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Siting First Aid Stations with LSCP on Toronto’s University Campus¶
Authors: Timothy Ellersiek, James Gaboardi
This notebook implements the location set covering problem (LSCP) to site first aid stations for Toronto’s University campus. LSCP determines locations such that as many demand points as possible are within the maximum service radius. By doing so, the number of locations required to cover all demand points is minimized. In contrast to the maximum covering location problem it does not require a maximum amount of locations as an input parameter. Hence, the use cases for LSCP are specific. One common scenario is finding suitable locations for hospitals in cities where the amount is governed by reachability preconditions to households in a certain amount of time.
In this notebook demand is described as buildings within Toronto’s University campus. Possible locations for the first aid stations are determined using Overpass API which fetches geometries directly from OpenStreetMap.org.
The routing matrix is computed using Open Source Routing Machine’s demo server API.
Prerequisites¶
To run this jupyter notebook locally, you will have to install the following additional libraries into your python environment.
[1]:
%config InlineBackend.figure_format = "retina"
%load_ext watermark
%watermark
Last updated: 2023-12-10T13:49:32.513924-05:00
Python implementation: CPython
Python version : 3.11.6
IPython version : 8.18.0
Compiler : Clang 15.0.7
OS : Darwin
Release : 23.1.0
Machine : x86_64
Processor : i386
CPU cores : 8
Architecture: 64bit
[2]:
import folium
import json
import numpy
import overpy
import pulp
import routingpy
from routingpy import OSRM
import shapely
from shapely.geometry import shape, Point
import spopt
from spopt.locate import LSCP
%watermark -w
%watermark -iv
Watermark: 2.4.3
pulp : 2.7.0
overpy : 0.6
spopt : 0.5.1.dev59+g343ef27
shapely : 2.0.2
routingpy: 1.2.1
json : 2.0.9
folium : 0.15.0
numpy : 1.26.2
Create overpy.Overpass() instance in order to make queries
[3]:
api = overpy.Overpass()
First of all we want to fetch all buildings in the University of Toronto boundary
[4]:
result = api.query(
"""
[out:json];
area[name="Toronto"]->.b;
way(area.b)[name="University of Toronto"];
map_to_area -> .a;
way[building="university"](area.a);
out center;
"""
)
Create a folium.Map() instance to visualize the data
[5]:
m = folium.Map(
location=[43.6624674, -79.3988052], tiles="cartodbpositron", zoom_start=15
)
Let’s add our results to the map!
[6]:
def get_centroid(pnt_info, xy=True):
"""Extract centroid from location information."""
if isinstance(pnt_info, overpy.Way):
x, y = pnt_info.center_lon, pnt_info.center_lat
if not xy:
x, y = y, x
else:
x, y = pnt_info["center"][1], pnt_info["center"][0]
return [x, y]
[7]:
get_tag = lambda w: w.tags.get("name", "n/a")
[8]:
university_buildings = []
university_buildings_locations = []
university_buildings_fg = folium.FeatureGroup("University Buildings").add_to(m)
for idx, way in enumerate(result.ways):
tag = get_tag(way)
loc_yx = get_centroid(way, xy=False)
folium.CircleMarker(
location=loc_yx,
radius=3,
fill=True,
fill_opacity=0.3,
popup=folium.Popup(tag, show=False),
color="blue",
).add_to(university_buildings_fg)
# building centroids
university_buildings.append({"name": tag, "center": get_centroid(way), "idx": idx})
university_buildings_locations.append(loc_yx)
n_uni_bld = len(university_buildings)
m
[8]:
Make this Notebook Trusted to load map: File -> Trust Notebook
And now we want to fetch some roads in the same area and return their center points which we can use for our potential locations for first aid stations
[9]:
result_roads = api.query(
"""
[out:json];
area[name="Toronto"]->.b;
way(area.b)[name="University of Toronto"];
map_to_area -> .a;
way["highway"~"secondary|tertiary|unclassified"](area.a);
out center;
"""
)
Similarly we want to add them to the map
[10]:
first_aid_stations = []
first_aid_stations_locations = []
first_aid_stations_fg = folium.FeatureGroup("First Aid Station Locations").add_to(m)
for idx, way in enumerate(result_roads.ways):
tag = get_tag(way)
loc_yx = get_centroid(way, xy=False)
folium.CircleMarker(
location=loc_yx,
radius=2,
fill=True,
fill_opacity=1,
popup=folium.Popup(tag, show=False),
color="black",
).add_to(first_aid_stations_fg)
# road midpoints
first_aid_stations.append(
{"name": tag, "center": get_centroid(way), "idx": idx + n_uni_bld}
)
first_aid_stations_locations.append(loc_yx)
m
[10]:
Make this Notebook Trusted to load map: File -> Trust Notebook
Computing the Distance (Routing) Matrix¶
Given our potential first aid station locations and the university buildings on the campus we want to compute our distance matrix. Instead of using simple euclidean distance we can make use of proper routes following roads which will yield more realistic travel times. The routing-py library helps us achieve this which features the ability to use different open source frameworks such as OSRM or Valhalla as well as 3rd party API’s such as Google Maps or HERE.
