CHARACTERIZATION OF THE FORCE VARIABLE IN
SYSTEMS NETWORK APPLICATIONS WITHIN A SOCIO , 7
ECONOMIC ENVIRONMENT
Thesis for the Degree Ph. D.
MICHIGAN STATE UNIVERSITY
PAUL 'BANKIT
1972
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This is to certify that the
thesis entitled
Characterization of the Force Variable
in Systems Network Applications
within a SocioEconomic Environment
presented by
Paul Bankit
has been accepted towards fulﬁllment
of the requirements for
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Ph.D. le ein Transportation
(76%,; 71mm",
Major professo/
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ABSTRACT
CHARACTERIZATION OF THE FORCE VARIABLE IN
SYSTEMS NETWORK THEORY APPLICATIONS
WITHIN A SOCIOECONOMIC ENVIRONMENT
By
Paul Bankit
Techniques originally developed to model and analyze
complex and interacting physical components have recently
been adapted for use in analyzing economic systems. These
techniques can be applied to any phenomenon which can be
identified as a collection of components interacting at
clearly defined interfaces, as long as the behavioral char
acteristics of these terms can be described mathematically
in terms of common flow and force variables. The vehicle for
analysis is a system model composed of a set of state equa
tions and an output vector.
This research focuses on the derivation of the force
variable as used in the economic sense. The application of
the element of demand to linear graph applications allows
the analyst to model and characterize the system being con—
sidered more completely and to provide strategies for control
and stability.
This study applies these techniques to the problem of
the requirement for a regional airport as a replacement for
existing local facilities. The resulting modeL depicts the
airport system as a combination of interacting flows and
forces. Operation of the system is simulated on the computer
and the sensitivity of the system to varying service levels
is studied. It is concluded that the force or demand
factor can be measured for use in systems application and
that the technique offers opportunities for socioeconomic
applications.
CHARACTERIZATION OF THE FORCE VARIABLE IN SYSTEMS
NETWORK APPLICATIONS WITHIN A SOCIOECONOMIC ENVIRONMENT
By
Paul Bankit
A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Marketing and Transportation
1972
[5’14
Copyright by
Paul Bankit
1972
'
ACKNOWLEDGEMENTS
The work contained herein is the result of the contri
butions of many peOple who have taught, assisted, and
encouraged the author through his lifetime. Words of thanks
are insufficient to express the appreciation for all their
efforts.
My gratitude is especially directed toward Dr. Frank
H. Mossman, committee chairman, for providing me with the
basic theoretical framework for implementing this research
His inspiration and material help made it possible.
I wish to thank Dr. Richard J. Lewis, committee mem
‘ber, for his invaluable critiques of the theoretical aspects
of this thesis. A special thanks to Dr. Leo G. Erickson for
providing meaningful insights into the relations encountered
in the research.
My greatest thanks must go to my dear patient wife,
Esther, whose long sacrifices, enduring patience, and untir
ing encouragement have been the prime factors in all my
SUCCESS .
ii
TABLE OF CONTENTS
Chapter Page
I. THE APPLICATION OF SYSTEMS NETWORK THEORY
TO A SOCIOECONOMIC PROBLEM . . . . . . . . . . 1
Introduction . . . . . . . . . . . . . . . . . 1
General Considerations 1
The Problem A
Problem Application 7
Systems Network Theory Considerations 8
Graph Methodology . . . . . . . . . . . . . . 12
The Research Objective . . . . . . . . . . . . 18
II. CHARACTERIZATION OF THE ”FORCE" VARIABLE . . . . 20
Introduction . . . . . . . . . . . . . . . . . 2O
Demand Factors For Air Travel . . . . . . . . 25
Distance—Mass Attraction . . . . . . . . . . . 26
DistanceCost . . . . . . . . . . . . . . . . 27
Distance Modal Choice . . . . . . . . . . . . 29
Modal Choice in the Research Cities . . . . . 30
The Service Level Ratio Effect . . . . . . . . 35
The "Force” Function . . . . . . . . . . . . . 37
Projecting the "Force" Function . . . . . . . 39
III. THE APPLICATION OF SYSTEMS NETWORK METHODOLOGY . 1+2
Introduction . . . . . . . . . . . . . . . . . U2
Structure of the System . . . . . . . . . . . A2
iii
Chapter Page
The Mathematical Relationships . . . . . . . . A7
Component Equations . . . . . . . . . . . . . A9
Characteristics of the Model . . . . . . . . . 55
Operation of the Model . . . . . . . . . . . . 59
Sensitivity of the Model to Service
Level Changes . . . . . . . . . . . . . . . 61
IV. EVALUATIONS AND CONCLUSIONS . . . . . . . . . . 68
General . . . . . . . . . . . . . . . . . . . 68
Significance of the Model . . . . . . . . . . 69
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . 75
APPENDICES
Appendix
A. Regional Airport Location Models . . . . . 80
B. Program SIMEQ . . . . . . . . . . . . . . 8A
iy
LIST OF TABLES
Table Page
1. Geographical Coordinates and Locations
of PrOposed MID Facility . . . . . . . . . . . l5
2. O & D Passengers Per Capita for Selected
Michigan Cities . . . . . . . . . . .,. . . . 22
3. Passengers per Capita Forecast for Selected
Michigan Cities . . . . . . . . . . . . . . . A1
A. System Service Level Sensitivity and Flows . . . 63
Figure
LIST OF FIGURES
Geographic Location of Proposed MID
Facility . . . . . .
Directed Graph for Research Cities
Typical Airline Service Offered at Cities
Proportions of Surface Travelers between
Detroit and Selected Michigan Cities
as a Function of Distance . .
Air Travel System "Force" Variable
Systems Graph For Component Model
Component Equation Matrix ..
Systems Component Equations
Forecast Air Travelers From Cities
vi
Page
1A
16
22
33
38
A6
50
56
72
CHAPTER I
THE APPLICATION OF SYSTEMS NETWORK THEORY
TO A SOCIOECONOMIC PROBLEM
Introduction
This dissertation represents an attempt to unite two
basic methodologies of electrical engineering and business
administration into an Operational procedure for use in en
hancing the decision making ability of the planner and
analyst. The two methodologies are systems network theory,
based.cm.the theory of electrical circuits, and economic
forecasting, a procedure from business administration. This
wan: can.then best be categorized as applied research in
that iizis prescriptive in.nature. It will attempt to pro
vide insight into an operational problem.
General Considerations
Modern society is dominated by a complex of networks
for the transmission of energy, the transportation of people,
the distribution of goods, and the dissemination of infor
mation. This complex consists of such diverse systems as
the telephone network, gas and oil pipelines, highway net
WOTkS, and the networks of computers serving as data banks
and remote processing units. The cost of the development
cﬁ‘these networks demands that they be rationally used and
new ones be intelligently planned and developed.
These networks have as the basis of their structure
the elements of branches, along which flows are transmitted,
and nodes, points where flows originate, are relayed, or are
terminated. These structural elements are combined into
mathematical entities called "graphs". The graph consists of
the connected branches and vertices.2’3
There are network applications such as Program Eval—
uation Research Technique (PERT) and the Critical Path Method
(0PM)!4 in which only flows are stated and the element of
"force" or demand is not present. These graphic techniques
are not suitable for characterizing the type of networks to
be<3onsidered in the context of this work. Additionally,
dynamic programming offers a solution to some network prob
especially those with time and state sequencing, but
lems,
heree‘too, the element of demand is not handled or treated.5
¥
lSee H. Frank and I. Frisch, Communication, Trans
mission and Transportation Networks (Reading, Mass., Addison
Wésley, 1971), Chapter‘I.
2See H. Koenig, Y. Tokad, and H. Kesavan, Analysis of
Discrete Physical Systems (New York, McGrawHill, 19677.
3See H. Frank and I. Frisch, Communication, Transmis
sion, and Transportation Networks (Reading, Mass., Addison
Wesley, 1971), Chapter 2.
