Principles of Modeling and Simulation in Teaching and Research1

OUR INDUSTRY T O D A Y Principles of Modeling and Simulation in Teaching and Research 1 D. R. M E R T E N S Animal and Dairy Science Department University of Georgia Athens 30602 ABSTRACT

reproduced in such a way that other researchers and teachers can duplicate the results and build upon its base. Finally, modelers should retain a healthy skepticism of models, especially their own, to ensure that they are evaluated adequately and used correctly.

A well defined mathematical model allows manipulation, condensation, interpretation, and utilization of quantitative information about complex problems. Modeling and simulation can provide a valuable adjunct to traditional research and teaching methods. Modeling aids the researcher and teacher by: (1) organizing information and crystallizing his thinking; (2) identifying new research and teaching areas and techniques; and (3) testing research and hypotheses. In addition to the attributes of modeling, simulation can aid the researcher and teacher by (1) quantifying experimental benefits and (2) predicting outcomes under new conditions, and assisting in deriving important experimental estimates of parameters. Although modeling and simulation are a combination of art and science, there are guidelines which can be used by the teacher or researcher to develop and utilize models: (1) define the problem and modeling goals, (2) observe and analyze the real system, (3) block diagram and synthesize the model, (4) mathematically formulate and implement the model, (5) process relevant data for variable and parameter estimates, (6) verify and validate the model, (7) improve the model, (8) accept it, (9) simulate results using the model, and (10) evaluate the simulation results. For adequate use of this research and teaching toot, its limitations and difficulties must be understood. Modeling and simulation require extensive communication between the modeler and his audience. Both the model and the data for its implementation must be

I NTRODUCTION

Received October 27, 1976.

1An invited paper at the 71st Annual Meeting of the American Dairy Science Association, June 20 to

23, 1976, Raleigh, NC.

The popularity of modeling and simulation in animal science research is evidenced by the 25 articles published in the Journals of Animal and Dairy Science between January, 1975, and June, 1976. In addition, there have been invited papers (20) and symposia (5, 21) which have discussed methods and uses of modeling. Surveying the literature suggests diversity in the use, and sometimes, abuse of this research tool. This raises the question, "Are there unifying principles for researchers, teachers, editors, and reviewers to utilize this tool and assess its value?" A review of biological mathematics (7, 22, 32, 34, 38, 40, 42), and modeling and simulation (1, 9 to 19, 23, 25 to 30, 33, 35, 37, 39, 41, 43 to 47) suggests that there are principles which can be beneficial in formulating and evaluating models. This presentation will attempt to (1) describe for nonmodelers what modeling involves and what it can do, and (2) standardize terminology and organize principles of model development and use. "Modeling" and "simulation" mean different things to different groups. A model is a simplified, abstract and idealized representation of reality based on an ordered set of assumptions. This is a broad definition that encompasses all types of models. This discussion will concern one type of model, the mathematical model. A mathematical model is defined as a set of equations that can be solved to predict changes that occur in the real world. It is a consequence of analytical efforts to abstract and define the real situation in precise mathematical terms. Thus, modeling, or model building, should be defined as the act of developing a mathematical model of a real situation.

1176

OUR INDUSTRY TODAY Modeling is easier to define than simulation. In the broadest sense, simulation could be described as duplicating the essence of a system or activity over time by artificial representation of the system or activity. To narrow this definition, simulation could be defined as the development and use of models to study the dynamics of a system (24). However, including modeling in the definition of simulation is confusing. As suggested by Wright (45), there are many instances when model building is the endpoint of scientific efforts; therefore, modeling should be a separate and distinct research or teaching tool. Separation of these two activities may encourage more definitive publication of the models themselves, rather than cursory coverage of the model with most emphasis placed on the analysis of model predictions. Therefore, simulation is the numerical operation of a dynamic mathematical model to obtain a solution. Simulation provides a specific result over a single time frame that is dependent upon the structure of the model and the value assigned to the variables and parameters. The detailed differences among models and their techniques of solution will be discussed in another section. Modeling and simulation introduce the philosophy of using mathematics to describe biological situations. Is mathematical description of these systems desirable or feasible? Some animal scientists might state that the biological sciences are so variable with such massive numbers of variables subject to such complex interactions that the use of mathematics to understand biology is inappropriate or impossible. However, it can be argued that intuitive and verbal theories are too ambiguous and open to misunderstanding. Thus, only precise, formal mathematical statement of the problem provides a clear and concise definition of its character. In addition, mathematical statement demands thorough and specific study of the situation which can lead to more productive analyses. A well-defined mathematical model, even if complex, is beneficial for rational and coherent discussion of a problem. A mathematical model can take data from a complicated, multivariate situation and precisely organize it in such a way that manipulation, condensation, and interpretation of the information can lead more easily to insight and logical analysis. However, the need for precise mathematical

