Simulation-based Optimization of a Real-world Travelling Salesman Problem Using an Evolutionary Algorithm with a Repair Function

Anna Syberfeldt, Joel Rogström & André Geertsen
International Journal of Artificial Intelligence and Expert Systems (IJAE), Volume (6) : Issue (3) : 2015 27
Simulation-based Optimization of a Real-world Travelling
Salesman Problem Using an Evolutionary Algorithm with a
Repair Function
Anna Syberfeldt anna.syberfeldt@his.se
Engineering Science
University of Skövde
Skövde, SE-54148, Sweden
Joel Rogström joel.rogstrom@his.se
Engineering Science
André Geertsen andre.geertsen@his.se
Engineering Science
Abstract
This paper presents a real-world case study of optimizing waste collection in Sweden. The
problem, involving approximately 17,000 garbage bins served by three bin lorries, is approached
as a travelling salesman problem and solved using simulation-based optimization and an
evolutionary algorithm. To improve the performance of the evolutionary algorithm, it is enhanced
with a repair function that adjusts its genome values so that shorter routes are found more
quickly. The algorithm is tested using two crossover operators, i.e., the order crossover and
heuristic crossover, combined with different mutation rates. The results indicate that the order
crossover is superior to the heuristics crossover, but that the driving force of the search process
is the mutation operator combined with the repair function.
Keywords: Evolutionary Algorithm, Simulation-based Optimization, Travelling Salesman
Problem, Waste Collection, Real-world Case Study.
1. INTRODUCTION
In this paper, we present a study of optimizing waste collection from households in Sweden. The
study was undertaken in collaboration with the AÖS (www.aos.skovde.se) waste management
organization. AÖS is responsible for collecting household waste in seven municipalities in West
Sweden from a total of approximately 120,000 bins. Each bin is emptied once or several times
during a 14-day period according to a predefined schedule. Currently, the bin lorry routes and
their scheduling are specified manually by a transportation planner. Manually producing efficient
routes and schedules is very difficult due to the vast number of bins combined with the many
parameters to be considered. Parameters include driver working hours and breaks, truck capacity
(in terms of both weight and volume), and time windows at offload facilities. Due to the complexity
of producing efficient routes and schedules, AÖS has expressed a need to improve its current
process by introducing an optimization tool, motivating this study.
We approach the waste collection problem as a travelling salesman problem, defined as finding
the shortest possible route visiting each existing node exactly once and finally returning to the
starting node [1]. Finding the shortest route between several nodes might seem simple, but is
classified as NP-hard in its simplest form [2]. With an NP-hard problem, the time needed to solve

the problem grows exponentially with the problem size – in this case, the number of bins. Finding
the optimal route is possible, but might take a very long time because all possible routes must be
evaluated to find the best one. Even small travelling salesman problems involve a huge number
of possible routes. For example, for a problem with only 15 bins to visit, there are 6,227,020,800
possible routes. One can easily understand that manually optimizing the waste collection problem
is virtually impossible, at least within a reasonable timeframe.
Due to the complexity of routing problems, optimization techniques that are not guaranteed to find
the optimal solution, but a sufficiently good one in a short time, are often used rather than exact
methods [3]. Evolutionary algorithms are a class of such inexact optimization techniques usable
to approach routing problems [4], and an evolutionary algorithm is used here. In the study, we
enhance an ordinary evolutionary algorithm with a repair function that considerably improves the
optimization performance. Unlike repair functions previously suggested in the literature, the
suggested repair function does not aim to transform invalid into valid solutions but focuses solely
on improving performance. To ensure valid solutions, a non-destructive crossover operator is
used instead of a destructive one.
To evaluate the solutions generated by an evolutionary algorithm, a mathematical function
representing the travelling salesman problem is usually used. In this study, we instead use a
simulation that models the real-world waste collection scenario to evaluate solutions in order to
ensure that the obtained results are valid in practice. We believe that that the combinatorial
relationships, uncertainty factors, and nonlinearities present in the waste collection scenario –
and in real-world scenarios in general – are far too complex to be effectively modelled analytically
In the next section, the concepts of evolutionary algorithms and optimization using simulations,
so-called simulation-based optimization, are described in detail for the convenience of readers
unfamiliar with them. Section 3 outlines the simulation model developed for the waste collection
scenario considered here. Section 4 describes the evolutionary algorithm used for optimizations
and its repair function. Section 5 presents the optimization experiments and Section 6 finally
outlines the conclusions of the study and presents possibilities for future work.
2. THEORETICAL BACKGROUND
This section presents the fundamentals of evolutionary algorithms (Section 2.1) and simulation-
based optimization (Section 2.2).
2.1. Evolutionary Algorithms
The basic idea of evolutionary optimization is to use computational models of evolutionary
processes in designing and implementing problem solving applications [5]. Based on Darwin’s
“survival of the fittest” concept, candidate solutions to a problem are iteratively refined. Generally,
an evolutionary algorithm consists of a genetic representation of solutions, a population-based
solution approach, an iterative evolutionary process, and a guided random search. The genetic
representation defines an individual solution as a set of genes.
In evolving a population of solutions, evolutionary algorithms apply biologically inspired
operations for selection, crossover, and mutation. The operators are applied in a loop, an iteration
of the loop being called a generation. The following pseudo code presents the basic steps of this
evolutionary process:
Initialize population
Evaluate the fitness of solutions in the population
repeat
Select solutions to reproduce
Form a new generation of the population through crossover and mutation
Evaluate the new solutions
until terminating condition

