UNDER UNCERTAINTY THE FUZZY COMPROMISE SUPPORT …€¦ · KEY WORDS Uncertainty fuzzy sets...
Transcript of UNDER UNCERTAINTY THE FUZZY COMPROMISE SUPPORT …€¦ · KEY WORDS Uncertainty fuzzy sets...
Eng Opt 1992 Vol 20 pp 21 43 1992 Gordon and Breach Science Publishers S.A
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DECISIONS UNDER UNCERTAINTY THE FUZZYCOMPROMISE DECISION SUPPORT PROBLEM
Q.-J ZHOU
Shp Engineering Department ABSAMERICAS 263 North Belt East
Houston Texas 77060 US.A
ALLEN
Janco Research Inc 4501 University Oaks Boulevard Houston Texas 77004 U.S.A
and
MISTREE
Systems Design Laboratory The George Woodruff School of Mechanical
Engineering Georgia Institute of Technology Atlanta Georgia 30332 U.S.A
Revised 28 March 1991 in final form 20 October 1991
compromise Decision Support Problem is used to improve an alternative through modification to
achieve multiple objectives However the compromise DSP requires precise numerical information to
yield rigorously accurate results In the early stages of conceptual design such precise information is often
unavailable For example design should be reliable manufacturable maintainable and cost-efficient
Although inherently vague each qualifier specifies an important attribute that the design must possess
Such vagueness may be modeled rigourously using the mathematics of fuzzy set theory In this paper we
introduce fuzzy formulation of the compromise DSP formulation which is particularly suitable for
modeling the decisions required in the early stages of engineering design We investigate the properties
of the fuzzy compromise DSP in the context of designing planar four-bar linkage
KEY WORDS Uncertainty fuzzy sets decision support compromise four-bar linkage
NOTATION
Is member of the set or is contained in
The intersection of
The union of sets
Indicates mapping from the set on the left to the set on the right
Is almost positive
sup The least upper bound
inf The greatest lower bound
max The largest of fuzzy sets
21
MYLAN - EXHIBIT 1029
22 Q..-J ZHOU ALLEN AND MISTREE
mm The smallest of or the intersection of fuzzy sets
Ax uAx represents the fuzzy set at the value whose grade of membership
is determined by the membership function
Ac The constants parameters associated with the capability of the system
with respect to the jth constraint on the system
Ad The constants parameters associated with the demand on the system
due to the jth constraint
Apk The constants parameters associated with the performance of the
system on the kth target
Atk Constants constants associated with the designers aspirations for the
kth target
C3.Ac3 linear or nonlinear capability associated with constraint that is
function of the parameters Ac3 and the system variables Italics are
used to indicate fuzziness
The extent of the cloud of fuzziness surrounding the main value of
fuzzy set This is numerically equivalent to the range of the membership
function
The fuzzifier associated with the grouped constraints
cc The fuzzifier associated with the parameters specifying the systems
capability in meeting the jth constraint
cd3 The fuzzifier associated with the demand due to the jth constraint
cpk The fuzzifier associated with the performance on the kth target
ctk The fuzzifier associated with the kth target
D3Ad3 Demand associated with the jth constraint Demand is function of
the system variables and parameters Ad3 Fuzzy demand is
denoted by italics
DC decision
Deviation variables used in the crisp non-fuzzy compromise DSP
formulation
possibility distribution
Hc The possibility distribution for the capability of meeting the jth con
straint
Hd3 The possibility distribution for the degree of compatibility of the system
associated with the demand from the jth constraint
IIPk The possibility distribution for the performance on the kth target
Htk The possibility distribution for the kth target
The possibility distribution representing the degree of compatibility of
the system with the constraints when the constraints are grouped
implies complete compatibility implies total incom
patibility
Total number of constraints in DSPTotal number of goals in DSPThe number of system variables in DSPThe main value of fuzzy set the value which is surrounded by
cloud of fuzziness
DECISIONS UNDER UNCERTAINTY 23
mc3 The main value of the fuzzy set which represents the systems capability
on the jth constraint
md3 The main value of the fuzzy set which represents the demand associated
with the jth constraint
mpk The main value of the fuzzy set which represents the performance
associated with the kth target
mtk The main value of thc fuzzy set which represents the kth target
PkApk Actual performance of system characterized by the system variables
