In this chapter, we have proposed methodologies that can help a DM in se-lecting a route that accommodates specific requirements, based on the user’s preferences. We have explored all of our approaches within the context of both objective space and decision space. This is important, as the solutions to these spaces can provide conflicting information. Our discussed approaches result in
Figure 7.12: Two examples of sets of alternative routes. Each set contains three routes.
different route suggestions. However, evaluating them is not trivial and must be performed in the future, in an in-depth analysis that considers the actual requirements of real DMs.
Figure 7.13: Found sets in the context of the complete solution set.
Length Ascent Time Smoothness
3 sets of alternative routes
2.88e+06 2.9e+06 2.92e+06 2.94e+06 2.96e+06 2.98e+06 3e+06
7000 8000 9000 10000 11000 12000 13000 14000 15000 16000 17000 18000
9.7e+07 9.8e+07 9.9e+07 1e+08 1.01e+08 1.02e+08 1.03e+08
345000 350000 355000 360000 365000 1
2 3 4 Route Set
8 Conclusion and Future Work
In this chapter, we conclude this thesis and present a framework that functions as a decision support for arbitrary pathfinding problems. We then give an overview of topics for future research in the field of multi- and many-objective pathfinding.
8.1 Conclusion
In this thesis, we have proposed the graph-based many-objective pathfinding problem, which can be used to model real existing problems. First, we analysed the related work; second, we proposed a benchmark suite that can generate various problem instances. Third, we have illustrated various problem represen-tations that EAs can handle and have analysed their differences in performance, quality and other metrics. In another chapter, we proposed several techniques to diversify the population in the decision and objective spaces, since the many-objective pathfinding problem can be deceptive. Additionally, we have proposed various techniques to support DMs by decreasing the number of presented solu-tions using methodologies from clustering. In the following secsolu-tions, we discuss the research questions that were formulated at the start of this study:
RQ 1 Which techniques exist to solve the MaOPF?
RQ 1.1 Which environmental classes are used and how do they differ?
RQ 1.2 Which state-of-the-art algorithms are used in the respective envi-ronments?
RQ 1.3 Can single-objective speed-up techniques be used to support a multi-objective approach?
In Chapter 3, we analysed the related work covering several techniques and approaches to various pathfinding problems with a range of environments. We pointed out that exact approaches, due to the NP-hardness of the problem, are usually applied on relatively small environments with few objective functions.
However, there are interesting speed-up techniques for single-objective pathfind-ing problems and their exact solution approaches. For instance,contraction hierarchiesis a technique for precomputing shortcuts upfront to greatly decrease
the query time. Such approaches can be also used during the meta-heuristic MOO in several phases, as we have shown. In the state of the art, various environments have been proposed which consist of different properties and, sometimes, non-linear constraints. Often, the algorithms were tailored to a specific environment and are not generally applicable in other settings. In the ar-ticles we reviewed , other approaches – such as particle swarm optimisation, ant colony optimisation, EAs and q-learning – have been proposed and evaluated.
RQ 2 Is there a significant difference between using problem and solution tailored representations rather than standard encodings?
In Chapter 5, we assessed different encoding schemes for the multi-objective pathfinding problem. We evaluated fixed-length and variable-length represen-tation schemes using our proposed benchmark suite. On smaller problem in-stances, the fixed-length approach was superior, but its performance deteriorated as the instances became larger. By contrast, the variable-length approach main-tained a certain quality level. From this experiment, we concluded that in most cases, a more natural encoding, i.e., variable-length, is suitable for such prob-lems. Considering only fixed-length approaches, we evaluated real-valued and binary encodings. We found that the latter outperformed the former.
The larger the instances, the more problems and challenges occurred in any encoding scheme that could be tackled using techniques from large-scale op-timisation. As the true Pareto-front is unknown for such large problems, it is difficult to assess the absolute performance of an algorithm. Although the results seem reasonably sound, they might be far from the global optimum. A way to circumvent this restriction is to compute the true fronts and sets using an exact approach, which would involve a large computational budget.
RQ 3 How should a scalable and variable benchmark test problem be designed to cover a wide variety of pathfinding problems?
RQ 3.1 Which real-world related objectives should be considered in the test problems?
In Chapter 4, we proposed a benchmark suite that enables researchers to cre-ate different pathfinding problem environments. Furthermore, we presented objective functions that can be used with such environments. The suite’s en-vironments can be populated with different constraining properties, such as obstacles in different variations, neighbourhood relations or elevation profiles.
These characteristics can also be found in other related works and resemble real-world pathfinding problems to a certain extent. The properties and settings can also be mapped to pathfinding problems from other domains. Our proposed objective functions are a suggestion. Nevertheless, other objective functions can be used with the same environment.
We also analysed several large maps of the benchmark and found that the optimal values of theascentobjective converged towards 0. The reason was that with larger instances more possibilities arise containing almost flat neighbours.
