30079_17.pdf

CHAPTER 17

DESIGN OPTIMIZATION-

AM OVERVIEW

A. Ravindran

Department of Industrial and Manufacturing Engineering

Pennsylvania State University

University Park, Pennsylvania

G. V. Reklaitis

School of Chemical Engineering

Purdue University

West Lafayette, Indiana

17.1 INTRODUCTION

353

17.4 STRUCTURE OF

OPTIMIZATION PROBLEMS

366

17.2 REQUIREMENTS FOR THE

APPLICATION OF

OPTIMIZATION METHODS

17.5 OVERVIEW OF

OPTIMIZATION METHODS

354

368

17.2.1 Defining the System

Boundaries 354

17.2.2 The Performance Criterion 354

17.2.3 The Independent Variables 355

17.2.4 The System Model

17.5.1 Unconstrained Optimization

Methods

368

17.5.2 Constrained Optimization

Methods

369

355

17.5.3 Code Availability

372

17.3 APPLICATIONSOF

OPTIMIZATION IN

ENGINEERING

17.6 SUMMARY

373

356

17.3.1 Design Applications

357

17.3.2 Operations and Planning

Applications

362

17.3.3 Analysis and Data

Reduction Applications

364

17.1 INTRODUCTION

This chapter presents an overview of optimization theory and its application to problems arising in

engineering. In the most general terms, optimization theory is a body of mathematical results and

numerical methods for finding and identifying the best candidate from a collection of alternatives

without having to enumerate and evaluate explicitly all possible alternatives. The process of optim-

ization lies at the root of engineering, since the classical function of the engineer is to design new,

better, more efficient, and less expensive systems, as well as to devise plans and procedures for the

improved operation of existing systems. The power of optimization methods to determie the best

case without actually testing all possible cases comes through the use of a modest level of mathe-

matics and at the cost of performing iterative numerical calculations using clearly defined logical

procedures or algorithms implemented on computing machines. Because of the scope of most engi-

neering applications and the tedium of the numerical calculations involved in optimization algorithms,

the techniques of optimization are intended primarily for computer implementation.

Mechanical Engineers' Handbook, 2nd ed., Edited by Myer Kutz.

17.2 REQUIREMENTS FOR THE APPLICATION OF OPTIMIZATION METHODS

In order to apply the mathematical results and numerical techniques of optimization theory to concrete

engineering problems it is necessary to delineate clearly the boundaries of the engineering system to

be optimized, to define the quantitative criterion on the basis of which candidates will be ranked to

determine the "best," to select the system variables that will be used to characterize or identify

candidates, and to define a model that will express the manner in which the variables are related.

This composite activity constitutes the process of formulating the engineering optimization problem.

Good problem formulation is the key to the success of an optimization study and is to a large degree

an art. It is learned through practice and the study of successful applications and is based on the

knowledge of the strengths, weaknesses, and peculiarities of the techniques provided by optimization

theory.

17.2.1 Defining the System Boundaries

Before undertaking any optimization study it is important to define clearly the boundaries of the

system under investigation. In this context a system is the restricted portion of the universe under

consideration. The system boundaries are simply the limits that separate the system from the re-

mainder of the universe. They serve to isolate the system from its surroundings, because, for purposes

of analysis, all interactions between the system and its surroundings are assumed to be frozen at

selected, representative levels. Since interactions, nonetheless, always exist, the act of defining the

system boundaries is the first step in the process of approximating the real system.

In many situations it may turn out that the initial choice of system boundary is too restrictive. In

order to analyze a given engineering system fully it may be necessary to expand the system bound-

aries to include other subsystems that strongly affect the operation of the system under study. For

instance, suppose a manufacturing operation has a point shop in which finished parts are mounted

on an assembly line and painted in different colors. In an initial study of the paint shop we may

consider it in isolation from the rest of the plant. However, we may find that the optimal batch size

and color sequence we deduce for this system are strongly influenced by the operation of the fabri-

cation department that produces the finished parts. A decision thus has to be made whether to expand

the system boundaries to include the fabrication department. An expansion of the system boundaries

certainly increases the size and complexity of the composite system and thus may make the study

much more difficult. Clearly, in order to make our work as engineers more manageable, we would

prefer as much as possible to break down large complex systems into smaller subsystems that can

be dealt with individually. However, we must recognize that this decomposition is in itself a poten-

tially serious approximation of reality.

17.2.2 The Performance Criterion

Given that we have selected the system of interest and have defined its boundaries, we next need to

select a criterion on the basis of which the performance or design of the system can be evaluated so

that the "best" design or set of operating conditions can be identified. In many engineering appli-

cations, an economic criterion is selected. However, there is a considerable choice in the precise

definition of such a criterion: total capital cost, annual cost, annual net profit, return on investment,

cost to benefit ratio, or net present worth. In other applications a criterion may involve some tech-

nology factors, for instance, minimum production time, maximum production rate, minimum energy

utilization, maximum torque, and minimum weight. Regardless of the criterion selected, in the context

of optimization the "best" will always mean the candidate system with either the minimum or the

maximum value of the performance index.