1177

statement does not diminish needs for descriptive information, intuition, and experience with the situation. The computer cannot transform a faulty model or inaccurate data into a useful observation for gaining insight or forming decisions. Mathematical expertise or computer power cannot substitute for knowledge of factors and their relationships to the problem or for accurate and appropriate data. Any nutritionist who has consulted with a mathematician or computer programmer can testify that although their assistance is helpful, it is not the essence of modeling ruminant digestion! Information about the components and their relationships is needed adequately to use inductive reasoning to generate hypotheses. Then deductive reasoning can use these hypotheses to form frameworks for predictions. Mathematics is a tool for biologists to build and manipulate these frameworks, thereby increasing our understanding of biological systems. Mathematical models are always simplifications of reality, and we should not lose sight of the holistic nature of biology which may never be described fully mathematically. Mathematics, as a tool, allows greater precision and simplicity in communicating ideas. The simplicity can belie the amount of rigorous thought needed to arrive at a solution while the precision of mathematics, which allows only right or wrong with no middle ground, allows the researcher no place to hide from his mistakes! The Value of Modeling and Simulation

Modeling and simulation can be a valuable adjunct to traditional research and teaching methods. Figure 1 illustrates relationships between modeling and traditional research approach. The suggestions of many authors (4, 11, 14, 23, 30, 31, 43, 45, 46) were incorporated into the five categories of interrelationships between modeling and traditional research. Emphasis will be on how modeling assists traditional methods although the converse situation is also true. It is hoped that focusing attention upon these advantageous interactions between modeling and traditional research will stimulate curiosity and interest in modeling as a research and teaching tool. One of the greatest advantages of modeling is that it forces the scientist to crystallize his thinking by providing a vehicle for organizing Journal of Dairy Science Vol. 60, No. 7

1178

MERTENS INFORMAT~ON

r

\,

v

/ \\i",.o7,7:;;, ~ i //'"

- -

!

FIG. 1. Illustration of the interaction between modeling and simulation and the traditional research method.

information. Detailed and systematic observation and analysis of the system to be modeled can lead to increased knowledge about it. Many times modeling actually will aid in defining the problem since the scientist must describe the factors and their relationships in clear and explicit form. Modeling can mobilize existing knowledge by incorporating a larger mass of relevant data into analysis of a problem. Computational powers of the computer easily can accommodate volumes of data and complex relationships. This can increase the efficiency of using all data to solve a research problem. Conversely, modeling can mobilize knowledge by requiring that complex systems be reduced to simple yet explicit statements. The computer uses simple mathematical statements that can be programmed easily in computer simulation languages. The reduction of a complex system to its simple parts and then to simple mathematical statements organizes information in new ways that remove clutter and reduce problems to manageable proportions. Many times this leads to greater knowledge about the system and to innovative research ideas. Journal of Dairy Science Vol. 60, No. 7

Modeling and simulation can provide a framework for developing a research program by identifying research areas and techniques that deserve scientific effort. Although modeling is not the only way to develop a research program, it does emphasize a central goal that prevents the program from sidetracking. Mathematical modeling forces the scientist to more rigorous and quantitative thinking that can lead to critical experiments needed to solve a problem. It will point out where information is lacking or inadequate, signaling the need for specific experiments. Sometimes synthesizing the model will indicate which variables are important and suggest which research topics deserve study and which technique will provide the information or critical test to solve the problem. In development of the model, hypotheses concern the relationship between model components and variables. Thus, the model can become a testable, although complicated, hypothesis. However, Passioura (31) has questioned if complicated models are testable. Testing the final output is only a weak test of the validity of the model while testing each subsystem is more rigorous hut more difficult. However, at least, the model can test hypotheses by reductio ad a b s u r d u m . Some hypotheses will be untenable, and new hypotheses will be developed to explain the phenomenon. Thus, modeling and simulation can test some hypotheses and lead to new hypotheses upon which experimentation can be based. Simulation can assist traditional research methods by providing quantitative assessment of the benefits and success of actual experimentation (4). For example, simulation may show that the magnitude of change of a particular variable is so small that detection of differences is impossible or requires extremely large sampling numbers. It also might indicate that changing a particular factor by several magnitudes will have little effect upon the final outcome. This information could be used by the researcher to decide how limited funds should be spent. Simulation also could provide the administrator with information concerning the ultimate benefit of research in an area which would aid his decisions of allocating resources. Finally simulation can predict outcomes that become a part of the data pool of general