. “See R. Miller, Schedule, Cost, and Profit Control
With PERT (New York, McGraw4Hill, 1963),Chapter 2 through A.
. 5See G. Hadley, Dynamic Programming (Reading, Mass.,
Addison4Wesley, l96A),ChEpters 1 through 3.
Mn.
N;
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Hus
5:
One network technology does however treat the factor
of propensity or "force". Systems Network Theory (SNT)
takes the graph of a network, models the characteristics,
and states these characteristics in the form of flow and
"force" variables so as to determine the interrelationships
of the parameters introduced in the modeling process.1
The use of Systems Network Theory in the description
and development of electrical networks is generally recog
nized. Electrical networks, telephone networks, power
systems, and some traffic flow applications have been com—
pleted utilizing the principles of the theory. There has
also been some work in the application of the theory in the
production and transportation areas.
The problem in previous applications has been the
development of suitable "force" variables, consistent with
known economic principles and their proper use in network
analysis. Systems network problems have been solved using
propensity (a term interchangeable with "force", but which
does not properly convey or denote the relationship in the
economic and business context) variables such as rate of
production cost of commodity,2 marginal cost of movement,3
lSee H. Koenig, Y. Tokad, and H. Kesavan, Analysis of
Discrete Physical Systems (New York, McGrawHill, I967),
Chapters 2 and 3.
2
See F. Mossman and J. Hynes, Systems Network Theory:
Applications to Distribution Problems TBraintree, Mass.
D. H} Mark, 1968),‘Chapter‘3.
3See J. P. Hynes, Motor Carrier Rates in a Normative
_§patial Environment (UnpuinShed Ph{D} Dissertation,
MiChIgan State University, 1971).
H
.v
I
l and others.
massenergy costs,
These applications have not satisfactorily expressed
the demand, push, or "force" which initiates, relays or ter
minates flows along branches. Because of the difficulty of
stating a satisfactory "force” variable, the use of systems
network theory has not been widespread and in fact has been
limited to academic applications in business. Such tech
niques as dynamic programming which use only flow variables
are more widely used and accepted but lack the ability to
state any measure of "force" or demand, and as such are also
limited in application.
The next level of programming is the use of simulation
techniques, which substitute heuristic or probabilistic
techniques in lieu of the determination of usable demand
factors. All these techniques are accomodations for lack of
ability to characterize and portray the demand factors of
the system being examined.
The Problem
The ability to characterize a suitable "force”
function is the greatest hurdle in the development and use
of systems network theory in the socioeconomic discipline.
The ability to characterize the "force" variable in the
electrical network applications is well defined by a series
of laws defining the relationships of flows and forces,
lSee H. Koenig, W. Cooper, and J. Falvey, "Engineering
for Ecological, Sociological, and Economic Compatibility",
IEEE Transactions on Systems, Man, and Cybernetics Vol. SMC2,
JU1Y 1972, PP 319331.
just as they are in the other physical science applications.1
The use of systems network theory as an analytical tool in
the study of economic processes has therefore not been an
area for extensive investigation because of the difficulty
of operationalizing the features of the technique, and
applying them to actual business and governmental problems.2
Review of previous works in the field do not yield
evidence of full use of the technique except in very re
stricted Situations, although there have been several
excellent efforts describing the technique. The prime ref
erence for the theory is that written by Koenig,3 which laid
A,5
the initial base for the work of Mossman and Hynes, who
lSee T. C. Koopmans and S. Reiter, "A Model of Trans
portation", in Activity Analysis of Production and Allocation,
ed. by T. C. KOOpmans (New YOrk: JOhn.Wiley & Sens, Inc.
1951). pp. 22259.
2Other linear graph applications within socioeconomic
systems are described in contemporary literature. For exam
ple see H. Koenig and T. Manetsch, Systems Analysis of the
Social Sciences (East Lansing, Michigan: College of Engineer
Ing, Michigan State University, 1966). (Mimeographed) or M.
Beckman, D. Christ, and M. Nerlove, Scientific Papers of
Tjallings C. Koopmans (New York, SpringerVéfIag, 1970),
pp. 184209.
3See H. Koenig, Y. Tokad, and H. Kesavan, Analysis of
Discrete Physical Systems (New York, McGrawHill, 1967), pp.
3434u21.
4
See F. Mossman and J. Hynes, Systems Network Theory:
Applications to Distribution Problems (Braintree, Mass.
19687; pp. 18437.
5See J. Hynes, Motor Carrier Rates in a Normative
Spatial Environment, (Unpublished Ph.D. dissertatIOn,
MIchigan State University, 1971).
have applied the theory and established some basic method
ology to the problems of transportation networks.
It appears that applications in the field of trans
portation offer opportunities for development because of
the similarities of flows along transportation links and
terminal throughput capabilities to the same applications in
electrical circuit theory. The transportation problem is
well known in linear programming models and the applications
of Mossman and Hynes have utilized linear programming to
implement systems network theory. The application of linear
programming to the theory limits the problem to static solu
tions and thus fails to provide adequate temporal sensitivity.
Transportation problem solutions as accomplished by the works
of Mossman and Hynes focused on the flow aspects of the
transportation network. Though considerable work was done
attempting to establish a propensity (force) variable suit
able for expression of the demands within the system, the
studies focused on marginal movement costs which were not
entirely satisfactory, especially in the operational sense.
Systems network theory has the ability to move from time
period to time period in the technique. It is in this area
that the research will develop and demonstrate a suitable
technique for the derivation of the "force" variable so
critical to socioeconomic applications. As Baumol points
out, "No matter how ingenious the economists circumlocutions
that have been employed, there has been no substitute devised
an,
.i‘
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9,,
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to replace the demand function."1
Problem Application
The area of transportation offers a particularly
fruitful research medium. Since flows and flow data are
available in standardized formats, a problem involving air
passenger transportation within the state of Michigan was
selected as a candidate for application of systems network
theory in an economic context.
An area of current interest is the possibility of
construction of an international regional airport facility
in the middlewestern states, occasioned by the rise of air
traffic volume both in the passenger and air cargo fields
and the saturation of present facilities in the area.2 The
various states are, of course, interested in the location of
such a facility within their borders. Since the construction
of such a facility is an extremely expensive undertaking, the
location and siting considerations are of critical importance.
To assist in the problem, a smaller scale problem using‘
techniques applicable to scaling could prove to be of assis
tance in the measurement of future traffic volumes and demand
service.
The area of investigation will be a systems network
application of the determination of the proper level of
1See W. J. Baumol, Economic Theory and erations
Anal sis, 2d, Ed. (Englewood CIIffs, N. 3., PrentIceHaIl,
I965I, pp. 210230.
2See "Facing the Airport Challenge", Avaition'Week
and Space Technology, November 15, 1971, p. 23}
y”
An
in
(I,
'Ill'll.‘
service for an air terminal facility intended for use as a
regional airport in the state of Michigan.1 The deter
mination of the "force" variable, an economic function
intended to depict the demand for services at the facility,
will be based on economic factors, and expressed as an index
number. This approach differs from other applications of
the propensity (force) variable, which have attempted to
express this parameter as a finite number, as derived in the
accounting sense. Accounting values, such as cost per unit,
have not been usable general statements of propensity. Index
numbers or ratios are more adaptable to the general case and
are freed of some constraints such as magnitude differences,
scale problems and quality or time differences.
Systems Network Theory Considerations
The use of systems network theory is based on the
ability to model and characterize the behavior of individual
entities and their environment and to assemble these behav
ior traits into systems. A system in our context is defined
as, "A coordinated collection of physical elements and con
ceptual linkages intended to serve a common purpose".2 While
1Examples of the type of planning criteria applied
to airport locations and sizing used in the long range plans
are prepared by the Federal government and adjusted to local
conditions by local governments. For example see: Federal
Aviation Administration, Airport Capacity Criteria Used in
Preparation of the National Airport Plan, U. S. Government
Pr n ng ce, washington, D. C. 1968, and Federal Avaition
Administration, Planning the State Airport System, U". S.
