ARCHIVES
FLIGHT TRANSPORTATION LABORATORY
REPORT R86-6
INTERACTIVE DYNAMIC AIRCRAFT
SCHEDULING AND FLEET ROUTING WITH THE
OUT-OF-KILTER ALGORITHM
JAN VAN COTTHEM
INTERACTIVE DYNAMIC AIRCRAFT SCHEDULING AND
FLEET ROUTING WITH THE OUT-OF-KILTER ALGORITHM
by
JAN VAN COTTHEM
ABSTRACT
A decision support system is introduced that automates dynamic aircraft
scheduling and fleet routing. Interactive graphics-based schedule construction and modification tools automate the dynamic scheduling of aircraft of
a single-type aircraft fleet and the out-of-kilter algorithm is employed to
automate their routing. With the out-of-kilter algorithm, either the minimum fleet size required to serve a complete schedule or the routes that
aircraft of a fleet of fixed size should serve in order to maximize fleet income can be determined. Since many scheduling scenarios can be easily
constructed and evaluated, fleet routing solutions that achieve planning
goals, while satisfying organizational constraints, can be quickly obtained.
As a result, significant improvements in aircraft scheduling and fleet routing
productivity and quality are possible.
ACKNOWLEDGEMENTS
Programming the software for the decision support system presented
in this thesis was largely a painstaking and solitary activity. Nonetheless,
I would like to thank the people who contributed in various ways to the
undertaking: Professor Simpson, who proposed additions to the software
and corrected my faulty reasoning, D.F.X Mathaisel, Research Engineer,
who demonstrated an intermediate version of the program to an airline,
Boon Chai Lee, graduate student, who in the course of many discussions
on the topic, made several helpful suggestions, and Dionyssios Trivizas,
graduate student, who introduced me to the LATex word processor.
LIST OF FIGURES
Figure 1 - The kilter diagram of arc ij. ....................
41
Figure 2 - Acceptable kilter states for forward and reverse arcs
to be included in the flow-augmenting path. . . . . . . . . 42
Figure 3 - Kilter states of arcs linking labelled and unlabelled nodes
of which the node multipiers will be changed. . . . . . . . 43
Figure 4 - Daily schedule plan schedule chart. . . . . . . . . . . . . . .44
Figure 5 - Multiple-day schedule plan schedule chart. . . . . . . . . . 45
Figure 6 - Network arc parameters by schedule plan
and optimization option. . . . . . . . . . . . . . . . .
*.
.
.46
Figure 7 - Mathematical formulations by schedule plan
and optimization option. .................
.. . 47
Figure 8 - Iteration scenario when specified fleet size
is less than maximum fleet size. .............
.. . 48
Figure
Segment number input. ..................
.. . 49
Figure
- Origin station code name input. ............
.. . 50
Figure
- Destination station code name input .........
.. . 51
Figure
- Flight segment value input. ...............
Figure
- Flight segment duration input. .............
Figure
- The flight segment rectangle is moved to
the appropriate position. ................
. . . .52
.. . 53
. . . .54
55
Figure
Flight segment 101 has been scheduled. ............
Figure
Flight segment characteristics display. . . . . . . . . . . . 56
Figure
Flight segment 101's value is increased. . . . . . . . . . . .57
Figure
A schedule plan consisting of four flight segments. . . . . 58
Figure 19 - Flight segment 104 is about to be selected
for rescheduling . .
- . . . . . . . . . . . . . . . . ..
*. .59
Figure 20 - Flight segment 104 is being rescheduled. . . . . . .
. . .
.60
Figure 21 - Flight segment 104 is about to be deleted. . . . . .
. . .
.61
Figure 22 - Flight segment 104 has been deleted. . . . . . . . ..
Figure 23 - A new time interval is selected . . . . . . . . . . . .
.. .62
. . .
.63
Figure 24 - The visible portion of the schedule chart
has been changed. . . . . . . . . . .
. . . . . . . . . .
Figure 25 - The schedule plan of Figure 23 is tabulated. . . . .
.
. . 64
.
Figure 26 - Station display for stations AAA and BBB. . . . . .
.
65
.. 66
Figure 27 - Flight segment 103 will be deleted. . . . . . . . . . . . . .67
Figure 28 - Flight segment 103 has been deleted. . . . . . . . .
.
.
.
68
Figure 29 - Flight segment 107 is being added to
the schedule plan. . . . . . . . . . . . . . . . . . . . . . . . 69
Figure 30 - The value of flight segment 107 is 350. . . . . . . . . . . .70
Figure 31 - The duration of the flight segment is 2.5 hrs. . . . .
.71
Figure 32 - Flight segment 107's departure time is selected. . . .
. .72
Figure 33 - Flight segment 107 has been rescheduled. . . . . . . . . . 73
Figure 34 - The "time fields" for the selection of
arrival or departure times. . . . . . . . . . . . . . . . . . .74
Figure 35 - Flight segment 101 is selected for rescheduling. . . .
.75
Figure 36 - The arrow is moved to the new departure time. . . .
.76
Figure 37 - Flight segment 101 has been rescheduled. . . . . . . . . .77
Figure 38 - The time axis of either AAA or BBB
can be translated. . . . . . . . . . . . . . . . . . . . . . . .78
Figure 39 - BBB's visible time axis interval will be
from 0800 to 1800. . . . . . . . . . . . . . . . . . . . . . . .79
Figure 40 - BBB's time axis has been translated. ............
.80
Figure 41 - Schedule plan resulting from previous
flight segment manipulations. . . . . . . . . . . . . . . . .81
Figure 42 - Daily schedule plan for fleet routing demonstrations ... .82
Figure 43 - Answer to the first optimization option. . . . . . . . . . .83
Figure 44 - Fleet size input for the fleet routing option. . . . . . . . .84
Figure 45 - Generic daily ownership cost (doc) input
for the fleet routing option. . . . . . . . . . . . . . . . . . .85
Figure 46 - Answer to the second optimization option
with fs=1 and doc=-200. . . . . . . . . . . . . . . . . . . .86
Figure
Flight of aircraft 001. . . . . . . . . . . . . . . . . . . . . .87
Figure
Answer to the second optimization option
with fs=4 and doc=-500. . . . . . . . . . . . . . . . . . . .88
Figure
Flights for aircraft in a fleet of four. . . . . . . . . . . ..
Figure
Flight of one aircraft. AAA-BBB-AAA
89
and
CCC-DDD-CCC also qualify. . . . . . . . . . . . . . . . .90
Figure 51 - Attempt to delete flight segment 109 leads
to an imbalanced flow in the network. . . . . . . . . . . .91
Figure 52 - After flight segment 109 has been "removed",
aircraft 001 is rerouted. . . . . . . . . . . . . . . . . . . . .92
Figure 53 - The flow bounds on a cycle arc are going
to be changed. . . . . . . . . . . . . . . . . . . . . . . . . .93
Figure
BBB's cycle arc is selected. . . . . . . . . . . . . . . . . . .94
Figure
BBB's cycle arc lower bound remains 0. . . . . . . . . . .95
Figure 56 - BBB's cycle arc upper bound becomes 2...........
.96
Figure 57 - The flow bounds on a flight arc are going
to be changed. . . . . . . . . . . . . . . . . . . . . . . . . . 97
Figure 58 - Flight segment 103 is selected. . . . . . . . . . . . . . . . .98
Figure 59 - Flight segment 103's flight arc lower bound
remains at 0. . . . . . . . . . . . . . . . . . . . . . . . . . .gg
Figure 60 - Flight segment 103's flight arc upper bound
becomes 2. . .
. .
. . . . . . . . . . . . . . . . . . . .. 100
Figure 61 - Two aircraft now overnight at BBB. . . . . . . . . . . . .101
Figure 62 - A "shuttle" schedule plan between AAA and BBB.. . . 102
Figure 63 - Two aircraft are needed to serve the shuttle. . . . . . . 103
Figure 64 - Only one aircraft seems to be required........... .104
Figure 65 - First part of a SAS schedule plan. ...............
105
Figure 66 - Second part of a SAS schedule plan. . . . . . . . . . . . 106
Figure 67 - Six aircraft are needed to serve all flight segments. . . . 107
Figure 68 - Answer to the second optimization option
with fs=6 and doc=-100. . . . . . . . . . . . . . . . . . . 108
Figure 69 - Aircraft flights with fs=6 (first part). . . . . . . . . . . .109
Figure 70 - Aircraft flights with fs=6 (second part). . . . . . . . . . 110
Figure 71 - The aircraft flights have been manually
rearranged (first part). . . . . . . . . . . . . . . . . . . . 111
Figure 72 - The aircraft flights have been manually
rearranged (second part) . . . . . . . . . . . . . . . . . . 112
Figure 73 - Answer to the second optimization option
with fs=5 and doc=-100. . . . . . . . . . . . . . . . . . . 113
Figure 74 - Aircraft flights with fs=5 (first part). . . . . . . . . . . . 114
Figure 75 - Aircraft flights with fs=5 (second part). . . . . . . . . . 115
Figure 76 - Aircraft flight with fs=1 (first part). . . . . . . . . . . . 116
Figure 77 - Aircraft flight with fs=1 (second part). . . . . . . . . . . 117
Figure 78 - Change in computation time with number of rotations. 118
Figure 79 - Change in computation time with fleet size. . . . . . . . 119
Figure 80 - Input of day of occurrence of a flight segment
of a multiple-day schedule plan. . . . . . . . . . . . . . . 120
Figure 81 - Flight segment 101 has been scheduled for day 14. . . . 121
Figure 82 - Flight segment 101's day of occurrence will be changed. 122
Figure 83 - Flight segment 101 is rescheduled for day 39. . . . . . . 123
Figure 84 - Flight segment 101 has been scheduled for day 39.. . . 124
Figure 85 - Station display for flight segment 101. . . . . . . . . . . 125
Figure 86 - One aircraft flies from AAA to FFF in 17 days. . . . . . 126
Figure 87 - Answer to the first optimization option. . . . . . . . . . 127
Figure 88 - Answer to the second optimization option. . . . . . . . 128
Figure 89 - Flight ZZZ-BBB-CCC has been selected,
but AAA-BBB-CCC is also possible. . . . . . . . . . . . 129
Figure 90 - Multiple-day schedule plan between days 1 and 12. .
.
