Dependency analysis in constraint negotiation

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 25, NO. 9, SEPTEMBER 1995 1301

Dependency Analysis in Constraint Negotiation Andrew Kusiak, Member, IEEE, and Juite Wang

Abstract- Problems in engineering design are constraint- oriented and often involve multiple perspectives. Designers must consider not only functional requirements of a product but also its life-cycle perspectives. Each perspective has its own set of constraints which may contain conflicting or unsatisfied requirements. A human designer cannot be aware of all constraints and design alternatives all the time. The objective of this paper is to develop a methodology to assist designers in negotiation of constraints. A network model is proposed to represent relationships among design variables. An algorithm is developed to derive dependencies between design variables and goals. Based on the dependencies obtained, design modifications are generated for resolving conflicts. In the second part of the paper, a fuzzy-logic-based approach is used to model imprecise dependencies between variables in the case when no sufficient quantitative information is available. The approach proposed can be used to increase the amount of information provided to the designers for making decisions.

I. INTRODUCTION

NGINEERING design is constraint-oriented and often E involves multiple perspectives [lo], [14], [ 171. Designers must consider not only the functional requirements of a product but also its life-cycle perspectives, such as manufac- turability, reliability, testing, and so on [ 191. Each perspective has its own set of constraints which may contain conflicting or unsatisfied requirements. A human designer cannot be aware of all alternatives and constraints all the time. When a conflict occurs, he/she may not know how to modify the design. It is important to have a mechanism to support the designer in detecting violated constraints, generating design modifications, and determining the impact of a design modification.

The role of constraints in design has led to research explor- ing the use of constraint-based representation to aid the design process, not only in checking the consistency of a design solution but also generating the solution. Serrano and Gossard [ 141 developed a graph-theoretic approach for evaluation of a set of constraints. A directed graph is used to represent dependencies among variables. Choy and Agogino [2] used monotonicity analysis [ 111 to determine active constraints and identify global monotonicity and degeneracies of a design problem. Rinderle and Krishnan [12] proposed an interval

Manuscript received March 13, 1994 revised October 7, 1994. This research was supported in part by The National Research Foundation, Grant DDM-9215259, and Rockwell International Corporation.

A. Kusiak is with the Department of Industrial Engineering, University of Iowa, Iowa City, IA 52242-1527 USA.

J. Wang is with the Department of Industxial Engineering, Feng Chia University, Taichung, Taiwan.

IEEE Log Number 9412701.

analysis approach with monotonicity and dominance principles to exploit regional properties of constraints, so that constraint reasoning can be facilitated by elimination of the inactive constraints. Kannapan and Marshek [9] introduced a design diagram to represent parametric design problems. They proposed a game-theoretic approach to resolve conflicts when a constraint is violated. Agrawal et al. [ l ] used an incidence matrix representing dependencies among design variables and constraints for computing dependent outputs given the value of input variables.

In most constraint management systems, it is assumed that the initial values of some design variables are determined by the corresponding perspectives and propagated to the remaining variables through a set of constraints [l], [9]. When a conflict occurs for a variable, a suitable value is selected for that variable and it is propagated backward to the variables that precede the conflicting variable. However, due to the existence of cycles and multipaths in a network of constraints, the backward chaining path is usually determined by ad-hoc heuristics [1], [9].

This paper develops an approach that analyzes dependencies among design variables and constraints to assist designers in negotiation of violated constraints. The dependency network is used to represent the qualitative and quantitative dependencies among design variables, constraints, and goals. When conflicts occur, the dependency analysis is performed to derive the dependencies between design variables and constraints as well as goals. Based on the dependencies obtained, design modification alternatives for negotiation of violated constraints are generated and the impact of the design modifications on the resolution of conflicts is determined. A feasible design modification is selected and its value is propagated forward to the remaining variables. The weakness of backward chaining approaches [1], [9] can be avoided.

In the second part of this paper, linguistic variables are used to represent dependencies between design variables in an uncertain and imprecise design environment. The fuzzy logic approach allows to represent dependencies for constraints that are not well understood and the relationship among variables that is vaguely known. Using the algorithm developed, the linguistic dependencies between variables and goals are obtained to assist designers in modifying a design.

This paper is organized as follows. Section I1 describes a network model for dependency analysis to support constraint negotiation. An example of spring design is used to illustrate the concept developed. Section 111 introduces a fuzzy logic approach to analyze dependencies among design variables. Section IV concludes the paper.

0018-9472/95$04.00 0 1995 IEEE

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1302 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 25, NO. 9, SEPTEMBER 1995

11. DEPENDENCY ANALYSIS IN CONSTRAINT NEGOTIATION

A. Problem Description A design problem can be described as a set of constraints

which are expressed by functional relations among design variables. In this paper, constraints include the following types:

Equations: Equations are normally used to represent engineering constraints (e.g., Hook’s law F = kz, where k is the spring rate and z is the compression length) and inequalites may represent, for example, design specifications or performance measures (e.g., the maximum stress ( T ) does not exceed the strength of the material ( T ~ ~ ) , T 5 T ~ ~ ) . In this paper, all inequality equations are in negative null form, i.e., h(z) 5 0. If the value of a constraint is positive, then the constraint is violated. Qualitative constraints: A qualitative constraint represents the relations among qualitative values defined by deKleer and Brown [4], e.g., “If the value of variable X increases, then the value of variable Y decreases.” Computer-based procedures: These constraints are im- plicit and are usually in the form of computer programs (e.g., design analysis or optimization programs) that take certain inputs and produce outputs. The computer-based procedures are treated as black boxes that capture the relations between input and output variables and can be considered as constraints. Influence rules: These constraints are conditional state- ments which specify a constraint under certain conditions, e.g., “if the material ASTh4229 is used, then the sheer module of the material is equal to 11.5 x lo6.”

The design variables involved in a design problem are

Decision variables: A designer is allowed to make in- dependent decisions on those variables. The subjective knowledge, experience, and desires of the designer are used to determine the value of decision variables. Design goals (objectives): A design goal is expressed with the other design variables and it is used to describe the idea of “best” design, e.g., minimizing the volume of a spring. Intermediate variables: The value of an intermediate variable is not directly determined by the designer, but by propagating the values of decision variables and design specifications through a set of constraints. In other words, a variable that is not a decision variable or design goal is an intermediate variable.

Given a set of design specifications, design can be viewed as a process of determining the values of decision variables and computing values of the intermediate variable and design goals. The design (the values of the three types of variables) is analyzed and evaluated from several perspectives. If there are no conflicting constraints or unsatisfied requirements, then the design generated is accepted. Otherwise, resolve conflicts and repeat the design problem-solving process (see Fig. 1).

