THE DEVELOPMENT OF A STATE'V‘SPACE MODEL
OF. AN AEROSPACE ENTERPRISE AND ITSEVALUATION: " '
AS A PRACTICAL MANAGEMENT TOOL I

Thesis for the' Degree of Ph. D.
MTC‘HJGAN STATE umvERsnv
RENE v. ELICARO A
1967

THE-1515

This is to certify that‘the

thesis entitled

THE DEVELOPMENT OF A STATE-SPACE MODEL
OF AN AEROSPACE ENTERPRISE AND ITS EVALUATION
AS A PRACTICAL MANAGEMENT TOOL

presented by

Rene V. Elicano

has been accepted towards fulfillment
of the requirements for

_P_h;D_-__ degree in Management

  
 
  

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Date November 17, 1967

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ABSTRACT

THE DEVELOPMENT OF A STATE -SPACE

MODEL OF AN AEROSPACE ENTERPRISE

AND ITS EVALUATION AS A PRACTICAL
MANAGEMENT TOOL

by Rene V. Elica'fio

This study constructs a detailed mathematical model of the internal
operations of an actual aerospace engineering and manufacturing enterprise .
It uses a methodology originally developed for analyzing system models of
electrical networks. This study evaluates the practical utility of this
technique in assisting the business executive.

The methodology used offers a formal process for generating a model
from a defined system structure and for analyzing the solution characteristics
of the model. This approach involves the use of linear graphs, flow and
unit cost variables and the development of a state model by reduction of the
algebraic and difference equations that characterize the system to a minimal
set. This model can be used for simulation and for stability and control
analysis .

The model in this study is essentially a direct costing accounting type
model. It follows product flow from receipt of order to shipment thru the
internal sectors of the firm. It shows the flow of resources and overhead

function services into the different product stages and imputes their cost

Rene V . Elica’fio

to the product.

Simulation based on the model is presented. The model is found to be
stable and state controllable. The control strategy to reduce the state to
zero is derived.

Considerable practical utility is found for both the methodology and the
model. The methodology provides a convenient formal and generalized
procedure for developing a model of the firm. This model of the firm can
be quite detailed without sacrificing compactness. This makes feasible a
real -time inquiry and simulation system whereby management can interro-
gate the model and get fast response time, without interfering with the
production programs that the computer is processing simultaneously.

The model has many managerial uses such as forecasting costs, load,
resource requirements or cash flow and analyzing total impact of alternative
decisions.

The most promising facet of this methodology is its generalized
analytical capabilities. As management pins down more of the true cause
and effect relationships in the firm, the control strategy from the model will

play an ever increasing role in guiding their key decisions .

THE DEVELOPMENT OF A STATE-SPACE MODEL
OF AN AEROSPACE ENTERPRISE AND ITS

EVALUATION AS A PRACTICAL MANAGEMENT TOOL

By

. I
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Rene V E lica’fio

A THE SIS

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

DOCTOR OF PHILOSOPHY

Department of Management

1967

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ACKNOWLEDGMENTS

The author wishes to express his appreciation to the many persons who
have assisted him directly or indirectly in making this study. Special thanks
are due to Mr. T. R. Rao, Dr. Richard Gonzales, Dr. Thomas Farrell, and

Dr. Herman Koenig for their invaluable advice and technical guidance .

ii

TABLE OF CONTENTS

ACKNOWLEDGMENTS.............

LISTOFTABLES.............

LISTOFFIGURES.............

LIST OF EXHIBITS O O O O O O O O O O O O O 0

Chapter

I.

II.

III.

INTRODUCTIONo-ooo.......

Purpose of the Study . . . . . . . . . .
Significance of the Study . . . . . . . . .
Scope of the Study . . . . . . . . . . . .
Methodology Used . . . . . . . . . . . .
Organization of the Study . . . . . . . .

DESCRIPTION OF THE BUSINESS ENTERPRISE

Size . .. . . . . . . . . . . . . . . .
Product Line . . . . . . . . . . . . . .
Market Environment . . . . . . . . . . .
Organization . . . . . . . . . . . . . .
Production . . . . . . . . . . . . . . .
Engineering . . . . . . . . . . . . . .
Contracts . . . . . . . . . . . . . . .
Manufacturing Service Functions . . . . . .
Other Service Functions . . . . . . . . .

CONSTRUCTION OF THE MODEL . . . . .

Alternative Formulations . . . . . . . . .
Building the Model . . . . . . . . . . . .
Component and Constraint Equations . . . . .
State Model and Output Vector . . . . . . .

iii

Page

ii

Vi

vii

O‘QNNH H

\l

15

15
16
19
33

Chapter Page
IV ANALYSIS OF THE MODEL 0 O O O O O O O O O O O O 4].
Parametric Data . . . . . . . . . . . . . . . . . 41
Heuristic Simulation . . . . . . . . . . . . . . . . 41

StabilityAnalysis.................60
contIOI AMIYSiS O O O O O O O O O O O O O O O O O 62

V SUMMARY AND CONCLUSIONS . . . . . . . . . . . 67
Summary....................67
Practical Utility to the Executive . . . . . . . . . . . 68

Future Development ofthe Model . . . . . . . . . . . 69

BIBLIOGRAPHY O O O O O O O O O O O O O O O O O O O O O 72

iv

10.

11.

12.

13.

14.

15.

16.

17.

LIST OF TABLES

Service and other overhead functions

Edges of proper tree . . . . . .

State, control and output variables of model

Input output parameters - . . .

Resource parameters - Basic line processes - Direct

resources........

Resource parameters - Basic line processes - Service

and other overhead functions

Service and other overhead functions - By type of resource .

Cost reduction parameters . . . .
Simplifying variables . . . . . .

Initial state values . . . . . . .

Given control vector values for heuristic simulation

Simulated states from initial state thru twelve time periods

Coefficients of characteristic polynomial of P .

Eigenvalues ofP . . . . . . . .

Coefficients of characteristic polynomial of H .

Eigenvalues ofH. . . . . . . .

Control signals to reduce state to zero

Page
1 9
21
34

42

43

44
48
49
50
56
57
58
61
61
63
64

65

Figure
1 .

2.

LIST OF FIGURES

Functional organization chart . . . . .

System graph of the model of the enterprise

Vi

Exhibit

LIST OF EXHIBITS

State model . . . . . . . . .
Output vector . . . . . . .
State model with parameter values

Output vector with parameter values

vii

Page
. 35
- 39
. 51

.54

CHAPTER I
INTRODUCTION

Purpose of the Study

 

The purpose of this study is to attempt to construct a mathematical
model of an actual aerospace engineering and manufacturing enterprise,
using a methodology originally developed for analyzing system models of
electrical networks. The study evaluates the practical utility of this
technique in assisting the business executive .

It has been the dream of many executives to have at their disposal,

a comprehensive and integrated computerized model of the entire firm,
facilitating the planning and forecasting tasks, simulating the over-all
impact of alternative decisions on a real -time basis, and utilizing
sophisticated techniques for developing optimal policies for the total firm
with its complex network of interacting relationships .

This study represents one phase of a long range program to provide
the management of the firm being analyzed with such a model, tailored to
its specific needs and characteristics and developed to the limits of
technological and economic feasibility.

Although the model constructed here is strictly a prototype, every
effort has been made not to compromise the accuracy and integrity of the
model. Company-private data has been modified without detracting from

the realism of the model.

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V

Significance of the Study

 

The use of mathematical models as a managerial tool is nothing new.
However, the majority of the practical applications have utilized either
straightforward heuristic trial -and-error simulation or highly specialized
analytical techniques . The approach evaluated in this study was chosen
because of its potential value as a more generalized analytical tool for
business models. If a realistic and comprehensive model of the internal
operations of a firm can be built without being unwieldy, and if this model
can be used not only to simulate the impact of alternative management
decisions but also to determine analytically the optimum policies to attain
specified objectives, then this methodology is of considerable practical
utility to the executive.

As best as can be determined, this study represents the first attempt
to apply this particular methodology to develop a detailed model of the
internal operations of an actual firm . A previous study by Mr. Linn Soule, 1
applied this approach to a firm but the model was limited to a black-box
cash-flow analysis which did not depict the inner workings of the firm .

Most other socio-economic applications have been in areas other than models

. . . . . 2
of the firm, such as a model of an educational institution.

 

lB. Linn Soule, "A Discrete State Deterministic System Model for Analysis

of the Firm, " (Unpublished Ph.D. Thesis, Michigan State University, 1967).

2Herman E . Koenig et al, A Systems Approach to Higher Education,
(East Lansing, Michigan: Michigan State University, 1966).

 

Scope of the Study

 

This study is concerned primarily with the generalized analytical capa-
bilities of the methodology used, as opposed to heuristic simulation. This is
not to imply that simulation is not applicable nor useful. On the contrary,
heuristic simulation has proven to be the most widely applicable and most
useful tool in the area of modeling. A recent unpublished survey3 made by
the IBM Corporation of its customers' applications of simulation, listed one
hundred forty-eight different uses in twenty-seven separate areas. In this
study, the term "heuristic simulation" is used in the sense of "cut-and -try"
simulation with a model that tries to describe reality.

Heuristic simulation will certainly play a major role in the over -all
modeling program of which this study is a part. However, since the field of
simulation is well-developed and well-travelled, this study limits itself to
demonstrating the feasibility of simulation as one of the facets of the method-
ology being evaluated.

Since the model constructed in this study is strictly a prototype, the
degree of detail depicted has been simplified to the point where the size of
the model is manageable but the realism of the model is not sacrificed.
Contingent on favorable findings of the study, the model can be easily expanded
to any desired degree of detail afterwards . Another limitation on the prototype
model is the ready availability of certain data. Fortunately this has not been

a major handicap due to the highly automated and sophisticated management

 

3IBM Corp., "Survey of Simulation Applications," (Unpublished Survey,
White Plains, N.Y., 1966).

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information and control system already existing in the firm being modeled.

