Exact and Heuristic Techniques for Solving the Aircraft
Rotation Problem
Roberto Pontes* Abilio Lucena Lucidio Cabral
Rio-Sul Linhas Aéreas
Universidade Federal do Rio de Janeiro - UFRJ
Universidade Federal da Paraiba - UFPB
rpontesbr@yahoo.com.br, abiliolucena@globo.com, lucidio@uol.com.br
May 15, 2002
Abstract
The Aircraft Rotation Problem refers to the designation of each airplane available at
airlines companies such that all planned flights are covered. This is a light
constrained problem since any aircraft can be assigned to an unlimited quantity of
duty work. However, there are many lower cost solutions that can be obtained by
simply adjusting the time of departure of a subset of flight legs, or even positioning
some aircraft by executing ferry flights between two near airports. We describe a
mathematical model that indicates the minimal changes to be done in time departure
and empty flights designations in order to decrease the total number of required
aircraft. An algorithm based on GRASP metaheuristic is also presented, for solving
very large problems, and some computational results are presented for a real world
problem at Rio-Sul Brazilian Airlines.
Keywords:
Topic Areas:
Aircraft Routing, Aircraft Rotation, GRASP
Global Optimization, Heuristics, Integer Programming, Resource
Allocation, Scheduling.
* Current Address: Rio-Sul Linhas Aéreas. Av Rio Branco 85 / 6° Andar. Rio de Janeiro, RJ. 20040-004. Brazil.
1
Introduction
Aircraft planning consists of several interconnected problems that are currently
solved separately. These include market modeling [12], fleet assignment [2] [10] [11],
aircraft rotation [6] [8] [9], crew pairing and scheduling [1] [4] [6]. In this paper, the
Aircraft Rotation Problem (ARP) [5] [7] is addressed. This problem is normally tackled at
the initial stages of the overall planning process and uses as input the output from the
Fleet Assignment Problem (FAP).
Several approaches have been applied to ARP. In [9] a constructive algorithm
conducted by the GRASP metaheuristic is presented. Exact methods can be found in
[6][7][8], and a dynamic programming strategy is showed in [11]. In this work, both
heuristic and exact approaches are addressed.
It is assumed that detailed flight planning is not yet complete, and that there is still
room for marginal changes, say in the suggested departure times of some the flights. Since
time slots for early morning flights are very narrow, not all flights are candidates for
changing suggested departure times. Furthermore, amongst candidate flights, time
windows may vary in width. It is also assumed that, whenever it is economically
attractive, new flight legs can be created to allow the repositioning of aircraft. As one
shall see this may represent having an aircraft free for a whole week in exchange for a
possibly unprofitable flight leg.
This paper is organized as follows. In Section 2, ARP is described in greater
detail. In Section 3, geographic and temporal connections constraints required to form a
connection network are discussed. In Section 4, an Integer Programming (IP) formulation
of the problem is presented. This formulation has shown to be capable of solving, under a
commercial IP solver, small to medium size instances of ARP. In Section 5, a GRASP
based procedure is described to solve larger instances of the problem. In Section 6,
computational results are presented for real world instances of the problem coming from
Rio-Sul Brazilian Airlines. Finally, in Section 7, some conclusions and suggestions for
future work close the paper.
2
Problem Description
Aircraft Rotation is one of the most important steps in the process of defining the
final set of flight legs that will be operated by an airline company. After a fleet is assigned
a set of flight legs, one must group together legs that will be operated by a same aircraft.
Basically, the problem consists in given a set of flights and a set of aircraft assigns this
last resource to all flights so those ground time requirements for each airport are met. This
is to be accomplished with available resources being used at minimum cost.
Flights must normally be connected in such a way that they do not violate certain
related constraints. These constraints are geographical and temporal in nature. For
example, the next flight leg in a rotation must have the same departure location as the
arrival location of the previous one. Additionally, the time between consecutive flights
must fall within a certain limit, called minimal connection time or ground connection
time. In this study, some of these general constraints are relaxed to allow the connection
of flight legs that, in principle, could not be connected. This is to be achieved through the
use of penalties and may lead to a reduction in the number of aircraft that would
otherwise be required to covering all flight legs.
Another important aspect relates with maintenance constraints. It is well known
that aircraft must undergo maintenance checks before a given limit on the amount of time
flown since the last check is reached. Opportunities to perform such checks only occur at
some of the potential connections available. Thus a sequence of flight legs must be
constructed in such a way that this constraint is not violated. In order to easily incorporate
the constraint in the framework being proposed, we shall assume that rotations are
assigned to non-specific aircraft. In this way, if a maintenance planner must stop a given
aircraft at a specific location, a dummy flight leaving and returning to that location should
be created and added to the set of flight legs to be covered. Finally, the rotation that is to
contain that task should be assigned to the specific desired aircraft.
3
Towards Global Optimization
The geographic and temporal connection constraints in ARP induce the creation of
a network of feasible connections. In this network nodes represent flight legs and arcs
represent legal connections between flights. The problem can thus be formulated as a
Minimum Cost Network Flow Problem.
