Integrated Airline Scheduling
4.3 Sequential Approach .1 Overview.1Overview
4.3.2 Solution Steps
4.3.2.2 Aircraft Maintenance Routing
104 4 Integrated Airline Scheduling The costs of each flight (arc) consist of operating and opportunity costs. Operat- ing costs include expenses directly related to the flight like costs for fuel, mainte- nance, landing fees etc. and can be easily obtained. For this application, the block hour costs multiplied by the block times are used (see page 66). Opportunity or spill costs are calculated by multiplying the number of spilled passengers with the fare they would have paid. This fare or yield is assumed to be given (see page 65). The number of spilled passengers is calculated by subtracting the assigned fleet type’s capacity from the unconstrained demand. In this application, the unconstrained de- mand is calculated using the schedule evaluation model presented in Sect. 4.2 with- out the spill and recapture step. Thus, the number of spilled passengers is the total of passengers demanding the flight as nonstop itinerary and as part of a connecting itinerary subtracting the capacity of the fleet type of the current flight arc. As a con- sequence, the connectivity of the flight schedule is incorporated in the opportunity costs.
Fig. 4.19 Splitting of nodes and distribution of arcs (Source: Gopalan and Talluri (1998a))
there is no optimization. Given the LOFs, the search for a valid solution repre- sents a network flow problem. A directed graph G= (V,E)represents the stations with aircraft staying at night (vertices V ) and the LOFs (arcs E). The number of arcs equals the number of aircraft of that fleet type, one aircraft is assigned to each LOF. V is partitioned into a set M of nodes representing maintenance sta- tions and in a set N=V\M representing non-maintenance stations. In this case, a valid maintenance routing is an Euler tour that includes no more than two nodes of N in succession. This Euler tour should be noted as three-day maintenance Eu- ler tour (3-MET). Assuming that the LOFs are connected and that the number of aircraft of one fleet type departing from one airport is equal to the number of ar- rival airports (flow balance constraint of the fleet assignment model), G is Eule- rian. Gopalan and Talluri (1998a) show that for all stations j∈N the number of arcs mojM going out to maintenance stations has to be greater than or equal to the number of arcs mijN coming in from non-maintenance stations to provide the exis- tence of a 3-MET. Assuming that this is true for G, a 3-MET in G can be found (if one exists) by searching for an Euler tour in a graph G derived from G. G is transformed into G by splitting each node j∈N into two nodes j and j and distributing the incoming and outgoing arcs of j between j and j as shown in Fig. 4.19. A number of(mojM−mijN) additional artificial arcs from j to j are included. Gopalan and Talluri (1998a) show that the existence of an Euler tour in Gimplies the extistence of a 3-MET in G. Thus, finding the Euler tour in Gwould result in the solution of the maintenance routing problem. The Euler tour can be found by a standard procedure (Bondy & Murty, 1978).
The described procedure represents a polynomial-time algorithm for finding a 3- MET. The 3-MET can only be found if the rotation is connected and the set of LOFs contains a 3-MET (mojM≥mijN). If the rotation is not connected or if mojM<mijN two heuristic methods presented by Gopalan and Talluri (1998a) must be applied to transform the LOFs (resp. G) to build a solvable problem. The Unlocker tries to con- struct connected LOFs, whereas the M-N Improver modifies the LOFs to meet the second condition. The use of heuristic methods to satisfy the maintenance require- ments is necessary because this problem is NP-hard (Gopalan & Talluri, 1998a).
Unlocker. If the LOFs of one fleet type are not connected, the situation is called a locked rotation. Each cycle of connected LOFs is denoted as component. In Fig. 4.20 an example of a locked (two components) and an unlocked (one com- ponent) rotation is given. The nodes represent the stations where the aircraft are
106 4 Integrated Airline Scheduling
(a) (b)
Fig. 4.20 Illustration of a locked (a) and an unlocked (b) rotation (Source: Gopalan and Talluri (1998a))
overnighting, the arcs represent the LOFs. In this example, a and f should represent the maintenance stations with a representing the balance check station, too. There are six aircraft and every aircraft can undergo the maintenance after three days.
However, because the graph is locked in figure (a), the aircraft flying between d, e and f will never visit the balance check station a making the routing invalid. In figure (b) the rotation is unlocked fulfilling the maintenance requirements. A graph can be unlocked by changing the flights within the LOFs, for example by swapping the tail assignment between two flights that depart from the same airport at the same time. For example, if the aircraft flying the LOF from b to c and the aircraft flying from d to e are on the ground at some station at the same time (after the turn time has elapsed), their assignment can be swapped leading to the unlocked rotation. In the following, three swapping methods are presented to convert a locked rotation into an unlocked rotation.12
1. In the first type of swap, tail numbers are switched between different LOFs within each fleet type, thus, the fleet assignment itself remains unchanged. If multiple components exist for one aircraft type, LOFs (one from each compo- nent) have to be found that intersect at one airport at the same time. Then, the flights of the LOFs following this intersection are changed between the LOFs, resulting in modified LOFs unlocking the graph.