and the parameters Apk fuzzy performance function is denoted
by italics
P1 Priority ranking factors for the achievement of the system goals used
in both the crisp and the fuzzy formulation DSP
TkAtk Target or aspiration level for system performance at the point defined
by the system variables and the parameters Atk fuzzy aspiration
level is denoted by italics
crisp vector of system variables
An achievement function representing the difference between system
performance and the designers goals for the system
PAX The membership function associated with the fuzzy set
DECISION SUPPORT IN THE VERY EARLY STAGES OF DESIGN
comprehensive approach called the Decision Support Problem DSP Technique14
is being developed and implemented at the University of Houston to provide support
for human judgment in designing an artifact that can be manufactured and main
tained Decision Support Problems provide means of modeling decisions en
countered in design manufacture and maintenance Formulation and solution of
DSPs provide means for making the following types of decisions
Selectionthe indication of preference based on multiple attributes for one
among several feasible alternatives
Compromise trade-ofl----the improvement of an alternative through modifica
tion
Hierarchicaldecisions that involve interaction between sub-decisions
Conditional--decisions in which the risk and uncertainty of the outcome are
taken into account
Compromise DSPs refer to class of constrained multiobjective optimization
problems that are used in wide variety of engineering applications Both selection
and compromise DSPs can be part of the hierarchical representation of an
engineering system which involves an ordered and directed set of DSPs where the
sequence of interactions among them is clearly defined Applications of these DSPs
include the design of ships damage tolerant structural and mechanical systems the
24 Q.-J ZHOU ALLEN AND MISTREE
design of aircraft mechanisms thermal energy systems design using composite
materials and data compression detailed set of references to these applications is
presented in Ref DSPs have been developed for hierarchical design coupled
selection-compromise compromise-compromise and selection-selection6 These con
structs have been used to study the interaction between design and manufacture7 and
between various events in the conceptual phase of the design process8 The compromise DSP is solved using unique optimization scheme called Adaptive Linear
Programming9 Other formulations of conditional decisions are described in Refs
For real-world practical systems all of the information for modeling systems
comprehensively and correctly in the early stages of the project will not be available
In the preliminary stages of engineering design there is great uncertainty about the
nature of the object that is being designed This uncertainty stems from vagueness
or imprecision of knowledge about the object being designed rather than from errors
in repeated measurements of the object being designed there can be no measurements
as the object does not exist yet Hence standard probabilistic approaches cannot
form an accurate mathematical representation of the object being designed However
both vagueness and imprecision can be modeled rigorously using fuzzy set theory13
Therefore we are investigating the incorporation of the mathematics of fuzzy sets
into methods being developed for use in the very early stages of design
In this paper we present theoretical model for the fuzzy compromise DSPs
followed by an example of their use non-linear kinematics problem involving the
minimization of the structural error in path-generating four bar linkage The
standard non-fuzzy crisp formulation of the compromise DSP is specific case of
the fuzzy compromise DSP Also the importance of being able to fuzzify constraints
and goals independently is shown
1.1 The compromise Decision Support Problem
compromise DSP is defined using the following descriptors system and deviation
variables system constraints and goals are defined by set of constant parameters
and system variables bounds on the system variables and deviation function The
compromise DSP its descriptors and its mathematical form have been described in
several publications39 and will therefore not be repeated here The generalized
formulation of the fuzzy compromise DSP that follows however has not been
published elsewhere and it reads as follows
Given
An alternative defined by the vector of independent system variables
system constraints which must be satisfied for an acceptable solution
C.Ac3 is the capability associated with the jth system constraint Acrepresents the constant parameters needed to characterize the capability associated
