# Optimization for Engineering Systems

Ralph W. Pike

Professor of Chemical Engineering and Systems Science

Louisiana State University

Copyright © 2001 Ralph W. Pike

## Preface

Optimization is a many faceted subject ranging from pure mathematics to automated manufacturing. Over time, there has been a flow of optimization concepts and algorithms from mathematics to applications in engineering design and operation of manufacturing plants. A classic example is the Simplex Algorithm of linear programming which is used in a wide variety of industrial and other applications.

The mathematics of optimization capitalizes on the structure of the problems to obtain formal proofs of global and local optimality and to develop efficient algorithms for locating best values of the economic model while satisfying constraints. The Simplex Algorithm and its extensions also illustrate this approach of using the mathematical form of all linear equations to find the global optimum. Other examples are geometric programming, in which the economic model and constraints are polynomials; convex programming, with a concave economic model and convex constraints; and dynamic programming, in which the stage structure is exploited by a series of partial optimizations.

This book stands at the interface of mathematics and industrial applications of optimization. The topics were selected for their breadth of application to the optimization of engineering systems, especially continuous ones. Moreover, the mathematics of optimization has been presented to provide a foundation for those methods that have proven successful in industrial applications.

An informal style of writing has been adopted as that best suited for most students by helping to eliminate mathematical tedium. In addition, a large number of simple examples have been included to emphasize the basics.

The material is structured to build on the students' knowledge of calculus and differential equations by beginning with the classical theory of maxima and minima to establish a base for the modern methods. The progression of topics was designed to add depth and breadth in the concepts and applications. Upon completion of the material the reader should have the necessary background for further reading of texts, monographs, and current research literature on the subject.

The text is a product of the author's experience in teaching and research in optimization, which includes developing and teaching a graduate course on optimization for the past twenty plus years to students in engineering, system science, and business administration. Also, some of the material, however, was developed for continuing education courses taught to practicing engineers.

This book is intended to serve as a text for a first year graduate course in engineering optimization. It is primarily aimed at engineers, but it could serve for a comparable course on operations research in business schools or a mathematical programming course in computer science that emphasizes applications.

The introductory chapter gives a brief historical perspective, the relation of this to other subjects, and an overview of the rationale for the order of presentation of the optimization methods. The second chapter covers analytical methods for unconstrained and constrained problems and serves as a foundation for the subjects that follow. The third chapter which investigates geometric programming, is presented as an extension of analytical methods that introduces the concept of a dual problem. The fourth chapter covers the most widely used optimization technique, linear programming, and includes an illustration of the application of a commercial code to an industrial plant. In the fifth chapter, single-variable search methods based on the minimax concept are given along with a FORTRAN program for Fibonacci search. These techniques used in conjunction with the multivariable search methods described in the sixth chapter, where constrained direct methods are emphasized. In the seventh chapter, the sequential partial optimization procedure of dynamic programming is developed, as are the concepts of resource allocation and optimization through time. The text concludes with a chapter on variational methods that give the important results for obtaining an optimum function rather than an optimum point.

The material provided here is more than adequate for a one semester course; in addition, references are given to books and journals on each chapter topic for further research. The idea was to include subjects that have proved to be valuable for industrial applications rather than to approach the text as a handbook. Moreover, the material was prepared with the idea that the users of optimization procedures would often be employing packaged computer programs such as the relatively sophisticated linear and nonlinear programming codes that are available on large computers, e.g. MPSX and MINOS V. References to available standard computer codes are also provided.

The author wishes to express his appreciation to Louisiana State University, Professor Lautaro Guerra of the Universidad Tecnica Federico Santa Maria, Valparaiso, Chile and the LSU Mining and Mineral Resources Research Institute for assistance in preparing the manuscript. Mr. Paul R. Lanoux prepared the MPSX solution to the refinery linear programming example with the assistance of Mr. Daniel Brignac, and Mr. Perry Bando of the Exxon Refinery in Baton Rouge provided some of the economic data for the linear programming model. Mr. Miguelangel R. Giammattei and Mr. Daniel M. Wu prepared the FORTRAN programs for the single and multivariable search methods. Also, Mr. Daniel M. Wu assisted in preparing the solutions to the problems and examples. The patient and careful preparation of the manuscript by Ms. Ana Elizabeth Lobos, Ms. Clara Marisol Lobos and Ms. Alycia B. Olano was invaluable in converting the draft of this book into a manuscript. Also, thanks are due to students, colleagues, and reviewers for their suggestions, including Mr. George P. Burdell of the Georgia Institute of Technology.

## Before You Begin

The purpose of this page is to clarify the user the arrangement of each chapter.

- Table of Contents page has the links for every Topic and sub Topic in each chapter.
- At the end of each topic or sub topic, There is a link called "top", Which takes you to the contents of that chapter.
- On each page of the chapter, there exist links to previous and following chapters, which take you to the respective chapter.
- In Problems Section of each chapter, At the end of each problem,There is a link called solution, which shows the solution to the corresponding problem.
- The Solution Manual consists of solutions to problems in the TextBook,These are arranged in chapter by chapter.