t041 - air cargo ﬂeet routing and timetable setting with multiple on-time demands

7/29/2019 T041 - Air cargo fleet routing and timetable setting with multiple on-time demands

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Air cargo fleet routing and timetable setting

with multiple on-time demands

Shangyao Yan *, Shin-Chin Chen, Chia-Hung Chen

Department of Civil Engineering, National Central University, Chungli 32054, Taiwan, ROC

Received 7 July 2004; received in revised form 3 January 2005; accepted 5 February 2005

Abstract

In this research we combine airport selection, fleet routing and timetable setting to develop an integrated

scheduling model. The objective is to maximize operating profit, given the related operating constraints.

The model is formulated as a mixed integer program that is characterized as NP-hard. We develop several

heuristics, and incorporate the use a mathematical programming solver, to solve the problem. To evaluate

the model and the solution algorithms, we perform a case study using real operating data from a majorTaiwan airline. The results show that the model and the solution algorithms could be useful for actual

operations.

2005 Elsevier Ltd. All rights reserved.

Keywords: Cargo; Airport selection; Fleet routing; Timetables; Timespace network

1. Introduction

According to predictions made by the International Civil Aviation Organization (ICAO), afterAD 2000, airlines in the Asian-Pacific area will continue to dominate the international air freight(cargo) market. In recent years, Taiwan, in the center of this region, has striven to develop itself asan Asian and Pacific air freight transportation hub. Moreover, air freight markets in this area

1366-5545/$ - see front matter 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.tre.2005.02.002

* Corresponding author. Tel.: +886 3 422 7151x34141; fax: +886 3 425 2960.

E-mail address: [email protected] (S. Yan).

www.elsevier.com/locate/tre

Transportation Research Part E 42 (2006) 409430
mailto:[email protected]:[email protected]


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have become more competitive than before, so under these circumstances, Taiwan s cargo carriers

have had to provide better service of less cost, in order to maintain their market competitiveness.Fleet routing and flight scheduling are essential to a carriers operations. The setting of a good

flight schedule cannot only enhance operating performance, but can also improve aircraft usage.Most cargo carrying airlines in Taiwan currently utilize a trial-and-error experience-based methodfor flight routing and scheduling, with the objective of maximizing the system profit or their mar-

ket share. First a set of airports is first selected with experience, then a timetable is drafted and aset of projected time periods are made that meet the cargo transport demands (i.e., the multipleon-time demands). The drafted timetable and the flight schedules are then adjusted according tofleet availability, approved flight quotas of airports/airport pairs, airport turn-around times, the

aircraft balance at each station, and other related operating costs/passenger revenues. This processis iterated manually, without optimization, from a systemic perspective. After adjustments, the

schedule is then checked for aircraft maintenance and crew scheduling viability, and minor revi-

sions may be made. Such a trial-and-error method is neither efficient nor effective, especially forlarge service networks.Much research by the air industry as well as academics has already been devoted to fleet routing

and flight scheduling problems. For example, Abara (1989) developed an integer linear program-ming model, formulated as a multi-commodity network flow problem, with fixed flight departure

times for fleet assignment. Dobson and Lederer (1993) developed a three level hierarchical processto study the competitive choice of flight schedules and airfares by airlines in a pure hub-and-spoke(with single hub) system. Hane et al. (1995) modified Abaras model so that it could solve daily

aircraft routing and scheduling problems (DARSP) without departure time windows. Clarkeet al. (1996), based on Hane et al.s basic model, tried to develop a fleet assignment model whichwould take aircraft maintenance and crew scheduling into consideration.

Yan and Young (1996) developed a set of network models that could help a carrier effectivelydesign short term flight schedules and fleet routs given a drafted timetable and other operatingconstraints. Their models had the advantage of being more systematic and efficient than the tra-

ditional trial-and-error method. Desaulniers et al. (1997) proposed two integer programmingmodels, a set partitioning type model and a time constrained multi-commodity network flowmodel, which could solve DARSPs utilizing a set of operational flight legs with known departuretime windows. To improve Yan and Youngs model, Yan and Tseng (2002) developed an inte-

grated scheduling model for multi-fleet routing and flight scheduling. The objective was to max-imize the system profit, given a fixed projected passenger demand and other operating constraints.They also developed a Lagrangian relaxation-based algorithm to efficiently solve the model.

Barnhart et al. (2002) considered airline fleet assignment problems involving profit maximizationand the assignment of different aircraft types to different flight legs. They proposed a new formu-

lation and solution approach that better captured network effects and generated superior solu-tions. Lohatepanont and Barnhart (2004) focused their attention on the steps of the airlineschedule planning process that involved schedule design and fleet assignment. They describedintegrated models and solution algorithms that could simultaneously optimize the selection offlight legs and the assignment of aircraft types to the selected flight legs.

The above-mentioned research on flight scheduling has mainly focused on passenger transpor-tation, which is fundamentally different from cargo transportation. In particular, the selection of

airports in a passenger service network usually involves long-term planning, but in cargo trans-

410 S. Yan et al. / Transportation Research Part E 42 (2006) 409430


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port, especially for Taiwans air carriers, this is not the case. To respond to significant rapid fluc-

tuations in demand, carriers must perform their airport selection, fleet routing and timetable set-ting to formulate short-term plans, while still considering demand and profit. Moreover,

passengers are more time sensitive than cargos. Too many transfers in a passenger service mayresult in a significant loss of customers, but cargos are not lost, provided they can be deliveredon time.

Research on freight transportation and fleet routing has been performed by many researchers.For example, Chan and Ponder (1979) reviewed the air freight industry with special reference tothe Federal Express Corporation. They outlined the characteristics of the industry and presenteda survey of different managerial practices. Chestler (1985) described the basic structure of a pure

hub-and-spoke network for air express carriers. Current et al. (1986) was the first to introducehierarchical network design problems. In subsequent research, Current et al. (1988) extended

the design problems to include transshipment facilities with fixed costs at the intersections of pri-

mary and secondary routes. Aykin (1995) addressed hub location and routing problems involvingthe joint determination of the location of interacting hubs and the types of service routes betweenthese points. They presented a mathematical formulation of the problem and a solution algo-

rithm. Jaillet et al. (1996) introduced three integer linear programming models for designingcapacitated networks and routing policies, without the assumption of a hub-and-spoke structure.

