Group Title: Department of Computer and Information Science and Engineering Technical Reports
Title: A Validation method using fuzzy simulation in an object oriented physical modeling framework
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Title: A Validation method using fuzzy simulation in an object oriented physical modeling framework
Series Title: Department of Computer and Information Science and Engineering Technical Reports
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Language: English
Creator: Kim, Gyooseok
Fishwick, Paul A.
Affiliation: University of Florida
University of Florida
Publisher: Department of Computer and Information Science and Engineering, University of Florida
Place of Publication: Gainesville, Fla.
Copyright Date: 1998
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Volume ID: VID00001
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A Validation Method using Fuzzy Simulation in an Object
Oriented Physical Modeling Framework

Gyooseok Kim and Paul A. Fi-lr-i, 1:

Computer and Information Science and Engineering Department
University of Florida


ABSTRACT
Object Oriented P!- -i1 1 Modeling (OOPM) is an object-oriented li!.,.l,_..-- for constructing 1i! i- 1 models
by emphasizing a clear framework to organize the geometry and dynamics of the models. An environment called
MOOSE ( !.1! .,,l ". ![!. Object Oriented Simulation Environment) is under development to explore this OOPM
concept. MOOSE provides a solid connection between blackboard models and software models in an unambiguous
way, capturing both static and dynamic semantics of objects. Even though this facility reinforces the relation of
"I-1! !" to "-1i, '. iii", an adequate validation technique for modeling processes has not yet been developed. In this
paper, we propose a validation method for the modeling process in MOOS. This method utilizes a fuzzy simulation
approach to encode uncertainty arisen from human reasoning process into computer simulation components.
Keywords: Physical Modeling, Validation, iF Simulation

1. INTRODUCTION
MOOSE (. ill !1,,i I Object Oriented Simulation Environment)1'2 is an enabling environment under the develop-
ment at University of Florida for modeling and simulation based on OOPM. OOPM extends object-oriented program
design with visualization and reinforces the relation of "--i i'. [ !" to "l' -' 'o iii". This permits a tight couple between
a model author and the modeling and simulation process through an interactive HCI (Human Computer Interface).
MOOSE consists of four i! ii .c components: Modeler, Translator, Engine and Scenario. Modeler interacts
with a model author via a GUI (Graphical User Interface) in a way to help the author make the valid conceptual
model of the -1, 11 Translator is a bridge between a model design and a model execution. It reads the output
from Modeler and builds the corresponding structures of the conceptual model with C++ code automatically,
therefore it ensures that the program is a valid representation of the conceptual model. Engine is a C++ program,
composed of Translator output plus runtime support, compiled and linked once, then repeatedly activated for Model
Execution. Scenario is a visualization-enabling GUI which interacts with Engine, and ,1 -11 ,- Engine's output
in a meaningful form so that the output of MOOSE can be validated against the author's expertise.
Even though this facility reinforces the relation of ", I' !I" to "l'''o i ii" in a natural way, i; adequate validation
technique for the modeling process, particularly between i,1! -1 .1 -1. lin- and conceptual models, and between
conceptual models and computerized (programmed) models has not yet been developed. Face validation4'5 by
domain experts is known as one of the validation techniques for the conceptual models. In this paper, we assume
that there is knowledge from a domain expert in the form of linguistic if-then rules. Our goal is to propose a validation
method that performs an automatic consistency checking between the expert's rule-based model and various I- -
of conceptual models in MOOS.
We organize this paper as follows: in Section 2, we propose the fuzzy set theory which is relevant to this research
followed by a comparison between a general modeling process and its counterpart of MOOS. In Section 3, we
propose a fuzzy simulation method that we can i !!.1. to validate the conceptual models of MOOS.
Other author information: (Send correspondence to G. Kim)
G. K.: Email: : Ill 1 1 .1. ..... 352-392-1435; Fax: 352-392-1414.
P.A.F: Email: fishwick@cise.ufl.edu; 1. ...... 352-392-1414; Fax: 352-392-1414










2. BACKGROUND
2.1. Fuzzy Set Theory
This section presents a review of the relevant aspects of fuzzy set theory which forms the basis of our fuzzy simulation.
The theory of fuzzy sets can be found in Refs. 6-11. F, sets i! i- be viewed as an attempt to deal with a I- I"-
of imprecision that arises when the boundaries of classes are not sharply defined. A fuzzy set A of a universe of
discourse X is characterized by a membership function PA : X -> [0, 1] which associates with each element x of X a
number pA(X) in the interval [0, 1] which represents the grade of membership of x in A.

