# Vehicle Routing with Time Windows (CVRPTW)Â¶

## Principles learnedÂ¶

• Add multiple list decision variables

• Use lambda expressions to define a recursive array

• Define a sequence of expressions

## ProblemÂ¶

In the capacitated vehicle routing problem with time-windows (CVRPTW), a fleet of delivery vehicles with uniform capacity must service customers with known demand and opening hours for a single commodity. The vehicles start and end their routes at a common depot. Each customer can only be served by one vehicle. The objectives are to minimize the fleet size and assign a sequence of customers to each truck of the fleet minimizing the total distance traveled such that all customers are served and the total demand served by each truck does not exceed its capacity.

## DataÂ¶

The instances provided come from the Solomon instances.

The format of the data files is as follows:

• The first line gives the name of the instance

• The fifth line contains the number of vehicles and their common capacity

• From the 10th line, each line contains the integer data associated to each customer, starting with the depot:

• The index of the customer

• The x coordinate

• The y coordinate

• The demand

• The earliest arrival

• The latest arrival

• The service time

## ProgramÂ¶

The LocalSolver model is an extension of the CVRP model. We refer the reader to this model for the routing aspect of the problem. The time-windows are handled as a prioritary objective. In case of early arrival, the truck has to wait for the opening hour of the customer. In case of late arrival, the lateness is measured and the total lateness has to be minimized. The solution is feasible when this cumulated lateness is zero.

In the model the ending times of each truck are defined as a recursive array. The arrival time is the maximum of:

• the end time of the previous visit plus the traveling time (equal to the distance in these instances). For the first visit it amounts to the traveling time from the depot (starting at t=0)

• the earliest allowed arrival at this customer

The ending time is merely this arrival time plus the service time for this customer. The arrival to the depot at the end of the tour occurs at the end time of the last visit plus the traveling time from this point to the depot.

Such a recursive definition is introduced with a function `(i, prev) => ...`. It allows defining the ith element of an array in function of i and of the (i-1)th element of the array. See our documentation on this topic for details.

The lateness at each visit is computed from the difference between the ending time of this visit and the latest allowed end for this customer. For the arrival at the depot, we compare the arrival time to the maximum horizon defined for the problem.

Finally we minimize in lexicographic order: the total lateness, the number of trucks used, and the total distance traveled by all the trucks.

