Analysis

Figure 7.4 shows the number of non-dominated solutions using different values for the angleα. Figure 7.5 shows the path set forα=102°, as the number of α-non-dominated solutions decreased only marginally after this value. After computing the solutions that were non-dominated usingα=102°, we obtained 12 solutions. Assessing the results visually, we concluded that one of the major routes via Lyon was classified as dominated by the others. This approach is, from a computational effort perspective, the one with the least effort, as computing the set requires only matrix operations.

Figure 7.5: Paths forα=102°.

Representative Identification

One approach to narrow down the set of solutions to a set containing interesting solutions is to cluster the initial set and identify the cluster’s medoids. A medoid serves as a representative of a cluster, as the medoid’s sum of distances to all other objects inside the cluster is minimal [SHR96].

Using this methodology, a DM can obtain an overview of the distribution of the solutions in each space. The DM can then make an informed decision that can guide further analysis. For instance, in interactive MOO, the DM is required to lead the optimisation in a direction that is most promising or interesting, in their expert opinion. It should be noted that the obtained clusterisation is not reflected in the respective other space and that solutions in a cluster in one space may not be in the same cluster in the other space.

Figure 7.6 shows the obtained medoids following the decision space clustering.

It is clearly visible – and was expected – they are well distributed on the map.

However, as evident in the lower image of the figure, the distribution in the objective space is suboptimal, not covering a wide range of it. A DM may gain a false impression by analysing only the path’s expression on the map.

If we take the clusterisation obtained in the objective space, the medoid com-putation results in the configuration depicted in Figure 7.7. In this case, the pattern is the opposite, namely, a well distributed set in the objective space, but close solutions on the map of Europe. Therefore, we conclude that a DM must not only take the information of one space into account but should always be presented with information about both spaces.

Robust and alternative routes

As shown in Section 7.3.1, the next step was to perform various cluster analysis techniques to find sets of routes that were either robust or resulted in alternative routes. Since for both approaches the first step of clustering is the same, we present and discuss the results first and then discuss the specific results for each approach.

Figure 7.6: Medoids obtained from decision space clustering.

In our analysis, we use complete linkage, as this approach tends to result in smaller clusters, whereassingle linkagetends to result in chaining. To obtain the optimal number of clusters in each space, we use the silhoutte coefficient. For the decision space, we obtainedκdec=6 as the optimal number of clusterswhereas we obtainedκob j=3 for the objective space. In the later analysis, we discovered that using this number resulted in very large cluster intersection. Therefore, we manually set it to κob j =7, which yielded better results. It should also be noted that during the cluster analysis, we simplified the routes by again employing the RDPA. We setεRDPA to a value for which the routes were still unique. Figure 7.8 illustrates the obtained clusters in the decision space, i.e., the map of Europe. Each cluster is assigned with a colour. In the lower figure, the corresponding objective values are shown. It is evident that the decision space clustering resulted in a fragmented objective space dataset, i.e., the clusterisation could not be transferred to the other space.

When performing the same analysis in the objective space, the results differed.

In Figure 7.9, on the upper panel, the paths are shown on the map clustered according to the objective space. The lower image again shows the same paths in the objective space, with the same colour coding.

Figure 7.7: Medoids obtained from objective space clustering.

After computing the clusters in both spaces, we applied the proposed techniques for route identification.

Robust Routes To obtain a set of robust paths, we employed the intersection methodology proposed in Section 7.3.1. This resulted in 18 sets of paths with a cardinality of 2 or higher. Other intersections resulting in 1 or zero paths were filtered out, as such sets do not provide evasion routes. In Figure 7.10, three such sets are shown. The lower panel of the figure shows the three clusters in the objective space. Here, the three sets are different in each objective; however, the red and green sets are close in the actual expression as a path on the map.

On the map, the difference is barely visible.

The intersection set with the largest cardinality is depicted in Figure 7.11. The paths are largely overlapping when analysed visually. However, as illustrated in the lower panel, the difference is evident in several single spots. For instance, in the area of Vichy, France, the routes do not overlap and result in unique expressions.

As a result, a potential driver may choose one of these clusters. The driver can follow any path of the cluster, knowing that it will result in similar objective

Figure 7.8: Clustered paths according to decision space clustering.

Length Ascent Time Smoothness

Objective Values clustered according to Decision Space

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 ²

1 3 4 5 6 Cluster

values. Nevertheless, the driver has more freedom in the actual traversal. Using such a set means having the option taking a slightly different route, while not deviating too much from the original route.

Alternative Routes For the alternative route identification, we applied the methodology proposed in Section 7.3.1. However, as this is a generalised approach, we introduced several additional process steps to narrow down the final solution set. After identifying the cluster intersections, we obtained sets of solutions that were contained in a cluster in both spaces. A starting route served as the baseline to identify alternative routes. We applied cone-dominance to the set and identified the solution that remained when usingα=180°; this was the starting route. Then, we obtainednroutes with the smallest distance according to the objective values in the corresponding objective cluster from which the baseline solution originated. In our case study, we chose n=2, resulting in three routes (baseline+n).

Figure 7.12 shows two examples. Each route is assigned a different colour. The routes have different expressions but similar objective values.

Figure 7.9: Clustered paths according to objective space clustering.

Length Ascent Time Smoothness

Objective Values clustered according to Objective Space

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

3 5 6 2 4 7 Cluster

In Figure 7.13, the corresponding objective values are visible in the context of all other solutions. The figure also shows a third set that we do not present in the decision space. Each colour refers to a set of routes. Route set 1 refers to the upper image in Figure 7.12, whereas set 2 refers to the bottom image. Set 3 is a set that is not shown in the decision space, and set 4 refers to all other routes in the solution set. It is evident that each route is close to the others within the set.

From a visual analysis it is apparent that several parts of the objective space are not covered by the set, because we chose to display only three sets. However, another problem arises here. It is not trivial which sets should be displayed to a DM. Interactive optimisation can be beneficial for such problems. It should be noted that the colours we used to represent the different paths on the maps (Figure 7.12) are independent of those used in the objective space diagram (Figure 7.13). On the map, the colours are used to distinguish the individual paths.

Sets of alternative routes are an example ofDPs. It is evident that the respective routes share several points and differ greatly in other areas. The points where routes diverge can be seen as DPs. A driver has to decide which expression, or route, to take. In our case, the alternatives overlap again at a later point. With

Figure 7.10: Three obtained robust clusters.

Length Ascent Time Smoothness

3 robust clusters

2.875e+06 2.88e+06 2.885e+06 2.89e+06 2.895e+06 2.9e+06 2.905e+06 2.91e+06

17200 17300 17400 17500 17600 17700 17800 17900 18000 18100

9.7e+07 9.75e+07 9.8e+07 9.85e+07 9.9e+07 9.95e+07

342000 344000 346000 348000 350000 352000 354000 356000 358000

1 2 3 Cluster

expert knowledge, an informed decision can be made when executing a certain route.