V
o
HD28
Dewey
.M414
no.
\U^-
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
A COMPOSITE ALGORITHM FOR THE
CONCAVE-COST LTL CONSOLIDATION PROBLEM
by
Anantaram Balakrishnan*
Stephen
WP #1669-85
C.
Graves**
June 1985
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
A COMPOSITE ALGORITHM FOR THE
CONCAVE-COST LTL CONSOLIDATION PROBLEM
by
Anantaram Balakrishnan*
Stephen
C.
Graves**
WP #1669-85
*
Krannert Graduate School of Management
West Lafayette, Indiana 47907.
**
A.
June 1985
,
Purdue University,
P. Sloan School of Management, Massacusetts Institute of
Technology, Cambridge, MA 02139.
AUG
9 1985
RECEIYSD
ABSTRACT
We consider the problem of routing LTL shipments from
distinct sources to destinations over
of scale
in
given network. Economies
transportation are possible from the consolidation of
several LTL shipments.
p
a
ecewi se-1 inear
,
We model
these economies of scale by
concave cost function for each arc.
formulate the problem as
a
mu
1
t
Thus,
a
we
-commod i ty network flow problem
with concave, pi ecewi se- 1 near arc costs.
We develop
a
composite
algorithm for generating both good lower bounds and heuristic
solutions, and report on computational experiences for networks
with general and special topologies.
Introduction
Tn
many
'real-world'
planning contexts, decision-makers must
contend with cost functions that exhibit strong economies of
While modeling such problems,
scale.
a
might be acceptable in some instances,
under considerat-on
strategic,
problems,
However,
nature.
in
it
operational,
is
locai
especially
concave
that arc.
of
the problem
or
long-term planning
for medium and
may be essential to account for the concavity of the
consider network problems
a
if
rather than tactical
cost function in the problem formulation.
is
linear approximation
p
i
ecew i se -
1
in
In
this paper,
we
which the cost of flow along each arc
nea r function of
the
total flow along
Such problems arise in transportation planning,
plant location and capacity expansion
computer networks,
planning, production
ann i ng/
p
i
ven t ory management,
and water
This problem can be formulated as
resource management.
design problem over
design
a
suitably defined network;
optimality would necessitate the use of
enumeration procedure.
a
a
network-
solving it to
branch-and - bound or
Our intent is to derive good lower and
upper bounds for the problem rather than to solve it optimally.
We develop and test
a
composite Lagrangean-based procedure that
exploits the special structure of this problem.
We first present
discuss
a
a
formal definition of the problem and
specific application,
namely the LTL {Less-than-
Truckload) consolidation problem,
area,
that motivated our work in this
and briefly review the literature on related minimum
concave-cost models.
We then use
a
I.
agrangean relaxation of the
integer programming formulation of the problem for three purposes
- to
generate good lower bounds, as the basis for
heuristic
a
We describe each of these
procedure, and for problem reduction.
segments in detail before reporting on our computational
experience with the algorithm on
problems
series of randomly generated
a
.
The concave-cost network flow problem (abbreviated as CCNFP)
that we consider involves routing multiple commodities on
its origin node and
We define each commodity by
network.
a
given
its
destination node, and by the flow amount required to be routed
The commodities can be routed
from the origin to the destination.
The cost of
via any number of intermediate transshipment nodes.
flow along each arc is
a
concave
ecewi se-
p
near function.
1
Therefore, for each arc (i,j) of the network,
the set of all
possible flows on that arc is partitioned into several
prespecified 'cost ranges'.
on
The incremental cost per unit of flow
the arc is constant within each of these ranges,
unit cost
is
lower for higher ranges.
Figure
cost curve for flow along arc (i,j).
total
figure will be defined
in
the next section.)
1
and this per
shows
a
typical
(The notation
in
the
The objective of the
CCNFP is to find the flow pattern that satisfies the commodity
requirements at minimum total cost.
Transportation planning
problem
facing
is
a
of
interest.
is
a
Consider,
natural context in which this
for
instance,
the problem
firm that must ship relatively small quantities of
finished goods (or components) from several widely dispersed
warehouses (or vendor locations) to its numerous customers (or
assembly plants).
One distribution strategy is to dispatch the
required flows directly from each warehouse to each customer.
Figure
1
:
Total cost of flow on arc (i,j)
Flow on arc (i,j)
X
This strategy Is usually not cost effective,
because
however,
it
does not exploit the economies of scale in transportation costs.
Typically,
freight carriers offer discounts that increase with the
volume shipped.
of
scale with
a
these economies
We assume here that we can model
concave,
p
ecew i s e -
1
near cost function.
Thus,
the shipping firm may be able to reduce its distribution costs by
routing its small shipments on non-direct routes on which the
benefits of the scale economies are available.
several warehouses may route
through
a
1
For instance,
ess - than- 1 ruck 1 oad
common warehouse (called
a
(LTL)
shipments
consolidation point) at which
These
the LTL shipments can be combined into truckloads.
truckloads can then be dispatched to another common warehouse
(called
a
breakbulk point) at which the truckload
'disassembled'
into the LTL shipments.
is
These LTL shipments may be
reconso 1 idated with other LTL shipments for dispatch to another
intermediate point, or they may now go direct to the customer
destination.
Provided that the intermediate points (e.g.
consolidation and breakbulk) are specified, then the problem
determine the best dispatching policy,
i.e.,
is
to
the route along which
each shipment (or commodity) goes in order to exploit the
transportation economies of scale to the maximum possible extent.
This transportation planning problem (or variants of it)
often referred to as the LTL consolidation problem.
is
We have
described the problem from the point of view of the shipper who
contracts for the services of one or more freight operators.
However,
carriers and freight operators who handle LTL shipments
(such as the package delivery services)
problem.
(See Powell and Sheffi,
1983.)
also face
In
a
similar
this case,
however.
the transportation cost function has
a
staircase structure that
reflects the fact that the carrier moves freight
truckjoads.
It
is
not clear whether
function would provide
a
discrete
in
piecewise-linear
concave,
good approximation to this cost function.
a
Branch-and-bound and dynamic programming are the two main
optimization approaches that have been discussed
literature
the
in
solving the general concave-cost network flow problem.
for
Zangwill
[1968]
exploits
characterization of extreme point
a
solutions for the single commodity problem to develop
dynamic
a
programming algorithm for the single source, multiple destination,
concave cost problem defined over an acyclic network.
