6.1. Comparison of Simulation Experiment Results
In order to clearly express the plan adjustment process of the TAS for trucks that cannot arrive at the port on time,
Table 1 shows the best appointment plan of a truck company and the actual arrival situation. Experiment 1 shows that (1,4,6,8) is the best booking time window of the truck, and the actual arrival situation is that the truck cannot reach the port at the first time window on time. Before the actual arrival of the truck, the TAS will change the appointment plan into (2,4,6,8). Experiment 2 is an appointment plan for two trucks. The best appointment plan for the first truck is (2,3). The actual arrival situation is that the first truck arrives at the fourth time window instead of the third time window. The best appointment plan for the second truck is (2,6,7), and the actual arrival situation is that the second truck arrive at the fourth time window instead of the second time window. The TAS changes the booking plans of the two trucks to (2,4) and (5,6,7). The fourth time windows will be allocated to the first truck, due to the first-come-first-served rule and the 1 quota of the fourth time window. Then, the second truck can only be reserved at the fifth time window. Experiment 3 is also an appointment plan for two trucks. The first truck arrives in port on time, and the second truck is late for two time windows, so the TAS delay the task of the second truck at the first time window by two later. Thus, we get the final appointment plan.
Nowadays, for trucks that cannot arrive at the port on time, the TAS will follow the “first come first” rule to arrange the trucks to enter the port after the truck arrives at the port. In order to verify the effectiveness of the redistribution model of the truck appointment scheme in this study in which trucks cannot arrive at port on time. We solve the small, medium, and large-scale problems, respectively, and compared the results with the traditional scheduling results in literature [
18]. The results are compared, as shown in
Figure 3.
The experimental parameter data are shown in the
Appendix A. For small-scale problems, the scale of goods is (4 Jobs 8 Jobs 11 Jobs 15 Jobs 18 Jobs 22 Jobs 26 Jobs 32 Jobs 36 Jobs 37 Jobs 50 Jobs 55 Jobs). The specific comparison results are shown in
Figure 3a. It can be seen from
Figure 3a that the total cost of the external truck dynamic appointment rescheduling model under uncertain arrival time is lower than that of the traditional scheduling model, which shows that the proposed model can be more perfect. Although the cost difference of individual cargo scale (for example, 8 Jobs, 15 Jobs, 26 Jobs) is small, it is mainly since the small-scale truck scheduling is relatively single, and the number of trucks that do not arrive at the port on time is relatively small, making the overall cost gap is relatively small. But for the small-scale problem, the proposed model is better than the traditional scheduling model.
For the medium-scale problem, the scale of goods is (257 Jobs 324 Jobs 417 Jobs 492 Jobs 569 Jobs 602 Jobs 663 Jobs), and the specific comparison results are shown in
Figure 3b. When the size of the goods becomes larger, the number of trucks arriving at the port on time will increase correspondingly, and the cost of re-scheduling will also increase correspondingly. It can be clearly seen from
Figure 3b that the target cost of the dynamic appointment rescheduling model for the uncertain arrival time is significantly lower than that of the traditional scheduling model. This shows that when the scale of the problem increases, the rescheduling model proposed in this study will more reasonably schedule the trucks that do not arrive at the port on time, which is more in line with the requirement of the trucks arriving at the port evenly.
For large-scale problems, the scale of goods is (701 Jobs 754 Jobs 827 Jobs 1032 Jobs 1782 Jobs 2438 Jobs), and the specific comparison results are shown in
Figure 3c. From the comparison results of large-scale problems, it can be clearly seen that the cost of the dynamic appointment rescheduling model for the next set of trucks with uncertain arrival time is significantly lower than the traditional scheduling cost. Especially when the quantity of goods reaches 1500 Jobs, scheduling costs will increase rapidly, since when the quantity of goods is large enough, the corresponding orders and the number of trucks will also increase. The uncertainty of the arrival time of trucks leads to large-scale adjustments of goods orders. This also shows that the proposed model is not only suitable for small and medium-scale problems, but also for large-scale problems. The dynamic appointment rescheduling model for trucks with uncertain arrival time can also be optimized for scheduling, and while comparing with the traditional scheduling model, the experimental results should be significantly better and more in line with the actual application requirements of the truck company and the port company.
