97-fold for cases with the minimum (+0 h/route) and the maximum (+2 h/route) transit time respectively. Compared to the cases without adding any transit time to each route, the relationship between neighborhood size and the average total time is closer to linearly proportional in the cases with longer transit time added to each route.
Without adding any transit time to each route, a disaster manager can evacuate everyone on the special needs registry in all sizes of neighborhood within 24 h (achieving the Satisfactory threshold) (Fig. 4). In addition, the smaller the neighborhood size, the more likely every resident on the special needs registry can be evacuated within 24 h (Satisfactory threshold) or 42 h (Borderline threshold). With a transit time of 2 h per route, only the smallest neighborhood can be fully evacuated by the end of 42 h. Of course, the trade-off is that dividing a city into smaller neighborhoods increases the number of neighborhoods to evacuate.
Social distancing. As expected, for any neighborhood size, algorithm, and transit time, stricter social distancing increases the total evacuation time (Fig. 5). For the strictest social distancing (2 people per vehicle), the evacuation time is less than 42 h only when transit time is zero. With milder social distancing (32 people per vehicle), the sensitivity of evacuation time to passenger limit is approximately the same in the DNN-based and One of the critical elements of an evacuation plan is the number of vehicles to allocate. A rough estimate for this can come from the ratio of the size of the special needs registry to the number of people evacuated by a single vehicle within the 42-h target window (Fig. 6). With stricter social distancing, the number of emergency vehicles increased. Stricter social distancing also increased the risk that one emergency vehicle could not fully evacuate one neighborhood within 24 or 42 h (Figs. 5 and 6).

Efficacy of DNN-based and non-DNN solutions.
Across all of the scenarios without additional transit time in each route, the DNN-based solution generated a shorter total evacuation time than the non-DNN solution in 66.67% of the cases (Fig. 7). The DNN-based solution required fewer routes than the non-DNN solution in 8.33% of the scenarios. On average, the DNN-based method took less time but used more routes. With additional transit time in each route, the advantage of the DNN-based algorithm decreased (Fig. 7). In general, the DNN-based solutions outperformed the non-DNN solutions, except when neighborhood size was very small ( DNN-based solutions outperformed non-DNN solutions by up to 40.18% (Fig. 7). The advantage of the DNN-based approach was largest with high-capacity vehicles, mild social distancing, and larger neighborhoods ( Fig. 7b-d). In visual terms, the DNN-based method outperformed the non-DNN method because its routing patterns were smoother loops compared to the non-DNN method, which wasted a lot of time going back and forth among houses with its more "spiky" routing ( Fig. 1

and the other figures in Supplementary information).
However, with the non-DNN approach, the vehicle always picked up passengers until it reached its capacity, while in some of the DNN-based solutions, the vehicle returned to the rescue center before reaching full capacity (see Supplementary Fig. S1). This led to the DNN-based solutions sometimes using more routes than the non-DNN solutions. An added transit time was effectively a penalty on the number of routes per neighborhood, leading to DNN-based solutions using up to 75.38% more time than the non-DNN solutions (Fig. 7a).  (Fig. 8). In 58.33% of the cases the difference is within ±10% . In 33.33% of the cases, DNN-based and non-DNN solutions used the same number of routes (Fig. 8b). In these cases, the transit time per route did not impact total evacuation time too much, and the advantage of the DNN-based solutions was expressed (Fig. 7b-d).
The advantage of deep reinforcement learning faded with smaller vehicles or stricter social distancing. When the vehicle capacity was close to the number of people in each household, the DNN-based solution performed about the same as or even worse than the sweep algorithm. It is not surprising that the DNN-based method did not work well when vehicle capacity was close to the household size. This is a situation known as the demand of a single node in the Capacitated Vehicle Routing Problem. If the vehicle collects two people from one house and then returns to the rescue center, the problem collapses to one of determining the shortest path between each node and the depot (Supplementary Fig. S6). The order of visiting houses might be different, but no version of a DNN-based approach would outperform a non-DNN approach.
We did not specifically train the DNN-based model for vehicle sizes close to the number of people in a house, largely because the fact that all solutions converge at some point indicates that this retraining could provide no more than marginal improvements. In addition, a DNN-based model is more complicated than is appropriate for these simple edge cases.
Across all scenarios, the DNN-based method required 1.67 more routes (20.96%) than the non-DNN solution but saved 0.73 h (14.78%), 0.1 h (6.86%), 0.93 h (2.11%), and 2.6 h (3.59%) on average for the cases with Although deep reinforcement learning can provide more efficient evacuation routing, the time performance boost from implementing the DNN-based solution was not large enough to fully compensate the time penalty from stricter social distancing without other compromises (Figs. 5 and 7). Evacuation plans can be modified to accommodate social distancing, but the modifications will require operational changes like increasing the number of vehicles (and decreasing the size of the neighborhood each serves) or extending the evacuation timeline. The magnitude of the required operational changes becomes larger as the social distancing becomes more aggressive.
Even if the addition of DNN-based evacuation plans can make a real contribution to efficiency and could, in critical cases, be the difference between a successful evacuation and one that is not completed in the allowed window, it is imperative for disaster managers to re-examine the evacuation timeline and incorporate additional disaster relief resources, such as more emergency vehicles, into any evacuation operation that requires social distancing.