[11]:
client = OSRM(base_url="https://router.project-osrm.org")
Isolate all location coordinates
[12]:
all_locations = university_buildings_locations + first_aid_stations_locations
all_locations = [[x, y] for y, x in all_locations]
all_locations[:5]
[12]:
[[Decimal('-79.3988052'), Decimal('43.6624674')],
[Decimal('-79.3994442'), Decimal('43.6643897')],
[Decimal('-79.3895107'), Decimal('43.6648522')],
[Decimal('-79.3971848'), Decimal('43.6625440')],
[Decimal('-79.3958563'), Decimal('43.6628344')]]
Define source and target indices
[13]:
source_indices = [i["idx"] for i in university_buildings]
target_indices = [i["idx"] for i in first_aid_stations]
[14]:
source_indices[-5:], target_indices[:5]
[14]:
([117, 118, 119, 120, 121], [122, 123, 124, 125, 126])
Generate routing matrix object
[15]:
osrm_routing_matrix = client.matrix(
dry_run=False,
locations=all_locations,
sources=source_indices,
destinations=target_indices,
profile="pedestrian",
)
Isolate route durations
[16]:
cost_matrix = numpy.array(osrm_routing_matrix.durations)
cost_matrix.shape
[16]:
(122, 35)
[17]:
cost_matrix
[17]:
array([[242.5, 132.1, 164.3, ..., 67.5, 95.5, 135.9],
[220.3, 109.9, 155.6, ..., 117.9, 46.9, 113.7],
[183.5, 134.5, 152.4, ..., 285.5, 222.4, 138.3],
...,
[243.1, 132.7, 178.4, ..., 97.9, 75.4, 136.5],
[181.5, 42.6, 88.3, ..., 196.5, 234.1, 46.4],
[136.1, 167.6, 207.2, ..., 260.6, 211.2, 171.4]])
Solving the LSCP¶
Last but not least we want to run the solver. We choose the pulp.COIN_CMD and set the distance cutoff to 2 minutes and 30 seconds afoot. Finally we loop over the solution and add the sited stations and the corresponding allocated buildings to the map with different radiuses.
[18]:
# maximum acceptable service duration, 2 minutes and 30 seconds
SERVICE_RADIUS = 150
# set the solver (for available solvers run ``pulp.listSolvers(onlyAvailable=True)``)
solver = pulp.COIN_CMD(msg=False)
Create and solve an LSCP instance
[19]:
lscp = LSCP.from_cost_matrix(cost_matrix, SERVICE_RADIUS)
lscp = lscp.solve(solver)
How many first aid sites are needed to entirely cover the university buildings?
