OPTIMUM DISTRIBUTION OF TRAFFIC
OVER A CAPACITATED STREET NETWORK
By
Charles Pinnell
Project 2-8-61-24
Freeway Surveillance and Control
Research Report 2 4-2
)
OPTIMUM DISTRIBUTION OF TRAFFIC
OVER A CAPACITATED STREET NETWORK
By
Charles Pinnell
Research Report Number 24-2
Freeway Surveillance and Control
Research Project Number 2-8-61-24
Sponsored by
The Texas Highway Department
In Cooperation with the
U. S. Department of Commerce Bureau of Public Roads
I
October
I
19 64
TEXAS TRANSPORTATION INSTITUTE
Texas A&M University
College Station Texas
I
GLOSSARY
CAPACITATED LINK - Network link with SJ:acific limitations on the total
amount of traffic that can pass over this link.
COPY - Street network having an input traffic volume at a single origin
node and corresponding volume outputs at various destination nodeso
E-TYPE CONVERSION - Form of numerical data output by which a decimal
fraction is exp~ssed as a power of ten. For example 0.1~ 04
equals 0.15 X 104 and O.l5E-04 equals 0.15 x 10"'4.
FREEWAY - High-type street or highway with full control of access and
no intersections at-grade.
ITERATION - Refers to the process of introducing a vector into the
basis of the multi-cop,r linear programming model.
LINK - One-way portion of street network connecting two adjacent nodes.
LINK TRAVEL COST - Time required to move over a given link from one
node to another.
MAJOR ARTERIAL - Urban street with intersections at-grade whose
function is to provide the through movement of tratfio.
MINIMUM PATH ROUTE - Route through a network from a given origin node
to a given destination node that requires the least amount of
travel time.
NODE - Point of intersection in a street network.
STREET NETWORK - System of nodes and links representing an urban street
system.
There are numerous linear programming ter.ms used in the dissertation
which are difficult to define in a brief manner. A reader not familiar
with this terminology is directed to references (10), (11), (20) and (21)
in the reference list.
1
INTR.ODUC TION
One of the most significant problems facing those responsible for
traffic movement in major urban areas is that of peak hour congestion.
"
Plans for adequate arterial street systems to cope with this problem ·
have been drafted but the required facilities are very costly and difficult to provide.
It will require many years to provide the necessary
facilities and during this time severe peak hour congestion will continua unless other mana are developed to cope with this problem.
A freeway is the major traffic carrier in an arterial street system
and is the facility which suffers most from peak hour congestion.
Only
a fraction of an over-all freeway system bas been developed in most urban
areas and this partial system contributes greatly to the peak hour
overload.
The high level of operation provided by a freeway often re-
sults in it attracting heavier traffic loads than it will ultimately be
required to handle when the entire freeway system is developed.
In order to ease the peak hour congestion problem there is a need
to effect maximum utilization of urban arterial systems which include
both freeways and at-grade facilities.
Thus there is a need for oper-
ational control during peak hours which would
11
spread11 the traffic load
over the entire arterial system.
Present operational technology in the traffic field does not extend to systems in a broad sense but rather is limited to individual
facilities.
This approach is inadequate as it does not consider the
2
interaction of the entire system or develop control measures which would
achieve optimum system operation.
Thus there is a great need to develop
a systems analysis approach to the problem of operating arterial street
ne~­
works.
Systems Analysis Problem
Figure l illustrates a typical master plan for a freeway system of a large
urban area. As previously noted
I
such a system exists only on paper and the
actual system presently being utilized may resemble that shown in Figure 2.
Thus a single freeway facility may serve a rather large area of a city and as
a result experience serious peak hour congestion.
The area S shown in Figure
2 represents a typical "corridor area" for which peak hour operational controls
are needed.
If an expanded view of area A of Figure 2 is taken and the combination of
freeway and at-grade arterials is shown as in Figl:lre 3 1 a basic network to
which operational controls may be applied is obtained.
There are two basic
approaches to operational controls which might be as follows:
l.
Traffic could be controlled along the freeway proper by regulating input
and output volumes at the various ramps and by controlling the traffic
on the freeway through signing and/or control measures.
2.
Traffic could be regulated over the entire system which would include
both the freeway and the at-grade arterials. With this approach, traffic
3
MASTER
PLAN
FOR
FREEWAY
Figure
I
SYSTEM
I
_,' ,.---
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\
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\
I
'I
,-------
' ...
........
I
\
'
I
',
\
\
\
\
(
I
...,---
I
I
II
1
I
------~------------~-------\ '>-
\
I
/
/
,"----------~:------~
____ 1_____
I
CBO }
'/
I
~""''Y'?····.-l~~~} , . .
I
'~~i~)Th, -------j~
I
I
\
\
',
I
.... , / . (
;
__ _
/
\
\
I
'
"
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'
',
I
'',
'.._
............
,I
I
I
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I
.,."
,~,
,
I
------------------
I
I
EXISTING
FREEWAY
Figure
2
SYSTEM
,/
""
I
LEGEND
EXISTING FREEWAY
---· PROPOSED FREEWAY
5
'
' '
''
' '\
\
\
\
\
AREA A
STREET
NETWORK
FOR
SYSTEM ANALYSIS
Figure
3
6
might be controlled or routed at or near its origin rather than at the
freeway and the operational controls required would be for the entire
system.
The first technique is already being applied in several surveillance projects
across the country and offers a feasible approach to the problem.
The second
technique would be a more difficult one but would result in better utilization of
the entire system of streets. In order to design a control system for an overall network, however 1 it would be necessary to know something of the desirable
or optimum operations of traffic over the given network.
The basic question that
must be answered initially is as follows:
Given a set of traffic inputs and the corresponding outputs for a street
network (i.e. - origins and destinations) how should traffic be routed over
this network to obtain optimum system operations.
This is the basic problem which will be considered in this report.
Existing Techniques
The need for adequate data from which freeway systems could be planned
and designed has led to the development of traffic assignment techniques.
These techniques deal with trip desires expressed in terms of origin and
destination requirements and a traffic network description defined by nodes 1
links and impedances.
Through the use of high speed digital computers 1 it
is possible to assign traffic volumes to a street network in a manner simulating
7
the daily loading of the system by individual drivers.
The technique of traffic assignment has advanced
r~pidly
since 19 52 at which
time it was stated (28}* that traffic assignment i.s considered to be more of an
art than a science. At the present time numerous sophisticated assignment programs exist which-form a basis for an approach to a systems analysis study of
arterial street operation.
The biggest problem in utilizing existing network assignment techniques,
however, is that they are designed to work with large systems and necessarily
lack some of the sensitivity that would be desired in a critical operational
analysis of a comparatively small system.
The basic procedure in an assignment method is to find a minimum path (in
terms of travel impedance} through the network from a given origin to a given
destination and then to assign the interzonal volumes to this path in some manner.
At the present, assignment methods fall into three general categories (28) which
are as follows:
1.
"All or Nothing" Assignment- all vehicles assigned to the path with the
least travel resistance.
2. Diversion Curve Assignment- total number of trips divided between two
routes depending upon the relative values of travel resistance on the two
routes.
3. Proportional Assignment - total number of trips divided between several
*
Numbers refer to articles in Reference List.
8
routes depending upon the relative values of travel resistance for
the several routes.
All of the above procedures may utilize capacity restraints which serve to
limit the amount of traffic which may be assigned to any given link.
The
manner in which tl}ese restraints are applied varies greatly and is a controversial subject {31).
Requirements for Analysis Technique
A desirable network analysis technique should possess the following
character! sties:
1. It should have the ability to determine an optimum means of distributing traffic over a network as well as the ability to simulate network traffic flow.
2. It should provide the ability to place specific capacity limitations
on any link in the system.
Critical study of existing assignment techniques indicates that they do not
have these desirable characterisitcs. A traffic assignment model developed
by the Traffic Research Corporation for Metropolitan Toronto (25) selects
as many as four alternate routes between a given origin and destination.
Vehicles are then assigned to those routes on a proportional basis which
varies as the inverse of the travel time on each route. A capacity restraint
feature is provided which increases link travel time as a function of assigned
volume. This feature would not enable one to hold link volumes at or below
prescribed values.
9
The procedure used by the Chicago Area Transportation Study (13)
might be termed an "all or nothing assignment." After finding a minimum
time path between an origin and destination, all trips between the origin
and destination are assigned to this path. After all trips in the system have
been assigned,
indiv~dual
link travel times are adjusted, new minimum paths
computed, and interzonal volumes are again assigned.
This iterative process
continues until the system stabilizes. There is no provision for minimizing
over-all system cost or for holding link volumes to predetermined levels.
The diversion curve technique of traffic assignment ( 14) also fails to provide a means of assuring optimum distribution of traffic or of holding link
volume at or below a predetermined level. In this technique a minimum time
path using only freeway links is computed for a trip between a given origin
and destination.
This same information is computed for a trip moving over the
at-grade arterial street system. A ratio of freeway travel time to at-grade
arterial travel time is determined and this value is used to select a percent
assignment to each type facility from a diversion curve.
Linear Programming Approach
The technique of linear programming which optimizes a linear criterion
function subject to a set of linear constraints appears to offer the required
approach to the operational analysis of traffic assignments. Such a method
offers an exact mathematical procedure for determining traffic distributions
which would minimize travel costs and permit restricting volume levels on
10
network links to desired values.
Two approaches to traffic network analysis using a linear programming
technique were found in the literature.
Heanue (23) proposed a linear pro-
gramming model which sought to minize system travel time subject to two
primary constraints. These constraints were:
(a) Total volume assigned to a network link must be less than or equal
to link capacity.
(b) Trips assigned to alternate routes between an origin and destination
must sum to total trip desire between the origin and destination.
The primary disadvantage of this model is that a set of alternate routes
between each origin and destination must be selected manually. Since an
extremely large number of alternate routes exist between each origin and destination in a large system, this choice would be difficult and would introduce
an arbitrary procedure into an otherwise exact method.
Charnes and Cooper ( ll) discuss a class of "coupled" linear programming
models which they propose has application to the capacitated transportation
problem with specific destinations.
This model is discussed further as a "multi-
copy" model in a presentation made by Charnes (8) at a symposiumon the Theory
of Traffic Flow conducted by General Motors in 19 59. Additional analyses of this
model were presented by Pinnell and Satterly (34).
This model satisfies the previously discussed requirements for a satisfactory
network analysis method.
The model associates a network description or "copy"
11
with each traffic origin yielding a
11
multi-copy" description of the traffic input
data.
The transportation problem when formulated as a general linear programming
problem has the disadvantage of requiring an extremely large number of constraint
equations and is not practical for solution by present computers. To avoid this
problem, Charnes states (11) that a change of variable is possible which permits
the large problem to be broken up into two smaller problems. The variable for
this problem is the convex combination of copy extreme points and the solution
becomes one of determing the proper 11 mixture 11 of individual copy extreme
point solutions.
The extreme points for each copy can be determined by a minimum path
procedure and the proper convex combination of these points is found by a
modified simplex technique.