with the jth constraint The capability can be linear or nonlinear function
DECISIONS UNDER UNCERTAINTY 25
DAd3 is the demand associated with the jth system constraint Adrepresents the constants needed to characterize the demand These constants are
some of the parameters characterizing the compromise DSP represents the system
variables
is the number of system goals which must be achieved to attain specified
target TkAtk Atk represents the constants necessary to specify the kth target
these constants are some of the parameters associated with the compromise DSP
PkApk is function specifying the performance associated with the kth
system goal Apk represents the constants needed to characterize the systems
performance on the kth target These constants are some of the parameters associated
with the compromise DSP
Find
The values of the independent system variables 1..The values of the non-negative deviation variables indicating the extent to which
the target values are attained and dk represent under-achievement and over-
achievement of the target where I.. such that and and
Satisfy
System Constraints is Equal to or Exceeds Demand
CAc3 DAdWith lower and upper bounds on the system variables
System Goals is Equal to or E.xceeds Performance
PkApk TkAtk 1.Minimize
deviation function quantifies the deviation of the system performance
PkApk from the ideal as defined by the set of target values TkAgk and
their associated priority levels There are two ways of representing the deviation
function
Preemptive Deviation Function
In the preemptive formulation the deviation function is
fd d1
where the functions of the deviation variables are ranked lexicographically
26 Q.-J ZHOU ALLEN AND MISTREE
Archimedean Deviation Function
min W1d W2d .. W2K1d W2KdK
The weights W1 reflect the importance of the achievement of the goals for
the design The weights must satisfy
k1and WkOforall
Only the Preemptive Case is considered here although the Archimedean formula
tion may be developed similarly
1.2 Thefuzzyforni of the compromise DSP
About 25 years ago Zadeh4 proposed mathematics of fuzzy or cloudy quantities
which are not describable in the terms of probability distributions Bellman and
Zadeh5 then developed procedure for fuzzy optimization Several teams began to
work in this area However usually they applied fuzziness uniformly to both goals
and constraints see Refs to 18 More recently Diaz9 has used fuzzy set theory
to develop multilevel fuzzy optimization procedure brief introduction to the
relevant aspects of fuzzy set theory follows further information is available in
Kandelt3
Fuzziness can be used as measure of complexity of model3 Fuzziness is
classified in three ways namely generality vagueness and ambiguity Generality
implies that fuzzy sets model several features or goals vagueness implies that the
boundaries are not precise and ambiguity that there is more than one distinguishable
subfeature i.e there is more than one local maximum
fuzzy number is characterized by main value and membership function
uAx which represents the grade of membership of in the fuzzy set The
membership function is assigned value of if is completely member of the fuzzy
set and value of if is not member of At present there is no mathematical
way of assigning shape to fuzzy membership function priori In this initial
formulation of the fuzzy compromise DSP the linear membership function in Eq
is used
Im-x1 mcx1cmc1andc1O/IAx
otherwise
fuzzy number is represented by its center and the width of the band of fuzziness
surrounding it
Amc
DECISIONS UNDER UNCERTAINTY 27
fuzzy possibility distribution is defined3 as Let be function of and let
take values in fx and possibility distribution function associated with
is fuzzy constraint on the values of that may be assigned to When
possibility distribution function is associated with constraint it may be thought
of as the degree of feasibility or the degree of compatibility of the design with
the constraints If it is associated with goal it may be thought of as the degree of
goal satisfaction
The extension principle permits the general extension of mathematical constructs
from nonfuzzy to fuzzy environment linear equation is24
yfXatXOTo create parallel fuzzy and non-fuzzy formulations of the compromise DSP it is
necessary to set A0 then
m0 c0H c1HX1 l.MThe extension principle can be used to define all types of fuzzy functions
DEVELOPMENT OF THE FUZZY COMPROMISE DSP
In this section the fuzzy form of the standard compromise DSP is developed The
standard DSP Section 1.1 is fuzzified and reformulated to result in fuzzy set of
