Integrated Airline Scheduling
4.3 Sequential Approach .1 Overview.1Overview
4.3.4 Experiments
4.3.4.1 Calibration
118 4 Integrated Airline Scheduling
imax=5, tw=60, pcnx=0.3, popt=0.3, pnew=0.1, poptimize=0.1.
The results of the different parameter settings focus on two key figures:
1. the objective value, and
2. the number of fitness evaluations until the best solution was found, representing a platform-independent quantification of the effort necessary to find the best solution.
For each scenario and setting, five optimization runs are conducted. The results of the different planning scenarios vary in their order of magnitude, because the sce- narios consist of different numbers of aircraft and airports. Thus, to find a parameter setting based on all scenarios, normalization and aggregation of the results are nec- essary. In this study, for a given parameter setting and scenario the deviation of the averaged results from the mean value of all parameter settings is used as an indi- cation of the current setting’s quality. Let fp,s denote the average fitness value of the five runs with parameter setting p∈P (P is the set of tested values for p) and scenario s∈S, then the average fitness value ¯fs for all settings for scenario s is calculated as:
f¯s=
p∈P∑ fp,s
|P| . (4.52)
The impact ip,sof setting p in scenario s is expressed as a relation with this average fitness:
ip,s= fp,s−¯fs
|¯fs| . (4.53)
Finally, the aggregation of all scenarios yields the setting p’s average impact ¯ipon solution quality:
¯ip= ip,s s∈S∑ip,s
. (4.54)
The following figures present the results ¯ip for different settings p for each pa- rameter as smoothed curves. The results of the number of fitness evaluations are calculated accordingly.13
Fig. 4.24 presents the calibration of the parameter imax. As was expected, an increase of the solution quality can be observed for higher values of imax. A higher imax gives the solution approach more time to find a better solution and to escape from local optima. However, if imaxis further increased, surprisingly solution quality
13The individual results of the scenarios including the absolute values are presented in Sect. C.1.1 in the appendix.
120 4 Integrated Airline Scheduling
Fig. 4.24 Aggregated cali- bration results for parameter imax
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1
0 5 10 15 20 25 30-0.5
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
Profit No. of evaluations
imax
Profit No. of evaluations
decreases. In future work, this observation has to be further investigated including additional experiments for confirmation.
Fig. 4.25 presents the calibration of the time window size tw. Increasing the time window size leads to higher computation times, since there is more freedom in planning. Solution quality reaches its maximum at approximately 30 minutes, fur- ther increasing this parameter results in lower solution quality. An explanation for this effect might be the assumption within the fleet assignment step of uniformly distributed demand within each time window. If this time window is long, large variations in the departure times will have an effect on passenger demand that is not detected by the fleet assignment step.
Fig. 4.25 Aggregated cali- bration results for parameter tw
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
0 10 20 30 40 50 60 70 80 90-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06
Profit No. of evaluations
tw
Profit No. of evaluations
Fig. 4.24 presents the calibration of the parameter pcnx. As can be observed, very small and high pcnx results in lower solution quality, best solutions were obtained with pcnx=0.05. If pcnx is close to 0, the Increase Connectivity step only applies very small changes to the schedule if it is infeasible. This might not be sufficient to obtain a feasible solution, which then has to be constructed using more rigor- ous operators that reduce solution quality. In contrast, if pcnx is set to high values, in each step many modifications to a schedule are applied independently of profit considerations and, thus, as a result solution quality is limited.
Fig. 4.26 Aggregated cali- bration results for parameter pcnx
-0.15 -0.1 -0.05 0 0.05 0.1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
Profit No. of evaluations
pcnx
Profit No. of evaluations
Fig. 4.27 Aggregated cali- bration results for parameter popt
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
Profit No. of evaluations
popt
Profit No. of evaluations
Fig. 4.28 Aggregated cali- bration results for parameter pnew
-0.15 -0.1 -0.05 0 0.05 0.1
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
Profit No. of evaluations
pnew
Profit No. of evaluations
Figures 4.27 and 4.28 present the calibration of the parameters popt and pnew. In general, the results are very similar. Parameter values around 0.25 yielded best results. The closer each parameter is to 0, the less the effect of the function using this parameter, and vice versa. Thus, the observed values for the best solution quality are presumed to result in the best compromise of a too low and a too high impact of each corresponding technique.
122 4 Integrated Airline Scheduling
Fig. 4.29 Aggregated cali- bration results for parameter poptimize
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12
Profit No. of evaluations
poptimize Profit No. of evaluations
Fig. 4.29 presents the calibration of the parameter poptimizefrom which the num- ber ioptimizemax of applications of each optimization method is determined. Increasing poptimize leads to more optimization steps in each iteration, thus, solution quality increases. The decreasing computation time might be explained by the reduced in- fluence of the other steps on the overall solution approach. The higher the number of optimization steps in one iteration, the more the overall schedule is determined. The degrees of freedom decrease (for example, there is less slack time), thus, there is less room for other solution steps to guide the solution towards their objectives that are not congruent with the operating profit and would increase computation time.
For each parameter, using the value at which the best fitness was achieved results in the following final parameter setting, that is used for the subsequent experiments and analyses:
imax=20, tw=30, pcnx=0.05, popt=0.25, pnew=0.25, poptimize=1.5.