multicriteria design optimization: procedures and applications

Multicriteria Design Optimization Procedures and Applications
With l7l Figures
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona
Prof. Dr.-Ing. Hans Eschenauer University of Siegen Research Laboratory for Structural Optimization at the Institute of Mechanics and Control Engineering 0-5900 Siegen Germany
Assoc. Prof. Dr. Eng. luhani Koski Dept. of Mechanical Engineering Tampere University of Technology P.O. Box 5Il S F-33 101 Tampere Finland
Assoc. Prof. Dr. hab. inz. Andrzej Osyczka Technical University of Cracow Institute of Machine Technology PL-31-155 Cracow Poland
ISBN 978-3-642-48699-9 ISBN 978-3-642-48697-5 (eBook)
DOl 10.1007/978-3-642-48697-5
Library of Congress Cataloging-in-Publication Data Multicriteria design optimization: procedures and applications [edited by] Hans Eschenauer, Juhani Koski, Andrzej Osyczka. Includes bibliographical references and indexes. ISBN 978-3-642-48699-9 I. Engineering design-Mathematical models. 2. Mathematical optimization. I. Eschenauer, Hans. II. Koski,Juhani.1I1. Osyczka, Andrzej. TA174.M85 1990 620'.00425-dc20 90-9931
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broad casting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions ofthe German Copyright Law of September9, 1965, in its current version and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin, Heidelberg 1990 Softcover reprint of the hardcover 18t edition 1990
The use of registered names, trademarks,etc. in this publication does not imply,even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
216113020-543210 - Printed on acid-free paper
To Gerda
To Anu
To Laura
The modern era of design optimization began about twenty years
ago with the recognition of the usefulness of mathematical programming
techniques. Methods based on mathematical programming were first
adapted to single-criterion optimum design problems. Now, more atten
tion is given to multicriteria modelling, as in many engineering appli
cations often several conflicting criteria have to be considered by the
designer.
Even though multicriteria optimization goes back as far as V. Pareto's
study in 1898, a greater interest in such fields as optimization theory,
operations research, and control theory was not aroused until the late
1960s. Since that time numerous studies on this topic have been
published. Most of them deal with the theory of decision making from
a general point of view whereas only a relatively small number of
publications can be found in the field of engineering design. Thus, the
aim of this book is to fill this gap and to provide the designer with a
new tool for solving the optimization problems in which several con
flicting and noncommensurable criteria are to be satisfied. In order to
get a representative survey of the current works, the editors asked for
contributions from some leading researchers so that a broad range of
applications could be gathered in a coherent volume.
In order to introduce the subject to the readers, Chapter 1 outlines
the background of multicriteria optimization, broadly describes the
relevant mathematical procedures, and also shows some real-life
examples which motivate the designer to apply multicriteria techniques.
The first part of this volume (Chapters 2-4) deals with multicriteria
optimization procedures. Chapter 2 presents the optimization procedure
SAPOP which provides the designer with a general tool for solving
structural optimum design problems. Most activities in multicriteria
design optimization concentrate on the application of interactive proce
dures. Chapter 3 outlines these procedures irrespective of their role in
the design process and also describes two software packages which
facilitate the interactive processes for optimum design. Knowledge-based
systems recently aroused great interest. Their use in multicriteria design
optimization is described in Chapter 4.
VIII Preface
The second part of this volume is devoted to the application of multi
criteria techniques to different design problems which are divided into
subject groups. The first group deals with mechanisms and dynamic
systems. Here, Chapters 5.1 and 5.2 are devoted to the problem of
optimal balancing of robot arms using counterweights and spring mecha
nisms. For the optimum design of spring balancing mechanisms, a
general method for dealing with computationally expensive objective
functions has been proposed. Optimization of automotive drive train
and multibody systems are discussed in Chapters 5.3 and 5.4. Chapter
5.5 shows a special method for finding a relationship between FEM
analysis and optimization procedures using regression models. The
second subject group explores aircraft and space technology problems.
In Chapter 6.1 multicriteria optimal layouts of aircraft and spacecraft
structures are discussed whereas Chapter 6.2 presents poblems of space
craft structures with emphasis on mass and stiffness. Multicriteria
optimization of machine tool systems is the subject of the thit"d group.
In Chapter 7.1 design problems of machine tool structures are presented,
and in Chapter 7.2 the optimum design of machine tool spindle systems
using a decomposition method is discussed. The fourth subject group
deals with metal forming and cast metal technology. In Chapter 8.1, a
multicriteria optimal control approach is applied to die designs for
symmetric strip drawing. Optimal layouts of heterogeneous thick-walled,
chilled cast-iron rollers are presented in Chapter 8.2, and a metal
forming process is optimized and simulated in Chapter 8.3. Problems
of civil and architectural engineering are considered in two chapters.
Chapter 9.1 presents the multicriteria optimization of concrete beams,
trusses, and cable structures, and in Chapter 9.2 multicriteria opti
mization techniques are applied to architectural planning. Finally, the
optimization of structures made of advanced materials is discussed.
Chapters 10.1, 10.2, and 10.4 deal with fibre-reinforced plate and shell
structures and ceramic components, respectively. In Chapter 10.3 multi
criteria optimization and advanced materials in telescope design are
presented.
The editors wish to express their appreciation to all authors for their
contributions and their cooperation in revising the chapters. We are
especially grateful to Ms Ursula Schmitz (Stud.Ass.) who has per
formed the type-setting of the book with great skill and efficiency.
She has also assisted as a translation editor for all chapters and tried
to meet the editors' requirements with much care and patience. In
Preface IX
preparing and organizing the publishing process, she did a splendid job.
We would also like to express our sincere thanks to Ms Birgit Holl
stein and Mr Michael Wengenroth for supervising the work on the
book in its final phase. Thanks are also due to Ms Petra Franke, Ms
Regina Knepper and Ms Henrike StrohbUcker who have done the drawing
of figures.
The editors wish to express their special thanks and appreciation to
Dr. R.D. Pat'bery (University of Newcastle/ Australia) for carefully proof
reading the typescl"ipt. On this occasion, Dr. Parbery would like to
thank the German Research Community <DFG) for sponsoring his stay
as a visiting professor at the University of Siegen.
The DFG is also owed a debt of gratitude for its sponsorship of
Professor A. Osyczka's eight-month-stay at the University of Siegen
where he did the main part of his work on the book.
Finally, we are indebted to Ms. E. Raufelder and Mr. A. von Hagen of
Springer Publishing Company, Heidelberg for the excellent cooperation.
Hans A. Eschenauer Juhani Koski Andrzej Osyczka
Siegen, Tampet'e, Cracow March 1990
CONTENTS
H.A. Eschenauer, J. Koski, A. Osyczka
1.1 Introduction .......................... .
Techniques ........................ .
the Design Process. . . . . . . . . . . . . . . . . . . . 3
Design Process. . . . . . . . . . . . . . . . . . . . 4
Optimization . . . . . . . . . . . . . . . . . . . 6
1.3 Components and Plants with their Objectives. .20
1.3.1 Optimum Design of Highly Accurate Parabolic
Antennas . . . . . . . . . . . . . . . . . . . . . . . 20
Collector. . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.4 Optimal Shape Design. . . . . . . . . . . . . . . 25
1.3.5 Optimal Layout of Tube-Flange Structures. .26
1.4 Conclusion .28
2 Optlrn.ization Procedure S.A.POP-.A. General Tool for ~ul ticri teria Structural Designs 35
M. Bremicker, H.A. Eschenauer, P. U. Post
2.1 Demands on an Optimization Procedure.
2.2 Structure of the Optimization Procedure.
2.2.1 Definitions ............. .
2.3.1 Problem Formulation and Input Data.
2.3.2 SAPOP Main Module ........... .
2.4 Optimization Modelling.
2.4.1 Design Models . . ..
2.4.2 Evaluation Models .
Reduced Line-Search Technique (QPRLT)' . . . .. .55
2.6 Comparison with other Structural Optimization Soft-
ware Systems .....
.58
.58
.65
.66
3.1 Introduction ................. . .71
3.2.2 Approach by Fandel .74
3.2.3 STEP-Method ...... .78
3.3.1 Basic Structure ......... . . . . . . . . . . 86
3.4 Software Package CAM OS. . . . . . . . . . . . . . . . 101
3.4.1 Optimization Algorithms Used in CAMOS . 102
3.4.2 Multicriteria Strategy Approaches . . . . . . . 104
3.4.3 Description of CAMOS . . . . . . . . . . . . . . 104
3.4.4 Interactive MO-Layouts of a Machine Tool
Spindle . 107
M. Balachandran, J.S. Gero
4.3 Role of a Knowledge-Based Approach in Multicriteria
Optimization . . . . . . . . . . . . . . . . . . . . . . . . . .. .119
4.4.1 Description and Representation of Optimization
Problems. . . . . . . . . . . . . . . . . . . . . . . . . 122
Problems ............................. 127
4.5.1 Recognition of Optimization Formulation . . . . . 133
4.5.2 Optimization Algorithm Selection . . . . . . . .. . 135
4.5.3 Knowledge-Based Control in Pareto-Optimal
Set Generation. . . . . . . 137
Sy & te:r.n.& ................... .... . · .. 151
5.1 Optimal Counterwelght Balancing of Robot Arms Using Multicriteria Approach . ......................... 151
J. Koski, A. Osyczka
5.1.3 Formulation of the Optimization Problem
5.1.4 Solution Method .....
5.1.5 Pareto-Optimal Designs
A. Osyczka, J. Zajac
of Industrial Robots ....................... 174
Balancing Mechanism . . . . . . . . . . . . . . . . . 181
· 183
5.3 On the Optimal Synthesis of an Automotive Drive Train . .. 184
F. Pfeiffer
5.3.2 The Mechanical Model . . . . . .... .
5.3.3 The Optimization Model . . . . . . . . .
5.3.4 The Solution Procedure
References ... 192
5.4 Modelling of Multibody Systems by Means of Optimiz- ation Procedures ... 193
H.H. MUller-Sian),
5.4.2 Adaptation of the Model as Optimization Procedure . 195
5.4.3 Example: Adapted Model for the Reflector of a
Parabolic Antenna . 199
5.4.4 Conclusion . 203
D.H. van Campen, R. Nagtegaai, A.J.G. Schoofs
5.5.1 Introduction ................... .
Response Variable .. . ............. .
