

In [ ]:

    
#    GNU LESSER GENERAL PUBLIC LICENSE
#    Version 3, 29 June 2007
#    Copyright (C) 2007 Free Software Foundation, Inc. <http://fsf.org/>
#    Everyone is permitted to copy and distribute verbatim copies
#    of this license document, but changing it is not allowed.

#    James Gaboardi, 2016

Automating Multiple Single-Objective Spatial Optimization Models for Efficiency and Reproducibility

James D. Gaboardi | Association of American Geographers 2016

Florida State University | Department of Geography

Outline

Background & Information

Models
- PMP
- PCP
- CentDian
- PMCP Method

Data & Processing

Solutions

Visualizations

Future Work
- COIN-OR

Set Path



In [1]:

    
import IPython.display as IPd

# Local path on user's machine
path = '/Users/jgaboardi/AAG_16/Data/'

Background & Information

$\Rightarrow$ Automating solutions for p-median, p-center, and p-centdian problems with p={1, 2, ..., n} facilities

$\Rightarrow$ Compare coverage and costs numerically, graphically, and visually

`PySAL 1.11.0` $\Rightarrow$ `PySAL.Network`

Python Spatial Analysis Library

[https://www.pysal.readthedocs.org]

Sergio Rey at Arizona State University leads the PySAL project. [https://geoplan.asu.edu/people/sergio-j-rey].

PySAL.Network was principally developed by Jay Laura at Arizona State Universty and the United States Geological Suvery. [https://geoplan.asu.edu/people/jay-laura]

"PySAL is an open source library of spatial analysis functions written in Python intended to support the development of high level applications. PySAL is open source under the BSD License." [https://pysal.readthedocs.org/en/latest/]

`Gurobi 6.5.0` $\Rightarrow$ `gurobipy`

Relatively new company founded by optimization experts formerly at key positions with CPLEX. [http://www.gurobi.com] [http://www.gurobi.com/company/about-gurobi]

Models

The p-Median Problem $\Rightarrow$ "minimum total cost" $\Rightarrow$ efficiency

The objective of the p-median problem, also know as the minimsum problem and the PMP, is to minimize the total weighted cost while siting [p] facilities to serve all demand/client nodes. It was originally proposed by Hakimi (1964) and is well-studied in Geography, Operations Research, Mathematics, etc. In this particular project the network-based vertice PMP is used meaning the cost will be calculated on a road network and solutions will be determined based on discrete locations. Cost is generally defined as either travel time or distance and it is the latter in the project. Population (demand) utilized as a weight at each client node. The average cost can be calculated by dividing the minimized total cost by the total demand.

For more information refer to references section.

Minimize

$\displaystyle {Z} = {\sum_{i \in 1}^n\sum_{j\in 1}^m a_i c_{ij} x_{ij}}$

Subject to

$\displaystyle\sum_{j\in m} x_{ij} = 1 ,$ $\forall i \in n$

$\displaystyle\sum_{j \in m} y_j = p$

$x_{ij} - y_j \geq 0,$ $\forall i \in n, j \in m$

$x_{ij}, y_j \in \{0,1\}$ $\forall i \in n , j \in m$

where

$i$ = a specific origin
$j$ = a specific destination
$n$ = the set of origins
$m$ = the set of destinations
$a_i$ = weight at each node
$c_{ij}$ = travel costs between nodes
$x_{ij}$ = the decision variable at each node in the matrix
$y_j$ = nodes chosen as service facilities
$p$ = the number of facilities to be sited

Adapted from:

Daskin, M. S. 1995. Network and Discrete Location: Models, Algorithms, and Applications. Hoboken, NJ, USA: John Wiley & Sons, Inc.

The p-Center Problem $\Rightarrow$ "minimum worst case" $\Rightarrow$ equity

The objective of the p-center problem, also know as the minimax problem and the PCP, is to minimize the worst case cost (W) scenario while siting [p] facilities to serve all demand/client nodes. It was originally proposed by Minieka (1970) and, like the PMP, is well-studied in Geography, Operations Research, Mathematics, etc. In this particular project the network-based vertice PCP is used meaning the cost will be calculated on a road network and solutions will be determined based on discrete locations. Cost is generally defined as either travel time or distance and it is the latter in the project.

For more information refer to references section.

Minimize

$W$

Subject to

$\displaystyle\sum_{j\in m} x_{ij} = 1,$ $\forall i \in n$

$\displaystyle\sum_{j \in m} y_j = p$

$x_{ij} - y_j \geq 0,$ $\forall i\in n, j \in m$

$\displaystyle W \geq \sum_{j \in m} c_{ij} x_{ij}$ $\forall i \in n$

$x_{ij}, y_j \in \{0,1\}$ $\forall i \in n, j \in m$

where

$W$ = the worst case cost between a client and a service node
$i$ = a specific origin
$j$ = a specific destination
$n$ = the set of origins
$m$ = the set of destinations
$a_i$ = weight at each node
$c_{ij}$ = travel costs between nodes
$x_{ij}$ = the decision variable at each node in the matrix
$y_j$ = nodes chosen as service facilities
$p$ = the number of facilities to be sited

Adapted from:

Daskin, M. S. 1995. Network and Discrete Location: Models, Algorithms, and Applications. Hoboken, NJ, USA: John Wiley & Sons, Inc.

The p-CentDian Problem

The p-CentDian Problem was first descibed by Halpern (1976). It is a combination of the p-median problem and the p-center problem with a dual objective of minimizing both the worst case scenario and the total travel distance. The objective used for the model in this demonstration is the average of (1) the p-center objective function and (2) the p-median objective function divided by the total demand. An alternative formulation is the p-$\lambda$-CentDian Problem, where ( $\lambda$ ) represents the weight attributed to the p-center objective function and (1 - $\lambda$) represents the weight attributed to the p-median objective function which was was proposed by Pérez-Brito, et al (1997).

For more information refer to references section.