feasible solutions but with crisp answer that is compromise DSP with fuzzy
system parameters and crisp system variables
2.1 System descriptors for the fuzzy compromise DSP
The general structure of the standard compromise DSP formulation presented in
Section 1.1 forms the basis for the formulation of the fuzzy compromise DSP System
descriptors of the standard and fuzzy compromise DSPs are in Table In the fuzzy
formulation the constant parameters in the goal and constraint equations may be
fuzzy in the standard formulation they are crisp In both cases the system variables
are not fuzzy they are crisp Thus in both the standard and fuzzy compromise DSPs
the solution to the design problem is crisp
Variables for the fuzzy compromise DSP The standard compromise DSP is
described in terms of system variables and deviation variables In the fuzzy formula
tion there are also crisp system variables
XXoXl...XL...XM_l XjO i1...M1where X0
Note that is defined in this way to emphasize the relationship between the crisp
and fuzzy compromise DSPs
28 Q.-J ZHOU ALLEN AND MISTREE
Table System descriptors of the standard and fuzzy compromise DSPs
Standard DSP Fuzzy DSP
Variables Variables
System Variables System Variables
Deviation Variables Possibility Distributions
System Constraints Fuzzy System Constraints
In terms of System Variables In terms of System Variables and
Possibility Distributions
Svs tern Goals Fuzzy System Goals
In terms of System Variables and In terms of System Variables and
Deviation Variables Possibility Distributions Hk
Deviation Function Deviation Function
Status minimizing Status maximizing
In terms of Deviation Variables In terms of and Hk
System constraints for the fuzzy compromise DSP In the standard compromise
DSP system constraints are described by system variables and crisp parameters In
the fuzzy compromise DSP fuzzy system constraints are described by system
variables and fuzzy parameters The crisp parameters in the constraint equation are
replaced by fuzzy numbers and the constraint equation becomes
CAcand the fuzzy demand
DJAdJ
Each of the system variables in is related to an He or Hd which measures its
compatibility with the constraints and thus the fuzzy constraint equation is24
C34mc cc Hc D.md cd Hdj1...J
The symbol means is fuzzily greater than or equal to20 For ease of solution
in the fuzzy formulation of the DSP all constraints must be rearranged so that the
left hand side is greater than or equal to the right hand side In very large problems
grouping of the fuzzy numbers is essential for calculation speed and solution
convergence Many authors choose to group everythingconstraints and goalsat single level of fuzziness We have chosen to describe the constraints as uniformly
fuzzy and have permitted the goals to be fuzzified individually Therefore
I-Ic Hd and cc cd
is defined on the interval when all constraints are crisp when
DECISIONS UNDER UNCERTAINTY 29
the constraints are maximally fuzzy Be substitution Eq becomes
C3.mc DmdIn the fuzzy compromise DSP formulation Eq replaces Eq As in the standard
DSP the capability and demand functions may be either linear or nonlinear
Fuzzy system goals Omitting the deviation variables Eq becomes
TkAtk PkApk 10
Eq 10 is valid if the target is equal to or greater than the performance if the designer
does not expect this to be the case larger target value must be selected Similarly
to the constraint equations performance is fuzzified by replacing the crisp number
Apk with fuzzy numbers Apk Fuzzy performance is then
PkApk or pkmpk CPk Hpk
and fuzzy target would be
TkAtk or Tkmtk ctk Htk
Thus the most general form of the fuzzy goal equation is
Tkmtk ctk Htk PkmPk CPk Hpk 11
Fuzzy decisions In fuzzy environment the feasible design space is determined by
the intersection of space bounded by fuzzy constraints and the aspiration space
representing the fuzzy goals fuzzy decision DC is the fuzzy set of alternative
solutions resulting from the intersection of the fuzzy constraints and the fuzzy
targets Therefore in its most general form the feasible design space is
DCAdc CAc TAt 12
and the grade of membership23 is
PDC PT
where JT denotes min /T or min HTI discussion of the rules
governing the mathematical manipulations of fuzzy sets can be found in Ref
For fuzzy optimization it is necessary to find the largest mm JUT
/1DC max minthus
Hdc max minHtk 1.. 13
30 Q.-J ZHOU ALLEN AND MISTREE
represents the level of fuzziness of all system constraints and measures the extent
to which the individual system constraints belong to fuzzy set of system constraints
is also the grade of system compatibility The larger the value of the more
completely the constraints are satisfied