By utilizing this mixing technique, it becomes possible to convert the
linear programming model to a practical computer program. This program could
utilize existing techniques such as the modified simplex program and the minimum
path algorithm. These two techniques could be combined so that the minimum
path algorithm furnishes individual copy solutions and the modified simplex technique method solves the "mixing problem. 11
It is felt that the linear programming model proposed by Charnes fits all the
requirements previously discussed and permits a very critical analysis of optimum
network traffic flow. It was decided therefore to utilize this model as an approach
to the study of optimum distribution of traffic through a capacitated network.
12
Once some basic knowledge is obtained regarding the optimum distribution of traffic through a network then it will be possible to consider the
operational controls necessary to affect such optimum operations. The research
work reported herein consisted of developing a computer program for the linear
programming model and- of utilizing the model to study optimum traffic distribution over a street network.
13
MATHEMATICAL FORMULATION
General Linear Programming Model
The linear programming technique for traffic network analysis as
proposed by Charnes and Cooper (8) has been termed a multi-copy model.
This terminology arises from the association of a network copy with each
traffic origin on the networkG
Figure 4A shows a sample traffic network
with traffic inputs and corresponding outputs (origins and destinations).
Figures 4B and 4C illustrate individual copies associated with the given
network.
By selecting the minimization of over-all travel time as the basic
criterion and utilizing the network copy concept, the network analysis
problem can be formulated as a linear programming problem.
Considering
an individual copy such as copy 1 of Figure 4, traffic distribution over
this network can be formulated as a linear programming problem as followsa
Minimize
where Cj
XCI(
j
subject to
l:
j
(1)
cj xj
= Travel cost (in time units) on link j
=Amount of traffic (number of vehicles) assigned to
link j on copy o(
Lj
eij
xr
=
E{
where ~ij = Incidence number for the j-th branch at the i-th
node (+1 for input, -1 for output, zero i f not
conne oted to node)
(2)
500
600
I
B
9
2
7
10
3
6
II
4
5
TRAFFIC
12
600
500
NETWORK
A
500
• I
8
9
2
7
10
3
6
II
4
5
-' 12
COPY
200
600
300
I
8
9
2
7
10
3
6
II
4
.. ===:I5
12
400
200
COPY 2
I
c
8
SAMPLE TRAFFIC NETWORK WITH COPY DESIGNATIONS
FIGURE
4
I-'
~
15
E7
= Influx or efflux at the i-th node on copy o<..
The previous constraint equations specify the Kirchhoff node conditions
(node input equals node output) for a given copy and assure
that the
desired origin-destination requirements for the copy are satisfied.
A link in each direction (only one in the case of one-way streets)
between adjacent nodes is provided to permit traffic flow in both
directions and to permit study of directional flow.
Similar formulations
can be made for each copy of the network.
Since each of the "copies" utilizes the same network (hence the
name
11
copy11 ) , the traffic flow for the several copies will be superim-
posed on various links of the system.
In order to insure that no single
link of the system will be overloaded, individual capacity limitations
can be placed on any specific link of the system.
This restraint is
formulated as follows:
Lc( x:-<J < Llj
(J)
where ~j =Capacity limitation (number of vehicles) on link j.
The inclusion of a capacity limitation on various branches of the
system provides a "coupling constraint" which ties the copies together.
The network distribution problem can then be formulated as a general
linear programming problem as follows:
Minimize
(4)
16
subject to
~Cij
0{
=
xj
E«
i
(5)
b.j
(6)
0
(7)
J
Lo(
xf
<
«;:::...
Xj-
The structure of this formulation is shown in Figure 5.
Multi-cow Mixing Model
A problem of model size is encountered at once when the distribution
problem is considered as a general linear programming problem.
This can
be readily illustrated by considering a sample network problem.
Assume
(see Figure 5) that the number of nodes (N)
(L)
=300
= 100,
the number of links
and that 20 copies (M) are required to describe the origin-
destination requirements.
If JO links are capacitated, then the total
size of the matrix involved is:
(N
X
M + K) by L X M
or
(100
X
20 + JO) by 300
X
30 = 2030 by 9000.
The magnitude of these numbers readily indicates that it is not
practical to treat the
probl~m
as a general linear programming problem.
In order to reduce the size of the problem from a computational stand-
point, Charnes and Cooper (11) have devised a special method which is
17
~
,.•• I
.
E.
o
I
L..-!_
rn.
j
...
...
.
~I
<-..!-
<
M • NUMBER OF COPIES
N • NUMBER OF NODES
rn.
)
L • NUMBER OF LINKS
K • NUMBER OF CAPACITATED LINKS
PROBLEM
TRAFFIC
LINEAR
STRUCTURE
NETWORK
PROGRAMMING
Figure
5
PROBLEM
18
termed a multi-copy 11 mixing 11 technique.
This technique provides a feasi-
ble approach to the solution of the traffic network linear programming
problem.
By using a change of variable, the unknown in the linear programming
problem is changed from the amount of traffic on a given branch for a
specific copy (xf) to the percent of a given extreme point solution.
Using vector and matrix notation, one may formulate the general linear
programming problem as follows:
Minimize
(8)
subject to
(9)
~)'i~ s:
o(
(10)
d
=1
( o< :::: 1,2, ••• m)
where Co( :::: Cost vector for copy o<. Individual element
cost on link j for copy o(.
7\,r..
cj is
the
j
== Solution vector for copy 0(.
Individual elenant )\.
is the number of vehicles assigned to link j on
copy o(.
Ac( == Matrix of incidence numbers for copy
~.
b~ ==Vector of node influxes or effluxes for copy ~.
Individual elemnt bj
node j on copy q(.
is the influx or efflux at
ko(
r~
==Matrix of structural coefficients ( 1 or 0 ) which
specifies the capacitated links on copy o<.
d
== Vector of capacity constraints.
Individual element dj
is the limiting capacity on link j.
19
Note - Vectors are considered as column vectors unless otherwise
indicated.
For example
c« T
is a row vector obtained by
the transpose of C Of.
Thus~ represents a solution to a given copy P{ and satisfies the
constraints
Therefore, /l o< is one element of the total solution vector ;t where
Assuming that the
.i\o( (individual
copy solutions) form a bo1.mded
non-empty convex set, one can then express
of extreme points as
ittl( as
a convex combination
follows~
(12)
with the condition
~
.,B~ fi~<{J :::
1
(13)
(1.4)
20
where
atA...P
~
~.p
.o
= The
~
-th extreme point solution on copy
~
= Percentage of the ,.t9 -th extreme point solution on copy
s~en
It can then be
binations or
11
o(.
that the total solution}\ consists of convex com-
mixtures 11 of individual extreme point copy
solutio~s.
The problem
Lm
Minimize
co<T /\.o<
(15)
o(=l
(16)
subject to
(17)
(18)
can be reformulated as a
11
mixing 11 problem as follows:
Minimize
(19)
subject to
(20)
=1
(21)
(22)
21
In the preceding fornulation,
A~
~ d...P
e
~
./3 =l
-~1
/-~
has been substituted for
anrl the conditions
(23)
have been incorporated as part of the constraint eq un. tiona.
The
c.onstraints
(25)
(26)
uro rodun,'lunt nnd omitted since E"'-H is a particular
'A.o<. which satisfies
these conditionfl.
'l'he vu.riable for the mixj.ng problem is
-C
d..{J
-
-
Cc< T
e
f:I.,P
JA..tt"f'' with
the cost
•
(27)
The vectors from the constraint equations will be made up of
(28)
whore
L~p =Vector of traffic assignments on capacitated links for
copy c<: and tho .,B -th extreme point solution. An
individual element L~ is the amount of traffic on
capacitated link k on copy c< for the -/3-th extreme
point solution
22
and the unity element from
= 1.
(29)
Solution Technique
The initial tableau for the problem can be set up
~
obtaining an
extreme point solution from each copy ( eot,/1) and ~ adding the necessary
slack or artificial vectors.
in Figure 6.
An example of the initial tableau is shown
This tableau contains the basis vectors for a feasible
solution to the problem.
If the inverse of the basis is obtained (B-1 ), the initial basic
feasible solution (P~) is found as follows:
P(0) 1 = B-l P(O).
(30)
The mixing problem is now in the form of a simplex tableau and
-
optimality checks can be obtained by considering individual "Zj - Cj"
expressions.
Lemke (12)
If a modified simplex method as developed by Charnes and
is used to progress toward an optimum solution, then an in-
dividual vector (Pj) to be checked as
11
come-in" vector may be expressed
as follows:
(31)
The vector W is an "evaluation vector" and is computed as follows:
23
COST
VECTOR
SOLUTION
VECTORS
Cu
C21
C31
P,,
P21
P31
V2,
v3,
CAPACITATED
VII
LINK I
CAPACITATED
V12
LINK 2
,.
V22
V32
• • •
• • •
• • Lie
• • I•
CMt
0
00
PMI
s,
A,
VMI
I
VM2
• • •
• • •
• • •
•
•
•
CAPACITATED
LINK K
COPY
I
VtK
V2K
V3K
I
• • •
-~~
0
SK
l
62
•
•
•
•
•
•
VMK
I
I
I
-
COPY
I
I
3
AK
I
I
2
p(O)
6,
I
•
•
•
I
COPY
.i.
• • •
•
•
•
•
I
•
•
•
•
•
•
COPY
M
I
I
CMI- COST OF FIRST SOLUTION ON COPY M
PMI- VECTOR FOR SOLUTION I ON COPY M
S1
-
SLACK
A1
-
ARTIFICIAL VECTOR
Vii- VOLUME ON LINK (j)
FROM COPY ( i )
VECTOR
INITIAL TABLEAU
MULTI-COPY MODEL
FIGURE
6
24
(32)
where
C
=Total cost
vector.
-
An individual element is Co</1 as previously
defined.
The W vector wiil be made up as follows:
(33)
where
uT
=Vector of
individual
uv
elements associated with the constraint
(34)
and b""o<. =
Individual CV element associated with the constraint
(35)
If an individual vector is checked as a "come-in" vector by con-
-
sidering its nzj - Cj" value, then
(36)
Letting
j
= oi..,B
(37)
and since
Pc<.p = [ Lq ,e , 0, ••• 0,1, 0, ••• 0]
(38)
25
and
W
(39)
T Pc<,e
then
zo(.,s -
-
oc<.,e =
uTL-<.fJ + ~-
-
oc( 11 ..
(40)
Rearranging (40) and using the relationships
(41)
and
(42)
then
(43)
•
For optimality, it is necessary and sufficient that for all o(
(44)
For non-optimality and the indication of a vector (Po<;') as a
vector, it is necessary and sufficient that for some
11 cc:>m~-in 11
c(
(45)
Thus at each stage, the value for
(46)
26
must be obtained.
