A DéSCRETE STATE {EETERMINISTiC SYSTEM MODEL
FDR ANALYSS OF THE FERE‘J’:

Thesis for the Degree of Ph. D;
MCHiGAfi STATE UNE‘JEHSQTY
B. LENN SGULE
1967

  
   

LIBRA R Y
h‘llClfigf '- '7’?”
UNVux‘AL)

  

This is to certify that the
thesis entitled

A DISCRETE STATE DETERMINISTIC SYSTEM MODEL
FOR ANALYSIS OF THE FIRM

presented by
B. LINN SOULE

has been accepted towards fulfillment
of the requirements for

_EIL_D_.._ degree in WENT

Date fl

0-169

 

 

 

 

73—737

 

ABSTRACT

A DISCRETE STATE DETERMINISTIC SYSTEM MODEL
FOR ANALYSIS OF THE FIRM

by B. Linn Soule

Techniques originally developed to model and analyze
complex systems of 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
characteristics of these components can be described mathe-
matically in terms of common flow and/or propensity variables.
The vehicle for analysis is a system model composed of a set
of state equations and an output vector.

The advantages of this methodology are twofold:

(1) precise procedural techniques are available for combining
the individual component models into an equation set suitable
for analysis, and (2) well developed analytical techniques

can be applied to the resulting system model. The system

model is formulated from two structural features of the sys-
tem: (1) the equations describing the unconstrained character-
istics of the components of the system, and (2) the constraints

imposed on the system by the interconnection pattern of its

B. Linn Soule

components as derived from a linear graph of the system.

The state model is a set of difference equations characteriz-
ing the internal states of the system as a function of the
inputs, and the output model is a set of algebraic equations
giving the system response as a function of the internal
states and the inputs.

Analysis of the system, as represented by the system
model, provides knowledge of its dynamic characteristics.

The limiting characteristics of the system can be examined
for stability, strategies can be derived to achieve specified
future objectives, and sensitivity analysis performed by
means of computer simulations.

This research applies these physical systems techniques
to a typical business firm. The resulting model depicts the
firm as a system of interacting flows of funds driven by the
sales of its products. The firm is perceived as a medium
for accumulating the resources necessary to produce sufficient
goods to satisfy this sales driver. The system is patterned
after a real firm and parameter values are derived from its
financial reports. The system as modeled is shown to be both
stable and controllable. Operation of the firm is simulated
on the computer and the sensitivity of profit and debt to
selected parameter changes is studied.

The study shows that a business firm can be modeled and

analyzed with these physical techniques. Controllability is

B. Linn Soule

shown to be a viable concept for application to the business
system. The validity and value of the analysis is determined
by the precision and detail with which the system is modeled,
which in turn depends on the descriptive detail of the com-
ponent models. More research is needed in the area of com-
ponent modeling; better techniques for modeling the com-
ponents of economic systems are necessary for this system

analysis methodology to become operational.

A DISCRETE STATE DETERMINISTIC SYSTEM MODEL

FOR ANALYSIS OF THE FIRM

BY

B. Linn Soule

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Management

1967

 

7/:7

fl

(9 Copyright by
B. LINN SOULE
1967

ii

ACKNOWLEDGMENTS

The author wishes to acknowledge the assistance of the
many persons who have so generously contributed both sub-
stantively and implicitly to the program of study and research
culminating in this dissertation. Both Professor Gonzalez,
chairman of the research committee, and Professor Mossman,
skillfully guided the research effort and provided the encour-
agement so essential to satisfactory completion of a program
of this type. The contribution of Professor Koenig, both as
a source of methodology and a continuing guide and evaluator,
is gratefully acknowledged. The aid of the industrial firm
which supported the research both financially and with busi-
ness data is greatly appreciated. The continuing nature of
the research and the proprietary nature of various data pre-
clude public disclosure of the specific firm at this time.

The continued encouragement of Clarence F. Hyde throughout
the course of this program and the unstinting tolerance and
devotion of my family have been vital to the completion of

this work.

iii

TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . iii

LIST OF TABLES. . . . . . . . . . . . . . . . . . . v

LIST OF FIGURES . . . . . . . . . . . . . . . . . . vi

LIST OF APPENDICES. . . . . . . . . . . . . . . . . vii
Chapter

I. INTRODUCTION. . . . . . . . . . . . . . . . 1

Systems Concepts. . . . . . . . . . . . . . 5

Research Design . . . . . . . . . . . . . . 6

Thesis Organization . . . . . . . . . . . . 7

II. MODEL OF THE FIRM . . . . . . . . . . . . . 9

Structure of the Firm . . . . . . . . . . . 9

Sector Models . . . . . . . . . . . . . . . 13

System Constraints. . . . . . . . . . . . . 20

The System Model. . . . . . . . . . . . . . 21

III. PROCESSING THE MODEL. . . . . . . . . . . . 25

Component Coefficients. . . . . . . . . . . 25

General Solution. . . . . . . . . . . . . . 50

Stability Analysis. . . . . . . . . . . . . 53

Control Strategy. . . . . . . . . . . . . . 55

Simulation. . . . . . . . . . . . . . . . . 41

IV. SUMMARY AND CONCLUSIONS . . . . . . . . . . 43

BIBLIOGRAPHY. . . . . . . . . . . . . . . . . . . . 47

APPENDICES. . . . . . . . . . . . . . . . . . . . . 51

iv

LIST OF TABLES

Page
Component Coefficients. . . . . . . . . . . . . 26
State Variables . . . . . . . . . . . . . . . . 27
Definitions and Base Values of Coefficients in
the System Model. . . . . . . . . . . . . . . . 31

LIST OF FIGURES

Figure Page
1. System graph of the firm depicting input-
output flows and the interconnection pattern
of the firm's two sectors. . . . . . . . . . . 10

2. Time paths of debt, equity, and working capi-
tal under control law. . . . . . . . . . . . . 58

3. Time paths of fixed assets under control law . 39

4. Sensitivity of profit to changes in work week
for linear sales . . . . . . . . . . . . . . . 65

6. Sensitivity of profit to changes in work week
for nonlinear sales. . . . . . . . . . . . . . 64

6. Sensitivity of debt to changes in work week
for linear sales . . . . . . . . . . . . . . . 65

7. Sensitivity of debt to changes in work week
for nonlinear sales. . . . . . . . . . . . . . 66

8. Sensitivity of profit to changes in tooling
for linear sales . . . . . . . . . . . . . . . 68

9. Sensitivity of profit to changes in tooling
for nonlinear sales. . . . . . . . . . . . . . 69

10. Sensitivity of debt to changes in tooling for
linear sales . . . . . . . . . . . . . . . . . 7O

11. Sensitivity of debt to changes in tooling for
nonlinear sales. . . . . . . . . . . . . . . . 71

12. Sensitivity of profit to changes in the ratio
of working capital to sales for linear sales . 73

13. Sensitivity of profit to changes in the ratio
of working capital to sales for nonlinear
sales. . . . . . . . . . . . . . . . . . . . . 74

14. Sensitivity of debt to changes in the ratio of
working capital to sales for linear sales. . . 75

15. Sensitivity of debt to changes in the ratio of
working capital to sales for nonlinear sales . 76

vi

LIST OF APPENDICES

Appendix Page
A. Program ANALYZE. . . . . . . . . . . . . . . 52
B. Sensitivity Analysis . . . . . . . . . . . . 61
C. Selected Computer Output . . . . . . . . . . 77

vii

CHAPTER I
INTRODUCTION

Throughout history attempts to describe and analyze the
natural phenomena of the universe have resulted in compre-
hensive mathematical descriptions of the phenomenon under
study. These scientific contributions to the general knowl-
edge about nature have increased the efficiency with which
man adapts his environment to his needs. The advent of high
speed digital computers and the promulgation of viable quan-
titative techniques have provided scientists with many power-
ful tools for the analysis and design of various types of
complex systems. This thesis applies some of these tools of
analysis to a business system in order to project future
effects of various changes in policy and system structure.
This effort is consistent with the generally increased empha-
sis on scientific approaches to business problems.

In the last three decades, the use of scientific methods
and quantitative techniques in business has accelerated
noticeably. The trend of modern managers has been to abandon
many rule-of—thumb practices in favor of more scientific
methods. This increased effort to quantify business factors
has led to a rather recent cleavage of management personnel.

Minas succinctly summarized this cleavage between the

management scientist, whom he called the formalist, and the
manager who uses heuristics as his dominant guide, in the
following manner:
The formalist is usually very much oriented towards
mathematics. . . . The realist is apt to rely more
heavily on "raw" observations than on formal devices
such as mathematics; he goes in for what might loosely
be called "clinical" procedures.1
The management scientist has borrowed heavily from other
disciplines and sciences in a continuing effort to optimize
business performance. McGuire attempts to validate this
interdisciplinary approach in the following statement:
Theories originally applicable in alien situations,
which were not meant specifically to be used to explain
business activities, often have provided insights and
explanations useful in business. If such "borrowed"
theories-~whatever their origin-~prove useful in explain—
ing or predicting business behavior, then, by all means,
let us borrow them.2
Material borrowed from another discipline or science
can be viewed as consisting of three classes: (1) content,
(2) techniques, and (5) theory.3 The content of a science,
consisting of Specific observations and measurements, is the
most specific of these three classes, and can be borrowed

quite easily, provided that care is used in transferring it

across the boundary separating the two disciplines.