Minimize

$ \displaystyle {W + {Z \over \sum_{i=1}a_i} \over 2}$

Subject to

$\displaystyle\sum_{j\in m} x_{ij} = 1,$ $\forall i \in n$

$\displaystyle\sum_{j \in m} y_j = p$

$x_{ij} - y_j \geq 0,$ $\forall i\in n, j \in m$

$\displaystyle W \geq \sum_{j \in m} c_{ij} x_{ij}$ $\forall i \in n$

$x_{ij}, y_j \in \{0,1\}$ $\forall i \in n, j \in m$

where

$W$ = the maximum travel cost between client and service nodes
$Z$ = the minimized total travel cost $\big({\sum_{i \in 1}^n\sum_{j\in 1}^m a_i c_{ij} x_{ij}}\big)$
$i$ = a specific origin
$j$ = a specific destination
$n$ = the set of origins
$m$ = the set of destinations
$a_i$ = weight at each node
$c_{ij}$ = travel costs between nodes
$x_{ij}$ = the decision variable at each node in the matrix
$y_j$ = nodes chosen as service facilities
$p$ = the number of facilities to be sited

Adapted from:

Halpern, J. 1976. The Location of a Center-Median Convex Combination on an Undirected Tree*. Journal of Regional Science 16 (2):237–245

The PMCP Method

$\Rightarrow$ solve the PMP and PCP to determine with optimal locations with equivalent [p]

$\Rightarrow$ "poor man's" p-CentDian Problem?

automated & efficient decision making for those who don't have access to multiple-objective capable solvers or the know-how to formulate a mutliple objectives into a single objective funtion

what it is:
- a comparision to determine equivalent site selection of single objective solutions
- probably best used with low cost sites
- a opportunity for finding optimal solutions without sacrificing either efficiency or equity

what it is not:
- an optimization solution with multiple objective functions
- capable of a true 'best solution' trade-off between efficiency and equity
- guaranteed to find identical solutions

Workflow for PMCP



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# Conceptual Model Workflow
workflow = IPd.Image(path+'/AAG_16.png')
workflow

Data & Processing

Process Imports



In [2]:

    
import pysal as ps
import geopandas as gpd
import numpy as np
import networkx as nx
import shapefile as shp
from shapely.geometry import Point
import shapely
from collections import OrderedDict
import pandas as pd
import qgrid
import gurobipy as gbp
import time
import bokeh
from bokeh.plotting import figure, show, ColumnDataSource
from bokeh.io import output_notebook, output_file, show
from bokeh.models import (HoverTool, BoxAnnotation, GeoJSONDataSource, 
                          GMapPlot, GMapOptions, ColumnDataSource, Circle, 
                          DataRange1d, PanTool, WheelZoomTool, BoxSelectTool,
                          ResetTool, MultiLine)
import utm
from cylp.cy import CyCbcModel, CyClpSimplex
import matplotlib.pyplot as plt
import matplotlib as mpl
%matplotlib inline
mpl.rcParams['figure.figsize']=8,8
output_notebook()









    



/Users/jgaboardi/Library/Enthought/Canopy_64bit/User/lib/python2.7/site-packages/matplotlib/font_manager.py:273: UserWarning: Matplotlib is building the font cache using fc-list. This may take a moment.
  warnings.warn('Matplotlib is building the font cache using fc-list. This may take a moment.')






    





    
        
        Loading BokehJS ...

Data preparation



In [3]:

    
import spanoptpy as sp



In [4]:

    
print dir(sp)









    



['__builtins__', '__doc__', '__file__', '__name__', '__package__', '__path__', 'author', 'spatan', 'spatopt', 'version']



In [10]:

    
from spanoptpy import AAG16_DataPrep



In [11]:

    
AAG16_DataPrep.data_prep()









    



---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-11-66137018cff6> in <module>()
----> 1 AAG16_DataPrep.data_prep()

/Users/jgaboardi/Desktop/spanoptpy/AAG16_DataPrep.py in data_prep()
      5 
      6     # Reproject the street network with `GeoPandas` --> Waverly  Hills
----> 7     STREETS_Orig = gpd.read_file(path+'Waverly_Trim/Waverly.shp')
      8     STREETS = gpd.read_file(path+'Waverly_Trim/Waverly.shp')
      9     STREETS.to_crs(epsg=2779, inplace=True) # NAD83(HARN) / Florida North

NameError: global name 'gpd' is not defined

What we have at this point



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Buff.plot()
STREETS.plot()
CLIENT.plot()
SERVICE.plot(colormap=True)

Call Client to Service Matrix Function



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# Call Client to Service Matrix Function
c_s_matrix()

Solutions

Solve all



In [ ]:

    
Gurobi_PMCP(len(SER), Ai, AiSum, All_Dist_MILES)

Visualizations

Draw PMP figure [p=1] large $ \rightarrow $ small [p=15]



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# Draw Network Actual Roads and Nodes
for e in ntw.edges:
    g.add_edge(*e)
nx.draw(g, ntw.node_coords, node_size=5, alpha=0.25, edge_color='r', width=2)

#PMP
size = 3000
for i,j in P_Med_Graphs.iteritems():
    size=size-120
    # p-Median
    P_Med = ps.open(path+'Results/Selected_Locations_Pmedian'+str(i)+'.shp')
    points_median = {}
    for idx, coords in enumerate(P_Med):
        P_Med_Graphs[i].add_node(idx)
        points_median[idx] = coords
        P_Med_Graphs[i].node[idx] = coords
    nx.draw(P_Med_Graphs[i], 
                points_median, 
                node_size=size, 
                alpha=.1, 
                node_color='k')

# Legend (Ordered Dictionary)
LEGEND = OrderedDict()
LEGEND['Network Nodes']=g
LEGEND['Roads']=g
for i in P_Med_Graphs:
    LEGEND['Optimal PMP '+str(i)]=P_Med_Graphs[i]
plt.legend(LEGEND, 
       loc='upper left', 
       fancybox=True, 
       framealpha=0.5, 
       scatterpoints=1)