[1981] have proposed
al.
dynamic programming procedure for the
a
Soland
concave-cost network flow problem.
general
describes
a
Erickson et
[1974]
branch-and-bound algorithm for the concave-cost plant
location problem;
this algorithm is derived as
a
special case of
method for minimizing separable concave functions over
polyhedral set.
Taha
[1973] has proposed
a
a
branch-and-bound
algorithm that employs
a
special cutting plane procedure for
finding the minimum of
a
concave function over
(that
is
Sodini
not necessarily
[1979] discuss
a
to
convex polyhedron
Gallo and
vertex-following algorithm for finding
local optimum for the general concave-cost mu
problem.
a
network flow polyhedron).
a
1
t
a
commod i ty flow
They report computational results for problems with up
48 nodes,
174 arcs and
5
commodities.
All
these papers deal
with problems having general^ concave-cost objective functions;
and,
with the exception of Soland [1974], all of them assume that
the networks are uncapac i tated
.
In
a
addition to this literature,
there are several other papers that deal with particular
8
application areas (Zadeh [1973],
[1974]
for communication
networks, Florian and Klein [1971], Love [1973], and Swoveland
[1975]
for production planning,
[1979]
for capacity expansion).
[1983],
Powell
[1985],
Lamar
and Fong and Rao
Finally,
Powell and Sheffi
and Lamar et al.
[1983],
and Luss
[1975],
have
[1984]
considered variants of the LTL consolidation problem that we just
These papers represent the problem as
described.
network design problem rather than as
Powell and Sheffi
problem.
a
fixed-charge
concave cost minimization
and Powell
[1983]
a
propose
[1985]
a
heuristic approach that trades off the level of service against
Lamar [1983],
the cost of the load plan.
on
and Lamar et al.
[1984],
solve the LTL problem as an uncapacitated
the other hand,
network design problem without any side constraints;
algorithm iteratively strengthens the
for""'.:
their
luLjon by adding valid
Inequalities to generate successively tighter lower bounds at each
node of the branch-and- bound tree.
Formulation of CCNFP
We define
node set and
a
A
the CCNFP on
is
a
given graph
D(k)
is
to be
e
N
is
(N,A)
e
N
is
(i,j)
where
G
e
from 0(k)
A we
to
D(k)
N
is
the
We have
the origin for commodity
the destination for commodity k,
shipped on
For each arc
=
the set of directed arcs defined on N.
set of K commodities where 0(k)
k,
G
and an amount
for each k=l.
2
...
dj^
K.
assume that the cost function is
piecewi se- 1 inear in the flow on the arc;
that
total flow over all commodities on arc (i.j),
is,
for
the cost
x
being the
is
r
r
F.
C.
+
.
1 J
for
r
X
(M.
t
X
.
1 J
- 1
r
1 J
.th
r
where
M. .1,
,
.
M.
the upper bound on flow in the
is
.
1
1
cost segment on arc
(i.j),
(M..
charge" for the r^^ cost segment,
flow in the r^h cost segment.
F..
0),
=
ij
and C..
We assume M.
(i.j).
the cost function
that
F
for
r
r+1
.
.
=
F
ij
>
r
.
.
+
iJ
continuous,
,r
C
.
-
.
C
iJ
r+1.
.
M
.
iJ
so
.
=
F
non-decreasing,
i.e.
r+
r
> C.
C.
ij
.
db-
where
,
is
R
Furthermore, we assume
r
.
.
IJ
Finally, we assume the cost function
1.
"fixed
the
the cost per unit
is
the number of cost segments for arc
is
is
i J
1
for all
>
.
ij
is
concave and
See figure
r.
1
for an example of the cost function.
To
formulate the CCNFP as
define the decision variables
m
a
kr
x.
xed as
.
i
nt
eger program we need to
the flow of commodity
k
in
i J
the r^h cost segment of arc
(i.j),
and
y.
as
.
a
that denotes whether the r^^ cost segment of arc
zero-one variable
(i,j)
is
used.
We can now formulate the CCNFP as
[CCNFP]
*
s
J
r
.
min
=
t
E
E C^(i: x"^^)
'J
'-^
k
(i.j) r
+
I
Z
(i.j) r
FiiVM
'J
'-^
(1)
.
^
j
r
+
dk
if
i
=
0(k)
-
dk if
i
=
D(k) V k.i
-^
otherwise
(2
10
kr
f
.
<
.
ij
d]^
y
r,k
V
(1
.
j
)
.
V
(
i
.
j
)
,r
(4)
V
(
i
,
j
)
.
r
(5)
V
(:
,
V
(
j
.
(3)
1 J
„r
- 1
r
1 J
1 J
kr
E XiJ
< M
E
<
.
y
.
1 J
y
.
.
1 J
1
(6)
j)
1
v^j
In this
e
10.
u,
x^;
>
the objective
formulation,
is
(1)
,r,k
j
to minimize
(7)
the total
flow cost subject to flow conservation constraints (2) at each
node for each commodity, and subject to constraints (3)
are necessary to define the concave,
function.
ecewi se-
p
-
(6)
that
near cost
1
Constraint set (3) are forcing constraints that ensure
that the fixed charge
F.
.
is
incurred
if
there is flow on the r^h
1 J
cost segment of arc {l,j).
Constraint sets
range for each cost segment on each arc.
(4)
and
define the
Due to our assumptions
that the cost function is concave and continuous,
sets are redundant;
(5)
these constraint
however, we retain them here since they are
not redundant in the Lagrangean relaxation that we will
consider.
The constraint set (6) specifies that at most one cost range is
chosen for each arc.
The solution procedure that we report on here Is based on
Lagrangean relaxation to [CCNFP].
We first use
relaxation to generate lower bounds; we find
the Lagrangean multipliers via
a
a
combination of
procedure and subgradient optimization.
the Lagrangean
"good" choice for
a
dual ascent
We also use the
Lagrangean relaxation to generate upper bounds.
a
We develop
a
11
Lagrangean-based heuristic procedure that transforms
solution to
local
feasible solution to [CCNFP].
a
Lagrangean
a
We can then apply a
improvement procedure to this feasible solution to obtain
a
Finally, we use the Lagrangean
locally-optimal solution.