OUR INDUSTRY TODAY information and can be used to derive estimates of parameters that cannot be derived by other methods (6). Simulation can provide information concerning the optimization of a system that would be prohibitively expensive to obtain under real situations. It can allow the assessment of new situations and changes in the systems that may be impossible to determine in any other way. Simulation also can assist interpretation of experimental results by deriving parameter values, such as efficiency coefficients, that are constrained by information already gained, yet difficult to determine without a mathematical model. Figure 1 indicates that although modeling and simulation can lead directly to knowledge and insight, they also can serve a much broader role in assisting the scientist in seeking knowledge through the traditional research method. Modeling can aid the researcher independent of its use during simulation. Often the modeler never reaches his goal of a valid model that truly can simulate the real situation. However, the knowledge gained during modeling and the improvement in research effectiveness easily can justify model building. Although the discussion of modeling and simulation thus far has been related to the researcher, it also has value for the teacher. At the graduate level, modeling and simulation as a teaching approach can illustrate the complexity of complete systems, yet provide a logical framework for attacking problems. This approach also has value for undergraduate education especially in management or applied nutrition where decision-making is an inherent objective of course work. Simulation is especially useful for teaching when incorporated into gaming simulation. In this instance the teacher develops the model and implements it to interactive computer use. This allows the student to inject his decision (via parameter or variable estimates) into the computer model for solution. Thus, the student gains immediate feedback concerning results of his decision. This provides him with experience at making decisions and reinforces correct decision-making procedures. Gaming simulation provides experience and knowledge about decisions without the need for understanding the intricacies of the model itself. Used in this fashion, models and gaming simulations are potentially useful teaching tools.

1179

Modeling and Simulation Applications

Modeling and simulation can be used to study three main categories of problems: (1) prediction or output analysis, (2) reception or input discovery, and (3) insight or transfer function identification. Prediction simply answers the question, "What happens if . . . . ?" This is the most common use of modeling. However, if it has many pitfalls, particularly if the model is not based upon fundamental principles which can increase the meaning and accuracy of extrapolation. At their worst, model predictions still should indicate the direction, and possibly the magnitude, of changes. Simulation allows the solution of predictive estimates even in complex models which are not tractable to mathematical or numerical analysis. However, it has limited ability to substitute for experimentation, if any at all. Simulation might substitute for experimentation in situations when: (1) research may be too expensive, (2) results cannot be achieved experimentally, or (3) when many factors must be considered simultaneously and experimentation would become uncontrollable. Even under these circumstances the results of simulation should be interpreted skeptically. Models used as a substitute for direct experimentation should be subjected to extreme critical review to determine: (1) that the logic and assumptions within each component and between components has been verified independently, (2) that parameter averages and ranges have considered all information available, and (3) that no important factor which could influence the result has been omitted. Our greatest concern should be that models predict results which are not verifiable and that the unsuspecting will give these results the same value as direct observation because they are published! The second category of problems studied by modeling, which is input discovery, attempts to answer the question, "Given an output, what input or combination of inputs can produce this result?" Many optimizing studies using simulation attempt to find the inputs that result in an optimal solution. This problem is simply the reverse of prediction and comments concerning prediction also apply to input analysis. However, input analysis is complicated by the fact that several combinations of inputs may obtain the same output. To be successful, Journal of Dairy Science Vol. 60, No. 7

1180

MERTENS

there is no substitute for knowledge of subject there must be a one input to one o u t p u t ratio matter and experience. Although intuition is during input analysis. Thus, all the other inputs must be known or estimated before a single needed, there still can be developed guidelines input can be related to a given output. How- useful for anyone who is contemplating modeling and simulation. These guides are not hard ever, this is not greatly different in nature from prediction where all the inputs are known or and fast rules for each model; however, even the most accomplished modeler will find that estimated to derive a single output. The third category of use of simulation may his thinking and assumptions will be clearer if be the most desirable, but is probably the most the pattern of model development is ordered. Figure 2 provides a schematic illustration of difficult. It is to gain insight into the functioning of the model and identify the type of model development and use. The methodology function that is operating in the real situation. of modeling is a cyclic and interactive process. This use attempts to answer the question, Models evolve as they are challenged at each "What is the mechanism operating in the real stage of development, and changes are made to world?" In this, simulation can be a valuable incorporate new knowledge and insight about adjunct to direct experimentation. Modeling the system being modeled and the model itself. Eventually, the model reaches development c a n be used to discover underlying interactions which is accepted as an accurate representation as shown by Romero et al. (36). Simulation can be used to test the reason- of the system, and the system can be simulated ableness of a model and sometimes can suggest and evaluated. Although many authors have which alternate hypothesis is implausible. It presented modeling procedures for engineering, also can point to worthwhile experiments that management, and economics (2 to 4, 9, 16, 18, can reject a model or provide information for 23, 30, 46), the following scheme was develforming new hypotheses. Finally, simulation oped for specific use by animal science teachers can provide insight about relationships within and researchers. 1. Define the problem and modeling goals. the model through sensitivity analysis. In sensitivity analysis, all parameters and variables are Clear statement of the problem and modeling held constant except one which is varied to goals is important to insure that the modelers determine the sensitivity of results attributable time and resources are not wasted needlessly. to that single variable. Sensitivity analysis can Some modeling goals may be broad, such as, provide insight concerning: (1) which variables "Evaluate current concepts of rumen nitrogen are unimportant and may be eliminated to metabolism and determine the adequacy of simplify the model, (2) which values are critical current data for testing current concepts and and require accurate quantitation, and (3) new hypotheses." However, the problem also which variables cause changes large enough to should be stated in specific terms that describe allow testing of model predictions with direct boundaries of the model, type of mathematical forms and techniques, and extent of simplificaexperimental observation. tion that is acceptable. Questions include: What Modeling and Simulation Methodology

