Developing Tools for the Team Orienteering Problem

A Simple Genetic Algorithm

João Ferreira

, José A. Oliveira

Guilherme A. B. Pereira

, Luis Dias

Fernando Vieira

, João Macedo

, Tiago Carção

, Tiago Leite

and Daniel Murta

Centre Algoritmi, Universidade do Minho, Braga, Portugal

Graduation in Informatics Engineering, Universidade do Minho, Braga, Portugal

Keywords: Routing Problems, Team Orienteering Problem, Optimization, Metaheuristics, Genetic Algorithm.

Abstract: Presently, the large-scale collection process of selective waste is typically expensive, with low efficiency

and moderate effectiveness. Despite the abundance of commercially available software for fleet

management, real life managers are only minimally helped by it when dealing with resource and budgetary

requirements, scheduling activities, and acquiring resources for their accomplishment within the constraints

imposed on them. To overcome these issues, we intend to develop a solution that optimizes the waste

collection process by modelling this problem as a vehicle routing problem, in particular as a Team

Orienteering Problem (TOP). In the TOP, a vehicle fleet is assigned to visit a set customers, while executing

optimized routes that maximize total profit and minimize resources needed. In this work, we propose to

solve the TOP using a genetic algorithm, in order to achieve challenging results in comparison to previous

work around this subject of study. Our objective is to develop and evaluate a software application that

implements a genetic algorithm to solve the TOP. We were able to accomplish the proposed task and

achieved interesting results with the computational tests by attaining the best known results in half of the

tested instances.

1 INTRODUCTION

In the last few decades, waste separation and

recycling have become critically important. They are

performed via an easy to accomplish daily routine

due to the existence of collection points, which are

usually placed near residential areas, with greater

concentration in city centres and more spread out in

the suburbs and rural areas.

Among waste collection companies, a common

problem is finding efficient methods for performing

the collection of separated waste, while using a

limited fleet of vehicles and obtaining the highest

possible profit. This problem may be addressed as a

variation of the well-known Vehicle Routing

Problem (VRP), described in the literature as the

problem of designing the least-costly routes from a

depot to a set of customers of known demand. The

routes must be designed so that each customer is

visited exactly once, without violating capacity

constraints and while aiming to minimize the

number of vehicles required and the total distance

travelled. However, the majority of real-world

applications require systems that are more flexible in

order to overcome some imposed constraints that

may lead to the selection of customers. To deal with

these changes, the Team Orienteering Problem

(TOP) models can be used. In the TOP, each

customer has an associated profit, and the routes

have maximum durations or distances. The choice of

customers is made by balancing their profits and

their contributions for the route duration or distance.

The objective is to maximize the total reward

collected by all routes while satisfying the time

limit.

The TOP is a fairly recent concept, first

suggested by Butt and Cavalier (1994) under the

name Multiple Tour Maximum Collection Problem.

Later, Chao et al. (1996) formally introduced the

problem and designed one of the most frequently

used sets of benchmark instances. In 2006, Archetti

et al. achieved many of the currently best-known

solutions for the TOP instances by presenting two

versions of Tabu Search, along with two

332

Ferreira J., Oliveira J., Pereira G., Dias L., Vieira F., Macedo J., Carção T., Leite T. and Murta D..

Developing Tools for the Team Orienteering Problem - A Simple Genetic Algorithm.

DOI: 10.5220/0004273801340140

In Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems (ICORES-2013), pages 134-140

ISBN: 978-989-8565-40-2

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

metaheuristics based on Variable Neighbourhood

Search (VNS). Other competitive approaches were

carried out by Ke et al. (2008), with two Ant Colony

Optimization (ACO) variations; Vansteenwegen et

al. (2009), with a VNS-based heuristic; and more

recently, Souffriau et al. (2010) designed two

variants of Greedy Randomized Adaptive Search

Procedure with Path Relinking.

The work presented in this paper is part of

experiments that are integrated in the R&D project

named Genetic Algorithm for Team Orienteering

Problem (GATOP), which was approved by the

Portuguese Foundation for Science and Technology

(Fundação para a Ciência e Tecnologia – FCT). It

involves five combined tasks to accomplish the

desired goal which is the development of a more

complete and efficient solution for several real-life

multi-level Vehicle Routing Problems (VRP), with

emphasis on the waste collection management. This

should be achieved by the implementation and

testing of heuristic and optimization strategies, in

close collaboration with demand forecasting,

transportation problems, simulation, and multi-

criteria decision models.

