Figure 4.2: Objectives (1) and (5) on an example path modelled by a graph [WM22b]
S
E
Objective 3: Elevation.The aggregated ascent of a solution path is represented by the third objective. Our proposed benchmark contains various possibilities for defining the elevation functionh(ni)which is defined on a nodeni. The ascent is calculated between two nodes in the graphe(ni,ni+1). The third objective
f3(p)is the sum of the elevations between all the nodes in a pathp:
f3(p) =
K−1
∑
i=1
e(ni,ni+1)
e(m,n) =
(h(n)−h(m), ifh(n)>h(m)
0, otherwise
(4.3)
This objective accounts to the amount of fuel consumption in a real-world application [WM22b].
Objective 4: Travelling time.The fourth objective represents the travelling time. For this calculation, we utilise the average velocity of two subsequent nodes defined by vmax(ni)+v2max(ni+1) for each nodeni and use the length of the path utilised in Objective 1 [WM22b]:
f4(p) =
K−1
∑
i=1
2d(ni,ni+1)
vmax(ni) +vmax(ni+1) (4.4) Objective 5: Smoothness. The smoothness or curvature of a path is mod-elled in the fifth objective. We measure smoothness by calculating the angle between three nodes on a path, as shown in Figure 4.2. The angle θ is ob-tained by extending the line between two nodes and measuring the angle to the third node. Similar to [ORK14, JQ10, HPVRFP15, DPMH15], we invert a·b=∥a∥∥b∥cos(θ):
f5(p) =
K−1
∑
i=2arccos
−−−→nini−1· −−−→ni+1ni
|−−−→nini−1| · |−−−→ni+1ni|
(4.5) Because we intend to minimise the objective values, the smaller smoothness value represents a straighter path [WM22b].
Figure 4.3: Three types of cells [WM22b]
2 4 6 8 10 12 14 16 18
X-Coordinate 2
4
6
8
10
12
14
16
18
Y-Coordinate
Figure 4.4: Examples of grid cell properties (dark to light colours represent high to low speed values) [WM22b]
(a)lakeobstacle
2 4 6 8 10 12 14 16 18
X-Coordinate 2
4
6
8
10
12
14
16
18
Y-Coordinate
(b)checkboard-obstacles
2 4 6 8 10 12 14 16 18
X-Coordinate 2
4
6
8
10
12
14
16
18
Y-Coordinate
designed to simulate block-like environments, and 2) the lake obstacle denotes a larges region which is not passable (Figure 4.4). For the checkerboard obstacles, we define every second cell to be an obstacle in bothxandydirections. The lake obstacle is defined as a circle on the grid. The circle radius is defined by a fraction of the x-size of the grid. We represent the checkerboard and lake obstacles as a variant of the square wave function and circle function, respectively (see Equations (4.8a) and (4.10)). In the tested instances, the lake obstacles are defined by a radius ofxmax/4. Figures 4.4a and 4.4b show the two obstacle types on an example instance of the benchmark problem. Figure 4.3 shows an example instance of size 19 [WM22b].
The corresponding equations are provided in Section 4.3.1.
For the elevation, we take four hill functions in the domain[−3,3], which will be scaled when applied to the grid with cell coordinates(x,y)represented by the node n in the path segment. For determining the corresponding height value h(x,y), the two cell coordinates must be scaled to the interval [−3,3], hence[{1,1},{xmax+1,ymax+1}]→[−3,3]and(x,y)→(xs,ys),{xs,ys∈R|−
3≤xs,ys≤3}. In the equation, we refer to (xs,ys) to represent the scaled coordinates:
hm(xs,ys) =3(1−xs)2e−x2s−(ys+1)2−10e−x2s−y2s (−x3s+xs/5−y5s)−1/3e−(xs+1)2−y2s h1(xs,ys) =5e−(xs+1.5)2−(ys+1.5)2
h2(xs,ys) =5e−(xs−1.5)2−(ys−1.5)2 h3(xs,ys) =5e−(xs−1.5)2−(ys+1.5)2
(4.6)
Figure 4.5: Different elevation profiles of the proposed benchmark [WM22b]
(a) Elevation profile fornh=M
-8 -6
3 3
-4 -2 0
2 2
2
Height
4 6
1 1
8 10
Y-Coordinate X-Coordinate
0 0
-1
-1 -2 -2
-3 -3
(b) Elevation profile fornh=1
0
3 3
2 2
1 1
Y-Coordinate X-Coordinate
0 0
-1 -1
2
-2 -2
-3 -3
Height
4 6
(c) Elevation profile fornh=2
0
3 3
2 2
1 1
Y-Coordinate X-Coordinate
0 0
-1 -1
2
-2 -2
-3 -3
Height
4 6
(d) Elevation profile fornh=3
0
3 3
2 2
1 1
Y-Coordinate X-Coordinate
0 0
-1 -1
2
-2 -2
-3 -3
Height
4 6
We chose these functions to represent different height settings on the grid.
