Eng Opt 1992 Vol 20 pp 21 43
`Reprints available directly from the publisher
`Photocopying permitted by license only
`
`1992 Gordon and Breach Science Publishers S.A
`Printed in the United Kingdom
`
`DECISIONS UNDER UNCERTAINTY THE FUZZY
`COMPROMISE DECISION SUPPORT PROBLEM
`
`Q.-J ZHOU
`Shp Engineering Department ABSAMERICAS 263 North Belt East
`Houston Texas 77060 US.A
`
`Janco Research Inc 4501 University Oaks Boulevard Houston Texas 77004 U.S.A
`
`ALLEN
`
`and
`
`MISTREE
`
`Woodruff School of Mechanical
`Systems Design Laboratory The George
`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
`the compromise DSP requires precise numerical
`achieve multiple objectives However
`information to
`yield rigorously accurate results In the early stages of conceptual
`design such precise information is often
`design should be reliable manufacturable maintainable and cost-efficient
`unavailable For example
`the design must possess
`Although inherently vague each qualifier specifies an important attribute that
`Such vagueness may be modeled rigourously using the mathematics of fuzzy set
`theory In this paper we
`fuzzy formulation of the compromise DSP
`formulation which is particularly suitable for
`introduce
`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
`
`sup
`
`inf
`
`max
`
`Is member of the set or is contained in
`The intersection of
`The union of sets
`mapping from the set on the left
`Indicates
`Is almost positive
`The least upper bound
`lower bound
`The greatest
`The largest of fuzzy sets
`
`21
`
`to the set on the right
`
`MYLAN - EXHIBIT 1029
`
`

`
`22
`
`mm
`Ax uAx
`
`Ac
`Ad
`
`Apk
`
`Atk
`
`C3.Ac3
`
`cc
`
`cd3
`cpk
`ctk
`D3Ad3
`
`DC
`
`Hc
`
`Hd3
`
`IIPk
`Htk
`
`Q..-J ZHOU
`
`ALLEN AND
`
`MISTREE
`
`The smallest of or the intersection of fuzzy sets
`whose grade of membership
`the value
`
`at
`
`represents the fuzzy set
`is determined by the membership function
`The constants parameters associated with the capability of the system
`to the jth constraint on the system
`with respect
`The constants parameters associated with the demand on the system
`due to the jth constraint
`The constants parameters associated with the performance of
`system on the kth target
`Constants constants associated with the designers aspirations for the
`kth target
`linear or nonlinear capability associated with
`constraint
`function of the parameters Ac3 and the system variables
`used to indicate fuzziness
`The extent of the cloud of fuzziness surrounding the main value of
`to the range of the membership
`fuzzy set This is numerically equivalent
`function
`The fuzzifier associated with the grouped constraints
`The fuzzifier associated with the parameters specifying the systems
`capability in meeting the jth constraint
`The fuzzifier associated with the demand due to the jth constraint
`on the kth target
`The
`fuzzifier associated with the performance
`The fuzzifier associated with the kth target
`Demand associated with the jth constraint Demand is
`and parameters Ad3 Fuzzy demand is
`the system variables
`denoted by italics
`decision
`Deviation variables used in the crisp non-fuzzy compromise DSP
`formulation
`possibility distribution
`The possibility distribution for the capability of meeting the jth con
`straint
`The possibility distribution for the degree of compatibility of the system
`associated with the demand from the jth constraint
`the performance on the kth target
`The possibility distribution for
`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
`incom
`implies total
`implies complete compatibility
`
`the
`
`that
`
`is
`
`Italics are
`
`function of
`
`patibility
`Total number of constraints in DSP
`DSP
`Total number of goals in
`The number of system variables in DSP
`the value which is surrounded by
`The main value of
`cloud of fuzziness
`
`fuzzy set
`
`

