# Facility Location (FLP)¶

## Principles learned¶

• Create a generic model that uses data

• Define the actual set of decision variables

• Use non-linearities: ternary operator, minimizing a sum of “min”

## Problem¶

The facility location problem is defined as follows: given a set of N locations and a transportation cost W between each pair of locations, select a subset S of p locations, which become facilities, in order to minimize transportation costs. The transportation cost for a location is defined as the distance to the closest facility, that is, the closest location in S. Thus, it aims to provide an optimal placement of facilities to minimize transportation costs. This problem is also known as the p-median problem.

## Data¶

The data files pmed1.in, pmed2.in, …, pmed40.in provided come from the OR-LIB. They are the 40 test problems from Table 2 of J.E.Beasley “A note on solving large p-median problems” European Journal of Operational Research 21 (1985) 270-273.

Each instance file contains the number of locations n, the number of edges in the original instance, the number of facilities to select p and the distance matrix computed from the original file using Floyd’s algorithm.

## Program¶

In this problem, once you know the locations that are in the subset S, you can deduce which facility in S is the closest to each location.

The general idea behind the model is than once one know the facilities that are in the subset S, one can deduce the distance between each location and the closest facility in S. There is no need to actually know which facility is the closest in the model, just the closest distance. It can still be obtained through post-processing if one need to know this information to print out the results.

Thus, the only decision variables are the booleans `x[i]`, equal to 1 if the ith location is in the subset S. The expressions `costs[i][j]` are created to store the transportation cost between location i and j. This cost is:

• edge weight between i and j if j is in the subset S, i.e is a facility

• infinity otherwise (represented by 2 times the maximum edge weight)

The transportation cost between each location i and the closest facility in S is represented by `cost[i]`: it is the minimum value among `costs[i][j]` for all j. The objective is then to minimize the sum of these transportation costs.

