Emergency situations that require the evacuation of urban areas can arise from man-made causes (e.g., terrorist attacks or industrial accidents) or natural disasters, the latter becoming more frequent due to climate change. As a result, effective and fast methods to develop evacuation plans are of great importance. In this work, we identify and propose the Bus Evacuation Orienteering Problem (BEOP), an NP-hard combinatorial optimization problem with the goal of evacuating as many people from an affected area by bus in a short, predefined amount of time. The purpose of bus-based evacuation is to reduce congestion and disorder that arises in purely car-focused evacuation scenarios. To solve the BEOP, we propose a deep reinforcement learning-based method utilizing graph learning, which, once trained, achieves fast inference speed and is able to create evacuation routes in fractions of seconds. We can bound the gap of our evacuation plans using an MILP formulation. To validate our method, we create evacuation scenarios for San Francisco using real-world road networks and travel times. We show that we achieve near-optimal solution quality and are further able to investigate how many evacuation vehicles are necessary to achieve certain bus-based evacuation quotas given a predefined evacuation time while keeping run time adequate.
Figures & tables
Figure 1: GREAT-EER Framework for the BEOP: a BEOP instance is transformed into a graph that encodes the pairwise shortest paths between all relevant nodes. Then, the GREAT encoder is used to process this edge-based graph. Finally, a pointer network iteratively selects nodes to visit and their order. The resulting tour is returned as the output.
Symbol
Description
General problem properties
n
number of evacuation nodes
V={0,1,…,n}
set of safe location (or “depot”) 0 and evacuation nodes
E={(u,v)∣u,v∈V,u=v}
set of directed edges
T
maximum evacuation time
K={1,…,k}
set of evacuation vehicles
Table 1: Notation for BEOP
Figure 2: Simplified visualization of the MDP for BEOP showing the different action cases.
Figure 3: Validation performance (evacuation quota) per epoch for GREAT-EER models with 100 evacuation points
Figure 4: Validation performance (evacuation quota) per epoch for GREAT-EER model with 20 evacuation points
Figure 5: Example evacuations generated by different solution methods in San Francisco given 1.5h evacuation time and 3 buses for evacuation with 50 seats each.
Instance
Gap w.r.t. Gurobi Bound(%)
Evac. Quota (%)
Runtimes (s)
Veh
TL
Cap
TW
G.-EER
Gurobi
Greedy
G.-EER
Gurobi
Greedy
G.-EER
Gurobi
Greedy
1
0.5
20
0
30.19
27.95
70.5
55.61
56.83
44.32
0.385
10054.9
0.003
1
0.5
20
0.3
30.36
28.56
70.0
54.66
56.12
43.6
0.338
9968.1
0.002
1
0.5
40
0
11.01
8.86
29.47
68.17
69.61
58.6
0.351
5580.3
0.003
1
0.5
40
0.3
12.57
9.08
31.64
66.77
69.05
57.25
0.351
5715.9
0.003
1
1
20
0
6.57
5.55
24.5
87.37
88.25
75.83
0.377
11047.5
0.004
Table 2: Comparison of GREAT-EER (G.-EER), Gurobi (30min time limit) and greedy baseline
Instance
Gap w.r.t. Gurobi Bound (%)
Evac. Quota (%)
Runtimes (s)
Veh
TL
Cap
TW
Gurobi
Gurobi
G&G
Gurobi
Gurobi
G&G
Gurobi
Gurobi
G&G
30min
1min
1 min
30min
1min
1min
30min
1min
1min
1
0.5
20
0
27.95
30.23
26.26
56.83
55.92
57.12
10054.9
568.8
529.9
1
0.5
20
0.3
28.56
28.98
28.79
56.12
55.96
55.58
9968.1
558.0
561.5
1
0.5
40
0
8.86
9.43
8.86
69.61
69.33
69.61
5580.3
328.2
338.2
1
0.5
40
0.3
9.08
9.08
9.95
69.05
69.05
68.34
5715.9
375.2
399.2
Table 3: Comparison of Gurobi (30min time limit), Gurobi (1min time limit) and Gurobi with warm GREAT-EER initialization “G&G” (1min time limit). Gaps w.r.t. the bound found by the Gurobi model with 30 minutes runtime.
Figure 6: Evacuation quotas in different BEOP settings (number of vehicles, evacuation time and vehicle capacity)
Figure 7: Visualization of the quota of people evacuated by bus depending on their distance (fraction of maximum evacuation time) to the safe location.