Given the possibilities of changes in the departure times of flights [10] and also
the permission to create repositioning flights, an enlarged network must be defined. This
network is constructed as follows:


Natural Connection Arcs (Type 1)– These are the arcs that connect flights
which obey suggested time and geographical constraints. Such arcs are
associated with connections requiring no suggested departure time changes.
They do not connect repositioning flights either. The cost of a type 1 arc is
zero;
Time Changes Arcs (Type 2) – Although the inbound and the outbound flight
legs in these arcs originate at a same location, the ground time available
(measured as the difference between the suggested departure time for the later
flight and the expected arrival time for the earlier one) is not sufficient to allow
the connection. Therefore, if such an arc is used it implies that suggested
departure times for either flights must have been changed. A type 2 arc cost
equals the sum of the deviations (in minutes) from the suggested departure
times involved;




Repositioning Flight Arcs (Type 3) – These arcs represent connections
between two flights arriving and departing from different airports. However,
sufficient time exists (even allowing for ground time requirements) for a
repositioning flight between the two locations. The cost of a type 3 arc equals
the duration of its repositioning flight;
Repositioning Flight plus Time Changes Arcs (Type 4) – These arcs represent
connections that will need a repositioning flight plus time departure changes.
A type 4 arc cost equals the duration of the repositioning flight plus the
deviations from the suggested departure times;
Arcs from Source Node (Type 5) – These arcs, which point outwards of a
dummy source node, are used to model the number of aircraft required to solve
the problem. Every type 5 arc has a cost of 1000 minutes, which is the
expected daily utilization of an aircraft.
Arcs to Sink Node (Type 6) – These are virtual arcs used to close rotations
initiated at the source node. Type 6 arcs have a cost of zero.
Denote by D=(V,A) the ARP network defined above. Vertex set V  {vi : i  I } of
D is indexed by I  {1,2,..., n  1, n  2} so that v n 1 and v n  2 identify, respectively, the
source and the sink nodes. Remaining vertices are associated with the n original flight
legs. Costs {cij : (i, j )  A} , as introduced above, are associated with every one of the arcs
in the network. Whenever applicable one may use the index set I  I \ {n  1, n  2} for
vertices associated with the original flight legs.
4
An ARP Formulation
Let {xij : (i, j )  A} be a set of binary 0-1 variables used to control the inclusion
( xij  1) or the exclusion ( xij  0) of an arc (feasible connection) between vertices (flight
legs) vi and v j . Real valued variables  i and  i , i  I are used to model, respectively,
deviation from suggested departure time and the norm of this deviation for (flight leg) vi .
These variables must therefore obey   i   i   i and 0   i   i , where  i is the
maximum allowed deviation (in either direction) from the suggested departure time for
the flight. Finally, suggest departure time are given by {si : i  I } A valid formulation for
ARP is given by:
4.1 Objective Function
Minimize
c
( i , j )A
4.2 Constraints
ij
xij
Flights Operation Guarantees.
x
ij
 1,
i  I
x
ij
 1,
j  I
( i , j ) A
( i , j ) A
Connection Feasibility:
si + ti xij - ti (1- xij )+ di £ s j + d j
"(i, j) Î A
Module of Deviation from suggested departure time
i   i
 i   i
i  I
i  I
Limits for the Variables
  i   i   i i  I
0  i   i
xij Î{0,1}
5.
i  I .
"(i,j)ÎA
The Constructive Algorithm
The formulation introduced in this paper was able to solve exactly ARP instances
involving about 1000 flight legs. This was achieved in a very straightforward manner
through the direct use of a Mixed Integer Programming (MIP) commercial solver
XPRESS-MP. Instances tested come from one of the smaller fleets at Rio Sul Airlines,
operating over a time horizon of one week. However, solutions are required at Rio Sul
Airlines for much larger ARP instances arising from the weekly operation of some of the
company’s larger fleets. Alternatively, they might also be associated with smaller fleets
operating over a time horizon in excess of one week. The company has thus requested a
heuristic algorithm to tackle these applications. Amongst the different candidate
metaheuristics previously used in the company, GRASP was selected for the task. Details
of the proposed algorithm follow.
In graph theoretical terms, a feasible ARP solution consists of a set of vertex
disjoint paths in D, each one originating at source vertex v n 1 and ending at sink vertex
v n  2 , and such that no vertex in V  V \ {v n 1 , v n  2 } is left unvisited. Our heuristic
constructs one path a time in a randomized greedy way.
5.1 Initializing and Closing a Path
A new path is initialized by choosing an arc (v n 1 , v j )  A , where v j is an yet
unvisited vertex in V with the earliest suggested departure time. This path will
eventually be closed (i.e. extended to vertex v n  2 when no yet unvisited visited vertex
in V exists to extend it.
5.2 Selecting an Arc to Extend the Path
Denote by vi  V the end vertex of the path being extended. Accordingly, denote
by Vi  V the set of all vertices v j  V for which and arc (vi , v j ) exists in A. The set of
candidate arcs (to extend the path) is thus given by A  {( vi , v j )  A : v j Vi } . Arc set A
can be partitioned into four sets in accordance with the different arc types defined in the
previous section. Subset Ai1  A is for type 1 arcs. Accordingly, Ai 2 , Ai3 and Ai 4 are
respectively for arcs of types 2, 3, and 4. An arc is chosen to extend the path by firstly
drawing one of the sets Ai1 , Ai 2 , Ai3 , or Ai 4 with probabilities 0.79, 0.16, 0.04 and 0.01,
respectively.
