notebook.community

Edit and run 





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#    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

Advisor:      David C. Folch

What's the point?

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

$\Longrightarrow$ Improve research transparency with efficiency and reproducibility

$\Longrightarrow$ Combine of multiple iterations of three problems for increased cost/benefit analysis

Set Path





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import IPython.display as IPd

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

Background & Information

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

$\Longrightarrow$ Combine solutions for p-median and p-center in a 'new' method mimicing the p-centdian problem

$\Longrightarrow$ 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 PM+CP Method

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

$\Longrightarrow$ "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 PM+CP





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# Conceptual Model Workflow
workflow = IPd.Image(path+'/AAG_16.png', width=700, height=700)
workflow



    


















    
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Data & Processing

I use Waverly Hills, a smallish neighborhood in Tallahassee, FL where quite a few professors at Florida State University own homes. The houses tend to be designed by well-known architects, but there are no sidewalks.

This shapefile was clipped from a US Census TIGER/Line file. Additionally, there are some good topological challenges in Waverly Hills that make for good test cases, as seen below.

Florida





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#FL
F = IPd.Image(path+'/F.png', width=600, height=600)
F



    


















    
Out[4]:

Tallahassee





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# Tallahassee
T = IPd.Image(path+'/T.png', width=600, height=600)
T



    


















    
Out[5]:

Waverly Hills





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# Waverly
W = IPd.Image(path+'/W.png', width=400, height=400)
W



    


















    
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GIS Challenges





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# Avon Circle
Avon_Cir = IPd.Image(path+'/Avon.jpg', width=400, height=400)
Avon_Cir



    


















    
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# Millstream Road
Millstream_Rd = IPd.Image(path+'/Millstream.jpg', width=400, height=400)
Millstream_Rd



    


















    
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Proof of concept for demonstration

$\Longrightarrow$ actual roads with interesting challenges

$\Longrightarrow$ constrained simulation of points that mimics an real-world facility location problem

Process Imports





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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
import matplotlib.pyplot as plt
import matplotlib as mpl
%matplotlib inline
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 ...

Define the function to calculate the cost matrix and convert to miles





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def c_s_matrix():  # Define Client to Service Matrix Function
    global All_Dist_MILES # in meters
    All_Neigh_Dist = ntw.allneighbordistances(
                        sourcepattern=ntw.pointpatterns['Rand_Points_CLIENT'],
                        destpattern=ntw.pointpatterns['Rand_Points_SERVICE'])
    All_Dist_MILES = All_Neigh_Dist * 0.000621371 # to miles

Define the function to solve the p-Median, p-Center, and p-CentDian problems





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def Gurobi_PMCP(sites, Ai, AiSum, All_Dist_Miles):
#**************************************************    
    # Define Global Variables
    global pydf_M           #--------- median globals
    global selected_M
    global NEW_Records_PMP
    global VAL_PMP
    global AVG_PMP
    #
    global pydf_C           #--------- center globals
    global selected_C
    global NEW_Records_PCP
    global VAL_PCP
    #
    global pydf_CentDian    #--------- centdian globals
    global selected_CentDian
    global NEW_Records_Pcentdian
    global VAL_CentDian
    #
    global pydf_MC          #--------- pmcp globals
    global VAL_PMCP
    global p_dens
    #***********************************************
    # Initiate Solutions   
    for p in range(1, sites+1): #--------- for all [p] in p = length(service facilities)

        # DATA
        # [p]        --> sites
        # Demand     --> Ai
        # Demand Sum --> AiSum
        # Travel Costs
        Cij = All_Dist_MILES
        # Weighted Costs
        Sij = Ai * Cij
        # Total Client and Service nodes
        client_nodes = range(len(Sij))
        service_nodes = range(len(Sij[0]))

        #*************************************************************
        # PMP
        t1_PMP = time.time()
        
        #     Create Model, Add Variables, & Update Model
        # Instantiate Model
        mPMP = gbp.Model(' -- p-Median -- ')
        # Turn off Gurobi's output
        mPMP.setParam('OutputFlag',False)

        # Add Client Decision Variables (iXj)
        client_var_PMP = []
        for orig in client_nodes:
            client_var_PMP.append([])
            for dest in service_nodes:
                client_var_PMP[orig].append(mPMP.addVar(vtype=gbp.GRB.BINARY,
                                                lb=0,
                                                ub=1,
                                                obj=Sij[orig][dest], 
                                                name='x'+str(orig+1)+'_'+str(dest+1)))

        # Add Service Decision Variables (j)
        serv_var_PMP = []
        for dest in service_nodes:
            serv_var_PMP.append([])
            serv_var_PMP[dest].append(mPMP.addVar(vtype=gbp.GRB.BINARY,
                                          lb=0,
                                          ub=1,
                                          name='y'+str(dest+1)))

        # Update the model
        mPMP.update()

        #     3. Set Objective Function
        mPMP.setObjective(gbp.quicksum(Sij[orig][dest]*client_var_PMP[orig][dest] 
                            for orig in client_nodes for dest in service_nodes), 
                            gbp.GRB.MINIMIZE)


        #     4. Add Constraints
        # Assignment Constraints
        for orig in client_nodes:
            mPMP.addConstr(gbp.quicksum(client_var_PMP[orig][dest] 
                            for dest in service_nodes) == 1)
        # Opening Constraints
        for orig in service_nodes:
            for dest in client_nodes:
                mPMP.addConstr((serv_var_PMP[orig][0] - client_var_PMP[dest][orig] >= 0))

        # Facility Constraint
        mPMP.addConstr(gbp.quicksum(serv_var_PMP[dest][0] for dest in service_nodes) == p)

        #     5. Optimize and Print Results
        # Solve
        mPMP.optimize()

        # Write LP
        mPMP.write(path+'LP_Files/PMP'+str(p)+'.lp')
        t2_PMP = time.time()-t1_PMP

        # Record and Display Results
        print '\n*************************************************************************'
        selected_M = OrderedDict()
        dbf1 = ps.open(path+'Snapped/SERVICE_Snapped.dbf')
        NEW_Records_PMP = []
        for v in mPMP.getVars():
            if 'x' in v.VarName:
                pass
            elif v.x > 0:
                var = '%s' % v.VarName
                selected_M[var]=(u"\u2588")
                for i in range(dbf1.n_records):
                    if var in dbf1.read_record(i):
                        x = dbf1.read_record(i)
                        NEW_Records_PMP.append(x)
                    else:
                        pass
                print '    |                                            ', var
            
        pydf_M = pydf_M.append(selected_M, ignore_index=True)
        
        # Instantiate Shapefile
        SHP_Median = shp.Writer(shp.POINT)
        # Add Points
        for idy,idx,x,y in NEW_Records_PMP:
            SHP_Median.point(float(x), float(y))
        # Add Fields
        SHP_Median.field('y_ID')
        SHP_Median.field('x_ID')
        SHP_Median.field('LAT')
        SHP_Median.field('LON')
        # Add Records
        for idy,idx,x,y in NEW_Records_PMP:
            SHP_Median.record(idy,idx,x,y)
        # Save Shapefile
        SHP_Median.save(path+'Results/Selected_Locations_Pmedian'+str(p)+'.shp')   

        print '    | Selected Facility Locations --------------  ^^^^ '
        print '    | Candidate Facilities [p] ----------------- ', len(selected_M)
        val_m = mPMP.objVal
        VAL_PMP.append(round(val_m, 3))
        print '    | Objective Value (miles) ------------------ ', val_m
        avg_m = float(mPMP.objVal)/float(AiSum)
        AVG_PMP.append(round(avg_m, 3))
        print '    | Avg. Value / Client (miles) -------------- ', avg_m
        print '    | Real Time to Optimize (sec.) ------------- ', t2_PMP
        print '*************************************************************************'
        print ' -- The p-Median Problem -- '
        print ' [p] = ', str(p), '\n\n'
        
        
        #******************************************************************************
        # PCP
        t1_PCP = time.time()
        
        # Instantiate P-Center Model
        mPCP = gbp.Model(' -- p-Center -- ')
        
        # Add Client Decision Variables (iXj)
        client_var_PCP = []
        for orig in client_nodes:
            client_var_PCP.append([])
            for dest in service_nodes:
                client_var_PCP[orig].append(mPCP.addVar(vtype=gbp.GRB.BINARY,
                                                lb=0,
                                                ub=1,
                                                obj=Cij[orig][dest], 
                                                name='x'+str(orig+1)+'_'+str(dest+1)))

        # Add Service Decision Variables (j)
        serv_var_PCP = []
        for dest in service_nodes:
            serv_var_PCP.append([])
            serv_var_PCP[dest].append(mPCP.addVar(vtype=gbp.GRB.BINARY,
                                          lb=0,
                                          ub=1,
                                          name='y'+str(dest+1)))

        # Add the Maximum travel cost variable
        W = mPCP.addVar(vtype=gbp.GRB.CONTINUOUS,
                    lb=0.,
                    name='W') 
        
        # Update the model
        mPCP.update()

        #     3. Set Objective Function
        mPCP.setObjective(W, gbp.GRB.MINIMIZE)

        #     4. Add Constraints
        # Assignment Constraints
        for orig in client_nodes:
            mPCP.addConstr(gbp.quicksum(client_var_PCP[orig][dest] 
                            for dest in service_nodes) == 1)
        # Opening Constraints
        for orig in service_nodes:
            for dest in client_nodes:
                mPCP.addConstr((serv_var_PCP[orig][0] - client_var_PCP[dest][orig] >= 0))

        # Add Maximum travel cost constraints
        for orig in client_nodes:
            mPCP.addConstr(gbp.quicksum(Cij[orig][dest]*client_var_PCP[orig][dest]
                                for dest in service_nodes) - W <= 0)
        
        # Facility Constraint
        mPCP.addConstr(gbp.quicksum(serv_var_PCP[dest][0] for dest in service_nodes) == p)

        #     5. Optimize and Print Results
        # Solve
        mPCP.optimize()

        # Write LP
        mPCP.write(path+'LP_Files/PCP'+str(p)+'.lp')
        t2_PCP = time.time()-t1_PCP

        # Record and Display Results
        print '\n*************************************************************************'
        selected_C = OrderedDict()
        dbf1 = ps.open(path+'Snapped/SERVICE_Snapped.dbf')
        NEW_Records_PCP = []
        for v in mPCP.getVars():
            if 'x' in v.VarName:
                pass
            elif 'W' in v.VarName:
                pass
            elif v.x > 0:
                var = '%s' % v.VarName
                selected_C[var]=(u"\u2588")
                for i in range(dbf1.n_records):
                    if var in dbf1.read_record(i):
                        x = dbf1.read_record(i)
                        NEW_Records_PCP.append(x)
                    else:
                        pass
                print '    |                                            ', var,  '         '
        pydf_C = pydf_C.append(selected_C, ignore_index=True)
        
        # Instantiate Shapefile
        SHP_Center = shp.Writer(shp.POINT)
        # Add Points
        for idy,idx,x,y in NEW_Records_PCP:
            SHP_Center.point(float(x), float(y))
        # Add Fields
        SHP_Center.field('y_ID')
        SHP_Center.field('x_ID')
        SHP_Center.field('LAT')
        SHP_Center.field('LON')
        # Add Records
        for idy,idx,x,y in NEW_Records_PCP:
            SHP_Center.record(idy,idx,x,y)
        # Save Shapefile
        SHP_Center.save(path+'Results/Selected_Locations_Pcenter'+str(p)+'.shp')   

        print '    | Selected Facility Locations --------------  ^^^^ '
        print '    | Candidate Facilities [p] ----------------- ', len(selected_C)
        val_c = mPCP.objVal
        VAL_PCP.append(round(val_c, 3))
        print '    | Objective Value (miles) ------------------ ', val_c
        print '    | Real Time to Optimize (sec.) ------------- ', t2_PCP
        print '*************************************************************************'
        print ' -- The p-Center Problem -- '
        print ' [p] = ', str(p), '\n\n'

        #******************************************************************************
        # p-CentDian
        
        t1_centdian = time.time()
        