# Title
plt.title('Waverly Hills\n Tallahassee, Florida', family='Times New Roman', 
      size=40, color='k', backgroundcolor='w', weight='bold')

# North Arrow and 'N' --> Must be changed for different spatial resolutions, etc.
plt.arrow(624000, 164050, 0.0, 500, width=50, head_width=125, 
          head_length=75, fc='k', ec='k',alpha=0.75,)
plt.annotate('N', xy=(623900, 164700), fontstyle='italic', fontsize='xx-large',
            fontweight='heavy', alpha=0.75)

Pandas PMP Data Frame



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qgrid.show_grid(pydf_M, show_toolbar=True)

Bokeh PMP [p vs. cost] trade off



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source_m = ColumnDataSource(
        data=dict(
            x=range(1, len(SER)+1),
            y=AVG_PMP,
            avg=AVG_PMP,
            desc=p_list,
            change=PMP_Avg_Diff))

TOOLS = 'wheel_zoom, pan, reset, crosshair, save'

hover = HoverTool(line_policy="nearest", mode="hline", tooltips="""
        <div>
            <div>
                
            </div>
            <div>
                <span style="font-size: 17px; font-weight: bold;">@desc</span> 
            </div>
            <div>
                <span style="font-size: 15px;">Average Minimized Cost</span>
                <span style="font-size: 15px; font-weight: bold; color: #ff4d4d;">[@avg]</span>
            </div>
            <div>
                <span style="font-size: 15px;">Variation</span>
                <span style="font-size: 15px; font-weight: bold; color: #ff4d4d;">[@change]</span>
            </div>
        </div>""")

# Instantiate Plot
pmp_plot = figure(plot_width=600, plot_height=600, tools=[TOOLS,hover],
           title="Average Distance vs. p-Facilities", y_range=(0,2))

# Create plot points and set source
pmp_plot.circle('x', 'y', size=15, color='red',source=source_m, 
                legend='Total Minimized Cost / Total Demand')
pmp_plot.line('x', 'y', line_width=2, color='red', alpha=.5, source=source_m, 
              legend='Total Minimized Cost / Total Demand')

pmp_plot.xaxis.axis_label = '[p = n]'
pmp_plot.yaxis.axis_label = 'Miles'

one_quarter = BoxAnnotation(plot=pmp_plot, top=.35, 
                            fill_alpha=0.1, fill_color='green')
half = BoxAnnotation(plot=pmp_plot, bottom=.35, top=.7, 
                            fill_alpha=0.1, fill_color='blue')
three_quarter = BoxAnnotation(plot=pmp_plot, bottom=.7, top=1.05,
                            fill_alpha=0.1, fill_color='gray')

pmp_plot.renderers.extend([one_quarter, half, three_quarter])

# Display the figure
show(pmp_plot)

Draw PCP figure [p=1] large $ \rightarrow $ small [p=15]



In [ ]:

    
# Draw Network Actual Roads and Nodes
for e in ntw.edges:
    g.add_edge(*e)
nx.draw(g, ntw.node_coords, node_size=5, alpha=0.25, edge_color='r', width=2)

#PCP
size = 3000
for i,j in P_Cent_Graphs.iteritems():
    size=size-150
    # p-Center
    P_Cent = ps.open(path+'Results/Selected_Locations_Pcenter'+str(i)+'.shp')
    points_center = {}
    for idx, coords in enumerate(P_Cent):
        P_Cent_Graphs[i].add_node(idx)
        points_center[idx] = coords
        P_Cent_Graphs[i].node[idx] = coords
    nx.draw(P_Cent_Graphs[i], 
                points_center, 
                node_size=size, 
                alpha=.1, 
                node_color='k')

# Legend (Ordered Dictionary)
LEGEND = OrderedDict()
LEGEND['Network Nodes']=g
LEGEND['Roads']=g
for i in P_Cent_Graphs:
    LEGEND['Optimal PCP '+str(i)]=P_Cent_Graphs[i]
plt.legend(LEGEND, 
       loc='upper left', 
       fancybox=True, 
       framealpha=0.5, 
       scatterpoints=1)

# Title
plt.title('Waverly Hills\n Tallahassee, Florida', family='Times New Roman', 
      size=40, color='k', backgroundcolor='w', weight='bold')

# North Arrow and 'N' --> Must be changed for different spatial resolutions, etc.
plt.arrow(624000, 164050, 0.0, 500, width=50, head_width=125, 
          head_length=75, fc='k', ec='k',alpha=0.75,)
plt.annotate('N', xy=(623900, 164700), fontstyle='italic', fontsize='xx-large',
            fontweight='heavy', alpha=0.75)

Pandas PCP Data Frame



In [ ]:

    
qgrid.show_grid(pydf_C, show_toolbar=True)

Bokeh PCP [p vs. cost] trade off



In [ ]:

    
source_c = ColumnDataSource(
        data=dict(
            x=range(1, len(SER)+1),
            y=VAL_PCP,
            obj=VAL_PCP,
            desc=p_list,
            change=PCP_Diff))
    
TOOLS = 'wheel_zoom, pan, reset, crosshair, save'
    
hover = HoverTool(line_policy="nearest", mode="vline", tooltips="""
        <div>
            <div>
                
            </div>
            <div>
                <span style="font-size: 17px; font-weight: bold;">@desc</span> 
            </div>
            <div>
                <span style="font-size: 15px;">Worst Case Cost</span>
                <span style="font-size: 15px; font-weight: bold; color: #00b300;">[@obj]</span>
            </div>
            <div>
                <span style="font-size: 15px;">Variation</span>
                <span style="font-size: 15px; font-weight: bold; color: #00b300;">[@change]</span>
            </div>
        </div>""")

# Instantiate Plot
pcp_plot = figure(plot_width=600, plot_height=600, tools=[TOOLS,hover],
           title="Worst Case Distance vs. p-Facilities", y_range=(0,2))