5.3 Sorting Arc Sets and Choosing an Arc
After an arc set is selected, one of its components must be chosen to extend the
path. This process is conducted by firstly sorting arcs in the chosen set. Arc sets are
sorted under different criteria. Arcs in set Ai1 are sorted in ascending order of suggested
departure times for the vertices (flight legs) they point to. Arcs in set Ai 2 are sorted in an
ascending order of their deviation from suggested departure times for the vertices they
point to. Arcs in set Ai3 are sorted in an increasing order of the duration of their
repositioning flights. Finally, arcs in set Ai 4 are sorted in an increasing order of
magnitude of the sum of the duration of their repositioning flights plus deviation from
suggested departure times. Once the chosen set is sorted, a candidate list with their first
30% elements is formed. An arc to extend the path is then randomly chosen from the
candidate list.
5.4 Closing a Path
A path is closed when no vertex other than vn  2 can be used to expand it.
5.5 Stopping Conditions
When no more paths can be created, a feasible ARP solution has been generated.
If the cost of this solution improves on the best previously obtained, best solution
generated is updated. The process is to be continued until some time or iterations count
criteria is met.
6. Computational Results
For the necessary tests we chose a small daily network flight from Rio-Sul Airlines,
in order to allow running both exact and heuristic approaches. All data is published in
appendix A to D to the present text. Doing so, we tried to make this study reproducible to
anyone who desires to submit to other models or heuristics. All tests run on a PC Pentium
650 MHz and 512 MB of RAM. Algorithms were written in straight C and the XPRESSMP version 12 was used to solve the integer model.
The daily flights group is composed by 107 regular flights. All possible arcs and
constraints generation consumed 200 seconds. The LP relaxation of the problem was
found in 12 seconds and the branch and bound algorithm found the integer solution in 300
seconds.
The heuristic algorithm run for 60 minutes. The best sub-optimal solution was
found in 384 seconds. All intermediate solutions are shown in Table 1:
Objective Value
17.010
17.008
17.007
16.544
16.319
16.313
16.260
16.240
16.184
16.137
Time (sec)
2
3
29
47
51
96
159
163
277
384
For this instance our approach show that we can save 4 aircraft simply creating 2
new flights between two near airports and changing 5 departure times involving only 11
minutes in the total, which represents a great improvement in relation to solution in which
departure time is fixed.
The flights network is presented in Appendix ‘A’; ground times are fixed in 20
minutes, except for airports SDU, CGH and BSB, where required time is 29 minutes.
Time flight between two airports is shown in appendix ‘B’.
7. Conclusions and Future Research
Aircraft are one of the most expensive resources in airline industry, and so, all
planning should be done around its economy. Because of combinatorial explosion, all
problems that precede aircraft rotation must be solved separately, however, it is
acceptable that some corrections of planing should be done when solving this part of the
problem.
Some scenarios obtained in this phase may help planners to make decisions and
correct previous decisions. In that way, allowing changes in flights as those suggested in
the present work can guarantee great savings for the airline company. The presented study
of case shows that we can save 4 aircraft simply creating 2 new flights between two near
airports and changing 5 departure times involving only 13 minutes in the total. Because in
this stage we have not fixed the flight network, we may think using ferry flights as
commercial flights, transporting passengers. Doing so, the solution becomes more
interesting, because we can use the original flights with little changes in its time schedule,
adding two new regular flights and getting four aircraft savings. That is really a great
effect with a reduced effort to obtain it.
The adoption of two approaches, exact and heuristics, seems always to be important
when dealing with combinatorial optimization, because of its unforeseeable behavior. The
algorithm guided by GRASP metaheuristic has reached very good solutions in a
reasonable time, and make the system reliable for future challenges. As future work a
local search procedure will be incorporated to improve the solution built at the final of a
GRASP iteration.
For future research, efforts have been done to idealize a complex model for solving
a complete flight planning. The idea is to group the three phases, marketing planning,
fleet assignment and aircraft rotation in a unique step. Planners will create as many flights
they want with a respective degree of importance and a model will maximize those needs
utilizing available aircraft number in each fleet.
References
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Problems: Random Pairing Generation and Strong Branching” Technical Report,
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[2] Götz S, Grothklags, Kliewer G and Tschoke S. "Solving the Weekly Fleet Assignment
Problem for Large Airlines", III Metaheuristics International Conference, 1999.
[3] Rezende, M. “ Greedy Randomized Adaptative Search Procedures (GRASP) ”.
Technical Report, ATT Labs Research, 1998.
[4] Rubin, J. “ A Technique for the Solution of Massive Set Covering Problems with
Application to Airline Crew Scheduling”. Transportation Science, 1973.
[5] Clarke L, Johnson E, Nemhauser G, Zhu ZX, “The Aircraft Rotation Problem”,
Annals of Operations Research 69: 33-46 1997.