D. Ci;. "' -.. 2.1: If f is an n-ary crisp function which is a mapping from a Cartesian product XY x ... x X, to a
space Y, and if A is a fuzzy set in XY x ... x X, which is characterized by a membership function pA (xi, * X1),
with xi, i =1,..., n, denoting a generic point in Xi, then extension principle 8 states that

f(A) = f( ( ,..., ,, /(X ,..., ,))
JX1X-*-xX,

= fA(Xi,...,Xn)/f(Xi,...,Xn) (1)

The membership function of A is expressed by

P.A(X1,. ..,X, ) = P 1 (2i) A PA2 (X2) A ... A PA. (Xn) (2)

where PAi, i = 1, .. ., n, is the membership function of Ai.
D, ;'. ',... 2.2: Let A and B represent two fuzzy numbers and let denote ;,i- of the four basic arithmetic
operations. Then, using the extension principle (1) under the assumption (2), we define a fuzzy set, A B on
R', where R' is the set of all real numbers, as

-.A*B(z) = "** jA(X) A PB(Y)), (3)

V z E R. Thus, for example, if A, B C R are two fuzzy numbers with respective membership functions pA(x)
and PB(y), then the four basic arithmetic operations, (i.e., addition, subtraction, multiplication and division)
give, for each x, y, z E R, the following results:

IA+B(z) = max,=x+y (l/(A () A PB (Y)). (4)

PiA-B() = max3=,=x-y(/A () A PB(y)). (5)
p1AxB(z) = ,(pA (x) A PB(y)). (6)
PA1 B(z) = max3= y (A (x) A PB (Y))- (7)
D, ;,i'.i,... 2.3: Let P be a compound statement of the I i, (X is A) (y is B), where X and Y are fuzzy
variables that take real numbers from some universal sets X, Y, respectively, A and B are fuzzy values on X,
Y, respectively and is a !i l lr, i !i (and) or a .1 -i.li l! i.1 (or).

When is a ..!ii.!n r.,!i the rule of conjunctive composition 9 states that P can be expressed by a
I"..--. 1ll' distribution 7r(x, y) which is defined by

{PAxB(x,y)/(x,y) x E X,y E Y}, (8)

where PAxB(X, y) denotes min t(P (X), B (y)) and x is the cartesian product.
When is a 1 -'ii i, ti .11 the rule of disjunctive composition 9 states that P can be expressed by a i ....~ 1 ~-l
distribution 7 (x,y) which is defined by Equation (8), where P AxB(X, y) denotes max(PA (x), B (y)).



















Operational / Analysis Conceptual
Validity Experimentation and Model
Data Modeling Validity
Validity
/ \\ '


Computerized omputerogramming Conceptual
Model - - Model
and Implementation


Computerized
Model
Verification

Figure 1. Modeling process and its relation to validation/verification


2.2. Modeling Process
Fig. 112,4,13 shows a general modeling process. The conceptual model represents the mathematical, logical or verbal
representation of the problem I! i I developed for a particular study, and the computerized model represents the
conceptual model implemented on a computer. The general purpose of the conceptual model validation depicted
in this figure is to validate the [1nili i1- ii- assumptions and theories. More specifically, the process is concerned
with whether this specific model's representation of the problem. I i being modeled and its structure, logic and
mathematical and causal relationships are reasonable for the intended use of the model.12 One of the primary
validation techniques used for this evaluation is face validation.4'5 Face validation involves having domain experts
evaluate the conceptual model to determine if they believe it is correct and reasonable for its purpose. This usually
means examining the flowchart or graphical model, or the set of model equations.
The counterpart of the above modeling process in MOOSE is depicted in Fig. 2. MOOSE supports 111 ii- different
I- of models1 including CODE, 1 S. I (1i i! I.i- I 1. Machine), I I;. (Functional Block Model), RBM (Rule Based
Model) and EC'N (I.1 1, ,i,.'Nal Constraint model) for the conceptual modeling process. Then, by translating the
conceptual model into C++ code, it constructs the computerized model. MOOSE does not yet !!I .1.. validation or
verification techniques.

