Tài liệu DESIGN OPTIMIZATIONAM OVERVIEW ppt

Ravindran Department of Industrial and Manufacturing Engineering Pennsylvania State University University Park, Pennsylvania 17.2.2 The Performance Criterion 354 17.2.3 The Independent V

17.1 INTRODUCTION

This chapter presents an overview of optimization theory and its application to problems arising inengineering In the most general terms, optimization theory is a body of mathematical results andnumerical methods for finding and identifying the best candidate from a collection of alternativeswithout 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 theimproved operation of existing systems The power of optimization methods to determie the bestcase 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 logicalprocedures 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

CHAPTER 17

DESIGN

OPTIMIZATION-AM OVERVIEW

A Ravindran

Department of Industrial and Manufacturing Engineering

Pennsylvania State University

University Park, Pennsylvania

17.2.2 The Performance Criterion 354

17.2.3 The Independent Variables 355

17.2.4 The System Model 355

OPTIMIZATION METHODS 368

17.5.1 Unconstrained OptimizationMethods 36817.5.2 Constrained OptimizationMethods 36917.5.3 Code Availability 372

17.6 SUMMARY 373

17.2 REQUIREMENTS FOR THE APPLICATION OF OPTIMIZATION METHODS

In order to apply the mathematical results and numerical techniques of optimization theory to concreteengineering 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 todetermine the "best," to select the system variables that will be used to characterize or identifycandidates, 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 theknowledge of the strengths, weaknesses, and peculiarities of the techniques provided by optimizationtheory

17.2.1 Defining the System Boundaries

Before undertaking any optimization study it is important to define clearly the boundaries of thesystem under investigation In this context a system is the restricted portion of the universe underconsideration 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 atselected, representative levels Since interactions, nonetheless, always exist, the act of defining thesystem 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 Inorder 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 Forinstance, 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 mayconsider it in isolation from the rest of the plant However, we may find that the optimal batch sizeand 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 expandthe system boundaries to include the fabrication department An expansion of the system boundariescertainly increases the size and complexity of the composite system and thus may make the studymuch more difficult Clearly, in order to make our work as engineers more manageable, we wouldprefer 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 tially serious approximation of reality

poten-17.2.2 The Performance Criterion

Given that we have selected the system of interest and have defined its boundaries, we next need toselect a criterion on the basis of which the performance or design of the system can be evaluated sothat 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 precisedefinition 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 energyutilization, 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 treatingmultiple competing objectives is to select one criterion as primary and the remaining criteria assecondary The primary criterion is then used as an optimization performance measure, while thesecondary criteria are assigned acceptable minimum or maximum values and are treated as problemconstraints 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 relativepriority on achieving these levels Goal programming treats these targets as goals to aspire for andnot as absolute constraints It then attempts to find an optimal solution that comes as "close aspossible" to the targets in the order of specified priorities Readers interested in multiple criteriaoptimizations are directed to recent specialized texts.'

17.2.3 The Independent Variables

The third key element in formulating a problem for optimization is the selection of the independentvariables that are adequate to characterize the possible candidate designs or operating conditions ofthe 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 andvariables whose values are fixed by external factors, lying outside the boundaries selected for thesystem 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 systemparameters On the other hand, the order in which the colors are sequenced is, within constraintsimposed 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 asfixed and those that are subject to fluctuations which are influenced by external and uncontrollablefactors For instance, in the case of the paint shop, equipment breakdown and worker absenteeismmay be sufficiently high to influence the shop operations seriously Clearly, variations in these keysystem parameters must be taken into account in the production planning problem formulation if theresulting optimal plan is to be realistic and operable

Second, it is important to include in the formulation all of the important variables that influencethe operation of the system or affect the design definition For instance, if in the design of a gasstorage system we include the height, diameter, and wall thickness of a cylindrical tank as independentvariables, but exclude the possibility of using a compressor to raise the storage pressure, we maywell obtain a very poor design For the selected fixed pressure we would certainly find the least costtank dimensions However, by including the storage pressure as an independent variable and addingthe compressor cost to our performance criterion, we could obtain a design that has a lower overallcost because of a reduction in the required tank volume Thus, the independent variables must beselected so that all important alternatives are included in the formulation Exclusion of possiblealternatives, 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 importantnot 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 ofequipment—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 heatexchanger 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 andshell passes, baffle spacing, header type, and shell dimensions would normally be considered in aseparate 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 systemperformance 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 problemvariables are related and the performance criterion is influenced by the independent variables Inprinciple, 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 operatedunder 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 independentvariable 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 tocarry out the study Models are typically used in engineering design because they offer the cheapestand 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 phenomenataking place in the system These equations will normally be supplemented by inequalities that defineallowable 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 beconsidered in calculating a design or in predicting the performance of an engineering system Quiteclearly the assembly of a model is a very time-consuming activity, and it is one that requires athorough understanding of the system being considered In simple terms, a model is a collection ofequations and inequalities that define how the system variables are related and that constrain thevariables to take on acceptable values