`
`DECISIONS UNDER UNCERTAINTY
`
`23
`
`The main value of the fuzzy set which represents the systems capability
`on the jth constraint
`The main value of the fuzzy set which represents the demand associated
`with the jth constraint
`the fuzzy set which represents the performance
`The main value of
`associated with the kth target
`thc fuzzy set which represents the kth target
`The main value of
`system characterized by the system variables
`Actual performance of
`and the parameters Apk
`fuzzy performance function is denoted
`by italics
`of the system goals used
`Priority ranking factors for the achievement
`in both the crisp and the fuzzy formulation DSP
`Target or aspiration level for system performance at
`and the parameters Atk
`by the system variables
`is denoted by italics
`level
`crisp vector of system variables
`function representing the difference between
`An achievement
`performance and the designers goals for the system
`The membership function associated with the fuzzy set
`
`system
`
`the point defined
`
`fuzzy aspiration
`
`mc3
`
`md3
`
`mpk
`
`mtk
`PkApk
`
`P1
`
`TkAtk
`
`PAX
`
`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
`that can be manufactured and main
`for human judgment in designing an artifact
`means of modeling decisions en
`tained Decision Support Problems provide
`countered in design manufacture and maintenance Formulation and solution of
`means for making the following types of decisions
`DSPs provide
`
`Selectionthe indication of
`feasible alternatives
`among several
`Compromise trade-ofl----the improvement of an alternative through modifica
`
`preference based on multiple attributes for one
`
`tion
`
`Hierarchicaldecisions
`
`Conditional--decisions
`
`taken into account
`
`involve interaction between sub-decisions
`
`that
`in which the risk and uncertainty of
`
`the outcome are
`
`to
`
`Compromise DSPs refer
`class of constrained multiobjective optimization
`problems that are used in wide variety of engineering applications Both selection
`of an
`and compromise DSPs can
`the hierarchical
`be
`part of
`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
`
`representation
`
`

`
`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
`DSPs have been developed
`presented in Ref
`for hierarchical design coupled
`compromise-compromise and selection-selection6 These con
`selection-compromise
`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 compro
`mise DSP is solved using
`unique optimization scheme called Adaptive Linear
`are described in Refs
`Programming9 Other formulations of conditional decisions
`
`For
`the information for modeling systems
`real-world practical systems all of
`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
`is being designed This uncertainty stems from vagueness
`nature of the object
`that
`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
`representation of the object being designed However
`form an accurate mathematical
`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
`theoretical model
`for the fuzzy compromise DSPs
`In this paper we present
`non-linear kinematics problem involving the
`followed by an example of their use
`four bar
`linkage The
`minimization of
`the structural error
`path-generating
`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
`
`in
`
`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
`here The generalized
`and will
`therefore not be repeated
`several publications39
`the fuzzy compromise DSP that
`follows however
`has not been
`formulation of
`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
`is the capability associated with the jth system constraint Ac
`C.Ac3
`represents the constant parameters needed to characterize the capability associated
`with the jth constraint The capability can be
`nonlinear function
`linear or
`
`