Government Printing Office, washington,‘D. C. ,‘l968.
2See A. M. Lee, Systems Analysis Frameworks (New
York, Wiley & Sons, 1970), p. 18.
the physical sciences were first to utilize the systems
approach to problems, the social sciences have also developed
a considerable experience in the approach. From the defini
tion and recognition of economic systems the procession to
use of physical science techniques in certain aspects of
economic inquiry such as this dissertation is a natural and
evolutionary step. Systems network theory builds on the
definition and is particularly suited for those economic
applications seeking to maximize benefits gained from the use
of scarce resources.1
The Axiom and Postulates of Systems Network Theory
Systems network theory, as previously mentioned, is
based on work done in the electrical engineering field.
This body of knowledge proceeds from the elementary laws of
electrical circuits developed by Kirchoff,2 and specifies
that there are two considerations in a circuit, voltage and
current, and that these factors are complementary in nature.
Flows and flow variables are analogous to electric current
and "force" is analogous to voltage and electrical pressure.
The electric power system depends on the complementary nature
1The reader is particularly directed to the work of
R. Handy and P. Kurtz entitled A Current Appraisal of the
Behavioral Sciences (Great Barrington, Mass., 1964, Behavioral
Research Council),for an excellent overview of the challenges
to application of systems sciences to the behavioral science.
2See G. Kirchoff, Uber die Auflosung der Gleichungen,
auf welche man bei der Untersuchungen der Linearn verteilupg
GaIvanischer Strome gerfuhrt wirdTEngliSh translation,
TiansactIOn ofIthe Institute of Radio Engineers, CTLEU, March
1958, pp. A7.
10
of voltage and current just as flows of goods and funds are
dependent on demand or "force" in a business firm. There
fore systems network theory is more suitable than other pro
gramming techniques for those applications where "force"
must be considered.
Limitations in the ability to define the "force"
variable are the primary stumbling blocks in the use of the
theory, but also provide the key to future applications.
The method of expressing the components of a system
as utilized in the systems network theory is the linear
graph.1 With this graph it is possible to model the system,
assemble the linkages, derive component equations to mathe
matically express the linkages, and finally to control the
system so as to measure performance and auxiliary effects.2
It may then be possible to stabilize and redesign the compon
ents so as to provide for optimized performance.
The fundamental axiom of systems network theory is
that a mathematical model of a closed system characterizes
the behavior of the system as an entity and independently of
how the component is interconnected with other components to
form a system.3 This axiom is further defined by a series
of postulates which characterize the depiction of the system.
1See H. Frank, and I. Frisch, Communication, Trans—
mission, and Transportation Networks (Reading,_Mass.,
Addison:Wesley, 1971),TChapter 2.
2
See H. Koenig, Y. Tokad, and H. Kesavan, Analysis of
Discrete Physical Systems,(New York, McGrawHill, 1967),
Chapter 1.
3See Koenig, etal, 2p. cit., p. 3
These postulates are:
Postulate I.
Postulate II.
Postulate III.
Postulate IV.
1Ibid, p. 5
2Ibid, p. 6.
3
Ibid, p. 111.
11
The pertinent behavior of character
istics of each nterminal as an
identified system structure are complet
ely identified by a set of nl equations
in nl pairs of oriented complementary
variables identified by an arbitrarily
chosen terminal graph.1
The systems graph is defined operation
ally as the collection of edges and
vertices obtained by coalescing the
vertices of the component terminal
graph in a one to one correspondence
with the way in which the terminals of
corresponding components are united to
form the system.2
The algebraic sum of the "force" var
iables implied by the oriented edges of
any circuit in the systems graph is
zero.3
The algebraic sum of the flow var
iables around a vertex and forming a
cutset sum to zero. (A cutset of a
12
connected graph is defined as a set of
edges having the property that when
these are removed they divide the graph
into two unconnected parts and no sub
set has the first property).1
These statements form the foundation of systems net
work theory and the base for the research to be accomplished
in this dissertation.
Graph Methodology
The development of the research entailed the organ
ization of a closed system encompassing the study cities,
all located in the state of Michigan. These cities are
Lansing (LAN), Flint (FNT), and TriCities (Midland, Bay
City and Saginaw (SGN)), which form the regional boundaries
for the proposed location of the new air terminal facility
(MID), and Detroit (DET) which serves as the hub city for
air traffic in the state. The three cities form the second
largest concentration of population and industry in the state
of Michigan, second only to Detroit, and generate considerable
amounts of air traffic. The city areas are organized into a
network graph. This graph will be the mathematical equival
ent of the actual relationships of the air traffic flowing
between the cities and will represent the air terminal
facilities presently operating, in addition a proposed air
terminal facility will be entered into the graph. The
1
Ibid, p. 113.
13
location of the proposed regional air terminal facility was
determined by constructing three centerof—gravity location
models. The first was an unweighted geographical centroid
location. The second model consisted of weighted locations
based on population and population changes from 1970 to the
year 2000. The third model located the facility on the
weighted basis of personal income for the three cities for
the years 1970 through 2000.1 Sensitivity of demand for air
travel at short distances requires careful consideration of
location for air terminal facilities.
The different locations are plotted and if the study
is utilized at a later time may be used in the determination
of the actual location of the terminal if desired. The
locations and geographical coordinates are shown at Figure l,
and Table 1. It should be noted that the different economic
and demographic growth rates for the three cities cause the
location to shift toward east northeast of the geographical
center.
The graphic representation of this scheme, shown as
directed graph with the appropriate flow directions is shown
at Figure 2. The flows as shown which provide the basis for
the directed state are the actual numbers of airline passen
gers who use the air terminal facility in Detroit as a point
to assemble for further travel to other destinations.
Some features of the directed graph are unidirectional
1Appendix A contains a complete description of the
location models used for the regional facility.
1A
FIGURE I
Geographic Location of Proposed MID Facility
I w 7'17,” Geographic Location of
I’I" ° ”N "3'; Proposed MID Facility
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15
TABLE I
Geographic Coordinates and Locations
of Proposed MID Facility
n
Geographic Center Coordinates
North Latitude. . A30 05’ 59"
West Longitude. . 8AO OA‘ 49"
GeoPerCenter Coordinates
2333 ngih West
1968 A30 03’ 1A" 8A0 01' 15"
1970 A30 05' 51" 8A0 01' 57"
1980 A30 05' 31" 8A0 01' 50H
1990 A30 05' 17" 8A0 01’ A1"
2000 A30 0A' 59" 8A0 01' 30H
GeoPICenter Coordinates
Year North West
1968 A30 0A' 15" 84° 00: AA"
1970 A30 0A' 12" 8A0 00' A3"
1980 A30 0A' 08H 8A0 00' A0”
1990 A30 0A' 28" 830 58' 16"
2000 A30 05 28 830 58? 16
16
FIGURE 2
Directed Graph for Research Cities
17
flows along the edges (the air routes between the city pairs).
In graph theory this type of graph, as opposed to an un
directed graph (one in which there exists bidirectional
flows along the edges), consists of a set of elements called
vertices and a set of ordered pairs of vertices called
directed edges. The set of vertices is denoted by the symbol
V and the set of directed edges by the symbol 8. The graph
is finite if both V and B are finite sets, as they are in
this case. It must be noted that there can at most be one
edge from any vertex (airport terminal facility) vi to any
other vertex v3.
The "force" variable applicable to each vertex (air
port terminal facility) is developed so as to reflect the
relative demand of each city pair. This variable is described
below with its development methodology. The importance of
the "force" variable in characterizing the total graph is of
primary importance at this point. One of the properties of
a graph, Postulate 3, is that all "force" variables sum to
zero around a circuit. A circuit is defined as a selected
set of connected branches or edges that form a unique closed
path in a systems graph; in other words a circuit is a set
of selected edges that form one and only one closed circular
path. The criticality of the term "directed graph" now is
paramount since the orientation of the edges around a circuit
provides the positive or negative value associated with each
"force" value.
