P-Center Problem

Authors: Germano Barcelos, James Gaboardi, Levi J. Wolf, Qunshan Zhao

Hakimi (1964) introduced the absolute center problem to locate a police station or a hospital such that the maximum distance of the station to a set of communities connected by a highway system is minimized. Hakimi (1965) generalizes the absolute center problem to the p-center problem. Given a set $X_p=\{x_1,\ldots,x_p\}$ of $p$ points in graph $G$, the distance $d(X_p, v_j)$ between $X_p$ and node $v_j$ is computed as $min_{i=1,\ldots,n}{w_jd(v_j, X_p)}$ is minimized.

Daskin (2013) proposed a Mixed Integer Programming formulation to discrete p-center problem written as:

$\begin{array} \displaystyle \textbf{Minimize} & W && (1)\\ \displaystyle \textbf{Subject to:} & \sum_{j}{y_{ij} = 1} & \forall i & (2) \\ & \sum_{j}{x_{j} = p} && (3) \\ & y_{ij} \leq x_{j} & \forall i,j & (4) \\ & W \geq \sum_{j}{d_{ij}y_{ij}} & \forall i & (5)\\ & x_{j} \in \{0,1\} & \forall j & (6) \\ & y_{ij} \geq 0 & \forall i,j & (7)\\ \end{array}$

$\begin{array} \displaystyle \textbf{Where:}\\ & & \displaystyle i & \small = & \textrm{index referencing nodes of the network as demand} \\ & & j & \small = & \textrm{index referencing nodes of the network as potential facility sites} \\ & & d_{ij} & \small = & \textrm{shortest distance or travel time between nodes} i \textrm{and} j \\ & & p & \small = & \textrm{number of facilities to be located} \\ & & x_j & \small = & \begin{cases} 1, \text{if locate at candidate site } j \\ 0, \text{otherwise} \\ \end{cases} \\ & & y_{ij} & \small = & \textrm{fraction of demand at node } i \textrm{that is served by a facility at node} j \\ & & W & \small = & \textrm{maximum distance between a demand node and the nearest facility} \\ \end{array}$

This tutorial solves PCenter using spopt.locate.PCenter instance that depends on a array 2D representing the costs between facilities candidate sites and demand points. For that it uses a lattice 10x10 with simulated points to calculate the costs.

In [1]:
from spopt.locate import PCenter
from spopt.locate.util import simulated_geo_points

import numpy
import geopandas
import pulp
import spaghetti
from shapely.geometry import Point
import matplotlib.pyplot as plt

Since the model needs a distance cost matrix we should define some variables. In the comments, it's defined what these variables are for but solver. The solver, assigned below as pulp.PULP_CBC_CMD, is an interface to optimization solver developed by COIN-OR. If you want to use another optimization interface as Gurobi or CPLEX see this guide that explains how to achieve this.

In [2]:
CLIENT_COUNT = 100 # quantity demand points
FACILITY_COUNT = 5 # quantity supply points

P_FACILITIES = 4

# Random seeds for reproducibility
CLIENT_SEED = 5 
FACILITY_SEED = 6 

solver = pulp.PULP_CBC_CMD(msg=False) # see solvers available in pulp reference

Lattice 10x10

Create lattice 10x10 with 9 vertical lines in interior.

In [3]:
lattice = spaghetti.regular_lattice((0, 0, 10, 10), 9, exterior=True)
ntw = spaghetti.Network(in_data=lattice)

Transform spaghetti instance to geopandas geodataframe.

In [4]:
street = spaghetti.element_as_gdf(ntw, arcs=True)

street_buffered = geopandas.GeoDataFrame(
    geopandas.GeoSeries(street["geometry"].buffer(0.2).unary_union),
    crs=street.crs,
    columns=["geometry"],
)

Plotting the network created by spaghetti we can verify that it seems a district with quarters and streets.

In [5]:
street.plot()
Out[5]:
<AxesSubplot:>

Simulate points in a network

The function simulated_geo_points simulates points inside a network. In this case, it uses a lattice network 10x10 created by using spaghetti package. Below we use the function defined above and simulate the points inside lattice bounds.

In [6]:
client_points = simulated_geo_points(street_buffered, needed=CLIENT_COUNT, seed=CLIENT_SEED)
facility_points = simulated_geo_points(
    street_buffered, needed=FACILITY_COUNT, seed=FACILITY_SEED
)

Plotting the 100 client and 5 facility points we can see that the function generates dummy points to an area of 10x10 which is the area created by our lattice created on previous cells.