Equationhm, also known as thepeaksfunction, has various hills and valleys.
Since this function is defined in the interval of[−3,3], we define the other three functions in the same interval. Each of the three other equations represents a hill on the landscape. In Figure 4.5, the linear combinations of the functions are depicted. Combining them yields various elevation characteristics of the problem instances. Finally, an instance can havehmora linear combination of the others as its elevation function. Therefore, we definehas:
h(x,y) =
∑nhi=1hi, ifnh∈ {2,3}
h3, ifnh=1 hm, ifnh=M
(4.7)
For the third objective, we aggregate positive slopes, as we want to focus on flat routes. Taking negative elevations into account too can result in a path containing a hill with a steep gradient, which is not beneficial for a bulky transportation. The fourth objective, expected delay, is defined byvmaxof two subsequent cells (see evaluation section Equations (4.11) and (4.12).)
All these variations of the properties are used in the name of a benchmark instance. The name starts with ASLETISMAC for the five objectives to be minimised: ascent, length, time, smoothness and accidents (expected delay).
Thereafter, the name includes the obstacle type, followed by the size (in X and Y directions), the elevation function (PM stands for the peaks function hmand the combination is set to Pnh), followed by the 2k-neighbourhood and finally the backtracking (B) property (T for true or F for false). For example, ASLETISMAC_CH_X10_Y10_P1_K2_BFdefines an instance with the checker-board obstacles, sized 10x10,nh=1 as the elevation function (one hill), four possible neighbours (K2), but no backtracking (BF) [WM22b]. For the values of delays (caused by accidents) in the second objective, we refer to real-world
Table 4.2: Integer sequences of possible number of paths from the north-western corner to the south-eastern corner one with no obstacles on oeis.org for the size of nxn [WM22b].
Benchmark type Integer sequence
K3,BF A001850
K3,BT A140518
K2,BF A000984
K2,BT A007764
statistical data (see Equation (4.2))1. We adopt the likelihood of encountering an accident from real-world data, depending on thevmaxof a certain cell. For instance, one is far more likely to have an accident when driving on streets located in a city, i.e., with a lowervmax, than on highways or country roads.
Therefore, we assume a smaller likelihood of encountering an accident with higher velocities. We also assume a large likelihood when the type of street changes, such as travelling on an access road or an exit road [WM22b].
4.2.1 Obtaining the True Pareto-front
We performed an exhaustive search on 272 benchmark instances with different obstacle types, sizes, elevation functions and neighbourhood metrics. To obtain the fronts, we performed a depth-first search (DFS) from the cell at the northern-west corner to the south-east corner cell. The larger the instances, the longer the DFS takes to complete. The most complex in terms of the number of possible paths we evaluated was the instanceASLETISMAC_NO_X14_Y14_PX_K3_BF, which had a size of 14x14, 4-neighbourhood and no backtracking. For this instance, there were 1 409 933 619 possible paths.
The number of possible paths is represented by specific integer sequences, obtainable at oeis.org. The numbers are shown in Table 4.2.
4.2.2 Benchmark Characteristics
The proposed benchmark has several specific characteristics. Regarding the decision space, we can define a fixed- or variable-length encoding of solutions.
Fixed encodings are suggested especially for theK2,BFinstances, as the allowed paths have the same length f1(p), i.e., f1(p) = ((xmax−1) + (ymax−1)). Using a variable-length approach can represent the problem as a combinatorial one.
For this purpose, one can use graph, real-value or integer-value representations.
In this case, the true Pareto-fronts of the test instances are disconnected and degenerate due to the discrete search space. In addition, the fronts are irregular and the different objectives have different scales. An interesting characteristic of this benchmark is that similar paths on the grid are not necessarily close in the objective space, implying that paths which differ in most of their nodes can lead to similar objective values. In theK2,BF instances, the challenges for algorithms depend on the chosen representation to find a feasible path, as the ratio of infeasible to feasible solutions is relatively high. In Section 4.3.4 and Appendix A.1.2, Figures 4.10 and A.2 to A.5 show several examples of obtained true Pareto-sets and fronts as well as algorithmic results [WM22b].
1. https://www.destatis.de/EN/Themes/Society-Environment/Traffic-Accidents/
_node.html