Fleet Assignment

Given a flight schedule and a set of aircraft, the fleet assignment problem determines which type of aircraft, each having a different capacity, should fly each flight segment to max-imize profitability, while complying with a large number of operational constraints. The objective of maximizing profitability can be achieved both by increasing expected revenue, for example, assigning aircraft with larger seating capacity to flight legs with high passenger demand, and by decreasing expected costs, for example, fuel, personnel, and maintenance.

Operational constraints include the availability of maintenance at arrival and departure sta-tions (i.e., airports), gate availability, and aircraft noise, among others. Assigning a smaller aircraft than needed on a flight results in spilled (i.e., lost) customers due to insufficient capacity. Assigning a larger aircraft than needed on a flight results in spoiled (i.e., unsold) seats and possibly higher operational costs.

Literature on the fleet assignment problem spans nearly 20 years. For interested readers, Sherali et al. [26] provide a comprehensive tutorial on airline fleet assignment concepts, models, and algorithms. They suggest future research directions including the considera-tion of path-based demands, network and recapture effects, and exploring the interacconsidera-tions between initial fleeting and re-fleeting. Additional review articles on fleet assignment include Gopalan and Talluri [27], Yu and Yang [28], Barnhart et al. [9], Clarke and Smith [19], and Klabjan [20].

Several of the earliest published accounts are by Abara [29], who describes a mixed integer programming implementation for American Airlines; Daskin and Panayotopoulos [30], who present an integer program that assigns aircraft to routes in a single hub and spoke network;

and Subramanian et al. [14], who develop a cold-start solution (i.e., a valid initial assign-ment does not exist) approach for Delta Airlines that solves the fleet assignassign-ment problem initially as a linear program using an interior point method, fixes certain variables, and then solves the resulting problem as a mixed integer program. Talluri [31] proposes a warm-start solution (i.e., a valid initial assignment does exist) that performs swaps based on the number of overnighting aircraft for instances when it becomes necessary to change the assignment on a particular flight leg to another specified aircraft type. Jarrah et al. [32] propose a re-fleeting approach for the incremental modification of planned fleet assignments. Hane et al. [33] employ an interior point algorithm and dual steepest edge simplex, cost pertur-bation, model aggregation, and branching on set-partitioning constraints with prioritized branching order. Rushmeier and Kontogiorgis [17] focus on connect time rules and incor-porate profit implications of connections to assign USAir’s fleets resulting in an annual benefit of at least $15 million. Their approach uses a combination of dual simplex and a fixing heuristic to solve a linear programming relaxation of the problem to obtain an ini-tial solution, which in turn is fed into a depth-first branch-and-bound process. Berge and Hopperstad [34] propose a model called Demand Driven Dispatch for dynamically assigning aircraft to flights to leverage the increased accuracy of the flight’s demand forecast as the actual flight departure time approaches.

Attempts have been made to integrate the fleet assignment model with other airline pro-cesses such as schedule design, maintenance routing, and crew scheduling. Because of the

interdependencies of these processes, the optimal solution for processes considered sepa-rately may not yield a solution that is optimal for the combined processes.

For example, integrated fleet assignment and schedule design models have the potential to increase revenues through improved flight connection opportunities. Desaulniers et al.

[35] introduce time windows on flight departures for the fleet assignment problem and solve the multicommodity network by branch-and-bound and column generation, where the col-umn generator is a specialized time-constrained shortest path problem. Rexing et al. [36]

discretize time windows and create copies of each flight in the underlying graph to repre-sent different departure time possibilities and then solve using a column generator that is a shortest path problem on an acyclic graph. Lohatepanont and Barnhart [37] present an inte-grated schedule design and fleet assignment solution approach that determines incremental changes to existing flight schedules to maximize incremental profits. The integrated sched-ule design and fleet assignment model of Yan and Tseng [38] includes path-based demand considerations and uses Lagrangian relaxation, where the Lagrangian multipliers are revised using a subgradient optimization method.

Integrated fleet assignment and maintenance, routing, and crew considerations have the potential for considerable cost savings and productivity improvements. Clarke et al. [39]

capture maintenance and crew constraints to generalize the approach of Hane et al. [33] and solve the resulting formulation using a dual steepest-edge simplex method with a customized branch-and-bound strategy. Barnhart et al. [40] explicitly model maintenance issues using ground arcs and solve the integrated fleet assignment, maintenance, and routing problem using a branch-and-price approach where the column generator is a resource-constrained shortest path problem over the maintenance connection network. Rosenberger et al. [41]

develop a fleet assignment model with hub isolation and short rotation cycles (i.e., a sequence of legs assigned to each aircraft) so that flight cancellations or delays will have a lower risk of impacting subsequent stations or hubs. Belanger et al. [42] present both a mixed-integer linear programming model and a heuristic solution approach for the weekly fleet assignment problem for Air Canada in the case where homogeneity of aircraft type is desired over legs sharing the same flight number, which enables easier ground service planning.

Models to integrate fleet assignment with passenger flows and fare classes adopt an origin–destination-based approach compared to the more traditional flight-leg approach.

Examples of fleet assignment formulations that incorporate passenger considerations include Farkas [43], Kniker [44], Jacobs et al. [45], and Barnhart et al. [46].