In the literature we found typically, that airport selection and frequency planning were consideredin a service network design. Both are related to long-term planning. Our research, however, fo-cuses in integrating airport selection, fleet routing and timetable together in short-term

operations.We note that, meta-heuristics have recently been employed to solve routing/scheduling prob-

lems, which are usually formulated as combinatorial optimization problems (e.g. see Gu and

Huang, 1994 or Brandao and Mercer, 1997). Only a few meta-heuristics have been applied tosolve pure (single commodity) network flow problems similar to the multi-commodity networkflow problem discussed below. For example, Palmer and Kershenbaum (1995) developed a genetic

algorithm (GA) that used Link and Node Biased (LNB) encoding method to solve an optimalcommunication spanning tree problem. Abuali et al. (1995) developed a GA that used a determi-nant encoding method to solve a probabilistic minimum spanning tree. Taguhi et al. (1998) pre-sented a GA, with a non-uniform mutation and an arithmetic crossover, that could solve optimal

flow assignment problems in computer networks. Yan and Luo (1998, 1999) employed the tabusearch (TS), threshold accepting (TA), and simulated annealing (SA) methods in their develop-ment of several advanced local search algorithms that could be used to solve bipartite transpor-

tation network problems. Yan et al. (in press) utilized a GA to develop a global search algorithmfor solving concave cost transshipment problems. However, the aforementioned meta-heuristics,

which were all developed for solving pure network flow problems without side constraints, are dif-ficult to apply to multi-commodity network flow problems.

In this research, on the basis of the carriers perspective, we develop an integrated schedulingmodel that combines airport selection, fleet routing and timetable setting, with the objective ofmaximizing the operating profit, given a set of projected cargo demands and the related operating

constraints. We also develop several heuristic algorithms to find solutions. Some airlines have al-ready introduced combi flights into their operations, meaning passengers and cargo are trans-

ported at the same time. However, the airline used as an example in this research has only one

S. Yan et al. / Transportation Research Part E 42 (2006) 409430 411


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type of cargo aircraft (B747-400F), so for simplicity, we have focused on pure cargo flight trans-

portation to construct the model. It is expected that this model will be a useful planning tool forcargo carriers to determine suitable service airports, fleet routes and timetables for short-term

operations with on-time demands.The scope of this research is confined to cargo fleet routing, flight scheduling and airport selec-

tion. Although, in practice, the related aircraft maintenance and crew scheduling processes must

be considered during scheduling, these processes are usually modeled separately to facilitate prob-lem solving (Teodorovic, 1986). For the studied Taiwan airline, the aircraft maintenance and crewconstraints are actually rather flexible, due to the use of stand-by crews and a progressive main-tenance policy. These processes are performed after the fleet routes and flights schedules have been

solved. Thus, to reduce problem complexity, as done in conventional research, e.g. Yan andYoung (1996) and Yan and Tseng (2002), we exclude these constraints in the modeling.

The rest of this paper is organized as follows: first, we introduce the model and develop several

solution algorithms to solve the proposed model. Then, a case study is conducted to evaluate theperformance of the model and the solution algorithms. Finally, some conclusions are offered.

2. Modeling approach

A timespace network technique is applied to construct an integrated scheduling model thatcombines airport selection, fleet routing and timetable setting, with the objective of maximizing

the operating profit. The major elements in the modeling, including the fleet-flow timespacenetwork, the multi-cargo-flow timespace networks, and the mathematical formulation, aredescribed below.

2.1. The fleet-flow timespace network

A timespace network, shown in Fig. 1, is established for single-fleet routing within a specifiedtime period (one week in this study) and specified locations. The horizontal axis represents theairport locations; the vertical axis stands for the time duration. All available airports are included.Nodes and arcs are the two major components in the network. Each node designates a spe-

cific airport and a specific time, while each arc represents an activity for an airplane, such as aflight leg, a ground holding period, or an overnight stay. The arc flows express the flow of air-planes in the network. Three types of arcs are defined below.

2.1.1. Flight leg arc

A flight leg arc represents a flight connecting two airports. All possible flight legs between thetwo airports in the network, within a reasonable block of time, are considered, as long as timeslots at the respective airports are available. Each flight leg arc contains information about thedeparture time, the departure airport, the arrival time, the arrival airport, and the operating cost.The time block for a flight leg is calculated as from the time when the airplane is prepared for this

flight leg to the time when this flight leg is finished. Basically, this time includes the time for inves-tigation prior to departure, fuelling, cargo loading and unloading, and flight time. The flight cost

is the arc cost. The arc flows upper bound is one, meaning that the flight leg can be served at most



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once. The arc flows lower bound is zero, implying that no airplane serves this flight leg. In addi-tion, the departure interval at an airport is adjustable to meet the carriers operating requirements.

Moreover, if a flight leg is served, then the two airports associated with that flight leg are used. Afixed cost, as well as a variable cost for using the two airports, should be imposed.

2.1.2. Ground arc

A ground arc represents the holding or the overnight stay of an aircraft at an airport in a timewindow. The arc cost, which includes the airport tax, the holding (or overnight stay) fee, the gate

use charge and other related costs, denotes the expenses incurred for holding an aircraft at an air-port in the corresponding time window. The arc flows upper bound is the apron capacity (or

infinity, if the capacity is large), indicating the maximum number of airplanes that can be heldat this airport during a specific time window. The arc flow s lower bound is zero, implying that

no airplane is held at this airport in this time window.