Within the GATOP project, the main task is to

solve the TOP and the development of heuristic

solutions based on a genetic algorithm (GA) is

suggested. The simplicity of a GA in modelling

more complex problems and its easy integration with

other optimization methods were the factors

considered for its choice. Therefore, we believe it

can be applied to solve the TOP, since it was also

used for the Orienteering Problem by Tasgetiren

(2002).

In this work we propose to solve medium-to-

large-scale TOP instances considering a time

constraint. We intend to verify whether it is possible

to develop a method, based on a GA, that optimizes

the TOP by achieving equal or better results as

presented in previous studies.

2 PROBLEM FORMULATION

The aim of the present study is to solve the TOP,

which means to develop a method that determines P

paths which start in the same location and have the

same destination, in order to maximize the total

profit made in each path, while respecting a time

constraint. Then, the generated paths are assigned to

a limited vehicle fleet, usually one path to each

available vehicle.

In this work we followed the mathematical

formulation presented in the work done by Ke et al.,

(2008). The objective function for the TOP is given

in equation 1, where n is the total number of

vertices, m is the number of vehicles available, the

value y shows if vertex i is visited or not by a

vehicle k, and finally, r is the reward associated to a

certain vertex i. The objective function consists of

finding m feasible routes that maximizes the total

reward or profit.

max 



∙











(1)

3 DEVELOPING TOOLS

3.1 The Genetic Algorithm

The Genetic Algorithm (GA) is a search heuristic

that imitates the natural process of evolution as it is

believed to happen to all the species of living beings.

This method uses nature-inspired techniques such as

mutation, crossover, inheritance and selection, to

generate solutions for optimization problems. The

success of a GA depends on the type and complexity

of the problem to which it is applied.

In a GA, the chromosomes or individuals are

represented as strings which encode candidate

solutions for an optimization problem, that later

evolve towards better solutions.

The GA evolutionary process starts off by

initializing a population of solutions (usually

randomly), which will evolve and improve during

three main steps:

 Selection: a portion of each successive

generation is selected, based on their fitness, in order

to breed the new, and probably better fit,

generations.

 Reproduction: the selected solutions produce

the next generation through mutation and/or

crossover, propagating the most crucial changes to

the future generations by inheritance.

 Termination: once a stopping criteria is met, the

generational process ends.

3.2 Algorithm Details

The GA we developed to solve the TOP takes into

account the main elements of the problem: the

customers and the vehicles.

There is a set of n customers that must be visited

by at most once and only by one vehicle. The

customers correspond to the vertices in a network

DevelopingToolsfortheTeamOrienteeringProblem-ASimpleGeneticAlgorithm

333

graph.

A fleet of m vehicles is available, and each

vehicle is given a priority list, randomly generated,

representing the preferred visiting order of

customers, figure 1.

Figure 1: Priority lists for 2 vehicles in an instance with 6

vertices.

In this GA, every chromosome is composed by m

sub-chromosomes, each one corresponding to a

different priority list of customers (figure 1). The

sub-chromosomes are composed by n genes, one for

each customer. The i gene of a chromosome

corresponds to the i-th (



) customer to be visited.

In other words, it is the customer in the i-th position

in a priority list. The allele is the index of a customer

in the service network and therefore indicates which

customer is visited in a given gene or position.

Consequently, the total number of genes in a

chromosome is equal to  (one priority list for

each vehicle).

The representation of a valid solution is the result

of a constructing process, where the vertices are

added to the routes by following the order they

appear in priority lists for each vehicle. During this

process, each vertex (or customer) v is chosen for a

route if:

 It is possible to go from the last added vertex to

v and from v directly to the final vertex, without

violating the time limit.

 Vertex v does not belong to another vehicle’s

route.

In figure 2 it is presented a graphic representation

of a feasible solution for a 6-vertices instance with

two vehicles, A (red line) and B (blue line).