The model constructed is symbolic and not iconic or analogue. In a
symbolic model, the components of what is represented and their inter-
relationship are given by symbols. Iconic models such as photographs and
sculptures "look like" what they represent. An analogue model represents
one set of properties, such as distance on a graph representing time or cost.

The model constructed is dynamic instead of static in that it can gener-
ate time paths of model variables . It describes how the system changes over
time. The model,at least the prototype version developed in this study, is
an aggregate or macroscopic one .

It is deterministic. The introduction of probability and Monte Carlo simu-
lation will come after this study. All equations are linear. Treatment of non-
linear relationships and their evaluation thru simulation will also come afterwards.

Since the internal operations of the firm are being modeled, the approach
taken is "structural" rather than "black box." The model is discrete . Hence
the equations used are difference rather than differential equations. Difference
equations parallel more closely the periodic batch-type reporting systems that
characterize industry.

Methodology Used

 

The methodology used is composed of the group of analytical techniques
described by Dr. Herman Koenig4 of Michigan State University in his mimeo-

graphed notes for a course on "Systems Analysis for Social Scientists".

 

4
Herman E . Koenig, "Systems Analysis for Social Scientists," (Unpublished
Class Notes, Michigan State University, 1966). (Mimeographed)

These techniques were developed originally for the analysis of system
models of electrical networks. This theory offers a formal process for
generating a model of the system from a defined system structure and for
analyzing the solution characteristics of the system model to simulate the
physical behavior of the system as a fimction of a change in the structural
features .

Like an electrical or mechanical system, a firm can also be conceptu-
alized as a system of interacting components. These components are
modular and are characterized by their behavior as measured at their inter-
faces with each other. Thus the components can be studied by themselves
or at any desired subsystem level.

Linear graph theory provides a convenient way of representing system
components and their interfaces by points (vertices) and connecting lines
(edges). The behavioral characteristics of a component can be specified by
a pair of complementary variables associated with each edge .

The X variable is called the propensity variable and is usually regarded
as the "cause" or "result" of the flow or Y variable. In mechanical processes
X and Y can be velocity and force respectively; in hydraulic processes,
pressure and flow rate; in electrical processes, voltage and current; in
a model of the firm, imputed cost and flow of units thru production.

The theory gives explicit procedures for developing a set of simultaneous
algebraic and/or differential or difference equations showing the inter-
dependence of a set of variables which characterize the observable behavior

of the system. These equations are composed of the system's component

characteristic equations and the constraint equations from the interconnection
pattern of the linear graph. The reduction of this set of equations to a
minimal set of first ordered difference or differential equations produces
the state model.

Solving the state model enables one to simulate its behavior over time.
Analytical procedures are furnished for testing whether or not the system
is stable, and whether or not any desired state or output can be reached by
manipulating the control variables . Optimal control techniques are also

available but will be re served for future work beyond this study.

Organization of Study

 

Chapter 11 describes the business enterprise being modeled and various
considerations in attempting to model it.

Chapter III narrates the construction of the model and problems
encountered. The linear graph as well as the matrices that form the state
model and output vector are developed.

Chapter IV treats the evaluation of the model. The derivation of
parameters from empirical data is described. Heuristic simulation is
demonstrated. Stability, state controllability, output controllability and
control strategies are analyzed.

Chapter V presents the summary and conclusions of the study. An
assessment is made of the practical utility of this methodology from the
standpoint of the business executive. Plans for further development of the

model are presented. The Bibliography follows Chapter V.

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CHAPTER II

DESCRIPTION OF THE BUSINESS ENTERPRISE

Size

 

The business enterprise being modeled is the largest division of a highly
diversified international corporation with yearly sales of $400 million. This
division accounts for about a fifth of corporate sales and employs about

four thousand people.

Product Line

 

Its products include reference and navigation systems for high per-
formance aircraft, missile guidance systems, spacecraft controls and
displays, etc. Its standard products can be categorized into gyroscopes,
indicators, amplifiers, computers, and accessories. The orders received
for these products can be divided into repeat business for old products and
new business for products which require substantial engineering design effort.
Spare parts orders contribute significantly to sales volume.

Besides the above hardware, pure engineering is also a major product
line . Customers frequently fund research and development studies and proto-
type hardware. Some examples are lunar rendezvous simulation studies or
development of a low cost inertial guidance system.

In order to check out the product , customers have to purchase aerospace
ground equipment (AGE). Such ground support equipment may be used, say,

by the maintenance crew on an aircraft carrier to check that the systems on

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the jet aircraft are "go" prior to take-off. Like the standard product line,
aerospace ground equipment orders can be classified into repeat and new
business .

Users of the products need operating instructions, maintenance manuals,
parts lists, and other technical data. The sale of this technical data consti-
tutes an attractive block of business .

Finally, these products wear out or are damaged and have to be returned
to the factory for repair or overhaul. These sales fall under the heading of

Service, Repair and Overhaul (SRO).

Market Environment

 

The majority of the sales are made to the Government either directly to
the Department of Defense and the National Aeronautics and Space Adminis-
tration or indirectly through Air Frame Prime Contractors . Most business
is won thru fixed -price competitive bidding. However, some contracts,
especially those of a developmental nature, are based on other terms such as
time -and-material, cost-plus-fixed-fee, cost-plus-incentive -fee, redetermin-

able price and fixed price with incentive provisions.

Organization

 

Figure 1 shows a functional organization chart. This organization is in
contrast to the project oriented organization typical of the huge air frame

prime contractors whose contracts are large enough to justify separate and

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10

self -sufficient project organizations.

Production

The business is characterized by highly sophisticated products, tight
quality requirements, simultaneous production of a large variety of models,
small quantity orders, very short lead times and competitive prices. Manu-
facturing processes can be classed into fabrication and assembly operations.
Due to the short-cycled lead times on most orders and the stringent
technological demands, many machined and sheetmetal parts are fabricated
in-house instead of purchased. The intricate assembly and calibration
operations are performed in environmentally controlled "clean rooms" . .

A Flexible Shop for non-production type fabrication and a Tool Room for
making and maintaining tools complement the Production facility.

The Production facility is run as a job shop. The wide array of products
enjoy a high degree of parts commonality. To achieve volume economies,
production job orders are not segregated by contract. To establish the parts
requirement schedule, the final unit schedules of all contracts are exploded
backward in time into their bills of material. To forecast or determine
status by contract, the parts status is allocated by contract as it is imploded

forward in time to final unit stage .

Engineering

 

Like the Production function described previously, the Engineering

1'1

function is an integral part of the basic line process of the firm. It can be
looked upon as a production shop turning out design specifications and draw-
ings for Manufacturing, and both software packages and prototype hardware
for customers. The Engineering services group provides facilities such as
a scientific computer, materials and environmental testing,and a model shop.

Besides line (direct labor) type engineering, other engineering activities
include company-funded research and development, production support and
sales support. Since the product technology often strains the state of the
art, close support from Engineering is needed to manufacture the products to
specifications, on schedule, and profitably. The highly technical nature of
the product also requires the engineers and contracts personnel to work

together in dealing with customers .

Contracts

The Contracts operations cover the Field Sales Offices, the Marketing
function, the Contract Administration function and the Support Logistics
function. The Field Sales Offices are the communication link with the
customer. All marketing campaigns are planned and implemented by the
Marketing group together with the sales support people from Engineering .
Market Research and Analysis is also the responsibility of the Marketing
Group. Once a sale is made, the Contracts Administration function processes
the order and assumes responsibility for all the tasks related to that order

till it is fulfilled. The Support Logistics function handles all sales of spare

12

parts, repair and overhaul services and technical data . The actual production

of the technical data is done within this group.

Manufacturing Service Functions

 

Production operations are supported by the Manufacturing Engineering,
Production Control and Plant Engineering functions .

Manufacturing Engineering takes the design specifications and drawings
from Engineering and develops the processing instructions and tooling to
build the product. It generates all manufacturing cost estimates. It
supports production by solving tooling, processing and other technical
problems that interfere with production. Where design problems exist, the
production support personnel from Engineering provide the necessary
assistance . Cost reduction of production costs is also a prime responsibility
of Manufacturing Engineering.

Production Control is responsible for the coordination and control of all
manufacturing activities, from receipt of order to shipment of product. The
procurement, manufacture and movement of all materials through the plant
are controlled by a plan and schedule developed by Production Control. This
function dispatches and expedites job 5, and assigns all priorities while
reporting on work progress . Production Control is responsible for inventory
control both from a planning and a physical control standpoint. It controls the
incorporation of engineering changes on all products. By acting as the sole
authorized source of all manufacturing commitments and other schedule

information, it functions as a communications control center between Manu-

 

l3

facturing and other operations . It forecasts, measures, and manipulates
the manpower and facility workload and capacity so that schedules can be
met with optimum efficiency and economy.
The Plant Engineering function is responsible for all plant layout, facility
and building maintenance, equipment installation, and planning and supervision

of brick-and ~mortar activities .

Other Service Functions

 

The remaining service functions are Purchasing, Quality, Finance,
Industrial Relations and Management Systems.

Purchasing is responsible for all outside procurement of materials, parts,
supplies, equipment and services . It develops and maintains adequate and
reliable sources of supply. Cost reduction of purchased material is a primary
responsibility of this function.

The Quality function is responsible for the quality assurance of product
at minimum cost through a continuing control function. It generates and
implements the procedures necessary to see that all quality and reliability
requirements are met. Other activities include measurement calibration and
standards control, quality assurance testing, quality audits, failure reporting,
and customer test correlation.

The Finance (or Accounting) function prepares budgets, develops financial

l4

objectives and prepares controls and reports for evaluating financial
performance. Its activities include payroll and timekeeping, accounts
receivable and payable, pricing, cost accounting, etc.

The Industrial Relations function is responsible for coordinating,
administering and maintaining all company-employee relationships . This
includes labor relations, recruiting, wage and salary administration,
employee benefit programs, coordination of employee training and manage-
ment development, security, plant protection and first aid .