Since~
is not a function of ,8 , the problem of finding
Max over
..,8 [ (uT K(l(- efT)
ec<.fJ +
o-J-
(47)
for a particular,copy (o<) is equivalent to solving the following:
Max
(UT f{f1(- cf'T) f?o<,l
subject to
Ao<
(48)
e«.,11 = b"(
(49)
eo<.s ~ o
(50)
or
Min
€ ('(,6'
eo(.,a = b o<
(51)
o.
(53)
(co<T - ij! lfo()
subject to
A0(
eCI(.,.~
(52)
The solution to the above formulation, however, is a minimum-path
solution to the copy network with the link costs given b,y
Link Cost Vector
(54)
Thus obtaining a minimum-path solution for a particular copy supplies a
€ «.,11
-
and a value for Ze<.p - 0«,8 •
If this value is greater than zero,
then the column vector for this extreme point may be used as a
vector.
11
come-in11
If the value is less than or equal to zero, another copy solu-
tion may be considered.
If a vector is brought into the basis, then new
values for W T and the "dummy" costs ( co(T - uTI(«.) may- be obtained.
Using these new costs, a newe~is generated and the corresponding
27
vector may be checked for
11
come-:in11 •
Thus the large over-all problem may be broken down 'into two smaller
problems for computational efficiency.
1.
These are:
The solution of a m:inimum-path problem for a given copy network 11
2.
The modified simplex procedure.
One other condition that must be considered is the occurrence of a
positive element in the uT vector.
This condition indicates that a
"slack vector" (a unity vector with a one in the position corresponding
to the positive uT element) would improve the solution and such a vector
must be
entered at this time.
The efficiency of the "mixing" technique is illustrated
ing the problem discussed on page 15.
qy
consider-
Whereas the solution of this ex-
ample as a general linear programmjng problem would require wor.K:tng-wftli ·
a 2030
ey 9000 matrix, the 11 mixing 11 model technique would require only
a 50th order "working" matrix.
The order of the mixing model matrix is
the number of copies plus the number of capacitated links.
Example Problem
In order to illustrate the model and to demonstrate the application
of the mathematics previously developed, the step-by-step solution of a
small network distribution problem will be presented and discussed.
The
simple four-node network with directional links and traffic requirements
shown in Figure 7 will be used for analysis.
The problem consists of
determining the optimum routing of traffic over this network with
28
70
30
10
50
60
NETWORK
COPY
DESCRIPTION
I
COPY
30
50
2
20
10
60
50
~-
LINK
CAPACITY
5
-
LINK
NUMBER
~
-
LINK TRAVEL COST
50
®-
"'G) -
NODE NUMBER
INPUT VOLUME
~- O~TPUT
VOLUME
10
NETWORK DESCRIPTION AND TRAFFIC DATA
EXAMPLE PROBLEM
FIGURE
7
29
capacity restrictions on links 3 and 9.
A step-by-step solution of this problem is as follows:
Step 1 - Determine minimum path solutions for each copy using actual
link costs.
Determine solution costs and solution vectors.
Set up initial basis and P(O) vector.
See Figure 8.
An
extremely high cost denoted by the letter M is assigned
to the artificial vectors.
Step 2 - Find inverse of initial basis and compute UJ T and P0 1 •
Set
up this information in a tableau as shown in Figure 9.
Step 3 - Compute new "dummy costs" for the network and find new
minimum path solution.
(Figure 10).
Note that the over-
load of links 3 and 9 results in these links receiving
high travel costs and that solution e1 2 does not utilize
these links.
"come-in.n
The vector P12 is formed and checked for
The large "Zj - Cj" indicates it would improve
the solution and a standard simplex computation for e indicates its
Step
11
come-in 11 point.
4 - Vector P12 is brought into the solution and the tableau
shown in Figure 11 results.
Dwnrny costs are computed and
the solution el3 is obtained with the corresponding
vector P13 •
The "Zj - Cj" for this.vector is negative,
which indicates it will not improve the solution and therefore copy 2 is now considered.
positive
nz j
-
c j" and 'oTill
Vector P22 has a large
be brought into the solution.
30
FIRST
MINIMUM
P,l
PATH- COPY I -
7-------
3
50
COST e1
= 50 (10) + 10 (6) .. 560
10
60
FIRST
MINIMUM
PATH- COPY 2-
30 VEH
30
20
./
I
I
~I
~1
COST
./
= 30 (7) + 50 (6) • 510
/
/
&-:----50
INITIAL
BASIS
AND
p(o) VECTOR
pll
p21
A,
A2
560
510
M
M
p (o)
0
50
-I
0
40
50
0
0
-I
20
I
0
0
0
I
0
I
0
0
I
INITIAL BASIS
EXAMPLE PROBLEM
FIGURE
8
31
INVERSE
OF
INITIAL
BASIS
0
0
0
0
0
-I
0
0
50
0
_,
50
0
0
-I.
B
•
-
W
= C B
VECTOR
-1
'
= t 560,510,M,M, l
[e-J
"r-M ,-M,560+50M,510+50M l
I
WT--+
P(o) BASIS
P,,
p21
0
0
I
0
I
PII
0
0
0
I
I
p21
-I
0
0
50
10
A,
0
-I
50
0
30
A2
-M
AI
Az
5 60 510 1070
-M
+ +
+
50M 50M 40M
CURRENT COST
INVERSE OF INITIAL BASIS
EXAMPLE PROBLEM
FIGURE
9
32
..----------------------------------------------
FOR
LINK
3
FOR
LINK 9
DUMMY
COST
DUMMY COST
= 6-~M
>+M
= 10- (-M) =10 +M
r-----------,--------------- -------------------------------4
50
10
60
T
p 12
• [ o, o ,t. o J
COST • 5o (8) +so (7) + to(6) • 8to
WP 12 -
C • 560 +5o M -810 • -250 +5o M
I
I
P(o) BASIS P,z
0
0
0
0
0
-I
0
0
50
10
A,
0
-I
50
0
30
Az
-M -M
0
Pu
0
P,z
0
560 510 1070
+
0
+
50M
10
"_!1 • I
-&-
= 3~-50
0
50
+-Z·-C·
J
J
INITIAL TABLEAU
EXAMPLE PROBLEM
FIGURE
-&
0
-250
+
150M 5bM 40M
P,z
•
~5 4 - -
33
I
p(o) BASIS pl3
I
Pzz
p22
0
~0
0
0
~5
Pn
0
0
~
0
0
0
I
I
p21
50
20
I
-I
0
0
50
10
A,
I
0
50
0
~5
I
-~o
0
-M
-5
I
0
pl2
510 1220
810
5dM IO+M
-e-
=
~Y2,/s
-e-
=
1
%o • Ys -
-/5
-330
-75
5dM
T
pl3.
[0,50,1,0]
c, 3
15 (5o) + 6 (lo) • 810
~
WP, 3 -C 13
• -75 +810-810• -75
60
30
20
T
P22 • [ o,2o,o,1 1
C22
•
WP 22
5o(6)+30(8)+I0(20)• 740
-
C22
• -100 +510 +5oM -740
• -330 +50 M
50
INVERSE OF SECOND BASIS
EXAMPLE PROBLEM
FIGURE II
1
-&=!/1•1
2
I
·
34
Step 5 - Vector P22 is brought into the solution and the tableau
shown in Figure 12 is obtained.
Solution
e23
is developed
and the corresponding vector P23 has a positive "Zj - Cj"•
Step 6 - Vector P23 is brought into the solution and the tableau
sho!'lll in Figure 13 obtained.
Solution
€ 24 is determined
and vector P24 is found to be an alternate solution.
Copy
1 is considered again and vector P14 is found to have a
positive ttz j - Cj."
Step 7 - Vector P is brought into the solution and the tableau
14
shown in Figure 14 is obtained. Vector P15 is found to
be an alternate solution as is vector P •
25
Thus an
optimum solution to the problem has been reached.
The final solution to the example problem is given b,y the P0 1 vector
in the tableau of Figure 14 and is as follows:
Taking the indicated percentages of the copy solutions in the final
basis and combining them yields the final optimum solution shown in .
Figure 15.
35
I
I
p(Ol BASIS p23
I
2
~50 /50
~ -~5 ~5
~0
0
0
-~c
0
0
I
-4.6
30
%
810
-5
Pu
20 -%5
Ys
p21
0
%
-&
• 'Y0/!5.
2
~
p22
0
~5
-&
• !.---0/!5 •
y3
I~
25
P12
I
%~
-& • ''Y2y% 5 • 1Y6
0
-~5C -.Yso
108
740 1266
_ _ffi_ - -
4
p23
20
3
/
/
/
E,
./
~
./
0
~
~23
0
N
/
v/
f6\
I
2
30
50
p
T
•
23
c23
•
[ 20, 0, 0, I ]
8(30)
WP23- C23 •
+
6(30)
+
- 4.6(20!
20(6)
+
•
540
740-540
•
108
INVERSE OF THIRD BASIS
EXAMPLE PROBLEM
FIGURE
I2
36
,
,
P(O) BASIS p24
0
1-----
I~
,__
__
~0
0
0
0
-3<3
I~
0
5/
/ 3
·-r--
~
I
~0
-Xo
-0
I ,
0
-I
'----
0
-
-~0
I
0
pll
25 ---15 p21
p23
IJ: pl2
25
-5
810 560 1230
-----
---
50
pl4
pl4
50
0
+-·-1 - - -
~·----·
0
0
5/3
0
I
-~
I
0
I
-&
= IC}f5~/3 =
2/§ -
-&
= 15~5~ =
3/5
0
--
-·--'--·---
[Aj.}ERNA TE
30
SOLUI@ij
20
T
p24 -
(50 ,0,0,1 J
C24 -
1 (3o)
+
WP24- C24 •
50
P 14
C14
T
=[
-
6 (so) • 510
-50 +560 -510-0
5o,o,I,OJ
6(60)
+ 6(50)
• 660
c:JPI4 - C 14 =- -so +slo -6so= +loo
INVERSE
OF
FOURTH
EXAMPLE
FIGURE
PROBLEM
13
BASIS
37
P(o') BASIS pl5
p25
0
20
0
~0
0
0
I~
/;o
0
0
-~5
2/.
5
pl4
50
0
0
0
0
I
I
p23
I
0
I
I
I
2/.
5
1/.
5
pl2
0
I
0
0
I
-~0 -~0
-3
-5
25 pll
810 600 1190
T
p 15 •[o,5o,l,oJ
c1s :.lo(so)+ 6(10). 560
WP15- C15 • -250 + e1o -560. o
60
ALTERNATE
SOLUTION
T
P2s • I 2o,o,o,1 I
c25 •
20 (6) + 6 (30) + e (30). 540
l.JP25- C25 • -60 + 600 -540 • 0
50
ALTERNATE
SOLUTION
INVERSE OF FINAL BA Sl S
EXAMPLE PROBLEM
FIGURE
14
38
0
0
50
<:.o
pll
( 10h5)
20
=
'/,.0
10
4
60
24
50
10
P,2 (1/5)
=
10
2
J.
