ESTADÍSTICOS DESCRIPTIVOS

To demonstrate the algorithm, it is applied to Problem 1 and the results are compared with the ‘No Action’ option. ‘No Action’ does not make any changes to the aircraft

allocation, the airline simply waits for A1 to be repaired and become available to fly.

Whilst this is a very unlikely option to take, it provides a benchmark for comparison as a baseline option. The minimum time between flights allowed in the original schedule is 30 minutes, though this includes some slack. After some experimentation, a more optimistic turn time of tmin = 25 minutes was selected for the IP, attempting to exploit the slack built into the system. A time discretisation of m = 5 minutes and a maximum delay of M = 180 minutes are used. This set of parameters implies 1975 flight delay arcs and 1435 ground arcs. The IP is set up and solved using the algorithm described in Section 3.4, with a maximum of 600 seconds used by the Gurobi Solver, with 8 processors available. This process finds five solutions, with different aircraft allocations and varying amounts of delay. Between four and six flights are delayed (so the simulation optimisation will have n⁺= 4, 5 or 6 variables) and no flights are cancelled. The IP performance measures for each of the solutions are shown in Table 5.3.1. The first five solutions are found quickly, each within 30 seconds. The next set of constraints chosen leads to an infeasible problem, and the next is not solved within the time limit. In solution 5, one flight is delayed beyond 2 hours and so includes some passenger compensation. No other flights require compensation meaning that the cost and delay objectives are closely correlated for all of these solutions.

Figure 5.3.1 shows the results of each stage of the process, using two of the ob-jectives, cost and tail number exchanges. Each solution has a corresponding colour.

The triangles represent the solutions found by the IP and evaluated within the IP.

This does not account for any stochastic elements of the system. The Pareto frontier between cost and reducing the number of schedule alterations is clear, as expected.

Table 5.3.1: IP solutions for Problem 1. Maximum allowable runtime was 600 seconds.

Solution Cost Tail Number Delay Cancellations Solution Time

(AC1000) Exchanges (minutes) (seconds)

1 15.00 30 300 0 4.23

2 15.00 15 300 0 15.67

3 15.75 12 315 0 12.61

4 16.50 8 330 0 25.66

5 25.25 4 495 0 18.71

These solutions are evaluated using the simulation to account for the uncertainty.

Each solution was simulated 1000 times using Common Random Numbers (CRN).

The mean is represented by the squares, with the interval going up to the empirical 0.95 quantile. The IP clearly underestimates the cost, which is not surprising as it removes stochasticity from the system. However, the ordering is very similar, helping to justify a multi-fidelity modelling approach.

Each of the solutions found by the IP is used as a starting point for the simulation optimisation process. Each starting point is simulated N_min^c times to provide an initial estimate of its mean and variance, before the algorithm begins. The linear models are built using n⁺+ bn⁺/2c + 1 design points and the quadratic models use 2n⁺+ bn⁺/2c + n⁺(n⁺− 1)/2 + 1 design points. The circles in Figure 5.3.1 show the means of the resulting solutions, with the interval again going up to the empirical 0.95 quantile. They are shifted horizontally for visual clarity, the aircraft allocations

Figure 5.3.1: Aircraft exchanges against cost estimates under different models for each solution. Interval is from mean to the 0.95 quantile.

remain the same. We see a substantial decrease in the mean as hoped for, though an increase in standard deviation also emerges.

Table 5.3.2 provides more detail on the performance of the solutions before and after the simulation optimisation. This confirms the decrease in mean, but increase in standard deviation. Furthermore, there is an improvement in the 0.9 quantile suggesting that the solutions are robust, though the tails appear to be longer, and the 0.95 quantiles are not consistently improved. The final column shows an estimate of the probability that the solution (x, d^∗) will lead to a lower cost than (x, d0), PI(d^∗).

As we have used CRN, the replications can be paired, as the same scenario is faced by both solutions. Let Gi(x, d0) and Gi(x, d^∗) be the cost observation of the solutions (x, d0) and (x, d^∗) using the i^thset of CRN, respectively. Then we can estimate PI(d^∗)

Table 5.3.2: The estimated means, ˆg(x, d), with 95% confidence interval halfwidths, standard deviations, ˆσ, 0.9 and 0.95 quantiles, ˆq0.9 and ˆq0.95, and the probability of improved cost ˆPI(d^∗), for the cost (AC1000) performance for the solutions.

Solution Initial Solution Improved Solution

ˆ

g(x, d0) σˆ qˆ0.9 qˆ0.95 ˆg(x, d^∗) σˆ qˆ0.9 qˆ0.95 PˆI(d^∗) No Action 37.3 (0.97) 15.7 58.9 66.0

1 18.4 (0.18) 2.89 21.0 23.7 13.5 (0.29) 4.60 19.0 22.6 0.949 where I(·) is the indicator function. The results suggest that there is improvement in over 85% of the replications, which adds evidence that the solutions keep some level of robustness but with a longer tail than the initial solutions (this will be discussed further in Section 5.4.1).

The Empirical Cumulative Distribution Functions (ECDFs) of the initial and re-sulting solutions based on the 1000 CRN replications are shown in Figure 5.3.2. The IP solutions appear quite robust across the 1000 replications, as shown by the ini-tial steep gradients. This suggests that the planned delays leave enough slack in the

Figure 5.3.2: ECDFs for Problem 1, based on 1000 CRN replications. Left plot shows solutions from the IP, the right shows solutions from the simulation optimisation.

Both are compared to taking no action at all. Colours match those in Figure 5.3.1.

new schedule to absorb most of the variability in turn times and repair time without much extra cost. This slack also prevents earlier take-off when possible, which the simulation optimisation appears to exploit. An ideal set of results for the simulation optimisation would see each ECDF moving to the left (reducing the mean) and having a steeper gradient (reducing the standard deviation) when compared to the ECDF of the corresponding IP solution. The results suggest that the simulation optimisation process improves the mean performance. However, increases in standard deviation are also seen. All solutions show an improvement in mean and standard deviation over the ‘No Action’ response, showing that improvements can be made using this solution method.

As the solutions faced the same 1000 scenarios (via CRN), the distributions of (Gi(x, d^∗) − Gi(x, d0)) can be estimated and are shown in Figure 5.3.3 for each plan.

Figure 5.3.3: ECDFs of the improvement over the IP solution for Problem 1, based on 1000 CRN replications. Colours match those in Figure 5.3.1.

This gives a more comprehensive view than the value of ˆPI(d^∗) (which is the value of the ECDF at 0). This emphasises the improvement in Plan 5 particularly. However, there is still a probability that the solution could perform worse than the original IP solution, though the tail of the distribution is not too long.

Tables 5.3.1 and 5.3.2 show that different solutions offer different characteristics.

Solution 3 has the lowest mean but is quite variable. However, solution 1 has the lowest standard deviation and the lowest 0.95 quantile but requires the largest number of changes to the schedule. Solution 5 offers far fewer schedule changes but is the most variable. Different airlines may have differing priorities on this trade-off, and will also take other considerations into account, such as the impact on crew. The use of a symbiotic simulation in this setting would allow this trade-off to be adapted for different decisions to reflect individual circumstances.