[6] Cordeau JF, Stojkovic G, Soumis F, Desrosiers J,”Benders decomposition for
simultaneous aircraft routing and crew scheduling”, Transportation Science 35: (4)
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[7] Gopalan R, Talluri KT, “Mathematical models in airline schedule planning: A
survey”, Annals of Operations Research 76: 155-185, Feb. 1998 .
[8] Bartholomew-Biggs MC, Parkhurst SC, Wilson SP, “Using DIRECT to solve an
aircraft routing problem”, Computational Optimization and Applications, 21: (3) 311323 Mar. 2002.
[9] Grosche T, Heinzl A, Rothlauf F., “A conceptual approach for simultaneous flight
schedule construction with genetic algorithms”, Applications of Evolutionary
Computing, Proceedings 2037: 257-267, 2001.
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dynamic fleet management, II: Multiperiod travel times”, Transportation Science 36:
(1) 40-54 Feb. 2002.
APPENDIX "A" – FLIGHTS NETWORK
001
002
003
004
005
006
007
008
009
010
011
012
013
014
015
016
017
018
019
020
021
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023
024
025
026
027
028
029
030
031
032
033
034
035
036
037
038
039
040
041
042
043
044
045
046
047
048
049
050
051
052
053
054
SL166
SL155
SL330
SL595
SL596
SL533
SL280
SL587
SL520
JH559
SL406
SL419
SL542
SL408
JH345
SL589
JH502
SL281
SL521
SL532
SL510
SL544
SL331
SL409
SL543
SL590
SL336
SL597
SL522
SL150
SL511
JH503
SL545
SL470
SL523
JH342
JH500
SL147
JH340
SL591
SL402
SL471
SL390
SL564
SL151
SL563
SL272
SL282
SL508
JH501
SL480
SL283
SL337
SL347
08:00
08:32
08:34
08:40
08:44
08:50
08:52
09:00
09:02
09:04
09:06
09:16
09:28
09:30
09:30
09:58
10:02
10:08
10:16
10:20
10:22
10:32
10:40
11:00
11:00
11:06
11:16
11:30
11:32
11:42
11:50
12:02
12:04
12:28
12:46
13:00
13:02
13:10
13:22
13:30
13:38
13:54
13:56
13:58
13:58
14:00
14:12
14:22
14:56
14:58
15:30
15:42
15:46
16:00
08:31
10:33
09:59
11:11
10:39
10:01
09:43
10:01
09:47
10:35
09:55
11:03
10:31
10:21
12:23
11:01
11:33
10:59
11:01
11:31
11:13
11:33
12:11
11:49
12:03
12:44
15:05
13:27
12:17
13:33
12:41
13:33
13:05
13:19
13:31
15:45
14:33
18:03
18:25
14:57
15:33
14:39
15:41
14:59
15:59
15:01
15:51
15:13
15:41
16:29
16:21
16:37
19:29
16:45
GYN
GYN
CGH
POA
CGH
BSB
CGH
PLU
SDU
BSB
SDU
FLN
CGH
SDU
SSA
PLU
CGH
CWB
PLU
PLU
CGH
CGH
VIX
CGH
PLU
CGH
CGH
POA
SDU
CGH
CWB
BSB
PLU
SDU
PLU
CGH
CGH
BSB
CGH
POA
CGH
CGH
CGH
CGH
GYN
PLU
CGH
CGH
SDU
BSB
SDU
CWB
REC
PLU
BSB
CGH
VIX
CGH
POA
PLU
CWB
CGH
PLU
CGH
CGH
CGH
PLU
CGH
CGH
CGH
BSB
CGH
SDU
BSB
CWB
PLU
CGH
SDU
CGH
POA
REC
CGH
PLU
GYN
CGH
CGH
CGH
CGH
SDU
SSA
BSB
BSB
JPA
CGH
POA
SDU
FLN
PLU
CGH
CGH
SJP
CWB
PLU
CGH
CGH
CGH
CGH
SDU
055
056
057
058
059
060
061
062
063
064
065
066
067
068
069
070
071
072
073
074
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076
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079
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082
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087
088
089
090
091
092
093
094
095
096
097
