Vehicle Routing Problem Solving System as a CRO-GRID Application
Hrvoje Gold, D.Sc, Tonči Carić, D.Sc, Ante Galić, B.Sc, Ivana Ćavar, B.Sc.
Faculty of Transport and Traffic Engineering
University of Zagreb
Vukelićeva 4, Zagreb, HR-10000, Croatia
Phone: +385-1-2380-222 Fax: +385-1-2314-415 E-mail: {hrvoje.gold | tonci.caric | ante.galic | ivana.cavar}@fpz.hr
Abstract - Transport logistics and fleet management problems
often fall into one class of the optimization problems. Finding
an optimal set of routes for group of vehicles in the transport
network under defined constraints is known as the Vehicle
Routing Problem (VRP). VRP represents the NP hard
problem. Solving of the VRP problem can be shortened by
parallelization of the time critical steps of the sequential
algorithm, decomposition of the problem and distribution of
its search space among the processors or by concurrent search
with different amounts of synchronism and cooperation
among sub-problems. The paper presents the description of
the developed VRP solving system Venera/Mars and the
discussion of results obtained by the implementation of the
heuristic algorithms for solving the practical VRP problems in
city logistics.
of corresponding vehicle. The requests from all the
customers should be fulfilled and all constraints satisfied.
In practice, the basic vehicle routing problem is
extended with constraints, for instance, on the allowed
capacity of the vehicle, length of route, arrival, departure
and service time, time of collection and delivery of goods.
The extended classes of VRP are Capacitated VRP
(CVRP), Distance Constrained VRP (DVRP), Distance
Constrained Capacitated VRP (DCVRP), VRP with Time
Windows (VRPTW), Backhauls VRP (VRPB), Pickup and
Delivery VRP (VRPPD), and Simultaneous Pickup and
Delivery VRP (VRPSPD), Fig. 1. The main goal in all
VRP problems is to obtain the minimal transportation cost.
Vehicle Routing Problem (VRP)
I. INTRODUCTION
Capacitated (CVRP)
One of the main aims of the 'CRO-GRID Applications'
project supported by the Ministry of Science, Education
and Sport of the Republic of Croatia [1] is to qualify the
academic community for the development of distributed
applications.
The application of ‘Optimization of Transport
Management’ is oriented towards the improvement of
organizing the transport of people and goods by solving the
practical vehicle routing problems using distributed /
parallel implementations of exact and (meta)heuristic
algorithms on cluster/grid infrastructure.
The vehicle routing problems represent NP problems
and the optimal solution cannot be reached in polynomial
time. Therefore, for the correct solution it is appropriate to
develop distributed implementations of exact algorithms
which can be solved on a computer cluster. In case of
solving practical vehicle routing problems that include a
number of special additional constraints, the approximate
solving methods should be used. The heuristic and metaheuristic algorithms provide approximate solutions
acceptable in practical application. By application of
distributed/parallel implementations it is possible to
achieve the accelerations of the performance of heuristic
and meta-heuristic algorithms.
II. VRP PROBLEM
Determining the optimal route that is used by a group of
vehicles in the course of serving a group of customers,
represents the vehicle routing problem [2]. Vehicles start
from the depot and to reach the customers they operate on
the transport network. The solution of the problem is the
set of routes. Each route has its first and last stop in a depot
Distance Constrained (DVRP)
Time Windows (VRPTW)
Backhauls (VRPB)
Pickup and Delivery (VRPPD)
Simultaneous Pickup & Delivery
(VRPSPD)
Fig. 1. Classification of vehicle routing problems
In the case when only one vehicle is serving the
customers and there are no additional constraints, the
vehicle routing problem is reduced to the travelling
salesman problem (TSP). Both problems are NP problems.
In the general case the exact solution cannot be found in
the polynomial time. For instance, to solve the TSP
problem with 10, 15, 18 and 20 customers, a single
contemporary computer with processing power of 109
operations/sec, needs less than 1 second, 21 seconds, 74
days, 77 years respectively.