Based on the Lagrangean lower
relaxation for problem reduction.
bound and the best-known upper bound, we try to reduce the problem
size by restricting the flow of certain commodities on certain
arcs through
"fathoming" test.
a
iiagrang§§n_B?l§2i§li2Il_2l_2CNFP
We form
Langrangean relaxation of [CCNFP] by dualizing the
a
flow conservation equations
(2)
using multipliers {v.}.
gives us the Lagrangean subproblem [SP(v)] as follows
%(k)
=
[SP(v)]:
(wlog we set
'^
z(v)
=
min
(c'".
E
E E
(i.j) r k
s.t.
+
v^
-
v*"
)
^
IJ
(3)
-
(7)
ij
x'"':
iJ
J
'-^
jj
(J
To solve
This
^
k
D(k)
LSP(v)] we first note that it separates by arc;
we have a Lagrangean subproblem
[SPjj(v)] for each arc
To solve
to
[SPi-j(v)]
we note that
that is
(i,j)
satisfy (6), at most one
e
y.
•^
must be set to one.
Consequently, we define [SP. .(v)] as the
subproblem for arc (i,j)
in
which y..
=
1;
that is.
1
A.
12
z..(v)=minrc..
[SP'.j(v)]
s
.
X..
+F..
8)
t
kr
X.J
Vk
dk
<
(9)
(10)
X
.
1
^kr
where C..
js
[
SP
^r
C..
=
k
v.
+
k
let r* be the
v.
-
minimum for arc (i,j).
j
j
(
V
)
if
!1
r
1 J
X
z. .(v)
O.then the solution to
<
r*
=
r
Vr
Vk,
!0
=
.
r*
?i
r*
i J
optimal solution to [SP. ,(v)]
.
and
range for which
else,
kr
.
z. .(v)
If
(11)
given by
is
]
Vk
>
.
J
Zii(v)
=
z. .(v).
z. .(v)
If
> 0,
if
r
then all the x- and y-
variables for arc (i,j) are set to zero.
We can solve
[SP..(v)] by
complexity O(KlogK) where
K
a
greedy algorithm that has
the number of commodities;
is
the
greedy algorithm requires that the commodities be first sorted
increasing order of
kr
C.
.
1
J
and then it sets the values of
in
x
sequentially as follows:
mln
=
x^":
ij
min
=
where
[
x
"*
]
=
{
'
{
du,
k-
[M^
L
dw,
M..
.
ij
Max {0,x}.
-
k-1
x^'^J^
E
ijJ
^^^
k-1
_
L
If
x
Ar
.
.
}
.
}
-
if
„kr
C
.
.
>
1 J
if
c'^^
<
the commodities have previously been
13
kr
C. .,
sorted In increasing order of
solving [SP^ (v)
.
0(K)
is
]
then the complexity for
.
We now show that the complexity for solving
0(KR
+
(i,j).
KlogK) where
To see this,
is
R
the maximum number of
we note that for
tSP|j{v)]
ranges on arc
given arc
a
the
(i,j)
commodities have the same ordering on each cost range.
if
C.
.
< C.
is
ks
C.
ij
c
.
C.
iJ
.
.
for two commodities k,X,
for any other range
from the definition of
kr
C.
.
s
r
?^
is
That is,
for some range r,
then
This follows immediately
.
since
,
1 J
(K
v.
1 J
is
-
i
1
K
,
,
V.)
-
X
X.
(v.
V.
-
1
J
J
independent of the range for any range
r
=
1,2,
...
As
R.
a
consequence of this observation, solving (SPj^j(v)] consists of
first sorting the commodities with complexity O(KlogK),
p
solving [SP..(v)j for r=l,
complexity 0(K).
0(KR
+
Thus,
2,
....
R,
where each instance has
the complexity for
[SPj[j(v)]
KlogK), and the complexity for [SP(v)]
where M is the number of arcs in
We note that
and then
is
is
0(MKR
+
G.
[SP(v)] does not satisfy the integrali_ty
Property (Geoffrion,
1974),
in
that the LP relaxation of
does not necessarily have an integer solution.
problem suggested by [SP{v)] will give
a
Thus,
the next section,
lower bound to [CCNFP]
we discuss two methods
solving this dual problem.
LSP(v)]
the dual
that is at least as good as solving the LP relaxation to
In
MKlogK)
[CCNFP]
for approximately
14
Generate on_of Lowe rBound St o_C CNF
[CCNFP], we
lower bounds on the optimal value of
To obtain
consider its dual problem
zp
[DP]
max z(v)
=
where z(v)
[SP(v)].
^
the solution value to the Lagrangean subproblem
is
our algorithm we do not solve
In
we attempt to find
rather,
procedure, and
an ascent
adjust the multipliers
v
[DP]
to
optimality;
near-optimal solution by using both
a
subgradient optimization procedure to
a
to
increase z(v).
this section we
In
describe the initialization of the multipliers, and the two
multiplier-adjustment procedures.
To
I,
v
in
i
t
ial
i
ze_the_mul
t
i^ej^iers
we note that the multiplier
,
analagous to the shortest path length from 0(k) to node
is
i
using some function of the fixed and flow costs as arc lengths.
For our work,
0(k)
to
we set v.
as
the length of the shortest path from
on graph G with arc
i
length
the number of cost segments on arc
R
R
C
(i.j)
+
R
(^jj/'^^jj)-
and M
p
.
=
^
initial choice of multipliers results in
which the flow on all arcs is zero,
f
where
.
Vi
=
.
ij
f
'^
.
=
ij
choice of
v
E
^
is
x'^':
iJ
then
.
,
,k
a
"here
E
dj^
.
R
This
k
solution to [SP(v)] in
i.e.
,
The optimal value for
is
[SP(v)]
for this
35
where
v
is
.
D
(
k
the length of the shortest path from 0(k)
to D(k)
)
using the arc lengths given above.
To
improve this initial choice of multipliers, we can use an
§§fI§Ql_Br2cedure
intent of which is to change iteratively the
the
,
multipliers so that z(v) increases monoton i ca 1
z
(
V
)
1
y
as
z(v)
E
Z
.
.(V
'
I
(i, J)
^k^D(k)
where Z|j(v) is the optimal value to [SPij(v)l
iterative strategy where we adjust
we employ an
remains unchanged for all
To keep z^j(v)
k
the difference v.
-
For
a
(i,j)
and
d
v
,
.
v
To
increase z(v)
so
that
z
j
j
(
v
increases for some
k.
unchanged, we note that [SPjj(v)] depends on
k
v.,
1
v..
We can write
.
rather than the actual values of
k
v.
and
i
J
particular choice of
k
we will
k
v.
