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81
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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
"Selecting Aircraft Routes for
Long-haul Operations:
A Formulation and Solution Method"
Anataram Balakrishnan
William Chien
T.
Richard T. Wong
MIT
Sloan
School
Working
July
Paper
#3047-89-MS
1989
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
311989
"Selecting Aircraft Routes for
Long-haul Operations:
A Formulation and Solution Method"
Anataram Balakrishnan
T.
William Chien
Richard T. Wong
MIT
Sloan
School
Working
July
1989
Paper
#3047-89-MS
AU6 3
1
1989
.
Selecting Aircraft Routes for Long-haul Operations
A Formulation and Solution Method
Anantaran Balakrishnan
Sloan School of Management
Massachusetts Institute of Technology
Ca«bridge,
MA 02139
T
.
William Chien
Baruch College
The City University of New York
New York. NY 10010
Richard
T.
Wong
Krannert Graduate School of Management
Purdue University
West Lafayette, IN 47907
Supported in part by the
Research Initiative grant
t
Office of Naval Research
NOOO 1 4 -86 -K-0689
under
University
ABSTRACT
In
this
paper
we
consider
the
routing of
main base to one or more terminal bases.
routing
decision
critical
becomes
long-haul
aircraft
from a
For these long-haul markets, the
route
because
profitability
must
be
evaluated for the extremely large number of feasible routes covered by the
operation.
In
addition,
"pickup-and-del ivery"
the
route
selection
characteristic
of
the
task
is
complicated by the
problem.
Therefore,
the
development of an efficient procedure for selecting good candidate routes
will
facilitate
the
more
profitable
timetables.
iterative
flight
We
scheduling process
define
an
aircraft
and
may
lead
routing problem
to
that
captures the important profit-generating factors (such as intercity traffic
estimates,
revenues,
selection decision.
operating costs and aircraft capacities)
in
the route
We formulate this problem as a mixed integer program,
and develop a Lagrangian-based solution procedure that exploits the special
structure of the problem.
Computational results for several test problems
indicate that the procedure is able to select a small number of profitable
candidate routes, and provide good bounds that confirm the near-optlmality
of the generated solutions.
1
INTRODUCTION AND LITERATURE REVIEW
1.
The aircraft routing decision is one of the most important components
in
overall
the
operational
process
scheduling process
flight
flights
timetable of
consists
phases:
two
of
schedule evaluation phase.
aircraft
routes
determined
taking
frequency
the
maximize
to
into
and
account
operating
profitable
The
flight
scheduling
phase
construction
schedule construction phase,
of
service
generated
from
the
a
set of
a
first
operation,
while
revenue
and
and
are
route
each
on
estimates
traffic
the
every
for
aircraft characteristics and operating costs, and
origin-destination pair,
some
profit
the
airline.
schedule
a
the
In
an
for
a
developing
of
restrictions.
construction
The
phase
is
completed
by
scheduling departure times and assigning aircraft to match the routing and
The resulting timetable is then examined by operating
frequency decisions.
for feasibility and other cost and performance considerations in
personnel
Any desired improvements are then fed back
the schedule evaluation phase.
into the construction phase, and a revised set of routes and the associated
The flight scheduling process iterates between
frequencies are determined.
these
two
phases
until
a
satisfactory
timetable
final
obtained
is
(Etschmaier and Mathaisel (1984)).
Early
research
applications where
on
the
the
aircraft
routing
problem
number of alternative routes
is
has
focused
on
relatively small,
and mainly used linear programming as the solution method (Dantzig (1963),
Kushige (1963), and Miller (1967)).
continuous
contain
variables;
fractional
These models represent frequencies as
consequently,
frequencies,
which
the
may
solutions
result
in
are
very
likely
suboptlmal
to
integer
solutions even after applying some sophisticated rounding procedures.
More
2
researchers
recently,
formulations
model
to
de Lamotte
instance,
have
using
started
and
solve
and Mathaisel
routing
aircraft
the
(1983)
integer
mixed
programming
problem.
For
used MPSX-MIP to solve problems
of moderate size faced by a small short-haul airline.
However,
for
the literature is still limited in terms of solution methods
larger-scale
For these airlines,
number
increases
the
faced
by
number
covered
cities
Walker-Powell
feasible routes.
of
dramatically
operations
their
by
cities
from
Sydney
London.
to
connections are only possible
airline
considerations.
several
sets
of
the
additional
imposed
also
in
instance,
For
called
cities,
The
to
cities
are
which covers
Indexed
so
that
direction of increasing order.
The
constraints
simplify
"slip
based
the
points",
crew
are
operational
on
scheduling
selected,
aircraft is required to stop at exactly one city in each set;
two
intermediate
stops
are
and
task,
every
flights may
Between two slip point sets, at
also terminate at one of the slip points.
most
discusses
(1970)
the Port Linkage Problem for Qantas Airline's Kangaroo Route,
26
carriers.
long-haul
the routing decision becomes complex because the large
intermediate
of
problems
routing
aircraft
permitted
for
each
flight.
Given
the
intercity demand, unit revenue per passenger carried, fixed operating cost
for
each origin-destination pair,
and aircraft capacity restrictions,
the
Port Linkage Problem seeks a profit maximizing set of routes from Sydney to
London (and/or to some slip points).
Richardson (1976) proposed
problem,
and
developed
a
a
They reported computational
Etschnaier and Richardson (1973) and
mixed integer programming formulation for this
Benders'
results
slip point sets, and at most 9 stops.
decomposition
for
algorithm
to
solve
it.
problems with up to 17 cities,
2
.
3
The multi-stop aircraft routing problem is related to the pickup-and-
delivery
routing
problem
disembarkation of
because
passengers
we
consider
must
embarkation
both
The pickup-and-
each intermediate stop.
at
and
delivery routing problem also arises in the context of school bus routing
(Bodin,
Assad
Golden,
and
Ball
tractor-trailer
and
(1983))
mixed loads (Ball, Golden, Assad and Bodin (1983)).
routing with
However, this problem,
which is considered more difficult than the pickup-only (or delivery-only)
problem,
unexplored
largely
still
is
researchers
by
(Golden
and
Assad
(1986))
routing
Since
aircraft
the
is
issue
central
in
schedule
the
construction phase, developing more efficient and effective route selection
procedures should result in more profitable timetables.
