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L162
BEBR
FACULTY WORKING
PAPER NO. 1463
Solving a Real World Highway Network
Design Problem Using Bilevel Linear Programming
Omar Ben-Ayed
Charles E. Blair
David E. Bo\ce
Hi*
9f
College of
Bureau
of
University
Commerce and Business Administration
Economic and Business Research
of Illinois. Urbana-Champaign
hit
BEBR
FACULTY WORKING PAPER NO. 1463
College of Commerce and Business Administration
University of Illinois at Urbana- Champaign
June 1988
Solving a Real World Highway Network Design Problem
Using Bilevel Linear Programming
Omar Ben-Ayed, Graduate Student
Department of Business Administration
Charles E. Blair, Professor
Department of Business Administration
David E. Boyce
Department of Civil Engineering, University of Illinois
Digitized by the Internet Archive
in
2011 with funding from
University of
Illinois
Urbana-Champaign
http://www.archive.org/details/solvingrealworld1463bena
.
SOLVING A REAL WORLD HIGHWAY NETWORK DESIGN PROBLEM
USING BILEVEL LINEAR PROGRAMMING
OMAR BEN-AYED and CHARLES
E.
BLAIR
Department of Business Administration
University of Illinois at Urbana-Champaign, IL 61820
DAVID
E.
BOYCE
Department of Civil Engineering
University of Illinois at Urbana-Champaign, IL 61801
(May 1988)
This paper is concerned with the solution of a real world Bilevel Linear
Program (BLP) of the highway Network Design Problem (NDP), based on empirical
data for Tunisia. The problem includes 2,683 variables (2,571 at the lower
level) and 820 constraints; it is far beyond the capacity of any existing
algorithm. A new algorithm is devised and the solution is found successfully
despite the NP-hardness of BLP and the large scale of the problem. The computational difficulties of BLP should not present a barrier against the effective
use of such a powerful optimization technique.
Bilevel
Linear Programming
(BLP)
is
a
mathematical model
of
organiza-
an
tional hierarchy in which two decision makers have to choose their strategies
from the polyhedron {(x,y):
who has
control over
lower decision maker
and,
x,
Ax + By < b;
x,
makes his decision
selects
y
[Bialas
y >
The upper decision maker,
0).
hence
first,
and Karwan
1982
fixing x before the
and
1984,
Bard
1983,
Ben-Ayed 1988]
The
ability
motivated
the
BLP
of
to
development
resource
allocation
planning
[Bracken
represent
of
[Cassidy
and
McGill
several
1971,
1973,
decentralized
applications
Fortuny-Amat
1974a
and
and
1974b,
decision-making
of
McCarl
Shere
including
model,
the
problems
1981],
military
and Wingate
1976,
Falk 1977] and government policy [Candler and Norton 1977, DeSilva 1978, Bialas
et
al.
1980,
Falk
and
McCormick
1982,
Candler
and Townsley
1982
and
1985].
2
attention
Recently,
been
has
making
decision
transportation
focused
on
applying
bilevel
problem
referred
to
formulations
Network
the
as
to
a
Design
Problem (NDP) [LeBlanc and Boyce 1986, Marcotte 1986, Ben-Ayed, Boyce and Blair
1988].
NDP
concerned with
is
optimal
the
improvement
of
transportation
a
network in order to minimize the system total costs.
Most
BLP
applications
lacking
the
practical
significance
majority of
overwhelming
limited to
are
theoretical
solving
of
real-world
the
real
problems
formulation,
problems.
Actually,
formulated
are
thereby
the
solved
and
as
single-level (single-objective) programs even when they are virtually bilevel.
This is mainly due to the inefficiency associated with BLP algorithms [Ben-Ayed
and Blair 1988], which presents a real barrier against the effective use of the
model.
In
trying
motivate
to
formulations
theoretical
real world problems,
and
applications
further
Ben-Ayed,
illustrative
small
the
Blair and Boyce [1988]
of
beyond
going
BLP
examples
interesting
to
presented
the
a BLP
formula-
tion of the interregional highway NDP based on empirical data for Tunisia. This
paper gives the solution to the problem formulated there. This problem includes
2,683 variables
(2,571 of which are lower) and 820 constraints, which makes it
too large to be solved by any of the existing algorithms.
Section
that
takes
1
presents
Section
In
2,
we
develop
a
advantage of the special structure of the problem.
reasons for having
a
deals
with
and
second
each
iteration we try to find
a
of
the
the
the
There are two
lower
problems
The
functions.
separately;
at
each
better compromise with the user, while including the
improvement
minimum computation effort.
presented and the results are analyzed.
some suggestions.
improvement
nonconvex
two
smallest possible number of nonconvex
solution with
new algorithm
first the user-optimized flow requirement
BLP formulation;
(user-equilibrium),
algorithm
the problem.
In
functions
Section
We conclude
in
the
3,
to get
the
last
the exact
solution
is
section with
The Problem to be Solved
1.
this
In
formulation resulting
section we give a short description of the
Blair and Boyce [1988]
from the empirical study conducted by Ben-Ayed,
on the
Tunisian interregional highway transportation network. We are concerned with
a
Each region is represented
system of roads connecting the regions of Tunisia.
by a centroid-node which is usually the largest city in the region. The traffic
entering and leaving a centroid-node is assumed to be that of the whole region;
every centroid-node
is
simultaneously an origin-node and
destination-node.
a
The other nodes in the network are intersections of roads generally correspondA road connecting any two cities
ing to intermediate cities.
is
called
a
link
and a sequence of links is called a route. We have empirical data on the number
of people who want to travel from each region to each other region.
a
traveller
conditions.
has
a
choice
Improvements
routes
of
various
in
makes
and
links
in
his
decision
network
the
Typically,
based
sill
on
road
cause
some
travellers to change their choice of routes. We will not provide details about
empirical
the
data,
the
construction
constraints in the formulation; however,
of
a
the
objective
functions
and
the
brief discussion on the inclusion of
the bilevel presentation is relevant to the object of this paper.
The
principal
variables
in
the
formulation
added capacity resulting from the improvement,
Units
per hour
(PCU).
combination of
The designers of
improvements
to
make.
a
are
the
traffic
flow
and
the
both measured in Passenger Car
road network wish to
Their objective
takes
find the best
into
account
the
cost of improvements and also factors which depend on the pattern of road use,
including system travel time, system operating cost and frequency of accidents.
The users of the network decide which routes
individual cost;
choices
that
fluence users
are
therefore,
optimal
choices,
to use primarily based on their
their choices do not necessarily coincide with the
for
the
however,
system
[Wardrop
by improving some
1952].
links
The
system
can
in-
and making them more
.
