*5.2. Analysis of Solution Process*

Moreover, we studied the time needed by Gurobi to find the first feasible solution ( ¯*t f irst*), the time after which the solution did not improve anymore (¯*t last*), the size of the Linear Program (LP)-Gap (*GapLP*), and the size of the final Gap (*Gapf inal*) when the time limit was reached. We calculated the LP-Gap as follows: *GapLP* = (*OFVopt* <sup>−</sup> *OFVrel*)/*OFVrel* , where *OFVopt* indicates the objective function value of the optimal solution and *OFVrel* indicates the LP relaxation solution's objective function value. With this value, we want to

describe the quality of the LP relaxation. A low LP-Gap indicates a good LP relaxation and a high value indicates a bad LP relaxation.

Since we have a time limit of 48 h, optimality cannot be proven for all instances. Therefore, the final gap (*Gapf inal*) indicates the gap between the upper and lower bound at the end of the time limit.

In addition, we performed a second numerical investigation in which the average number of equally optimal solutions (*N*¯ *opt*) was determined for the small instances. We consider solutions as equally optimal if they have the same objective function value as the optimal solution but differ in structure. Gurobi searches for all solutions with the same objective function value as the optimal solution and counts the number. Gurobi stops when all solutions are found or if the total number of equally optimal solutions is higher than 100,000 . Due to the time limit, such a computation is not possible for the instances in the medium and large instance classes. Because of the upper limit, the average value of the equally optimal solutions *N*¯ *opt* is underestimated.

For all described metrics, the average values of 81 instances for each instance graph are shown in Table 6.


**Table 6.** Analysis of solution process.

A first feasible solution is found on average in less than one second in each instance class. This shows that the problem's difficulty does not lie in finding a feasible solution. This is not surprising as a feasible (but expensive) solution can always be constructed by equipping each link with an ITU and then installing just one PSU.

Compared to the total computation time, a solution that is no longer improved is found after a relatively short time, particularly for small and medium-sized instances. Even for the hard-so-solve large instances, in most of the computation time, no improvement of the incumbent solution occurs. Figure 10 shows the upper and lower bound course stemming from the Branch-and-Bound process for one exemplary instance of a large-sized instance of structure A. We observe that a feasible solution is found very quickly. The upper bound, i.e., the objective function value of the best solution found so far, improves very strongly at first, then only very slightly. After about 3 h, the upper bound no longer improves. The lower bound slowly approaches the upper bound in the remaining ten hours until the optimization terminates with the proof of optimality of the Branch-and-Bound procedure used by Gurobi.

**Figure 10.** Solution Process of one Instance.

Table 6 shows that there are many equally optimal solutions for the instances of the small instance class. The high number of equally optimal solutions can be a reason for the long time needed to prove optimality. A large number of equally optimal solutions leads to many nodes to be visited in the Branch-and-Bound process, even if one optimal solution may already have been found. One reason for the high number of equivalent optimal solutions lies in the design of the instances. The links often have the same length and thus the same energy contributions and investments. Hence, many different infrastructure designs lead to the same energy contributions and investments. The structure also highly influences the number of equally optimal solutions and the computation time. If a structure is symmetrically built, many different infrastructures lead to the same energy contributions and investments. This is especially the case for structure C in our study. We observe the highest number of equally optimal solutions for this structure in the small instance class. But above all, we observe the highest computation times in all instance classes.

In addition, and making things worse, symmetric solutions occur, especially when the link length is reduced. Figure 11 shows an example of a symmetric solution. There is an intermediate node between nodes *i*1 and *i*2. Let all links have the same length. In this case, both represented solutions are equivalent because they contribute the same energy to each service request and have the same investment. In any route segment that is not interrupted by an intersection and in which the link lengths are the same, each solution is equivalent in that the sum of the ITUs built in either direction is identical. In the example, the sum of ITUs in each direction equals 1. Consequently, with smaller link lengths, more symmetries are expected since there are significantly more such route sections.

i1 i2 i1 i2

**Figure 11.** Example of a symmetric solution.

As expected, the final gap increases with increasing problem difficulty. Again, we observe that instances of structure C are more difficult to solve than those of the other two structures. Larger instances mean that it is more likely that the time limit will be reached. For this reason, the final gap is larger with increasing problem size.

The mean value of the LP-Gap has similar sizes for all structures (between 40% and 60%). However, the LP-Gap varies extremely between the individual instances. As Table 7 shows, the most influential factor is the energy ratio. The table shows the average values of the LP-gap for all instances together with the respective energy ratio *ei ec* . For an energy ratio of 1.2, the average LP-Gap is only 5%, for a ratio of 3.24 it increases to 26% and for a ratio of 10 even to 115%. For a high energy ratio, it is sufficient to only partially equip links with an ITU, especially if the links are very long. In the solution of the LP-relaxation of the DICP, these links are "utilized" with very small fractionals such as 0.1. Therefore, the LP-Gap is very poor (far away from the ideal of 0%). However, if the energy ratio is low, many ITUs must be built, and partial equipment is not useful. Thus the LP-Gap is smaller, reducing the numerical effort of the Branch-and-Bound process performed by Gurobi. Unfortunately, this numerically attractive situation is not very interesting from a practical point of view. It corresponds to a situation in which ITUs would need to be installed almost everywhere, which is exactly what one would like to avoid in the attempt to find an economically efficient dynamic charging infrastructure.

**Table 7.** Influence of energy ratio on the LP-Gap.