In
this paper we
address the multiple (homogeneous) aircraft routing problem faced by longGiven
carriers.
haul
destination pair,
traffic
aircraft
estimates
and
operating costs,
revenues
for
each
origin-
aircraft capacities,
and
the
aircraft routing problem seeks a set of good candidate routes from a main
base to one or more terminal bases in order to naxinize total profit.
assume that,
for
long-haul
operations,
connections are only possible from
This
nodes.
practical
vast
restriction
on
the
lower
cities can be indexed so that
indexed nodes
direction
of
to
connections
higher
is
indexed
considered
and even desirable because long-haul markets usually cover very
geographic
relatively long.
areas
and
the
distances
traffic
between
interaediate
cities
are
We note here that the aircraft routing problem requires
Joint consideration of demand selection
of
the
We
demand
to
serve
between
(i.e.,
each
determining the proportion
origin-destination
pair)
and
routing of aircraft in order to achieve the profit maximization objective.
4
In
context,
this
demand selection decision arises because of
the
limited
aircraft capacity and differences in the profitability of different origin-
destination pairs.
The
rest
of
We first present a
paper is organized as follows:
this
formal definition of the long-haul aircraft routing problem, and formulate
it
as
solve
mixed
a
optimization-based procedures
tight
bounds
upper
In
Section
optimally,
problem
the
program
integer
to
to
evaluate
focus
we
good
select
quality
the
Lagrangian relaxation for this purpose.
relaxation
scheme
for
the
mixed
Rather
2.
on
devising
candidate
of
Section
integer
than attempting to
3
these
efficient
routes and obtain
results.
We
use
describes our Lagrangian
programming
We
formulation.
develop an efficient procedure that exploits the special problem structure
to solve the Lagrangian subproblems
a
(and obtain upper bounds),
and discuss
Lagrangian-based heuristic which constructs good candidate routes (lower
bounds).
Section
In
4
we
present
computational
results
for
several
randomly generated problems with up to 26 cities for the single-aircraft
case,
and
23
cities
for
the
4-alrcraft case.
Finally,
in
Section
5,
we
suggest possible extensions of the model.
2.
PROBLEM DEFINITION AND FORMULATION
The aircraft routing problem that we consider involves routing a fleet
of aircraft
of
the same
type
(i.e.,
aircraft having the same capacity and
operating costs) from a main base, through several intermediate cities, to
one or more
terminal
bases.
As
indicated earlier,
the cities
the bases) are indexed so that connections are possible from city
j
only if
i
<
j.
(including
1
to city
The maximum (forecasted) intercity traffic demand (number
5
of potential passengers)
is known,
h < k)
need
not
be
and is denoted as
satisfied.
However,
d^^j^.
earns
airline
The
a
the entire traffic demand
revenue
passenger it transports from origin h to destination
traveling on segment (i.j). i.e., directly from city
denoted
as
As
f^i-
we
demonstrate
of
rj^j^
every
for
We assume that the
k.
the operating cost of
total operating cost is separable by route segments;
and
(with
for each origin-destination (0-D) pair (h,k)
to city j,
is fixed,
our
can
i
later,
model
also
accommodate variable operating costs that depend on the traffic volume on
each
For
leg.
development.
simplicity,
we
variable
these
ignore
costs
The aircraft routing problem consists of
(1)
in
our model
determining the
number of passengers to transport between each 0-D pair (subject to maximum
demand
traffic
constraints),
assigning the selected traffic to each
(ii)
aircraft (subject to aircraft capacity restrictions), and (iii) routing the
aircraft.
The problem objective is to maximize total profit.
Every
aircraft
model.
airline
routes;
imposes
we
do
not
some
additional
include all
operating
constraints
revenues,
operating
cost,
and
aircraft
such as traffic
capacity,
that
influence the underlying routing decision of a long-haul carrier.
the
analysis
of
our
basic
aircraft
routing model
should
the
factors,
it
model
captures
can serve as
some
a
of
the
most
relevant
most
However,
provide
insights and information for the actual route selection decision.
since
its
such detailed restrictions in our
Instead, we focus only on those essential factors,
demand,
on
useful
First,
profit-determining
preliminary screening procedure to assess the
market profitability of the regions in which the airline wishes to operate.
Secondly, the routes selected by the model represent potentially profitable
routes that deserve further examination in the iterative flight scheduling
.
6
Finally,
process.
benchmark
profit
the
indicated
by
model
this
serve
can
as
a
cost-effect analysis of other operational and service-level
for
considerations
aircraft
The
G:(N,A)
problem
defined
is
consisting of n nodes in the set
belonging
arcs
routing
intermediate
We
a
Nodes
A.
and
cities
segments.
specifies
to
bases,
the
and
arcs
loss
represent
correspond
of
directed
a
network
and at most n(n-l)/2 directed
network
of
without
assume,
N,
over
various
potential
to
generality,
the
that
the
flight
airline
Otherwise, we connect the last terminal
unique terminal base.
base to the remaining bases using dummy arcs (that are directed towards the
last base) with zero cost.
1
The nodes are Indexed from
and n denoting the main base and terminal base,
1
with indices
to n,
respectively;
indices
2
to n-1 denote the intermediate cities.
The network contains a directed arc
from
and direct
city
i
possible.
directed
city
to
With
path
j
only if
this
from
node
i
network
1
to
<
j,
representation,
node
directed arc or flight segment from
We
n.
to
i
j
;
will
service
an
from
aircraft
use
(i,j)
to
i
to
is
j
route
is
denote
a
the
(h.k) denotes the 0-D pair with
origin city h and destination city k, and <h,k> represents the "commodity"
(which
will
be
explained
between
later)
0-D
pair
The
(h.k).
input
parameters for the problem are as follows:
n
number of cities (including the bases) covered by the operation,
V
number of aircraft (of the same type) at the main base,
b
aircraft capacity,
d^l^
maximum (forecasted) traffic demand for 0-D pair (h,k),
r^^ unit revenue for 0-D pair (h,k),
1
<
h < k < n.
