Optimal combination of operators in Genetic Algorithmsfor VRP problems

International Journal of Modern Research in Engineering and Technology (IJMRET)
www.ijmret.org Volume 3 Issue 1 ǁ January 2018.
w w w . i j m r e t . o r g I S S N : 2 4 5 6 - 5 6 2 8 Page 75
Optimal combination of operators in Genetic Algorithmsfor VRP
problems
Borja Vicario Medrano1
, Iván Colodro Sabell 1
, Javier López Ruiz 1
, Antonio
Moratilla Ocaña 1
, Eugenio Fernández Vicente1
1
(Computer Science, University of Alcalá, Spain)
ABSTRACT : The well-known Vehicle Routing Problem (VRP) consist of assigning routeswith a set
ofcustomersto different vehicles, in order tominimize the cost of transport, usually starting from a central
warehouse and using a fleet of fixed vehicles. There are numerousapproaches for the resolution of this kind of
problems, being the metaheuristic techniques the most used, including the Genetic Algorithms (AG). The
number of approachesto the different parameters of an AG (selection, crossing, mutation...) in the literature is
such that it is not easy to take a resolution of a VRP problem directly. This paper aims to simplify this task by
analyzing the best known approaches with standard VRP data sets, and showing the parameter configurations
that offer the best results.
KEYWORDS -Optimization, Vehicle Routing Problem, Genetic Algorithms, Operators.
I. INTRODUCTION
At present, the usual operation of freight
transport operators isbased on the distribution of
productsfrom a base of operations or central
warehouse to different destinations or
customers.There are many variants that make the
problem more or less complex, such as
optimization of the load, conditions that impose
incompatible loadson the same vehicle, restrictions
impose by customers, routes prohibited by law to
vehicles with certain products, availability of
workers and vehicles... In short, we could say that
there are a large number of problems of this type,
and that they areincludedinproblems that are
known as Vehicle Routing Problem (VRP).
1. Vehicle Routing Problem
In general, we can say that a VRP consist
of determining a set of minimum cost routes
starting and ending in the depots, with a set of
dispersed customers and a fleet of vehicles
given.Manyextensions of VRP have emerged over
the years, among which the most discussed are the
following:
 CVRP (Capacitated VRP): each vehicle
has a limited capacity.
 MDVRP (Multi-Depot VRP): the seller
uses multiple depots to supply customers.
 PRVP (Periodic VRP): orders can only be
carried out on certain days.
 SDVRP (Split Delivery): customers can
be supplied by different vehicles.
 SVRP (Stochastic VRP): some values
such as the number of customers, their
demands, service time or travel time are
random.
 VRPB (VRP with Backhauls): customers
can return products.
 VRPPD (VRP with Pick-Up and
Delivering): customers have the option of
returning some goods to the depot.
 VRPSF (VRP with Satellite Facilities):
vehicles can be supplied without returning
to the warehousein other auxiliaries during
the route.
 VRPTW (VRP with Time Windows):
each customer must be attended in a
certain time window.
Fig. (1): example of VRP. Source: own elaboration.

We can affirm that the VRP and all the
extensions listed previouslyare a generalization of
the well-known Travel Salesman Problem (TSP)
and, therefore, are included in the combinatorial
optimizationproblems, which makes it, from the
point of view of computationalcomplexity, one of
the most complex because it is of the NP-Complete
type: it is not possible to solve them in polynomial
time (Balinzki and Quandt, 1964; Garvin et al,
1957; Toth et al, 2002).
For its resolution, there are a lot of
techniques that can be classified into three
maincategories: exact, heuristic and metaheuristic
methods. Exact methods are efficient in problems
up to 50 deposits(Azi et al., 2010)due to
computational time constraints, and we classify
them into three groups: direct tree search, dynamic
programming and linear and integer programming.
On the other hand, heuristic methods provide us
with procedures that obtain acceptable solutions
through a limited exploration of the search space.
Within these methods we have constructive
methods, insertion heuristics and elementary
allocation methods. A review of these can be found
at (Olivera, 2004). Finally, metaheuristic
techniques, developed in the late 1990s, perform a
search procedure to find acceptable solutions
through the application of domain independent
operators that modify intermediate solutions guided
by the fitness of their objectivefunction. These
include Neural Networks, Tabu Search, Genetic
Algorithms or Ants Algorithms, among others. A
review of these can be seen in (Contardo, 2005).
2. Genetic Algorithms
One of the most widely used metaheuristic
techniques in VRP problems, and object of this
paper, are the Genetic Algorithms that, in a basic
way, obtain solutions by using concepts coming
from the world of biology such as crossing and
mutation as well as the natural rules of self-repair
and adaptation of living beings. The use of genetic
algorithms in optimization problems such as VRP
has become very popular in recent years, as they
often offer successful results in real applications
(Reeves, 2003).
This technique differs from others in four
basic aspects:
1. They work with a coding of parameters
(or genotype) and not with the parameters
themselves (phenotype), so that each
solution (member of the population) is
represented by a vector called
chromosome, in which each of its
components (gene) represents a parameter
of the solution.
2. They search from a population of
solutions and not from a single solution,
which, according to (LeBlanc, 1999),
ensures the exploration of a largeportion
of the solutions space and avoids the fall
in local optimal.
3. They use the information from the
evaluation of the fitness function to guide
the search, not auxiliary knowledge.
4. They use both probabilistic and non-
deterministic transition rules.
In a general way, the process of a Genetic
Algorithm is as follows:
 Perform an adequate representation or
coding of individuals
 Select an initial population of individuals
 As long as the completion condition is not
satisfied…
o Select two membersfrom the
population for crossing.
o Cross these members with a
certain probability.
o Mutate the two descendants with
a certain probability.
o Evaluate the new generated
individuals.
o Insert new individuals into the
population.
II. APPROACH
One of the problems associated with the
use of Genetic Algorithms is that the representation
of each individual that composes the population,
the size of the population, the strategies of crossing
and mutation, and the rest of the different
parameters of the algorithm, differ according to the
authors and type of problems to be addressed, so
that its use is not as easy as it would be to be
desired and, therefore, it needs experienced

