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 In...- t s ._— .- “ 3 I g L 'B A“? 1"? .P. Y n ." ' I” 31"!1 ' .13»; rpm 451$]: IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII ,_ 51 Y ' I 3 1293 10 This is to certify that the thesis entitled Characterization of the Force Variable in Systems Network Applications within a Socio-Economic Environment presented by Paul Bankit has been accepted towards fulfillment of the requirements for I I Ph.D. le ein Transportation (76%,; 71mm", Major professo/ Date W 2 0-7639 (Ma/3 5‘ LIL ULARSP 76 ABSTRACT CHARACTERIZATION OF THE FORCE VARIABLE IN SYSTEMS NETWORK THEORY APPLICATIONS WITHIN A SOCIO-ECONOMIC 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 socio-economic applications. CHARACTERIZATION OF THE FORCE VARIABLE IN SYSTEMS NETWORK APPLICATIONS WITHIN A SOCIO-ECONOMIC 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 SOCIO-ECONOMIC 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 Distance-Cost . . . . . . . . . . . . . . . . 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 SOCIO-ECONOMIC 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, McGraw-Hill, 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; ‘1 .1 N- 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, McGraw-Hill, 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. mass-energy 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 socio-economic 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. SMC-2, JU1Y 1972, PP- 319-331. 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. 222-59. 2Other linear graph applications within socio-economic 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, Springer-VéfIag, 1970), pp. 184-209. 3See H. Koenig, Y. Tokad, and H. Kesavan, Analysis of Discrete Physical Systems (New York, McGraw-Hill, 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 socio-economic applications. As Baumol points out, "No matter how ingenious the economists circumlocutions that have been employed, there has been no substitute devised an, .i‘ '1 9,, -.V I. ’1 (II " l A" 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., PrentIce-HaIl, I965I, pp. 210-230. 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 Unter-suchungen der Linearn verteilupg GaIvanischer Strome gerfuhrt wirdTEngliSh translation, TiansactIOn ofIthe Institute of Radio Engineers, CTLEU, March 1958, pp. A-7. 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, McGraw-Hill, 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 n-terminal as an identified system structure are complet- ely identified by a set of n-l equations in n-l 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 cut-set sum to zero. (A cut-set 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 Tri-Cities (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 center-of—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 uni-directional 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 1‘ , , .4 I): I n4 . M "II . " --- swm‘. . i - "hm :4 . 7' "M. u / V ‘Hfi'hhn ., Geo Center .. \ ”M "’2 ~ N .r Geo PI Center (2000) ‘ . I '1 Cxii'fit'.’ :3; 9—101, ‘7" J k; ...- Geo Per Center (2000) 1"” I I in mm- 0.- _ I .. I m INUIA' p3 ‘JA m\ V I I I ll nvtgugn‘ I": «s w ".‘H I \ . - - , : l \ . . 333 A ZI \ I - a \ K _ . . ‘\ \ I ' . W S . ~ ’ ”“ng m- : “t. may ,. ' I ,,, i |l a x, Y ‘ .9. ul'lh ”ch ”offing , I" I ”a... ‘0 " ‘ low" 1 L _ '21 .,. ., tion: I ——.-\ .1 ‘ fl . A I I 1 I S , ru 1 \ r ' A \ M c, "‘3‘: \V.§'..', u, :7 ma ‘v In I In v .u: In: ‘ "" o“ 'l " u ‘ 1 In - I / H u I L f ‘ - 4 , '|. ' v -— P v ‘ u- .T" _ "5°. u nuisa- _ u’ _ a . " In» \ I 14 ‘ s .\ ' . A M ' 8 Y ‘ ' - na- ' ' _ I A r , , 2 . , a z ’ 1 ' ‘ U 2 I ‘ ' at r g. \ x A ~ ~ .._ m. _ . Mnc’r‘ 9 , \\, 'v ‘_ ‘3“: | . k n b...‘. ,._ nu. "" u ‘ A d.) . mu ‘ ' I -| . A “ “1° ‘ ’ " 'm o ‘ ~ " / . n hang-h. g , I- k ’ 3°" ‘ own"? .,. < t A . ' as u w ”I , 2 . ""“ a" . 5‘ . ._ v ‘ . ., 'I gt. .. a . / _ m . w, . , .. . t W? s r In _._ no . I . . K . .' -1 I I .1— 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 bi-directional 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 City-Saginaw. 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 socio-economic 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 she's! qu". 0‘ no- I-‘b ’A‘a choir . a... .. a 8" v ., .- ..E‘.. mu; ‘Ar ‘IVu 19 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, McDonnell-Douglas DC-8 and DC-lO, and Lockheed Tri-Star. 