. 130
Figure 91 - Answer to the second optimization option
with fs=3 and doc=-300. .....................
131
Figure 92 - The flights selected for the three aircraft......... .132
Figure 93 - The flights selected for two aircraft............
.133
Figure 94 - Schedule chart with maintenance arc. ............
134
Figure 95 - Schedule plan consisting of two flights.......... .. 135
Figure 96 - Aircraft are rerouted after a maintenance
-
stopover has been introduced...............
136
Figure 97 - Loss of service on a flight segment
when maintenance stopover is too long. . . . . . . . . . 137
Figure 98 - The network data structure consists of a node
linked list and an arc linked list. . . . . . . . . . . . . . 138
Contents
1
introduction
11
2
The Out-of-kilter Algorithm
13
2.1
The Minimum Cost Flow Problem ..............
13
2.2
The Out-of-kilter Algorithm .......................
14
2.3
Fleet Routing With The Out-of-kilter Algorithm . . . . . .
18
3
Interactive Dynamic Aircraft Scheduling and Fleet Routing
,
26
3.1
The IASP for .the Daily Schedule Plan ................
26
3.2
Fleet Routing with the Daily Schedule Plan .........
30
3.3
Provision for Maintenance Stopovers ................
34
3.4
The IASP for the Multiple-day Schedule Plan . . . . . . . .
35
3.5
Fleet Routing with the Multiple-day Schedule Plan
36
. . . .
4 Conclusions
38
A Figures
40
B Data Structure
139
Chapter 1
introduction
The development of a working aircraft schedule plan' within an air
transportation organization is a formidable task. A schedule plan has to be
constructed that not only achieves the goals of geographic, frequency and
departure time planning,but also satisfies the constraints of fleet size and
maintenance and crew schedules.
The development of a working schedule plan is an iterative process that
tries to route aircraft onto flights of a schedule, meet the planning goals and
satisfy operational constraints. Presently, the development occurs manually, whereby aircraft are routed on a schedule chart. Since adjustments to
the schedule plan or the aircraft routes can require the erasure of existing
flights and the addition of new ones onto the chart, the iterative construction of a schedule plan can be time-consuming.
After a working schedule plan has been finalized, many occurrences can
require rescheduling of flights and rerouting of aircraft: maintenance and
'Definitions of schedule-related terms are given on p.18.
crew schedule changes, adverse weather conditions, or new demands for service. This rescheduling is called dynamic scheduling and usually occurs
under time-pressure.
Because even dynamic scheduling can be time-consuming, the need has
been recognized for a computer-based system that would facilitate the development of working schedule plans. However, since the computer is still
less versatile than the human mind in making decisions, the development
of an efficient man-computer interface has been focused upon [1]. Such an
interface has been referred to as a decision support system (DSS) 12,3].
In order to gain acceptance, a DSS should, ideally, simulate tried and
true schedule construction and modification methods. As part of an evolutionary increase in the effectiveness of a DSS, however,it is desirable to
also include operations research models for optimization of fleet routing. In
this work, a DSS is presented that combines both features. It employs interactive graphics-based schedule construction and modification tools and
optimization routines based on the out-of-kilter algorithm [4,5,6,7 for fleet
size determination and fleet routing. The use of the out-of-kilter algorithm
for fleet routing was already proposed by Simpson [8] in 1969.
The following chapter covers respectively the theory of the out-of-kilter
algorithm, its application to fleet routing, the tools for schedule plan construction and modification, and examples of fleet routing optimization.
Chapter 2
The Out-of-kilter Algorithm
2.1
The Minimum Cost Flow Problem
The minimum cost flow problem is a linear programming problem which
has the following formalization:
minimize
E cijz;;
isjEA
subject to
: zik ikEA
, zki = 0
VEN
kiEA
0 4, ix,,uq;
WjEA
Vij
,where zig is the amount of flow in arc ij, cii is the cost a unit of flow
incurs in ij, 4, and u,, are respectively the lower and upper bounds on z;;,
and A is the set of arcs and N is the set of nodes in the graph of the network.
The optimal solution to the minimm cost flow problem is a set of flows
in the arcs of a network which has a minimum total cost. In addition, the
flows in this set are conserved at every node of the network and remain between the upper and lower bounds on the flows in the arcs of the network.
The optimal solution to the minimum cost flow problem is obtained when
any of the following conditions are satisfied for every arc in the network:
k = ci;
+h
- IL < 0
when x; = ugg
Ziy
> 0
when zig = t5
k= 0
when 153;ziy<_ugy
,where Il; is the simplex multiplier for node i. An arc that satisfies
one of the above optimality conditions is said to be in-kilter, otherwise,
it is out-of-kilter. Since Zij appears as a single variable in the optimality
conditions, the kilter state of the arc ij can be conveniently represented
in a Z5 vs zij or kilter diagram. The mapping of the kilter states corresponding to the optimality conditions for arc ij then results in a kilterline
of the form shown in Figure 1 (see Appendix A).
2.2
The Out-of-kilter Algorithm
An algorithm has been developed that modifies the arc flows and node
multipliers in order to bring every arc of the network in-kilter; it is called
the out-of-kilter algorithm.
The out-of-kilter algorithm requires an initial set of values for the arc
flows and the node multipliers; usually, all of these values are equal to zero,
but this need not be the case. Indeed, the values of the arc flows and node
multipliers of a problem that differs only slightly from the one studied are
also acceptable. After the arc flows and node multipliers have been
initialized, the algorithm searches for all out-of-kilter arcs in the network
and attempts to bring them in-kilter, one by one. There are two algorithms
in the out-of-kilter algorithm which can accomplish this task: one that adjusts the arc flows and one that modifies the node multipliers.
The former algorithm is a labelling algorithm that seeks to change the
flow in an arc ij that is out-of-kilter. Depending on the value of Zi,,
either increases or decreases the flow from i to
flow-augmenting path from respectively
j
j.
In both instances, a
to i or i to
j
has to exist in
the network in order for the flow to be conserved at i and
labels - i and
j
it
j.
The node
- are by convention reversed in the first case, so that the
flow augmenting path always goes from i to j.
The arcs that are included in the flow-augmenting path can be forward
or reverse. A forward arc is included in the path if either
'eg > 0 and z,,> t;
or
Egy < 0 and z,, > uq
and a reverse arc is added if either
Egg < 0
or
and zig > uq
Eg > 0 and z;; > i.
These acceptable kilter states are shown in Figure 2.
Since no arc that is already in-kilter should be brought out-of-kilter and
no distance between an out-of-kilter arc's position in the kilter diagram and
its kilter line should be increased, flow changes of too great a magnitude
in the arcs of the flow-augmenting path are prohibited. To this end, the
labelling algorithm assigns two labels to every new node of the path. One
label is the label of the node that precedes the new node in the path from i
(with, in addition, a + or - that indicates whether the new arc is forward
or reverse). The other label is an integer whose value is the maximum additional amount of flow that can be sent along the path without moving any
arc in the path beyond its kilter line. This flow increment is the smaller of
the previous flow increment and the maximum flow increment possible in
the new arc of the path. The new arc's maximum flow increment equals:
for a forward arc ij:
1, - xi when Egg > 0
u,; - xi when Eg <; 0
for a reverse arc ij
4,
when
gij> 0
z;; - ugg when
Egg < 0.
xig -
When the endnode j of the out-of-kilter arc ij has been labelled, a flowaugmenting path has been found (a breakthrough has occurred) and the
value of j's flow increment label is added to the flow in every arc of the
flow augmenting path. Hereafter, all label pairs of the nodes of the path are
discarded and the kilter state of ij is determined once more. The labelling
algorithm is applied to ij until ij is either in-kilter or no flow-augmenting
path is found (a non-breakthrough has occurred).
In the latter case, the second algorithm is employed to aid in bringing
the out-of-kilter arc in-kilter. This algorithm changes the value of the node
multipliers of the labelled nodes whose incidents arcs link labelled and unlabelled nodes by an amount M which is the minimum of:
-The minimum of Egg for all forward arcs ij linking labelled and
unlabelled nodes for which Eg > 0 and tq<xi < ug.
-The minimum of |eq | for all reverse arcs ij linking labelled and
unlabelled nodes for which Eg < 0 and t4 < xzjgujg.
-Infinity.
The first two cases are shown in Figure 3. The new values of the reduced
costs of the forward and reverse arcs will therefore be, respectively, ?Eg-M
and Z,,+M. As a result, at least one arc's new position in its kilter diagram
will be on the horizontal segment of the kilter line and no arc's position will
be beyond this segment. If M has an infinite value, however, no feasible
flow exists and the out-of-kilter algorithm then terminates. If infeasibility
has not occurred, the second algorithm will have prepared at least one
more node for labelling by the first algorithm - thereby extending the flowaugmenting path - or the out-of-kilter arc will have been brought in-kilter.
In the latter case, the out-of-kilter algorithm searches for a new out-ofkilter arc; otherwise, it restarts the labelling process from node i, since
changes in the node multipliers may have affected the effective capacity of
the flow-augmenting path.
Eventually, the out-of-kilter algorithm either has brought all out-ofkilter arcs in-kilter, in which case a minimum cost flow has been found, or
has not been able to find a flow augmenting path for an out-of-kilter arc,
in which case no feasible flow can exist in the network, due to a conflict
between flow constraints.
2.3
Fleet Routing With The Out-of-kilter
Algorithm
The out-of-kilter algorithm can be applied to various network-related
mathematical models of realistic problems when the cost parameters of the
minimum cost flow objective function are changed, or arcs or arc parameters are added to the original network. Consequently, the out-of-kilter
algorithm can solve some of the most common network flow problems:
maximum flow, shortest path, assignment, transportation and transshipment problems. This fact clearly demonstrates its power and usefulness.
On the following pages, the application of the out-of-kilter algorithm to
mathematical models of two common and closely related air transportation
problems is discussed:
- Find the minimum size of an aircraft fleet, so that all flight
segments of the schedule plan are served.
- Find the optimal flights (which accomodate the flight segments of
highest "value") over a schedule plan for an aircraft fleet of fixed
size.
A flight segment of a schedule plan is a non-stop connection between
two airfields at a paricular date and time and a flight represents the sequence of flight segments assigned to a single aircraft for service; the geographic trace of a flight is called a route. The schedule plan is then a
collection of flight segments that are to be served, or can be served, within
a planned time horizon.