098
099
100
101
102
103
104
105
106
107
SL598
SL568
SL403
SL273
SL391
JH343
SL481
JH506
SL405
SL569
SL152
JH546
SL572
SL482
JH341
SL592
JH507
SL528
JH550
SL537
SL576
SL573
SL599
JH547
SL153
SL483
SL412
SL529
SL584
SL514
JH344
SL577
SL536
JH551
SL332
SL492
SL586
SL530
SL515
SL593
SL585
SL518
SL413
SL407
SL167
SL531
JH558
SL493
SL333
SL588
SL594
SL154
SL418
16:00
16:02
16:08
16:10
16:12
16:12
17:00
17:02
17:14
17:32
17:44
17:56
17:58
18:30
18:50
18:54
19:00
19:02
19:04
19:22
19:26
19:34
19:46
20:00
20:04
20:08
20:10
20:14
20:26
20:30
20:48
21:02
21:04
21:04
21:20
21:30
21:30
21:34
21:45
21:52
22:04
22:16
22:22
22:24
22:42
22:48
22:56
23:02
23:04
23:06
23:18
23:22
23:32
17:55
17:03
17:59
17:51
17:55
18:57
17:45
18:33
19:57
18:33
19:35
19:27
18:59
19:21
00:19
20:45
20:31
19:47
20:35
20:33
20:27
20:35
21:41
21:31
22:05
20:53
21:29
20:59
21:27
21:21
23:39
22:03
22:13
22:35
22:31
22:21
22:31
22:19
22:37
23:19
23:05
23:07
23:31
23:09
23:13
23:33
00:27
23:47
00:15
00:07
01:57
01:13
01:19
CGH
CGH
POA
SJP
FLN
SSA
CGH
CGH
CGH
PLU
CGH
CGH
CGH
SDU
JPA
CGH
BSB
SDU
CGH
BSB
CGH
PLU
POA
BSB
GYN
CGH
CGH
PLU
CGH
CGH
CGH
PLU
PLU
BSB
CGH
SDU
CGH
SDU
CWB
POA
PLU
CGH
FLN
CGH
BSB
PLU
CGH
CGH
VIX
CGH
CGH
CGH
CGH
POA
PLU
CGH
CGH
CGH
CGH
SDU
BSB
CGH
CGH
GYN
BSB
PLU
CGH
CGH
POA
CGH
PLU
BSB
PLU
PLU
CGH
CGH
CGH
CGH
SDU
FLN
SDU
PLU
CWB
SSA
CGH
BSB
CGH
VIX
CGH
PLU
PLU
CGH
CGH
CGH
CWB
CGH
SDU
GYN
SDU
BSB
SDU
CGH
PLU
POA
GYN
FLN
APPENDIX "B" – FLIGTH TIME BETWEEN AIRPORTS
BSB
BSB
BSB
CGH
CGH
CGH
CWB
CWB
CWB
FLN
FLN
GYN
GYN
JPA
JPA
PLU
POA
REC
SDU
SJP
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
CGH
JPA
SDU
CWB
PLU
SJP
GYN
REC
VIX
POA
SSA
POA
SSA
REC
VIX
SJP
SDU
SDU
SJP
VIX
091
197
107
051
058
049
114
283
125
042
225
173
143
013
181
067
129
215
079
110
BSB
BSB
BSB
CGH
CGH
CGH
CWB
CWB
FLN
FLN
FLN
GYN
GYN
JPA
PLU
PLU
POA
REC
SDU
SSA
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
CWB
PLU
SJP
FLN
POA
SSA
JPA
SDU
GYN
REC
VIX
REC
VIX
SDU
POA
SSA
SJP
SJP
SSA
VIX
125
071
066
105
097
171
293
078
142
300
136
210
118
226
155
113
119
242
130
097
BSB
BSB
BSB
CGH
CGH
CGH
CWB
CWB
FLN
FLN
GYN
GYN
JPA
JPA
PLU
PLU
POA
REC
SDU
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
FLN
POA
SSA
GYN
REC
VIX
PLU
SJP
JPA
SDU
JPA
SDU
PLU
SJP
REC
VIX
SSA
SSA
VIX
152
185
125
095
246
087
095
060
311
087
216
109
198
251
187
044
267
075
048
BSB
BSB
BSB
CGH
CGH
CWB
CWB
CWB
FLN
FLN
GYN
GYN
JPA
JPA
PLU
POA
POA
REC
SJP
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
GYN
REC
VIX
JPA
SDU
FLN
POA
SSA
PLU
SJP
PLU
SJP
POA
SSA
SDU
REC
VIX
VIX
SSA
031
191
109
255
049
028
062
208
114
088
076
054
352
085
045
342
177
169
169
APPENDIX "C" – OPTIMAL STANDARD AIRCRAFT ROTATION
Objective Value: 20.000
Route
SL406
SL409
SL470
SL471
SL480
JH506
SL537
SL536
SL167
01
09:06
11:00
12:28
13:54
15:30
17:02
19:22
21:04
22:42
09:55
11:49
13:19
14:39
16:21
18:33
20:33
22:13
23:13
SDU
CGH
SDU
CGH
SDU
CGH
BSB
PLU
BSB
CGH
SDU
CGH
SDU
CGH
BSB
PLU
BSB
GYN
Route
SL520