The solutions in the initial population are usually generated randomly. In each generation, a
proportion of the solutions in the population is selected to breed offspring for the next generation
of the population. Solutions are selected based on their fitness, representing a quantification of
their optimality. Solutions with high fitness typically have a higher probability of being selected,
but to prevent premature convergence, it is common that a small proportion of solutions with
worse fitness is also selected. From the solutions selected, new solutions are created to form the
next generation of the population. To create each new solution, two parent solutions are chosen
and, through mating (called crossover), an offspring is produced (Figure 1).
3 7 5 1
B1 B2 B3 B4
6 2 5 7
B1 B2 B3 B4
Crossover point
3 7
B1 B2
5 7
B3 B4
6 2
B1 B2
5 1
B3 B4
Parents
Children
FIGURE 1: Example of a Crossover.
Occasionally, the new solution can undergo a small mutation in order to maintain high diversity in
the population and avoid local minima. A mutation usually involves changing an arbitrary part of a
solution with a certain probability, for example, slightly changing the size of the buffer in Figure 2.
FIGURE 2: Example of a Mutation.
When the new population is formed, the average fitness will have generally increased. The
process of evolving generations continues until a user-defined termination criterion has been
fulfilled, for example, that the best solutions in the last n generations have not changed or that a
certain time has passed.
2.2.Simulation-based Optimization
In simulation-based optimization, the evolutionary algorithm uses a simulation model to evaluate
the fitness of solutions [6]. The simulation model is a representation of the real-world scenario
that mimics reality as closely as possible, including stochastic events as well. Simulation-based
optimization is an iterative process (Figure 3): the evolutionary algorithm generates a solution (in
this case, a set of routes) and feeds it to the simulation model, which computes the fitness (in this
case, the driving time). Based on the evaluation feedback obtained from the simulation model, the
optimization algorithm generates a new set of parameter values and the generation–evaluation
process continues until a user-defined stopping criterion is satisfied. Such a criterion may, for
example, be that a certain amount of time has passed or that specific objective values have been
achieved.

FIGURE 3: Simulation–optimization Process.
The next section describes the simulation model developed for the waste collection scenario
considered here.
3. SIMULATION MODEL
To ensure that the optimized routes are valid in practice, a simulation is built that mimics the real-
world scenario as closely as possible. The simulation model is based on data from the Swedish
national road database, which holds information about all roads in Sweden. The information is
quite detailed and covers not only the length and speed limit of each road but also, for example,
traffic rules, road width, and whether the road is surfaced with asphalt or gravel. The database
also contains geographical information that makes it possible to render the roads graphically.
Although the Swedish national road database is huge, it is straightforward to use as it follows an
open standard that defines how to read and extract data [7].
The next two subsections outline the basic structure of the simulation model (Section 3.1) and
describe the graph used to navigate the roads in the model (Section 3.2).
3.1. Basic Structure
To represent the roads, the simulation model uses a road network. The road network comprises
two main elements, nodes and links. Nodes are the logical endpoints that connect links to each
other, while a link is a logical representation of a road. A link owns a set of link parts each of
which describes the link at different moments in time. If changes are made to a road, the link is
never changed; instead, the underlying link parts are replaced with newer parts that describe the
link from that moment on. Each link part owns a time interval that describes when this link part is
active.
To connect the nodes, links, and link parts, ports are used. A port is simply a connection between
two elements, and every node owns a set of node ports while every link owns a set of link ports.
A node connects to a link through a connection between one of its node ports and one of the
link’s link ports. A link’s various link parts connect to one another using the link’s link ports.
Furthermore, every link port holds information about its relative distance from the link’s starting
point. Note that a node does not have to connect to the end points of a link but only to a link port.
This concept is illustrated in Figure 4, in which three nodes each connect to the same link. The
concept of a link comprising its link parts is illustrated in Figure 5, in which the greyed-out link part
represents an older part of the road that has been removed. The link still represents the same
road, but its underlying representation has been changed. The figure also illustrates how the link
parts are connected through link ports.