In [7]:
fig, ax = plt.subplots(figsize=(6, 6))
street.plot(ax=ax, alpha=0.8, zorder=1, label='streets')
facility_points.plot(ax=ax, color='red', zorder=2, label='facility candidate sites ($n$=5)')
client_points.plot(ax=ax, color='black', label='clients points ($n$=100)')
plt.legend(loc='upper left', bbox_to_anchor=(1.05, 1))
Out[7]:
<matplotlib.legend.Legend at 0x7fb7cc2556d0>

Transform simulated points to real points

To use cost matrix or geodataframes we have to pay attention in some details. The client and facility points simulated don't belong to network, so if we calculate the distances now we are supposed to receive a wrong result. Before calculating distances we snap points to the networok and then calculate the distances.

Below we snap points that is not spatially belong to network and create new real points geodataframes.

In [8]:
ntw.snapobservations(client_points, "clients", attribute=True)
clients_snapped = spaghetti.element_as_gdf(
    ntw, pp_name="clients", snapped=True
)

ntw.snapobservations(facility_points, "facilities", attribute=True)
facilities_snapped = spaghetti.element_as_gdf(
    ntw, pp_name="facilities", snapped=True
)

Now the plot seems more organized as the points belong to network. The network created is plotted below with facility points and clients points:

In [9]:
fig, ax = plt.subplots(figsize=(6, 6))
street.plot(ax=ax, alpha=0.8, zorder=1, label='streets')
facilities_snapped.plot(ax=ax, color='red', zorder=2, label='facility candidate sites ($n$=5)')
clients_snapped.plot(ax=ax, color='black', label='clients points ($n$=100)')
plt.legend(loc='upper left', bbox_to_anchor=(1.05, 1))
Out[9]:
<matplotlib.legend.Legend at 0x7fb7cc189910>

Calculating the cost matrix

Calculate distance between clients and facilities.

In [10]:
cost_matrix = ntw.allneighbordistances(
    sourcepattern=ntw.pointpatterns["clients"],
    destpattern=ntw.pointpatterns["facilities"],
)

The expected result here is a Dijkstra distance between clients and facilities points, so we our case an array 2D 100x5.