0f the salient points of systems theory, the
18
requirement that in a closed system all the "force" or demand
values sum to zero around a circuit is very important. In
the case of the systems graph utilized in this study an
additional vertex was required so as to retain the properties
of a closed system. This is required since some travelers
originating and terminating at the research cities do not
move through the Detroit air terminal. The additional vertex
(AOD) serves as the origin and destination for all travelers
moving to and from the four cities to points outside the
system. Thus the systems graph as shown in Figure 2 describes
all airline travelers to and from the base cities of Lansing,
Flint, and Bay CitySaginaw. The inclusion of the additional
vertex also serves as a device allowing the circuits to meet
the systems network criteria specified in the four postulates
and the basic axiom.
The Research Objective
The objective of the study is the development of a
suitable "force" variable capable of depicting the demand
for air passenger transportation originating and terminating
at the three base cities, and to portray a suitable method
for deriving this function in socioeconomic applications
in order to exploit the advantages of systems network theory.
The "force" variable centers around the service levels of
airline passenger seats utilized at the base cities to the
Detroit air terminal facility and the availability of
alternate surface travel options for the traveler. The
development of the "force" variable is considered in the
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second chapter of this dissertation.
The third chapter develops the systems network model
utilized to test the applicability of the "force" variable
to the airport feasibility study. The model is designed to
forecast the flows and demand conditions for airline travel
developed for the time period 1968 to 2000, and the effect
of the operation of a regional air terminal facility on the
traffic flows between the new facility, the three base cities
and Detroit. It generates the total demand (force) for air
travel at the regional facility and provides a methodology
for comparison of total demands for airline travel with and
without the proposed facility.
The fourth chapter synthesizes the conclusions drawn
from the investigation, and describes the implications and
limitations of the investigation for planning, and other
possible applications.
CHAPTER II
CHARACTERIZATION OF THE "FORCE" VARIABLE
Introduction
This chapter is concerned with the deveIOpment of the
"force" variable function used to characterize the demand for
air travel in the base cities used in this study. The three
base cities and Detroit form an excellent example of the
"hub and Spoke" airline transportation system used for
domestic air carrier transportation in the United States.
Truck line service is provided between high density (hub)
airports and is characterized by use of such aircraft as the
Boeing 707 and 7A7, McDonnellDouglas DC8 and DClO, and
Lockheed TriStar. Feeder service is provided by airlines
between hub airports and medium and low density airports.
This service is characterized by the use of aircraft such as
the Boeing 727, McDonnellDouglas DC9 and Convair AAO air
craft. This system is designed to provide air service to
cities which could not support full trunk line service to and
from major pOpulation centers. This is accomplished by pro
viding feeder service to larger population centers, which
then perform the assembly function of pooling small groups
of passengers from remote locations into groups of sufficient
size to support trunk line service to major population
centers. The Federal Aviation Administration (FAA) plan, for
20
21
the period 1971 through 1980, shows development and funding
for thirty hub airports and 1A9 medium density air carrier
airports.1 The three base cities in this study all
qualify as medium density airports, and Detroit is classed
as a high density hub airport.
Since the hub airports perform the assembly and dis
assembly function within the system, the number of origin
destination passengers per capita is higher than that seen
at the Spoke cities, reflecting the numbers of passengers
moving through the hub city for further movement to other
destinations.
A graphic diSplay of this arrangement is depicted in
Figure 3, which shows that actual origindestination traffic
for large centers of population exceeds the national average
while smaller cities show a lower than average of traffic
per capita. The diagram (figure 3) is constructed such that
the actual amount of air travel per capita originating at
cities of various population levels shown on the ordinate
and the forecast amount of air travel shown on the horizontal
line are drawn at the same scale. Any city which has a level
of air passenger traffic equal to the national average fore
cast would be shown on the A50 line. The A50 line has the
1High density (hub) airports are those that enplane
over one million passengers annually, while medium density
airports are those that will enplane from 50,000 to one million
passengers annually. For an excellent description of the
planning funding of the national airport system see, The
National Aviation System Flap, Ten Year Plan,_ 19711986,
Department of:Transpoftation, Federal AviatIOn Administration
(Washington, 1970).
22
Figure 3
Typical Airline Service Offered at Cities
Air Travel Consumption Line
(Actual Travelers
from City)
Naﬂonal
Average
Spoke Regional
Cities I Airpori
Cities (Medium Density/‘
w/o SVC ‘ Airports)
Hub Cities
(Hi h DensirLAirporteII
/ FORECAST AIR TRAVELERS FROM CITIES
ACTUAL AIR TRAVELERS FROM CITIES (NATIONAL AVERAGEI/l
l/Air Travelers per capita (l968 equals LO?
/ 0 8 D pox/person
/
Source: Civil Aeronautics Board, 1968.
I
I
I
I /
II
23
property that the indicated forecast level of air travel,
measured by the vertical distance of the point from the
horizontal axis is exactly equal to 100 percent of its actual
level of air travel on a national basis as measured by the
horizontal distance of the point from the vertical axis. For
instance in 1968, the Detroit air terminal had a total of
6,823,960 origindestination passengers move through the
facility. During that year the national average of air
travelers was 1.07 per capita, the Detroit facility had an
average of 1.3A origindestination passengers per capita.1
Detroit would be located in the hub city section of Figure 3.
Conversely, the Michigan cities which provide transfer
passengers to the Detroit air terminal facility had the
levels of passengers per capita listed in Table 2 below.
This may be due in large part to the travelers within the
TABLE 2
O & D Passenger Over Capita for
Selected Michigan Cities
PBpulation O & DIPassengers O—ED_=
City (000's), ,per capita Passengers
Jackson 1A9 .09 1A,6l6
Flint A87 .33 158,95A
Lansing 361 .71 25A,511
Bay CitySaginaw 372 .75 268,805
Grand Rapids 51A .83 A2A,255
Detroit 5,015 1.3A 6,823,960
Source: TMinhiganTDepartment OTCommerce, State Aeronautics
Commission, 1968 Data.
lSee Ori inDespination Survey of Domestic Airline
Passenger Tra ic, 1968: Civil Aeronautics Board, washington,
D}C.
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system electing to drive to the hub city rather than utiliz
ing the local air service.
It is apparent that the base cities analyzed in this
study and shown in Table 2 above have lower than average air
line passengers per capita than the national average of 1.07
for 1968. The Table shows that as city size increases the
rate of airline travel increases also. This corroborates the
information shown in Figure 3. The Figure also shows that
there is a range of city size that should have a level of
service equal to the national average (regional airports).
It would seem then that at this particular city size the
economies of service on the part of airlines which provide
access to the city are such that a sufficient number of
passengers are generated so that trunk line operations to
hub cities is feasible. This exact relationship is limited,
of course, to the cities depicted in the system being analyzed.
The element of distance is also not considered in the graph,
but will be shown as a vital element in the relationship of
demand for the modes of travel. The intersection of the air
travel demanded at the Michigan system cities is an indication
of the number of travelers and combined cities sizes that are
required at a single facility to ensure a supply of air
service equal to the national average, that is the level of
service which would enable trunk line service direct to and
from major hub cities. The total system demand is balanced
by the increased demand at the hub city resulting from the
origination of air travel at the hub city by those travelers
25
who elected to use surface travel for the first leg of their
journey rather than utilize local air service.
Demand Factors for Air Travel
Several investigators have attempted to develop the
empirical relationships of demand for air travel.1 One
such effort considered and determined the effects of distance
as it concerned the attraction between two cities.2 The
authors related three considerations of the distance factor,
which they called the primary reason for utilizing air travel,
they were:
1) The distance between two peOple may be related to
the probability that an occasion for communication
between them will arise. As we move out from any
point in the economy, the variety of demands that
can be satisfied rises as the distance from the
point increases; the selfsufficiency of larger
areas, other things being equal, is greater than
that of smaller areas.
