Model Building in Mathematical Programming

H. Paul Williams

• Publisher: Wiley
• Year: 2013
• Print ISBN: 9781118443330
• eBook ISBN: 9781118506172
• Edition: 5
• Topic: Business
• Subtopic: Management
• Index: Business

The 5th edition of Model Building in Mathematical Programming discusses the general principles of model building in mathematical programming and demonstrates how they can be applied by using several simplified but practical problems from widely different contexts. Suggested formulations and solutions are given together with some computational experience to give the reader a feel for the computational difficulty of solving that particular type of model. Furthermore, this book illustrates the scope and limitations of mathematical programming, and shows how it can be applied to real situations. By emphasizing the importance of the building and interpreting of models rather than the solution process, the author attempts to fill a gap left by the many works which concentrate on the algorithmic side of the subject.

In this article, H.P. Williams explains his original motivation and objectives in writing the book, how it has been modified and updated over the years, what is new in this edition and why it has maintained its relevance and popularity over the years: http://www.statisticsviews.com/details/feature/4566481/Model-Building-in-Mathematical-Programming-published-in-fifth-edition.html

Part I

Chapter 1

Introduction

1.1 The concept of a model

Many applications of science make use of models. The term ‘model’ is usually used for a structure that has been built with the purpose of exhibiting features and characteristics of some other objects. Generally, only some of these features and characteristics will be retained in the model depending upon the use to which it is to be put. Sometimes, such models are concrete, as is a model aircraft used for wind tunnel experiments. More often, in operational research, we will be concerned with abstract models. These models will usually be mathematical in that algebraic symbolism will be used to mirror the internal relationships in the object (often an organization) being modelled. Our attention will mainly be confined to such mathematical models, although the term ‘model’ is sometimes used more widely to include purely descriptive models.

The essential feature of a mathematical model in operational research is that it involves a set of mathematical relationships (such as equations, inequalities and logical dependencies) that correspond to some more down-to-earth relationships in the real world (such as technological relationships, physical laws and marketing constraints).

There are a number of motives for building such models:

1. The actual exercise of building a model often reveals relationships that were not apparent to many people. As a result, a greater understanding is achieved of the object being modelled.

2. Having built a model it is usually possible to analyse it mathematically to help suggest courses of action that might not otherwise be apparent.

3. Experimentation is possible with a model, whereas it is often not possible or desirable to experiment with the object being modelled. It would clearly be politically difficult, as well as undesirable, to experiment with unconventional economic measures in a country if there were a high probability of disastrous failure. The pursuit of such courageous experiments would be more (though not perhaps totally) acceptable on a mathematical model.

It is important to realize that a model is really defined by the relationships that it incorporates. These relationships are, to a large extent, independent of the data in the model. A model may be used on many different occasions with differing data, for example, costs, technological coefficients and resource availabilities. We would usually still think of it as the same model even though some coefficients have been changed. This distinction is not, of course, total. Radical changes in the data would usually be thought of as a change in the relationships and therefore the model.

Many models used in operational research (and other areas such as engineering and economics) take standard forms. The mathematical programming type of model that we consider in this book is probably the most commonly used standard type of model. Other examples of some commonly used mathematical models are simulation models, network planning models, econometric models and time series models. There are many other types of model, all of which arise sufficiently often in practice to make them areas worthy of study in their own right. It should be emphasized, however, that any such list of standard types of model is unlikely to be exhaustive or exclusive. There are always practical situations that cannot be modelled in a standard way. The building, analysing and experimenting with such new types of model may still be a valuable activity. Often, practical problems can be modelled in more than one standard way (as well as in non-standard ways). It has long been realized by operational research workers that the comparison and contrasting of results from different types of model can be extremely valuable.

Many misconceptions exist about the value of mathematical models, particularly when used for planning purposes. At one extreme, there are people who deny that models have any value at all when put to such purposes. Their criticisms are often based on the impossibility of satisfactorily quantifying much of the required data, for example, attaching a cost or utility to a social value. A less severe criticism surrounds the lack of precision of much of the data that may go into a mathematical model; for example, if there is doubt surrounding 100 000 of the coefficients in a model, how can we have any confidence in an answer it produces? The first of these criticisms is a difficult one to counter and has been tackled at much greater length by many defenders of cost–benefit analysis. It seems undeniable, however, that many decisions concerning unquantifiable concepts, however, they are made, involve an implicit quantification that cannot be avoided. Making such a quantification explicit by incorporating it in a mathematical model seems more honest as well as scientific. The second criticism concerning accuracy of the data should be considered in relation to each specific model. Although many coefficients in a model may be inaccurate, it is still possible that the structure of the model results in little inaccuracy in the solution. This subject is mentioned in depth in Sections 4.2 and 6.3.

At the opposite extreme to the people who utter the above criticisms are those who place an almost metaphysical faith in a mathematical model for decision-making (particularly if it involves using a computer). The quality of the answers that a model produces obviously depends on the accuracy of the structure and data of the model. For mathematical programming models, the definition of the objective clearly affects the answer as well. Uncritical faith in a model is obviously unwarranted and dangerous. Such an attitude results from a total misconception of how a model should be used. To accept the first answer produced by a mathematical model without further analysis and questioning should be very rare. A model should be used a...

Table of contents

• Cover
• Title Page
• Copyright
• Dedication
• Preface
• Part I
• Part II
• Part III
• Part IV
• References
• Author Index
• Subject Index