While this finding may seem interesting from an optimisation approach, it impacts actual implementations of the benchmark. Such small floating-point values can be an issue, depending on the programming language, and may result in inaccuracies. Comparing different algorithms can be difficult in light of this issue. Solutions are either to accept this inaccuracy or to scale the value to an
integer by multiplication with a factor. Other approaches include specialised techniques to cope with such small numbers.
We have shown how a variable and scalable benchmark can be defined and have assessed the environments as well as the objective functions. Furthermore, we applied them on real-world data.
RQ 4 How can the geometrical properties of a path be assessed?
RQ 4.1 How can differences from other paths be measured?
RQ 5 How can these properties be exploited for the optimisation process?
RQ 6 Can such properties be used to increase the diversity of the resulting solution set?
In one of our studies, we explained that diversification techniques that are ap-plied in the objective space can be detrimental during optimisation. Pathfinding problem often have the characteristic of close solutions in the decision space being far from each other in the objective space, and vice versa. Therefore, emphasising isolated solutions and unexplored objective space areas can lead to a non-diverse set of solutions in the decision space. However, a DM is often interested in such a set. To overcome this issue, in Chapter 6 we proposed utilising path similarity measurements. The geometrical properties of paths are incorporated in the selection process of the EA. Specifically, we measured the similarities (or differences) between paths by computing their respective Fréchet distances. Furthermore, we evaluated alternative measurements, such as the Hausdorff distance or DTW. Each of these possibilities poses specific advantages and disadvantages regarding the accuracy or computational effort.
In a first approach, a stochastic measurement of tendency in a set of paths was computed, and the path with the highest value (the most isolated path) was determined as the selection candidate. As this method needed in a large com-putational budget, we then proposed a technique to determine neighbourhood relations in a path set. We suggest contracting each path to a single represen-tative point and measuring the signed distance to the beeline that connects the common first and last point of each path. Our proposed approaches reduced the computational effort and increased the quality of the solution set. Furthermore, the resulting solution set had an increased diversity, which can be beneficial for a DM’s decision.
RQ 7 What performance indicator (PI) can be used to evaluate the algorithm’s performance?
In this thesis, in most cases we have employed well-known state-of-the-art performance indicators, such as IGD+, IGD, IGDX and hypervolume. The IGD+was used in most studies, as the computation of the hypervolume results in large computation times when using many objectives. However, we are aware that IGD+is only weakly Pareto-compliant. In addition to the standard indicators that are obtained in the objective space, we used IGD’s counterpart in the decision space, i.e., IGDX. As the distance function, we utilised the path similarity measurement that was proposed in Chapter 6, specifically the Fréchet distance. However, a major drawback is that theδF of two paths can be equal to another pair of paths, although they are different from a visual perspective.
Using DTW can solve this issue but can also require a larger computational effort.
RQ 8 How to reduce the number of solutions that are presented to a decision-maker?
In Chapter 7, we proposed several approaches to narrow down a result set of an algorithm to the most interesting solutions that can be presented to a DM. Such techniques are necessary because a many-objective pathfinding problem can result in many non-dominated solutions, which is barely comprehensible for a DM. Again, we employed similarity measurements. We discussed a technique using clustering to determine robust solution sets as well as sets of alternative solutions. To compute such sets, we also incorporated information from the objective space where we had applied clustering techniques. Intersecting the resulting clusters resulted in the respective sets. In addition, we suggested using cone-dominance as an alternative dominance criterion to decrease the cardinality of the solution set. We tested our approaches on real-world data, including determining a set of non-dominated solution paths from Warsaw to Madrid, which is one of the major truck roads in Europe. Four objectives were minimised and only one run was conducted to resemble an optimisation task that may occur in this setting in an actual company. From the result set, we could determine several sets of alternative solutions as well as robust solution sets.
Furthermore, the former were an example of decision points, i.e., points where an alternative solution can be chosen. Finally, we represented each subset using the respective medoid that was computed with the proposed path similarity measurement. These representatives can be presented to a DM.
In summary, we generated techniques and approaches to increase the perfor-mance of EAs applied to the multi-objective pathfinding problem. These method-ologies can be used in other approaches too and are dedicated to a specific aspect of the underlying problem. However, a drawback of this thesis is that it lacks a comparison with existing pathfinding approaches. There were two reasons for this lack. As outlined in Chapter 3, in most cases it was not trivial to reimple-ment the approaches or recreate the environreimple-ments used. Furthermore, comparing approaches across programming languages is not trivial. With the proposed benchmark, we created a baseline problem set with well-defined environment characteristics and objective functions. We want to encourage the research community to use it.
Another perspective that was covered only in part is the constraint handling. In reality, several problems are highly constrained and special techniques must be used. In Chapter 5, we implemented a trivial approach of constraint handling by penalising solutions that were outside the desired domain. Although that is a working approach, more sophisticated methodologies are missing from this thesis. In most of our other studies, we decreased the search space by omitting such nodes that are inaccessible. In reality, this information can be hidden or accessed only if the respective node is traversed. In our studies, neglecting such areas can be classified as preprocessing oroffline constraint handling. Finally, we extracted the feasible search space area with problem knowledge.