FIGURE 4: Three nodes connecting to the same link using the port system.
FIGURE 5: A link and its underlying link parts.
3.2. Navigation Graph
To navigate the road network, a navigation graph is used in the simulator. This navigation graph
is a directed logical graph in which a set of nodes is connected to other nodes through edges. In
implementation, an edge can only be connected to two nodes. The edge’s beginning and ending
nodes denote the direction of the edge. As each link is connected to a node through a port, it is
straightforward to create edges between each port using the information contained in the link
part. In other words, each link part is converted to an edge between two nodes in the navigation
graph. This is illustrated in Figure 6, in which nodes A, B, and C are connected using edges
instead of links and link parts. A directed graph is used because not all nodes are connected to
each other in both directions. For instance, some roads allow traffic in only one direction. The
direction of travel is also important as some bin lorries can only empty receptacles on the right-
hand side of the road.
FIGURE 6: The same graph elements as in Figure 4, but in the navigation graph format.
An edge’s identity is defined by its beginning and ending nodes, its underlying link, and the port
ID used when connecting to its nodes. Using this approach makes it possible to separate two
edges that connect the same nodes, as they are constructed from different links and ports. It also
makes it possible to differentiate loops in different directions, as each loop will connect to different
beginning and ending ports. Examples of both scenarios are illustrated in Figure 7, in which
nodes A and B are connected through two different edges, and node A is connected to itself
through a loop. As every node is connected to a link through a port, and each port stores its
relative distance from the link’s beginning and end, it is possible to calculate the length of each
edge using the relative distance and the link’s entire length.

FIGURE 7: Examples of edge cases in the navigation graph.
The next section describes the evolutionary algorithm that uses the simulation model to evaluate
generated routes.
4. EVOLUTIONARY ALGORITHM
This section describes the configuration of the evolutionary algorithm implemented to solve the
optimization problem: first the genetic representation (Section 4.1), then the evolutionary
operators (Sections 4.2–4.4), and finally the repair function (Section 4.5) is described.
4.1. Genome
To represent solutions, a combinatorial genome is used. Combinatorial genomes are commonly
used when dealing with travelling salesman problems as they allow a simple, yet powerful,
genetic representation of the problems [8]. In the implemented genome, each gene represents an
edge to be visited by the bin lorry. All edges are visited in the order in which they appear in the
genome, making the genome a direct representation of a route. This implies that all genes of the
genome must be unique, and also that the crossover and mutation operators must never replace
or remove any genes.
As the route has specific beginning and ending points, they must be represented in the genome
and taken into account by the crossover and mutation operators. When building the genome, the
first entry is therefore always the starting point and the last entry is always the ending point. To
preserve these points, both the crossover and mutation operators operate on a subset of the
genome, ignoring the first and last entries.
4.2. Selection Operator
As the selection operator, binary tournament selection [9] is chosen. In binary tournament
selection, two lists are first created, each containing references to all individuals in the population.
Both lists are then shuffled, in the present case using the Fisher–Yates shuffle method [10]. By
comparing each element at the same index in both lists and choosing the best element, we select
individuals for recombination. This selection operator is used because it is fairly simple, fast and
easy to implement, and fair. The best solution in a population is guaranteed to be selected at
least once and at most twice.
4.3. Crossover Operator
Two crossover operators are implemented and tested for the problem: the ordered crossover [11]
and heuristic crossover [8]. Both operators are specifically designed for combinatorial problems
and therefore only change the order of the genes, never removing values or introducing new ones
into the genome. The ordered crossover is an older operator and is fairly simple. It is fast and
efficient, but it operates in a black box manner and uses no problem-specific information. The
heuristic crossover, in contrast, is more complex and uses problem-specific knowledge to guide
the crossover towards better results.
The order crossover works as follows: Two crossover points are randomly chosen in the parents.
The genes between the points in the first parent are copied into the second child, and the genes
from the second parent are copied into the first child. The remaining genes not covered by the
crossover are copied from the first parent into the first child in the order in which they appear. The
same is done for the second parent and second child. An example of an order crossover is given
in Figure 8.