To solve the VRP problem the exact, heuristic and metaheuristic algorithms are in use. The exact algorithms, based
on linear programming techniques are limited to solving
simple practical problems. In special cases when the set of
possible solutions can be reduced the branch, cut and price,
column generation and Lagrange relaxation algorithms are
often used.
With heuristic algorithms, e.g. Clark and Wright, Sweep,
the acceptable solutions can be achieved relatively fast.
The algorithms of improving the heuristic solutions, e.g., 2opt, try to further optimize an already existing solution by
route modifications. The meta-heuristic methods, e.g.
simulated annealing, genetic algorithms, tabu search, ant
colonies, simulate natural processes and they have proven
as very successful optimization methods. In order to solve
the practical VRP problems the procedures are used which
select the starting routes by construction techniques, and
then improve the solutions by linking up of the heuristic
and meta-heuristic methods.
The optimized routes have a significant impact on the
transport organization by reducing the costs and travelling
duration. Practical experiences from literature show that by
improving the transport organization by optimizing the
routes a reduction of transportation costs of even up to
20% may be achieved. The main users are the carriers who
perform the delivery and collection e.g. the transport
companies, municipal services, post, courier services.
III. VRP SOLVING ENVIRONMENT
In order to realize the established goal the VRP solving
environment that consists of a database with models of test
and real-world VRP problems, database of exact and
heuristic VRP algorithms as well as VRP solving system
has been determined, Fig. 2.
Distributed/parallel versions of VRP algorithms are
executed on cluster/grid network.
IV. VRP SOLVING SYSTEM VENERA
In order to set and visualize the results of the vehicle
routing problem, the interactive graphical user interface
Venera [3] has been developed, tested, and used for the
solution of practical problems. In the current version,
Venera has the task of insuring the necessary tools for
solving the capacitated vehicle routing problem, solving of
the vehicle routing problem with time windows, research,
experimenting, and testing of heuristic methods and
algorithms.
For the implementation of heuristic algorithms the traffic
technologist-oriented programming language Mars was
developed and tested on the solving of VRP problems from
the literature and from practice. Mars interpreter is
integrated in the interactive graphical programming
environment Venera. Venera and Mars have been
developed in the object programming language Java.
The main window of Venera, Fig. 3, is divided into three
parts, one for the table presenting customer data (1), part
for table containing data on the vehicle (2), and a part for
graphical presentation of the problem (3). Programming
window (4) consists of an editor of the source code and a
window for displaying system information and messages
while the program is running.
DATABASE OF TEST
AND PRACTICAL
VRP PROBLEMS
DATABASE OF EXACT
AND HEURISTIC
VRP ALGORITHMS
1
TRANSPORTATION GIS
VRP SOLVING SYSTEM
CLUSTER/GRID
Fig. 2. Vehicle routing problem solving environment
Since each VRP problem is specific and depends on the
stated service and transport conditions, the database of test
and practical VRP problems assists the user in
classification of his problem. The database of exact and
meta(heuristic) VRP algorithms is the collection of VRP
solvers implemented in sequential or distributed/parallel
form. For solving the practical VRP problems the
environment is augmented by geographic information
system of transport network with traffic attributes.
3
2
4
Fig. 3. Graphical user interface of VRP solving system
The table with the customer data shows the index, the
location by means of x and y coordinates, request for
delivery, the earliest time of the start of serving, the latest
time of the end of serving and the expected time of serving
duration. The table containing data on the vehicles shows
the index, colour of the vehicle (route), occupancy,
remaining capacity, length of the travelled path and the
route recording. Apart from the setting of the problem, the
graphical interface allows editing of the customer data and
manual design of the vehicle route. If necessary, the form
and appearance of display can be adapted by tools supplied
inside the interface.
As a means of staying close to the real problems, the
possibility is provided of using the traffic matrix of
distances between the customers (examples from literature
assume Euclidian distances between two points in a twodimensional space). It is also possible to load problems
written in the format of standard Solomon's test of VRP
tasks. There is a possibility to test and compare the results
obtained by one's own algorithms with the already known
solutions and results from the literature.