Responses ...................... .
5.5.5 Computer Program for Experimental Design
5.5.6 Application
5.5.7 Conclusion
6.1 Multicriteria Optimal Layouts of Aircraft and Spacecraft Structures
G. Kneppe
6.1.1 Introduction
.205
.207
.213
.217
.218
.218
.226
.227
. .229
.. 229
.229
.230
.232
Satellite Structure
6.1.5 Conclusion
References . . . . .
.238
.242
.243
6.2 Multicriteria Design of Spacecraft Structures with Special Emphasis on Mass and Stiffness . . ..... 244
H. Baier
6.2.4 Solution Strategies ............. .250
6.2.6 Conclusion .258
7.1. Application of Multicriteria Optimization Methods to Machine Tool Structural Design
M. Yoshimura
7.1.2 Competitive and Cooperative Relationships between
Evaluative Factors . . . . . . . . . . . . . . 266
7.1.4 Application Examples. .274
its Applicatlon to Machine Tool Spindle Design . . . . . . . . . 282
J. Montusiewicz, A. Osyczka, J. Zamorski
7.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 282
7.2.3 Application of the Strategy to the Design of a
Machine Tool Spindle System with Hydrostatic
Bearings ................................ 291
8.1 Optimal Die Design for Symmetric Strip Drawing. .303
W. Stadler
8.1.4 Conclusion
M. Bremicker, H. Eschenauer, H.- W. Wodtke
8.2.1 Introduction .................. .
8.2.3 Structural Analysis ........... .
8.2.5 Optimization Results
References .....
.. 303
. .304
.308
.317
.317
. .... 319
.319
.321
.325
.331
.331
.335
.337
.337
8.3 A Mechanical Model for the Optimization and Simulation of the Metal Farming Process in Roller Levelling of Sheets 339
L. Henrich, K. Schiffner
.340
.344
References . . . . . . . . . . . . . . . . . . . . .......... 352
XVIII Contents
9.1 Multicriteria Optimization of Concrete Beams, Trusses,
... 355
9.1.3 Optimization of Isostatic Trusses ........... .360
9.1.4 Optimization of Cable Structures. . . . . . . . . . . . . 364
9.1.5 Further Applications of Muticriteria Optimization. .371
References . . . . . . . . . . . . . . . . . . . . . . . . . .. . 372
Programming ................................ 376
9.2.2 Dynamic Programming ........... .
.376
.378
.381
9.2.5 Conclusion .............................. 394
10.1.1 Introduction ............................ 397
10.1.3 Objective Functions - Design Variables - Constraints 405
10.1.4 Examples of Application 407
10.1.5 Conclusion 414
S. Ada/i, K.j. Duffy
10.2.3 Multicriteria Design Problem
K.-H. Stenvers
10.3.2 Program System for Structural Analysis and
Optimization ................... .
References .. . . . . . . . . . . . . . . . . .
.418
.420
.421
.424
.427
.427
J. Koski, R. Si/vennoinen
Components . . . . . . . . . . . . . . . . . . . . .448
10.4.5 Design of a Ceramic Piston Crown. .456
10.4.6 Conclusion .461
Su.bJect Index. .469
LIST OF CONTRIBUTORS
S. Adali Faculty of Engineering, Dept. of Mechanical Engineering, King George V Ave., Durban 4001, SOUTH AFRICA H. Baier Dornier-System GmbH, Dept. MEB, Postfach 1360, 07990 Friedrichsha fen 1, FRG M. Balachandran Dept. of Architectural Science, University of Sydney, NSW 2006, AUSTRALIA M. Bremicker University of Michigan, 2212 G.G. Brown Building, Ann Arbor, Mi 48109, USA D.H. van Campen Eindhoven University of Technology, Building WHoog-01.144, P.O. Box 513, 5600 MB Eindhoven. NETHERLANDS H.A. Eschenauer Research Laboratory for Applied Structural Optimization at the Institute of Mechanics and Control Engineering, University of Siegen, Paul-Bo natz-Str .. 05900 Siegen, FRG W. Fuchs Lemmel'Z- Werke KGaA, Dept. TEF, Postfach 1120, 05330 Konigswinter 1, FRG J.S. Gero Dept. of Al'chitectural Science, University of Sydney, NSW 2006, AUSTRALIA L. Henrich Institute of Mechanics and Control Engineering, University of Siegen, Paul-Bonatz-Str., 05900 Siegen, FRG S. Jendo Polish Academy of Sciences, Institute of Fundamental Technological Research, 00-049 Warszawa, Swietokrzyska 21, POLAND G. Kneppe MBB Hubschrauber und Flugzeuge, CAE/Informationssysteme LWD 132, Postfach 801160, 08000 MUnchen 80, FRG J. Koski Tampere Univel'sity of Technology, P.O. Box 527, 33101 Tampere, FINLAND J. Montusiewicz Technical University of Lublin, 20-618 Lublin, ul. Nadbystrzycka 38, POLAND
XXII List of Contributors
B.B. MWler-Slany Research Laboratory for Applied Structural Optimization, University of Siegen, Paul-Bonatz-Str., D 5900 Siegen, FRG R. Nagtegaal Eindhoven University of Technology, Building WHoog-0t.144, P.O. Box 513, 5600 Eindhoven, NETHERLANDS A.Osyczka Technical University of Cracow, 31-155 Cracow, ul. Warszawska 24, POLAND F. pfeiffer Institute of Mechanics, TU MUnchen, Postfach, D 8000 MUnchen 2, FRG P.u. Post Festo KG, Dept. EF-BE, Ruiterstr. 82, D 7300 Esslingen 1, FRG M.A. Rosenman Dept. of Architectural Science, Univel"sity of Sydney, NSW 2006, AUSTRALIA R. Silvennoinen Tampere University of Technology, P.O. Box 527, 33101 Tampere, FINLAND E. Schiifer Research Laboratory for Applied Structural Optimization, University of Siegen, Paul-Bonatz-Str., D 5900 Siegen, FRG K. Schiffner Institute of Mechanics and Control Engineering, University of Siegen, Paul-Bonatz-Str., D 5900 Siegen, FRG AJ.G. Schoofs Eindhoven University of Technology, Building W-Hoog-0t.144, P.O. Box 513, 5600 MB Eindhoven, NETHERLANDS W. Stadler San Francisco State University, Division of Engineering, 1600 Holloway Ave., San Francisco, California 94132, USA K.B. Stenvers Krupp Industrietechnik, GmbH, Dept. Systems Technology, Postfach 141960, D 4100 DUisburg 14, FRG B.W. Wodtke Research Laboratory for Applied Structural Optimization, University of Siegen, Paui-Bonatz-Str., D 5900 Siegen. FRG M. Yoshimura Dept. of Precision Engineering, Kyoto University, Sakyo-ku, Kyoto 606, JAPAN J. Zamorsk1 Technical University of Cracow, 31-155 Cracow, ul. Warszawska 24, POLAND J. Zajac Technical University of Cracow, 31-155 Cracow, ul. Warszawska 24, POLAND
LIST OF SYMBOLS
Note: The following list is restricted to the most important subscripts,
notations and letters in the book. Further terms are explained
in the text.
Scalar quantities are printed in roman letters, vectors in boldface,
tensors or matrices in capital letters and in boldface.