Execution:
localsolver cvrptw.lsp inFileName=instances/R101.25.txt [lsTimeLimit=] [solFileName=]
```use io;

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

if (inFileName == nil) throw usage;

computeDistanceMatrix();
}

/* Declare the optimization model */
function model() {

// All customers must be visited by exactly one truck

for [k in 0...nbTrucks] {
local c <- count(sequence);

// A truck is used if it visits at least one customer
truckUsed[k] <- c > 0;

// The quantity needed in each route must not exceed the truck capacity
routeQuantity[k] <- sum(sequence, j => demands[j]);
constraint routeQuantity[k] <= truckCapacity;

// End of each visit
endTime[k] <- array(0...c, (i, prev) => max(earliestStart[sequence[i]],
i == 0 ?
distanceDepot[sequence[0]] :
prev + distanceMatrix[sequence[i - 1]][sequence[i]]) + serviceTime[sequence[i]], 0);

// Arriving home after max horizon
homeLateness[k] <- truckUsed[k] ?
max(0, endTime[k][c - 1] + distanceDepot[sequence[c - 1]] - maxHorizon) :
0;

// Distance traveled by truck k
routeDistances[k] <- sum(1...c,
i => distanceMatrix[sequence[i-1]][sequence[i]])
+ (truckUsed[k] ?
(distanceDepot[sequence[0]] + distanceDepot[sequence[c - 1]]) :
0);

// Completing visit after latest end
lateness[k] <- homeLateness[k] + sum(0...c,
i => max(0, endTime[k][i] - latestEnd[sequence[i]]));
}

// Total lateness, must be 0 for a solution to be valid
totalLateness <- sum[k in 0...nbTrucks](lateness[k]);

// Total number of trucks used
nbTrucksUsed <- sum[k in 0...nbTrucks](truckUsed[k]);

// Total distance traveled (convention in Solomon's instances is to round to 2 decimals)
totalDistance <- round(100 * sum[k in 0...nbTrucks](routeDistances[k])) / 100;

// Objective: minimize the lateness, then the number of trucks used, then the distance traveled
minimize totalLateness;
minimize nbTrucksUsed;
minimize totalDistance;
}

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

/* Write the solution in a file with the following format:
* - number of trucks used and total distance
* - for each truck the customers visited (omitting the start/end at the depot) */
function output() {
if (solFileName == nil) return;
local outfile = io.openWrite(solFileName);

outfile.println(nbTrucksUsed.value, " ", totalDistance.value);
for [k in 0...nbTrucks] {
if (truckUsed[k].value != 1) continue;
outfile.print(customer + 1, " ");
}
outfile.println();
}
}

skipLines(inFile, 4);

// Truck related data

skipLines(inFile, 3);

// Depot data
depotIndex = line[0].toInt();
depotX = line[1].toInt();
depotY = line[2].toInt();
maxHorizon = line[5].toInt();

// Customers data
i = 0;
while (!inFile.eof()) {
line = inLine.split();
if (count(line) == 0) break;
if (count(line) != 7) throw "Wrong file format";
customerIndex[i] = line[0].toInt();
customerX[i] = line[1].toInt();
customerY[i] = line[2].toInt();
demands[i] = line[3].toInt();
serviceTime[i] = line[6].toInt();
earliestStart[i] = line[4].toInt();
// in input files due date is meant as latest start time
latestEnd[i] = line[5].toInt() + serviceTime[i];
i = i + 1;
}
nbCustomers = i;

inFile.close();
}

function skipLines(inFile, nbLines) {
for [i in 0...nbLines]
}

// Compute the distance matrix
function computeDistanceMatrix() {
for [i in 0...nbCustomers] {
distanceMatrix[i][i] = 0;
for [j in i+1...nbCustomers] {
local localDistance = computeDist(i, j);
distanceMatrix[j][i] = localDistance;
distanceMatrix[i][j] = localDistance;
}
}

for [i in 0...nbCustomers] {
local localDistance = computeDepotDist(i);
distanceDepot[i] = localDistance;
}
}

function computeDist(i, j) {
local x1 = customerX[i];
local x2 = customerX[j];
local y1 = customerY[i];
local y2 = customerY[j];
return computeDistance(x1, x2, y1, y2);
}

function computeDepotDist(i) {
local x1 = customerX[i];
local xd = depotX;
local y1 = customerY[i];
local yd = depotY;
return computeDistance(x1, xd, y1, yd);
}

function computeDistance(x1, x2, y1, y2) {
return sqrt(pow((x1 - x2), 2) + pow((y1 - y2), 2));
}
```
Execution (Windows)
set PYTHONPATH=%LS_HOME%\bin\python
python cvrptw.py instances\R101.25.txt
Execution (Linux)
export PYTHONPATH=/opt/localsolver_12_0/bin/python
python cvrptw.py instances/R101.25.txt
```import localsolver
import sys
import math

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

def main(instance_file, str_time_limit, output_file):
#
#
nb_customers, nb_trucks, truck_capacity, dist_matrix_data, dist_depot_data, \
demands_data, service_time_data, earliest_start_data, latest_end_data, \

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

# Sequence of customers visited by each truck
customers_sequences = [model.list(nb_customers) for k in range(nb_trucks)]

# All customers must be visited by exactly one truck
model.constraint(model.partition(customers_sequences))

# Create LocalSolver arrays to be able to access them with an "at" operator
demands = model.array(demands_data)
earliest = model.array(earliest_start_data)
latest = model.array(latest_end_data)
service_time = model.array(service_time_data)
dist_matrix = model.array()
for n in range(nb_customers):
dist_depot = model.array(dist_depot_data)

dist_routes = [None] * nb_trucks
end_time = [None] * nb_trucks
home_lateness = [None] * nb_trucks
lateness = [None] * nb_trucks

# A truck is used if it visits at least one customer
trucks_used = [(model.count(customers_sequences[k]) > 0) for k in range(nb_trucks)]
nb_trucks_used = model.sum(trucks_used)

for k in range(nb_trucks):
sequence = customers_sequences[k]
c = model.count(sequence)

# The quantity needed in each route must not exceed the truck capacity
demand_lambda = model.lambda_function(lambda j: demands[j])
route_quantity = model.sum(sequence, demand_lambda)
model.constraint(route_quantity <= truck_capacity)

# Distance traveled by each truck
dist_lambda = model.lambda_function(