Discussion
Artificial intelligence has been employed to solve various hard problems in operations research, computer science, business, healthcare, and other fields. We showed how human-centered AI techniques can augment the efficiency of an evacuation, but its benefit decreases and eventually disappears with stronger requirements for social distancing. The findings from our research are relevant to other disaster evacuations that are based on www.nature.com/scientificreports/ a registry of people to be collected. The efficiency improvements that come from implementing a DNN-based solution can be substantial, but they may not compensate for the extra time required for adding social distancing to each evacuation route. In general, the DNN-based solutions were not useful for cases where vehicle capacity was close to the size of a single household (node). In typical CVRP simulations and benchmarking research 42,52 , the vehicle capacity is much larger than the demand of each node. In future work, the DNN-based model could be retrained for scenarios with low-capacity vehicles. However, when the vehicle goes to only one house and then returns to the rescue center, all possible solutions collapse to the same performance. Furthermore, with low-capacity vehicles, the total time both DNN-based and non-DNN methods need is far from achieving the threshold, and a modest performance boost from a DNN-based solution would be unlikely to change this pattern, especially in larger neighborhoods (Fig. 5d).
On the other hand, when vehicle capacity is much larger than the size of each household, multiple households can be combined in a route, and it is non-trivial to come up with an efficient routing strategy. In cases like these, the DNN-based model can reinforce itself by learning from various possible solutions. It is not guaranteed that a DNN-based method can always outperform a non-DNN method, especially in settings without sufficient data (possible routing solutions) for learning and reinforcing the DNN-based model. The limitation of DNN-based method can be a part of the planning for evacuation routing, suggesting cases where DNN-based approaches have and do not have the potential to be helpful.
To accommodate the challenges in disaster evacuations during a pandemic, the approach used in this study could be extended in many ways. For example, both DNN-based and non-DNN methods could be modified A transit time to a central evacuation center, required in any real-world situation, tends to decrease the advantage of an efficient routing algorithm (Fig. 7). For our analysis, it creates an additional burden on the DNN-based approach. This arises from the fact that the DNN algorithm is attempting to minimize the time collecting people, without regard to the number of routes. But because transit adds a fixed time to each route, it penalizes any solution that requires additional routes. In future work, it may be possible to address this through multi-objective optimization (minimizing both the evacuation time and the number of routes) or to design test neighborhoods with depots located at some distance.
When social distancing is required, evacuations become more complicated, and they require additional resources. Disaster managers will need to accommodate these complexities both through capitalizing on tools for more efficient routing and in other aspects of the evacuations. Specifically, they may want to explore options for (1) extending the evacuation timeline, (2) increasing the number of evacuation vehicles that serves each neighborhood, or (3) partitioning the evacuation area into a larger number of smaller neighborhoods, each served by a single vehicle. In any particular setting, the preferred option will be determined by local resources and constraints. In every location, however, it will be wise to plan ahead for the possibility of an evacuation with social distancing.
One of the durable lessons from COVID-19 is that governments, NGOs, and community members should have all-hazard emergency operations and evacuation plans that consider the interactive effects of both a pandemic and a wide range of possible disasters. Disaster managers should also take advantage of every efficiency they can find, including DNN-based vehicle routing. Consistent with the FEMA suggestion that jurisdictions incorporate social distancing into emergency evacuation during the COVID-19 pandemic 53 , this study investigates the impacts of social distancing on an evacuation timeline and the number of emergency vehicles required to evacuate a city.

Methods
Dataset and problem formulation. Vehicle Routing Problems (VRPs) in the context of pre-disaster evacuation are both critical and complex 54 . Although we cannot include all real-world constraints, we can roughly formulate the problem of pre-disaster evacuation as the most studied variant of VRPs, the Capacitated Vehicle Routing Problem (CVRP) with a rescue center (typically called depot) and a given set of n houses denoted nodes or customers, N = {1,2,...,n} 42,55 .
Traditionally, there are homogeneous vehicles K with the same capacity C, where K = {1, 2, . . . , |K|} , C > 0. In this study, we calculate total evacuation time based on one emergency vehicle for each neighborhood (with all neighborhoods the same size and being evacuated in parallel), so K is the same vehicle for every route in K in a neighborhood, and |K| is the number of routes required to evacuate each neighborhood. The number of people on the special needs registry to be picked up (typically referred to as demand) from each house is d j , where 0 < d j ≤ C, j ∈ N. Let E = {e = {i, j} = {j, i} : i, j ∈ {0, 1, 2, . . . , n}, i � = j} be the edge set 55 . An emergency vehicle moving from house i to j incurs travel cost c ij for {i, j} ∈ E or evacuation time in this study 42,55 . We define an integer www.nature.com/scientificreports/ 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 j [55][56][57] . The vehicle starts from the rescue center, goes to a subset of houses S ⊆ N to pick up people, and finally returns to the rescue center before or until the vehicle capacity C is reached 55 . 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 follows [55][56][57] .
According to the parameters we defined earlier [55][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 center 55,56,58 . 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 problems 50,51 . 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) 59-63 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 announced 43,44 . 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 cities 43 . Finally, k∈K n i=0 n j=1,i� =j k∈K i∈S j∈S,i� =j www.nature.com/scientificreports/ the city government will bring evacuees back to their homes or local shelters once it becomes safe to return to New Orleans 43 . As of February 23, 2021, the special needs registry for New Orleans included approximately 4000 individuals 64 . 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 × 3 km) 65 . The land area of New Orleans is 438.8 km 2 ( 169.42 mile 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 × 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 Bureau 66 , 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 256 42 . 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 investigate 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 guidelines 53 into the investigation of the impacts of social distancing on emergency evacuation. In light of FEMA's guidance on Recommended Evacuee Queuing and Boarding Process 53 , 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 shelters 43,67 . 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 management 68,69 . 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 algorithms 70 with greedy rollout baseline to the attention-based Transformer model 71 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 in 42 to our scenarios of evacuation with social distancing.