The fuzzy preemptive deviation function In the standard DSP the objective is to
minimize the deviation of the performance from the target In the fuzzy compromise
DSP the objective is to maximize the compatibility of the possibility distributions
and Htk as required by Eq 14 Thus in the fuzzy DSP formulation fuzzy
deviation function is maximized fuzzy preemptive deviation function is shown in
Eq 14
max H1.. 14
where the possibility distributions are ranked lexicographically10
The fuzzy Archimedean deviation function This function is stated as follows
max WHtand Wk represent the weights reflecting designers desire to achieve con
straints or certain goals more than others for the constraints and the kth target
respectively
2.2 The fuzzy compromise Decision Support Problem
The fuzzy compromise DSP is obtained from the standard compromise DSP
presented in Section 1.1 by replacing constraint Eq with Eq goal Eq with
Eq 11 and Eq with Eq 14
Given
An alternative defined by the vector of independent system variables which
is crisp vector
system constraints that must be satisfied for an acceptable solution
Estimated fuzzifiers membership functions associated with the goals and
constraints
CAc is the fuzzy capability associated with thejth system constraint Acare the fuzzy parameters needed to characterize the capability associated with the
jth constraint The capability can be linear or nonlinear function of any
type or degree of convexity
D.Ad is the fuzzy demand associated with thejth system constraint Adare the fuzzy constants needed to characterize fuzzy demand and represents the
system variables Hd is the fuzzy possibility distribution of the demand
DECISIONS UNDER UNCERTAINTY 31
is the number of system goals to be achieved to reach specified fuzzy target
TkAtk Atk are the fuzzy constants needed to specify the kth target target need
not be function of the system variables but the most general case is given here
PAp is the fuzzy performance on the kth system goal Apk are the fuzzy
constants needed to characterize performance
Find
The values of the independent system variables crisp X1 1..The maximum degree of compatibility of all system constraints
The maximum degree of satisfaction desired for each target Htk
1...KandOHtk
Satisfy
Fuzzy system constraints is Equal to or Exceeds Demand
Cmc D3.md1
Fuzzy System Goals is Equal to or Exceeds Performance
Tkrntk ctk Htk Pkmpk CPk Hpkk1...K 11
Bounds For
For the possibility distributions
Htk Hpk Hd1
j1 and k1...K
Maximize
Fuzzy preemptive deviation function
max H1 .. Hk 14
32 Q.-J ZHOU ALLEN AND MISTREE
Vague or imprecise information may be modelled explicitly using the fuzzy
compromise DSP However in spite of the vagueness in the problem statement
crisp nonfuzzy solution is obtained Moreover the standard crisp formulation
of the DSP is specific case of the more general fuzzy form If all fuzzifiers
are set to zero in Eqs and it then all fuzzy sets are replaced by their main values
and the fuzzy DSP reduces to the crisp DSP
DESIGN OF FOUR-BAR PATH-TRACING LINKAGE
To understand the fuzzy compromise DSP better planar four-bar path-tracing
linkage problem is studied This is highly non-linear problem with multiple
objectives that is difficult to solve using standard formulations2122 However it is
ideal for fuzzy compromise DSP Although the results are clear we do not focus
on specific solutions to the four-bar linkage problem but instead use it to demonstrate the fuzzy compromise DSP
3.1 four-bar linkage for path generation
Problem statement planar four-bar path generating linkage is to be designed
Figure It is composed of four links connected by four pin joints The links are to
be rigid and of uniform cross-sectional area This linkage must be capable of tracing
given path specified by set of accuracy points the prescribed path well-
designed linkage would be able to touch each point precisely It must also satisfice
transmission angle characteristics The system variables that must be determined are
as follows the length of the input link L1 the length of the floating link L2 the
length of the output link L3 the length of the fixed link L4 the length of the coupler
link L5 the size of the coupler angle coordinates of the ground pivot X0 Y0 and
the inclination of the ground link with the horizontal 61Constraints include all those used in traditional design
To permit efficient force transfer to the output link the transmission angle
must lie between Pmin to /tmax for all angles 02 during the rotation of the input link
The linkage must allow complete rotation of the input link and therefore must
satisfy Grashofs criterion
Practical considerations bind the coupler locus to the region defined by