`
`DECISIONS UNDER UNCERTAINTY
`is the demand associated with the jth system constraint Ad
`DAd3
`represents the constants needed to characterize the demand These constants are
`some of the parameters characterizing the compromise DSP
`represents the system
`
`25
`
`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
`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
`
`is
`
`Find
`
`The values of the independent system variables
`
`1..
`The values of the non-negative deviation variables indicating the extent to which
`and dk represent under-achievement and over-
`the target values are attained
`I..
`and
`and
`such that
`where
`achievement of the target
`
`Satisfy
`
`System Constraints
`
`is Equal
`
`to or Exceeds Demand
`
`DAd
`With lower and upper bounds on the system variables
`
`CAc3
`
`System Goals
`
`is Equal
`
`to or E.xceeds Performance
`
`PkApk
`
`TkAtk
`
`1.
`
`Minimize
`
`quantifies the deviation of the system performance
`deviation function
`from the ideal as defined by the set of target values TkAgk
`PkApk
`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
`the design The weights must satisfy
`
`of the goals for
`
`k1
`
`and WkOforall
`
`Only the Preemptive Case is considered here although the Archimedean formula
`tion may be developed similarly
`
`1.2
`
`Thefuzzyforni of the compromise DSP
`
`fuzzy set
`
`brief introduction to the
`
`information is available
`
`in
`
`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
`teams began to
`procedure for fuzzy optimization Several
`work in this area However usually they applied fuzziness uniformly to both goals
`to 18 More recently Diaz9 has used fuzzy set theory
`and constraints see Refs
`fuzzy optimization procedure
`to develop
`multilevel
`theory follows further
`aspects of
`relevant
`Kandelt3
`measure of complexity of model3 Fuzziness is
`Fuzziness can be used as
`and ambiguity Generality
`classified in three ways namely generality vagueness
`fuzzy sets model several
`the
`implies that
`features or goals vagueness
`implies that
`boundaries are not precise and ambiguity that there is more than one distinguishable
`subfeature i.e there is more than one local maximum
`is characterized by main value and
`fuzzy number
`membership function
`uAx which represents the grade of membership of
`The
`in the fuzzy set
`is completely member of the fuzzy
`value of
`membership function is assigned
`is not member of
`there is no mathematical
`set and
`At present
`value of
`fuzzy membership function
`In this initial
`shape to
`priori
`way of assigning
`formulation of the fuzzy compromise DSP the linear membership function in Eq
`is used
`
`if
`
`if
`
`Im-x1 mcx1cmc1andc1O
`
`/IAx
`
`otherwise
`
`fuzzy number is represented by its center
`surrounding it
`
`and the width of the band of fuzziness
`
`Amc
`
`

`
`DECISIONS UNDER UNCERTAINTY
`
`27
`
`be
`
`function of
`
`is
`
`it
`
`fuzzy possibility distribution is defined3 as Let
`and let
`fx and
`associated with
`possibility distribution function
`take values in
`When
`that may be assigned to
`on the values of
`fuzzy constraint
`it may be thought
`is associated with
`constraint
`possibility distribution function
`of as the degree of feasibility or the degree of compatibility of the design with
`it may be thought of as the degree of
`is associated with
`goal
`the constraints If
`goal satisfaction
`The extension principle permits the general extension of mathematical constructs
`fuzzy environment
`linear equation is24
`from nonfuzzy to
`
`yfXatXO
`To create parallel fuzzy and non-fuzzy formulations of the compromise DSP it
`then
`to set A0
`
`necessary
`
`is
`
`m0
`
`c0H
`
`c1HX1
`
`l.M
`
`The 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
`fuzzy set of
`compromise DSP with fuzzy
`crisp answer
`feasible solutions but with
`that
`is
`system parameters and crisp system variables
`
`in
`
`2.1
`
`System descriptors for the fuzzy compromise DSP
`The general structure of the standard compromise DSP formulation presented
`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
`
`in
`
`XXoXl...XL...XM_l
`where X0
`
`XjO i1...M1
`
`is defined in this way to emphasize the relationship between the crisp
`Note that
`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
`System Variables
`Deviation Variables
`
`System Constraints
`In terms of System Variables
`
`Svs tern Goals
`In terms of System Variables and
`Deviation Variables
`
`Deviation Function
`
`Status minimizing
`In terms of Deviation Variables
`
`Variables
`System Variables
`Possibility Distributions
`
`Fuzzy System Constraints
`In terms of System Variables and
`Possibility Distributions
`
`Fuzzy System Goals
`In terms of System Variables and
`Possibility Distributions Hk
`
`Deviation Function
`Status maximizing
`and Hk
`In terms of
`
`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
`by system
`are described
`variables and fuzzy parameters The crisp parameters in the constraint equation are
`replaced by fuzzy numbers and the constraint equation becomes
`CAc
`
`and the fuzzy demand
`
`DJAdJ
`
`is related to an He or Hd which measures its
`Each of the system variables in
`compatibility with the constraints and thus the fuzzy constraint equation is24
`
`C34mc cc Hc D.md cd Hd
`j1...J
`
`means is fuzzily greater than or equal to20 For ease of solution
`The symbol
`in the fuzzy formulation of the DSP all constraints must be rearranged so that
`the
`to the right hand side In very large problems
`left hand side is greater than or equal
`the fuzzy numbers is essential
`speed and solution
`for calculation
`grouping of
`and goals
`convergence Many authors choose to group everythingconstraints
`single level of fuzziness We have chosen to describe the constraints as uniformly
`at
`fuzzy and have permitted the goals to be fuzzified individually Therefore
`cc cd
`
`and
`
`I-Ic
`
`Hd
`
`is defined on the interval
`
`when
`
`all constraints are crisp when
`
`