lambda i: model.at(dist_matrix, sequence[i - 1], sequence[i]))
dist_routes[k] = model.sum(model.range(1, c), dist_lambda) \
+ model.iif(c > 0, dist_depot[sequence[0]] + dist_depot[sequence[c - 1]], 0)

# End of each visit
end_time_lambda = model.lambda_function(
lambda i, prev:
model.max(
earliest[sequence[i]],
model.iif(
i == 0, dist_depot[sequence[0]],
prev + model.at(dist_matrix, sequence[i - 1], sequence[i])))
+ service_time[sequence[i]])

end_time[k] = model.array(model.range(0, c), end_time_lambda, 0)

# Arriving home after max horizon
home_lateness[k] = model.iif(
trucks_used[k],
model.max(
0,
end_time[k][c - 1] + dist_depot[sequence[c - 1]] - max_horizon),
0)

# Completing visit after latest end
late_lambda = model.lambda_function(
lambda i: model.max(0, end_time[k][i] - latest[sequence[i]]))
lateness[k] = home_lateness[k] + model.sum(model.range(0, c), late_lambda)

# Total lateness
total_lateness = model.sum(lateness)

# Total distance traveled
total_distance = model.div(model.round(100 * model.sum(dist_routes)), 100)

# Objective: minimize the number of trucks used, then minimize the distance traveled
model.minimize(total_lateness)
model.minimize(nb_trucks_used)
model.minimize(total_distance)

model.close()

# Parameterize the solver
ls.param.time_limit = int(str_time_limit)

ls.solve()

#
# Write the solution in a file with the following format:
#  - number of trucks used and total distance
#  - for each truck the customers visited (omitting the start/end at the depot)
#
if output_file is not None:
with open(output_file, 'w') as f:
f.write("%d %d\n" % (nb_trucks_used.value, total_distance.value))
for k in range(nb_trucks):
if trucks_used[k].value != 1:
continue
# Values in sequence are in 0...nbCustomers. +1 is to put it back in
# 1...nbCustomers+1 as in the data files (0 being the depot)
for customer in customers_sequences[k].value:
f.write("%d " % (customer + 1))
f.write("\n")

# The input files follow the "Solomon" format

for i in range(4):
next(file_it)

nb_trucks = int(next(file_it))
truck_capacity = int(next(file_it))

for i in range(13):
next(file_it)

depot_x = int(next(file_it))
depot_y = int(next(file_it))

for i in range(2):
next(file_it)

max_horizon = int(next(file_it))

next(file_it)

customers_x = []
customers_y = []
demands = []
earliest_start = []
latest_end = []
service_time = []

while True:
val = next(file_it, None)
if val is None:
break
i = int(val) - 1
customers_x.append(int(next(file_it)))
customers_y.append(int(next(file_it)))
demands.append(int(next(file_it)))
due = int(next(file_it))
stime = int(next(file_it))
# in input files due date is meant as latest start time
latest_end.append(due + stime)
service_time.append(stime)

nb_customers = i + 1

# Compute distance matrix
distance_matrix = compute_distance_matrix(customers_x, customers_y)
distance_depots = compute_distance_depots(depot_x, depot_y, customers_x, customers_y)

return nb_customers, nb_trucks, truck_capacity, distance_matrix, distance_depots, \
demands, service_time, earliest_start, latest_end, max_horizon

# Computes the distance matrix
def compute_distance_matrix(customers_x, customers_y):
nb_customers = len(customers_x)
distance_matrix = [[None for i in range(nb_customers)] for j in range(nb_customers)]
for i in range(nb_customers):
distance_matrix[i][i] = 0
for j in range(nb_customers):
dist = compute_dist(customers_x[i], customers_x[j],
customers_y[i], customers_y[j])
distance_matrix[i][j] = dist
distance_matrix[j][i] = dist
return distance_matrix

# Computes the distances to depot
def compute_distance_depots(depot_x, depot_y, customers_x, customers_y):
nb_customers = len(customers_x)
distance_depots = [None] * nb_customers
for i in range(nb_customers):
dist = compute_dist(depot_x, customers_x[i], depot_y, customers_y[i])
distance_depots[i] = dist
return distance_depots

def compute_dist(xi, xj, yi, yj):
return math.sqrt(math.pow(xi - xj, 2) + math.pow(yi - yj, 2))

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

instance_file = sys.argv[1]
output_file = sys.argv[2] if len(sys.argv) > 2 else None
str_time_limit = sys.argv[3] if len(sys.argv) > 3 else "20"

main(instance_file, str_time_limit, output_file)
```
Compilation / Execution (Windows)
cl /EHsc cvrptw.cpp -I%LS_HOME%\include /link %LS_HOME%\bin\localsolver120.lib
cvrptw instances\R101.25.txt
Compilation / Execution (Linux)
g++ cvrptw.cpp -I/opt/localsolver_12_0/include -llocalsolver120 -lpthread -o cvrptw
./cvrptw instances/R101.25.txt
```#include "localsolver.h"
#include <cmath>
#include <cstring>
#include <fstream>
#include <iostream>
#include <vector>

using namespace localsolver;
using namespace std;

class Cvrptw {
public:
// LocalSolver
LocalSolver localsolver;

// Number of customers
int nbCustomers;

// Capacity of the trucks
int truckCapacity;

// Latest allowed arrival to depot
int maxHorizon;

// Demand for each customer
vector<int> demandsData;

// Earliest arrival for each customer
vector<int> earliestStartData;

// Latest departure from each customer
vector<int> latestEndData;

// Service time for each customer
vector<int> serviceTimeData;

// Distance matrix between customers
vector<vector<double>> distMatrixData;

// Distance  between customers and depot
vector<double> distDepotData;

// Number of trucks
int nbTrucks;

// Decision variables

// Are the trucks actually used
vector<LSExpression> trucksUsed;

// Cumulated lateness in the solution (must be 0 for the solution to be valid)
LSExpression totalLateness;

// Number of trucks used in the solution
LSExpression nbTrucksUsed;

// Distance traveled by all the trucks
LSExpression totalDistance;

Cvrptw() {}

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

// Sequence of customers visited by each truck
for (int k = 0; k < nbTrucks; ++k) {
}

// All customers must be visited by exactly one truck

// Create LocalSolver arrays to be able to access them with an "at" operator