1
<
h < k < n,
f|j
(non-negative)
fixed routing cost per aircraft on flight segment
(i.j)
€A.
To formulate the aircraft routing problem as a mixed integer program
we define three sets of decision variables:
Suj^
number of passengers transported from origin city h to destination city
k.
y^^ number of aircraft traveling on flight segment (i,j),
hk
x^^ number of passengers originating at city h and destined for city k who
l<h<i <j<k<n.
travel on flight segment (i,j),
s-variables
The
determines
demand
to
0-D
these
for
aircraft
base
which
represent
pairs
0-D
will
pairs
routing variables)
the
passengers
terminal
will
a
set
city
h
and
of
for
decision
selection
how much
and
satisfied.
be
Finally,
base.
originating at
served,
be
form
demand
the
traffic
the
of
y-variables
The
aircraft routes
each 0-D pair
from
(h,k),
destined for city k as
a
that
(or,
the main
we
treat
separate
commodity (denoted as commodity <h,k>); the x^^ variable denotes the amount
of commodity <h,k> transported on segment
Since traffic demand is
(i,j).
only a forecasted value representing the expected number of passengers for
each 0-D pair,
these
the
decision
flow-based
mixed
routing problem:
s
and x variables are assumed to be continuous.
variables,
integer
we
formulate
programming
the
following
formulation
[ARP]
Using
multi-commodity
for
the
aircraft
[ARP]
maximize
subject
If
-
Ir,,s,,
(1]
y
lo:
Shk
^
<-
for all [hM],
[2]
for all <hM>.
P)
for all
(4]
i=h
+Sf^,
if
0.
if
h<i<k
-s^,
if
i=k
i-1
k
I xf
Ixf^ =
-
5
j-iM
j-h
ly,,
by
<
Ixf5
(i.j).
V
<-
(5)
J-2
i-1
Yy
Yy =0
-
for 2 <
for
xf5
y
Range of
.
s,j
>
0,
i
<
n-1.
<hJo,
[i.j],
>
for all <ia;>.
(i.j],
integer
for all
all
(6]
[7]
and
[8)
[9]
(i.j].
l<h<i<j<k<a
indices:
The objective is to maximize total profit, which is defined in (1) as
the
total
term).
revenue
Constraint
(the
first
(2)
ensures
term)
that
minus
the
the
routing costs
(the
second
amount of traffic served between
each O-D pair does not exceed the corresponding demand.
Constraints
(3)
9
commodity flow conservation equations.
are
the
net
flow
equal
at
every
other
Constraint
leaving origin
flight
leave the main base.
one
trip
that
Here,
during
(5)
and entering destination k
outflow
number
of
passengers
exceed
the
total
Constraints
transported
capacity of
the
model
the
and
(5)
commodity.
this
for
(6)
specifies that at most V aircraft can
implicitly assume that each aircraft makes
we
the
the
not
segment.
Constraint
h
the
transported between 0-D pair (h,k);
total
the
should
segment
traveling on
s^j^
equals
inflow
that
states
aircraft movements.
most
the
node
(4)
any
aircraft
at
<h,k>
the number of passengers
to
through
commodity
of
they set
Essentially,
Constraint
planning horizon.
(6)
specifies
that the number of aircraft entering each intermediate city must equal the
number
aircraft
of
constraint
(i,j)
(4)
(7)
leaving
Finally,
city.
that
impose
we
which specifies that passengers cannot travel on
unless we schedule aircraft on this segment.
also imposes the same restriction.
in the integer formulation
[ARP]
Hence,
However,
.
forcing
the
segment
a
Observe that constraint
constraint (7) is redundant
they are not redundant when we
relax the Integrality restriction on the y variables; thus,
including them
in the formulation improves the linear programming relaxation bound.
Notice
directions
without
that,
our
assumptions
about
(from lower to higher indexed nodes only),
permissible
the routing problem
A subtour
formulation requires additional constraints to prevent subtours.
is a directed cycle that does not include one or both bases;
are not
feasible routes.
problem
that
is
defined
However,
over
a
add
explicit
subtour
thus,
subtours
when we maximize profit for a routing
general
subtours that have positive net revenue.
must
flight
elimination
network,
the
model
may
select
To prevent this phenomenon,
constraints;
these
we
constraints
10
increase the formulation size and make the problem more difficult to solve.
thus obviating the
Our node indexing assumption makes the network acyclic,
need for these additional constraints in our aircraft routing model.
The
aircraft
problem
using
and
number
passengers
continuous
single-commodity
relaxation
sets
two
variables
of
(LP)
also
can
formulated
be
in
several
For instance, Etschmaier and Richardson (1973) formulate
alternative ways.
the
problem
routing
decision
of
variables
single-commodity
traveling
each
flight
more
compact,
on
formulation
is
variables:
integral
that
represent
Although
segment.
the
this
programming
linear
its
routing
(The LP bound obtained using their
bound is very loose.
formulation for one small test problem resulted in 120% gap between the LP
and Integer solutions;
the LP solution coincided with the integer solution
using our formulation.)
The size of the mixed integer program for
number of nodes
(with all
190
increases.
For
possible directed arcs
Integer
variables,
7505
for a 20-city complete network
instance,
(i,j),
<
i
continuous
[ARP] grows rapidly as the
j),
the formulation contains
variables,
and
9425
constraints.
The size is more than doubled when we consider a 26-city problem, for which
the
formulation
will
contain
variables,
and 24400 constraints.
relatively
difficult
enumeration
methods.
to
solve
In
integer
325
variables,
continuous
20800
Mixed integer programs of this size are
optimally
particular,
using
the
branch-and-bound
imbedded
or
other
"plckup-and-delivery"
characteristic of the problem complicates the task of evaluating the large
number
of
feasible
feasible routes;
a
26-city
routes
(for
instance,
a
20-city
problem
has
262,144
the number of feasible routes increases to 16.777,216 for
problem)
.