Therefore, two routes, one for each vehicle, were

calculated:

 Route A – The solution contains vertices 1-6-3-

4, and these vertices become automatically

unavailable for vehicle B. Vertices 5 and 2 are not

visited because the time limit for the route is

exceeded.

 Route B – Since the route determined for

vehicle A already picked up four vertices from the

priority list of vehicle B, the next available vertex to

be added is vertex 5, concluding the route since

adding vertex 2 means disrespecting the time

constraint.

Figure 2: Representation of a possible solution for 2

vehicles in a 6-vertices instance.

A fitness function is used in the genetic

algorithm to evaluate the overall fitness of a

chromosome, which corresponds to the total reward

collected in a reached solution. The quality of this

information highly influences the reproductive

process.

The implemented GA follows a very simple and

common operation structure:

Begin

P ← Initialize (population);

S ← Selection (P);

While stop criteria not met do:

O ← Reproduction (S);

P ← O + ΔP

S ← Selection (P);

End

Where P is the population of potential solutions

(chromosomes) at the moment, S is the group of

solutions within P, selected to produce the offspring

O, which is the next generation of solutions. As for

ΔP

, it represents a residual part of P that contains

non-selected solutions, used to promote

diversification in the reproductive process and avoid

early convergence to local optimal solutions.

During the reproduction phase, two procedures

take place: crossover and mutation. The crossover

operation involves the combination of two

chromosomes in order to obtain a better one by

inheriting the good genes from each parent. In our

algorithm the process goes by removing a block of

consecutive genes from one parent solution and

placing it in a random position within the other

parent solution. The block of genes has a random

size, and the insertion position may vary randomly.

After the insertion, the repeated vertices in the

solution are removed to maintain the correct size of

the sub-chromosomes.

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The schematic in figure 3 shows how the

crossover procedure is done. In this example, the

crossover occurs between two chromosomes, A and

B, with two sub-chromosomes each. At first, a block

of consecutive genes is removed from B. Then, the

same block is inserted in A within a randomly

chosen sub-chromosome, in an arbitrary position.

The removal of repeated vertices starts in the

position where the sub-chromosome of A was

modified, therefore preventing the removal of the

new added vertices. The new generated solution is

then ready to struggle for a spot in the next

generation.

Figure 3: The crossover procedure.

In the other reproduction method, mutation, a

valid solution can alter its own genetic code by

randomly modifying the genes order. In this case we

decided to perform a simple position switch by first

selecting one vertex (or gene) from one of the

priority lists and exchange with the vertex right after

that selected one, in the same list. An example on

how this process works can be observed in figure 4.

The individuals are selected to be reproduced by

Crossover and Mutation according to a triangular

distribution, where the higher the fitness of a

chromosome, the higher the probability it has to take

part in the reproduction process. In our algorithm

there are parameters that set the number of

mutations and crossovers that may occur during each

reproduction phase (optimization cycle).

Figure 4: The mutation procedure.

Other parameters allow changing the number of

randomly generated chromosomes at each new

iteration, and the number of immune chromosomes

(unchangeable between iterations). It is also possible

to set the algorithm to just accept crossovers and/or

mutations that improve the group of solutions. In the

case of just accepting improving crossovers, the

offspring of two chromosomes is measured in terms

of fitness, and if it is more fit than one of its parents,

then the new chromosome is kept, otherwise it is

discarded.

All the above mentioned parameters will

determine the global efficiency of the algorithm in

respect to a specific problem, which in this case is

the Team Orienteering Problem (TOP).

3.3 Software Developed

A JAVA software application was developed to

implement the algorithm described previously. The

application has a simple, yet functional, Graphical

User Interface (GUI), with a handful of options that

allows parameters of the GA to be adjusted and help

obtain better results in a specific problem or TOP

instance. Although aware of this, we decided to keep

the same settings during all the tests so that a

comparison with other research authors could be

done, as well as an overall evaluation of our GA.

Figure 5: The “Solution Viewer” representing routes while

the application solves an instance.

The developed software application can load

different instances and it is possible to set the

stopping condition. The other previously described

parameters can also be changed through the menu.

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There is also a visualization element that represents

the current loaded instance (figure 5), where the

circles correspond to customers in the TOP.

We consider the software is able to run a series

of experiments on some well-known test instances.

