A method for resolving the consistency problem between
rule-based and quantitative models using fuzzy simulation
Gyooseok Kim and Paul A. Fi-lr-i, 1:
Computer and Information Science and Engineering Department
University of Florida
Given a ,1!i -;1 .1 -i. i, there are experts who have knowledge about how this -. ii, operates. In some cases,
there exists quantitative knowledge in the form of deep models. One of the main issues dealing with these different
I of knowledge is "!- ..- does one address the difference between the two model I I'" each of which represents a
different level of knowledge about the -. "" We have devised a method that starts with 1) the expert's knowledge
about the - I 1i1 and 2) a quantitative model that can represent all or some of the behavior of the -- -1. i- This
method then ;,,lii-1 the knowledge in either the rule-based -- -I. i, or the quantitative -I. i i to achieve some degree
of consistency between the two representations. Through checking and resolving the inconsistencies, we provide a
way to obtain better models in general about -1. ,!1- by exploiting knowledge at all levels, whether qualitative or
Keywords: Rule-based model, Quantitative model, Knowledge acquisition cycle, FiL simulation, Consistency
checking and resolving
Given a I '1 -1, 1 -- -. i i knowledge about the -I. ii, is often obtained from experts in the form of rules. Although the
rule-based model is occasionally associative or shallow in nature, this model can easily capture human heuristic and
problem solving knowledge in an. ~ It. w ii way.' In some cases, there exists a quantitative model which represents
all or part of the behaviors of the i,1! 1 - -. ii This model provides deeper and more theoretical knowledge when
expert -- -. i developers want to find solutions for technical problems.2'4'3
Assuming the above two different model I- i" for the identical -1. 11, some important questions can arise: how
much do the models I II and how can one resolve the ....... ... By 1 ii i- to answer these questions, we
obtain benefits to expert -I. ,!- using simulation and benefits to simulation modeling using expert knowledge.4-11
Especially, when expert -1 111 researchers are -1 r 1 i the acquisition of deep knowledge from an expert or validating
the expert's knowledge against quantitatively compiled knowledge, the first I- I of benefits can be obtained from
simulation models.4-8 The benefit from the reverse direction is also obtained when simulation model validations are
performed during the simulation modeling process with the aid of the expert knowledge.7,'911
One way of handling the inconsistencies between the expert's level of qualitative knowledge and the lower level
of deep knowledge is to form a knowledge acquisition (.,, 1, as in Fig. 1(a).2 Approaches to creating model bases are
discussed within the context of computer simulation.12 For example, the model base represents compiled knowledge
about i! i~1- domains such as the mathematical queuing model for waiting line problems. If a match is found, then
shallow rules are generated by means of qualitative or quantitative simulation based on this deep model. Since the size
of the shallow rules resulted from the deduction process is usually too big for a human to study and validate against
the original expert's rules, induction methods can be i! !1. ,1 to obtain a more comprehensible and generalized set
For domains in which the rules from experts contain in ii, linguistic terminologies whose boundaries are not
exact, we need a way to encode this vagueness into computer simulation. For such cases, we can use either qualitative
or quantitative simulation with fuzzy set .,,. p1 -' '' for the deduction process. However, the well known problem
Other author information: (Send correspondence to G. Kim)
G. K.: Email: ': . ll .1 1. 1. ....... 352-392-1435; Fax: 352-392-1414.
P.A.F: Email: fishwick(cise.ufl.edu; !. ...... 352-392-1414; Fax: 352-392-1414
(a) A knowledge acquisition cycle (b) A knowledge acquisition cycle using fuzzy simulation
Figure 1. A knowledge acquisition cycle
in using the qualitative methods is the possibly generating of spurious behaviors of the -1, during the reasoning
process.5,'2 Moreover, in order to get a compressed and generalized set of fuzzy rules, additional methods such as
fuzzy induction or fuzzy -- -I. i-, identification ri. II! -1-" ' should be adopted. Consequently, forming the above
knowledge acquisition cycle to !!ii!. fuzzy set concepts requires a series of, 1!t!hl ll tasks.