Execution:
localsolver facility_location.lsp inFileName=instances/pmed1.in [lsTimeLimit=] [solFileName=]
```use io;

function input() {
local usage = "Usage: localsolver facility_location.lsp "
+ "inFileName=inputFile [solFileName=outputFile] [lsTimeLimit=timeLimit]";

if (inFileName == nil) throw usage;

inFile.readInt(); // Skip number of edges

w[i in 0...N][j in 0...N] = inFile.readInt();
wmax = max[i in 0...N][j in 0...N] (w[i][j]);
}

/* Declare the optimization model */
function model() {
// One variable for each location: 1 if facility, 0 otherwise
x[i in 0...N] <- bool();

// No more than p locations are selected to be facilities
constraint sum[i in 0...N] (x[i]) <= p;

// Costs between location i and j is w[i][j] if j is a facility or 2 * wmax if not
costs[i in 0...N][j in 0...N] <- x[j] ? w[i][j] : 2 * wmax;

// Cost between location i and the closest facility
cost[i in 0...N] <- min[j in 0...N] (costs[i][j]);

// Minimize the total cost
totalCost <- sum[i in 0...N] (cost[i]);
minimize totalCost;
}

/* Parametrize the solver */
function param() {
if (lsTimeLimit == nil) lsTimeLimit = 20;
}

/* Write the solution in a file with the following format:
* - value of the objective
* - indices of the facilities (between 0 and N-1) */
function output() {
if (solFileName == nil) return;
local solFile = io.openWrite(solFileName);
solFile.println(totalCost.value);
for [i in 0...N : x[i].value == 1]
solFile.print(i, " ");
solFile.println("");
}
```
Execution (Windows)
set PYTHONPATH=%LS_HOME%\bin\python
python facility_location.py instances\pmed1.in
Execution (Linux)
export PYTHONPATH=/opt/localsolver_12_0/bin/python
python facility_location.py instances/pmed1.in
```import localsolver
import sys

with open(filename) as f:
return [int(elem) for elem in f.read().split()]

#
#
# Number of locations
N = next(file_it)
next(file_it) # Skip number of edges
# Size of the subset S of facilities
p = next(file_it)

# w: Weight matrix of the shortest path between locations
# wmax: Maximum distance between two locations
wmax = 0
w = [None] * N
for i in range(N):
w[i] = [None] * N
for j in range(N):
w[i][j] = next(file_it)
if w[i][j] > wmax:
wmax = w[i][j]

return N, p, wmax, w

def main(instance_file, output_file, time_limit):
N, p, wmax, w = read_instance(instance_file)

with localsolver.LocalSolver() as ls:
#
# Declare the optimization model
#
m = ls.model

# One variable for each location: 1 if facility, 0 otherwise
x = [m.bool() for i in range(N)]

# No more than p locations are selected to be facilities
opened_locations = m.sum(x[i] for i in range(N))
m.constraint(opened_locations <= p)

# Costs between location i and j is w[i][j] if j is a facility or 2 * wmax if not
costs = [None] * N
for i in range(N):
costs[i] = [None] * N
for j in range(N):
costs[i][j] = m.iif(x[j], w[i][j], 2 * wmax)

# Cost between location i and the closest facility
cost = [None] * N
for i in range(N):
cost[i] = m.min(costs[i][j] for j in range(N))

# Minimize the total cost
total_cost = m.sum(cost[i] for i in range(N))
m.minimize(total_cost)

m.close()

# Parameterize the solver
ls.param.time_limit = time_limit

ls.solve()

#
# Write the solution in a file with the following format:
# - value of the objective
# - indices of the facilities (between 0 and N-1)
#
if output_file is not None:
with open(output_file, 'w') as f:
f.write("%d\n" % total_cost.value)
for i in range(N):
if x[i].value == 1:
f.write("%d " % i)
f.write("\n")

if __name__ == '__main__':
if len(sys.argv) < 2:
print("Usage: python facility_location.py instance_file [output_file] [time_limit]")
sys.exit(1)

instance_file = sys.argv
output_file = sys.argv if len(sys.argv) >= 3 else None
time_limit = int(sys.argv) if len(sys.argv) >= 4 else 20
main(instance_file, output_file, time_limit)
```
Compilation / Execution (Windows)
cl /EHsc facility_location.cpp -I%LS_HOME%\include /link %LS_HOME%\bin\localsolver120.lib
facility_location instances\pmed1.in
Compilation / Execution (Linux)
g++ facility_location.cpp -I/opt/localsolver_12_0/include -llocalsolver120 -lpthread -o facility_location
./facility_location instances/pmed1.in
```#include "localsolver.h"
#include <fstream>
#include <iostream>
#include <sstream>
#include <vector>

using namespace localsolver;
using namespace std;

class Facilitylocation {