Since the value of modeling and simulation can be established, the question arises, "How are models developed?" Model building is both a science and an art. It is an art because of the intuitive process that is important in formulating the mathematical framework and manipulating its logic. Science is needed to provide information from which the model is constructed and to which it is compared for validation. However, intuition plays a role in deciding which assumptions and simplifications are allowable and which relationships are important. To make these intuitive judgments, Journal of Dairy Science Vol. 60, No. 7

i ~m,E ~ A~,~LYZE , THE S r S r E ~ ~

#o~ut.t~ a ,~PtEU~nr uO~L

t~OV~

~ ~r \

VALII~A

~

ft

PaOCESS ~*r~ ~Oa E S r t u , r E S

FIG. 2. Schematic design of the cyclic process of model development leading to use of the model during simulation.

OUR INDUSTRY TODAY result is needed from the model? Should the model be dynamic or probabilistic? What is the scope of the problem? What basis should the model be? How will it be used? What data are required and available for building and testing the model? What solution method is desired, analytical, numerical, simulation? Sometimes it is necessary actually to begin formulating the model before these questions can be answered. However, it is easier to formulate the model and justify the appropriateness of simulation if the questions are pondered beforehand. 2. Observe and analyze the system. Although most of us will model a system with which we are familiar, it is nonetheless important that we critically concentrate on the system to be modeled. What are the components of the system? How do they interact? What are the inputs and outputs? How are they measured? What factors influence the situation? What factors can be modified? 3. Diagram and synthesize the model. It is easiest to develop the model in parts. Not only does this aid in model building, but it also will help in analyzing the results and discussing the model with colleagues and in publications. First, block diagram the model including all variables thought to influence the system. Although the block diagram simply may represent the components and their relationships, it is important that all variables be included in this literal or graphic model even though many of them will be ignored to simplify the model. Including all variables will aid in determining inherent assumptions when the real situation is simplified for implementation. This cannot be overstressed. If all assumptions are not listed, it is easy to forget these and conclude that the model is an exact replica of reality. Second, synthesize the model by formulating hypotheses which can relate components and simplify the model by assumptions and by lumping components together which have a similar effect. It is most desirable to formulate hypotheses from fundamental principles from existing evidence. Think about these fundamental principles. Many times you will be surpised how few alternate hypotheses there are which can explain the results. Then these few can be tested. Often mathematical or numerical analysis of the whole model will eliminate some hypotheses leaving only a few that need testing by critical experimentation. For lumping and

1181

simplifying the situation, assumptions should be listed for future reference. Since it is difficult to determine which factors can be neglected or simplified, there are many dead ends. There can be some consolation that most of the thinking in formulating unproductive models can be used in the analysis of more appropriate models in the future. 4. Formulate and implement the model. Formulate the model by rendering the hypotheses explicit and testable through mathematical equations. Since the model is developed in small parts, these mathematical equations usually will be expressed simply and easily. Thus, an extensive mathematical background may be helpful, but it is not necessary for modeling. Implementing the model for computer simulation is likewise relatively easy. There are many computer simulation languages available for implementation of models which need no prior training in programming. If analytical implementation of the model is desired, three additional steps are useful. Write down simple relationships between components. Some of these will be so obvious as to seem trivial; however, they are the building blocks needed for solution. A list of these relationships would include: equations of conservation (sum = parts or input = output), equilibria, definitions, proportionalities, assumptions, and derived equations. Next arrange the equations in tabular form by writing the equations in the righthand column, with a column for each of the terms in all of the equations. By placing a check in the correct column for each term in each equation, a ready reference table provides a list of all equations which contain that term. Finally, the equations can be combined to find the solution. 5. Obtain and process data for estimates of parameters and variables. Many times this step proves to be the most time consuming and frustrating. All too often one will find that data presumed to exist is not available or that information is in the literature in such a way that it is useless to anyone with a different hypotheses from that of the original author. Many times incomplete information is provided; therefore, logical decisions concerning the use of the data cannot be made. It is not uncommon to find nutrition papers in the literature that do not have the feedstuff composition of the ration, its chemical composiJournal of Dairy Science Vol. 60, No. 7