Therefore,
we
focus
on
developing
an
efficient
11
procedure to identify good (possibly suboptimal) demand selection decision
candidate
and
routes
using
Lagrangian
a
which
scheme
relaxation
we
introduce in the next section.
THE SOLUTION METHOD
3.
A
number
successful
of
applications
approach have been reported in the literature.
relaxation
Lagrangian
the
of
For instance,
Lagrangian-
based algorithms have been used effectively to solve the traveling salesman
problem
(Held and Karp (1970)),
Fisher
and
Jaikumar
(Fisher,
(1979),
Nemhauser
numerous
its
discuss
(1981)
applications.
"complicating"
and
(1977)),
the
Van Wassenhove
and
Fisher
and
the facility location problem
constraints,
obtain
a
subproblem
special structure.
objective
that
can
consists
value
solved
be
serves
Shapiro
the
in
certain
objective
The purpose of this relaxation is
efficiently
as
an
upper
because
of
its
the relaxed problem's
For a given set of multipliers,
function
removing
of
incorporating them
and
(1974),
problem
Lagrangian relaxation approach and
method
function using Lagrangian multipliers.
to
assignment
Geoffrion
(1980)).
the
The
generalized
(Cornuejols,
bound
(for
maximization
objective functions) on the optimal profit of the original problem.
In
our Lagrangian relaxation-based solution procedure, we dualize the
commodity flow conservation equations (3) using Lagrangian multipliers Uihk
for all <h,k>,
cost
to
h <
transport
destination
k.
i
<
k.
each
Intuitively,
passenger
u^
from
represents the imputed routing
origin
The relaxed problem, denoted as [RP]
h,
,
through
city
is given below.
1,
to
12
[RP]
I r^s,,
= maximize
Zrp
^f.^y
-
subject to
II
^
(ij3
(h.k)
[2],
c[^jX[>5
tij)<hji>
-
(4]
[9].
Where,
J'hk
+
<^
cf^ = u|*
-
u]'''
Thk =
For
a
-
^
for
a^l
tWc],
for all <lik>.
given set of Lagrangian Bultipliers
the
,
[i.j].
relaxed problen
[RP]
further decomposes into a demand selection subproblem (containing only the
s
variables)
(containing
and
only
allocation/route
capacity
a
the
and
x
The
variables).
y
selection
subproblem
selected
Lagrangian
multiplier values determine the subproblem objective function coefficients,
namely,
the
"adjusted"
be
added
demand
to
the
selection
and
f^j^
the
imputed
intercity
Observe that if the application context imposes
transportation costs c"^.
variable operating costs
revenues
unit
(in addition to the fixed costs),
imputed
cost
subproblem
hk
c"^.
can
be
As
we
solved
show
In
the
these costs can
inspection.
by
section
next
the
On
the
other
hand,
the capacity allocation/route selection subproblem can be transformed
into
a
minimum-cost
shortest path problem
can
be
Solving
solved
the
flow
for
problem
the
efficiently
demand
for
the
multiple-aircraft
single-aircraft
using
selection
and
case.
specialized
capacity
case
or
a
Both these problems
network
flow
algorithms.
allocation/route
selection
.
13
subproblems gives an upper bound to [ARP]
To improve the upper bound, we
.
iteratively update the Lagrangian multipliers using the subgradient search
method.
We also apply
a
Lagrangian-based heuristic to construct feasible
solutions that provide lower bounds to [ARP]
These different components of the Lagrangian relaxation approach are
described
in
describes
both
greater
demand
the
selection subproblems,
subgradient
the
multipliers.
detail
the
in
selection
following
and
the
procedure
search
that
we
use
to
3.1
allocation/route
capacity
and their solution methods.
Section
sections:
Section 3.2 discusses
update
the
Lagrangian
we present a Lagrangian-based heuristic that
In Section 3.3,
constructs locally optimal solutions to [ARP].
Solving the Lagrangian Subproblems
3.1.
The demand selection subproblea
the
,
abbreviated as [DSS]
,
consists of only
variables, and is defined below:
s
[DSS]
maximize
^ Thk^hk
^0)
subject to:
<
s^
<
dj^
for all
Observe that [DSS] can be solved by inspecting the sign of
is.
if
f^
is positive for 0-D pair
(h,k),
(11]
(lUc].
fj^j^.
That
this 0-D pair is "selected", and
the entire demand for this 0-D pair is transported; otherwise, no passenger
14
for
this 0-D pair
is
transported.
Therefore,
the optimal solution to the
subproblem [DSS] is:
^hk-
if fhk >
0.
if fhk < 0.
'hk
for all 0-D pairs (h.k).
The
allocation/route
capacity
selection
subproblem
,
denoted
[CARSS], contains only the x and y variables and is defined as follows:
[CARSS]
minimize
^rv
X ^ c|%|^
+
^
(12]
subject to:
Sy,j
<
V
J-2
n
1-1
[13)
as
15
capacity on each segment of
the
"commodities"
route among the
the
<h,k>
(restricted by (15) and (16)).
Observe
requirements,
be
does not have commodity flow conservation
[CARSS]
since
that
the optimal allocation of capacity among the commodities can
separately
determined
for
each
aircraft traveling on arc (i.j).
minimum
cost
corresponding
arc.
that
there
are
The optimal capacity allocation,
flow
aircraft
this
to
Suppose
can
be
m(<
V)
and the
determined
by
solving the following continuous knapsack problem [CKP"j]:
[CKP^j]:
mmimize
6l^ =
^ c^h^)
mf„
+
[19]
subject to:
^xf\
^
This
^'^ ^
(20)
for all <lUo.
<Jhk
[21]
Sort
the
commodities
in
increasing order of
Consider each commodity <h,k> in sequence from this list.
negative,
(dj^l^)
mb
continuous knapsack problem can be solved efficiently using the
following greedy procedure:
Cjj.