LSExpression demands = model.array(demandsData.begin(), demandsData.end());
LSExpression earliest = model.array(earliestStartData.begin(), earliestStartData.end());
LSExpression latest = model.array(latestEndData.begin(), latestEndData.end());
LSExpression serviceTime = model.array(serviceTimeData.begin(), serviceTimeData.end());
LSExpression distMatrix = model.array();
for (int n = 0; n < nbCustomers; ++n) {
}
LSExpression distDepot = model.array(distDepotData.begin(), distDepotData.end());

trucksUsed.resize(nbTrucks);
vector<LSExpression> distRoutes(nbTrucks), endTime(nbTrucks), homeLateness(nbTrucks), lateness(nbTrucks);

for (int k = 0; k < nbTrucks; ++k) {
LSExpression c = model.count(sequence);

// A truck is used if it visits at least one customer
trucksUsed[k] = c > 0;

// The quantity needed in each route must not exceed the truck capacity
LSExpression demandLambda =
model.createLambdaFunction([&](LSExpression j) { return demands[j]; });
LSExpression routeQuantity = model.sum(sequence, demandLambda);
model.constraint(routeQuantity <= truckCapacity);

// Distance traveled by truck k
LSExpression distLambda = model.createLambdaFunction(
[&](LSExpression i) { return model.at(distMatrix, sequence[i - 1], sequence[i]); });
distRoutes[k] = model.sum(model.range(1, c), distLambda) +
model.iif(c > 0, distDepot[sequence[0]] + distDepot[sequence[c - 1]], 0);

// End of each visit
LSExpression endTimeLambda = model.createLambdaFunction([&](LSExpression i, LSExpression prev) {
return model.max(earliest[sequence[i]],
model.iif(i == 0, distDepot[sequence[0]],
prev + model.at(distMatrix, sequence[i - 1], sequence[i]))) +
serviceTime[sequence[i]];
});

endTime[k] = model.array(model.range(0, c), endTimeLambda, 0);

// Arriving home after max horizon
homeLateness[k] =
model.iif(trucksUsed[k], model.max(0, endTime[k][c - 1] + distDepot[sequence[c - 1]] - maxHorizon), 0);

// Completing visit after latest end
LSExpression lateLambda = model.createLambdaFunction(
[&](LSExpression i) { return model.max(0, endTime[k][i] - latest[sequence[i]]); });
lateness[k] = homeLateness[k] + model.sum(model.range(0, c), lateLambda);
}

// Total lateness
totalLateness = model.sum(lateness.begin(), lateness.end());

// Total number of trucks used
nbTrucksUsed = model.sum(trucksUsed.begin(), trucksUsed.end());

// Total distance traveled (convention in Solomon's instances is to round to 2 decimals)
totalDistance = model.round(100 * model.sum(distRoutes.begin(), distRoutes.end())) / 100;

// Objective: minimize the lateness, then the number of trucks used, then the distance traveled
model.minimize(totalLateness);
model.minimize(nbTrucksUsed);
model.minimize(totalDistance);

model.close();

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

localsolver.solve();
}

/* Write the solution in a file with the following format:
*  - number of trucks used and total distance
*  - for each truck the customers visited (omitting the start/end at the depot) */
void writeSolution(const string& fileName) {
ofstream outfile;
outfile.open(fileName.c_str());

outfile << nbTrucksUsed.getValue() << " " << totalDistance.getDoubleValue() << endl;
for (int k = 0; k < nbTrucks; ++k) {
if (trucksUsed[k].getValue() != 1)
continue;
// Values in sequence are in 0...nbCustomers. +1 is to put it back in 1...nbCustomers+1
// as in the data files (0 being the depot)
for (int i = 0; i < customersCollection.count(); ++i) {
outfile << customersCollection[i] + 1 << " ";
}
outfile << endl;
}
}

private:
// The input files follow the "Solomon" format
ifstream infile(fileName.c_str());
if (!infile.is_open()) {
throw std::runtime_error("File cannot be opened.");
}

string str;
long tmp;

int depotX, depotY;
vector<int> customersX;
vector<int> customersY;

getline(infile, str);
getline(infile, str);
getline(infile, str);
getline(infile, str);

infile >> nbTrucks;
infile >> truckCapacity;

getline(infile, str);
getline(infile, str);
getline(infile, str);
getline(infile, str);

infile >> tmp;
infile >> depotX;
infile >> depotY;
infile >> tmp;
infile >> tmp;
infile >> maxHorizon;
infile >> tmp;

while (infile >> tmp) {
int cx, cy, demand, ready, due, service;
infile >> cx;
infile >> cy;
infile >> demand;
infile >> due;
infile >> service;

customersX.push_back(cx);
customersY.push_back(cy);
demandsData.push_back(demand);
latestEndData.push_back(due + service); // in input files due date is meant as latest start time
serviceTimeData.push_back(service);
}

nbCustomers = customersX.size();

computeDistanceMatrix(depotX, depotY, customersX, customersY);

infile.close();
}

// Compute the distance matrix
void computeDistanceMatrix(int depotX, int depotY, const vector<int>& customersX, const vector<int>& customersY) {
distMatrixData.resize(nbCustomers);
for (int i = 0; i < nbCustomers; ++i) {
distMatrixData[i].resize(nbCustomers);
}
for (int i = 0; i < nbCustomers; ++i) {
distMatrixData[i][i] = 0;
for (int j = i + 1; j < nbCustomers; ++j) {
double distance = computeDist(customersX[i], customersX[j], customersY[i], customersY[j]);
distMatrixData[i][j] = distance;
distMatrixData[j][i] = distance;
}
}

distDepotData.resize(nbCustomers);
for (int i = 0; i < nbCustomers; ++i) {
distDepotData[i] = computeDist(depotX, customersX[i], depotY, customersY[i]);
}
}

double computeDist(int xi, int xj, int yi, int yj) {
return sqrt(pow((double)xi - xj, 2) + pow((double)yi - yj, 2));
}
};

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

const char* instanceFile = argv[1];
const char* solFile = argc > 2 ? argv[2] : NULL;
const char* strTimeLimit = argc > 3 ? argv[3] : "20";

try {
Cvrptw 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 Cvrptw.cs /reference:localsolvernet.dll
Cvrptw instances\R101.25.txt
```using System;
using System.IO;
using System.Collections.Generic;
using localsolver;