It is important to note that within the context of the optimization methods, only one critrion or

performance measure is used to define the optimum. It is not possible to find a solution that, say,

simultaneously minimizes cost and maximizes reliability and minimizes energy utilization. This again

is an important simplification of reality, because in many practical situations it would be desirable

to achieve a solution that is "best" with respect to a number of different criteria. One way of treating

multiple competing objectives is to select one criterion as primary and the remaining criteria as

secondary. The primary criterion is then used as an optimization performance measure, while the

secondary criteria are assigned acceptable minimum or maximum values and are treated as problem

constraints. However, if careful considerations were not given while selecting the acceptable levels,

a feasible design that satisfies all the constraints may not exist. This problem is overcome by a

technique called goal programming, which is fast becoming a practical method for handling multiple

criteria. In this method, all the objectives are assigned target levels for achievement and a relative

priority on achieving these levels. Goal programming treats these targets as goals to aspire for and

not as absolute constraints. It then attempts to find an optimal solution that comes as "close as

possible" to the targets in the order of specified priorities. Readers interested in multiple criteria

optimizations are directed to recent specialized texts. 1 ' 2

17.2.3 The Independent Variables

The third key element in formulating a problem for optimization is the selection of the independent

variables that are adequate to characterize the possible candidate designs or operating conditions of

the system. There are several factors that must be considered in selecting the independent variables.

First, it is necessary to distinguish between variables whose values are amenable to change and

variables whose values are fixed by external factors, lying outside the boundaries selected for the

system in question. For instance, in the case of the paint shop, the types of parts and the colors to

be used are clearly fixed by product specifications or customer orders. These are specified system

parameters. On the other hand, the order in which the colors are sequenced is, within constraints

imposed by the types of parts available and inventory requirements, an independent variable that can

be varied in establishing a production plan.

Furthermore, it is important to differentiate between system parameters that can be treated as

fixed and those that are subject to fluctuations which are influenced by external and uncontrollable

factors. For instance, in the case of the paint shop, equipment breakdown and worker absenteeism

may be sufficiently high to influence the shop operations seriously. Clearly, variations in these key

system parameters must be taken into account in the production planning problem formulation if the

resulting optimal plan is to be realistic and operable.

Second, it is important to include in the formulation all of the important variables that influence

the operation of the system or affect the design definition. For instance, if in the design of a gas

storage system we include the height, diameter, and wall thickness of a cylindrical tank as independent

variables, but exclude the possibility of using a compressor to raise the storage pressure, we may

well obtain a very poor design. For the selected fixed pressure we would certainly find the least cost

tank dimensions. However, by including the storage pressure as an independent variable and adding

the compressor cost to our performance criterion, we could obtain a design that has a lower overall

cost because of a reduction in the required tank volume. Thus, the independent variables must be

selected so that all important alternatives are included in the formulation. Exclusion of possible

alternatives, in general, will lead to suboptimal solutions.

Finally, a third consideration in the selection of variables is the level of detail to which the system

is considered. While it is important to treat all of the key independent variables, it is equally important

not to obscure the problem by the inclusion of a large number of fine details of subordinate impor-

tance. For instance, in the preliminary design of a process involving a number of different pieces of

equipment—pressure vessels, towers, pumps, compressors, and heat exchangers—one would nor-

mally not explicitly consider all of the fine details of the design of each individual unit. A heat

exchanger may well be characterized by a heat-transfer surface area as well as shell-side and tube-

side pressure drops. Detailed design variables such as number and size of tubes, number of tube and

shell passes, baffle spacing, header type, and shell dimensions would normally be considered in a

separate design study involving that unit by itself. In selecting the independent variables a good rule

to follow is to include only those variables that have a significant impact on the composite system

performance criterion.

17.2.4 The System Model

Once the performance criterion and the independent variables have been selected, then the next step

in problem formulation is the assembly of the model that describes the manner in which the problem

variables are related and the performance criterion is influenced by the independent variables. In

principle, optimization studies may be performed by experimenting directly with the system. Thus,

the independent variables of the system or process may be set to selected values, the system operated

under those conditions, and the system performance index evaluated using the observed performance.

The optimization methodology would then be used to predict improved choices of the independent

variable values and the experiments continued in this fashion. In practice most optimization studies

are carried out with the help of a model, a simplified mathematical representation of the real system.

Models are used because it is too expensive or time consuming or risky to use the real system to

carry out the study. Models are typically used in engineering design because they offer the cheapest

and fastest way of studying the effects of changes in key design variables on system performance.