2) Distance is related to price and may be taken as
proxy for the cost of the trip.
3) Distance is related to the competitive position
of different modes of travel; in particular, there
is likely to be no advantage in traveling by air
instead of by some other means of transportation
Some examples of work accomplished in the area are
given below. P. Cherington, "The Domestic Market for Air
TranSportation", li ht Forum (Sponsored by Connecticut Gen
eral Life Insurance ompany), July 1962, pp. l—lO. "Benefits
from a National Air Service Guide", an excerpt from testimony
by G. Burnard before the U.S. Senate, Committee on Commerce,
Review of the Local Air Carrier Industry, Washington, D.C..
USGPO, ‘1966, pp. 3359337”. "The‘EconomIEs of Convenient Air
line Service" , Tijdschrift voor Vervoerswetenschap, No.3,
1966, Netherlands Institute of Transport, pp. 2171233 (re
printed in Passenger Transport Michigan State University
Business Stﬁdies, 1968).
2See J. Lansing, J. Liu, and D. Suits, "An Analysis
of Interurban Air Travel", Quarterly Journal of Economics,
(February 1961; pp. 8795.
26
if the distance is less than some minimum number
of miles. Beyond that distance, the time saved
by air over the other modes may be expected to be
roughly proportional to the number of miles to be
covered. There is reason to suppose that the
proportion of all travel which is by air will be
close to zero for very short distances and tend to
increase with distance.
These three factors of distancemass attraction, distance
cost and distance—modal choice constitute the source of
demand for airline travel, in fact for all travel. They also
form the basis for the derivation of the "force" variable
utilized in this dissertation.
DistanceMass Attraction
The understanding of the attraction of pOpulation
masses for retail sales is well known, and is generally so
widely accepted as a measure of the power of trading.area
that it has been granted the cognomen of a "law". First
propounded by Reilly over forty years ago, the "law of retail
gravitation" expresses the relationships of city size and
distance as they pertain to the ability of trading centers
to attract patrons.2
The relationship is a linear one with
attraction a direct function of the ratio of population and
an inverse relationship of the square of the distance sepa
rating the two cities. This relationship is also expressed
in the distancemass attraction statement quoted in the pre
ceding paragraph. Our statement also states that as the
19p.p;3., p. 89.
2See W. Reilly, The Law of Retail Gravitation (New
York, William J. Reilly, 1931).
27
size of the city increases the selfsufficiency increases
and that smaller cities are less able to support themselves
and provide necessary service. This implies something other
than a straight line relationship, possibly some curvilinear
increasing function as city size increases. A revision of
Reilly's original formulae by Converse substantiated the in
ability to express a straight line relationship when city
size differences exceeded multiples of twenty, the predictive
power of the original statement is reduced.1 There are limits
then, which must be observed in the application of the Reilly
Converse formulae, the comparison of retail power must be
used only for cities of similar sizes. This would obviate
the use of such a model for use in this "force" variable
determination, since there is such a large diSparity in the
sizes of Detroit and the three base cities. We must search
further for a suitable determinant.
DistanceCost
Distance and cost relations are well established in
the field of tranSportation where, as in the statement of the
cost and price relationship above, there is a direct correla
tion. As distance increases the cost of moving that distance
increases directly, though there are instances where there
are discounts, called rate tapering, for trips of extended
lSee P. Converse, A Study of Retail Trade Areas in
East Central Illinois, Business Studies Number 2 (Urbana,
IllindiS., The Univeristy of Illinois l9A3).
and
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lengths.l While in the traditional depiction of demand price
is the determinant of quantity, the demand for travel must
contend with competitive modes, time, convenience, service,
accessibility, safety, and price. For these reasons the
factor of price or cost does not lead us nearer to a suitable
measure of "force" without considerable adjustment for the
aforementioned factors and for the heterogeneity of the
individual traveler and his utilitypreference values. In
addition, because of the prevailing differences in average
fares per mile for different citypairs, principally because
of the availability of lower priced coach service on some
routes, there are other possible abattoirs which trap the
investigator using price alone as a measure of demand for
air travel. The difference in citypair markets, pOpulation,
income, and tastes in these markets also mitigate against
the use of price alone.2 In addition, where different types
of carriers (trunk, local, or third level) Operate on the
same route segment or within the system being analyzed the
apparent differences of equipment, times and connections
along with price may very well consist of the full aspects
of what could be called the costprice aSpect of the "force"
lSee D. Pergrum, Transportation EconomicsAgnd Public
Policy, (Homewood, Ill., Richard'D. Irwin Inc., 1968).
2For an example of a study of the elasticity of air
fares and demand for air travel on a national basis see S.
Brown, and W. Watkins, "The Demand for Air Travel: A
Regression Study of TimeSeries and CrossSectional Data in
the U.S. Domestic Market", Paper  A7th Annual Meeting,
Highway Research Council, WaShington, D. C., January 16,
1968.
29
variable. For this reason (the incalculability of the price
aspect) this phase of the typical demand function was not
selected for use directly in the "force" variable construction.
DistanceModal Choice
While the factor of price is not directly utilized
in the analysis for the derivation of the elements of the
demand function, the factor of modalchoice certainly con
tains as one of its elements the factor of price or cost.
The traveler, in making his choice of mode, takes into account
the elements of time, cost, and distance when making the
decision prior to initiating travel. An important element
appears to be travel time differences and the traveler
attaches a value to this time, just as he measures the value
of accessibility to the terminal for each mode, the schedule
convenience, vehicle delay at the terminal, and average
station wait. This timevalue concept then forms the basis
for measuring the total cost of the blocktimes for each
mode of transportation utilized in reaching his destination.
A comparison, planned or unconscious, is made by the traveler
and the decision made as to the choice of mode to be utilized
in traveling. It would seem that travelers are willing to
accept certain penalty costs in order to save time and that
when these penalty costs exceed a certain level, the less
expensive mode is selected. One study on the time—value
coverage principle states that most air travelers are willing
to accept a $2 to $3 penalty cost just to save one hour
traveling by air, over and above a slower but cheaper means
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on the surface.1
Modal Choice in the Research Cities
The aspects of modal choice decisions in the research
cities for air travel required to move through the Detroit
facility from the base cities was clearly demonstrated in a
study of land use in the Detroit metrOpolitan area accom
plished in 1968. This study, which measured the anticipated
land requirements for various industrial needs, services,
highways, and population, estimated the composition of air
travelers moving through the Detroit air terminal facility.
This was accomplished through the use of a passenger survey
conducted by all the airlines serving the city.2
This sub
study of the Detroit Regional Transportation and Land Use
Study (TALUS) provided much valuable input for the data used
in this study.3
Passenger information taken from respondents
originating at the three base cities substantiated the modal
choice factor previously stated.
The travelers from these cities and their choice of
modes used to move to the Detroit facility was in general
agreement with the statement that there was some minimum
lSee R. Rice, "Time and Cost in Carrier Competition."
Passenger TranSportation., Edited by S. C. Hollander, East
LdnSing, Michigan.,‘MSU Business Studies, 1968, pp. llAll7.
2See Travel Patterns & Characteristics of Airline Pas—
sengers, Detroit MetropOIitan Airport, 1968, Wayne County Road
Commission (Detroit, Michigan, 1969).
See Detroit Regional TranSportation and Land Use Study.
Southeast Midhigan COuncil onGovernmentS (Detroit, Miéh.,
1969). The survey referenced in 2 above was accomplished by a
joint effort of the two agencies listed and was not included
in the TALUS publication.
31
distance at which almost no demand for air travel could be
generated for local air terminal. A graphical portrayal of
this distance effect on modal choice is shown at Figure A.