V. PROGRAMMING LANGUAGE MARS
Apart from the basic numerical and character data types
and language constructions, the programming language
Mars also contains data types related to the data that are
used in setting and solving VRP problems, as well as
special statements and functions for local searches and
minimization (maximization) of the objective function [4].
In this way the heuristic algorithm programming is
significantly simplified and the code is clear and short.
The basic transport data types in the language Mars are
CUSTOMER, VEHICLE and ARC. The structure of type
CUSTOMER consists of the following data: customer
index, customer location coordinates, delivery/collection
demand, earliest allowed time of beginning the serving, the
latest allowed time of ending the serving, duration of
serving and number of visits to the customer. The transport
type VEHICLE contains data on the vehicle index, total
and remaining vehicle capacity, total travelled path and
customer location at which the vehicle is positioned. The
type ARC consists of: arc length, index of the starting and
destination customer and vehicle index which is the arc
owner. The values of these variables are usually set before
running the program in the very working environment by
means of the graphical interface or by loading the concrete
VRP problem from the database.
For the needs of the program it is possible to increase
the number of transport variables and to declare them
subsequently. Thus declared variables represent auxiliary
variables, rather than the content of the VRP problem that
is being solved. These subsequently declared transport
variables are accessed by means of the name stated in the
declaration, and their values are assigned to them only
during the running of the program. In case one wants to run
a block of statements, e.g. within the statement SELECT or
FOR loop, one can avoid declaration of the auxiliary
variables, since these two programming structures, if
necessary, carry out independently the initialization and
declaration of the variables. Such approach allows fast
interaction between the users and the system as well as
obtaining of the answer to simple queries without having to
declare the variables.
The type of variables CUSTOMER and VEHICLE
indicate numerical data which are closely related to the
definition of VRP, CVRP, DVRP and VRPTW problems.
The grammar of the language is formalized in EBNF
(Enhanced Backus Naur Form) notation. Fig. 4 shows the
statement for building of a route between two customers
ADDARC and statement to delete the arc DELARC
described by syntactic diagrams.
ADDARC
vehicle
DELARC
,
customer
,
customer
arc
Fig. 4. Syntax of statement ADDARC and DELARC
in language MARS
The basic language constructions, e.g. IF, DO, WHILE,
FOR, denote meanings that are similar to those of the
majority of programming languages. The statements and
functions such as MOVE, ADDARC, DELARC,
ADDVEHICLE,
DELVEHICLE,
DISTANCE,
ALLVISITED, SOLVED etc. serve for the communication
with the VRP problem objects.
A special place belongs to the declarative statement
SELECT which acts as query, and not as an element of the
sequential structure of algorithm step description. The form
of the query statement is of similar syntax as the statement
of the same name for selection in SQL (Structured Query
Language) – a language for making queries to the database
management system. The task of the selection statement is
to search all the customers, vehicles or arcs that satisfy the
stated logical expressions. For all the selected variables,
additional search may be requested in order to minimize or
maximize the stated objective functions.
Such structure of search conditions is extremely useful in
the development and coding of the heuristic algorithms.
The statement SELECT automates the search within a part
of the neighbourhood which is described in the WHERE
part of the statement SELECT. By using the options
MINIMISE and MAXIMISE precisely the variable is
selected which is contained in the selected part of the
neighbourhood, and at the same time provides the desired
minimal or maximal result of the objective function. An
example of a simple query that will select the nearest
customer c to vehicle v is presented in Fig. 5.
SELECT CUSTOMER c, VEHICLE v WHERE
c.INDEX != v.POSITION.INDEX
MINIMISE DISTANCE(c, v.POSITION)
ENDSELECT
Fig. 5. Query structure in language MARS
VI. HEURISTIC VRP ALGORITHMS
Constructive heuristic methods such as the Nearest
Neighbour Heuristic (NNH), the Nearest Addition
Heuristic (NAH), the Sweep method, Sweep-NAH method,
the Sweep-Farthest Addition Heuristic (FAH) and the
Clark and Wright method, along with the 2OPT or local
search e.g. Global Best (GB), First Best (FB) on large
neighbourhood generated by  interchange mechanism [5]
written in Mars, using standard test data from the literature
set and visualized in the Venera, have shown the results
close to the optimal results from the literature.