professionals to design algorithms that are
appropriate for each case.
In the case of coding, there are several
variants (binary, integer, real...), although the most
commonly used coding is binary (Holland, 1975).
With respect to population size, different studies
relating to population size have determined that, for
chromosomes with binary coding and a length "L",
this size grows exponentially with the size of the
chromosome (Goldberg, 1989). Under these
assumptions, we can find work that suggests, based
on scientific evidence, that a population size
between L and 2L is sufficient to be successful in
solving problems(Alanderk, 1992).
The selection operators most frequently
used and analyzed in this work are:
 Tournament Selection (TS): Selects the
best fit individual from a subset of the
population.
 Roulette Wheel Selection (RWS): Places
the individuals in a roulette distributing
portions according to the fitness of each
one, following the idea of normalized
fitness.
 Stochastic Universal Sampling (SUS): It
bases its idea on the roulette of the
previous method, establishing a system of
equidistant marks on the roulette to make
a single turn and obtain from a single spin
the positions for each mark.
There are many crossing operators that
have been proposed, the most widely used and
analyzed in this work are the following:
 Partially Mapped Crossover (PMX): a part
of the string representing one parent is
matched with an equal part of the other
parent's string, exchanging the remaining
information.
 Cycle Crossover (CX): Creates a
descendant from the parents, so that each
position is occupied by the corresponding
element of one of the parents.
 Order Crossover (OX): builds descendants
by choosing a sub tour of one parent and
preserving the relative order of the other
parent's cities.
With respect to the mutation operators
most used, and object of analysis, we have:
 Displacement Mutation (DM): starts by
selecting a random substring. It is
extracted from the tour and inserted in a
random place.
 Exchange Mutation (EM): randomly
selects two individuals in the population
and changes them.
 Simple Inversion Mutation (SIM): Selects
randomly two cut-off points on the string,
then reverses the substrings between them.
Only by using the previous operators
would we have 27 possible combinations or
configurations of the genetic algorithm to use in
practice. If we take into account that there are
many other operators, along with the other
parameters to select, the possibilities increase
significantly, which makes it practically
unmanageable for someone who has no experience
with a genetic algorithm in real applications.
In order to clarify the selection of one
operator or another (regarding selection, crossing
and mutation) a genetic algorithm is programmed
together with the operators mentioned above, and
an exhaustive study of their use is carried out with
standard test databases to analyze the pertinence of
using one or the other.
The analysis begins with the preparation
of a set of tests to evaluate the way in which the
developed implementation works. During this
phase the three selection operators, the three
crossover operators and the three mutation
operators explained above will be taken to establish
the necessary combinations between them all.
Prepared execution blocks will be repeated for each
of the combinations that have been set, and will be
taken as results the cost of solving the problem,
contributed as the sum of distances of the routes of
the best individual, and the execution time it has
taken the algorithm to finish. It will also take into
consideration the chromosomes obtained as
solutions to the problem and the cut-off points that
mark the separation between the different routes
contained in the chromosome. At the end of the
tests for a data set, the solution with the shortest
distance will be searched and its chromosome and

