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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
17.2.1 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
17.5.2 Constrained Optimization
17.5.3 Code Availability
17.3.1 Design Applications
17.3.2 Operations and Planning
17.3.3 Analysis and Data
Reduction Applications
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.
ISBN 0-471-13007-9 © 1998 John Wiley & Sons, Inc.
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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
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.
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
^"""^^-^ ^
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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