The Management Systems function has a two -fold responsibility. First,
it coordinates and controls all the firm's activities relative to major new
product programs. Since the organization of the firm is functionally oriented,
program -oriented control has to come from this activity. Second, Manage-
ment Systems is responsible for all scientific management technology, systems
analysis, programming, and computer operations. One of its prime objectives
is the design and implementation of a highly computerized, integrated and

sophisticated management information and control system. One step towards

this is a long range program for the development of a model of the firm.

CHAPTER IH

CONSTRUCTION OF THE MODEL

Alternative Formulations

 

A firm is such a complex multi-faceted entity that any attempt to achieve
a truly comprehensive model is fraught with frustration.

To capture systematically the elements that characterize this firm, each
sector was analyzed as to its function; the decisions it made; its inputs and
outputs in terms of data, labor, material, facilities and funds; and the source
and destination of each input and output. This mass of information represented
the basic raw material to be molded into a model of the firm.

However, the shape of this model depends on its purpose. An all-purpose
model would be so unmanageable that it would become a no-purpose model.
One approach often visualized by the executive is to develop an integrated auto-
mated management information system which will not only perform the actual
functions of the firm but also act as a model by processing upon request
hypothetical instead of actual data . There is nothing conceptually wrong with
this approach. Unfortunately the building of such a system is a long, tedious
and sometimes economically unfeasible task. And the increased cost of a
computer large and fast enough not only to process the programs using the
actual operating data but also to simulate the firm with hypothetical data and
provide management with the answers in an acceptable waiting time, will

tend to discourage most firms from this approach to a model of the total firm.

15

16

The above approach has been used successfully for specific parts of the
firm. For example, the firm being studied has used the regular machine load
forecasting program to compute the impact of potential or hypothetical contracts .
However to model the total firm usefully and economically, the model would
have to limit itself to the truly significant factors and relationships .

Specific parts of a firm are easier to model than the total firm . Examples
are job shop simulators for testing alternative sequencing rules, queuing
models, inventory control models, marketing games, or manufacturing games .

Another approach to modeling a firm is to model specific aspects of the
total firm. Some examples are a communications model, a decision-making
model, a cash flow model, a paperwork flow model, an accounting model, etc .

Thus in specifying a model of the total firm, one must decide on the
objectives of the model, the perspective to take, the important sectors, factors
and relationships in the firm that should be included in the model and the degree

of detail to portray.

Building the Model

 

For the prototype model in this study, it was decided to build what was
essentially a direct costing accounting -type model. This model follows the flow
of the product from receipt of order to shipment thru the various sectors that
represent the basic line processes of the firm. The model depicts the flow of

resources into the different product stages either directly or indirectly thru

17

the service and other overhead fimctions. As these resources and services
go into the product, their costs are imputed to the product cost. The two
complementary variables specifying the behavioral characteristics of each
component are the flow volume of the units and resources as the Y variable
and the imputed cost of these flows as the X variable . To avoid repetition,
the objectives of the prototype model will be deferred to the last chapter
when its uses are discussed.

An initial attempt was made to build product price into the model. In
other industries prices are determined after production by adding a fixed
mark-up to costs or prices are set by market demand and supply or by a
price leader, etc . In this industry pricing is done before the contract is won
and competitive bidding is the predominant method of government procurement.
Unfortunately competitive pricing data is one of the few areas where available
empirical data leaves something to be desired. It was decided not to bog down
the prototype model with the development of a bid strategy model. This can
be done later on. Therefore product price is excluded from the model. It
is felt that this does not jeopardize the usefulness of the prototype model
provided sales input volume is expressed as a function of the previous period's
product cost besides other factors . This relationship implies that lower costs
will permit lower prices which will win more contracts. Furthermore, a cost
model is quite pertinent to management's goal of measuring, forecasting,

controlling and minimizing costs.

18

To keep the size of the model manageable, the product line is aggregated
into seven categories, namely technical data, repair and overhaul work, new
ground support equipment business, repeat ground support equipment business,
engineering software and hardware, new standard product contracts and
repeat standard product contracts. Subsequent models should expand the
standard product category to the various product families .

The manufacturing processes are compressed into the two stages of
fabrication and assembly .

Several factors are found which constitute the control variables used
by management. These include company-funded research and development,
the marketing budget, the sales support budget, the investment in facilities,
inventory and equipment (especially automation equipment like computers),

A and investment in service function manpower. These can all be classed
into cost reduction activities and sales promoting activities .

Twelve service and other overhead functions are built into the model.
These are administration, flex shop and tool room, finance, industrial
relations, plant engineering, management systems, production control,
quality, manufacturing engineering, purchasing, contracts and engineering
support.

Basic resources are reduced to labor, material and facilities. Since
the enterprise under study is a division of a corporation, the funds it uses
to pay for these resources are borrowed from the corporate office . A fixed
charge is levied by the corporate office on its divisions to cover both interest

and corporate overhead costs .

19

Figure 2 is the system graph of the model of the enterprise. The numbers
in the circles are used to identify the edges of the graph. For example Y6_25(n)
refers to the flow on the edge from the circle marked 6 to the circle marked 25 .
This is the number of units of standard product flowing out of the fabrication
sector to the assembly sector during time n. X6_25(n) is the imputed cost of
each of these units at that point in time. Table 1 lists the service and other
overhead functions. In line with keeping down the size of this prototype model,
the basic unit of time chosen is a quarter.

Table l . - Service and other overhead functions.

A B Function

60 80 Administration

61 81 Flex Shop 81 Tool Room
62 82 Finance

63 83 Industrial Relations

64 84 Plant Engineering

65 85 Management Systems
66 86 Production Control

67 87 Quality

68 88 Manufacturing Engineering
69 89 Purchasing

70 9O Contracts

71 91 Engineering Support

Component and Constraint Equations

 

The following section shows the development of the component equations

20

 

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21

describing the characteristic relationships between system components .
Linear graph theory provides mechanical procedures for forming constraint
equations. These are relationships which must exist between the system
variables by virtue of their structural interconnections in the system linear
graph. These mechanical procedures involve selecting a "proper tree" in

the linear graph and developing "fundamental cut set and circuit equations"
(which are the constraint equations). By reducing the component and constraint
equations to a minimal set of first ordered difference equations, the state
model is attained.

Table 2 lists the "edges" of the "proper tree" selected. In the following
sections, all input unit costs are taken as positive numbers and output unit
costs as negative numbers . This convention is necessary from the standpoints
of both the methodology used and sound accounting practice .

Table 2. - Edges of proper tree .

16-6 3-24
17-1 8-30
18-2 9-31
19-3 4-32
20-9 5-33
21-4 j-k (j = 27 to 33; k = 34 to 40)
22-5 k-l (k = 34 to 40; l = 41 to 47)
6-25 54-m (m = 12/p, 13/p, l4/p)
10-27 (p = A/l to A/ll, 1 to 11)

1-23 (A = 60 to 71)

22

Table 2. - (continued)

7-26 A/q-B/q (q=1 to ll)(A=60 to 71)(B=8O to 91)
11-28 49-3 (8 = 50 to 53)
2-29

Incoming Orders Sector

 

The per unit cost of incoming orders XIS-i (for i = 16 to 22) is expressed
as a direct function of the previous period's per unit factory cost Xm-k
(for m = 10, ll, 2, 8, 9, 4, 5 and k = 27 to 33). This indicates that the cost of
acquiring business decreases as lower factory costs make this task easier.

The volume of incoming business YIS-i (for i = 16 to 22), is determined
partly by sales promoting activities, partly by the previous period's shipment
volume Yj-48 (j = 41 to 47), partly by the previous period's cost of sales
Xj-48 (j = 41 to 47) and partly by some known trend Fj (j = 41 to 47). This
trend allows for the basic expansion or contraction of the market.

The component and constraint equations are given below:
Xi-p(n+l) = 'Ki-p Xm_k(n)

(i = 16 to 22; p = 6,1, 2,3,9,4,S; m = 10,11, 2,8,9,4,5;

k = 27 to 33) (l) to (7)

Y15_i(n+l) = KIIOWII Fj(n)+Kk-1Lk'lz(n)+Pj‘48Yj'48(n)+Qj-48Xj"48(n)

(i = 16 to 22; j = 41 to 47; k = 27 to 33; l = 34 to 40) (8) to (14)
Cut Set Eqns:
Y15_i(n) = Yi-m(n) (i = 16 to 22; m = 6, l, 2, 3, 9, 4, 5) (15) to (21)

Circuit Eqns:

X15_i(n) = -Xi_m(n) (i = 16 to 22; m = 6, 1, 2, 3, 9, 4, 5) (22) to (28)

23

Engineering Sector

 

Most of the component equations for the engineering sector are quite
similar to those for the other sectors in the basic line process, namely the
Service, Repair and Overhaul Sector, the Technical Data Sector, the Aero-
space Ground Support Equipment Sector, the Standard Product-Fabrication
Sector and the Standard Product-Assembly Sector. Consequently the
detailed explanation of the seven common forms of component equations
will not be repeated in describing those sectors. Only equations unique to
those sectors will be explained.

In this prototype model all the sectors use one time period for processing
the product. Thus the volume of product flowing into the sector in a given
time period is equal to the volume flowing out in the next time period. This
first common form of component equation is illustrated by equations (74) to
(76) for the Engineering Sector.

At each of these sectors, the support of the service and other overhead
functions flows into the product. To simplify this prototype model, the number
of service units flowing into the product is considered equal to the number of
product units being serviced. Thus the X variable for these service functions
is really their cost per unit of product. So the second common form merely
equates flow of services to flow of products, as in equations (29) to (64).

Similarly labor, material and facilities are incorporated directly into the
product. The heterogeneous nature of the products and processes make it
necessary to equate the number of units of material and of facilities flowing

into the product, to the number of product units. This third common form is

24

illustrated by equations (68) to (73).

The flow of labor can be conveniently expressed in terms of manhours.
Within the different functions, labor rates are reasonably uniform. Thus
the fourth form of component equation specifies the number of labor hours
per unit of product. (Equations 65 to 67)

The fifth form (Equations 77 to 112) breaks down the cost of the service
functions per unit of product into its fixed and variable elements. The sixth
form (Equations 113 to 115) does the same thing with the cost of facilities per
unit of product.

