Integrated Airline Scheduling
4.5 Evaluation
4.5.2 Experimental Verification
were presented in figures 4.66 and 4.70.35The results clearly indicate that the GA- based simultaneous approach still outperforms the extended sequential approach, although it produced more profitable schedules compared to its first application (at the cost of significantly increased effort).
To validate the results, a standard Student’s t-test is conducted for the results on the profit of both approaches.36The null hypothesis H0is that the observed differ- ences in the profit values are random. Hα says that the differences are a result of the solution approach. The critical t-value for p=0.995 is 3.335, for p=0.999 it is 4.501. The results presented in Table 4.10 for the five scenarios show that the t- values always exceed the critical t-value of the level of significance. Thus, H0can be rejected on the 99.9%-level for all experiments except for scenario D, for which H0
can be rejected on the 99.5%-level. As a consequence, the simultaneous approach significantly produces more profitable schedules than the sequential approach while requiring considerably less effort to obtain these schedules.
162 4 Integrated Airline Scheduling
(a) (b) (c)
Fig. 4.73 Resulting hub-and-spoke network structure
hub-and-spoke network. Then, market sizes are chosen so that the resulting network should reflect a triangular-shaped route network (three routes).
Hub-and-Spoke Structure. Three experiments on three different market structures are conducted. In the first experiment (a), market sizes are zero for all city-pairs ex- cept for routes to or from Dortmund (DTM), London Heathrow (LHR), and Madrid Barajas (MAD). Thus, it should result in a hub-and-spoke network with DTM, LHR, and MAD as hubs. In the second experiment (b), the demand for routes to and from MAD is also set to zero (hub-and-spoke network with DTM and LHR as hubs);
and in the third experiment (c) only DTM generates demand with all other city- pairs having no demand (single-hub network). The following figure 4.73 presents the network structures of the optimal schedules obtained for each experiment. The hub-and-spoke network structure is clearly visible for each experimental setup.
To illustrate the search process of the GA and its continuous modification of the network structure towards the final network, the following Fig. 4.74 presents net- works that were processed during the optimization. As an example, the single-hub network is chosen. Based on one optimization run, the presented networks corre- spond to the best schedule in the population after initialization, 2% iterations (com- pared to the total number of evaluations needed for the complete run), 10%, 25%, and 50% iterations. Because the initialization of the GA run is conducted randomly with every decision variable having the same selection probability, the route net- work is spread over all available airports at the beginning. Then, the network con- tinuously shifts towards the single-hub network with DTM as hub representing the optimal network structure for the given input data.
Routes. The following figures present the progress of the search process for the scenario in which market sizes between only three airports are given (DTM, LHR, MAD). Thus, a triangular-shaped route network should reflect the best network structure. In Fig. 4.75, the development from a randomly initialized route network towards the optimal solution is clearly visible.
(a) initialization (b) 2% iterations (c) 10% iterations
(d) 25% iterations (e) 50% iterations (f) 100% iterations Fig. 4.74 Illustration of the search process resulting in a single-hub network
(a) initialization (b) 2% iterations (c) 10% iterations
(d) 25% iterations (e) 50% iterations (f) 100% iterations
Fig. 4.75 Illustration of the search process resulting in a network consisting of three routes
164 4 Integrated Airline Scheduling
Fig. 4.76 Profit and number of required fitness evalua- tions for different numbers of aircraft
-100000 -50000 0 50000 100000 150000 200000
0 20 40 60 80 100 120
0 20000 40000 60000 80000 100000 120000 140000 160000
Profit No. of evaluations
No. of aircraft Profit No. of evaluations
Fig. 4.77 Profit and number of required fitness evalua- tions for different sizes of the airport set
0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06 1.6e+06
0 50 100 150 200 250 300 350 42000 44000 46000 48000 50000 52000 54000 56000 58000 60000 62000 64000
Profit No. of evaluations
No. of airports Profit No. of evaluations
4.5.2.2 Number of Aircraft
The basic configuration of scenario D includes 20 aircraft of two different fleets of equal size. In the following, results of experiments are presented that use different numbers of aircraft. For example, if more aircraft are available, more passengers can be carried, possibly resulting in higher profits. On the other hand, if there are too many aircraft, their operational costs might exceed the revenues. Thus, finding the optimal fleet size for a given scenario is not easy and represents an important planning problem that can be supported by the simultaneous planning approach pre- sented in this study.
The following Fig. 4.76 presents the operating profit and the number of required fitness evaluations for different fleet sizes. Following the basic configuration of sce- nario D, the distribution between both fleet types is kept at equal size. A peak in this figure can be identified, indicating the optimal fleet size. This number represents the best trade-off between too many aircraft causing high operational costs and too few aircraft reducing the number of passengers that can be transported.
4.5.2.3 Number of Airports
Every scenario considered in this study consists of a fleet composition and a set of airports. The solution approach then is allowed to choose only those airports that are included in this given set. This restriction should reflect the decision of some airlines to exclude certain airports from their schedules. In the following Fig. 4.77, results of experiments on different sizes of the airport set are presented. If the number of airports is increased, solution quality is higher, since there is more freedom in planning and more profitable flights can be selected. It is at its maximum if all 320 available airports can be selected for scheduling. However, the higher degree of freedom in search also requires more effort to obtain the final solution, thus, the number of fitness evaluations also grows with an increasing number of airports available for scheduling.
4.5.2.4 Fleet Types
In the following, an example is given on how different fleet types affect operating profit. Using the same number of aircraft, schedules are optimized with different fleet types. For clarity, in these experiments, the total fleet of the scenario only con- sists of the fleet type currently under investigation, thus, in each run 20 aircraft of the same fleet type are given. Besides the two types already used in the preceding exper- iments, four additional types from the 38 possible types (see Sect. A) are examined.
They are selected with respect to differences in their operational characteristics (seat capacities). The total set of fleet types then consists of:37
• Fairchild Dornier 328JET (FRJ), seat capacity: 32
• Canadair Regional Jet 700 (CR7), seat capacity: 70
• Boeing 737-800 (738), seat capacity: 161
• Airbus A330-200 (332), seat capacity: 243
• McDonnell Douglas MD11 (M11), seat capacity: 279
• Boeing 747-400 (744), seat capacity: 373
Fig. 4.78 presents the profit for the different fleet types (seat capacities in paran- theses). In general, for higher capacities of the fleet types, the profit decreases and becomes negative. As Fig. 4.79 illustrates, this is most likely the result of the re- duced seat load factor. For the given demand in the scenario, these fleet types are oversized and the small number of passengers results in revenues too low to com- pensate the higher operating costs. These results correspond to reality, since airlines usually do not use large fleet types like the McDonnell Douglas MD11 or the Boe- ing 747-400 for flight service within Europe. The smallest aircraft type (FRJ) has the highest seat load factor, because it is easy to fill each aircraft with passengers.
However, the profit of this fleet type is lower than for larger aircraft, indicating that the number of potential passengers is higher than the capacities offered.
37Fleet codes are in parentheses.
166 4 Integrated Airline Scheduling
Fig. 4.78 Profit for different fleet types
-400 -300 -200 -100 0 100 200
FRJ (32) 738 (161) 332 (243) M11 (279) 744 (373)
Profit
Scenario
Fig. 4.79 Seat load factor (SLF) for different fleet types
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
FRJ (32) 738 (161) 332 (243) M11 (279) 744 (373)
SLF
Scenario