4
When deciding on these improvements,
attractive than others.
preferences
influence the users'
to
on
to minimize
total
system costs.
the upper decision maker chooses the amount of improvement to
Thus in our BLP,
make
in order
the system tries
the
each of
different
(links
roads
in
the
network)
while the
decision maker chooses how much traffic there will be on each road.
approach
is
used by LeBlanc and Boyce
[1986],
and Ben-Ayed,
lower
A similar
Boyce and Blair
[1988]
In
addition to the user-optimized flow requirement,
BLP formulation allows
the analysis of nonconvex piecewise linear functions [Ben-Ayed, Blair and Boyce
1988].
A typical
improvement function might be:
f(Z) = 10Z
< Z < 200
= 2000 + 3(Z-200),
= 2900 + 8(Z-500),
where
Z
the amount of
is
improvement.
difficult
to
Note
that
analyze.
neither
is
However,
link and
convex
function
the
Z
< 500
(1)
500 > 500.
improvement of a
f(Z)
200 <
nor
can
f(Z)
the cost of that
is
concave,
be
making
thus
included
in
the
objective of a BLP with an appropriate lower objective. The technique
is
it
upper
based
on the following:
f(Z) = F(Z,W) = 10Z
The
variables
Wm
are
-
7W X + 5W 2
controlled
,
where W^MAXfO, Z-200}
by
the
lower
trying to make the W m as small as possible,
>
Z-500 guarantee that W m are
problem;
,
W 2 =MAX(
the
,
lower
Z-500]
.
objective
and the constraints W^ > Z-200, W 2
least upper bounds.
This technique can be used
for any piecewise linear function.
Two drawbacks should be mentioned. First,
the number of
lower variables,
are many piecewise
and this
linear functions
in
the method described does increase
increase may be considerable if there
the problem.
Second,
the resulting BLP
5
is
typically
objectives;
distinguished
conflict
the
by
between
its
upper
and
lower
there are many variables which the upper objective wants to be as
large as possible,
while the lower objective wants them as small as possible.
Experience suggests this type of BLP
is
difficult to solve.
Including user-optimized flow and nonconvex improvement
problem obtained by Ben-Ayed,
Blair
Boyce
and
[1988]
is
functions,
synthesized
the BLP
in
the
following:
MIN Z F(Z,W) + G(C)
where
[C,
C,
W,
X and Y]
solve:
MIN F(W) + G(C)
subject to:
=
%(¥)
(2)
H 2 (X,Y) =
H 3 (Z,C,X) >
H 4 (Z,C,X) >
H 5 (Z,W) >
H 6 (Z) >
Z,
where
the
regional
link.
function F
system
is
travel
the
cost.
C,
C,
W,
X,
improvement
The
vector
Y >
The
cost.
X
is
the
function G
amount
of
is
the
inter-
traffic on each
Any traveller from one origin-node to a destination-node has a choice of
different routes to use,
corresponding to different sets of links. We use the
components of the vector Y to indicate the number of people using each route.
These numbers determine the numbers of users of each link, shown by the vector
X;
the
H2(X,Y)
relationship
while
specified nodes
the
is
between
X
requirement
and
that
Y
is
given
specified
given by the constraints
by
the
numbers
equation
of
H^Y) referred
people
constraints
go
between
to as the conserva-
6
tion of flow conditions. F is the lower objective of nonconvex improvement cost
The components
functions.
the vector
of
are
Z
amounts of
the
improvement on
each of the different links. W is the vector of the second-stage variables used
in
computing the improvement cost functions as described previously in (1).
gives
vectors
the
interregional travel costs on each link for the user.
the
of
system
G
and C are
C
second-stage variables giving total travel costs on each link for
and
respectively.
the user,
H3(Z,C,X)
and H^(Z,Z,X)
are
the
ine-
quality constraints needed to the formulation of convex travel cost functions
and the effect on improvement on decreasing those costs for the system and the
user,
respectively.
constraints.
Hg(Z)
description of
the
functions
detailed
Appendix
correspond
Hr(Z,W)
are
variables
improvement
and
the
constraints.
limit
functions
is
provided
linear
A
in
more
the
the total number of variables (see Appendix A)
2,683 and the total number of constraints is 820,
bound
piecewise
A.
In the above formulation (2),
is
the
nonconvex
the
to
(improvement
constraints
limit
constraints
excluding upper and lower
and
nonnegativity
const-
raints). Those numbers are partitioned as follows:
Variables
112
112
112
112
104
36
Constraints
(excluding bound constraints)
(X)
(C)
(C)
(Z)
(W^)
(Wo)
(Y)
2,095
120 (conservation of flow conditions )
112 (link flow definition constraints)
224 (system travel costs formulation)
224 (cumulative user costs formulation)
68 (convex improvement costs formulation)
72 (nonconvex functions formulation)
2,683
The
among
Z
the
functions
are
820.
2,683
all
only
the
variables
variables,
have
3
2,571
pieces;
two
controlled
are
by
lower.
constraints
the
The
are
upper problem;
nonconvex
needed
by
therefore
improvement
each
link.
cost
The
convex improvement functions require only one constraint per link because they
.
all consist of two pieces only.
solution
ideal
An
(lower
bound)
minimization problem
BLP
any
for
can
be
obtained by ignoring the lower objective function and solving the problem as an
call
we
If
LP.
solution
(Z°,
equal
is
to
bound), we substitute
C
C
,
Y
,
F(Z ,W
)
,
W
+ G(C
the
)
F(Z ,W
by
Z
C°,
C°,
X°,
+ G(C
)
W°)
Y°,
the
obtained
get
an
To
).
solution,
incumbent
solution
obtained
solution,
the
incumbent
(upper
If we call
and solve the lower LP problem.
Z
ideal
the
solution
is
(X
equal
,
to
Let us refer to the gap between incumbent and ideal solutions
).
as:
gap = (incumbent
-
ideal)/ideal
The gap for the BLP problem (2) is larger than 111 % [Ben-Ayed 1988]; this high
value,
mainly due
heuristic
gorithms,
to
algorithms
the nonconvex
extremely
functions
difficult,
formulation,
Heuristic
impossible.
not
if
the task of
makes
such as the Parametric Complementary Pivot algorithm (PCP)
Karwan-Shaw 1980,
[Bard 1983],
and Bialas-Karwan 1984]
al-
[Bialas-
and the Grid Search Algorithm (GSA)
lead to very unsatisfactory results when applied to BLP problems
with combinatorial nature,
such as Knapsack problems or nonconvex formulation
problems [Ben-Ayed and Blair 1988].
The problem is even harder for branch and bound or related implicit enumeration
1981,
1982,
is
techniques
Bialas
[Falk
and
1973,
Karwan
Gallo
1982,
and
Fortuny-Amat
1977,
Ulkiicu
Papavassilopoulos
1982,
Candler
and
and
McCarl
Townsley
Bard and Falk 1982, Bard and Moore 1987]. The Bard-Moore algorithm, which
supposed to be one of the most efficient,
does not
accept
a
problem with
more than 100 lower variables; our problem has more than 25 times this number.