3. A PROPOSED FUZZY SIMULATION METHOD
Using the fuzzy simulation method introduced in this section, the face validation process discussed in the previous
section can be automated, thereby contributing to validate the conceptual models in MOOS1. A prerequisite for this
process is that there should exist an expert's validated rule set associated with appropriate membership functions for
the -- -. iin of interest. Given that this condition is satisfied, the fuzzy simulation method can perform consistency
check between the conceptual model and computerized model as shown in Fig. 3. In the fuzzy simulation method,
every vertex in the fuzzy number is issued independently to the simulation function, and the outputs of the simulation
are mapped into the most closely matched fuzzy linguistic value by a linguistic approximation. In this way, we obtain
rules from CODE, 1 S., FBM, Et^ N and RBM by, 1,''l i, the fuzzy simulation, and through consistency checking
against the expert's rules, we can identify ,!!- inconsistency due to an inadequate conceptual model of MOOSE or
an improperly programmed or implemented conceptual model on the computer. To make this validation available,
we have focused on the following tasks.














Problem
Entity






Conceptual
Experimentation
and
Physical Modeling






Computerized Model Translation Conceptual Model
Model -
(C++) (FSM, FBM, EQN)


Figure 2. Modeling process in MOOSE

Problem
Entity



Conceptual
and
Physical Modeling
Experimentation



Conceptual Model


Expert's rules
Computerized Model Translation
Model ---------------------- CODEFSAFBM,
(C-) EQN, RBM



Consistency Checking
using Fuzzy Simulation


Figure 3. Consistency checking in MOOSE


1. Developing a user interface for accepting the expert's fuzzy rules as one of the conceptual models in MOOS.

2. Making consistency-checking facility available between a computerized model and the expert's rules by using
fuzzy simulation.

3. Developing a user interface via human intervention for resolving a inconsistency.


In this paper, we discuss mainly the fuzzy simulation approach that we i 1I' ,1- ,1 for checking consistency in Task 2.
In the fuzzy simulation approach, we are able to calculate a confidence factor for each rule by !! i 11i membership
degrees of fuzzy sets in the expert's rule premise and issuing them to simulation components. Then we compare
against the confidence factor from the expert. This quantitative measurement provides us with useful information
such as the most inconsistent rule and the amount of knowledge discrepancy as a whole. This facility serves to
construct a convenient environment for resolving inconsistency in Task 3.











Table 1. Notation

Notation Usage
MFfuzy Fuzzy Membership Functions generated by fuzzy simulation
MFpremise Membership Functions of fuzzy value in rule premise
MFconseq Membership Functions of fuzzy value in rule consequence
CFexpert C," ",' 1... Factor presented by an expert
CFfu zy C, ; 1, ... Factor calculated by fuzzy simulation


3.1. Formats of Expert Rules as Input
The input of fuzzy simulation is a collection of expert rules. In what follows, we assume that the three following
canonical forms of rules are presented by the expert.

IF X is A THEN y is B (CF)

IF X is (A, A2) THEN y is B (CF)

IF (X is A) (y is B) THEN Z is C (CF),

where X, Y and Zare fuzzy variables that take real numbers from universal sets X, Y Z, respectively, A, A1 and
A2 are fuzzy values on X, B and C are fuzzy values on Y and Z, respectively, CF is a .... I, .... factor in the rule
consequence given that the premise conditions are satisfied, and are arithmetic (+, -, x or +) and logic (or or
and) operators.
In what follows, we call the first I- i of rule simplex rules, and the other two I- i, of rule compound rules.
These two I- |, of rule are handled in different way by the fuzzy simulation method discussed in next section. For
-ii!I 1. i- the notation in Table 1 will be used. The premise parts of the last two canonical I- i, of rules can be
combined to make a more complex rule such as IF (X is (A1 + A2)) or (Y is (B1 + B2)) THEN Z is C.

3.2. Fuzzy Simulation
The Fi simulation method is capable of simulating the expert rules using quantitative models. For each expert
rule, this method takes the premise part and its MFpremise, and through simulation it generates a conclusion
associated with CFf uy. \\ iii, the intention of comparing this result against the expert's counterpart, the fuzzy
simulation method is forced to derive the same conclusion that the expert presented, but with possibly different
CFfuzzy from CFexpert. When the expert rule is simplex, fuzzy simulation involves one simulation by taking each
element within the MFpremise. In contrast, when the rule is compound, we obtain an intermediate fuzzy set by
'I 'll i_ the extension principle8 or the rule of conjunctive or disjunctive composition 9 prior to sampling.

