Systems Analysis and Modeling: A Macro-to-Micro Approach with Multidisciplinary Applications

By Donald W. Boyd

Systems Analysis and Modeling presents a fresh, new approach to systems analysis and modeling with a systems science flavor that stimulates systems thinking. After introducing systems modeling principles, the ensuing wide selection of examples aptly illustrate that anything which changes over time can be modeled as a system. Each example begins with a knowledge base that displays relevant information obtained from systems analysis. The diversity of examples clearly establishes a new protocol for synthesizing systems models.

• Macro-to-micro, top-down approach
• Multidisciplinary examples
• Incorporation of human knowledge to synthesise a systems model
• Clear and concise systems delimitation
• Complex systems using simple mathematics
• "Exact" reproduction of historical data plus model generated secondary data
• Systems simulation via systems models

• Language: English
• Publisher: Academic Press
• Release date: Oct 19, 2000
• ISBN: 9780080518398

Donald W. Boyd

Donald W. Boyd holds B.S. and M.S. degrees in electrical engineering, a B.S. degree in mathematics, and a Ph.D. in agricultural economics from Montana State University.

Book preview

1

Preface

This book, in its first edition, is not garnished by bells and whistles that are deferred to later editions. Its no-nonsense style conveys sincerity, a welcome feature in an introductory edition, and it has received effusive compliments from peer reviewers. Furthermore, the author believes that overembellishment detracts from the message of the book—its content should be the foremost attraction. The book is directed toward seniors, graduate students, and professionals whose career goals or practice involves fields that are systems oriented. These include the following areas:

The list is not exhaustive, and each area can be expanded into subareas (macro to micro), as, for example, Population Modeling, ranging from global, ethnic populations to microbial populations encountered in biofilm engineering.

A new and unique application of systems analysis and modeling was pioneered by the author as an outgrowth of a four-year research project (1968–1972) that produced for Montana an operative, state water resources planning model, first among the 50 states. Analytically, systems detail can be represented by a spectrum ranging from micro to macro. His macro-to-micro (Mtm) approach stands in contrast to the conventional approach that starts with micro forms, infinitesimal entities in differential equations that necessitate integration from micro to macro (mtM) to achieve application. Differential equations induce infinitesimal displacements as explicit functions of time. In general, solution of these equations by integration yields nonlinear dynamic forms, ill suited to systems modeling. Except for the simplest of examples, integration is possible only if one is willing to make simplifying assumptions—for example, linear approximations, time invariant coefficients, reduced number of variables, and so on. Thus, systems analysis and modeling that follow the mtM approach are fraught with approximation errors.

In contrast, the Mtm approach reverses the procedure by synthesizing an initial model at the macro-detail end of the spectrum. Mental grasp is easy and leads to successively more micro levels of detail to the extent required. In further contrast, Mtm models consist of linear equations that yield finite displacements as implicit functions of Δt. This book introduces a unique modular system of linear balance equations and linear dynamic forms that precisely models the dynamic characteristics of the physical system. Trajectories through space and time exhibited by mass and/or their phenomena, whether linear/nonlinear or deterministic/stochastic, are accurately reproduced by Mtm models.

Ultimate accuracy is limited by the precision capability inherent to the combination of hardware and software used for computation. Model synthesis follows the same pattern at each level within the hierarchical, Mtm sequence. Namely, the physical system is broken down into subsystems, each subsystem is broken down into components of supply and demand, and each component is assigned one-to-one correspondence with variables of the model. Variables are broken down into levels and rates. Primary data values are available from nature as levels, obtained by measuring or counting. Rates are calculated by differencing pairs of levels separated by Δt increments of time.

Surprisingly, Mtm modeling does not depend on primary data to produce a model, but rather, it depends on systems analysis and knowledge acquisition to produce a knowledge base. Secondary data are synthesized by the model to produce a complete data base. Reverse regression is used in the absence of primary data: Given the coefficients (derived from the systems knowledge base), find the data. Data synthesis by the model stands in obvious contrast to forward regression: Given the data (derived from historical observations), find the coefficients. The model must be validated for each intended application. Validation is achieved by employing Turing Type tests and various simulation tests such as perturbation

Because Mtm systems models are linear (not linearized or piecewise linear), fast and efficient solutions are obtained by using standard programming software that features matrix multiplication, inversion, and data handling. Having common matrix structures, interacting systems are readily linked so that accurate, multi-system solutions are possible. Furthermore, models can be optimized using standard linear or quadratic programming algorithms.