The purpose of this paper is to develop a methodology to analyze and utilize dependencies among design variables to support the design decision making process. A dependency

classified into three types:

Dependency analysis

11 Redesign

Specifications

Initial design

1 A Manufacture I 1

Analysis/ Evaluation

I-- Acceptable *

1 yes &ne

Fig. 1. The design problem-solving process.

between variables is the effect of change in a value of one variable on another variable and is classified into two types: qualitative dependency and quantitative dependency. The qualitative dependency determines a direction of change of a variable affected by another variable. The quantitative dependency is the rate of change of a variable affected by another variable. As conflicts are detected, the dependency analysis is used to identify the relationship between decision and conflicting variables by reasoning with the set of constraints, as well as to suggest altematives for modification of the design in order to resolve conflicts.

The importance of dependency knowledge in design modifications has been identified in parametric design. Dixon e? al. [5] utilized dependencies between inputs and performance variables stored in a look-up table to perform a hill-climbing search. Ishii and Barkan [7] proposed a sensitivity analysis expert system that represents dependency knowledge with production rules to assist designers in the iterative design process. Both approaches are based on monotonicity analysis and explicitly represents the dependencies between decision and performance variables, which are obtained from a sim- ulation model or are estimated by the designers. A similar approach was also proposed by So0 and Wang [15]. To date, no formal dependency theory has been developed.

Section 1I.B introduces the network representation of dependency knowledge. An algorithm to obtain dependencies between decision and performance variables is presented in Section 1I.C. The dependency analysis to support constraint negotiation is presented in Section 1I.D.

B. Dependency Network

It is not sufficient to satisfy a set of constraints; it is important to represent and track dependencies in the set of constraints and evaluate the qualitative and quantitative impact of decision variables on other design variables (Finger ef al. 1990). In this paper, a dependency network is used to represent the qualitative and quantitative relationship among design variables. For a constraint, if variable b is dependent on variable a, then an arc is drawn from a to b. Thus, a set

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KUSIAK AND WANG: DEPENDENCY ANALYSIS IN CONSTRAINT NEGOTIATION 1303

of constraints can be represented as a dependency network. The network not only provides a clear representation but also improves the efficiency of the inference process. A formal definition of the dependency network is provided next.

Dejinition: Dependency network. Dependency network is a four-tuple

G = (V, E , 0, Q)

where: V is a set of vertices representing a set of inequality

constraints VC and design variables that are classified into three types:

V, VI Vp E

R Q

The qualitative dependency between variables, based on

Definition: Qualitative dependency. Let G = (V, E , R , Q) be a dependency network. The

qualitative dependency 6ab E R , where a, b E V and ( a , b) E E. is defined as:

a set of decision variables a set of intermediate variables a set of design goals is a set of directed edges, E is a subset of the set

is the qualitative dependency of each arc, is the quantitative dependency of each arc.

(V x V ) ,

Wellman [18], is defined next.

+ -

0 ? otherwise

if (aa = + and db = +)

if ( d a = + and db = -)

db = 0, when da E {+, -,0,?}

or ( a a = - and db = -)

or ( d a = - and db = +) dab =

“dz = +” means that the variable z is increasing. “dz = -” or “dz = 0” means that the variable z is decreasing or remains unchanged, respectively. If the sign of change of variable cannot be determined, the value ‘?’ is assigned to dz [4], [ 181. The qualitative dependency between two variables is ‘+’ if the directions of change of the two variables are identical. The qualitative dependency between two variables is ‘-’ if the directions of change of the two variables differ. If a variable is not influenced by another variable, then the qualitative dependency between them is ‘0.’ Otherwise, qualitative dependency between two variables is ‘?.’ When the relationship between two variables is represented with a mathematical equation, the qualitative dependency between these two variables can be determined by the derivative.

Two types of qualitative inferences, parallel and serial inference, defined by Wellman [ 181, are used for reasoning the dependency on the network. The parallel and serial inference are analogous to the qualitative algebra defined by deKleer and Brown [4].

Definition: Parallel inference. Let G = (V, E , 0, Q) be a dependency network. Variables

a ,b , c E V and, 6,,,6bC E R , where both a and b have a directed edge incident to variable c, and a and b are not directly linked (see Fig. 2(a)). The result of parallel inference 6,, @ 6bc

6ac 63 6bc 6ab 8 8bC (a) (b)

Fig. 2. Two types of inferences: (a) parallel, (b) serial.

Fig. 3. An example of ambiguity of pure qualitative algebra.

is as follows:

+ I + ? + ?

? / ? ? ? ?

Definition: Serial inference. Let G = (V, E , R, Q) be a dependency network. Variables

a , b,c E V and, 6ab,6bC E 0, where variable a is incident to variable b and variables b is incident to variable c (see Fig. 2(b)). The result of serial inference 8 6bc that reasons from variable a to c via b is as follows:

Sa b q-f+ + 0 0 0 0

? ? ? Q ?

The result of the qualitative inference produced is some- times vague. Consider the dependency network in Fig. 3. Applying the parallel and serial inference, the qualitative dependency between variable w1 and 213 is determined to be unknown (=?).

In engineering design, it is important to know not only the qualitative but also the quantitative relationship between design variables. Using the quantitative dependency in the inference process, the vagueness of the qualitative inference is resolved. It also allows to determine the change required of a decision variable for conflict resolution.

The proposition presented next is used to derive the quantitative relationship between design variables.

Proposition I: Let a and b be two design variables. The relationship between a and b is represented as “b 0: an;’’ i.e., b is approximately proportional to a to the power of n. The rate of change between a and b is expressed with the following

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equation [ 151 :

Ab Aa

Equation (1) can be interpreted as: "if a changes r%, then b will change nr%.'' In this paper, the quantity n is used to measure the quantitative dependency between variables a and b.

Definition: Quantitative dependency. Let G = (V ,E,R,Q) be a dependency network and

variables a, b E V ( G ) and the directed edge ( a , b ) E E(G). The quantitative dependency '$ab is defined as:

The quantitative dependency can be derived from the current design knowledge, handbooks, or domain experts [5], [7]. If the relationship between variables is represented with a mathematical equation, for example,

Y = f(z1, zz,. . . , zn)

the quantitative dependency between variable zi and y can be estimated based on (1) as shown:

where xg and yo are the current values of variables zi and y, and Ay is obtained by perturbing variable xi by

Analogously, the parallel and serial inference can be also ( A Z ~ / X ~ ) x 100%.

defined for the quantitative dependency:

Parallel inference: $ac + +bc

Serial inference: x '$bc.