As
for
efficient.
any
A
large
new
NP-hard
problem,
the
use
algorithm taking advantage of
of
the
general
special
problem has to be devised to solve the problem at hand.
algorithms
is
not
structure of the
2.
A Proposed Algorithm
Formulation (2) shows that there are two separate lower objective functions.
In fact,
each of the two objectives has its own constraints that are indepen-
dent of the constraints of the other objective, provided that
Z
are fixed (they
are fixed by the upper objective). This can be better seen when the BLP problem
is
(2)
restated as follows:
MIN Z >
F(Z,W) + G(C)
(3)
such that {W) solve (4):
MIN F(W)
subject to:
(4)
H 5 (Z,W) >
W >
and {C, C, X and Y) solve (5):
MIN G(C)
subject to:
(5)
H(X,C,C,X,Y) >
C,
where H(X,C,C,X,Y)
H 3 (Z,C,X) > 0,
>
substitutes
H 4 (Z,C,X) > 0,
Y >
X,
C,
constraints
the
and H 6 (Z) >
=
H 1 (Y)
0,
H 2 (X,Y)
=
0,
0.
The first lower problem (4) corresponds to the nonconvex functions formulation and the other
corresponds to the equilibrium problem.
lower problem (5)
Because of the NP-hardness of BLP,
it
more convenient to take advantage of
is
the structure of the problem and solve it as two separate smaller problems.
As
we
previous
situation
(5);
the
mentioned
in
our
discussion
the
BLP
corresponding
different
for
the
section,
is
travel
cost
functions
BLP
of
of
to
(4)
is
corresponding to
the
cost
improvement
system
and
functions
difficult.
the
the
in
the
However,
the
equilibrium problem
user
are
very similar
9
[Ben-Ayed,
Blair and Boyce 1988]. The coefficients of the two functions always
have the same sign,
and are even fairly proportional;
therefore the use of BLP
heuristics is safe for this second part.
tries to identify a solution that
Our approach to satisfy (5)
good for
is
both the upper and lower objectives by using a convex combination of the two:
tG(C) + (l-t)G(C)
so that we are looking at the problem:
MIN Z F(Z,W) + tG(C) + (l-x)G(C)
such that {Z, C, C, X and Y)
solve:
H(X,C,C,X,Y) >
Y >
X,
C,
C,
Z,
and such that (W)
solve:
(6)
MIN F(W)
subject to:
H 5 (Z,W) >
W >
In general,
polyhedron
BLP solution (x ,y
a
(Ax°
+ By
lower solution y
x
.
In our case,
be the case that,
the
to
<
x
b;
is
)
y° >
,
0.
0};
called feasible if it belongs to the
and satisfies
the optimality of the
lower problem when the upper variable x is
the
fixed at
for a feasible solution to (6) to be feasible to (2),
for the given choice of the upper variables
lower variables
clearly be the case
to minimize
are such as
for
t
=
for some
the same will be true for any
lower objective
for which this
x
<
t
T
<
t
is
.
G.
This will
as
pos-
feasible,
then
small
the case.
the solution to (6)
,
the values of
Z,
To make the upper objective as
0.
sible, we want to find the largest
We will assume that if,
the
it must
is
This is not always true, but counter-
10
and a simulation suggests this
easy to construct,
examples are not
Granted this assumption, the optimal
assumption.
which
by a binary search,
much quicker than
is
call it
t,
x
a
safe
can be located
,
grid search,
a
is
as
used in Bard
[1983].
For a given
(6)
t,
is still a BLP
with too many lower variables to be solved
We approach this problem by trying to identify links
in a straightforward way.
on which no improvement will be made. For those links, we replace the nonlinear
improvement cost function f(Z) by aZ + B, chosen so that aZ +
We
Z.
are
temporarily assuming that the
costs
for
those
< f(Z)
/3
links
we
for all
think are
unlikely to be improved are less than they really are. This produces a smaller,
more tractable, problem.
If the solution to this smaller problem does not use a
cheaper cost it certainly would not want an improve-
link with the simplified,
In the case where the small problem does recommend an
ment with the real cost.
improvement based on the simplified cost, we must solve a new problem with the
real cost as part of the formulation.
Thus we solve a sequence of problems in
which links with real costs are added when it seems necessary, but not otherin practice the process terminates
wise. Fortunately,
a
(no new links added) with
relatively small number of links showing improvement.
Let us call S the set of the links having convex or linear cost improvement
cost
functions,
functions,
from
T
belonging
to
set
of
links
the
the subset
T
links
as
candidates
for
their three-piece improvement cost functions f(Z)
Let
(Z 1
,
X1
,
functions f(Z)
C
1
,
in
C
1
,
(6)
Y1
,
improvement
for which
Z
containing the remaining elements of
considered
are
nonconvex
having
the subset of T containing all
and T
0,
the
T
W1 )
be
the
improvement
cost
are different
T.
and
The
links
must
have
included in the formulation.
solution of the LP where all nonlinear
are substituted by aZ +
J3
,
for each link of T.
the following BLP problem with relatively few lower variables:
We obtain
11
MIN Z
Z
+
agS F a (Z a ,W ma )
such that
Z
+
a£T1 F a (Z a ,W ma )
C,
[Z,
Z
a£T2 (a Q Z a +l3 a ) + xG(C) + (l-x)G(C)
X and Y} and {W ma
C,
,
m=l,M a
,
solve:
aeS}
H(X,C,C,X,Y) >
Y >
X,
C,
C,
Z,
H5a (Za> Wma) *
W ma *
and such that {W ma
MIN
E
(7)
°
m=l,M a
,
Z
aeTl
°
acT 1
,
}
solve:
m=l,Ma W ma
subject to:
W ma
"
Z
a -
W ma *
where
Ma
number
the
is
function of link
cost on link
Let us call (Z 2
link
j
such that
X
2
C
,
2
a
having
Z
C
,
2
,
Y2
W 2 ) the solution of (7).
,
does not belong to T
has to be subtracted from T
links
improvement
cost
Hc a gives the corresponding constraints.
,
j
piecewise
the
in
F a is the upper objective function of nonconvex improvement
a;
and,
a;
°-
breakpoints
of
"^ma
?
a
,
added to T
positive
1
,
")
and Z-
different from
is
then
0,
j
and (7) has to be resolved until all
elements
are
If there exists a
of
T
1
in
,
which
case
the
exact
solution of (6) is found.
We have constructed from the main BLP problem (2)
formulations:
Problem (7)
solution
is
a
is
parameterized BLP (6) and a BLP with few lower variables (7).
solved to give a feasible and optimal solution to (6).
user-optimized
(i.e.
t
can be tried.
decreasing the value of
For any solution (Z 1
two different values.
the
solves
fixed at its optimal value for (7)),
value of
two other secondary BLP
lower
LP problem
(5)
If this
when
Z
is
then it is feasible for (2) and a higher
Otherwise, we continue our search for the optimum by
t.