The Mtm approach has undergone thorough testing over the many years since its inception. Mtm methods were integrated into the engineering courses taught by the author and in 1981, formed the basis for introducing a graduate course in systems modeling.

Although providing much descriptive information about systems and systems analysis, the book’s main purpose is quantitative—synthesis of systems models. This book should be on the desk of anyone engaged in building systems models and especially those in engineering, business, economics, biology, agriculture, and federal agencies that oversee the nation’s resources. Furthermore, the methodology is highly suitable to development and installation of regional, executive planning models that are operated on-line in real time via satellite telemetry between earth data collection stations and a supercomputer site. Such installations have the potential for monitoring traffic, floodwaters, and forest fires through the use of systems models and interacting these models with resources models such as water, forest, game populations, energy, and labor pool.

The book is suitable for seniors and graduates of any engineering major and also for other majors that provide a background of at least two years of quantitative work. Desirable prerequisites include (1) an introductory course in statistics that covers regression analysis, (2) a working knowledge of matrix algebra, and (3) computer familiarity. Calculus is used in some places, but it is not essential in implementing the Mtm approach.

Part I provides an introduction to systems analysis and modeling. Chapter 1 introduces the systems philosophy on which the Mtm approach is based. Chapter 2 introduces the principles and dynamic forms that uniquely shape the Mtm approach. Chapter 3 introduces a simple example for which a systems model is defined, calibrated, and tested.

Part II, spanning Chapters 4 through 7, provides numerous examples of deterministic models, drawing from physics, numerical methods, ballistics, inventory theory, and corporate strategy.

Part III details applications to stochastic modeling via Chapters 8 through 13, with examples from ergonomics, glaciology, water resources planning, forestry, manufacturing processes, time series, operations research, and large-scale systems.

Chapters 1, 2, and 3, with selections from Parts II and III, are adequate for a 3-credit semester course. If modeling projects are included, the book will serve two semesters. Practical exercises are provided to emphasize key concepts, verify details slighted in the narrative, suggest library assignments, and to encourage mastery of Mtm concepts via a hands-on approach.

Appendixes are provided with each quantitative chapter. They contain data and graphs so that model results can be reproduced by the student before attempting his/her own applications. An instructor’s solutions manual is available for use with the text; however, some exercises have open-ended solutions that encourage student resourcefulness and innovation.

Finally, the author wishes to acknowledge the encouragement received from colleagues and students throughout the years that this work was being compiled. Portions of the materials submitted by graduate students Larry Cawlfield, Greg Goltz, Terry Ross, and Robert Wehrman were included in the book, and special appreciation is extended to them.

Part I

Introduction to Analysis and Modeling

CHAPTER 1

Systems Analysis and Model Synthesis

1.1 INTRODUCTION

Two or more components comprise a system if they interact within a common domain to achieve a common goal or function. In general, a system is not isolated but rather is linked to adjacent systems through its inputs and outputs. Consequently, systems are dynamic, adjusting internally in response to external disturbances. A systems model consists of two or more interacting components. An effective systems model is capable of linking with models of adjacent systems. To possess such specific properties, a model must be designed, following a procedure that in many ways resembles engineering design. Engineering design depends on a growing body of knowledge. Knowledge begins with engineering education and continues to grow through practical experience. Usually, engineering design involves the following elements:

• Analysis Design object is differentiated into subcomponents and their interrelationships. Specifications are formulated.

• Synthesis Alternative systems designs are considered. Subcomponents are integrated into a functional whole to produce tentative designs.

• Testing Design specifications are verified for the chosen alternative, performance is monitored for validity, and functionality is checked for credibility with the user.

• Implementation Design object is installed and becomes operational for the user.

Recycling to previous steps generates new alternatives and leads to improvements in the design.

Similarly, model design depends on a growing body of knowledge. Each system domain is characterized by knowledge items. Knowledge items are acquired through study (analysis) of the system. Relevant items are synthesized into a systems model. Even a cursory study reveals that systems are characterized by triads. For example, this paragraph just introduced a systems triad of knowledge acquisition, systems analysis, and model synthesis. Furthermore, knowledge acquisition is a process that involves three entities, a general information triad composed of source, medium, and message. Here are some examples of information triads:

• Textbook author, paper, theme

• Tracks animal, snow, a bear passed this way yesterday

• Fossil record prehistoric being, bone, a tyrannosaur lived here 100 million years ago

• Physical universe designer, design, axioms of existence

• Knowledge base Table 1.1 displays a triad of information triads.