SL521
SL522
SL523
SL508
SL569
SL584
SL585
02
09:02
10:16
11:32
12:46
14:56
17:32
20:26
22:04
09:47
11:01
12:17
13:31
15:41
18:33
21:27
23:05
SDU
PLU
SDU
PLU
SDU
PLU
CGH
PLU
PLU
SDU
PLU
SDU
PLU
CGH
PLU
CGH
Route
SL587
SL544
SL545
SL402
SL403
JH550
JH551
SL588
03
09:00
10:32
12:04
13:38
16:08
19:04
21:04
23:06
10:01
11:33
13:05
15:33
17:59
20:35
22:35
00:07
PLU
CGH
PLU
CGH
POA
CGH
BSB
CGH
CGH
PLU
CGH
POA
CGH
BSB
CGH
PLU
Route
SL419
SL564
SL347
SL528
SL529
SL530
SL531
04
09:16
13:58
16:00
19:02
20:14
21:34
22:48
11:03
14:59
16:45
19:47
20:59
22:19
23:33
FLN
CGH
PLU
SDU
PLU
SDU
PLU
CGH
PLU
SDU
PLU
SDU
PLU
SDU
Route
JH345
SL282
SL283
SL405
SL514
SL515
SL594
05
09:30
14:22
15:42
17:14
20:30
21:45
23:18
12:23
15:13
16:37
19:57
21:21
22:37
01:57
SSA
CGH
CWB
CGH
CGH
CWB
CGH
CGH
CWB
CGH
CGH
CWB
CGH
POA
Route
SL166
JH559
SL590
SL591
SL598
SL599
SL518
06
08:00
09:04
11:06
13:30
16:00
19:46
22:16
08:31
10:35
12:44
14:57
17:55
21:41
23:07
GYN
BSB
CGH
POA
CGH
POA
CGH
BSB
CGH
POA
CGH
POA
CGH
CWB
Route
SL330
SL331
JH342
JH343
SL576
SL577
SL493
07
08:34
10:40
13:00
16:12
19:26
21:02
23:02
09:59
12:11
15:45
18:57
20:27
22:03
23:47
CGH
VIX
CGH
SSA
CGH
PLU
CGH
VIX
CGH
SSA
CGH
PLU
CGH
SDU
Route
SL595
SL150
SL151
SL481
SL482
SL412
SL413
08
08:40
11:42
13:58
17:00
18:30
20:10
22:22
11:11
13:33
15:59
17:45
19:21
21:29
23:31
POA
CGH
GYN
CGH
SDU
CGH
FLN
CGH
GYN
CGH
SDU
CGH
FLN
CGH
Route
SL155
SL336
SL337
SL483
SL492
JH558
09
08:32
11:16
15:46
20:08
21:30
22:56
10:33
15:05
19:29
20:53
22:21
00:27
GYN
CGH
REC
CGH
SDU
CGH
CGH
REC
CGH
SDU
CGH
BSB
Route
SL596
SL597
SL390
SL391
SL592
SL593
10
08:44
11:30
13:56
16:12
18:54
21:52
10:39
13:27
15:41
17:55
20:45
23:19
CGH
POA
CGH
FLN
CGH
POA
POA
CGH
FLN
CGH
POA
CGH
Route
SL533
SL543
JH500
JH507
SL332
SL333
11
08:50
11:00
13:02
19:00
21:20
23:04
10:01
12:03
14:33
20:31
22:31
00:15
BSB
PLU
CGH
BSB
CGH
VIX
PLU
CGH
BSB
CGH
VIX
CGH
Route
SL542
SL563
SL568
SL573
SL586
12
09:28
14:00
16:02
19:34
21:30
10:31
15:01
17:03
20:35
22:31
CGH
PLU
CGH
PLU
CGH
PLU
CGH
PLU
CGH
PLU
Route
JH502
JH503
JH546
JH547
SL407
13
10:02
12:02
17:56
20:00
22:24
11:33
13:33
19:27
21:31
23:09
CGH
BSB
CGH
BSB
CGH
BSB
CGH
BSB
CGH
SDU
Route
SL280
SL281
JH340
JH341
14
08:52
10:08
13:22
18:50
09:43
10:59
18:25
00:19
CGH
CWB
CGH
JPA
CWB
CGH
JPA
CGH
Route
SL589
SL152
SL153
SL154
15
09:58
17:44
20:04
23:22
11:01
19:35
22:05
01:13
PLU
CGH
GYN
CGH
CGH
GYN
CGH
GYN
Route
SL510
SL511
SL572
16
10:22 11:13 CGH CWB
11:50 12:41 CWB CGH
17:58 18:59 CGH PLU
Route 17
SL408 09:30 10:21 SDU CGH
SL272 14:12 15:51 CGH SJP
Route 19
JH501 14:58 16:29 BSB CGH
JH344 20:48 23:39 CGH SSA
Route 20
SL273 16:10 17:51 SJP CGH
SL418 23:32 01:19 CGH FLN
Route 18
SL532 10:20 11:31 PLU BSB
SL147 13:10 18:03 BSB BSB
APPENDIX "D" – OPTIMAL DYNAMIC AIRCRAFT ROTATION
Objective Value: 16.129
Route
SL280
SL281
NEW
SL523
SL508
NEW
JH546
JH547
SL518
01
08:52
10:08
11:28
12:46
14:56
16:01
17:56
20:00
22:16
09:43
10:59
12:29
13:31
15:41
17:02
19:27
21:31
23:07