Two types of schedule plans will be converted into mathematical models
that are suitable for solution with the out-of-kilter algorithm. The first type
of schedule plan, the daily schedule plan, consists of flight segments that
are to be served on a daily basis and whose flights (rotations) usually1 originate and terminate at the same airfield, where the aircraft will overnight.
The second type of schedule plan, the multiple-day schedule plan, contains flight segments that have been planned for service during a period of
at least one day. The aircraft of a fleet assigned to this schedule plan can
therefore overnight at stations along their route and do not have to return
to the station at which their flight originated.
The value of a flight segment can have several meanings, depending
on the nature and circumstances of the flight segment or the objectives of
the people who have an interest in it. The value of a commercial flight
segment, for example, might be its revenue, whereas the value of a military
flight segment might be directly related to the priority of its cargo.
'An exception is presented on p.31.
One important restriction to the application of the models is that neither one can satisfy the optimization criteria for a fleet of aircraft of mixed
type. A fleet routing problem for a mixed-type aircraft fleet will therefore
have to be decomposed into subproblems according to each aircraft type.
The type of schedule plan determines the form of its equivalent network,
which in turn influences the identity of the mathemathical model. Arcs,
nodes, lower and upper bounds and flow costs on the arcs are the building
blocks of a network that corresponds to a schedule plan. Flight, ground
and cycle arcs and station event nodes provide the framework of the
network. Here, a station event corresponds to an arrival' or departure of
a flight segment and a ground arc connects two successive station events.
In the case of a daily schedule plan, the cycle (overnight) arc links the last
station event to the first at each station in the schedule plan. The arcs and
nodes of this schedule plan can be graphically represented as a schedule
chart, as shown in Figure 4. In the case of the multiple-day schedule plan,
the network also contains arcs that connect the "first" station events with
a source node and arcs that connect the "last" station events with a sink
node. One cycle arc connects the sink node with the source node and there
are no overnight arcs in this network, which is shown in Figure 5.
The optimization problem and the type of schedule plan determine the
mathematical model and the values of the parameters of the arcs in the
network. The arc parameters are presented in Figure 6 and the mathematical model formulations in Figure 7.
2 In
practice, the arrival time is converted into an earliest time of departure, or "ready"
time.
The values of the arc parameters for the first optimization problem determining the minimum fleet size - are identical for the daily and multipleday schedule plans. The "value" of a flight arc is negative, reflecting its
revenue or negative cost. Since the first optimization problem is actually
reduced to the counting of the minimum number of aircraft needed to serve
all flight segments of the schedule plan, both the lower and upper bound
on the flow in the flight arcs are equal to one, so that exactly one aircraft will serve each flight segment. As is shown in Figure 7, for the daily
schedule plan network the sum of all cycle arc flows equals the fleet size:
E cxzi =
ijEA
E
z,,. For the multiple-day schedule plan network,no
ijEcycle arc
overnight arcs exist and the fleet size is consequently equal to the flow in
the cycle arc ts, zt,.
The mathematical models for the second optimization problem - finding
the optimal flights in a schedule plan for an aircraft fleet of fixed size - are
not as straightforward as the previous mathematical models; a few concepts
need to be introduced before they can be explained:
-The daily ownership cost of an aircraft, doc, is a measure of
the daily expense associated with the ownership and operation of
an aircraft; it has a positive value.
-The take of an aircraft:
E
cixji , where ij is an arc in
ijEflight arC
the network representing a flight segment of the aircraft's flight.
-The net take of an aircraft:
E
ci zij + doe
ijfligMs arcs
-The net take of an aircraft fleet is the sum of the takes of all
aircraft in the fleet:
-for the daily schedule plan:
cijzi + doc (
E
ijEflight
-for the multiple-day schedule plan:
zij)
ijEcyle arc
area
E
c,,zxi + doc zg
ijEflight are*
-The maximum doc is that doc for which the net take of the
aircraft fleet becomes zero.
For the second optimization problem, the objective is to maximize the
net take of the aircraft fleet. In order to describe the mathematical models
in minimum cost form, the objective functions in Figure 7 are the negative
of the fleet net takes given above. Both mathematical models now contain
the additional constraint that the fleet size cannot exceed a given number,
FS. As is shown in Figure 6, the lower and upper bounds on the flow in
the cycle arcs of the daily schedule plan network, t, and uc, can be set to
values that reflect restrictions imposed on the number of aircraft that can
overnight at the airfields. The default values are respectively 0 and oo, so
that an indefinite number of aircraft can overnight at each airfield. The
lower and upper bounds on the flight arcs, if and u, can also be changed,
so that several aircraft can simultaneously serve a given flight segment. For
the multiple-day schedule network, u, determines the maximum fleet size,
while the flight arc flow bounds allow for not more than one aircraft to
serve each flight segment. For both networks, the daily ownership cost is
assigned to the cycle arc(s).
As was described above, the solution to the first optimization problem
consists of the minimum fleet size necessary to serve all flight segments of
the schedule plan. It is obtained by employing the out-of-kilter algorithm
once and summing the flows in all the cycle arcs.
The solution to the second optimization problem consists of the fleet size
and the flights assigned to each aircraft in the fleet. The solution method
for the fleet size determination is based upon the fact that the net take
of each aircraft in the fleet has to be positive; if there are aircraft whose
take is less than the daily ownership cost, they are to be removed from
the aircraft fleet and the aircraft in the remainder of the fleet can then be
assigned new flights, which may include individual flight segments of the
discarded flights if these increase their take. Since the daily ownership cost
thus indirectly sizes the aircraft fleet, it assumes the role of control variable in the iteration procedure that maximizes the fleet's net take, while
satisfying the fleet size constraint.
The iteration procedure arises from the use of a Lagrange multiplier
technique and is explained with the aid of the following scenario. Suppose
that an initial doc, doc1, results in a fleet size fs, larger than FS, as shown
in Figure 8. This situation can occur when FS is less than the fleet size
necessary to serve all flights in the schedule plan. Since fs decreases with
increasing doc, doc1 is added to doc until fs is less than or equal to FS. At
doc3, fs is less than FS and doc3 is therefore decreased by an amount equal
to one-half of the original increase in an attempt to reach the FS mark.
Since fs is still less than FS at doc4, doc4 is again decreased, so that doc2
is reached anew. Now, however, only one-quarter of the original increase
(one-half of the previous decrease) is added to doc2. In this case, the iteration procedure terminates at doc5 because the termination criterion is
now satisfied: fs is equal to FS.
If the aircraft fleet is larger than is necessary to serve all flight segments in the schedule plan, it is decreased to the maximum fleet size for
the given schedule plan before the iteration procedure is started. In section 3.2, it is shown that this procedure results in a significant reduction
in computation time, because the solution is obtained after fewer iterations.
Upon termination of the iteration procedure described above, the out-ofkilter algorithm has determined the fleet size and the set of flight segments
of the schedule plan that are served, but not the aircraft flights. The latter
are determined in a final employment of the out-of-kilter algorithm. First,
the arc flows and node multipliers are reinitialized and the cycle arc costs
are set to the final value of doc. Then, as each arc of the network is brought
in-kilter for the first time, the set of flight segments that are brought inkilter along with it are assigned a unique label. The time-ordered sequence
of those flight segments then forms the flight of a single aircraft in the fleet.
In this section, the determination of the minimum fleet size and the
set of flights corresponding to a given fleet size was presented as a pair of
optimization problems for two schedule plans, a daily and a multiple-day
schedule plan. A solution to these problems with the out-of-kilter algorithm
led to the conversion of the optimization problems and schedule plans into
four distinct mathematical models.
Whereas the solution to the former
optimization problem could be determined after a single employement of
the out-of-kilter algorithm, the latter optimization problem dictated an iterative solution to the fleet size constraint, based on the variation of the
generic daily ownership cost of the aircraft in the fleet. Aircraft flights came
into existence as batches of flight segments simultaneously brought in-kilter.
In the following chapter, the working of the schedule planning/fleet
routing DSS will be discussed.
Chapter 3
Interactive Dynamic Aircraft
Scheduling and Fleet Routing
The PASCAL procedures that collectively solve both optimization problems exist as two sets, one for the daily schedule plan and one for the
multiple-day schedule plan, and each set is imbedded within a copy of an
interactive aircraft scheduling program (IASP) that has been developed
under Version 3 of the Apple Macxl PASCAL Workshop. The features of
the IASP and the optimization problem solvers will be described hereafter.
Appendix B explains the data structure behind the networks that represent
the schedule plans.
3.1
The IASP for the Daily Schedule Plan
The IASP allows for the dynamic construction and modification of a
schedule plan because menu options can be moused to activate PASCAL
procedures that add, delete, move, or modify flight segments displayed on a
schedule chart; presently, a maximum of 104 flight segments can be created.
Figures 9 through 22 animate the construction of a simple daily schedule
plan and the alteration of flight segment characteristics.
Upon selection of the add option, the IASP prompts for the flight segment's number, origin code name, destination code name, value and duration, as may be seen in Figures 9 through 131. After this information has
been input, a rectangle that depicts the flight segment appears and can be
moved with the mouse to the desired position on the schedule chart. For
any particular departure time, the vertical position on the schedule chart
is unimportant. The minutes past the hour of the departure time are displayed in the upper-righthand corner, as shown in Figure 14. Figure 15
presents a flight segment that departs from AAA at 2 pm and arrives at
BBB at 5 pm and has a value of 100.
In order to alter any flight segment characteristic, both the modify option and the flight rectangle must be moused; a new display then appears
as presented in Figure -16. If, for example, the flight segment's value is to
be increased to 200, the value bar is moused and the new value is input, as
is shown in Figure 17.
Figure 18 presents a schedule plan consisting of four flight segments between stations AAA, BBB and CCC. Any flight segment can be moved or
deleted by mousing both the corresponding option and the flight segment
rectangle itself. The result of selecting each option is shown in Figures 19
through 22; flight segment 104 is first moved and then deleted.
1Although the cost parameter of a flight segment is negative, the value of the flight segment
is input as a positive number,corresponding to the inflow of 'value".