In [11]:
cost_matrix
Out[11]:
array([[12.60302601,  3.93598651,  8.16571655,  6.04319467,  5.65607701],
       [13.10096347,  4.43392397,  8.66365401,  6.54113213,  5.15813955],
       [ 6.9095462 ,  4.2425067 ,  2.47223674,  0.34971486,  5.34955682],
       [ 2.98196832,  7.84581224,  3.45534114,  3.57786302,  6.25374871],
       [ 7.5002892 ,  6.32806975,  4.55779979,  6.43527791, 11.75939222],
       [ 0.60209077, 11.42987132,  5.03940023,  7.16192211,  9.8378078 ],
       [ 5.37335867,  6.20113923,  2.43086927,  4.30834738,  9.6324617 ],
       [ 5.40801577,  5.41976478,  3.02929369,  1.15181557,  4.85108725],
       [ 3.68807115,  8.51585171,  2.12538061,  4.24790249,  7.94717417],
       [14.22503627,  4.60274429,  9.78772681,  7.66520493,  4.98931924],
       [10.32521229,  4.99225179,  7.38272288,  9.260201  , 14.58431531],
       [ 6.65436171,  7.98732222,  5.59685112,  3.719373  ,  2.58135531],
       [11.55510375,  1.11193575,  7.11779429,  5.37988496, 10.70399927],
       [10.90832519,  1.75871431,  6.47101573,  6.02666352, 11.35077783],
       [ 9.29354019,  9.53424036,  7.14376926,  5.26629115,  0.05782317],
       [11.25279502,  3.57498553,  6.81548556,  4.69296368,  6.01707799],
       [ 6.14400601, 11.47696651,  9.08649542,  7.2090173 ,  3.09171102],
       [10.43008909,  2.23695041,  5.99277963,  6.50489962, 11.82901393],
       [ 1.79838406, 11.13134457,  4.74087347,  6.86339535,  9.53928104],
       [ 2.93052752,  7.89725303,  3.50678194,  3.62930382,  6.30518951],
       [11.55272282,  6.21976231,  8.61023341, 10.48771153, 15.81182584],
       [ 8.83964081,  3.66742137,  5.89715141,  7.77462952, 13.09874384],
       [ 4.11777697,  9.45073748,  7.06026638,  5.18278826,  7.11794005],
       [ 8.69768642,  8.63527408,  5.75519701,  7.63267513, 12.95678945],
       [ 8.2652832 ,  6.56249735,  4.79222739,  2.66970551,  3.02956617],
       [ 1.71437731,  9.6185832 ,  3.2281121 ,  5.35063398,  8.02651967],
       [ 4.30308213,  6.52469842,  2.13422733,  2.25674921,  5.95602089],
       [ 9.31612329,  8.64908379,  6.25861269,  4.38113458,  0.94297974],
       [ 2.86540683, 13.69318738,  7.30271629,  9.42523817, 12.10112386],
       [ 8.95995574,  2.29291624,  4.52264628,  6.4001244 , 11.72423871],
       [10.54288208,  7.87584258,  6.10557262,  3.98305074,  1.71622094],
       [ 8.58885878,  8.74410173,  5.64636937,  7.52384749, 12.8479618 ],
       [ 2.51163835, 12.82132215,  6.43085106,  8.55337294, 11.22925863],
       [ 5.19213144,  5.63564912,  0.75482198,  1.74285727,  7.06697159],
       [ 4.1276352 , 13.2053253 ,  6.81485421,  8.93737609, 11.61326178],
       [ 3.99217608,  6.83560448,  0.44513338,  2.94281263,  6.75645905],
       [ 5.88198594, 11.21494644,  8.82447535,  6.94699723,  5.35373109],
       [ 8.24225403,  4.58552653,  3.80494457,  1.68242269,  5.006537  ],
       [10.89255004,  6.22551054,  6.45524058,  4.3327187 ,  3.36655299],
       [ 6.58504851, 11.91800902,  9.52753792,  7.6500598 ,  2.65066851],
       [ 5.44204086,  8.77500136,  6.38453026,  4.50705215,  3.79367617],
       [ 5.56289993,  7.26488062,  4.87440953,  2.99693141,  3.6728171 ],
       [ 7.96716366, 10.86061689,  8.4701458 ,  6.59266768,  1.26855337],
       [ 7.9603294 ,  5.3726311 ,  5.01783999,  6.89531811, 12.21943243],
       [ 8.68198919,  4.65097132,  5.73949978,  7.6169779 , 12.94109221],
       [ 9.06064716,  8.39360767,  6.00313657,  4.12565845,  1.19845586],
       [15.325265  ,  4.65822551, 10.88795554,  8.76543366,  6.08954798],
       [ 3.51444772,  7.81851278,  1.95175718,  3.92572094,  7.77355074],
       [ 3.33469883, 14.16247938,  7.77200828,  9.89453017, 12.57041585],
       [ 4.46482284,  6.36295772,  1.40731225,  2.0950085 ,  5.79428018],