The seventh form (Equations 116 to 118) expresses the cost of the product
leaving the sector as the sum of product cost when it entered the sector and
the costs of all resources and services that flowed into the product while it
was in the sector.

YB/i-i(n) = Yj-i(n) (B = 80 to 91) (i = I, 2, 3; j = l7, l8, 19) (29) to (64)
Y12/1_i(n) = K12/1_in_i(n) (i = I, 2, 3; j = 17, 18, 19) (65) to (67)
Yr-i(n) = Yj-i(n) (r = l3/i, l4/i)(i = 1, 2, 3; j = l7, l8, 19) (68) to (73)
Yj-i(n) = Yi_k(n+l) (i = l, 2, 3; j = 17, 18, 19; k = 23, 29, 24) (74) to (76)
XB/i-i(n) = FB/i(n)-KB/i_in_i(n)(B = 80 to 91) (i = l, 2, 3; j = 17, 18, 19)
(77) to (112)

Xl4/i-i(n) = Fl4/i(n)'Kl4/i-1Yj-i(n) (i = l, 2, 3; j = 17, 18, l9)(113) to (115)

_ 9! 14/1
Xi_k(n+1) — -Xj-1(n)-g;8OXB A_i(n)-I;713/ixr__i(n)-1<12 /i-1X12 /i-i(n)

(i = 1,2, 3; j = 17, 18,19; k = 23, 29,24) (116) to (118)

25

Cut Set Eqns:
Yi-k(n) = Yl-m(n) (i = l, 2, 3; k = 23, 29, 24; l = 23, 43, 24; m = 7, 48, 8)
(119) to (121)
Circuit Eqns:
Xi-k(n) = Xk-p(n) (i = l, 3; k = 23, 24; p = 7, 8) (122) to (123)

Service, Repair 8: Overhaul Sector 81 Technical Data Sector

 

Yj-i(n) = Yi_k(n+l) (j = 21, 22; i = 4, 5; k = 32, 33) (124) to (125)
YB/i_i(n) = Yj_i(n) (B = 80 to 91) (1 = 4, 5; j = 21, 22) (126) to (149)
Y12/i-i(n) = K12/1_1Yj_i(n) (i = 4, 5; j = 21, 22) (150) to (151)
Yr-i(n) = Yj-i(n) (r = l3/i, 14/i) (i = 4, 5; j = 21, 22) (152) to (155)

xB/Hm) = FB/i(n)-KB/i_in_i(n) (B = 80 to 91) (1 = 4, 5; j = 21, 22)
(156) to (179)

X14 /1-1(n) = F14/i(n)-K14/1-1Yj_i(n) (1 == 4. 5; j = 21. 22) (180) to (181)
91

14/
Xi_k(n+l) = “Xj_i(n)'z XB/i‘iul)‘ :1 Xr-i(n)-K12/1-1X12/1-i(n)
B=8O '

r=l3/i

(i = 4, 5; j = 21, 22; k = 32, 33) (182) to (183)
Cut Set Eqns:
Yi-k(n) = Y1_48(n) (i = 4, 5; k = 32, 33; l = 46, 47) (184) to (185)

Aerospace Ground Support Equipment Sector

 

Yj-i(n) = Yi-k(n+1) (j = 24, 20; i = 8, 9; k = 30, 31) (186) to (187)

Yl2/i-i(n) = KIZ/i-in-i(n) (i = 8, 9; j = 24,20) (212) to (213)

26

Yr-i(n) = Yj-i(n) (r = 13/1, 14/1) (1 = 8, 9; j = 24, 20) (214) to (217)
XB/i-i(n) = FB/i(n)'KB/i-1Yj-i(n) (B = 80 to 91) (i = 8, 9; j = 24, 20)
(218) to (241)
Xl4/i-i(n) = Fl4/i(n)’K14/i-1Yj-i(n) (i = 8, 9; j = 24, 20) (242) to (243)
91 14/1
X._ (n+1) = -X._-(n)- X -_.(n)- X _.(n)-K ._.X -_-(n)
1 k j 1 [£80 8/1 1 1253/1 r 1 12/1 1 12/1 1

(i = 8, 9; j = 24, 20; k = 30, 31) (244) to (245)
Cut Set Eqns:
Yi-k(n) = Y1_48(n) (i = 8, 9; k = 30, 31; l = 44, 45) (246) to (247)

Standard Product-Fabrication Sector

 

Yj_i(n) = Yi-k(n+l) (j = 16,23; 1 = 6,7; k = 25,26) (248) to (249)
YB/i_i(n) = Yj-i(n) (B = 80 to 91) (1 = 6,7; j = 16, 23) (250) to (273)
1712/1461) = K12/1_1Yj_i(n) (1 = 6, 7; j = 16, 23) (274) to (275)
Y1._i(n) = Yj_i(n) (r = 13/1, 14/1) (1 = 6,7; j = 16,23) (276) to (279)
XB/i-i(n) = FB/i(n)‘KB/i-1Yj-i(n) (B = 80 to 91) (1 = 6, 7; j = 16, 23)

(280) to (303)
91 14/7
X _ (n+l)= -X _ (n)- X _ (n)- X _ (n)-K _ X _ (n)
7 26 23 7 Béso 8/7 7 213/7 r 7 12/7 7 12/7 7
(306)
91 lf/é
X6-25(n+1) = X16-6(n)'Z XB/6-6(n)' Xr-6(n>‘K12/6-6X12/6-6(n)
B=80 r=13/6
(307)
Cut Set Eqns:

Yi-k(n) = Yk-m(n) (i = 6, 7; k = 25, 26; m = 10, 11) (308) to (309)

27

Circuit Eqns:

Xi-k(n) = Yk-m(n) (i = 6, 7; k = 25, 26; m = 10, 11) (310) to (311)
The equations (307) and (371) for the product cost leaving both this sector
and the standard product-assembly sector differ from those in the preceding
sectors in that the costs and savings of cost reduction activities are imputed
into the "repeat" standard product business flowing thru the fabrication and
assembly sectors. The cost reduction investment per product unit is

segregated into its fixed and variable elements.

Standard Product-Assembly Sector

 

YJ_i(n) = Yi-k(n+1) (j = 25, 26; i = 10, 11; k = 27, 28) (312) to (313)
YB/i-i(n) = Yj-i(n) (B = 89 to 91) (i = 10, 11; j = 25, 26) (314) to (337)
Y12/1_i(n) = KIZ/i-in-i(n) (i = 10, 11; j = 25, 26) (338) to (339)
Yr-i(n) = Yj-i(n) (r = l3/i, l4/i) (i = 10, 11; j = 25, 26) (340) to (343)
XB/i-i(n) = FB/i(n)'KB/i-in-i(n) (B = 80 to 91) (i = 10, 11; j = 25, 26)
(344) to (367)

Xl4/i-i(n)=F 14/i (n)- K149/li_iY j"161) (i = 10,11; j- 25,26) (368) to (369)

14/11
X11-28(1“+1)="‘X26-11(I1)'?7_ XB/11-11(11)'Z Xr-11(n)
B=80 r=l3/ll
' K-12/11-11X9112/11- 1191) (370)
/1410
X10- 27‘1““): ‘Xzs- 10(11) 7 X8 10-10(11) - (BI‘K _
B_80 / 121 13/10 Xr 10 12/10 10

X12/10-10(n)+X5-10(n) X52-1o(n)K52-1oY25_10(n) (371)
Cut Set Eqns:

Yi-k(n)= 1_48(n) (i = 10, 11; k = 27, 28; l = 41, 42) (372) to (373)

28

Sales Promoting Activities Sector

 

These activities are called sales promoting rather than sales promotion
since the latter term is associated with advertising campaigns and the like.

In the aerospace industry, advertising plays a relatively minor role. Sales
promoting activities include visits to Defense agencies and airframe prime
contractors by marketing and engineering personnel, technical presentations,
and distribution of technical literature . Also included is the company-funded
research and development which has a definite effect on sales volume.

The impact of these activities on sales has already been specified in the
description of the Incoming Orders Sector.

These activities are one of the controls by which management can manipu-
late the system. An early version of the model used a separate control variable
for each product line. To condense the model these variables have been reduced
to one control variable Z(n) which is the total investment in sales promoting
activities. The cost of Z(n) is allocated to each product line by the factor Li-k'

In the next chapter, the procedure will be shown for solving for the values
of Z(n) and the other control variables, which will enable the firm to attain any
desired cost and sales volume objectives .

Xi-k(n) = Li-kZ(n) (i = 27 to 33; k = 34 to 40) (374) to (380)
where Z(n) = Unknown f(n) (control variable)

Cut Set Eqns:

Yi-k(n) = Y1_48(n) (i = 27 to 33; k = 34 to 40; l = 41 to 47) (381) to (387)

(Xi-1.01% 0.1x1_48<n»

29

Corporate Overhead 81 Interest Sector

 

Since the enterprise modeled is a division of a firm whose corporate office
provides all the funding required, Xi-k represents the overhead charge paid to
the corporate office for interest, dividends and maintenance of that office . It
is expressed as a function of the accumulated product cost.