Xmax min and maxThe linkage should have minimum structural error That is at the specified
accuracy points the deviation of the actual path X1 Y1 from the prescribed path
Xx should be minimum Further the overall structural error must also be
minimized
In real linkage the path followed by the coupler often deviates somewhat from the
prescribed path For complete rotation of the input link an estimate of the accuracy
DECISIONS UNDER UNCERTAINTY 33
Prescribed Path Actual Path
L5
L3
L2 L4
92o L1
Yo
x0
Expanded View of the Path
Prescribed Path Actual Path
Path EITorlY4 41
Figure Path-generating four-bar linkage showing the system variables
of the path generated by the coupler is obtained by taking the sum of the deviations
of the actual path from the prescribed path this is referred to as the structural error
of the linkage
set of accuracy points are specified along the desired path to compare
the prescribed path and actual path At each of the specified coordinates along
the path X1 the differences between the coordinates 1Y51 Y1I are
summed to obtain the structural error The difference between the coordinates
IY YI at the ith position is the path error at that point The objectives in
kinematic synthesis are to minimize the structural error in the linkage and to achieve
34 Q.-J ZHOU ALLEN AND MISTREE
minimum path error at certain pre-specified accuracy points through appropriate
choice of system parameters consistent with the constraints imposed on the de
sign
3.2 The four-bar linkage problem The standard non-fuzzy compromise DSP
The mathematical formulation of constraints and goals is based on the kinematic
analysis of the four-bar linkage and linkage performance20
Given
Accuracy points on the prescribed path X1 1..Lower and upper limits on transmission angle /2mjfl and I1max
Spatial bounds on the coupler locus min and maxPosition of ground pivot X-axis X0 and Y-axis Y0
System variables Units
Fixed Link
Input Link L2
Output Link L3
Floating Link L4
Coupler Link L5
Coupler angle
Inclination of fixed link to horizontal 61
Satisfy
System constraints
Grash ofs criterion for crank-rocker linkages must be satisfied2
L1L2L3L4L2L1
L2L4L1 L22 L3 L42 15
The value the transmission angle must lie between tmjfl and Pmax
L1 L22 2L3L4Lmin
L1 L22 2L3L4iUmax 16
Where mifl and /tmax are the lower and upper bounds on the transmission angle
DECISIONS UNDER UNCERTAINTY 35
The coupler locus must lie within the space defined by Xmj Xmax min and max
X0 L2 cos02J L5 coscx 3j Xmjn
Xmax X0 L2 COS02J L5 coscx 3j
Yo L2 SIflO2j L5 sin 03j min
kmax L2 sin023 L5 sinx 03j 17
System goals
The path error at the accuracy points should be minimum
Y0 L2 sin021 L5 sinx 03.1 01/Y1 dj 1.0 18
Y0 L2 sin023 L5 sino 033 91/Y53 1.0 19
Y0 L2 sin925 L5 sin 035 O1/Y5 1.0 20
The structural error should be minimum at points and
IY YI 0.0 212.5
Bounds
On link lengths
Lmin Lmax
On coupler angle
min max
Ground pivot position X0
X0 X0 Xomax
Ground pivot position Y0
Omin Omax
Inclination of fixed link
1min 0i 0imax 22
36 Q.-J ZHOU ALLEN AND MISTREE
Minimize
Preemptive formulation For convenience the pseudo-preemptive form of the devia
tion function is used
Pl13 P14d 23
Goals 13 are to minimize the path error at the accuracy points Eqs 1820 Theyare assigned equally high priorities Goal is to minimize the structural error
Eq 21 designer has decided that it is more desirable for Goals 13 to be satisfied
than for Goal to be satisfied
P113 P14
where indicates preference
The problem is solved using the DSDES software9
3.3 The four-bar linkage problem The fuzzy compromise DSP
Four aspects of the fuzzy formulation of the four-bar linkage problem will be
investigated to determine their influence on the results
CASE The effect of introducing fuzziness into the design of four-bar linkage
Three cases are used to assess the effect of introducing fuzziness into the formulation
CASE Al Is crisp non-fuzzy and uses the standard compromise DSPformulation
CASE A2 Is partially fuzzy compromise DSP in which only the goals are
fuzzy i.e the problem is antisymmetric
CASE A3 Is completely fuzzy compromise DSP with both fuzzy goals and
fuzzy constraints
CASE Al is standard DSP Using CASE Al as basis CASES A2 and A3 are
fuzzified using the rules given in Zhou20 The fuzzy formulation for CASE A3 is
presented in Table CASE A2 is combination of CASES Al and A3 crisp
constraints are used as in CASE Al and fuzzy goals are used as in CASE A3 The
fuzzifiers in both CASES A2 and A3 are set arbitrarily to of the values of the
main values The results are presented in Table
In Table L1 are system variables In CASE Al d7 and dIare deviations from thejth goal In CASES A2 and A3 Hjj 1.. represent the