1. Subscripts and Notations
Nabla operator with respect to a vector x
difference
quantity a is valid under the assumptions b,c
all elements a have the attribute A
quantity a equals quantity b
A is a sufficient condition for B
x is an element of set M
x is not an element of set M
set M is a subset of set N
union of set M and set N
intersection of set M and set N
Euclidean norm of a vector x
absolute value of a scalar x
substitute problem
sensitivity matrix
concrete area
1) pay-off table matrix
systematic departure (bias) from the true physical relation
ship
moisture concentration
damping matrix
1) expected value
2) Young's modulus
i-th unit vector
objective function, objective function vector, j-th coordinate
function of the vector f ( j=l, ... , m )
criterion map
reduced gain, gain of an ideal parabolic antenna
vector of inequality ·constraints, j-th inequality constraint
( j=1, ... , p )
( i=l, ... , q)
criterion
2) stiffness matrix reduced to plane stresses
vector of degrees of freedom
set of all real numbers
n-dimensional Euclidean vector space
3) excitation vector
transformation matrix
1) state variable vector
2) displacement vector
2) weight
coefficient or factor
complete Pareto-optimal solution set
vector of IR n , design variable vector, i-th design variable
1) vector of input variables
2) vector of coded or standardized input quantities
compromise (suitable) solution
vector
slack variables
subspace
1) analysis variables from linear transformation
2) vector of output or response variables
point of the functional-efficient boundary, minimal point
vector of the demand level, j-th demand level
set of feasible solutions
3. Greek Letters
vector of unknown coefficients
curvature of the middle surface
coefficients of the thermal stress tensor
coefficients of the hygrothermal stress tensor
r
(3
8
coefficients of the shear strain tensor
direction of minimization step
physical stresses
stress tensor
modal matrix
MOTIVATION
1.1 Introduction 1.1.1 On the Historical Development of Optimization Techniques
A well-known statement of the energy principles says that among
all possible displacements the actual displacements make the total
potential energy an absolute minimum. This means that the application
of the principle of minimum potential energy leads to the fundamental
equations of the boundary value problem in the theory of elasticity.
The principles of mechanics go back to the 17th century. They allowed
the formulation of classical problems in numerous fields of the natu
ral and engineering sciences by means of the calculus of variation
[1,2]. G.W. Leibniz (1646-1716) and L. Euler (1707-1783) found a suitable
mathematical tool for finding the extreme values of given functions
by introdUcing the infinitesimal calculus, with which it is possible to
carry out an integrated treatment of energy principles in all fields of
mechanics with application to dynamics of rigid bodies, general elastic
ity theory, analysis of load supporting structures (frames, trusses,
plates, shells), the theory of buckling, the theory of vibrations, etc.
Some very interesting examples from the field of classical mechanics
are the "curve of the shortest falling time" ("brachistochrone") and the
isoperimetric problem investigated by J. Bernoulli (1655-1705) and D.
Bernoulli (\700-1782). Another important problem is that of the "smallest
resistance of a body of revolution" solved by Sir I. Newton (\643-1727).
With the principle of least action and the integral principles J.L. de
Lagrange (\736-1813) and W.R. Hamilton (1805-1865) contributed to the
perfection of the calculus of variation which still serves as the basis
for various optimization methods. A further important application of
minimal principles is to introduce special approach functions. Instead
of solving the governing differential equations together with the boun-
2 Multicriteria Optimization - Fundamentals and Motivation
dary conditions, often a mathematically difficult task, the problem can
be interpreted as finding functions which satisfy some or all boundary
conditions and minimize the potential energy and the complementary
energy, respectively. Useful approximation methods based on the vari
ational principles of mechanics were devised by Lord Rayleigh (1842-1919),
W. Ritz (1878-1909), B.G. Galerkin (1871-1945), and others.
In a first application on "Optimum Structural Design" variational
methods have been treated by J.L. de Lagrange, T. Clausen (1801-1885),
and B. de Saint-Venant (1797-1886). The investigations of finding out
optimal designs of one-dimensional structures under various loadings
should be mentioned here. Typical examples are the bar subjected to
buckling loads or the cantilever beam under single load or dead weight,
for which optimal cross-sectional shapes were found by means of the
calculus of variation. For this purpose, optimality criteria are derived
in terms of necessary conditions, e.g. Euler's equations in the case of unconstrained problems. If constraints are considered, Lagrange's
multiplier method is applied. This corresponds to the solution of an
isoperimetric problem [3].
Fig. 111 shows an interesting comparison between the natural shape
of a tree branch and the optimal shape design of a cantilever beam
under dead weight. W. Stadler treated this problem using structural
control and multicriteria optimization techniques [71 Nowadays, some
al
a) A tree branch as a natural shape
b) A cantilever beam under dead weight
1.1 Introduction 3
researchers in the field of applied structural optimization use vari
ational principles for their investigations [4]. In developing optimization
procedures together with mathematical p,"ogramming methods, the
"optimality criteria"- method can be effectively integrated into the prob
lem system.
Process
procedures into the practical design phase:
1) Increasing the quality and quantity of products and plants and at
the same time reducing costs and thereby being competitive.
2) Fulfilling the permanently increasing specification demands as well
as considering reliability and safety, observing severe pollution
regulations and saving energy and raw materials.
3) Introducing inevitable rationalization measures in development and
design offices <CAD, CAE) in order to save more time for the staff
to work creatively.
solutions" can only be met if
- sufficient time is available for the development of alternative
solutions,
- mathematical optimization methods are available and applicable.
Although there are numerous activities in establishing CAD-work
stations, the two first points are only partially fulfilled. In addition,
many branches of industry have reservations concerning structural opti
mization. This is due to the yet unrealized demand of practitioners for
reliable, efficient and robust methods which are simple in application
and furthermore proven to be problem-independent. It is a major con
cern of research in the field of design optimization to meet these
practical demands.
The rapid development of efficient computer systems allows the inte
gration of an optimization procedure into the process of engineering de
sign [401 This requires detailed mathematical-physical modelling for
any structural analysis of a technical design problem and coupling it
4 1 Multicriteria Optimization - Fundamentals and Motivation
with a suitable optimization algorithm as well. Both are combined via
the problem-programs of optimization modelling for tasks in engi
neering design (see Chapter 2).
For all these tasks of design optimization mentioned above, it is
equally essential to note that the application of optimization theories
in the design process depends upon the fundamental aspects of the
technical problems.
1.1.3 Multicriteria Optimization 88 a Strategy in the Design Process
Nowadays, a multitude of new constructions and their corresponding
designs require much closer attention because more than one main
criterion corresponding to the aspects 1) and 2) in Section 1.1.2 is given
respectively. Such optimization problems for multipJe criteria are
called either Vector or Multicriteria Optimization ProbJems. With
reference to V. Pareto (1848-1923), the French-Italian economist and
sociologist, who established an optimality concept in the field of econ
omics based on a multitude of objectives, i.e. on the permanent conflict
of interests and antagonisms in social life, this special field of opti
mization is also called Pareto optimization [5] .
The application of multicriteria optimization (MO) to problems in
structural mechanics or technology in general took quite a long time.
It was W. Stadler who referred to the scientific application of Pareto's
optimality concept in the 1970's for the first time and who published
several papers especially on natural shapes [6,7]. Since the end of the
seventies, vector optimization has been more and more integrated into
problems of optimal designs in the papers of a number of scientists
(e.g. 18-16]).
Within the scope of a design process a designer and an analyst
have to figure out which dimensions and shapes will fulfill certain
main criteria of a structure in the best possible way. From common
experience it is known, however, that all these demands can rarely be
met simultaneously. Competing objectives require an estimation of
the importance of the individual objectives by weighing them corres
pondingly. The application of vector optimization as a strategy in a
design process is particularly suitable for decision making during
single phases of development of new constructions or components.
Fig. 1/2 shows the appropriate phases of such a process. In order to
System specifications - Loads - Objecti yes - Constraints - Design Variables - Bounds
Il Feasibility stud)' - Topo log) - Geomef.r~
- Versions , ( Decision
final s~ stem - Specification for
a selected version , ( Decision
, ( Decision
t
1.1 Introduction S
MO -Tool
Global behaviour of the entire system. Compliance with the most important demands
I
Globa l behav iour of the ent.ire system and the main assemblies
I
Global behaviour of the main s;.,s t e m . Loca l beha ,'iour of the subsys t ems (assemblies'
I Local behav iour of t he subsystems , Compliance with demands of subsys t ems (si ngle components!. Interacth<e MO
I Global and local beha\ iou r of the final ... ersion including all assemblies and components
Fig. 1/2. Use of multic riteria techniques during the several phases of
a design process
obtain statements concerning the global behaviour of different design
versions , it is advantageous to find a number of Pareto-optimal or
functional-efficient solutions (see Section 1.3), The particular local
behaviour of a st.·uctural component or a group of structural compo
nents (assemblies) can efficiently be found via an interactive design
process (s ee Chapter 3),
1.2 Mathematical Fundamentals
1.2.1 General Definitions and Notations in Scalar Optimization
The objective of design optimization is to select the values of the
design variables xi (j=I .. ... n) under consideration of various constraints
in such a way that an objective function f=f(x) attains an extreme value.
This can be expressed in the abbreviated form
min {f(x): h(x) = O. g(x) sO} xE IR n
(1-0
with IRn the set of real numbers. f an objective function. x E IRn a vector
of n design variables. g a vector of p inequality constraints. h a vector
of q equality constraints (e.g .. system equations for the determination
of stresses and deformations). and X: = {x E IR n : h(x) = O. g(x) sO}
the "feasible" domain where s has to be interpreted for each individual
component.
An additional problem in design optimization is that the objective
function(s) and the constraints are generally nonlinear functions of the
design variable vector x E IRn. The continuity of the functionals as
well as of their derivatives is usually assumed (Fig.1I3).
Before treating multicriteria optimization problems some relevant
definitions and conditions from scalar optimization will be considered.
X2
1.2 Mathematical Fundamentals 7
The subset XC IRn will be given as the domain of definition. U (x*) E
describes the s-neighbourhood of the point x*; i.e. the number of all
points x whose distance from x* is smaller than s > O. The distance
is given by the Euclidean metric.