C. Reduction of the Dependency Network

The dependency network provides means for representing and reasoning about the dependencies between design variables. Using the serial or parallel inference, a dependency network can be reduced to a simpler form that characterizes the qualitative and quantitative dependencies between decision variables and design constraints (or goals). A reduction algorithm is developed in this section. The reduced network provides important information for generating design modification alternatives. In this paper, a triangle represents an inequality constraint. An octagon is used to represent a design goal. A square box represents a decision variable. A circle represents an intermediate variable. In addition, a diamond represents a design specification.

Before presenting the algorithm, three propositions are introduced.

Proposition 2: (Removal of a trivial design variable). A design variable can be removed from a dependency

network, if it has no successors, or it does not connect to any goal or constraint, Le., a variable w E (V,P U VC) (see Fig. 4).

This means that design variables that do not contribute to the performance of a system are trivial and can be removed from the dependency network.

Fig. 4. Removal of three trivial design variables w j . 0 7 , and ua

Fig. 5 . Substitution of design variable v. - \WY .",

Fig. 6. Removal of an intermediate variable.

Proposition 3: (Substitution of a design variable). If a variable w E Vp is preceded and succeeded by other

variables, then connect all of its predecessors to its successors and remove the arcs between the variable w and its successors (see Fig. 5).

The original design problem does not change after a substitution is made.

Proposition 4: (Removal of an intermediate variable). If a variable w E VI located on a directed path from a

decision variable to a constraint (or goal), then the intermediate variable may be removed, provided that the qualitative and quantitative dependency between the decision variable and constraint (or goal) can be estimated (see Fig. 6).

Proposition 4 implies that if the qualitative and quantitative dependencies between decision variables and constraints (or goals) are obtained, then the removal of the intermediate variables will not affect the quality of decisions of a designer.

Definition: Reduced dependency network. Let G = (V, E , $2, Q) be a reduced dependency network,

where:

V E

R Q

is a set of design variables, V = V, U Vp U VC is a set of directed edges, E is a subset of the set

is the qualitative dependency of each arc, is the quantitative dependency of each arc.

(VD x (VP u VC)),



Let pred(v ) be a set of variables that precede variable w and succ(v) be a set of variables that succeed variable w. The propositions 2-4 are used in the reduction algorithm presented next.

Algorithm: (Reduction of a dependency network). Input: Dependency network G = (V, E , 0, Q ) . Output: The output is in terms of the qualitative dependency

6ab and quantitative dependency G a b on the reduced network G’ = (V’,E’,n’,Q’), where variables a E V, and b E (Vp U V c ) , where V, is the set of decision variables, Vp is the set of design goals, and VC is th,: set of inequality constraints.

Procedure: (see bottom of this page). In Step 1, all trivial design variables are deleted. In Step

2, for a design variable that is preceded by a constraint or goal, the corresponding constraints (or goals) are expressed in terms of their predecessors. Step 3 determines the qualitative and quantitative dependencies between decision variables and constraints (or goals). For each design constraint (or goal) preceded by an intermediate variable, the reduction procedure based on Proposition 4 is performed and the paths examined are labeled. In Step 4, each intermediate variable and the arcs incident to it are removed. In the worse case, Steps 1 and 2 involve O( [VI) operations. Step 3 is performed in at most O( IVI( x IVpI) operations, because each constraint or goal has at most lV11 predecessors as intermediate variables. Step 4 is performed in at most O(lV~l(lVl - 1)) operations, because

each intermediate variable has at most IVI - 1 edges. The overall time bound of this algorithm is 0(lVl2).

After the network reduction, the constraint or goal that is reachable from a decision variable is connected by an arc and the corresponding qualitative and quantitative dependency is also obtained. Proposition 5 is used to generate all possible design altematives for conflict resolution, given the design modification required. The impact of the design modifications on the related constraints (or goals) is also determined.

Proposition 5: Let G = (V, E , 0, Q ) be a reduced dependency network, consisting of a set of n decision variables x1,x2, ,x, E VD, a set of f constraints u1, u2, ... , uf E Vc, and a set of m design goals v 1 , w 2 , ~ ~ ~ , v , E VP, where the qualitative value of each variable ranges over {+, -, 0, ?}. A feasible qualitative solution for the reduced network includes the qualitative values of all variables that satisfy qualitative constraints (3) and (4) (see Fig. 7):

( d m 8 &,,ut) @ (ax2 8 8Z2,7L,) @ = auz, fori = 1 , . . . , f

( d m c3 = d v j , f o r j = l , . . . , m .

63 (ax, @J SZn,,%)

@ (ax2 8 &,,,J @ . . . 63 (ax, 8 &*,,J (3)

(4)

The network representation greatly reduces the number of altematives required to explore in the implementation of a procedure for qualitative constraint checking. First of all, the qualitative constraints related to the design goals and

Step 1: For all w E V, such that w is trivial, do

Step 2: For all v E VP, such that succ(v) # 4, do

/* Proposition 2 */

/* Proposition 3 */ v = v - {w};

Begin E = E + ( (u1,u~)lul E pred(v) and u2 E succ(v)}, E = E - { ( ~ , ~ 2 ) 1 ~ 2 E S U C C ( W ) } ;

End

Begin Step 3: For all w E (Vp U Vc) , do /* Proposition 4 *I

Initialize the set of edges examined Mark = 4; While there exists u1 E pred(w) n VI and u2 E pred(u l ) , such that

Begin ( u 2 , ~ 1 , ~ ) 6 Mark, do

If (u2,w) E E , then /* Case 1: */ /* Update the dependencies */

/* Case 2: */ /* Add a new arc (u2, v) */

S,,,, = &,,, C!3 (6,,,,, 8 S,,,,), = h,,, + (TL,,,~ x ?Itu,,,); I* On the arc (u2,w) *I

If (u2, v ) 6 E , then E = E + ( (212 , V)},

L,, = SU,,,, 8 &L,,,, $U,,V = ~ z L Z , z L l x Gu1,v ;

Mark = Mark u {(u2,u1,v)); /* Mark the paths (u2, ul, v) */ End

End

Begin Step 4: For all u E VI, do

E’= E - ( (w,u)Iw1 ~ p r e d ( u ) } - {(w2,u)Iw2 E succ(u)} , V’ = v - {u};

End

1306 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 25, NO. 9. SEPTEMBER 1995

Decision variables or specifications

I I Constraints

Goals

Fig. 7. (3) and (4).

The reduced dependency network corresponding to the constraints

Fig. 9. The dependency network for the spring design problem.

60 lb

V 51.5 in

Fig. 8. An example of spring design.