From the preceding discussion, we observe that a problem suitable for the application of ization methodology consists of a performance measure, a set of independent variables, and a modelrelating the variables Given these rather general and abstract requirements, it is evident that themethods of optimization can be applied to a very wide variety of applications We shall illustratenext a few engineering design applications and their model formulations.

optim-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 shouldkeep in mind that the optimization step is but one step in the overall process of arriving at an optimaldesign 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 newoperating plan will be obtained only after solving a series of optimization problems, the solution toeach of which will have served to generate new ideas for further system structures In the interest ofbrevity, the examples in this section show only one pass of this iterative cycle and focus mainly onpreparations for the optimization step This focus should not be interpreted as an indication of the

ENGINEERING DESIGN I RECOGNITION OF NEEDS AND RESOURCES -^><^CISIONS><i-

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

Op-of the following example is simply to illustrate the wide variety but common form Op-of the optimizationproblems 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 thedesign of separate pieces of equipment to the preliminary design of entire production facilities Forpurposes of optimization the shape or structure of the system is assumed known and optimizationproblem reduces to the selection of values of the unit dimensions and operating variables that willyield 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: oreand flux are charged to the unit, are treated for a specified time period, and then are discharged Thiscyclic 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, D0, and a time interval

O2 - J1) of high demand rate, D1 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 afixed 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 tankmust be designed A number of design possibilities can be considered In the simplest case, onecould select the oxygen plant capacity to be equal to D1, the high demand rate During the low-demand interval the excess oxygen could just be vented to the air At the other extreme, one couldalso 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 designscould involve some combination of venting and storage of oxygen The problem is to select theoptimal design

Formulation The system of concern will consist of the O2 plant, the compressor, and the storagetank The BOF and its demand cycle are assumed fixed by external factors A reasonable performanceindex for the design is the total annual cost, which consists of the oxygen production cost (fixed andvariable), the compressor operating cost, and the fixed costs of the compressor and of the storage

Fig 17.2 Oxygen demand cycle.

Fig 17.3 Design of oxygen production system.

vessel The key independent variables are the oxygen plant production rate F (Ib O2/hr), the

com-pressor and storage tank design capacities, H (hp) and V (ft3), respectively, and the maximum tank

pressure, p (psia) Presumably the oxygen plant design is standard, so that the production rate fully

characterizes the plant Similarly, we assume that the storage tank will be of a standard designapproved for O2 service

The model will consist of the basic design equations that relate the key independent variables

If /max is the maximum amount of oxygen that must be stored, then using the corrected gas law

we have

V = % - z (17.1)

M p where R = the gas constant

T = the gas temperature (assume fixed)

z = the compressibility factor

M = the molecular weight of O2

From Fig 17.1, the maximum amount of oxygen that must be stored is equal to the area under

the demand curve between t l and t 2 and D 1 and F Thus,

/^x = O )1- F X f2- O (17.2)Substituting (17.2) into (17.1), we obtain

y = (P1-FX^r1)Jg;

M p

The compressor must be designed to handle a gas flow rate of (D1 - F)(t 2 ~ I 1 )Jt 1 and to

compress it to the maximum pressure of p Assuming isothermal ideal gas compression,3

g_ C P1- F X ^ I j T / p N

*1 k A \Po/

where ^1 = a unit conversion factor

k 2 = the compressor efficiency

P — the O delivery pressure

In addition to (17.3) and (17.4), the O2 plant rate F must be adequate to supply the total oxygen

demand, or

D0J + D1(J2 - f,)

F > — — (17.5)

? 2Moreover, the maximum tank pressure must be greater than the O2 delivery pressure,

P ^ Po (17.6)

The performance criterion will consist of the oxygen plant annual cost,

Q($/yr) = a, + a 2 F (17.7) where a v and a 2 are empirical constants for plants of this general type and include fuel, water, andlabor costs

The capital cost of storage vessels is given by a power-law correlation,

C2($) = ^V* 2 (17.8)where ^1 and b 2 are empirical constants appropriate for vessels of a specific construction

The capital cost of compressors is similarly obtained from a correlation,

The compressor power cost will, as an approximation, be given by

b 5 t,H where b 5 is the cost of power

The total cost function will thus be of the form,

Annual cost = a, + a 2 F + dfaV* 2 + b 3 H b4 } + Nb 5 I 1 H (17.10) where N = the number of cycles per year

d = an appropriate annual cost factor

The complete design optimization problem thus consists of the problem of minimizing (17.10),

by the appropriate choice of F, V, H, and p, subject to Eqs (17.3) and (17.4) as well as inequalities

(17.5) and (17.6)

The solution of this problem will clearly be affected by the choice of the cycle parameters (N,

D 0 , D 1 , J1, and t 2 ), the cost parameters (a l , a 2 , b l -b 5 , and d), as well as the physical parameters (T,

P 0 , Ic 2 , z, and M).