2.1.3. Cycle arc

A cycle arc represents the continuity between two consecutive planning periods. It connects theend of one period to the beginning of the next period for each airport. The arc cost is the cost of

holding an aircraft overnight, and is similar to the ground arc cost, but with an additional over-night charge. The upper bound and lower bound of the arc flow are set as the same as those of theground arcs.

Station - 1 Station - kStation - 3Station - 2

(1) 4:00 (day1)

8:00 (day1)

12:00 (day1)

(3)

16:00 (day1)

20:00 (day1)

24:00 (day1)

24:00 (day7)

20:00 (day7)

(2)

Fig. 1. Fleet-flow timespace network.



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2.2. The cargo-flow timespace network

The timespace network technique is also applied to formulate cargo movement corresponding

to specified times (one week in this study) and locations, as shown in Fig. 2. Each cargo-flow timespace network represents a specific OD pair from the origindestination table (known as the ODtable). Such networks are designed to correspond to the fleet-flow timespace network, so as to

facilitate problem solving. In addition, it may be necessary for cargos with the same or differentODs to be delivered within different lengths of time, which means that the time window for deliv-ery for an OD pair may be less than the length of the airline schedule (i.e., a week in this study).According to the time sensitivity of the cargos handled by the airline, we divide the time windows

into three types (one day, four days, and one week). The horizontal and vertical axes are definedto be the same as those in the fleet-flow timespace network. A node, here, also represents an air-

port at a specific time; however, an arc designates a cargo movement activity. Altogether, there

are three types of arcs defined.

Station -1 Station -2 Station -3 Station-k

0 :00 (1)

4 :00 (1)

8 :00 (1)

12 :00 (1)

16 :00 (1)

20 :00 (1)

24 :00 (1)

4:00 (2)

8:00 (2)

12 :00 (2)

16:00 (2)

20 :00 (2)

OD -time-pair : ( 1,2 ) ---- ( the 1st layer)

OD- time -pair : ( 1,3 ) ---- (the2rdlayer)

OD- time -pair : ( m,m ) ---- (the nth layer)

0:00(1)

4:00(1)

8:00(1)

12 :00 (1)

16 :00 (1)

20:00(1)

24:00(1)

4:00 (2)

8:00 (2)

12:00(2)

16 :00 (2)

20 :00 (2)

24:00(2)

4:00(3)

8:00(3)

16:00(7)

20:00(7)

24 :00 (7)

::::::

24 :00 (2)

(1)

(2)(3)

(1) delivery arc (2) holding arc (3) demand arc

Fig. 2. Cargo-flow timespace network.



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2.2.1. Delivery arc

A delivery arc represents the transportation of the cargo from one airport to another on a spe-cific flight. The transportation time is the same as the corresponding time block in the fleet-flow

timespace network for the associated flight. The arc cost is a variable cost for handling the cargo,per unit weight, on that flight, which is, in general, very small compared to the flight cost. The arcflows upper bound is the aircrafts capacity (denoted as the weight capacity for the studied air-

line). The arc flows lower bound is zero, indicating that no cargo from the corresponding OD pairis delivered on the associated flight.

2.2.2. Holding arc

A holding arc indicates the holding of a cargo in a specific time window. The arc cost is theholding cost (or penalty) for this time window. However, if the arc just happens to connect either

the departure or the arrival station of the corresponding OD pair, the arc cost is then zero, be-

cause whether a cargo is held before or after the delivery is usually not decided by the airline. Nev-ertheless, in practice the arc cost is adjustable. That is, if, in some special cases, a holding period atthe O/D airport needs to be considered, then a suitable holding cost can be imposed. The arc

flows upper bound is the stations cargo service capacity (or infinity, if the capacity is relativelylarge), meaning the maximum amount of cargo (in weight units appropriate for the studied air-

line), that can be accommodated at this airport. The arc flow s lower bound is zero, showing thatno cargo from the corresponding OD pair is held at the airport during this time window.

2.2.3. Demand arc

A demand arc, associated with an OD pair, connects the arrival station at the last time to thedeparture station at the first time. It denotes the actual service demand for this OD pair. The arc

cost is a negative value for cargo per unit weight delivered.1 The arc flows upper bound is theprojected demand for this OD pair. The aim is to maximize profit, which implies that, for thisOD pair, not all cargos will necessarily be served. The arc flow s lower bound is zero, meaning

that none are served. The trip demand for a specific OD pair can be flexibly divided into severaltransportation time intervals, according to the actual delivery requirements. For example, thetime length of the corresponding cargo-flow timespace network could be shorter for an OD pairincluding express deliveries. In contrast, the time length could be longer for a cargo that is less

time sensitive. The time lengths are adjustable.

2.3. Notation and symbols used in the model

Before introducing the model formulation, we list the notation and symbols used:Xij the arc(i,j) flow in the fleet networkYnij the arc(i,j) flow in the nth cargo networkWi a decision variable, which equals 1 if station i is served, and 0 otherwise

1 The cargo rate structure that an airline charges a forwarder or shipper is, in general, in decreasing ladder form (i.e., a

concave function in terms of cargo accumulation). However, the amount of cargo transported on a flight is far larger

than the charge to an individual customer. Consequently, in the planning scheduling stage, an average cargo fare per

unit weight is usually used.