3.2.1. Simplex Rules
Fl Simulation Algorithm
Consider a simplex rule of the I |" IF X is A THEN y is B. Then the algorithm for fuzzy simulation is:

1. Let a fuzzy simulation component such as a parameter p be defined as a fuzzy set A, where

A = A(X21)/X21 + IA(X2) /2 + ... + A(X.)/Xn.

Assume the element of A is identified by brackets (i.e., A[2] = x2).
2. Forj E 1,2,...,n:
(a) Let p[j] = .lr1[
(b) i. LATEE REAL
(c) obtain (pB(yJ)/yJ)(te)














05 ---- 05---

x x .,
X 2 X3 X4 Xs -VV23 yV4Ys V5 xxx y23y45

Fuzzy I I I I Fuzzy
S Simulation Simulation


(a) All members of A support the conclusion B with full confidence (b) None of the members of A support
the conclusion B

Figure 4. All members or none of members support the conclusion


3. calculate CFfuzzy,

where 1. !IULATE REAL denotes simulation using real arithmetic, yj,j = 1,... ,n denotes real values on Y,
and and t, is the end time for the simulation.
During '1. IULATE REAL, the correlated uncertainty method requires that when we replace p with a real
number whose membership degree is d, we should replace other fuzzy simulation components with real numbers
whose membership degrees are also d. This procedure involves a two-step process of searching membership
degree of p and then using this degree to drive the elements of other fuzzy sets. In what follows, 1. !ULATE
REAL involves this operation.

*Calculation of CFfuzy
Just as CFexpert is presented by expert, we need a way to obtain CFfuzy from fuzzy simulation. By doing
this, we benefit from the comparison of the two rules in terms of their CF values. However, since the derivation
of the CFexpert involves a -111,i., live opinion as well as certain amount of iL,,. i; Iifl there is no theoretical
formulation to calculate the CFf,,Uy whose derivation process is exactly the same as that of the CFepert. Our
solution is to define an equation in such a way that its result agrees with human intuition as much as possible.
We used a weighted average method to create such an intuition. Given a simplex rule, we define the CFfuzy
by using the weighted average method


CyFf Y E (A(Xj) X (9)
Ej'= l[tA(Xj)

where xj,j = 1,2,..., n, denote real values on X in the fuzzy set A,
and yj,j = 1, 2,..., n, denote real values on Y obtained from simulation using xj. The validity of calculating
CFfuzzy using weighted average method can be easily shown in Fig. 4 and Fig. 5. CFfuzu, using Equation
(9) is 1.0 and 0.0 for Fig. 4(a) and Fig. 4(b), respectively. The results exactly match our intuition. When
the CF falls into some range between the above two extreme cases (i.e., 0.0 and 1.0) as shown in Fig. 5, we
can intuitively that the greater CF we get, each member in A supports the conclusion B with a higher
confidence. Using (9), the CFfuzzy for Fig. 5(a) is 0.75 and the CFfuzzy for Fig. 5(b) is 0.475.

3.2.2. Compound Rules with Arithmetic Operations
Ft Simulation Algorithm
Consider a compound rule of the I" IF X is (A1 A2) THEN y is B, where is one of the four basic
arithmetic operators. Then the algorithm for fuzzy simulation is:


1. Apply Equation (3) to the rule premise.


















XI X234X5


Fuzzy
Simulation



(a) Members of Asupport the conclusion Bwith higher confidence
compared to the case of (b)


XI X234X5


y, y2 Y3 Y4 Ys


Fuzzy
Simulation



(b) Members of A support the conclusion B with less confidence
compared to the case of (a)


Figure 5. Some members support conclusion
Al # A2


0 1 2 3 4 5 6


0 1 2 3 4 5 6


(a) (b)

Figure 6. Two fuzzy sets for addition


2. Let Z be a resulting intermediate fuzzy set, and let a fuzzy simulation component such as a parameter p
be defined as a fuzzy set Z, where

Z = Pz(Zl)/Zl + pz(ZZ2)/z2 ... +- [Z(Zn)/Z.