Xi-k(n) = Ki-k(xj -i(n)+Xp-j(n»

(i = 34 to 40; k= 41 to 47; j = 27 to 33; p = 10,11,2,8,9,4,5)

(Ki-k = 0.06) (388) to (394)
Cut Set Eqns:

Yi-k(n) = Yk_48(n) (1 = 34 to 40; k = 41 to 47) (395) to (401)

Cost of Sales Sector

 

Cost of sales is the grand total of all previous costs that went into the
product including the corporate charge .
Circuit Eqns:
Xp- j(n)1-X j_k(n)+-Xk_i(n) = -Xi_48(n)
(p= 10,11,2,8,9,4,5; j = 27 to 33; k = 34to 40; i = 41 to 47)
(402) to (408)

Resources Sector

 

Labor cost per manhour and material cost per product unit are known
functions of time.
X12/1_i(n) = Known F12/1(n) (1 = A/l to A/ll, l to 11)
(A = 60 to 71) (409) to (551)

X13/i-i(n) = Known F13/i(n) (i = l to 11) (552) to (562)

30

Cut Set Eqns:
Y54_I(n) = Yr-i(n) (r = 12/i, l3/i, l4/i) (i = A/l to A/ll, l to 11)
(A = 60 to 71) (563) to (991)
Circuit Eqns:
X54_r(n) = 'Xr-i(n) (r = 12/1, l3/i, 14/i) (i = A/l to A/ll, l to 11)
(A = 60 to 71) (992) to (1420)

Service 81 Other Overhead Functions Sector

 

Equations (1421) to (1552) indicate the number of labor hours per unit of
service. Equations (1553) to (1816) equate the units of measure of material
and facilities flow and service flow. Equations (1817) to (1948) show what
per cent material costs constitute of the total imputed service cost. Equations
(1949) to (2080) merely state that the total imputed service cost is the sum of
the costs of the labor, material and facilities used by the service function.
Y12/1_i(n) = K12/1_1Yi_j(n) (i = A/l to A/ll; j = B/l to B/ll)

(A = 60 to 71; B = 80 to 91) (1421) to (1552)
Yr/i-i(n) = Yi-j(n) (r = 13, 14) (i = A/l to A/11;j = B/l to B/ll)
(A = 60 to 71; B = 80 to 91) (1553) to (1816)
X13/i-i(n) = 'K13/i-1Xi-j(n) (i = A/l to A/11;j = B/l to B/ll)
(A = 60 to 71; B = 80 to 91) (1817) to (1948)
Xi_j(n) = -X

(n)-X (n)-K

13/1-1 14/1-1 12/1-1X12/1-1(n)

(i = A/l to A/11;j = B/l to B/ll)

(A = 60 to 71; B = 80 to 91) (1949) to (2080)

31

Cut Set Eqns:

Yi-j(n) = Yj-k(n) (i = A/l to A/11;j = B/l to B/ll; k = l to 11)
(A = 60 to 71; B = 80 to 91) (2081) to (2212)

Circuit Eqns:

Xi-j(n) = -Xj_k(n) (i = A/l to A/ll; j = B/l to B/ll; k = l to 11)
(A = 60 to 71; B = 80 to 91) (2213) to (2344)

Cost Reduction Activities Sector

 

The cost reduction activities together with the sales promoting activities
comprise the control vector of the model.

In the description of the Incoming Orders Sector, cost of sales are
specified as having an inverse causal relationship with sales input volume
(i.e., the lower the costs, the more contracts won). In the real world, this
is not a stable absolute relationship because competitors are also striving to
reduce costs. To make this prototype model realistic without expanding it
to include a sector on competitive firms, a compensating simplification must
be built into the model.

This simplification involves treating all company-funded research and
development strictly as sales promoting activities and not as cost reduction
activities. In other words the assumption is made that this research-and-
engineering increases sales by developing new or improved products and
enhancing in-house technical capabilities. However, its effect in reducing
product cost is approximately just enough to maintain the current cost position
relative to those of competitors . Furthermore this eliminates double-counting

of its impact on sales.

32

Therefore cost reduction activities are limited in this model to the manu-
facturing engineering and purchasing efforts relative to the repeat standard
product business flowing thru the fabrication and assembly sectors. The
resulting savings are separated into fixed and variable elements.
X 49_ i(n) = Unknown f(n) (control variables)

(1 = 52, 53) (2345) to (2346)
X49-j(n) = M49—j“K49-j(N49-j'x49-1(n»

(j = 50, 51; i = 52, 53) (2347) to (2348)
Cut Set Eqns:
Y49_i(n) = Yi-j(n) (i = 50 to 53; j = 10, 6, 10, 6) (2349) to (2352)
Circuit Eqns:
X49_i(n) = Xi-j(n) (i = 50 to 53; j = 10, 6, 10, 6) (2353) to (2356)

Simplifying Variables

 

To simplify the matrix entries, let:

D V

m-i = Lk-m m-i
Vj-k = Q1-48 Vm-i
Vk_m = (Kk_m -vj_k) Lk_m (1 = 41 to 47; j = 10, 11, 2, 8, 9,4, 5;

k=27to33;m=34to40)

33

V49-52 = (K49-SOI'l
V49-53 = “(49-51"1
V49-50 = K49-50 N49-so 'M49-50
V49-51 = K49-51 N49-51 “M49-51

91
F -(n)= F -(n)
V/J 8}:ng B/J

Fr/j(n) = Fl3/j(n) + F14/j(n) + Kl2/j-jF12/j(n) (j = l to 11)
91
V1 - z; KB/j-j+Kl4/j-j (1- 15-17 to 15-22, 7 26, l 23, 3 24,
B-80
j= 1, 2, 3,9, 4, 5,11,7, 8)
V 91 K K K (1 6 25 10)
= + . . + = - , j:
1 =80 B/j-j 14/1-1 52-10
V-91 K K K 1-1516 '-6
1‘3280 Ian-1+ 14/1-1+ 53-6 ( - ‘ » J- >

State Model and Output Vector

 

Table 3 lists the state, control and output variables of the model. The
state variables are sufficient to specify any aspect of the system. They are
the flow and imputed cost variables wherever a time lag occurs in the model.
The other model variables can be expressed as linear functions of the state
variables. The control variables are the sales promoting and cost reduction

activities. The output variables are the final costs of product shipped.

34

Table 3. - State, control and output variables of model.

 

State Variables = Y15_i(n) (i = 16 to 22)
Yj_48(n) (j = 41 to 47)
Y25-10(n)

Y26-11(n)

Yrs-791)

Y24-8(n)

X64591)

X7-26(n)

X1430“)

X3-24(n)

Xk_1(n) (k = 10, 11, 2, 8, 9, 4, 5;1= 27 to 33)

Xi-p(n) (i = 16 to 22; p = 6, 1, 2, 3, 9, 4, 5)

 

Control Variables Z(n)
X49_j(n) (j = 52, 53)
Output Variables = Xj-48(n) (j = 41 to 47)

 

By reducing the preceding equations to a minimal set of first ordered
difference equations, the state model is derived. Exhibit 1 shows the state
model. Its basic form is:

171/(ii-i-l)= P (f(n) + QE(n) + SF(n)
where (,1 is the state vector, P is the transition matrix, Q is the excitation
matrix, E is the control vector, F is a vector of known functions of time and
S is its coefficient matrix. Thus next period's state is expressed as a function

of the current state, the control vector and known functions of time .

35

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39

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hfinovflu
ownomflu
mVImen
wwntha
$7ch-
Nvumea
ETA-MD.-

 

 

CHAPTER IV
ANALYSIS OF THE MODEL

Parametric Data

Ready availability of empirical data was one of the factors considered in
the design of the model. All parametric data were estimated from actual
empirical data . Tables 4 thru 9 list the parametric data . Exhibits 3 and 4

show the substitution of the parametric data into the state model and the

output vector respectively.

Heuristic Simulation

In order to use the model for heuristic simulation, it is necessary to

solve for t); (n). This solution is:

F
Y (n) = PnLlV (O)+ [Pm-1Q I’m—2Q . . . . PDQ] E(O)

 

 

- 13(1)
+ G(n)
LE(n-l)
where G(n) = [Pm s Pn-ZS . . . . PCS] [F(0) '
F(l)
l..:"(n-l)J

 

 

Tables 10 and 11 give the initial state values and the control vector values
required by the above formulas. Table 12 shows the results of the simulation

from the initial state thru twelve periods of time.

41

42

Table 4. - Input output parameters.

Input Function Rel. to Units Rel. to Cost

F .(n) = Shipped of Sales
j a 4]- b(n) Pj_43 Qj_43
41 6900 172 .05 -.017
42 4300 107 . 04 - . 01
43 2800 70 .04 -.028
44 130 3 .02 -.001
45 1000 25 .02 -.01
46 500 12 .02 -.005
47 200 5 .02 -.002

Rel. to Sales % of Total
Prom. Sales Prom.

1‘ 1 Kk-l Lk-l
27 34 - 6.9 .3
28 35 - 3.8 .4
29 36 -11.2 .06
30 37 - .4 .06
31 38 - 4 .06
32 39 - 2 .06
33 40 - .8 .06

Input Cost Rel. to Factory Cost

K = .001 (i=16to 22;p=6,1,2,3,9,4,5)

i'P

43

Table 5. - Resource parameters - Basic line processes - Direct resources.

Direct Labor Direct Direct Material Direct Facility
Cost /Hr . Labor Cost /Unit Cost/Unit
F12/1(n) = Hrs/Unit FIB/1m) Fl4/i(n)

1 a + b(n) K12/i-i a + b(n) a + b(n) K14/1-1
1 5.00 .04 34.0 95.00 10.00 197.00 .25 .04
2 5.00 .04 20.6 65.00 7.00 25.70 .15 .004
3 4.87 .04 90.0 165.00 4.00 63.00 .50 .04
4 3.00 .02 100.0 300.00 15.00 26.00 .15 .023
5 3.65 .03 12.0 3.00 .04 5.50 .05 .014
6 2.96 .02 17.4 27.65 .26 635.00 4.00 .030
7 2.96 .02 17.4 27.65 .26 635.00 4.00 .030
8 2.95 .02 61.0 565.00 20.00 19.00 .14 .005
9 2.95 .02 61.0 565.00 20.00 19.00 .14 .005

10 3.00 .02 36.0 137.95 .92 1650.00 7.00 .130

11 3.00 .02 36.0 137.95 .92 1650.00 7.00 .130

44

Table 6. - Resource parameters - Basic line processes - Service and other
overhead functions .