We have developed a method as shown in Fig. l(b), where we've !,1!' ,1 a fuzzy simulation approach4 6 for
directly encoding uncertainty arising from human linguistic vagueness into simulation components as well as for
utilizing quantitative models for the deduction process. Since this method uses a linguistic mapping process to map
simulation inputs and outputs into fuzzy linguistic values that were also used by experts, direct comparison is possible
without an additional induction step.
Our method consists of two phases: 1)consistency checking phase, and 2)resolving phase. In the consistency
checking phase, experts provide various levels of estimates for a fuzzy set and then, through fuzzy simulation and
incremental optimization over the error surface, fuzzy set boundary vertices are created to fill in the expert's knowl-
edge. Currently, we've implemented an approach where the estimates are presented in the form of central points.
For quantitative comparison between the two knowledge, quantitative measures have been formulated to gauge the
sources and the degree of inconsistency. The final products of this stage are rules derived from quantitative models,
approximate fuzzy membership functions for those rules, and the amount of inconsistency against the expert's rules.
If the amount of inconsistency shows ',i ~1l a reasonable range, the resolving phase is necessary. In this phase,
human intervention is present: either expert rules (including the definitions of fuzzy numbers) or simulation model
components are reevaluated or modified to reduce the amount of inconsistency. Even at this point, the quantita-
tive measures mentioned above help them identify and revise the most inconsistent component rapidly and :i! i1
the effectiveness of that modification, thereby allowing the two .l!!, i Ii levels of knowledge to gradually reach a
consensus with high resolution. The knowledge acquisition cycle presented here forms a more potentially organized
framework that resolves the inconsistency between two knowledge sources in an ItI !i. ii and -- -1, 11, 11i manner.
We will first discuss the fuzzy set theory which is relevant to this paper and its relation with computer simulation.
Then, in section 3 and 4, we present our method in detail, and illustrate an application of the proposed method with
a simple example. I i !! in chapter 5 we present a research direction in the future.
2.1. Fuzzy set theory
The theory of fuzzy sets can be found in Refs. 20-23. Generally speaking, fuzzy sets 11i be viewed as an attempt
to deal with a 1- 1 of imprecision which 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,. C"! '",,. 2.1: A fuzzy set A of the universe of discourse X is convex if and only if for all Xi, X2 in X
pA(Axi + (1 A)x2 > Min(pA(Xl), PA(X2))
where A E [0, 1].
D,. "; 2.2: A fuzzy set A of the universe of discourse X is called a normal fuzzy set if 3xi E X, pA(xi) = 1.
D, i "', 2.3: A fuzzy number is a fuzzy subset in the universe of discourse X that is both convex and normal.
To simplify the representation of fuzzy sets, a finite fuzzy subset, A, of X is expressed as
A = PA(xl21)/x2 +A (x2)/x2 + .. + piA(xn)/2,, or A = E 1 PA i(xi)/xi,
where + sign denotes the union rather than the arithmetic sum.
If the fuzzy subset, A, is not finite, A i, i- be represented in the form A = f, ipA(x)/x in which the integral sign
stands for the union of the fuzzy singletons p A(x)/x.
D. C."! '.. 2.4: The complement of A is denoted by A and is defined by
A (1- pA(X))/X. (1)
The operation of complementation corresponds to negation.
D,. "'! ".,. 2.5: The union of fuzzy sets A and B is denoted by A U B and is defined by
AUB= f (PA(x) VPB(X))/x. (2)
where V is maximum operator.
D. ;,.! '..i 2.6: The intersection of fuzzy set A and B is denoted by A n B and is defined by
AnB = (A (x) AB (x))/. (3)
where A is minimum operator.