'12
0
50
pl4 (2;5)
=
0
j1>-
10
4
24
30
20
30
p23 ( I )
20
=
30
30
50
60
FINAL SOLUTION
EXAMPLE PROBLEM
FIGURE
l5
39
COMPUTER PROGRAM DEVELOPMENT.
In order to study the optimum distribution of traffic using a linear programming model, it was necessary to develop a computer program which
would perform the arithmetic operations of the problem. A computer
program to perform this technique was coded by staff m€mbers of the Data
Processing Center of Texas A&M University. This coding was written in
Fortran II with the exception of one subroutine which was written in FAP
(Fortran Assembly Program).
The logic, testing and operation of this
program will be discussed· in the following sections of this chapter.
Program Logic
The basic logic of the program is illustrated in the block diagram
shown in Figure 16. The program utilizes three basic operations in proceeding toward an optimum solution.
These are as follows:
( 1) Computation of a minimum time path through the network.
(2) Modified simplex technique of checking optimality at each stage.
(3) Gaussian elimination technique of bringing new vectors into the
basis.
These three basic steps are repeated until an optimum solution is reached
on all copies.
Initial Program
The initial program was not designed to handle a large network in
INPUT :
NETWORK
DESCRIPTION, CAPACITY
LIMITATIONS AND
TRAFFIC REQUIREMENTS
YES
CONSTRUCT INITIAL BASIS
INVERT INITIAL
B'ASIS
(FIND B- 1)
CHANGE ORIGINAL CAPACIT.tm:D
LINK COSTS TO NEW
DUMMY COSTS
FIND MINIMUM
PATH TREE FOR COPY
« AND TOTAL COST
OF THIS SOLUTION
FIND DUMMY COSTS
FOR CAPACITATED LINKS
ColT= U T Kol
BRING Po~.s INTO BASIS
WITH GAUSSIAN
ELIMINATION
FIND SMALLEST
e = ~~,?Jl
. . . .______"~ - ....,,
PROGRAM
SET UP
SLACK VECTOR
LOGIC
FIGURE16
,l:>.
0
41
order to insure that all main functions of the program could be
~rformed
within core storage of the I.B.M. 709, a stored program digital computer
with 32,768 words of core storage.
The following size limitations were
established for the initial program:
1.
The main matrix was limited to an order of 20, which is determined by the sum of the number of copies plus the number of
capacitated links.
2.
The number of copies was restricted to 15.
3. The network was limited to a total of 400 links.
4. The number of output nodes per copy was limited to 15.
These limitations did not seriously restrict the size of test problem
that could be run and avoided the possibility of exceeding core storage.
The initial computer program had 12 subroutines which performed the
following functions:
SUBROUTINE START - Subroutine to initialize all arrays and read all
input data.
Input data consist of a complete network description,
individual copy flow data, capacitated link numbers and their capacities.
See Figure 17 in Appendix A for flow diagram.
SUBROUTINE TREE - Subroutine to assign traffic on one or all
copies~
It is called initially with key of 1 and then finds minimum paths
and makes traffic assignments on all copies. When called with a
key of 2, the subroutine finds a minimum path for a specific copy
and assigns traffic to this path.
A flow diagram for this sub-
routine is shown in Figure 18 in Appendix A.
42
SUBROUTINE PATH - This subroutine finds the minimum path route
through a network from a given input node to all other nodes.
The
technique used in this subroutine is a modification of the procedure
first described by Moore (32).
in performing
i~s
functions.
Subroutine PATH is called by TREE
A flow diagram of this subroutine is
shown in Figure 19 in Appendix A.
SUBROUTINE KOST - This subroutine sets up the initial main matrix,
the true cost vector and the P(o) vector.
It also computes the
indices which specify the (+) or (-) ones to be placed in the
initial matrix.
See Figure 20 in Appendix A.
SUBROUTINE MULT - Subroutine to accomplish the various matrix and
vector multiplications.
See Figure 21 in Appendix Ae
SUBROUTIIE CHKINT - Subroutine to compute the P(o) Prime and Omega
vector for a copy solution being checked for entry into the basis.
This subroutine utilizes subroutine MULT in this computatione
After
computation of the Omega vector, this vector is checked for a positive element and the need to enter a slack vector into the basis.
See Figure 22 in Appendix A for a flow diagram.
SUBROUTINE LOGIC - Subroutine to compute the "dummy costs" for the
capacitated links of the network and to transfer these costs to the
corresponding links.
See Figure 23 of Appendix A.
SUBROUTHE NEWVAL - Subroutine to utilize the "dummy costs" found
by subroutine LOGIC in computing a new minimum path solution for a
specific copy.
After finding the minimum path and assigning traffic
to it, the solution cost, the solution vector P(j) and Z(j) are
43
computed.
See Figure 24 of Appendix A.
SUBROUTINE OTHVAL- If a check of Z(j) - C(j) in the main program
indicates a "come-in" vector, this subroutine finds the entry point
of this vector into the basis by comparing elements of a computed
P(j) prime vector and the P(o) prime vector to deter.mine a minimum
Theta value.
It then brings the new vector into the basis by a
Gaussian Elimination technique and computes a new cost vector.
Alternately, subroutine OTHVAL is called when the Zj - Cj check
indicates an alternate solution.
This subroutine then finds the
entry point into the basis for the alternate solution.
See Figure
25 of Appendix A.
SUBROUTINE SLACK - Whenever a positive element is detected in the
Omega vector by subroutine CHKINT, this subroutine is called to
bring a slack vector into the basis.
The slack vector comes into
a position corresponding to the positive Omega element.
See
Figure 26 of Appendix A.
SUBROUTINE SHELL - This subroutine computes the convex combinations
(or percentages) of all copy solutions in the basis.
It then com-
bines the traffic volumes from the various copies for final output.
See Figure 27 of Appendix A.
SUBROUTINE PRNT - This subroutine outputs the answers in their
final form.
See Figure 28 in Appendix A.
A flo\ol' diagram of the main control program illustrating the calling
of the various subroutines is shown in Figure 29 in Appendix A.
44
Program Tests
Two basic test problems were utilized in the initial checkout of
the computer program.
The first problem run (Test Problem 1) was the
network problem previously illustrated in Figure 7.
Test Problem 2 con-
sisted of a 12-copy 1 3 5-node network problem with 4 capacitated links.
The network description and data on traffic inputs and outputs are shown
in Figure 3 0.
These problems offered the advantage of a known solution in each
case.
Test Problem l was worked out manually as an example and Test
Problem 2 was solved manually in a paper by Pinnell and Satterly.
These.
problems provided good checkout material and after some relatively minor
modifications to the original program it was possible to duplicate the manual
results obtained for both of the problems.
After initial checkout of the program 1 a second phase of testing was
initiated to ascertain the ability of the program to handle a wide range of
input data over networks of various descriptions and to study the optimum
distribution of traffic.
The work of this phase was accomplished by the use
of six additional problems numbered for reference as Test Problems 3 1 4, 51
6 1 71 and 8.
Test Problem 3 - This problem was devised principally to test the
ability of the program to handle the case where a network link was capacited
at· several different levels. A two-copy problem with four capacitated links
was developed for this purpose.
Figure 31 illustrates the network descrip-
tion and traffic requirements for this problem.
Capacitating a link at several levels is necessary to provide a
piecewise linear approximation to a travel time.
This curve as
45
110
230
70
'~--~--~t~~=2~0--~~
7
roo
-r:=r--v--
60\,
21
6
230
LINK CAPACITY
LINK TRAVEL COST
NETWORK DESCRIPTION AND TRAFFIC DATA
TEST PROBLEM 2
FIGURE
JO
46
80
8
6
r+-----'---"!.
- ' - - - - - - - 1 a >+---·--'--"_L_-----< 1 - - . 20
!50
NETWOR!·< DESCRIPTION AND TRAFFIC DATA
TEST PROBLEM 3
FIGURE
JJ.
47
developed for freeways resembles the one shown in Figure 32.
The piece-
wise linear approximation to this nonlinear curve is slightly in error
but improves in accuracy as the number of linear approximations is
increased.
As one level ·of capacity is satisfied, each vehicle utilizing the
next level of capacity must be given a time penalty large enough to represent its effect on the entire traffic stream.
this consider
th~
As an explanation of
following example:
Assume the following conditions exist:
For volumes from 0 to 100 vehicles per hour
Travel time is 5 minutes per vehicle
For volumes between 100 and 300 vehicles per hour --Travel time is 8 minutes per vehicle
For volumes between 300 and 500 vehicles per minute --Travel time is 15 minutes per vehicle
At 100 vehicles per hour total travel time is 500 veh/m.ln while
at 300 vehicles per hour the total travel time is 2400 veh/min.
Thus vehicles added in the 100-JOO volume range should have a
time cost of 2400-500/200 or 9.5 minutes per vehicle.
The vehicles added in the 300-500 volume range will be assigned a
time cost of 7500-2400/200 or 25.5 minutes per vehicle.
The previous example would be represented by a network description
like that shown in Figure 33.
48
(/)
w
f-
:;)
z
4000
:E
w
....J
u
-
I
w 3000
>
w
'
..J
~
a::
w
a..
LINEAR
2000
f-
(/)
0
u
..J
w
1000
?'
>
<(
0::::
f..J
<(
f-
0
t-
::;..---
/
::;....-
0
0
400
800
1200
I GOO
VOLUME -VEHICLES PER LANE
PIECEWISE LINEAR APPROXIMATION
OF TRAVEL
COST
FIGURE
32
CURVE
2000
49
___
..,I
CAPACITY
= 100
VEHICLES
}----------------~
TRAVEL
TIME
= 5 MIN
2
..
r--~
1 VEI-l
CAPACITY = e>o
TRAVEL
TIME = 0
A
CAPACITY
TRAVEL
=
CAPACITY = 200 VEHICLES
TRAVEL
TIME :: 9. 5 MIN I VEH
8
ex:)
TIME = 0
c
NETWORK
THREE
CAPACITY = 200 VEHICLES
TRAVEL
CODING
TIME = 25.5 MIN I VEH
FOR
LINK
0
WITH
LEVELS OF CAPACITY
FIGURE
33
Since Test Problem 3 involved numerous calculations when solved
manually, a solution was obtained by use of' the LP/90 program of the
Texas A&M University Data Processing Center.
The LP/90 program is a
comprehensive system for linear programming developed by C.E.I.R., Inc.
(7).
In order to utilize this system, it was necessary to provide a
general linear programming formulation of the problem.
is as follows:
Minimize
subject to
This formulation
50
(1)
xll - xl
(2)
X1 - X15 - x2
(3)
x2- x3
(4)
x3 + x
(5)
X4 + X13 ·
(6)
x 5 + x12
(7)
x6- x7
(8)
x7 + x2o - x8
(9)
x8 - x9
14
(10)
X9 - X1o - X11
(11)
X5 - X2o - xl8 + X19
(12)
xl8 - xl9 - X13 - X16 + X17
(13)
xl6 - xl7 - xl4 + xl5
= -50
=0
= 10
=0
=0
= 100
= 20
=0
= -80
=0
=0
=0
=0
(14)
x3
~
70
(15)
xl2
~
20
(16)
X13
~
20
(17)
x14
~
20
(18)
xj
~
0
The criterion function is a summation of individual link costa times
the number of vehicles assigned to that link, which produces a total
travel cost.