Assume the element of Z is identified by brackets (i.e., Z[2] = z2).
3. Forj E 1,2,...,n:
(a) Let p[j] = Z[j].
(b) 'i. LATEE REAL
(c) obtain (pB(Yj)/yj)(te)
4. calculate CFfuzzy.

* Calculation of CFfuzzy
Given a compound rule with arithmetic operations, we define the CFfuzz by using the weighted average
method


CFf,


(10)


where zj,j = 1, 2,..., n, denote real values on a fuzzy set resulted from arithmetic operation, A1 A2, yj,j
1, 2,..., n, denote real values on Y obtained from simulation using zj.


y, y,yy, Y4


y j=l(-tAj*A2( Z) X rB(yj))
Ej=1 PA1*A2(Z)














----- Y --L-- -- .--- ._ L

--I-|-- -- -- -- -- -- -----|---|-----^



S2 3 4 5 6 7 8 9 10 11 12 13



SIMULATE REAL


10 --



06 --
05


y
25 29 32 42 51 57
tttrtlr


Goo


Figure 7. F, simulation using compound (addition) rule

Example
Let's assume that we want to perform fuzzy simulation using the following rule, IF X is (A1 + A2) THEN Y
is B, where A1 and A2 are defined by Fig. 6 (a) and Fig. 6 (b). Then by ;,,'I i;!, Equation (4) defined by

PA1+A2 () = max1=A1+A2 (PA (x) A PA2 ()),
we can obtain the following set of equation for intermediate fuzzy set Z.

pz(1) = (0A0.3)V(0A0.1)=0,
[z(2) = (0 A 0.6) V (0.1 A 0.3) V (0.3 A 0) = 0.1,
pz(3) = (0 A 1) V (0.1 A 0.6) V (0.3 A 0.3) V (0.8 A 0) = 0.3,


[z(13) = (0.3 A 0) V (0 A 0.1) = 0.


Fig. 7 shows the fuzzy set Z and the result of fuzzy simulation using Z, where the result is arbitrarily made
for illustration purposes. Using Equation (10), we can calculate CFfEuzy by

CFfup = (0.1 x 1.0) + (0.3 x 0.6) 0.06
CfuY 0.1 + 0.3 + 0.3 + 0.6 + 0.8 + 1.0 + 0.7 + 0.7 + 0.3 + 0.2 + 0.1


3.2.3. Compound Rules with Logic Operations
F, Simulation Algorithm
Consider a compound rule of the i" IF (X is A) (y is B) THEN Z is C, where denotes ;oI logical
operator. Then the algorithm for fuzzy simulation is:

1. If is and operator, then apply the rule of conjunctive composition (D. 6"',!.. 2.3) to rule premise and
calculate a I .. ..-i;1 1il distribution r(x, y). If is or operator, then ; I'1 il! rule of disjunctive composition
(D, ;,./ ,.. 2.3) to rule premise and calculate a I...--l1.i 1, distribution -r(x,y).
2. Let fuzzy simulation components such as p and q be defined as fuzzy sets A and B, respectively, where

A = pA(IX)/Xi + A(X2)/X2 +*.. * +A(X)/Xr,
B = tB(Yl)/y1+ iJ B(Y2)/y2 r *J... B(Yn)/I .

Assume the elements of A and B are identified by brackets (i.e., A[2] = x2 and B[2] = y2).

















T(xy) 44 59 70
01 /(11)
06 /(1,2)
1 /(1,3)
01 /(2,1) SIMULATE
06 (2,2)
06/(2,3)
01/(3,1)
01 / (3,2) *0
01/(3,3)


Figure 8. FT- simulation using compound (' ,,!i! ,, 1r;! rule

3. For i E 1,2,...,m
For j 1,2,...,n
Let p[i] = A[i].
Let q[j] = B[j].
'i. LATEE REAL
obtain (pc(zij)/zij)(te)
4. calculate CFfuzzy,
where m and n are the number of elements in A and B, respectively.

Notice that in the rule defined above, the universal sets of the fuzzy variables A and B are not identical.
Otherwise, instead of 7r(x,y), we can get a more simplified fuzzy set as an intermediate set by D, C"i', 2.5
and D, ;. ',' .1 2.6 for *1-.1 Iti ii. .1, and' .iij l lii, respectively.

* Calculation of CFfuzy
Given a compound rule with logic operations, CFfuzy is defined by using the weighted average method

-21 E Eji (1tuA*B(Zi, yj) X c(ij))
EC, =t A*B(i,) (11)

where denotes an logical operator.