The basic fleet assignment model (FAM) can be described as a multicommodity flow problem with side constraints defined on a time-space network and solved as an integer program using branch-and-bound. The time-space network has a circular time line rep-resenting a 24-hour period, or daily schedule, for each aircraft fleet at each city. Along a given time line, a node represents an event: either a flight departure or a flight arrival. Each departure (arrival) from the city splits an edge and adds a node to the time line at the depar-ture (arrival + ground servicing) time. A decision variable connects the two nodes created at the arrival and departure cities and represents the assignment of that fleet to that flight.

Mathematically, the fleet assignment model may be stated as follows, which is an adap-tation of Hane et al. [33] without required through-flights (i.e., one-stops). The objective is to minimize the cost of assigning aircraft types to flight legs as given in Equation 6.1. To be feasible, the fleet assignment must be done in such a way that each flight in the sched-ule is assigned exactly one aircraft type (i.e., cover constraints as given in Equation 6.2), the itineraries of all aircraft are circulations through the network of flights that can be repeated cyclically over multiple scheduling horizons (i.e., balance constraints as given in Equation 6.3), and the total number of aircraft assigned cannot exceed the number available in the fleet (i.e., plane count constraints as given in Equation 6.4). Additional inequality

Airline Optimization 6-9 constraints may be incorporated to address such issues as through-flight assignments, main-tenance, crew, slot allocation, and other issues.

min 

The mathematical formulation requires the following notation with parameters:

C = set of stations (cities) serviced by the schedule, I = set of available fleets,

S(i) = number of aircraft in each fleet for i ∈ I, J = set of flights in the schedule,

O(i) = set of flight arcs, for i ∈ I, that contains an arbitrary early morning time (i.e., 4am, overnight),

N = set of nodes in the network, which are enumerated by the ordered triple {iot}

consisting of fleeti ∈ I, station o ∈ C, and t = takeoff time or landing time at o.

t⁻= time precedingt, t⁺= time followingt,

{i_otn} = last node in a time line, or equivalently, the node that precedes 4 am, {iot1} = successor node to the last node in a time line, and decision variables:

X_iodt=X_ij = 1 if fleeti is assigned to the flight leg from o to d departing at time t, and 0 otherwise;

Y_iott⁺= number of aircraft of fleeti ∈ I on the ground at station o ∈ C from time t to t⁺. 6.2.3 Aircraft Routing

The Federal Aviation Administration (FAA) mandates some safety requirements for aircraft.

Those are of four types:

• A-checks are routine visual inspections of major systems, performed every 65 block-hours or less. A-checks take 3–10 h, and are usually performed at night at the gate.

• B-checks are detailed visual inspections, and are performed every 3 months.

• C-checks are performed every 12–18 months depending on aircraft type and operational circumstances, and consist of in-depth inspections of many systems.

C-checks require disassembly of parts of the aircraft, and must be performed in specially equipped spaces.

• D-checks are performed every 4–5 years, perform extensive disassembly and struc-tural, chemical, and functional analyses of each subsystem, and can take more than 2 months.

Of these checks, A and B are performed on the typical planning horizon of fleet planning and crew scheduling, and must therefore be incorporated into the problem.

Aircraft routing determines the allocation of candidate flight segments to a specific aircraft tail number within a given fleet-type while satisfying all operational constraints, including maintenance. Levin [47] is the first author to analyze maintenance scheduling. One simplified formulation of the maintenance problem, following Barnhart et al. [40], is the following. We define a string as a sequence of connected flights (i.e., from airporta to airport b, then from b to c, until a final airport) performed by an individual aircraft. Moreover, a string satisfies the following requirements: the number of block-hours satisfies maintenance requirements, and the first and last nodes in the sequence of airports are maintenance stations. A stringk has an associated costck. LetS be the set of all augmented strings. For every node v, I(v) andO(v) denote the set of incoming and outgoing links, respectively. The problem has two types of decision variables:x_sis set to 1 if strings is selected as a route for the associated fleet, and 0 otherwise.y_mis the number of aircraft being serviced at service stationm. The problem then becomes:

min 

s∈S

csxs (6.7)

subject to: 

i∈S

xs= 1 for all flightsi (6.8)



j∈O(v) j∈s

x_s− 

j∈I(v) j∈s

x_s+y_O(v)− y_I(v)= 0 for all stationsv (6.9)

wherey ≥ 0, x = 0/1.

The problem can be formulated as a multicommodity flow problem (e.g., Cordeau et al.

[48]), as a set partitioning problem (Feo and Bard [49]; Daskin and Papadopoulos [30]), and employing eulerian tours (Clarke et al. [50], Talluri [31], and Gopalan and Talluri [51]).

Recent work includes that of Gabteni and Gronkvist [52], who provide a hybrid column generation and constraint programming optimization solution approach, Li and Wang [53], who present a path-based integrated fleet assignment and aircraft routing heuristic, Mercier et al. [54], who solve the integrated aircraft routing and crew scheduling model using Benders decomposition, and Cohn and Barnhart [55], who propose an extended crew pairing model that integrates crew scheduling and maintenance routing decisions.