The graph shows that at a distance of forty miles almost
all travelers elected to utilize surface transport rather
than using the available airline service connecting the city
of Jackson and the Detroit air terminal facility. The effect
of distance at greater ranges is also shown by the proportion
of travelees choosing to fly from Grand Rapids rather than
drive to the Detroit airport. From this diSplay of infor
mation, one can infer that the distance of forty miles con
stitutes the indifference point at which the timevalue of
air service is overcome by the less expensive means of
driving to the hub air terminal facility. There is, of
course, the long distance end of the Spectrum which shows
that almost no one would choose to drive to the hub air
terminal facility when the distance exceeds 180 miles. This
modal choice on the part of the traveler constitutes an
important factor in the demand for air travel.
From Figure A one may also infer that there is some
fixed waiting time (the distance of forty miles) associated
with air travel which the traveler can save by using surface
means to get to the hub facility. This waiting time, when
equated to a driving Speed of sixty miles per hour by auto,
is approximately equal to the time to park an auto, be
ticketed, and wait the thirty minutes asked by airlines of
passengers.
32
If this is the case the traveler then makes a decision
as to the savings afforded by air travel and as the distance
to the hub airport increases a greater prOportion then elect
to travel by air. Not all the travelers attach the same
value to time, but the relationship in our example in Figure
A is linear, that is, the same proportion elect to change
modes per mile of distance. This is not surprising since it
reflects the changing elasticity of demand for airline travel
as distance increases. The demand for air travel becomes
inelastic since the largest prOportion of travelers have
already been switched over to the airline mode thus it takes
a larger time savings to cause the last numbers to move into
the air travel mode.
This time ratio (surface driving time over air travel
time) is critically important to the demand for air travel
because of the higher cost of air travel compared to the per
ceived cost of driving an owned auto. Since the traveler
already possesses the means to move by surface, the additional
cost of air travel over and above the cost of surface move
ment must be accompanied by a real time savings. As the
time ratio in favor of air travel increases the number of
travelers choosing the air travel mode increases after the
forty mile distance and out to the 180 mile distance. This
travel time ratio rather than the utilization of either cost,
time, or convenience alone shows the demand for air travel,
since it embodies all the elements as perceived by the
traveler. It also depicts the relationship shown in the
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third distance factor (distance—modal choice) described
earlier. The distancemodal choice factor as stated above
also affects the first two distance factors of mass and
cost as described through the prOportionality of travel
assigned to each mode by the fixed time and Speed difference
of air to surface travel. In the city combinations being
investigated in this dissertation the travel time ratio,
surface travel time divided by air travel time, ranges from
a factor of 2.AO to a factor of 3.31. This ratio is derived
from highway travel times from the center of the Spoke city
to the Detroit airport for the surface mode and equated to a
speed of sixty miles per hour, and from the actual flight
time from the Spoke city airport to the Detroit airport as
published in the airline schedules. The surface travel
time becomes the numerator of the ratio and the flight time
the denominator. Thus the ratio is able to express changes
in technology in either of the two modes. Improved highways
which would lower surface travel times would adversely affect
the demand for air travel, while improvement in flight times
sould increase the demand for air travel. In the relatively
short distance included in this system, the fixed component
of the air travel mode (the forty mile minimum distance) as
perceived by the traveler represents the greatest area for
improvement since it represents to the traveler the greatest
single time factor. As will be shown this factor is depicted
in the portrayal of the demand factor selected for this
systems problem, and thus is considered automatically when
35
stating the relative demands for the surface and air travel
modes, as used in the determination of systems flows and
"forces".
The Service Level Ratio Effect
The actual service levels at the Spoke cities reflects
the relative demand between the several cities and with the
national system airline. Service level is stated as the
number of origindestination passengers per capita, a
standard Civil Aeronautics Board (CAP) term. An origin
destination passenger is defined as a single boarding and
disembarkation with no immediate stops enumerated. As dis
tance increases the demand for airline service increases
because of the timevalue savings accrued by the traveler
when using air tranSport. This is manifested by a greater
demand per capita at greater distances and a lower demand at
short travel distances, due to the higher proportion of
travelers electing to fly rather than drive at the longer
distances. This effect is reflected in the service level
Offered at the local Spoke city air terminal facility, not
in the number of aircraft arrivals and departures, but in
the actual passenger boardings which in the long run depict
the actual service level. The total system demand is
balanced in this case by increased demand at the hub city
resulting from the origination of air travel at the hub city
by those travelers who elected to use surface travel for the
first leg of their journey rather than utilize local air
service. This shows the individual preference and
36
timevalue associated with the distance between the origi
nating spoke city and the hub city. In order to establish a
reference point the national average for the year being con
sidered is utilized as a base value and the service levels
at the air terminal facilities being studied is computed as
a percentage of that value, in the case of Detroit in 1968
the value assigned to the service is equal to 1.3A, while
that associated with Lansing is .71. From these two values,
a service level ratio has been constructed using the hub
city (Detroit) value as a denominator and the Spoke city (in
this case, Lansing) value in the numerator. The resulting
service level ratio is a number less than one for all
Detroit/Spoke city combinations i.e., LansingDetroit within
the "hub and spoke" system of Detroit and its Spoke cities.
An additional constraint on this value is that it be less
than l.OO/l.3A, that is that the city being considered has
an associated service level equal to the national average,
which would indicate that it is capable of being independent
of the air terminal facility located in Detroit, having no
need of the assembly function accomplished by the Detroit
facility for its' Spoke cities. This service level ratio
as stated comprises the second element of the ”force"
function utilized in the system study. It relates the "force”
function that is equal to the demand for air travel at the
base cities being considered in this dissertation to the
travel time ratio as perceived by the traveler.
PAC mule ..___ _ _._
37
The "Force" Function
The two elements of travel time ratio and service
level ratio comprise the "force" function as developed for
use in this application of systems network theory. Together
they express the relationships of demand or "force" between
the system base cities and Detroit. The ”force" function is
graphically shown at Figure 5. It depicts that the service
level required at a spoke airport in the system is related
to the travel time ratio by the following relationship.
F = a + b + CT2
F = Service Level Ratio (Spoke city/
Detroit) (Force Value)
T = Travel Time Ratio (Surface Driving
Time/Air Flight Time).
In this form the "force" equation is:
F = .003 — .306T + .151T2
This form of the equation depicts the fact that demand for
air travel is equal to zero at travel times ratios less
than 2.0, increasing rapidly then increasing at a decreasing
rate as the service level ratio approaches .75 (the ratio of
l.00/l.3A). This is, of course, the ratio of independence
from use of the Detroit air terminal facility.
From the graph one can ascertain that the area to the
right and below the "force" curve contains the feasible area
for service at spoke airports. This is a level lower than
that offered presently but within the modal choice possibility
constrained by the travel time ratio which fixes the limit of
traveler decisions.
38
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39
Changes in the travel time ratio will cause the "force"
curve to shift either to the left or right. Improvements in
air service through reduced travel times causes a shift to
the left reducing the fixed component (minimum distance at
which the traveler first begins to choose air travel) of the
air travel demand. Likewise, improvements in surface trans
portation enabling the traveler to reach the air terminal
facility in less time will cause the "force" curve to move
to the right, thus reducing demand for air travel at the
affected Spoke city.
For the purposes of this dissertation, it is assumed
that there will be no improvements in the travel time ratio
(that is to say that there will be no high magnitude changes
in the other mode). This allows only a vertical shift in
the "force" curve due to changes in the relative growth
rates at the various spoke cities over the period being con
sidered. Rather the changes in technology that are foreseen
will be incremental improvements in both the surface and air
travel modes, resulting in only slight travel time changes
and little or no ratio value changes.
Projecting the "Force" Function
The "force" or demand function values in the "hub and
Spoke" system as constituted in this study are based on the
relative values of the local demand for air travel and com
pared to the national average per capita. In the closed
system design used in the systems network technique each
edge possesses unique "force" and flow variable values. The
“0
”force” variable is directionally oriented on the basis of
passenger flow, in order to fulfill one of the postulates of
systems network theory (Postulate IV) requiring flows around
circuits to sum to zero.