public class Cvrptw : IDisposable
{
// LocalSolver
LocalSolver localsolver;

// Number of customers
int nbCustomers;

// Capacity of the trucks
int truckCapacity;

// Latest allowed arrival to depot
int maxHorizon;

// Demand for each customer
List<int> demandsData;

// Earliest arrival for each customer
List<int> earliestStartData;

// Latest departure from each customer
List<int> latestEndData;

// Service time for each customer
List<int> serviceTimeData;

// Distance matrix between customers
double[][] distMatrixData;

// Distances between customers and depot
double[] distDepotData;

// Number of trucks
int nbTrucks;

// Decision variables

// Are the trucks actually used
LSExpression[] trucksUsed;

// Cumulated lateness in the solution (must be 0 for the solution to be valid)
LSExpression totalLateness;

// Number of trucks used in the solution
LSExpression nbTrucksUsed;

// Distance traveled by all the trucks
LSExpression totalDistance;

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

{
}

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

void Solve(int limit)
{
// Declare the optimization model
LSModel model = localsolver.GetModel();

trucksUsed = new LSExpression[nbTrucks];
LSExpression[] distRoutes = new LSExpression[nbTrucks];
LSExpression[] endTime = new LSExpression[nbTrucks];
LSExpression[] homeLateness = new LSExpression[nbTrucks];
LSExpression[] lateness = new LSExpression[nbTrucks];

// Sequence of customers visited by each truck
for (int k = 0; k < nbTrucks; ++k)

// All customers must be visited by exactly one truck

// Create LocalSolver arrays to be able to access them with an "at" operator
LSExpression demands = model.Array(demandsData);
LSExpression earliest = model.Array(earliestStartData);
LSExpression latest = model.Array(latestEndData);
LSExpression serviceTime = model.Array(serviceTimeData);
LSExpression distDepot = model.Array(distDepotData);
LSExpression distMatrix = model.Array(distMatrixData);

for (int k = 0; k < nbTrucks; ++k)
{
LSExpression c = model.Count(sequence);