,
X1
,
C
,
C
,
Y1
,
W 1 ),
we have to distinguish between
The first value is that of the upper objective function
12
equal to F(Z,W) + G(C);
of formulation (2),
is
the actual cost
us
call
includes
actual
improvement
improvement
cost
cost
on the other side,
actual travel cost,
XJ
the
fCZ 1 )
into
Z
1
The first value
1
feasible solution;
it
1
of Z;
let
system travel
1
actual
the
in
or nonconvex.
The
obtained after we solve the lower LP
costs;
CJ
,
other words,
in
YJ
,
A1
and plug
)
equal
is
to
corresponds to a potential optimal, but not
allows
ideal
find
to
problem (2). The second value A 1 corresponds to
optimal, solution;
Z
convex,
obtain the solution (XJ, C J
,
convex system travel
the
+ G(CJ).
necessarily
fixed at
Z
is
Z
and actual
obtained by plugging
is
improvement functions f(Z), whether they are linear,
problem (5) with
The second value
.
incurred by the system given a specific value
A1
.
actual
The
cost.
A1
it
1
let us call it
a
solutions
to
the BLP
feasible, but not necessarily
it allows to find incumbent solutions
for (2).
The proposed algorithm is an iterative procedure that tries at each iteration
to
reduce
the
between
gap
solution
ideal
procedure terminates when the gap is brought below
the
number
gives
a
of
iterations
general
exceeds
description
provided by Appendix
a
fixed
the
of
limit
iterative
incumbent
and
a
N.
desired value
The
following
procedure;
more
e,
x
A.
-
Solve (6)
Test whether solution of (6)
is feasible to (2)
B.
Inf easib! Le
Feaj sible
1
r
Li 1
.
Mn1.
iomn*
i
t
i
1«-~„_
iai ej
6
.
PO
\j£.
.
M«1,>.e
~
n-
_~,.11
M .~
oiua
iici
chart
details
(Using Binary Search)
Start with t=1
or when
flow
B:
Searching for
The
solution.
are
13
in which (6)
The most difficult step in the algorithm is step A,
solved. Using the concepts explained above in this section,
is
has to be
the solution of (6)
obtained as shown in the following flow chart:
Fixed
t
Start with T X =0, T 2 =T
Al.
A2.
Solve (7)
Test whether new elements
should be added to T
A3.
New elements added
A/.
M~r»
Tl
J.
Several
cient.
improvements
can be
r^r.T
new
K
T2
(
C\
;
added
r.~1,r~^
added to the algorithm to make
it
more effi-
The starting ideal and incumbent solutions can be found in a more effi-
cient way.
A better ideal solution for (2)
x=l
obtained after substituting at
there
,
No element?
is
no situation better
can be obtained when solving the LP
all nonlinear functions f(Z) in (6) by ccZ+B;
for the system than having cheap improvement and
system-optimized flow.
Also,
problem is solved as
BLP must be lower than the actual cost incurred when
simpler
technique
a
is
complex techniques.
used;
the
otherwise,
Therefore,
to
cost
actual
incurred by the system when the
there
obtain
an
is
no
justification
incumbent
solution,
for
we
can
a
using
just
solve the problem as an LP; we approximate improvement cost functions by their
best linear fit,
(Z°,
X°,
C°,
C°,
ignore the lower objective and solve the problem.
Y°)
the obtained solution,
A
is
Let us call
the value of the starting
incumbent solution. Using those techniques makes the gap between starting ideal
14
solution and starting incumbent solution as
low
as
instead of the
%
3
111
%
found earlier in the previous section.
since the procedure is iterative,
Moreover,
more efficient to use the
is
iterations in order to make the next one easier
information from the previous
to solve.
it
One way to do this is to introduce to the problem the constraint:
L < F(Z,W) + G(C) < U
where L and U
the
are
values
current
the
of
and
ideal
incumbent
solutions,
respectively.
the solution of step
Similarly,
assume that we solved (7),
Let us
W
9
may require solving (7)
2
obtained the solution (Z
more than once.
9
X
,
9
C
,
9
C
,
9
,
Y
9
,
and have to add a new element and resolve (7). We use the fact that the new
)
cost will be at most as low as the value we got for the objective when we used
cheap linear improvement costs (a-Z-
9
+
J3-)
for the links
to be added to T
j
1
;
which gives a lower bound for the solution. An upper bound can also be found by
noticing
that
incurred
by
new
the
the
system
cost
is
at
for
an
added
least
capacity
between upper and lower bounds, when step
+ Bj)
-
high
as
2
as
equal
actual
the
9
to
Z
.
has to be repeated,
cost
The
9
)
difference
E^ti
is
f(Z
(
a i^i
9
f(Z 2 ).
Finally,
feasible solution to
a
a
BLP problem
obtained by finding the
is
optimal solution to the lower problem after substituting values for the upper
variables.
In our
any optimal solution (Z 1
case,
X1
,
,
C1
,
C
,
Y1
,
W 1 ) of the
BLP problem (6) satisfies the optimality requirement of W 1 to the lower problem
(A);
problem (2)
(5).
(Z 1
therefore
if
it
Even when
solution (Z k
,
Xk
C1
,
,
C1
Y1
,
W1 )
,
feasible
a
is
solution of
the
BLP
satisfies the optimality requirement for the second problem
the
,
X1
,
Ck
solution
,
Ck
,
Yk
,
(Z 1
,
X1
,
C1
,
C1
,
Y
,
W1)
is
not
feasible
a
new
W k ) can be obtained, with feasibility guaranteed.
15
It
known from the theory of Linear Programming
is
Chapter 10 pp.
1983,
x >
solution
generate
x
of
LP1,
solutions
feasible
b'
x > 0),
,
where b and
Ax = b,
(MIN ex,
are different,
b'
of LP2 that has the same nonbasic variables
basic feasible solution x
optimal
that given two LP problems LP1:
Ax =
(MIN ex,
and LP2:
0}
148-168]
instance Chvatal
for
[see
is
for
solution
optimal
an
BLP
problem
(2),.
of
LP2.
which
is
as
any
the
Therefore
to
equivalent
to
generating optimal solutions to the lower problem, we can solve BLP problem (7)
with the additional
(XJ,
CJ
,
CJ
YJ
,
)
requirement
that
all
nonbasic variables
of the lower problem (5) with
slack and surplus variables be locked.
Z
fixed at
Z
1
of
the
solution
and all nonbasic
This is added to the algorithm as shown
in Appendix B.