Indirect Cost /Unit

F B Mn) '
1 B a + b(n) KB/i -i
1 80 490 1.46 .08
81 15 .11 .001
82 75 .32 .01
83 104 .18 .02
84 31 .18 .003
85 41 .28 .003
86 94 .68 .006
87 43 .30 .003
88 55 .38 .004
89 34 .25 .002
90 38 .29 .002
91 133 .90 .01
2 80 115 .95 .007
81 10 .07 .001
82 24 .21 .001
83 14 .11 .001
84 15 .12 .001
85 21 .18 .001
86 70 .44 .009
87 22 .19 .001
88 28 .25 .001
89 19 .16 .001
90 22 .19 .001
91 78 .58 .007
3 80 426 3.43 .63
81 39 .27 .09
82 80 .76 .03
83 47 .42 .04
84 47 .43 .03
85 85 .68 .13
86 168 1.60 .06
87 75 .71 .03
88 95 .90 .04
89 85 .59 .20
90 103 .70 .25

91 270 2.11 .45

Table 6. - (continued)

81
82
83
84
85
86
87
88
89
9O
91

81
82
83
84
85
86
87
88
89
9O
91

81
82
83
84
85
86
87
88
89
9O
91

45

Indirect Cost /Unit

395
27
87
43
51
84

167
72
91
59
69

208

457
89

123
80
54
87

234

106

136

N
NVOW
Oflml—I

l—I :—
MNNOOWNHmcb—I

._-

+

Fla/10‘) =

b(n)

3.25
.26
.72
.40
.41
.64

1.52
.67
.86
.56
.66

2.00

4.13
.33
.92
.51
.52
.81

1.93
.85

1.09
.71
.84

2.54
.13
.01
.02
.01
.01
.02
.06
.02
.03
.02
.02
.08

KB/i-i

.14
.002
.03
.006
.02
.04
.03
.01
.01
.005

.015

46

Table 6. - (continued)

Indirect Cost /Unit

FEM“) =
i B a + b(n) KB/i_i
7 80 20 .13 .001
81 1 .01 --
82 9 .02 .001
83 8 .01 .001
84 1 .01 --
85 2 .02 --
86 13 .06 .001
87 9 .02 .001
88 10 .03 .001
89 2 .02 --
90 2 .02 --
91 15 .08 .001
8 80 252 1.66 .08
81 14 .13 .001
82 44 .37 .007
83 23 .20 .003
84 22 .21 .001
85 34 .32 .002
86 84 .77 .007
87 37 .34 .003
88 48 .44 .004
89 29 .28 .001
90 34 .33 .001
91 104 1.02 .002
9 80 252 1.66 .08
81 14 .13 .001
82 44 .37 .007
83 23 .20 .003
84 22 .21 .001
85 34 .32 .002
86 84 .77 .007
87 37 .34 .003
88 48 .44 .004
89 29 .28 .001
90 34 .33 .001

91 104 1 .02 .002

47

Table 6. - (continued)

Indirect Cost /Unit

138/101) ‘

i B a + b(n) KB /i-i

10 80 34 .27 .001
81 2 .02 --
82 13 . 06 .001
83 10 .03 .001
84 3 .03 --
85 5 .05 --
86 20 .13 .001
87 12 .05 .001
88 14 .07 .001
89 4 .04 ~-
90 5 .05 --
91 17 .17 --

11 80 34 .27 .001
81 2 .02 --
82 13 .06 .001
83 10 .03 .001
84 3 .03 --
85 5 .05 --
86 20 .13 .001
87 12 .05 .001
88 14 .07 .001
89 4 .04 --
90 5 .05 --

91 17 .17 --

48

Table 7 . - Service and other overhead functions - By type of resource.

Indirect Labor Cost/Hr Labor Material
F12/i _i(n) Hrs /Unit Cost/Unit
K12/1-1 K13/1—1
1 A a + b(n)

1 to 11 60 4.70 .03 9.4 .32
61 3.00 .02 1.5 .20
62 2.80 .02 4.1 .21
63 3.10 .02 1.3 .51
64 3.00 .02 2.8 .07
65 4.50 .03 1.4 .53
66 2.80 .02 8.2 .24
67 3.15 .02 3.5 .17
68 4.70 .03 3.3 .15
69 4.00 .03 1.9 .34
70 4.10 .03 2.5 .26
71 5.00 .04 5.2 .26

49

Table 8. - Cost reduction parameters.

K52-10
K53-6

K49-50
K49-51
N49-50
N49-51
M49-50

M49-51

-220

-165

-200

-150

.001

.001

50

Table 9. - Simplifying variables.

34
35
36
37
38
39
4O

HOOQNOU‘hfiWNI—I

I—IH

49-52
49-53
.V49-50

49-51

41
42
43
44
45
46
47

15-16
15-17
15-18
15-19
15-20
15-21
15-22
6-25
7-26
1-23
3-24

j k

10 27

11 28

2 29

8 30

9 31

4 32

5 33

FI/1(n)
a + b(n)
462.00 11.61
193.70 7.97
666.30 8.10
626.00 17.15
52.30 .45
714.15 4.61
714.15 4.61
763.95 21.36
763.95 21.36
1895.95 8.64
1895.95 8.64
.. .1

.. .1

+2

-+1.5

1.06
1.06
1.06
1.06
1.06
1.06
1.06

V1
.038
.183
.036
.202
.117
.336

1.582
.137
.136
.037
.117

In-i

IDm-i

.318
.424
.064
.064
.064
.064
.064

1153
438
1520
1353
1822
92
92
725
725
139
139

V/j(n)
+ b(n)

5.33
3.45
12.60
11.95
15.18
.43
.43
6.07
6.07
.97
.97

51

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32-5
32-3,
33-";
Auvmv-_*>
Auvuu-m~>
Adv—n-m—>
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32:“;
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No.

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one.

 

j
:3 n-2x
73 Tax
:45 o-Sx
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:5 «.2x
73 72x
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:13 3.?
:3 2-2x

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:13 n-0x
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53

a 3 38.: J
336 + no.3:
Acid + 8.2.2
58.: + 3.2:
58.: + 8.9:
33.. + 26:.
33.. + 2;:
396+ 8.2
32.2 + 8.08
32... + 868
38.: 2.2:
33.: + 8.3+
32.6 + on:
336 + «.2
38.... + n2.
38... + m2.
335 + no
336 + 3
52.2 + 22
53.: + 22
“58.2 + 22
39.... + a?
386 + 2:
3m. + :8
32 + 8m
3va + 82
3n + 9..
32 + 83
A58: + 83
£th + 8%

 

v:

N.+

._- ‘a—v—p r‘

 

ON

2

5N

ON

mm

vn

an

«N

am

Aumssflnoov . .m ”mafia

ON 2

2

2

o-

m.—

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54

. 839, 3388.3 8:3 H30? 3930 .. .w ”mam

:5 9sz
3 Tax
3 0-2x
:5 To;
Aqv leqx
3 T:
E 0-2x
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ARV ~Mnox
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33-3 . a -

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Table 10. - Initial state values.

5000
2200
1000
140
900
500
200
4500
2000
900
130
850
400
180
\/(0)== 4700
2100
2150
135
-4000
-4500
-1000
- 1000
-1000
—1000
-1000
-1700
-1300
- 1200
- 400

i-uv—w—al—‘I—a;

56

Table 11. - Given control vector values for heuristic simulation.

E(O) =

E(l) =

E(2) =

13(3) =

12(4) =

5(5):

E(6) =

-l70
- 20
- 15

-270
- 25
- 20

-360
- 3O
- 25

-430
- 35
- 30

-56O
- 40
- 35

-710
- 45
- 4O

-5 60
- 40
- 35

57

E(7) =

E(8) =

E(9) =

E(10)=

E(ll)=

-430
- 35
- 30

-360
- 3O
- 25

270
- 25
20

-360
- 30
- 25

-430
- 35
- 3O

58

Table 12. - Simulated states from initial state thru twelve time periods.

WVOWACDNH

D—‘D—Il—Il—Il—Il—tD—I—i
\IOU‘QQJNF-‘OO

18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36

7404
4588
2920

136
1010

523

210
4700
2100
1000

135

900

500

200
5000
2150
2200

140
-3091
-3049
- 595
-1873
-1383
-1810
-1557
- 612
-1926
-1206
-2157

t—t—Ii—ni—nv—nq;

7809
4867
3073

140
1034

545

219
5000
2150
2920

140
1010

523

210
7404
2200
4588

136
-1972
-3679
- 537
-3656
-1397
-1831
-1557
- 526
-1935
- 783
-2179

NNf-‘NP-‘OO

8202
5105
3282

144
1063

569

228
7404
2200
3073

136
1034

545

219
7809
4588
4867

140
-1568
-3691
- 543
-3706
-1421
-1852
-1557
- 518
-1431
- 750
-2198

NNF—‘b-P-l-‘I-b

8646
5320
3405

148
1090

589

237
7809
4588
3282

140
1063

569

228
8202
4867
5105

144
-1515
-2872
- 546
-3751
-1444
-1873
-1558
- 510
-l392
- 723
-2216

NNl-‘b-lkl-‘hli

9108
5729
3570

154
1119

617

248
8202
4867
3405

144
1090

589

237
8646
5105
5320

148
-1463
-2806
- 553
-3796
-1468
-1895
-1560
- 500
-1362
- 700
-2236

NNF-‘Ql-‘OON

9610
6075
3745

161
1148

648

261
8646
5105
3570

148
1119

617

248
9108
5320
5729

154
-1403
-2753
- 558
-3843
-1492
-1915
-1558
- 488
-1337
- 643
-2256

NNF‘vhl-‘OO

'Table 12. -(conmhnued)

ooxloxcnqxooms—

l—‘l-II—‘l—ib-ll—‘l—Il—I
\IO‘Cflr-AOONl-‘OO

18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36

9495
5965
3721

161
1169

642

259
9108
5320
3745

154
1148

648

261
9610
5729
6075

161
-1337
-2708
- 563
-3889
-1515
~1934
-1553
- 474
-1270
- 596
-2276

NNI—‘hbl-‘OJ

9423
5884
3711

161
1192

639

258
9610
5729
3721

161
1169

642

259
9495
6075
5965

161
-1264
~2596
- 574
-3935
-1540
-1964
-1572
- 483
-1216
- 634
-2297

NNNth-‘OO

9477
5902
3733

162
1215

652

259
9495
6075
3711

161
1192

639

258
9423
5965
5884

161
-1298
-2504
- 586
-3984
-1565
-l994
~1589
- 492
-1263
- 666
-2318

NNNu-hl—‘OJ

59

10

9457
5888
3742

163
1238

653

260
9423
5965
3733

161
1215

652

259
9477
5884
5902

162
-1326
-2577
- 596
-4032
-1589
-2019
-1602
- 494
-1303
- 679
-2338

NNthl-‘OO

11

9811
6126
3873

168
1265

676

269
9477
5884
3742

162
1238

653

260
9457
5902
5888

163
-1332
-2638
- 607
~4079
-1614
-2048
-1616
- 501
-1322
- 699
-2359

NNNt-bl—‘OO

12

10130
6335
3990

172
1292
696
278
9457
5902
3873
163
1265
676
269
9811
5888
6126
168

-1352

-2664

- 613

-4127

-1637

-2069

-1617

- 493

-1347

- 672

-2379

NNNI-Dhl-‘CD

60

As stated earlier, the treatment of simulation in this study is limited to

a demonstration of its feasibility with the methodology being evaluated.