        # Instantiate P-Center Model
        mPcentdian = gbp.Model(' -- p-CentDian -- ')
        
        # Add Client Decision Variables (iXj)
        client_var_CentDian = []
        for orig in client_nodes:
            client_var_CentDian.append([])
            for dest in service_nodes:
                client_var_CentDian[orig].append(mPcentdian.addVar(vtype=gbp.GRB.BINARY,
                                                lb=0,
                                                ub=1,
                                                obj=Cij[orig][dest], 
                                                name='x'+str(orig+1)+'_'+str(dest+1)))

        # Add Service Decision Variables (j)
        serv_var_CentDian = []
        for dest in service_nodes:
            serv_var_CentDian.append([])
            serv_var_CentDian[dest].append(mPcentdian.addVar(vtype=gbp.GRB.BINARY,
                                          lb=0,
                                          ub=1,
                                          name='y'+str(dest+1)))

        # Add the Maximum travel cost variable
        W_CD = mPcentdian.addVar(vtype=gbp.GRB.CONTINUOUS,
                    lb=0.,
                    name='W') 
        
        # Update the model
        mPcentdian.update()

        #     3. Set Objective Function
        M = gbp.quicksum(Sij[orig][dest]*client_var_CentDian[orig][dest] 
                    for orig in client_nodes for dest in service_nodes)
        
        Zt = M/AiSum
        
        mPcentdian.setObjective((W_CD + Zt) / 2, gbp.GRB.MINIMIZE)

        #     4. Add Constraints
        # Assignment Constraints
        for orig in client_nodes:
            mPcentdian.addConstr(gbp.quicksum(client_var_CentDian[orig][dest] 
                            for dest in service_nodes) == 1)
        # Opening Constraints
        for orig in service_nodes:
            for dest in client_nodes:
                mPcentdian.addConstr((serv_var_CentDian[orig][0] - 
                                      client_var_CentDian[dest][orig] 
                                      >= 0))

        # Add Maximum travel cost constraints
        for orig in client_nodes:
            mPcentdian.addConstr(gbp.quicksum(Cij[orig][dest]*client_var_CentDian[orig][dest]
                                for dest in service_nodes) - W_CD <= 0)
        
        # Facility Constraint
        mPcentdian.addConstr(gbp.quicksum(serv_var_CentDian[dest][0] 
                                          for dest in service_nodes) 
                                          == p)

        #     5. Optimize and Print Results
        # Solve
        mPcentdian.optimize()

        # Write LP
        mPcentdian.write(path+'LP_Files/CentDian'+str(p)+'.lp')
        t2_centdian = time.time()-t1_centdian

        # Record and Display Results
        print '\n*************************************************************************'
        selected_CentDian = OrderedDict()
        dbf1 = ps.open(path+'Snapped/SERVICE_Snapped.dbf')
        NEW_Records_Pcentdian = []
        for v in mPcentdian.getVars():
            if 'x' in v.VarName:
                pass
            elif 'W' in v.VarName:
                pass
            elif v.x > 0:
                var = '%s' % v.VarName
                selected_CentDian[var]=(u"\u2588")
                for i in range(dbf1.n_records):
                    if var in dbf1.read_record(i):
                        x = dbf1.read_record(i)
                        NEW_Records_Pcentdian.append(x)
                    else:
                        pass
                print '    |                                            ', var,  '         '
        pydf_CentDian = pydf_CentDian.append(selected_CentDian, ignore_index=True)
        
        # Instantiate Shapefile
        SHP_CentDian = shp.Writer(shp.POINT)
        # Add Points
        for idy,idx,x,y in NEW_Records_Pcentdian:
            SHP_CentDian.point(float(x), float(y))
        # Add Fields
        SHP_CentDian.field('y_ID')
        SHP_CentDian.field('x_ID')
        SHP_CentDian.field('LAT')
        SHP_CentDian.field('LON')
        # Add Records
        for idy,idx,x,y in NEW_Records_Pcentdian:
            SHP_CentDian.record(idy,idx,x,y)
        # Save Shapefile
        SHP_CentDian.save(path+'Results/Selected_Locations_CentDian'+str(p)+'.shp')   

        print '    | Selected Facility Locations --------------  ^^^^ '
        print '    | Candidate Facilities [p] ----------------- ', len(selected_CentDian)
        val_cd = mPcentdian.objVal
        VAL_CentDian.append(round(val_cd, 3))
        print '    | Objective Value (miles) ------------------ ', val_cd
        print '    | Real Time to Optimize (sec.) ------------- ', t2_centdian
        print '*************************************************************************'
        print ' -- The p-CentDian Problem -- '
        print ' [p] = ', str(p), '\n\n'
        
        #******************************************************************************
        # p-Median + p-Center Method
        
        # Record solutions that record identical facility selection
        if selected_M.keys() == selected_C.keys() == selected_CentDian.keys():
            
            pydf_MC = pydf_MC.append(selected_C, ignore_index=True) # append PMCP dataframe
            p_dens.append('p='+str(p)) # density of [p] 
            VAL_PMCP.append([round(val_m,3), round(avg_m,3), 
                             round(val_c,3), round(val_cd,3)]) # append PMCP list
            
            # Instantiate Shapefile
            SHP_PMCP = shp.Writer(shp.POINT)
            # Add Points
            for idy,idx,x,y in NEW_Records_PCP:
                SHP_PMCP.point(float(x), float(y))
            # Add Fields
            SHP_PMCP.field('y_ID')
            SHP_PMCP.field('x_ID')
            SHP_PMCP.field('LAT')
            SHP_PMCP.field('LON')
            # Add Records
            for idy,idx,x,y in NEW_Records_PCP:
                SHP_PMCP.record(idy,idx,x,y)
            # Save Shapefile
            SHP_PMCP.save(path+'Results/Selected_Locations_PMCP'+str(p)+'.shp')
        else:
            pass

Reproject the street network with GeoPandas





In [12]:



    


# Waverly  Hills
STREETS_Orig = gpd.read_file(path+'Waverly_Trim/Waverly.shp')
STREETS = gpd.read_file(path+'Waverly_Trim/Waverly.shp')
STREETS.to_crs(epsg=2779, inplace=True) # NAD83(HARN) / Florida North
STREETS.to_file(path+'WAVERLY/WAVERLY.shp')
STREETS[:5]



    


















    
Out[12]:










  
    

      
      FULLNAME
      LINEARID
      MTFCC
      RTTYP
      geometry
    

  
  
    

      0
      N Ride
      110158908901
      S1400
      M
      LINESTRING (621388.7127543514 163519.180833002...
    

    

      1
      N Ride
      110158908902
      S1400
      M
      LINESTRING (621970.5489300904 163446.280840687...
    

    

      2
      S Ride
      110158908913
      S1400
      M
      LINESTRING (621390.8512775075 163260.113274841...
    

    

      3
      Walton Dr
      110158976711
      S1400
      M
      LINESTRING (623030.6549003529 165573.807212860...
    

    

      4
      Fontaine Dr
      110159019423
      S1400
      M
      LINESTRING (623368.1251134621 165788.033358145...

Plot Waverly Hills





In [13]:



    


STREETS.plot()



    


















    
Out[13]:








<matplotlib.axes._subplots.AxesSubplot at 0x103e55a50>

Instantiate Network and read in WAVERLY.shp





In [14]:



    


ntw = ps.Network(path+'WAVERLY/WAVERLY.shp')
shp_W = ps.open(path+'WAVERLY/WAVERLY.shp')

Create Buffer of 200 meters





In [15]:



    


buff = STREETS.buffer(200)  #Buffer
buff[:5]



    


















    
Out[15]:








0    POLYGON ((621465.3167118202 163719.7320321212,...
1    POLYGON ((622011.323892185 163231.3835929646, ...
2    POLYGON ((621448.7888749669 163451.5130565979,...
3    POLYGON ((623119.6454389954 165752.7642512885,...
4    POLYGON ((623171.3152202072 165844.5275350949,...
dtype: object

Plot Buffers of Individual Streets





In [16]:



    


buff.plot()



    


















    
Out[16]:








<matplotlib.axes._subplots.AxesSubplot at 0x118facbd0>

Create a Unary Union of the individual street buffers





In [17]:



    


buffU = buff.unary_union  #Buffer Union
buff1 = gpd.GeoSeries(buffU)
buff1.crs = STREETS.crs
Buff = gpd.GeoDataFrame(buff1, crs=STREETS.crs)
Buff.columns = ['geometry']
Buff



    


















    
Out[17]:










  
    

      
      geometry
    

  
  
    

      0
      POLYGON ((622700.1341171626 163125.4208353004,...

Plot the unary union buffer





In [18]:



    


Buff.plot()



    


















    
Out[18]:








<matplotlib.axes._subplots.AxesSubplot at 0x1195c8110>

Create 1000 random ppoints within the bounds of WAVERLY.shp





In [19]:



    


np.random.seed(352)
x = np.random.uniform(shp_W.bbox[0], shp_W.bbox[2], 1000)
np.random.seed(850)
y = np.random.uniform(shp_W.bbox[1], shp_W.bbox[3], 1000)  
coords0= zip(x,y)
coords = [shapely.geometry.Point(i) for i in coords0]
Rand = gpd.GeoDataFrame(coords)
Rand.crs = STREETS.crs
Rand.columns = ['geometry']
Rand[:5]



    


















    
Out[19]:










  
    

      
      geometry
    

  
  
    

      0
      POINT (623427.8190332755 166275.3062495199)
    

    

      1
      POINT (622697.2031300153 162992.9652048317)
    

    

      2
      POINT (621922.7038578516 164917.050600741)
    

    

      3
      POINT (621615.3609543599 164243.1879802518)
    

    

      4
      POINT (623454.7605480383 163733.8321171276)

Plot the 1000 random





In [20]:



    


Rand.plot()



    


















    
Out[20]:








<matplotlib.axes._subplots.AxesSubplot at 0x1195df210>

Create GeoPandas DF of the random points within the Unary Buffer





In [21]:



    


Inter = [Buff['geometry'].intersection(p) for p in Rand['geometry']]
INTER = gpd.GeoDataFrame(Inter, crs=STREETS.crs)
INTER.columns = ['geometry']
INTER[:5]



    


















    
Out[21]:










  
    

      
      geometry
    

  
  
    

      0
      POINT (623427.8190332755 166275.3062495199)
    

    

      1
      ()
    

    

      2
      POINT (621922.7038578516 164917.050600741)
    

    

      3
      POINT (621615.3609543599 164243.1879802518)
    

    

      4
      ()

Plot the points within the Unary Buffer





In [22]:



    


INTER.plot()



    


















    
Out[22]:








<matplotlib.axes._subplots.AxesSubplot at 0x11b582550>

Add only intersecting records to a list





In [23]:



    


# Add records that are points within the buffer
point_in = []
for p in INTER['geometry']:
    if type(p) == shapely.geometry.point.Point:
        point_in.append(p)
point_in[:5]



    


















    
Out[23]:








[<shapely.geometry.point.Point at 0x11880d1d0>,
 <shapely.geometry.point.Point at 0x11b5d23d0>,
 <shapely.geometry.point.Point at 0x11b5d24d0>,
 <shapely.geometry.point.Point at 0x11b5d26d0>,
 <shapely.geometry.point.Point at 0x11b5d27d0>]

Keep the first 100 for clients and the last 15 for service facilities





In [24]:



    


CLIENT = gpd.GeoDataFrame(point_in[:100], crs=STREETS.crs)
CLIENT.columns = ['geometry']
SERVICE = gpd.GeoDataFrame(point_in[-15:], crs=STREETS.crs)
SERVICE.columns = ['geometry']
CLIENT.to_file(path+'CLIENT')
SERVICE.to_file(path+'SERVICE')



    











In [25]:



    


CLIENT[:5]



    


















    
Out[25]:










  
    

      
      geometry
    

  
  
    

      0
      POINT (623427.8190332755 166275.3062495199)
    

    

      1
      POINT (621922.7038578516 164917.050600741)
    

    

      2
      POINT (621615.3609543599 164243.1879802518)
    

    