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Cij the arc(i,j) cost in the fleet networkTnij the arc(i,j) cost in the nth cargo networkFi a fixed cost for choosing station i, either as a departure airport or an arrival airport

Vi a variable cost at station i to handle cargos per unit weight, including loading andunloading

n, N the nth OD pair and the set of all ODsA, NF the set of all arcs and nodes in the fleet networkBn, NPn the set of all arcs and nodes in the nth cargo networkAF the number of available airplanes in the fleet networkFF the set of all flight leg arcs in the fleet network

CF the set of all cycle arcs in the fleet networkBFn the set of all demand arcs in the nth cargo networkSa the set of flight arcs associated with the ath station

S

ab

the set of flight arcs that connect the ath station to the bth stationQa the approved flight quota at the ath stationQab the approved flight quota that connects the ath station to the bth stationK the aircraft capacity (including a planning load factor)SA the set of all stations

SP the set of airport pairs with an approved flight quotaUij the arc(i,j) flows upper bound in the fleet networkUnij the arc(i,j) flows upper bound in the nth cargo networkB a very large value

2.4. The model formulation

The model, given the fleet-flow and the cargo-flow time space networks introduced above, isformulated as a mixed integer program. The objective of this model is to flow all aircraftand cargos simultaneously, in all networks at a minimum cost considering the cargo handling

cost at the selected airports. Since the revenue from the cargo-flow networks is in the form ofa negative cost, this objective is equivalent to the maximization of profit. There are several

other issues that need to be carefully considered, such as: the number of aircraft requiredshould not exceed the number of available airplanes; the accumulation of flights for a certainperiod at a specific airport/airport pair should not exceed the available quota; the amount of

cargo carried on a flight should not exceed the capacity of that aircraft; and the airport se-lected should be able to handle cargo. Consequently, four types of side constraints are neces-sary for problem formulation: (1) the sum of the cycle arc flows in the fleet-flow network

should not be greater than the number of available airplanes; (2) the sum of all individualflights for each airport/airport pair should not exceed its approved flight quota; (3) the sumof all delivery arc flows corresponding to the same flight should not exceed the flight arc flowmultiplied by the airplane capacity; and (4) the sum of all cargo network arc flows correspond-

ing to an origin/destination airport should be less than or equal to a very large value times thebinary variable value (either one or zero) associated with this airport. The model is formulatedas follows:



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Min ZX

ij2A

Cijxij X

n2N

X

ij2Bn

Tnijynij X

i2SA

Fiwi X

n2N

X

ij2BFn

ynijVi Vj 1

s:t::X

j2NF

xij X

k2NF

xki 0 8i 2 NF 2

X

j2NPn

ynij X

k2NPn

ynki 0 8i 2 NPn; 8n 2 N 3

X

ij2CF

xij 6 AF 4

X

ij2Sa

xij 6 Qa 8a 2 SA 5

X

ij2Sab

xij 6 Qab 8ab 2 SP 6

Xn2N

yn

ij6 Kxij 8ij 2 FF 7

0 6X

n2N

X

ij2BFn

ynij 6 Bwi 8i 2 SA 8

xij 6 1 8ij 2 FF 9

0 6 xij 6 Uij 8ij 2 A 10

0 6 ynij 6 Unij 8ij 2 Bn; 8n 2 N 11

xij 2 I 8ij 2 A 12

wi 2 f0; 1g 8i 2 SA 13

The model is formulated as a mixed integer multiple commodity network flow problem, in which

the objective is to minimize the system cost. Constraints (2) and (3) ensure flow conservation atevery node in each fleet/cargo network. Eq. (4) denotes that the number of airplanes used inthe fleet network should not exceed the available number of airplanes. Eq. (5) ensures that the

sum of all flights at each station does not exceed its approved quota. Eq. (6) ensures that thesum of all flight arcs connecting the ath station to the bth station does not exceed the approved

flight quota. Eq. (7) keeps the cargo delivery volume within the aircrafts carrying capacity. Eq. (8)is used to determine whether a station is used for cargo serving or not. That is, if station i is usedfor serving cargos, then Wi = 1; otherwise Wi = 0. Eq. (9) indicates that each flight is served atmost once. Eqs. (10) and (11) hold all the arc flows within their bounds. Eq. (12) ensures the inte-

grality of the airplane flows. Eq. (13) indicates that each airport selection decision is binary.2

3. Solution method

The model is formulated as a mixed integer program that is characterized as NP-hard (Garey

and Johnson, 1979). It is almost impossible to optimally solve a realistically large problem within

2 If a carrier has to provide a minimum number of flights for certain station-pairs, due to a specific marketing strategy,

then a constraint can be introduced for each of the station-pairs, to ensure that the sum of the associated flight leg arc

flows is greater than or equal to the minimum number of flights.

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a limited time. For example, in numerical tests the all-stop heuristic (Section 4.1) took 110668.77 sto solve a small-scale problem with only eight stations and four airplanes. Therefore, we must de-velop a family of heuristics to solve large-scale problems. The heuristics take into account the

stops required by the cargo to be transported, as described below:

(1) Non-stop heuristic: as shown in Fig. 3a, all cargos are delivered using non-stop flights thatgo directly from their point of origin to their destination. In this heuristic, the delivery arcs in

each cargo network need only be set from their associated origin to their associated destina-tion. Other delivery arcs are removed from the cargo network. The modified model, the non-

stop network, can then be optimally solved.(2) One-stop heuristic: as shown in Fig. 3b, all cargos are transported using non-stop and/or

one-stop flights. In this heuristic, the delivery arcs in each cargo network need only be setfrom the associated origin to all other stations and from all other stations to the associated

destination. Other delivery arcs are removed from the cargo network.3 The modified model,the one-stop network, can then be optimally solved.

(3) All-stop heuristic: as shown in Fig. 3c, all cargos are transported using non-stop, one-stop,

and/or multiple stop flights. In this heuristic, all possible delivery arcs between two airportsin each cargo network are set.4 The modified model, the all-stop network, can then be opti-

mally solved. Obviously, the problem size of the all-stop network is larger than that of the

one-stop network, which is larger than that of the non-stop network.(4) Mixed-stop heuristic: Cargos are transported using non-stop, one-stop, and/or multiple stopflights, according to the OD distance. Although the all-stop network is useful for finding themost effective transport plan, its problem scale may be too large to solve. To suitably reduce

the problem size of the all-stop network, the delivery arcs in this heuristic, in each cargo net-work, are set according to their OD distance. In particular, in accordance with real practice,

origin destination

(a) Non-stop network

origin destination

(b) One-stop network

origin destination

(c) All-stop network

Fig. 3. Heuristic Networks.