[20]:
lscp.problem.objective.value()
[20]:
3.0
Add the results to the map
[21]:
sited_first_aid_stations_fg = folium.FeatureGroup("Sited First Aid Stations").add_to(m)
allocated_buildings_demand_fg = folium.FeatureGroup("Allocated Buildings").add_to(m)
solution = {}
colors = [
"darkcyan",
"mediumseagreen",
"saddlebrown",
"lightskyblue",
"darkgoldenrod",
"coral",
"mediumvioletred",
"blueviolet",
"mediumorchid",
]
client_marker_size = 20
numpy.random.seed(0)
for i in range(len(first_aid_stations)):
if lscp.fac2cli[i]:
color_choice_idx = numpy.random.choice(range(len(colors)))
color = colors.pop(color_choice_idx)
# selected candidate facility sites
folium.CircleMarker(
location=first_aid_stations_locations[i],
radius=30,
fill=True,
popup=folium.Popup(f"Sited First Aid Station (index: {i})", show=True),
color=color,
).add_to(sited_first_aid_stations_fg)
# client locations with associated facility symbology
client_marker_size -= 4
solution[i] = []
for j in lscp.fac2cli[i]:
solution[i].append(university_buildings[j])
folium.CircleMarker(
location=university_buildings_locations[j],
radius=client_marker_size,
fill=True,
popup=f"Building allocated to First Aid Station with index: {i}",
color=color,
).add_to(allocated_buildings_demand_fg)
folium.LayerControl().add_to(m)
m
[21]:
Make this Notebook Trusted to load map: File -> Trust Notebook
It is important to note that the modeling solution is a direct result of the input cost matrix (seconds of travel) obtained from routingpy.OSRM. Although the solution set and decisions may not appear to optimal, the formulation of and solution to the LSCP are correct based on the input matrix.
[22]:
solution
[22]:
{6: [{'name': 'Sidney Smith Hall',
'center': [Decimal('-79.3988052'), Decimal('43.6624674')],
'idx': 0},
{'name': 'Robarts Library',
'center': [Decimal('-79.3994442'), Decimal('43.6643897')],
'idx': 1},
{'name': 'Koffler Student Services Centre',
'center': [Decimal('-79.3968996'), Decimal('43.6591667')],
'idx': 6},
{'name': 'Daniels Building',
'center': [Decimal('-79.4007115'), Decimal('43.6596491')],
'idx': 7},
{'name': 'Warren Stevens Building',
'center': [Decimal('-79.4010325'), Decimal('43.6627537')],
'idx': 8},
{'name': 'Ramsay Wright Building',
'center': [Decimal('-79.3990600'), Decimal('43.6632684')],
'idx': 13},
{'name': 'Clara Benson Building',
'center': [Decimal('-79.4002584'), Decimal('43.6629833')],
'idx': 14},
{'name': 'Fisher Rare Book Library',
'center': [Decimal('-79.3989770'), Decimal('43.6640748')],
'idx': 15},
{'name': 'Simcoe Hall',
'center': [Decimal('-79.3959493'), Decimal('43.6607473')],
'idx': 18},
{'name': 'n/a',
'center': [Decimal('-79.3981669'), Decimal('43.6591511')],
'idx': 23},
{'name': 'John H. Daniels Faculty of Architecture, Landscape & Design',
'center': [Decimal('-79.3980011'), Decimal('43.6586788')],
'idx': 24},
{'name': 'The Terrence Donnelly Centre for Cellular and Biomolecular Research',
'center': [Decimal('-79.3929327'), Decimal('43.6601354')],
'idx': 25},
{'name': 'Lassonde Mining Building',
'center': [Decimal('-79.3934676'), Decimal('43.6595254')],
'idx': 26},
{'name': 'Leslie Dan Pharmacy Building',
'center': [Decimal('-79.3917767'), Decimal('43.6599499')],
'idx': 27},
{'name': 'Wallberg Building',
'center': [Decimal('-79.3952383'), Decimal('43.6591926')],
'idx': 28},
{'name': 'Sanford Fleming Building',
'center': [Decimal('-79.3952159'), Decimal('43.6601368')],
'idx': 32},
{'name': 'Centre for Biological Timing and Cognition',
'center': [Decimal('-79.3993038'), Decimal('43.6628627')],
'idx': 34},
{'name': 'The Jones and Edna Davenport Chemical Research Building',