public:
// Number of locations
int N;

// Size of the subset S of facilities
int p;

// Weight matrix of the shortest path between locations
vector<vector<int>> w;

// Maximum distance between two locations
int wmax;

// LocalSolver
LocalSolver localsolver;

// Decisions variables
vector<LSExpression> x;

// Objective
LSExpression totalCost;

ifstream infile;
infile.open(fileName.c_str());

int tmp;
infile >> N;
infile >> tmp; // Skip number of edges
infile >> p;

w.resize(N);
wmax = 0;
for (int i = 0; i < N; ++i) {
w[i].resize(N);
for (int j = 0; j < N; ++j) {
infile >> w[i][j];
if (w[i][j] > wmax) {
wmax = w[i][j];
}
}
}
}

// Declare the optimization model
void solve(int limit) {
LSModel m = localsolver.getModel();

// One variable for each location: 1 if facility, 0 otherwise
x.resize(N);
for (int i = 0; i < N; ++i)
x[i] = m.boolVar();

// No more than p locations are selected to be facilities
LSExpression openedLocations = m.sum(x.begin(), x.end());
m.constraint(openedLocations <= p);

// Costs between location i and j is w[i][j] if j is a facility or 2 * wmax if not
vector<vector<LSExpression>> costs(N);
for (int i = 0; i < N; ++i) {
costs[i].resize(N);
for (int j = 0; j < N; ++j) {
costs[i][j] = m.iif(x[j], w[i][j], 2 * wmax);
}
}

// Cost between location i and the closest facility
vector<LSExpression> cost(N);
for (int i = 0; i < N; ++i) {
cost[i] = m.min(costs[i].begin(), costs[i].end());
}

// Minimize the total cost
totalCost = m.sum(cost.begin(), cost.end());
m.minimize(totalCost);
m.close();

// Parameterize the solver
localsolver.getParam().setTimeLimit(limit);

localsolver.solve();
}

/* Write the solution in a file with the following format:
* - value of the objective
* - indices of the facilities (between 0 and N-1) */
void writeSolution(const string& fileName) {
ofstream outfile;
outfile.open(fileName.c_str());

outfile << totalCost.getValue() << endl;
for (int i = 0; i < N; ++i) {
if (x[i].getValue() == 1)
outfile << i << " ";
}
outfile << endl;
}
};

int main(int argc, char** argv) {
if (argc < 2) {
cerr << "Usage: facility_location inputFile [outputFile] [timeLimit] " << endl;
return 1;
}

const char* instanceFile = argv;
const char* solFile = argc > 2 ? argv : NULL;
const char* strTimeLimit = argc > 3 ? argv : "20";

try {
Facilitylocation model;
model.solve(atoi(strTimeLimit));
if (solFile != NULL)
model.writeSolution(solFile);
return 0;
} catch (const exception& e) {
cerr << "An error occurred: " << e.what() << endl;
return 1;
}
}
```
Compilation / Execution (Windows)
copy %LS_HOME%\bin\localsolvernet.dll .
csc Facilitylocation.cs /reference:localsolvernet.dll
Facilitylocation instances\pmed1.in
```using System;
using System.IO;
using System.Collections.Generic;
using localsolver;

public class FacilityLocation : IDisposable
{
// Number of locations
int N;

// Size of the subset S of facilities
int p;

// Weight matrix of the shortest path beween locations
int[][] w;

// Maximum distance between two locations
int wmax;

// LocalSolver
LocalSolver localsolver;

// Decision variables
LSExpression[] x;

// Objective
LSExpression totalCost;

// List of facilities
List<int> solution;

public FacilityLocation()
{
localsolver = new LocalSolver();
}

{
{
N = int.Parse(tokens);
p = int.Parse(tokens);

w = new int[N][];
wmax = 0;
for (int i = 0; i < N; ++i)
{
w[i] = new int[N];
for (int j = 0; j < N; ++j)
{
w[i][j] = int.Parse(tokens[j]);
if (w[i][j] > wmax)
wmax = w[i][j];
}
}
}
}

public void Dispose()
{
if (localsolver != null)
localsolver.Dispose();
}

// Declare the optimization model
public void Solve(int limit)
{
localsolver = new LocalSolver();
LSModel model = localsolver.GetModel();
x = new LSExpression[N];

// One variable for each location: 1 if facility, 0 otherwise
for (int i = 0; i < N; ++i)
x[i] = model.Bool();

// No more than p locations are selected to be facilities
LSExpression openedLocations = model.Sum(x);
model.Constraint(openedLocations <= p);

// Costs between location i and j is w[i][j] if j is a facility or 2 * wmax if not
LSExpression[][] costs = new LSExpression[N][];
for (int i = 0; i < N; ++i)
{
costs[i] = new LSExpression[N];
for (int j = 0; j < N; ++j)
costs[i][j] = model.If(x[j], w[i][j], 2 * wmax);
}

// Cost between location i and the closest facility
LSExpression[] cost = new LSExpression[N];