3 There is no delivery arc pointing to the associated origin or emanating from the associated destination.4 Similar in the one-stop network, there is no delivery arc pointing to the associated origin or emanating from the

associated destination.



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short-haul OD cargo networks are designated as non-stop networks. Similarly middle-haul/long-haul OD cargo networks are designated as one-stop/all-stop networks. The modifiedmodel, the mixed-stop network, can then be optimally solved. This model can effectively pro-vide multi-stop/non-stop flights for long-haul OD cargos, one-stop/non-stop flights for mid-

dle-haul OD cargos, and non-stop flights for short-haul OD cargos.

Solve all-stop networks using

CPLEX

Set L1 and L2 to represent

boundary distances

Modify the selected networks

from non-stop networks to

one-stop networks

Converges?

Solve the non-stop network

using CPLEX, and compare

the objective value with the

lower bound

No

Converges?

Set the lower bound solution

obtained from CPLEX

No

Converges?

Satisfied?

No

No

Modify the selected networks

from one-stop networks to all-

stop networks.

Yes

Converges?

Satisfied?

No

Use the best solution obtained

above as the final solution

Yes

Yes

YesYes

Yes

Start

End

Solve the modified problem,

and compare the objective

value with the lower bound

Check all cargo networks with

an OD distance greater than L1

have been modified into one-

stop networks

Solve the modified problem,

and compare the objective

value with the lower bound

Check that all cargo networks

with an OD distance greater

than L2 have been modified

into all-stop networks

No

Fig. 4. Flowchart of the solution method.



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(5) Improved mixed-stop heuristic: Unlike the four heuristics that use the mathematical

programming solver, CPLEX, to directly solve the associated networks, in this heuristic, werepeatedly modify and solve a series of networks using CPLEX, to approximate a near-

optimal solution, an improvement on the mixed-stop heuristic. The flowchart of the solu-tion method is shown in Fig. 4, and the steps are listed as follows:

Step 1: Solve the all-stop network in a pre-set time using CPLEX. If the solution is occurringwithin the specified error tolerance, then stop. Otherwise, set the lower bound solutionobtained from CPLEX as the lower bound and go to step 2.

Step 2: Solve the non-stop network using CPLEX, then compare the objective value with thelower bound. If it falls within the specified error tolerance, then stop. Otherwise, go tostep 4.

Step 3: Set L1 and L2 to represent the respective boundary distances defining short-haul and

middle-haul flights, as well as middle-haul and long-haul flights (L1 < L2).Step 4: Sort the cargo networks according to increasing OD distances. Choose a number p1 of

non-stop cargo networks, sequentially from the top, with OD distances greater thanL1. The p1 value which results in good solution quality can be tested. In practice, a car-rier may find a number ofp1 values, as in this research, to be tested. Modify the networks

selected from the non-stop networks to be one-stop networks. Solve the modified prob-lem using CPLEX and compare the objective value with the lower bound. If it fallswithin the specified error tolerance, then stop. If all cargo networks with an OD distance

greater than L1 have been modified into one-stop networks, then go to step 5. Otherwise,return to step 4.

Step 5: Sort the cargo networks according to increasing OD distance. Choose a number p2 of

one-stop cargo networks, sequentially from the top. Their OD distance should be greaterthan L2.5 Modify the networks selected from the one-stop networks to be all-stop net-works. Solve the modified problem using CPLEX and compare the objective value with

the lower bound. If it falls within the specified error tolerance, then stop. If all cargo net-works with an OD distance greater than L2 have been modified into all-stop networks,then go to step 6. Otherwise, return to step 5.

Step 6: Use the best solution obtained above as the final solution.

4. Numerical tests

To test how well the model and the solution algorithms may be applied in the real world, weperformed a case study using operational data from a major Taiwan airline. To build and to solvethe models, we used the C computer language, coupled with the CPLEX 7.1 mathematical pro-gramming solver, to develop the solution algorithms. The tests were performed on a Pentium

4-1.8G with 1 Gb of RAM in the environment of Microsoft Windows 2000. We first used theoperating data to build a mathematical model, and then applied the solution algorithms to solve

the problems. Finally, we performed several sensitivity/scenario analyses.

5 Similar to the setting of p1, the suitable p2 value should be tested.



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4.1. Model tests and result analyses

The numerical tests were mainly based on data obtained from a major Taiwan airlines South-

east Asian operations during 2002. Eight cities were served by four B747-400F airplanes, eachwith a carrying capacity of 100 metric tons. All the cost parameters, cargo fare rates and otherinput, such as the flight times, the distance between two stations, the approved flight quota for

each airport/airport pair, the available time slots at each airport, and the ground handling time,were set based on the actual operating data, as well as Taiwan government regulations, with rea-sonable simplifications.

The heuristics developed included non-stop, one-stop, all-stop, mixed-stop and improved

mixed-stop heuristics. After testing to find suitable L1 and L2 values that would result in goodsolution quality, L1 and L2 were set to be 1500 km and 3000 km.6 In other words, flights with

OD distances of less than 1500 km, between 1500 km and 3000 km, or greater than 3000 km

would be characterized as short-haul, middle-haul or long-haul flights, respectively. In addition,in step 4 of the improved mixed-stop heuristic, we chose three non-stop cargo networks (with ODdistances greater than 1500 km) in each round, to modify into one-stop networks. In step 5 of the

improved mixed-stop heuristic, we chose one one-stop cargo network (with an OD distance great-er than 3000 km) in each round, to modify into an all-stop network.

The problem sizes handled by each heuristic, with the exception of the improved mixed-stopheuristic, are described in Table 1.7 In the tests, we set the convergence gap (error tolerance) tobe 0.05 and used CPLEX to solve the problems. All the problems were solved to within an errorgap of 0.05.