cut-off points will be used to make a graphical
representation of the solution obtained.
To contrast the results obtained, the
established tests will be run for four standard
datasets defined by different authors and used over
the last few years by the entire scientific
community. A description of these can be found at
http://neo.lcc.uma.es/vrp/vrp-instances/. The four
specific instances used are:
 A-n32-k5.vrp
 B-n43-k6.vrp
 E-n76-k7.vrp
 att-n48-k4.vrp
III. RESULTS
Once the tests have been carried out on the
data instances mentioned, repeating these tests a
statistically significant number of times, we obtain
the results.In each case study the enclosed figures
with Cartesian plan showa representation of the
location of the destinations (central warehouse and
clients) according to their coordinates on a
Cartesian axis.
1. First case study: A-n32-k5
Based on the results, the selection operator
that obtains the best solutions in this case is the TS,
followed by the RWS and SUS, with a more than
remarkable difference between them. However, this
difference is not so pronounced in the analysis of
crossing and mutation operators. The crossing
operator that seems to get the best results is the
OX, followed closely by the PMX and a little
further away by the CX. On the other hand, the
mutation operator that behaves better is EM,
although the SIM operator also approaches results.
The DM operator remains behind in the trend of the
other two types of mutations.
Fig. (2): Cartesian plan with the location of the
destinations of case study A-n32-k5. Source: own
elaboration.
2. Second case study: B-n43-k6
The first case study discussed above is not
sufficient to reach the conclusions that motivate
this work, so a new case study is then carried out
with the second of the study data. This new
analysis aims to consolidate the assumptions made
in the first case and clarify those ideas that could
not be demonstrated, in addition to adding some
experimental richness to the work by having a
battery of tests with a greater volume of results.
Following this criterion, the mentioned
data are taken as the set of initial conditions that
will use the genetic algorithm to solve the VRP
problem, with a greater volume of data than the one
used in the previous case, motivated by a greater
number of destinations. This implies that the size of
the chromosome will be larger, as well as the size
of the population, so that each iteration of the
algorithm will increase the time required for the
execution and memory occupation of the data
structures used.
destinations of case study B-n43-k6. Source: own
elaboration.
The results obtained show a pattern
similar to the one seen in the previous case study.
The selection operator with the best results is again
TS, although it should be noted that in some cases
the RWS obtains similar solutions. On the other
hand, the SUS operator continues to show results
too far behind expectations. In the case of the
crossover operator, the OX operator gets the best
solutions, followed closely by the PMX and then
the CX. The best mutation operator appears to be
EM, followed by SIM and DM operators.
3. Third case study: E-n76-k7

The data obtained so far are beginning to
show clear indications of the operators'
contribution to the execution of the algorithm, but
analysis is still needed to provide a wider range of
information in order to reach the desired
conclusions. Therefore, a new case study will be
carried out with the third of the four datasets
selected for this work. It should be noted that in
each new case study, initial data are used that form
more complicated problem configurations to be
optimally solved.
This new initial data set represents a
considerable increase in the size of the problem due
to the increased number of destinations. This
means that chromosome and population sizes will
be larger, so test runtime and memory occupancy
will increase significantly. This case presents a
better distributed layout of the locations on the
plan, as shown in Fig4, contrary to what happened
in the previous case where the facilities were
grouped into areas.
destinations of case study E-n76-k7. Source: own
elaboration.
In this case study with a large amount of
data, the great differences that already existed in
previous cases with respect to certain combinations
of operators are again highlighted. However, the
comparison of solutions obtained by other
combinations of operators is adjusted more closely
and it is difficult to determine which one works
best, so the tendency of solutions to approach the
best solution will prevail. The selection operator
with the best overall performance for all cases is
the TS, closely followed by the RWS. The SUS
operator gets poorer quality results, so it can be
ruled out as a potential selection operator of choice.
In the case of the crossing operator, the OX
operator brings back the best solutions, but the
PMX also provides very close results on average.
The CX operator also sometimes gets good results,
but on other occasions he is ranked as the worst of
the three. In terms of mutation, the EM operator
continues to stand out, followed by SIM and DM.
4. Fourth case study: att-n48-k4
To finalize this analysis of the genetic
algorithm and its operators, a last case study will be
carried out with the fourth dataset. Their
particularity lies in the positioning of the
installations on the plane, so that this time the
coordinates have a much wider range of values,
having greater distances that is why the distance
units in which the journeys are measured are
significantly greater and will test the capacity of
the algorithm to find solutions with a good total
distance value.
The initial placement is reflected in Fig. 5
with the same representation system used in the
previous cases to make the graphical presentation
of the map. The destinations are not as dispersed as
in the previous case study, but are sufficiently
dispersed not to consider area dispersion as in the
second case study.
destinations of case study att-n48-k4. Source: own
elaboration.
In this case, the results collected show a
different trend from what we have seen so far. On
this occasion, the selection operator that seems to
stand out is the RWS, although it is true that in
some cases the TS improves the solution or is very
close to the data of the previous one. Crossover and
mutation operators require another type of analysis
that focuses more on their own results in order to
determine the actual influence these operators have
on the result obtained by the algorithm. It can be
said for the crossover operator that the OX
generally still performs well in relation to what the
other crossover operators achieve, while for the
mutation operator the EM is the best performing

