Introduction

There are many problems where the variables are not divisible as fractions. Some examples are the number of operators that can be assigned to jobs, the number of airplanes that can be purchased, the number of plants operating, and so on. These problems form an important class called integer programming problems. Further, several decision problems involve binary variables that take the values 0 or 1. If some variables must be integers and others allowed to take fractional values, the problem is of the mixed integer type. Discrete programming problems are those problems where the variables are to be chosen from a discrete set of available sizes, such as available shaft sizes and beam sections, engine capacities, pump capacities.

In integer programming problems, treating the variable as continuous and rounding off the optimum solution to the nearest integer is easily justified if the involved quantities are large. Consider the example of the number of barrels of crude processed where the solution results in 234566.4 barrels. This can be rounded to 234566 barrels or even 234570 without significantly altering the characteristics of the solution. When the variable values are small, such as the number of aircraft in a fleet, rounding is no longer intuitive and may not even yield a feasible solution. In problems with binary variables, rounding makes no sense as the choice between 0 or 1 is a choice between two entirely different decisions. The following two variable example provides some characteristics of integer requirements.

Introduction

Optimization is the process of maximizing or minimizing a desired objective function while satisfying the prevailing constraints. Nature has an abundance of examples where an optimum system status is sought. In metals and alloys, the atoms take positions of least energy to form unit cells. These unit cells define the crystalline structure of materials. A liquid droplet in zero gravity is a perfect sphere, which is the geometric form of least surface area for a given volume. Tall trees form ribs near the base to strengthen them in bending. The honeycomb structure is one of the most compact packaging arrangements. Genetic mutation for survival is another example of nature's optimization process. Like nature, organizations and businesses have also strived toward excellence. Solutions to their problems have been based mostly on judgment and experience. However, increased competition and consumer demands often require that the solutions be optimum and not just feasible solutions. A small savings in a mass-produced part will result in substantial savings for the corporation. In vehicles, weight minimization can impact fuel efficiency, increased payloads, or performance. Limited material or labor resources must be utilized to maximize profit. Often, optimization of a design process saves money for a company by simply reducing the developmental time.

Introduction

Finite elements is a well known tool for analysis of structures. If we are given a structure, such as an airplane wing, a building frame, a machine component, etc., together with loads and boundary conditions, then finite elements can be used to determine the deformations and stresses in the structure. Finite elements can also be applied to analyze dynamic response, heat conduction, fluid flow, and other phenomena. Mathematically, it may be viewed as a numerical tool to analyze problems governed by partial differential equations describing the behavior of the system. Lucien Schmit in 1960 recognized the potential for combining optimization techniques in structural design [Schmit 1960]. Today, various commercial finite element packages have started to include some optimization capability in their codes. In the aircraft community, where weight and performance is a premium, special codes combining analysis and optimization have been developed. Some of the commercial codes are listed at the end of Chapter 1. The methodology discussed in this chapter assumes that a discretized finite element (or a boundary element) model exists. This is in contrast with classical approaches to optimization of structures that directly work with the differential equations governing equilibrium and aim for an analytical solution to the optimization problem.

This book is a revised and enhanced edition of the first edition. The authors have identified a clear need for teaching engineering optimization in a manner that integrates theory, algorithms, modeling, and hands-on experience based on their extensive experience in teaching, research, and interactions with students. They have strived to adhere to this pedagogy and reinforced it further in the second edition, with more detailed explanations, an increased number of solved examples and end-of-chapter problems, and source codes on multiple platforms.

The development of the software, which parallels the theory, has helped to explain the implementation aspects in the text with greater insight and accuracy. Students have integrated the optimization programs with simulation codes in their theses. The programs can be tried out by researchers and practicing engineers as well. Programs on the CD-ROM have been developed in Matlab, Excel VBA, VBScript, and Fortran. A battery of methods is available for the user. This leads to effective solution of problems since no single method can be successful on all problems.

The book deals with a variety of optimization problems: unconstrained, constrained, gradient, and nongradient techniques; duality concepts; multiobjective optimization; linear, integer, geometric, and dynamic programming with applications; and finite element–based optimization. Matlab graphics and optimization toolbox routines and the Excel Solver optimizer are presented in detail.

Introduction

Determination of the minimum of a real valued function of one variable, and the location of that minimum, plays an important role in nonlinear optimization. A one-dimensional minimization routine may be called several times in a multivariable problem. We show later that at the minimum point of a sufficiently smooth function, the slope is zero. If the slope and curvature information is available, minimum may be obtained by finding the location where the slope is zero and the curvature is positive. The need to determine the zero of a function occurs frequently in nonlinear optimization. Reliable and efficient ways of finding the minimum or a zero of a function are necessary for developing robust techniques for solving multivariable problems. We present the basic concepts involved in single variable minimization and zero finding.

Theory Related to Single Variable (Univariate) Minimization

We present the minimization ideas by considering a simple example. The first step is to determine the objective function that is to be optimized.