 

lJ. Sayer Minas, "Formalism, Realism and Management
Science," Management Science, III (October, 1956), p. 12.

 

2Joeseph W. McGuire, Interdisciplinary Studies in
Business Behavior (Cincinnati, Ohio: South-Western Publish-
ing Company, 1962), p. 8.

 

3Michael Halbert, The Meaning and Sources of Marketing
Theory (New York: McGraw-Hill Book Company, 1965), pp. 10—15.

Techniques include both measurement and analysis; borrowing
from this class is often labeled adaptation of methodology.4
Techniques have been broadly adapted and extensively utilized
by the management scientist in pushing back the frontiers of
business knowledge. The main thrust of this thesis is adapta-
tion of these two classes of material to the entity known as
the business firm.

The sheer scope and complexity of the operations of a
large business firm contribute significantly to sub-optimiza-
tion in decision making. The concept of "system" is a viable
one in the physical universe, and should provide the manager
of an economic system with a powerful and comprehensive tool.
The pervasive nature of the system concept is aptly presented
by Boulding's comments on the emergence of a general theory
of a system:

General Systems Theory is the skeleton of science in

the sense that it aims to provide a framework or struc-

ture of systems on which to hang the flesh and blood

of particular disciplines and particular subject matters
in an orderly and coherent corpus of knowledge.5

Systems Concepts

Engineers and physical scientists have long been able to

describe with mathematical precision the characteristics of

 

4Ibid.

5Kenneth E. Boulding, "General Systems Theory-~The Skeleton
of Science," Management Science, II (April, 1956), p. 208.

most of the components of the systems with which they deal.
The hydraulic concepts of pressure and flow, and the electri—
cal concepts of voltage and current have provided the physical
scientist with the common denominator for study of their
system components and a means for characterizing and classi-
fying component behavior. Using the scientific knowledge of
components thus generated, the engineer has been able to syn-
thesize systems designed to meet specific objectives.

In the nineteenth century Kirchoff6 developed a metho-
dology for analyzing electrical networks with known physical
components by calculating currents and voltages in all parts
of the network. Kirchoff's analytical methods provided a
rigorous methodology for formulating, for any electrical net—
work, a system of mathematical equations called a model. He
introduced the basic concepts of linear graph theory for
model formulation, which have since been formalized by mathe-
maticians such as Whitney7 and electrical engineers such as

Seshu and Reed.8 They have recently been generalized by

 

6G. Kirchoff, "Uber die Auflosung der Gleichungen, auf
welche man bei der Untersuchungen der Linearen Verteilung
Galvanisher Strome gefuhrt wird," English translation,
Transaction of the Institute of Radio Engineers, CT-5 (March,
1958), pp. 4-7.

7H. Whitney, "Non-separable and Planar Graphs,“ Trans-
actions of the American Mathematical Society, XXXIV (1952),
pp. 339-562.

8Sundaram Seshu and Myril B. Reed, Linear Graphs and
Electrical Networks (Reading, Massachusetts: Addison Wesley
Publishing Company, 1961).

 

Koenig, Blackwell,9 and others, in their application to
essentially the entire class of physical systems. These
concepts coupled with more recent developments form much of
the foundation for a discipline known as Systems Science.

The system model utilized in analysis and control is
formulated from two major structural features of the system
under study: (1) the equations describing the unconstrained
characteristics of the components of the system, and (2) the
constraints imposed on the system by the interconnection
pattern of its components as derived from a linear graph of
the system. The sets of equations characterizing these two
features of the system together constitute the system model
and form the basis for analysis of the system.

A study of the system model can lead to considerable
insight regarding the Operational characteristics of the
system. In particular, the stability of the system can be
examined to determine its limiting characteristics and
strategies can be derived to achieve a specified objective
at a future point in time.

The applicability of this discipline to socio-economic
systems has been advocated for several years by Professor
Koenig through a series of graduate courses in Systems Analy-

sis for Social Scientists offered by him in the College of

 

9Herman E. Koenig and William A. Blackwell, Electro—
mechanical System Theory (New York: McGraw-Hill Book Company,

 

1961).

Engineering at Michigan State University. His class notes10
from these courses are the primary reference source for this

thesis.
Research Design

The focus of this study is a large modern corporation
manufacturing and selling a consumer product in the United
States. This corporation serves as the pattern for the system
model and as a source of data for analysis. The firm is viewed
in its most basic form as an economic system which collects
the capital assets required to produce and sell an economic
good. The basic objectives of the study are threefold:

1. Develop a model of the firm as a system of inter—
acting components or sectors.

2. Examine the system for stability and controllability.

5. Describe quantitatively the way in which the firm
reacts to its environment, and trace the time path
of its key variables by computer simulation.

The first step in developing a system model is to identi-
fy the system, that is, within the level of aggregation de-
sired, the components of the system must be identified and
their characteristics described mathematically. In this study

the firm is perceived as a complex system of financial trans-

actions which take place both internally and externally.

 

loHerman E. Koenig, Unbound class notes for the three
course sequence entitled "Systems Analysis for Social Scien-
tists" (College of Engineering, Michigan State University,
1965-66 school year). (Mimeographed.)

These resulting flows of money accumulate pools of assets
at various places in the firm. The flows are evaluated at
discrete points in time, and the dynamic characteristics of
the process are presented by a set of difference equations
called a state model.

Data for empirical determination of parameter values
are obtained by analysis of the firm's financial reports.
Sensitivity of the model to a selected set of parameters is
investigated by simulating operation of the firm for twenty-
five years and introducing various incremental changes in
the coefficients of the component equations. The transition
matrix of the state model is examined to determine system
stability, and a general strategy for controlling the system
state is calculated. A large scale digital computer is used

to calculate the specific system values required for analysis.

Thesis Organization

This thesis follows the natural evolution of the research
it relates; beginning with a brief description of the systems
analysis methodology, proceeding to development of the model
and its use for analysis, and ending with the conclusions
drawn from the research.

A system model of the firm is developed in Chapter II
using the methodology outlined in this chapter. Chapter II
begins with a linear graph of the system conceptualized to

represent the "anatomy" of the business operation. Component

behavior is modeled mathematically and system constraints
applied, resulting in two sets of equations: (1) a set of
difference equations characterizing the internal states of
the system as a function of the inputs, and (2) a set of
algebraic equations giving the output as a function of the
internal states and the inputs.

Chapter III relates the effort to derive numerical values
for the system model matrices, and the results of the stability
and control analysis. Several computer simulations are
described in Appendix B, and selected results are shown
graphically.

The Bibliography represents a comprehensive and thorough
search of the literature. Because of the uniqueness of the
particular methodology used in this study there is a dearth
of directly pertinent literature. However, much material felt
to be of a generally related and constructively pertinent
nature is included.

The complete computer program used for analysis, along
with a brief description, is listed in Appendix A, and Appen-

dix C contains a selected sample of computer output.

CHAPTER II

MODEL OF THE FIRM

The system being examined in this study is a typical
firm operating in the United States. A macro view is taken
of the firm and its money flows are aggregated into a mini-

mum set of decision parameters.

Structure of the Firm

The firm as conceived consists of two basic sectors:
(1) production, and (2) finance. The production sector is
assumed to have three interfaces with its environment (the
surrounding economy) and the finance sector has two. The
production sector provides the salable goods that result in
a flow of funds into the firm in the form of cash sales.
The flow at one of the two remaining environmental interfaces
represents the cost of Operating the firm, and the other is
the payment of federal income taxes. The production sector
of the firm reduces the gross sales income by the sum of the
cash cost of operation and federal taxes, and passes the
remainder to the finance sector, in the form of profit and
depreciation. It is assumed to possess exactly the proper

quantity of fixed assets to satisfy production requirements.

10

The finance sector of the firm provides the assets with
which the production sector manufactures the goods the firm
sells, and maintains sufficient working capital to support
current operations. It has two sources of funds from which
to meet these requirements: (1) cash flow from the production
sector, and (2) environmental sources. It is assumed to pro-
vide exactly the quantity of fixed assets and working capital
required, and the funds available from the environmental
sources are without limit; that is, they make available what-
ever amount of funds the finance sector demands. Let the
structure of the firm as perceived above be represented by

the graph shown in Figure 1, having ten edges as indicated.

/

 

 

 

> d / 1 f
1 / j...
A\Y2 / (Y6
Y8 /
l
C Y1 y g V '
fi/ __ I ‘ f f '
Nf I Nd,Ne,Nw
Y4 I Y4'
I
b
\/Ys /
./
4Le //
./
Production Sector Finance Sector

Figure 1.-—System graph of the firm depicting input-
output flows and the interconnection pattern of the
firm's two sectors.

11

The arrows show the assumed directions of flow of goods or
funds. Reversed flow will be indicated in the solution set
by negative values. Each of the end edges 1, 2, 5, 6, and 7,
represent flows to or from the economic environment, and
the remainder are intra-firm flows.