       [11.20742649,  1.459613  ,  6.77011704,  5.72756222, 11.05167653],
       [11.15442417,  5.67335639,  6.71711471,  4.59459283,  3.91870714],
       [ 5.17021584,  5.65756471,  0.73290638,  2.6103845 ,  7.93449881],
       [ 5.54400588, 10.87696639,  8.48649529,  6.60901717,  5.28490286],
       [ 5.28695668,  8.04600382,  2.34446727,  4.22194539,  9.5460597 ],
       [ 7.33259845,  6.66555896,  4.27508786,  2.39760974,  2.92650457],
       [ 8.08642618, 10.74135437,  8.35088328,  6.47340516,  1.14929085],
       [ 7.97403829,  2.85374226,  3.53672884,  4.96095042, 10.28506473],
       [ 5.04455411,  6.2884064 ,  2.1020647 ,  3.97954282,  9.30365713],
       [ 8.05520721,  3.2777533 ,  5.1127178 ,  6.99019592, 12.31431023],
       [ 8.033197  ,  3.2997635 ,  5.09070759,  6.96818571, 12.29230002],
       [ 4.88391014,  5.94387041,  3.55339931,  1.6759212 ,  4.62480712],
       [ 3.38092176,  9.44685879,  6.32341117,  5.17890958,  7.85479527],
       [ 5.83945489,  5.17241539,  2.78194429,  0.90446618,  4.41964814],
       [10.25764123,  4.57013932,  5.82033178,  3.69780989,  5.02192421],
       [ 3.16471551,  8.168245  ,  1.7777739 ,  3.90029578,  7.59956747],
       [ 8.83620663,  8.49675387,  5.89371722,  7.77119534, 13.09530965],
       [ 7.60754658,  6.94050708,  4.55003599,  2.67255787,  2.65155644],
       [ 4.14555919,  9.4785197 ,  7.0880486 ,  5.21057048,  5.09015784],
       [ 7.24126831,  4.57422881,  2.80395885,  0.68143697,  5.01783472],
       [ 5.70322513,  8.53100569,  2.76073572,  4.63821384,  9.96232815],
       [ 9.27617639,  9.55160416,  7.16113307,  5.28365495,  0.04045936],
       [ 2.5651854 , 11.39296595,  5.00249486,  7.12501674,  9.80090243],
       [14.22296519,  3.5559257 ,  9.78565573,  7.66313385,  6.03613783],
       [ 8.33806089,  2.48971967,  3.90075143,  5.77822955, 11.10234386],
       [14.34079531,  3.51301476,  9.90348585,  7.78096397,  7.91830771],
       [ 7.55811406,  6.89107456,  4.50060346,  2.62312535,  2.70098897],
       [ 9.54667188,  8.87963238,  6.48916129,  4.61168317,  0.71243114],
       [ 6.99771477,  3.83006578,  2.56040532,  2.43788343,  7.76199775],
       [10.85478728,  4.18774778,  6.41747782,  4.29495594,  5.40431574],
       [ 6.89563349,  8.43732701,  3.95314408,  5.8306222 , 11.15473651],
       [12.29945454,  3.63241504,  7.86214508,  5.7396232 ,  5.95964848],
       [ 6.57929244,  6.75366806,  3.63680304,  5.51428115, 10.83839547],
       [ 8.35675866,  8.47102189,  6.0805508 ,  4.20307268,  1.90234436],
       [11.26183   ,  1.40520949,  6.82452055,  5.67315871, 10.99727302],
       [ 6.92663397,  8.25959447,  5.86912337,  3.99164526,  2.30908306],
       [ 6.97410775,  3.8536728 ,  2.53679829,  3.96088096,  9.28499527],
       [10.00715257,  8.82062799,  6.94964198,  4.92783614,  0.77143554],
       [ 8.83013405,  7.9976465 ,  5.77262346,  3.89514534,  1.59441703],
       [ 6.69445759,  4.63850292,  2.86823295,  4.74571107, 10.06982539],
       [ 2.60649588, 11.43427644,  5.04380534,  7.16632722,  9.84221291],
       [ 9.01806225,  4.31489826,  6.07557284,  7.95305096, 13.27716527],
       [ 7.49191577,  3.84104474,  4.07077477,  5.94825289, 11.2723672 ],
       [ 7.80056437,  9.53239613,  4.85807497,  6.73555308, 12.0596674 ],
       [ 8.85156915,  2.48139135,  4.71112139,  6.58859951, 11.91271382],
       [10.04988811,  2.61715138,  5.61257866,  6.8851006 , 12.20921491],
       [ 3.68039673,  7.65256378,  1.26209268,  3.38461456,  7.08388625],
       [10.04984807,  7.28311243,  7.10735867,  8.98483678, 14.3089511 ],
       [ 8.34309643, 10.48468413,  8.09421303,  6.21673491,  0.8926206 ],
       [14.48203148,  3.65425093, 10.04472202,  7.92220014,  7.77707154]])