Definition 1.1 Global and local minima
i. A point x* E X is a global minimum. if and only if
(!-2a)
:Ii ii. A point x E X is called a local minimum point of f on X, if and
only if for some s
f (x*) :> f( x) (t-2b)
The value f*= f (x*) is accordingly called a local (relative) minimum.
Definition 1. 2 Conve.,ity
i. A subset X of IRn is convex if and only if
(t-3a)
for each xl'x2 E X and for each real number 0:> [l:> 1.
ii. A real-valued function f on the convex subset X is convex on X if
and only if
(1-3b)
for each x1,x 2 E X and for each 0 :> [l :> 1.
Theorem 1.1 Optimality conditions for unconstrained minimum problems
i. Necessa/:1 Condition:
(1-4b)
is positive definite, f(x) has a local minimum at x*.
Theorem 1.2 Conditions for constrained minimum problems
For the dete,-mination of optimality conditions, the Lag,-angian function
is now introduced [23]
q p
L(x,«,(3) = f (x) + L cx. h.(x) + L (3. g.(x) , i= 1 I I j= 1 J J
(1-5)
ments yield the following conditions:
i. Necessary conditions for a local minimum
The Kuhn-Tucker conditions [24] are applied to test local optimality
at a point x*
q p V L(x*) = Vf(x*) + L cx* V h .(x*) + ~ (3. *V g.(x*) = 0
i=1 i I j =1 J J
and h.(x*) = 0 = 1, ... q (1-6) I
,
~(x*) :>; 0 j = 1....,p
(3~g. (x*) = 0 (3~ ~ 0 = 1, ... ,p J J J
ii. Sufficient conditions
Fo,- problems for which f(x) is convex, the equality constraints are
linear, and the inequality constraints are convex functions, i.e. for
so-called convex p,-oblems, the Kuhn-Tucker conditions are also suffi
cient conditions (see [23]).
Fig. 1I-t. shows a geometric interp,-etation in the presence of three
inequality constraints. According to the constraints (1-6), the points A
and B in Fig. 114 satisfy the following conditions:
1. At point A -> 'i7f(x*)= (3*'i7g (x*) + (3*'i7g (x*). 1 1 3 3
(1-7a)
The gradient does not lie in the cone «(31 < 0) set by the gradients of
the constraint functions; A is not a minimum point because the function
value can be reduced within the feasible domain.
2. At point B (t-7b)
The considered point B is a local optimum because there is no direction
within the feasible domain in which the function value can be reduced.
consideration of three inequality constraints
1.2.2 The Multicriteria Programming Problem
In problems with mUltiple criteria one deals with a design variable
vector x which fulfills all constraints and renders the m components
of an objective function vector f(x) as small as possible . A completion
of the problem (1-1) yields the vector optimization problem:
min (f(x) : h(x) = 0, g(x) s; 0 l XEIR"
(1-8)
A characteristic feature of such optimization problems with multiple
criteria is the appearance of an objectil'e conflict, i.e. none of the
feasible solutions allows the simultaneous minimization of all objectives ,
or the i ndi v idual sol u tions of each sing Ie objecti ve fu nction differ.
Consequently, the subject of multicriteria optimization deals with all
kinds of conflicting problems.
Definition 1.3 Convexity of MO
A multicriteria optimization problem on IR m is convex if and only if
(a) the components of the objective function vector f(x) are convex,
(b) the components of the vector of the inequality constraints g(x)
are convex, and
(c) the components of the vector of the equality constraints hare
affine-linear functions of x .
Definition 1.4 Functional-efficiency or Pareto-optimality ([6,25,30])
A vector x* E X is Pareto-optimal resp. p-efficient or functional-efficient
for the problem (1-8), if and only if there is no vector X E X with the
characteris tics
and (1-9)
f.(x) < f. (x*) for at least one j E {1, ... ,m} . J J
For all non-Pareto-optimal vectors, the value of at least one objective
function fj can be reduced without increasing the functional values of
the other components. Fig. 115 shows a mapping of the two-dimen
sional design space X C IR2 into the criterion space Y C IR2 where the
Pareto-optimal solutions lie on the curved section AB. Solutions of
nonlinear vector optimization problems can be found in different ways.
By defining so-called substitute problems these are normally reduced
to scalar optimization problems.
The problem
min p[f(x)] (1-1Oa) xEX
is a substitute problem if there exists x E X* sLich that
p[f(x)] = min p[f(x)] . xEX
(1-1Ob)
The function p is called a preference function or a substitute objective
function OJ" a criterion of control effectiveness (the last term is mainly
used in control engineering) [8,1O-12,15.26J. It is obviously important to study whether the solutions X of the
substitute problems are Pareto-optimal with respect to X and to the
set of objective functions ft, ... ,fm' i.e. that a point y=f(x) actually
lies on the efficient boundary ay * [6,11].
A number of publications have dealt with various methods fOJ" trans
forming vector optimization problems into substitute problems [11-16,
32,38]. In the following these transformation ru les will be denoted
"strategy" when referring to the optimization procedure. Since the prob
lem-dependence of the various strategies may be highly relevant, it
is of interest to test their efficiency and thus their preference behav
iour on typical structures [16].
Design Space X Criterion Space Y
Fig. 115. Mapping of a feasible set into the criterion space
Some of the strategies used at"e described below:
a) Method of Objective Weighting
Objective weighting is obviously one of the most usual substitute
models for vector optimization problems. It permits a preference
formulation that is independent of the individual minima for positive
weights; it also guarantees that all points will lie on the efficient
boundary for convex problems. The pt"eference function or utility func
tion is here determined by the linear combination of the criteria fl' ... ,f m
togethet" with the cOl"responding weighting factors wt, ... ,wm :
In
p[f( x)] := 2 [w. f. (x)] = w T f (x) , j=l J J
XEX. 0-11)
m 2 w. = 1 .
J j=l
It is possible to genet"ate Pat"eto-optima for the odginal problem (1-8)
by vat"ying the weights Wj in the preference function. In engineering and
in economics this approach has been applied for quite some time [9,27,
28]. The deficiency of this stategy in structural optimization has been
discussed for example in [39].
bJ Method of Distance Functions
Distance functions are frequently applied and also lead to a scalarization
of the vector optimization problem. A decision maker specifies a so
called demand level vector 1 = (Y, ... ,Y )T with the objective function t m
value to be achieved in the best possible way. In design optimization
this corresponds to a set of assumed specification values or demands
for the single objective functions. The respective substitute problem
then reads
j=1 J 1 " r " 00, X EX, (1-12)
where the variation of I' meets various interpretations of the "distance"
between the demand levels 1 and the functional-efficient solutions. The
following distance functions are most frequently used:
r = 1: p[f(xl] = ~ ! f.(xl-Y! ' j = t J J
r = 2: p[f(xl] = [ ~ (f.(xl _ y.)2]V2 j = 1 J J
r - 00: p[f(x) ]
(H3cl
The choice of a demand level may cause problems. Therefore, Fig. 1/6
qualitatively gives the solutions of the substitute problem for various
demand levels. It shows that the choice of 11 yields a solution x of
the substitute pt-oblem for which y 1 = f (x) E c)Y * is efficient with
respect to Y. The choice of 1 2 , howevet-, yields an y 2 E c) Y * not lying
on the efficient boundary, and with the choice of the inner point 1 3 ,
the respective solution y 3 is not an efficient point of the boundary of Y.
The use of distance functions is subject to the following disadvan
tages [11]:
1. The selection of "wrong" demand levels 1 will lead to nonefficient
solutions (Fig. 1/6),
2. The selection of "correct" or "valid" demand levels 1 requires know
ledge of the individual minima of the m objective functions f/xl.
j=I, ... ,m which is not easy to achieve with nonconvex problems.
f 1min
Fig. 1/6. Solution of the substitute problem for various demand levels
The methods of the distance functions can also be parametrized to
generate Pareto-optima for the original problem (1-8), For example in
[38] several possibilities for choosing the parameters and their relations
to Pareto-optima have been considered in detail.
c) Method of Constraint Oriented Transformation (Trade-off Method)
Retransformation of the vector optimization problem into a scalar sub
stitute problem may also be achieved by minimizing only one objective
function with all others bounded [11,12]):
p [f (x)] = XEX
j=2, ... , m
Thus, f 1 is called the main objective, and f 2" " ,f m are called secondary
01' side objectives. The given problem can be interpreted in such a way
that when minimizing fl the other components are not allowed to
exceed the values y 2 , ... , Y m ' The dependence of the solution on the
selection of these constraint levels for the two-dimensional case is
shown in Fig . 117.
Fig. 1/7. Solution of a constraint-oriented transformation depending
on the constraint level
The main objective function f1 is generally one for which no a priori
estimation of an upper limit Yt is available.
If the const)'aint levels are taken as equality constraints, and if other
constraints a)'e not considered, the problem corresponds to the minimi
zation of the )'espective Lagrangian fu nction [2<1]
m L(x,cx) ;= f1 (x) + L cx.[f.(x)-Y.],
j=2 J J J (1-15)
which is used in this case as a preference function. The necessary
optimality criteria corresponding to the Kuhn-Tucker conditions (1-6)
without inequality constraints are
c}L = ~ + f cx . ~~ ~ 0, c}x i c}x i j=2 J i
i = 1, ... , n ,
( 1-16a)
( 1-16b)
They are the basis fo)' calculating the optimal values for xt"",xn and
those of the adequate Lagrange mUltipliers cx z '"'' cx m' The introduction
of the abbreviations
~ jj j = 2
m In L<x,a) L rx. f. (x) - L rx.Y· =
j=2 J J j=2 J J
(1-18)
Thus. the expression (1-18) corresponds locally to the substitute prob
lem with objective weighting if one disregards the normalization of the
weighting factors and the additive parametel- C. which are irrelevant
for solving the problem.