'center': [Decimal('-79.3977963'), Decimal('43.6613289')],
'idx': 35},
{'name': 'McLennan Physical Laboratories',
'center': [Decimal('-79.3985019'), Decimal('43.6610126')],
'idx': 36},
{'name': 'McLennan Physical Laboratories',
'center': [Decimal('-79.3985063'), Decimal('43.6604892')],
'idx': 37},
{'name': 'Cody Hall',
'center': [Decimal('-79.3975700'), Decimal('43.6606429')],
'idx': 38},
{'name': 'Physical Geography Building',
'center': [Decimal('-79.3965267'), Decimal('43.6603861')],
'idx': 50},
{'name': 'McLennan Physical Laboratories',
'center': [Decimal('-79.3981462'), Decimal('43.6605703')],
'idx': 51},
{'name': 'D. L. Pratt Building',
'center': [Decimal('-79.3948160'), Decimal('43.6596078')],
'idx': 52},
{'name': 'Central Steam Plant',
'center': [Decimal('-79.3978616'), Decimal('43.6599806')],
'idx': 53},
{'name': 'n/a',
'center': [Decimal('-79.3972158'), Decimal('43.6600916')],
'idx': 54},
{'name': 'n/a',
'center': [Decimal('-79.3992100'), Decimal('43.6642407')],
'idx': 56},
{'name': 'n/a',
'center': [Decimal('-79.3994414'), Decimal('43.6647887')],
'idx': 57},
{'name': 'Bancroft Hall',
'center': [Decimal('-79.3998739'), Decimal('43.6609086')],
'idx': 58},
{'name': 'Faculty Club',
'center': [Decimal('-79.4005248'), Decimal('43.6611294')],
'idx': 59},
{'name': 'Koffler House',
'center': [Decimal('-79.4006308'), Decimal('43.6605786')],
'idx': 61},
{'name': 'Lash Miller Laboratories',
'center': [Decimal('-79.3979944'), Decimal('43.6618573')],
'idx': 62},
{'name': 'Anthropology Building',
'center': [Decimal('-79.3984688'), Decimal('43.6598666')],
'idx': 63},
{'name': 'Newman Centre',
'center': [Decimal('-79.3981613'), Decimal('43.6644228')],
'idx': 64},
{'name': 'School of Graduate Studies - Office of the Dean',
'center': [Decimal('-79.3971708'), Decimal('43.6620125')],
'idx': 72},
{'name': 'Macdonald-Mullard House',
'center': [Decimal('-79.3970259'), Decimal('43.6618029')],
'idx': 73},
{'name': 'n/a',
'center': [Decimal('-79.3878370'), Decimal('43.6608131')],
'idx': 75},
{'name': 'Department of Obstetrics and Gynaecology',
'center': [Decimal('-79.3880876'), Decimal('43.6606226')],
'idx': 76},
{'name': 'Banting Institute',
'center': [Decimal('-79.3886232'), Decimal('43.6605821')],
'idx': 77},
{'name': 'Fields Institute for Research in Math Science',
'center': [Decimal('-79.3975564'), Decimal('43.6588127')],
'idx': 79},
{'name': 'Engineering Annex',
'center': [Decimal('-79.3954298'), Decimal('43.6596061')],
'idx': 80},
{'name': 'Haultain Building',
'center': [Decimal('-79.3936379'), Decimal('43.6599224')],
'idx': 81},
{'name': 'Tanz Neuroscience Building',
'center': [Decimal('-79.3918974'), Decimal('43.6603578')],
'idx': 82},
{'name': 'McMurrich Building',
'center': [Decimal('-79.3929260'), Decimal('43.6612699')],
'idx': 83},
{'name': 'Medical Sciences Building',
'center': [Decimal('-79.3933729'), Decimal('43.6607748')],
'idx': 84},
{'name': 'Centre of Criminology, School of Public Policy & Governance',
'center': [Decimal('-79.3929617'), Decimal('43.6619386')],
'idx': 85},
{'name': 'Graduate Students Union',
'center': [Decimal('-79.4003077'), Decimal('43.6608112')],
'idx': 105},
{'name': 'Bahen Centre for Information Technology',
'center': [Decimal('-79.3974539'), Decimal('43.6597085')],
'idx': 107},
{'name': 'Cumberland House',
'center': [Decimal('-79.3959338'), Decimal('43.6594572')],
'idx': 108},
{'name': 'Lash Miller Laboratories',
'center': [Decimal('-79.3984263'), Decimal('43.6615978')],
'idx': 112},
{'name': 'Convocation Hall',
'center': [Decimal('-79.3954478'), Decimal('43.6607760')],
'idx': 113},