Equations (1) through (13) are node equations, equations
(14) through (17) represent capacity limitations and equation (18) imposes non-negativity restraints on the variables.
The results of the LP/90 solution to this problem were identical to
51
those obtained with the Multi-Copy program.
Test Problem 4 - Test Problem 4 was obtained by adding three additional copies to Test Problem 2.
The same 35 node network was utilized
but additional traffic inputs were added.
This provided an extension of
the number of copies considered and considerable overload of the capacitated links.
The network and traffic data are shown in· Figure 3 4.
Test Problem 5 - Problem 5 again utilized the 3 5 node network and its
basic objective was to provide a larger study problem which could be verified by LP/9 0.
The 8 copy, 10 capacitated link problem used is shown in
Figure 3 5.
Solution of this problem by LP/90 presented some difficulty because
of the number of constraint equations involved when formulated as a
general linear programming problem.
For each copy, 35 node equations
are required and the network size introduces 108 unknowns.
The main
matrix then must have an order of 280 by 864. When the capacitated link
restrictions are added the total formulation has 29 1 rows and 8 7 5 columns.
The coding of this information required more than 3000 cards which
presented a very sizeable card punching job.
This job was simplified,
however, by the use of a data conversion program and the IBM 1401.
Test Problem 6 - This problem (see Figure 36) was used to study special
effects of data scaling on the linear programming solution.
Test Problem 7 - This problem provided a larger system than could be
handled by the initial program and was used as a test problem for the
final program.
The problem consisted of 10 copies and 20 capacitated
links and required a 30 by 30 basic matrix.
The network description and
52
r---------------------------------------------------------------------------~
300
t
10
70,
80,
~-.--------------,~-------------",.---,~------·---····-,~------------"'~
~7+-----------~~a~----------~21r------------~2~2L___________~35
~r-------------{f;o}-------------{ 19;+------------{241)-------------\'33
~
H
120~ 4
II
ao"H
H
M
25
32 . . . 700
18
'i~-------....(~J-----------..{t -----------.....J26J------------....{'E
100'-
eo'-.
so"
IIV:2:¥---------~v::l3't-----------rl~6,-.-- -------v-2--~:7~----------~30
,.j--'-1'-------------"1-"4'---·-----·-_____,..._15.....____________..,)28 _ _ _ _ _ _ _ _ __..,2:_:;;9
100
320
NOTE -TRAVEL TIMES AND CAPACITY RESTRAINTS
ARE
THE SAME AS TEST PROBLEM 2
NETWORK DESCRIPTION AND TRAFFIC DATA
TEST PROBLEM 4
FIGURE Jh
1500
600
t
,..
600
600
'
_1ll
I
8
7l
21
22
20
23
35
~
9
6
:---
[1
1200
h
~
tf
fi7\
117\
"1-i
~
. 117\
25
\!V
®
122\
24
H
18)
l@QQJ
\..!V
12
\gJ
t'-
122\
19
1200,
II)
~
\!!!
(22\
10
till
... 4 )
~
~
122\
5
34
[@QQj
~
r-
~
®
32 --.3000
~
\!!!
~
31
r-
[900]
\BI
~
2
~
600
I
600
141__
--·
NOTE -
'
15
(28(______ _______
!
TRAVEL TIME • 37 ON ALL
LINKS NOT OTHERWISE INDICATED
NETWORK
30
27
16)
13)
600
'
---------'
29
1.500
DESCRIPTION AND TRAFFIC DATA
TEST PROBLEM 5
FIGURE
J)
Ul
w
18000
7000
7000
'
7000
~
t
8
7
'\,
21
22
35
20)
23
34
J..-
:.-
00
6
9
r-
C-.
6
13000~
't---1
J..19
~
~
!?
13)
2
~
H
~
1'-
~
~
7000
NOTE -
~
.
'
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LINK TRAVEL TIMES
FOR PROBLEM 5
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NETWORK DESCRIPTION AND TRAFFIC DATA
TEST PROBLEM 6
FIGURE
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55
traffic inputs for this problem are shown in Figure 3 7.
Test Problem 8 - This problem was designed to stlfdY run times for
a large network using the final program.
The network consisted of 2 56
nodes and 960 links 1 which approaches the network size limitations of
the final program.
Problems Encountered
In developing the computer program for the linear programming model,
problems of machine roundoff and slow conversions were encountered.
These
problems affected the accuracy of the output and the machine time required
for problem solution. A detailed discussion of how these problems were
treated is presented in reference (3 5).
Final Program
In order to handle a network analysis problem of the type encountered
in a real system, it was necessary to make provisions in the initial program
to handle much larger problems.
This was done by placing the link data for
individual copy loadings on tape instead of storing it in core.
This provided
more room in core for elements of a larger matrix and permitted the descrip-:tion of larger networks.
In addition 1 a terminate-restart procedure was incorporated to allow
a break in execution on a long problem. With this technique available,
it was possible to divide an extremely long problem into several short runs
on the computer.
56
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TEST PROBLEM 5
NETWORK DESCRIPTION AND TRAFFIC DATA
TEST PROBLEM 7
fiGURE
J7
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57
Except for the above noted changes 1 the logical flow of the prog
is essentially the same.
i.e.
1
The program divisions
a main program and 12 subroutines.
;arne as l
Many of the original subroutines
required no alternation other than changes in DIMENSION and COMMON
statements I which were changed uniformly throughout the program.
The data input method was changed slightly to facilitate an improved
minimum path procedure.
The format of the output remained the same as
for the initial program.
A total of four scratch tapes {three for the initial program) are used in
the normal operation of the final program in addition to the regular inputoutput tapes. An additional tape is required for the intermediate dump when
the terminate-restart option is utilized.
The use of one additional tape
operation by the final program did not appreciably affect run time on the Test
Problems considered.
The limitations of the final program were as follows:
11 Oth
order basic matrix
50
copies
60
capacitated links
1000
network links
An analysis of the core storage required for a problem of the above magnitude
explains these limitations.
A total of 29,151 storage locations are needed
for such a problem compared to the 32 I 768 locations available on a machine
58
with a 32K memory.
The 3616 unused storage locations were reserved for
additional modifications or additions to the program which might later be
necessary.
Run Time
One of the most significant items for evaluation is that of the amount of
computer run time required to obtain the solution to a given problem.
This
run time is related to a number of variables such as size of network, number
of copies, number of capacitated links, and degree of overloading capacitated links.
The most significant of these variables are ( 1) size of the
network and (2) the number of copies.
The most time consuming operation in the program, because of its
repetitive nature 1 is that of determining a minimum path from an input node
to all output nodes. As the network size increases, the time required for
this operation increases also.
Similarly
1
as the number of copies increases,
the minimum path determinations also increase. Large networks with a great
number of copies thus require a considerable amount of machine time to compute the necessary minimum paths.
The 256 node and 960 link network of Test Problem 8 was utilized in
connection with a two copy and five capacitated link system to study the
run time on a large system.
The minimum path calculations were timed
to obtain -a measure of run time required for this operation.
Table 14 shows
59
the findings of this run.
Thus it was found that for a 256 node system, approximately one
minute of machine time is required for each minimum path. The number of
minimum paths that must be calculated for any given program is a function
of the number of copies and the degree to which the capacitated links
are overloaded and is impossible to predict accurately.
The time required to determine a minimum path becomes a problem as
the network size, the number of copies and the number of capacitated
links increase. Improvement of computation speed on the path subroutine
would significantly reduce over-all run time.
TABLE 14
RUN TIME FOR MINIMUM PATH DETERMINATIONS - TEST PROBLEM 8
Path Number
Time Required
1
1. 04 minutes
2
1. 02 minutes
3
1. 04 minutes
4
1. 04 minutes
5
1. 04 minutes
6
1. 02 minutes
7
1. 01 minutes
8
1. 01 minutes
9
1. 04 minutes
Avg. Time ==
1. 03 minutes
60
PROGRAM OUTPUT
On a normal run, four separate types of data are output as follows:
1.
Data on the progress of the problem solution,
2.
Final solution data.
3.
Individual traffic loadings for all copies in the final basis.
4. Link volumes for loaded links in the network.
Solution Progress
Three types of program output are possible as a means of following
the progress of the solution.
illustrated in Figure 38.
1.
The first type, called nor.mal output is
Data on the following items are furnished:
Current Basis - Both the copy and extreme point solution number
are provided.
Thus, for example 6002 means 6-th copy and 2nd
extreme point solution.
A positive integer less than 1001
represents a slack vector and a negative integer represents an
artificial vector.
2.
Z(j) minus C(j) values -Separate values of Z(j) - C(j) are
listed along with the copy and extreme point solution number.
3.
Type of Vector Entered - This will be either a solution vector
or slack vector.
4.
The entry position in P(o)' is also indicated.
Current Cost - The current cost of the solution after the
entry of the indicated vector is provided.
In order to locate sources of program error it was necessary to
output more information than the normal output.
Two other types of
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output were used for this purpose - an intermediate output illustrated
in Figure 39 and a detailed output as shown in Figure 40.
For the detailed output, the following data are available:
1.
Vector entered and position.
2.
Current Cost and Z(j) - C(j) value.
3· Minimum paths from input nodes to all other nodes •
4o Elements of the cost vector showing costs of each cop,r
solution in the current basis •
5. Elements of the entering copy or P(j) vector.
6. Elements of the primed entering cop,r or P(j)' vector.
7. Elements of the P(o) 1 or solution vector.
8. Elements of the current main matrix.
9. Elements of the Omega vector.
The intermediate output eliminates the minimum path and matrix data.
For all cases the data are output after each iteration of the
problem. With detailed output it is possible to follow each specific
step of the solution.
Solution Outl2JJjt
After convergence, the final solution data shown in Figure 41 are
output.
This provides information on the final basis, the cost of each
copy solution in the final basis, the solution vector, the Omega vector
and the total cost of the optimum solution.
In addition, data on individual copy loadings (Figure 42) and
total link loadings (Figure 43) are available.
The individual copy
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DATA- DETAILED
FIGUR£
40
OUTPUT
PAGE
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FINAL SOLUTION
FIGURE
LQ
DATA
(J)
Ul
66
PAt;t
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INDIVIDUAL
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8
67
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TOTAL LINK
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FIGURE
3J
teEIA r.cARESPCNOihG ENTRY POINTS-
ld
68
loadings show the links in the copy solution, the traffic assigned to each
link and the percent of the copy solution that will be utilized. In the
case of a slack vector in the final solution, it will be indicated as shown
in Figure 42.
The principal
~nswer
loading data in Figure 43.
to a given problem is shown by the total link
The proper percentages of the link loadings on
each copy are obtained and combined to give total traffic on each loaded
link of the system.