1182

MERTENS

zion, the bodyweight of the animals, or their intake. It is common to find summarization tables in the literature which take as much space as the listing fo each individual observation. Yet the summarization data provided only limited information while the detailed data would be much more desirable to the simulator or anyone who has an alternate explanation for the experimental results. Realizing that there are limitations of space in our journals, perhaps they should adopt use of "Supplementary Materials" available as phot o c o p y or microfiche as in the Journal of Agricultural and F o o d Chemistry. However, as a minimum, our editors and reviewers should ask that the following information be in the paper or be capable of calculation from values in the paper: dry matter consumption (absolute amounts/day), nutrients and energy in the diet (including amino acid contents for nonruminants), and major influencing factors such as body weight of the animal, production, breed or strain, and environmental conditions. In collecting and processing data, averages and extremes should be recorded, and the following questions should be answered. What is the accuracy of the data? Were there uncontrolled conditions in the experiments which could affect the values? Can data from different experiments be combined? Are constants in the model really constant? Care is needed in lumping values into constants to be sure that all values lumped together are really constant. For example, in the model of Burroughs et al. (8) three values are lumped to form the factor 1.04 which is used to convert TDN to microbial protein. Are these values constant? 6. Verify and validate the model. Although the terms verification and validation often are used interchangeably, they are two distinct processes of model testing. Verification is defined as the process of establishing the correctedness and accuracy of the model to behave as the modeler conceived it (3). It refers to the accuracy with which the mathematical formulation and computer programming are accomplished. Do the mathematical equations accurately represent the real, observed relationships? Do the assumptions in accepting mathematical description agree with relevant theory, experience, and general knowledge? Is the mathematical logic correct? Is the model written in the correct form and sequence? Is every Journal of Dairy Science Vol. 60, No. 7

symbol precisely defined and consistently used? Are equations in simplest form? Is the computer implementation of the model correct? These errors can be debugged by the followed methods: (1) follow each component through the entire model logic, (2) force rare events to occur including the substitution of 0 and oc values when appropriate, (3) check dimensions to make sure they are correct, (4) program the logic to provide each calculated value in sequence, and make certain the values are reasonable, and (5) hand calculate results twice with two different sets of values. Not all these techniques can or need to be used with a given model. The second area of evaluating the m o d e / i s validation. Validation indicates that the model is capable of being justified for use in representing reality. It encompasses establishing the accuracy of model assumptions and conceptualization of reality (3, 47). Although, models should be capable of rigorous experimental testing (14), it is actually difficult, if not impossible, completely to validate a model (31). The most objective validation procedure is to compare model prediction with real world observations. Ideally, validation would involve comparing model predictions, under circumstances not used to synthesize the model, with experimental evidence specifically collected to test rigorously the model. Less rigorous validation procedures include: (1) splitting data to use half for mode/ synthesis and the other half for validation, (2) observing if average outputs are predicted when average values for inputs are used, (3) determining if reasonable outputs are obtained when extreme input variables are used, and (4) determining if interaction values between subunits of the model reflect real world observations. No doubt the first model will have some shortcomings which must be corrected bringing one to the next step. 7. Improve the model. There are two ways to alter the model: (1) change parameter estimates or (2) change the model logic structure. The danger exists at this stage that changes will become curve-fitting, perhaps destroying the entire basis for the model. Any change should be compared to the original hypotheses to determine if it is consistent. It is best to have several alternate hypotheses when the model was developed. Then this step of model development is simply one of selecting