<
If c^^ is
hk
set x^^ equal to the minimum of commodity <h,k>'s traffic demand
and the remaining aircraft capacity.
Stop if the aircraft capacity
is exhausted or all remaining commodities have non-negative imputed routing
cost Cjj.
Notice
aircraft
that
travel
dj^jj,,
on
arc
which
(i.j),
denotes
is
the
convex
minimum
in
n
routing
since
the
"cost"
when
m
commodities are
,
16
selected
non-decreasing order of c^^.
in
increases
decreases.)
can
incremental
the
solve
After computing d^
[CARSS]
by
flow
arc
each
on
segment.)
flight
for all arcs
^^
and m
(i,j)
If
represent
the
on each arc.
d^^^^
number
the
of
resulting minimum
aircraft
of
=
cost
is
node n,
to
in [CARSS],
(Recall that,
aircraft
we
V,
1
1
m
passengers
additional
selecting
of
finding a minimum-cost flow from node
with the convex cost function
the
benefit
number
the
(As
assigned to
negative,
that
the
then
directed paths in the optimal minimum-cost flow solution define an optimal
set of aircraft routes,
flow
the
on
with the number of aircraft on each route equal to
corresponding directed
the
path;
otherwise,
no
route
is
initiated.
We next demonstrate how to transform the minimum convex cost problem
into
ordinary network
an
Figure
1
original
gives
small
a
network
flow problem over
example
contains
3
suitably modified network.
a
illustrating
nodes
and
transformation.
this
directed
3
arcs.
We
The
set
up
a
modified network for the equivalent minimum-cost flow problem as follows:
To model the convex cost function on each original arc (l,j).
V parallel directed arcs from node
(d|j^
....
is
-
V.
d^
u_i).
j
Thus,
respectively,
to node
1
for
the
j
with costs equal to d^.^ and
first
and the m
arcs,
the cost on each parallel directed arc from node
1
m
=
1
to node
j
the marginal cost of assigning an additional aircraft to segment (i,j).
Each directed arc carries an upper limit on flow of
the parallel
two
we introduce
arcs,
dummy directed
have costs equal
to
1
unit.
In addition to
the modifies network also contains a dummy node D,
arcs
0,
(1,D)
and
(n,D).
These two dummy directed arcs
and upper limits on flow equal to V.
cost flow problem is solved from node
1
and
to node D.
The Inimum-
If the minimum cost is
.
17
route
no
zero,
selected;
is
otherwise,
optimal
the
flows
arc
define
an
optimal set of aircraft routes for subproblem [CARSS]
insert Fig.
For
is
single-aircraft case,
the
necessary for each arc
Observe
(i,j).
shortest
contain
length
problems
path
directed
any
of
can
be
cycles
shortest
the
even
1
path
to node n using d^jj as the length for
though
solved
(and,
is
d^-j,
can alternatively be solved by
[CARSS]
(i,j),
that
about here
since only one cost coefficient,
finding a shortest path from node
arc
1
lengths
arc
easily
hence,
may
since
the
any negative
negative,
the
shortest
be
negative,
the
network does not
cycles).
path
If
defines
the
an
optimal aircraft route; otherwise, no route is initiated.
3.2. Generating Upper Bounds for [ARP]
For
given
any
function value,
bound to
vector
Zpp(U),
[ARP].
of
U
the
of
Lagranglan
multipliers,
Lagranglan problem
[RP]
the
objective
provides an upper
To find the best upper bound we must solve the following
dual problem:
[DP]
min Zpp(U)
U
Our
method
attempts
find
to
a
near-optimal
solution
to
[DP]
by
using a
hk
subgradient search method to update the multipliers u^
and improve Zpp(U).
hk
Since the multipliers u"
serve as imputed variable costs for transporting
passengers from origin
through city
.hk
h,
i,
to destination k, we initialize
18
if i=h
'-^hk / 2.
=
"i*^
(^hi ^ Sik)
I
[
where,
fixed
costs
if h <
<
i
for all
k
(h.k)
if i=k.
2.
the shortest path distance from node p to node q using the
is
g
/
''hk
b,
/
arc
as
f^g
lengths.
Effectively,
the
initialization scheme
prorates the fixed routing costs to each passenger traveling from an origin
city
<
k.
h,
through an intermediate city
i
to a destination city k,
,
for h
<
i
At both the origin city h and the destination city k, we use half the
unit revenue for 0-D pair
(h.k),
and treat it as a negative variable cost
for transporting one passenger from city h to city k.
Since our
equations
and
— hk
in
(3)
from
Xj^j
relaxation scheme dualizes the commodity flow conservation
[ARP]
[CARSS]
,
solutions
the
may
not
conserve
outflow of commodity <h,k> at each node
direction
procedure
for
flow
i,
Lagrangian
the
obtained from subproblem
h <
at
i
£
each
k,
node.
The
[DSS]
excess
defines a subgradient
multipliers
hk
Uj^
.
Our
solution
iteratively updates the Lagrangian multipliers using a standard
subgradient
(1974),
changing
s^j
search
and Fisher
method
(1981)),
(see,
for
example,
Held,
Wolfe
and
Crowder
solves the revised Lagrangian subproblems,
and
computes the new subgradients until either the number of iterations exceeds
a
prespecified limit or the step-size multiplier value falls below a given
threshold.
We
note here
that
our relaxation scheme does not satisfy the
Integrality Property (Geoffrion 1974), and may,
therefore, provide tighter
bounds than the LP relaxation of [ARP].
3.3.
A Lagrangian-Based Heuristic for [ARP]
The Lagrangian solutions,
although not necessarily feasible to [ARP],
19
do
provide
useful
Lagrangian-based
information
heuristic
uses
the
capacity.
procedure
Recall
attempts
that
Then,
[ARP].
better
to
[CARSS]
to
to
we apply a local
utilize
aircraft
the
define either a single route or a set of multiple routes.