Copy Solutions
Since any network problem whose combined minimum path flow overloads any given capacitated link will produce a "mixture .. of copy solutions,
it is of interest to consider the multiple copy solutions.
These solutions
represent a diversion from minimum path flow and correspond to the proportional
assignment technique developed by Irwin, et al. From the listing of traffic
flow on each of the copy solutions in the final basis (Figure 42), the
diverted traffic and the routes it follows can be observed.
Omega Vector
The Primal of the multi-copy problem is formulated as follows:
subject to
):
1
69
The criterion function for the dual to the problem· is as follows:
Max wl dl + w2 dz + • •. + w n dn + w n+l + w n+2 + ••• + wk
where di = capacity limitation on link (i)
w1 = the i-th variable of the dual solution
n = number of capacitated links
k = sum of copies plus capacitated links
At an optimum solution the total cost of the solution (G) is given by
G
Since
then
:11
C P(O)'.
P(O)' = B-l P(O)
G = C B-l P(O).
-1
but w is defined as C B
Therefore G = w P( 0)
•
or
Thus the Omega vector contains the dual variables of the optimum solution
and provides evaluators for the capacitated links.
Consideration of the first n elements of the Omega vector make it
possible to determine which capacitated links contribute the greatest
detrimental effect to the network flow. For example, consider the ex-
70
ample problem (Figure 7) which had two capacitated links and the following Omega vector when an optimum solution was
re~ched.
w = [-3, -5, 810, 600]
w1 is associated with link 3 and w2 with link 9. If it were possible
to increase the link capacity values, the greatest improvement in the
total network flow costs would result from changing link 9 •.Thus 1 the
final output provides a valuable tool to analyze the capacitated links of
a traffic network.
The results obtained from the solution from the various test problems
indicate that the desired answer can be obtained by the multicopy program
and that it provides insight to the distribution problem.
The program con-
siders each origin to destination requirement and the routing of this trip
concerning capacity requirements and over-all system travel times.
The output of the computer program for an optimum solution to the
problem shows how traffic moving from each input node to all output nodes
should be routed. These optimum routings can then be compared to the
minimum path routings to find that traffic which should be diverted from
the freeway by some type of operational control.
Figure 44 shows an example of the traffic diversion previously
discussed,
The first drawing in Figure 44 shows the minimum path that
traffic would follow from a given input node to the specified output nodes
if there were no capacity restraints.
However 1 the solution of the problem
71
by the optimization technique produced the combination of routings shown
in the next three drawings.
It would be impossible to route traffic exactly as indicated by the
solution shown in Figure 44 but it would be possible to affect some diversion
of traffic from the <Jenera! area of the city represented by the input node
in question. Thus from the solution it would be possible to pinpoint the
specific origins of traffic which should be diverted.
This diversion could
possibly be accomplished through ramp closures, one-way street operation,
advisory signing, etc.
In most cases the traffic which must be diverted from the freeway is
short trip traffic. However, the ability of the at-grade facility to move
traffic would also be reflected in the assignment to the freeway.
Traffic
. moving relatively short distances in areas where a poor supporting street
system exists might be more adversely affected than traffic moving much
greater distances over better at-grade facilities.
72
-----¢
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COPY
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COPY
3-10 25%
TRAFFIC DIVERSION BY COPIES
FIGURE
ld.f
73
OPTIMUM DISTRIBUTION OF TRAFFIC
Solution Required
Before it is possible to develop a control system to operate a street
network, the basic question of which traffic will be controlled must be answered.
The freeway links of the network have the greatest potential for traffic movement
and thus it is extremely important that their ability to move traffic be maintained
at a high level. The basic question can therefore be narrowed to ask what
traffic must be diverted from the freeway to maintain its high level of efficiency
and to provide for an optimum assignment of traffic over the system.
An algorithmic approach which was tried might be stated as follows:
1. Close off all capacitated links or assign them a very high cost and
search a minimum path of each origin to all destinations on all copies.
2.. Compute the time cost of each of the above minimum paths and rank.
these in order with the highest cost path first followed by the second
highest cost path and so forth.
, 3. Assign traffic to the network considering the previously developed
ranking. Allow traffic to utilize the capacitated links of the network
until the indicated capacity of some link is reached. When the capacity
of a link is reached this link is removed from the system and the assignment is continued until all traffic has been assigned.
The advanced procedure was programmed by Blumentritt and it provided a
very rapid assignment technique. It was found that this algorithmic technique
produced the correct answer to test problem 2 and approximated very closely
the correct answer to test problem 4. It was found 1 however 1 that it would be
74
necessary to introduce some refinements into the technique in order to
duplicate the multi-copy solution over a wide range of problems. Work
with this algorithmic approach is being continued and it offers a good approach
to studying optimum distribution of traffic over a street network. The advantage
of the algorithmic .assignment technique is that it can be programmed to run
much faster than the multi-copy model and can deal with much larger traffic
networks than is possible with a linear programming solution.
Algorithmic Assignment
A technique which shows a great deal of promise for the study of optimum
traffic distribution is an algorithmic assignment to a given network which would
approach the linear programming solution produced by the multi-copy model.
Such techniques are being studied by Blumentritt (2} and some progress has been
made toward obtaining successful solutions by this method.
The basic element in the algorithmic assignment is the individual movements
between a given origin and destination. It is possible to consider the effect or
cost of a trip between a given origin and destination if this trip cannot be made
over the capacitated
freeway.
link~
of the system or, in the case of a real problem, the
Those trips which sustain the most adverse effect are then assigned
to the capacitated links in that order until link capacity is reached.
The remain-
ing trips must then be diverted from their minimum cost path and would become
the subjects of a traffic control program.
75
CONCLUSIONS AND RECOMMENDATIONS
Conclusions
The research reported herein has concerned itself with the development
of a computer program to provide a linear programming solution for the
optimum distribution of traffic over portions of a street network.
This program
was developed, evaluated and its application to the problem of developing
operational controls for street networks was studied.
The following conclu-
sions were reached as a result of this research.
l.
The multi-copy linear programming model can be adapted to a computer
program which will provide a solution to the optimum distribution
problem for small networks.
2.
The solution to the optimum distr lbution problem provides information
which would be of significant value in planning and designing a
traffic control system to operate an arterial street network.
The origin
of trips which should be diverted from the freeway can be determined
and this information utilized in the design of the control system.
3. The multi-copy program was found to be limited to the following
conditions:
50 input nodes
6 0 capacitated links
300 network nodes
1000 links
In addition to the network limitations the problems of slow convergence
76
and long machine times indicated the undesirability of utilizing
the linear programming solution when dealing with large networks.
4.
Preliminary investigations indicate that it is possible to duplicate
the linear programming solution of the optimum distribution problem
by an
alg.~rithmic
technique. Such a technique offers the decided
advantage of a much faster solution on the computer and the ability
to handle a large network.
5. The multi-copy program is not applicable to normal traffic assignments
for urban planning but is intended only as a tool for the critical
analysis of small street networks. In this capacity it could be a
useful technique for planning operational controls for freeway-major
arterial systems.
The concept of the optimum distribution of traffic is one which
might well be applied to the general traffic assignment problem.
It appears that the algorithmic technique could be developed to permit
traffic assignment to urban street systems on a optimum distribution
basis.
Recommendations
The following recommendations for additional research seem pertinent:
1. The development of the algorithmic technique for solving the network
problem in a much faster manner appears to have much promise.
Studies
which would compare the results of multi-copy solutions to algorithmic
77
solutions and lead to the development of a satisfactory assignment
technique seem highly warranted.
2. After development of the algorithmic technique so that large networks
could be studied with a reasonable amount of machine time it would
be desirable to study the optimum distribution of traffic over actual
systems.
To do this it would be necessary to break a system up into
links and nodes and to arrive at the origin-destination requirements
for this network.
78
RE:FERENCES
1.
Berry, D. s., "Evaluation of Techniques for Determining Overall
Travel Time", Proceedings of the Highway Research Board, PP•
429-440, 1952.
2.
Blumentritt, Charles w., "Algorithmic Techniques For Traffic
Network Analysis," Unpublished Report, Texas A&M University,
1963.
3.
Brokke, G. E. , 11 As signing Traffic to a Highway Network", Public
Roads, PP• 77-87, 1958.
4.
Brown, R.
"Expressway Route Selection and Vehicular Usage",
Bulletin No, 16, Highway Research Board, pp. 12-21, 1948.
5.
Bureau of Public Roads, "Electronic Computer Program for Assignment of Traffic to Street and Freeway Systems", Division of
Development, Bureau of Public Roads, 1959.
M.,
6. Carroll, J, Douglas, Jr., 11 A Method of Traffic Assignment to an
Urban Network",
64-71.
Bulletin~~
Highway Research Board, PP•
7.
CEIR Inc., LP/90 Usage Manual - /! Comprehensive Computing Syatem
for Linear Programm~, July, 1961,
8~
Charnes, A., and Cooper, W. w., 11 Multicopy Traffic Network
Models", Proceedings of £mposill!J1 QU Theocr .Qf Traffic Flow,
General Motors Corporation, PP• 85-96, 1959.
9.
Charnes, A., and Cooper, W. W., "External Principles for Simulating Traffic Flow in a Network", Proce~ings !fational ,Academy
Q! Science, 1958, pp. 201-204.
10.
Charnes, A., Cooper, W. w., and Henderson, Introguction !Q Linear
Programming, John Wiley & Sons, Inc., New York, 1953.
11. Charnes, A., and Cooper, W. w., Management M~ and !ngustrial
Applications Q! Linear Programming, Volume II, John Wiley and
Sons, Inc., pp. 785-796.
12.
Charnes, A., and lemke, C. E., 11 A Modified Simplex Method For
Control of Round-off Error in Linear Programming, 11 Proceed~ Qt the Association for Computing Machinery, Pittsburgh,
May 1952, PP• 97-98.
79
13.
Chicago Area Transportation Study.
Chicago, 1960.
Final Report, Volume II,
14. Cleveland, D. E., "A Study of Highway Traffic Assignment",
Doctoral Dissertation, A&M College of Texas, 1962.
o.,
R.,
l5e
Covault, D.
and Roberts, R.
"The Influence of On-Ramp
Spacing on Traffic Flow on the Atlanta Freeway and Arterial
Street System", Paper presented at the 42nd Annual Highway
Research Board Meeting, 1963.
16.
Dantzig, C. B., "On the Shortest Route Through a Network",
Management Science, 1960, pp. 187-190.
17.
Dantzig, G. B., "Maximization of a Linear Function of Variables
Subject to Linear Inequalities", Chapter XXI of Koopmans (27).
18.
DeRose, F., "The Developnent and Evaluation of the John c. Lodge
Freeway Traffic Surveillance and Control Project", presented
to the Annual Meeting of the Highway Research Board, 1963.
19.
Forbes, T. W., "Speed, Headway and Volume Relationships on a
Freeway", Proceedings of the Institute of Traffic Engineers,
PP• 103-126, 1951.
20.
Fuller, Leonard
1963.
21.