The "force" function that drives the systems model is
the air travel demand generated at each city in the system.
This "force" value is based on the forecast growth in pop
ulation and personal income during the period being considered
(1968 to 2000). These factors are the same ones used by the
CAB for estimating the number of passengermiles per capita
on a national basis.1 Adapting the general formulation to g
the local system, the forecast measures the passengers per
capita growth rate as a function of time, based on the
changes in personal income and population. In addition, the
formulation provides for growth in acceptance of air travel.
The formula thus developed is:
PPCi =. PPCi_1 + .0725(l.3Peq_+ 1.1 PIi+.3AR)
PPCi = Passengers per capita
Peri = Population Growth Rates per annum
PIi = Personal Income Growth Rate per annum
Ar = Growth in acceptance of air travel
(equals .3 per annum)
i = Forecast year
The factors in this formulation were applied to the
research cities in this analysis and a forecast of air
1
See footnote 1 on page 28 (Brown and Watkins)., The
authors develop the rationale for this formulation for pas
senger—miles on a national basis using regression analysis.
ui
traveler rates was constructed for the years 1970, 1980,
1990, and 2000. These forecasts are used in establishing
the particular "force" value for each city in the systems
model, and are shown in Table 3 below.
TABLE 3
Passengers Per Capita Forecast
for Selected Michigan Cities
City 1970 1980 1990 2000
Flint 1.09 1.36 1.u1 1.u6
Lansing 1.06 1.31 1.36 1.u0
TriCities 1.06 1.28 1.35 1.39
Detroit 1.37 1.46 1.53 1.57
VA.“I
CHAPTER III
APPLICATION OF SYSTEMS NETWORK METHODOLOGY
Introduction
This chapter is concerned with the systems model of
the traffic flows and system "forces" in the airport system
formed by the research cities. The model provides a view
of the air travel system and establishes the interrelation—
ship of the flows and forces within the system and for all
origins and destinations outside the four city system being
analyzed.
Structure of the System
The system is conceived as having two sectors:
(1) the presently Operating airline system, and (2) the
proposed regional airport system. The present system is
composed of the local airports and the air and highway links
with the Detroit facility. The prOposed regional airport
system is composed of the highway links between the research
cities and the regional airport and the air link with the
Detroit terminal facility. Each of the systems had its
interfaces with the national airport system.
The base cities are considered to generate a "force"
or demand for air travel, stated and measured in passengers
per capita, and through this demand a flow of air passengers
42
'U‘ '..
43
from the city is produced. This flow of passengers is
measured in passengers per year. The model differentiates
between the travelers driving from the local city to the
hub airport at Detroit and those utilizing the local air
service. This differentiation is shown in the model by one
path (edge) for surface travel and one path (edge) for air
travel between the spoke city and the hub city (e.g. Flint
and Detroit).
Each of the airports in the system has an interface
with the national airport system. This is evidenced in the
actual system by the airline connections with other des
tinations than the hub city of Detroit.
The proposed regional airport is included in the
system and is Joined to the three base cities and Detroit
by a similar set of edges linking the base cities. The
"force" or demand for this vertex (airport) is considered
to be a function of the three base cities.
The demand generated at the base cities is transferred
from the local air terminal facilities to the new regional
airport, thus leaving the local airport without airline
service. This transfer is affected when the relative demand
for airline service at the three base cities reaches a
critical level.
The systems model, as designed, Operates on an
iterative basis being a state sequential mixedparameter
type. This type of model is characterized by its ability
to provide successive time period information based on prior
"J
3
1
I
99
i
)
at
system states. Mixedparameter refers to the fact that the
model utilizes a hybrid state equation to describe the
actions of the system. In this type of model the state
equation uses both "force" and flow information in the
description of the action of the model. The state equation
in this study is a discretetime type in that the variables
in the state equation are linear functions of time dependent
functions and current systems variables.
The state equations for this model take the form:
S(n/l) = P'S (n) / Q’E(n)
where E
n is an integer representing discrete
points in time.
S(n) is a vector of system variables at
time n.
P & Q are constant matrices containing
system values.
E (n) is a vector of variables dependent
on time.
The model is observed at discrete points in time, in
our case on a yearly basis. The output as the state vector
at time n becomes a portion of the inputs to the system for
ensuing time period. This recursive cycling is dependent
also on the "force" and flow of the present time period.
In this way the model represents the actual system in
operation, where forecast system states in one time period
depend to some degree on the conditions prevailing in the
previous time period.
The systems model also incorporates an output equation
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ii ’
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_
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s
‘ a
P)
n...
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uy‘
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45
system which contains the system outputs at time n in order
to more closely observe the interactions and changing con
ditions within the model structure. This output vector
takes the form:
R(n) = M‘S(n) / N'E(n)
where
R(n) is the output vector containing
variables of interest in the systems model
at time n.
M & N are constant matrices containing
system values.
I suer 'QtLl 5.11;"
Both the state equation section of the model and the
output vector section of the model utilize identical inputs.
This provision allows the lateration of a minimum set of
input variables in order to change internal conditions
within the model and to simplify the simulation of changing
economic conditions external to the system being analyzed.
The equations for both the state and output sections
of the systems model are derived from the characteristics
of the linkage between each of the system graph vertices.
The systems graph utilized in this study differs slightly
from the general systems graph shown in Chapter 1, Figure 2.
The systems graph of the airportcity linkages is shown at
Figure 6. The model consists of 28 separate edges, each
represented by complementary "force" and flow. The state
equations provide primary information on the system condi
tions. The model deve10ped in this study utilizes all
system outputs in the response equations for ease of in
Spection and determination of the intrasystem relationships.
46
FIGURE 6
22
IO
4—3
7 26
System Graph For Component Model
19‘"
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tn
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I.
47
The Mathematical Relationships
Each of the cities being analyzed in this study
possesses different "force" and flow characteristics. The
complementary relationship between "force" and flow variable
allows us to specify one of the variables if the other is
known or can be calculated.
The equation in the systems model are of two different
types called (1) dissipative components and (2) dynamic
components. The equations which are of the dissipative
type are considered as passive relationships within the
model. That is, they do not provide for changing relation
ships within the characterization of the description of the
edge being modeled. These dissipative equations are of the
type:
Y(n) is the flow value at time n.
X(n) is the ”force" value at time n.
a is a constant parameter indicating the
relationship between Y and X.
The equation for edge 10 in the systems graph is:
Y (n) = 9u3x (n),
10 l
where
Y equals the number of air travelers
lO driving from SGN to Detroit.
X equals the "force" value for SGN
l
943 equals the transformation coefficient
for the edge.
The equation states that the annual flow of passengers along
48
the edge is equal to 943 times the "force” value developed
at SGN in the time period.
The a parameter indicates the linear relationship
between the flow and "force" variables describing the edge
in the model. It is equal to the slope of the line measuring
the characteristics of the flow and "force" variable. This
linearity can be considered to be true within a limited range.
Another characteristic of this type of equation is
that a change in sign of one variable signals a change in
sign of the other variable. In the case of the model in
this study, changes in sign of the "force" variable would
amount to the change in direction of flow between the cities
being analyzed.
The second type of equation used in the system model
is the dynamic component. This type of equation is char
acterized by the time aspect, the flow variable being .
dependent at subsequent time states on the values of the
"force" and flow component values in previous time stages.
This type of equation is of the form:
Y(n/l) = PY(n) / QX(n)
where
Y is the flow variable.
X is the "force" variable.
In the above equation the flow variable takes on the values
of the flow and "force" variables of the previous time
period, the flow variable of that time period being dependent
on the values in preceding periods.
’ 2H“. . v
'rJ
49
The equation for edge 5 in the systems graph is:
Y5(n/l) = .07 Y5(n) / H913 Xl(n) / 71.5
X (n),
3
where
Y equals the number of air travelers who
5 depart the SGN airport for destinations
other than DET. :,
X equals the "force" value for SGN. E Jim”
1 f
X3 equals the "force" value for DET.