# Create plot points and set source
pcp_plot.circle('x', 'y', size=15, color='green', source=source_c,
                   legend='Minimized Worst Case')
pcp_plot.line('x', 'y', line_width=2, color='green', alpha=.5, source=source_c,
                   legend='Minimized Worst Case')

pcp_plot.xaxis.axis_label = '[p = n]'
pcp_plot.yaxis.axis_label = 'Miles'

one_quarter = BoxAnnotation(plot=pcp_plot, top=.35, 
                            fill_alpha=0.1, fill_color='green')
half = BoxAnnotation(plot=pcp_plot, bottom=.35, top=.7, 
                            fill_alpha=0.1, fill_color='blue')
three_quarter = BoxAnnotation(plot=pcp_plot, bottom=.7, top=1.05,
                            fill_alpha=0.1, fill_color='gray')

pcp_plot.renderers.extend([one_quarter, half, three_quarter])

# Display the figure
show(pcp_plot)

Draw CentDian figure [p=1] large $ \rightarrow $ small [p=15]



In [ ]:

    
# Draw Network Actual Roads and Nodes
for e in ntw.edges:
    g.add_edge(*e)
nx.draw(g, ntw.node_coords, node_size=5, alpha=0.25, edge_color='r', width=2)

#CentDian
size = 3000
for i,j in P_CentDian_Graphs.iteritems():
    size=size-150
    P_CentDian = ps.open(path+'Results/Selected_Locations_CentDian'+str(i)+'.shp')
    points_centdian = {}
    for idx, coords in enumerate(P_CentDian):
        P_CentDian_Graphs[i].add_node(idx)
        points_centdian[idx] = coords
        P_CentDian_Graphs[i].node[idx] = coords
    nx.draw(P_CentDian_Graphs[i], 
                points_centdian, 
                node_size=size, 
                alpha=.1, 
                node_color='k')

# Legend (Ordered Dictionary)
LEGEND = OrderedDict()
LEGEND['Network Nodes']=g
LEGEND['Roads']=g
for i in P_CentDian_Graphs:
    LEGEND['Optimal CentDian '+str(i)]=P_CentDian_Graphs[i]
plt.legend(LEGEND, 
       loc='upper left', 
       fancybox=True, 
       framealpha=0.5, 
       scatterpoints=1)

# Title
plt.title('Waverly Hills\n Tallahassee, Florida', family='Times New Roman', 
      size=40, color='k', backgroundcolor='w', weight='bold')

# North Arrow and 'N' --> Must be changed for different spatial resolutions, etc.
plt.arrow(624000, 164050, 0.0, 500, width=50, head_width=125, 
          head_length=75, fc='k', ec='k',alpha=0.75,)
plt.annotate('N', xy=(623900, 164700), fontstyle='italic', fontsize='xx-large',
            fontweight='heavy', alpha=0.75)

Pandas CentDian Data Frame



In [ ]:

    
qgrid.show_grid(pydf_CentDian, show_toolbar=True)

Bokeh CentDian [p vs. cost] trade off



In [ ]:

    
source_centdian = ColumnDataSource(
        data=dict(
            x=range(1, len(SER)+1),
            y=VAL_CentDian,
            obj=VAL_CentDian,
            desc=p_list,
            change=CentDian_Diff))
    
TOOLS = 'wheel_zoom, pan, reset, crosshair, save'
    
hover = HoverTool(line_policy="nearest", mode="vline" ,tooltips="""
        <div>
            <div>
                
            </div>
            <div>
                <span style="font-size: 17px; font-weight: bold;">@desc</span> 
            </div>
            <div>
                <span style="font-size: 15px;">Center Median Cost</span>
                <span style="font-size: 15px; font-weight: bold; color: #3385ff;">[@obj]</span>
            </div>
            <div>
                <span style="font-size: 15px;">Variation</span>
                <span style="font-size: 15px; font-weight: bold; color: #3385ff;">[@change]</span>
            </div>
        </div>""")

# Instantiate Plot
centdian_plot = figure(plot_width=600, plot_height=600, tools=[TOOLS,hover],
           title="Center Median Distance vs. p-Facilities", y_range=(0,2))

# Create plot points and set source
centdian_plot.circle('x', 'y', size=15, color='blue', source=source_centdian,
                   legend='Center Median')
centdian_plot.line('x', 'y', line_width=2, color='blue', alpha=.5, source=source_centdian,
                   legend='Center Median')

centdian_plot.xaxis.axis_label = '[p = n]'
centdian_plot.yaxis.axis_label = 'Miles'

one_quarter = BoxAnnotation(plot=pcp_plot, top=.35, 
                            fill_alpha=0.1, fill_color='green')
half = BoxAnnotation(plot=pcp_plot, bottom=.35, top=.7, 
                            fill_alpha=0.1, fill_color='blue')
three_quarter = BoxAnnotation(plot=pcp_plot, bottom=.7, top=1.05,
                            fill_alpha=0.1, fill_color='gray')

centdian_plot.renderers.extend([one_quarter, half, three_quarter])

# Display the figure
show(centdian_plot)

Draw PMCP figure



In [ ]:

    
# Draw Network Actual Roads and Nodes
for e in ntw.edges:
    g.add_edge(*e)
nx.draw(g, ntw.node_coords, node_size=5, alpha=0.25, edge_color='r', width=2)

size = 500
shape = 'sdh^Vp<8>'
counter = -1
for i,j in PMCP_Graphs.iteritems():
    if int(i) <= len(SER)-1:
        counter = counter+1
        pmcp = ps.open(path+'Results/Selected_Locations_PMCP'+str(i)+'.shp')
        points_pmcp = {}
        for idx, coords in enumerate(pmcp):
            PMCP_Graphs[i].add_node(idx)
            points_pmcp[idx] = coords
            PMCP_Graphs[i].node[idx] = coords
        nx.draw(PMCP_Graphs[i], 
                    points_pmcp, 
                    node_size=size,
                    node_shape=shape[counter],
                    alpha=.5, 
                    node_color='k')
    else:
        pass
    
