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

Reprints available directly from the publisher Printed in the United Kingdom

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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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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



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


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



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


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


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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under uncertainty the fuzzy compromise support …€¦ · key words uncertainty fuzzy sets...

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