Regarding the reliability of the algorithm, there are no heuristics involved
finding
incumbent
the
solutions.
and
the
ideal
algorithm for those solutions
are
exact.
in
the
solution
However,
it
it
is
gives
is
the
actual
As
long as
optimum
values
the
within
possible that one finds bad examples
the convergence is not satisfactory;
The
by
the
algorithm converges,
the
accuracy
for this
in such a case the
given
desired.
algorithm where
algorithm ends with
a
large gap between upper and lower bounds, but does not give a wrong solution.
3.
Solution of the Real World Problem
When trying to solve the BLP problem (7) we faced a major constraint, since
we could find no reliable BLP software that could handle problems this
Fortunately,
(7)
can be transformed into an equivalent Mixed Integer Program-
ming problem (MIP),
to the number of
large.
with the number of zero-one variables in the latter equal
lower variables in (7).
of MIP does not mean that it
is
It
should be emphasized that the use
easier to solve.
MIP and BLP are both NP-hard
problems and are equally difficult to solve; however, MIP is older and benefits
:
16
availability of good software to solve it.
from the
Zero/One Optimization Method
Programming system.
1986 by R.
the
at
Marsten
E.
the
extension of the XMP Linear
Marsten from the Department of Management and Information Systems
University of Arizona.
[1981].
The
Center
description of the XMP
A
subroutines
for
ZOOM
of
to be
and
XMP
are
library
used
is
by
our
given in
program
The computational study,
solved.
to be
was conducted on the CRAY X-MP/24 supercomputer of
presented in this section,
National
an
is
ZOOM and XMP are both written in Fortran 77 and revised in
whenever an LP or an MIP needs
the
which is
(ZOOM),
One such package
Supercomputing Applications
at
the
University
of
Illinois at Urbana-Champaign.
First, we show how to transform BLP (7) into MIP. The constraint:
W m = MAX{(Z
is
-
q m ),
0)
equivalent to:
either: W m =
which can be equivalently restated,
-
Z
qm
or:
,
Wm =
after introducing zero-one variables V ma
,
as
Wm
-
Z <
-q
m + MV m
,
-W m + Z < q m + MV m
(8)
Wm *
where M is
a
large number (it
"(l-V,
is
"W
m *
^-V
sufficient to make M larger than the maximum
possible road improvement).
The formulation of (7) as MIP is obtained by replacing the lower problem in
(7)
by the constraints
(8)
for each
link
a.
trivial because of the nonnegativity of W
m
;
The constraints W m < M(l-V m )
we take them out,
following problem to be solved whenever we need to solve (7):
are
and obtain the
17
« IN Z
Z
+
aeS F a( Z a' W ma )
such that
E
+
aeTl F a^a' W ma )
(Z,
C,
aeT2 (°a Z a +6 a)
and {W ma
X and Y}
C,
Z
,
m=l,M a
,
+
*
G
^
+ (l-t)G(C)
solve:
aeS)
H(X,C,C,X,Y) >
C,
Z,
X,
C,
Y >
H 5 a( Z a> W ma) *
(9)
and such that {W ma
"
W ma +
W ma
Z
Z
"
m=l,M a
,
a
"
aeT
)
solve:
MV ma * ^ma
"
a
,
MV ma * "^ma
*ma + MV ma * M
V ma
The technique used to transform (7)
used
by
Fortuny-Amat
McCarl
and
(0,
e
1).
into MIP
The
[1981].
doubled the number of zero-one variables
is
different from the technique
use
of
this
latter
would
have
(requiring a zero-one variable
for
each lower variable and a zero-one variable for each lower constraint).
The large size of the problem (2)
included in the formulation.
of routes
be used,
mainly due to the high number (2,095)
Since not all those routes are going to
the problem can be simplified by keeping the most attractive ones and
discarding
routes
is
the
rest.
We
will
solve
the
problem with
the
reduced
number
(we refer to this problem as the small version of the problem),
of
and we
solve also the original problem with all routes included; next we will compare
the solutions of the two problems.
The attractive routes are found by solving
both the user equilibrium problem (5) and the system equilibrium problem (same
as
(5)
except that G(C)
is
replaced by G(C)) as we increase the values of the
entries of the trip matrix to make the users and the system choose their next
desirable
routes.
When
solving
improvement cost because of
its
the
system
equilibrium,
effect on the routes
we
choice.
vary
also
the
The accumulated
number of routes chosen by both the system and the user reaches 389 as the flow
18
fluctuates between the original trip matrix and three times that matrix. Those
averaging more than three routes per origin destination pair are
routes,
389
the ones to be included in the small version of the problem as compared to the
2,095 included in the large version.
About
minutes
15.25
space
storage
computing
CPU
time
were
and
required
version of the problem. At iteration
million words
1.4
0,
obtain
to
solutions.
actual
incurred by the
in the
flow
solution
the obtained gap is
Section
in
2
system
(namely
1
257,037.230)
with T=l,
the MIP
obtained
shows
small
111.36 %;
starting
TflAX
as
13
anc*
-'-
new
a
T
MIN
incumbent
= ^'
When we
the list of
1,
for improvement, we use the solution as a new better
the gap decreases to 3.03
chart),
the
better
get
to
first step (A1-A2 in the flow chart) of iteration
ideal solution;
of
word)
we solve the problem as an LP and we use the
Then we start the iteration
links that are candidates
the
proposed
At the same iteration,
cost
solution.
find,
techniques
the
use
we
solution
the
per
we compute starting ideal and incumbent
solutions using the ordinary techniques of BLP;
next,
bits
(64
is
(9)
that
%.
In the second step
(A3-A4-A2 in
solved with only six integer variables.
no more
integer
variables
need to
be
The
added;
a
slightly improved ideal solution is found for the problem (new gap equals 2.93
%)
.
the
The feasibility of the solution is tested in step
flow
solution
is
not
user-optimized.
obtained when
We
testing
generate
feasibility;
(B
3
feasible
a
a
new
in the flow chart);
solution
incumbent
using
the
solution
is
obtained and the gap is reduced to 2.57 % at the end of the first iteration. At
iterations
Iteration
2
4,
and
3
where
there
x
=
is
no
.125,
improvement at the end of step
i^x
3,
procedure is stopped at Iteration
steady state;
all
steps.
The
Iteration
final
5
success
gives
=
in
-250
improving the
and t^IN = 0,
when generating
5
a
incumbent solution.
gives
another slight
new feasible solution. The
because the solution seems to be reaching
exactly the same solutions as
a
Iteration 4 in
solution retained incurs a cost to the system equal to
19
256115.006.
The small version of the problem is not hard to solve even on an ordinary
However,
computer.
large version
the
is
more demanding;
required about 4
it
minutes CPU time and 2.7 million words computing storage space for running one
No more iterations were needed because the solution,
single iteration.
at the
end of the first iteration, was the same as the solution obtained by the small
version of the problem at that point; which suggests that our choice for the
attractive routes was successful.
