spopt.locate.LSCP¶

class spopt.locate.LSCP(name: str, problem: LpProblem)[source]¶

Implement the Location Set Covering Problem (LSCP) optimization model and solve it. A capacitated version, the Capacitated Location Set Covering Problem – System Optimal (CLSCP-SO), can also be solved by passing in client client demand and facility capacities.

The standard LSCP, as adapted from [CM18], can be formulated as:

\[\begin{split}\begin{array}{lllll} \displaystyle \textbf{Minimize} & \displaystyle \sum_{j \in J}{Y_j} && & (1) \\ \displaystyle \textbf{Subject To} & \displaystyle \sum_{j \in N_i}{Y_j} \geq 1 && \forall i \in I & (2) \\ & Y_j \in \{0,1\} && \forall j \in J & (3) \\ & && & \\ \displaystyle \textbf{Where} && i & = & \textrm{index of demand points/areas/objects in set } I \\ && j & = & \textrm{index of potential facility sites in set } J \\ && S & = & \textrm{maximum acceptable service distance or time standard} \\ && d_{ij} & = & \textrm{shortest distance or travel time between locations } i \textrm{ and } j \\ && N_i & = & \{j | d_{ij} < S\} \\ && Y_j & = & \begin{cases} 1, \textrm{if a facility is sited at location } j \\ 0, \textrm{otherwise} \\ \end{cases} \end{array}\end{split}\]

The CLSCP-SO, as adapted from [CM18] (see also [CS88]), can be formulated as:

\[\begin{split}\begin{array}{lllll} \displaystyle \textbf{Minimize} & \displaystyle \sum_{j \in J}{Y_j} && & (1) \\ \displaystyle \textbf{Subject to} & \displaystyle \sum_{j \in N_i}{z_{ij}} = 1 && \forall i \in I & (2) \\ & \displaystyle \sum_{i \in I} a_i z_{ij} \leq C_jx_j && \forall j \in J & (3) \\ & Y_j \in {0,1} && \forall j \in J & (4) \\ & z_{ij} \geq 0 && \forall i \in I \quad \forall j \in N_i & (5) \\ & && & \\ \displaystyle \textbf{Where:} && i & = & \textrm{index of demand points/areas/objects in set } I \\ && j & = & \textrm{index of potential facility sites in set } J \\ && S & = & \textrm{maximal acceptable service distance or time standard} \\ && d_{ij} & = & \textrm{shortest distance or travel time between nodes } i \textrm{ and } j \\ && N_i & = & \{j | d_{ij} < S\} \\ && a_i & = & \textrm{amount of demand at } i \\ && C_j & = & \textrm{capacity of potential facility } j \\ && z_{ij} & = & \textrm{fraction of demand } i \textrm{ that is assigned to facility } j \\ && Y_j & = & \begin{cases} 1, \text{if a facility is located at node } j \\ 0, \text{otherwise} \\ \end{cases} \end{array}\end{split}\]

Parameters:

namestr: The problem name.
problempulp.LpProblem: A pulp instance of an optimization model that contains constraints, variables, and an objective function.

Attributes:

namestr: The problem name.
problempulp.LpProblem: A pulp instance of an optimization model that contains constraints, variables, and an objective function.
fac2clinumpy.array: A 2D array storing facility to client relationships where each row represents a facility and contains an array of client indices with which it is associated. An empty client array indicates the facility is associated with no clients.
cli2facnumpy.array: The inverse of fac2cli where client to facility relationships are shown.
aijnumpy.array: A cost matrix in the form of a 2D array between origins and destinations.

__init__(name: str, problem: LpProblem)[source]¶

Initialize.

Parameters:

namestr: The desired name for the model.

Methods

`__init__`(name, problem)	Initialize.
`check_status`()	Ensure a model is solved.
`client_facility_array`()	Create a 2D array storing client to facility relationships where each row represents a client and contains an array of facility indices with which it is associated.
`facility_client_array`()	Create a 2D array storing facility to client relationships where each row represents a facility and contains an array of client indices with which it is associated.
`from_cost_matrix`(cost_matrix, service_radius)	Create an `LSCP` object based on a cost matrix.
`from_geodataframe`(gdf_demand, gdf_fac, ...)	Create an `LSCP` object from `geopandas.GeoDataFrame` objects.
`solve`(solver[, results])	Solve the `LSCP` model.

facility_client_array() → None[source]¶

Create a 2D array storing facility to client relationships where each row represents a facility and contains an array of client indices with which it is associated. An empty client array indicates the facility is associated with no clients.

Returns:

None

classmethod from_cost_matrix(cost_matrix: array, service_radius: float, predefined_facilities_arr: array = None, demand_quantity_arr: array = None, facility_capacity_arr: array = None, name: str = 'lscp')[source]¶

Create an LSCP object based on a cost matrix. A capacitated version of the LSCP (the CLSCP-SO) can be solved by passing in the demand_quantity_arr and facility_capacity_arr keyword arguments.

Parameters:

cost_matrixnumpy.array: A cost matrix in the form of a 2D array between origins and destinations.
service_radiusfloat: Maximum acceptable service distance.
predefined_facilities_arrnumpy.array (default None): A binary 1D array of service facilities that must appear in the solution. For example, consider 3 facilites ['A', 'B', 'C']. If facility 'B' must be in the model solution, then the passed in array should be [0, 1, 0].
demand_quantity_arrnumpy.array (default None): Amount of demand at each client location.
facility_capacity_arrnumpy.array (default None): Capacity for service at each facility location.
namestr, (default ‘lscp’): The name of the problem.