105 lb

T

constraints that are intended to modified are checked based on generate-and-test strategy. If the alternative (the set of selected decision variables or specifications) generated is feasible, then the values of decision variables or specifications selected are propagated to the other design goals and constraints for determining the impact of design modification.

D. Design Example The reduced network is used to assist a designer in gener-

ating design modification altematives (strategies) in constraint negotiation. The strategies generated include:

1) Selection of decision variables to resolve conflicts. 2) Determination of the direction of change of decision

variables selected. 3) Determination of the rate of change of decision variables

selected. 4) Determination of the impact of design changes on the

remaining design variables and constraints. Based on the strategies generated, the designer selects an ap-

propriate strategy and determines the modification required for resolving conflicts. The new values of decision variables are propagated through the entire dependency network. A spring design problem is used to illustrate the concept developed [8].

Example I : A designer designs a helical spring which is required to exert an initial force ( F ) of 60 lb at a length that cannot exceed Li = 2.5 (see (31) in Appendix A), and a compression force (F,) of 105 lb at a length that is Ax = 0.5 in shorter. The spring must fit inside a 1.5 in diameter hole ( H ) (see (39) in Appendix A and Fig. 8). All the constraints used next are listed in Appendix A.

The designer specifies the wire diameter (d ) , spring diameter (D), and spring material (Mat ) to determine the number of coils ( N ) and the free length ( L f ) necessary to satisfy the functional requirements (31). The design has to satisfy reliability constraints, including the stress constraint (34) and buckling constraint (36) (see Appendix A). Assume that the largest diameter of a drill available is D, = 1 in. If the spring size is larger than 1 in, then an additional boring operation is required. The cost of boring operation is proportional to the radius ( H ) of a hole (39).

Consider an initial design with the spring material, oil- tempered steel (ASTM 229, S, = 210,000 psi), d = 0.15, and D = 1.35. The values of variables d and D are propagated through the constraints (22)-(40) (shown in Appendix A) to the remaining variables. The solution to the spring design problem is shown in Appendix B.

The design obtained is infeasible, because the stress constraint (34), which states that the stress produced ( T , ) should not exceed the maximum allowable stress (rmaX) of spring material selected, is violated. The dependency network corresponding to the spring design problem is shown in Fig. 9. In fact, the network in Fig. 9 represents the direction of constraint propagation. The design specifications might have to be negotiated with other perspectives in order to resolve conflicts.

The qualitative and quantitative dependencies between variables are listed in Table I. For example, in constraint (27), the compression length (L,) is dependent on the compression force (F,) and spring rate ( I C ) . Thus, an arc is drawn from variables F, and IC to variable L,, respectively. The qualitative dependency between variables F, and L,, and IC and L, are shown in (5).

~ F , , L , =Sign [ 51 ] = + and aFs (F,O,ko)

where (F:, IC') and the current values of variables F, and I C . According to (2), the quantitative dependency between

variables IC and L, is obtained by perturbing the value of

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TABLE 1 QUALITATIVE AND QUANTITATIVE DEPENDENCIES IN THE NETWORK

Qualitative Quantrtative depcndencydepMjcny

( 6 " b ) (Vd) -1.3333

-0.9091

+ 2.3333

+ 1 .woo -0.9091

+ 0.5953

+ 0.4W7

+ I .oooo -2.4869

+ I .woo + I .oooo

-0.0581

+ I.ooW

+ 0.6217

+ 1 .m

-0.9091

-0.9091

+ 1 .wM

+ 4 6410

4.9091

-2.4869

Qualitative Quantitarive -- (6,) (V*)

-0.1823

+ 1wM

+ 0.9000

+ 0 1000

+ I .woo + 2 . 1 m

+ I .woo

+ 2.0712

5.5523

- 1.6235

-0.5021

+ 5.8564

4.8564

+ 1.5696

-0.56%

-2 2500

+ 3.2500

+ 45000

-3 5000

+ I .woo -1.ooW

variable IC by 10%:

L y - LE -0.1061 1 x -

= -0.9091

where LE is the current value of the variable L, and L r w is the new value obtained by perturbing the value of variable IC by ,r%.

The reduced network generated by the algorithm char- acterizing the relationship between decision variables (and specifications) and constraints (and goals) is shown in Fig. 10. The dependencies between decision variables (and specifications) and constraints (and goals) for the network in Fig. 10 are shown in Table 11. For example, the reduction algorithm discussed in Section 1I.C determines the qualitative and quantitative dependency between variable d and constraint h2 in (6) and (7):

Sd,h2 = [[[(SC,K, 8 S K s , ~ s ) '8 Sd,C] @ h d , ~ , ] '8 S ~ , , h a ]

@ [6Tmax,hz '8 S d , ~ m a x l

= [[[(- '8 +) '8 -1 €3 -1 CB -1 @ [- €3 -1 ='? (6) Gd,hz = [ [ [ ( $ C , K , $ K , , T ~ ) '$d,C] -t- $ d , ~ ~ ] x ' $ ~ ~ , h ~ ]

+ [$Tmax,hz $d,Tmax]

= [[[(-0.0581 x 1) x (-0.9091)] + (-2.4869)] x 5.85641 + [-4.8564 x (-0.1823)]

= -13.3692 (7)

From (6), it is observed that the qualitative dependency between variable d and constraint h2 is unknown. However, the vague qualitative dependency can be clarified by the quantitative dependency (7). The qualitative dependency between

V Design constraints Specificaitons and the goal

Fig. 10. The reduced dependency network Corresponding to Example 1.

TABLE 11 THE QUALITATIVE AND QUANTITATIVE

DEPENDENCB FOR THE REDUCED NETWORK

Qualitative Quantitative depSldary'j?=j=Y

(&*) (Vd) -2.1381

+ 1.2218

+ 2.0712

+ 1.0157

-2.5524

+ 5 8564

4.8564

- 13.3692

+ 5.5158

4.0352

+ 1.5696

0.1737

0.1184

-0.8955

Qualitative Quantitscivc -Y depnjary

(6.3) (Vd) + 0.8742

+ 3.2500

-2.2500

+ 2.0455

-3 5"

+ 4.5000

40909

-I.ooW

+ 0.9oOo

+ 1.2121

+ 0 8264

-2.1212

+ 6.7410

- I .4869

variable d and constraint h2 is redefined in (8):

bd,hz = - (8)

To resolve the violation of the stress constraint (34), the value of the left hand side of constraint (34) has to decrease; Le., dhz = -. Based on the qualitative constraints (3) and (4) and dependency information from Table 11, constraint (9) has to be satisfied:

dMat @ S1\.iat,hz) @ (ad '8 &,h2) CB (ao '8 S D , ~ ~ ) @ (8Fs '8 SF^,^^) = aha

@ (aFs '8 +) = -.