Table
1
gives a summary of the intermediate and final results related to the
solution of the real world problem; the complete presentation of those results
is
provided
Appendix E
by
in
Ben-Ayed
includes
The Table
[1988].
only
the
iterations and the steps resulting in an improvement of the gap (increase of
the ideal solution or decrease of the incumbent solution).
the table that,
is
as
is
It
can be seen from
usually the case for iterative algorithms, most progress
made during the first iterations while convergence is slower as the number
of iterations
increases.
The first two columns of the Table are very different
from the others because of the instability of the "maximum" variables W in the
absence of the corresponding
lower objective
function;
the negative value of
the improvement in the first column is a consequence of this
fourth
row of
the Table specifies whether
when a solution
is
incumbent,
incurred by the system (A 1 ),
just
to
column,
the
value
of
the
the
rows
5,
a
solution is
6
and
7
ideal
function
(0
1
).
the solution has a tendency to stabilize;
or
Starting
The
incumbent;
cost
refer to the actual
and when the solution is ideal,
objective
instability.
those rows refer
from
the
the numbers of used
third
links,
used routes and improved links are almost the same in all six last columns.
Several
results
can be
inferred from the computational experience and the
solution of the real world problem.
and
accuracy,
the
Concerning the tradeoff between efficiency
equivalence of the solutions
of
the
large version
and the
.
20
small version shows that a complex problem can be simplified without affecting
optimality
the
However
solution.
the
of
simplification must
the
be
done
carefully and efficiently; other simplifications, such as reducing the problem
to a single-level could lead to bad results.
The actual cost incurred by the system when the problem is solved as an LP
exceeds
the
thousands
of
actual
incurred
cost
Tunisian Dinars.
difference as high as
it
when
There
a
BLP
solution
are
two
reasons
supposed to be.
is
First,
used
is
not
for
922.224
by
having
this
the travel cost at stable
flow is a one-piece linear function that does not depend on the added capacity.
However,
if stable flow, which is the overwhelming proportion of the flow,
does
not depend on the added capacity, which is the only instrument in the model for
the
system to influence the user's
system to affect
choice of the routes,
user-optimized equilibrium
the
is
the
ability of the
almost paralyzed.
Second,
the limitation of the added capacity resulted in low improvement costs.
that improvement costs are about fifty times smaller than
be seen from Table
1
travel
introduction of
costs.
improvement
The
costs
It can
did
not
give
the
much
sophisticated
saving
formulation of
BLP
compared
as
to
the
simpler
the
LP
formulation because those costs are already low according to the data of the
problem.
The
availability
ability
of
of
realistic
problem would definitely
the
problem when
BLP
solved
to
data;
lead to
as
a
give
better
providing
a
BLP
results
more
conditioned
is
representative
data
by
the
to
our
larger difference between the solution of
and
its
solution when
solved
as
a
single
level
Another harmful simplification would be the inclusion of strict capacities
on the links,
in
our
even when the overflow is very small on all links, as is the case
solution.
Imposing
strict
capacities
results
in
the
violation
of
Wardrop's first condition and therefore the collapse of the equilibrium, which
is one of the bases of the problem.
Baene
MEDITERRANEAN
Cape Bon
Pwntslkna
(ITALY)
boh
Peleg/t
(fTALYI
MEDITERRANEAN
SEA
100 K.loT^irri
Figure
1
The Links to Bo Improved
.
22
Summary of the Results
Iteration
1
1
1
4 & 5
4
-
-
-
1
2
3
4
1.000
1.000
1.000
1.000
1.000
1.000
1.000
ideal
incumb.
incumb.
ideal
ideal
incumb.
incumb
incumb.
Travel Cost
255328.
273736.
252428.
245152.
245320.
252577.
251724.
251680.
Improvement
-124325
3153.
4609.
4329.
4410.
4410.
4424.
4435.
Total Cost
131003.
276889.
257037.
249481.
249730.
256987.
256148.
256115.
278
279
276
275
276
276
275
275
99
100
97
97
97
97
97
97
18
18
14
14
14
14
14
15
3.03
2.93
2.91
2.57
2.56
Step
Value of
t
Solution
Routes Used
Links Used
Linkslmproved
-
Gap (%)
96.21
111.36
Table
final solution,
In the
used
by
redundant
values
of
inter-regional
the
in
275
inter-regional
cannot
one
network or
entries
capital,
improvements
all
to
predicted by
its
a
major
happen
neighboring
trip
whether
lack
of
matrix.
15
those
are not
are
links
use
is
due
The
study
to
is
low
more
those are shown on the map in Figure
be
to
tell
their
the
in
concerned with the links to be improved;
Almost
.125
routes and 97 links are used and 15 links need
traffic;
corresponding
the
<
1
Among the 112 links included in the empirical study,
to be improved.
1.
1
on
cities.
the
This
roads
connecting Tunis,
result
can
the
intuitively
be
simple look at the trip matrix [Ben-Ayed, Blair and Boyce 1988];
the trips originating or ending in Tunis are 56 % of the sum of all the trips
of the matrix.
SETEC
A similar recommendation was made by the Plan Directeur Routier
exists
of
Tunis will be highly congested by the year 2000 if no improvements are made.
In
[
1982
contrast,
pp.
6-8]
stating that
the
roads
corresponding to the
the entries of the trip matrix corresponding to regions
such as the
.
23
which results in no improvement even for unpaved
southern ones are very low,
roads
The Links to
Link
1
3
4
6
10
12
14
19
20
26
28
30
73
77
Improved
Flow
Existing
Capacity
Added
Capacity
Improvement
Cost
2144
2537
2112
2015
1980
1052
306
1056
2275
2537
2010
1370
1637
1115
1055
814
448
271
447
1173
1637
1370
1370
862
1055
1370
774
899
997
960
1166
604
592
394
297
615
484
1517
2112
1772
2058
90
B<2
Limit on Added
Capacity
887
899
1083
1070
1460
2046
1354
2046
1101
899
887
887
1496
1070
887
84
22
35
609
1101
899
640
147
1250
717
687
89
463
245
362
60
394
156
179
Table 2
In
be
considering the specific link improvement, Table
improved the
interregional
flow (60 % of the total
existing capacity),
gives for each link to
capacity available to this
existing
the
flow,
2
the added capacity,
its
cost and
the maximum added capacity allowed by the formulation. The improvement required
for the
the
link connecting nodes 4 and 29
surface
connecting
this
of
nodes
29
gives
link
and
it
very small;
exactly
which does
28
is
not
same
the
need
a
slight improvement of
capacity
as
the
although
improvement
link
it
is
closer to the capital. All other improvements are more significant and three of
them,
namely those on links
allowed by the formulation.
new
more
capacity;
upgrading
however
them
the
to
3,
decision
and 26,
require the maximum added capacity
In each of those links the flow is higher than the
detailed
three
20
or
maker
study
four
is
lane
needed
highways.
responsible
for
include
to
Link
the
90
system
possibility of
the
is
also
does
not
congested;
add
more
24
although
capacity
possibility
the
has
he
do
to
The
so.
reason
is
that
the
congestion on those roads is less expensive than the addition of more capacity.