In general, the model will be composed of the basic material and energy balance equations,

engineering design relations, and physical property equations that describe the physical phenomena

taking place in the system. These equations will normally be supplemented by inequalities that define

allowable operating ranges, specify minimum or maximum performance requirements, or set bounds

on resource availabilities. In sum, the model consists of all of the elements that normally must be

considered in calculating a design or in predicting the performance of an engineering system. Quite

clearly the assembly of a model is a very time-consuming activity, and it is one that requires a

thorough understanding of the system being considered. In simple terms, a model is a collection of

equations and inequalities that define how the system variables are related and that constrain the

variables to take on acceptable values.

From the preceding discussion, we observe that a problem suitable for the application of optim-

ization methodology consists of a performance measure, a set of independent variables, and a model

relating the variables. Given these rather general and abstract requirements, it is evident that the

methods of optimization can be applied to a very wide variety of applications. We shall illustrate

next a few engineering design applications and their model formulations.

17.3 APPLICATIONS OF OPTIMIZATION IN ENGINEERING

Optimization theory finds ready application in all branches of engineering in four primary areas:

1. Design of components of entire systems.

2. Planning and analysis of existing operations.

3. Engineering analysis and data reduction.

4. Control of dynamic systems.

In this section we briefly consider representative applications from the first three areas.

In considering the application of optimization methods in design and operations, the reader should

keep in mind that the optimization step is but one step in the overall process of arriving at an optimal

design or an efficient operation. Generally, that overall process will, as shown in Fig. 17.1, consist

of an iterative cycle involving synthesis or definition of the structure of the system, model formulation,

model parameter optimization, and analysis of the resulting solution. The final optimal design or new

operating plan will be obtained only after solving a series of optimization problems, the solution to

each of which will have served to generate new ideas for further system structures. In the interest of

brevity, the examples in this section show only one pass of this iterative cycle and focus mainly on

preparations for the optimization step. This focus should not be interpreted as an indication of the

ENGINEERING DESIGN

RECOGNITION OF

NEEDS AND RESOURCES

-^><^CISIONS><i-

PROBLEM DEFINITION •*

^^<DECISIONS>^-

MODEL DEVELOPMENT •*

^^<^DECIS!ONS>^-

^"""^^-^ ^

\ ANALYSIS \*

<DECISIONS>^-

-^<CDECISIONS>^-

OPTIMIZATION

COMPUTATION *-<DKISIONS>^

Fig. 17.1 Optimal design process.

dominant role of optimization methods in the engineering design and systems analysis process. Op-

timization theory is but a very powerful tool that, to be effective, must be used skillfully and intel-

ligently by an engineer who thoroughly understands the system under study. The primary objective

of the following example is simply to illustrate the wide variety but common form of the optimization

problems that arise in the design and analysis process.

17.3.1 Design Applications

Applications in engineering design range from the design of individual structural members to the

design of separate pieces of equipment to the preliminary design of entire production facilities. For

purposes of optimization the shape or structure of the system is assumed known and optimization

problem reduces to the selection of values of the unit dimensions and operating variables that will

yield the best value of the selected performance criterion.

Example 17.1 Design of an Oxygen Supply System

Description. The basic oxygen furnace (BOF) used in the production of steel is a large fed-

batch chemical reactor that employs pure oxygen. The furnace is operated in a cyclic fashion: ore

and flux are charged to the unit, are treated for a specified time period, and then are discharged. This

cyclic operation gives rise to a cyclically varying demand rate for oxygen. As shown in Fig. 17.2,

over each cycle there is a time interval of length t l of low demand rate, D 0 , and a time interval

O 2 - J 1 ) of high demand rate, D 1 . The oxygen used in the BOF is produced in an oxygen plant.

Oxygen plants are standard process plants in which oxygen is separated from air using a combination

of refrigeration and distillation. These are highly automated plants, which are designed to deliver a

fixed oxygen rate. In order to mesh the continuous oxygen plant with the cyclically operating BOF,

a simple inventory system shown in Fig. 17.3 and consisting of a compressor and a storage tank

must be designed. A number of design possibilities can be considered. In the simplest case, one

could select the oxygen plant capacity to be equal to D 1 , the high demand rate. During the low-

demand interval the excess oxygen could just be vented to the air. At the other extreme, one could

also select the oxygen plant capacity to be just enough to produce the amount of oxygen required

by the BOF over a cycle. During the low-demand interval, the excess oxygen production would then

be compressed and stored for use during the high-demand interval of the cycle. Intermediate designs

could involve some combination of venting and storage of oxygen. The problem is to select the

optimal design.

Formulation. The system of concern will consist of the O 2 plant, the compressor, and the storage

tank. The BOF and its demand cycle are assumed fixed by external factors. A reasonable performance

index for the design is the total annual cost, which consists of the oxygen production cost (fixed and

variable), the compressor operating cost, and the fixed costs of the compressor and of the storage

Fig. 17.2 Oxygen demand cycle.

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