OUR INDUSTRY TODAY the best alternative and guards against unreasoned curve-fitting. 8. Accept the model as an adequate representation of the real world. If accepted, some understanding of the system already has been gained. In addition, one can feel confident that prediction is possible in the sense that changes in the situation can be followed logically through the model and related to reality. The importance of accepting the model before simulation cannot be overstressed. This should be a distinct step in modeling and simulation to emphasize that the simulator must have confidence in his model before he uses it to generate results for analysis and commentary. This step often is just implied in simulations; however, scientists should justify clearly their acceptance of a model before they use it and publish the results. 9. Simulate. Realistic input values can be used to produce quasi-observations which represent the real system within model assumption constraints. Parametric simulation where only one value is allowed to vary will provide insight into variable relationships and also may provide information for developing future critical experiments and further refining the model. 10. After simulation experiments, results are evaluated and discussed. However, unlike real world observations, results of simulation experiments must be interpreted in reference to the assumptions of the model. This does not detract necessarily from the conclusions but serves to reinforce that simulations are abstractions of reality and should be considered carefully before conclusions are made. Although the use of the guidelines will not guarantee success in modeling and simulation, it is hoped they provide a logical sequence which will aid the researcher or teacher in using these techniques. It is also important that some principles be developed which also aid the colleague or student who is attempting to understand or use our models. At this juncture, we need to focus our attention on the limitations of modeling so we do not create a monster that devours our data and ideas and converts them into meaningless or erroneous results. Perhaps these problems or pitfalls in modeling should be mentioned in closing. The first problem facing the scientist using modeling and simulation is communicating the model to others so they can understand and

1183

accept the results. The interaction time and scope between the simulator and his audience must not be too large. It would be beneficial to publish description of a model that is separate from the paper on simulation analysis. Most modeling papers present the result of simulation experiments without providing a clear presentation of the model itself. Perhaps modeling papers should be published in sequences. This would allow adequate space for discussing the model components, their assumptions, derivation, and justification, with ample documentation concerning how conclusions were reached. Too many model papers leave the reader speculating about the basis for author decisions. Although their decisions may be well grounded, by not stating the basis for them their conclusions are in doubt. It is also important that assumptions of the model be listed and justified. Several recent papers using modeling have discussed some assumptions, yet let major assumptions such as the use of linear programming as a solution technique go without comment. Finally the author and reviewers should ask if there is adequate information in the presentation to allow exact duplication. This is critical if we are to advance science by building upon previous models. Researchers other than the authors must be able to duplicate the models. Thus, communication of modeling results is an important problem which must be solved for this tool to be useful. A second problem in modeling is acquiring information to form estimates of parameters. The problems of literature studies have been discussed; however, the responsibility of the simulator to state clearly and distinctly how be arrived at his estimates is paramount. Surely publication space imposes limits on the extent of review possible; however, there must be a balance between economy of publication and dissemination of information. We must remember that simulation results depend not only on the model framework hut also on the estimates of parameters. Most model values are not documented, and the reader must depend on intuition or rely on faith in the author to determine if values are reasonable. This situation should be remedied. Perhaps a system for microfilming or photocopying this documentation should be available to interested scientists (at cost) as in the Journal of Agricultural and Food Chemistry. Journal of Dairy Science Vol. 60, No. 7

1184

MERTENS

The third problem a model builder must confront is fitting the model to a mathematical framework. Rarely is the scope of mathematics the problem. Usually the modeler is limited by his knowledge and intuition of mathematics. He also is faced with the decision as to what simplifying assumptions to make. He may create a model that is manipulated easily yet meaningless or develop a plausible model that is too complex to formulate and manipulate or provides solutions which arc untestable. The problem becomes one of balancing insight against reality and validation. Fourth, the scientist using modeling must face the problem of evaluating his model. It is easy to state that a model is valuable only so long as it represents real situations and yet very difficult to determine just how closely it does mimic reality. At best, models are an idealization of what we think the real world is or what we think it should be. Although they may be simplifications of reality, they must be relevant to it. All models will contain assumptions for the sake of simplicity because no model can contain all the variables needed to describe the real world. However, we constantly must keep in mind that the model is only as good as its assumptions. It is only a conceptualization of the biological situation but may never describe it completely. Some suggestions for testing models have been given earlier. However, there is no ultimate test. The modeler only can make certain that his model includes all important variables and that it can predict all known observations it was designed to mimic. Finally, the model builder always should remain skeptical of models, especially his own. We tend to become protective of that which we create, and this is especially true of a modeler who has spent much time and mental effort bringing his model into the world. However, we should remember that there is no unique correspondence between a model and reality. The only criteria for judging a model is its ability to duplicate the behavior of the real system. If we develop alternate hypotheses, as well we should to advance science, one of the models necessarily must fail. If we can begin always to develop alternate hypotheses, perhaps the trauma of rejection will not be as great or taken as personally. Historically, as a science matures, it progresses to a hypothetico-deductive philosophy where Journal of Dairy Science Vol. 60, No. 7