Thus, as an example of the standard test problem E05105e with 50 customers and 5 vehicles, using the two-phase
Sweep-NAH algorithm, the deviation achieved in the
Venera environment compared to the best result described
in literature amounts to 1.7%, and the result is presented in
Fig. 6. The points are customers index and arrows show
calculated directions of vehicles for each route displayed
on the Venera graphical user interface. To each route is
assigned a different colour.
ADDVEHICLE capacity
ENDFOR
ENDPROCEDURE
PROCEDURE DeleteUnusedVehicles
FOR i=VEHICLECOUNT-1 TO 0 STEP -1
IF VEHICLE(i).CAPACITY =
VEHICLE(i).CAPACITYLEFT
DELVEHICLE VEHICLE(i)
ENDIF
ENDFOR
ENDPROCEDURE
PROCEDURE DeleteAllVehicles
FOR i=VEHICLECOUNT-1 TO 0 STEP -1
DELVEHICLE VEHICLE(i)
ENDFOR
ENDPROCEDURE
Fig. 7. Clark and Wright algorithm in language Mars
Fig. 6. The solution of standard test example E051-05e
For the example E076-10e with 75 customers and 10
vehicles applying the Clark and Wright algorithm the
deviation amounts to 6% compared to the best published
solution. The example of the Clark and Wright algorithm
realization in Mars is presented in Fig. 7.
'----------------------------------------------'
' CLARK AND WRIGHT ALGORITHM
'
'----------------------------------------------'
CLEAR
AddVehicles CUSTOMERCOUNT, 160
CUSTOMER depot = CUSTOMER(0)
NUMBER end, end0=0
FOR i = 1 TO CUSTOMERCOUNT-1
ADDARC VEHICLE (i-1), depot, CUSTOMER (i)
ADDARC VEHICLE (i-1), CUSTOMER (i), depot
ENDFOR
DO
SELECT ARC arc1, ARC arc2 WHERE
EQUAL CUSTOMER (arc1.TO, depot)
EQUAL CUSTOMER (arc2.FROM, depot)
NOT EQUAL
VEHICLE(arc1.VEHICLE, arc2.VEHICLE)
arc2.VEHICLE.CAPACITY arc2.VEHICLE.CAPACITYLEFT <
arc1.VEHICLE.CAPACITYLEFT
MAXIMISE arc1.LENGTH + arc2.LENGTH
DISTANCE(arc1.FROM, arc2.TO)
ENDSELECT
IF FOUND (arc1)
MESSAGE "MAX saving -> " + arc1.FROM.INDEX +
" on " + arc2.TO.INDEX
end = 0
DO
SELECT ARC arc3 WHERE
NOT EQUAL ARC (arc3, arc2)
EQUAL VEHICLE (arc3.VEHICLE,
arc2.VEHICLE)
ENDSELECT
IF FOUND (arc3)
ADDARC arc1.vehicle, arc3.FROM, arc3.TO
DELARC arc3
ELSE
end = -1
ADDARC arc1.VEHICLE, arc1.FROM, arc2.TO
DELARC arc1 DELARC arc2
ENDIF
LOOPIF end = 0
ELSE end0=-2
ENDIF
LOOPIF end0 = 0
DeleteUnusedVehicles
PROCEDURE AddVehicles(NUMBER number, NUMBER
capacity)
DeleteAllVehicles
FOR i=0 TO number-1
In the first part of the program the FOR structure creates
as many routes as there are customers, and each route
consists only of the depot and one customer. After the
initialization the first SELECT statement selects those
routes that need sufficient capacity of one of the vehicles in
order to combine the routes and at the same time to
maximally save the length of the total travelled distance by
the vehicle. The next SELECT statement combines two
routes and reduces the number of the necessary vehicles by
one. Selection and combining of routes continues until is
possible to gain a savings.