-
v.
determine for each arc
J
(i,j)
how much we can change
1
k
J
particular, we consider decreasing
without changing ZjWv).
'
k
v.
k
v.
-
(effectively,
k
k
k
Increasing v.) so as to be able to increase v^,, ,.
D(k)
We define u.
k
as an acceptable amount by which we can decrease v.
1
u.
.
to
-
k
We
v..
J
indicate first how to
1 J
use the
For
u.
a
.
to adjust
given
k
the multipliers.
suppose we have
k
we determine the adjustment to v.,
1
u.
.
call
for all arcs
it
.
ij
J
defer discussion on how to determine
In
k
6.,
i
Then
(i,j).
such that
ZjWv)
-^
16
remains unchanged and
increased by the maximum amount;
is
v
this is given from the solution to the following LP:
max
D(k
-k
,k
&.-6.
s.t.
1
J
D(k)
on G with arc
0(k)
to
i
,
j
'
just the dual of
is
shortest path problem from 0(k) to
a
k
lengths u.
where we set
i
V
1
=
*0(k)
But this
k
<u..
k
6.
.;
to be
6q(>^j
is
the shortest distance from
zero.
shortest path problem, we then update
v
k
Having solved this
by v
k
k
k
:=
^
^^
"^^^
•
*i
ascent procedure iterates in this fashion over the set of
commodities until no more improvement
in
z(v)
we first note tnat if f..
To determine u..
1
1
k
v.
k
v. will
-
1
k
=
u
change Zi<{v);
k
f
.
for
thus,
k
f
=
.
0,
=
d
u
i J
1 j
,
we set
"^
the determination of
however,
then any
i J
-"
J
When
0.
d.
K
=
J
J
change to
possible.
is
k
^
not
is
J
immed ia te
as
For
=
f
suppose that the optimal solution for [SP^Wv)]
0,
-^
1
occurs in range
,kr
C.
=
.
C.
ij
.
1 J
Zjj(v).
As
+
V
a
k
Since
r.
-
V
k
f
> 0;
.
=
.
1
to
V
J
it
is
easy to conclude that
we may set
else,
f
consequence, an upper bound on
k
u_
unfortunately, though, setting
V
0,
-c'".)
ij
may reduce
[SPjj(v)] may now occur in
z
,•
i
-J
a
(
v
)
to
kr
C
(
.
.
ij
k
u.
=
d
is
k
and decrease
kr
C^.
equ i va 1 ent ly
;
,
setting
since the optimal solution to
range beyond
r.
To correct
for this
17
possibility, we determine
permissible value for
a
u.
by the
.
following procedure:
k
Resolve [SPij{v)] with (v.
a)
-
k
v.)
set equal
to
-C
r
.
.
.
Define Zjj as the optimal value
Set
b)
u
.
=
.
C
1 J
kr
.
_
Zj
j
(
v)
^ij
.
1 J
We can show that
k
we decrease v.
if
1
-
k
v.
by up
k
u.
to
then the
.,
1 J
3
solution value of [SPjj(v)] remains at Zij(v).
second procedure for increasing the value of z(v) by
A
multiplier adjustment
is
subgrad i entogt i_m za t i on
i.
We define the
.
subgradient {w.} for the Lagrangean function by
w
k
E
.
1
where
f
k
ij
.
=
.
T x
^
f*^
kr
ij
.
.
is
.
-
H
f^.
.
if
i
=
0(k)
if
i
it
0(k)
if
i
=
D(k)
solution to [SP(v)].
a
,
D(k)
Rather than use the
subgradient to update the multipliers, we use
a
weighted
'
k
subgradient {w.), as given by Crowder (1976), which
average of previous observations of the subgradient.
is
a
smoothed
For the
current iteration we compute the weighted subgradient by
"k
w
.
:
=
1
where
a
w
k
.
i
is
subgradient
+
a
'k
w
.
1
the prespecified
is
"discount factor."
used to adjust the multipliers by
Now,
the weighted
18
V
k
:
.
=
V
1
where
is
=
e
.
the step size given by
-
z
\
Z
(
V
)
Iwl 1^
the solution value to the dual
an upper bound on
is
|jw||
X
w
e
1
1
z
"k
+
.
J
e
E
k
the Euclidean norm for the weighted subgradient,
is
(0,2)
is
the prespecified step-size multiplier.
Implementation,
[CCNFPJ;
problem zq,
z
our
will be the value of the best known solution to
the specification of
step-size multiplier
a
will
\
the discount factor
be given
in
a
and of the
the section on
The subgradient procedure continues
computational tests.
Iteratively until
In
and
stopping criterion
is
triggered;
this will
also be specified in the section on computational tests.
L§gI!§nS§3n2i3§§d_Heuristl^c_for_CCNFP
conjunction with the generation of lower bounds to
In
[CCNFP],
with
we will
search for good feasible solutions.
We do this
Lagrangean-based heuristic that can be applied for any
a
choice of multipliers
for instance,
v
and its corresponding solution to
we can apply the heuristic after
[SP(v)];
the completion of
the ascent procedure or after any iteration of the subgradient
optimization procedure.
The structure of the heuristic algorithm Is as follows:
St
eg_
1^
:
Construct an initial feasible solution using the
Lagrangean solution corresponding to the current set of
multipliers
v.
i
19
S t.e£_2
For each commodity
:
k
e
K:
there an alternate routing from 0(k)
Is
along which commodity
k
to D(k)
can be routed and the total
cost decreased?
yes,
If
find the best alternate routing,
i.e.
one
which decreases total cost most.
Next k;
which when rerouted results in
Let k* be the commodity,
the maximum decrease in total cost.
Steg_3:
If
rerouting does not decrease total cost for any
Otherwise,
commodity, STOP.
return to Step
reroute commodity k* and
2.
Steps 2-3 represent
local
a
improvement procedure that is
equivalent to the algorithm proposed by Gallo and Sodini (1979)
for networks with general
concave costs.
"alternate routing" for commodity
solution of
k
in
The determination of an
step
2
requires the
shortest path problem from 0(k) to D(k) on
a
G
with
arc costs that reflect the routings of the other commodities.