alternate hypotheses are tested. Modeling, like other areas of sciences, passes through three stages. Stage I is the natural history period in which uncritical observations are recorded and phenomena described. However, man soon begins to ask the questions why or how? Thus, Stage II arrives which is characterized by inductive reasoning which generates ideas of a general nature based on controlled, scientific experimentation. However, it soon becomes evident that there arc many possible inductive reasons which explain the observation. Thus, there is a need to determine what is the best explanation. Stage II1 science attempts logically to arrive at the solution by deduction from testable hypotheses. This stage is characterized by (a) devising alternate hypotheses, (b) devising crucial experiments, each of which excludes one or more of the hypotheses, (c) conducting experiments to obtain a clear result and (d) recycling the procedure to refine the possibilities. Some researchers might suggest that their area of science is in Stage 111; yet how many scientists actually write down alternate hypotheses and design crucial experiments to exclude one or more of them. Most models in the literature suggest Stage I or II development. They attempt to produce untested observations or describe inductively a system. Although these are laudable achievements, the development of hypothetico-deductive reasoning is critical for modeling and simulation. It is easy in modeling to develop a whole scheme of interactions which are not grounded in realistic or testable hypotheses. These models can cause more confusion than insight. In addition, these models may not increase predictive capabilities beyond empirical equations. It is easy to realize the limitation of extrapolation of regression equations; however, the problem is infinitely worse when we use a model that is not grounded in fundamental principles. We should remember that the best models are simple, have wide applicability if true, and are rejected easily by critical experiments. Thus, the philosophy of science is pragmatic, models are only as valuable as long as they function to predict correctly all observations. Knowledge is never absolute because we cannot prove, only disprove. It is hoped that this presentation has provided some principles of the description, use, limitations, and philosophy of modeling that can be used in order that

OUR INDUSTRY TODAY m o d e l i n g a n d s i m u l a t i o n can p r o c e e d to adv a n c e science a n d e d u c a t i o n . If we can avoid t h e pitfalls a n d f o l l o w t h e s e guides, m o d e l i n g a n d s i m u l a t i o n will b e c o m e an increasingly p o p u l a r a n d p o w e r f u l t o o l to solve p r o b l e m s in research a n d t e a c h i n g .

REFERENCES

1 Agrawal, R. C., and E. O. Heady. 1972. Operations Research Methods for Agricultural Decisions. The Iowa State University Press, Ames. 2 Anderson, J. R. 1972. An overview of modeling in agricultural management. Rev. Marketing Agr. Econ. 40:111. 3 Anderson, J. R. 1974. Simulation: Methodology and application in agricultural economics. Rev. Marketing Agr. Econ. 42: 3. 4 Anderson, J. R., and J. B. Dent. 1972. Systems simulation and agricultural research. J. Aust. Int. Agr. Sci. 38:264. 5 Baldwin, R. L. (Chairman). 1974. Application of simulating modeling techniques in animal science. 14th Annual Ruminant Nutrition Conference. Fed. Proc. 33:175. 6 Baldwin, R. L., and N. E. Smith. 1971. Application of a simulation modeling technique in analyses of dynamic aspects of animal energetics. Fed. Proc. 30:1459. 7 Bailey, N. T. J. 1967. The Mathematical Approach to Biology and Medicine. John Wiley & Sons, New York. 8 Burroughs, W., D. K. Nelson, and D. R. Mertens. 1975. Protein physiology and its application in the lactating cow: The metabolizable protein feeding standard. J. Anita. Sci. 41:933. 9 Coldicott, P. R. 1971. Principles of Systems Analysis. Macdonald & Co., London. 10 Dantzig, G. B. 1963. Linear Programming and Extensions. Princeton University Press, Princeton, New Jersey. 11 Dent, J. B., and J. R. Anderson. 1971. Systems, management and agriculture. Page 3 in Systems Analysis in Agricultural Management, J. B. Dent and J. R. Anderson, eds. John Wiley & Sons Australasia Pty. Ltd., Sydney. 12 Dent, J. B., and H. Casey. 1967. Linear Programming and Animal Nutrition. J. B. Lippincott Co., Philadelphia, Pennsylvania. 13 DeWit, C. T., and J. Goudriaan. 1974. Simulation of Ecological Processes. Centre for Agricultural Publishing and Documentation, Wageningen. 14 DeWit, C. T. 1969. Dynamic concepts in biology. Page 9 in The Use of Models in Agricultural and Biological Research. J. G. W. Jones, ed. The Grassland Research Institute, Hurley, Maidenhead, Berkshire, England. 15 Eidman, V. R. 1971. Agricultural Production Systems Simulation. Oklahoma State University, Stillwater, OK. 16 Forrester, J. W. 1968. Principles of Systems (2nd ed.). Wright-Allen Press, Inc., Cambridge, Massa-

