In the work described in Chapter 7, several other disruption problems were solved.
This section briefly reports the simulation optimisation results of these examples.
These problems are based on a fleet of 116 aircraft. In each case, the recovery window is until the end of the day, and so the problem size decreases as the start time gets later. The IP was set up with strict constraints on the position of aircraft at the end of the day, applying the return node constraint (3.3.10) to the entire fleet.
A summary of the disruptions is shown in Table 5.6.1. The disruptions are caused by either an aircraft being grounded for several hours due to a technical issue (Days 1, 3 and 4) or weather events causing high airport congestion (Days 2 and 6) or closure (Day 5). Days 1, 3 and 4 all involve hub airports of varying size, LGW, Geneva Airport (GVA) and MAN, allowing flexibility in the solution as multiple aircraft come
Table 5.6.1: Description of each problem and simulated cost performance (AC1000) of No Action and IP solutions, with 95% confidence interval halfwidths in parentheses.
Problem Disrupted Time No Action IP Solution
Aircraft g(x, d)ˆ qˆ0.95 ˆg(x, d0) qˆ0.95
Day 1 1 16:00 61.76 (1.01) 75.61 13.96 (0.12) 17.47 Day 2 2 9:40 41.09 (0.66) 53.77 32.84 (0.48) 38.27 Day 3 1 8:45 154.64 (1.67) 176.21 6.75 (0.83) 16.00 Day 4 1 13:20 165.3 (2.65) 210.75 26.3 (0.39) 30.45 Day 5 2 10:40 28.77 (0.58) 34.97 22.56 (0.49) 27.87 Day 6 5 10:10 22.61 (0.78) 29.81 23.51 (0.76) 28.55
through the hub airports. The repair time distribution for each is a Γ(100, 3), and a delay of 370 minutes is applied in the IP. For the weather affected days, busy airports or periods where weather was more likely to cause congestion were chosen. Day 2 occurs at AMS, one of Europe’s busiest airports. Day 5 occurs at a small hub airport, Nantes Atlantique Airport (NTE). Even though this is not a busy airport, the period of closure leads to a backlog when the airport reopens. The Day 6 disruption occurs at Paris Orly Airport (ORY). It is expected to affect five aircraft directly, some of which have not yet arrived at ORY. As in Problem 3, the flights directly disrupted by the weather are not changed and only subsequent flights are included in the IP. In all problems, the turn time distributions were identical to those in Problems 2 and 3.
Table 5.6.1 also shows the simulation-based cost comparison between not taking
any action and the IP solution. From these we can see that altering the aircraft allocation through the IP has the largest effect in the cases where an aircraft is grounded. This is not surprising as not making any changes to the schedule means that every subsequent flight of the disrupted aircraft will be delayed by several hours, each incurring high compensation costs. In reality, this would not happen and so the ‘No Action’ benchmark is less relevant. As the Day 1 disruption occurs in the afternoon, there are fewer future flights to be impacted, resulting in a smaller overall cost. In the weather related cases (Days 2, 5 and 6), where more flights are delayed but by a smaller amount, there seems more scope for recovery by using the buffer built into the schedule without schedule change. Whilst the IP can make positive aircraft exchanges, the impact is smaller. In the case of Day 6, this even leads to a negative impact. This could be due to the estimated ready times for disrupted aircraft being too conservative, leading to unnecessary delays.
Due to limited computational resources, we have only selected one IP solution from each problem to perform the simulation optimisation on, with 50 macro-replications for Day 1 and 25 for the others. As in Section 5.4, the results were simulated 1000 times using CRN to produce estimates of their mean and 0.95 quantile performance in cost. The less extensive results from these problems are shown in Figures 5.6.1, 5.6.2 and 5.6.3. The overall picture is consistent with the results of Section 5.4: the simulation optimisation almost always reduces the mean costs from the IP based solutions but is not always successful in reducing the 0.95 quantile. In all but Day 5, the STRONG algorithm was able to reduce both the mean and 0.95 quantile of the solution for all starting seeds.
However, on Day 5 there are three macro-replications that fail to find a noticeable improvement. Figure 5.6.3 shows that these solutions do not get worse (as perhaps suggested by the histogram format in Figure 5.6.1). On two occasions the algorithm finds a solution with only a slight improvement in mean and a larger 0.95 quantile.
Perhaps more concerning is that in one case of Day 5 the algorithm repeatedly rejects proposed solutions and so ends up where it began (seen as a dot at the intersection of the red lines in Figure 5.6.3). It may be that the objective function in this case is dominated by the variance of the costs or that the trust-region size does not shrink quickly enough to make use of a quadratic model, i.e., γ0 is too large and ˜∆ too small, though this has not been fully investigated.
Note that on Day 3, the IP rearranged the aircraft allocation in such a way that no delays were required. Therefore, the simulation optimisation was not needed. We believe that there will be disruptions of this kind, particularly at hub airports where there are often many options for rearranging aircraft allocations.
Figure 5.6.3 also shows the relationship between the mean and 0.95 quantile. None of these problems appear to have the conflicting relationship between quantile and expected value that was observed in Problem 1 (Section 5.4.1). The relationship looks stronger for the weather related problems, Days 2, 5 and 6. The reason behind this has not been explored.
Figure 5.6.1: Histograms of the mean cost for the simulation optimisation solution in the different problems. The red line is the mean of the starting solution from the IP.
Figure 5.6.2: Histograms of the cost 0.95 quantile for the simulation optimisation solution in the different problems. The red line is the 0.95 quantile of the starting solution from the IP.
Figure 5.6.3: Mean against 0.95 quantile for each problem. Red lines indicate the performance of the starting solution from the IP. The blue lines show the performance when no action is taken.