// A truck is used if it visits at least one customer
trucksUsed[k] = c > 0;

// The quantity needed in each route must not exceed the truck capacity
LSExpression demandLambda = model.LambdaFunction(j => demands[j]);
LSExpression routeQuantity = model.Sum(sequence, demandLambda);
model.Constraint(routeQuantity <= truckCapacity);

// Distance traveled by truck k
LSExpression distLambda = model.LambdaFunction(
i => distMatrix[sequence[i - 1], sequence[i]]
);
distRoutes[k] =
model.Sum(model.Range(1, c), distLambda)
+ model.If(c > 0, distDepot[sequence[0]] + distDepot[sequence[c - 1]], 0);

// End of each visit
LSExpression endTimeLambda = model.LambdaFunction(
(i, prev) =>
model.Max(
earliest[sequence[i]],
model.If(
i == 0,
distDepot[sequence[0]],
prev + distMatrix[sequence[i - 1], sequence[i]]
)
) + serviceTime[sequence[i]]
);

endTime[k] = model.Array(model.Range(0, c), endTimeLambda, 0);

// Arriving home after max_horizon
homeLateness[k] = model.If(
trucksUsed[k],
model.Max(0, endTime[k][c - 1] + distDepot[sequence[c - 1]] - maxHorizon),
0
);

// Completing visit after latest_end
LSExpression lateLambda = model.LambdaFunction(
i => model.Max(endTime[k][i] - latest[sequence[i]], 0)
);
lateness[k] = homeLateness[k] + model.Sum(model.Range(0, c), lateLambda);
}

// Total lateness
totalLateness = model.Sum(lateness);

// Total number of trucks used
nbTrucksUsed = model.Sum(trucksUsed);

// Total distance traveled (convention in Solomon's instances is to round to 2 decimals)
totalDistance = model.Round(100 * model.Sum(distRoutes)) / 100;

// Objective: minimize the lateness, then the number of trucks used, then the distance traveled
model.Minimize(totalLateness);
model.Minimize(nbTrucksUsed);
model.Minimize(totalDistance);

model.Close();

// Parametrize the solver
localsolver.GetParam().SetTimeLimit(limit);

localsolver.Solve();
}

/* Write the solution in a file with the following format:
*  - number of trucks used and total distance
*  - for each truck the customers visited (omitting the start/end at the depot) */
void WriteSolution(string fileName)
{
using (StreamWriter output = new StreamWriter(fileName))
{
output.WriteLine(nbTrucksUsed.GetValue() + " " + totalDistance.GetDoubleValue());
for (int k = 0; k < nbTrucks; ++k)
{
if (trucksUsed[k].GetValue() != 1)
continue;
// Values in sequence are in 0...nbCustomers. +1 is to put it back in 1...nbCustomers+1
// as in the data files (0 being the depot)
for (int i = 0; i < customersCollection.Count(); ++i)
output.Write((customersCollection[i] + 1) + " ");
output.WriteLine();
}
}
}

public static void Main(string[] args)
{
if (args.Length < 1)
{
Console.WriteLine("Usage: Cvrptw inputFile [solFile] [timeLimit]");
Environment.Exit(1);
}
string instanceFile = args[0];
string outputFile = args.Length > 1 ? args[1] : null;
string strTimeLimit = args.Length > 2 ? args[2] : "20";

using (Cvrptw model = new Cvrptw())
{
model.Solve(int.Parse(strTimeLimit));
if (outputFile != null)
model.WriteSolution(outputFile);
}
}

{
if (line == null)
return new string[0];
return line.Split(new[] { ' ' }, StringSplitOptions.RemoveEmptyEntries);
}

// The input files follow the "Solomon" format
{
{
string[] splitted;

splitted = SplitInput(input);
nbTrucks = int.Parse(splitted[0]);
truckCapacity = int.Parse(splitted[1]);

splitted = SplitInput(input);
int depotX = int.Parse(splitted[1]);
int depotY = int.Parse(splitted[2]);
maxHorizon = int.Parse(splitted[5]);

List<int> customersX = new List<int>();
List<int> customersY = new List<int>();
demandsData = new List<int>();
earliestStartData = new List<int>();
latestEndData = new List<int>();
serviceTimeData = new List<int>();

while (!input.EndOfStream)
{
splitted = SplitInput(input);
if (splitted.Length < 7)
break;
int due = int.Parse(splitted[5]);
int service = int.Parse(splitted[6]);

latestEndData.Add(due + service); // in input files due date is meant as latest start time
}

nbCustomers = customersX.Count;