To evaluate the performance of the heuristics, we utilized the current timetable (24 non-stopflights and 4 one-stop flights) and fleet routing (4 airplanes) of the studied airline with the pro-

jected cargo transportation demand. In particular, we first fixed the values of the fleet-flowtimespace network variables, given the timetable and fleet routing. Then we used CPLEX tosolve a simplified cargo transportation model. For simplicity, the results obtained are referred

to as actual operations.Table 2 shows the test results. As shown in Table 2, OBJ represents the system cost for the best

feasible solution obtained, which is equal to a negative profit, because the revenue from the cargo-flow networks is calculated in the form of a negative cost. For example, the all-stop heuristic, as

shown in Table 2, has a system cost of 26,413,886 which is equal to a profit of 26,413,886. Forsimplicity, we use the system cost to later interpret the test results. Best Node represents the bestobjective function value of all the unexplored nodes in the branch-and-bound tree, and serves as

the lower bound of the problem. Gap represents the gap between OBJ and Best Node.The improved mixed-stop heuristic yielded the best solution, with an objective value of26,712,087. The mixed-stop heuristic was next, with an objective value of 26,510,486, followedby the one-stop heuristic, with an objective value of26,472,688. The all-stop heuristic performedslightly worse than the one-stop heuristic, with an objective value of26,413,886, followed by theactual operation, with an objective value of 24,433,012. The non-stop heuristic performed most

6 The search for the most suitable L1 and L2 is performed similar to that for finding the most suitable p value; See step

4 in the improved mixed-stop heuristic. L1 and L2 are adjustable in other applications.7 The problem size handled by the improved mixed-stop heuristic changes in each round.



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poorly, with an objective value of 2,402,5318. The results show that a mixture of non-stopflights, one-stop flights and multiple-stop flights, given by the improved mixed-stop heuristic, pro-vided the most profitable schedule, while the purely non-stop flights given by the non-stop heu-

ristic will yielded the least profitable schedule.The results also show that, except for the non-stop heuristic, the objective value of the other

four heuristics were all better than actual operating value by at least 8.03%, which shows that,

from a systematic optimization perspective the proposed heuristics are superior to the currenttrial-and-error method. Although the non-stop heuristic is designed for optimization analysis,it yielded an objective value that was worse than the actual operating value. This may be a resultof the fact that the non-stop heuristic does not take into account one-stop flight operations, which

are relatively non-systemic compared to the trial-and-error method currently used, in the South-east Asia network.8

It is also found that except for the non-stop heuristic, the other four heuristics all yielded 46one-stop flights. The all-stop heuristic also yielded a two-stop flight. The objective values of thesefour heuristics are close to each other, the greatest difference being 1.1%. The reason is that the

8 According to the optimality theory, the more flexible the model, the better the solution obtained. Consequently, the

all-stop network should provide the best solution, if all networks are optimally solved. However, due to the 5% error

gap set for convergence in CPLEX, in the tests the all-stop heuristic did not yield better solutions than the improved

mixed-stop heuristic, the mixed-stop heuristic or the one-stop heuristic.

Table 2

Test results

Heuristic Actual operations Non-stop One-stop All-stop Mixed-stop Improved

mixed-stop

OBJ (NT$) 24,433,012 2,402,5318 26,472,688 26,413,886 26,510,486 26,712,087Best Node (NT$) 25,031,259 24,963,600 27,675,212 27,734,232 27,680,854 27,869,619Gap (%) 2.39% 3.91% 4.54% 5.00% 4.41% 4.33%

Number of iterationsfor running CPLEX 859 30 14,578 15,200 26,788 13,365

Computation time (s) 91.43 4.92 3072.16 110668.77 39748.61 16843.33

Number of stations 8 8 8 8 8 8

Fleet size 4 3 3 3 3 3

Frequency (flights/week) 28 29 28 28 28 28

Average load factor (%) 67.18% 65.52% 68.18% 67.86% 68.18% 68.18%

Transfer rate (%) 3.32% N/A 9.52% 5.74% 9.52% 9.52%

Service rate (%) 99.15% 99.52% 100.00% 99.52% 100.00% 100.00%

One-stop flight 4 N/A 6 4 6 6

Two-stop flight 0 N/A 0 1 0 0

Table 1

Problem size

Heuristic Non-stop One-stop All-stop Mixed-stop

# Variables 5142 19,614 43,734 25,104# Constraints 4757 9737 9737 7841



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test was based on the carriers Southeast Asia service network, where most distances between air-

ports are short. Therefore, a combination of non-stop and one-stop flights would be most effectivefor most cargos. More than one-stop flights are not significantly effective in this area, which is why

only one two-stop flight is employed in the all-stop heuristic solution. If a larger flight networkneeds to be tested, then more multiple-stop flights should be used.

The all-stop heuristic used more computation time than did the others, since it dealt with the

most complicated network. For example, as shown in Table 1, the number of variables in theall-stop network was 2.22 times that in the one-stop network. The numbers of constraints in boththese networks were the same. However, the computation time required by the all-stop networkwas 36.02 times of that required by the one-stop network, showing that for NP-hard problems, the

computation time will increase substantially when the number of variables increases. Althoughthe all-stop network incorporated the most information, its objective was not the best, due to

the convergence gap set in CPLEX. A similar result was also found for the mixed-stop heuristic.

Of the five heuristics, the improved mixed-stop heuristic not only had a better objective valuethan the other heuristics, but was also less time-consuming than the all-stop heuristic and themixed-stop heuristic (requiring only 15% of the all-stop heuristics and only 42% of the mixed-stop

heuristics computation time). As a result, the improved mixed-stop heuristic was superior to theothers in terms of solution quality and computational efficiency.