      3
      POINT (621862.673209806 163415.9461312958)
    

    

      4
      POINT (621659.2900576409 163168.6459518427)
    

  




















In [26]:



    


SERVICE[:5]



    


















    
Out[26]:










  
    

      
      geometry
    

  
  
    

      0
      POINT (623558.8536816949 166234.7359778296)
    

    

      1
      POINT (623016.3125997729 166225.5051476926)
    

    

      2
      POINT (621521.47511704 164019.1317467149)
    

    

      3
      POINT (621613.5815189295 163400.6942524033)
    

    

      4
      POINT (623159.364408439 165123.3394007629)

Instaniate non-solution graphs to be drawn





In [27]:



    


g = nx.Graph() # Roads & Nodes
g1 = nx.MultiGraph() # Edges and Vertices
GRAPH_client = nx.Graph() # Clients 
g_client = nx.Graph() # Snapped Clients
GRAPH_service = nx.Graph() # Service
g_service = nx.Graph() # Snapped Service

Instantiate and fill Client and Service point dictionaries





In [28]:



    


points_client = {} 
points_service = {}

CLI = ps.open(path+'CLIENT/CLIENT.shp')
for idx, coords in enumerate(CLI):
    GRAPH_client.add_node(idx)
    points_client[idx] = coords
    GRAPH_client.node[idx] = coords
    
SER = ps.open(path+'SERVICE/SERVICE.shp')
for idx, coords in enumerate(SER):
    GRAPH_service.add_node(idx)
    points_service[idx] = coords
    GRAPH_service.node[idx] = coords

Simulate weights for Client Demand





In [29]:



    


# Client Weights for demand
np.random.seed(850)
Ai = np.random.randint(1, 5, len(CLI))
Ai = Ai.reshape(len(Ai),1)
AiSum = np.sum(Ai) # Sum of Weights (Total Demand)

Instantiate Client .shp





In [30]:



    


client = shp.Writer(shp.POINT) # Client Shapefile
# Add Random Points
for i,j in CLI:
    client.point(i,j)
# Add Fields
client.field('client_ID')
client.field('Weight')
counter = 0
for i in range(len(CLI)):
    counter = counter + 1
    client.record('client_' + str(counter), Ai[i])
client.save(path+'Simulated/RandomPoints_CLIENT') # Save Shapefile

Instantiate Service .shp





In [31]:



    


service = shp.Writer(shp.POINT) #Service Shapefile
# Add Random Points
for i,j in SER:
    service.point(i,j)
# Add Fields
service.field('y_ID')
service.field('x_ID')
counter = 0
for i in range(len(SER)):
    counter = counter + 1
    service.record('y' + str(counter), 'x' + str(counter))
service.save(path+'Simulated/RandomPoints_SERVICE') # Save Shapefile

Snap Client and Service points to the network





In [32]:



    


# Snap
Snap_C = ntw.snapobservations(path+'Simulated/RandomPoints_CLIENT.shp', 
                     'Rand_Points_CLIENT', attribute=True)
Snap_S = ntw.snapobservations(path+'Simulated/RandomPoints_SERVICE.shp', 
                     'Rand_Points_SERVICE', attribute=True)

Create lat/lon lists of snapped service coords





In [33]:



    


# Create Lat & Lon lists of the snapped service locations
y_snapped = []
x_snapped = []
for i,j in ntw.pointpatterns['Rand_Points_SERVICE'].snapped_coordinates.iteritems():
    y_snapped.append(j[0]) 
    x_snapped.append(j[1])

Instantiate snapped Service .shp





In [34]:



    


service_SNAP = shp.Writer(shp.POINT) # Snapped Service Shapefile
# Add Points
for i,j in ntw.pointpatterns['Rand_Points_SERVICE'].snapped_coordinates.iteritems():
    service_SNAP.point(j[0],j[1])
# Add Fields
service_SNAP.field('y_ID')
service_SNAP.field('x_ID')
service_SNAP.field('LAT')
service_SNAP.field('LON')
counter = 0
for i in range(len(ntw.pointpatterns['Rand_Points_SERVICE'].snapped_coordinates)):
    counter = counter + 1
    service_SNAP.record('y' + str(counter), 'x' + str(counter), y_snapped[i], x_snapped[i])
service_SNAP.save(path+'Snapped/SERVICE_Snapped') # Save Shapefile

What we have thus far with imported & simulated data





In [35]:



    


mpl.rcParams['figure.figsize']=5,8
# Draw Graph of Roads
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)

# Draw Graph of Snapped Client Nodes
g_client = nx.Graph()
for p,coords in ntw.pointpatterns['Rand_Points_CLIENT'].snapped_coordinates.iteritems():
    g_client.add_node(p)
    g_client.node[p] = coords
nx.draw(g_client, ntw.pointpatterns['Rand_Points_CLIENT'].snapped_coordinates, 
        node_size=75, alpha=1, node_color='b')

# Draw Graph of Snapped Service Nodes
g_service = nx.Graph()
for p,coords in ntw.pointpatterns['Rand_Points_SERVICE'].snapped_coordinates.iteritems():
    g_service.add_node(p)
    g_service.node[p] = coords
nx.draw(g_service, ntw.pointpatterns['Rand_Points_SERVICE'].snapped_coordinates, 
        node_size=200, alpha=1, node_color='c')

# Draw Graph of Random Client Points
nx.draw(GRAPH_client, points_client, 
    node_size=20, alpha=1, node_color='y')

# Draw Graph of Random Service Points
nx.draw(GRAPH_service, points_service, 
    node_size=75, alpha=1, node_color='w')

# Legend (Ordered Dictionary)
LEGEND = OrderedDict()
LEGEND['Network Nodes']=g
LEGEND['Roads']=g
LEGEND['Snapped Client']=g_client
LEGEND['Snapped Service']=g_service
LEGEND['Client Nodes']=GRAPH_client
LEGEND['Service Nodes']=GRAPH_service
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')

# 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)
plt.annotate('<| ~ .25 miles |>', xy=(623200, 163600), 
             fontstyle='italic', fontsize='large', alpha=0.95)

plt.show()

Call Client to Service Matrix Function





In [36]:



    


# Call Client to Service Matrix Function
c_s_matrix()

Create Lists to fill index and columns of GeoPandas Data Frames





In [37]:



    


# PANDAS DATAFRAME OF p/y results
p_list = []
for i in range(1, len(SER)+1):
    p = 'p='+str(i)
    p_list.append(p)
y_list = []
for i in range(1, len(SER)+1):
    y = 'y'+str(i)
    y_list.append(y)

Instantiate GeoPandas Dataframes





In [38]:



    


pydf_M = pd.DataFrame(index=p_list,columns=y_list)
pydf_C = pd.DataFrame(index=p_list,columns=y_list)
pydf_CentDian = pd.DataFrame(index=p_list,columns=y_list)
pydf_MC = pd.DataFrame(index=p_list,columns=y_list)

Create PMP, PCP, and CentDian solution graphs





In [39]:



    


# p-Median
P_Med_Graphs = OrderedDict()
for x in range(1, len(SER)+1):
    P_Med_Graphs["{0}".format(x)] = nx.Graph()
    
# p-Center
P_Cent_Graphs = OrderedDict()
for x in range(1, len(SER)+1):
    P_Cent_Graphs["{0}".format(x)] = nx.Graph()
    
# p-CentDian
P_CentDian_Graphs = OrderedDict()
for x in range(1, len(SER)+1):
    P_CentDian_Graphs["{0}".format(x)] = nx.Graph()

Instantiate lists for objective values and average values of all models





In [40]:



    


# PMP
VAL_PMP = []
AVG_PMP = []

# PCP
VAL_PCP = []

# CentDian
VAL_CentDian = []

# PMCP
VAL_PMCP = []
p_dens = [] # when the facilities for the p-median & p-center are the same

Solutions

Solve all





In [41]:



    


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



    


















    






*************************************************************************
    |                                             y10
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  1
    | Objective Value (miles) ------------------  252.628622209
    | Avg. Value / Client (miles) --------------  1.01866379923
    | Real Time to Optimize (sec.) -------------  0.236118793488
*************************************************************************
 -- The p-Median Problem -- 
 [p] =  1 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e+00, 1e+00]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 1e+00]
Presolve removed 1600 rows and 1500 columns
Presolve time: 0.13s
Presolved: 101 rows, 16 columns, 1515 nonzeros
Variable types: 1 continuous, 15 integer (15 binary)
Found heuristic solution: objective 2.8439770
Found heuristic solution: objective 2.6194347
Found heuristic solution: objective 2.2724425
Presolve removed 9 rows and 9 columns
Presolved: 7 rows, 108 columns, 657 nonzeros

Presolve removed 7 rows and 108 columns

Root relaxation: objective 1.619386e+00, 6 iterations, 0.00 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0    1.61939    0    4    2.27244    1.61939  28.7%     -    0s
H    0     0                       1.7981720    1.61939  9.94%     -    0s
     0     0     cutoff    0         1.79817    1.79817  0.00%     -    0s

Explored 0 nodes (8 simplex iterations) in 0.16 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 1.798172038792e+00, best bound 1.798172038792e+00, gap 0.0%

*************************************************************************
    |                                             y10          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  1
    | Objective Value (miles) ------------------  1.79817203879
    | Real Time to Optimize (sec.) -------------  0.290205001831
*************************************************************************
 -- The p-Center Problem -- 
 [p] =  1 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e-05, 5e-01]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 1e+00]
Presolve removed 1600 rows and 1500 columns
Presolve time: 0.13s
Presolved: 101 rows, 16 columns, 1615 nonzeros
Variable types: 1 continuous, 15 integer (15 binary)
Found heuristic solution: objective 1.6074385
Presolve removed 1 rows and 100 columns
Presolved: 15 rows, 17 columns, 45 nonzeros


Root relaxation: objective 1.408418e+00, 1 iterations, 0.00 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

*    0     0               0       1.4084179    1.40842  0.00%     -    0s

Explored 0 nodes (1 simplex iterations) in 0.14 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 1.408417919012e+00, best bound 1.408417919012e+00, gap 0.0%

*************************************************************************
    |                                             y10          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  1
    | Objective Value (miles) ------------------  1.40841791901
    | Real Time to Optimize (sec.) -------------  0.295914888382
*************************************************************************
 -- The p-CentDian Problem -- 
 [p] =  1 



*************************************************************************
    |                                             y12
    |                                             y15
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  2
    | Objective Value (miles) ------------------  178.435962925
    | Avg. Value / Client (miles) --------------  0.719499850505
    | Real Time to Optimize (sec.) -------------  0.121235847473
*************************************************************************
 -- The p-Median Problem -- 
 [p] =  2 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e+00, 1e+00]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 2e+00]
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)
Found heuristic solution: objective 1.9504745
Found heuristic solution: objective 1.9092634
Found heuristic solution: objective 1.8309455

Root relaxation: objective 1.091514e+00, 1348 iterations, 0.03 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0    1.09151    0  409    1.83095    1.09151  40.4%     -    0s
H    0     0                       1.5822633    1.09151  31.0%     -    0s
H    0     0                       1.4568885    1.09151  25.1%     -    0s
     0     0     cutoff    0         1.45689    1.45689  0.00%     -    0s

Cutting planes:
  Gomory: 1
  MIR: 6
  Zero half: 4

Explored 0 nodes (3729 simplex iterations) in 0.23 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 1.456888501112e+00, best bound 1.456888501112e+00, gap 0.0%

*************************************************************************
    |                                             y4          
    |                                             y7          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  2
    | Objective Value (miles) ------------------  1.45688850111
    | Real Time to Optimize (sec.) -------------  0.332101106644
*************************************************************************
 -- The p-Center Problem -- 
 [p] =  2 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e-05, 5e-01]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 2e+00]
Presolve time: 0.02s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)
Found heuristic solution: objective 1.3653839