`
`DECISIONS UNDER UNCERTAINTY
`
`29
`
`C3.mc
`
`the constraints are maximally fuzzy Be substitution Eq
`Dmd
`As in the standard
`replaces Eq
`In the fuzzy compromise DSP formulation Eq
`DSP the capability and demand functions may be either linear or nonlinear
`Fuzzy system goals Omitting the deviation variables Eq
`becomes
`
`becomes
`
`TkAtk
`
`PkApk
`
`10
`
`to or greater than the performance if the designer
`Eq 10 is valid if the target is equal
`larger target value must be selected Similarly
`this to be the case
`does not expect
`to the constraint equations performance is fuzzified by replacing the crisp number
`Apk with fuzzy numbers Apk Fuzzy performance is then
`or pkmpk CPk Hpk
`
`PkApk
`
`and
`
`fuzzy target would be
`or Tkmtk ctk Htk
`
`TkAtk
`
`Thus the most general
`Tkmtk ctk Htk
`
`form of the fuzzy goal equation is
`PkmPk CPk Hpk
`
`11
`
`Fuzzy decisions In
`the intersection of
`
`the feasible design space is determined by
`fuzzy environment
`space bounded by fuzzy constraints and the aspiration space
`fuzzy decision DC is the fuzzy set of alternative
`representing the fuzzy goals
`and the fuzzy
`the fuzzy constraints
`solutions resulting from the intersection of
`form the feasible design space is
`Therefore in its most general
`
`targets
`
`DCAdc
`
`CAc
`
`TAt
`
`12
`
`and the grade of membership23 is
`
`PDC
`
`PT
`
`where
`
`discussion of
`
`the rules
`
`or min HTI
`JT denotes min /T
`governing the mathematical manipulations of fuzzy sets can be found in Ref
`is necessary to find the largest mm
`JUT
`For fuzzy optimization it
`max min
`
`/1DC
`
`thus
`
`Hdc
`
`max min Htk
`
`1..
`
`13
`
`

`
`30
`
`Q.-J ZHOU
`
`ALLEN AND
`
`MISTREE
`
`represents the level of fuzziness of all system constraints and measures the extent
`fuzzy set of system constraints
`to which the individual system constraints belong to
`the more
`is also the grade of system compatibility The larger the value of
`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
`fuzzy preemptive deviation function is shown in
`deviation function is maximized
`Eq 14
`
`max H1..
`
`14
`
`where the possibility distributions are ranked lexicographically10
`The fuzzy Archimedean deviation function This function is stated as follows
`
`max WHt
`
`and Wk represent
`designers desire to achieve con
`the weights reflecting
`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
`goal Eq
`presented in Section 1.1 by replacing constraint Eq
`with Eq
`with
`Eq 11 and Eq
`with Eq 14
`
`Given
`
`An alternative defined by the vector of
`crisp vector
`
`is
`
`independent system variables
`
`which
`
`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 Ac
`are the fuzzy parameters needed to characterize the capability
`jth constraint The capability
`nonlinear
`be
`can
`linear or
`type or degree of convexity
`D.Ad
`is the fuzzy demand associated with thejth system constraint Ad
`needed to characterize fuzzy demand and
`are the fuzzy constants
`represents the
`system variables Hd is the fuzzy possibility distribution of the demand
`
`function of any
`
`associated with the
`
`