# Legend (Ordered Dictionary)
LEGEND = OrderedDict()
LEGEND['Network Nodes']=g
LEGEND['Roads']=g
for i in PMCP_Graphs:
    if int(i) <= len(SER)-1:
        LEGEND['PMP/PCP == '+str(i)]=PMCP_Graphs[i]
plt.legend(LEGEND, 
       loc='upper left', 
       fancybox=True, 
       framealpha=0.5, 
       scatterpoints=1)

# Title
plt.title('Waverly Hills\n Tallahassee, Florida', family='Times New Roman', 
      size=40, color='k', backgroundcolor='w', weight='bold')

# North Arrow and 'N' --> Must be changed for different spatial resolutions, etc.
plt.arrow(624000, 164050, 0.0, 500, width=50, head_width=125, 
          head_length=75, fc='k', ec='k',alpha=0.75,)
plt.annotate('N', xy=(623900, 164700), fontstyle='italic', fontsize='xx-large',
            fontweight='heavy', alpha=0.75)

Pandas PMCP Data Frame



In [ ]:

    
qgrid.show_grid(pydf_MC, show_toolbar=True)

Bokeh PMP & PCP [p vs. cost] comparision



In [ ]:

    
TOOLS = 'wheel_zoom, pan, reset, crosshair, save, hover'

bokeh_df_PMCP = pd.DataFrame()
bokeh_df_PMCP['p'] = [int(i[2:]) for i in p_dens]
bokeh_df_PMCP['Total Obj. Value'] = [VAL_PMCP[x][0] for x in range(len(VAL_PMCP))]
bokeh_df_PMCP['Avg. Obj. Value'] = [VAL_PMCP[x][1] for x in range(len(VAL_PMCP))]
bokeh_df_PMCP['Worst Case Obj. Value'] = [VAL_PMCP[x][2] for x in range(len(VAL_PMCP))]
bokeh_df_PMCP['CentDian Obj. Value'] = [VAL_PMCP[x][3] for x in range(len(VAL_PMCP))]

plot_PMCP = figure(title="Optimal PMP & PCP Selections without Sacrifice", 
                        plot_width=800, plot_height=600, tools=[TOOLS], y_range=(0,2))

plot_PMCP.circle('x', 'y', size=5, color='red', source=source_m, legend='PMP')
plot_PMCP.line('x', 'y', 
                 color="#ff4d4d", alpha=0.2, line_width=2, source=source_m, legend='PMP')

plot_PMCP.circle('x', 'y', size=5, color='green', source=source_c, legend='PCP')
plot_PMCP.line('x', 'y', 
                 color='#00b300', alpha=0.2, line_width=2, source=source_c, legend='PCP')

plot_PMCP.circle('x', 'y', size=5, color='blue', source=source_centdian, legend='CentDian')
plot_PMCP.line('x', 'y', 
                 color='#3385ff', alpha=0.2, line_width=2, source=source_centdian, legend='CentDian')

plot_PMCP.circle_x(bokeh_df_PMCP['p'], 
                 bokeh_df_PMCP['Avg. Obj. Value'], 
                 legend="Location PMP=PCP for PM+CP", 
                 color="#ff4d4d",
                 fill_alpha=0.2,
                 size=15)

plot_PMCP.circle_x(bokeh_df_PMCP['p'], 
             bokeh_df_PMCP['Worst Case Obj. Value'], 
             legend="Location PCP=PMP for PM+CP", 
             color='#00b300',
             fill_alpha=0.2,
             size=15)

plot_PMCP.circle_x(bokeh_df_PMCP['p'], 
             bokeh_df_PMCP['CentDian Obj. Value'], 
             legend="Location CentDian = PMCP", 
             color='#3385ff',
             fill_alpha=0.2,
             size=15)

plot_PMCP.xaxis.axis_label = '[p = n]'
plot_PMCP.yaxis.axis_label = 'Miles'

one_quarter = BoxAnnotation(plot=plot_PMCP, top=.35, 
                            fill_alpha=0.1, fill_color='green')
half = BoxAnnotation(plot=plot_PMCP, bottom=.35, top=.7, 
                            fill_alpha=0.1, fill_color='blue')
three_quarter = BoxAnnotation(plot=plot_PMCP, bottom=.7, top=1.05,
                            fill_alpha=0.1, fill_color='gray')

plot_PMCP.renderers.extend([one_quarter, half, three_quarter])

show(plot_PMCP)

Convert Service Facilities Back to Longitude/Latitude for Google Maps Plots



In [ ]:

    
points = SERVICE
points.to_crs(epsg=32616, inplace=True) # UTM 16N
LonLat_Dict = OrderedDict()
LonLat_List = []

for i,j in points['geometry'].iteritems():
    LonLat_Dict[y_list[i]] = utm.to_latlon(j.xy[0][-1], j.xy[1][-1], 16, 'N')  
    LonLat_List.append((utm.to_latlon(j.xy[0][-1], j.xy[1][-1], 16, 'N')))

Service_Lat_List = []
Service_Lon_List = []
    
for i in LonLat_List:
    Service_Lat_List.append(i[0])
for i in LonLat_List:
    Service_Lon_List.append(i[1])

Create Lists of Selected Locations for Google Maps Plot



In [ ]:

    
# p-Median Selected Sites
list_of_p_MEDIAN = []
for y in range(len(y_list)):
    list_of_p_MEDIAN.append([])
    for p in range(len(p_list)):
        if pydf_M[y_list[y]][p_list[p]] == u'\u2588':
            list_of_p_MEDIAN[y].append([p_list[p]])
            
# p-Center Selected Sites
list_of_p_CENTER = []
for y in range(len(y_list)):
    list_of_p_CENTER.append([])
    for p in range(len(p_list)):
        if pydf_C[y_list[y]][p_list[p]] == u'\u2588':
            list_of_p_CENTER[y].append([p_list[p]])
            
# p-CentDian Selected Sites
list_of_p_CentDian = []
for y in range(len(y_list)):
    list_of_p_CentDian.append([])
    for p in range(len(p_list)):
        if pydf_CentDian[y_list[y]][p_list[p]] == u'\u2588':
            list_of_p_CentDian[y].append([p_list[p]])