The display can also be translated along the time axis of the schedule
chart. To this end, the time axis option is moused and the "time rectangle" is moved; only hourly translations can be made. The time axis option
display is shown in Figure 23 (the time interval of the portion of the schedule chart currently displayed is painted black) and the display resulting
from the translation of the time rectangle is shown in Figure 24.
Mousing the schedule option results in a tabular display of the schedule plan, which may be seen in Figure 25.
With the station option, all arrivals and departures at two airfields of
the schedule plan can be seen. In addition, new flight segments between
those airfields can be added and existing flight segments can be deleted and
moved. The modify, schedule and end options all return the display to
the original schedule chart.
Figure 26 presents the station display for stations AAA and BBB. In
order to delete a flight segment, the delete option is selected and the
mousebutton is pressed when the mouse arrow is positioned over the flight
segment number of either the arrival or departure arrow, as shown in Figures 27 and 28.
In order to add a flight segment, its number, duration and value are
input and its arrival or departure time is selected by positioning the mouse
arrow in the desired "time field" and mousing at the desired time. This
sequence of events is shown in Figures 29 through 33. Flight segment 107
departs from AAA at 1530 and arrives at BBB at 1730. In Figure 34, the
four time fields are indicated; mousing in any one is sufficient to schedule
a flight segment between AAA and BBB. If the departure time is DT and
the arrival time is AT, then mousing in time field
1 fixes the AT of BBB to AAA,
2 fixes the DT of AAA to BBB,
3 fixes the AT of AAA to BBB,
4 fixes the DT of BBB to AAA.
A flight segment's arrival or departure time can be moved by selecting
the slide option, marking the arrival or departure arrow, as before, and
mousing at the new arrival or departure time, as shown in Figures 35, 36
and 37, where the departure time of flight segment 101 has been moved to
1410.
With the time axis option, the visible portion of each station's time
window can be changed independently.
It suffices to select the desired
station option and move the time rectangle to its new position, as was explained before. In Figures 38, 39 and 40, station BBB is selected and the
visible portion of its time axis is translated.
As a result of the flight segment manipulations described above, the
schedule chart display now appears as shown in Figure 41.
The IASP offers an array of tools with which a schedule plan can be
constructed and modified. It also employs the schedule plan as an input to
both optimization options. Hereafter, the input to and output from both
optimization options will be described and evaluated for the daily schedule
plan.
3.2
Fleet Routing with the Daily Schedule
Plan
The input-output features of the optimization options will be identified
with the aid of the schedule plan presented in the form of the schedule chart
of Figure 42.
Upon selecting the first optimization option (1), the display becomes as
given in Figure 43. Four aircraft are needed to serve all flight segments and
they overnight at stations AAA, BBB, and DDD. The response time of .2
sec demonstrates the high speed with which the network is constructed and
the minimum fleet size problem is solved.
The second optimization option (2) requires as input the maximum
number of aircraft in the fleet and the daily ownership cost of the aircraft,
in addition to the schedule plan. As is shown in Figures 44 and 45, the flight
of one aircraft witha doc of -200 is to be found
2.
The first part of the so-
lution is presented in Figure 46. The aircraft's net take is 360 - 200 = 160,
2
Again, the cost parameter of a cycle arc is positive, but the doc is input as a negative
number to account for the outflow of "value".
its maximum doc is 360 and it overnights at station AAA. The message
"grey flights are flown" points to the second part of the solution, which can
be seen after mousing. The grey (black here) flight segment rectangles that
are tagged with the aircraft number belong indeed to the flight aircraft 001
can serve, as may be seen in Figure 47. Aircraft 001's take is the highest
take a single aircraft can achieve for this schedule plan. Figures 48 and 49
confirm the answer of the first optimization option: four aircraft are needed
to serve all flight segments in the schedule plan. Since the doc was set at
-500, which is higher than the maximum doc of -265, the fleet net take
is negative. Although flight DDD-CCC-DDD has the same total value as
flight BBB-CCC-BBB, the progam assigns one aircraft to each flight. As
a result, flights that are not included in the final schedule plan, but could
be preferrable to flights of equal total value that are included, should be
checked for. An example is given in Figure 50. If only one aircraft is available, the program selects flight EEE-FFF-EEE, but either one of the other
two flights may be preferrable.
Under realistic conditions, some flight segments may become unservicable at certain moments. Rather than deleting from the schedule plan the
flights whose flight segments cannot all be served, the IASP offers the opportunity to "remove" those flight segments and reoptimize the fleet routing
for the modified schedule plan. An obvious step would be to employ the
delete option to this end; however, if, for example, flight segment 109 of
Figure 47 is thus deleted, the display becomes as shown in Figure 51, because of a violation of the flow conservation constraint at stations CCC and
EEE: their number of arrivals does not equal their number of departures.
Instead of deleting the non-servicable flight segments, the modify option
should be employed; it suffices to select the value bar and input "out" to
"remove" the flight segment from the schedule plan3 . The flight segment
will then be displayed in light gray on the schedule chart. If flight segment
109 has become unservicable, a single aircraft will be rerouted from the
optimal flight shown in Figure 47 to the one shown in Figure 52.
With the u,l option, the flow bounds on any flight arc or cycle arc can
be adjusted. If the number of overnight aircraft at an airfield has to be
restricted, the word "stat" (short for station) and the station code name
and the upper and lower bounds have to be input, as shown in Figures 53
through 56 for station BBB, where the upper bound is set to 2. The number
of aircraft that can simultaneously serve a flight segment can be changed
by respectively inputting "fit", selecting the flight segment rectangle and
inputting the upper and lower bounds on the flow in its corresponding flight
arc, as shown in Figures 57 through 60 for flight segment 103. If both flight
segment 103 and 104 have an upper bound equal to 2 and the fleet size is
set to 5, the solution to option 2 is as shown in Figure 61. Two aircraft
now overnight at station BBB and serve flight segments 103 and 104.
Figure 62 presents a "shuttle" schedule plan between stations AAA and
BBB. The solution to option 2 reveals that two aircraft are needed to serve
both flight segments, as shown in Figure 63 (the doc was -100). In Figure
64, however, all flight segments have been labelled with '001'. The reason
is that they were all brought in-kilter simultaneously and therefore labelled
'in'
reinserts the flight segment into the schedule plan.
accordingly. For any schedule plan, the presence of "shuttles", if not immediately visible on the schedule chart, will always be noticeable from the
discrepancy between the actual fleet size and the maximum label number
of the flight segments in the schedule plan.
The schedule plan discussed above has served to explain the characteristics of the IASP and the optimization options; a final, somewhat more
realistic schedule plan, will now be presented. This schedule plan will also
serve as a test case for the performance measurement of the optimization algorithms for the daily schedule plan. The schedule plan is shown in Figures
65 and 66 and has been derived from a portion of an actual Scandinavian
Airline System flight schedule. Each row of flight segments represents a
complete rotation. Since there are six rotations, six aircraft should be required in order to serve all flight segments of the schedule plan. The solution
to option 1 confirms the fleet size and the overnight stations and is shown
in Figure 67; it was found after .8 sec. The solution to option 2 - with an
input of six aircraft for the fleet size and -100 for the doc - also confirms
the above result, but includes rotations different from the ones that were
apparent. This solution is presented in Figures 68, 69 and 70. The flight
segments belonging to the same aircraft can now be manually rearranged
so that they appear on the same horizontal line, as shown in Figures 71 and
72. The fleet net take for an aircraft fleet that flies the original rotations or
the ones found by the optimization procedures will naturally be the same;
if, however, one aircraft breaks down, it may not be obvious which rotations should be served so as to minimize the loss in take; the optimization
algorithms will provide that answer. For the solution shown in Figures 73,
74 and 75, the fleet size was set to five, while the doc was kept at -100. The
fleet net take has decreased to 3815, which is attributable to the unservicable flight. It is worth noting that the flights shown in Figures 74 and 75
are different from those in Figures 69 and 70 because the reoptimization
has reallocated the flight segments to the aircraft of the fleet. If only one
aircraft was available, it would serve the flight shown in Figures 76 and 77.
The SAS schedule plan example has been used to determine the performance of the software. Figure 78 shows that the computation time grows
linearly with the schedule plan size. Figure 79 shows the variation of computation time with fleet size for a fixed schedule plan. The computation
time for fleet sizes smaller than the maximum fleet size (6) varies from two
to three times the computation time for the maximum fleet size case, because the number of iterations required to obtain a solution varies with the
fleet size. The computation time for fleet sizes larger than the maximum
fleet size is, however, constant and approximately equal to the minimum
computation time, because the fleet size is automatically decreased to the
maximum fleet size before the iteration is started.
3.3
Provision for Maintenance Stopovers
Maintenance checks, regular or unexpected, are performed on all operating aircraft. They are represented as maintenance arcs in the daily
schedule plan network, which force aircraft to undergo maintenance at a
particular station and time. A maintenance arc is depicted in Figure 94;
the cost of the arc is set to zero because maintenance accounting is normally
distinguished from flight service accounting. The addition of a maintenance
arc should not change the flow in the arcs whose finish time is less than the
start time of the maintenance arc if an out-of-kilter flow solution already
existed for the original network, this to avoid rerouting of the aircraft prior
to the maintenance stopover.
Figures 95 and 96 demonstrate the rerouting that can occur after a
maintenance stopover that has been introduced into a schedule plan. In
Figure 95, the flights of two aircraft have been determined with the second
optimization option. One flight includes flight segments 101 and 102, the
other includes flight segments 103 and 104. In Figure 96, a maintenance
stopover of 3 hours at station BBB - depicted by segment M01 - was added
and the flow bounds of segments 101 and M01 were set to 1 with the u,l
option. All flight segments are still served, but the flights have changed.
Whereas one flight now consists of flight segments 102 and 103, the other
consists of flight segments 101 and 104 and the maintenance stopover at
station BBB.
Figure 97 shows that aircraft 001 is not able to serve flight segment 104
if the maintenance stopover lasts until a time later than the departure time
of the flight segment.
3.4
The IASP for the Multiple-day Schedule
Plan
The flight segments of a multiple-day schedule plan are distinguished,
not only by the characteristics they have in common with the daily schedule plan flight segments, but also by their day of occurrence. With this
program, flight segments can be scheduled within a time horizon of 999
days. The day for which the flight segment is scheduled simply has to be
input as part of the input of the flight segment characteristics, as shown in
Figure 80. In Figure 81, flight segment 101 has been scheduled for day 14.