+ ( 3 M a t '8 -) @ (ad @ -) @ (i3D 8 +) (9)

From (9), fifteen qualitative strategies for conflict resolution and the corresponding impact on the five constraints and


TABLE III QUALITATIVE STRATEGIES FOR CONFLICT RESOLUTION

Conflicting Dsision COnMint variable Spee. Conseaint Goat

Suarcgy h d D Ma F, tt, 4 h, 4 1 0 0 0 . + + o o o +

0 . 0 . - - + - - + 0 - 0 0 ? ? + - - + + o o 0 - - + - + +

+ - 0 - ? ? + - ? + o o + 0 0 0 0 0 0 0 0 0 + - + + o o o + o . + 0 - - + - - + 0 - + ? ? + - - +

+ o + ? ? + - + + + o + 0 - - + - + +

+ . + 0 - - + - ? + I5 - + - + - ? ? + - ? +

* infeasible strategy

the design goal are generated based on the generate-and-test technique [16] and listed in Table 111.

The strategies that are infeasible are identified in Table I11 by an asterisk (*) . For example, the four qualitative strategies 4,5, 12, and 13, that increase the value of hs (constraint (39)), are infeasible, because constraint (39) is bounded. They can be removed from further consideration.

If the designer intends to reduce the volume of the spring, then strategy 8, which suggests to use a material with higher strength (e.g., A S T M 4 0 1 ) , can be applied to resolve the violated constraint h2 without increasing the volume of the spring. If the designer wants to reduce the size of the spring, then strategies 2, 3, 10, and 11 may be used to reduce the manufacturing cost. Note that the strategies that involve design specifications require negotiation with other perspectives (e.g., variable F, in strategy 1).

A qualitative strategy that does not violate a bounded constraint (e.g., constraint (34)) does not have to be feasible in a quantitative sense, because the result of implementing this strategy (design modification) may violate other constraints. A quantitative analysis is required to estimate the impact of design modification. To allow a systematic search for a feasible design modification, the set of feasible qualitative strategies is represented with a tree (see Fig. 11). Each level in the tree indicates strategies with the same number of decision variables and specifications, e.g., level 1 includes strategies 1, 3, and 8, each with one variable or specification. For example, if strategy 3 involving variable D is selected and cannot resolve the conflict, then either strategy 2 (including variable F,) or strategy 1 0 (including variable Mat) is selected. The depth- first search is used to explore the solution space.

Using the quantitative dependency in Table 11, the value of change required for conflict resolution is determined. For example, using (1) and the quantitative dependencies in Table 11, the rate of change of spring diameter (D) involved in strategy 3 is determined in (10).

-$D,hz - * Ah2 - A D -19510.50 D 19510.50 A D A D D D

=5.5158 x - + - = -18.13%. (10)

The quantitative analysis for the three strategies 1, 3, and 8 is shown in Table IV. From the result of quantitative analysis in Table IV, it is known that three strategies are feasible. If the objective is to minimize the volume of spring, then

Root

Level 1 1 +*

Level 2 2 9 1 0 6

d h 4 d

Level 3 7 1 1 14

1 variable or specification

2 variables or specifications



Fig. 1 1 . The search tree for qualitative strategies.

TABLE N THE RESULT OF QUANTITATIVE ANALYSIS IN CONFLICT RESOLUTION

Dsi ion variable Spec. ConSUSkd OT I d SUaWgY D d Ma F, h, 4 h, h, &

1 0 0 0 -17.08% -1.65 0.00 -2.56 -4 00 -2.00 0.00 0 34 3 -18.13% 0 0 0 -0.98 0.00 -2.14 -5.63 -0.37 -0.22 0 3 1 8 0 0 astm401 0 -2.55 -1.21 -2.M) -4.00 -2324.9 0.00 0 25

strategy 8 is the best choice. For the strategies in level n, the desired rate of changes of n - 1 variables (or specifications) have to be given by the designer in order to determine the required rate of changes of the last variable and the impacts on the corresponding design goals and constraints. The implementation of the system uses an interactive approach to prune the search space. The designer is allowed to specify the variables that he/she does or does not prefer to modify. The variables that are unwilling to change will be removed from further consideration. In addition, the strategies investigated before will not be explored again.

The prototype system is implemented in Prolog [16] and integrated with a C language code.

111. A FUZZY-LOGIC-BASED APPROACH FOR DEPENDENCY ANALYSIS

A. Representation of Dependencies with Linguistic Variables

In many engineering design problems, dependencies among variables are not well defined. The ambiguity of pure qualitative algebra can be remedied with quantitative dependencies discussed in Section I1 or with the fuzzy logic [21], [22] discussed in this section. Even though the quantitative dependency may be estimated by domain experts, the fuzzy logic approach provides a more flexible and natural way to represent the quantitative dependency among design variables.

The dependency between variables is described as a linguistic variable characterized by a quintuple ($, T(+), U, G, M ) , where

+: the linguistic variable “dependency” defined according to (1):

TB Ablb +=<=n(lla, where T A and TB is the rate of change of variable a and b, respectively.

KUSIAK AND WANG: DEPENDENCY ANALYSIS IN CONSTRAINT NEGOTIATION

. ” -.1( PS PM PL NS NM

1309

PS PM PL PSorNS NS NL PM PL PL PM NS, PS NLwNM PL PL PL PL PL

PSorNS PM PL NS NM NL NM PSorNS PL NM NL NL

?

TABLE V THE ADDITION OF TWO LINGUISTIC TERMS

I r ,er - I PS PM PL NS NM NL

NL

- Fig. 12. The membership function of the linguistic variable “dependency.”

T($): the set of names of linguistic terms of $; T($) = {PL, PM, PS, NL, NM, NS}, where

PL: positive large PM: positive same PS: positive small NL: negative large NM: negative same NS: negative small

U : a universe of discourse; U = [-20,201 G: a syntactic rule for generating the terms in the term

M : a semantic rule that assigns a meaning, i.e., a fuzzy set, to the terms.

The following membership functions of the linguistic terms

set T ( $ )

are defined (see Fig. 12):

The linguistic term “PS” (“NS”) means that if the value of linguistic variable $ is positive (negative) and its value is close to zero, then the value of the membership function “PS” (“NS”) is 1. The linguistic term “PM’ (“NM”) means that if the value of variable $ is near 1 (-1), then the value of membership function “PM’ (“NM’) is 1. Similarly, if the variable $ is greater than 2, then the value of membership function “PL” (or “NL”) becomes 1.