Root relaxation: objective 9.384803e-01, 959 iterations, 0.02 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0    0.93848    0  313    1.36538    0.93848  31.3%     -    0s
H    0     0                       1.3108682    0.93848  28.4%     -    0s
H    0     0                       1.1642570    0.93848  19.4%     -    0s
     0     0    0.98114    0  356    1.16426    0.98114  15.7%     -    0s
     0     0    0.98266    0  379    1.16426    0.98266  15.6%     -    0s
     0     0    1.00716    0  462    1.16426    1.00716  13.5%     -    0s
     0     0    1.00822    0  472    1.16426    1.00822  13.4%     -    0s
     0     0    1.00925    0  519    1.16426    1.00925  13.3%     -    0s
     0     0    1.00955    0  460    1.16426    1.00955  13.3%     -    0s
     0     0    1.01074    0  423    1.16426    1.01074  13.2%     -    0s
     0     0    1.01100    0  493    1.16426    1.01100  13.2%     -    0s
     0     0    1.01201    0  492    1.16426    1.01201  13.1%     -    0s
     0     0    1.01202    0  493    1.16426    1.01202  13.1%     -    0s
     0     0    1.01223    0  468    1.16426    1.01223  13.1%     -    0s
     0     0    1.01289    0  452    1.16426    1.01289  13.0%     -    0s
     0     0    1.01313    0  499    1.16426    1.01313  13.0%     -    0s
     0     0    1.01316    0  514    1.16426    1.01316  13.0%     -    0s
     0     0    1.01337    0  508    1.16426    1.01337  13.0%     -    0s
     0     0    1.01339    0  507    1.16426    1.01339  13.0%     -    0s
     0     0    1.01374    0  490    1.16426    1.01374  12.9%     -    0s
H    0     0                       1.1104573    1.01374  8.71%     -    0s
     0     0    1.01419    0  523    1.11046    1.01419  8.67%     -    0s
     0     0    1.01450    0  492    1.11046    1.01450  8.64%     -    0s
     0     0    1.01455    0  509    1.11046    1.01455  8.64%     -    0s
     0     0    1.01466    0  493    1.11046    1.01466  8.63%     -    0s
     0     0    1.01466    0  494    1.11046    1.01466  8.63%     -    0s
     0     0    1.01466    0  494    1.11046    1.01466  8.63%     -    0s
     0     2    1.01466    0  494    1.11046    1.01466  8.63%     -    0s

Cutting planes:
  Gomory: 3
  MIR: 5
  Zero half: 8

Explored 32 nodes (3159 simplex iterations) in 0.88 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 1.110457339150e+00, best bound 1.110457339150e+00, gap 0.0%

*************************************************************************
    |                                             y7          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  2
    | Objective Value (miles) ------------------  1.11045733915
    | Real Time to Optimize (sec.) -------------  1.10769510269
*************************************************************************
 -- The p-CentDian Problem -- 
 [p] =  2 



*************************************************************************
    |                                             y6
    |                                             y9
    |                                             y15
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  3
    | Objective Value (miles) ------------------  136.143483242
    | Avg. Value / Client (miles) --------------  0.548965658235
    | Real Time to Optimize (sec.) -------------  0.107157945633
*************************************************************************
 -- The p-Median Problem -- 
 [p] =  3 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e+00, 1e+00]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 3e+00]
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)
Found heuristic solution: objective 1.5785868
Found heuristic solution: objective 1.4568885

Root relaxation: objective 8.591137e-01, 912 iterations, 0.02 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0    0.85911    0  338    1.45689    0.85911  41.0%     -    0s
H    0     0                       1.3729803    0.85911  37.4%     -    0s
H    0     0                       1.2590943    0.85911  31.8%     -    0s
H    0     0                       1.1257531    0.85911  23.7%     -    0s
     0     0     cutoff    0         1.12575    1.12575  0.00%     -    0s

Cutting planes:
  Gomory: 1
  MIR: 1

Explored 0 nodes (2761 simplex iterations) in 0.13 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 1.125753136820e+00, best bound 1.125753136820e+00, gap 0.0%

*************************************************************************
    |                                             y6          
    |                                             y9          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  3
    | Objective Value (miles) ------------------  1.12575313682
    | Real Time to Optimize (sec.) -------------  0.226248979568
*************************************************************************
 -- The p-Center Problem -- 
 [p] =  3 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e-05, 5e-01]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 3e+00]
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)
Found heuristic solution: objective 1.1092692

Root relaxation: objective 7.343400e-01, 739 iterations, 0.02 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0    0.73434    0  369    1.10927    0.73434  33.8%     -    0s
H    0     0                       1.0374863    0.73434  29.2%     -    0s
H    0     0                       0.8373594    0.73434  12.3%     -    0s
     0     0    0.74354    0  411    0.83736    0.74354  11.2%     -    0s
     0     0    0.74705    0  427    0.83736    0.74705  10.8%     -    0s
     0     0    0.75609    0  342    0.83736    0.75609  9.71%     -    0s
     0     0    0.75609    0  287    0.83736    0.75609  9.71%     -    0s
     0     0    0.77291    0  292    0.83736    0.77291  7.70%     -    0s
     0     0    0.77304    0  293    0.83736    0.77304  7.68%     -    0s
     0     0    0.78962    0  371    0.83736    0.78962  5.70%     -    0s
     0     0    0.78994    0  343    0.83736    0.78994  5.66%     -    0s
     0     0    0.79100    0  383    0.83736    0.79100  5.54%     -    0s
     0     0    0.79102    0  384    0.83736    0.79102  5.53%     -    0s
     0     0    0.79115    0  385    0.83736    0.79115  5.52%     -    0s
     0     0    0.79192    0  369    0.83736    0.79192  5.43%     -    0s
     0     0    0.79215    0  377    0.83736    0.79215  5.40%     -    0s
     0     0    0.79238    0  375    0.83736    0.79238  5.37%     -    0s
     0     0    0.79238    0  386    0.83736    0.79238  5.37%     -    0s
     0     0    0.79240    0  387    0.83736    0.79240  5.37%     -    0s
     0     0    0.79261    0  372    0.83736    0.79261  5.34%     -    0s
     0     0    0.79264    0  384    0.83736    0.79264  5.34%     -    0s
     0     0    0.79294    0  433    0.83736    0.79294  5.31%     -    0s
     0     0    0.79298    0  400    0.83736    0.79298  5.30%     -    0s
     0     0    0.79388    0  405    0.83736    0.79388  5.19%     -    0s
     0     0    0.79388    0  406    0.83736    0.79388  5.19%     -    0s
     0     0    0.79395    0  396    0.83736    0.79395  5.18%     -    0s
     0     0    0.79400    0  403    0.83736    0.79400  5.18%     -    0s
     0     0    0.79417    0  390    0.83736    0.79417  5.16%     -    0s
     0     0    0.79418    0  389    0.83736    0.79418  5.16%     -    0s
     0     0    0.79419    0  390    0.83736    0.79419  5.16%     -    0s
     0     2    0.79419    0  390    0.83736    0.79419  5.16%     -    0s

Cutting planes:
  Gomory: 3
  MIR: 9
  Zero half: 11

Explored 18 nodes (2973 simplex iterations) in 0.76 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 8.373593975272e-01, best bound 8.373593975272e-01, gap 0.0%

*************************************************************************
    |                                             y6          
    |                                             y9          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  3
    | Objective Value (miles) ------------------  0.837359397527
    | Real Time to Optimize (sec.) -------------  0.926659107208
*************************************************************************
 -- The p-CentDian Problem -- 
 [p] =  3 



*************************************************************************
    |                                             y6
    |                                             y9
    |                                             y12
    |                                             y15
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  4
    | Objective Value (miles) ------------------  113.400992358
    | Avg. Value / Client (miles) --------------  0.457262065961
    | Real Time to Optimize (sec.) -------------  0.113733053207
*************************************************************************
 -- The p-Median Problem -- 
 [p] =  4 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e+00, 1e+00]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 4e+00]
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)
Found heuristic solution: objective 2.1103205
Found heuristic solution: objective 1.8656435
Found heuristic solution: objective 1.8309455

Root relaxation: objective 7.619977e-01, 1001 iterations, 0.03 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0    0.76200    0  316    1.83095    0.76200  58.4%     -    0s
H    0     0                       1.4589251    0.76200  47.8%     -    0s
H    0     0                       1.4492867    0.76200  47.4%     -    0s
H    0     0                       1.4396841    0.76200  47.1%     -    0s
     0     0    0.81016    0  313    1.43968    0.81016  43.7%     -    0s
     0     0    0.81016    0  323    1.43968    0.81016  43.7%     -    0s
H    0     0                       1.4395274    0.81016  43.7%     -    0s
H    0     0                       1.4286353    0.81016  43.3%     -    0s
     0     0    0.84044    0  326    1.42864    0.84044  41.2%     -    0s
H    0     0                       1.3773513    0.84044  39.0%     -    0s
     0     0    0.86119    0  355    1.37735    0.86119  37.5%     -    0s
     0     0    0.86426    0  327    1.37735    0.86426  37.3%     -    0s
H    0     0                       1.3448528    0.86426  35.7%     -    0s
H    0     0                       1.2773295    0.86426  32.3%     -    0s
     0     0    0.86931    0  249    1.27733    0.86931  31.9%     -    0s
     0     0    0.86931    0  249    1.27733    0.86931  31.9%     -    0s
H    0     0                       1.2689454    0.86931  31.5%     -    0s
H    0     0                       1.0419579    0.86931  16.6%     -    0s
H    0     0                       1.0388639    0.86931  16.3%     -    0s
     0     0    0.87521    0  242    1.03886    0.87521  15.8%     -    0s
     0     0    0.91169    0  159    1.03886    0.91169  12.2%     -    0s
     0     0    0.94070    0  113    1.03886    0.94070  9.45%     -    0s
H    0     0                       0.9408986    0.94070  0.02%     -    0s

Explored 0 nodes (8909 simplex iterations) in 0.39 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 9.408985900044e-01, best bound 9.408985900044e-01, gap 0.0%

*************************************************************************
    |                                             y4          
    |                                             y6          
    |                                             y12          
    |                                             y13          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  4
    | Objective Value (miles) ------------------  0.940898590004
    | Real Time to Optimize (sec.) -------------  0.482060909271
*************************************************************************
 -- The p-Center Problem -- 
 [p] =  4 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e-05, 5e-01]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 4e+00]
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)
Found heuristic solution: objective 0.9313797

Root relaxation: objective 6.508189e-01, 528 iterations, 0.01 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0    0.65082    0  249    0.93138    0.65082  30.1%     -    0s
H    0     0                       0.8177119    0.65082  20.4%     -    0s
H    0     0                       0.7267475    0.65082  10.4%     -    0s
     0     0    0.66575    0  273    0.72675    0.66575  8.39%     -    0s
     0     0    0.66575    0  243    0.72675    0.66575  8.39%     -    0s
     0     0    0.68720    0  241    0.72675    0.68720  5.44%     -    0s
     0     0    0.69032    0  213    0.72675    0.69032  5.01%     -    0s
     0     0    0.69233    0  226    0.72675    0.69233  4.74%     -    0s
     0     0    0.70014    0  177    0.72675    0.70014  3.66%     -    0s
     0     0    0.70037    0  177    0.72675    0.70037  3.63%     -    0s
H    0     0                       0.7059598    0.70037  0.79%     -    0s
     0     0    0.70073    0  216    0.70596    0.70073  0.74%     -    0s
     0     0    0.70129    0  215    0.70596    0.70129  0.66%     -    0s
     0     0    0.70129    0  228    0.70596    0.70129  0.66%     -    0s
     0     0    0.70129    0  153    0.70596    0.70129  0.66%     -    0s
     0     0    0.70135    0  169    0.70596    0.70135  0.65%     -    0s
     0     0    0.70298    0  171    0.70596    0.70298  0.42%     -    0s
     0     0    0.70320    0  203    0.70596    0.70320  0.39%     -    0s
     0     0    0.70378    0  198    0.70596    0.70378  0.31%     -    0s
     0     0    0.70380    0  203    0.70596    0.70380  0.31%     -    0s
     0     0    0.70384    0  203    0.70596    0.70384  0.30%     -    0s
     0     0    0.70384    0  123    0.70596    0.70384  0.30%     -    0s
     0     0    0.70384    0   59    0.70596    0.70384  0.30%     -    0s
     0     0     cutoff    0         0.70596    0.70596  0.00%     -    0s