Gass, Saul I., Linear Programming Methods and Applications,
McGraw-Hill Book Company, Inc., 1958.
22.
Gomory, R. E., and Hu , T. C., "An Application of Generalized
Linear Programming to Network Flows", Research Report RC 319,
International Business Machines Corporation, September,
1960.
23.
Heanue, Kevin E., and Covault, Donald o., "The Developnent of
Criteria for Closing a Freeway Ramp Using Linear Programming",
Committee Report, Theory of Traffic Flow, Highway Research
Board, January, 1962.
24•
Irwin, N.
and von Cube, H. G., 11 Capacity Restraint in MultiTravel Mode Assignment Programs, P~per presented to the Origin
and Destination Committee of the Highway Research Board,
January, 1962o
25.
Irwin, N. A., Dodd, Norman and von Cube, H. G., "Capacity Restraint
in Assignment Programs", Bulletin ?.YL, Highway Research Board,
pp. 109-127.
E.,
Basic Matrix Theory, Prentice Hall, Inc.,
A.,
80
26.
Keese, C. J., Pinnell, c., and McCasland, W. R., "A Study of Freeway Traffic Operations", Bulletin ill, Highway Research Board,
PP• 73-132.
27.
Koopmans, T. C., Editor, "Activity Analysis of Production and
Allocation", Cowles Commission Monograph 13, John Wiley & Sons,
Inc., New York, 1951.
28.
Martin, B. v., Memmott, F. w., and Bone, A. J., Principles and
Technigues of Predicting Future Demand for Urban Area Transportation, Research Report No. 38, Joint Highway Research
Project, Massachusetts Institute of Technology, 1961.
29.
May, A. D., Athol, P. J., Parker, W. L. and Rudden, J. B.,
"Development and Evaluation of Congress Street Expressway
Pilot Detection System", presented at the Annual Meeting of
the Highway Research Board, 1963.
30.
McCracken, Daniel D., ! Guide to FORTRAN Programming, John Wiley
and Sons, Inc., New York, 1961.
31.
Mertz, William L., "Review and Evaluation of Electronic Computer
Traffic Assignment Programs", Bulletin w_, Highway Research
Board, PP• 94-105.
32.
Moore, E. F., "The Shortest Path Through a Maze", Proceedings of
an International Symposium QU the Theory: of Switching Part
II, April, 1957.
33.
Moskowitz, K., "Research on Operating Characteristics of Freeways",
Procee~ Q£ the Institute of Traffic Engineers, pp. 85-110,
1956.
34.
Pinnell, Charles and Satterly, G. T., "Systems Analysis Technique
for the Evaluation of Arterial Street Operation", Paper presented at the Annual Meeting of the American Society of Civil
Engineers, Detroit, Michigan, October, 1962.
35.
Pinnell, Charles, "Development of a Linear Programming Technique
for a Systems Analysis Study of Arterial Street Operation" 1 Doctoral
Dissertation, Texas A&M University, 1964.
36.
Pollack, M., "The Maximum Capacity Through a Network", Operations
Research, 1960, pp. 733-736,
37.
Saaty 1 Thomas L., Mathematical Methods of Operations Research
McGraw-Hill Book Company, Inc. , 19 59.
. 81
)8..
Schneider, M., "Traffic Distribution on a Road Networ~ Chicago Area
Transportation Study Publication 66538, May, 1961.
39.
Smock, Robert B., 11 An Iterative Assignment Approach to Capacity
Restraint on Arterial Networks'', paper presented at the Annual
Meeting of the Highway Research Board, January, 1962.
40.
Texas Highway Department, Highway Planning Survey,
Traffic Survey, Houston, Texas, 1951.
41.
Tomazinis, A. R., 11 A New Method of Trip Distribution in an Urban
Area", Paper Number 15, Penn Jersey Transportation Study, 1962.
42 ..
Voorhees, A. M., 11 A General Theory of Traffic Movement", Proceed~gs Institute of Traffic Engineers, 1955.
43.
Wattleworth, Joseph A., "Peak-Period Control of a Freeway SystemSome Theoretical Considerations", Report 9, Expressway Surveillance Projec~ Illinois Department of Highways, August,
1963.
~
Freeway
A., and Shuldiner, Paul W., "The Inclusion of
Left Turns in Traffic Prediction Models of the Charnes-Cooper
Type", Paper presented at the Amerlcan Society of Civil
Engineers Annual Meeting, October, 1962.
44. Wattleworth, Joseph
45.
Webb, G., and Moskowitz, K., "California Freeway Capacity Study1956 11 , Proceedings of the Highway Research Board, Volume 12_,
PP• 587-642, 1957.
82
APPENDICES
83
CALLING
CALL
SEQUENCE :
START (NOV,K,NOKAP)
NOV = NUMBER OF "LINKS
K = NUMBER OF COPIES
NOKAP = NUMBER OF CAPACITATED
LINKS.
REWIND TAPES
2 , 3 I AND 4
CALL LIBRARY
ROUTINE ERASE
FOR ARRAY
I Nl T IALIZAT ION
INPUT : NETWORK
DESCRIPTION, INPUT VOLUME
DATA,. OUTPUT VOLUME
DESCRIPTION, CAPACITATED
LINKS,
FLOW DIAGRAM
SUBROUTINE START
FIGURE
17
84
CALLING
SEQUENCE :
CALL TREE
KOPY , KEY, IND, LINK , NO)
KOPY = NUMBER OF COPIES
KEY = I OR 2, WHICH SPECIFIES ENTRY POINT
IND = COPY NUMBER UNDER CONSIDERATION
LINK = NUMBER OF LINKS
NO = NUMBER OF NODES MINUS TWO
KEY = I
BUILDS MINIMUM PATHS FOR THE
INITIAL BASIS
KEY = 2 BUILDS THE MINIMUM PATH FOR
I TH COPY.
r
I
THE
KEY =
N_G_E_ NETWORK-C-OS_T_S
OM FIXED POINT TO
FLOATING POINT
-~~-----------~
~J0--@
................-
...........................
~-----~--
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __..J
85
(A)
//
/
//
J
IS
/
THE Nlll ELEMENT
,
rc\'!'_~K
OF THE MIN I MUM
' '-,,.._
\_' /joG
PATH ARRAY EQUAL T 0
//
-JHE K!!l OUTPUT NODE /
. · 0 F THE 1".1! C 0 P Y / /
·--.
7
.. /
./'
-(0)
____ ______ __ __ ,~4fNo
204
(KEY • 2)
FLOW
DIAGRAM
SUBROUTINE
TREE
Bu__ _ _ _ _ _ _ _ _ ____.l
c___ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _fi_G_UA~la.~-
86
-----------------------------
IS
THE
DESTINATION OF
THIS MINIMUM PATH
EQUAL TO ITS
ORIGIN?
~YE~S4r~j";2
120
FOR COPY (I), LOAD ALL LINKS
OF THE MINIMUM
PATH
OF
THIS OUTPUT NODE WITH THE
VOLUME OF THE OUTPUT NODE
B
KEY= 2
NO
E
IS
THIS
LAST
THE
COPY?
l
YES
~----ADD ROUNDING
FACTOR
TO FLOATING POINT COSTS
OF THE NETWORK
DE~CRIPTION AND TRUNCATE
(KEY= I)
FLOW DIAGRAM
SUBROUTINE TREE
FIGURE
L _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
-
18-C
----------------------------1
87
r---------------·-------------------,
CALLING
CALL
ESTABLISH INPUT NODE
AS REFERENCE NODE.
SEQUENCE :
PATH (I, LINK, NO)
WHERE l• COPY NUMBER
LINK= NUMBER OF LINKS
NO • NUMBER OF NODES MINUS TWO.
PLACE CONNECTING NODES
AND CUMULATIVE COSTS
IN SCRATCH TABLE.
REMOVE SCRATCH TABLE
ENTRY THAT RETURNS
TO REFERENCE NODE.
SCAN SCRATCH TABLE
FOR MINIMUM COST
CONNECTING
NODE.
IS
THIS NODE
ALREADY IN THE
SEQUENCE
TABLE 7
YES
REFERENCE NODE
IS
NOW THE LAST ENTRY
IN SEQUENCE.
NO
PLACE THIS NODE IN
THE SEQUENCE TABLE AND
REMOVE FROM SCRATCH TABLE
HAS A
PATH BEEN
BUlLT TO ALL OTHER
NODES IN THE
SYSTEM?
NO
FLOW DIAGRAM
SUBROUTINE
FIGURE
----·----···----------·---- --·-·
-·--
PATH
19
--·--------------------------'
88
CALLING SEQUENCE :
CALL KOST (NOV , K , NOKAP)
NOV • NUMBER OF LINKS
K • NUMBER OF COPIES
NOKAP • NUMBER OF CAPACITATED
LINKS
SE1 FIRST NOKAP ELEMENTS
OF THE P(O) VECTOR EQUAL
TO THE CAPACITIES OF THE
CAPACITATED LINKS
SET UP FIRST NOKAP
ROWS OF MATRIX
NO
DID
THE INITIAL
ASSIGNMENT
OVERLOAD THE
CAPACITATED
LINK
~
NO
IIH
I) PLACE POSITIVE SLACK
IN ARRAY
2) SET CORRESPONDING
POSITION OF COST VECTOR • 0
I) PLACE NEGATIVE SLACK IN
ARRAY
2) SET CORRESPONDING POSITION
OF COST VECTOR EQUAL TO
10000 (HIGH M COST )
t---------------
FLOW DIAGRAM
SUBROUTINE
FIGURE
KOST
20-A
89
SET FIRST K POSITIONS
OF BASIS VECTOR = COPY
NUMBER AND FIRST ITERATION
.......,--------~!_______,.
I) SET REMAINDER OF P (0)
VECTOR ELEMENTS = I
2) COMPLETE REMAINDER OF
ARRAY BY PLACING UNIT
ELEMENTS IN CORRESPONDING
COPY POSITIONS
,-------_]_~-------'
ADD TO EACH COST VECTOR
ELEMENT
THE COST OF
THE MINIMUM PATH FOR
THIS COPY
11
WRITE~ PiOllvECToR;;]N
TAPE 2
2) WRITE COPY LOADINGS
ON
TAPE
1··---·--
I
END FILE 2
REWIND 2
I
j
1
RETURN
FLOW DIAGRAM
SUBROUTINE KOST
FIGURE
20-B
90
CALLING SEQUENCE :
CALL MUL T (KEY, SUM, ICOLA)
KEY" I
ICOLA = ORDER OF MATRIX
MUTIPL Y COST VECTOR
BY MATRIX TO OBTAIN
OMEGA ROW
IS
THE ABSOLUTE
VALUE OF ANY
OMEGA ELEMENT
LESS THAN
SET THAT
ELEMENT = 0
,~ I.E-04!