[CARSS]
Our
non-zero y solutions obtained from subproblem
the
that
solution
optimal
construct an initial feasible solution for
improvement
solutions.
constructing feasible
for
In the
following discussions, we will focus on the case when multiple routes are
obtained from [CARSS].
A step-by-step description of the heuristic follows:
Step
1:
Set
them.
Separate the set of routes into Vq single routes and number
m=l
Recall
.
that
the
optimal
solution
to
equivalent
the
minimum-cost flow problem defines a set of aircraft routes, with the number
of aircraft on each route equal
the flow on the corresponding directed
to
The heuristic first obtains Vq single routes by tracing through each
path.
directed path and identifying the flow on the directed path.
Step
2:
can
equal
to
initial
supply
eunounts
To get a good initial supply amount
[CARSS]
that
Obtain
be
served by
the minimum of
<h,k> to node k,
the
set
from
s^j^
the
routes obtained in step
of
the demand for 0-D pair (h,k),
solutions
from
for each 0-D pair (h.k)
outflow of <h,k> from node
the
x
h,
1,
we set
the
s^
Inflow of
and the aircraft capacity.
That is,
s°, =
min
{
I
j-h*l
x,j
'
.
Ix
j-h
^
,
.
dj^
.
This supply assignment is preferable to using the
subproblem
[DSS],
since
this
subproblem's
b
s
).
solution obtained from
"all-or-nothing"
solution
20
characteristic
traffic demand
Step
aircraft
value,
the
dj^j^
capacity
capacity.
,
Let
.
served by the m
(H.Kj^j,
on
some
Step
information
route
m
6
(h.k)
values
s^j^
the
constraints
can be
for
little
much
how
on
the
of
maxlfflum
to actually transport.
For
3:
that
(h,k)
provides
capacity is violated,
feasibility
(H.K)^
denote
the
respect
with
Although each individual
route.
segments
the
of
possible
this
for
may
route
exceed
situation
to
0-D pairs
of
set
satisfies the capacity constraint,
checks
3
verify
s^j^
the sum of
the
aircraft
if
aircraft
and,
reduces the supply amounts to those 0-D pairs that
result in the smallest decrease in profit, until the capacity constraint is
met on each segment of the route.
for the 0-D pair
Step
Compute
4^:
aircraft
capacity by
correspondingly
necessary.
Step
possible
denote the revised supply amount
improvement
For each 0-D pair
.
heuristic
the
s^j^
(h,k) 6 (H,K)^.
aircraft capacity
demand,
Let
computes
the
increasing the
reducing
(h.k)
marginal
profit
amount
this
other
If profit improves for none of the 0-D pairs,
Revise
5:
sj^j^
reallocating
of
to
to
performed.
After revising the corresponding supply amounts,
Step
update
an
improved
produce
Xf
6:
{s^j^}
4
,
The
if
most
profitable
reallocation
step
in
4
Is
the heuristic
to check for other possible improvements.
"i^V q.
Stop
increase
capacity
negative
pairs,
by reallocating the aircraft capacity to the most
commodity
returns to step
and
0-D pair,
0-D
the
go to step 6.
profitable
.
reallocating
by
that has unsatisfied
(H,K)|||
supply amount
supply
the
&
profit
in
.
Otherwise
,
check profitability of this route
by 1, and go to step 3.
allocation.
profit.
In
this
However,
case,
we
the
,
So far we have obtained
entire
route
simply discard
may
this
still
route,
21
reduce y^j by the amount of aircraft flow for each (l.j) on this route, and
set
for all
to
Sjjj^
o
{sPj^>
hk'
amounts
1
^hk
6
(h.k)
Finally,
(H.K)„,.
initial
the
set of supply
is revised by setting
<—
""^^
<
s^^
-
s^^.
for all
)
(h.k)
€
(H.K)„.
The heuristic returns to step 3 to examine the next route.
our
In
every
implementation,
iterations)
10
multiplier
reaches
to
a
heuristic
the
improve
lower
the
prespecified
is
periodically
applied
bound
threshold
until
value.
(once
step-size
the
Thereafter,
the
heuristic is applied after every subgradient iteration.
COMPUTATIONAL RESULTS
4.
Since
nature,
actual
we
problem
data
tested
the
reported
in
difficult
is
solution procedure
Etschmaier
randomly-generated problems.
larger
(in
attempted
terms
number
of
problems
obtain
to
on
the
Richardson
and
due
single-aircraft
categorized
generated
generator.
20,
23,
5
15
on
12-city
several
The test problems reported here are not only
of
described
into
and
(1973),
cities
in
the
and
aircraft)
literature,
than
but
reflect the size of the problems that arise in practice.
were
confidential
its
to
classes.
problem
For
each
also
the
previously
more
closely
The test problems
problem
class,
we
different instances using different seeds for the random number
Problem classes
1
to
4
are
and 26 cities, respectively.
problems with 17,
20,
23,
single-aircraft problems with 17,
Problem classes 5 to 8 are 2-aircraft
and 26 cities,
to 12 are 3-alrcraft problems with 17,
20,
respectively.
23,
Problem classes 9
and 24 cities, respectively.
22
problem classes
Finally,
20,
13,
and 15 are 4-alrcraft problems with 17,
14,
and 23 cities, respectively.
Given
the
desired
number
(including the
cities
of
bases),
the
test
problem generator first randomly locates the required number of nodes on a
100 X 100 grid,
and numbers the nodes in increasing order of the Euclidean
All our test problems are complete,
distance from the origin (0,0).
Demand for each 0-D
the possible directed arcs.
the networks contain all
pair was generated from a uniform (0,100) distribution.
Finally,
revenue
on
for
each 0-D pair,
and
the
fixed
i.e.,
routing cost
the unit
each arc were
generated as follows:
^hk
=
^'^hk
f^j = 30
where
EU
is
*
the
aircraft capacity,
""
UNIFORM(0,10)
EU^j
+
UNIFORM(O.IOO)
Euclidean
b,
,
distance
,
for all
(h,k).
for all
(i,j).
between
node
p
node
and
q.
The
was set equal to 100 for all the test problems.