Explored 0 nodes (2083 simplex iterations) in 0.43 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 7.059598493583e-01, best bound 7.059598493583e-01, gap 0.0%

*************************************************************************
    |                                             y6          
    |                                             y9          
    |                                             y12          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  4
    | Objective Value (miles) ------------------  0.705959849358
    | Real Time to Optimize (sec.) -------------  0.577835798264
*************************************************************************
 -- The p-CentDian Problem -- 
 [p] =  4 



*************************************************************************
    |                                             y6
    |                                             y9
    |                                             y10
    |                                             y12
    |                                             y15
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  5
    | Objective Value (miles) ------------------  102.664903993
    | Avg. Value / Client (miles) --------------  0.41397138707
    | Real Time to Optimize (sec.) -------------  0.110842943192
*************************************************************************
 -- The p-Median Problem -- 
 [p] =  5 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e+00, 1e+00]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 5e+00]
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)
Found heuristic solution: objective 2.1103205
Found heuristic solution: objective 2.0703495
Found heuristic solution: objective 2.0155480

Root relaxation: objective 7.212557e-01, 988 iterations, 0.02 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0    0.72126    0  260    2.01555    0.72126  64.2%     -    0s
H    0     0                       1.8669800    0.72126  61.4%     -    0s
H    0     0                       1.5276038    0.72126  52.8%     -    0s
H    0     0                       1.4209713    0.72126  49.2%     -    0s
H    0     0                       1.2974999    0.72126  44.4%     -    0s
     0     0    0.73314    0  265    1.29750    0.73314  43.5%     -    0s
     0     0    0.74922    0  241    1.29750    0.74922  42.3%     -    0s
H    0     0                       1.2667513    0.74922  40.9%     -    0s
H    0     0                       0.9614777    0.74922  22.1%     -    0s
H    0     0                       0.9408986    0.74922  20.4%     -    0s
     0     0    0.83262    0  131    0.94090    0.83262  11.5%     -    0s
     0     0    0.83262    0  106    0.94090    0.83262  11.5%     -    0s
H    0     0                       0.9217976    0.83262  9.67%     -    0s
     0     0     cutoff    0         0.92180    0.92180  0.00%     -    0s

Cutting planes:
  Gomory: 1
  Zero half: 1

Explored 0 nodes (6477 simplex iterations) in 0.28 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 9.217975808173e-01, best bound 9.217975808173e-01, gap 0.0%

*************************************************************************
    |                                             y4          
    |                                             y6          
    |                                             y7          
    |                                             y9          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  5
    | Objective Value (miles) ------------------  0.921797580817
    | Real Time to Optimize (sec.) -------------  0.38795208931
*************************************************************************
 -- The p-Center Problem -- 
 [p] =  5 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e-05, 5e-01]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 5e+00]
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)
Found heuristic solution: objective 0.8373682

Root relaxation: objective 5.933325e-01, 494 iterations, 0.01 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0    0.59333    0  213    0.83737    0.59333  29.1%     -    0s
H    0     0                       0.7129892    0.59333  16.8%     -    0s
H    0     0                       0.6844683    0.59333  13.3%     -    0s
     0     0    0.60142    0  201    0.68447    0.60142  12.1%     -    0s
     0     0    0.60142    0  230    0.68447    0.60142  12.1%     -    0s
     0     0    0.61223    0  213    0.68447    0.61223  10.6%     -    0s
H    0     0                       0.6843145    0.61223  10.5%     -    0s
     0     0    0.61913    0  167    0.68431    0.61913  9.53%     -    0s
     0     0    0.62372    0  160    0.68431    0.62372  8.85%     -    0s
     0     0    0.62401    0  222    0.68431    0.62401  8.81%     -    0s
     0     0    0.62401    0  222    0.68431    0.62401  8.81%     -    0s
     0     0    0.62412    0  222    0.68431    0.62412  8.80%     -    0s
     0     0    0.62442    0  213    0.68431    0.62442  8.75%     -    0s
     0     2    0.62442    0  213    0.68431    0.62442  8.75%     -    0s

Cutting planes:
  Gomory: 4
  MIR: 1

Explored 18 nodes (1423 simplex iterations) in 0.33 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 6.843145099126e-01, best bound 6.843145099126e-01, gap 0.0%

*************************************************************************
    |                                             y6          
    |                                             y9          
    |                                             y10          
    |                                             y12          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  5
    | Objective Value (miles) ------------------  0.684314509913
    | Real Time to Optimize (sec.) -------------  0.519908905029
*************************************************************************
 -- The p-CentDian Problem -- 
 [p] =  5 



*************************************************************************
    |                                             y3
    |                                             y6
    |                                             y10
    |                                             y12
    |                                             y13
    |                                             y15
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  6
    | Objective Value (miles) ------------------  94.5669128279
    | Avg. Value / Client (miles) --------------  0.381318196887
    | Real Time to Optimize (sec.) -------------  0.126721143723
*************************************************************************
 -- The p-Median Problem -- 
 [p] =  6 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e+00, 1e+00]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 6e+00]
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)
Found heuristic solution: objective 2.1103205
Found heuristic solution: objective 1.7981720
Found heuristic solution: objective 1.7748490

Root relaxation: objective 7.038147e-01, 798 iterations, 0.02 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0    0.70381    0  261    1.77485    0.70381  60.3%     -    0s
H    0     0                       1.6026288    0.70381  56.1%     -    0s
H    0     0                       1.4568885    0.70381  51.7%     -    0s
H    0     0                       1.4495653    0.70381  51.4%     -    0s
H    0     0                       0.7186486    0.70381  2.06%     -    0s
     0     0     cutoff    0         0.71865    0.71865  0.00%     -    0s

Cutting planes:
  Gomory: 1
  MIR: 2

Explored 0 nodes (1890 simplex iterations) in 0.11 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 7.186486151187e-01, best bound 7.186486151187e-01, gap 0.0%

*************************************************************************
    |                                             y4          
    |                                             y6          
    |                                             y7          
    |                                             y9          
    |                                             y12          
    |                                             y14          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  6
    | Objective Value (miles) ------------------  0.718648615119
    | Real Time to Optimize (sec.) -------------  0.217437982559
*************************************************************************
 -- The p-Center Problem -- 
 [p] =  6 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e-05, 5e-01]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 6e+00]
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)
Found heuristic solution: objective 0.8222154

Root relaxation: objective 5.511881e-01, 345 iterations, 0.01 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0    0.55119    0   68    0.82222    0.55119  33.0%     -    0s
H    0     0                       0.5516980    0.55119  0.09%     -    0s
     0     0     cutoff    0         0.55170    0.55170  0.00%     -    0s

Cutting planes:
  MIR: 1

Explored 0 nodes (346 simplex iterations) in 0.06 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 5.516979567454e-01, best bound 5.516979567454e-01, gap 0.0%

*************************************************************************
    |                                             y4          
    |                                             y6          
    |                                             y9          
    |                                             y10          
    |                                             y12          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  6
    | Objective Value (miles) ------------------  0.551697956745
    | Real Time to Optimize (sec.) -------------  0.202598810196
*************************************************************************
 -- The p-CentDian Problem -- 
 [p] =  6 



*************************************************************************
    |                                             y3
    |                                             y4
    |                                             y6
    |                                             y10
    |                                             y12
    |                                             y13
    |                                             y15
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  7
    | Objective Value (miles) ------------------  87.3193388308
    | Avg. Value / Client (miles) --------------  0.352094108189
    | Real Time to Optimize (sec.) -------------  0.107014894485
*************************************************************************
 -- The p-Median Problem -- 
 [p] =  7 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e+00, 1e+00]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 7e+00]
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)
Found heuristic solution: objective 2.1103205
Found heuristic solution: objective 2.0992328

Root relaxation: objective 6.923302e-01, 682 iterations, 0.02 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0    0.69233    0   30    2.09923    0.69233  67.0%     -    0s
H    0     0                       0.9942685    0.69233  30.4%     -    0s
H    0     0                       0.9374837    0.69233  26.2%     -    0s
H    0     0                       0.7174374    0.69233  3.50%     -    0s
     0     0     cutoff    0         0.71744    0.71744  0.00%     -    0s

Cutting planes:
  Gomory: 3

Explored 0 nodes (1377 simplex iterations) in 0.08 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 7.174373546020e-01, best bound 7.174373546020e-01, gap 0.0%

*************************************************************************
    |                                             y4          
    |                                             y5          
    |                                             y6          
    |                                             y9          
    |                                             y10          
    |                                             y12          
    |                                             y14          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  7
    | Objective Value (miles) ------------------  0.717437354602
    | Real Time to Optimize (sec.) -------------  0.19621014595
*************************************************************************
 -- The p-Center Problem -- 
 [p] =  7 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e-05, 5e-01]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 7e+00]
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)
Found heuristic solution: objective 0.8209395

Root relaxation: objective 5.323701e-01, 342 iterations, 0.01 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0    0.53237    0   68    0.82094    0.53237  35.2%     -    0s
H    0     0                       0.5438722    0.53237  2.11%     -    0s
     0     0    0.53545    0  120    0.54387    0.53545  1.55%     -    0s
     0     0    0.53571    0  115    0.54387    0.53571  1.50%     -    0s
H    0     0                       0.5365451    0.53571  0.16%     -    0s
     0     0     cutoff    0         0.53655    0.53655  0.00%     -    0s

Cutting planes:
  Gomory: 1

Explored 0 nodes (647 simplex iterations) in 0.12 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 5.365451355982e-01, best bound 5.365451355982e-01, gap 0.0%

*************************************************************************
    |                                             y4          
    |                                             y6          
    |                                             y9          
    |                                             y10          
    |                                             y12          
    |                                             y13          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  7
    | Objective Value (miles) ------------------  0.536545135598
    | Real Time to Optimize (sec.) -------------  0.298075914383
*************************************************************************
 -- The p-CentDian Problem -- 
 [p] =  7 



*************************************************************************
    |                                             y3
    |                                             y4
    |                                             y5
    |                                             y6
    |                                             y10
    |                                             y12
    |                                             y13
    |                                             y15
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  8
    | Objective Value (miles) ------------------  83.7381742802
    | Avg. Value / Client (miles) --------------  0.337653928549
    | Real Time to Optimize (sec.) -------------  0.122640132904
*************************************************************************
 -- The p-Median Problem -- 
 [p] =  8 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e+00, 1e+00]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 8e+00]
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)
Found heuristic solution: objective 2.7112002
Found heuristic solution: objective 2.4632524
Found heuristic solution: objective 2.2724425

Root relaxation: objective 6.923302e-01, 500 iterations, 0.01 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0    0.69233    0    8    2.27244    0.69233  69.5%     -    0s
H    0     0                       0.9008560    0.69233  23.1%     -    0s
H    0     0                       0.8317502    0.69233  16.8%     -    0s
H    0     0                       0.6923302    0.69233  0.00%     -    0s

Explored 0 nodes (982 simplex iterations) in 0.06 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 6.923302409058e-01, best bound 6.923302409058e-01, gap 0.0%

*************************************************************************
    |                                             y1          
    |                                             y4          
    |                                             y5          
    |                                             y6          
    |                                             y9          
    |                                             y10          
    |                                             y12          
    |                                             y14          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  8
    | Objective Value (miles) ------------------  0.692330240906
    | Real Time to Optimize (sec.) -------------  0.158569097519
*************************************************************************
 -- The p-Center Problem -- 
 [p] =  8 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e-05, 5e-01]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 8e+00]
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)
Found heuristic solution: objective 0.8063274