All edge flows are measured in units of dollars per
year. The flows yi, i = 1,2,5,3',4,4',...,8, identified by

the edges of the system graph are Specifically

y; = Sales y4' = Capital expenditure

y2 = Federal income tax y5 = Cash cost of operation
y3 = Net profit '§6 = Flow of outside funds
y3a= Cash flow y7 = Dividends

'§4 = New capital items y8 = Depreciation

Each of these flows can, if desired, be regarded as a vector.
The fixed assets of the production sector are repre-

sented by the variable NE, which is called an internal state
of the production sector. Similarly the variables Nd’ Ne’
and NW representing respectively, debt, paid in capital, and
working capital are internal states of the finance sector.
All of these internal states are measured in dollar units.
Specifically, these internal states, which also can be re-
garded as vectors if desired, are defined as

Nd = Total corporate debt

Ne = Total paid in capital
NE = Net fixed assets
N = Net working capital

12

Standard accounting procedures are used for measuring
financial flows and balances. Since the year is a commonly
used time interval in accounting practice, and as a planning
horizon, all flows refer to an aggregate flow over a time
period of one year, and the variable n is used to represent
points in time at regular twelve month intervals. The state
variables are measured at the beginning of the interval n.
The set of variables defined above represents the critical
items found in a firm's Balance Sheet, Profit and Loss, and
Source and Application of Funds statements, usually presented
in the annual financial report.

In more generalized models representing actual flows of
goods or physical units of resources, a complementary variable
which reflects unit values or prices can be introduced. This
variable, analogous to the physical concept of propensity,
or pressure, must satisfy the generalized Kirchoff circuit
postulate.l In applications which utilize this concept it is
possible to relate policy parameters to prices as well as to
flows. In such a development the dollar value is the product
of the two variables. No attempt is made in this study to

include these effects.

 

1An excellent reference for explanation of this postulate
and amplification of general systems analysis methodology is
the book by Koenig, Tokad, and Kesavan entitled Analysis of
Discrete Physical Systems (New York: McGraw-Hill, 1967).
Chapters 4 and 5 are especially appropriate for economic sys-
tems.

15

Sector Models

A model of the system is developed by first considering
each sector of the firm independent of any constraints im-
posed by the interconnections between the sectors; that is,
strictly in terms of the variables associated with that sector.
The model so constructed can also be used in the context of
other constraints with equal validity. The mathematical
equations characterizing the unconstrained behavior of the

sectors of the system follow.

Edge 1. This is the edge in the system representing the
annual sales income of the firm. It is considered to be the
flow driver for the system and is therefore taken as the

general function of time

yl(n) = F(n) (2-1)

Edge 2. This edge represents the annual federal income
tax paid by the firm. The tax is based on the gross profit
of the firm and is assessed at a rate T. Since gross profit
is determined by subtracting total cost of operations from

sales income, the tax component equation is

y2(n) = T[y1(n)-y5(n)-ya(n)] (2—2)

Edge 5. Net profit is represented in the system by
edge 5. Net profit equals gross profit minus tax and the

resulting equation is

14
y3(n) = yl(n)-y2(n)-y5(n)-y8(n) (2‘3)

Edge 4. The flow on this edge is the total annual ex-
penditure for various capital items used in the production
process. Due to the extreme range of types of assets and
wide differences in their characteristics, it is worthwhile
to consider this edge as a vector. The capital assets of
the firm are considered to be of three basic types: (a) tools,
(2) machinery, and (3) buildings. These are represented
respectively as components y41, Y4g, y43 of the vector y,.

This equation attempts to capture the management decision
process regarding the procurement of capital goods, and ap-

pears as

 

:41“); ”R1 — Po ‘—
§4<n) = y42(n) = 1/P Ra/Ke y1(n+1) - Nf2(n) (2-4)
Y43(n) Rs/Ka Nf5(n)

 

 

 

 

 

Product obsolescence requires the firm to continuously re-
tool, based on predicted sales volume in the n+1 time period,
regardless of present tooling capacity. The tooling component
of the capital expenditure vector is therefore determined en-
tirely by the sales forecast and the size of the production
parameter R1. The coefficient P is the selling price of the
product and converts the sales dollar volume into physical
units. The coefficient R1 is the number of dollars per pro—

duction unit spent for capital production goods of the ith

1

type; it converts the dollar value of the i type of fixed

15

f2 and Nf5 are

respectively net fixed machinery assets and net fixed build-

asset into production units. The states N

ing assets, and K2 is the percentage of maximum machine
capacity utilized in normal operations.

The vector components of'N represent the dollar value

f
of different classes of capital assets which can be converted
into units of production capacity by a representation such

as Nfi/Ri; this relates the dollar value of the ith type of
asset to the maximum number of units it can produce. When
this maximum value is Operated on by Ki it is converted to

a normal, or desired utilization level. The capital expendi-
ture needed for the ith type of asset can be expressed
basically as the difference between present capacity and fore-

casted sales volume. This difference, when operated on by

R .

l and Ki results in the dollar expenditure required to equate

that asset's need with its present value. The basic form

for the machinery component in this system is
Y42(n) = R2/K2 [Yi(N+l)/P - (K2/R2)Nf2(n)] (2-4a)

The decision rule for adding to building capacity is based

on the concept of slack between present capacity to contain
machinery and anticipated machinery needs in the n+1 period.
If present building capacity is greater than the predicted
machinery need at n+1, there is no need for building addition

at time n. The basic form for the building component is

Y43(n) = R3 [Yi(n+1)/K2P - Nf5(n)/R3] (2-4b)

16

Simplifying these equations and adding yil yields the result-

ing component equation 2-4.

Edge 5. Cash cost includes all operating expenses except
depreciation, and in this system is visualized as consisting
of three basic components: (1) fixed costs, (2) variable
costs, and (5) interest expense. Interest expense is kept
as a separate item so that the impact on cost of various debt
positions can be analyzed. The fixed cost portion is related
to the scale of plant involved in the production process
and the variable cost is a function of production volume.
Applying an average interest rate to the outstanding debt
yields the interest cost for the period. Since this model
considers three types of fixed assets, these three separate

types are included in the model as

YS(n) = CllN (n) + Clng2(n) + ClgN )

f1 fsin

+ C2y1(n) + RsNd(n) (2—5)

Edge 6. The flow of funds between the firm and the en-
vironment is a vector represented by edge number six. The
two components of the vector are: (1) debt, represented by
yel, and (2) equity, represented by Yea- Outside funds are
needed whenever those generated internally from operations
are not sufficient. The funds generated internally during
period n are called cash flow and are represented by edge
3' in the system graph. Cash flow enters into the working

capital pool Nw, and the amount of Nw at time n is the sum

17

total of internal cash available to the firm. Basically
there are two uses of funds within the firm: (1) capital
expenditures of the three types, and (2) cash for Operating
expenses during the period. Operating expenses are a func—
tion of the sales level during the period and capital expendi—
tures are determined by the flow on edge four. In this model
outside tooling funds are selected entirely from debt sources
while funds for machinery and buildings are divided between
debt and equity. When additional working capital is required,
this need is allocated between debt and equity also. It is
only the change in Nw that is allocated in this manner, the
final composition of the pool cannot be specified. The re-

sulting equation for capital expenditure is therefore

_ ( ) Yei(n) [—I F1 F2 _Y4J.(nT
y n = =
6 Yegin) i9 1-F1 1-F2 y42(n)

Y43<nlj

 

 

+ Wy1(n) - Nw(n) (2-6)
l-G l-G

Edge 7. Dividend payments to the owners of the firm
are based strictly on the amount of cash flow during the

period. This relationship is formulated as
y7(n) = Dsy3'(n) (2-7)

Edge 8. Depreciation in the firm is based on the value

of the fixed assets measured at the beginning of period n and

18

the capital expenditures during the period. The acquisition
of assets throughout the year is assumed to be normally dis-
tributed with a mean depreciation period of six months;
consequently, depreciation can be calculated on new assets
at one-half the annual rate for its class. When the three
classes of assets have depreciation rates D1, D2, and D3,
and the declining balance method of depreciation is used,

the depreciation equation is

7" “‘7
y8(n) = E1 D2 D3 Nfl(n

Nf2(n)

 

Bis”:

(2-8)
+ [Ea/2 D2/2 Ds/é] FT4iin)

Y42in)

 

 

Y4s(n)

Id

As this system is perceived there are six internal
state variables: (1) net debt, (2) total paid in capital,
(5) net working capital, (4) net tooling assets, (5) net
machinery assets, and (6) net building assets. These internal
states appear as variables in equations 2-4, 2-5, 2-6, 2-8,

and can be expressed as a function of the flows as shown next.

Net debt. In determining the debt position of the firm
it is assumed that some fixed percentage of the outstanding
debt is repaid each periOd. Any new debt incurred is then

added to give the resultant net debt in the n+1 time period.