{'name': 'Rosebrugh Building',
'center': [Decimal('-79.3932695'), Decimal('43.6600629')],
'idx': 114},
{'name': 'Mechanical Engineering Building',
'center': [Decimal('-79.3938368'), Decimal('43.6600562')],
'idx': 115},
{'name': 'Myhal Centre',
'center': [Decimal('-79.3965625'), Decimal('43.6607288')],
'idx': 117},
{'name': 'Robarts Common',
'center': [Decimal('-79.4000387'), Decimal('43.6642971')],
'idx': 119},
{'name': 'Medical Sciences Building',
'center': [Decimal('-79.3934363'), Decimal('43.6613361')],
'idx': 120}],
20: [{'name': 'J. M. Kelly Library',
'center': [Decimal('-79.3895107'), Decimal('43.6648522')],
'idx': 2},
{'name': 'E. J. Pratt Library',
'center': [Decimal('-79.3912921'), Decimal('43.6663202')],
'idx': 9},
{'name': 'Carr Hall',
'center': [Decimal('-79.3904148'), Decimal('43.6652328')],
'idx': 11},
{'name': 'Muzzo Family Alumni Hall',
'center': [Decimal('-79.3901376'), Decimal('43.6647637')],
'idx': 12},
{'name': 'Flavelle House',
'center': [Decimal('-79.3939081'), Decimal('43.6661332')],
'idx': 20},
{'name': 'Falconer Hall',
'center': [Decimal('-79.3939344'), Decimal('43.6667134')],
'idx': 21},
{'name': 'Isabel Bader Theatre',
'center': [Decimal('-79.3923649'), Decimal('43.6672514')],
'idx': 29},
{'name': 'Birge Carnegie',
'center': [Decimal('-79.3929187'), Decimal('43.6670922')],
'idx': 30},
{'name': 'Odette Hall',
'center': [Decimal('-79.3886433'), Decimal('43.6663535')],
'idx': 40},
{'name': 'Cardinal Flahiff Building',
'center': [Decimal('-79.3885826'), Decimal('43.6650353')],
'idx': 41},
{'name': 'Maritain House',
'center': [Decimal('-79.3899690'), Decimal('43.6656445')],
'idx': 42},
{'name': '2',
'center': [Decimal('-79.3898803'), Decimal('43.6654428')],
'idx': 43},
{'name': 'Sorbara Hall Student Residence',
'center': [Decimal('-79.3888806'), Decimal('43.6657524')],
'idx': 46},
{'name': 'Brennan Hall',
'center': [Decimal('-79.3897654'), Decimal('43.6663937')],
'idx': 48},
{'name': 'Burwash Dining Hall',
'center': [Decimal('-79.3917361'), Decimal('43.6674639')],
'idx': 49},
{'name': 'Munk School of Global Affairs',
'center': [Decimal('-79.3985410'), Decimal('43.6674676')],
'idx': 55},
{'name': 'Rotman School of Management and Rotman Commerce',
'center': [Decimal('-79.3992073'), Decimal('43.6669050')],
'idx': 66},
{'name': '162 St. George Street',
'center': [Decimal('-79.4000373'), Decimal('43.6670826')],
'idx': 69},
{'name': 'Elmsley Hall',
'center': [Decimal('-79.3902908'), Decimal('43.6666687')],
'idx': 89},
{'name': 'Old steam plant',
'center': [Decimal('-79.3902904'), Decimal('43.6661749')],
'idx': 90},
{'name': 'Toronto School of Theology',
'center': [Decimal('-79.3906171'), Decimal('43.6647518')],
'idx': 91},
{'name': 'Multicultural History Society of Ontario',
'center': [Decimal('-79.3905489'), Decimal('43.6645719')],
'idx': 92},
{'name': 'Jesuits in English Canada',
'center': [Decimal('-79.3904015'), Decimal('43.6642662')],
'idx': 93},
{'name': 'Goldring Student Centre',
'center': [Decimal('-79.3924051'), Decimal('43.6677618')],
'idx': 94},
{'name': 'Loretto College',
'center': [Decimal('-79.3900057'), Decimal('43.6673727')],
'idx': 97},
{'name': 'Law House',
'center': [Decimal('-79.3898938'), Decimal('43.6678167')],
'idx': 103},
{'name': 'Stephenson House',
'center': [Decimal('-79.3898109'), Decimal('43.6678347')],
'idx': 104},
{'name': 'Regis College',
'center': [Decimal('-79.3902796'), Decimal('43.6640045')],
'idx': 106},
{'name': 'Victoria College',
'center': [Decimal('-79.3919559'), Decimal('43.6669263')],
'idx': 109},
{'name': 'Emmanuel College',
'center': [Decimal('-79.3926504'), Decimal('43.6667142')],
