Integrated Airline Scheduling
4.4 Simultaneous Approach .1 Overview.1Overview
4.4.3 Experiments
4.4.3.1 Calibration
One advantage of the adaptive control of the search operators is the reduction of the number of parameters to be set for each metaheuristic. However, each metaheuristic presented previously is still controlled by some parameters that have to be calibrated.
In this section, experiments on different parameter settings are presented to find a final calibration for each metaheuristic. Then, the solution quality of all three calibrated models is compared to decide about the search strategy (TA, rGA, or GA) for further analysis and comparison with the sequential airline scheduling approach.
The calibration process is conducted in analogy to the calibration of the sequen- tial approach (see Sect. 4.3.4.1). Different parameter settings are examined by ap- plying each metaheuristic to five different planning scenarios (see Appendix B). For each parameter setting and scenario, five runs are conducted. The results presented in the following thus represent averages of these five runs. In analogy to the results
26Additional parameters are N=5 for the adaptive control of the search parameters and tinit=25 for initialization and some search operators.
144 4 Integrated Airline Scheduling
Algorithm 9. Standard Steady-State Genetic Algorithm 1: choose parameters:
2: population size n
3: pconvto determine convergence of the population 4: create initial population S0with n numbers of solutions s 5: calculate fitness value f(s)for each s∈S
6: calculate pofor all operators o∈O 7: calculate pRand pL
8: iteration i=0 9: repeat 10: i=i+1
11: if random(0,1)<pRthen
12: select recombination-based search operator o∈R according to po
13: choose two solutions s1and s2randomly 14: create solution s∗ofrom s1and s2
15: else
16: select local search operator o∈L according to po
17: create neighboring solution s∗o
18: end if
19: calculate new fitness value f(s∗o) 20: replace the worst solution in S by s∗o
21: update pofor all operators o∈O 22: update pRand pL
23: determine solution ˆs∈S with maximum fitness 24: calculate average fitness ¯f(S)of population 25: until f(s)ˆ −¯f(S)<pconv·f(ˆs)
from the sequential approach, they include fitness values as a measure of solution quality and the number of fitness evaluations until the best solution was found as a platform-independent quantification of the effort to solve the problem. All values are normalized to aggregate among all scenarios (see page 119).27
Threshold Accepting. The TA algorithm was implemented as presented in algo- rithm 7. This specification uses four parameters to control the search process:
• the initial threshold T ∈[0,1],
• the threshold reduction step size r<T ,
• the number of iterations idecreasebetween the threshold reductions, and
27The individual results of the scenarios including the absolute values are presented in Sect. C.1.2 in the appendix.
• the maximum number of iterations imax>idecreaseat the end of the algorithm when T=0 (then the search process represents a local hill climbing algorithm that does not accept inferior solutions).
The impact of the individual parameters is tested by solving the airline scheduling problem with different parameter combinations. In each setting, one parameter is set to different values while the others remain constant. For the constant parameters, the following setting is chosen as the basic setting:
T =0.2, r=0.005, idecrease=20,
imax =500.
In the following, four diagrams are presented, each illustrating the results for the experiments on one parameter.
Fig. 4.49 presents the calibration of the initial threshold T . The solution quality decreases for low and high values of T , it is highest for values around 0.2 – 0.25.
Thus, these values seem to represent the best compromise between a random search (high T ) and a hill-climbing technique that does not accept inferior solutions during search (low T ).
Fig. 4.50 presents the calibration of the parameter r. The smaller r, the higher the resulting solution quality. If r is low, the threshold is reduced very slowly, allowing an explorative search. However, the computational effort also increases. Surpris- ingly, for values of r very close to 0, a decrease in solution quality is observed, which is confirmed by additional experiments on these values.
Fig. 4.51 presents the calibration of the parameter idecrease. In general, solution quality increases with higher idecrease. The higher idecrease, the more search steps are performed before the threshold is further reduced. This allows the exploration of more solution space during optimization. As Fig. 4.51 clearly shows, the higher solution quality is obtained at the cost of increased computational effort.
Fig. 4.49 Aggregated cali- bration results for parameter T
-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
Fitness No. of evaluations
T
Fitness No. of evaluations
146 4 Integrated Airline Scheduling
Fig. 4.50 Aggregated cali- bration results for parameter r
-0.04 -0.02 0 0.02 0.04 0.06 0.08
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 -0.2 -0.1 0 0.1 0.2 0.3 0.4
Fitness No. of evaluations
r
Fitness No. of evaluations
Fig. 4.51 Aggregated cali- bration results for parameter idecrease
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12
0 100 200 300 400 500 600 700 800-1 -0.5 0 0.5 1 1.5 2
Fitness No. of evaluations
idecrease
Fitness No. of evaluations
Fig. 4.52 Aggregated cali- bration results for parameter imax
-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4
0 200 400 600 800 1000 1200 1400 1600 1800 2000-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
Fitness No. of evaluations
imax
Fitness No. of evaluations
Fig. 4.52 presents the calibration of the parameter imax. This parameter comes into play when the threshold is set to T=0. Then, the TA represents a hill-climbing technique that stops after imax iterations without increase of solution quality. In- creasing imaxleads to a higher solution quality, since the hill-climbing technique has more attempts to escape from local optima.