19
The resulting difference equation is
Nd(n+1) = (1 - R4)Nd(n) + Yei(n) (2-9)

Paid in capital. Total paid in capital at n+1 is repre-
sented simply as the sum of the amount at time n and any new

capital paid in during the period, thus
Ne(n+1) = Ne(n) + y52(n) (2—10)

Net working capital. If R5 represents the imputed ef-
fective rate of interest earned on the total working capital

pool resulting in interest income to the firm,2

and R4Nd is
the annual debt installment, the working capital at the n+1

period is

Nw(n+1) = (1+R5)Nw(n) + y3'(n) + yel(n)

+ Y62(n) ‘ Y4i(n) - Y42(n) - y43(n)

" Y7(n) - R4Nd(n) (2—11)

The difference equation for this state variable is similar
to the accounting statement "Source and Application Of Funds."
In addition to the entries of that statement, since the dif-
ference equation represents the cumulative working capital

pool, it includes the value of the pool at the beginning of

the time period.

 

2The total working capital pool and an imputed interest
rate are used because the rapid and large changes in the
actual amounts producing interest income at any point in time
make it impossible to track exactly.

20

Net fixed assets. The net fixed assets can be repre—

 

sented for the n+1 time period as the sum of the depreciated
values at the beginning of period n and the properly depre-
ciated expenditures made during the period. The three-com-

ponent vector of assets is

 

 

 

 

 

__ __. __. __ __ ___
Nf1(n+1) 1-D1 o o Nf1(n)
'Nf(n+1) = Nf2(n+1) = O 1-D2 O Nf2(n)
_§E3(n+S:, __o o 1-?31.§E5(Ii
rZ-Dl/z o o ‘7‘ F},l(flf
+ o 1—02/2 O Y42(n) (2-12)
0 O 1-D3/2 y43(n

 

 

 

 

System Constraints

When the components of the system are united at their
interfaces, the terminal variables of the component models
are no longer independent, but are constrained uniquely in
a manner determined by the interconnection pattern depicted
by the system graph shown in Figure 1. The constraint equa-
tions are develOped from this graph by summing the flows of
funds at the vertices of the graph. The seven interfaces,
and the common reference point, in this system result in a
total of eight vertices. Since edges 1, 2, 5, 6, and 7, are
end edges their corresponding end vertices are eliminated
from consideration in the constraint equations. At vertex b

the constraint equation is

y4(n) - y4-(n) = 0 (2—15)

At vertex a the constraint equation is

y3(n) + ya(n) - y3'(n) = 0 (2-14)

These two constraint equations are linearly independent as

is evident when they are written in matrix form

1 o { -1 o 1 Ham—7
| _.
O 1 l O -1 O y4(n)
7;:Tn) = 0 (2-15)
Y4'in)
Y8(P)

 

 

The System Model

When the set of linear constraints in equation 2-15 is
substituted into the component models, a system model is

Obtained. The result after some algebraic reduction is

 

 

 

 

 

(Nd (n+1) Val O -G 0 -Fl ~15: ”Nd (n_)T
Ne (n+1) O 1 Q4 0 ca 05 Ne (n)
NW (n+1) = Q8 0 R5 d9 Q10 all NW (n)
Nf1(n+1) O O O as O Nf1(n)
Nf2(n+1) O O O 0 -d6 0 Nf2(n)
_§f5(n+{lj _O_ O O O O -3:. _Nf3(nL

F7721? T31?

Q13 Q16

+ 014 yl(n) + 020 Yi(n+1) (2-16)

0 317

O 018

.9 .1 _£ue_

 

 

 

 

 

 

 

 

 

 

#Y2 (qu -—:TRe O 0 'Pi “E2 -BS—_ TN; (g;
Y3 (n) B7Rs O O 54 BS 56 Ne (n)
Y4I(n) O O O O O 0 NW (n)
Y42(n) O O O O -1 O Nf1(n)
Y43(n) = O O O O O -1 Nf2(n)
y5 (n) R6 0 0 C11 C12 C13 Nf5(n)
Ysi(n) O O -G 0 -F1 -F2 J —_
Y62(n) O O 074 0 C12 C13
y7 (n) 13567126 0 O ClgDS 0710135 (11le

Lya (nld ___ O O D; as @7_J

p. __ __ __
T58 ’59
”fi7B8 filo

O Ri/P

O Rg/Kgp

+ 0 y1(n) + R3/K2P Y1(n+1) (2—17)

C2 0

0&2 015

0:3 0616
’955758 Bil
__P __ __Bia __

 

 

 

 

Where the coefficients in this model are determined in the

development as

01-7

0is

011 0
C11 1
C112
061 3
011 4
C11 5
C11 6
<11 7
C11 8
Q1 9
0120
51

52

B4
(35
Be

25

1-D1
D2/2

D3/2

R6(1—D5)(T-1)-R4
(1-DS)(T(c11+Dl)-c11)
(1-D5)(T(C12+D2/2)-C12)
(1-Ds)(T(C13+D3/2)-Cis)

GW

(1-G)W
(1-D5)(1-C2)(1-T)+W
(1/P)(R1+F1R2/K2+F2R3/K2)
(1/PK2)(R2(1-F1)+R3(1-F2))
(Ri/P)(1-D1/2)
(Ra/PK2)(1-D2/2)
(Ra/PK2)(1-D3/2)
T(1-D5)/(2PK2)(D1R1K2+D2R2+D3R3)
T(C11+DI)

T(C12+D2/2)

T(C13+D3/2)

( a9/1-D5) -D1

(am/i-Ds) ”0,6

(all/l-Ds) “Q7

24

B7 = T-1
58 = 1’C2
59 = Qeo/l-DS

610 = 020(T-1)/(T(1-D5))
511 = DSQeO/l'DS

612 = Geo/(T(1-D5))

Using matrix notation this model can be written in the

form

Y(n+1) PT(n) + Qlyl(n) + QaYl(n+l) (2-18)

R(n) MT(n) + le1(n) + N2y1(n+1) (2-19)

The matrix P is called the transition matrix, the matrices
Q1 and 02 are called excitation matrices, and R(n) is called
the output, or response, vector.

Note that the coefficients in the system model are
dependent upon, and uniquely determined by, the coefficients

in the sector models.

CHAPTER III

PROCESSING THE MODEL

The practical value of any model is determined by the
accuracy with which it portrays the behavioral character-
istics of the system it represents. The possible studies
that can be made by use of the model range from simulation
to control. Before any such analyses can be carried out,
however, numerical values must be assigned to the coefficients
in the final system model. These coefficients are calculated
from the numerical values of the coefficients in the com-
ponent models by the relations following equations 2-16 and
2-17. In this chapter coefficients are evaluated, the stabil-
ity of the system is investigated, a control strategy is

developed, and a simulation program is outlined.

Component Coefficients

The coefficients are evaluated from a data base corres-
ponding to a typical large firm operating in the United States,
as taken from its financial statements. The three primary
statements used are: (1) Profit and Loss, (2) Balance Sheet,
and (5) Source and Application of Funds. Those coefficients
that cannot be calculated from these reports are imputed.

Coefficients so obtained are given in Table 1.

26

Table 1.—-Component Coefficients

 

 

Coeffi- Base

cient value Definition

C11 1.978 Overhead cost factor of tooling

C12 1.978 Overhead cost factor of machinery

C13 1.978 Overhead cost factor of buildings

C2 0.540 Ratio of variable cost to sales

D1 0.500 Depreciation rate for tooling

D2 0.250 Depreciation rate for machinery

D3 0.100 Depreciation rate for buildings

D5 0.150 Ratio of annual dividends to cash flow

Fl 0.400 Ratio of debt to equity in machinery
expenditures

F2 0.100 Ratio of debt to equity in building
expenditures

G 0.600 Ratio of debt to equity in working
capital change

Kl $7.500x106 Slope of linear sales function

K2 0.476 Ratio of rated machine capacity to
maximum

P $2.500x103 Product unit sales price

R1 $6.000x10 Tooling expenditure per production unit

R2 $1.440x102 Machinery expenditure per production
unit

R3 $5.600x10 Building expenditure per production
unit

R4 0.100 Ratio of annual debt repayment to total
debt

R5 0.005 Interest rate achieved on total working
capital

R6 0.050 Interest rate on total debt

T 0.480 Federal tax rate on gross income

W 0.160 Ratio of working capital to sales

 

27

The initial values of the state variables derived from

the financial statements are listed in Table 2.

Table 2.-—State Variables

 

 

State Variable Initial Value Definition
Nd $ 4.25 x 106 Corporate debt
Ne $11.98 x 108 Paid in capital
Nw $ 9.80 x 106 Working capital
Nf1 $ 5.04 x 106 Tooling
Nf2 $12.09 x 106 Machinery
Nf3 $ 5.02 x 106 Buildings

 

Development of the values for the coefficients in Table 1
is described in the order in which they appear in the eight

component equations.

Sales (K1). Both linear and non-linear sales functions
are used in the analysis. The slope of the linear function,
K1 in this model, is determined by a simple linear regression
using historical sales data. The number of years used in
the regression was chosen to correspond to what is felt to
be the "modern era," and would typify the near future

characteristics of the firm and its industry.

Federal tax rate (T). The tax rate is given by federal

statute. While it has some variation based on gross profit,

28

for a large firm only very slight error is introduced by

considering it to be constant.