Root relaxation: objective 5.170791e-01, 306 iterations, 0.01 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0    0.51708    0  117    0.80633    0.51708  35.9%     -    0s
H    0     0                       0.5287194    0.51708  2.20%     -    0s
H    0     0                       0.5254779    0.51708  1.60%     -    0s
     0     0    0.52280    0   49    0.52548    0.52280  0.51%     -    0s
     0     0    0.52280    0   46    0.52548    0.52280  0.51%     -    0s
     0     0    0.52425    0   48    0.52548    0.52425  0.23%     -    0s
     0     0    0.52435    0   48    0.52548    0.52435  0.21%     -    0s
     0     0     cutoff    0         0.52548    0.52548  0.00%     -    0s

Cutting planes:
  Gomory: 2

Explored 0 nodes (457 simplex iterations) in 0.09 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 5.254779370978e-01, best bound 5.254779370978e-01, gap 0.0%

*************************************************************************
    |                                             y4          
    |                                             y5          
    |                                             y6          
    |                                             y8          
    |                                             y9          
    |                                             y10          
    |                                             y12          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  8
    | Objective Value (miles) ------------------  0.525477937098
    | Real Time to Optimize (sec.) -------------  0.23900103569
*************************************************************************
 -- The p-CentDian Problem -- 
 [p] =  8 



*************************************************************************
    |                                             y3
    |                                             y4
    |                                             y5
    |                                             y6
    |                                             y8
    |                                             y10
    |                                             y12
    |                                             y13
    |                                             y15
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  9
    | Objective Value (miles) ------------------  80.74360394
    | Avg. Value / Client (miles) --------------  0.325579048145
    | Real Time to Optimize (sec.) -------------  0.0997478961945
*************************************************************************
 -- The p-Median Problem -- 
 [p] =  9 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e+00, 1e+00]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 9e+00]
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)
Found heuristic solution: objective 2.0992328
Found heuristic solution: objective 2.0125124
Found heuristic solution: objective 2.0091626

Root relaxation: objective 6.923302e-01, 406 iterations, 0.01 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

*    0     0               0       0.6923302    0.69233  0.00%     -    0s

Explored 0 nodes (618 simplex iterations) in 0.04 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 6.923302409058e-01, best bound 6.923302409058e-01, gap 0.0%

*************************************************************************
    |                                             y1          
    |                                             y4          
    |                                             y5          
    |                                             y6          
    |                                             y9          
    |                                             y10          
    |                                             y12          
    |                                             y13          
    |                                             y14          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  9
    | Objective Value (miles) ------------------  0.692330240906
    | Real Time to Optimize (sec.) -------------  0.138150930405
*************************************************************************
 -- The p-Center Problem -- 
 [p] =  9 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e-05, 5e-01]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 9e+00]
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)
Found heuristic solution: objective 0.6410806

Root relaxation: objective 5.103203e-01, 222 iterations, 0.00 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0    0.51032    0   26    0.64108    0.51032  20.4%     -    0s
H    0     0                       0.5103251    0.51032  0.00%     -    0s

Explored 0 nodes (222 simplex iterations) in 0.04 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 5.103251159507e-01, best bound 5.103203326948e-01, gap 0.0009%

*************************************************************************
    |                                             y4          
    |                                             y5          
    |                                             y6          
    |                                             y8          
    |                                             y9          
    |                                             y10          
    |                                             y12          
    |                                             y13          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  9
    | Objective Value (miles) ------------------  0.510325115951
    | Real Time to Optimize (sec.) -------------  0.177711963654
*************************************************************************
 -- The p-CentDian Problem -- 
 [p] =  9 



*************************************************************************
    |                                             y3
    |                                             y4
    |                                             y5
    |                                             y6
    |                                             y8
    |                                             y9
    |                                             y10
    |                                             y12
    |                                             y13
    |                                             y15
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  10
    | Objective Value (miles) ------------------  78.6731180279
    | Avg. Value / Client (miles) --------------  0.317230314628
    | Real Time to Optimize (sec.) -------------  0.102800130844
*************************************************************************
 -- The p-Median Problem -- 
 [p] =  10 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e+00, 1e+00]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 1e+01]
Found heuristic solution: objective 2.81913
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)

Root relaxation: objective 6.923302e-01, 322 iterations, 0.01 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

*    0     0               0       0.6923302    0.69233  0.00%     -    0s

Explored 0 nodes (466 simplex iterations) in 0.03 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 6.923302409058e-01, best bound 6.923302409058e-01, gap 0.0%

*************************************************************************
    |                                             y1          
    |                                             y4          
    |                                             y5          
    |                                             y6          
    |                                             y9          
    |                                             y10          
    |                                             y12          
    |                                             y13          
    |                                             y14          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  10
    | Objective Value (miles) ------------------  0.692330240906
    | Real Time to Optimize (sec.) -------------  0.128736972809
*************************************************************************
 -- The p-Center Problem -- 
 [p] =  10 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e-05, 5e-01]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 1e+01]
Found heuristic solution: objective 2.05011
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)

Root relaxation: objective 5.049770e-01, 185 iterations, 0.00 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

*    0     0               0       0.5049770    0.50498  0.00%     -    0s

Explored 0 nodes (185 simplex iterations) in 0.03 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 5.049769752478e-01, best bound 5.049769752478e-01, gap 0.0%

*************************************************************************
    |                                             y3          
    |                                             y4          
    |                                             y5          
    |                                             y6          
    |                                             y8          
    |                                             y9          
    |                                             y10          
    |                                             y12          
    |                                             y13          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  10
    | Objective Value (miles) ------------------  0.504976975248
    | Real Time to Optimize (sec.) -------------  0.317002058029
*************************************************************************
 -- The p-CentDian Problem -- 
 [p] =  10 



*************************************************************************
    |                                             y1
    |                                             y2
    |                                             y3
    |                                             y4
    |                                             y5
    |                                             y6
    |                                             y9
    |                                             y10
    |                                             y12
    |                                             y13
    |                                             y15
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  11
    | Objective Value (miles) ------------------  77.3823534807
    | Avg. Value / Client (miles) --------------  0.312025618874
    | Real Time to Optimize (sec.) -------------  0.0958559513092
*************************************************************************
 -- The p-Median Problem -- 
 [p] =  11 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e+00, 1e+00]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 1e+01]
Found heuristic solution: objective 2.46144
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)

Root relaxation: objective 6.923302e-01, 295 iterations, 0.00 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

*    0     0               0       0.6923302    0.69233  0.00%     -    0s

Explored 0 nodes (441 simplex iterations) in 0.03 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 6.923302409058e-01, best bound 6.923302409058e-01, gap 0.0%

*************************************************************************
    |                                             y1          
    |                                             y3          
    |                                             y4          
    |                                             y5          
    |                                             y6          
    |                                             y9          
    |                                             y10          
    |                                             y12          
    |                                             y13          
    |                                             y14          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  11
    | Objective Value (miles) ------------------  0.692330240906
    | Real Time to Optimize (sec.) -------------  0.126229047775
*************************************************************************
 -- The p-Center Problem -- 
 [p] =  11 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e-05, 5e-01]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 1e+01]
Found heuristic solution: objective 1.8681
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)

Root relaxation: objective 5.023746e-01, 175 iterations, 0.00 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

*    0     0               0       0.5023746    0.50237  0.00%     -    0s

Explored 0 nodes (175 simplex iterations) in 0.02 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 5.023746273704e-01, best bound 5.023746273704e-01, gap 0.0%

*************************************************************************
    |                                             y1          
    |                                             y2          
    |                                             y3          
    |                                             y4          
    |                                             y5          
    |                                             y6          
    |                                             y9          
    |                                             y10          
    |                                             y12          
    |                                             y13          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  11
    | Objective Value (miles) ------------------  0.50237462737
    | Real Time to Optimize (sec.) -------------  0.165828943253
*************************************************************************
 -- The p-CentDian Problem -- 
 [p] =  11 



*************************************************************************
    |                                             y1
    |                                             y2
    |                                             y3
    |                                             y4
    |                                             y5
    |                                             y6
    |                                             y8
    |                                             y9
    |                                             y10
    |                                             y12
    |                                             y13
    |                                             y15
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  12
    | Objective Value (miles) ------------------  76.453088462
    | Avg. Value / Client (miles) --------------  0.308278582508
    | Real Time to Optimize (sec.) -------------  0.101176023483
*************************************************************************
 -- The p-Median Problem -- 
 [p] =  12 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e+00, 1e+00]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 1e+01]
Found heuristic solution: objective 3.04114
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)

Root relaxation: objective 6.923302e-01, 220 iterations, 0.00 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

*    0     0               0       0.6923302    0.69233  0.00%     -    0s

Explored 0 nodes (307 simplex iterations) in 0.03 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 6.923302409058e-01, best bound 6.923302409058e-01, gap 0.0%

*************************************************************************
    |                                             y1          
    |                                             y3          
    |                                             y4          
    |                                             y5          
    |                                             y6          
    |                                             y9          
    |                                             y10          
    |                                             y11          
    |                                             y12          
    |                                             y13          
    |                                             y14          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  12
    | Objective Value (miles) ------------------  0.692330240906
    | Real Time to Optimize (sec.) -------------  0.120443105698
*************************************************************************
 -- The p-Center Problem -- 
 [p] =  12 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e-05, 5e-01]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 1e+01]
Found heuristic solution: objective 2.19009
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)

Root relaxation: objective 5.005011e-01, 176 iterations, 0.00 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

*    0     0               0       0.5005011    0.50050  0.00%     -    0s

Explored 0 nodes (176 simplex iterations) in 0.03 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 5.005011091877e-01, best bound 5.005011091877e-01, gap 0.0%

*************************************************************************
    |                                             y1          
    |                                             y2          
    |                                             y3          
    |                                             y4          
    |                                             y5          
    |                                             y6          
    |                                             y8          
    |                                             y9          
    |                                             y10          
    |                                             y12          
    |                                             y13          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  12
    | Objective Value (miles) ------------------  0.500501109188
    | Real Time to Optimize (sec.) -------------  0.165493011475
*************************************************************************
 -- The p-CentDian Problem -- 
 [p] =  12 



*************************************************************************
    |                                             y1
    |                                             y2
    |                                             y3
    |                                             y4
    |                                             y5
    |                                             y6
    |                                             y8
    |                                             y9
    |                                             y10
    |                                             y11
    |                                             y12
    |                                             y13
    |                                             y15
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  13
    | Objective Value (miles) ------------------  75.7366934931
    | Avg. Value / Client (miles) --------------  0.305389893117
    | Real Time to Optimize (sec.) -------------  0.0976669788361
*************************************************************************
 -- The p-Median Problem -- 
 [p] =  13 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e+00, 1e+00]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 1e+01]
Found heuristic solution: objective 3.04114
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)

Root relaxation: objective 6.923302e-01, 224 iterations, 0.00 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

*    0     0               0       0.6923302    0.69233  0.00%     -    0s

Explored 0 nodes (318 simplex iterations) in 0.03 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 6.923302409058e-01, best bound 6.923302409058e-01, gap 0.0%

*************************************************************************
    |                                             y1          
    |                                             y3          
    |                                             y4          
    |                                             y5          
    |                                             y6          
    |                                             y8          
    |                                             y9          
    |                                             y10          
    |                                             y11          
    |                                             y12          
    |                                             y13          
    |                                             y14          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  13
    | Objective Value (miles) ------------------  0.692330240906
    | Real Time to Optimize (sec.) -------------  0.122086048126
*************************************************************************
 -- The p-Center Problem -- 
 [p] =  13 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e-05, 5e-01]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 1e+01]
Found heuristic solution: objective 2.16651
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)

Root relaxation: objective 4.990568e-01, 170 iterations, 0.00 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

*    0     0               0       0.4990568    0.49906  0.00%     -    0s

Explored 0 nodes (170 simplex iterations) in 0.02 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 4.990567644923e-01, best bound 4.990567644923e-01, gap 0.0%

*************************************************************************
    |                                             y1          
    |                                             y2          
    |                                             y3          
    |                                             y4          
    |                                             y5          
    |                                             y6          
    |                                             y8          
    |                                             y9          
    |                                             y10          
    |                                             y11          
    |                                             y12          
    |                                             y13          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  13
    | Objective Value (miles) ------------------  0.499056764492
    | Real Time to Optimize (sec.) -------------  0.172690153122
*************************************************************************
 -- The p-CentDian Problem -- 
 [p] =  13 