In
Step
we construct
1
the solution to
[SP(v)]
need to determine
To do this,
Ak
=
{
a
kr
.}
feasible solution that coresponds to
for the current value of v.
path from 0(k)
to D(k)
In
effect we
for each commodity k.
we define
(i, j)
e
A
:
E x^'l
r
where {x.
a
is
>
}
,
-^
the current solution to
[SP(v)].
set of arcs with positive flow for commodity k.
or more paths
from 0(k)
to
Thus,
If
A^ is the
there is one
D(k) defined by arc set A^
,
then the
20
heuristic assigns one of them to
0(k)
to D(k),
(a)
then the path for
is
k
Aj^
defines no paths from
selected as follows:
When the heuristic algorithm is applied immediately after
an ascent procedure,
path from 0(k)
(b)
If
k.
to
commodity
using
D(k).
is
k
u.
.
as
routed on
arc
shortest
a
lengths.
When the heuristic algorithm is applied at an
intermediate stage of the subgradient procedure,
commodity
is
k
routed on the same path that was used
in
the previous application of the heuristic algorithm.
Note that in our implementation we will always generate
a
heuristic solution before using the subgradient optimization
procedure.
Hence,
in
there will always be
step (b)
a
"previous
application."
Problem Reduction
The purpose of problem reduction is to deduce,
based on the
current best upper bound and on the Lagrangean subproblem. whether
or not commodity k must flow on
optimal solution to [CCNFP].
particular arc (i,j)
a
in
an
When we can determine this, we
effectively reduce the size of the problem, and should see an
improvement in the value of the lower bound to [CCNFP];
the
problem reduction may also suggest an improvement to the upper
bound
.
Our basic strategy for problem reduction is similar to that
used to prune the enumeration tree in
procedure.
To test
if
commodity
k
a
must
branch -and -bound
(or must not)
flow on arc
21
we solve the Lagrangean subproblem with the current
(i,j),
multipliers but with the additional constraint
E
x'^':
(12a)
=
''
r
(or E X.J
If
=
{12b)
dk)
the new Lagrangean objective function,
than the value of the incumbent
not)
flow on arc (i,j).
kr
Ex..
r-
or E
(
to
both
=
Thus,
z,
z*
it
then commodity
k
,
is
must
greater
(or must
we can add the constraint
,
(13a)
di,
x"^"^
call
=
(13b)
)
[CCNFP] and its Lagrangean subproblem.
this reduction stems from the fact that z*
[CCNFP] with the additional constraint
is
The validity of
lower bound on the
a
(12).
there is an
If
optimal solution to the [CCNFP] that does not violate constraint
(12),
then we must have z* <
But if z*
z.
then all optimal
z,
>
solutions violate constraint (12), and thus we can add (13) to
tighten the formulation.
We note that the Lagrangean subproblem with an additional
constraint such as (12)
is
easily solved using
the algorithm outlined earlier.
Also,
in
this problem reduction
procedure we exploit the observation that there
solution to [CCNFP]
path from 0( k)
If
in
to D( k
minor variant to
a
which each commodity
k
is
an
flows on
optimal
a
single
)
we know that commodity
k
must flow on arc
(i,j),
then we
22
also know that
on arcs
for
(A.j)
I
i^
A
Also if
i.
Consequently,
i.
then
incident to it,
(t,i)
must flow on arc
k
flow via node
we know that k must
only one arc
for
cannot flow at all on arcs (i.A)
k
A
j
7^
then
(i.j),
if
node
must also flow on
k
or
i
has
(l,i)
second type of problem reduction attempts to determine
whether commodity
must flow through node
k
an
in
optimal
this we solve the Lagrangean
To do
solution to [CCNFP].
1
subproblem with the additionaJ constraints
J
1 J
r
(14)
kr
If
E
T.
I
r
=
x'r.
the new Lagrangean objective function z* exceeds the value of
then commodity
the incumbent z,
finding
useful when, as
is
incident to it;
and/or on arc
(
Jt
,
i
)
This
i.
result of previous reductions,
a
this case,
in
must flow via node
node
incident from it and/or one arc (X.i)
(i.j)
has exactly one arc
k
commodity
k
must flow on arc
(i,j)
.
Similarly, we can attempt to determine whether commodity
must not flow through node
i.
Here we need to solve
a
Lagrangean
subproblem in which we force commodity
k
to flow through node
then we conclude that
k
must not flow through
Again if z*
>
z,
and thus we add constraints
A
(14)
to
k
i.
i
the formulation.
third type of problem reduction is based on the network
topology rather than the solutions to the [CCNFP] and its
Lagrangean subproblem.
commodity
k
Consider the reduced arc set on which
can or must flow;
either from 0(k) to node
1
or
if
for some node
from node
1
i
there is no path
to D(k),
then
k
cannot
i
23
flow via node
(14)
to
1
in
an optimal
the formulation of
Thus, we add constraints
solution.
[CCNFP] and its Lagrangean subproblem.
The various problem reduction steps can be performed once we
have an upper bound
and for any choice of multipliers.
z
Furthermore, we can attempt the various reductions iteratively
improvement is possible.
until no additional
Oy^H^i^ewof _Com20s j_te_Algor i t hm
In
the previous sections we have presented
procedures for generating both
[CCNFP],
a
piece these procedures together into
a
set of
lower bound and an upper bound to
and for improving these bounds.
generating
a
indicate next how we
We
composite algorithm for
a
good solution to [CCNFP],
as well
as
an ex post
assessment of the closeness of this solution to the optimum.
The
composite algorithm consists of seven steps, as given below.
INITIALIZATION
SteE_l:
For each commodity
k
e
K
and every node
length of the shortest path from 0(k)
c^.
+
as the arc
(
f'^./m'^.
length,
i
e
set v.
N,
to node
i
equal
to the
using
)
where
R
is
the number of cost segments on arc
(i. J).
SteE_2:
INITIAL ASCENT
Apply the multiplier adjustment procedure until no more
improvement is possible.
Ste2_3:
INITIAL HEURISTIC
Construct an initial feasible solution from the results of the
Initial Ascent.
Find a locally optimal solution by iteratively
improving the routing of individual commodities.
24
Stee_4:
SUBGRADIENT OPTIMIZATION and INTERMEDIATE HEURISTIC
Apply the subgradient method to update the multipliers.
Periodically apply the heuristic procedure using the current set
of multipliers to find a feasible solution, and then reroute the
commodities to get a locally optimal solution.
SteB_5:
FINAL ASCENT
Starting with the best multipliers found in Step
multiplier adjustment method.
Ste2_6:
4,
apply the
FINAL HEURISTIC
Apply the heuristic procedure using the current set of multipliers
to find a feasible solution, and then reroute the commodities to
get a locally optimal solution.