Capital expenditure coefficients (K2,P,R1,R2,R3). Since
a machine can run an absolute maximum of one hundred sixty-
eight hours per week, the value for K2 is set by what the
firm considers to be its standard work week. The rated capa-
city for the firm under study was determined by considering
a standard work week of eighty hours. The product price P is
established by dividing gross sales income by the number of
units sold. The figure for capital assets per production unit
can be approximated from data contained in the balance sheet,
and summary production figures. By comparison of data from
similar situations and consultation with business represent-
atives, the total assets of the firm are allocated by type
in the following way: sixty percent machinery, twenty—five

percent tooling, and the remaining fifteen percent buildings.

Cash cost of operation coefficients (C11,C12,C13,C2,R6).
Depreciation is removed from the total cost of operation to
determine actual cash cost. The average rate of interest on
the total debt, designated R5, can be calculated from the
total interest expense figure and total debt. After depreci-
ation and interest expense are subtracted from the total cost
of Operation, the remaining cost can be allocated between
fixed and variable costs. This allocation is forty percent

fixed cost and sixty percent variable. Using this fixed cost

29

figure and the firm‘s fixed assets, the value of the vector
Cl can be found. All three types of fixed assets contribute
to fixed cost in the same manner, and thus all Cli, i = 1,2,5,
are taken as identical to Ci. The remaining sixty percent

of cost (considered variable) can be expressed as a function

of sales, thereby yielding the value of C2.

Depreciation rates (D1,D2,D3). This firm uses the de-

 

clining balance method of depreciation at twice the straight
line rate. Tool assets are taken to have an effective life
of four years, resulting in a declining balance rate for D1
of fifty percent per year. The rates for D2 and D3 are
determined in a similar way, with their lives respectively

calculated as eight years and twenty years.

Capital expenditure coefficients (F1,F2,W,G). The co-
efficients F1, F2 and G are complex intra-firm decision para—
meters and can only be estimated. A value for W can be
calculated from the gross sales and net working capital

figures.

Dividend rate (D5). The value for this parameter can

 

be established by relating actual dividends paid with their

corresponding cash flows.

Debt retirement rate and interest rate on working
capital (R4,R5). Two additional coefficients are introduced

in the state equations. The debt retirement rate R4 can be

50

approximated by comparing the current portion of the long
term debt with the total debt of the firm. Current and
anticipated money market conditions must also be considered
in establishing this coefficient value. It is difficult to
arrive at an accurate value for R5, the effective rate Of
interest on total working capital, resulting in interest
income. As used in this state model the value of R5 must be
imputed from some estimate of total interest income compared
with net working capital.

The base values Of the component coefficients as listed
in Table 1 are used to calculate base values for the co-
efficients in the system model. The results are given in

Table 5 on the following page.

General Solution

While the discrete form of the state model can be solved
for any specific point in time by recursive substitution,
this method is tedious for large values of n. It is helpful,
both for ease of solution and for system analysis, to show
the solution explicitly as a function of time. Applying the

linear transformation
6(n) = Y(n) - 02Y1(n) (5—1)
the state model

Y(n+1) = PY(n) + Qly1(n) + Qayl(n+1) (3-2)

51

Table 5.--Definitions and Base Values of Coefficients in the
System Model

 

 

 

Coefficient Definition Base Value
Q1 1‘R4 0.900
02 F1-1 -0.600
03 F2-1 —0.900
04 G—1 —0.400
as 1—Dl 0.500
06 D2/2 0.125
07 D3/2 0.050
08 R5(1-D5)(T-1)-R4 —0.125
mg (1-D5)(T(C11+D1)-C11) -0.686
010 (1-D5)(T(C;2+D2/2)-C12) -0.845
all (1—D5)(T(C13+D3/2)-C13) —0.874
012 GW 0.096
013 (1-G)W 0.064
ml, (1-D5)(1-C2)(1-T)+W 0.568
Q15 (1/P)(R1+F1R2/K2+F2R3/K2) 0.075
are (1/PK2)(R2(1-F1)+R3(1-F2)) 0.100
Q17 (Rl/P)(1-D1/2) 0.018
ala (Ra/PK2)(1-D2/2) 0.106
ale (R3/PK2)(1-D3/2) 0.029
020 T(1—Ds)/(2PK2)(DlRlK2+02R2+D3R3) 0.009
Bl T(CII+D1) 1.190
52 T(C12+D2/2) 1.012
83 T(C13+D3/2) 0.972
64 (as/i—Ds)-Dl -1.288
65 (ale/i-D5)-a6 -1.085
Be (all/i-D5)—d7 -1.055
37 T-1 -0.520
58 1-02 0.460
89 020/1-D5 0.011
BIO 020(T-1)/(T(1-D5)) -0.055
311 Dsgeo/l-Ds 0.001

512 Cbo/(T(1-D5)) 0.023

 

52

reduces to

m(n+1) = Pa(n) + Qayl(n) (5—3)

where the new matrix 03 is

03 = P02 + 01 (5-4)

Substituting equation 5-5 recursively into itself, the

general solution is

 

 

a(k) = Pka(0) + [Pk-103 . . . 0;] y1(0)
y1(1)
Yi(k-1) (5-5)
__ .4
k-1 . . . . .
Where P 03 . . . 03 is a matrix of order 3 x k, 3 being

the order of the state vector Y, and k the number of time
periods involved. Substituting the linear transformation
5-1 into equation 5-5 gives the solution in terms of the

original state vector, as

Y(k) = PkY(o) - (930; — pk‘103)y1<0)
k-1 _ _
+ Qay1(k) + méi Pk m 1st1(m) (5-6)

Using this general solution the state vector at any time k
can be calculated if the initial state Y(0), and the vector

of inputs y1(n), n=0,1,..I,k are known.

55
Stability Analysis

There are several ways in which stability of linear time
invariant systems can be defined mathematically. One defini—
tion is based on the concept that if a stable system is
disturbed a small amount from an equilibrium state it should
either return to that state or stay within some pre-assigned
finite region near the equilibrium state. This definition,
given originally by Lyapunovl applies to both linear and non-
linear systems. The equilibrium states of the linear system
T(n+1) = PY(n) are stable in the sense of Lyapunov if:2

1. The magnitude of all the zeroes of the minimal

polynomial m(l) of P are less than or equal to one,

and

2. All zeroes of the minimal polynomial of magnitude
one, if any, are simple.

If either of these conditions is violated the system has an
unbounded solution. The solution is asymptotically stable

if and only if the magnitudes of all the eigenvalues of P are
less than one. With caution the characteristic polynomial
can be used for the analysis since it has the same zeroes as
the minimal polynomial, although perhaps of different multi—

plicity.

 

1M. A. Lyapunov, "Problem General de las Stabilite du
Mouvement," Annals of Mathematical Studies, XVII (Princeton,
N. J.: Princeton University Press, 1947).

aKoenig, Tokad and Kesavan, Analysis of Discrete
Physical Systems, p. 248.

54

Stability of the system in this study is investigated
by examining the eigenvalues of the P matrix. In general
the characteristic polynomial of any real matrix P of order

n is of the form

D(?\) = |)\U-P I: in + min-1 + pain—2 + . . . +pn (5—7)

Since the coefficients pi are all real numbers, the zeroes
of D(%) are either real or occur in complex conjugate pairs.
For the system under study here the characteristic polynomial

can be Obtained by expansion of the determinant of the matrix

F;:Ih 0 G 0 F1 F2__-
0 l-l -a4 0 -d2 -Q3
[AU-P] = -08 0 A-Rs -09 -alo -Q11 (5-8)
0 O O x—as O O
0 0 O O A+a6 O
O 0 O 0 0 A+a7
___ __J

 

 

In factored form the characteristic polynomial is

DIA) = [(A-1)(7\-C15) (A+Qe)(7\+a’7)l

[l2 - (R5+al))\ + (ale—QBGH (IS-9)

Substituting the base values for the parameters from Table 1

and Table 5 into equation 5—9 the eigenvalues are

55

A1 = 1

k2 = as = . 500

A3 = -06 = -.125

A4 = ~07 = -.050 (5-10)
A5,x6== (ch + R5)/2

 

:t[ J (01+R5)2 - 4(01R5—deG)]/2

= .808, .097

Since the magnitude of all the eigenvalues is less than or
equal to one, the system is stable, however it is not
asymptotically stable since one eigenvalue is on the unit

circle.
Control Strategy

Modern control theory provides a theoretical basis for
determining the input controls required to take a system
from one state to another in a finite number of time inter-
vals. The controls determined to perform this function are
called the control law, or control strategy. The control
problem is the inverse of the problem of calculating the
system state T at time k as a function of the initial con-
dition T(0) and the time vector of inputs Y1. When determin-
ing a control strategy the final state is specified along
with the initial conditions, and the inputs required to
achieve that final state are calculated. Specifically, the
system under study is said to be state controllable if an

arbitrary state T(0) can be reduced to zero (or any other

56

value) by a finite vector sequence of inputs y1(n), n=1,
2, . . . , k, in a finite number of time intervals k.
Let the general solution for the state model, equation

5-6, be re—written here in the condensed matrix form

T(k) = PkT(O) - Q*y1(0) + 02y1(k)

+ MYi(i) 7 i=1I-o-.k-1 (5-11)

where 0* represents the six element column vector

Png - Pk-le, M represents the product matrix Pk-ZQs...Qs

of order six by k—1, and y1(i) is the time vector of inputs
of order k—1. The matrix M can be converted to order six

by k by including Q2 as its kth column. This adds y1(k) to
Y1 as the kth row, increasing its order also, and maintaining

the conformability of the two matrices. The resulting equa-

tion now appears as

y(k) = pkv<0) - Q*y1(0) + p*y1k (3-12)
The matrix P* is of order six by k and is defined as

P = [P 03 P 03 - - - 03 02] (5-15)

Given the boundary states T(0) and T(k), and the flow driver
at time zero, equation 5—12 can be solved for the time
sequence of scalar inputs Yik- The only requirement for
solution is that the matrix P* must be nonsingular. The
minimum number of time intervals required for control is

determined by the order of the state model. Since the system

57

model developed in this study is Of order six, the state
vector can be controlled in six intervals if P* is non-
singular.

