Applied Optimization with MATLAB Programming - Hardcover

Venkataraman, P.

Hardcover

ISBN 10: 047008488X ISBN 13: 9780470084885

Publisher: Wiley, 2009

Synopsis

Technology/Engineering/Mechanical

Provides all the tools needed to begin solving optimization problems using MATLAB®

The Second Edition of Applied Optimization with MATLAB® Programming enables readers to harness all the features of MATLAB® to solve optimization problems using a variety of linear and nonlinear design optimization techniques. By breaking down complex mathematical concepts into simple ideas and offering plenty of easy-to-follow examples, this text is an ideal introduction to the field. Examples come from all engineering disciplines as well as science, economics, operations research, and mathematics, helping readers understand how to apply optimization techniques to solve actual problems.

This Second Edition has been thoroughly revised, incorporating current optimization techniques as well as the improved MATLAB® tools. Two important new features of the text are:

• Introduction to the scan and zoom method, providing a simple, effective technique that works for unconstrained, constrained, and global optimization problems

• New chapter, Hybrid Mathematics: An Application, using examples to illustrate how optimization can develop analytical or explicit solutions to differential systems and data-fitting problems

Each chapter ends with a set of problems that give readers an opportunity to put their new skills into practice. Almost all of the numerical techniques covered in the text are supported by MATLAB® code, which readers can download on the text's companion Web site www.wiley.com/go/venkat2e and use to begin solving problems on their own.

This text is recommended for upper-level undergraduate and graduate students in all areas of engineering as well as other disciplines that use optimization techniques to solve design problems.

About this title

About the Author

P. Venkataraman, PhD, is an associate professor in the Mechanical Engineering Department, Rochester Institute of Technology, Rochester, New York.

From the Back Cover

Technology/Engineering/Mechanical

Provides all the tools needed to begin solving optimization problems using MATLAB®

The Second Edition of Applied Optimization with MATLAB® Programming enables readers to harness all the features of MATLAB® to solve optimization problems using a variety of linear and nonlinear design optimization techniques. By breaking down complex mathematical concepts into simple ideas and offering plenty of easy-to-follow examples, this text is an ideal introduction to the field. Examples come from all engineering disciplines as well as science, economics, operations research, and mathematics, helping readers understand how to apply optimization techniques to solve actual problems.

This Second Edition has been thoroughly revised, incorporating current optimization techniques as well as the improved MATLAB® tools. Two important new features of the text are:

• Introduction to the scan and zoom method, providing a simple, effective technique that works for unconstrained, constrained, and global optimization problems

• New chapter, Hybrid Mathematics: An Application, using examples to illustrate how optimization can develop analytical or explicit solutions to differential systems and data-fitting problems

Each chapter ends with a set of problems that give readers an opportunity to put their new skills into practice. Almost all of the numerical techniques covered in the text are supported by MATLAB® code, which readers can download on the text's companion Web site www.wiley.com/go/venkat2e and use to begin solving problems on their own.

This text is recommended for upper-level undergraduate and graduate students in all areas of engineering as well as other disciplines that use optimization techniques to solve design problems.

From the Inside Flap

Technology/Engineering/Mechanical

Provides all the tools needed to begin solving optimization problems using MATLAB®

The Second Edition of Applied Optimization with MATLAB® Programming enables readers to harness all the features of MATLAB® to solve optimization problems using a variety of linear and nonlinear design optimization techniques. By breaking down complex mathematical concepts into simple ideas and offering plenty of easy-to-follow examples, this text is an ideal introduction to the field. Examples come from all engineering disciplines as well as science, economics, operations research, and mathematics, helping readers understand how to apply optimization techniques to solve actual problems.

This Second Edition has been thoroughly revised, incorporating current optimization techniques as well as the improved MATLAB® tools. Two important new features of the text are:

• Introduction to the scan and zoom method, providing a simple, effective technique that works for unconstrained, constrained, and global optimization problems

• New chapter, Hybrid Mathematics: An Application, using examples to illustrate how optimization can develop analytical or explicit solutions to differential systems and data-fitting problems

Each chapter ends with a set of problems that give readers an opportunity to put their new skills into practice. Almost all of the numerical techniques covered in the text are supported by MATLAB® code, which readers can download on the text's companion Web site www.wiley.com/go/venkat2e and use to begin solving problems on their own.