Finally, one can state that the considered "Trade-off"-formulation yields
the Pareto-optimal set of solutions if one critedon is replaced by a
sequence of inequality constraints. Therefore. this stl-ategy is sometimes
called "multi-constraint" or "bound formulation" (see [4-, 4-1]).
d) Method of Min-Max Formulation
Besides the preference functions described above. the min-max formu
lation plays a vel-) important role in solving substitute problems. It
is based on the minimization of relative deviations of the single objec
tive functions from the respective individual minimum [12,32].
For the interpretation of a min-max fOl-mulation three given objective
functions with the domain of definition xl S X S x2 are considered
(Fig. 118), If the extrema f. are established separately for each objective J
function (critedon), the desired solution is that x which results in the
smallest value of the maximum deviation of all objective functions.
Therefore, the scalar substitute problem can be defined according to
the min- max formulation as follows:
p[f(x)] = max [z.(x)] j=l .... ,m J
with
(t-19a)
(I-l9b)
Fig. VB. Preference function of a min-max formulation in the one
dimensional case
In [.:/.], a weIl-known modification of equation (1-18) is applied for
p.'actical computations. It consists of the minimization of a new variable
xn+t (comparable to a slack variable, e.g. see [31]) while simultaneously
considering the additional const.'aints:
X E X j = 1, ... , m . <1-20)
Equation 0-20> is especiaIly appropriate for nonlinear optimization
problems using effectively the inequality constraints in the optimization
algorithms <e.g. method of sequential linearization) [21].
For the min-max formulation (1-20) a geometric inte"pretation can
be given on the basis of the hypothesis that all inequality const.'aints
z.<x) - xn+1 5. 0 (j=I, ... ,m) are active within the min-max optimum X, J
i.e. Zj<X)- xn+\ = O. Without going into detailed proof here, it can
be stated that there will be a parameter graph within the hyperspace
IRm from which one can conclude that the optimal solution point f(x)
must lie on a line within the criterion space. Therefore, it yields a
fit'st geometric position for f(i). The second one results from mini
mizing the distance (r=2, Euclidean norm) between the reference point
f and any random point of the line in the criterion space. It can be
shown that this precisely corresponds to the minimization of xn+l'
The min-max optimum can therefore be interpreted as the intersection
of a line in the space with the functional-efficient solution set ay -:lE.
Fig. 1/9 shows this behaviour for two objective functions.
These investigations show that there are certain interdependencies
between the min-max fOJ'mulation and the method of distance functions.
J=1.2 fj
~f T 1
Fig. 1/9. Geometric interpretation of a min-max formulation for two
criteria
Starting from the general distance formulation according to (1-13), the
min-max formulation for r--;>co (Chebyshev metric) results in
p[f(x)] = max If.(x)-Yl. j = 1 ..... In J J
XEX ( 1-21)
with the components of the demand level vector}.. If the minima r. J J
of the individual objective function components are selected as compo-
nents fo.' the demand level vecto.'. and if every objective function is
related to the respective r. . then the distance function formulation is J
t.'ansfOl'med into the min-max formulation in accordance with Eq. (1-19).
The min-max formulation described above yields the compromise
solution x considering all objective functions with equal priority. But
if the single objectives have to meet a special order or if the functional
efficient solution set X" is of great importance for the decision maker.
the min-max formulations can be modified or extended (see [16.21,34]):
- Min-Mav Formulation with Objective Weighting The introduction of dimensionless weighting factors w j ;:: 0 transforms
the substitute problem (1-19) into
p[f(x)] = max [w.z.(x)], j=1 ..... m J J
XEX, (1-22)
where Zj (x) denotes the j-th relative deviation as in (1-19). The
weighting factors describe the priority of the single objective functions.
Thus. it is possible to select definite compromise solutions from
f-IXI-T] p[f(x)] = max [ w. ~
j=1.2 J fj
consideration of different weighting-factor relations
random fields of functional- efficient sets. Mo,-eover, the variation of
w. allows the establishment of the ,-epresentative solution set. A J
similar modification also exists for Eq. (1-20)
j = 1, ... ,m. (1-23)
Fig. 1110 shows the geometric interpretation of Eq. (1-23) for the two
dimensional case. It is obvious that depending on the ratio Wt/W2 of
the two weighting factors one obtains different compromise solutions
describing the whole functional-efficient boundary.
- Min-Ma, .... Formulations Presuming a Demand-Level Vector
If the definition of the relative deviations in (1-19b) is not based
on the individual minima f. but on the given components y. of the J _ J
demand level vector with the characteristics Yj = f j' one can get
analogous substitute problems to (1-22) and (1-23). However, the
problem formulation does not guarantee that all inequality constraints
become active at the solution point x. In other words that they can
be regarded as equality constraints. Only if all inequality constraints
become active, the solution vector X lies on the intersection of the
line in the space with the functional- efficient solution set ClY*.
The difference with respect to the previously mentioned formulation
is illustrated in Fig.1Ill. If the line passing through the point y and
defined by the relation w t /w 2 intersects the functional-efficient
boundary, the intersection point is also the compromise solution. If
J=1.2 J Yj
vector Y
there is no intersection point , the point corresponding to f 1 or to f 2
is the solution depending on the ratio w/w 2 .
The special selection of a demand-level vector Y =0 along with
omitting the division by y. within the relative deviation z.(x) yields a J J
further modification of the min-max formulation a formulation
p[f(x)) = max [w . f.(x)], j=I . .... m J J
XEX (1-24)
1.2.3 The Multicriteria Control Problem [35. 36]
As mentioned in the preceding section. some of the optimum design
problems can be modelled using an optimal control approach [35,36].
Let the state X E A C IR n be controlled by means of a control vector
u( ·): [to,tt] - U c IR r in the state equation
X = g(x,U) (1-25)
with x(to ) E X corresponding to the initial set and x(t1) E X corre
sponding the terminal set and with xn = t, the independent variable,
so that gn(x,u) = I. Furthermore,
g( . ): A x U - B (open) C IR n (1-26)
is the velocity function and U is the control constraint set, the set of
all possible values of u(·). It is usual to assume that u(') belongs to a
nonempty set F of admissible controls. A criterion map f(·); F - IRn is
defined in terms of the component integrals
tl
where
goi (.) ; A x U - C i (open) C IRn. i=I, ... ,m . (1-28)
The state space IRn is augmented with
(1-29)
where y € IRm is the criterion space and go=(gol' .... gom >. Let u(·) € F.
and x(·) be a corresponding solution of the state equation (1-25), and
let s(·) be a solution of (1-29) corresponding to the pair (x('), u(·».
The attainable criteria set is then defined by
(1.30)
Multicriteria control problems can be stated as finding an "optimal"
control u*(·) € F for f(u(·» subject to u(·) E F.
1.3 Components and Plants with their Objectives
In this chapter several examples illustrate the advantages of multi
criteria optimization techniques in decision making during the planning
and the design process of complex components and plants.
1.3.1 Optimum Design of Highly Accurate Parabolic Antennas [17,18]
A practical application of the optimization strategies and procedures
is to figure out the layout of the main components for highly accurate
focusing parabolic antennas. Antennas can be defined as so-called
wavetype transducers. As transmitting antennas they transform cable
1.3 Components and Plants with their Objectives 21
gUided high- frequency energy into wave types convenient for an ex
tension into free space, and as receiving antennas they retransform the
energy taken from free space into cable guided waves. Moreover, one tries to achieve a transformation from one condition into the otheJo
with least possible losses in order to get optimal antenna gain. The
transmission and reception of waves in the dm- , cm-, and mm-range
(micro-wave range) are usually realized by means of parabolic reflectors
based on the laws of geometrical optics.
The rays radiated from the focus of a paraboloid during transmission
are reflected on its surface and leave the mirror as parallel, in-phase
rays. This process is reversed for wave reception. ThE' in-phase condition
of the rays essentially depends on the existence of an accurate para
bolic surface. As the ray reception is analogous to that of optical
astronomy, radio astronomists usually call their parabolic antennas
Fig. 1/12. View of a 30-m-radio telescope for millimeter-wave range
(Max-Planck Institute for Radio Astronomy, Bonn, FRG)
"radio telescopes" in contrast to the "mirror telescopes" in optical
astronomy. Ideally, all incident rays should intersect in the focus
assuming an ideal surface as exact as possible in any given position.
It is obVious that due to this demand, the reflector and its supporting
structure are the most impo)·tant components of a movable parabolic
antenna. In practice, however, such a highly accurate surface is hardly
attainable. Fig. 1112 shows the latest radio telescope for the mm-wave
range (MRT) with 30 m aperture diameter developed, designed and
manufactured by two German companies (Krupp Industrietechnik, Duisburg and MAN, Gustavsburg) and ordered by the Max- Planck
Association, Munich (FRG).