`
`DECISIONS UNDER UNCERTAINTY
`
`31
`
`TkAtk
`not be
`
`is the number of system goals to be achieved to reach
`specified fuzzy target
`Atk are the fuzzy constants needed to specify the kth target
`target need
`but the most general case is given here
`function of the system variables
`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
`k1...K
`
`Hpk
`
`11
`
`Bounds For
`
`For the possibility distributions
`
`Htk Hpk Hd1
`k1...K
`
`and
`
`j1
`
`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
`is obtained Moreover
`crisp nonfuzzy solution
`the standard crisp formulation
`of the DSP is
`specific case of the more general
`fuzzy form If all
`and it then all fuzzy sets are replaced by their main values
`are set to zero in Eqs
`and the fuzzy DSP reduces to the crisp DSP
`
`fuzzifiers
`
`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
`to solve using standard formulations2122 However
`that
`fuzzy compromise DSP Although the results are clear we do not focus
`to demon
`on specific solutions to the four-bar linkage problem but instead use it
`strate the fuzzy compromise DSP
`
`objectives
`
`ideal for
`
`is difficult
`
`it
`
`is
`
`3.1
`
`four-bar linkage for path generation
`
`It
`
`Problem statement
`four-bar path generating linkage is to be designed
`planar
`is composed of four links connected by four pin joints The links are to
`Figure
`be rigid and of uniform cross-sectional area This linkage must be capable of tracing
`set of accuracy points the prescribed path
`given path specified by
`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
`link L1 the length of the floating link L2 the
`as follows the length of the input
`link L3 the length of the fixed link L4 the length of the coupler
`length of the output
`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 61
`Constraints include all
`those used in traditional design
`
`To permit efficient
`link the transmission angle
`force transfer to the output
`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
`Xmax min and max
`The linkage should have minimum structural error That
`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
`
`locus to the region defined by
`
`is at
`
`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
`
`Figure
`
`Path-generating four-bar
`
`linkage showing the system variables
`
`Path EITorlY4
`
`41
`
`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
`are specified along the desired path to compare
`set of accuracy points
`coordinates along
`the prescribed path and actual path At each of the specified
`X1 the differences between
`coordinates 1Y51 Y1I are
`the
`the path
`coordinates
`summed to obtain the structural error The difference between the
`IY YI at
`that point The objectives
`the ith position is the path error at
`kinematic synthesis are to minimize the structural error in the linkage and to achieve
`
`in
`
`

`
`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
`min and max
`Spatial bounds on the coupler
`locus
`Position 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
`L1L2L3L4
`L2L1
`
`L2L4
`L1
`
`L22
`
`L3
`
`L42
`
`The value the transmission angle
`
`must lie between tmjfl and Pmax
`
`L1
`
`L22
`
`L1
`
`2L3L4Lmin
`
`L22
`
`2L3L4iUmax
`
`Where mifl and /tmax are the lower and upper bounds on the transmission angle
`
`15
`
`16
`
`

`
`DECISIONS UNDER UNCERTAINTY
`
`35
`
`The coupler locus must lie within the space defined by Xmj Xmax min and max
`
`17
`
`18
`
`19
`
`20
`
`21
`
`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
`
`System goals
`
`The path error at
`
`the accuracy points
`
`should be minimum
`
`Y0
`
`Y0
`
`Y0
`
`L2 sin021
`
`L2 sin023
`
`L2 sin925
`
`L5 sinx
`
`03.1
`
`01/Y1
`
`dj
`
`1.0
`
`L5 sino
`
`033
`
`91/Y53
`
`L5 sin 035
`
`O1/Y5
`
`1.0
`
`1.0
`
`and
`
`0.0
`
`The structural error should be minimum at points
`
`IY YI
`
`2.5
`
`Lmin
`
`Lmax
`
`Bounds
`
`On link lengths
`
`On coupler angle
`
`min
`
`max
`
`X0 X0
`
`Xomax
`
`Omin
`
`Omax
`
`Ground pivot position X0
`
`Ground pivot position Y0
`
`Inclination of fixed link
`
`1min
`
`0i
`
`0imax
`
`22
`
`