This text is recommended for upper-level undergraduate and graduate students in all areas of engineering as well as other disciplines that use optimization techniques to solve design problems.

Excerpt. © Reprinted by permission. All rights reserved.

Applied Optimazation with MATLAB Programming

By P. Venkataraman

John Wiley & Sons

Copyright © 2008 P. Venkataraman
All right reserved.
ISBN: 978-0-470-08488-5

Chapter One

INTRODUCTION

Optimization is, in essence, a search for the best objective when operating within a set of constraints. For example you want the lightest car that can at least deliver standard performance and possess good safety features while being powered a hybrid engine. Maybe you want to generate maximum shareholder return on investment while meeting the 10 percent sales growth and holding inventory at the minimum. Sometimes you want to minimize commuting time by choosing alternate routes during peak traffic periods. You can come up with several examples of this kind from everyday experience. What you have done is loosely define an optimization problem that needs a solution. The procedure by which you will establish a solution to the above examples uses optimization techniques. A second read through these sentences will suggest you are really talking about designs that are qualified by the words lightest, maximum, minimize. All of these, and many others like it can be described by the word optimum. In addition, these designs must be within a certain envelope or satisfy certain conditions or must be limited, if they are to be acceptable. These are the constraints on the design. With this basic definition you can identify problems of design optimization. Most of the book is about formally expressing these kinds of problems and looking at several techniques that can be employed to establish the solution.

Optimization is an essential part of design activity in all major disciplines. These disciplines are not restricted to engineering. In product development, competition demands producing economically relevant products with embedded quality. Today, globalization demands that additional dimensions such as location, language, and expertise must also merit consideration as new constraints in the development process. Improved production and design tools coupled with inexpensive computational resources have made optimization an important part of the process. Even in the absence of a tangible product, optimization ideas provide the ability to define and explore problems while focusing on solutions that subscribe to some measure of usefulness. Generally, the use of the word optimization implies the best result under the circumstances. This includes the particular set of constraints on the development resources, current knowledge, market conditions, and so on. The ability to make the best choice is a perpetual desire among us all. The techniques that are used in optimization are also used for obtaining solutions to nonlinear problems in many disciplines-so the subject has an attraction to a wider audience from many fields.

In this book, optimization is often associated with design, be it a product, service or a strategy. Aerospace design was among the earliest disciplines to embrace optimization in a significant way driven by a natural need to lower the tremendous cost associated with carrying unnecessary weight in aerospace vehicles. Minimum mass structures are the norm. It forms part of the psyche of every aerospace designer. Just recently, Boeing introduced the Dreamliner to the world. The aircraft uses more than 50% plastic in its structure, replacing metals, without compromising on safety. Today, an emerging discipline is multidisciplinary optimization (MDO) of greater significance to aerospace designers, where structural design and aerodynamics are combined to produce an optimal vehicle shape. A good example is the Boeing blended wing-body design. Saving on fuel through trajectory design was another problem that suggested itself. Very soon, the entire engineering community could recognize the need to define solutions based on merit.

Recognizing the desire for optimization and actually implementing it has taken some time to mature. In the past, optimization was usually attempted only in those situations where there were significant penalties for generic designs. The application of optimization demanded large computational resources. In the nascent years of digital computation, these were available only to large national laboratories and programs. These resources were necessary to handle the nonlinear problems that are associated with engineering optimization. As a result of these constraints most of the everyday products were designed without regard to optimization.

Today, it is inconceivable that current replacement products, such as the car, the house, the desk, the pencil, and so on, are not designed optimally in some sense or another. The most obvious contemporary example of the use of optimization manifests in the same streamlined bubblelike look in all cars from all manufacturers. Minimizing drag coefficient, particularly through software, improves fuel consumption. This is an easier option for meeting fuel consumption standards without requiring a change in the engine performance, which has proved very stubborn.