In principle, we could solve this problem by eliminating V and H from (17.10) using (17.3) and

(17.4), thus obtaining a two-variable problem We could then plot the contours of the cost function

(17.10) in the plane of the two variables F and p, impose the inequalities (17.5) and (17.6), and

determine the minimum point from the plot However, the methods discussed in subsequent chaptersallow us to obtain the solution with much less work For further details and a study of solutions forvarious parameter values the reader is invited to consult Ref 4

The preceding example presented a preliminary design problem formulation for a system sisting of several pieces of equipment The next example illustrates a detailed design of a singlestructural element

con-Example 17.2 Design of a Welded Beam

Description A beam A is to be welded to a rigid support member B The welded beam is to consist of 1010 steel and is to support a force F of 6000 Ib The dimensions of the beam are to be

selected so that the system cost is minimized A schematic of the system is shown in Fig 17.4

Formulation The appropriate system boundaries are quite self-evident The system consists of the beam A and the weld required to secure it to B The independent or design variables in this case are the dimensions h, I, t, and b as shown in Fig 17.4 The length L is assumed to be specified

at 14 in For notational convenience we redefine these four variables in terms of the vector ofunknowns x,

Fig 17.4 Welded beam.

x = [X 1 , Jt2, X 3 , x 4 ] T = [h, /, t, b] T

The performance index appropriate to this design is the cost of a weld assembly The major costcomponents of such an assembly are (a) set-up labor cost, (b) welding labor cost, and (c) materialcost:

Welding Labor Cost: C 1 Assume that the welding will be done by machine at a total cost of

$10 per hour (including operating and maintenance expense) Furthermore, suppose that the machinecan lay down 1 in.3 of weld in 6 min Therefore, the labor cost is

c , = ( 10 1)(^V 6 2^W=i (AU

1 \ hr/ \60min/ \ in.3/ w \in.3/ w

where V w = weld volume, in.3

Material Cost: C2

C2 = C 3 V w + C4V5where C3 = $/volume of weld material = (0.37)(0.283)($/in.3)

C4 - $/volume of bar stock - (0.17)(0.283)($/in.3)

V 8 = volume of bar A (in.3)

From the geometry,

Note all combinations of Jt1, X 2 , X 3 , and X 4 can be allowed if the structure is to support the loadrequired Several functional relationships between the design variables that delimit the region offeasibility must certainly be defined These relationships, expressed in the form of inequalities, rep-resent the design model Let us first define the inequalities and then discuss their interpretation.The inequities are:

where r d = design shear stress of weld

T(JC) = maximum shear stress in weld; a function of x

cr d = design normal stress for beam material

CT-(JC) = maximum normal stress in beam; a function of Jt

PC(X) — bar buckling load; a function of Jt

DEL(X) = bar end deflection; a function of x

In order to complete the model it is necessary to define the important stress states

Weld stress: T(X) After Shigley,5 the weld shear stress has two components, T' and T", where T' is the primary stress acting over the weld throat area and T" is a secondary torsional stress:

T' = FfV^x 1 X 2 and T" = MRIJ with M = F[L + (x 2 /2)]

R = {(xl/4) + [(X 3 + ^)/2]2}1/2

J = 2(0.707Jt1Jt2[JtI/12 + (X 3 + Jt1) II) 2 }}

where M = moment of F about the center of gravity of the weld group

/ = polar moment of inertia of the weld group

Therefore, the weld stress r becomes

T(X) = [(T')2 + 2rV cos 6 + (r")2]172

where cos B = x 2 /2R.