`
`36
`
`Minimize
`
`Q.-J ZHOU
`
`ALLEN AND
`
`MISTREE
`
`Preemptive formulation For convenience
`tion function is used
`
`the pseudo-preemptive form of the devia
`
`Pl13
`
`P14d
`
`23
`
`the accuracy points Eqs 1820 They
`Goals 13 are to minimize the path error at
`are assigned equally high priorities Goal
`is to minimize the structural error
`Eq 21 designer has decided that
`is more desirable for Goals 13 to be satisfied
`than for Goal
`to be satisfied
`
`it
`
`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
`investigated to determine their influence on the results
`
`be
`
`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
`
`Is crisp non-fuzzy and
`
`uses
`
`the standard compromise DSP
`
`CASE Al
`formulation
`CASE A2
`partially fuzzy compromise DSP in which only the goals are
`fuzzy i.e the problem is antisymmetric
`CASE A3
`completely fuzzy compromise DSP with both fuzzy goals and
`fuzzy constraints
`
`Is
`
`Is
`
`is
`
`standard DSP Using CASE Al as
`CASE Al
`basis CASES A2 and A3 are
`fuzzified using the rules given in Zhou20 The fuzzy formulation for CASE A3 is
`CASE A2 is
`of CASES Al and A3 crisp
`in Table
`combination
`presented
`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
`are system variables In CASE Al d7 and dI
`In Table
`L1
`In CASES A2 and A3 Hjj 1..
`are deviations from thejth goal
`represent
`In CASE A3
`degree of the satisfaction of thejth goal
`is the grade of constraint
`compatibility The solution in CASE A3 is superior
`to that of CASE A2 because it
`feasibility and higher degrees of goal satisfaction
`has
`greater grade of constraint
`
`the
`
`

`
`DECISIONS UNDER UNCERTAINTY
`
`37
`
`Table
`
`The mathematical
`
`formulation of the fuzzy four-bar
`
`linkage problem CASE Al
`
`Accuracy
`
`points on the prescribed path Xs Ys
`Chosen Points XP YP1
`Ground Pivot X0 Y0
`fuzzifIcrs cc ith constraint
`Constraint
`here cc41
`cc61
`Goal Fuzzifiers cg jth goal
`P1 Pl1_3 P14
`021 03i correspond to accuracy points
`02Pi 03P1 correspond to chosen points
`mm mx /1min Pmax min max
`Lmini Lmam
`
`jth fuzzilier in that constraint
`
`YJ
`Y1
`
`L1 L2 L3 L4 L5
`H1 H2 H3 H4
`
`cc11H L22
`
`cc21HL1
`
`L22
`
`cc2 3H COSmj
`cc3iHL
`cc3 2HL1
`cc33H COSMmaj
`cc4 1H L2 cos623
`cc5 1H L2 cos021
`cc1H L2 sin023
`cc7 1H L2 sin021
`
`cc12HL3
`cc2 1HL
`
`L42
`
`L22
`
`01/Xs3
`L5 cosz
`033
`01/Xs1
`L5 cos
`031
`01/Ys3
`L5 sin 033
`01/Ys1
`L5 sin 03
`
`1.2
`
`1.2
`
`GIVEN
`
`FIND
`
`SATISFY
`
`CONSTRAINTS
`Grashofs
`
`Criteria
`
`Transmission
`
`Angle
`
`Coupler
`
`Locus
`
`BOUNDS
`
`0.8
`
`0.8
`
`On link lengths
`Limin
`On coupler angle mjfl
`Inclination of
`
`Ljma
`max
`
`fixed link
`
`Possibility
`Distributions
`
`1nifl
`
`0j
`
`H1 H2 H3 H4
`
`GOALS
`
`Path
`
`Error
`
`Structural
`
`Error
`
`MAXIMIZE
`
`Y0
`Y0
`Y0
`
`0.185
`
`L2 sin021
`L2 sin022
`L2 sin023
`
`-4- L5 sin 031
`L5 sin 032
`L5 sinx
`032
`
`cg1H1Ys1
`cg2H2Ys3
`cg3H3Ys5
`
`cg4H4 ABS L2 sin02
`0k
`
`YP1
`H3
`
`H2
`
`P14H4
`
`11
`
`L5 sin
`P1H Pl13H1
`
`see Table 3a The fuzzy CASES A2 and A3 converge to
`crisp CASE Al
`
`solution faster than the
`
`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
`function Using CASE A3 as the basic model
`in the achievement
`the deviation
`
`

