Integrated Airline Scheduling
4.4 Simultaneous Approach .1 Overview.1Overview
4.4.3 Experiments
4.4.3.2 Analysis
150 4 Integrated Airline Scheduling
Fig. 4.58 Number of re- quired fitness evaluations for the different search strategies
10000 20000 30000 40000 50000 60000 70000 80000
A B C D E
No. of evaluations
Scenario
TA rGA GA
focusing only on one type of operators. This finding is not surprising, since most problems of practical importance inherit properties applicable to both search con- cepts, local and recombination-based search (Michalewicz & Fogel, 2000,Rothlauf, 2006a). To validate the results, an unpaired t-test is conducted.29The null hypoth- esis H0is that the observed differences in the fitness values are random. Hα says that the differences are a result of the model specification. The critical t-value for p=0.975 is 2.306. The results shown in Table 4.8 for the three models and five scenarios show that the t-values always exceed the critical t-value of the level of significance. Thus, H0can be rejected on the 97.5%-level. The GA represents the search strategy that works best using the presented airline scheduling approach.
Table 4.8 t-values for the validation of the search strategy comparison Scenario
Models A B C D E
TA vs. rGA 6.405 20.136 13.222 13.637 3.277
TA vs. GA 7.057 79.604 16.775 16.127 2.400
rGA vs. GA 10.554 17.641 5.645 6.251 4.811
Table 4.9 Key figures of airline schedules constructed with the simultaneous planning approach
Scenario
Key Figure A B C D E
Profit 571,812 485,775 130,384 165,556 125,880
(20,556) (3,086) (12,019) (9,894) (7,466)
SLF 0.344 0.501 0.219 0.495 0.316
(0.019) (0.008) (0.008) (0.018) (0.011)
No. of passengers 5,535 4,442 2,551 2,493 1,801
(113.33) (88.37) (68.13) (64.53) (224.69)
No. of flights 114 125 70 72 38
(2.88) (3.71) (2.61) (3.08) (3.77)
No. of fitness evaluations 69,832 46,909 69,550 43,569 36,580
(9,168) (3,196) (14,477) (9,626) (15,899)
Total no. of evaluations 70,723 47,857 71,249 44,721 38,682
(9,417) (2,682) (14,251) (9,120) (16,354)
Fig. 4.59 Trend of fitness values of all five optimiza- tion runs of scenario A
-600000 -400000 -200000 0 200000 400000 600000 800000
0 10000 20000 30000 40000 50000 60000 70000 80000
Fitness
No. of evaluations
the stability of the metaheuristic solution approach. Scenario A yielded the highest fitness values, although the seat load factor was best for scenario B. Compared to the uncalibrated genetic algorithm (using the basic parameter setting), the obtained profit values represent an average increase of 52.67%. The number of required fit- ness evaluations is on average 21.64% higher.
The following figures focus on the solution process. They present results of ex- periments on scenario A as an representative example for all scenarios.31Aggregat- ing the results of the five runs of one scenario or even among the different scenarios would result in meaningless diagrams, since each individual run requires a different number of fitness evaluations leading to different plots of the progress subject to the number of evaluations.
Fig. 4.59 plots the fitness of the best solution in each population. It shows the typ- ical progress for a GA. The progress or improvement of the best solution is highest
31The results of all scenarios are presented in Sect. C.2.2.2 in the appendix.
152 4 Integrated Airline Scheduling
Fig. 4.60 Trend of seat load factors (SLF) of all five optimization runs of scenario A
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
0 10000 20000 30000 40000 50000 60000 70000 80000
SLF
No. of evaluations
Fig. 4.61 Trend of numbers of flights of all five opti- mization runs of scenario A
100 110 120 130 140 150 160 170
0 10000 20000 30000 40000 50000 60000 70000 80000
No. of flights
No. of evaluations
at the beginning of the GA and continuously decreases during optimization. There are two reasons for this: first, as in every metaheuristic, the search operators have much room for improvements, since the early solutions inherit random elements due to the random initialization; second, the population converges during the GA run, reducing the potential capability of the recombination-based operators with their rather large modifications to solutions, and leaving only room for local (and, thus, small) search steps.