If it has to be rescheduled, the day option should be selected and a new
day input, as demonstrated in Figures 82, 83 and 84, where flight segment
101 has been rescheduled for day 39.
Figure 85 presents the station display for flight segment 101. The day
for which the flight segment is scheduled now marks both the arrival and
departure arrows.
3.5
Fleet Routing with the Multiple-day
Schedule Plan
Figures 86, 87 and 88 show the solution to respectively option 1 and
option 2 for a flight scheduled for one aircraft that has a doc of -100. The
flight starts at station AAA at day 12 and ends 17 days later at station
FFF. The overnight stations of the flight are not explicitly shown, but can
be inferred from the days of occurrence of succesasive flight segments.
When two aircraft compete for the same flight segment and the take of
either aircraft would be the same, the program assigns the flight segment
to one aircraft, as shown in Figure 89, where the fleet size was set to one.
Flight segments that are not included in the final solution, but may be
preferrable, should therefore be inspected.
Figure 90 presents a schedule plan set to occur between days 1 and 12.
The schedule plan consists of three flights, as shown in Figures 91 and 92
(the doc was -100) . The solution was obtained after barely .3 sec. If only
two aircraft were available, their flights would become as indicated in Figure 93. The flight of aircraft 001 is now different from the one of Figure 92
because the aircraft had access to the flight segments of the "third" aircraft.
In this chapter, schedule construction and modification with the IASP
and fleet routing with imbedded optimization procedures based upon the
out-of-kilter algorithm have been described. It has been shown that a schedule plan is easily built and modified and that optimal flights and minimal
fleet sizes are, in most cases, quickly obtained. Maintenance stopovers, if
carefully scheduled, can be integrated into a schedule plan with a minimal
loss of regular service. The usefulness of the solutions to the optimization
problems, however, ultimately have to be considered relative to constraints
and demands other than those related to the schedule plan itself. This fact
will be expanded upon in the conclusion presented hereafter.
Chapter 4
Conclusions
Computer-based schedule plan construction and modification tools, in
combination with out-of-kilter-based fleet routing optimization functions
create an effective decision support system for dynamic scheduling. Efficient
dynamic scheduling is achieved because of the straightforward interaction
between keyboard and mouse inputs and graphically displayed information.
The fleet routing optimization procedures provide answers quickly, but
their quality and applicability has to be assessed:
-Flight segments not included in the solution may be preferrable to flight
segments of equal value that are included.
-Schedule plan or fleet size changes may lead to a complete reconfiguration of the aircraft flights. It would be necessary to reconfigure
maintenance and crew schedules accordingly.
-The proposed overnight stations in a daily schedule plan may not be
desirable, but constraints can be imposed to produce the desired
overnighting.
The need for assessing the quality of fleet routing optimization solutions
is unavoidable and defines and restricts the role of a decision support system. This does not represent a real drawback, however, for many scheduling
scenarios can be quickly staged and interpreted in succession, so that a realistic solution can be found which is fairly close to the "blind" optimum.
Since many alternatives can be evaluated, such a solution will certainly be
better than one that is obtained manually within the same time-frame.
The functions of the DSS presented in this work can be expanded and
refined. For example:
-The manual rearrangement of flight segments on the schedule chart
according to aircraft number can be automated.
-The input of maintenance segment characteristics can be separated
from the input of the flight segment characteristics; flow bounds on
the maintenance arcs can then be set automatically. The fixation of
flows in segments that finish at times before a given time can also
be automated.
-Software that correctly labels the flights of aircraft for shuttle plans
can be developed.
The software of this DSS, written in PASCAL, should be easily transferrable to computer systems with more advanced graphics features and
larger database capacities. In addition, this DSS should also be useful to
organizations other than those in the air transportation field, because the
segment rectangles and their characteristics can be easily redefined.
Appendix A
Figures
C
.
0
X..
1J
kilterline
Figure 1 - The kilter diagram of arc ij
C..
1")
.
1I
9
9
9
.
I ,
,
,
I
I
i I I
. . .. .reverse
I
I
.* .
e
e
.
e
S ..
. .
.
i
I
I
S
I
I
I
I
I
I
I
I
I
I
.1
. forward
.
..
0
I.-
I
I
Figure 2 - Acceptable kilter states for forward and reverse arcs
to be included in the flow-augmenting path'
42
C.
forward
Ox
If
I
I
I
,reverse
I
II
I
I
,
f
I
I
Figure 3 - Kilter states of arcs linking labelled and unlabelled nodes
of which the node multipiers will be changed
station
cycle
arc ,
service
arc
station event
Figure 4 - Daily schedule plan schedule chart
source
cycle
service
station event
-
sink arc
sink
Figure 5 - Multiple-day schedule plan schedule chart
Minimum Fleet Size
Fleet Routing
ground arts:
cycle arcs:
flight arcs:
I = 0,u = oo, = 0
I = 0,u = co,c = 1
I = 1,u = Ic = value
I = 0,u = oo,c = 0
I = ,(0), u = u,(oo), c = doc
I I (0), u = uy(1), = value
ground arcs:
cycle arcs:
flight arcs:
source,sink arcs:
I
I
I
I = 0, u =
Optimization
Problem
Schedule Plan
Daily
Multiple-day
O,u =
= 0,u =
= 1,u =
I = O,u =
=
oo,C = 0
oo,c = 1
1,C = value
oo,
= 0
0, c = 0
= doe
u,,c
t = 0,u =
0, u = 1,c = value
I
I = 0,u = oo,c = 0
I =lower bound
u =upper bound
C =cost
doc =daily ownership cost
Figure 6 - Network arc parameters by schedule plan
and optimization option
Optimization
Problem
Fleet Routing
Minimum Fleet Size
Schedule Plan
min
E
min
Xii
-[
= 0
zxi
xi -
E
Daily
<_li<zi;<ui;
0
ci.xi + doc(
>j
ijE flight
x;
ijEcycle are,
ViEN
arc,
I
ijEcycle arc,
xi;FS
ijEccdl are
VijEA
1: zik ik
Vi EN
xzu = 0
hi
VijEA
min
Multiple-day
min
it.
E x,
ik
-
= 0
zti
ki
Xi -
L
E
ViEN
-[
(:
cizij + doc t.]
ijE flight are#
zts<FS
(
zik -
x 1, = 0
xzi = 0
ViEN
#i
LXit
-
its =
0
Xj- xt*
0
it
Z
zit -
Xs = 0
it
VijcEA
0<ti;<xi;<ui;
a =source
Figure 7 - Mathematical formulations by schedule plan
and optimization option
t =sink
fs
fs2
FS
fs4
fs3
- ~ ~
-
docl
doc4
doc2
doc3
doc5
Figure 8 - Iteration scenario when specified fleet size
is less than maximum fleet size
11
TIME
AX IS
12
DELETE
13
14
ADD
15
SLIDE
16
17
MODIFY
Figure 9 - Segment number input
18
SCHEDULE
2
1
UL
101
SEGMENT NO.:
19
20
STATION
21
END
AAA
ORIGIN:
12
13
14
16
17
Figure 10 - Origin station code name input
19
20
21
DESTINATION:
11
BBB
12
13
14
16
17
Figure 11 - Destination station code name input
19
20
VALUE:
100
13
14
15
16
17
Figure 12 - Flight segment value input
19
20
21
FLYING TIME:
300
13
14
15
16
17
Figure 13 - Flight segment duration input
18
19
20
SELECT DEPARTURE TIME ....
11
12
13
14
MOUSE TO CONTINUE
15
16
18
Figure 14 - The flight segment rectangle is moved to
the appropriate position
19
20
21
11
12
13
14
15
16
17
18
0100
AAA
"o,
0
BOB
0
Figure 15 - Flight segment 101 has been scheduled
19
20
21
SELECT FLIGHT DATA ELEMENT TO BE CHANGED
FLYING TIME
=
0300
Figure 16 - Flight segment characteristics display
I
NEW VALUE:
SEGMENT
&
200
NUMBER
ORIGIN
DES'TINATION
Figure 17 - Flight segment 101's value is increased
OPTIMIZATION OPTIONS
U,L
11
16
oloo
AAAE
012 0
10
851
06
TIME
DELETE
ADD
1
18
19
sf#
488
IMMMM913i
2
20
21
* AAA
to
0204
Ii
BOB
rCCL
Ite
SLIDE
MODIFY
U
SCHEDULE
AXIS
Figure 18 - A schedule plan consisting of four flight segments
STATION
END
SELECT SEGMENT TO BE MOVED....MOUSE DURING FLYING TIME
11
12
13
14
15
16
17
18
0100
AAA
101
o6
TIME
DELETE
40
(
21
AAA
10
.-
888
iI
Zi
ADD
20
020
0120
to
88
0 30
I 2
8
5
19
SLIDE
MODIFY
23
SCHEDULE
AXIS
Figure 19 - Flight segment 104 is about to be selected
for rescheduling
STATION
END
SELECT NEW DEPARTURE TIME FOR FLIGHT SEGMENT
12
13
AAA
14
-
15
0104
101
101
-
17
88
012
06
16
-2c
C
Figure 20 - Flight segment 104 is being rescheduled
18
20
19
AAA
21
15
13
12
AAA
AA
1
m-
- I
01601
16
18
pas
in,
.lot!
oil
888
06
21
Figure 21 - Flight segment 104 is about to be deleted
19
20
21
13
AAA
14
15
18
88
Cdc
Figure 22 - Flight segment 104 has been deleted
19
AAA
03
0r
o5
o6
07
0S
0
10
010oom
M01
AAAM
IL,
15
13
11
it
-
13
16
14
15
16
1?