Today, you would definitely explore procedures to optimize your investments by tailoring your portfolio. You would optimize your business travel time by appropriately choosing your destinations. You can optimize your commuting time by choosing your time and route. You can optimize your necessary expenditure for living by choosing your day and store for shopping. You can buy software that will optimize your connection to the Internet. You can have affordable access to resources that will allow you to perform all these various optimizations. Seeing the variety of problems that need to generate optimum solutions, the study of optimization is actually more of a tool that can be applied to a variety of disciplines. If so, all of the optimization problems from many disciplines should be described in a common way.

The partnership between design and optimization activity is still more often found in engineering. This book recognizes that connection. Much of the problems used for illustrations and practice are from engineering, primarily mechanical, civil, and aerospace design. Nevertheless, the study of optimization, particularly applied optimization, is not an exclusive property of any specific discipline. It involves the discovery and design of solutions through appropriate techniques associated with the formulation of the problem in a specific manner. This can be done for example, in economics, chemistry, and business management. A Google search on "optimization of", at the time of this writing, resulted in 128 million hits, the first six of them related to optimization in digital circuits, trading system, immunomagnetic separation, object-oriented programming, risk measures, and chemical process.

1.1 OPTIMIZATION FUNDAMENTALS

Optimization was described as the process of search for the solution that is more useful than several others. Qualitatively, this assertion implicitly recognizes the necessity of choosing among alternatives. This book deals with optimization in a quantitative way. This means that an outcome of applying optimization techniques to the problem, design, or service must yield numbers that will define our solution-in other words, numbers or values that will characterize the particular design or service. Quantitative description of the solution requires a quantitative description of the problem itself. This description is called a mathematical model. The design, its characterization, and its circumstances must be expressed mathematically prior to the application of the optimization methods. In many situations, coming up with a mathematical model will prove to be very challenging. The development of a suitable mathematical model presupposes knowledge of content in the particular design area that the optimization problem is being formulated. Consider the design activity in the following cases:

1. New consumer research, with deference to the obesity problem among the general population, suggests that people should drink no more than about 0.25 liter of soda pop at a time. The fabrication cost of the redesigned soda can is proportional to the surface area, and can be estimated at $1.00 per square centimeter of the material used. A circular cross-section is the most plausible, given current tooling available for manufacture. For aesthetic reason, the height must be at least twice the diameter. Studies indicate that holding comfort requires a diameter between 5 and 8 cm. Create a design that will cost the least.

2. Design a cantilevered beam, of minimum mass, carrying a point load F at the end of the beam of length L. The cross-section of the beam will be in the shape of the letter I (referred to as an I-beam). The beam should be sufficiently strong in bending and shear. There is also a limit on its deflection.

3. MyPC Company has decided to invest $12 million in acquiring several new component placement machines to manufacture different kinds of motherboards for a new generation of personal computers. Three models of these machines are under consideration. Total number of operators available is 100 because of the local labor market. A floor-space constraint needs to be satisfied because of the different dimensions of these machines. Additional information relating to each of the machines is given in Table 1.1. The company wishes to determine how many of each kind is appropriate to maximize the number of boards manufactured per day.

4. The first-order differential equation (t + 1)dy/dt - (t + 2)y = 0, subject to the initial condition y(0) = 1, can be solved analytically using the power series method. Set up an alternate procedure using optimization.

This list represents four problems that will be used to define the elements of problem formulation. Each problem requires information from the specific area or discipline it refers to. To recognize or design these problems assumes that the designer is conversant with the particular subject matter. Such kinds of problem are quite common, and they are expressed in far greater detail than formulated here. The problems are kept simple to focus on optimization issues. The last example is included to show that optimization ideas and techniques can migrate very well to solve standard problems in a nontraditional manner. In fact, with creative formulation, you should be able to solve most problems through optimization. Optimization also provides an opportunity to establish useful values for designs that are unique or one of a kind.

1.1.1 Elements of Problem Formulation

In this section, we will introduce the formal elements of the optimization problem. Please keep in mind that optimization presupposes the knowledge of the design rules for the specific problem, primarily the ability to describe the design in mathematical terms. For engineering problems this means that the designer is aware of the relevant physics from the particular area, including all of the expressions and techniques used in mathematical analysis of the design. The terms in optimization include design variables, design parameters, and design functions. Traditional design practice-that is, design without regard to optimization, includes all of these elements, except it is not necessary to formally recognize them as such. It is also a good idea to recognize that optimization is a procedure for searching the best design among candidates, each of which can produce an acceptable product.