Parent1 1 2 3 4 5 6
Parent2 4 6 5 2 3 1
Child1 1 6 5 2 3 4
Child2 6 2 3 4 5 1
FIGURE 8: Example of an order crossover.
The heuristic crossover (also known as the greedy crossover) uses a white box approach
employing problem knowledge. The crossover includes the following steps: First, a random gene
is chosen as the starting point for the crossover. The selected neighbouring genes in both parents
are then identified, and thereafter any neighbours already present in the child are sorted out. If
one or more neighbours remain, a heuristic is used to estimate the cost of travel between the
selected gene and the remaining neighbours. The lowest-cost neighbour is added to the child’s
genome and is set as the next selected gene. If all neighbours of the selected gene exist in the
child, one unvisited gene is randomly chosen as the next selected gene. The procedure is
repeated until the genome has been filled and the child is complete. This complete process is
repeated twice to produce two children.
4.4. Mutation Operator
As the mutation operator, the displacement operator [12] is used. This operator is well suited for
combinatorial problems as it manipulates only the order of elements in the genome. The
displacement operator works by randomly generating two offset points in the genome and then
shifting all the elements between the two points in a randomly selected number of steps. The
operator is demonstrated in Figure 9.
Displacement 1 2 3 4 5 6
1 5 6 2 3 4
FIGURE 9: The displacement operator demonstrated.
4.5. Repair Function
A pattern emerged when implementing and testing the evolutionary algorithm. Initially, the routes
resulting from the optimization often bypassed unvisited route stops on the way to other stops.
This led to various weaving patterns in which a route would bypass an unvisited stop several
times before it was finally visited (i.e., the bin was emptied). As it is not rational to drive past an
unvisited stop, a constraint was considered specifying that the sum of bypassed unvisited stops
must be zero. However, the act of detecting passed stops lends itself to fixing the problem, as we
know that travelling between stops can entail bypassing at least one unvisited stop. Instead of
adding a constraint, a solution was found in which the genome was reordered so that any
unvisited stop bypassed when travelling between two stops would be placed between these
stops. This approach was demonstrated to improve the performance of the optimization
significantly.
The repair function is illustrated in Figure 10. In the example, the genome is in form [1, 5, 4, 6, 8],
each number representing an edge in the graph. Asking the simulator for the shortest path
between the first and second genes would return the following path: [1, 3, 4, 5]. By iterating
through the path, we can find any unvisited stops bypassed. In the example given, the path
bypasses edge number four. To fix this, we can rearrange the genome in such a way that we visit
that edge before visiting edge number five. The genome after the fix would have the form [1, 4, 5,
6, 8] and the resulting route would be considerably shorter.

FIGURE 10: Graph illustrating the repair function.
It can be mentioned that repair functions have previously been described in several different
studies in the literature (see for example [13-17]). The difference between the existing functions
and the one suggested in this study is that the existing ones aim to transform invalid solutions into
valid ones, while the suggested one aims to improve the general performance of the algorithm.
The previously described repair functions are mainly used to compensate for a destructive
crossover, while in this study a non-destructive crossover is used and the problem of invalid
solutions is thereby eliminated.
The next section describes the results of optimizations using the evolutionary algorithm.
5. OPTIMIZATION
This section first describes the scope of the problem being optimized (Section 5.1), then presents
the exact algorithm configuration (Section 5.2), and finally outlines the optimization results
(Section 5.3).
5.1. Problem Scope
To test the optimization, one of the seven municipalities served by AÖS is selected. The selected
municipality includes approximately 17,000 unique bins spread over an area of approximately 40
× 50 kilometres. For the reader to grasp the problem, the geographical locations of the bins are
plotted on maps shown in Figures 11 and 12 (the latter zooms in on part of the former).

FIGURE 11: Garbage bins included in the optimization.
FIGURE 12: Zoomed-in view of the area corresponding to the black square in Figure 11.