4 COMPUTATIONAL RESULTS

A series of computational experiments were

conducted in order to test the performance of our

GA. The computational experiments were performed

on 24 of the 320 benchmark instances published by

Chao et al. (1996). The instances were chosen in a

semi-random way within four different sets, in order

to introduce diversity and different degrees of

difficulty while testing our algorithm. We compared

our results to the ones obtained by Chao et al.

(1996), hereafter referred to as CGW, the results

achieved by Tang and Miller-Hooks, hereafter

referred to as TMH, and also the results produced by

the algorithms presented by Archetti et al. (2006),

hereafter referred to as AHS. The tests were run on a

laptop computer with an Intel Pentium Core 2 Duo

2.53GHz processor and 4GB of RAM.

The chosen stopping criterion was the number of

iterations to run the algorithm for each instance. This

value was set to 100,000 iterations. There are 100

vertices in the first set, 66 in the second set, 64 in the

third, and 102 in the fourth set of instances.

In each set of instances, the location and score of

each vertex is identical. An instance is characterized

by a number of vehicles, varying between 2 and 4,

and by a time limit (Tmax). The best-known solution

values for the tested instances were produced by the

AHS algorithms.

In order to execute the tests, the following

configuration for the GA was set:

 Total Population: 55

 30 Crossovers per iteration

 10 Mutations per iteration

 10 Randomly Generated Individuals

 5 Immune Individuals

 Only accepts crossovers with improvement

 Accepts all the results from mutations

We ran our algorithm ten times on each instance.

The results obtained in the tested instances with our

GA algorithm, hereafter referred to as GATOP-1,

are presented in Table 1, with the best scores

displayed in bold print. The values fmin and fmax

are respectively denoted as the minimum and

maximum value obtained with the fitness function.

In the TOP, the fitness value corresponds to the

total reward collected or profit made with all the

determined routes (one for each available vehicle).

The value fmin can be considered a guaranteed

value, reflecting the overall robustness of the

algorithm, along with the average fitness value. The

value fmax represents the ability of the algorithm to

reach good solutions. It is obtained by running the

algorithm more than once (ten times in our

experiments) and taking benefit of the randomness,

resulting in a larger computational time.

Computational time of a run, with the default

settings, rarely exceeds 8 minutes in the most

difficult instances.

As can be observed in Table 1, on 13 of the 24

instances, our algorithm achieved the same value as

AHS. Comparing to the other authors, our scores

were equal to TMH in 6 instances and equal to

CGW in 4. In addition, GATOP-1 outperformed

TMH eight times and CGW eleven times.

Table 1: Results achieved with GATOP-1 in the selected

benchmark instances.

Avg fmin fmax

p4.2.e 566.80 514 601 618 593 580

p4.2.n 978.40 894 1056 1171 1150 1112

p4.3.f 552.40 519 573 579 579 552

p4.3.i 696.00 624 778 807 785 798

p4.4.l 774.40 730 812 880 875 847

p4.4.q 941.90 876 1010 1161 1124 1084

p5.2.e 180.00 180 180 180 180 175

p5.2.m 829.50 800 860 860 860 855

p5.3.h 260.00 260 260 260 260 255

p5.3.v 1357.00 1310 1395 1425 1410 1400

p5.4.o 676.50 660 690 690 680 675

p5.4.t 1076.00 1035 1160 1160 1100 1160

p6.2.j 913.20 882 948 948 936 942

p6.2.n 1192.80 1164 1242 1260 1260 1242

p6.3.i 630.50 606 642 642 612 642

p6.3.l 961.80 936 1002 1002 990 972

p6.4.k 526.80 522 528 528 522 546

p6.4.n 1030.80 948 1068 1068 1068 1068

p7.2.e 290.00 290 290 290 290 275

p7.2.r 924.10 863 1027 1094 1067 1082

p7.3.g 344.00 344 344 344 344 338

p7.3.s 907.30 839 974 1081 1061 1064

p7.4.i 366.00 366 366 366 359 338

p7.4.t 920.40 839 953 1077 1067 1066

CGW

Inst

ance

GATOP‐1

AHS TMH

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In respect to the best scores, GATOP-1 fell

behind on 12 instances, with a maximum error of

151 and an average error of 57.7. The results

produced with GATOP-1, in terms of consistency,

are worse than the results achieved by AHS. In some

instances, since there is a considerable gap between

the fmin and fmax values, and also between fmax and

the average values (Avg column in Table 1). In

addition , in some of the tested benchmark instances,

our GA scored considerably less than the best

solution for each of those instances, marked in bold

print (Table 1). We consider the gap between our

scores and the best scores occurred for three reasons.