2. The first swap mechanism might not be able to unlock the graph because it might be difficult to find an intersection possibility. Thus, the second mecha- nism changes the fleet type of some flights to unlock the graph. With this swap, only flights within an LOF might be changed, leaving the fleet composition at the end of each LOF unchanged. By changing only these flights, the fleet composition at overnight stations remains unchanged. In addition, changes are rather small, because for each swap the LOFs directly affected are changed;
changes are not carried on into connected LOFs via the overnight stations.
3. In the third swapping type the equipment type composition of overnighting air- craft might be changed, thus, increasing the total number of affected flights by the fleet assignment change.
12Examples for the swapping methods can be found in Gopalan and Talluri (1998a).
If a rotation is locked, the three swap mechanisms are applied in the order presented.
Swaps are only allowed if they do not result in new locked rotations of the affected fleets. Each swap has to be feasible. Especially when changing the fleet assignment, different block times of different fleet and operational limitations have to be taken into account. The objective of all three swap mechanisms is to unlock the graph rather than to increase profit. Thus, unlocking the graph might result in less profit because for example a small fleet type needed to be assigned to a high demand flight.
M-N Improver. If in a connected LOF-graph each node in N has fewer arcs coming in from N nodes than arcs going out to M nodes, a 3-MET does not exist, because there are at least three LOFs in succession that do not include maintenance stations at the arriving nodes. The objective of the M-N improver is to fulfill this requirement for each node in N by swapping pairs of edges. Assuming mojM<mijNfor node j∈N and that edge e comes into j from j1∈N. Edge e will be swapped with an edge e originating at a node in M. If the terminating node k of eis a N node, it must satisfy mokM−mkNj ≥1 to fulfill the M-N constraint in k. Assuming that e is an edge going out to a node j2∈N, a swap of e and eis allowed if there is a path to all other nodes from the origin nodes of e and e, because then the swap does not create a locked rotation.
Application. The model was implemented in the integrated planning approach as presented in the previous section. For the construction of the LOFs the FIFO-rule was used. However, several enhancements were included to better fit this model into the overall planning process and to increase the chance of finding a feasible solution.
For example, the model assumes that the number of LOFs corresponds to the num- ber of available aircraft. However, in this approach changes to the schedule are made in the fleet assignment and optimization step, possibly leading to different numbers of LOFs and aircraft. If aircraft of a fleet type remain unassigned after the construc- tion of the LOFs, new flights for this fleet type are created and included into the schedule. These flights are included in those markets that have the highest market size after realizing all currently scheduled flights (including flights of competing airlines). If these LOFs are not connected (more than one component), the flights are chosen to connect the different components. If the number of LOFs exceeds the number of aircraft of a fleet type, flights need to be removed from the schedule.
This task is accomplished by using the operator Flight Choice (presented on page 114) that assigns the attribute optional to the flights (see page 103) and by restarting the fleet assignment. All three swap mechanisms presented by Gopalan and Talluri (1998a) can only unlock a situation with multiple components of the same fleet type if this is possible with the current flights, thus, the Unlocker does not modify the flights except for the assigned fleet. Because this limits the probability of finding a feasible solution, additional unlocking steps are included (Extended Unlocker) that are applied when the Unlocker presented by Gopalan and Talluri (1998a) fails. The objective is to increase the number of potential positions that allow the swap of air- craft between different LOFs to connect them to one component (two aircraft of two different components have to be on the ground at the same airport at the same
108 4 Integrated Airline Scheduling time). This is accomplished by including new flights into the schedule. First, ad- ditional flights are inserted to connect two components. If additional flights cannot be included because there is not enough time left between the existing flights, the available time is increased by deleting other flights. The number of flights deleted is increased until sufficient connecting flights can be included (at maximum, all flights of one LOF are deleted to allow the unlocking). Like the Unlocker of Gopalan and Talluri (1998a), any modification is only allowed if all constraints are satisfied (cur- few restrictions, airport operating hours, turn times, operational restrictions etc.).
Although this procedure might result in large changes to the schedule with reduced profit, it has to be included to obtain feasible solutions.
The maintenance routing model assumes that each LOF begins with a flight de- parting after 2 a.m. The routing for each fleet type then is constructed by connecting these LOFs via the overnighting stations to a single circle. However, because the maintenance routing algorithm only considers the origin and destination airport of each LOF when connecting, it might be possible that an LOF starting at 2 a.m. is attached to an LOF ending after 2 a.m., if for example the last flight of the first LOF departs before and arrives after 2 a.m. Although not violating the constraints of the maintenance routing algorithm, this situation would lead to an infeasible routing sequence. In such cases, the routings have to be modified. If there is an LOF that exceeds the departure time of the following LOF, the amount of time that has to be saved to produce a feasible solution is removed from idle ground times between the flights of the LOF. If there is not enough ground time available, two succeeding flights (with the smallest market size) are replaced by one direct flight. To produce a feasible solution, any constraints (curfew restrictions, airport operating hours, fleet ranges, etc.) are taken into account when applying the changes.