.07, #913, and 71.5 are the transformation
coefficients for the edge.
The equation states that the passenger flow in the next e—*
time period is equal to .07 times the flow in this period
plus 4913 times the "force" value at SGN and 71.5 times the
"force" value developed at DET in the present time period.
These two types of component equations form the
model of the system being analyzed in this study and it
now remains to specify the relationships of these equations
in the model.
Component Equations
The component equations for the systems model used in
this study can be grouped into three like sets, differing
only in the coefficients used in the determination of the
relationship of the "force" and flow variables. Each of
the three base cities utilizes the same pattern of edges to
describe its relationship within the airport system. The
equation matrix shown in Figure 7 summarizes the systems
graph shown in Figure 6.
Figure 7
Component Equation Matrix
50
E D G E
Passengers
transferring
ﬁfsm Hub to
27
assengers
transferring
to Hub from
MID
28
"Force" for
air travel
Passengers
leaving
system
7,261
Passengers
driving
from MID
23
21
19
Passengers
driving
to MID
2O
24
22
Passengers:
from
Hub
l6
17
18
Passengers
flying to
Hub
13
15
lb.
Passengers
driving
to Hub
10
ll
ll
12
Flint
Lansing
TriCities
Detroit
A I I O
1/ Edge 26 collects and separates those flows which originate
base cities.
in the three
~19
51
Edges l, 2, 3, and u prescribe the external "force"
values for each of the cities in the system. The complemen
tary flow variable is equal to the number of passengers
moving into the system. These edges correspond to the
drivers of the system providing energy for system operation.
These equations take the form:
3 (n) .. F(n) “1
l, 2, 3, 4
These system "force" variables are taken as known functions ‘
of time and are derived from economic forecasts of air i
passenger traffic as developed in the previous chapter. E__
The edges 5, 6, 7, and 8 are the edges which represent 6
the air travelers leaving the system at the base cities.
The equation for these edges take the form:
Y (n/l) = PY (n) / QX (n) where
i i i
i = 5, 6, 7, and 8.
These edges form a portion of the state equation system and
provide the recursive sector of the model. These equations
state that the flow in the subsequent time period is a
function of the flow in this time period and the "force" in
this time period. In the use of this type of equation in
the model, the (differential) demand or "force" is utilized
in the equation reflecting changes in growth patterns.
The edges 10, 11, and 12 represent the travelers who
choose to drive to the hub city rather than to utilize the
available local air service. The proportion of travelers
electing to use this mode at any given distance in the
52
system is shown in Figure A. The edges represent the differ
ence in the "force" function value and the actual demand
for air travel originating at the city to which the edge
pertains. These edges take the form:
(Y. = G'X.
.¥5~th_ _J
where
i = 10, 11, or 12.
J = l, 2, or 4, dependent on the city
from which the edge originates.
G = constant parameter relating the
"force" at city J times the pro
portion driving to the hub air
port. This value may be unique
on each edge.
These edges are of the dissipative type and indicate the
direct transfer of "force" and flow within a single time
period.
Edges 13, 14, and 15 correspond to the travelers
flying between the base city and Detroit. The "force"
values on these edges are the system "force" values derived
in the previous chapter. These edges are of the same type
as edges 10, 11, and 12, and have the same mathematical
relationships. These edges transmit the "force" developed
at the base cities to the Detroit terminal facility for
further movement into the national airport system.
Edges 16, 17, and 18 correspond to the return flow
of travelers from the Detroit terminal facility to the base
cities. These edges combine the travelers driving and
flying. It is assumed that the prOportions of driving and
‘0
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at:
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b v
A14
and
a .
53
flying travelers returning to the base city are the same
as those who moved to the Detroit facility. These edges
are of the same type as the outbound edges from the base
cities (10, 11, 12, 13, 14, and 15), and possess the same
property of transmitting the flows without any loss of
"force".
The edges 19, 21, and 23 are the edges which corres
pond to the travelers who will drive from the proposed
regional airport to the base cities. These edges are also
of the dissipative type directly transmitting the "forces"
and flows generated at the base cities and returned through
the regional airport.
Edges 20, 22, and 24 are the edges which correspond
to the travelers driving to the prOposed regional airport,
and are of the dissipative type as are edges 19, 21, and 23.
These edges directly transmit the flows of passengers from
the base cities. When the regional airport (MID) is
activated these edges will accomplish the same activities
as the combination of the edges which radiate from the base
cities during the time the local air service is in action.
For example, at the FNT facility edges 11, 14, and 6
account for the outbound flow of travelers from the city
while the local airport is in operation; when the regional
facility is activated local service will be terminated and
all airline service will be offered at the regional facility
for the city and edge 20 will perform the function of
moving travelers outbound from the Flint area by surface to
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the regional facility.
Edges 27 and 28 correspond to the travelers who will
utilize the MID facility as a transfer point to and from
the Detroit facility. It is assumed in this study that if
the level of service offered at the MID facility can be
increased to a critical level there will be some trade—off
of service between the Detroit facility and the proposed
MID terminal and that there may be some transfer of passen
gers between the facilities for further movement in the
national airport system.
Edge 26 in the systems graph is designed to total
the travelers moving through the Detroit facility and that
originate at the base cities. The equation for this edge
is of the dissipative type.
The two remaining edges (9 and 25) in the component
model are of primary interest in the system and as such are
included in the state equations of the systems model. These
edges correspond to the travelers who will enter and leave
the three city complex formed by the establishment of a
regional airport serving these cities. The equations for
these edges are of the dynamic type and are expressed as
follows:
Y (n/l) = PY (n) / QX / QX / QX / OX
25 25 1 2 3 a
X (n/l) = PX (n) / QX / 0x / QX — QX
9 9 l 2 3 u
where
Q is a constant value relating the flow variables
to systems "forces".
Pf.
.
1!?
4 u—
n
...v .
I
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an...
5“"? 9
vJ.~._
a, ‘r
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55
These equations are combined into the systems inter
face model, the state equation sector and the output vector
as functions of the systems input vector, and capable of
solution by conventional mathematical methods. The set of
component equations are shown in Figure 8.
Characteristics of the Model
The system of equations, depicted in Figure 8 have
certain standard characteristics that are of interest for
possible applications in the socio—economic environment.
The transition matrix (labeled P in the example of the state
matrix shown earlier) has the characteristic that, if the
connectivity of the system allows, this matrix may be raised
to the power of the number of time periods in the iteration
to arrive at a solution for the state equation. That is if
there are twenty time periods to be considered in the problem
solution the transition matrix may be raised to the twentieth
power directly by matrix algebra and a solution arrived at
immediately.'
The model developed in this study does not have that
capability since it is an identity matrix having the property
that when raised to any power the value of the matrix is
still one, or equal to the original value. This character
istic of the model as used herein is due to the fact that
there is no feasible air linkage between the three base
cities. The "force" function developed for the system shows
that at the distances encountered in the system between the
base there is no feasible level of air service that can be
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developed to overcome the time ratios for surface and air
travel. However, this characteristic of the transition
matrix is an important feature, despite the fact that there
was no usable way to utilize this feature in the present
model.
Additionally only one edge in the output vector is
stated in terms of the state variables. This edge (27)
provides the energy for increasing the transfer of travelers
to and from the MID facility based on the changing "force"
levels of the system when the MID facility is activated.
Activation of the MID facility was of concern in
the study; the switching of the system from the hub and
spoke mode to the regional mode was accomplished through the
use of an external constraint equation. Switching of the
system was accomplished to the regional mode occurs when
the combined "force" values of the three base cities equalled
the "force" generated at the Detroit terminal.
This external decision equation accomplishes the
switching of the systems state and output equations and is
of the "on or off" type. The decision equation is written
in the following manner:
IfXZX /X /X thenY ,Y ,andY equal 0,
3 1 2 4 20 22 24
butifX