18
17
19
10
18
Ieop0
Ida
1,8
21
22
23
19
20
Z1#
o
01
21
AAA
Oil
88E
1Ce
06
TIM'E
AXIS_
DELETE
ADD
SLIDE
MODIFY
SCHEDULE
Figure 23 - A new time interval is selected
STATION
END
os
16
17
18
19
20
21
22
23
24
01
02
0230
888
Iet
IAan
'to
DELETE
to-
ADD
SLIDE
MODIFY
SCHEDULE
Figure 24 - The visible portion of the schedule chart
has been changed
STATION
END
SCHEDULE DATABASE
FLT#
ORIG
DEST
DEPT
ARRV
101
AAA
BBB
1145
1450
102
BBB
AAA
1640
1920
103
BBB
COJ
1306
1421
Cn'
Figure 25 - The schedule plan of Figure 23 is tabulated
STATIM ACI'IVITY
11
12
13
14
15
16
17
18
19
20
21
20
21
102
888
AAA
0lI
11
12
13
14
15
l01
A
16
17
18
2
BBB
I
C(CI
103
AAA
102
Figure 26 - Station display for stations AAA and BBB
19
14
15
20
102
IO
AAA
1 [1
101
12
13
16
17
18
BBB
Figure 27 - Flight segment 103 will be deleted
19
20
STATIM ACrIVITY
13
16
14
17
18
20
I0CZ
Bs
I
AAA
OB
11
|
14
18
15
101
AAA
IO
BBB
I II
~I
1 110
I
R
Figure 28 - Flight segment 103 has been deleted
1
SEGlMNT NO.:
107
13
14
16
17
18
20
I.
1*2
I
I
U
I
I
1
AAA
Be.
I
I.
I
15
i
16
1o
II
BBB
AAA
Figure 29 - Flight segment 107 is being added to
the schedule plan
21
r
13
15
16
17
20
21
20
21
102
I88
AAA
101
13
1
14
15
I
BBB
||
A~1011
Figure 30 - The value of flight segment 107 is 350
FLYING TIME:
11
230
12
13
14
15
17
20
I
21
1
AAA
101i
.1
12
13
14
15
17
16
18
19
20
101
A41
BBB
I
1
I
AAA
1021
Figure 31 - The duration of the flight segment is 2.5 hrs
I
21
14
15
21
16
I02
I
AAA
OI
12
13
16
17
18
BBB
I oz
Figure 32 - Flight segment 107's departure time is selected
19
20
STATIM ACrIVITY
13
16
19
20
21
I
I
lot
888
I.
I
T
1 -~
I
AAA
r
.
4
BOB
;s
W8
101
.1
12
1
13
14
15
16
18
17
lot
I o?
AAAI
BBB
lAAA
I02
Figure 33 - Flight segment 107 has been
scheduled
I
W
_________
STATICN ACTIVITY
11
13
14
16
17
18
AAA
BBB
Figure 34 - The "time fields' for the selection of
arrival or departure times
20
21
16
15
18
lot
sob
I
AAA
48
Bee
47
5B
1*7
11
12
13
14
to)
I
BBB
18
15
I 7
Ann
I~
AA
12 1
Figure 35 - Flight segment 101 is selected for rescheduling
1
20
12
lot
980
AAA
17
12
13
16
17
18
BBB
Figure 36 - The arrow is moved to the new departure time
19
STATIm1 ACIIVITY
11
13
15
16
17
20
lo_
I
I ______________
I
I
I
IW1w~
I______________ £
I
AAA
40
OB
Fos
I0I
13
r
107
15
101
107
AAA
I
I
BBB
Ilotl
I
-
Figure 37 - Flight segment 101 has been rescheduled
I
21
13
16
14
2
20
131 1617 18
l~ I
.
AAA
47
Bag
888
foi
107
15
16
17
BBB
Figure 38 - The time axis of either AAA or BBB
can be translated
20
21
05
0.
12
Or
*6
13
10
o7
o
0
14
12
I
15
,
K'
014
16
17
18
19
I
20
. 102oZ
I
AA
4
BOB
I,
BB8
197
101
12
13
16
17
lot
20
10o
AAAi
&
£
1~
I
11
I
A_______
BBB
II2
Figure 39 - BBB's visible time axis interval will be
from 0800 to 1800
I
21
STATION ACTIVITY
13
11
14
15
20
19
16
loz
1
F
I
-
56
AAA
40
S7
BB
I @1
08
09
107
14
11
15
107
lei
AAA1
FIA~
I~
Ip
BBB
____
___
I
I
.
___
___
1
*I
Figure 40 - BBB's time axis has been translated
-
AA
OPTIMIZATION OPTIONS
11
12
UL
14
13
15
16
18
17
q0
ADD
SLIDE
21
AAA
20
30
30
DELETE
20
19
r1
or08
it17
TIME
AXIS
2
021;0
0100
A inA
1
MODIFY
SCHEDULE
Figure 41 - Schedule plan resulting from previous
flight segment manipulations
STATION
END
DDD
20
16
15
12
CCL
4
amo
mm
00
CC'
-
AAAA~,o,0900
AAA
-I
TIME
AXIS
DELETE
00
01
888
ADD
@0
-
CCC
DO
SLIDE
e9
00
000
-0
ao
20
MODIFY
4o
000
AAA
Pog
EE,
"mI~
(CL
21
0 DOD
o
00
00
00
2
1
UL
OPTIMIZATION OPTIONS
tIs
SCHEDULE
STATION
Figure 42 - Daily schedule plan for fleet routing demonstrations
END
fleet size:
004
overnight stations:
DDD (01) , BBB(01)
, AAA (02)
nouse to continue
Figure 43 - Answer to the first optimization option
NO.
OF A/C:
11
0100
DOD
I
000S
00
o
00
c cc
M
010
t
00
TIME
AXIS
00
DELETE
.
e0
ADD
o
00 cc(
006
M ccc
30
SLIDE
000
888
o
00
o
A
00030,0
90
S866
AAS
00
00
20
ccc
00
IDDP
I',
00
o
BOI
AA o
21
Of10
10cc
00
20
19
18
17
16
15
14
13
12
2
1
UL
1
0
EEEJ
to
MODIFY
L,
SCHEDULE
Figure 44 - Fleet size input for the fleet routing option
g0
444
0
io
STATION
END
DAILY A/C COST:
11
-200
12
UL
13
15
14
010
li
Ppp
00
16
18
17
I
@0
00
Ccc
toll
00
to
0
00
TIME
00
09o00
17
DELETE
0~
ovi
o
8
CccA
00000
00
21
pop
.1
00
?
20
Vito
to
AnA
2
D0
00
0100
oo
19
CLc
00
896
1
o
ADD
EE
3o
SLIDE
.o
MODIFY
o
y
SCHEDULE
AXIS
Figure 45 - Generic daily ownership cost (doc) input
for the fleet routing option
o
STATION
END
fleet size:
001
fleet net take:
max doc:
0160
-360
max fleet size:
overnight stations:
001
AAA(01)
grey flights are flown
nouse to continue
Figure 46 - Answer to the second optimization option
with fs=1 and doc=-200
OPTIMIZATION OPTIONS
UL
0100
to
DD
BOB
1c08
TIME
AXIS
DELETE
020
IsA9
000
0
09
00
0eg
CCC
00
00
00
00
0 00
9e5
7.0
c C
0
ADD
0
90
00090
88B
-7
Sao88
00
0o
AAA
DOD
00oc
CCC
00
ARA
2
010
C<.
Do
00
1
00
20
SLIDE
):C
20
MODIFY
Figure 47 - Flight of aircraft 001
EE F
40
110
%
SCHEDULE
AAA
0al
STATION
END
fleet size:
004
fleet net take:
-0940
max doc:
-0265
DDD (01) , BBB(01) , AAA (02)
overnight stations:
muse to continue
grey flights are flown
Figure 48 - Answer to the second optimization option
with fs=4 and doc=-500
OPTIMIZATION OPTIONS
11
12
UL
13
14
V11O
let
000
(
AAA
0e
00
or1
I
00o
090
TIME
AXIS
to '
DELETE
C
ADD
00 t
oe
00
o" 3
0
00
00
or
I
20
21
o1
30
SLIDE
NW >
to
e
AAA
0090
EE
I0
2
o 140
MODIFY
ODD
0C
.
0)90
CCC
Ia
1o
r00
19
00
ct
0090
966
18
L
m2
tSoo
00
17
2
010
0
00
16
CCL
00
Be
AA
15
1
SCHEDULE
Figure 49 - Flights for aircraft in a fleet of four
O
AAA
o
STATION
END
OPTIMIZATION OPTIONS
12
UL
13
14
*oo
101
AAAR
18
15
6220
598B
00
00
0
0
00
CCC
09
00
ADD
30
FFF
EEE
DELETE
21
oc
1 og
TIME
AXIS
20
AAA
DOD
00
19
pc
00
of
(CC-
.1
0
SLIDE
EEE
00
61
1
MODIFY
00
SCHEDULE
Figure 50 - Flight of one aircraft. AAA-BBB-AAA and
CCC-DDD-CCC alo qualify
STATION
END
inbalanced stations:
EEE-,
CCC
inouse to continue
Figure 51 - Attempt to delete flight segment 109 leads
to an imbalanced flow in the network
OPTIMIZATION OPTIONS
UL
14
13
1
2
1
o i 3a
CC(
I,___I
VDD
00
0
01ID
0'+
C CC
00
00
00
O0
-L
000
o
or 00
CCC
6 300
00
-
00
AAA
oo
TIME
AXIS
Er
DELETE
0to
-0
00
001
SLIDE
to
MODIFY
00
04
04
Scc
&as
ADD
00
oc
(to
I- E
, LI
SCHEDULE
Figure 52 - After flight segment 109 has been "removed",
aircraft 001 is rerouted
$so
AAA
00
STATION
END
STAT
STAT/ELT:
11
12
UL
13
14
15
DD
16
17
.2
1
18
19
C
20
21
OD
.6
1
vi<<.
610 08
ol 00
0o
000
oe
00
0100O02a
CCC
A q
*0
AAA
10
TIME
AXIS
oo0
00
I~
00
g86
20
DELETE
I
ADD
so
90.0
4o
CCC
I
00
3
SLIDE
91
E
go4
MODIFY
i1
4Aq
00
SCHEDULE
Figure 53 - The flow bounds on a cycle arc are going
to be changed
AAA
STATION
END
STAAME:
2
UL
BBB
19
15
14
21
20
01 '0
-~ =-I
DOD
60
-'4
00
00
o
vi 00
0
ccc
3Bo
00
00
02 ao
to'M
CCC.
AAAA
00
AAA
AIS
xe
t 7? E
DELETE
00
B6
ADD
3oC
o
EM0
SLIDE
ot
90
EmEl
to
MODIFY
1050
4.