One of them may relate to a non-optimal balance

between the genetic operators (crossovers,

mutations), the number of new solutions randomly

generated, and the number of immutable solutions

(auto-immune individuals).

We performed the tests with a fixed parameter

configuration, but we also checked if, by

proportionally increasing the parameters values, the

algorithm would perform better, which in fact did

occur but not in a significant way and at the expense

of greater computational time. Therefore, improved

results may be produced once a better configuration

for the GA is determined.

Another reason that might explain some failures

with GATOP-1 is the way of constructing feasible

solutions. Before the algorithm evaluates the fitness

of a chromosome, the process of solution

construction takes place. It begins by generating the

route for the first vehicle, following the priority list

present in the first sub-chromosome. The customers

are consecutively added to the route, respecting the

ordered list. Once the time limit is reached, the next

route starts to be generated. This method can restrict

the evolution of the solution, and this process could

be improved if it is performed in a parallelized

system, so that all the routes are generated at the

same time, allowing them to achieve better profits.

We concluded that the last reason for failure

could be the lack of another genetic operator in the

algorithm. During the tests with GATOP-1, we

observed the best solution values achieved and also

their graphical representations, and in some cases we

noted optimization deficits in the generated routes.

For example, we can consider a part of a route where

six customers are visited, following the order given

by the blue lines in figure 6. The time (t) between

the vertices is also given. By calculating the total

time of the current route, we get the value 15.6, but

it is possible to lower this to 10.6 simply by

following a different visiting order of customers, for

example, going from vertex 1, then to vertex 3, then

2, and finally to vertex 4. So when rearrangement is

possible and it decreases the total time consumption

of the routes, then the new route should be kept

instead of the initial one, since the fitness or total

reward is maintained (same customers visited). It

can also happen that the time saved by the improved

route is enough to add one or more customers to it,

and consequently increase the total profit to be

collected.

Figure 6: Example situation of an under-optimized route.

Figure 7: Improving an under-optimized route with a

special mutation process.

Our algorithm, GATOP-1, is not able to optimize

situations such as the given example with its current

features. To overcome this problem, we promote the

development of a new and specific mutation

operator that can perform rearrangements to the

routes between their validation (construction of a

feasible solution) and their fitness evaluation. In

order to perform properly, the new mutation

operator should execute a series of swaps between a

group of random yet consecutive genes in a route.

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While executing the swaps, unfeasible solutions

might be produced, but they should be kept and

mutated until a maximum number of swaps are

attained. In figure 7, the special mutation process is

presented, where consecutive simple swaps take

place to optimize the route in figure 6.

5 CONCLUSIONS

In this paper we presented a solution method for the

TOP based on a genetic algorithm (GA). Based on

our results, we can state that our GA is fairly

efficient and attained the best-known solutions in

half of the tested instances.

With these experiments, we were able to improve

our knowledge in the TOP while aiming for the

purposes of the GATOP project. Future research

should focus on the development and testing of other

methods to apply in a GA to solve the TOP.

The overall performance of the developed GA is

fairly acceptable, but there are some small, yet

observable, inconsistencies that lie specifically in the

gap between the highest and the lowest scores we

achieved during the tests. We believe that by

avoiding the suggested sources of error, our results

will improve significantly in our lowest scores. The

usage of dynamic parameters to set the behaviour of

the evolution process within the genetic algorithm

can be interesting idea to explore. That way, the

algorithm would be able to adapt to its own current

performance and try to improve it actively.

ACKNOWLEDGEMENTS

This study was partially supported by the project

GATOP - Genetic Algorithms for Team

Orienteering Problem (Ref PTDC/EME -

GIN/120761/2010), financed by national funds by

FCT / MCTES, and co-funded by the European

Social Development Fund (FEDER) through the

COMPETE – Programa Operacional Fatores de

Competitividade (POFC) Ref FCOMP-01-0124-

FEDER-020609.

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