4.
Conclusions
BLP remains a little known technique,
rarely used despite all its potential
capabilities. The computational difficulties of BLP may not be the main reason
of
infrequent applications;
its
as Mixed Integer Programming,
a great
BLP is a new model;
advantage
software
of
powerful
its
that
uses
there are other optimization techniques,
that are not easier to solve but are widely used.
effort is needed to help people know it and take
formulations.
structure
data
such
and
Developing
includes
that
efficient
reliable
and
heuristics
several
and
branch and bound algorithms would have considerable effect on increasing the
use of BLP.
The
conducted
study
empirical
research
this
in
demonstrated
that
BLP
is
tractable and can be efficiently used to formulate and solve even complex real
world
problems
network.
problems
such
the
as
design
and
improvement
the
The complexity of BLP should not be
as
single- levels
;
the
solution
a
happen to be far away from the actual optimum,
computation
time
account
special
the
may
not
be
worth
structure of
problem
single-level
might
and the gain in simplicity and
taking
into
loss
in
optimality.
problem
at
hand
is
always possible to solve the
the
the
transportation
a
reason for formulating bilevel
the
to
of
powerful computers available to us nowadays, it
By
and making use of
the
BLP problem and get a better solution than the one obtained when approximating
it
by a single
always
gives
a
level.
For any BLP problem,
solution
that
is
better
algorithm
the use of any related
than,
or
at
worst
the
same
as,
the
solution obtained when ignoring the lower objective and solving the problem as
a
single-level.
In
other words,
the
relative
inefficiency of
BLP
algorithms
does not justify the formulation of bilevel problems as single-levels.
25
Appendix
A.
Definition of the Variables and the Functions in Formulation (2)
F(Z,W) =
E
a=liU2
F a (Z a> W ma ) =
I
a=ljll2 ((b la Z a +b 0a ) +
=
Z
a=l,112
Z
m=l,Ma W ma
^
C a-- 4c 0a k a)
I a ( W ma)
F(W) =
^
a=lj ii2
W)
=
Z
a=l,112 G aC C a)
=
Z
a=l,112
G(C) =
E
a=l,U2 G a (C a )
=
Z
a=l,112 ^a
H 1 (Y) =
is
is
reRod Y r ~ u od
defined as:
for each link
Xa
H 3 (Z,C,X) >
is
H 4 (Z,C,X) >
is
defined as:
Z
"
for each link
is
a:
"
<W X a
+
-
4k a ) * °
ca
"
c
la( x a
+
-
4k a ) +
defined as:
la
"
<W z a
tela
"
^Oa) Z a + <ha
(
c
+ d
la * °
for each link a:
^la^a
a *
defined as:
-
+
-
4k a)
4k a)
for each link
W ma
H^(Z) >
ar Y r ~ °
Ca
C
is
a:
5
r=l,2095
C a * c a< x a +
H 5 (Z,W) >
m=1>Ma (b m+lja -b ma )W ma
for each origin-destination pair od:
defined as:
Z
H 2 (X,Y) =
E
defined as:
"
Z
a
+
"
a:
%,a ^
for each link
m=1 M a
°
>
a:
- q
>
Z„
'a
^3a
3r
Z,
C,
C,
W,
X,
Y >
is
defined as: for each link
Z
a>
X a'
C a'
a:
^a> W ma - °
for all reR od
m=l,M.
:
Yr >
where Cq 3 and c^ a are the slopes of the system travel cost on link a at stable
and unstable flows,
user travel
cost
at
respectively;
c
Qa and c la are the slopes of the cumulative
stable and unstable flows,
respectively;
d la
and d la are
26
intercepts
the
cumulative user,
curve has
link
in
pieces
2
travel
cost
unstable
at
flow
system
the
of
and
the
respectively; M fl is the number of breakpoints in the improve-
function
cost
ment
the
of
and
M fl
a;
when the curve is
is
when it has
2
pieces;
3
bg a
is
linear,
the
1
when the
intercept of the
first piece of the improvement cost function on link a; b ma is the slope of the
piece delimited by q m _^
equal to q3 a
capacity of
and q ma
a
m=l,M a +l, qQ a being equal to zero and q^ a +^
,
the maximum improvement allowed by the model; k fl is the existing
,
link
cost on link
a;
Xa
a;
C_
link
a;
C
is
fl
the system total travel
the cumulative user total travel
is
a
the flow on
is
cost on link
the number of passenger car units (PCU) added to the capacity of link
the maximum of (Z a ~q ma ) and
Y
0;
the flow on route
is
r;
R
j
is
a;
a;
Z
is
a
W_ a is
the set of all
routes from origin o to destination d; u Q(^ are the entries of the trip matrix;
6
ar is binary number equal to
when link a belongs to route
1
r,
and equal to
otherwise.
B.
Description of the Algorithm
When describing the algorithm, we use T, T
and T
1
,
as defined in Section 2:
Iteration
•
Find a current ideal solution,
current incumbent solution and calculate
a
the current gap for BLP problem (2).
•
Call Termination test.
•
Set:
•
Start iteration
i
Iteration
Step
•
1,
x
MAX =
1,
t
min
=0,
1=1.
I.
I
T1 =
Set:
=
<f>,
T 2 = T.
1
Solve
the
functions
LP
by
obtained
aZ
+ 6
after
for
all
substituting
links
to
find
all
the
nonlinear
links
of
improvement
T
that
are
.
27
candidates for improvement.
•
Get the solution (Z 1
•
For every link
subtract
Step
•
X1
,
of T
j
C
,
1
,
C
1
Y1
,
W 1 ).
if the corresponding Zi
,
from T 7 and add it to T 1
j
,
is
not equal to zero,
.
2
Solve the BLP problem (7) with the set of lower variables equal to {W ma
m=1
b m+l,a- b ma < °>
2
>
•
Get the solution (Z 2
•
If there are elements
Subtract from T
Repeat Step
9
asTl )'
5
X2
,
j
C2
,
,
C2
Y2
,
,
W 2 ).
such that Z-
of T
is positive,
i
all elements
j
and add them to T x
2
Else:
The optimal solution of (6) is obtained
End If.
•
If
=
x
1
2
and
value of current ideal solution, then:
>
Update ideal solution and calculate new gap
Call termination test
End If.
Step
3
•
Solve the lower LP problem (5) with
•
Get the solution (X 3
•
If A
9
<
C3
,
C3
,
,
Z
equal to
Z.
Y3 )
value of current incumbent solution, then:
Update incumbent solutions and calculate new gap
Call termination test
End If.