I NL 1 NL NLorNM ? NL NL V a y N L I

TABLE VI THE MULTIPLICATION OF TWO LINGUISTIC TERMS

PM PM NM PL PM PL veryPL NM NL v c y N L NS v q N S NS NM NM NS NM NL PS PM PL NL NS NL NL PS PL PL

Fig. 13. A mechanical structure.

For the six primary linguistic terms, different terms can be generated by using linguistic hedges [21]. Currently only the term “very” is used in this paper:

M(veryz) = ( x ) ~ , wherez E {PS,PM,PL,NS, NM, NL}

Other linguistic hedge, such as more or less, can be included.

B. Qualitative Algebra Based on Fuuy Logic The reduction algorithm presented in Section I1 can be

also used for dependency analysis with fuzzy logic. The only modification made is that the qualitative algebra has been replaced with fuzzy logic (Zadeh 1973, 1983). The fuzzy logic is represented with rules listed in Tables V and VI.

The fuzzy inference is represented as follows:

IFTA = AandTB = B THEN T A * TB = C, T A = A* or @3

where * denotes an operator @

rg = B*

T A * T B = C * .

When the conditional statement of a rule A = A* and B = B*, the conclusion C* := C. When the condition is A # A* and B # B*, then the generalized modus ponens is used to derive the value of T A * TB [21], [22]. The latter procedure is summarized in Appendix C. The example presented next illustrates the concepts discussed above.

Example 2: Consider the design of mechanical structure to support an overhanging vertical load (see Fig. 13) [20].


Decision Variable

wAe i

Fig. 14. The linguistic dependency network.

h i m goal U W", W C D

NLvNM PM PM

PM PM very (PS v NS)

L W r

PL 7 PM PM PM NM

v: the disjunctive operator

between variables and is obtained from the network in Fig. 14 as follows:

1ClwAB ,u = $wA, , W c D 8 [($wcD , w ~ D 8 $ ~ c D , u ) @ $wcD ,ul

$wAB,m = PM @ [(PM 8 PM) @ NM] = P M 8 [PM @ NM] = PM 8 [very (PS V NS)]

The following set of design variables is used to define this =very (PS v NS). mechanical structure:

L: the length of the horizontal member CD. L1 : the length between the end point of the member CD

and the attached point of members AB and CD. WAB: the width of member AB. WCD: the width of member CD. t: the height of member AB and CD. W: vertical loading on the member CD. WAB: weight of the member AB. WCD: weight of the member CD. 6': the angle between the member AB and wall. a: the bending stress on the member CD. p: the density of the material used for the members. T : the ratio of the length BD to the length of member

CD. The designer wants to minimize the bending stress (0) of the

member CD in order to avoid the damage of both members. It is known that the bending stress (a) is the function of variables L, Li, W, WCD, WCD, and t in (17):

a = f l ( L , L1, W, W C D , WCD, t ) . (17)

The relationship between design variables WCD, WCD, WAB, L1 and other variables is also represented as functions ( 18)-(2 1):

W C D = f Z ( L , p , W C D , t ) (18) W C D = f3(WAB) (19) WAB = f4(L, Li, P, WAB, t) (20)

L1 = f 5 ( 4 .) (21)

Assume that the functions (17)-(21) (quantitative dependency) are not exactly known. The linguistic terms defined previously are used to define the dependency between variables as shown in Fig. 14.

Applying the fuzzy version of the reduction algorithm to the network in Fig. 14, the relationship between decision variables and goals is shown in Table VII. For example, using the reduction algorithm presented in Section 11, the dependency

" \

It is observed from Table VI1 that in order to decrease the value of stress (a) of member CD, one may decrease the value of variables L,W, and T or increase the value of variable t. The rate of change of variable a may be roughly estimated. For example, if the value of variable W changes lo%, then the value of variable a also changes about 10%. If the value of variable WAB changes lo%, then the variable a changes far less than 10% and the value of variable WAB increases about 10%. However, the direction of change of variable a can not be determined. The value of a decision variable that is highly sensitive to a performance variable should be carefully determined. In this example, variable L has the largest effect on variable a, however, the influence of change of variable L on variable WAB cannot be determined unless more quantitative information is obtained.

The approach discussed in this section can be used to increase the quality of decisions made regarding the performance of design alternatives, e.g., at the early stage of a design process. It is appropriate for applications when an exact solution can not be determined due to the lack of quantitative data. To obtain the exact solution, more comprehensive quantitative data is needed. Finally, the fuzzy logic algebra appears to be more general than the pure qualitative algebra for dependency analysis. The linguistic dependency that falls between qualitative and quantitative dependency appears to be useful for dependency analysis, when the constraints are not well defined.

IV. CONCLUSION In this paper, an approach based on qualitative reasoning to

analyze dependencies among design variables was developed to assist designers in negotiation of constraints. The qualitative and quantitative relationship between design variables was represented with a dependency network. An algorithm was developed to derive dependencies between decision variables and constraints (or goals) for conflict resolution. Based on the dependencies obtained, design alternatives were generated. In the second part of the paper, a fuzzy-logic-based depen-


dency analysis was developed to generate design alternatives that assists designers in making decisions when no sufficient quantitative information is available. It is observed that the fuzzy-logic-based approach is suitable to characterize the imprecise dependency information in design.

The future research is to extend the network model developed in this paper to the conflict resolution in feature-based design [3]. The network model developed may be also useful in representing the intent of a designer or other life-cycle perspectives.

APPENDIX A LIST OF PARAMETERS, VARIABLES, AND

CONSTRAINTS OF THE SPRING DESIGN

Design Parameters and Variables: d D C c u b

c l b

FS F A X

IC Ks G

s, r , rmax N L, LC L f Li DO v (in3) Bf Bf lim

H M C cd Dr cb h

(in) helical spring wire diameter. (in) mean helical spring diameter. spring index. upper limit of spring index (= 13). lower limit of spring index (= 7). (lbs) solid force. (lbs) compression force. (in) difference of compression lengths of compression forces F and F,,,. (lbshn) spring rate. Wahl spring factor. (psi) share modulus of spring material (G = 11.5 x lo6, when material ASTM 229 is used). (psi) tensile strength of spring material (S , = 210,000 psi, when material ASTM 229 is used). (psi) maximum stress for the spring. (psi) maximum stress for spring material. number of spring coils. (in) solid length of a spring. (in) maximum compression length. (in) free length of a spring. (in) installed length of a spring. (in) the size of a spring. volume of helical spring. horizontal buckling factor for a spring. upper limit of the horizontal buckling factor for a spring (= 4). (in) the maximum size of the spring. (dollars) the total drilling cost. (dollars) unit drilling cost. (in) drill diameter. (dollar) unit boring cost. (in) drilling depth.