VII. TRANSPORT GIS
The solving of the vehicle routing problem is a
demanding mathematical problem, but mathematical
methods and models alone are not sufficient for its
practical solution. It is necessary to combine the VRP
model with the real environment of a transport company
whose transport organization needs to be improved, e.g. by
suggesting changes of transport routes.
As a system for managing spatial data, the Geographic
Information System (GIS) appears as one of the obligatory
interfaces between the methods developed on the VRP
mathematical model, underlying real transport network and
the actual working environment of the transport company
with all its specific operations. The GIS system, composed
of a set of programming tools that are used for the input,
storage, manipulation, analysis and presentation of spatial
data, links all the spatial data within the applied coordinate
system, e.g. geographic latitude and longitude.
In order to input and prepare the real data on the
locations of customers as well as to visualize the existing
and the resulting routes on a digital map, special GIS
application Miranda has been developed. The current
transportation layer of the network of urban traffic routes is
expanded by a layer of actual constraints acting on the
allowed traffic flow, Fig. 8. Also, since by setting the
actual VRP problems the addresses of collection or
delivery locations are most usually known, rather than their
geographical coordinates (latitude/longitude), the address
module of the geo-information system has been used. The
address module performs the transformation of addresses
into adequate digital map coordinates.
For the needs of transforming the actual traffic route
network into adequate graph used for the solving of the
VRP problem, the layers of the digital map are
supplemented by a layer of the shortest paths between the
given locations.
The shortest path calculation is performed by modified
Dijkstra algorithm. While calculating the shortest path, the
application takes into consideration the direction of the
streets and the allowed turnings at intersections and
generates an asymmetric matrix of the shortest distances
between all the given locations. This allows the solution of
practical VRP problems. Since the real transport network
represents an incomplete graph, a fully connected directed
graph was constructed and used in the calculation of the
shortest path, as the first approximation of the real
transport network. Non existing connections like prohibited
directions, e.g. one-way street, are added in the form of
bidirectional arcs but one direction is always avoided in
route due to assigned weight (large number) which are
penalised by objective function.
∞
1
∞
∞
∞
3
2
∞
∞
4
5
∞
Fig. 8. Transport network layer, traffic data layer
and virtualy connected directed graph
Since the accuracy of the solution of the practical VRP
problem is directly related to the accuracy of the traffic
data on the digital map, it is necessary that the traffic layer
data are precise and valid. Updating of the data is an
ongoing process, and the accuracy of the data in the spatial
and attribute base of the digital map, along with the
collected data from the field, can be supplemented by the
simulation and visualization of the existing and calculated
routes obtained during the solving of the VRP problem.
The digital map with the transport network and traffic
layer was provided for the needs of the project by the
company Mireo d.o.o.
VIII. THE SOLUTION OF A PRACTICAL PROBLEM
For the needs of solving the practical VRP problems,
cooperation was established with the municipal waste
collection company Čistoća d.o.o. from Zagreb, and the
garbage collecting routes were optimized for the district of
Novi Zagreb. Based on the work orders and by processing
the data from the statistical reports, the data on the
available vehicles and their capacities, delivery places and
the existing methods of vehicle routing while garbage
collecting were gathered. 169 places of garbage collection
in the region of Novi Zagreb were included. The garbage
collection in the cities is a municipal service activity, which
from the aspect of selecting the optimal routes, can be very
well described as an Asymmetric Capacitated Vehicle
Routing Problem (ACVRP).
In ACVRP problem all the customers and their orders
are known in advance, the vehicles are identical, and the
common starting point is at the central depot. The only
constraint that exists is the capacity of individual vehicles.
The objective function expresses the request for
minimizing the overall cost (e.g. weight function of the
number of vehicles and their total length or time) in serving
all the customers.