•
If A 2 =
(Z 2
2
,
X2
Set:
x
(or A 2
,
C
2
,
MIN "
2
~
C2
,
Y2
x
T
>
then:
),
,
=
W 2 ) is feasible
(
T
MAX +
T
)/ 2
then:
:
28
Else:
Try to improve feasible solution
MAX =
T,
Start iteration
I.
Set:
x
t
= (x
MIN + t)/2
End If.
•1 =
•
1
+
1.
Termination Test
•
If
I
> N,
then:
Number of iterations exceeded before reaching desired accuracy
Incumbent is the best solution found
Stop
Else if gap
< e:
Incumbent solution is optimal within the specified accuracy
e
Stop
End If.
•
Return.
Try to improve feasible solution
•
Solve
(6)
while
locking
nonbasic
the
variables
surplus variables) of the LP problem (5) solved with
•
Get the solution (Z 4
•
If
<
,
X4
,
C
4
,
C
4
,
Y
4
,
(including
Z
=
Z
slack
and
.
W 4 ).
value of current incumbent solution, then:
Update incumbent solution and calculate new gap
Call termination test
End If.
References
BARD, J. F. 1983. An Efficient Point Algorithm for a Linear Two-Stage Optimization Problem. Operations Research 31, 670-684.
29
BARD,
and J. E. FALK 1982. An Explicit Solution to the Multi-Level
J.
F.
Programming Problem. Computers and Operations Research 9, 1, 77-100.
,
BARD,
J. F. and J. T. MOORE 1987. A Branch and Bound Algorithm for the Bilevel
Programming Problem. Department of Mechanical Engineering, University of
Texas, Austin, TX.
BEN-AYED, 0. 1988. Bilevel Linear Programming Analysis and Application to the
Network Design Problem Ph.D. Thesis. Department of Business Administration. University of Illinois, Urbana-Champaign.
:
.
BEN-AYED 0. and C. E. BLAIR 1988. Computational Difficulties of Bilevel Linear
Programming. Submitted to Operations Research
.
BEN-AYED 0., D. E. B0YCE and C. E. BLAIR 1988. A General Bilevel Linear
Programming Formulation of the Network Design Problem. Transportation
Research B22.
BEN-AYED 0., C. E. BLAIR and D. E. B0YCE 1988. Construction of a Real World
Bilevel Linear Program of the Highway Network Design Problem. Submitted to
Transportation Research
.
BIALAS, W. F. and M. H. KARWAN 1982.
tion on Automatic Control AC-27,
BIALAS, W. F. and M. H. KARWAN
Science 30, 8, 1004-1020.
1984.
On Two-Level Optimization.
211-214.
IEEE Transac-
1,
Two-Level Linear Programming.
Management
BIALAS, W. F.
M. H. KARWAN and J. P. SHAW 1980. A Parametric Complementary
Pivot Approach for Two-Level Linear Programming. Research Report No. 80-2,
Operation Research Program, Department of Industrial Engineering, State
University of Buffalo.
,
BRACKEN
J.
and J. T. McGILL 1973. Mathematical Programs with
Problems in the Constraints. Operations Research 21, 37-45.
Optimization
BRACKEN J. and J. T. McGILL 1974a. Defense Applications of Mathematical
Programs with Optimization Problems in the Constraints. Operations
Research 22, 1086-1096.
BRACKEN J. and J. T. McGILL 1974b. A Method for Solving Mathematical Programs
with Nonlinear Programs in the Constraints. Operations Research 22, 10971101.
CANDLER W. and R. NORTON 1977. Multilevel Programming and Development Policy.
World Bank Staff Working Paper 258, IBRD, Washington, DC.
CANDLER, W. and R. T0WNSLEY 1982. A Linear Two-Level
Computers and Operations Research 9, 1, 59-76.
Programming
Problem.
CANDLER, W. and R. TOWNSLEY 1985. Optimizing Subsidy Payments in Linear
Programming. IEEE Proceedings of the International Conference on Cybernetics and Society 930-934.
,
CASSIDY, R. G., M. J. L. KIRBY and W. M. RAIKE 1971. Efficient Distribution of
Resources Through Three-Levels of Government. Management Science 17, 8,
,
30
B-462-B-473.
CHVATAL,
1983.
V.
Linear Programming
,
Freeman, New York.
DESILVA, A. H. 1978. Sensitivity Formulas For Nonlinear Factorable Programming
and Their Application to the Solution of an implicitly Defined Optimization Model of U.S. Crude Oil Production D.Sc. Thesis, George Washington
University, Washington, DC.
.
FALK,
E.
J.
1973.
A
Linear Max-Min Problem.
Mathematical Programming
5,
169-
188.
FALK,
E.
J.
1977.
Quarterly 24,
FALK,
A
3,
Nonconvex
Max-Min
Naval
Problem.
Research
Logistics
441-450.
J. E.
and G. P. McCORMICK 1982. Mathematical Structure of the InternaU.S. Department of Energy Report DOE/NBB-0025
tional Coal Trade Model
Washington DC.
.
FORTUNY-AMAT, J. and B. McCARL 1981. A Representation and Economic Interpretation of a Two-Level Programming Problem. Journal of Operational Research
Society 32, 783-792.
GALLO, G. and A. ULKUCU 1977.
Mathematical Programming 12,
Bilinear
Programming:
An
Exact
Algorithm.
173-194.
LEBLANC, L. J. and D. E. BOYCE 1986. A Bilevel Programming Algorithm for Exact
Solution of the Network Design Problem with User-Optimal Flows. Transportation Research 20B, 3, 259-265.
MARCOTTE, P. 1986. Network Design Problem with Congestion Effects:
Bilevel Programming. Mathematical Programming 34, 142-162.
A
Case of
R.
1981. The Design of the XMP Linear Programming Library.
E.
Transactions on Mathematical Software 7, 4, 481-497.
MARSTEN,
PAPAVASSILOPOULOS, G. P. 1982. Algorithms for Static Stackelberg Games
Linear Costs and Polyhedra Constraints. Proceedings of the 21st
Conference on Decision and Control 2 and 3, 647-652.
ACM
with
IEEE
,
SETEC Economie et SOTINFOR 1982. Plan Directeur Routier Rapport General Tomes
Direction des
1
et 2, Republique Tunisienne, Ministere de 1 Equipement
Ponts et Chausees, Sous-Direction des Etudes.
.
'
,
K. and J. WINGATE 1976. Allocation of Resources to Offensive Strategic
Weapon Systems. Naval Research Logistics Quarterly 23, 1, 85-93.
SHERE,
WARDROP
J.
ings
,
1952. Some Theoretical Aspects of Road Traffic Research. ProceedInstitution of Civil Engineering, Part II, 1, 325-378.
G.
ECKMAN
INDERV
INC.
JUN95