Bar Bending Stress: cr(x) The maximum bending stress can be shown to be equal to

0-(Jt) - 6FLIx 4 Xl Bar Buckling Load: P c (x) If the ratio tlb = Jt3/Jt4 grows large, there is a tendency for the bar

to buckle Those combinations of Jt3 and Jt4 that will cause this buckling to occur must be disallowed

It has been shown6 that for narrow rectangular bars, a good approximation to the buckling load is

4.Qi3Vl^r X3 EI-]

P < (X) ~ L- L 2 l V « J where E = Young's modulus = 30 X 106 psi

/ - Vi2Jt3Jt45

a = 1 AGx 3 Xl

G = shearing modulus = 12 X 106 psi

Bar deflection: DEL(x) To calculate the deflection assume the bar to be a cantilever of length

L Thus,

DEL(x) = 4FL /Exlx

The remaining inequalities are interpreted as follows

£3 states that it is not practical to have the weld thickness greater than the bar thickness g 4 and

g 5 are nonnegativity restrictions on X 2 and X 3 Note that the nonnegativity of Jc1 and X 4 are implied

by #3 and g 7 Constraint g 6 ensures that the buckling load is not exceeded Inequality g 1 specifiesthat it is not physically possible to produce an extremely small weld

Finally, the two parameters r d and cr d in ^1 and g 2 depend on the material of construction For

1010 steel T d = 13,600 psi and cr d = 30,000 psi are appropriate.

The complete design optimization problem thus consists of the cost function (17.13) and thecomplex system of inequalities that results when the stress formulas are substituted into (17.14)through (17.21) All of these functions are expressed in terms of four independent variables.This problem is sufficiently complex that graphical solution is patently infeasible However, theoptimum design can readily be obtained numerically using the methods of subsequent sections For

a further discussion of this problem and its solution the reader is directed to Ref 7

17.3.2 Operations and Planning Applications

The second major area of engineering application of optimization is found in the tuning of existingoperations We shall discuss an application of goal programming model for machinability data op-timization in metal cutting.8

Example 17.3 An Economic Machining Problem with Two Competing Objectives

Consider a single-point, single-pass turning operation in metal cutting wherein an optimum set ofcutting speed and feed rate is to be chosen which balances the conflict between metal removal rateand tool life as well as being within the restrictions of horsepower, surface finish, and other cuttingconditions In developing the mathematical model of this problem, the following constraints will beconsidered for the machining parameters:

Constraint 1: Maximum Permissible Feed.

where / is the feed in inches per revolution /max is usually determined by a cutting force restriction

or by surface finish requirements.9

Constraint 2: Maximum Cutting Speed Possible If v is the cutting speed in surface feet per

^max = maximum spindle speed available on the machine

Constraint 3: Maximum Horsepower Available If Pmax is the maximum horsepower available

at the spindle, then

In optimizing metal cutting there are a number of optimality criteria that can be used Suppose weconsider the following objectives in our optimization: (i) maximize metal removal rate (MRR), (ii)maximize tool life (TL) The expression for MRR is

A Goal Programming Model

Goal programming is a technique specifically designed to solve problems involving complex, usuallyconflicting multiple objectives Goal programming requires the user to select a set of goals (whichmay or may not be realistic) that ought to be achieved (if possible) for the various objectives It thenuses preemptive weights or priority factors to rank the different goals and tries to obtain an optimalsolution satisfying as many goals as possible For this, it creates a single objective function thatminimizes the deviations from the stated goals according to their relative importance

Before we discuss the goal programming formulation of the machining problem, we should cuss the difference between the terms "real constraint" and "goal constraint" (or simply "goal") asused in goal programming models The real constraints are absolute restrictions placed on the behavior

dis-of the design variables, while the goal constraints are conditions one would like to achieve but arenot mandatory For instance, a real constraint given by

X 1 + X 2 = 3

requires all possible values of Jc1 + X 2 to always equal 3 As opposed to this, if we simply had a

goal requiring X 1 + X 2 = 3, then this is not mandatory and we can choose values of Jt1, X 2 such that

Jc1 + Jc2 ^ 3 as well as Jc1 + Jt2 < 3 In a goal constraint positive and negative deviational variablesare introduced as follows:

Jc1 + Jc2 + d\ - d\ = 3, d l9 d\ > O Note that if d\ > O, then Jc1 + Jt2 < 3, and if d\ > O, then Jc1 + Jc2 > 3 By assigning suitable

preemptive weights on d j~ and d ± , the model will try to achieve the sum Jc1 + X 2 as close as possible

to 3

Returning to the machining problem with competing objectives, suppose that management siders that a given single-point, single-pass turning operation will be operating at an acceptableefficiency level if the following goals are met as closely as possible

con-1 The MRR must be greater than or equal to a given rate M 1 (in.3/min)

2 The tool life must equal T 1 (mm).

In addition, management requires that a higher priority be given to achieving the first goal than thesecond

The goal programming approach may be illustrated by expressing each of the goals as goalconstraints as shown below Taking the MRR goal first,

I2vfd c + di - dl = M 1

where (I 1 represents the amount by which the MRR goal is underachieved, and d J" represents any

overachievement of the MRR goal Similarly, the TL goal can be expressed as

^77^ + K - ^ = T