Integrated Airline Scheduling
4.3 Sequential Approach .1 Overview.1Overview
4.3.4 Experiments
4.3.4.2 Analysis
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.
Table 4.7 Key figures of airline schedules constructed with the sequential planning approach scenario
Key Figure A B C D E
Profit 450,629 325,927 -60,166 51,849 97,774
(59,668) (16,536) (20,219) (20,835) (7,345)
SLF 0.300 0.369 0.172 0.387 0.267
(0.012) (0.021) (0.017) (0.023) (0.027)
No. of passengers 5,947 3,493 1,682 2,144 2,057
(616.36) (183.13) (65.06) (48.15) (270.72)
No. of flights 136 97 61 73 49
(10.55) (8.50) (3.19) (3.97) (4.90)
No. of fitness evaluations 212,886 36,772 154,618 32,265 48,621 (183,070) (11,386) (85,735) (12,894) (35,137) Total no. of evaluations 369,150 112,090 260,428 50,335 83,369 (236,341) (25,747) (137,999) (10,100) (29,406)
No. of iterations 19 24 36 9 20
(12) (5) (17) (3) (5)
appearing to be different could include very similar flight programs (for example, flights are assigned to a different fleet type or another rotation), limiting the expla- nation of a very detailed presentation.14In general, as the low standard deviations indicate, the results are stable. Most variations exist in the duration of the complete optimization run, which is measured by the last three rows of Table 4.7. Scenario A resulted in the highest profit values, although the seat load factor was best for scenario D. Scenario C even results in an operational loss (although it required the most attempts to improve solution quality, since the number of iterations is highest).
Compared to the uncalibrated model (basic parameter setting), the obtained profit values represent an average increase of 19.26%. The number of required fitness evaluations is on average 38.39% higher than with the uncalibrated model. Differ- ences in the order of magnitude between the number of iterations and the number of fitness evaluations are the result of the different specifications of the scenarios: the more aircraft and airports are available in a scenario, the more fitness evaluations are necessary in each iteration. It has to be emphasized that a meaningful inter- pretation of the absolute values of these indicators is not possible, since for each scenario the competition and the set of airports were chosen randomly (and, thus, can represent markets with low airline travel demand) and the market size estimates represent only poor approximations of the real demand. In reality, airlines usually have average SLF of about 75% ICAO (2006).15
The following figures focus on the solution process. They present results of ex- periments on scenario A as a representative example for all scenarios.16Because the
14Individual results are presented in tables in Sect. C.2.1.1.
15For example, when optimizing airline schedules for scenario D but using past passenger num- bers for selected city pairs as market sizes instead of the estimates (see the discussion on 75), a SLF of 0.630 was obtained.
16The results of all scenarios are presented in Sect. C.2.1.2 in the appendix.
124 4 Integrated Airline Scheduling
Fig. 4.30 Trend of profits of all five optimization runs of scenario A
150000 200000 250000 300000 350000 400000 450000 500000 550000
0 5 10 15 20 25 30 35
Profit
Iteration
Fig. 4.31 Trend of seat load factors (SLF) of all five optimization runs of scenario A
0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32
0 5 10 15 20 25 30 35
SLF
Iteration
number of iterations is different between the individual runs for each scenario and between the scenarios, a meaningful aggregation among the individual runs is not possible.
Fig. 4.30 plots the profit for the five different runs of scenario A. As is clearly visible, the number of iterations varies among the different runs, the shortest run required less than five iterations until it terminated. However, it yielded the best solution quality. Besides a general growth from start to end (except for one run), the optimization progress shows a very unstable trend. There are large variations (peaks and drops) in the profit even between succeeding iterations. As will be shown later in Fig. 4.34, this is most likely the result of the maintenance routing steps and the related supportive functions.
The following three figures show similar characteristics. They plot the SLF (Fig.
4.31), the number of flights (Fig. 4.32), and the number of passengers (Fig. 4.33) as smoothed curves. The number of flights and the number of passengers have the same trend. Thus, if there are more flights, more passengers are transported. Ex- cept for one run, the number of flights increases in the beginning. This trend re- sults from the basic initialization of the schedule, in which only a basic schedule
Fig. 4.32 Trend of numbers of flights of all five opti- mization runs of scenario A
105 110 115 120 125 130 135 140 145 150 155
0 5 10 15 20 25 30 35
No. of flights
Iteration
Fig. 4.33 Trend of num- bers of passengers of all five optimization runs of scenario A
3500 4000 4500 5000 5500 6000 6500 7000
0 5 10 15 20 25 30 35
No. of passengers
Iteration
is created, which is then successively extended by the solution steps during the iterations.
To investigate the contribution of the different solution steps of the sequential planning approach, the following Fig. 4.34 plots the profit on a more detailed level (for each solution step for each iteration). For clarity, the results of only one run of scenario A are presented as a representative example.17 To better understand the shape of the plot, Fig. 4.35 presents an excerpt of Fig. 4.34, focusing on iter- ations 3-5. As this figure illustrates, the drops in operating profit result from the application of the maintenance routing algorithm. Thus, the results from the pre- ceding fleet assignment usually contain locked rotations and/or are infeasible with respect to the three-day maintenance requirement. Because the repair mechanisms of the maintenance rounting step do not consider the operating profit when modify- ing the current schedule, the drops can easily be explained.18A profit decrease in
17Similar figures for the other runs and for the other scenarios are presented in Sect. C.2.1.2 in the appendix.
18However, the additional repair mechanisms developed in this study take market sizes into ac- count when applying changes to the schedule. For example, if a flight needs to be inserted, this is accomplished for the market with the highest passenger demand (under consideration of demand already satisfied by existing flights).
126 4 Integrated Airline Scheduling
Fig. 4.34 Profit contribution by individual solution steps (scenario A, run3)
50000 100000 150000 200000 250000 300000 350000 400000 450000 500000
Profit
Iteration
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Fig. 4.35 Profit contribution by individual solution steps between iteration 3 and 5 (scenario A, run3)
100000 150000 200000 250000 300000 350000 400000 450000 500000
Fleet Assignment Maintenance Routing Slack Reduction Airport Removal Airport Optimization Fleet Assignment Maintenance Routing Slack Reduction Airport Removal Airport Optimization Fleet Assignment Maintenance Routing Slack Reduction Airport Removal Airport Optimization
Profit
Iteration 3 Iteration 4 Iteration 5
the fleet assignment step (for example in iteration 5 in Fig. 4.35) can be explained by the objective function of this step, which is to minimize costs (operating and spill costs). Because revenues are not taken into account, minimizing operating costs can contradict maximizing profit. For example, if no flights at all are conducted, operat- ing costs are minimized without earning operating profit.