degree of the satisfaction of thejth goal In CASE A3 is the grade of constraint
compatibility The solution in CASE A3 is superior to that of CASE A2 because it
has greater grade of constraint feasibility and higher degrees of goal satisfaction
DECISIONS UNDER UNCERTAINTY 37
Table The mathematical formulation of the fuzzy four-bar linkage problem CASE Al
GIVEN Accuracy points on the prescribed path Xs YsChosen Points XP YP1Ground Pivot X0 Y0Constraint fuzzifIcrs cc ith constraint jth fuzzilier in that constraint
here cc41 cc61Goal Fuzzifiers cg jth goalP1 Pl1_3 P14
021 03i correspond to accuracy points YJ02Pi 03P1 correspond to chosen points Y1
mm mx /1min Pmax min maxLmini Lmam
FIND L1 L2 L3 L4 L5H1 H2 H3 H4
SATISFY
CONSTRAINTSGrashofs cc11H L22 cc12HL3 L42Criteria
Transmission cc21HL1 L22 cc2 1HLAngle cc2 3H COSmj
cc3iHL cc3 2HL1 L22cc33H COSMmaj
Coupler cc4 1H L2 cos623 L5 cosz 033 01/Xs3 1.2
Locus 0.8 cc5 1H L2 cos021 L5 cos 031 01/Xs1cc1H L2 sin023 L5 sin 033 01/Ys3 1.2
0.8 cc7 1H L2 sin021 L5 sin 03 01/Ys1
BOUNDS
On link lengths Limin LjmaOn coupler angle mjfl maxInclination of
fixed link 1nifl 0j
Possibility
Distributions H1 H2 H3 H4
GOALS
Path Y0 L2 sin021 -4- L5 sin 031 cg1H1Ys1Error Y0 L2 sin022 L5 sin 032 cg2H2Ys3
Y0 L2 sin023 L5 sinx 032 cg3H3Ys5Structural
Error 0.185 cg4H4 ABS L2 sin0211
L5 sin 0k YP1
MAXIMIZE P1H Pl13H1 H2 H3 P14H4
see Table 3a The fuzzy CASES A2 and A3 converge to solution faster than the
crisp CASE Al
CASE Effect of ranking goals in the four-bar linkage problem
The focus of this study is on the ranking the values and the distribution of rankings
in the achievement function Using CASE A3 as the basic model the deviation
38 Q.-J ZHOU ALLEN AND MISTREE
Table Results of Case Study
Solutions
Variable Al A2 A3
L1 5.142 8.196 8.622
L2 1.042 0.752 0.694
L3 5.605 10.0 10.0
L4 3.859 9.400 10.0
L5 0.918 0.737 0.685
0.202 0.482 0.0004
01 4.324 4.013 4503
d1 0.151d2 0.233d3 0.866d4 0.146
0.9999
H1_3 0.985 0.9999
H4 0.980 0.9998
Convergence to the solution
Al A2 A3
of Cycles 20 13
Cycle Reached SoIn 19 13
function is modified by using the weights
CASE BI PV P1 P11_3 Pl4
CASE B2 P12 Pl P11....3 P14
CASE B3 Pt3 P1 Pl1..3 P14 52CASE B4 Pt4 P1 Pl3 P14
where
pjk is the vector of weights for CASEPP is the weight of the jth goal
The system variables obtained for CASES B2 and B4 are similar see Table 4a
However their weighting vectors have little in common P12 and PI4
Apparently the goal weights alone do not have clear influence on
constraint compatibility or goal satisfaction The rate of convergence for this case
study is also shown in Table
CASE Effect of the size offuzzfiers in the four-bar linkage problem
The effect of fuzzifiers on the solution is studied in this section Four sets of fuzzifiers
are inserted into the basic fuzzy formulation CASE A3 The fuzzifiers are expressed
DECISIONS UNDER UNCERTAINTY 39
Table Results of Case StudySolutions
Variable BI B2 B3 B4
L1 8.622 9.998 9.996 7.972
L2 rn 0.694 0.601 0.585 0.683
L3 10.0 10.0 10.0 10.0
L4 10.0 7.620 10.0 7.218
L5 0.685 0602 0.558 0.669
0.0004 0.241 4.618 0.0255
01 4.503 4.751 0.120 4.821
0.9999 0.9980 0.9961 0.9999
H1_3 09999 0.9981 0.9961 0.9999
H4 0.9998 0.9980 0.994 0.9998
4.999 4.990 4.979 4.999
Convergence to the solution
BI B2 B3 B4
of Cycles 13 20
Cycle Reached Soln 13 13
as percentage of the corresponding main values The sets of fuzzifiers are used in
the basic formulation CASE A3 is generalized fuzzifier for the constraints
corresponding to all cc13 in Table
CASE Cl The set of fuzzifiers cgj3 Cg4 0.5
CASE C2 The set of fuzzifiers Cg_3 Cg4
CASE C3 The set of fuzzifiers Cg_3 Cg4 23CASE C4 The set of fuzzifiers Cg4 16
where represents the fuzzifiers associated with all constraints
Cgj3 are the fuzzifiers for goals to
Cg4 are the fuzzifiers for goal
The results are in Table In this case the constraint with the smallest fuzzifier is
best satisfied in the solution Yet when the fuzzifiers are larger the overall constraint
compatibility and degree of goal satisfaction are greater Also in this problem as
shown in Table Sb the smaller the fuzzifier the faster the convergence
CASE Antisymmetric structure offour-bar linkage problem
In many fuzzy programming techniques constraints and goals are treated identically
and fuzzified uniformly68 these formulations are symmetric However in the fuzzy
compromise DSP constraints and goals can be treated separately that is the problem
40 Q.-J ZHOU ALLEN AND MISTREE
Table Results of Case Study
Solutions
Variable Cl C2 C3 C4
L1 9.098 2.716 10.0 10.0
L2 1.894 0.507 0.792 1.198
L3 10.0 3.084 8.907 9.882
L4 7.182 2.3 13 7.328 7.798
L5 1.355 0.571 0.802 0.827
0.1009 0.013 5.092 4.234
01 3.595 5.329 0.0004 0.0070