Step?:
PROBLEM REDUCTION
Using the best multipliers and Incumbent solution, apply the
problem reduction steps until no more reduction is possible.
Recalculate the optimal Lagrangean solution for the reduced
problem.
At the end of
this algorithm,
it
may be possible to improve
further both the upper and lower bounds by repeating steps 4-7.
For instance,
Step
7,
if
there were
a
significant problem reduction from
then it may be possible to get an improved set of
multipliers for the reduced problem by reapplying the subgradient
method.
Indeed, we found in our computational tests that
significant improvement was often possible from doing this.
25
Compu ta t_i ona l^_Resu j^t s
We tested the composite algorithm for the
[CCNFP] on two
types of problems:
(1)
General networks that have arbitrary topologies and
demand patterns;
(2)
Three-layer networks,
in
which origins and destinations
are distinct and do not serve as transshipment nodes,
and every origin-destination path passes through exactly
two intermediate nodes.
For both problem types, we randomly generated
for each of
5
5
problem instances
The number of variables
problem sizes.
in
the
IP
formulations of these problems varied from 168 binary and 1680
continuous variables to 1488 binary and 89,280 continuous
We discuss the computational
variables.
results for the general
networks first.
GENERAL NETWORKS
This type of problem has an arbitrary network configuration
and every node of the given network is
destination.
Transshipment
is
a
candidate origin and/or
allowed at all nodes of the
network
To generate
a
test problem, we first specify the number of
randomly
nodes, number of commodities, and the arc density.
We
locate the nodes of the network on
a
100x100 grid.
We then
randomly select for each commodity
a
unique or i g i n -des t ina t i on
pair.
If
the Euclidean distance between the origin and
destination
is
below
a
given threshold value (initially set at
26
50),
a
new origin-destination pair is chosen.
If
no such pair
exists, we repeat the procedure after Jowering the threshold
distance.
specify such
We
a
threshold distance in order to
increase the likelihood that the optimal route for each commodity
contains more than one arc; consequently,
several commodities are
likely to share arcs in the optimal routing,
problems more difficult to solve.
To
rendering the
select the arcs of the
network, we use the specified arc density as the probability that
we select
the arc connecting each node pair.
problem has
feasible solution, we add
a
destination arcs,
5
origin-
experiments, we tested the [CCNFP]
network sizes,
increasing size.
'direct'
that the
for each commodity.
For our computational
algorithm on
To ensure
labeled GENl to GENS in order of
For each network size,
we generated
5
different
instances by changing only the seed for the random number
generator.
categories
Table
1
shows the network sizes for each of the
.
Table
1:
General Network Problem Size Parameters
5
27
For each test problem we set the demand for all commodities
to
be
the same,
for each arc,
for all arcs.
so
say,
1
demand unit.
specify the cost function
To
we first set the number of cost ranges to be four
The widths of each of the
that the optimal
cost ranges were chosen
solution contains, with high probability,
several arcs that operate in the
'higher'
exploit the economies of scale.
Table
(in demand units)
4
for the
5
2
cost ranges in order to
gives the range widths
problem categories.
Range_Widths_X2n_demand_unitsJ^_for_Generai_Networks
Tabie_22
1
Range
28
We assume the fixed cost F.
.
for the first range in zero for ail
1 J
arcs.
A
preprocessing routine executed at the end of the graph
generation procedure identifies, for each commodity
that do not lie on any path from 0(k)
to
n(k).
k,
all
arcs
This procedure
reduces the upper limit of flow (and the number of ranges,
if
necessary) on all such arcs.
We coded the composite algorithm in FORTRAN on
We did not use any special
computer.
a
PRIME 850
Djikstra's
data structures.
[1959] algorithm was used to find shortest paths in the granh.
initiate the algorithm, we must specify
parameters,
to
To
several control
such as the maximum number of subgradient
iterations
executed, the intial value of the step size multiplier, and
be
so on.
We now describe
these settings below.
(1)
The maximum number of subgradient iterations for the
(The subgradient procdure
initial run was set at 100.
might terminate earlier if either the step size becomes
too small or the Lagrangean subproblem solution is
Subsequent continuation runs of 100
primal feasible.)
subgradient iterations each were initiated if the gap
between the best Lagrangean lower and the best upper
bound was relatively large and If no significant problem
reduction was achieved In the initial run.
(2)
the first run, the step size multiplier X was
We do not adjust the step size
initialized to 2.0.
multiplier during the first 20 iterations or until the
first improvement in z(v) is realized, whichever occurs
earlier; thereafter, the step size multiplier was halved
whenever the Lagrangean objective function value did not
In
improve for 10 consecutive subgradient iterations.
subsequent runs, \ was initialized to half the initial
value of the previous run (i.e., to 1.0 in the second
Again, the step
run, 0.5 in the third run, and so on).
size multiplier was not adjusted during the first 20
iterations or until an improvement in z(v); thereafter,
it was halved after 5 consecutive 'no improvement'
Iterations
In
.
29
initiated once every 10
If the initial heuristic
subgradlent iteration s
solution obtained fro m the c urrent Lagrangean solution
solution derived during the
is identica] to the i n t a
previous execution of the he uristic procedure, then the
Also,
loca] improvement alg or i thm is not applied.
recall that, for each commod ity k, some path from 0(k)
to D(k) using arcs wi th pos i tive flow in the current
Lagrangean solution i s se 1 ec ted as the initial routing
Our a 1 gor i thm terminates the heuristic
for commodity k.
procedure if no such path ex ists for more than 10
otherwise, for all
percent of the commod i t i es
commodities that cann o t be r outed in this manner, the
algorithm uses the pr evi ous initial routing.
The heuristic procedu re was
(3)
.
i
i
]
;
After some initial experimentation, we set the
discounting factor for computing the weighted
subgradient equal to 0.2 for all the runs.
(4)
Table
3
shows the average,
minimum and maximum values of
different performance measures for each of the five problem sizes.
We now interpret some of the key figures.
We express all
indicators pertaining to the lower and upper bounding components
of
the algorithm as
obtained
a
percentage of the best upper bound that was
.
Qua]_it]^_of_the_fi_nal,_Lagrangean_l_ower_bounds
On average, over all the 25 problem instances, the final lower
bound as a percentage of the best upper bound was 98,3 percent.