*************************************************************************
    |                                             y1
    |                                             y2
    |                                             y3
    |                                             y4
    |                                             y5
    |                                             y6
    |                                             y8
    |                                             y9
    |                                             y10
    |                                             y11
    |                                             y12
    |                                             y13
    |                                             y14
    |                                             y15
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  14
    | Objective Value (miles) ------------------  75.1421439649
    | Avg. Value / Client (miles) --------------  0.302992515988
    | Real Time to Optimize (sec.) -------------  0.0979490280151
*************************************************************************
 -- The p-Median Problem -- 
 [p] =  14 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e+00, 1e+00]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 1e+01]
Found heuristic solution: objective 3.04114
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)

Root relaxation: objective 6.923302e-01, 192 iterations, 0.00 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

*    0     0               0       0.6923302    0.69233  0.00%     -    0s

Explored 0 nodes (192 simplex iterations) in 0.02 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 6.923302409058e-01, best bound 6.923302409058e-01, gap 0.0%

*************************************************************************
    |                                             y1          
    |                                             y3          
    |                                             y4          
    |                                             y5          
    |                                             y6          
    |                                             y7          
    |                                             y8          
    |                                             y9          
    |                                             y10          
    |                                             y11          
    |                                             y12          
    |                                             y13          
    |                                             y14          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  14
    | Objective Value (miles) ------------------  0.692330240906
    | Real Time to Optimize (sec.) -------------  0.118752002716
*************************************************************************
 -- The p-Center Problem -- 
 [p] =  14 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e-05, 5e-01]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 1e+01]
Found heuristic solution: objective 2.21301
Presolve time: 0.01s
Presolved: 1701 rows, 1516 columns, 6015 nonzeros
Variable types: 1 continuous, 1515 integer (1515 binary)

Root relaxation: objective 4.976614e-01, 139 iterations, 0.00 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

*    0     0               0       0.4976614    0.49766  0.00%     -    0s

Explored 0 nodes (139 simplex iterations) in 0.02 seconds
Thread count was 4 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 4.976613784467e-01, best bound 4.976613784467e-01, gap 0.0%

*************************************************************************
    |                                             y1          
    |                                             y2          
    |                                             y3          
    |                                             y4          
    |                                             y5          
    |                                             y6          
    |                                             y8          
    |                                             y9          
    |                                             y10          
    |                                             y11          
    |                                             y12          
    |                                             y13          
    |                                             y14          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  14
    | Objective Value (miles) ------------------  0.497661378447
    | Real Time to Optimize (sec.) -------------  0.176685810089
*************************************************************************
 -- The p-CentDian Problem -- 
 [p] =  14 



*************************************************************************
    |                                             y1
    |                                             y2
    |                                             y3
    |                                             y4
    |                                             y5
    |                                             y6
    |                                             y7
    |                                             y8
    |                                             y9
    |                                             y10
    |                                             y11
    |                                             y12
    |                                             y13
    |                                             y14
    |                                             y15
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  15
    | Objective Value (miles) ------------------  74.615226966
    | Avg. Value / Client (miles) --------------  0.300867850669
    | Real Time to Optimize (sec.) -------------  0.0964210033417
*************************************************************************
 -- The p-Median Problem -- 
 [p] =  15 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e+00, 1e+00]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 2e+01]
Found heuristic solution: objective 3.04114
Presolve removed 1701 rows and 1516 columns
Presolve time: 0.00s
Presolve: All rows and columns removed

Explored 0 nodes (0 simplex iterations) in 0.01 seconds
Thread count was 1 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 6.923302409058e-01, best bound 6.923302409058e-01, gap 0.0%

*************************************************************************
    |                                             y1          
    |                                             y2          
    |                                             y3          
    |                                             y4          
    |                                             y5          
    |                                             y6          
    |                                             y7          
    |                                             y8          
    |                                             y9          
    |                                             y10          
    |                                             y11          
    |                                             y12          
    |                                             y13          
    |                                             y14          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  15
    | Objective Value (miles) ------------------  0.692330240906
    | Real Time to Optimize (sec.) -------------  0.104492902756
*************************************************************************
 -- The p-Center Problem -- 
 [p] =  15 


Optimize a model with 1701 rows, 1516 columns and 6115 nonzeros
Coefficient statistics:
  Matrix range    [2e-03, 3e+00]
  Objective range [1e-05, 5e-01]
  Bounds range    [1e+00, 1e+00]
  RHS range       [1e+00, 2e+01]
Found heuristic solution: objective 2.20619
Presolve removed 1701 rows and 1516 columns
Presolve time: 0.00s
Presolve: All rows and columns removed

Explored 0 nodes (0 simplex iterations) in 0.01 seconds
Thread count was 1 (of 4 available processors)

Optimal solution found (tolerance 1.00e-04)
Best objective 4.965990457876e-01, best bound 4.965990457876e-01, gap 0.0%

*************************************************************************
    |                                             y1          
    |                                             y2          
    |                                             y3          
    |                                             y4          
    |                                             y5          
    |                                             y6          
    |                                             y7          
    |                                             y8          
    |                                             y9          
    |                                             y10          
    |                                             y11          
    |                                             y12          
    |                                             y13          
    |                                             y14          
    |                                             y15          
    | Selected Facility Locations --------------  ^^^^ 
    | Candidate Facilities [p] -----------------  15
    | Objective Value (miles) ------------------  0.496599045788
    | Real Time to Optimize (sec.) -------------  0.153763055801
*************************************************************************
 -- The p-CentDian Problem -- 
 [p] =  15

Calculate and record percentage decrease





In [42]:



    


# PMP Total
PMP_Tot_Diff = []
for i in range(len(VAL_PMP)):
    if i == 0:
        PMP_Tot_Diff.append('0%')
    elif i <= len(VAL_PMP):
        n1 = VAL_PMP[i-1]
        n2 = VAL_PMP[i]
        diff = n2 - n1
        perc_change = (diff/n1)*100.
        PMP_Tot_Diff.append(str(round(perc_change, 2))+'%')

# PMP Average
PMP_Avg_Diff = []
for i in range(len(AVG_PMP)):
    if i == 0:
        PMP_Avg_Diff.append('0%')
    elif i <= len(AVG_PMP):
        n1 = AVG_PMP[i-1]
        n2 = AVG_PMP[i]
        diff = n2 - n1
        perc_change = (diff/n1)*100.
        PMP_Avg_Diff.append(str(round(perc_change, 2))+'%')
        
# PCP
PCP_Diff = []
for i in range(len(VAL_PCP)):
    if i == 0:
        PCP_Diff.append('0%')
    elif i <= len(VAL_PCP):
        n1 = VAL_PCP[i-1]
        n2 = VAL_PCP[i]
        diff = n2 - n1
        perc_change = (diff/n1)*100.
        PCP_Diff.append(str(round(perc_change, 2))+'%')

# p-CentDian
CentDian_Diff = []
for i in range(len(VAL_CentDian)):
    if i == 0:
        CentDian_Diff.append('0%')
    elif i <= len(VAL_CentDian):
        n1 = VAL_CentDian[i-1]
        n2 = VAL_CentDian[i]
        diff = n2 - n1
        perc_change = (diff/n1)*100.
        CentDian_Diff.append(str(round(perc_change, 2))+'%')

# PMCP       
PMCP_Diff = []
counter = 0
for i in range(len(VAL_PMCP)):
    PMCP_Diff.append([])
    for j in range(len(VAL_PMCP[0])):
        if i == 0:
            PMCP_Diff[i].append('0%')
        elif i <= len(VAL_PMCP):
            n1 = VAL_PMCP[i-1][j]
            n2 = VAL_PMCP[i][j]
            diff = n2 - n1
            perc_change = (diff/n1*100.)
            PMCP_Diff[i].append(str(round(perc_change, 2))+'%')

Data Frames adjust





In [43]:



    


# PMP
pydf_M = pydf_M[len(SER):]
pydf_M.reset_index()
pydf_M.index = p_list
pydf_M.columns.name = 'Decision\nVariables'
pydf_M.index.name = 'Facility\nDensity'
pydf_M['Tot. Obj. Value'] = VAL_PMP
pydf_M['Tot. % Change'] = PMP_Tot_Diff
pydf_M['Avg. Obj. Value'] = AVG_PMP
pydf_M['Avg. % Change'] = PMP_Avg_Diff
pydf_M = pydf_M.fillna('')
#pydf_M.to_csv(path+'CSV')  <-- need to change squares to alphanumeric to use

# PCP
pydf_C = pydf_C[len(SER):]
pydf_C.reset_index()
pydf_C.index = p_list
pydf_C.columns.name = 'Decision\nVariables'
pydf_C.index.name = 'Facility\nDensity'
pydf_C['Worst Case Obj. Value'] = VAL_PCP
pydf_C['Worst Case % Change'] = PCP_Diff
pydf_C = pydf_C.fillna('')
#pydf_C.to_csv(path+'CSV')  <-- need to change squares to alphanumeric to use

pydf_CentDian = pydf_CentDian[len(SER):]
pydf_CentDian.reset_index()
pydf_CentDian.index = p_list
pydf_CentDian.columns.name = 'Decision\nVariables'
pydf_CentDian.index.name = 'Facility\nDensity'
pydf_CentDian['CentDian Obj. Value'] = VAL_CentDian
pydf_CentDian['CentDian % Change'] = CentDian_Diff
pydf_CentDian = pydf_CentDian.fillna('')
#pydf_CentDian.to_csv(path+'CSV')  <-- need to change squares to alphanumeric to use

# PMCP
pydf_MC = pydf_MC[len(SER):]
pydf_MC.reset_index()
pydf_MC.index = p_dens
pydf_MC.columns.name = 'D.V.'
pydf_MC.index.name = 'F.D.'
pydf_MC['Min.\nTotal'] = [VAL_PMCP[x][0] for x in range(len(VAL_PMCP))]
pydf_MC['Min.\nTotal\n%\nChange'] = [PMCP_Diff[x][0] for x in range(len(PMCP_Diff))]
pydf_MC['Avg.\nTotal'] = [VAL_PMCP[x][1] for x in range(len(VAL_PMCP))]
pydf_MC['Avg.\nTotal\n%\nChange'] = [PMCP_Diff[x][1] for x in range(len(PMCP_Diff))]
pydf_MC['Worst\nCase'] = [VAL_PMCP[x][2] for x in range(len(VAL_PMCP))]
pydf_MC['Worst\nCase\n%\nChange'] = [PMCP_Diff[x][2] for x in range(len(PMCP_Diff))]
pydf_MC['Center\nMedian'] = [VAL_PMCP[x][3] for x in range(len(VAL_PMCP))]
pydf_MC['Center\nMedian\n%\nChange'] = [PMCP_Diff[x][3] for x in range(len(PMCP_Diff))]
pydf_MC = pydf_MC.fillna('')
#pydf_MC.to_csv(path+'CSV')  <-- need to change squares to alphanumeric to use

Create Graphs of the PMCP results





In [44]:



    


# Create Graphs of the PMCP results
PMCP_Graphs = OrderedDict()
for x in pydf_MC.index:
    PMCP_Graphs[x[2:]] = nx.Graph()

Visualizations

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

  • The following is 15 distinct solutions in one plot. The largest circle is 1 candidate selction and the smallest 15 are all candidates. With a transparency of 0.1 the candidate site selections are placed on top of each other making the sites that are selected more often darker. It will be repeated in the following solutions.




In [45]:



    


#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')

# 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)

# Legend (Ordered Dictionary)
LEGEND = OrderedDict()
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)
plt.annotate('<| ~ .25 miles |>', xy=(623200, 163600), 
             fontstyle='italic', fontsize='large', alpha=0.95)



    


















    
Out[45]:








<matplotlib.text.Annotation at 0x11d3aad10>

Pandas PMP Data Frame

  • Interactive dataframe including an index of candidate facility density (p=n) and facility ID in columns (y#).