In the results, 8 stations and 3 airplanes were used for all five heuristics. The non-stop heuristicprovided 29 flights, while the other four heuristics provided 28 flights. In addition, the averageload factor for the non-stop heuristic was 65.52%, which was lower than for the other four heu-

ristics and for actual operations. We also found that effective cargo transfers not only led to animprovement in the average load factor but also the cargo service rate. For example, all theone-stop, mixed-stop and improved mixed-stop heuristics yielded the highest transfer rate,

9.52%, the highest average load factor, 68.18% and the highest cargo service rate, 100%, meaningthat the resources had been most effectively used.9 The other two heuristics (the non-stop and theall-stop heuristics) and actual operations yielded worse results, in particular lower transfer rates

(none, 5.74% and 3.32%, respectively), lower load factors (65.52%, 67.86% and 67.18%, respec-tively) and lower cargo service rates (99.52%, 99.52% and 99.15%, respectively), meaning that re-sources were relatively less effectively used.

Finally, the fleet flows obtained above could not yet be directly put into practice without iden-tifying each airplane path in the fleet networks. The flow decomposition method ( Yan and Young,

1996) was applied to trace the path of each airplane. An example of the three aircraft routes isshown in Fig. 5.

4.2. Sensitivity/scenario analyses

To understand the influence of the model parameters on the solution, we performed a sensitiv-ity analysis of the available fleet size, the cargo demand (OD), and the fixed station cost all of

9 The results, except for the objective value, from the mixed-stop and the improved mixed-stop heuristics were all the

same. We traced the routes and found that these two routes, except for the transfer times of some cargos, were the same.

In particular, the cargo holding costs was what made the objective values be different.



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which are essential inputs to the model.10 For simplicity, we used the two better heuristics, that is,

the mixed-stop and the improved mixed-stop heuristics, for the analyses. To evaluate the perfor-mance of the proposed heuristics for solving middle/large scale problems, we also performed a sce-

nario analysis.

Fig. 5. Aircraft route example.

10 Sensitivity analyses of other factors can be similarly performed, but are left for future research.



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4.2.1. Available fleet size

As shown in Fig. 6, the objective value of the mixed-stop heuristic decreased from 20,427,560to 26,497,732 (i.e., a profit improvement of 29.72%), when the available fleet size increased from1 to 3 airplanes.11 However, when the available fleet size increased from 3 to 4 airplanes, the opti-

mal fleet size remained at 3 airplanes. Although the objective value changed slightly, from26,497,732 to 26,510,486 (a difference of 0.05%), this was due to the convergence gap set inCPLEX. The situation was found to be similar for the improved mixed-stop heuristic.

4.2.2. Cargo demand

To evaluate the potential for a future increase in cargo demand, we performed a cargo demandsensitivity analysis. As shown in Fig. 7, the objective value of the mixed-stop heuristic decreasedwhen the cargo demand increased, however, the decrement tended to be smaller. For example, theobjective value decreased from 26,510,486 to 30,813,412 (i.e., a profit improvement of16.23%), when the cargo demand increased from 100% to 120%. However, when the cargo de-

mand increased from 180% to 200%, the objective value decreased from 40,238,048 to41,514,180 (i.e., a profit improvement of 3.17%). The latter is significantly smaller than the for-mer.12 Similar situations were also found for the improved mixed-stop heuristic.

4.2.3. Stations fixed cost

The fixed cost for operating each station was estimated based on the airlines annual report. As

shown in Fig. 8, the objective value of the mixed-stop heuristic increased when the fixed cost in-creased, meaning that the carriers profit decreased. When every stations fixed cost increased from100% to 140%, the objective value increased from 26,510,486 to 26,171,030 (i.e., the profit de-creased by 1.3%), meaning that the profit was relatively uninfluenced by the stations fixed cost. A

similar situation was found for the improved mixed-stop heuristic, except its objective values were

-28000000

-26000000

-24000000

-22000000

-200000000 1 2 3 4 5

Available fleet size

Objectivevalue(NT$)

Mixed-stop heuristic

Improved mixed-stop heuristic

Fig. 6. Sensitivity analysis of the available fleet size.

11 All available airplanes were used up.12 The service rate decreased as the demand increased. This decrease was a result of the fact that the sum of all flights at

each station exceeded the approved quota, so that additional cargos could not be serviced.



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slightly better. Both analyses showed that the stations fixed cost was not sensitive to the modeled

objective value.

4.2.4. Problem scales

To evaluate the performance of the proposed heuristics for middle/large scale problems, we fur-

ther tested seven problem instances with different scales ranging from 10 cities, 10 airplanes and 30OD pairs, to 24 cities, 30 airplanes and 60 OD pairs. For each problem instance, we added a num-

ber of cities, airplanes and OD pairs to our original problem. The distance and the flight times foreach new city-pair were randomly set, based on the original flight network. The available timeslots and the ground handling time at each new airport were randomly set, in reference to the ori-

ginal problem. Based on the original OD demands, as well as the original/new fleet size, we ran-domly set the cargo demand for each OD pair. In particular, the cargo demand for each OD pair

was suitably increased with the fleet size. Similarly, the flight quota for every airport/airport pairwas increased when the associated OD demand increased. The other cost parameters and cargofare rates were randomly selected in reference to the original problem. The parameters used in

-28000000

-27000000

-26000000

-25000000

-2400000080% 100% 120% 140% 160%

Station fixed cost (%)

O

bjectivevalue(NT$)



Fig. 8. Sensitivity analysis of the stations fixed cost.

-45000000

-40000000

-35000000

-30000000

-25000000

80% 100% 120% 140% 160% 180% 200% 220%

Cargo demand (%)

Objectivevalue(NT$)



Fig. 7. Sensitivity analysis of the cargo demand.