for (int i = 0; i < N; ++i)
cost[i] = model.Min(costs[i]);

// Minimize the total cost
totalCost = model.Sum(cost);
model.Minimize(totalCost);

model.Close();

// Parameterize the solver
localsolver.GetParam().SetTimeLimit(limit);
localsolver.Solve();

solution = new List<int>();
for (int i = 0; i < N; ++i)
if (x[i].GetValue() == 1)
}

/* Write the solution in a file with the following format:
* - value of the objective
* - indices of the facilities (between 0 and N-1) */
public void WriteSolution(string fileName)
{
using (StreamWriter output = new StreamWriter(fileName))
{
output.WriteLine(totalCost.GetValue());
for (int i = 0; i < N; ++i) {
if (x[i].GetValue() == 1)
output.Write(i + " ");
}
output.WriteLine();
}
}

public static void Main(string[] args)
{
if (args.Length < 1)
{
Console.WriteLine("Usage: Facilitylocation inputFile [outputFile] [timeLimit]");
Environment.Exit(1);
}
string instanceFile = args;
string outputFile = args.Length > 1 ? args : null;
string strTimeLimit = args.Length > 2 ? args : "20";

using (FacilityLocation model = new FacilityLocation())
{
model.Solve(int.Parse(strTimeLimit));
if (outputFile != null)
model.WriteSolution(outputFile);
}
}
}
```
Compilation / Execution (Windows)
javac Facilitylocation.java -cp %LS_HOME%\bin\localsolver.jar
java -cp %LS_HOME%\bin\localsolver.jar;. Facilitylocation instances\pmed1.in
Compilation / Execution (Linux)
javac Facilitylocation.java -cp /opt/localsolver_12_0/bin/localsolver.jar
java -cp /opt/localsolver_12_0/bin/localsolver.jar:. Facilitylocation instances/pmed1.in
```import java.util.*;
import java.io.*;
import localsolver.*;

public class Facilitylocation {
// Number of locations
private int N;

// Size of the subset S of facilities
private int p;

// Weight matrix of the shortest path between locations
private int[][] w;

// Maximum distance between two locations
private int wmax;

// LocalSolver.
private final LocalSolver localsolver;

// Decisions variables
private LSExpression[] x;

// Objective
private LSExpression totalCost;

private Facilitylocation(LocalSolver localsolver) {
this.localsolver = localsolver;
}

private void readInstance(String fileName) throws IOException {
try (Scanner input = new Scanner(new File(fileName))) {
N = input.nextInt();
input.nextInt(); // Skip number of edges
p = input.nextInt();

w = new int[N][N];
wmax = 0;
for (int i = 0; i < N; ++i) {
for (int j = 0; j < N; ++j) {
w[i][j] = input.nextInt();
if (w[i][j] > wmax)
wmax = w[i][j];
}
}
}
}

private void solve(int limit) {
// Declare the optimization model
LSModel model = localsolver.getModel();

// One variable for each location: 1 if facility, 0 otherwise
x = new LSExpression[N];
for (int i = 0; i < N; ++i) {
x[i] = model.boolVar();
}

// No more than p locations are selected to be facilities
LSExpression openedLocations = model.sum(x);
model.constraint(model.leq(openedLocations, p));

// Costs between location i and j is w[i][j] if j is a facility or 2 * wmax if not
LSExpression[][] costs = new LSExpression[N][N];
for (int i = 0; i < N; ++i) {
for (int j = 0; j < N; ++j) {
costs[i][j] = model.iif(x[j], w[i][j], 2 * wmax);
}
}

// Cost between location i and the closest facility
LSExpression[] cost = new LSExpression[N];
for (int i = 0; i < N; ++i) {
cost[i] = model.min(costs[i]);
}

// Minimize the total cost
totalCost = model.sum(cost);
model.minimize(totalCost);

model.close();

// Parameterize the solver
localsolver.getParam().setTimeLimit(limit);
localsolver.solve();
}

/* Write the solution in a file with the following format:
* - value of the objective
* - indices of the facilities (between 0 and N-1) */
private void writeSolution(String fileName) throws IOException {
try (PrintWriter output = new PrintWriter(fileName)) {
output.println(totalCost.getValue());
for (int i = 0; i < N; ++i) {
if (x[i].getValue() == 1)
output.print(i + " ");
}
}
}

public static void main(String[] args) {
if (args.length < 1) {
System.err.println("Usage: java Facilitylocation inputFile [outputFile] [timeLimit]");
System.exit(1);
}

String instanceFile = args;
String outputFile = args.length > 1 ? args : null;
String strTimeLimit = args.length > 2 ? args : "20";

try (LocalSolver localsolver = new LocalSolver()) {
Facilitylocation model = new Facilitylocation(localsolver);