To demonstrate the concept of control the strategy re-
quired to drive the initial state 7(0) to zero in six time
intervals is calculated. The computer program listed in
Appendix A performs this function for the base coefficient
values of Table 1 and initial state value of Table 2. The

resulting control strategy is

y1(1) = 7.2713 x 108
YI(2) = -1.4102 x 109
y1(5) = 2.8178 x 108
yl(4) = 9.3591 x 107 (5—14)
y1(5) = 4.7499 x 106
y1(6) = 4.7780 x 104

The negative sales in this strategy, forced by selection of

a final state of zero, is equivalent to paying customers to
take the product of the firm, and of course is not a feasible
occurrence. The final state Of zero is used only to demon-
strate the control principle; in practice, any final state

can be specified. The results Of this control law are shown

in Figures 2 and 5 which trace the system state variables
during the six period control cycle. These graphs show that

a minimum time control strategy of this type, when implemented,
may subject the state variables to extreme variations. Other

control techniques are available which minimize these

Million dollars

58

 

 

 

 

 

+200—
+|00—
AD
v"§
/ \
I 0 ix\ fix“.
c) x:\ T "4§::--
\;\q\ ‘gxy
\‘b .\. -.-O’ /
-I00— " /
/
\l
O
-200l—
-500;— ---- NE
. —— ”0
NW
1 1 l l I
0 I 2 3 4 5
Period

Figure 2.--Time paths of debt, equity and working
capital under control law.

Million dollars

59

 

 

 

 

+ 50*- °
I'O\
’9.» o ..
of. ” "7.2..
.\ ’0‘“
\“\O/”I
\ ’
\ I
\‘ ’I
0
- IOO- """ NF3
NFZ
———NH
-|50 —
1 l l l
O l 2 3 4
Peflod

Figure 5.--Time paths of fixed assets under control

law.

 

4O

potentially harmful excursions by extending the control
cycle over a longer time period. These Optimal techniques
which allow boundaries to be placed on the state variables
during control are beyond the scope of this thesis.

Using the same base coefficient, and initial state and
sales values, a general control strategy can be developed.
This strategy is expressed as an explicit function of the

desired final state at k=6. This strategy is
- _i
Ylk = (9*) 1[WM] - (P*) [Pkw(0) + Q*y1(0)l (5-15)

or numerically

682.5 -1265.5 -2.6 1552.2 -457.0 5581.1
-946.4 1628.2 100.5 -2582.2 1564.5 -7476.9
165.9 -275.8 -79.5 756.2 -1605.4 5987.6
Ylk = \i’(k)
62.0 -100.4 7.5 240.8 469.7 -1697.4
5.6 -5.2 5.6 15.6 0.5 -4.1
0.1 -0.1 0.1 0.5 ~1.4 59.6y

 

H u_

7.2713 x 108
—1.4102 x 109
2.8178 x 108
- 9.5591 x 107 (5—16)

4.7499 x 106

 

 

4.7780 x 104

41
Simulation

McMillan and Gonzalez3 summarize simulation as an experi-
mental problem solving technique applied to problems of
system design or analysis. The simulation performed in this
study is a portrayal of the Operations of a firm as reflected
by a numerical solution to the mathematical model of the firm.
Operation of the firm is simulated for twenty-five periods,
equivalent to twenty-five years Of operation. The initial
state conditions of Table 2 are used as the starting point
along with sales of 100 million dollars at time zero. Values
at discrete points in time are calculated by recursive sub-
stitution of the state model of equation 2-16 into itself.
Response flows are calculated from the output model of equa-
tion 2-17.

Using the base coefficients in Table 1, identical simu-

lations are run for both linear and nonlinear changes in

sales with time. The specific functions used are

ylln) = Kln + 100 x 106 (5-17)
and

Yi(n) = (Kln+108) + [107sin(2wn/5)] + (107x) (5-18)

 

3Claude McMillan and Richard F. Gonzalez, Systems
Analysis; A Computer Approach to Decision Models (Homewood,
Illinois: Richard D. Irwin, Inc., 1965), p. 16.

42

where 107x is a random noise factor. The nonlinear sales
function begins with the same zero values as does the linear
function, and is a combination of a linear function, a sinu-
soidal function with a ten year period, and a random noise
factor. The noise is normally distributed with unit variance,
and both the periodic and random functions have maximum
amplitudes of ten million dollars.

The effects on the system of changes in policy or environ-
mental conditions can be simulated as described above.
Sensitivity analyses of three system parameters is described
in Appendix B and the results are shown graphically.

The computer program used for system analysis and simu-
lation is listed in Appendix A, along with a brief description,
and Appendix C contains a selected portion of the computer

printout.

CHAPTER IV

SUMMARY AND CONCLUSIONS

The three primary objectives of the study as stated in
Chapter I are:

1. Develop a model of the firm as a dynamic system.

2. Examine the system for stability and controllability.

5. Simulate the response of the firm to its environment
and policy parameters.

A typical business entity, conceptualized as two inter-
acting sectors characterized by flows of funds, is described
in Chapter II. These two sectors are modeled by sets of
algebraic and difference equations as though they were uncon-
strained. System constraints obtained from a graph of the
system are applied to these models to derive the state and
output equations of the system, referred to collectively as
the system model.

In Chapter III numerical parameter values are estab-
lished for the system model from analysis of the financial
reports of a large production firm operating in the United
States. The quantitative model resulting from the intro-
duction of this real data is used for stability and control
analysis, and computer simulations are prescribed for
investigating the sensitivity of system response to para-

meter changes. The system as perceived is shown to be both

45

44

stable and controllable. Specific sensitivity analyses are
reported in Appendix B.

The research has successfully demonstrated the applica—
tion of systems techniques to a specific type Of socio-
economic system—-a business firm. It has shown that a man-
made system of this type can be viewed within the same con-
ceptual framework as a natural system, and that meaningful
results can be achieved by applying to these synthetic systems,
procedural techniques originally developed for application to
physical phenomena. Several conclusions result from the
research.

This model, although gross, is a vehicle for analysis of
the firm, and provides explicit knowledge of important de-
cision parameters. A more detailed level of analysis can be
approached if the system components are sufficiently magnified
and refined; increased detail in the component models,
achieved by a lower scale of aggregation, will increase the
detail of the resulting analysis. Since the two sectors of
the firm are modeled independently of constraints, each can
be removed from the system and studied separately. This
property of systems analysis methodology, when applied to the
model developed in this study, should lead to considerable
refinement in the component models.

While a manufacturing firm forms the basis of this re-
search, the resulting model is not limited to such narrow

applications. The generalized nature of the model allows its

45

use in any type of business situation characterized by money
flows, however, to adapt it to any specific system, that
system's parameter values must be evaluated by empirical
observations.

Techniques for analysis are well developed and highly
articulate within the physical sciences, and are directly
applicable to business systems. The control concept has
special appeal for the business manager; to know what system
inputs would be required to reach a pre-assigned state would
have considerable predictive value, and would greatly assist
in performance of the planning function. If the control
strategy does involve a feasible input set, then the manager
might well gain competitive advantage by utilizing it.

Simulating the Operation of the firm provides a valuable
experimental device to test the results of both internal and
external changes. If the sensitivity of the system to
changes in various parameters is known, they can be given
special consideration. A system model of this form is not an
essential item for simulation; there are many other ways to
model the firm which are well suited to simulation, however,
using the system model in this way is a bonus which accrues
to the system analyst after the model is developed.

While the research has demonstrated that many of the
techniques used in analysis of physical systems have appli-
cation in the realm of business systems, it has also shown

that the resulting analysis can be no better than the system

46

model itself, which in turn depends on the quality of the
component models. The weakest link in the chain of analysis
of business systems is the ability to identify the system
components and model them in a mathematically tractable form.
If business components could be classified as neatly and
described with the precision of physical components, the
value of the discipline to the business environment would be
greatly enhanced.

