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1999 | Buch

Mathematical Modeling and Optimization

An Essay for the Design of Computer-Based Modeling Tools

verfasst von: Tony Hürlimann

Verlag: Springer US

Buchreihe : Applied Optimization

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Über dieses Buch

Computer-based mathematical modeling - the technique of representing and managing models in machine-readable form - is still in its infancy despite the many powerful mathematical software packages already available which can solve astonishingly complex and large models. On the one hand, using mathematical and logical notation, we can formulate models which cannot be solved by any computer in reasonable time - or which cannot even be solved by any method. On the other hand, we can solve certain classes of much larger models than we can practically handle and manipulate without heavy programming. This is especially true in operations research where it is common to solve models with many thousands of variables. Even today, there are no general modeling tools that accompany the whole modeling process from start to finish, that is to say, from model creation to report writing. This book proposes a framework for computer-based modeling. More precisely, it puts forward a modeling language as a kernel representation for mathematical models. It presents a general specification for modeling tools. The book does not expose any solution methods or algorithms which may be useful in solving models, neither is it a treatise on how to build them. No help is intended here for the modeler by giving practical modeling exercises, although several models will be presented in order to illustrate the framework. Nevertheless, a short introduction to the modeling process is given in order to expound the necessary background for the proposed modeling framework.

Inhaltsverzeichnis

Frontmatter

Introduction

Chapter 1. Introduction
Abstract
Observation is the ultimate basis for our understanding of the world around us. But observation alone only gives information about particular events; it provides little help for dealing with new situations. Our ability and aptitude to recognize similarities in different events, to distil the important factors for a specific purpose, and to generalize our experience enables us to operate effectively in new environments. The result of this skill is knowledge, an essential resource for any intelligent agent.
Tony Hürlimann

Foundations of Modeling

Frontmatter
Chapter 2. What is Modeling?
Abstract
This work is about computer-based mathematical modeling tools. But before we can implement these tools, we have to understand modeling and, above all, mathematical modeling. The purpose of this chapter is to give a precise definition of the term model. The overview will begin with general, unspecified notions, and then proceed to more formal concepts. Finally, a short historical digression will be presented to suggest further arguments for the importance of mathematical modeling.
Tony Hürlimann
Chapter 3. The Modeling Life Cycle
Abstract
In this chapter, the modeling life cycle will briefly be presented. The goal is not to give detailed information for the modeler on how to create and maintain models, but rather to offer a short survey for the modeling tool implementor in order to make her aware of the different tasks in the model building process. Modeling tools should not only support the solution process,but the whole life cycle of a model as well.
Tony Hürlimann
Chapter 4. Model Paradigms
Abstract
The objective of this chapter is to present a brief — by no means complete — overview of different model types and paradigms. It is certainly not the goal to give a thorough account of all types of mathematical models and their solution procedures. That would by itself fill several volumes. I shall present only certain model types and approaches that from my point of view seem important — especially in the light of the ongoing vast unification in mathematical modeling that is taking place at different levels today.
Tony Hürlimann

A General Modeling Framework

Frontmatter
Chapter 5. Problems and Concepts
Abstract
From the more mathematical aspects of modeling, we now switch to the implementation aspects of computer-based modeling tools. Therefore, this chapter is more relevant for the designer of modeling tools than for the modeler or the model user. The previous chapters detail important prerequisites for understanding what modeling is, what components are necessary to implement modeling tools, and what model types should be part of the framework. In a word, from mathematics we switch to computer science. From a computer science point of view, this book could be viewed as the presentation of a large software project: The first four chapters describe the functionality and the “software requirement catalogue” from the user’s point of view. This chapter summarizes these requirements and presents the formal specifications. Finally, Part III presents a concrete implementation; the LPL system.
Tony Hürlimann
Chapter 6. An Overview of Approaches
Abstract
Different approaches and modeling tools have been proposed. A quick tour is given which makes no claim of being complete. Rather the aim is to get an idea of what can be considered as the present state of mathematical modeling.
Tony Hürlimann
Chapter 7. A Modeling Framework
Abstract
After having investigated and criticized various approaches in the last chapter, it is time to present my own view of a computer-based modeling framework. As one might expect, it has been inspired from many sources. The different algebraic languages have had a certain influence, although at least the initial versions of LPL were developed completely independently from any other algebraic language. Structured Modeling too, influenced me, helping to shape the point of view presented in this chapter. More conceptual and implementation specific aspects were influenced by computer science and especially programming language design, as taught by Wirth [Wirth 1996], [Wirth/Gutknecht 1992].
Tony Hürlimann

LPL — An Implemented Framework

Frontmatter
Chapter 8. The Definition of the Language
Abstract
LPL is not a commercial system which needs to compete with other systems. The language is relatively simple and small. There are three reasons why LPL has been developed. In the mid-eighties, when the LPL project began, there were virtually no modeling tools available for personal computers. Yet, people at our Institute of Informatics had to manage real-life, large models. LPL and other tools have been developed and have been used since for this purpose. The second reason was to have computer-based tools for teaching modeling in operations research. I always found it to be an anachronism to teach modeling in OR without a computer. After all, wasn’t OR a child of the advent of computers?
Tony Hürlimann
9. The Implementation
Abstract
In this Chapter, a concrete implementation of LPL, together with the user interface of the Windows NT (and 95) version, is briefly presented. It basically consists of two modules: the compiler-interpreter system which implements the language, and the user interface which implements the communication between the user and the language. The compiler-interpreter system — subsequently called kernel — is completely independent of the user interface and can be used without any changes on other platforms or in other applications. This design principle turned out to be very beneficial when different versions of LPL on MS/DOS, Macintosh and Unix-SUN were produced (see Figure 9–1).
Tony Hürlimann
Chapter 10. Selected Applications
Abstract
In this Chapter, several interesting models and their formulation in LPL are presented. The examples have been chosen in order to give a “round-trip” of LPL and to highlight several stimulating aspects not found in other languages, especially in the field of modeling logical problems. Each problem is briefly described in natural language followed by a formal statement of the model structure. Comments are given on the model structure and the LPL code when appropriate from the point of view of modeling language specification. All problems can be found on the LPL-site (see Availability of the LPL System).
Tony Hürlimann
11. Conclusion
Abstract
Computer-based modeling is not an “ad hoc science” where we could apply the slogan “everything goes”; on the contrary, it is, like software engineering, a discipline which must follow certain criteria of quality. These criteria are reliability to ensure correctness and robustness of models, and transparency to enhance model extendibility, reusability and compatibility. Reliability can be achieved by a unique notation, in which models are coded, by a formal specification of its semantics, and by introducing types, units and other checking mechanisms, in order to enforce domain consistencies. Transparency can be obtained by flexible decomposition techniques which fractionates the whole model into easily maintainable software modules and components.
Tony Hürlimann
Backmatter
Metadaten
Titel
Mathematical Modeling and Optimization
verfasst von
Tony Hürlimann
Copyright-Jahr
1999
Verlag
Springer US
Electronic ISBN
978-1-4757-5793-4
Print ISBN
978-1-4419-4814-4
DOI
https://doi.org/10.1007/978-1-4757-5793-4