2%
SCHEDULE
Figure 54 - BBB's cycle arc is selected
AAA
00
STATION
END
11
12
14
13
16
15
01~o
19
18
17
P1
0
0ok
Te
ccc.
,
00
TIME
AXIS
Do
00
090
100VI
BO 6
7
AAA
20
DELETE
CCC
o0
ADD
AA
-
A0oCCC
no
21
20
D
00O
o
2
1
UL
0
ILMER:
30
SLIDE
Zo
MODIFY
0
001
Ef 6
9
4o
1
25
SCHEDULE
Figure 55 - BBB's cycle arc lower bound remains 0
AA
00
STATION
END
16
17
20
if T0
Duo
21
0i o0
Ccc
00
2
1
UL
2
UPPER:
ODD
00
00
00
oto
00 n
00
00
Cte
Do1s0
021o
AAA
00
A
Milo
A 7
O0~O
ova38
DO
TIME
AXIS
DELETE
ADD
= CCC
ofma
1+ 1oi
ew
90
00
.
2
3o
SLIDE
106
A4AA
00
MODIFY
SCHEDULE
Figure 56 - BBB's cycle arc upper bound becomes 2
STATION
END
STAT/FLT:
11
LqT
12
13
14
Oo
15
16
17
It
.o
18
19
i
oCC
00
00
21
oJO
6100
00
00
o 0
0
CCC
~
0000s
oo0
A
lo
DELETE
20
oc0
$o<0
A76
2
000
#tooo
TIME
AXIS
1
UL
ADD
so
SLIDE
00 :0
ze
MODIFY
6i
v2
o
2
SCHEDULE
Figure 57 - The flow bounds on a flight arc are going
to be changed
0
IE
tAn
STATION
END
SELEC'W
SEGMEN1 4T TO BE MODIFIED . . . MOUSE DURING FLYING TIME
11
12
15
16
18
17
0I
b ob
20
19
21
to
Cce00
0100
k.
-
to
090
of
TIME
AXIS
DELETE
Io.
o 00oi
AAA
ccc
I
-M
000
>O
AA
AAA
ccc
00
ADD
Mccc
30
SLIDE
ON
to0
10
jIU LEE
-0
MODIFY
Figure 58 - Flight segment 103 is selected
SCHEDULE
0090
*0
STATION
MA
END
LOWER:
11
0
12
UL1
13
14
at
ODD
g
15
16
17
18
cccc
666~~/
it
00
00
et ooo
o
it
CCC
0*
TIME
AXIS
00
to
00
ADD
00
90
00100
Z0
DELETE
Ic
o
01
06
21
0
<C88
0
e 10
20
DDD
00
0o
AAA7
19
ol 0o
_
,
A
2
CCC
N
SLIDE
0000
EEf
:o
o2
4.
MODIFY
SCHEDULE
Figure 59 - Flight segment 103's flight arc lower bound
remains at 0
AAA
AA
00
STATION
END
11
e
o 03
_
zy
cc
DD
21
20
19
18
17
16
15
14
13
12
2
1
UL
2
UPPER:
ooCDC
063
<C
13
00
0
TIME
AX IS
to
DELETE
1
%
0
CCc
Do
30
00
ADD
00
Do
00
666
1e?
00
00
DoCCC
c
l
A Ao
AAA
6
0
SLIDE
MODIFY
EEF
4a
A
11
25
SCHEDULE
Figure 60 - Flight segment 103's flight arc upper bound
becomes 2
4A0
0010
16
20
AA
00
STATION
END
fleet size:
005
fleet net take:
1760
max doc:
0252
max fleet size:
005
bvernight stations:
DDD(01),
BBB(02),
AAA(02)
grey flights are flown
Figure 61 - Two aircraft now overnight at BBB
nouse to continue
17
13
AAA
19
688
BOB
Figure 62 - A "shuttle" schedule plan between AAA and BBB
21
fleet size:
002
fleet net take:
0060
max doc:
max fleet size:
overnight stations:
-0130
002
BBB(01), AAA(01)
grey flights are flown
Figure 63 - Two aircraft are needed to serve the shuttle
nouse to continue
15
16
17
18
1 2.
AAA
.ei
00
588
* I
886
AAA
.el00
8
00
_
o@
e*
e.
Figure 64 - Only one aircraft seems to be required
19
20
21
OPTIMIZATION OPTIONS
U,L
1
2
.. I
05
09
06
13
oil
Pof
0 IIf~5*
5,
So
o
0,0
L4L
C PHi
If
35-
C
0C
rE4
4g
OilS
bus
r
fePH
00
to
o0 4-0
'5',
cp#
;o
oil*
0i
C ?qt
to
C
DELETE
Ille
1r Oil
ADD
rn
AW
SLIDE
r
ole
FIRN
00
TIME
AXIS S
I <<<
to'f
ol
fOSL
to
0lI5*
U
iR
Isr
ePs'
FAo
20
Is
00
4 PH
FK
Im
30
o01S4
oIQ
00
14
r~i
AA
I.Ak
off*
413JU
5-5-
MODIFY
SCHEDULE
Figure 65 - First part of a SAS schedule plan
STATION
EN
OPTIMIZATION OPTIONS
U,L
14
13
18
15
seo*
Ito
IMa
PO
~ ei f o OSL.
£~
oE0
35"
00
38
16
FA
ARM
I.
r.
jm
.
aP"
=06
11512e
6
LS
DELETE
ADD
2o
Ott
it
Cue
TIME
AXISIIII
U
0
600
to
'95s
1010
Dos
i
15
0okI@
CP1 A"
c PH
4o
e'
e its3
0o)-0
e
A
-r
00
9-f
00
*.
I
004s,
C PH M4
CIh
4
0g
to
2
20
19
.e
o Itt.
003
-1
1
$5
4
35
SLIDE
MODIFY
SCHEDULE
Figure 66 - Second part of a SAS schedule plan
STATION
END
fleet size:
006
overnight stations:
OSL(01),
FPA(01),
CPII(02),
BGO(01),
AlM (01)
rnouse to continue
Figure 67 - Six aircraft are needed to serve all flight segments
fleet size:
006
fleet net take:
4335
max doc:
max fleet size:
overnight stations:
-622
006
OSL (0 1) , FRA (0 1) , CPH (0 2) , BGO (0 1) , Arlq (0 1)
grey flights are flown
mouse to continue
Figure 68 - Answer to the second optimization option
with fs=6 and doc=-100
OPTIMIZATION OPTIONS
1
U,L
I
I
05
09
06
oil
C PH
00
o l't-S*
001
00?..
6w
f
-
o.3
*.
0
o$Z.f
C PH
ee5
I*
r411
foS
00
2
oo
35*too4
FR
031
*A
C PH
CPl1
UOSL
Zo
U
£0
0
elf
10lo
060
olr
"r
IS.
'4s
,Q- ij.5T
5',
~
*o6
to00!
.115
oo5
off
0I1
*I
00
Otto
DIP
00
DELETE
41
C Pit
ot*
C
TIME
AX IS
IIAN
L
10
* if*
NEA-13m
to00Oc
ADD
SLIDE
MODIFY
SCHEDULE
Figure 69 - Aircraft flights with fs=6 (first part)
STATION
END
4(<
Ses95
,CH
01
C
419
1=
35 o 6
00
Fi*
o
its0
oAAt
01'a i o ao .
***'te
ef
TIME
AXIS
0ol
CPH
I
DELETE
01;
qI
CLelo
of
o
6'0
ree
W
tJ3
e
vo6
0
eoo'6
0
-e
oito
0
21
0
5 o0
elo
2
20
19
V f' US
oi5
0
ios o%
18
17
16
15
14
13
12
11
1
UL
OPTIMIZATION OPTIONS
f
o
20
35
**I
,
I
oil ?
001ot f
D SS
6
eicvita
LS
0ol
ADD
55
SLIDE
4f
MODIFY
SCHEDULE
Figure 70 - Aircraft flights with fs=6 (second part)
STATION
END
OPT IMIZATION OPTIONS
04
05
1
U,L
09
06
2
10
of#*
C Plf
,Vito
-Io u
oP il
ite
~4P.N
Off
5
o
V
0114
LGW
oe
so
'iii
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ze
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156
-
oe3
3.
I PR
II
r
55
A4~
00
oils.
04
1.<<.
*j
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KI~1U 6
2.
r
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it.
o
OSL -
-oleo
00*ofi e
cPn
2.do
C efi
I Dos
V.3
if eel
10
w
oleo
W5
oftL5
141
CPM
TIME
AX IS
DELETE
ADD
SLIDE
006
to
MODIFY
po
o(
*e
SCHEDULE
Figure 71 - The aircraft flights have been manually
rearranged (first part)
'U
o.5
I
0.1
0100o
2.o
@125
0
6
Os'-
SS' 0
STATION
END
OPTIMIZATION OPTIONS
11
12
UL
13
14
Otto
15
oft
if~~~I
25ool
)5
20
21
g
'AAt
30
**00o
8_
CPH
**
MI
45
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*0 4An
66
T
3f 0
o003
Ds
4a'to
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oi
CPH
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of
Do%"
..
$
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eiryosy
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6
oo4
!
860
'S
.
lW
r
07e 1e*44
ADD
(PH
otis
76 P5
to
Ire
toee)
to
gg
lv-05
of 0
DELETE
PH
FAA
46e
o*f
u1
-5 0e2 55
'if p
TIME
AXIS
19
ol1
(PH
-PM
18
AQ4ct
$
i
3M o.1
t
to oot
CPH
17
2
lie
oIt
NL
16
1
SLIDE
eFr
o
0e" &I
MODIFY
SCHEDULE
Figure 72 - The aircraft flights have been manually
rearranged (second part)
STATION
END
0-4
grey flights are flown
TIME
AXIS
DELETE
ADD
SLIDE
MODIFY
SCHEDUL
Figure 73 - Answer to the second optimization option
with fs=5 and doc=-100
OPTIMIZATION OPTIONS
I
05
2
- -
09
06
015
C Pf1
00
13
o off f
L 6w
fo
Vol
*0
4.
009-*
cP1
00
14
0140i
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S's.
CPm
J3M
05
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DCPH~
7
0)jL
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1
U,L
K((
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0~L
;r
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2045
oIZf
FAA
~460
00
to
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Zof
90001
S7
01
eggs
b'us
F0
3'
4
is
lme
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005
set
oil*
00
OIRN
0?
oo3
C Pig
F (gotI
55
to
erPH
I
.110
.03
TIME
Ax is
DELETE
ADD
SLIDE
5*5~
MODIFY
Ao.