In 4 out of the 25 instances, this gap was zero, while the largest
As might be expected, this gap seems to be
gap was 5.4 percent.
greater for the larger problems.
Ef £ ec tl^venes
sof theLagrangean^basedheur i_s 1
1
c
In all but 4 instances, the Lagrangean-based heuristic improved
On average, the Lagrangeanupon the initial heuristic solution.
based heuristic improved the solution by 2.3 percent while the
maximum improvement was 9.0 percent.
Ef f ec t
iyenessof _the_2n_i
t i_a_l_i
zat i_on_2r2cedure
The initial lower bound as a percentage of the best upper bound
was 46.8 percent on average, with a high and low of 58.3 percent
As the problem size increases, the
and 39.6 percent respectively.
percentage gap between the initial lower bound and the best upper
bound seems to increase.
31
EfffCtiyeness_of_the_iTiul^t2Bli§£_§^iy5jt!n§Ql -Procedure
The percentage improvement in the Lagrangean lower bound due to
the Id t_ia j^ascent phase was markedly higher than the improvement
The
brought about by subsequent multiplier adjustment phases.
improvements
the
initial
maximum
caused
by
minimum
and
average,
ascent and subsequent ascent phases are
Initial
Ascent
Average improvement
Minimum improvement
Maximum Improvement
22.9
13.2
27.7
Subsequent
Ascent
.
1
0.0
0.3
out of the 25 problem instances, no improvement was achieved
The percentage improvement due
the subsequent ascent phases.
seem
to
depend on the problem size.
does
not
ascent
Initial
to
In
in
7
E§l£2I!D§Q£§_2f_lb?_§ilkgIl§^i£nt_2rocedure
the problem size increases, more subgradient iterations are
required before the procedure terminates (due to small step size).
As a percentage of the best upper bound, the average, minimum, and
in z(v) are 28.7 percent, 18,8 percent and
maximum improvements
respectively.
The subgradient procedure
percent,
35,3
consistently increased the Lagrangean lower bound by more for the
largest problem size GENS; however, for the other four problem
sizes, there is no discernible relationship between this
percentage improvement and the problem size.
As
lIf?ct2yeness_of_the_2roblem_reducti_on_2I2cedure
The extent to which problems are reduced depends on the absol^ute
Dl^SQitii^? of ^he gap between the Lagrangean lower bound and the
On average,
best upper bound, rather than on the percentage gap.
of
original
eliminated
10.6
percent
the
3 P£§EE2ces s i ng routine
kr
To measure the effectiveness of the
flow' variables x
1
J
Lagrangean-based problem reduction phase, we use the following
indicator
:
percent of free flow variables at end of algorithm
PR
=
1
-
percent of free flow variables after Preprocessing
This index was 48.5 percent, on average, over all problem
Instances.
This procedure fixed al_l the variables for 2 out of
the 25 problem instances, while it did not fix any variable for
instance
1
.
Comgutati^onalregui^rements
The figures for CPU times in Table 3 show that the computational
requirements grow very rapidly as the problem size increases.
Notice also that, for larger problems, the problem reduction phase
32
In
requ i r es m ore time than the other components of the algorithm.
considerably,
if
for
reduced
have
been
this time could
ret ros pec t
larger pro blems we had specified a maximum limit of 200 or 300
subgra di en t iterations in the initial run, rather than the ]00
For almost all the instances of
limit that we used.
i tera t ion
had
to run the composite algorithm
prob] ms G EN4 and GENS, we
three time s before the improvement in the Lagrangean value began
In all these cases, the improvement in the first
to tap er o ff
two ru ns w as insufficient to lead to any significant problem
Instead,
reduc t i on and thus, entailed substantial wasted effort.
or
300
say
200
continue
for
if we had allowed the first run to
reduction
could
problem
subgra d 1 en t iterations, the total time for
total
time
for
the
2/3rds,
and
have b een reduced by approximately
significantly.
decreased
subgra d i en t optimization would also have
Additi ona 1 savings could have been achieved by employing efficient
sort in g ro utines (required for solving the Lagrangean subproblems
at eac h St age), special data structures and updating procedures,
and an ef f icient shortest path subroutine.
,
.
THREE-LAYER NETWORKS
This,
problem type
is
a
special case of the CCNFP in which
nodes of the network are classified into
(1)
nodes.
4
types
Source
-
Consolidation points, Breakbulk points, and
Destination nodes,
transshipment
(2)
is
permitted only at consolidation and
breakbulk points, and
the network contains only three categories of arcs:
(3)
Consolidation arcs, Conso 1 ida t ion-Breakbu
Ik arcs,
Source-
and
Breakbulk-Des t inat ion arcs.
Thus,
as
the origins and destinations are distinct and do not serve
intermediate nodes.
three
of a
'layers'
of
Every commodity must be transported across
the network.
Figure
typical network of this type.
2
shows the configuration
33
Figure
SOURCE
nodes
2
;
Example of a Three-layer Network
CONSOLIDATION
points
BREAKBULK
points
DESTINATION
nodes
34
This type of network
is
of
interest as
a
model for
consolidating and routing LTL shipments, as described at the
Consolidation points are the nodes at which incoming LTL
outset.
shipments from the various sources are consolidated into
truckloads before being dispatched to the breakbulk points.
the breakbulk nodes,
incoming truckloads are
At
sorted
'broken',
destination-wise, and forwarded {perhaps as LTL shipments) to
their respective destinations.
(i.e.,
p
iecewi se- inear concave cost functions) on all arcs of the
network.
it
We permit economies of scale
Note that,
although only
3
types of arcs are permitted,
possible to model direct source-to-destination links, by
is
This type of
introducing dummy consolidation and breakbulk nodes.
model
is
for operational
very useful when,
requires that no shipment
is
reasons,
the
load plan
transshipped at more than two
Intermediate points.
To generate the test problems,
we specified
(a)
the number of source, consolidation, breakbulk, and
destination nodes, denoted as ng, n^, ng, and n^
The specified number of commodities must
respectively.
be between Max [ng, np] and ns*n[3 to ensure that each
source and destination node is utilized and that all
commodities have distinct origin-destination pairs; and
(b)
consolidationthe density of the source -conso 1 da t i on
breakbulk, and br eakbu 1 k-des t i na t on arcs, denoted as
•^SC- df;p, and dgp, respectively.
i
,
i
we generated a random network on
Then,
a
100x100 grid for each
test problem by randomly locating
-
-
source nodes in the [0,20]
x
[0,100]
rectangle,
consolidation and breakbulk points in the [20,50] x [0,100]
[50,80] x [0,100] rectangles, respectively, and
and
-
destination nodes
in
the
[80,100]
x
[0,100]
rectangle.
