Airline Scheduling Process
2.3 Aircraft Scheduling .1 Problem.1Problem
2.3.2 Solution Models
Aircraft routing and scheduling models were the earliest OR-models of airline plan- ning. First approaches were developed in the 1950s (Ferguson & Dantzig, 1956b;
Ferguson & Dantzig, 1956a) and 1960s (Dantzig, 1963; Miller, 1967). Most of these models consider simplified flight schedules where the number of alternative routes is small. The minimization of the fleet size necessary to fly the schedule was one com- mon objective in those approaches (Pollack, 1974; Gertsbach & Gurevich, 1977).
2.3.2.1 Fleet Assignment
The complexity of the basic daily fleet assignment problem is studied by Gu et al.
(1994). The authors analyze the structure of the solution as a function of the number of fleets; they observe that the complexity of the feasibility problem for two fleets is unknown and for three fleets it is NP-complete.
Most models to solve the daily fleet assignment problem are formulated as large multi-commodity flow problems with side constraints defined on a time-expanded network (Abara, 1989; Hane et al., 1995; Clarke et al., 1996; Barnhart & Talluri, 1997; Kim & Barnhart, 1997). The network contains flight arcs corresponding to flight legs, ground arcs corresponding to aircraft on the ground and overnight arcs corresponding to overnighting aircraft, the aircraft correspond to the commodities, a count line is used to determine the number of aircraft in use (see Fig. 2.5).
A typical mathematical formulation is given in the paper by Hane et al. (1995).
However, as the number of integer variables is large, it can be difficult and time- consuming to find optimal integer solutions. These problems are often severely degenerated, which leads to poor performances of standard integer linear program- ming (LP) techniques. Hane et al. (1995) discuss various methods to decrease the size of the problem, including variable aggregation, cost perturbations, dual sim- plex with steepest-edge pricing, and intelligent branch-and-bound strategies. Most approaches to the fleet assignment problem focus on different techniques to re- duce complexity when solving the formulated model (Sriram & Haghani, 2003).
2.3 Aircraft Scheduling 23 For example, in one of the first models, Abara (1989) uses the simplex method, fractional variables are then rounded to obtain an integer solution.
Daskin and Panayotopoulos (1989) present an integer program to solve the fleet assignment problem for a hub-and-spoke network with a single hub. They propose a Lagrangian relaxation of the problem and combine it with heuristics for converting the Lagrangian solutions into primal feasible solutions.
Berge and Hopperstad (1993) address the re-fleeting problem. The mathemati- cal programming formulation they describe is similar to the daily fleet assignment formulation, except for the aircraft count constraint, which is no longer required because the aircraft are positioned at the beginning of the planning horizon. The number of aircraft of each type present at each station at the beginning of the plan- ning horizon ist fixed. They present two heuristic solution approaches: one heuristic solves a sequence of single-commodity flow problems, and the other begins with a feasible assignment and performs multiple profit-improving aircraft swaps.
A weekly fleet assignment model is solved by Kliewer and Tsch¨oke (2000). The authors use a simulated annealing (SA) approach to deal with the higher complexity.
Kliewer (2000) combines this approach with a demand model: Once one tempera- ture level is completed, the current fleeting is sent to the market model. The market model then forecasts the number of passengers on each itinerary based on the given fleet assignment, and the simulated annealing algorithm continues.
In most fleet assignment models spill costs are leg-based. Thus, it is assumed that capacity is constrained only on the leg for which the estimate is being made and unconstrained on every other flight leg. In consequence, estimates of recap- tured revenue are achieved without knowledge of capacity or passenger flow on the flight network. Barnhart et al. (2002) develop a fleet assignment model based on O&D-itineraries using a a branch-and-price approach. This model is capable of capturing network effects and more accurately estimating spill and recapture of pas- sengers. Moreover, the authors include demand and fares for different fare classes in their model. Theoretically branch-and-price offers the best chance of finding a so- lution that is close to the optimum, but column generation requires the solution of a constrained shortest path problem which can be both memory and time consuming.
Moreover, its application requires significant customization of the IP solver and best reduced cost columns may improve the LP value but not the IP value (Klabjan et al., 2001b). Relaxations, heuristic procedures or integration of domain-knowledge are common practices to support or replace these decisions (Anbil et al., 1992; Chu et al., 1997).