CGH
CWB
CGH
PLU
SDU
PLU
CGH
BSB
CGH
CWB
CGH
PLU
SDU(+3)
PLU
CGH
BSB
CGH
CWB
Route
SL589
SL150
SL151
SL481
SL528
SL529
SL492
JH558
04
09:58
11:42
13:58
17:00
19:02
20:14
21:30
22:56
11:01
13:33
15:59
17:45
19:47
20:59
22:21
00:27
PLU
CGH
GYN
CGH
SDU
PLU
SDU
CGH
CGH
GYN
CGH
SDU
PLU
SDU
CGH
BSB
Route
SL330
SL331
SL282
SL283
SL572
SL573
SL586
07
08:34
10:40
14:22
15:42
17:58
19:34
21:30
09:59
12:11
15:13
16:37
18:59
20:35
22:31
CGH
VIX
CGH
CWB
CGH
PLU
CGH
VIX
CGH
CWB
CGH
PLU
CGH
PLU
Route
SL520
SL521
SL522
SL563
SL598
SL599
SL407
10
09:02
10:16
11:32
14:00
16:00
19:46
22:24
09:47
11:01
12:17
15:01
17:55
21:41
23:09
SDU
PLU
SDU
PLU
CGH
POA
CGH
PLU
SDU
PLU
CGH
POA
CGH
SDU
Route
JH345
SL402
SL403
SL514
SL515
SL154
13
09:30
13:38
16:08
20:30
21:45
23:22
12:23
15:33
17:59
21:21
22:37
01:13
SSA
CGH
POA
CGH
CWB
CGH
CGH
POA
CGH
CWB
CGH
GYN
Route
SL155
SL336
SL337
JH344
16
08:32
11:16
15:46
20:48
10:33
15:05
19:29
23:39
GYN
CGH
REC
CGH
CGH
REC
CGH
SSA
Route
SL408
SL409
SL470
SL471
SL480
SL405
SL584
SL585
SL418
02
09:30
11:00
12:28
13:54
15:30
17:14
20:26
22:04
23:32
10:21
11:49
13:19
14:39
16:21
19:57
21:27
23:05
01:19
SDU
CGH
SDU
CGH
SDU
CGH
CGH
PLU
CGH
CGH
SDU
CGH
SDU
CGH
CGH
PLU
CGH
FLN(+2)
Route
SL587
SL544
SL545
SL564
SL347
SL482
SL483
SL530
SL531
03
09:00
10:32
12:04
13:58
16:00
18:30
20:08
21:34
22:48
10:01
11:33
13:05
14:59
16:45
19:21
20:53
22:19
23:33
PLU
CGH
PLU
CGH
PLU
SDU
CGH
SDU
PLU
CGH
PLU
CGH
PLU
SDU
CGH
SDU
PLU
SDU
Route
SL166
JH559
SL590
SL591
SL152
SL153
SL594
05
08:00
09:04
11:06
13:30
17:44
20:04
23:18
08:31
10:35
12:44
14:57
19:35
22:05
01:57
GYN
BSB
CGH
POA
CGH
GYN
CGH
BSB
CGH
POA
CGH
GYN
CGH
POA
Route
SL595
JH500
JH501
JH506
SL537
SL577
SL493
06
08:40
13:02
14:58
17:02
19:22
21:02
23:02
11:11
14:33
16:29
18:33
20:33
22:03
23:47
POA
CGH
BSB
CGH
BSB
PLU
CGH
CGH
BSB(-4)
CGH
BSB
PLU
CGH
SDU
Route
SL596
SL597
SL390
SL391
SL576
SL536
SL167
08
08:44
11:30
13:56
16:12
19:26
21:04
22:42
10:39
13:27
15:41
17:55
20:27
22:13
23:13
CGH
POA
CGH
FLN
CGH
PLU
BSB
POA
CGH
FLN
CGH
PLU
BSB
GYN
Route
JH502
JH503
SL568
SL569
JH550
JH551
SL588
09
10:02
12:02
16:02
17:32
19:04
21:04
23:06
11:33
13:33
17:03
18:33
20:35
22:35
00:07
CGH
BSB
CGH
PLU
CGH
BSB
CGH
BSB
CGH
PLU
CGH
BSB
CGH
PLU
Route
SL542
SL543
JH342
JH343
SL412
SL413
11
09:28
11:00
13:00
16:12
20:10
22:22
10:31
12:03
15:45
18:57
21:29
23:31
CGH
PLU
CGH
SSA
CGH
FLN
PLU
CGH
SSA
CGH
FLN
CGH
Route
SL533
SL532
SL147
JH507
SL332
SL333
12
08:50
10:20
13:10
19:00
21:20
23:04
10:01
11:31
18:03
20:31
22:31
00:15
BSB
PLU
BSB
BSB
CGH
VIX
PLU
BSB(+1)
BSB
CGH
VIX
CGH
Route
SL419
SL272
SL273
SL592
SL593
14
09:16
14:12
16:10
18:54
21:52
11:03
15:51
17:51
20:45
23:19
FLN
CGH
SJP
CGH
POA
CGH
SJP
CGH(+1)
POA
CGH
Route
SL406
SL510
SL511
JH340
JH341
15
09:06
10:22
11:50
13:22
18:50
09:55
11:13
12:41
18:25
00:19
SDU
CGH
CWB
CGH
JPA
CGH(-2)
CWB
CGH
JPA
CGH
APPENDIX "E" – GRASP SUBOPTIMAL DYNAMIC AIRCRAFT ROTATION
Objective Value: 16.137
Route
SL166
JH559
SL409
SL470