1185

chuset~s. 17 Garfinkel, D. 1965. Computer simulation in biochemistry and ecology, tn Theoretical and Mathematical Biology, T. H. Waterman and H. J. Morowitz, eds. Blaisdell Publishing Co., New York. 18 Gordon, G. 1969. System Simulation. PrenticeHall, Inc., Englewood Cliffs, New Jersey. 19 Heady, E. O., and W. Chandler. 1958. Linear Programming Methods. Iowa State College Press, Ames. 20 Joandet, G. E., and T. C. Cartwright. 1975. Modeling beef production systems. J. Anim. Sci. 41:1238. 21 Jones, J. G. W. 1969. The Use of Models in Agricultural and Biological Research. The Grassland Research Institute, Hurley, Maidenhead, Berkshire, England. 22 Lotka, A. J. 1965. Elements of Mathematical Biology. Dover Publishers, Ind., New York. 23 Martin, F. F. 1968. Computer Modeling and Simulation. John Wiley & Sons, Inc., New York. 24 McLeod, J. 1968. Simulation is wha-a-at? Page 3 in Simulation, J. McLeod, ed. McGraw-Hill Book CO., New York. 25 Mihran, G. A. 1972. Simulation: Statistical Foundations and Methodology. Mathematics in Science and Engineering. Vol. 92. Academic Press, New York. 26 Milhorn, H. T. 1966. The Application of Control Theory to Physiological Systems. W. B. Saunders & Co., Philadelphia, Pennsylvania. 27 Millan, C. M., and R. F. Gonzalez. 1968. Systems Analysis: A computer approach to decision models. Richard D. Irwin, Inc., Homewood, Illinois. 28 Milsum, J. H. 1966. Biological Control Systems Analysis. McGraw-Hill Book Co., New York. 29 Naylor, T. H. (ed.). 1969. The Design of Computer Simulation Experiments. Duke University Press, Durham, North Carolina. 30 Naylor, T. H., J. L. Balintfy, D. S. Burdick, and C. Kong. 1966. Computer Simulation Techniques. John Wiley & Sons, New York. 31 Passioura, J. B. 1973. Sense and nonsense in crop simulation. J. Aust. Inst. Agr. Sci. 39:181. 32 Rajagopal, P. 1974. Mathematics and biology. Page 261 in Mathematical Problems in Biology. P. van den Driessche, ed. Springer-Verlag, New York. 33 Reitman, J. 1971. Computer Simulation Applications. John Wiley & Sons, New York. 34 Riggs, D. S. 1963. The Mathematical Approach to Physiological Problems. The M.I.T. Press, Cambridge, Massachusetts. 35 Roberts, F. S. 1976. Discrete Mathematical Models. Prentice-Hall, Inc., Englewood Cliffs, New Jersey. 36 Romero, J. J., R. Canas, R. L. Baldwin, and L. J. Kong. 1976. Lactational efficiency complex of rats: Provisional model for interpretation of energy balance data. J. Dairy Sci. 59:57. 37 Rosen, R. 1970. Dynamical Theory in Biology. Vol. 1. Stability Theory and Its Application. Wiley-lnterscience, New York. 38 Rubinov, S. 1. 1975. Introduction to Mathematical Biology. Wiley-lnterscience, New York. Journal of Dairy Science Vol. 60, No. 7

1186

MERTENS

39 Shapiro, H. M. 1969. Input-output models of biological systems: Formulation and applicability. Comput. Biomed. Res. 2:430. 40 Simon, W. 1972. Mathematical Techniques for Physiology and Medicine. Academic Press, New York. 41 Smith, J. 1968. Computer Simulation Models. Hofner Publishing CO., New York. 42 Stibitz, G. R. 1966. Mathematics in Medicine and the Life Sciences. Year Book Medical Publishers, Inc., Chicago, Illinois. 43 Sutler, R. E., and R. J. Crom. 1964. Computer models and simulation. J. Farm Econ. 46:1341.

Journal of Dairy Science Vol. 60, No. 7

44 Thornley, J. H. M. 1976. Mathematical Models in Plant Physiology. Academic Press, New York. 45 Wright, A. 1971. Farming systems, models and simulation. Page 17 in Systems Analysis in Agricultural Management. J. B. Dent and J. R. Anderson, eds. John Wiley & Sons Australasia Pry. Ltd., Sydney. 47 Van Horn, R. 1969. Validation. Page 232 in The Design of Computer Simulation Experiments. T. H. Naylor, ed. Duke University Press, Durham, North Carolina. 46 Zeigler, B. P. 1976. Theory of Modeling and Simulation. John Wiley & Sons, New York.