The first generation model resulting from this research
is acknowledged to be crude; it deals only with first order
effects and grossly aggregated system components. Systems
analysis, when Operational, has the potential for far greater
analytical power than demonstrated here; however, this re-
search shows the feasibility and merit of the concept and
the need for continuing effort to develop an Operational

methodology of pragmatic value in the business world.

BIBLIOGRAPHY

Books

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New York: McGraw-Hill Book Company, 1955.

Boulding, K. E. A Reconstruction of Economics. New York:
John Wiley and Sons, Inc., 1950.

Bonini, C. P. Simulation of Information and Decision
Systems in the Firm. Englewood Cliffs, N.J.: Prentice-
Hall, Inc., 1965.

Brown, R. G., and Nilsson, J. W. Introduction to Linear
Systems Analysis. New York: John Wiley and Sons, Inc.,
1962.

Busacker, R. G., and Saaty, T. L. Finite Graphs and Net-
works; an Introduction with Applications. New York:
McGraw—Hill Book Company, 1965.

Ellis, D. O., and Ludwig, F. J. Systems Philosophy.
Englewood Cliffs, N. J.: Prentice-Hall, Inc., 1962.

Faddeeva, V. N. Computational Methods of Linear Algebra.
Translated from the Russian by Curtis D. Benster.
New York: Dover Publications, Inc., 1959.

Gantmacher, F. R. The Theory of Matrices. Vol. I.
New York: Chelsea Publishing Company, 1960.

Hahn, W. Theory and Application of Lyapunov's Direct Method.
Translated by S. H. Lehnigk and H. H. Hosenthien.
Englewood Cliffs, N.JL: Prentice—Hall, Inc., 1965.

Harary, F., and Norman, R. Z. Graph Theory as a Mathematical
Model in Social Science. Ann Arbor, Mich.: University
of Michigan Institute for Social Research, 1955.

Huggins, W. H., Flagle, C. D., and Roy, R. H. Operations
Research and Systems Engineering. Baltimore, Md.:
Johns Hopkins Press, 1960.

Koenig, D. Theories der Endlichen und Unendlichen Graphen.

New York: Chelsea Publishing Company, 1950.

47

-E«-
$0.13.:
15441 .I o-
r.” ‘

 

 

48

Koenig, H. E., and Blackwell, W. A. Electromechanical
System Theory. New York: McGraw-Hill Book Company,
1961.

Koenig, H. E., Tokad, Y., and Kesavan, H. K. Analysis Of
Discrete Physical Systems. New York: McGraw-Hill
Book Company, 1967.

LaSalle, J., and Lefschetz, S. Stability by Lyapunov's
Direct Method with Applications. New York: Academic
Press, Inc., 1961.

Leontief, W. The Structure of the American Economy 1919-
1959. 2nd ed. New York: Oxford University Press, 1951.

Leontief, W. et al. Studies in the Structure of the American
Economy. New York: Oxford University Press, 1955.

Lorens, C. S. Flowgraphs for the Modeling and Analysis of
Linear Systems. New York: McGraw—Hill Book Company,
1964.

McGuire, J. W. Interdisciplinary Studies in Business
Behavior. Cincinnati, Ohio: South-Western Publishing
Company, 1962. '

McMillan, C., and Gonzalez, R. F. Systems Analysis, A Com-
puter Approach to Decision Models. Homewood, Illinois:
Richard D. Irwin, Inc., 1965.

Naylor, T. H. et al. Computer Simulation Techniques.
New York: John Wiley and Sons, Inc., 1966.

Nord, 0. C. Growth of a New Product: Effects of Capacity-
Acquisition Policies. Cambridge, Mass.: The M. I. T.
Press, 1965.

Orcutt, G. H. Microanalysis of Socio-Economic Systems.
New York: Harper & Brothers, 1961.

. Simulation in Social Science: Readings.
"Simulation of Economic Systems." Englewood Cliffs,
N. J.: Prentice-Hall, Inc., 1962.

Packer, D. W. Resource Acquisition in Corporate Growth.
Cambridge, Mass.: The M. I. T. Press, 1964.

Routh, E. Dynamics of A System of Rigid Bodies. London,
England: Macmillan, 1877.

Schwarz, R. J., and Friedland, B. Linear Systems.
New York: McGraw—Hill Book Company, 1965.

49

Seely, S. Dynamic Systems Analysis. New York: Reinhold
Publishing Company, 1964.

Seshu, S., and Reed, M. B. Linear Graphs and Electrical
Networks. Reading, Mass.: Addison Wesley Publishing
Company, Inc., 1961.

Stubberud, A. R. Analysis and Synthesis of Linear Time-
Variable Systems. Berkeley, California: University
of California Press, 1964.

Zadeh, L. A., and Desoer, C. A. Linear System Theory.
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Articles and Periodicals

Boulding, K. E. "General Systems Theory--The Skeleton of
Science," Management Science, II (April, 1956), 197-208.

Ellis, J. B., Koenig, H. E., and Milstein, D. N. "Physical
Systems Analysis of Socio-Economic Situations,"
(Abstract) Bulletin of The Operations Research Society
Of America, XXII (Fall 1964).

Evans, W. D., and Hoffenberg, M. "The Inter-Industry Rela-
tions Study for 1947," Review of Economics and Statistics,
XXXIV (February, 1952), 97-145.

Forrester, J. W., "Common Foundations Underlying Engineering
and Management," IEEE Spectrum (September, 1964), 66-77.

Frame, J. S. "Matrix Functions and Applications," IEEE
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Frame, J. S., and Koenig, H. E. "Matrix Functions and Appli—
cations, Part III," IEEE Spectrum, I (May, 1964), 100-109.

Gilbert, E. G. "Controllability and Observability in Multi-
variable Control Systems," Journal of the Society of
Industrial Applied Mathematics, Series A: Control, I
(1965). '

Goldman, M. R., Marimont, M. L., and Vaccara, B. N., "The
Interindustry Structure of the United States, A Report
on the 1958 Input-Output Study," Survey of Current
Business (November 1964).

50

Koenig, H. E., "Mathematical Models of Socio-Economic
Systems: An Example," IEEE Transaction on Systems
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Koenig, H. E., and Tokad, Y. "State Models of Systems of
Multi—Terminal Linear Components," IRE International
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Leontief, W. "Quantitative Input-Output Relations in the
Economic System of the United States," Review of
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Lyapunov, M. A. "Probleme General de las Stabilite du
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Miller, J. G. "Toward a General Theory for the Behavioral
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Unpublished Material

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Ellis, J. B. "Description and Analysis of Socio-Economic
Systems by Physical Systems Techniques," Unpublished
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State University, 1965.

APPENDICES

51

APPENDIX A

PROGRAM ANALYZE

The computer program used in this research, written in
5600 FORTRAN language, is listed in this appendix. It is
not a general program, but is written specifically to satisfy
the requirements of the research described by this thesis.
All of the format statements are grouped at the beginning of
the program, following the declarations. The initial state
conditions of the system are listed next, followed by the
statements which read coefficient values from data cards and
calculate parameter values. The control strategy to drive
the system state to zero at n = 6 is next calculated by calls
to the subroutines for multiplying matrices, and computing a
matrix inverse. After the control strategy is printed out,
nineteen simulations of twenty-five years each are executed,
the first using the base coefficient values and the next
eighteen reading new values from the data deck. A twenty-
five year simulation is executed for both a linear and a non—
linear sales function by calling two subroutines; LINSALES
for the linear values, and RANSALES for nonlinear. Subroutine
RANSALES uses the same upward trend line as LINSALES but adds
to this a sinusoid with maximum amplitude of ten million

dollars and a period of ten years. The random number generator

52

55

in the computer is used to superimpose on this a random noise
factor normally distributed between zero and ten million
dollars with unit variance and a mean of five million dollars.
This random sales function is repeatable since the statement
CALL RANFSET specifies the point of beginning and the same
random sequence is returned in each succeeding simulation.

Subroutine MATMULT used for matrix multiplication was
adapted from the computer library, as was the subroutine
MATINV, used by the main program to compute the inverse of
the product matrix PSTAR required for determination of the
control strategy.

The first eight pages of output from this program are

shown in Appendix C.

54

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APPENDIX B

SENSITIVITY ANALYSIS

Effects of management decisions or changes in environ-
mental conditions such as tax rates can be simulated by
changing the magnitudes of the coefficients in the component
equations. Three experiments with parameter sensitivity are
described in this appendix and graphs Of the resulting profit
and debt positions are shown. The three experiments are
conducted with the coefficients K2, R1, and W, which are re-
spectively the factors dealing with the work week, tooling,
and working capital. The system model used for these simu-
lations allows unrestricted bidirectional flow in all com-
ponents. Negative flow on the profit edge y3 is equivalent
to an Operating loss, and negative debt, shown by the internal

state variable N is interpreted as a fund of money the firm

d’
has loaned to others, resulting in interest income to the

firm rather than interest expense. Negative debt occurs only
after all outstanding debt has been repaid and excess funds
are generated from operations. Twenty-five years of Operation
are simulated for each parameter value, for both the linear

and nonlinear sales functions, resulting in four separate

graphs for each experiment.