Stability Analysis

 

In lay terms, stability analysis seeks to ascertain whether the model by
virtue of its internal characteristics will explode over time or will remain
stable. Lyapunovl defined a stable system as one which,if disturbed by a
small amount from an equilibrium state, would either return to that state
or stay within some pre -assigned finite region near the equilibrium state.
Using this definition, stability can be determined by solving for the eigen-
values of the minimal polynomial of the transition matrix P. The system is
stable if the modulus of each eigenvalue of multiplicity one is less than or
equal to one and the modulus of each eigenvalue of multiplicity two or greater
is less than one . (One can also use with proper care the eigenvalues of the
characteristic polynomial which are the same as those of the minimal
polynomial but may have different multiplicity.)

Table 13 lists the coefficients ofthe characteristic polynomial of P.
Table 14 lists the eigenvalues of P. Since the moduli of the eigenvalues are

all less than one, the system is asymptotically stable.

 

1A. M. Lyapunov, "Le Probleme General de la Stabilite du Mouvement, "
Annals of Mathematical Studies, XVII (Princeton, N.].: Princeton University
Press, 1949).

 

Table 13. - Coefficients of characteristic polynomial of P.

\OOONC‘uUlI-F-OJNo—A

HHHHH
QWNHO
It...

16.
17.
18.

Table 14. - Eigenvalues of P.

1.
2.
3.
4.
5.
6.
7.
8.

9.
10.
ll.
12.
13.
14.
15.
l6.
17.
18.

0.0 0
0.1032110 x 10
0.7146900 x 10"1
0.3916918 x 10'1

07376387 x 10'2
05312974 x 10'2
02799382 x 10'2
0.2978753 x 10‘3
0.3107028 x 10‘3
0.3022769 x 10'4
01416625 x 10'4
03944258 x 10‘5
0.5427271 x 10‘6
0.1864992 x 10'6

02013224 x 10‘7

06039891 x 10’8
0.5575385 x 10’9
0.1444537 x 10"9

REAL

0.3198519 x 10’1
0.3455048 x 10‘1
0.3455048 x 10"1

0.2253598 x 10'2
0.2253598 x 10‘2

02732943 x 10'1
02732943 x 10'1
08777088 x 10"1
01061778 x 10°
01431792 x 10°
01504633 x 10°
01504633 x 10°
05440717 x 10'1
0.5161469 x 10'1
0.5161469 x 10'1
0.4689423 x 10‘1
0.4689423 x 10'1
0.1461591 x 10°

61

19. 01360869 x 101°
20. 01812394 x 10'11
21. 0.3152086 x 10‘12
22. 0.9166884 x 10‘15
23. 04320720 x 10‘14
24. 0.2060674 x 10‘15

25. 0.1587409 x 10:
26. 02072171 x 10

16
17

27. 0.1015041 x 10'18
28. 0.7953481 x 10‘20
29. 0.1621995 x 10‘21
30. 0.5053931 x 10"22
31 . 0.1074626 x 10'24

32. 0.8071177 x 10'

25

33. 02377468 x 10‘28
34. 0.8525294 x 10'28
35. 0.1769470 x 10’28

36. 08107572 x 10'

IMAGINARY

0.0
0.3762221 x 10-

1

03762221 x 10"1
0.5944792 x 10"1
05944792 x 10‘1

0.4771433 x 10-

1

04771433 x 10'1
01478515 x 10‘15

0.2268977 x 10’

15

0.1946615 x 1035
01385099 x 10
0.1385099 x 10‘2

06100480 x 10'

14

0.8587146 x 10'1
08587146 x 10'1
0.8579135 x 10'1
08579135 x 10'1
0.1814614 x 10‘2

30

MODULUS

9.3198520 x 10‘1
0.5108001 x 10‘1
0.5108001 x 10'1
0.5949062 x 10‘1
0.5949062 x 10‘1
0.5498686 x 10'1
0.5498686 x 10'1
0.8777093 x 10'1
0.1061778 x 100
0.1431792 x 10°
0.1504698 x 100
0.1504698 x 10°
0.5440717 x 10'1
0.1001897 x 10°
0.1001897 x 10
0.9777130 x 10'1
0.9777130 x 10‘1
0.1461704 x 10°

Table 14. - Eigenvalues of P. (continued)

REAL IMAGINARY MODULUS
19. 0.1461591 x 10° 01814614 x 10'2 0.1461704 x 10°
20. 0.1521912 x 10° 0.9834735 x 10'11 0.1521912 x 10°
21. 0.2026185 x 10° 06848959 x 10‘13 0.2026186 x 10
22. 0.1228949 x 10° 0.1228949 x 10° 0.1737997 x 10°
23. 0.1228949 x 10° 01228949 x 10° 0.1737997 x 10°
24. 01228948 x 10° 0.1228949 x 10° 0.1737996 x 10°
25. 01228948 x 10° 01228949 x 10° 0.1737996 x 10°
26. -O.1364051 x 10° 0.2362606 x 10° 0.2728103 x 10°
27. 01364051 x 10° 02362606 x 10° 0.2728103 x 10°
28. 01879226 x 10° 0.3254915 x 10° 0.3758452 x 10°
29. 01879226 x 10° 03254915 x 10° 0.3758452 x 10°

62

15

0.2026187 x 10°
0.2728103 x 10°
0.3758452 x 10°
0.4575791 x 10°
0.4575791 x 10°
0.4575791 x 10°
0.4575791 x 10

0.4475919 x 10"
0.3633910 x 10'14
01030937 x 10'14
0.5503974 x 10"16
04575791 x 10°
0.4575791 x 10°
04575791 x 10°

30. 02026187 x 10°
31. 0.2728102 x 10°
32. 0.3758452 x 10°
33. 0.4575791 x 100
34. 0.1613792 x 10'11
35. 0.1482293 x 10'11
36. -0.l482293 x 10"11

Control Analysis

 

A system is said to be state controllable if by means of a series of control
signals, it can be brought from any initial state to any desired state in a finite
period of time. The number of control signals required is equal to the order
of the system model, in this case thirty-six. Since there are three control
variables, twelve time periods are required to reach a desired state.

To find out whether the system is state controllable, it is necessary to
compute . H.

".

'11 "
H=1P Q PIOQ...P°Q;
L i

H in this case is a 36 x 36 matrix. If H is non-singular, the system is state

controllable .

63

Table 15 lists the coefficients of the characteristic polynomial of H.
Table 16 lists the eigenvalues of H . These results indicate that H is non-
singular and hence the system is state controllable.

To find the values of the thirty-six control signals needed to reach a
desired state, the following formula is used:

ES = H"1 [31) desired - P12 \f) (o) - C(12)]
where E s is the thirty-six control signals (twelve sets of three each) to bring
the system to the desired state . The other variables have been defined
previously. This formula is actually the formula for solving the state model,
with the control vector on the left side of the equation.

A common way of demonstrating state controllability is to show that the

system can be reduced from any initial state to a final state of zero (though

the final state could just as easily be any desired state).

Table 15. - Coefficients of characteristic polynomial of H.

1- -0.8721831 x 10’l 19. 04496994 x 10'42
2- 0-2810221 x 10‘2 20. 0.5456394 x 10"42
3- 0-2870016 x 10': 21. 04993221 x 10’43
4. 0.6446587 x 10"8 22. 05114684 x 10-44
5- 05279475 x 10' 23. 04077664 x 10'45
6. ”0.2848693 x 10'-11 24. 0.3406399 X 10-46
7. -0.5126255 x 10-14 25_ -0.2600429 x 10-47
8- 0372930?- X 10'” 26. 0.2024554 x 10‘48
9. 08353666 x 10:32 27. -0.1513900 x 10-49
1°- '°-7212567 x 10 30 28. 0.1141644 x 10'5°
ll . 0.3169615 X 10:31 29. '0.8439314 x 10'52
12. '0.1121086 X 10_33 30. 0.6260988 X 10‘53
13- 0-7454431 X 10 31. 04596795 x 10'54
14- *0-2619676 x 10‘“ 32. 0.3379925 x 10‘55
15. 0.1675823 x 1033 33. -0.2471257 x 10-56
16- '0-5235852 x 10 38 34. 0.1807917 x 10'57
17. 0.2995573 X 10- 35. ”0.1318572 X 10'58

13- '0-3366049 x 10'4” 36. 0.9618951 x 10'60

Table 16. - Eigenvalues of H.

p...
OOOOVGCflt-D-OONH
.......

I—r—ai—r—t—t—s
O‘Ult-FBOONH
......

17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.