With PCenter.from_cost_matrix we model the PCenter problem to minimize the maximum distance between a client and a facility point using 4 facility sites and the cost matrix calculated previously.

In [12]:
pcenter_from_cost_matrix = PCenter.from_cost_matrix(cost_matrix, p_facilities=P_FACILITIES)
pcenter_from_cost_matrix = pcenter_from_cost_matrix.solve(solver)

Expected result is an instance of PCenter.

In [13]:
pcenter_from_cost_matrix
Out[13]:
<spopt.locate.p_center.PCenter at 0x7fb7cc02b8e0>

Using GeoDataFrame

In [14]:
clients_snapped
Out[14]:
id geometry comp_label
0 0 POINT (2.00000 8.85562) 0
1 1 POINT (2.00000 9.35355) 0
2 2 POINT (5.00000 6.16214) 0
3 3 POINT (7.76544 5.00000) 0
4 4 POINT (3.00000 1.75230) 0
... ... ... ...
95 95 POINT (0.00000 4.30248) 0
96 96 POINT (6.00000 3.42781) 0
97 97 POINT (2.20274 0.00000) 0
98 98 POINT (7.40431 10.00000) 0
99 99 POINT (0.00000 8.73462) 0

100 rows × 3 columns

With PCenter.from_geodataframe we model the PCenter problem to minimize the maximum distance between a client and a facility point using 4 facility sites and geodataframes without calculating the cost matrix previously.

In [15]:
pcenter_from_geodataframe = PCenter.from_geodataframe(
    clients_snapped, 
    facilities_snapped, 
    "geometry", 
    "geometry", 
    p_facilities=P_FACILITIES,
    distance_metric="euclidean"
)
pcenter_from_geodataframe = pcenter_from_geodataframe.solve(solver)

Expected result is an instance of PCenter.

In [16]:
pcenter_from_geodataframe
Out[16]:
<spopt.locate.p_center.PCenter at 0x7fb81c6dc0d0>

Plotting the results

The cell below describe the plotting of the results. For each method from PCenter class (from_cost_matrix, from_geodataframe) there is a plot displaying the facility site that was selected with a star colored and the points covered with the same color. Sometimes the demand points will be colored with not expected colors, it represents the coverage overlapping.

In [17]:
from matplotlib.patches import Patch
import matplotlib.lines as mlines

dv_colors = [
    "darkcyan",
    "mediumseagreen",
    "cyan",
    "darkslategray",
    "lightskyblue",
    "limegreen",
    "darkgoldenrod",
    "peachpuff",
    "coral",
    "mediumvioletred",
    "blueviolet",
    "fuchsia",
    "thistle",
    "lavender",
    "saddlebrown",
] 

def plot_results(model, facility_points):
    arr_points = []
    fac_sites = []
    
    for i in range(FACILITY_COUNT):
        if model.fac2cli[i]:

            geom = client_points.iloc[model.fac2cli[i]]['geometry']
            arr_points.append(geom)
            fac_sites.append(i)

    fig, ax = plt.subplots(figsize=(6, 6))
    legend_elements = []

    street.plot(ax=ax, alpha=1, color='black', zorder=1)
    legend_elements.append(mlines.Line2D(
        [],
        [],
        color='black',
        label='streets',
    ))

    facility_points.plot(ax=ax, color='brown', marker="*", markersize=80, zorder=2)
    legend_elements.append(mlines.Line2D(
        [],
        [],
        color='brown',
        marker="*",
        linewidth=0,
        label=f'facility sites ($n$={FACILITY_COUNT})'
    ))

    for i in range(len(arr_points)):
        gdf = geopandas.GeoDataFrame(arr_points[i])

        label = f"coverage_points by y{fac_sites[i]}"
        legend_elements.append(Patch(facecolor=dv_colors[i], edgecolor="k", label=label))

        gdf.plot(ax=ax, zorder=3, alpha=0.7, edgecolor="k", color=dv_colors[i], label=label)
        facility_points.iloc[[fac_sites[i]]].plot(ax=ax,
                                marker="*",
                                markersize=200 * 3.0,
                                alpha=0.8,
                                zorder=4,
                                edgecolor="k",
                                facecolor=dv_colors[i])
        
        legend_elements.append(mlines.Line2D(
            [],
            [],
            color=dv_colors[i],
            marker="*",
            ms=20 / 2,
            markeredgecolor="k",
            linewidth=0,
            alpha=0.8,
            label=f"y{fac_sites[i]} facility selected",
        ))

    plt.title("P-Center", fontweight="bold")
    plt.legend(handles = legend_elements, loc='upper left', bbox_to_anchor=(1.05, 1))

PCenter built from cost matrix

In [18]:
pcenter_from_cost_matrix.facility_client_array()
plot_results(pcenter_from_cost_matrix, facility_points)

PCenter from geodataframes

In [19]:
pcenter_from_geodataframe.facility_client_array()
plot_results(pcenter_from_geodataframe, facility_points)