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, McDonnell-Douglas DC-9 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 origin-destination 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,_ 1971-1986, 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 origin-destination 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 origin-destination 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—E-D-_= City (000's), ,per capita Passengers Jackson 1A9 .09 1A,6l6 Flint A87 .33 158,95A Lansing 361 .71 25A,511 Bay City-Saginaw 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 in-Despination Survey of Domestic Airline Passenger Tra ic, 1968: Civil Aeronautics Board, washington, D}C. .u‘ .n u! ~. \ J- a. 2A 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 self-sufficiency 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. 87-95. 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 distance-mass 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. Distance-Mass 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 distance-mass 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 self-sufficiency 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. Distance-Cost 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 I. a so? Iv.- In: no 0 .00! 1. F“ -\U 28 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 utility-preference values. In addition, because of the prevailing differences in average fares per mile for different city-pairs, 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 city-pair 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 cost-price 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 Time-Series and Cross-Sectional 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. Distance-Modal 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 modal-choice 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 time-value concept then forms the basis for measuring the total cost of the block-times 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 (an .\~ .744 a . f p-A \. Flu 30 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. llA-ll7. 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 time-value 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 33 cm. mom. 60536600 noom 3:30 ago? 3:035 *0 3:05: o 3 3:5 .522 023.3 28 :otoo c333 2226.; oootsm ac degraded woz<._.m_o madmmbm 0: ON. 00. CG Om 0? ON A _ . _ _ - fi 4 — 1 4 — q d — umagaom luemenow reusing 10;] Argo an” or bumla Slalom”, 1w 5d 0/. 3A third distance factor (distance—modal choice) described earlier. The distance-modal 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 origin-destination 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 time-value 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 time-value 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., Lansing-Detroit 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 T.<\ooot:mv 0231 u}; ..m> senses: Empmzm Hm>oRB ham L m mmbUHm 0.. (MID RAH) (The node) "30305, 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 passenger-miles 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 Tri-Cities 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 mixed-parameter 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. Mixed-parameter 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 discrete-time 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 "u: ' II‘ t of! ii ’ ;. on to" «tn. V‘y an a'.“ V. ‘1‘;- _ s.._ ”n s ‘ a P) n... T'u. uy‘ v 4/. 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 sue-r '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 airport-city 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‘" v.“ tn .--- Our“ .‘u 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 Tri-Cities 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 “‘1 at: .v 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 -v- “-4 nh‘ .,‘ vuu 1,. d.. D (II an. 4" (a .“v 54 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 -:.r. an... 5“"? 9 vJ.~._ a, -‘r .. '.- I‘ ,1 u. *. n... . . . a "4‘ 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 1-- v...-. 56 l J H H H ox so. o o o o o ox Hmm mssm mmmm mm» o . so. o o o o mus H.moH o o + as o o so. o o o u ms mHos o o s» o o o so. o o s» m.soH msw: o ms o o o o so. o ms m.Hs o mHms z hm» o o o o o so. ms sowamm cowpoSUm mpopm mQOHpoddm pcmsomSoo mempmhm w osswflm 57 ax mx mx 2 Hx O momH me Hom o mmmH o o mmm o mmsH Hmw HO CI) 0] OOOOOOLDCUOJOOOOOO o oomm o so: sm: o oomm o owm o mHsm o o + mwml. ZJ 1% mm» w% s% % rmw (I) Ln OOOOOOOOOOOOOOOOCDO I OOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOO IIIIIIJ o o o O o O o o O o O O O O o o o O O O O o O O O O O o o o O O O o O o wms sms oms ems mms mms Hms oms mHs st sHs oHs mHs :Hs mHs mHs HHs oHs Moscow QOHpmsom mmcommmm .poscflpcOo : .m osswflm 58 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