Fig. 4.53 Aggregated cal- ibration results for param- eter pconv of the selecto- recombinative genetic algo- rithm (rGA)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2
0.005 0.01 0.015 0.02 0.025-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15
Fitness No. of evaluations
pconv
Fitness No. of evaluations
For each parameter, the value at which the best fitness was achieved is used for the subsequent experiments. Thus, the complete final parameter setting is as presented in the following:
T =0.25, r=0.005, idecrease=650,
imax =1500.
Selecto-Recombinative Steady-State Genetic Algorithm. Algorithm 8 describes the specification of the selecto-recombinative GA. This algorithm uses the following two parameters:
• population size n,
• parameter pconvto determine the convergence of the population to terminate the algorithm. pconvrepresents a percentage of the fitness value of the best solution, if the difference between the fitness of the best solution and the average fitness of the population is smaller than this value, the algorithm is terminated.
The following two figures 4.53 and 4.54 present the result on different settings for these two parameters. The plotted values are the values of the best solution from every population. The standard setting is chosen as in the following:
n=50, pconv=0.01.
Fig. 4.53 presents the calibration of the parameter pconv. The smaller pconv, the higher the required convergence of the population before the algorithm terminates.
To reach this convergence, many search steps are necessary, each possibly creating a better solution. This results in overall better solution quality. On the other hand, more search steps require more computational effort. Both effects can clearly be observed in Fig. 4.53.
148 4 Integrated Airline Scheduling
Fig. 4.54 Aggregated cal- ibration results for pa- rameter n of the selecto- recombinative genetic algo- rithm (rGA)
-5 -4 -3 -2 -1 0 1 2 3
0 50 100 150 200 250 300-1
-0.5 0 0.5 1 1.5
Fitness No. of evaluations
n
Fitness No. of evaluations
Fig. 4.55 Aggregated cali- bration results for parameter pconvof the standard genetic algorithm (GA)
-5 -4 -3 -2 -1 0 1
0.005 0.01 0.015 0.02 0.025-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2
Fitness No. of evaluations
pconv
Fitness No. of evaluations
Fig. 4.54 presents the calibration of the parameter n. Vice versa to pconv, increas- ing n yields better solution quality, since more solutions are processed. The more solutions, the higher the chance of the search steps to find a better solution. In ad- dition, convergence is more difficult to achieve because more solutions enter the calculation of the average fitness of the population. As Fig. 4.54 shows, for increas- ing n the fitness value asymptotically approximates a maximum value, whereas the required number of schedule evaluations constantly increases.
The final parameter setting used in the following for rGA is:
n=200, pconv=0.00125.
Standard Steady-State Genetic Algorithm. The specification in algorithm 9 of the standard GA corresponds to the algorithm 8 except for the incorporation of the local search operator. Thus, the parameters are the same as in the selecto-recombinative GA and the same basic setting is used. The following two figures 4.55 and 4.56 present the result for the standard GA on different settings for these two parameters.
Fig. 4.56 Aggregated cali- bration results for parameter n of the standard genetic algorithm (GA)
-6 -5 -4 -3 -2 -1 0 1 2 3
0 50 100 150 200 250 300
-1 -0.5 0 0.5 1 1.5
Fitness No. of evaluations
n
Fitness No. of evaluations
Fig. 4.57 Fitness values for the different search strategies
0 100000 200000 300000 400000 500000 600000
A B C D E
Fitness
Scenario
TA rGA GA
The results for the standard GA are almost the same as for the selecto- recombinative GA. The final parameter setting for the GA is:
n=200, pconv=0.00125.
Strategy Selection. The parameter settings above should lead to high quality solu- tions for each search strategy. To decide among these strategies, each strategy with its calibrated parameters is applied to the different planning scenarios. The resulting (average) fitness values and number of required fitness evaluations for each strategy and scenario are presented in the following figures 4.57 and 4.58.28
For all scenarios, the GA yielded the highest solution quality. Except for scenar- ios A and E, the selecto-recombinative GA produced better results than TA. Because the GA and rGA use populations of solutions, they require significantly more fit- ness evaluations than TA, which processes only one solution. These results indicate that a combined local and recombination-based search outperforms search strategies
28The individual results of the scenarios including the absolute values are presented in Sect.
C.1.2.4 in the appendix.
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