Integrated Airline Scheduling
4.3 Sequential Approach .1 Overview.1Overview
4.3.2 Solution Steps
4.3.2.1 Fleet Assignment and Flight Scheduling
Model. The fleet assignment and flight scheduling is conducted following the model by Rexing et al. (2000). This model is based on the approach by Hane et al. (1995) and integrates the daily fleet assignment with the flight scheduling into one single model. Given a flight schedule, a time interval around the actual departure time of each flight is specified. The model is then allowed to change the departure time in the interval and assigns a fleet type. Within each time interval, demand is assumed to be constant and independent of the actual departure time and fleet assigned. If the time window size is small (for example 30 minutes), changes in the departure time will be small, limiting demand variations. The objective is to minimize the direct operating costs and opportunity costs. Opportunity costs exist if potential passengers must be left behind because the demand for a flight is higher than the assigned fleet’s capacity. The general constraints of the fleet assignment problem presented on page 18 are included in this model.
The fleet assignment problem is represented as a network flow problem consist- ing of nodes, ground arcs and flight arcs. Each fleet type is represented by a separate network containing all airports in consideration. The nodes represent departures or
arrivals at an airport at a specific time. The nodes at each airport are ordered by their time, thus every node has a direct predecessor and a direct successor. A time line is constructed by connecting these nodes by ground arcs, representing an air- craft resting on the ground between its arrival and subsequent departure. Flights are represented as flight arcs connecting the corresponding nodes. The arrival node of each flight arc is placed at the ready time (block time + turn time) of the flight. To guarantee flow balance in the daily fleet assignment, ground arcs connect the last and the first node at every airport. The number of aircraft in use can be calculated by summing up all flows on all (flight and ground) arcs at one point in time (count time). Fig. 4.17 illustrates one example of a flight network with two airports.
Fig. 4.17 Two-airport flight network (Source: Rexing et al. (2000))
By introducing time windows for each flight, the flight scheduling problem can be included into this representation of the problem. A time window specifies how much the original departure time is allowed to vary. The additional flexibility in choosing departure times is expected to result in a more efficient (less expensive) schedule.
For example, two flights are assigned to two different aircraft because the ready time of the first flight is later than the departure time of the second flight. If the departure time (and thus arrival/ready time) of the first flight can be shifted forward and the departure time of the second flight can be delayed, it might become possible to assign both flights to the same aircraft. In this model formulation, the time window is split up in intervals (for example five-minute intervals) and copies of the original flight (arc) are placed at every interval, representing alternative departure times (see Fig. 4.18). The model is then allowed to choose only one copy of each set, fixing
Fig. 4.18 Two-airport flight network with time win- dows (Source: Rexing et al.
(2000))
the final departure time for each flight. Introducing time windows is quite simple, however, the resulting model becomes computationally expensive. One parameter to control the difficulty of the problem is the number of copies per flight and the time window size. Because it is assumed that there is no variation of the demand within each time window, the size of each window should be kept small to comply
102 4 Integrated Airline Scheduling with this assumption, however, limiting the degree of freedom for flight scheduling in this combined model.
After constructing the flight networks for each fleet, the fleet assignment problem can be solved using the following formulation.
Minimize:
k∈K
∑ ∑
i∈F
∑
n∈Nik
cikxnik (4.45)
Subject to:
k∈K
∑ ∑
n∈Nik
xnik =1 ∀i∈F (4.46)
k∈K
∑ ∑
n∈Nik
b1lniknnik+
∑
g∈Gk
b2lgkygk =0 ∀l∈Lk,∀k∈K (4.47)
i∈F
∑ ∑
n∈Nik
d1nikxnik+
∑
g∈Gk
d2gkygk ≤Sk ∀k∈K (4.48) ygk ≥0 ∀g∈Gk,∀k∈K (4.49) xnik∈ {0,1} ∀i∈F,∀n∈Nik, (4.50)
∀k∈K Parameters:
F = set of flights K = set of fleets
Sk = number of aircraft of fleet k
Gk = set of ground arcs in fleet k’s network Lk = set of nodes in fleet k’s network
Nik = set of arc copies of flight i in fleet k’s network
|Nik| = number of arc copies of flight i with fleet type k cik = cost to fly flight i with fleet type k
b1lnik=
⎧⎪
⎨
⎪⎩
1 if copy n of flight i begins at node l in fleet k’s network
−1 if copy n of flight i ends at node l in fleet k’s network 0 otherwise
b2lgk =
⎧⎪
⎨
⎪⎩
1 if ground arc g begins at node l in fleet k’s network
−1 if ground arc g ends at node l in fleet k’s network 0 otherwise
d1nik =
1 if copy n of flight i crosses the count time in fleet k’s network 0 otherwise
d2gk =
1 if ground arc g crosses the count time in fleet k’s network 0 otherwise
Decision Variables:
xnik =
1 if copy n of flight i is flown by fleet k 0 otherwise
ygk = number of aircraft on ground arc g in fleet k’s network
Equation 4.45 is the objective function minimizing the costs of assigning aircraft types to the flight arcs (operating costs and opportunity costs). Constraint 4.46 re- quires that each flight is covered by exactly one fleet by allowing only one copy of flight arcs to be chosen. Constraint 4.47 ensures the flow balance at each node, constraint 4.48 limits the number of the available aircraft.
Application. The model was implemented in the overall sequential planning ap- proach as presented in the previous section. Network preprocessing steps suggested by Rexing et al. (2000) to prune the problem before constructing and solving the LP matrix were not implemented, because preliminary tests showed that these steps are not necessary for the problem instances examined in this study (Barth, 2005). In addition, the optimization steps of the integrated approach represent the computa- tionally demanding part during application.
The fleet assignment problem cannot be solved if the number of available aircraft is not sufficient to perform all flights given. Thus, it is necessary to remove flights from the schedule. Therefore, an additional attribute optional is introduced and as- signed to every flight that might be removed from the schedule. If a flight is optional, constraint 4.46 is changed for this flight in order to allow that no copy of flight arcs at all is covered, resulting in the deletion of the flight from the schedule. Because the fleet assignment model tries to minimize the costs, usually an optional flight is removed from the schedule if no other constraint (for example flow balance) is vi- olated. Furthermore, to meet maintenance restrictions sometimes it is necessary to assign certain flights to specific fleet types and to prevent the presented model from changing this assignment. For example, one fleet type is assigned only to flights that do not depart or arrive at a suitable maintenance station for this type. In such cases, the attribute maintenance assigned to those flights indicates that copies of the flight arc are allowed with the fleet type assigned in the previous planning cycle resulting in a similar fleet assignment. Both attributes optional and maintenance are set out- side of the presented flight scheduling and fleet assignment model by the supportive functions presented later (see page 113).
Because Rexing et al. (2000) show that narrow departure times for all copies of one flight cause an extreme increase in the problem size, often without the benefit of providing a substantially better solution than broader departure times would, the number of flight arc copies in this approach is limited to five. The time window size in minutes is set by the parameter tw with all copies of flight arcs evenly distributed within this interval. Flight arcs that violate airport operating hours or curfew restric- tions are not included. Following the suggestion in the paper written by Rexing et al.
(2000), the count time is set to a time which is crossed usually only by wrap around arcs and a few flight arcs. In this approach, 03:00 a.m. is used as count time.
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.