KEY • 3
ENTER
MULTIPLY
B·l
BY
THE
P(J) VECTOR TO GET
THE PRIME OF THAT
VECTOR
KEY= 2
MULTIPLY COST VECTOR BY
Po TO GET THE OVERALL
COST OF THE PRESENT
SOLUTION
KEY= 4
~EL~
L TIPL Y THE OMEGA
W TIMES THE
TERING COPY VECTOR
[]
GET THE Z(J)
RETURN
FLOW DIAGRAM
SUBROUTINE MULT
FIGURE
21
91
CALLING SEQUENCE :
CALL CHKINT (KEY, KEYA, NOKAP)
KEY
II
I
ENTER
NOKAP= NUMBER OF CAPACITATED LINKS.
IS
THERE A
POSITIVE ELEMENT
IN THE OMEG
ROW?
KEY= 2
KEYA
=
I
ENTER
FLOW DIAGRAM
SUBROUTINE
FIGURE
---····
CHKINT
22
---·--
-- ----
---
-·
-------~-------
92
CALLING
CALL
SEQUENCE
LOGIC ( NOKAP)
NOKAP =NUMBER OF CAPACITATED
LINKS.
ENTER
PLACE
NEW DUMMY
COST ON THE CAPACITATED
LINKS OF THE
SPECIFIED
COPY.
FLOW DIAGRAM
SUBROUTINE LOGIC
FIGURE
23
93
CALLING
SEQUENCE :
CALL NEWVAL (KOPY, LINK, N, ND)
KOPY =NUMBER OF COPIES
LINK • NUMBER OF LINKS
N ., COPY NUMBER UNDER CONSIDERATION
ND • NUMBER OF NODES MINUS Z
REMOVE ' DUMMY COSTS'
FROM CAPACITATED LINKS
AND REPLACE
WITH
ACTUAL TRAVEL
COSTS.
-~
SUM TOTAL COST
FOR THIS C~~ (C )
SET THE NOKAP +N-th
POSITION OF THE ENTERING
VECTOR P (J) EQUAL TO ONE.
(OBTAIN Zj)
(.)Pj = Zj
FLOW DIAGRAM
SUBROUTINE NEWVAL
FIGURE 2lf-
94
CALLING
SEQUENCE
CALL OTHVAL (KOPY,LINK,IND)
KOPY = NUMBER OF COPIES
LINK = NUMBER OF LINKS
IND = POSITION THAT VECTOR (OR SLACK)
WILL BE ENTERED (DETERMINED IN
THIS SUBROUTINE)
YES
IS
NORDER
WHERE NORDER IS
THE ORDER OF THE
MATRIX?
It
>N_O_ _ _ _~
DETERMINE
SMALLEST
POSITIVE 9.1. AND SET
IND = 1
YES
ENTER
BY
P; INTO THE BASIS
GAUSSIAN
ELIMINATION
(COMPUTE NEW
OMEGA ROW)
FLOW DIAGRAM
SUBROUTINE OTHVAL
FIGURE
25
95
CALLING
SEQUENCE :
CALL SLACK (KOPY, LINK, NOKAP)
KOPY =NUMBER OF COPIES
LINK= NUMBER OF LINKS
NOKAP = NUMBER OF CAPACITATED
LINKS.
DETERMINE THE LARGEST
POSITIVE ELEMENT IN THE
OMEGA ROW
SET THE P(J) VECTOR
ELEMENT CORRESPONDING TO
THE POStnVE ABOVE = I
INCREASE NUMBER OF
SLACK ITERATIONS BY I
(ENTER THIS SLACK
INTO BASIS)
WRITE ZERO LINK LOADINGS
ON TAPE 3 FOR THIS
BASIS VECTOR
WRITE OUTPUT TAPE 6 :
ENTRY POINT OF THIS SLACK
AND OUTPUT CURRENT BASIS
DESCRIPTION
FLOW
DIAGRAM
SUBROUTINE
--------
FIGURE
SLACK
26
---------------------------------~
96
~T0
CALLING SEQUENCE
CALL SHELL (K,NOV,NOKAP)
S L L ERAUS
FOR COPY
_OAD ARRAY
K = NUMBER
OF
NOV = NUMBER
.----I___~3
COPIES
OF
LINKS
KOKAP = NUMBER OF CAPACITATED
LINKS
READ TAPE 2,
ENTERING COPY VECTOR AND
ALL LINK LOADINGS
-1 REL21
J
TAPE 3
T EOF
).:READ
V
Y>-E_S_ _..,
TAP~NE-COPY-
-1
L·TERAT~~""-~rc;~e~-~·-"'""sJ
/ls~--
NO
1------<
/THIS COP~
IN THE FINAL;>
---
l --r~-------~~~r~=
~-----
SUM ALL LINK LOADINGS
ACCORDING TO THE AMOUNT
SPECIFIED BY THE P(O) ELEMENT
FOR COPIES IN THE FINAL BASIS
APPLY ROUNDING FACTOR
-----------r
BASISV
AND FIX RESULTS
YES
~
TO COPY LOAD ARRAY
FLOW DIAGRAM
SUBROUTINE SHELL
FIGURE
27
97
CALLING
SEQUENCE
( KOPY , LINK , NOKAP)
KOPY = NUMBER OF COPIES
LINKS= NUMBER OF LINKS
NOKAP : NUMBER OF CAPACITATED
LINKS .
.-[j{-R-1-TE
P_~[~~_NU~_BE§J
NOTE : ALL
STATEMENTS
[WRiTEPROBLEMHEADINGJ
TO OUTPUT
WRITE
REFER
TAPE 6 ·
~-L- ----------- ~-----___l___ ----, WRITE COS~~CTOfU
~ITE FINAL BASIS
DESCRIP~
~AULL~-\
KEY=J
FLOW DIAGRAM
SUBROUTINE PRNT
FIGURE 28-A
'------------------------------------------------'
98
r------------------------------------------,
WRITE
COPIES
IN FINAL BASIS
WRITE MATRIX POSITION, COPY
NUMBER, CORRESPONDING
P ( o)'
ELEMENT FOR EACH COPY IN
FINAL BASIS. ALSO LIST
RESPECTIVE
NON- ZERO
LINK LOADINGS.
WRITE FINAL
CONVEX
COMBINATION OF NON-ZERO
LINK LOADINGS FOR ALL COPIES
ARE
THERE
ANY ALTERNATE
SOLI,ITIONS?
NO
WRITE INDICATION THAT
ALTERN ATE SOLUTIONS
WERE FOUND
>--~ NO
REWIND 2
REWIND 3 k - - - - - - - - - - - '
REWIND 4
FLOW DIAGRAM
SUBROUTINE PRNT
FIGURE
28-B
---------------------------------~
99
~--~~--=----
INPUT CONTROL CARD
( NUMBER OF
COPIES, NUMBER OF LINKS, NUMBER OF
CAPACITATED LINKS, NUMBER OF NODES,
ITERATION FACTOR CONVERGENCE LIMIT,
TITLE OF PROBLEM, INTERMEDIATE OUTPUT
CONTROL NUMB_=-:ER_:_:_·.:_l- . - - - - -
~CE
__ ] ____ _
]OUTPUT
- ~---
] ..
L_IQ__NEW PAGE
~LiZE-COUNTIN~
-~~g$_
\g~~
LL
EE
y =I
YES
[ ~~~~~r~~g~·]
7
AVE 10
CONSECUTIVE
ENTRIES BEEN
MADE INTO THE SAME BASIS
POSITION OR HAVE 10
CONSECUTIVE SLACKS
BEEN INTRODUCED?
N_Q__
-----@
CALL MATINVJ
(LIBRARY MATRIX
INVERT ROUTINE
~~
~MATRIX >r~§____ _
__
', SINGUL~Y
1000
104 _NO
CALD
CHKINT
---K_EY = 2
l___ - - ·-··
------------
N= N +I
OfX N~:I:REMENTI
ALN
B__ J ____
NITIALI>E
PY ITERATIONS
NUMBER I
to
FLOW DIAGRAM
MAIN CONTROL PROGRAM
FIGURE 2. 9 -A
100
DID .
FINDS~~~- ;~~~',,,
YES
NOKAP~
ELEMENT IN FIRST
POSITIONS OF OMEGA ROW,
WHERE NOKAP = NUMBER
~OF CAPACITATED
~"LINKS?
108
'>
NO
T(N)= IT(N)+I
REMENT CURRENT
Y ITERATION BY I
~~~~;~~~t:~~·J
L
THIS POINT
-·..-----·----- · - - - - - -
'
DOES·,,"'
THIS SUM "-,
EXCEED THE ORDER~­
OF THE MATRIX TIMES
THE INPUT
ITERATION
FACTOR ':1
OUTPUT: NO FEASIBLE
SOLUTION FOUND IN
--ITERATIONS
---- ----------------
----------,
OUTPUT THE FOLLOWING:
I) MINIMUM PATH
2) COST VECTOR
3) ARRAY
-------------14) ENTERING COPY VECTOR
5) PRIMED ENTERING COPY
VECTOR
6) P- rERO VECTOR
7) OMEGA VECTOR
FLOW DIAGRAM
MAIN CONTROL PROGRAM
29-B
----- - - - - - - - - - - - - - - - - - - - - - - - - - - '
FIGURE
________________
..._
--
-------
101
THIS IS AN
ALTERNATE
SOLUTION
CALL
OTH VAL
LINK=O
WRITE TAPE 4•COPY
AND ITERATION
NUMBER, AND POSI·
TION WHERE THIS
VECTOR COULD
ENTER BASIS
HAS
ANYTHING
ENTERED
THE BASIS IN
NOKOP
TTEMPTS
?
218
YES
OUTPUT Zj,Cj,Zj-Cj,
AN 0 CURRENT COPY
ITERATION
FLOW DIAGRAM
MAIN CONTROL PROGRAM
FIGURE
29-C
102
--------------------------~
(ENTER THIS
VECTOR INTO BASIS)
WRITE COPY AND
ITERATION NUMBER;
LINK LOADINGS FOR
THIS COPY ON TAPE 3
PUT POINT OF ENTRY,
URRENT COST, ~j- Cj I
AND NEW BASIS
SUM ALL
ITERATIONS
TO THIS POINT
DOE~'',"
THIS SUM
"'~
NO
'>-----<
EXCEED MAXIMUM
ITERATION
/
CRITERION?
IS "'--"
INTERMEDIATE
OUTPUT INFORMATION
DESIRED FOR THIS
ENTRY?
--~
YES
OUTPUT THE FOLLOWING :
I) ENTERING COPY VECTOR
2) PRIMED ENTERING COPY
VECTOR
3) P- i!ERO VECTOR
4) OMEGA VECTOR
FLOW DIAGRAM
MAIN CONTROL PROGRAM
FIGURE
29-D
----- ---------- - - - - - - - - - - - - - - - - - - - - - - - l
PUBLICATIONS
Project 2-8-61-24
Freeway Surveillance and Control
1. Research Report 24-l, , Theoretical Approaches to the Study and Control of
Freeway Congestion, by Donald R. Drew.
2.
Research Report 24-2, ''Optimum Distribution of Traffic Over a Capacitated
Street Network, by Charles Pinnell.