* Example
Let's assume that we want to perform fuzzy simulation using the following compound rule, IF (X is A) and
(3 is B) THEN Z is C, where A and B are defined as A = small = 1/1 + 0.6/2 + 0.1/3 and B = ... ,,. =
0.1/1 + 0.6/2 + 1/3. Then by .,il,- i;,- the rule of conjunctive composition, the predicate, (X is A) and (Y is
B), yields the following i ..--. l1ii distribution:

7(x,y) = {[PlAandB(x1, Y1)(x1, yI)], [IAandB(x1, Y2)(x1, y2)],
= [ AandB(X1, Y3)(X1 Y3)], [AandB(x2, Y1)/(x2, YI),
S, [PA andB (3, 3)/(X33y 3) }
= {[0.1/(1, 1)], [0.6/(1, 2)], [1/(1, 3)], [0.1/(2, 1)], [0.6/(2, 2)], [0.6/(2, 3)],
[0.1/(3,1)], [0.1/(3, 2)], [0.1/(3,3)]}
Let's assume that we have the result as shown in Fig. 8 by performing fuzzy simulation on this 7r(x, y). Using
Equation (11), we can calculate CFfuzzy by
(1.0 x 0.5) + (0.1 x 0.8) + (0.6 x 0.3)
CFfu3 = 1.0+0.1 6= 0.44
1.0 + 0.1 0.6












rule editor


very_eryshort zeroer ryvery_little 0 900000
very_short zero => very_llttle 0.600000
short zero => little 0.600000
slightly moderate zero => little 0.600000
moderate zero => medium 0 600000
slghtly_long zero => slightlyjuch 0 900000
long zero => slightly much 0.500000
very_long zero => much 0.900000
long very_short => much 0.500000
slightly moderate very long => medium 0.800000
moderate short => slightlymuch 0.500000
shoM t long => little 0.500000
very_long veryery_short => much 0.700000
lhghtly_long slightly_moderate => much 0 60000
very short slightly_long => little 0 900000
very_very_short moderate => very_very_little 0.900000













very very short vr ve0 1 ryjte t__

very_short 5 very little 39|

short 10 little

slightly_moderate medium 57 65 73

moder-ate 1 I4 lslightly_much

slightlyjlong 19 much

long I 20 23

verylong Li_ 23 23

Reset Rules Reset MFs Evauate Exit


evaluation button ---


Membership
function
editor


Figure 9. A GUI for human intervention for resolving inconsistency


3.3. Resolving Inconsistency

For the case where the amount of inconsistency is out of range after executing fuzzy simulations, we have developed an
advising facility which suggests all expected rules from fuzzy simulations. \\ il 1 this information, the model author
starts to resolve the inconsistency. For this process, either the expert rules (including CF,,pert and membership
function definitions) or the simulation parameters can be modified interactively. Every time these modification
happens, the fuzzy simulation is reinvoked with visual aids so that the user can easily recognize the effect of the
modification. Fig. 9 shows the GUI we developed for the human intervention to resolve the inconsistency.


4. CONCLUSION

As we discussed in the previous sections, the consistency between two I I," of models (expert's rules and conceptual
models in MOOSE) can be measured by the difference between the CF presented by an expert and the CF calculated
from fuzzy simulation on each rule. This gives the model author of MOOSE useful information, such as which
components of the conceptual models should be further investigated. Whenever the inconsistency is detected, the
quantitative measure mentioned above helps the human author identify and revise the most inconsistent component
rapidly and :i 1- 1- the effectiveness of that modification, thereby allowing the two models to gradually reach a
consensus with high resolution. Consequently, by incorporating the proposed method into MOOSi. we can obtain a
benefit from validating the simulation models against the expert's knowledge.











ACKNOWLEDGE EN IENTS
We would like to thank the following funding sources that have contributed towards our study of modeling and
implementation of the MOOSE multimodeling simulation environment: GRCI Incorporated (Gregg Liming) and
Rome Laboratory (SI. .. Farr) for web-based simulation and modeling, as well as Rome Laboratory (Al Sisti) for
multimodeling and model abstraction. We also thank the Department of the Interior under a contract under the
ATLSS Project (Don DeAngelis, University of Miami). \\ ih!..,i their help and encouragement, our research would
not be possible.

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