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Table 3

Test results for middle/large-scale problems

Non-stop One-stop All-stop Mixed-stop Improved

mixed-stop10 Cities, 30 OD pairs and 10 airplanes

# Variables 6508 36,987 60,257 24,782

# Constraints 5843 12,209 12,209 9219

OBJ (NT$) 101,243,344 105,914,328 10,571,3709 106,533,160 106,732,979Best Node (NT$) 106,153,768 110,535,352 110,613,301 110,380,315 110,599,832Gap (%) 4.85 4.36 4.63 3.61 3.49

Computation time (s) 4.53 15.34 2348.56 354.16 100.09

10 Cities, 30 OD pairs and 15 airplanes

# Variables 6508 36,987 60,257 24,782

# Constraints 5843 12,209 12,209 9219




# Variables 6508 36,987 60,257 24,782

# Constraints 5843 12,209 12,209 9219




# Variables 13,902 85,480 222,075 65,017

# Constraints 12,957 28,119 28,119 20,708




# Variables 13,902 85,480 222,075 65,017

# Constraints 12,957 28,119 28,119 20,708




# Variables 13,902 85,480 222,075 65,017

# Constraints 12,957 28,119 28,119 20,708


Computation time (s) 8.36 138.23 30560.70 2007.11 602.41(continued on next page)



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the heuristics and CPLEX were set the same as in the previous test. Furthermore, the computa-tional capability was upgraded. The tests were performed on a Pentium 4-3.2G with 1.5 GB ofDDR RAM in the environment of Microsoft Windows XP. The CPLEX version was upgradedfrom 7.1 to 8.1.

Table 3 shows the test results for different problem instances. The results similar to that of the

original smaller problem, show that the proposed heuristics, except for the all-stop heuristic, couldefficiently solve middle/large-scale problems. For example, as shown in the 16 city, 42 OD pair and

20 airplane scenario, the improved mixed-stop heuristic yielded the best solution, with an objec-tive value of261,146,491. The non-stop heuristic performed the worst, with an objective value of244,625,040. The other three heuristics, the mixed-stop heuristic, the all-stop heuristic and theone-stop heuristic, yielded similar objectives within 0.12% (= j(257,951,790 257,645,875)/257,951,790j of each other). However, the all-stop heuristic was significantly more time-consum-ing. For example, the all-stop network required a computation time 15.22 times that of the mixed-stop network and 50.73 times that of the improved mixed-stop network. The other problem

results are similar and not discussed here.13

5. Conclusions

In the past, most research on airline scheduling has been focused on passenger transport, whichis fundamentally different from air cargo transport. A short-term scheduling model has not yet

been developed for air cargo transportation. In this research, based on Taiwan air carrier data,we developed a novel integrated scheduling model combining airport selection, fleet routing

and timetable setting, with the objective of maximizing the operating profit, subject to the relatedoperating constraints. Such a model is expected to be a useful planning tool, with which cargocarriers can determine suitable servicing airports, fleet routes and timetables for short-term oper-

ations. Network flow techniques are employed to construct the model, which includes multiplecargo- and fleet-flow networks. In the cargo-flow networks, unlike in past research, we construct

Table 3 (continued)

Non-stop One-stop All-stop Mixed-stop Improved

mixed-stop

24 Cities, 60 OD pairs and 30 airplanes# Variables 30,902 235,684 584,260 151,326

# Constraints 27,296 60,620 60,620 42,241

OBJ (NT$) 336,685,344 380,294,470 ** 381,628,506 381,972,634Best Node (NT$) 350,772,576 394,005,874 394,330,000 393,701,862 39,368,5142Gap (%) 4.18 3.48 ** 3.16 2.97

Computation time (s) 9.89 1358.72 259,807 6392.56 1985.24

: The problem size handled by the improved mixed-stop heuristic changes in each round.

**: The all-stop heuristic could not find a feasible solution in three days (259,200 s).

13 To save space, other results are not discussed here. The reader may contact the authors for the detail.



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multiple OD-time-pair timespace networks on the basis of the desired delivery times. The model

is formulated as a mixed integer program that is characterized as NP-hard. We develop severalheuristics with the use of a mathematical programming solver to solve the problem.

To evaluate the model and the solution algorithms, a case study based on real operating datafrom a major Taiwan cargo carrier is performed. The results show that in terms of solution qualityand computational efficiency the improved mixed-stop heuristic out-performs the others. The re-

sults also show that the use of suitable cargo transfers not only reduces the operating cost but alsoincreases profitability. To understand how the essential parameters influenced the solution, severalsensitivity analyses are also performed. From the results, we see that the improved mixed-stopheuristic performs better than the mixed-stop heuristic, in each test. Additional testing of mid-

dle/large-scale problems is performed. The results, similar to that for the small problem, show thatthe proposed heuristics, except the all-stop heuristic, can all efficiently solve middle/large-scale

problems. Since those test problems are artificial, practical problems should next be tested for

the carrier to evaluate the proposed heuristics in actual operations. The model, the solution algo-rithms, the case study, and the sensitivity/scenario analyses should all be useful as reference mate-rial, to help an airline to determine optimal short-term cargo fleet routing and flight scheduling.

Although the preliminary test results show that the proposed heuristics, except the all-stop heu-ristic, have potential to be used for solving middle/large-scale problems, the heuristics can be suit-

ably modified to solve larger-scale problems, for larger airlines. For example, if the heuristics asproposed cannot efficiently solve large-scale problems, then modern meta-heuristic techniques(e.g. the tabu search method, threshold accepting method, or genetic algorithm), lagrangian relax-

ation or column generation, may be employed to develop a more efficient algorithm. This could bea direction of future research. The extension of single-fleet routing to multi-fleet routing, and theincorporation of other routing constraints (for example, maintenance and crew scheduling) or

other objectives involved in actual operations, could also be directions for future research. Here,we used cargo flights for the input data. Some airlines have introduced combi flights. How tocombine passenger flights and cargo flights into an integrated flight scheduling model could also

be a topic of future research. Finally, the proposed model and solution techniques could be ap-plied to other transport industries, for example, container ship scheduling or bus scheduling.

Acknowledgments

This research was supported by a grant (NSC-92-2211-E-008-046) from the National Science

Council of Taiwan. We thank the airline for providing the test data and their valuable opinions.

We also thank the two anonymous referees and the editor for their helpful comments and sugges-tions on the presentation of the paper.

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