The reflector consisting of single adjustable panels (Fig. 1113) suppor
ted on a rear spatial framework is deformed by dead weight, by wind,
and by temperature loads . Furthermore, there are manufacturing tole
rances as well as measuring and adjusting faults during the positioning
of the reflector surface. Due to these systematic and statistical diffe
rences, the phases of the individual rays will be different. Part of the
energy will be diffused and radiated towards other directions. According
panel surface
~----------d----------~
Fig. 1/13. Design of a parabolic reflector with circular aperture in
panel surfaces
1.3 Components and Plants with theit· Objectives 23
to Ruze, the reduced gain G can be described by a Gaussian error
equation [17]
= ? e - (4rrCl/)..) -
The relation GIGo expresses the "efficiency" of an antenna, and
(1-31 )
(1-32)
is the "gain of an ideal parabolic antenna with d = apertul'e diameter,
A = wavelength, (j = standard deviation or root mean square value
(rms-value). 1] = sul'face efficiency.
The rms-value (j is defined as a measure for the surface accuracy. It
is determined by the method of least squares with a "bestfit"-surface
being described by a set of n given points of the deformed and imper
fect reflector surface [tSl As the efficiency of a parabolic antenna
substantially depends on the surface accuracy, the rms-value plays the
most important role besides the weight of the design of an antenna.
Both objectives must be fulfilled in the best possible way. They are
used as criteria in this bicriteria optimization problem.
1.3.2 Optimal Layout of a Novel Solar Energy Collector [19]
As a further example, a special type of a concentrated solar energy
collector, the so-called "Rear-Focus Collector", is considered (see Fig.
1114-), It consists of several frustum-type reflector shells linked together
by two intersecting ribs. The focus of the rays and accordingly the
absorber are located behind the collector. The system efficiency 1] of the
concentrating collectors depends on the geometry, the shape accuracy
of the reflector. and the tracking en'or. These aspects have a substantial
influence on the two relevant quantities of 1]. the concentration factor
and the intercept factor. For this collector, unlike a similarly designed
one developed by A. Spyridonos in the early seventies, the optimal
arrangement of the single shells of the collector are determined by
means of the mathematical programming. Apart from one objective
function representing the system efficiency, the volume is included as a
further one.
24 Multicriteria Optinlization - Fundarnentals and Motivation
Fig. 1/14. Front view of the Rear Focus Collector (University of
Calgary, Canada)
The supporting behaviour of shells can be considerably improved by
shape optimization without incl'easing the weight, This is especially
impoJ'tant for constructions in satellite technology because it is exactly
this field of technology which has enormous demands on light con
structions with high reliability.
TheJ'efoJ'e, a method fOJ' the optimal layout of the middle sudace and
the wall thickness distdbution of satellite tanks was developed. In
this case the weight of the tank has to be minimized while at the
same time the volume has to be maximized and the feasible stJ'esses
have to be fulfilled. In ordeJ' to avoid buckling problems, negative
stresses should not be permitted. Fig. 1115 shows a view of such
satellite tanks.
t!M 2 TOf"M !...to OR
Fig. 1/15. View of satellite tanks (MBB-ERNO, Bremen, West Germany)
1.3.4 Optimal Shape Design of a Conveyer Belt Drum [21,22]
Efficient conveyers are necessary for extensive soil shifting operations
in open mining. Here, large rubber-belt conveyel"S could prove to be
successfully used. According to Fig. tlt6a, the conveyer belt drum , the
track supporting roller, and the belt are essential components of belt
conveyers. A conveyer belt drum consists of the cylindrical drum shell
(t) and the bottom (2) (Fig. \/\6bl. Drum bottom and shaft (4) are
connected by a clamping ring (3) .
Development in this field is characterized by continuously increasing
demands on conveying capacity, conveying track, and opel"ational safe
ty which leads to enlarged distances between the axes and the conveyer
belt width. As the criteria can be sufficiently realized by essentially
larger tension fOI"ces, the stresses in the belt drums are inevitably
enlarged, too. It was attempted to reduce these stresses by extending
the wall thickness and by implementing ribs . But these measures
often led to an extreme increase in weight so that damage and
failure could not be avoided. The optimal shape design of a conveyer
belt drum was treated with the dil"ect method of shape optimization.
26 Multicdteda Optimization - Fundamentals and Motivation
bl Detail A
1 Drum shell 4 2 Drum bottom 3 Clamping ring 4 Shaft
Fig. 1/16. Sketch of the belt conveyer a) complete system
b) conveyer belt cylinder
On the basis of a given midsurface contour dcp) , the optimal wall thick
ness distribution t(cp) had to be determined in a way that the criteria
of optimization "minimal weight W " and "minimal reference stress
Cl ref max" were fulfilled as well. Constraints were specified in terms
of other limitations.
Steel engineering in various areas of "structural engineering" (crane
technology, steel engineering, "offshore"-techniques, piping construc
tions), circular cylindrical shells are often employed as structural units
connected with other elements (e.g. plates>. The main parts of a slewing
crane (Fig. 1I17a) are the overhang beam (1) , the tubular column (2) ,
and the slewing ball bearing (3)' The bottom flange (Fig.1I17b) which
is welded to the tubular column is connected to the foundation with
anchor bolts . With the force F at the overhang beam the column and
also the flange connection are loaded by an axial force and a bending
moment.
al
bl
t
F
b) bottom flange
, f
ds
In the design phase, particu larl) the shape of the region near the edges is
important. As there are stress concentrations near the edges the problem
is to find out suitable dimensions for the flange leading to a mini
mization of stl'ess concentrations but additional weight reduction.
These two competing objectives lead to a multicriteria optimization
problem.
Some results are presented in Fig. 1118. The conflict between the
criteria f 1(x) ~ W(x) (weight) and f2 (x) ~ o(x) (stress concentration)
is given . Des igns with low stresses give relatively high weight values.
The sensitivity of the flange height x1= h is much higher than the inner
diameter x 2 = d i of the flange. During the variation only h is changed
by the optimization algorithm. If h=h max is reached, d j varies as well.
The results discussed here are adequate and very important not only
for decision making on this particular design problem of the investi
gated connection but also for all other examples.
h "' .75 ho
10-2 10- 1 10 0 10 1
Fig. 1/18. Conflict between the two criteria weight and max. stress
00' W 0 specification values
w 2/w t ratio of weighting factors
1.4 Conclusion
This first chapter is a presentation of the fundamentals of scalar and
multicriteria optimization and illustrates the necessity of application
of multicriteria optimization techniques to develop and to layout
components and stl'uctures by means of some real-life examples. The
applicaton of MO-techniques is primarily due to the fact that today
the manufacturing of machines does not only require a minimization of
costs but also observes objectives such as shape accuracy and
reliability. Such problems are defined as "optimization problems with
multiple objectives" (multicriteria optimization).
The objectives which are mostly competitive and nonlinear do not
lead to one solution point for the optimum but rather to a "functional
efficient" (Pareto-optima/) solution set, i.e. the decision maker selects
the most efficient compromise solution out of such a set. The use of
preference fUnctions transforms the multicriteria optimization problem
References 29
into a scalar substitute problem. This so-called optimization strategy is
a basic part of optimization modelling (see Chapter 2). For the trans
formation a number of preference functions such as objective weighting,
distance functions, constraint-oriented transformation (trade-off
method) and min-max formulation have been analysed and tested. It
can be shown that the efficiency of the single preference functions
depend both on the problem and on the adaptation to certain opti
mization algorithms.
The examples from industrial practice given in Section 1.3 show how
important it is for the designer to get a tool for decision making in
the design pt"ocess, especially when there is more than one criterion
to be fulfilled. A large number of possible multicriteria formulations
which go fat" beyond these examples is presented in the second pat"t
of the book (Applications).
design can be summarized as follows:
(j) Multicriteria modelling very well reflects the design process in
which usually several conflicting objectives have to be satisfied.
(ij) The designer has the possibility to explore a broader range of
altemative solutions than with single criteria models for which
the solution is immediately fixed after the problem-formulation.
(ijj) Multicriteria formulation provides a basis for explicit trade-off
between conflicting objectives or interests.