35
The selection of the orJgJn-destination pair for each commodity
was random,
except for modifications to ensure that all source and
destination nodes are used and that no origin-destination pair
To generate
assigned to more than one commodity.
the arcs
is
for the
network, we identified for each source node the (dg^nQ) closest
consolidation points and included the corresponding sourceSimilarly, each destination
consolidation arcs in the network.
node is connected to the (dp^nR) closest breakbulk points.
This
choice reflects the characteristic of practical problems in which
each source and destination is typically connected to
a
few of the
closest transshipment (consolidation and/or breakbulk) points.
The consolidation-breakbulk arcs are chosen randomly with
probability d^BFor each commodity,
it
is
necessary to check
the current
if
graph has at least one path from the commodity's origin to its
destination,
order to ensure that the problem is feasible.
in
some commodity
k
does not have any path from 0(k)
appropriate consolidation-breakbulk arc
D(k)
to
,
If
an
randomly added to
is
ensure feasibility.
For our computational
different sizes,
generated
5
experiments, we generated problems
for each of which we
labeled LTLl to LTL5,
problem instances.
Table
in
4
specifies the problem
sizes for each category.
All other problem parameters
structure, and range widths
-
-
for the demand,
variable cost
were identical to those used for
generating the general CCNFP test problems.
We used the general CCNFP algorithm,
without any
5
36
modifications for exploiting the special LTL structure,
the 25 test instances of the three-layer problem.
solution parameters
in
the CCNFP algorithm,
e.g.
All
to
solve
the
the maximum
number of subgradient iterations per run, had the same values used
for solving general networks.
Table
sizes.
of
As
presents the summary statistics for the five problem
5
before,
we consider
the performance of each component
the composite procedure in turn.
All
the improvements
in
the
upper and lower bounds are evaluated in terms of percentages of
the best final upper bound.
Qual^_ity_of_the_fi,na2_L§gI§DS?§Il_i2wer_bound
The average value of the final lower bound as a percentage of the
best upper bound was 99.6 percent while the largest gap was 2.5
percent.
In 19 out of the 25 problem instances, there was no gap
between the final lower bound and the best upper bound, indicating
that the optimal solution had been found in all these cases.
l££§cti^yeness_of_the_Lagrangean-based_heuristj^c_procedure
In all but 3 instances, the Lagrangean-based heuristic improved
upon the intlal heuristic solution.
As a percentage of the best
upper bound, the Lagrangean-based heuristic improved the initial
solution by an average of 6.8 percent; the maximum improvement was
24.8 percent
.
l£f §£liy§0§s sof _the__i n
i
t i.a
1.
i^za t
i^on_procedure
The average, minimum, and maximum values of the initial lower
bound as a percentage of the best upper bound were 89.6 percent,
83.8 percent, and 95.4 percent respectively.
For this class of
problems, therefore, the multiplier initialization method seems to
be very effective.
37
Table
4
;
Problem
Size
No. of SOURCE nodes
Three-layer network problem size para^ieters
LTLl
LTL2
LTL3
LTL4
LTL5
38
TABLE 5
Summary statistics for test runs
on Three-layer Network problems
Problem
class
LTLl
LTL2
LTL3
LTL4
LTL4
SITE
39
IIf§£liHeness_of_the_multi22ier_adjustment_method
before, the initial ascent phase gives significantly better
improvements to the Lagrangean lower bound than the subsequent
The average, minimum, and maximum
multiplier adjustment phases.
values of the improvement in the lower bound as a percentage of
the best upper bound are
As
Initial
Ascent
Average increase
Minimum increase
Maximum increase
45
1.11
7 73
4
.
.
Subsequent
40
COMPARISON OF RESULTS FOR GENERAL AND THREE-LAYER PROBLEMS
The three-layer problems that we tested were obviously easier
to
solve using our algorithm than the general network problems.
The performance of every component of the algorithm was superior
for the three-layer problems.
First,
the preprocessing routine
was significantly more effective for these problems
(75.7 percent
reduction compared to 10.6 percent for general networks).
of
the significant reduction
stage,
the
problem size at the preprocessing
initial lower bound was on average 89.6 percent of the
best upper bound,
problems.
in
Because
as compared
to
only 46.8 percent for general
Since the percentage gap between the lower and upper
bound is relatively small from the beginning,
the ascent and
subgradient phases did not improve the bounds as much for three
layer problems as for general network problems.
gaps were smaller, more reduction was possible,
layer problems required fewer iterations
time).
Finally,
Also,
since the
and the three-
(and hence
less CPU
the Lagrangean-ba sed heuristic procedure also
performed better for this class of problems (improving the initial
heuristic solution by 6.8 percent compared to 2.3 percent).
the final
values,
set of Lagrange multipliers were closer to the optimal
the Lagrangean-based initial
solutions might have been
better than for general network problems.
of
Since
Thus,
the effectiveness
each component of the composite algorithm contributes to
increasing the effectiveness of subsequent phases.
41
CONCLyDING_REMARKS
this paper we developed
a
composite procedure for finding
good upper and lower bounds for
a
special case of
In
problem that Is of considerable practical
combining
mu
a
procedure,
a
1
t
i
p
1 i
er -ad j us tmen t
network design
importance.
procedure with
problem reduction phase, and
a
a
a
By
subgradient
Lagrangean-ba sed
heuristic algorithm, we were able to solve fairly large concaveThis composite algorithm exploits the
cost network flow problems.
special structure of the CCNFP, and the computational results
confirm the usefulness of
a
strategy that combines lower and upper
bounding schemes, which are based on the partial optimization of
some related problems.
Also,
the performance of
such algorithms
seems to improve substantially when the networks have certain
special structures.
See Balakrishnan
(1985)
for more details on
the algorithm and on the computational experiments.
(1985) also explores several
Balakrishnan
possible extensions to the algorithm;
in
particular, he considers the use of
in
the Lagrangean,
a
more complex subproblem
and the development of an ascent procedure that
permits Zjj(v) to change.
ACKNOWLEDGEMENT
This research was supported by
a
grant from the Center for
Transportation Studies, Massachusetts Institute of Technology,
Cambridge
,
MA
42
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?
25
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