'idx': 110},
{'name': 'Goldring Centre',
'center': [Decimal('-79.3983851'), Decimal('43.6669768')],
'idx': 111},
{'name': 'n/a',
'center': [Decimal('-79.3987786'), Decimal('43.6673873')],
'idx': 118}],
21: [{'name': 'Sidney Smith Hall',
'center': [Decimal('-79.3988052'), Decimal('43.6624674')],
'idx': 0},
{'name': 'Robarts Library',
'center': [Decimal('-79.3994442'), Decimal('43.6643897')],
'idx': 1},
{'name': 'Sir Daniel Wilson Residence (University College)',
'center': [Decimal('-79.3971848'), Decimal('43.6625440')],
'idx': 3},
{'name': 'University College',
'center': [Decimal('-79.3958563'), Decimal('43.6628344')],
'idx': 4},
{'name': 'Toronto Magnetic and Meteorological Observatory',
'center': [Decimal('-79.3946172'), Decimal('43.6631765')],
'idx': 5},
{'name': 'Koffler Student Services Centre',
'center': [Decimal('-79.3968996'), Decimal('43.6591667')],
'idx': 6},
{'name': 'Warren Stevens Building',
'center': [Decimal('-79.4010325'), Decimal('43.6627537')],
'idx': 8},
{'name': 'E. J. Pratt Library',
'center': [Decimal('-79.3912921'), Decimal('43.6663202')],
'idx': 9},
{'name': 'Northrop Frye Hall',
'center': [Decimal('-79.3921389'), Decimal('43.6664141')],
'idx': 10},
{'name': 'Carr Hall',
'center': [Decimal('-79.3904148'), Decimal('43.6652328')],
'idx': 11},
{'name': 'Muzzo Family Alumni Hall',
'center': [Decimal('-79.3901376'), Decimal('43.6647637')],
'idx': 12},
{'name': 'Ramsay Wright Building',
'center': [Decimal('-79.3990600'), Decimal('43.6632684')],
'idx': 13},
{'name': 'Clara Benson Building',
'center': [Decimal('-79.4002584'), Decimal('43.6629833')],
'idx': 14},
{'name': 'Fisher Rare Book Library',
'center': [Decimal('-79.3989770'), Decimal('43.6640748')],
'idx': 15},
{'name': 'Munk School of Global Affairs & Public Policy',
'center': [Decimal('-79.3966877'), Decimal('43.6650246')],
'idx': 16},
{'name': 'Claude T. Bissell Building',
'center': [Decimal('-79.3993354'), Decimal('43.6650287')],
'idx': 17},
{'name': 'Simcoe Hall',
'center': [Decimal('-79.3959493'), Decimal('43.6607473')],
'idx': 18},
{'name': 'Edward Johnson Building',
'center': [Decimal('-79.3945745'), Decimal('43.6666055')],
'idx': 19},
{'name': 'Flavelle House',
'center': [Decimal('-79.3939081'), Decimal('43.6661332')],
'idx': 20},
{'name': 'Falconer Hall',
'center': [Decimal('-79.3939344'), Decimal('43.6667134')],
'idx': 21},
{'name': 'Joseph L. Rotman Faculty of Management',
'center': [Decimal('-79.3984429'), Decimal('43.6653599')],
'idx': 22},
{'name': 'n/a',
'center': [Decimal('-79.3981669'), Decimal('43.6591511')],
'idx': 23},
{'name': 'John H. Daniels Faculty of Architecture, Landscape & Design',
'center': [Decimal('-79.3980011'), Decimal('43.6586788')],
'idx': 24},
{'name': 'Leslie Dan Pharmacy Building',
'center': [Decimal('-79.3917767'), Decimal('43.6599499')],
'idx': 27},
{'name': 'Wallberg Building',
'center': [Decimal('-79.3952383'), Decimal('43.6591926')],
'idx': 28},
{'name': 'Isabel Bader Theatre',
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{'name': 'n/a',
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{'name': '162 St. George Street',
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{'name': 'Innis College',
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{'name': 'Tanz Neuroscience Building',
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{'name': 'Loretto College',
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{'name': 'n/a',
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{'name': 'Medical Sciences Building',
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