# PMCP Selected Sites
list_of_PMCP = []
for y in range(len(y_list)):
    list_of_PMCP.append([])
    for p in p_dens:
        if pydf_MC[y_list[y]][p] == u'\u2588':
            list_of_PMCP[y].append(p)

Google Maps Plot



In [ ]:

    
map_options = GMapOptions(lat=30.4855, lng=-84.265, map_type="hybrid", zoom=14)

plot = GMapPlot(
    x_range=DataRange1d(), y_range=DataRange1d(), map_options=map_options, title="Waverly Hills")

hover = HoverTool(tooltips="""
        <div>
            <div>
                
            </div>
            <div>
                <span style="font-size: 30px; font-weight: bold;">Site @desc</span> 
            </div>
            <div>
                <span> \b </span>
            </div>
            <div>
                <span style="font-size: 17px; font-weight: bold;">p-Median: </span> 
            </div>
                <div>
                <span style="font-size: 15px; font-weight: bold; color: #ff4d4d;">@p_select_median</span>
            </div>
            <div>
                <span> \b </span>
            </div>
            <div>
                <span style="font-size: 17px; font-weight: bold;">p-Center</span> 
            
            <div>
                <span style="font-size: 14px; font-weight: bold; color: #00b300;">@p_select_center</span>
            </div>
            <div>
                <span> \b </span>
            </div>
            <div>
                <span style="font-size: 17px; font-weight: bold;">p-CentDian</span> 
            </div>
            <div>
                <span style="font-size: 14px; font-weight: bold; color: #3385ff;">@p_select_centdian</span>
            </div>
            <div>
                <span> \b </span>
            </div>
            <span style="font-size: 17px; font-weight: bold;">PMCP Method</span> 
            </div>
            <div>
                <span style="font-size: 14px; font-weight: bold; color: 'gray';">@p_select_pmcp</span>
            </div>
        </div>""")

source_1 = ColumnDataSource(
        data=dict(
        lat=Service_Lat_List,
        lon=Service_Lon_List,
        desc=y_list,
        p_select_center=list_of_p_CENTER,
        p_select_median=list_of_p_MEDIAN,
        p_select_centdian= list_of_p_CentDian,
        p_select_pmcp=list_of_PMCP))

#source_2 = ColumnDataSource(
#    data=dict(
#        xs=line1x,
#        ys=line1y))

facilties = Circle(x="lon", y="lat", size=10, fill_color="yellow", fill_alpha=0.6, line_color=None)

#streets = MultiLine(xs="xs", ys="ys", line_width=20, line_color='red')
#plot.title = "Waverly"

plot.add_glyph(source_1, facilties)
#plot.add_glyph(source_2, streets)

plot.add_tools(PanTool(), WheelZoomTool(), BoxSelectTool(), ResetTool(), hover)
output_file("gmap_plot.html")
show(plot)

Future Work & Vision

$\Longrightarrow$ Provide more accessible alternatives to spatial optimization & GIS

$\Longrightarrow$ Develop a `python` library to bring together spatial analysis & spatial optimization in one package [`spanoptpy`], potentially incorporating:

QGIS	PySAL	NetworkX	Pandas & QGrid	GeoPandas	NumPy / SciPy	Shapely	Bokeh	CyLP	PostgreSQL & PostGIS
GIS	network analysis	network analysis	data frames	geo dataframes	scientific computing	geometric objects	visualizations	optimization	DBMS

$\Longrightarrow$ Need `PySAL.Network` to be able to handle larger networks

$\Longrightarrow$ Intergration with `PostgreSQL` & `PostGIS` for data storage

$\Longrightarrow$ Develop functionality within a `Linux` environment

$\Longrightarrow$ `scipy.spatial.cKDTree(dist_matrix)`

$\ast$ `query_ball_point()` for close neighbors of the selected sites

$\Longrightarrow$ Master `CyLP` from `COIN-OR` or develop an open-source optimization suite

interface with CLP, CBC, CGL
[ http://mpy.github.io/CyLPdoc/ ]
relatively steep learning curve
Computational Optimization Infrastructure for Operations Research
[ http://www.coin-or.org ]

$\ast$ CyLP example

Minimize

$ 3x + 2y $

Subject To

$ x \geq 3$

$ y \geq 5$

$ x - y \leq 20$



In [ ]:

    
# Gurobi
m = gbp.Model()
m.setParam('OutputFlag', False) 
x = m.addVar(vtype=gbp.GRB.CONTINUOUS, name='x')
y = m.addVar(vtype=gbp.GRB.CONTINUOUS, name='y')
m.update()
m.setObjective(3*x + 2*y, gbp.GRB.MINIMIZE)
m.addConstr(x >= 3)
m.addConstr(y >= 5)
m.addConstr(x - y <= 20)
m.optimize()
#m.write('path_m.lp')
print m.objVal
print m.getVars()

# CyLP
s = CyClpSimplex()
x = s.addVariable('x', 1)
y = s.addVariable('y', 1)
s.addConstraint(x >= 3)
s.addConstraint(y >= 5)
s.addConstraint(x - y <= 20)
s.objective = 3 * x + 2 * y
s.primal()
#s.writeLp('path_s')
print s.objectiveValue
print s.primalVariableSolution
print 'Gurobi & CLP Objective Values match? --> ', m.objVal == s.objectiveValue

COIN warning that will be debugged in the next release

System Specs



In [ ]:

    
import datetime as dt
import os
import platform
import sys
import bokeh
import cylp

names = ['OSX', 'Processor ', 'Machine ', 'Python ','PySAL ','Gurobi ','Pandas ','GeoPandas ',
         'Shapely ', 'NumPy ', 'Bokeh ', 'CyLP', 'Date & Time']
versions = [platform.mac_ver()[0], platform.processor(), platform.machine(), platform.python_version(),
            ps.version, gbp.gurobi.version(), pd.__version__, gpd.__version__, 
            str(shapely.__version__), np.__version__, 
            bokeh.__version__, '0.7.1', dt.datetime.now()]
specs = pd.DataFrame(index=names, columns=['Version'])
specs.columns.name = 'Platform & Software Specs'
specs['Version'] = versions
specs # Pandas DF of specifications

email $\Longrightarrow$ jgaboardi@fsu.edu

GitHub $\Longrightarrow$ https://github.com/jGaboardi/AAG_16



In [ ]:

    
IPd.HTML('https://github.com/jGaboardi')

References

Behnel, S., R. Bradshaw, C. Citro, L. Dalcin, D. S. Seljebotn, and K. Smith. 2011. Cython: The best of both worlds. Computing in Science and Engineering 13 (2):31–39.
Bokeh Development Team. 2014. Bokeh: Python library for interactive visualization.
Church, R. L. 2002. Geographical information systems and location science. Computers and Operations Research 29:541–562.
Church, R. L., and A. T. Murray. 2009. Business Site Selections, Locational Analysis, and GIS. Hoboken, NJ, USA: John Wiley & Sons, Inc.
Conde, E. 2008. A note on the minmax regret centdian location on trees. Operations Research Letters 36 (2):271–275.
Current, J., M. S. Daskin, and D. A. Schilling. 2002. Discrete Network Location Models. In Facility Location Applications and Theory, eds. Z. Drezner and H. W. Hamacher, 81–118. New York: Springer Berlin Heidelberg.
Dan Fylstra, L. Hafer, B. Hart, B. Kristjannson, C. Phillips, T. Ralphs, (Matthew Saltzman, E. Straver, (Jean-Paul Watson, and H. G. Santos. CBC. https://projects.coin-or.org/Cbc.
Daskin, M. 2013. Network and Discrete Location: Models, Algorithms and Applications 2nd ed. Hoboken, NJ, USA: John Wiley & Sons, Inc.
Daskin, M. S. 2008. What You Should Know About Location Modeling. Naval Research Logistics 55 (2):283–294.
GeoPandas Developers. 2013. GeoPandas. http://geopandas.org.
Gillies, S., A. Bierbaum, and K. Lautaportti. 2013. Shapely.
Gurobi. 2013. Gurobi optimizer quick start guide.
Hagberg, A. A., D. A. Schult, and P. J. Swart. 2008. Exploring network structure, dynamics, and function using NetworkX. Proceedings of the 7th Python in Science Conference (SciPy 2008) (SciPy):11–15.
Hakimi, S. L. 1964. Optimum Locations of Switching Centers and the Absolute Centers and Medians of a Graph. Operations Research 12 (3):450–459.
Hall, J., L. Hafer, M. Saltzman, and J. Forrest. CLP.
Halpern, J. 1976. The Location of a Center-Median Convex Combination on an Undirected Tree. Journal of Regional Science 16 (2):237–245.
Hamacher, H. W., and S. Nickel. 1998. Classification of location models. Location Science 6 (1-4):229–242.
Horner, M. W., and M. J. Widener. 2010. How do socioeconomic characteristics interact with equity and efficiency considerations? An analysis of hurricane disaster relief goods provision. Geospatial Analysis and Modelling of Urban Structure and Dynamics 99:393–414.
Horner, M. W., and M. J. Widener. 2011. The effects of transportation network failure on people’s accessibility to hurricane disaster relief goods: A modeling approach and application to a Florida case study. Natural Hazards 59:1619–1634.
Hunter, J. D. 2007. Matplotlib: A 2D graphics environment. Computing in Science and Engineering 9 (3):99–104.
Lima, I. 2006. Python for Scientific Computing Python Overview. Marine Chemistry :10–20.
Lougee-Heimer, R. 2003. The Common Optimization INterface for Operations Research. IBM Journal of Research and Development 47 (1):57–66.
Marcelin, J. M., M. W. Horner, E. E. Ozguven, and A. Kocatepe. 2016. How does accessibility to post-disaster relief compare between the aging and the general population? A spatial network optimization analysis of hurricane relief facility locations. International Journal of Disaster Risk Reduction 15:61–72.
McKinney, W. 2010. Data Structures for Statistical Computing in Python. In Proceedings of the 9th Python in Science Conference, 51–56.
Miller, H. J., and S.-L. Shaw. 2001. Geographic Information Systems for Transportation. New York: Oxford University Press.
Millman, K. J., and M. Aivazis. 2011. Python for scientists and engineers. Computing in Science and Engineering 13 (2):9–12.
Minieka, E. 1970. The m-Center Problem. SIAM Review 12:38–39.
Owen, S. H., and M. S. Daskin. 1998. Strategic facility location: A review. European Journal of Operational Research 111 (3):423–447.
Pérez, F., and B. E. Granger. 2007. IPython: A system for interactive scientific computing. Computing in Science and Engineering 9 (3):21–29.
Pérez-Brito, D., J. A. Moreno-Pérez, and R.-M. Inmaculada. 1997. Finite dominating set for the p-facility cent-dian network location problem. Studies in Locational Analysis (August):1–16.
Pérez-Brito, D., J. A. Moreno-Pérez, and I. Rodrı́guez-Martı́n. 1998. The 2-facility centdian network problem. Location Science 6 (1-4):369–381.
QGIS Development Team. Open Source Geospatial Foundation Project. 2016. QGIS Geographic Information System.
ReVelle, C. S., and R. W. Swain. 1970. Central facilities location. Geographical Analysis 2 (1):30–42.
Rey, S. J., and L. Anselin. 2010. PySAL: A Python Library of Spatial Analytical Methods. In Handbook of Applied Spatial Analysis, eds. M. M. Fischer and A. Getis, 175–193. Springer Berlin Heidelberg.
Shier, D. R. 1977. A Min–Max Theorem for p-Center Problems on a Tree. Transportation Science 11:243–52.
Suzuki, A., and Z. Drezner. 1996. The p-center location problem in an area. Location Science 4 (1-2):69–82.
Tamir, A., J. Puerto, and D. Pérez-Brito. 2002. The centdian subtree on tree networks. Discrete Applied Mathematics 118 (3):263–278.
Teitz, M. B., and P. Bart. 1968. Heuristic Methods for Estimating the Generalized Vertex Median of a Weighted Graph. Operations Research 16 (5):955–961.
Tong, D., and A. T. Murray. 2012. Spatial Optimization in Geography. Annals of the Association of American Geographers 102 (6):1290–1309.
Towhidi, M., and D. Orban. 2011. CyLP.
US Census Bureau. 2015. TIGER/Line® Shapefiles and TIGER/Line® Files. U.S. Census Bureau Geography. https://www.census.gov/geo/maps-data/data/tiger-line.html.
Walt, S. van der, S. C. Colbert, and G. Varoquaux. 2011. The NumPy Array: A Struture for Efficient Numerical Computation. Computing in Science & Engeneering 13:22–30.
William, R. 1971. The M-center problem.