To further validate the performance of the proposed model, experiments are carried out when the truck arrival time follows Poisson distribution. The 95% confidence interval [
19] of the rescheduling cost are plotted in
Figure 3a–c.
It can be seen from the simulation results that the total cost of the truck dynamic appointment rescheduling model under the uncertain arrival time in this study is much lower than the cost of the traditional scheduling model. When the arrival time follows Poisson distribution, the total cost of the proposed model is still highly probabilistically lower than the traditional model. Another reason is that the arrival time of the truck will be reconfirmed before, and the TAS can arrange the corresponding appointment plan in advance, saving a lot of time and costs.
According to the results, a continuous trendline for the proposed scheduling model is obtained through polynomial fit. The polynomial is
f(
x) = 9.469 × 10
−7 x3 − 5.687 × 10
−5 x2 + 7.986
x + 552.2. The original experiment results and the polynomial fit curve are plotted in
Figure 4. Another twenty inputs are tested and the test results are also given in
Figure 4. It can be seen that this polynomial can basically reflect the rescheduling cost’s changing trend with job numbers. The test results roughly follow this fitting curve.
In order to highlight the superiority of the proposed model, the number of trucks in the port is compared with the traditional scheduling model, as shown in
Figure 5. The horizontal axis is the time window of the port in one day, and the vertical axis is the number of trucks in the port. Five time points are selected in each time window to count the number of external trucks in port. Similar to
Figure 3, experiments are carried out when the time window follows normal distribution. The 95% confidence interval of the truck number is plotted in
Figure 5.
By comparing with the traditional scheduling model, it can be seen that the proposed model under uncertain arrival time can better schedule the flow of vehicles, and the peak traffic of trucks in the port is also significantly reduced since the trucks arrive at the port more evenly. Even if the time window follows the normal distribution, the proposed model still has a high probability to outperform the traditional model. Since the impact of the morning and evening rush hours on the arrival time of trucks are considered, it can be clearly seen that the number of trucks in the port during the urban traffic peak period is significantly reduced, which increases the fault-tolerant rate of truck companies and trucks and relieves the congestion of the port. The whole working process is more in line with the needs of actual operations, and the actual application value of the TAS has been greatly improved.
6.2. Algorithm Comparison
In order to verify the effectiveness of the algorithm, we selected five medium-scale calculation examples, numbered Z1, Z2, Z3, Z4, Z5, and compared them with the algorithms in literatures [
4,
20]. The specific results are shown in
Table 2. It can be seen that the double-chain real genetic algorithm is significantly faster than the other two algorithms, and it is more in line with the timeliness requirements of the TAS to reschedule the appointment plan. At the same time, the target value of the optimal solution of the proposed algorithm is also lower than the target value obtained by other algorithms, which is more in line with the requirements of reducing the re-scheduling cost of the TAS.
In order to verify the performance of the double-chain real quantum genetic algorithm. Based on the parameters of Z5, the double-strand real quantum genetic algorithm (Abbreviated as Algorithm 1), tabu search algorithm (Abbreviated as Algorithm 2) [
18] and genetic algorithm (Abbreviated as Algorithm 3) [
19] are used to obtain the solution. The algorithm runs 20 times continuously. The box plot comparison of fitness value, convergence algebra, and convergence time is shown in
Figure 6.
It can be seen from
Figure 6 that the performance of the double-chain real quantum genetic algorithm is significantly better than the other two algorithms, especially the convergence time has a more obvious advantage. Thus, the TAS can reschedule the appointment in time to meet the comprehensive needs of trucks and ports.