Table 1.1

Information Triads of the Knowledge Base

Knowledge items are expressed in the message column, and are classified as hard or soft. For hard knowledge to be relevant, the message must consist of true facts that are applicable. Whether from scientific inquiry or statistical observation, a fact must be treated as a binary variable, a hypothesis to be tested. Facts can be established as true or false because they have the property of being observable by objective application of the physical senses. On the other hand, soft knowledge is subjective, not open to observation, and consists of heuristics: experience and rules of thumb. Knowledge items may be sought from each source to form the knowledge base utilized in systems modeling. Because a systems model synthesizes information out of a blend of hard and soft knowledge, its initial status is tentative. Therefore, each application of the model becomes a hypothesis to be tested. Just as in statistical hypotheses testing, two types of error are possible: (1) rejecting a true application (hypothesis) and (2) accepting a false application (hypothesis) [27].

Although Chapter 1 echoes the title of this book, its purpose is merely to provide an introduction to systems analysis and modeling. Beginning with a macro overview in Chapter 1 and principles in Chapter 2, ensuing chapters introduce finer detail, thus paralleling the method by which a newspaper journalist presents the news: headline, sketch, and back page (another system triad!). The book shows how to employ a macro-to-micro (Mtm) approach to systems analysis and modeling. The next section further explores the concept of system triads.

1.2 THREEFOLD NATURE OF THE UNIVERSE

Figure 1.1 displays an orthogonal axes system representing the physical universe of space, mass, and time. Macro analysis is extended in the micro direction by increasing detail along one or more of the axes of space, mass, and time. Subjectively defined, unit quantifiers provide for scaling along each axis.

Figure 1.1 Fundamental Units of Extension

Entities L, M, and T are used by scientists in an axiomatic sense to represent the existence of space, mass, and time, respectively, as components of the physical universe. Their axiomatic nature may be symbolized by the universal quantifier, ∃, meaning there exists:

Axiomatic entities are accepted as the starting point in any formal system of reasoning. Such entities retain their abstract nature unless they are coupled to observation and experience. Observed as a system, the universe is a triad of space, mass, and time, each entity also being a triad. Figures 1.2 and 1.3 portray this threefold nature of the universe.

Figure 1.2 The Universe as a Triad

Figure 1.3 Space, Mass, and Time Triads

1.2.1 Space

Geometrically, space is described in terms of length, width, and height, as depicted in Figure 1.3. Demarcation of space, to the exclusion of mass and time, is senseless in systems analysis. Typical unit quantifiers of space are universe, galaxy, solar system, Earth, hemisphere, continent, country, state, county, township, section, subsection, acre, plot, …, increment (Δx, Δy, Δz), infinitesimal (dx, dy, dz). Consequently, any system selected for Mtm analysis exists within some containing unit of geometric space. Furthermore, the size or scale of a system is most frequently classified with respect to the magnitude of its domain—that is, the amount of space circumscribed by the system.

Mtm concepts are applicable to systems of any scale, ranging from the microscopic, cellular level (e.g., microbial populations) to the macroscopic global level (e.g., human populations) and beyond to stellar systems. A macroscope [14] is the inverse of a microscope, obscuring all but general features of a system. For instance, a large-scale system impacts one or more human institutions and is extensive relative to the systems scale. Examples are the economy, government, health care, and so forth. Figure 1.4 presents the systems scale.

Figure 1.4 The Systems Scale

1.2.2 Mass

Physical objects are observable because of the properties of mass: matter (in) motion (producing) phenomena. Refer to Figure 1.3. Mass is the measure of difficulty by which an object can be accelerated [23]. Less precisely, mass is the quantity of matter in an object expressed in units of weight, for example: megaton, kiloton, ton, kilogram, gram, centigram, milligram, …, Δm, dm. Matter conforms to a triad of natural states: gas, liquid, and solid. Motion resulting from constant acceleration is a triad expressed in terms of distance, velocity, and acceleration. From cosmological physics, phenomena are the result of a triad of primary fields: electric, magnetic, and gravitational.

Because of its physical properties, mass does not exist independently of space and time. Mass is related to space (volume) by its density: ρ = m/V. Mass (atomic) is also time (velocity) dependent, appreciable only at velocities comparable to the velocity of light [18].