Returns:

spopt.locate.coverage.LSCP

Examples

>>> from spopt.locate import LSCP
>>> from spopt.locate.util import simulated_geo_points
>>> import geopandas
>>> import numpy
>>> import pulp
>>> import spaghetti

Create a regular lattice.

>>> lattice = spaghetti.regular_lattice((0, 0, 10, 10), 9, exterior=True)
>>> ntw = spaghetti.Network(in_data=lattice)
>>> streets = spaghetti.element_as_gdf(ntw, arcs=True)
>>> streets_buffered = geopandas.GeoDataFrame(
...     geopandas.GeoSeries(streets["geometry"].buffer(0.2).unary_union),
...     crs=streets.crs,
...     columns=["geometry"]
... )

Simulate points about the lattice.

>>> demand_points = simulated_geo_points(streets_buffered, needed=100, seed=5)
>>> facility_points = simulated_geo_points(streets_buffered, needed=5, seed=6)

Snap the points to the network of lattice edges.

>>> ntw.snapobservations(demand_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
... )

Calculate the cost matrix from origins to destinations.

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

Create and solve an LSCP instance from the cost matrix.

>>> lscp_from_cost_matrix = LSCP.from_cost_matrix(
...     cost_matrix, service_radius=8
... )
>>> lscp_from_cost_matrix = lscp_from_cost_matrix.solve(
...     pulp.PULP_CBC_CMD(msg=False)
... )

Get the facility lookup demand coverage array.

>>> for fac, cli in enumerate(lscp_from_cost_matrix.fac2cli):
...     print(f"facility {fac} serving {len(cli)} clients")
facility 0 serving 0 clients
facility 1 serving 63 clients
facility 2 serving 85 clients
facility 3 serving 0 clients
facility 4 serving 58 clients

classmethod from_geodataframe(gdf_demand: GeoDataFrame, gdf_fac: GeoDataFrame, demand_col: str, facility_col: str, service_radius: float, predefined_facility_col: str = None, demand_quantity_col: str = None, facility_capacity_col: str = None, distance_metric: str = 'euclidean', name: str = 'lscp')[source]¶

Create an LSCP object from geopandas.GeoDataFrame objects. Calculate the cost matrix between demand and facility locations before building the problem within the from_cost_matrix() method. A capacitated version of the LSCP (the CLSCP-SO) can be solved by passing in the demand_quantity_col and facility_capacity_col keyword arguments.

Parameters:

gdf_demandgeopandas.GeoDataFrame: Demand locations.
gdf_facgeopandas.GeoDataFrame: Facility locations.
demand_colstr: Demand sites geometry column name.
facility_colstr: Facility candidate sites geometry column name.
service_radiusfloat: Maximum acceptable service distance.
predefined_facility_colstr (default None): Column name representing facilities are already defined. This a binary assignment per facility. For example, consider 3 facilites ['A', 'B', 'C']. If facility 'B' must be in the model solution, then the column should be [0, 1, 0].
demand_quantity_colstr: Column name representing amount of demand at each client location.
facility_capacity_arrstr: Column name representing capacity for service at each facility location.
distance_metricstr (default ‘euclidean’): A metric used for the distance calculations supported by scipy.spatial.distance.cdist.
namestr (default ‘lscp’): The name of the problem.

Returns:

spopt.locate.coverage.LSCP

Examples

>>> from spopt.locate import LSCP
>>> from spopt.locate.util import simulated_geo_points
>>> import geopandas
>>> import numpy
>>> import pulp
>>> import spaghetti

Create a regular lattice.

>>> lattice = spaghetti.regular_lattice((0, 0, 10, 10), 9, exterior=True)
>>> ntw = spaghetti.Network(in_data=lattice)
>>> streets = spaghetti.element_as_gdf(ntw, arcs=True)
>>> streets_buffered = geopandas.GeoDataFrame(
...     geopandas.GeoSeries(streets["geometry"].buffer(0.2).unary_union),
...     crs=streets.crs,
...     columns=["geometry"]
... )

Simulate points about the lattice.

>>> demand_points = simulated_geo_points(streets_buffered, needed=100, seed=5)
>>> facility_points = simulated_geo_points(streets_buffered, needed=5, seed=6)

Snap the points to the network of lattice edges and extract as GeoDataFrame objects.

>>> ntw.snapobservations(demand_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
... )

Create and solve an LSCP instance from the GeoDataFrame objects.

>>> lscp_from_geodataframe = LSCP.from_geodataframe(
...     clients_snapped,
...     facilities_snapped,
...     "geometry",
...     "geometry",
...     service_radius=8,
...     distance_metric="euclidean"
... )
>>> lscp_from_geodataframe = lscp_from_geodataframe.solve(
...     pulp.PULP_CBC_CMD(msg=False)
... )

Get the facility lookup demand coverage array.

>>> for fac, cli in enumerate(lscp_from_geodataframe.fac2cli):
...     print(f"facility {fac} serving {len(cli)} clients")
facility 0 serving 0 clients
facility 1 serving 0 clients
facility 2 serving 100 clients
facility 3 serving 0 clients
facility 4 serving 0 clients

solve(solver: LpSolver, results: bool = True)[source]¶

Solve the LSCP model.

Parameters:

solverpulp.apis.LpSolver: A solver supported by pulp.
resultsbool (default True): If True it will create metainfo (which facilities cover which demand) and vice-versa, and the uncovered demand.

Returns:

spopt.locate.coverage.LSCP