Quantitative Relationship Between Variables of Design Perspectives

design perspective are as follows: Design Perspective: The design constraints for the spring

D C = - d

0.615 C

K s = l + -

Gd4 8D3k

N = -

L, = ( N + 2 ) d FS L, =IC

Lf = L , + Lc D o = D + d

r2d2DN V=- 4

Reliability Perspective:

8F,D 7rd3 rs = - KS

Manufacturing Perspective:

APPENDIX B A SESSION OF DEPENDENCY ANALYSIS FOR SPRING DESIGN

***** This is a test program for spring design ***** >help.

Command Function input(variable, value) evaluate evaluate set of constraints

assign value to variable

list suggest([change(Constr~nt, [DUM) qualitative-evaluation

list the result of computation suggest a set of qualitative strategies remove strategies that are qualitatively infeasible estimate the impact of a strategy and desired modification on related constraints and goals

quantitative-analysis(Suategy, Mod)

> input(spring-dia. 1.35) > input(wire-dia, 0.15). > input(material).

* Select the spring material:

The following spring material are: Material The maximum allowable stress astm227 90269.02 astm229 94750.12

F, - F I C = - AX

astm232 104463.48 astm228 114030.91 astm401 116585.54

I: asun229.


> evaluate. * Specifications

H (Space limit) Clb (C lower limit ) Cub (C upper limit ) B h (Buckling l i t ) Li (Install Length) Dx (Compression length) Fs (Solid force) F (Compression force)

* Decision variables Mat (Spring material) d ( W i t diameter) D (Spring diameter)

Do (Spring size) Bf (Buckling factor) T (Solid toque) Lf (Free length) Lc (Compression length) Ls (Solid length) N (Number of coil) Ks (Wahl factor) C (Spring factor) k (Spring rate) Tmax (Max toque)

* Design

Value 1 S O 7.00 13.00 4.00 2.50 0.50 105.00 60.00

Value astm229(linax=94750.12) 0.15 1.35

Value 1 S O 1.45 114260.61 1.96 1.17 0.79 3.29 1.07 9.00 90.00 94750.12

* G o a l ValW V (Spring volume) 0.25

* Status Constraint Value Adjustment hl: Lf-(F/k)-Li<=o -1.21

# h 2 T - T m a x d 19510.50 h3: Bf-Bflim<=O -2.55 b4: C-Cuk=O -4.00 h5: Clb-C<=O -2.00 h6: Do-H<?, , 0.00 The symbol # indicates the violated constraints

> suggest([cbange(h2, [-I)]).

*** strategy 1 *** * The direction of change for conflict resolution:

* The direction of change of design specification:

* The direction of change of decision variable:

Variable Direction of change



h2: T-Tmax<=O

solid-force (Fs)

springmaterial (Mat) 0 wire-dia (d) 0 springdia (D) 0

springvolume (V) + h 6 Do-H<=O 0 h 4 C-Cube0 0 h5: CIbC<=O 0 hl: Do-H<=O + h3: Bf-Bflim<=O +

* The direction of change of goal and constraint: Variable Direction of change

*** Strategy 2 *** * The W o n of chanac for conflict resolution:

Variable hz: T-Tmax<=O

* The direction of change Variable

solid-force (Fs) * The direction of change

Variabk springmaterial (Mat) wk-dia (d) springdia (D)

* The direction of change Variable

springvolume (V) h6: Do-H<=O h4: C-Cuk=O h5: Clb-C<=O h 1 : Do-H<=O h3: Bf-Bflim<=O

Direction of change

Direction of change of design specification:

0 of decision variable:

Direction of change 0 0

of goal and constraint: Dinctioa of change

+ +

........ e*: ~ t r a & i ' l 4 :*: * The direction of change for conflict resolution:

* The direction of change of design specificauon:

Variable Directlon of change h 2 T-Tmax<=O

Vahble Direction of change solid-force (Fs) 0

* The direction of change of decision variable: Variable Direction of change

springmaterial (Mat) + wire-dia (d) + springdia (D)


springvolume (V) + h6: Do-H<=O ? h 4 C - C u k d h5: Clb-C<=O hl: Do-H<=O h3: Bf-Bflim<=O

+

*** Strategy 15 ***

h 2 T-Tmax<=O

* The direction of change for conflict resolution:

* The direction of change of design specification:

* The direction of change of decision variable:


Variable Direction of change solid-force (Fs)

Variable Direction of change springmaterial (Mat) + wire-dia (d) + springdia (D)


springvolume (V) + h6: Do-H<=O ? h4: C-Cub<=O + hS: Clb-C<=O hl: Do-H<=O ? h3: Bf-Bflim<=O 7

> qualitative-evaluation.

* The strategy 4 is infeasible, due to constraint h6 being bounded. * The strategy 5 is infeasible, due to constraint h6 being bounded. * The strategy 12 is infeasible, due to constraint h6 being bounded. * The strategy 13 is infeasible, due to constraint h6 being bounded.

> quantitative-analysis(strategy3).

* The required rate of change for spring diameter (D) is: -18.13% The new value is 1.105.

Constraint Rateofchange Value h3: Bf-Bflim<=O -0.16 -2.14 hl : Do-H<=O -0.18 -0.98 h5: Clb-C<=O -0.82 -0.37 h4: C-Cub<=O 0.41 -5.63

h2: T-Tmax<=O -1.00 0.00 hl: Do-H<=O -0.22

The estimated impact:

Goal Rate of change Value springvolume (V) 0.27 0.31

> quantitative-analysis(strategy 1)

The new value is 87.066.

constraint Rateofchange Value h3: Bf-Bflirn<=O 0.01 -2.56 hl : Do-H<=O 0.37 -1.65 h5: ClbC<=O 0.00 -2.00 h4: C-Cub<=O 0.00 -4.00 h2: T-Tmax<=O -1.00 0.00 hl: Do-H<=O 0.00

springvolume (V) 0.36 0.34

* The required rate of change for solid force (Fs) is: -17.08%

The estimated impact:

Goal Rate of change Value

APPENDIX C THE GENERALIZED MODUS PONENS OF FUZZY REASONING

Consider the set of n rules used in fuzzy algebra:

IFTA = All andTB = A12, THENTA * TB = c1 . . . ... . . . . . . . . . . . .

I F ~ A = Ail andrB = Ais, THENTA * TB = Ci . . . . . . . . . . . . . . . . . .

