# Redesigning the Wheel for Systematic Travelling Salesmen
## 3 Investigations
### General Behaviour  
The general behaviour shall be illustrated by using the instance C316K.0. If other instances show different characteristics for certain settings, this will be mentioned explicitly. Candidate selection is restricted to the five closest candidates, the maximum edge distance between points Q and R is set to maxEdgeDist = 100, parallel processing is turned off, and the one-level structure is used. Figure 8 shows the progress of optimization and marks special features.  
The initial tour with a length of 201 871 162 is provided by DoLoWire (30 s). After an initial loop over all points P using 2-opt permutations according to Section 2.4.1, the tour length is reduced to 200 615 308 by 3845 successful permutations. This reduction is possible because DoLoWire does not optimize the initial tour across cluster boundaries. Starting with the second loop, 3-opt operations (see Section 2.4.2) are also enabled. Table 9 represents the entire process in numbers. The time measurements have a precision of one second in all experiments.  
With each loop, the number of successful permutations decreases until a certain threshold is reached. Then the next \(k\)-opt processing is activated and so on. The last loop simply verifies that no further improvement is possible and the processing terminates after  
investigated. It affects the course of optimization and the time required to process each loop over P as can be seen in Figure 9.  
For each parameter set, there are two curves depending on the loop number: one shows the current tour length and the other shows the elapsed time \(t\).  
The magenta curve (maxEdgeDist = 5) has two kinks at \(loop=5\) and \(loop=10\). At these locations, the next \(k\)-opt permutations are activated. At the same loop positions, the time curve also has kinks because enabling more complex permutations increases the time needed for one loop over all P positions. Once the majority of successful permutations is performed, the duration per loop remains nearly constant until the set of enabled permutations is further expanded.  
The smaller the maximum edge distance is chosen, the smaller the improvements per loop are, because some helpful permutations are not checked. At the same time, the duration per loop is shorter, i.e., the algorithm is faster. This leads to a flatter slope of the time curves and the optimizer can perform more loops in the same time. For very low values of maxEdgeDist, the optimization converges to unfavourable local minima. For example, with maxEdgeDist = 5, the algorithm stops after 20 loops, which are processed in three minutes; when using maxEdgeDist = 200, a local minimum is reached after 18 loops and approximately 73 min. Basically, we can see that the time needed to complete a certain loop is a function of maxEdgeDist, which is actually not a big surprise because the maxEdgeDist determines the number of inspected R positions (see Listing 2 line 10).  
The behaviour within the first five loops is always the same because only 2-opt and 3-opt permutations are performed here, which do not depend on maxEdgeDist. Increasing maxEdgeDist from 100 to 200 only has a minor effect on the achieved improvements per loop. For other instances, it can be observed that values higher than maxEdgeDist = 100 rarely benefit the optimization process when using tight time constraints.