The solution procedure was coded in FORTRAN and implemented on an IBM
3083 computer.
subproblem
networks;
For the single aircraft case,
using
the
algorithm
discussed
for multiple aircraft problems,
in
we solved the shortest-path
Lawler
for
(1976)
acyclic
we use a modified version of the
NETFLO code (Kennington and Helgason (1980)) to solve the minimum-cost flow
subproblem.
structures;
Our
implementation
does
not
employ
with improvements in data organization,
any
special
data
the solution procedure
could perhaps solve larger problems.
Our
implementation
of
step-size multiplier to 2.0,
objective
function value
the
subgradient
and halves
does not
procedure
the multiplier
if
initializes
the
the Lagrangian
improve for 20 consecutive
iterations.
23
solution
The
exceeds
The
procedure
1000,
local
the
or
improvement
terminates
when
either
number
the
of
Iterations
step-size multiplier reduces to a value below 0.01.
heuristic was
applied at
first
the
iteration,
and
once every 10 iterations until the step-size multiplier reached a value of
thereafter,
0.1;
applied at every iteration.
the heuristic was
test problem we recorded
the
(i)
Initial upper bound
(obtained by solving
the Lagrangian problem with the initial multipliers),
bound when the procedure terminates,
all heuristic solutions.
LB
*
bounds,
the
UB
(11)
the best upper
the best lower bound among
(iii)
We use the performance measure *GAP
and
LB
are
respectively
the
best
=
upper
(UB - LB)
/
lower
and
to evaluate the quality of the solutions.
Table
of
where
100%,
and
For each
15
1
summarizes the performance of the solution procedure for each
problem classes.
Notice that the algorithm was able to reduce
the gap between the upper and lower bounds from an Initial value of 50.93%
to
a
final
value of 1.65% on average, and the largest final %GAP was only
3.22% among the 75 test problems.
These statistics show the effectiveness
of the subgradient method for improving the upper bounds.
the
following
generated
subproblem
by
interesting
the
[CARSS]
method.
characteristic
For
every
of
problem
the
We also observed
heuristic
instance
that
solutions
we
tested,
generated only a few alternative routes for step-size
multiplier values below 0.1, and the best feasible solution to each problem
instance always resulted from one of the routes In this small set.
the
solution procedure
seems
to
select
a
small
Hence,
number of good candidate
routes from the fairly large number of possible routes in the problem.
insert Table
1
about here
24
figures
The
requirements
grow
increases.
for
CPU
number
the
as
However,
times
Table
in
rate than the problem size.
computational
that
aircraft
number of
the
increases at a slower
requirement
computational
the
and/or
nodes
of
show
1
the size of a 26-city complete
For instance,
network problem is more than twice the size of a 20-city complete network
problem
terms
in
formulation.
times
for
constraints
l-aircraft
case
1.75
and
times
the
in
problem
requirement was only 1.88
2-aircraft
the
for
case.
increasing the number of aircraft seems to increase the CPU time only
Also,
marginally.
about
instance,
For
procedure.
For
subproblem accounted for about
single-aircraft
accounted
cases.
requirement
increases
only
to 4.
2
records the breakdown of total CPU time among the 3 submodules
2
solution
the
computational
the
on average when the number of aircraft increases from
383b
Table
of
and
on average the computational
Yet,
the
variables
decision
of
about
for
1/3
of
1/5
the
of
feasible
CPU time
minimum-cost
the
for
shortest
the
total
the
CPU time
total
the
constructing
Finally,
1/4
to
solving
problems;
solving
instance,
flow
the
subproblem
multiple-aircraft
requires
solutions
for
path
increasing
an
proportion of the total CPU time as the number of nodes and/or the number
of
aircraft
constructs,
solution
network
every
in
for
code
generates
the
some
to
only
We
note
iteration,
subproblem
for
iteration,
NETFLO
increases.
computational
retain
the
limited
In
flow
time
initial
number
that
of
retrospect,
problem
may
be
network.
an
initial
since
the
remains
saved
if
we
had
routes
underlying
in
modified
since
(for
code
feasible
same
the
Furthermore,
alternative
NETFLO
modified
the
fresh network and
a
[CARSS].
minimum-cost
here
each
the
[CARSS]
step-size
multiplier values below 0.1), we could use the solution from one iteration
25
thus saving some additional computational time.
to initialize the next,
insert Table
2
about here
We also tested our solution procedure on the 12-city problem listed in
Etschmaier and Richardson (1973).
exactly one
city
slip
the
in
To ensure that our solution will contain
point
set
(cities
7
and 8),
large fixed routing cost on arcs that skip these nodes
with
the
set
direct
12)
6
<
i
(i.e.,
flights
and 9
arc
7
1
<
j
<
12),
(7,8)).
The
Therefore,
in
large
(1
routing costs
to 6)
between city
7
arcs
(i,j),
on
these arcs make
and following cities
(9
If an aircraft stops at
aircraft must stop at either
city
7,
the large routing cost
and city 8 prevents a subsequent stop at city 8.
Therefore,
our solution will contain exactly one city in the slip point set (cities
and
8).
However,
we
relaxed
to
order to serve customers originating in
to 6 and destined for cities 9 to 12,
or city 8.
(i.e.,
and on arcs interconnecting the nodes of
between the preceding
unprofitable.
cities
city
<
1
we assigned a
Etschmaier
and
Richardson's
7
original
restriction that aircraft can stop at most twice before and after the slip
point stop.
Table
3
compares our solution with Etschmaier and Richardson's
schedule.
insert Table 3 about here
Observe that by relaxing the maximum number of stops restriction, we
obtained a solution with $1078 higher profit.
This value is essentially an
opportunity cost for imposing the upper limit on number of stops.
Notice
26
that
also
optimal
the
route
our solution by applying a drop heuristic,
cities
three
problem
that
least
illustrates
decreases
the
problem can be derived from
the original
for
which deletes one of the first
flexibility
and
result
The
profit.
the
usefulness
of
this
model
our
of
test
in
selecting good candidate routes which must satisfy some specific operating
Similar
constraints.
unique
operational
procedures
considerations
can
employed
be
to
account
constructing aircraft
in
for
routes
other
for
a
specific airline.