REAL

0.9686768 x 10‘1
0.9686768 x 10"1
0.6419629 x 10‘1
0.6419629 x 10‘1

0.2917200 x 10'1
0.2917200 x 10'1
07076509 x 10'2

07076509 x 10:
-0.4337440 x 10

04337440 x 10‘1
07851803 x 10::
07851803 x 10

01112903 x 100
01112903 x 10°
01404754 x 108
01404754 x 10
01648769 x 100
01648769 x 100
01833575 x 10°
01833575 x 10°
01949226 x 10°
01949226 x 10°
01988646 x 10°
0.1509723 x 10°
0.1509723 x 10°
0.1705385 x 100
0.1705385 x 10°
0.1840671 x 10°
0.1840671 x 10°
0.1909888 x 100
0.1909888 x 10°
0.1261210 x 10°
0.1261210 x 100
06650921 x 10°
0.5650247 x 10°

05650247 x 100

64

IMAGINARY

0.1662307 x 100
01662307 x 10°
0.1817901 x 10°
01817901 x 100
0.1910860 x 10°
01910860 x 10°
0.1937758 x 10°
01937758 x 10°
0.1896941 x 10°
01896941 x 10°
0.1788594 x 10
01788594 x 100
0.1614823 x 100
01614823 x 10°
0.1379801 x 10
01379801 x 10°
0.1090015 x 10°
01090015 x 10°

0.7546777 x 10“1
07546777 x 10'1

0.3861605 x 10"1
03861605 x 10‘1
03524585 x 10‘14

0.1185385 x 10°
01185385 x 100

0.8792925 x 10'1
08792925 x 10‘1

0.5410934 x 10‘

05410934 x 10'1
0.1826744 x 10‘1
01826744 x 10’1

0.1449206 x 10°
01449206 x 10°

01929737 x 10‘”
01114763 x 10'11

0.1114763 x 10'

MODULUS

0.1923955 x 100
0.1923955 x 10°
0.1927922 x 10

0.1927922 x 10°
0.1932999 x 10°
0.1932999 x 100
0.1939050 x 10°
0.1939050 x 10(0)
0.1945899 x 10

0.1945899 x 10°
0.1953350 x 10°
0.1953350 x 10°
0.1961175 x 108
0.1961175 x 10

0.1969057 x 10°
0.1969057 x 10°
0.1976506 x 10°
0.1976506 x 103
0.1982810 x 10

0.1982810 x 108
0.1987110 x 10

0.1987110 x 100
0.1988646 x 103
0.1919480 x 10

0.1919480 x 10

0.1918722 x 10°
0.1918722 x 100
0.1918555 x 10

0.1918555 x 10°
0.1918604 x 10°
0.1918604 x 10°
0.1921159 x 10°
0.1921159 x 10°
0.6650921 x 10°
0.5650247 x 103
0.5650247 x 10

65

Table 17 shows the twelve sets of control vector signals that will bring the
state to zero.

Table 17. - Control signals to reduce state to zero.

1 2 3
1 0.1398637 x 109 0.9657004 x 106 0.4602871 x 107
2 0.2353126 x 106 0.5304721 x 105 0.3534450 x 106
3 0.1527203 x 106 0.3557549 x 109 0.6364067 x 107
4 0.8184403 x 106 0.3478202 x 107 01090784 x 105
5 0.2431166 x 10?0 0.1254539 x 105 07918559 x 10‘71

6 0.1793574 x 10 0.5466162 x 10 0.2431576 x 10
7 0.6201062 x 108 0.2293626 x 107 0.3894586 x 105
8 0.1160093 x 10 0.2311828 x 105 0.3908741 x 105
9 0.4612551 x 105 0.2330217 x 10 0.3834192 x 109
10 0.9635799 x 107 0.3343301 x 105 04534569 x 106
11 09012514 x 104 04100174 x 104 03637680 x 104
12 08405845 x 103 05680206 x 104 01074879 x 105

The significance of reduction to state zero is purely mathematical since
an executive would hardly want to drive his firm out of business . Since the
model is based on relationships and empirical parameters which exist within
a realistic range, manipulation of the model is meaningful if kept within this
range. Thus control strategy should be concerned with the attainment of
realistic sales and profit goals.

A prototype model such as the one in this study, with its simplification
of factors and relationships, may generate impractical control strategies .
Future work on more advanced and comprehensive models will cope with
this problem.

The techniques of optimal control,which will be employed in future studies,
give specific consideration to the magnitudes of the control variables imposed

by the control strategy and the magnitudes of the state variables during the

66

control interval.

Other variables which are linear functions of the state variables, such
as total cost of sales, can be specified as the outputs which one may wish
to bring to desired levels in a minimum amount of time thru a series of
control signals. A system in which this is possible is said to be output
controllable. To test for output controllability, one must compute K thus:

K = [ MPQ MQ N]
where M and N are the coefficient matrices of the state vector and control
vector respectively in the expression for the output vector. P is the transition
matrix and Q, the excitation matrix. If K is non-singular, the system is out-
put controllable. The output control equation is a restatement of the formula
for solving the output equation.

Unfortunately, with the simplified structure of the prototype model, K is
singular and hence the system is not output controllable . This result is not
particularly catastrophic . Since the model is state controllable and the output
vector is a linear function of the state vector, management can still specify target
costs of sales and find the control signals that will achieve them. As future
versions of the model become more complex and comprehensive, the model will
very probably become output controllable .

The intriguing area of optimal control is deferred to future studies since

this would represent a major undertaking in itself.

CHAPTER V
SUMMARY AND CONCLUSIONS

Summa

This study constructs a detailed prototype mathematical model of the
internal operations of an actual aerospace engineering and manufacturing
enterprise . It uses a methodology originally developed for analyzing
system models of electrical networks. This study evaluates the practical
utility of this technique in assisting the business executive . It is concerned
primarily with the generalized analytical capabilities of the methodology used.

The methodology used offers a formal process for generating a model
from a defined system structure and for analyzing the solution characteristics
of the model. This approach involves the use of linear graphs, flow and
unit cost variables,and the development of a state model by reduction of the
algebraic and difference equations that characterize the system to a minimal
set. This model can be used for simulation and for stability and control analysis.

Chapter II gives a detailed description of the firm being modeled. It is
essentially a large job shop with a wide range of highly technical precision
products for use on high performance aircraft, missiles and spacecraft, sold
mainly to meet Government defense or space needs.

Chapter 111 describes the construction of the model. It is essentially a

direct costing accounting type model. This formulation is more salable to

67

68

management because its correspondence to the actual system can be readily
understood. It also makes the model more useful for management by providing
them with a structure with which they are familiar and comfortable . The
model follows product flow from receipt of order to shipment thru the internal
sectors of the firm. It shows the flow of re sources and overhead function
services into the different product stages and imputes their cost into the
product.

Chapter IV discusses the analysis of the model. Simulation based on the
model is presented. The model is found to be stable and state controllable.

The control strategy to reduce the state to zero is derived .

Practical Utility to the Executive

 

Considerable practical utility is found for both the methodology and the
model. Relative to other techniques, the methodology provides a convenient
formal and generalized procedure for developing a model of the firm .

This model of the firm can be quite detailed without sacrificing compact-
ness . This aspect is very pertinent. A model that requires considerable
computer memory storage and processing time to run, will not be used
frequently if at all by the executive . Its usefulness diminishes if the executive
cannot get sufficiently rapid response time and if he knows that while the
model is being processed, the regular production work on the computer is
being jeopardized . Ideally the executive will want to interrogate and manipu-
late the model in conversational mode . Such a real ~time inquiry and simulation

system is feasible only if the model is compact enough to fit in a memory

69

partition of a multi-programming computer and require an amount of
computing that is not too voluminous to permit an acceptable response time .

The model has many managerial uses such as forecasting costs, load,
resource requirements or cash flow and analyzing total impact of alternative
decisions . The fact that the model automatically takes into account all the
critical interactions of the firm, assists the executive in an area where he
has traditionally felt impotent .

Since one can determine from the model each component cost and flow
volume at any time period, this data can be translated easily to any accounting
conventions, e.g., charging costs to a time period different from that in which
they were incurred. Computer output from the model can be structured in the
terms and formats of the management reports normally used by the executive .

The model is also useful for training purposes. It can be employed as a
management game simulating the total firm or, if desired, just specific sectors .

The most promising facet of this methodology is its generalized analytical
capabilities . As management pins down more of the true cause and effect
relationships in the firm, the control strategy from the model will play an ever

increasing role in guiding their key decisions.

Future Development of the Model

 

The route for future development of the model to take is toward greater
detail, more variables, more sectors, more interrelationships, more realism

and in general more complexity and sophistication.

70

Future models should introduce a finer breakdown of product lines . The
next model should at least specify the major product families within the
standard product line .

The basic unit of time should be reduced to a month. This will introduce
a greater number of time lags into the process. This in turn can be handled
by dummy sectors, or preferably by bringing in the various departments that
make up each sector.

Modeling sectors external to the firm, such as competition or the labor
market and linking these to the model of the firm, should provide interesting
results.

More factors and interrelationships such as scrap and reject flows can
be incorporated into the model. The introduction of price into the model will
be difficult and require considerable data accumulation but it will enhance the
realism and usefulness of the model significantly. More control variables
can be used. An intensive search must be made for the true cause and effect
relationships in the firm. Non-linear relationships such as the impact of
research and development on transition coefficients can be evaluated by
heuristic simulation.

A very promising approach for handling non-linear relationships thru
linear analytical techniques is to divide the firm into profit centers wherever
possible . Each of these profit centers can have independently derived imputed
profits or losses which would then be brought together for the entire firm in

the output vector or objective function.

71

The future development of the model should include the application of
optimal control techniques . If successful, the model will become an invaluable
tool for management. If price is part of the model, the objective function may
be maximum profit.

All the above suggestions will entail considerable time and money as well
as trade-offs in the size, response time and analyzability of the model.

However, this study has shown that the approach evaluated is of much
practical utility to the executive and bears great promise as a generalized
analytical tool for building and studying models of firms. Therefore the

investment in future development is definitely justified.

BIBLIOGRAPHY

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Bonini, C. P. Simulation of Information and Decision Systems in the Firm.
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Bowman, E . H. and Fetter, R. B. Analysis for Production and Operations
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Brown, R. G. and Nilsson, J. W. Introduction to Linear Systems Analysis.
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Koenig, H. E. and Blackwell, W. A. Electromechanical System Theory.
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74

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75

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