Automating Multiple Single-Objective Spatial Optimization Models for Efficiency and Reproducibility

James D. Gaboardi | Association of American Geographers 2016

Florida State University | Department of Geography

Outline

Set Path

Background & Information

PySAL 1.11.0 $\Rightarrow$ PySAL.Network

Python Spatial Analysis Library

Gurobi 6.5.0 $\Rightarrow$ gurobipy

Models

The p-Median Problem $\Rightarrow$ "minimum total cost" $\Rightarrow$ efficiency

$\displaystyle {Z} = {\sum_{i \in 1}^n\sum_{j\in 1}^m a_i c_{ij} x_{ij}}$

$\displaystyle\sum_{j\in m} x_{ij} = 1 ,$ $\forall i \in n$

$\displaystyle\sum_{j \in m} y_j = p$

$x_{ij} - y_j \geq 0,$ $\forall i \in n, j \in m$

$x_{ij}, y_j \in \{0,1\}$ $\forall i \in n , j \in m$

The p-Center Problem $\Rightarrow$ "minimum worst case" $\Rightarrow$ equity

$W$

$\displaystyle\sum_{j\in m} x_{ij} = 1,$ $\forall i \in n$

$\displaystyle\sum_{j \in m} y_j = p$

$x_{ij} - y_j \geq 0,$ $\forall i\in n, j \in m$

$\displaystyle W \geq \sum_{j \in m} c_{ij} x_{ij}$ $\forall i \in n$

$x_{ij}, y_j \in \{0,1\}$ $\forall i \in n, j \in m$

The p-CentDian Problem

$ \displaystyle {W + {Z \over \sum_{i=1}a_i} \over 2}$

$\displaystyle\sum_{j\in m} x_{ij} = 1,$ $\forall i \in n$

$\displaystyle\sum_{j \in m} y_j = p$

$x_{ij} - y_j \geq 0,$ $\forall i\in n, j \in m$

$\displaystyle W \geq \sum_{j \in m} c_{ij} x_{ij}$ $\forall i \in n$

$x_{ij}, y_j \in \{0,1\}$ $\forall i \in n, j \in m$

The PMCP Method

$\Rightarrow$ solve the PMP and PCP to determine with optimal locations with equivalent [p]

$\Rightarrow$ "poor man's" p-CentDian Problem?

Workflow for PMCP

Data & Processing

Process Imports

Data preparation

What we have at this point

Call Client to Service Matrix Function

Solutions

Solve all

Visualizations

Draw PMP figure [p=1] large $ \rightarrow $ small [p=15]

Pandas PMP Data Frame

Bokeh PMP [p vs. cost] trade off

Draw PCP figure [p=1] large $ \rightarrow $ small [p=15]

Pandas PCP Data Frame

Bokeh PCP [p vs. cost] trade off

Draw CentDian figure [p=1] large $ \rightarrow $ small [p=15]

Pandas CentDian Data Frame

Bokeh CentDian [p vs. cost] trade off

Draw PMCP figure

Pandas PMCP Data Frame

Bokeh PMP & PCP [p vs. cost] comparision

Convert Service Facilities Back to Longitude/Latitude for Google Maps Plots

Create Lists of Selected Locations for Google Maps Plot

Google Maps Plot

Future Work & Vision

$\Longrightarrow$ Provide more accessible alternatives to spatial optimization & GIS

$\Longrightarrow$ Develop a python library to bring together spatial analysis & spatial optimization in one package [spanoptpy], potentially incorporating:

$\Longrightarrow$ Need PySAL.Network to be able to handle larger networks

$\Longrightarrow$ Intergration with PostgreSQL & PostGIS for data storage

$\Longrightarrow$ Develop functionality within a Linux environment

$\Longrightarrow$ scipy.spatial.cKDTree(dist_matrix)

$\ast$ query_ball_point() for close neighbors of the selected sites

$\Longrightarrow$ Master CyLP from COIN-OR or develop an open-source optimization suite

$\ast$ CyLP example

System Specs

Thanks for your time! Please share with your students and local government GIS departments, etc.!

email $\Longrightarrow$ jgaboardi@fsu.edu

GitHub $\Longrightarrow$ https://github.com/jGaboardi/AAG_16

References

`PySAL 1.11.0` $\Rightarrow$ `PySAL.Network`

`Gurobi 6.5.0` $\Rightarrow$ `gurobipy`

$\Longrightarrow$ Develop a `python` library to bring together spatial analysis & spatial optimization in one package [`spanoptpy`], potentially incorporating:

$\Longrightarrow$ Need `PySAL.Network` to be able to handle larger networks

$\Longrightarrow$ Intergration with `PostgreSQL` & `PostGIS` for data storage

$\Longrightarrow$ Develop functionality within a `Linux` environment

$\Longrightarrow$ `scipy.spatial.cKDTree(dist_matrix)`

$\ast$ `query_ball_point()` for close neighbors of the selected sites

$\Longrightarrow$ Master `CyLP` from `COIN-OR` or develop an open-source optimization suite