References
Gattingen, Heidelberg: Springer 1972
Stuttgart: Birkhauset" 1979
Mathematiques et Astronomiques I (1849-1853) 279-294
[4] Bends0e, M.P.; Olhoff, N.; Taylor, J.E.: A Variational Formulation
for Multicriteria Structural Optimization. Joumal of Struc
tural Mechanics, Vol. 11, No.4, 1983
[5] Pareto, V.: Manual of Political Economy. Translation of the French
edition (1927) by A.S. Schwier. London-Basingslohe: The McMillan
Press Ltd., 1971
Optimality. In: Marzolio/Leitmann (edsJ: Multicriterion
Decision Making. CISM Courses and Lectures.Berlin, Heidelberg,
New York: Springer 1975
(1978) 169-217
Tragwerken insbesondere bei mehrfachen Zielen. Dissertation,
TH Darmstadt, 1978
University of Technology Publication, Tampere 1979
[10] Eschenauel', H.: tiber die Optimierung hochgenauer TI'agstruk
turen. Karl- Marguerre-Gedachtnisband, Schriftenreihe "THD
Wissenschaft und Technik", (1980) 89-101
[11] Sattler, H.-J.: Ersatzprobleme fUr Vektoroptimierungaufgaben
und ihre Anwendung in del' Strukturmechanik. Dissertation,
Universi tat-GH-Siegen, 1982
York, Chichester, Brisbane, Toronto: John Wiley, 1984
[13] Radford, A.D.; Gero, J.S.; Roseman, M.A.; Balachandran, M.:
Pareto Optimization as a Computer-Aided Design Tool. In: Gero,
J.S. (ed'): Optimization in Computer-Aided Design. North
Holland Amsterdam, New York, Oxford: Elsevier Science Publi
shing Company (1984) 47-80
Sons (1984) 459-481
Atrek, Gallagher, Zienkiewicz (ed.): New Directions in Optimum
Structural Design. Chichester, New York, Brisbane, Toronto,
Singapore: J. Wiley & Sons (1984) 483-503
References 31
Structul'al Optimization of Engineer-ing Designs. DFG-Report of
the Research Laboratory for Applied Structural Optimization,
Univel'sity of Siegen, May 1985
[17] Eschenauer, H.: Parabolantennen fUr Satellitenfunk und Radio
astronomie im Millimeterwellenbereich-Forderungen und Auf
gaben an den Ingenieur. In: Kreuzel', H., Bonfig, K. W.: Entwick
lungen del' siebziger Jahre, Gerabronn: Hohenlohel' Druck- und
VerIagshaus, (1978) 531-54-9
Accurate Focusing Systems. In: W. Stadler: Application of Mul
ticriteria Optimization in Engineel'ing and the Sciences. Plenum
Publishing Corporation. (1988) 309-354-
[19] Eschenauer, H.; Vermeulen. P.: Contribution to the Optimization
of a Novel Solar Energy Collector. ZFW, Bd. to, H.3, (1986)
190-198
of Structural Optimization 1, (1989) 171-180
[21] Kneppe, G.: Dil'ekte Losungsstrategien zur Gestaltsoptimierung
von FHi.chentragwerken. Dissertation, Universiti:it-GH-Siegen, 1985
[22] Eschenauer, H.; Kneppe. G.: Min-Max-Formulierungen als Strate
gie in del' Gestaltsoptimierung. ZAMM 6 (1985) T344--T34-5
[23] Pierre, D.A.: Lowe, M.J.: Mathematical Programming via Augmen
ted Lagrangian. London: Addison-Wesley, 1975
[24-] Kuhn, H.W.; Tucker, A.W.: Nonlinear Progl'amming. Proceedings
of the 2nd Berkeley Symposium on Mathematical Statistics and
Probability, University of California, Berkeley, California, 1951
[25] Hettich, R.: Charakterisierung lokalel' Pareto-Optima. Optimiza
tion and Operations Research. In: Oettli, W.; Ritter, K. (eds,):
Lecture Notes in Economics and Mathematical Systems. No. 117.
Berlin: Springer-Verlag, (1976) 127-141
3-26
Vieweg- Verlag, 1979
In: Lecture Notes in Economics and Mathematical Systems. No. 76.
Bedin: Spl'inger-Verlag, 1972
[29] Sattler, H.J.: Eine Herleitung der Zielgewichtung in der Vektor
optimierung aus einer Abstandsfunktionsformulierung. In: Zeit
schrift fUr Angewandte Mathematik und Mechanik. ZAMM 62
(1982) T382-T384 [30] Charnes, A.; Cooper, W. W.: Management Models and Industrial
Application of Linear Programming. Vol. 1. New York: Wiley 1961
[31] Fox, RL.: Optimization Methods for Engineering Design. London:
Addison-Wesley, 1971
blems for Engineering Design, Computational Methods in Applied
Mech. and Eng., Vol. 15, (1978) 309-333
[33] Osyczka, A.: An Approach to Multicritel'ion Optimization for
Structural Design, Proceedings of International Symposium on
Optimum Structural Design, University of Arizona, 1981
[34] Osyczka, A.: Multicriterion Optimization for Engineering Design,
In: Gero, J.S. (ed): Design Optimization. New York: Academic Press
Inc., (1985) 193-227
[35] Stadler, W.: Multicriteria Optimization in Engineering and in the
Sciences. New York and London: Plenum Press. 1988
[36] Stadler, W.: Multicriteria Optimization in Mechanics (A Survey).
Applied Mechanics Rewievs, Vol. 20, (1984) 1442-1471
[37] VOl-Guideline 2212: Systematisches Suchen und Optimieren
Konstruktiver Losungen. VDI-Handbuch Konstruktion, DK631:
658,512,2 (083,132)
[38] Koski, J.; Silvennoinen, R.: Norm Methods and Partial Weighting
in Multicriterion Optimization of Structures. International Journal
for Numerical Methods in Engineering. Vol. 24 (1987) 1101-1121
[39] Koski, J.: Defectiveness of Weighting Method in Multicriterion
Optimization of Structures. Communications in Applied Numerical
Methods, Vol. 1 (t985) 333-337
[40] Eschenauer, H.; Post, P.U.; Bremicker, M.: Einsatz del' Optimie
rungsprozedur SAPOP zur Auslegung von Bauteilkomponenten.
Bauingenieur 63, II (1988) 515-526
[41] Rozvany, G.I.N.: Structural Design via Optimality Criteria. 001'
drecht/Bostonl London: Kluwer Academic Publishers, 1989
PART I
2 OPTIMIZATION PROCEDURE SAPOP - A GENERAL TOOL FOR MULTICRITERIA STRUCTURAL DESIGNS
M. Bremicker, H.A. Eschenauer, P. U. Post
2.1 Demands on an Optimization Procedure
As presented in Chapter 1, it is an important goal of engineering
activities to improve and optimize technical designs, structural assem
blies and stl'uctural components. The task of stl'uctul'al optimization is
to SUppOI't the engineer in searching fOl' the best possible design alter
natives of specific structul'es. The "best possible" or "optimal"' structul'e
here applies to that structure which mostly corresponds to the designer's
desired concept and his objectives meeting at the same time operational,
manufactul'ing and application demands. Compared with the "Tl'ial and
Error"-method generally used in engineering practice and based on an
intuitive empirical approach, the determination of optimal solutions by
applying mathematical optimization procedures is more reliable and
efficient. These procedures can be expected to be more frequently
applied in industl'ial practice. In order to apply structural optimization
methods to an optimization task, both the design objectives and the
J'elevant constraints must be expl'essed by means of mathematical func
tions. One example of a design objective is the demand fOl' the maximum
degree of stiffness of a stl'ucture which can be described by the objective
"minimization of the maximum structural deformation". The design
val'iables al'e the parameters of the structure, for example the CI'OSS
sectional and geometl'ical quantities, which should be selected in a way
that the objective function can be minimized by considel'ing additional
conditions. These conditions or constraints al'e equality and inequality
equations which include the mathematical formulation of demands such
as permissible stresses, stability critel'ia etc. The formulation of the
scalar design pl'Oblem is generally given by (1-0:
Min {f(x) I h(x) = 0 ; g (x) :s; 0 } . xEIRn
(2-1)
The solution of optimization problems requires software systems
which are easy to use, provide sufficient efficiency, and are available for
practical application. Several optimization algorithms should be linked
to structural analysis procedures in a suitable manner by means of
optimization model processors [1,2,45].
In general, a software system should meet the following requirements:
- possibility of selecting the suitable optimization algorithm for an
optimization problem from a number of efficient methods,
- use of different methods for structural analysis such as finite and
analytical methods,
- application of automatic design and evaluation models (pre- and post
processors) for a wide range of standard problems in optimization
modelling; simple integration of special optimization models if re
quired,
rent program modules,
without comprehensive implementation work,
optimization tasks by applying efficient algorithms (e.g. sensitivity
analysis of FE-structures, solution methods for linear and nonlinear
equation systems etc.),
systems), utilization of modern programming techniques (parallel
computing>,
mentation.
2.2.1 Definitions
Before describing an optimization procedure and its practical realiz
ation some of the terms frequently used in this chapter shall be defined
(see Figs. 211-213):
Optimization algorithm : mathematical procedure for constrained/
unconstrained optimization (optimality critel"ia
methods. mathematical programming methods),
optimization problem,
problems to simplified substitute problems or
smaller subproblems, respE'ctively,
or, in accordance with the given requirements,
of several design objectives.
into one scalar substitute objective function,
Constraints mathematically formulated design requirements
which are not covered by the objective
function(s),
behaviour (mathematical-physical modell,
Analysis variables structural parameters which can be varied
during optimization computations,
Initial design initial values of the design variables at the
beginning of the optimization process,
Design model
analysis variables.
tion concepts,
objective function and constrain t values under
consideration of optimization strategies,
mode\(s>.
When dealing with a structural optimization problem, it is recommen
dable to proceed following the "Three-Columns Concept" [1] (Fig. 2/2.>'
38 2 Optimization P."ocec\ure SAPOP
r----, Data I Designer t--------, L.. • Input -,-- I
Optimal
Design
x·
I
('-'---'-- -'-'--'-'-i . Transformed i I AnalysIs
Variables .i J Transformation L LI Design Model I i Variables
Z I I Z - x IDe~ I x - y I I y
I Variables i i x i i i i i i OPTIMIZATION i Structural
Optimization
Algorithm i MODEL i i i I i i i jl i
f,p.g I Evaluation I i U
Preference Function! L Model I I State
Model
Fig 2/1. Structure of an optimization loop
The first step is the theoretical formulation of the optimization problem
taking into account all relevant demands on the structure. The next step
involves the solution of the subproblems "structural modelling" and
"optimization modelling". From the third column an optimization algo
rithm is selected and linked with the structural and the optimization
model to form an optimization procedure. In the following a detailed
description of the columns is given.