because the SIM brings too many ups and downs
and the chosen one has more constant values.
IV. EVALUATION OF RESULTS
After performing the necessary tests to
examine the behavior of the genetic algorithm, it is
time to carry out an evaluation process to
determine which operators are the ones who bring
the best qualities to the algorithm in order to
achieve the best possible performance. Based on
the results obtained in the tests carried out for four
case studies with different initial datasets, a
relatively common pattern can be observed in the
value of the solutions, and it may be noted that the
TS operator generally obtains better results,
followed by the RWS and SUS operators. In the
case of the crossing operator, the quality order
would be defined by the operators OX, PMX and
CX. Finally, the mutation operator that adapts best
is EM, and then SIM and DM operators in that
order of choice.
These conclusions are preliminary
statements obtained from a first visual evaluation to
compare previous results. To corroborate this
proposal, it is necessary to carry out a more
exhaustive comparison of the results in order to
obtain reliable and generalized data on the
implication that each operator has in the execution
of the genetic algorithm. Based on the results
obtained in the tests, a proportionality relationship
has been established between the best solutions
provided by the algorithm and the number of
solutions related to certain selected operators.
Based on this, the combination of operators that
best adapts to the execution of the genetic
algorithm can be determined in order to obtain the
best possible results, as we can see in the graphical
results in Figure 6. This combination is composed
of the Tournament Selection Operator (TS), Order
Crossing Operator (OX) and Exchange Mutation
Operator (EM).
To finalize the evaluation of the results
and reach the objective of this paper, a relationship
will be established between the conclusions
obtained in terms of total distances and runtimes. It
had been established that the best combination of
operators analyzing distances is the Tournament
Selection Operator (TS), Order Crossing Operator
(OX) and Exchange Mutation Operator (EM). On
the other hand, the combination of operators with
the best performance according to runtimes was
formed by the Tournament Selection Operator
(TS), the Partially Mapped Crossing Operator
(PMX) and the SimpleInversion Mutation Operator
(SIM).
Fig. (6): Ratio of best distance results for selection,
crossover and mutation operators. Source: own
elaboration.
Comparing both conclusions, the first
important observation that can be made is that the
TS selection operator is the best of these operators
when it comes to achieving a good level of
algorithm efficiency and fits perfectly into any of
the possible operator combinations. The next step
is to find the right crossover and mutation
operators, since the conclusions obtained are
different in terms of distances and runtimes.
Analyzing the results on the crossing
operators in more detail, the best operator
minimizing distances is the OX, followed by the
PMX and CX. However, the order of choice
changes to PMX, CX and OX by analyzing in
terms of runtimes. The OX operator gets the best
results in terms of distance, but has the
67%
33%
TS
RWS
SUS
25%
69%
6% PMX
OX
CX
3%
64%
33%
DM
EM
SIM

disadvantage of being the operator that consumes
longer runtime. On the other hand, the PMX
operator is the one with the best times and is the
second operator in terms of distances, being close
to the first one in quality. Following this reasoning,
the PMX operator presents considerably better
times than the OX and, although in terms of
distances it is slightly worse, the differences are
minimal. In this way, it can be said that the PMX
operator is best suited for the algorithm to maintain
efficiency criteria.
Similarly for mutation operators, the best
operator, determined by distances, is EM, followed
by SIM and DM. However, the choices are
different if the runtime is taken into account,
choosing the SIM, MS and DM operators in that
order. Continuing with this idea, the EM operator
achieves the best distance results, while the SIM is
a little behind in performance. On the other hand,
both operators get good runtimes, slightly better
SIM results. With this, in view of the provision of
very similar times, it is recommended to evaluate
the distances obtained in the results to achieve the
desired efficiency, so that the operator that best fits
this requirement is EM.
V. CONCLUSION
The resolution of VRP (Vehicle Routing
Problem) is satisfactorily addressed through the use
of Genetic Algorithms. However, there are a large
number of approaches to the different operators
used. After extensive testing with different
datasets, we can say that the ideal combination for
best algorithm performance is composed of the
Tournament Selection Operator (TS), the Partial
Correspondence Crossing Operator (PMX) and the
Exchange Mutation Operator (EM).
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