Despite these dependencies, mass, as a quantifier for matter, satisfies the continuity or conservation principle expressed as a mass balance equation [30]. The same is true for related physical phenomena such as momentum, energy, and electricity. Flow of mass through time is modeled as a conservative system, portrayed in Figure 1.5, and defined as follows:

Figure 1.5 Conservation of Mass

X1 = initial level of mass

X2 = terminal level of mass

X3 = mass input rate

X4 = mass output rate

At the end of each Δt period of time, the mass balance equation requires that mass be conserved, that is, neither created nor destroyed:

1.2.3 Time

Time is a physical property and varies with respect to mass, acceleration, and gravity. Two objects exhibiting mass and separated by space are also separated by time. Time is related to space (distance) by the velocity of matter in motion: v = S/t. Thus, time completes the most basic physical triad, that of the universe. Time, also a triad, future, present, and past, is symbolized by the third triangle in Figure 1.3. Quantifiers of time are aeon, …, millennium, century, decade, year season, month, week, day, hour, minute, second, …, Δt, dt.

The future consists of differentiated events and can be regarded in two different ways:

• Deterministic The future is a repetition of the past, and events come as no surprise.

• Nondeterministic The future is not a repetition of the past and events are differentiated only under risk or uncertainty.

Projecting from the present into the future is called forecasting. Projecting from the present into the past is called backcasting. Both projections are attended with exponentially increasing error whose magnitude becomes astronomical when projecting beyond the near present. Errors associated with time are as follows:

The future contains differentiated bumps and is divided from the past by the present. On the other hand, the past contains integrated bumps, past occurrences that have been blended with the passage of time through the entropic process of change. To illustrate, future precipitation will materialize in the present (be differentiated) as storms. With the integration of precipitation over time into past occurrences, the storms as bumps are lost. Storms become integrated into macro bodies of water such as snowpack, streams, rivers, and eventually the ocean. Periodic measurement (monthly, daily, hourly, etc.) captures the result of this physical integration process. In contrast, analysts traditionally start with differential equations and the infinitesimal, dt, that describe flow at a point, as, for example, Darcy’s law [21], and laboriously perform mathematical processes of integration to obtain approximate flow models and solutions. Why not allow the physical system to perform its natural process of integration to obtain the desired level of detail? Great ease and accuracy of solution is possible through circumscribing and modeling the physical system at a macro level. Linear equations, corresponding to the circumscription, model the natural process by integrating rates into levels over unit time, Δt.

Perception of time relates to duration and is found to vary widely among individuals. Thus, for objectivity, time is a process that requires calibration. Customary references to time involve triads: 12/7/41, month day hour, day hour minute, hour minute second. Does motion of a pendulum mark time? Yes, but its calibration is meaningless unless there is an observer. Likewise, the expanse of space and the properties of mass are unrecognized without an observer. Although space, mass, and time exist independently of any observer, their existence invites observation, thus leading to scientific inquiry.

Time, as similar to space, can be represented as an expanse. Mass is distributed by motion over both space and time. This suggests that a system is most basically characterized as the flow of mass over space and time. Time, as also similar to mass, can be represented as a flow. Which way does time flow? From the past through the present into the future? Or from the future through the present into the past? Thought provokingly, if the latter is true, then the future is older than the past, for it will take longer for the more distant future to arrive at the present! However, Einstein theorized that time is a vector field over space and furthermore that the distinction between past, present, and future is only an illusion, however persistent [16]. He further explained that time is not at all what it seems. It does not flow in only one direction, and the future exists simultaneously with the past.

Time measurements require a triad: an observer, a reference point, and a moving (usually rotating) object. Examples are sundials, hourglasses, clocks, and celestial bodies. Although time is axiomatic, viewing time from the basis of systems analysis reveals some interesting working definitions that may be expressed as follows:

Relational Definition

Present time is a process that converts one unit of future time into one unit of past time at an absolute rate (frequency) of one unit (cycle) per 2π radians.

A radian is the angle subtended by an arc whose length is equal to the radius. Figure 1.6 expresses this relationship.

Figure 1.6 Relational Definition of Time

The process of converting future into past is schematically portrayed by a finite time wheel with unity radius, as seen in Figure 1.7. The wheel is rolling at an absolute frequency of ω = 1/2π cycles per radian along the expanse of time.

Figure 1.7 The Time Wheel

Absolute Definition

The absolute unit of time is defined by the reciprocal of absolute frequency: one time unit = 1/ω = 2π radians per cycle.

Consequently, the absolute unit of time is represented by the circumference of a time wheel with unit radius. Time measurements are relative to the absolute unit of time.

Example

Let 52 weeks (one year) constitute one cycle. Then (2π radians) × k = 52 weeks per cycle, where k is a proportionality constant. k = 52/2π weeks/(radian cycle) is defined as the radius, R. Thus, one year is represented by the circumference of a time wheel having radius 52/2π: C = 2π R = 2π · (52)/(2π) = 52 weeks per