ComputeDistanceMatrix(depotX, depotY, customersX, customersY);
}
}

// Compute the distance matrix
private void ComputeDistanceMatrix(
int depotX,
int depotY,
List<int> customersX,
List<int> customersY
)
{
distMatrixData = new double[nbCustomers][];
for (int i = 0; i < nbCustomers; ++i)
distMatrixData[i] = new double[nbCustomers];

for (int i = 0; i < nbCustomers; ++i)
{
distMatrixData[i][i] = 0;
for (int j = i + 1; j < nbCustomers; ++j)
{
double dist = ComputeDist(
customersX[i],
customersX[j],
customersY[i],
customersY[j]
);
distMatrixData[i][j] = dist;
distMatrixData[j][i] = dist;
}
}

distDepotData = new double[nbCustomers];
for (int i = 0; i < nbCustomers; ++i)
distDepotData[i] = ComputeDist(depotX, customersX[i], depotY, customersY[i]);
}

private double ComputeDist(int xi, int xj, int yi, int yj)
{
return Math.Sqrt(Math.Pow(xi - xj, 2) + Math.Pow(yi - yj, 2));
}
}
```
Compilation / Execution (Windows)
javac Cvrptw.java -cp %LS_HOME%\bin\localsolver.jar
java -cp %LS_HOME%\bin\localsolver.jar;. Cvrptw instances\R101.25.txt
Compilation / Execution (Linux)
javac Cvrptw.java -cp /opt/localsolver_12_0/bin/localsolver.jar
java -cp /opt/localsolver_12_0/bin/localsolver.jar:. Cvrptw instances/R101.25.txt
```import java.util.*;
import java.io.*;
import localsolver.*;

public class Cvrptw {
// LocalSolver
private final LocalSolver localsolver;

// Number of customers
int nbCustomers;

// Capacity of the trucks
private int truckCapacity;

// Latest allowed arrival to depot
int maxHorizon;

// Demand for each customer
List<Integer> demandsData;

// Earliest arrival for each customer
List<Integer> earliestStartData;

// Latest departure from each customer
List<Integer> latestEndData;

// Service time for each customer
List<Integer> serviceTimeData;

// Distance matrix
private double[][] distMatrixData;

// Distances between customers and depot
private double[] distDepotData;

// Number of trucks
private int nbTrucks;

// Decision variables

// Are the trucks actually used
private LSExpression[] trucksUsed;

// Distance traveled by each truck
private LSExpression[] distRoutes;

// End time array for each truck
private LSExpression[] endTime;

// Home lateness for each truck
private LSExpression[] homeLateness;

// Cumulated Lateness for each truck
private LSExpression[] lateness;

// Cumulated lateness in the solution (must be 0 for the solution to be valid)
private LSExpression totalLateness;

// Number of trucks used in the solution
private LSExpression nbTrucksUsed;

// Distance traveled by all the trucks
private LSExpression totalDistance;

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

private void readInstance(String fileName) throws IOException {
}

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

trucksUsed = new LSExpression[nbTrucks];
distRoutes = new LSExpression[nbTrucks];
endTime = new LSExpression[nbTrucks];
homeLateness = new LSExpression[nbTrucks];
lateness = new LSExpression[nbTrucks];

// Sequence of customers visited by each truck.
for (int k = 0; k < nbTrucks; ++k)

// All customers must be visited by exactly one truck

// Create LocalSolver arrays to be able to access them with an "at" operator
LSExpression demands = m.array(demandsData);
LSExpression earliest = m.array(earliestStartData);
LSExpression latest = m.array(latestEndData);
LSExpression serviceTime = m.array(serviceTimeData);
LSExpression distDepot = m.array(distDepotData);
LSExpression distMatrix = m.array(distMatrixData);

for (int k = 0; k < nbTrucks; ++k) {
LSExpression c = m.count(sequence);

// A truck is used if it visits at least one customer
trucksUsed[k] = m.gt(c, 0);

// The quantity needed in each route must not exceed the truck capacity
LSExpression demandLambda = m.lambdaFunction(j -> m.at(demands, j));
LSExpression routeQuantity = m.sum(sequence, demandLambda);
m.constraint(m.leq(routeQuantity, truckCapacity));

// Distance traveled by truck k
LSExpression distLambda = m
.lambdaFunction(i -> m.at(distMatrix, m.at(sequence, m.sub(i, 1)), m.at(sequence, i)));
distRoutes[k] = m.sum(m.sum(m.range(1, c), distLambda), m.iif(m.gt(c, 0),
m.sum(m.at(distDepot, m.at(sequence, 0)), m.at(distDepot, m.at(sequence, m.sub(c, 1)))), 0));