Column 1: Structural model
Any structural optimization requires the mathematical determination of
the physical behaviour of the structure. In the case of mechanical
systems, this refers to the typical structural response subject to static
and dynamic loading such as deformations, stresses, eigenvalues, etc.
Furthermore, information on the stability behaviour (buckling loads) has
to be determined. All state variables required for the objective function
and constraints have to be provided. The structural calculation is carried
out using efficient analysis procedures such as the finite element method
or transfer matrices methods. In order to ensure a wide field of applic
ation. it should be possible to adapt several structural analysis methods.
2.1 Demands on an Optirnization Procedure 39
Column 2: Optimization modelling
From an engineer's point of view, this column is the most important
one of the optimization procedure. First of all, the analysis variables
which are to be changed during the optimization process are selected
from the structural parameters. The design model including variable
linking. variable fi:\.ing. shape functions etc. provides a mathematical
link between the analysis variables and the design variables. In order to
increase efficiency and improve the convet'gence of the optimization,
the optimization problem is adapted to meet the special requirements
of the optimization algorithm by transforming the design variables into
transformation variables. By using this approach, it is e.g. possible to
almost linearize the stress constraints of a sizing optimization problem.
Additionally, objective functions and constraints have to be determined
by procedures that evaluate the structural response or state variables.
\Vhen formulating the optimization model, the engineer has to consider
the demands from the fields of design, material, manufacturing, assem
bly and operation.
OPTIMIZATION PROCEDURE
I Oeslgn I I Matf'rlal I I Manufacturing I I Auembl) I '-.. ......... '" ¥
r Structur-al MuJf'1 Optlmlzlltlon Model I Optimlz.atlon Algorithms
I I I j j Anal}tical Discrete
"an.' J 1 r l 1 ~bthematical Special Methods Methods forme'!.- Transfor- Design D('sign Anal}sis Programming Methods 'l mal;on J -l Mod.1 J 'I - Rayligkl/Ritz - Finite Element Variables z _:0( Variables " _) VariableS! - S('C]uential - OC- Mtothods
Method Unearization - Galerkln - Discrete
- Exact Element I E"aluation I Quadratic Solution Method State Objectivt'S Programing - Dvnamic
Variables "
Method Gradients - Stochastic Optimization
- Transfer OptimiZation Strat~gles
V I r v.eto< I r u'g' Scal. I Sh.pe
V Optimization System Function
Optimization Decomposition Optimization
I
Column 3: Optimization algorithms
problems. These algorithms are iterative procedures which, proceeding
from an initial design xo ' generally provide an improved design variable
vector xk as a result of each iteration k. The optimization is terminated
if a breaking-off criterion responds during an iteration. Numerous
studies have demonstrated that the selection of the optimization algo rithm is problem-dependent. This is particularly important for a reliable
optimization and a high level of efficiency (computing time, rate of
convergence>. If, for example, all iteration results have to lie within the
feasible domain, an algorithm that iterates within the feasible domain
(e.g. generalized reduced gradients (GRG» should be applied.
2.3. Basic Ideas of the Procedure SAPOP
On the basis of the "Three-Columns Concept" and on the )'equirements
mentioned above, the software system SAPOP (Structural Analysis Program and Optimization Procedure, Fig. 2/3,) was developed. It
INPUT
Fawco3 I' l8---IDIWSAPI
IrORCE I [DiS"="I ~
lSAP
POST
WEIGHT DISPlACE 1JISPl..:E STRESS NlPlAT STRESS ElGEN EIGEN ElJ(KlE FALCRIT ~
'viiPST OBJWEI COT ......... DISTANCE
GRAPH
consists of three independent parts communicating with each other via
a Data Management System (data base>. Each of these parts is divided
into individual blocks connected by standardized intel"faces to ensure
the largest possible modularity. Each block contains a number of inter
changeable modules. When carrying out an optimization computation
only those modules which are actually needed are linked together.
2.3.1 Problem Formulation and Input Data
The input system is used to prepare the input data provided by the user
to be stored on a database. All quantities required to describe the struc
tLII"al and the optimization model as well as the parameters for control
ling the optimization pmcess are edited hel"e. The user has to provide at
least two different data items. The data fi Ie OPTDA T incl udes all data
necessary to control the optimization process as well as the initial
values of the design variables and the input quantities for the formu
lation of the optimization model. The data file STRDAT includes the input
data fot" the software system which is applied to the structural analysis.
The module MINBA includes a band-width-minimizer for FE-stnlctlll"es.
The input data for a multilevel optimization using a decomposition
stl"ategy are provided by the file DECDAT; COMDAT supplies the
relevant material specifications of fibre-reinforced composites. In future,
the user will be supported by an expert system EXPERT when generating
the input data.
The optimization computation is actually carried out by the SAPOP
main module MAIN. First of all, an initialization phase is run, and sub
sequently the optimization is stal"ted via the ONE-SYSTEM module of
the DECOM block. Two decomposition strategies (cutting force method
FORCE and deformation method DISPLACE [12]) allow to optimize large
structures by optimizing substructures. A number of different optimiz
ation algorithms can be called by the DECOM-modules. Apart from the
seven mathematical programming methods. an optimality criteria pl"oce
dure (stress- ratio method) is available. It is also possible to couple
42 2 Optimization Procedure SAPOP
different algorithms by means of a series connection (serial hybrid
approach), Among others, the following algorithms can be applied:
COMBOX EXTREM SEQLI2
Direct Search Algorithm by Jacob [4-],
SEQuentialUnear Optimization Extended Version [2-6],
Variable Metric Method for Constrained Optimization
Including Watch Dog Technique [7],
Nonlinear Program with Quadratic Une Search [8],
Generalized REduced G.·adient Algorithm [9-11],
Quadratic Programming with Reduced Line-Search
Technique [12,13],
Lagrange Penalty Method for NonLinear Problems [14],
Optimality CRITeria Method (Stress Ratio Method) [15].
For each iteration the actual values of the objective function and
constraints are required, and for mos,t of the algorithms the gradients
have to be calculated with regard to the transformation variables. The
control program FUNC for structural analyses and the control programs
for sensitivity analyses (gradient calculations, Section 2.4.3) are called
via the interface module COMBIN. Module FDYN is an algorithm for
solving time-dependent optimization problems [41].
The transformation module TRANS shifts the transformation variables
into design variables. The subsequently called PRE-processor contains
different design models used to determine the analysis variables from
the design variables. The design model SIZE includes variable linking
and variable fixing for cross-section optimization (sizing). SHAPE, GEOM
and MESH modules can be used for shape and geometry optimization
tasks. As far as composite designs are concerned, the module COMP
transforms design variables into layer thicknesses and ply angles of a
fibre composite lamina, and the corresponding mateJ"ial characteristics
(elasticity, stiffness, thermal and hygrothermal coefficients) are calcu
lated. If a special design model is to be used to solve an optimization
problem, a corresponding program module can be included. or the entire
preprocessor can be exchanged.
The structural analysis is now carried out using the updated analysis
variables. These are part of the structural parameters of the mathema
tical-mechanical model which describes the physical behaviour of the
2.3 Basic Ideas of the Procedure SAPOP 43
actual structure. Systems of algebraic or differential equations are
solved by using efficient numerical methods. At present, the following
structural analysis methods a.'e available in SAPOP:
SAPV-2
ORSAB
LSAP
NLPLAT
PAFEC
ANSYS
Method. The modules for linear displacement, stress
and eigenvalue analysis are integrated in SAPOP;
Orthotrope Rotationsschalen unte.' allgemeiner Belast
ung, Transfer-Matdx Method for arbitrary loaded iso
tl'Opic and orthotropic shells of revolution [1,46,47];
Laminated Shell Analysis Program [17], Finite Difference
Method for anisotropic shells of fj bel' composite
material;
Difference Method for anisotl'Opic composite plates with
time-dependent material behaviour and imperfections; Pt'ogram for Automatic Finite Element Calculation [49];
Finite-Element-Pl'Og.'am of Swanson Analysis Systems.
Apart from these programs, the user can link his own st.'uctu.'al
analysis modules to SAPOP. Thus, it is possible to deal with structures
using analytical calculations or to deal with any examples from inte.'
disciplinary fields. For the latter, however, other pre- and postproces
sors a.'e usually .'equit'ed in orde.' to formulate the design and evaluation
model.
The computed state vadables a.'e transferred to the postprocessor in
orde.' to determine the objective function{s) and constraints. Modules
are available for computing weight as well as stress, defo.'mation,
and eigenvalue evaluations. In the case of composite structures the
failure criteria of laminate composites are determined [20-23].
Multicriteria optimization problems are solved by transforming the
objective function into a scalar substitute function {preference fUnctions,
see (1-10) to (1-24)).
If the range of performace of the postprocessor is not sufficient for a
special application, user-defined programs for the fomlUlation of objec
tive functions and constraints can be linked via standa.'dized interfaces.
The actualized objective functions and constraints a.'e t.'ansfen'

multicriteria design optimization: procedures and applications

Documents