`
`38
`
`Q.-J ZHOU
`
`ALLEN AND
`
`MISTREE
`
`Table
`
`Results of Case Study
`Solutions
`
`Variable
`
`Al
`
`A2
`
`A3
`
`L1
`
`L2
`L3
`L4
`L5
`
`01
`
`d1
`
`d2
`
`d3
`
`d4
`
`H1_3
`H4
`
`5.142
`
`1.042
`
`5.605
`
`3.859
`
`0.918
`
`0.202
`
`4.324
`0.151
`0.233
`0.866
`0.146
`
`8.196
`
`0.752
`
`10.0
`
`9.400
`
`0.737
`
`0.482
`
`4.013
`
`8.622
`
`0.694
`
`10.0
`
`10.0
`
`0.685
`
`0.0004
`4503
`
`0.985
`
`0.980
`
`0.9999
`
`0.9999
`
`0.9998
`
`Convergence
`
`to the solution
`
`of Cycles
`Cycle Reached SoIn
`
`Al
`
`20
`
`19
`
`A2
`
`A3
`
`13
`
`13
`
`function is modified by using the weights
`
`PV P1 P11_3 Pl4
`CASE BI
`CASE B2 P12 Pl P11....3 P14
`CASE B3 Pt3 P1 Pl1..3 P14 52
`CASE B4 Pt4 P1 Pl3 P14
`
`where
`
`is the vector of weights for CASE
`pjk
`PP is the weight of the jth goal
`
`The system variables obtained for CASES B2 and B4 are similar see Table 4a
`in common P12
`and PI4
`their weighting vectors have little
`However
`Apparently the goal weights alone do not have
`clear
`constraint compatibility or goal satisfaction The rate of convergence
`study is also shown in Table
`
`influence on
`
`for this case
`
`CASE
`
`Effect of the size of fuzzfiers 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 Study
`Solutions
`
`Variable
`
`BI
`
`B2
`
`B3
`
`B4
`
`L1
`L2 rn
`L3
`L4
`
`L5
`
`01
`
`H1_3
`H4
`
`8.622
`
`0.694
`
`10.0
`
`10.0
`
`0.685
`
`0.0004
`
`4.503
`
`0.9999
`09999
`
`0.9998
`
`4.999
`
`9.998
`
`0.601
`
`10.0
`
`7.620
`0602
`
`0.241
`
`4.751
`
`0.9980
`
`0.9981
`
`0.9980
`
`4.990
`
`9.996
`
`0.585
`
`10.0
`
`10.0
`
`0.558
`
`4.618
`
`0.120
`
`0.9961
`
`0.9961
`
`0.994
`
`4.979
`
`Convergence
`
`to the solution
`
`of Cycles
`Cycle Reached Soln
`
`BI
`
`13
`
`13
`
`B2
`
`B3
`
`7.972
`
`0.683
`
`10.0
`
`7.218
`
`0.669
`
`0.0255
`
`4.821
`
`0.9999
`
`0.9999
`
`0.9998
`
`4.999
`
`B4
`
`20
`
`13
`
`as
`
`percentage of the corresponding main values The sets of fuzzifiers are used in
`the basic formulation CASE A3
`for the constraints
`generalized fuzzifier
`corresponding to all cc13 in Table
`
`is
`
`CASE Cl
`The set of fuzzifiers
`CASE C2 The set of fuzzifiers
`CASE C3 The set of fuzzifiers
`CASE C4 The set of fuzzifiers
`
`cgj3 Cg4
`
`0.5
`
`Cg_3 Cg4
`Cg_3 Cg4 23
`Cg4
`
`16
`
`where
`
`represents the fuzzifiers associated with all constraints
`to
`Cgj3 are the fuzzifiers for goals
`Cg4 are the fuzzifiers for goal
`
`In this case the constraint with the smallest fuzzifier
`The results are in Table
`best satisfied in the solution Yet when the fuzzifiers are larger the overall constraint
`compatibility