l1 0.970 0.996 0.9999 0.9991
H1 0.965 0.995 0.9999 0.9986
Fl20.953 0.994 0.9999 0.9985
H3 0.946 0.993 0.9999 0.998
H4 0.969 0.996 0.9999 0.9990
4.803 4.971 4.999 4.994
Convergence to the solution
Cl C2 C3 C4
of Cycles 10 13
Cycle Soin Reached 10 13
may have an antisymmetric structure In this section we investigate the differences
between symmetric and antisymmetric structures and attempt to determine the
advantages of each type of formulation The results of this investigation are presented
in Table
CASE Dl Has symmetric structure The fuzzifiers in CASE Cl are used
CASE D2 Has an antisymmetric structure This case also uses the fuzzifiers
from CASE Cl
CASE D3 Has symmetric structure The fuzzifiers in CASE C2 are used
CASE D4 Has an antisymmetric structure and also uses the fuzzifiers from
CASE C2
For this problem the grades of constraint compatibility are higher in the antisym
metric CASES D2 and D4 than in the symmetric ones Dl and D3 This implies
that using an antisymmetric structure provides more accurate model of this
problem Fewer synthesis cycles are required to reach solution in the symmetric
cases than in the antisymmetric ones thus the solution converges more rapidly
SOME OBSERVATIONS
Based on this example some general observations are made These observations are
intended to indicate directions for further investigation and should not be extra
DECISIONS UNDER UNCERTAINTY 41
Table Results of Case Study
Solutions
Variable Dl D2 D3 D4
L1 5.142 8.196 8.622 8.426
L21.042 0.752 0.694 0.4978
L35.605 10.0 10.0 10.0
L4 ml 1859 9.400 10.0 10.0
L5 0.918 0.737 0.685 0.571
0.202 0.482 0.0004 0.032
Oj4.324 4.013 4.503 4.886
0.996 0.999 0.9905 0.9999
H1_3 0.985 0.9999
H4 0.980 0.9998
Convergence to the solution
Dl D2 D3 D4
of Cycles 13
Cycle Reached Soln 13 13
polated beyond the domain of this problem From Case Study it is apparent
that formulating fuzzy compromise DSP can be useful and appropriate in certain
cases The fuzzy compromise DSPs converge faster than the standard compromise
DSPAs seen from Case Study the size of the fuzzifiers has great influence on the
solution obtained However fuzzifier size is often subjective It is recommended that
small fuzzifiers be used for constraints while comparatively larger ones are used for
goals If possible small fuzzifiers should be assigned to the most important con
straints/goals It would be worthwhile for an astute design manager to spend
resources investigating and refining the important design constraints and goals so
that they have the smallest possible fuzzifiers However the smaller the fuzzifier the
smaller the chance of reaching solution because the solution must lie within the
intersection of constraints and goals On the other hand if the fuzzifiers are too large
as in CASE C4 good solution ceases to exist Large fuzzifiers require that the
system be good in too great range solution to the problem can be obtained
only by sacrificing some degree of goal satisfaction or constraint compatibility The
importance of the fuzzifier size also explains why changing the weights of the goals
has so little apparent effect The location of the solution in the design space is
determined by the extent of fuzzification Cases with larger fuzzifiers converge more
slowly This may be because as the size of the fuzzifier increases the number of
participating active constraints also increases and therefore the number of vertexes
in the feasible space increases Therefore it takes more cycles to reach solution To
generalize from this problem the rate of convergence is inversely related to fuzzifier
size
42 Q.-J ZHOU ALLEN AND MISTREE
The choice between symmetric structure and an antisymmetric one must reflect
the designers needs Usually solution obtained from an antisymmetric formula
tion has higher degree of goal satisfaction while solution obtained from symmetric
formulation converges faster For standard design problem the antisymmetric
formulation is preferred because of its greater reliability but for an on-line problem
the symmetric formulation may be desirable because of its rapid rate of convergence
Acknowledgements
This work was completed at the University of Houston During her graduate studies Q.-J Zhou was
supported by her family and funds generated by the Systems Design Laboratory from industry The cost
of computer time was underwritten by the University of Houston We gratefully acknowledge the financial
support of our corporate sponsor the B.F Goodrich Company to further develop the Decision Support
Problem Technique We gratefully recognize the valuable suggestions offered by Sridhar Srinivasan for
improving this paper
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