IF T A = Ani and TB = An2, THEN T A * rg = C,


The fuzzv conditional statement can be defined as a fuzzv r 111 P. J. PaDalambros and D. J. Wilde, Principles of Optimal Design. New - - . .I ~- York damhridge Univ. Press, 1988.

[12] J. R. Rinderle and V. Krishnan, “Constraint reasoning in concurrent design,” in ASME Int. Con$ on Design Theory and Methodology, Chicago,

relation with the Cartesian product of three fuzzy sets. Assum- ing that a user supplies the facts T A = A; and T B = A ; . The procedure to obtain the value of conclusion is summarized next [22].

Step 1: Compute the degree of consistency between an-

IL, Sept. 16-19, 1990, pp. 53-62. [13] J. R. Rinderle and V. Krishnan, “Constraint reasoning in concurrent

design,” in The Second Int. Con$ on Design Theory and Methodology, Chicago, Illinois, Sept. 1990, pp. 53-62.

tecedent of a rule A,, and a fact A;. The degree of consistency is defined as:

?‘A,, = SUP ( P A , , (u) A PA; (u)), U

i = l , . . . , n a n d j = 1,2.

_ _ [ 141 D. S&rano and D. -Gossard, “Constraint management in conceptual

design,” in Knowledge Based Systems in Engineering Planning and Design, Computational Mechanics, D. Sriram and R. A. Adey, Eds. New York, 1987, pp. 211-224.

[15] V.-W. So0 and T.-C. Wang, “Integration of qualitative and quantitative reasoning in iterative parametric mechanical design,” in ArtiJcial Intel- ligence in Engineering Design, Analysis, and Manufacturing, vol. 6, no. 2;pp. 95-1019, 1992:

[16] L. Sterling and E. Shapiro, The A n of Prolog. Cambridge, MA: MIT where PA. . (u) and PA* (u) are the membership functions of Press, 1986.

[17] D. G. Ullman, T. G. Dietterich, and L. A. Stauffer, “A model of the mechanical design process based on empirical data,” Art$ Intell. Eng.

- 3 ‘ ‘ ., , ,

Aij and A;, respectively. Step 2: Compute the overall degree of consistency for each

Design, Analysis, and Manufacturing, vol. 2, no. 1, pp. 33-52, 1988. [18] M. P. Wellman, Formulation of Tradeoffs in Planning Under Uncer-

tainty. [19] R. I. Winner, J. P. Pennell, H. E. Bertrand, and M. M. G. Slusarczuk,

San Mateo, CA: Morgan Kaufmann, 1990.

“The role of concurrent engineering in weapons system acquisition,” IDA Report R-338, Institute for Defense Analyses, Alexandria, Virginia,

[20] K. L. Wood and E. K. Antonsson, “Computations with imprecise parameters in engineering design: Background and theory,” ASME J.

step 3: Consider as a weight. The desired expression for C* is as follows: 1988.

c* = (71 A c,) V . . . V (m A Cn), where

PT*Ac,(u) =? ‘ iAPC,(u) , i = 1 , . . . , n .

REFERENCES

[I] R. Agrawal, G. L. Kinzel, R. Srinivasan, and K. Ishii, “Engineering constraint management based on an occurrence matrix approach,” ASME J. Mech. Design, vol. 115, no. 1, pp. 103-109, 1993.

[2] J. K. Choy and A. M. Agogino, “SYMON: Automated symbolic monotonicity analysis system qualitative design optimization,” in Proc. ASME Int. Computers in Engineering Conf . Chicago, IL. Julv 24-26. - “

1986, pp. 305-310. 131 M. R. Cutkosky and J. M. Tenenbaum, “A methodoloav and computa-

II .~

tional framework for concurrent product and process design,” Mecha- nism and Machine Theory, vol. 25, no. 3, pp. 365-381, 1990.

[4] J. deKleer and J. S. Brown, “A qualitative physics based on confluences,” Artificial Intelligence, vol. 24, no. 1-3, pp. 7-84, 1984.

[5] J. R. Dixon, A. Howe, P. R. Cohen, and M. R. Simmons, “Dominic I: Progress toward domain independence in design by iterative redesign,” Engineering with Computers, vol. 2, no. 3, pp. 137-146, 1987.

[6] S. Finger, M. S. Fox, D. Navinchandra, F. B. Prinz, and J. R. Rinderle. “Design fusion: A product life-cycle view for engineering designs,” in IFIP WG 5.2 Workshop on Intelligent CAD, Univ. of Cambridge, UK, Sept. 1990.

[7] K. Ishii and P. Barkan, “Rule-based sensitivity analysis-A framework for expert systems in mechanical system design,” in Expert Systems in Computer-Aided Design, J. S . Gero, Ed. New York: North-Holland,

[8] R. C. Juvinall and K. M. Marshek, Fundamentals of Machine Component Design. New York: Wiley, 1991.

[9] S. M. Kannapan and K. M. Marshek, “An approach to parametric machine design and negotiation in concurrent engineering,” A. Kusiak, Ed., Concurrent Engineering: Automation, Tools, and Techniques. New York John Wiley, 1993, pp. 509-533.

[IO] B. D. McGinnis and D. G. Ullman, “The evolution of commitments in the design of a component,” ASME J. Mech. Design, vol. 114, no. 1, pp. 1-7, 1992.

1987, pp. 179-202.

kech . Design, vol. 111, no. 4, pp. 616-625, 1989. [2t] L. A. Zadeh, “Outline of a new approach to the analysis of complex

systems and decision processes,” IEEE Trans. Syst. Man Cyber., vol. SMC-3, no. 1, pp. 28-44, 1973.

[22] -, “The role of fuzzy logic in the management of uncertainty in expert systems,” Fuzzy Sets and Systems, vol. 11, no. 3, pp. 199-227, 1983.

Andrew Kusiak (M’90) is Professor and Chair- man of the Department of Industrial Engineering, University of Iowa, Iowa City. He is interested in engineering design and manufacturing. Dr. Kusiak has authored and edited numerous

books and published research papers in joumals sponsored by various societies, including AAAI, ASME, IEEE, IIE, ORSA, and SME. He serves on the editorial boards of a number of journals, edits book series, and is the Editor-in-Chief of the Journal of Intelligent Manufacturing.

Juite Wang received the B.S. degree in industrial engineering from the National Tsing Hua Univer- sity, Taiwan, in 1985, and the M.S. and Ph.D. degrees in industrial engineering from the Uni- versity of Iowa, Iowa City, in 1990 and 1994, respectively. He is an Associate Professor of In- dustrial Engineering of the Feng Chia University, Taiwan. His interests are in applications of artificial intelligence in design and manufacturing.





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