SL471
SL480
SL481
SL482
SL483
SL492
JH558
01
08:00
09:04
11:00
12:28
13:54
15:30
17:00
18:30
20:08
21:30
22:56
08:31
10:35
11:49
13:19
14:39
16:21
17:45
19:21
20:53
22:21
00:27
GYN
BSB
CGH
SDU
CGH
SDU
CGH
SDU
CGH
SDU
CGH
BSB
CGH
SDU(+4)
CGH
SDU
CGH
SDU
CGH
SDU
CGH
BSB
Route
SL406
SL510
SL511
SL564
SL347
SL528
SL529
SL530
SL531
02
09:06
10:22
11:50
13:58
16:00
19:02
20:14
21:34
22:48
09:55
11:13
12:41
14:59
16:45
19:47
20:59
22:19
23:33
SDU
CGH
CWB
CGH
PLU
SDU
PLU
SDU
PLU
CGH(-2)
CWB
CGH
PLU
SDU
PLU
SDU
PLU
SDU
Route
SL533
SL532
JH503
SL272
SL273
JH550
JH551
04
08:50
10:20
12:02
14:12
16:10
19:04
21:04
10:01
11:31
13:33
15:51
17:51
20:35
22:35
BSB
PLU
BSB
CGH
SJP
CGH
BSB
Route
SL280
SL281
JH500
JH501
SL405
SL584
SL585
07
08:52
10:08
13:02
14:58
17:14
20:26
22:04
09:43
10:59
14:33
16:29
19:57
21:27
23:05
Route
SL520
SL589
JH340
JH341
SL514
SL515
SL594
10
09:02
09:58
13:22
18:50
20:30
21:45
23:18
Route
SL419
NEW
SL563
SL568
SL569
JH344
Route
JH502
SL147
JH507
SL586
Route
SL521
SL522
SL523
SL508
NEW
SL572
SL573
SL418
03
10:16
11:32
12:46
14:56
16:01
17:58
19:34
23:32
11:01
12:17
13:31
15:41
17:02
18:59
20:35
01:19
PLU
SDU
PLU
SDU
PLU
CGH
PLU
CGH
SDU
PLU
SDU
PLU
CGH
PLU
CGH
FLN
PLU
BSB(+1)
CGH
SJP(-1)
CGH
BSB
CGH
Route
SL330
SL331
JH342
JH343
SL576
SL577
SL493
05
08:34
10:40
13:00
16:12
19:26
21:02
23:02
09:59
12:11
15:45
18:57
20:27
22:03
23:47
CGH
VIX
CGH
SSA
CGH
PLU
CGH
VIX
CGH
SSA
CGH
PLU
CGH
SDU
Route
SL595
SL150
SL151
JH506
SL537
SL536
SL167
06
08:40
11:42
13:58
17:02
19:22
21:04
22:42
11:11
13:33
15:59
18:33
20:33
22:13
23:13
POA
CGH
GYN
CGH
BSB
PLU
BSB
CGH
GYN
CGH
BSB
PLU
BSB
GYN
CGH
CWB
CGH
BSB
CGH
CGH
PLU
CWB
CGH
BSB
CGH(+4)
CGH
PLU
CGH
Route
SL587
SL544
SL545
SL402
SL403
SL412
SL413
08
09:00
10:32
12:04
13:38
16:08
20:10
22:22
10:01
11:33
13:05
15:33
17:59
21:29
23:31
PLU
CGH
PLU
CGH
POA
CGH
FLN
CGH
PLU
CGH
POA
CGH
FLN
CGH
Route
SL542
SL543
SL282
SL283
SL152
SL153
SL154
09
09:28
11:00
14:22
15:42
17:44
20:04
23:22
10:31
12:03
15:13
16:37
19:35
22:05
01:13
CGH
PLU
CGH
CWB
CGH
GYN
CGH
PLU
CGH
CWB
CGH
GYN
CGH
GYN
09:47
11:01
18:25
00:19
21:21
22:37
01:57
SDU
PLU
CGH
JPA
CGH
CWB
CGH
PLU(-5)
CGH(+4)
JPA
CGH
CWB
CGH
POA
Route
SL155
SL590
SL591
SL598
SL599
SL518
11
08:32
11:06
13:30
16:00
19:46
22:16
10:33
12:44
14:57
17:55
21:41
23:07
GYN
CGH
POA
CGH
POA
CGH
CGH
POA
CGH
POA
CGH
CWB
Route
SL596
SL597
SL390
SL391
SL592
SL593
12
08:44
11:30
13:56
16:12
18:54
21:52
10:39
13:27
15:41
17:55
20:45
23:19
CGH
POA
CGH
FLN
CGH
POA
POA
CGH
FLN
CGH
POA
CGH
13
09:16
11:32
14:00
16:02
17:32
20:48
11:03
12:33
15:01
17:03
18:33
23:39
FLN
CGH
PLU
CGH
PLU
CGH
CGH
PLU
CGH
PLU
CGH
SSA
Route
SL408
SL336
SL337
SL332
SL333
14
09:30
11:16
15:46
21:20
23:04
10:21
15:05
19:29
22:31
00:15
SDU
CGH
REC
CGH
VIX
CGH
REC
CGH
VIX
CGH
Route
JH345
JH546
JH547
SL407
15
09:30
17:56
20:00
22:24
12:23
19:27
21:31
23:09
SSA
CGH
BSB
CGH
CGH
BSB
CGH
SDU
16
10:02
13:10
19:00
21:30
11:33
18:03
20:31
22:31
CGH
BSB
BSB
CGH
BSB
BSB
CGH
PLU