Otto
4IA13
to
eel
SCHEDULE
Figure 74 - Aircraft flights with fs=5 (first part)
STATION
END
I
OPTIMIZATION OPTIONS
11
UL
12
16
003
1
19
2
21
o yf
I
"0
'f-c
'0
0
oef
~%
00 ce)q.5
35 ve3
U
to
LS4W
it
i
00
D(US
( PH
SFO
A
T005
60
£90 12.
00
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I FAA
10
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415
ov"o
049"
ovi
,
'p0f
**e.
wfoo
|412
0f
c'o
TIME
AX IS
oct
CP
A"
DELETE
20
4 fl
'..7f
355
oie
1
Vol
860
.
I
opa
A9.0
3$.i
of t
Or
I
0(10
CP
L
Ite
LI '
Oog
ADD
SLIDE
'
MODIFY
SCHEDULE
Figure 75 - Aircraft flights with fs=5 (second part)
STATION
END
05
04
06
OSL
CPHE
'Pb,'
8GO
Figure 76 - Aircraft flight with fs=1 (first part)
rKA
OPTIMIZATION OPTIONS
11
12
UL
13
14
j3AX
c~
pit
157
PH
5-06
00
~
0
0(1.
,5
o
oil
TIME
AXIS
I
DELETE
*If
Otisc
45c
I
CPH
.5 **'
ADD
20
21
C PM
ot
Vio
5e
19
A
o ei
offfdi
$?
DHus
A"1I
o
'F
E
-60
M5M|'
00
6T
Ir
2.
18
2
It4
o
*0
17
s)U
Io
o.'t-&
0 *04
*
16
L 4; kf
"h3
11
= j 5os35
35
15
1
35
i5
SLIDE
l oo
t1Co
2
16O
el
'
o9
t
e
n
Ij
q W
4f
MODIFY
SCHEDULE
Figure 77 - Aircraft flight with fs=1 (second part)
STATION
END
t
(seC)
9
14rotations
1
2
3
4
5
6
Figure 78 - Change in computation time with number of rotations
118
t
(sec)
27 -
24
21
18
15
12
9
*0
-0
6
3
1
2
3
4
5
6
7
8
9
1Q
Figure 79 - Change in computation time with fleet size
. 119
TIME
AX IS
DELETE
ADD
SLIDE
MODIFY
SCHEDULE
--..--
Figure 80 - Input of day of occurrence of a flight segment
of a multiple-day schedule plan
STATION
END
11
12
13
14
15
ety-
44A
Jo
16
17
18
67-o
let
06
o3
Figure 81 - Flight segment 101 has been scheduled for day 14
19
20
21
12
13
14
16
18
17
19
66
Figure 82 - Flight segment 101's day of occurrence will be changed
20
21
39
11
12
4AY
13
14
15
16
17
18
oletto
AnAse
T'
Figure 83 - Flight segment 101 is rescheduled for day 39
19
20
21
Bob
Figure 84 - Flight segment 101 has been scheduled for day 39
STATIO
AC1IVITY
13
16
17
16
17
19
AAA
II
11
12
1581
'eI
Bob
15
18
013
1 01
AAA
I
IBB
Figure 85 - Station display for flight segment 101
2
11
'12
13
17
14
AAA
all.
01
00
0C 1
o210
00
DO
00
00
DDD
0w I
IO9
Olo
00
0
.
FFF
ODD
1
00
Figure 86 - One aircraft flies from AAA to FFF in 17 days
19
TIME
AXIS
DELETE
ADD
SLIDE
MODIFY
SCHEDI
Figure 87 - Answer to the first optimization option
fleet size:
001
fleet net take:
0300
max doc:
-400
001
max fleet size:
grey flights are flown
Figure 88 - Answer to the second optimization option
nouse to continue
Figure 89 - Flight ZZZ-BBB-CCC has been selected,
but AAA-BBB-CCC is also possible
OPTIMIZATION OPTIONS
DAY
2
15
13
DD
1
127
o 00
fim
005
030
55S
333
)
ISKJO
o00
INN
of 0
mI4m
373
010
103to
IsI
KMK 0to00
DKKKN
FFF
V2
KKK
'1
o3
12.0
35-
00
KKK
FFF
=L1
QQq
00
TIME
AXIS
DELETE
ADD
SLIDE
MODIFY
SCHEDULE
Figure 90 - Multiple-day schedule plan between days 1 and 12
STATION
END
Figure 91 - Answer to the second optimization option
with fs=3 and doc=-300
OPTIMIZATION OPTIONS
DAY
15
14
04.o
ot
000
16
18
2
20
19
21
3o
Vol
eo
ve
ce L
3jj
00
1
55
0
0to|
5
of
ofo
6
001
NNNv
FFF
00
KKKWU
33 " >3
'to
D
003
lo
00
0o1
6
eUc
qqcK
-
FF9:
3%
0I
-og
TIME
AXIS
DELETE
ADD
SLIDE
MODIFY
SCHEDULE
Figure 92 - The flights selected for the three aircraft
00
STATION
END
OPTIMIZATION OPTIONS
11
12
ol
00
DAY
13
14
15
61'0
Vel
*wL
16
17
19
20
2
21
000
00
0
o0
18
1
1
FFF
003
NNN
KX
K
7333
I
00
0o2.o
elo
0lso
05 Lo9
00
35 *2.
'e
-
AXISK
TIME
AXIS
DELETE
ADD
SLIDE
MODIFY
0K
001000
0O9
lo
oo
ol
o6
olot
So
ool
SCHEDULE
Figure 93 - The flights selected for two aircraft
35
0.
STATION
END
station
cycle
arc ,
time
service
arc
station event-w
maintenance arc
Figure 94 - Schedule chart with maintenance arc
134
OPTIMIZATION OPTIONS
UL
13
AAA
14
C3+1
15
1
2
20
19
21
688
.4.1±
*I o
AAA
TIME
AXIS
DELETE
-O
00
04
ADD
1
58
O
1.
Do
SLIDE
MODIFY
SCHEDULE
Figure 95 - Schedule plan consisting of two flights
0
.
-O
q4A
64
STATION
END
OPTIMIZATION OPTIONS
11
12
UL
13
14
15
16
17
18
888
e
AAA
t
**0 .
AA
o
1
19
2
20
21
AAA
.
BBb
AAA
Sopes
TIME
AXIS
DELETE
ADD
SLIDE
MODIFY
SCHEDULE
STATION
Figure 96 - Aircraft are rerouted after a maintenance
stcpover has been introduced
END
OPTIMIZATION OPTIONS
11
12
UL
13
AAA
14
15
16
17
18
2
20
21
AAA
2.t04
010
o. 00
888AAA
AAA
SSB
DELETE
19
88I
it
@100
TIME
AXIS
1
ADD
Soo
SLIDE
ini_8
MODIFY
SCHEDULE
Figure 97 - Loss of service on a flight segment
when maintenance stopover is too long
STATION
END
decreasing
station
code names
and station
event times
start
finish
flight arc
station X
start
---
finish
ground arc
base
NODE LIST
sta-t
finish
cycle
arc
base
Figure 98 - The network data structure consists of a node
ARC LIST
138
linked list and an are linked list
Appendix B
Data Structure
There is only one network that corresponds to a single type of schedule
plan, but there are many data structures that can represent a network.
The linked list, being a dynamic data structure, is an obvious choice here
considering the demands of a dynamic scheduling environment; it not only
conserves memory space, but also allows for rapid access to its records. A
node linked list and an arc linked list that are composed of records of the
format given below describe the network of a schedule plan:
node: record with
-a pointer to the next record
-the station code name
.-the time of arrival or departure
-the day of arrival or departure
-the simplex multiplier
-the first node label: a pointer to the preceding node
in the flow-augmenting path
-the sign of the first node label
139
-the second node label: the maximum allowable
flow in the flow-augmenting path
-a pointer to the arc record of which this node
represents one station event
arc: record with
-a pointer to the next arc record
-a pointer to a node containing information on the start
of this arc's flight segment
-a pointer to a node containing information on the finish
of this arc's flight segment
-the day of occurrence of the flight segment
-the lower bouad on the arc flow
-the upper bound on the arc flow
-the flight arc value or the aircraft daily ownership cost
-the arc flow
-the number of the corresponding flight segment
-the arc label: the number of the flight to which this
flight segment belongs
The node records are ranked alphabetically and sequentially; one node
precedes another in the linked list if it has an alphabetically higher station
code name or if it has the same station code name, but a later event time.
Ground and cycle arc records, identical to flight segment arc records, can
be created and linked to the arc list only after the node list has been created because:
140
-The start and finish fields of the ground arc records have to point
to consecutive event pairs at each station in the node list.
-For the daily schedule plan, the cycle arcs have to point respectively to the last and the first event at each station.
-For the multiple-day schedule plan, the source and sink arcs have
to point to all last and first stations events.
The node linked list and arc linked list of the daily schedule plan and
the relationship between their records are presented in Figure 98.
141
Bibliography
[1] Etschmaier, M.M. and Mathaisel, D.F.X. Aircraft Scheduling:
the State of the Art, Proc. of XXIV AGIFORS Symposium,
Strasbourg, France, 1984.
[2] Keen, P.G. and Morton M.S. Decision Support Systems An Organizational Perspective, Addison-Wesley, Reading, Mass., 1978.
[3] Sprague,R.H. and Carlson, E.D. Building Effective Decision Support Systems,
Prentice-Hall, New Jersey, 1982.
[4]
Ford, L.R. and Fulkerson, D.R. Flows in Networks, Princeton U. Press,
New Jersey, 1962.
[5] Basaraa, M.S. and Jarvis, J.J. Linear Programming and Network Flows,
Wiley, New york, 1977.
[6] Lawler, E.L. Combinatorial Optimization: Networks and Matroids,
Winston, 1976.
[7] Magnanti,T.L. and Golden, B.L. Network Optimization, MIT,
Cambridge, Mass., 1980.
142
[8] Simpson, R.W. Scheduling and Routing Models for Airline Systems,
FTL-report R68-3, MIT, Cambridge, Mass., 1969.
143