The transport management may be improved by
changing the route plans alongside keeping the existing
place of collection. In case there is intention and sufficient
financial means for changing the collection places, new
collection places could be proposed regarding the changes
of the density of population and the newly formed
distribution of the collection places.
The solving procedure of the practical VRP problem by
using the Venera/Mars system combines the processes of
data collection, data processing by means of GIS tools,
analyses of the current routes, interactive generation of the
new route plans by using heuristic methods and
presentation of results.
Thus, in the procedure of solving the set problem, first a
simulation of the existing routes was performed. By
entering the data from the work orders into Miranda, by
calculating the distances matrix, by transferring the
collected and calculated data into the working environment
of the Venera, and by running the simulation program of
driving according to the route plan from the work orders
entered in the programming language Mars, the data for the
existing routes were obtained.
For the calculation of the new route plans, several
constructive heuristic algorithms and their chains were
implemented in Mars, and performed in Venera, in order to
find the initial solution (NNH, NAH, Sweep, FAH, Clark
and Wright) and made improvements with additional local
search (2OPT, , GB, FB) constrained with selected
number of iterations (1E5, 1E6).
All the obtained results indicated that in order to realize
the required routes one vehicle less would be sufficient
compared to the initially used number of vehicles. In case
of applying the algorithms NAH and GB-1E5 with the
reduction in the number of vehicles, also the totally
travelled path necessary to visit the collecting points is
reduced by 29% compared to the totally travelled path
according to the current work orders.
While entering the data and using the available digital
map, several errors have been noted. Difficult to see by the
naked eye, there are errors in the map construction which
caused traffic inaccessibility of some minor areas on the
map. The inaccessibility was reflected in the impossibility
to calculate the shortest path to certain locations. The error
was identified and eliminated while analyzing the
asymmetric matrix of distances and try to explain the poor
results in the operation of certain algorithms.
The complexity of modelling and solving the practical
VRP problems regarding a number of special requirements
that are brought by the actual problems, as well as the need
for ever shorter solving times, imposes additional
requirements on the approximate solving methods as well.
Therefore, the distributed and parallel processing on the
computer cluster/grid [6] in the form of parallelization of
the time critical steps of the sequential part of the heuristic
algorithm, decomposition of the problem and distribution
of the search space between the processor or concurrent
search with different forms of synchronization and
cooperation between sub-problems, represents an adequate
framework in the optimization of the transport
management.
and the totally travelled path were achieved, compared to
the initially found values.
With the aim of allowing the development of distributed
applications for solving of the VRP problems, the Venera
system is expanded by new graphical interface that will
allow setting of the problem and visualizing of the solution
on the local computer, whereas the solving itself will be
carried out on the computer cluster network. The
development of new data types, e.g. multiple depots,
garages, as well as new Mars instructions for distributed
programming are underway.
IX. CONCLUSION
REFERENCES
For the solution of the problems of distribution or
collection of goods, the descriptions of technological
processes need to be brought nearer to the description of
the mathematical model. Apart from the case of fewer
number of vehicles, the mathematical combinatory
optimization problem, can not be generally and exactly
solved within an acceptable period of time. Practical
solutions are obtained by approximation methods that are
usually of heuristic nature. The combination of the
geographic information system, mathematical methods and
databases, improves significantly the utility value of the
existing optimization methods, i.e. algorithms, since the
verification of the new and the simulation of the existing
route plans can be performed fast.
For the needs of solving the practical VRP problems, an
interactive system for VRP problem solving was
developed, composed of the graphical user interface and a
language interpreter to perform heuristic algorithms
recorded in the traffic technologist-oriented programming
language Mars.
The operation of the system was successfully tested by
solving a number of standard VRP examples from the
literature, with the best obtained result indicating slight
deviation from the optimal result from the literature.
In solving the ACVRP problem of optimizing the waste
collecting routes of the municipal waste collection
company, the real data about the vehicle fleet were used,
and with the help of a digital map the data on the collection
places and the direction of streets. By applying the
heuristic algorithms, the savings in the number of vehicles
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