Three trucks are available for emptying the selected bins, and each of them empties on average
approximately 550 bins per day. This means that a unique route should be created for each truck
and each workday during a 14-day period (excluding Saturdays and Sundays). If successful, the
optimization might free one or several trucks for one or several days, which would save the waste
management organization considerable money.
5.2. Algorithm Setup
The choice of parameters of an evolutionary algorithm has a distinct impact on the performance
of the algorithm and thereby also on the final results [18, 19]. No single set of parameter values is
universally optimal; rather, the parameters must be configured for each individual problem. In the
study we therefore evaluate various settings of the crossover and mutation operators, as
presented in the next subsection. In all presented experiments, the population size is set to 100
and the total number of iterations of the evolutionary algorithm is set to 10,000.
5.3. Results
The results of the experiments are presented in Table 1. Each combination of parameter values is
tested ten times, the presented results representing an average of these ten runs. As can be
seen in the table, the order crossover is superior to the heuristic crossover. It is also clear that the
higher the mutation rate, the better the results. The reasons for and implications of these results
are further discussed in Section 6. Due to lack of existing repair functions with equivalent focus, it
has not been possible to perform benchmarking against any other solution previously presented
in the literature. As mentioned in Chapter 4.5, existing repair functions have a completely different
purpose than the suggested one and a performance comparison is therefore not meaningful.

Crossover
Mutation
rate
Average minimum
fitness
Average minimum iterations
before local optimum is found
Order 1 2917.1 8920
Order 0.8 2938 8390
Order 0.9 2943.6 8460
Order 0.6 2958.1 9170
Order 0.7 2962.6 8490
Order 0.5 2977.8 9290
Order 0.4 3007.7 9380
Order 0.3 3034.4 8910
Order 0.2 3050.1 8930
Order 0.1 3160.8 9280
Heuristic 0.5 3163.2 6890
Heuristic 0.6 3167.5 3110
Heuristic 0.9 3168.6 5110
Heuristic 0.8 3182 5600
Heuristic 0.4 3194.5 4330
Heuristic 1 3211.5 5280
Heuristic 0.3 3218.7 1990
Heuristic 0.7 3223.1 6540
Heuristic 0.2 3230.6 2260
Heuristic 0.1 3249.6 3940
Heuristic 0 3260.4 610
Order 0 3885.9 100
TABLE 1: Results from experiments (ordered by average minimum fitness).
The routes resulting from the most successful optimization runs were presented to domain
experts in the waste management organization, who carefully studied and evaluated them. The
domain experts rated the improvement at about 25% when comparing the optimized routes with
the routes created manually by the transportation planner, who can instead concentrate on other
work tasks.
6. CONCLUSIONS AND FUTURE WORK
This study presented the optimization of waste collection, considering a problem including
approximately 17,000 garbage bins. The problem was approached using an evolutionary
algorithm and simulation-based optimization. To improve the performance of the evolutionary
algorithm, it was enhanced with a repair function that adjusts genome values so that shorter
routes are found more quickly. The algorithm was tested with two crossover operators, the order
crossover and heuristic crossover, combined with different mutation rates.
The results of the optimization experiments clearly indicate that the order crossover produces
much better results than does the heuristic crossover, at least as long as the mutation rate is
above zero. This is likely because the heuristic crossover does not introduce into the genome
changes as radical as does the order crossover. The order crossover, however, only works well
when the repair function is used, as the operator introduces many changes into the genome.
These changes might have negative effects, but these are immediately eliminated by the repair

function. The repair function is therefore of great importance and significantly influences the
results. It can be noted that experiments with the repair function indicate that its importance
increases with the size of the routes, i.e., the more points to be visited, the more value it
produces.
Another interesting finding from the optimization experiments is that when the mutation rate is set
to zero and the order crossover is used, the search stagnates after only 100 iterations. This
indicates that the mutation rate has a much greater impact on the search than does the
crossover, likely because the repair algorithm overwrites many of the changes made by the
crossover. The repair algorithm tends to group together genes that are near each other, and as
the mutation operator shifts groups of genes around in the genome, these genes can be assumed
to work well together.
It is concluded from the study that the combination of mutation operator and repair function is the
most important factor for rapid search progress and successful optimization results. In the future,
it would be interesting to test additional crossover operators combined with the repair function to
see whether the optimization results can be improved even further.
It would also be interesting to consider a dynamic rather than static approach to garbage
collection. A problem is static if all transport requests are known when the routes are constructed
[20], which was the case here. In a situation in which transport requests may be added while
trucks are running the routes, meaning that routes have to be changed “on the fly”, the problem is
instead dynamic. In reality, the problem is already dynamic as customers often overfill their bins
and need immediate emptying outside the ordinary schedule. Dynamic problems are considerably
more difficult to tackle as the data change constantly and the optimizations must be performed in
real time, putting additional demands on the algorithm used.
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