Design Variables: Design variables are entities that define a particular design. The values of a complete set of these variables will establish a specific design. In the search for the optimal design, the values of these entities will change over a prescribed range, hence the tag variables. The number of these design variables used to be very significant in the early days of optimization, with the recommendation that the set be as small a possible. This prohibition was related to available computational resources and is no longer a limitation in applied optimization. The type of these variables, continuous, or discrete, or integer, or mixed, is important in identifying and setting up the quantitative optimal design problem and the optimization procedure. It is crucial that this choice capture the essence of the object being designed and at the same time provide a quantitative characterization of the design problem. We will develop the elements of optimization assuming the variables are continuous. In applied mathematical terminology, design variables serve as the unknowns of the problem being solved. Using an analogy from the area of system dynamics and control theory, they are equivalent to defining the state of the system-in this case, the state of design. Typically, design variables can be associated with describing the size of the object, like length and height. In other cases, they may represent the number of items. The choice of design variables is the responsibility of the designer guided by intuition, expertise, and knowledge. There is a fundamental requirement to be met by this set of design variables. They must be linearly independent. This means that you cannot establish the value of one of the design variables viathe values of the remaining variables through basic arithmetic (scaling or addition) operations. For example, in a design having a rectangular cross-section, you cannot have three variables representing the length, height, and area. If the first two are prescribed, the third is automatically established. In complex designs, these relationships may not be very apparent. Nevertheless, the choice of the set of design variables must meet the criteria of linear independence for applying the techniques of optimization. Many of these techniques are borrowed from linear algebra, where this property is necessary for a solution. From a practical perspective, the property of linear independence identifies a minimum set of variables that can completely describe the design. This is significant because the effort in obtaining the solution varies as an integer power of the number of variables, and this power is typically greater than two.

The set of design variables is identified as the design vector. This vector will be column vector in this book. In fact, all vectors are column vectors in this book, unless indicated otherwise. The length of this vector, which is n, is the number of design variables in the problem. The design variables can express different dimensional quantities in the problem, but in the mathematical model, they are distinguished by the lowercase x for the variable and the uppercase X for the vector. All the techniques of optimization in the book are based on the generic mathematical model. The subscript on x, for example, [x.sub.3] represents the third design variable, which may be the height of an object in the characterization of the product. This abstract model is only necessary for mathematical convenience. This book will refer to the design variables in one of four ways:

1. [X]: (the square parenthesis defines a vector) the vector of design variables.

2. X or x (without subscripts): referring to the vector again, omitting the square brackets for convenience if appropriate.

3. [[[x.sub.1], [x.sub.2], ..., [x.sub.n]].sup.t]: indicating the vector through its elements. Note the transposition symbol [sup.t] to identify it as a column vector.

4. [x.sub.i], i = 1, 2, ..., n: referring to all the elements of the design vector.

The above notational convenience is extended to all vectors in the book. Once again, vectors refer to a collection or a related set of values.

Design Parameters: In this book, design parameters identify constants that will not change as different designs are generated and compared during optimization. Expressing the design and its properties requires more than design variables. Please be aware that many texts use the term design parameters to represent the design variables we defined earlier and do not formally recognize design parameters as defined here. The principal reason is that parameters have no role to play in determining the optimal design. They are significant in the discussion of modeling issues. The book will draw attention to these issues when the occasion arises. Examples of parameters include material property, applied loads, and choice of shape. The parameters in the generic mathematical model are recognized similar to the design vector, except that we use the character p. Therefore [p.sub.1]], P, p, [[p.sub.1], [p.sub.2], ..., [p.sub.q]] represent th Note the length of the parameter vector is q. It is important to restate that except in the discussion of modeling, the parameters will not be explicitly referred, as they are primarily predetermined constants in the evaluation of the design.

(Continues...)

Excerpted from Applied Optimazation with MATLAB Programmingby P. Venkataraman Copyright © 2008 by P. Venkataraman. Excerpted by permission.
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Edition details

• Publisher: Wiley
• Publication date: 2009
• ISBN 10: 047008488X
• ISBN 13: 9780470084885
• Binding: Hardcover
• Language: English
• Edition number: 2
• Number of pages: 544