Fig. 4.60 presents the SLF for the five runs of scenario A. It depends on the results in figures 4.61 and 4.62, which plot the number of flights in the best schedule of each population and the total number of passengers expected to travel on these flights. Not surprisingly and as indicated by the progress of the fitness values, the SLF increases during optimization. This increase is a result of an increase in the number of passengers and a reduction of the number of total flights. Apparently, in each GA run unprofitable flights are removed from the schedule.32Since the number of flights is higher in the beginning of each run and assuming that the average block time of all flights remained constant, the final schedules must include some idle
32Since the fitness function of the GA considers the connectivity of the schedules, the removal of flights does not depend on their individual passenger demand but also takes their function as legs of connecting itineraries into account.
Fig. 4.62 Trend of num- bers of passengers of all five optimization runs of scenario A
1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
0 10000 20000 30000 40000 50000 60000 70000 80000
No. of passengers
No. of evaluations
Fig. 4.63 Application prob- ability of recombination- based operators for all five optimization runs of scenario A
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0 10000 20000 30000 40000 50000 60000 70000 80000
Probability
No. of evaluations
ground times. Thus, an additional increase in profit seems possible if the number of airports available for planning is increased to allow the scheduling of additional (profitable) routes. In scenario E this effect can be observed (see Fig. C.26). After removing unprofitable flights, the number of flights in the schedule increases. Since the number of passengers also increases, the SLF remains constant, however, the overall fitness grows.
The GA represents a self-adaptive procedure, thus, another interesting observa- tion is the extent of application of the different search operators during the opti- mization run. Fig. 4.63 presents the (smoothed) share of the recombination-based operators during the optimization. In general, the different runs show a similar search behavior. The selection probability for recombination-based operators is at its minimum at the start of the optimization. This could be a random result, since in some runs from other scenarios (see Fig. C.28) the initial probability is at the max- imum value. However, in all runs recombination becomes the main search operator after the starting phase. Then, its application probability continuously decreases un- til the end of the optimization. In the final phase of each run, larger variations in the probability exist. The continuous shift from recombination-based to local search can be explained by the different characteristics of the search concepts. Recom- bination represents a global search operator which is useful for the exploration of
154 4 Integrated Airline Scheduling
Fig. 4.64 Application prob- ability of the different vari- ants of the recombination- based search operators (scenario A, run 3)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0 10000 20000 30000 40000 50000 60000 70000
Probability
No. of evaluations rec1P rec2P recString
Fig. 4.65 Application prob- ability of the different vari- ants of the local search operators (scenario A, run 3)
0.05 0.1 0.15 0.2 0.25 0.3
0 10000 20000 30000 40000 50000 60000 70000
Probability
No. of evaluations locDelGT locChgFA locInsGT locChgApt locChgRot locInsApt locDelApt locChgAptMS locChgAptDist
the search space. The better the solutions of the population and the more the pop- ulation converges, the more the search has to concentrate on good regions within the search space (exploitation). This is accomplished by the local search operators.
Then, in the final phase when each operator reaches its limit with regard to solution improvement, there is no clear advantage of one search concept, leading to the larger variation of the number of applications of each operator type.
The same variation at the end of the optimization can be observed in Fig. 4.64, which plots the application of the different variants of the recombination-based search operators. For clarity, the results of the best run from scenario A are plot- ted as a representative example.33In general, after the initialization the application probability stabilized at approximately the same value for each operator variant.
Thus, during the optimization each recombination operator is chosen with the same probability.
Fig. 4.65 presents the application of the different variants of the local search operator. There is no clear indication of an advantageous variant, since all operators are used during the optimization to a different and fluctuating extent. However, on
33Similar figures for the other runs and for the other scenarios are presented in Sect. C.2.2.2 in the appendix.
average there is a trend that three local search operators are of less priority during search: locDelGT, locInsGT, and locInsApt.