Further information on the fleet assignment problem including models and ap- proaches is published by Sherali et al. (2006). In this paper, the authors present a tutorial including basic and integrated fleet assignment models on a detailed level.
2.3.2.2 Aircraft Routing
A common formulation for the aircraft routing problem is a network circulation problem with side constraints where exact and heuristic algorithms are applied to find feasible subtours. Solutions are considered to be feasible if each aircraft
overnights at an appropriately equipped maintenance station at least every three or four days (Barnhart & Talluri, 1997). Thus, flight connections during the day are fixed and only overnights are allowed as maintenance opportunities (Gopalan &
Talluri, 1998a). Because this may lead to aircraft rotations that are not able to fulfill the three- or four-day maintenance requirement, swapping techniques for the flights are necessary to unlock a rotation. Talluri (1998) develops a model for the four-day aircraft maintenance routing problem. Several heuristics and one exact approach are proposed to solve this problem. Furthermore, the mathematical complexity regard- ing the four-day routing problem and the three-day routing problem is investigated.
Bard and Cunningham (1987) consider the single through flight assignment prob- lem. When through values are not considered, the aircraft routing problem is usually reduced to a feasibility problem (Cordeau et al., 2001; Klabjan et al., 2002). How- ever, if through flight assignment and maintenance routing are solved separately, the latter problem is constrained by the results of the first, making it more difficult to find an optimal or even feasible solution (Gopalan & Talluri, 1998b). For exam- ple, since the pair of flight legs that constitute the through flight must be flown by the same aircraft, less freedom at the routing phase for the design of efficient rout- ings that meet maintenance requirements is provided. Clarke et al. (1997) present an aircraft routing problem under consideration of through revenues and maintenance constraints. The objective is to build rotations that are profitable measured by the sum of through values of routing flights through airports, operationally attractive in terms of a single rotation for each fleet, and satisfy maintenance requirements by allowing aircraft to visit maintenance stations regularly for a sufficient length of time. In this approach, all connections between flights are used as options for main- tenance (instead of using only overnight connections). Moreover, this approach can handle different types of maintenance requirements. The problem is formulated as an asymmetric traveling salesman problem with side constraints and is solved by using Lagrangian relaxation and heuristics.
Generally, in aircraft routing models capacity constraints at maintenance sta- tions are not considered because A-checks usually only require minor inspections.16 Some aircraft maintenance models consider the more intensive but less frequent balance-check. In order to ease scheduling operations, this constraint is met by per- forming the balance-check every n days whether there is one rotation of n aircraft in the fleet (Barnhart & Talluri, 1997; Gopalan & Talluri, 1998a).
Feo and Bard (1989) consider the maintenance location problem which involves finding the minimum number of maintenance stations required to meet the main- tenance requirement for a proposed flight schedule. Stops during the day are not considered as maintenance opportunities. This problem is formulated as a mini- mum cost, multi-commodity network flow problem with integer restrictions on the variables. Because the size of the formulation is too large to optimize, they give a two-phase heuristic that begins by generating aircraft assignments to match flight requirements. A probabilistically set covering heuristic is then used to locate the maintenance stations.
16Duffuaa and Andijani (1999) present an integrated simulation model for the planning of main- tenance operations for a maintenance station.
2.4 Crew Scheduling 25 Lan et al. (2006) solve the aircraft maintenance routing problem to create robust schedules. The objective is to find an aircraft routing that is less susceptible to flight delays and does not propagate a single disruption through the following flights. The problem is formulated as a mixed-integer programming problem with stochastically generated inputs, the objective function is to minimize the expected total propagated delay for the selected routings.
Sarac et al. (2006) solve an operational aircraft maintenance routing problem.
In this problem, a routing for individual aircraft to maintenance stations has to be found on a daily basis. In contrast to traditional maintenance routing approaches, this approach does not construct a regular maintenance schedule but focuses on the operational excecution of maintenance, which might be affected by stochas- tic events. In addition, maintenance resource availability constraints are taken into account. Based on the remaining legal flying hours without violating maintenance restrictions, the aircraft have to be rerouted to appropriate maintenance stations with enough maintenance hours and maintenance slots. The problem is formulated as a set-partitioning problem and solved using a branch-and-price approach.