// End of each visit
LSExpression endTimeLambda = m.lambdaFunction((i, prev) -> m.sum(
m.max(m.at(earliest, m.at(sequence, i)),
m.sum(m.iif(m.eq(i, 0), m.at(distDepot, m.at(sequence, 0)),
m.sum(prev, m.at(distMatrix, m.at(sequence, m.sub(i, 1)), m.at(sequence, i)))))),
m.at(serviceTime, m.at(sequence, i))));

endTime[k] = m.array(m.range(0, c), endTimeLambda, 0);

LSExpression theEnd = endTime[k];

// Arriving home after max_horizon
homeLateness[k] = m.iif(trucksUsed[k],
m.max(0,
m.sum(m.at(theEnd, m.sub(c, 1)), m.sub(m.at(distDepot, m.at(sequence, m.sub(c, 1))), maxHorizon))),
0);

// completing visit after latest_end
LSExpression lateLambda = m
.lambdaFunction(i -> m.max(m.sub(m.at(theEnd, i), m.at(latest, m.at(sequence, i))), 0));
lateness[k] = m.sum(homeLateness[k], m.sum(m.range(0, c), lateLambda));
}

totalLateness = m.sum(lateness);
nbTrucksUsed = m.sum(trucksUsed);
totalDistance = m.div(m.round(m.prod(100, m.sum(distRoutes))), 100);

// Objective: minimize the number of trucks used, then minimize the distance traveled
m.minimize(totalLateness);
m.minimize(nbTrucksUsed);
m.minimize(totalDistance);

m.close();

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

localsolver.solve();
}

// Write the solution in a file with the following format:
// - number of trucks used and total distance
// - for each truck the customers visited (omitting the start/end at the depot)
private void writeSolution(String fileName) throws IOException {
try (PrintWriter output = new PrintWriter(fileName)) {
output.println(nbTrucksUsed.getValue() + " " + totalDistance.getDoubleValue());
for (int k = 0; k < nbTrucks; ++k) {
if (trucksUsed[k].getValue() != 1)
continue;
// Values in sequence are in 0...nbCustomers. +1 is to put it back in 1...nbCustomers+1
// as in the data files (0 being the depot)
for (int i = 0; i < customersCollection.count(); ++i) {
output.print((customersCollection.get(i) + 1) + " ");
}
output.println();
}
}
}

// The input files follow the "Solomon" format
private void readInputCvrptw(String fileName) throws IOException {
try (Scanner input = new Scanner(new File(fileName))) {
input.nextLine();
input.nextLine();
input.nextLine();
input.nextLine();

nbTrucks = input.nextInt();
truckCapacity = input.nextInt();

input.nextLine();
input.nextLine();
input.nextLine();
input.nextLine();

input.nextInt();
int depotX = input.nextInt();
int depotY = input.nextInt();
input.nextInt();
input.nextInt();
maxHorizon = input.nextInt();
input.nextInt();

List<Integer> customersX = new ArrayList<Integer>();
List<Integer> customersY = new ArrayList<Integer>();
demandsData = new ArrayList<Integer>();
earliestStartData = new ArrayList<Integer>();
latestEndData = new ArrayList<Integer>();
serviceTimeData = new ArrayList<Integer>();

while (input.hasNextInt()) {
input.nextInt();
int cx = input.nextInt();
int cy = input.nextInt();
int demand = input.nextInt();
int due = input.nextInt();
int service = input.nextInt();

latestEndData.add(due + service);// in input files due date is meant as latest start time
}

nbCustomers = customersX.size();

computeDistanceMatrix(depotX, depotY, customersX, customersY);

}
}

// Computes the distance matrix
private void computeDistanceMatrix(int depotX, int depotY, List<Integer> customersX, List<Integer> customersY) {
distMatrixData = new double[nbCustomers][nbCustomers];
for (int i = 0; i < nbCustomers; ++i) {
distMatrixData[i][i] = 0;
for (int j = i + 1; j < nbCustomers; ++j) {
double dist = computeDist(customersX.get(i), customersX.get(j), customersY.get(i), customersY.get(j));
distMatrixData[i][j] = dist;
distMatrixData[j][i] = dist;
}
}

distDepotData = new double[nbCustomers];
for (int i = 0; i < nbCustomers; ++i) {
distDepotData[i] = computeDist(depotX, customersX.get(i), depotY, customersY.get(i));
}
}

private double computeDist(int xi, int xj, int yi, int yj) {
return Math.sqrt(Math.pow(xi - xj, 2) + Math.pow(yi - yj, 2));
}

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

try (LocalSolver localsolver = new LocalSolver()) {
String instanceFile = args[0];
String outputFile = args.length > 1 ? args[1] : null;
String strTimeLimit = args.length > 2 ? args[2] : "20";

Cvrptw model = new Cvrptw(localsolver);