5.
scheduling is one of the most important operational decisions
Flight
for
airline.
an
iterations
The final
the
of
long-haul carriers,
phase
route
timetable of flights is the result of several
construction
and
evaluation
route
phases.
For
the aircraft routing problem in the route construction
complex
very
is
SUMMARY
because
selecting
a
set
good
of
candidate
routes
requires evaluating a combinatorially large number of feasible routes, and
incorporating
the
Therefore,
the
development
candidate
routes
"pickup-and-delivery"
will
of
an
facilitate
characteristic
efficient
procedure
iterative
the
flight
of
for
the
problem.
selecting good
scheduling process
and eventually lead to a more profitable timetable.
this
In
paper
we
have
discussed
a
version of
the
aircraft
routing
problem that captures some of the most important profit-determining factors
in the
route selection decision faced by a long-haul carrier that operates
in a base-to-base environment.
cities
are
mixed
integer
aircraft
visited
in
increasing order of node indices.
programming
problem,
and
Our model makes the common assumption that
formulation
developed
a
for
Lagranglan
the
multiple
We presented a
(homogeneous)
relaxation-based
solution
.
27
procedure
results
show
allocations
large,
exploits
that
that
that
this
the
procedure
the
of
optimality
of
Computational
problem.
aircraft
generates
1-3%
within
are
random test problems.
problem serves as
structure
routes
relatively
several
for
traffic
and
The proposed analysis of the aircraft routing
preliminary screening procedure to assess the market
a
profitability of different regions, identify potentially profitable routes,
and facilitate cost-effect analysis of other operational and service level
policies
In addition to the basic problem setting described in this paper,
model
accommodate
can
practice.
For
operating
some
instance,
the
in
this
constraints
that
may
section we
have
illustrated a
previous
arise
in
method to modify the cost matrix in order to ensure that each route serves
exactly one city in the slip point set.
To
comply with other operating
constraints such as fixed number of stops between slip point sets, we can
apply a drop/add heuristic
to the best
solution for the basic model.
We
note that although the Lagrangian upper bounds are still valid for the more
restricted
problem,
additional
operating
hence,
the
accommodate
addition
to
lower
constraints
bound.
variable
the
expect
we
As
operating
fixed
cost
will
we
increase
because
usually
decrease
the
noted
costs
per
to
%GAP
the
our
earlier,
(that
depend on
aircraft
on
each
imposing
profit,
model
traffic
flight
and
easily
can
volume)
In
segment.
Application contexts where more than one terminal base exits can be solved
using a suitable network representation.
modeled
by
replicating
original one-way network.
each
node,
i.e.,
Round-trip operations can also be
adding
a
mirror
Image
of
The solution procedure can then be applied
the
.
,
28
to
the expanded network with modified demand,
and operating cost
revenue,
matrices
Several
extensions
investigation.
First,
variants
and
basic
our
of
model
can accommodate
while our model
further
merit
fixed as variable
operating costs,
all
flight segment.
Additional test results for problems with variable costs
our
test
a
fixed cost for each
testing of alternative heuristic methods and
Empirical
might be useful.
problems assume only
multiplier adjustment schemes
also worth pursuing.
is
assumes that all aircraft are of the same type.
to use aggregate routing variables
Second,
our model
This assumption enables us
(the y variables)
in formulation
[ARP]
instead of detailed binary variables to trace the route for each individual
Modeling heterogeneous aircraft fleet (with different operating
aircraft.
costs
and
capacities),
on
the
other
hand,
will
require
a
detailed
formulation that destroys some of the special structure in our model.
solution
well.
algorithm might
We
method's
expect,
possibly
however,
computational
Finally,
because
requirements
(measured in terms of *GAPs)
aircraft types.
that
extend
will
to
of
will
this
the
more
complex
model
Our
as
increased complexity the
increase
and
its
performance
deteriorate for problems with multiple
relaxing the node indexing assumption (i.e., the
assumption that flights are permitted only in the direction of increasing
node indices) might increase the model's applicability to contexts (such as
short-haul routes) where the cities are not completely ordered.
Again this
extension increases the model complexity greatly, and may require different
solution approaches.
Chien (1987) explores some of these assumptions.
.
29
ACKNONLEDGENENTS
The authors wish to thank the two anonymous referees and the Associate
Editor for their heipful comments on the presentation of the paper.
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Captions
Table
1:
Computational Results for Test Problems
Table
2:
Breakdown of Total CPU Time
Table
3:
Comparison of Solutions for the 12-City Test Problem
Fig.
1:
Transforming [CARSS] into
a
Minimum-Cost Flow Problem
)
Figure
1
Original Network
capaci ty
cost
1
,d 131
1' (<^13in-^l,3.in-l)
1
,
(dj3y dj
1
121
,
1
/^
l'<'^12m-<^1.2,m-l)
!H
)
12V °1
,
2
,
V-1
)/
2
^
•
<
<^2
-d
y_j
3
231
3di-^2
,
3
.
m-
1
>
)
\
M
3
l,(d23v ^2,3.V-1^
V.O
Modi fled Minimum-Cost Network
Table
1
1
Aircraft
(
Table 1
continued
3
Legend
Aircraft
)
Table
1
Aircraft
No.
of Nodes
20
23
17
UB^excl. Sp2)
2
67 2%
20 6%
63 8%
SP
FEAS^
12.2%
14
.
.
2
17
UB(excl. NETFLO'*)
NETFLO
FEAS
.
21.4%
.
8%
Aircraft
61 2%
22 1%
16. 7%
.
.
26
58 0%
23 7%
18 3%
.
.
.
Table
3
Solution from Etschmaier and Richardson (1973
Optimal Route:
Profit
:
1-3-5-8-9-12
10312
Passenger Distribution:
MIT LIBRARIES DUPL
3
TDflO
1
0DS70432
2