61

I: GI“

 

 

 

62

The coefficient K2 (the ratio Of the firm's standard
work week to the maximum hours possible) is varied around
the base value in three steps of ten percent plus and minus.
Increases in the work week correspond to management decisions
to meet increased sales through increased use of present
production assets. Since increasing the work week may re-
quire overtime or bonus pay to production workers, this
increase in the variable cost of Operation is reflected in
the simulation by increasing the size of the coefficient C2
in the cash cost of Operation equation ys. If labor makes
up about one-half of the variable cost, C2 is increased in
three related steps of five percent each. Decreasing the
factor K2 is the converse of the above management decision;
however, if the firm has a guaranteed minimum wage, the
variable cost will not be reduced below the base value, and
C2 will remain unchanged. The graphs of profit and debt for
these parameter changes are shown in Figure 4 through Figure
7. These graphs show that maximum profit is achieved by
continuing the present eighty hour work week and meeting in—
creased production requirements through capital expenditures.
The firm continues to earn a profit with all lower asset
utilization rates except the lowest one considered in the
experiment (K2=0.552). For this lowest value of K2 increas-
ing losses are incurred and increased amounts must be borrow—

ed, resulting in an undesirable Operating condition.

in million dollars

Profit

+|5

+lO

+5

Figure 4.--Sensitivity of profit to changes in work

65

 

 

 

 

Curve K2 Cg

I .3332 .540
2 .3808 .540
3 .4284 .540
4 .4760 .540
5 .5236 .567
6

7

 

 

 

 

 

l

l

 

 

 

IO

week for linear sales.

Period

IS

20

25

in million dollars

Profit

64

 

 

 

+I5

 

 

 

 

 

+I0

+5

 

 

l l l l
0 5 l0 IS 20 25

Peflod

 

 

Figure 5.-~Sensitivity of profit to changes in work
week for nonlinear sales.

in million dollars

Debt

65

 

 

 

 

 

 

 

 

 

 

 

 

I
+507—
+257~ 2
07— 3
7
6
Curve K2 C2 5
-25— l .3332 .540 4
2 .3808 .540
3 .4284 .540
4 .4760 .540
5 .5236 .567
6 .57l2 .594
7 .6l88 .62I
-505.
1 1 1
O 5 IO I5
Period

Figure 6.--Sensitivity of debt to changes in work
week for linear sales.

in million dollars

Debt

+50

+25

66

 

 

 

 

 

 

 

 

 

 

 

25

0
Curve K2 C2
I .3332 .540
_ 25 2 .3808 .540
3 .4284 .540
4 .4760 .540
5 .5236 .567
6 .57l2 .594
7 .6 I 88 .62 I
- 50
l l l
5 l0 IS 20
Penod

Figure 7.--Sensitivity of debt to changes in work
week for nonlinear sales.

67

The second experiment is with the coefficient R; which
represents the firm's annual tooling level, and directly
influences sales through styling changes. This coefficient
is varied around the base value of sixty dollars in steps of
ten, twenty, and thirty percent, while holding the other
assets per unit, R2 and R3, constant. Each change in R1 is
assumed to result in an equal percentage change in the normal
change in sales (K1 in the sales function). The resulting
graphs of profit and debt during the twenty-five years of the
simulation are shown in Figure 8 through Figure 11. Although
increased tooling expenditure does cause increased sales
income, the maximum profit position is reached when only
minimum expenditures are made. This condition also results
in minimum borrowing requirements, and would be the recom-
mended strategy for the firm to follow under these conditions.

Experiment number three deals with the ratio of working
capital to sales (coefficient W in the system model). There
are many things that might affect W, examples are changes in
policies relating to current liabilities, or changes in
accounts receivable. Policy changes in the credit offered
to customers generally influence the sales volume in a direct
manner; increased leniency in credit terms increases sales,
and more stringent credit policies reduce sales. Policy
czhanges of this type are simulated by varying W in steps of
ten, twenty, and thirty percent above and below the base

value. Customer reaction is reflected by introducing

 

in million dollars

Profit

 

+|5-

+IO

+
(II

 

 

 

 

 

 

 

 

 

 

2
3 l
4 I}
6
7
Curve R. K.-=-l05
— I 42 5. II
2 48 5.84
3 54 6.57
4 60 7.30
5 66 8.03
6 72 8.76
_ 7 78 9.49
l l L l
0 5 IO IS 20
Period

Figure 8.--Sensitivity of profit to changes in tool-

ing for linear sales.

in million dollars

Profit

69

 

+l5

+IO

+5

 

 

 

 

 

 

 

 

 

 

 

Curve R. K.-:-l06
0 P I 42 5| I
2 48 5.84
3 54 6.57
4 60 7.30
5 66 8.03
6 72 8.76
7 78 9.49
_ 5 .—
I I I I
O 5 IO IS 20
Pefiod

25

Figure 9.--Sensitivity of profit to changes in tool-

ing for nonlinear sales.

in million dollars

Debt

+40

+20

-20

-4o

-60

7O

 

 

 

 

 

 

 

 

 

 

 

 

 

Curve R. K.-:—I05
I 42 5.II
2 48 5.84
3 54 6.57
4 60 7.30
5 66 8.03
6 72 8.76
7 78 9.49
I I I
I0 I5 20
Penod

25

Figure 10.--Sen8itivity of debt to changes in tooling
for linear sales.

in million dollars

Debt

71

 

 

 

 

 

 

 

 

 

 

 

+40—
+20-
O _
_ 20 _
-4O " Curve Fi' K|+|06
l .42 5.II
2 48 5.84
3 54 6.57
4 60 7.30
-60 — 5 66 8.03
6 72 8.76
7 '78 9J¢9
I 1 l l
O 5 K) IS 20
Peflod

25

Figure 11.--Sensitivity of debt to changes in tool-
ing for nonlinear sales.

“l

 

_..-'

 

 

 

72

corresponding changes in the normal change in sales (K1 in
the sales function). The graphs of Figure 12 through Figure
15 show the sensitivity of profit and debt to these policy
changes. These graphs show that this firm can achieve
greater profit by reducing the stringency of their credit
policies, thus increasing sales volume and accepting the
attendant increases in working capital requirements. This
position, however, does increase the debt level and extend

the point in time when the firm becomes debt-free.

in million dollars

Prof”

75

 

 

 

 

 

 

 

 

 

 

 

 

7
6
+I5P— 4
3
2
l
+ I0 —
+ 5 —
Curve w K.+I06
0" I .n2 5.”
2 J28 1184
3 J44 I557
4 J60 'r30
5 J76 I103
6 J92 8J6
7 .208 9 .49
l l l l
O 5 IO IS 20 25
Pefiod

Figure 12.--Sensitivity of profit to changes in ratio
of working capital to sales for linear sales.

in million dollars

Profit

74

 

 

 

 

 

 

 

 

 

 

 

 

+I5 -
+|O _
+5 -
Curve w KL-i-IOS
0 *- l JI2 5.II
2 J28 5.84
3 .I44 6.57
4 J60 7.30
5 J76 8.03
6 .I92 8.76
7 .208 9.49
l l l l
O 5 IO I5 20 25
Pefiod

Figure 15.--Sensitivity of profit to changes in ratio

of working capital to sales for nonlinear sales.

in million dollars

Debt

75

 

 

 

 

 

 

 

 

 

 

 

+|O '—
_ 7
O 6
5
‘5
‘6
-IO l— 4 ‘\
3
2
I
Curve W K.-!-l06
I .II2 5.II
2 J28 5.84
3 .I44 6.57
4 J60 7.30
‘30 "' 5 .I76 8.03
6 .I92 8.76
7 .208 9.49 w\
l l l l 7
O 5 IO I5 20 25
Peflod

Figure 14.--Sensitivity of debt to changes in the ratio
of working capital to sales for linear sales.

"Hr—H.-

 

 

in million dollars

Debt

76

 

 

 

 

 

 

 

 

 

 

 

 

+l0 I-
0 ~— g
5
—I0 e 4
3
2
I
_20 .—
Curve W Ki-i-IO6
I .n2 5JI
2 J28 584
3 .I44 (L57
4 J60 'r30
-30 *- 5 .I76 8.03
6 J92 :876
7 .208 9.49
L l I | _
O 5 l0 IS 20 25

Penod

Figure 15.--Sensitivity of debt to changes in the ratio
of working capital to sales for nonlinear sales.

APPENDIX C

SELECTED COMPUTER OUTPUT

This appendix contains the first eight pages of computer
printout from program ANALYZE listed in Appendix A. The
first page shows the coefficient values used in the control
strategy computation and the resulting parameter values.
Matrices P, 01, and 02 in the system model are shown next
along with the powers of P from two through six. The product
matrix which operates on the input vector, called PSTAR, is
shown next and its inverse is also shown. The result of the
test of the accuracy of the inverse is shown to be a unit
matrix with accuracy to seven significant figures. Inter-
mediate steps in computation of the control strategy are
printed out next, including the transient values of the state
and sales boundary conditions, and the error signal in the
sixth time period for the desired result of zero. The time
vector of inputs to drive the state to zero at n = 6 is
listed next. The last three pages of the appendix show the
coefficient values and the results of the first simulations

with base values and linear and nonlinear sales functions.

77

78

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