Evacuation time
time window fraction
Capacity 30
Capacity 40
Capacity 50
Capacity 60
0.5 h
0
0.26 / 0.20
0.29 / 0.23
0.31 / 0.24
0.33 / 0.25
0.3
0.26 / 0.20
0.28 / 0.23
0.31 / 0.23
0.32 / 0.24
0.6
0.25 / 0.19
0.27 / 0.21
0.30 / 0.22
0.31 / 0.23
1 h
0
0.46 / 0.37
0.51 / 0.39
0.54 / 0.43
0.57 / 0.44
0.3
0.45 / 0.36
0.50 / 0.38
0.53 / 0.42
0.56 / 0.44
0.6
0.44 / 0.36
0.48 / 0.38
0.52 / 0.42
0.55 / 0.43
Table 4: Results for n=1 vehicles. Entries show GREAT-EER mean evacuation quota and greedy mean evacuation quota.
Evacuation time
time window fraction
Capacity 30
Capacity 40
Capacity 50
Capacity 60
0.5 h
0
0.45 / 0.36
0.50 / 0.39
0.53 / 0.41
0.56 / 0.43
0.3
0.44 / 0.36
0.49 / 0.39
0.52 / 0.41
0.55 / 0.43
0.6
0.43 / 0.34
0.48 / 0.37
0.51 / 0.40
0.53 / 0.42
1 h
0
0.73 / 0.62
0.79 / 0.68
0.83 / 0.72
0.85 / 0.73
0.3
0.72 / 0.62
0.78 / 0.66
0.82 / 0.71
0.84 / 0.71
0.6
0.71 / 0.60
0.76 / 0.64
0.80 / 0.69
0.81 / 0.69
Table 5: Results for n=2 vehicles. Entries show GREAT-EER mean evacuation quota and greedy mean evacuation quota.
Evacuation time
time window fraction
Capacity 30
Capacity 40
Capacity 50
Capacity 60
0.5 h
0
0.59 / 0.49
0.65 / 0.54
0.69 / 0.57
0.72 / 0.60
0.3
0.59 / 0.48
0.65 / 0.53
0.68 / 0.57
0.72 / 0.59
0.6
0.58 / 0.47
0.63 / 0.51
0.66 / 0.56
0.69 / 0.58
1 h
0
0.89 / 0.79
0.95 / 0.84
0.97 / 0.88
0.98 / 0.90
0.3
0.88 / 0.79
0.93 / 0.83
0.96 / 0.86
0.98 / 0.88
0.6
0.87 / 0.77
0.92 / 0.80
0.94 / 0.85
0.96 / 0.86
Table 6: Results for n=3 vehicles. Entries show GREAT-EER mean evacuation quota and greedy mean evacuation quota.
Evacuation time
time window fraction
Capacity 30
Capacity 40
Capacity 50
Capacity 60
0.5 h
0
0.71 / 0.59
0.77 / 0.67
0.81 / 0.70
0.83 / 0.72
0.3
0.70 / 0.59
0.76 / 0.65
0.80 / 0.69
0.82 / 0.70
0.6
0.68 / 0.57
0.74 / 0.64
0.78 / 0.69
0.80 / 0.70
1 h
0
0.97 / 0.91
0.99 / 0.95
1.00 / 0.97
1.00 / 0.98
0.3
0.97 / 0.89
0.99 / 0.94
1.00 / 0.97
1.00 / 0.97
0.6
0.96 / 0.88
0.99 / 0.91
0.99 / 0.94
1.00 / 0.95
Table 7: Results for n=4 vehicles. Entries show GREAT-EER mean evacuation quota and greedy mean evacuation quota.
Instance
Evacuation quota (%)
Vehicles
Time Limit
GREAT-EER
Greedy
1
0.5
33.03
24.64
1
56.61
45.18
1.5
72.33
61.37
2
0.5
55.50
42.68
1
83.74
72.21
Table 8: Average evacuation quota (over 10 instances) of GREAT-EER vs. greedy in the hazard zone setting. Higher is better.
Figure 8: Example evacuations generated by GREAT-EER and greedy in San Francisco given 1.5h evacuation time with hazard zones
Instance
Evacuation quota (%)
Vehicles
Time limit
GREAT-EER
Greedy
3
0.5
43.03
35.68
1
69.73
61.60
1.5
87.40
78.82
2
97.25
90.82
4
0.5
52.35
44.10
Table 9: Average evacuation quota (over 10 instances) of GREAT-EER and greedy for instances of size 200. Higher is better.
Figure 9: Evacuation quotas by stochastic GREAT-EER model and deterministic plans generated by GREAT-EER
The Bredesen Center for Interdisciplinary Research and Graduate Education · Department of Mechanical and Aerospace Engineering, University of Tennessee