In [46]:



    


qgrid.show_grid(pydf_M, show_toolbar=True)

Bokeh PMP [p vs. cost] trade off

  • Interactive Cost/Benefit graph with info. tags




In [47]:



    


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)



    


















    








    











    
Out[47]:







<Bokeh Notebook handle for In[47]>

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





In [48]:



    


#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')

# 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)

    
# Legend (Ordered Dictionary)
LEGEND = OrderedDict()
for i in P_Cent_Graphs:
    LEGEND['Optimal PCP '+str(i)]=P_Cent_Graphs[i]
plt.legend(LEGEND, 
       loc='upper left', 
       fancybox=True, 
       framealpha=0.1, 
       scatterpoints=1)

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

# 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)
plt.annotate('<| ~ .25 miles |>', xy=(623200, 163600), 
             fontstyle='italic', fontsize='large', alpha=0.95)



    


















    
Out[48]:








<matplotlib.text.Annotation at 0x120e8fd90>

Pandas PCP Data Frame





In [49]:



    


qgrid.show_grid(pydf_C, show_toolbar=True)

Bokeh PCP [p vs. cost] trade off





In [67]:



    


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="hline", 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)



    


















    








    











    
Out[67]:







<Bokeh Notebook handle for In[67]>

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





In [51]:



    


# 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')

# 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)

# Legend (Ordered Dictionary)
LEGEND = OrderedDict()
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')

# 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)
plt.annotate('<| ~ .25 miles |>', xy=(623200, 163600), 
             fontstyle='italic', fontsize='large', alpha=0.95)



    


















    
Out[51]:








<matplotlib.text.Annotation at 0x104b81490>

Pandas CentDian Data Frame





In [52]:



    


qgrid.show_grid(pydf_CentDian, show_toolbar=True)

Bokeh CentDian [p vs. cost] trade off





In [68]:



    


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="hline" ,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)



    


















    








    











    
Out[68]:







<Bokeh Notebook handle for In[68]>

Draw PMCP figure





In [54]:



    


size = 3000
for i,j in PMCP_Graphs.iteritems():
    size = size-1200
    if int(i) <= len(SER)-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,
                    alpha=.5, 
                    node_color='k')
    else:
        pass
    
# 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)
    
# Legend (Ordered Dictionary)
LEGEND = OrderedDict()
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')

# 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)
plt.annotate('<| ~ .25 miles |>', xy=(623200, 163600), 
             fontstyle='italic', fontsize='large', alpha=0.95)



    


















    
Out[54]:








<matplotlib.text.Annotation at 0x10ea6c950>

Pandas PMCP Data Frame





In [55]:



    


qgrid.show_grid(pydf_MC, show_toolbar=True)

Quad Comparision

  • Visual comparison of the solutions to all problems




In [56]:



    


mpl.rcParams['figure.figsize']=8,11
# Quad Graphs
# p-Median
P_Med_Graphs_Quad = OrderedDict()
for x in range(1, len(SER)+1):
    P_Med_Graphs_Quad["{0}".format(x)] = nx.Graph()
    
# p-Center
P_Cent_Graphs_Quad = OrderedDict()
for x in range(1, len(SER)+1):
    P_Cent_Graphs_Quad["{0}".format(x)] = nx.Graph()
    
# p-CentDian
P_CentDian_Graphs_Quad = OrderedDict()
for x in range(1, len(SER)+1):
    P_CentDian_Graphs_Quad["{0}".format(x)] = nx.Graph()

# PM+CP
PMCP_Graphs_Quad = OrderedDict()
for x in pydf_MC.index:
    PMCP_Graphs_Quad[x[2:]] = nx.Graph()

#############################################################################    

plt.figure(1)

plt.subplot(221)
# Draw Network Actual Roads and Nodes
for e in ntw.edges:
    g.add_edge(*e)
nx.draw(g, ntw.node_coords, node_size=2, alpha=0.1, edge_color='r', width=2)
# PMP
size = 700
for i,j in P_Med_Graphs_Quad.iteritems():
    size=size-50
    # p-Median
    P_Med_Quad = ps.open(path+'Results/Selected_Locations_Pmedian'+str(i)+'.shp')
    points_median_quad = {}
    for idx, coords in enumerate(P_Med_Quad):
        P_Med_Graphs_Quad[i].add_node(idx)
        points_median_quad[idx] = coords
        P_Med_Graphs_Quad[i].node[idx] = coords
    nx.draw(P_Med_Graphs_Quad[i], 
                points_median_quad, 
                node_size=size, 
                alpha=.1, 
                node_color='k')
# Title
plt.title('PMP', family='Times New Roman', 
      size=30, color='k', backgroundcolor='w', weight='bold')
# 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)
plt.annotate('<|~ .5 miles |>', xy=(623200, 163600), 
             fontstyle='italic', fontsize='large', alpha=0.95)

################################################################################

plt.subplot(222)
# Draw Network Actual Roads and Nodes
for e in ntw.edges:
    g.add_edge(*e)
nx.draw(g, ntw.node_coords, node_size=2, alpha=0.1, edge_color='r', width=2)
# PCP
size = 700
for i,j in P_Cent_Graphs_Quad.iteritems():
    size=size-50
    # p-Center
    P_Cent_Quad = ps.open(path+'Results/Selected_Locations_Pcenter'+str(i)+'.shp')
    points_center_quad = {}
    for idx, coords in enumerate(P_Cent_Quad):
        P_Cent_Graphs_Quad[i].add_node(idx)
        points_center_quad[idx] = coords
        P_Cent_Graphs_Quad[i].node[idx] = coords
    nx.draw(P_Cent_Graphs_Quad[i], 
                points_center_quad, 
                node_size=size, 
                alpha=.1, 
                node_color='k')
# Title
plt.title('PCP', family='Times New Roman', 
      size=30, color='k', backgroundcolor='w', weight='bold')
# 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)
plt.annotate('<|~ .5 miles |>', xy=(623200, 163600), 
             fontstyle='italic', fontsize='large', alpha=0.95)

###############################################################################

plt.subplot(223)
# Draw Network Actual Roads and Nodes
for e in ntw.edges:
    g.add_edge(*e)
nx.draw(g, ntw.node_coords, node_size=2, alpha=0.1, edge_color='r', width=2)
# CentDian
size = 700
for i,j in P_CentDian_Graphs_Quad.iteritems():
    size=size-50
    P_CentDian_Quad = ps.open(path+'Results/Selected_Locations_CentDian'+str(i)+'.shp')
    points_centdian_quad = {}
    for idx, coords in enumerate(P_CentDian_Quad):
        P_CentDian_Graphs_Quad[i].add_node(idx)
        points_centdian_quad[idx] = coords
        P_CentDian_Graphs_Quad[i].node[idx] = coords
    nx.draw(P_CentDian_Graphs_Quad[i], 
                points_centdian_quad, 
                node_size=size, 
                alpha=.1, 
                node_color='k')
# Title
plt.title('CentDian', family='Times New Roman', 
      size=30, 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)
plt.annotate('<|~ .5 miles |>', xy=(623200, 163600), 
             fontstyle='italic', fontsize='large', alpha=0.95)

###################################################################################

plt.subplot(224)
# Draw Network Actual Roads and Nodes
# PM+CP 
size = 700
for i,j in PMCP_Graphs_Quad.iteritems():
    size = size - 300
    if int(i) <= len(SER)-1:
        counter = counter+1
        pmcp_Quad = ps.open(path+'Results/Selected_Locations_PMCP'+str(i)+'.shp')
        points_pmcp_Quad = {}
        for idx, coords in enumerate(pmcp_Quad):
            PMCP_Graphs_Quad[i].add_node(idx)
            points_pmcp_Quad[idx] = coords
            PMCP_Graphs_Quad[i].node[idx] = coords
        nx.draw(PMCP_Graphs_Quad[i], 
                    points_pmcp_Quad, 
                    node_size=size,
                    alpha=.75, 
                    node_color='k')
    else:
        pass
for e in ntw.edges:
    g.add_edge(*e)
nx.draw(g, ntw.node_coords, node_size=2, alpha=0.1, edge_color='r', width=2)
# Legend (Ordered Dictionary)
LEGEND = OrderedDict()
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=.25,
        scatterpoints=1)    
# Title
plt.title('PM+CP', family='Times New Roman', 
      size=30, color='k', backgroundcolor='w', weight='bold')
# 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=1)
plt.annotate('<|~ .5 miles |>', xy=(623200, 163600), 
             fontstyle='italic', fontsize='large', alpha=0.95)

plt.show()

Bokeh PMP & PCP [p vs. cost] comparision

  • Graphical comparison of the solutions to all problems




In [57]:



    


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)



    


















    








    











    
Out[57]:







<Bokeh Notebook handle for In[57]>

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





In [58]:



    


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])

Google Maps Overlay

Create Lists of Selected Locations for Google Maps Plot





In [59]:



    


# 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 [60]:



    


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))

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

plot.add_glyph(source_1, facilties)

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



    


















    








    











    
Out[60]:







<Bokeh Notebook handle for In[60]>

Future Work & Vision

$\Longrightarrow$ 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 PuLP & COIN 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) $\Longrightarrow$ query_ball_point() for approx. neighbors

$\Longrightarrow$ PuLP from COIN-OR

$\ast$ PuLP example with CBC, Gurobi, and CPLEX

Maximize

         $ 4x + 2y $

Subject To

         $ 2x - 3y \geq 25$

         $ 5x + 2y \leq 100$





In [73]:



    


import pulp

x = pulp.LpVariable("x", 0, cat='INTEGER')
y = pulp.LpVariable("y", 0, cat='BINARY')

prob = pulp.LpProblem("myProblem", pulp.LpMaximize)
prob += 2*x - 3*y >= 25
prob += 5*x + 2*y <= 100
prob += 4*x + 2*y

# CBC
status = prob.solve()
cbc = round(pulp.value(prob.objective), 5)

# GUROBI
status = prob.solve(solver=pulp.GUROBI(msg = 0))
guro = round(pulp.value(prob.objective), 5)

# CPLEX
status = prob.solve(solver=pulp.CPLEX(msg = 0))
cplex = round(pulp.value(prob.objective), 5)

print 'obj. --> ', cbc 
print 'x -----> ', pulp.value(x)
print 'y -----> ', pulp.value(y)


if pulp.LpStatus[status] == 'Optimal':
    print '\nSolved to optimality'
print '\nCOIN-OR CBC, Gurobi, and CPLEX solutions equal to 5 significant digits?'
print cbc == guro == cplex



    


















    






obj. -->  81.57895
x ----->  18.4210526316
y ----->  3.94736842105

Solved to optimality

COIN-OR CBC, Gurobi, and CPLEX solutions equal to 5 significant digits?
True

System Specs





In [62]:



    


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

names = ['OSX', 'Processor ', 'Machine ', 'Python ','PySAL ','Gurobi ','Pandas ','GeoPandas ',
         'Shapely ', 'NumPy ', 'Bokeh ', 'PuLP', '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__, pulp.VERSION, dt.datetime.now()]
specs = pd.DataFrame(index=names, columns=['Version'])
specs.columns.name = 'Platform & Software Specs'
specs['Version'] = versions
specs # Pandas DF of specifications



    


















    
Out[62]:










  
    

      Platform & Software Specs
      Version
    

  
  
    

      OSX
      10.11.3
    

    

      Processor
      i386
    

    

      Machine
      x86_64
    

    

      Python
      2.7.10
    

    

      PySAL
      1.11.0
    

    

      Gurobi
      (6, 5, 0)
    

    

      Pandas
      0.17.1
    

    

      GeoPandas
      0.1.1
    

    

      Shapely
      1.5.13
    

    

      NumPy
      1.9.2
    

    

      Bokeh
      0.11.1
    

    

      PuLP
      1.6.1
    

    

      Date & Time
      2016-03-14 02:27:46.750664

Thanks for your time and please share!

email $\Longrightarrow$ jgaboardi@fsu.edu

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





In [63]:



    


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



    


















    
Out[63]:









  
    

    
    
    
    
    

    
    

    
    
    
    
    
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