Let A and B represent two fuzzy numbers and let denote .i, of the four basic arithmetic operations. Then we
define fuzzy set, A B on R', where R' is a set of all real numbers, as
f A*B(Z) = ""'. ipA(x) A PB(Y)), (4)
for all z E R. Thus, for example, if A, B C R are two fuzzy numbers with respective membership functions piA(X)
and pB(y), then the four basic arithmetic operations, i.e., addition, subtraction, multiplication and division, give for
each x, y, z E t the following results:
PA+B(z) = maXz=x+y (A (zx) A PB.(y)). (5)
PA- B(Z) = max3=x -y (A (x) A PB (Y)). (6)
f1-AxB(z) = ,'i.. ,(PA (x) A PB(y)). (7)
PA1* B(z) = (mapx3;=-y (PA(x) A PB (y)). (8)
2.2. Fuzzy set theory in computer simulation
Probability based methods are useful when most of the uncertainty can be effectively described through the use of
large data sets and their associated moments. However, experts often do not think in probability values, but in terms
such as much, usually, always, sometimes, etc. In domains where estimation or measurement of probabilities is not
amenable, fuzzy set theory offers an alternative.24 Here, we can use ;~i- I' ".- of fuzzy number, such as an interval-
valued fuzzy number, a triangular fuzzy number, a trapezoidal fuzzy number or a general discrete (or continuous)
fuzzy number depending on the degree of uncertainty. Owing to the extension ri t 1'l'- in the fuzzy set theory,
1j..5rf *- mathematical structures can be made fuzzy. Here is a sample of how this relates to simulation. We can
make fit 1) a state variable value including initial conditions, 2) parameter values, 3) inputs and outputs, 4)
model structures, and 5) algorithmic structures.
Table 1. Notations
MFpremrise Membership Function of fuzzy value in rule premise.
MFconseq Membership Function of fuzzy value in rule consequence.
RUT F Expert's simplex rule.
RUT F Expert's compound rule.
CFexpert C," "''; ... Factor presented by an expert.
CFfuzzy C, 1 ; .... Factor calculated using fuzzy simulation.
In order to simulate mathematical models using the fuzzy set concept, three kinds of fuzzy simulation approaches
have been reported: Qualitative Simulators (i.e., Qua.Si13), F., Qualitative Simulation (i.e., Fusim14), and three
methods (Monte Carlo, Uncorrelated U .... !'"i,.. and Correlated U...., ".!.'i of fuzzy simulation introduced by
1 i-l!- i !: 6 While the first two kinds of fuzzy simulation are useful when there is not enough information to simu-
late quantitatively, the third kind takes linguistic information from the expert and performs computer simulation
,... .,' ',, on continuous and discrete event models. This approach is similar to the sampling method used in
Monte Carlo simulation, except that fuzzy variables are used so that the sampling technique Ltil, i Rules or FSA
(1 [ii.- i ti.- Automata) can be extracted from these quantitative models through linguistic mappings, and these
results can be validated directly against the expert domain knowledge. The fuzzy simulation method we present is an
extension version of the correlated ......'* .i', method. For more information of the correlated I...' ..i,!"'. method,
see Refs. 4-6.
3. A PROPOSED METHOD
In this chapter, we propose a method for resolving the inconsistencies between the expert's rules and the quantitative
models. As we discussed, our method consists of two phases: consistency checking and resolving '....1.... ," ., While
the first phase is done through an automatic process, the second phase is performed semi-automatically. In this
section, we will focus on the first phase, since the part of algorithm presented in this section with human interaction
can cover the second phase as well. Before exploring the algorithm, we must first introduce the input of the algorithm
and two important usages of fuzzy simulation that we've ii *I,!' 1
3.1. Format of Expert Rules as Input of Proposed Method
In what follows, we assume that the format of expert rules is one of the following two I- i" The input of the
proposed method is a collection of the expert's rules below, with conclusions from the same fuzzy variable:
IF (X is A1) THEN (T is B); CF; CLA,; CLB
IF (xi is A1) OP (X2 is A2) OP, ..., OP (xn is A,,)
THEN (T is B); CF; CL1; CLA2; ...; CLA,; CLB
Xi, i = 1, 2,..., n, and Y are fuzzy variables that take real numbers from some universal set X, Y respectively,
Ai, i = 1, 2, ...,n, and B are fuzzy values on X, Y respectively,
CF is a ... ;, 1... factor in the rule consequence given that the premise conditions are satisfied,
OP is a fuzzy logic (or or and) or fuzzy arithmetic (+, -, x or ) operator, and
CLAi, i = 1, 2,..., n, and CLB are expert's ....- ;. ... levels on the fuzzy values in each rule.
The two I- i" of rules above are called complex rules and compound rules respectively. The value of CL can be
a center point estimate, an interval estimate, an approximate fuzzy number or a complete fuzzy number depending
on the expert's confidence level on the linguistic term he used. In this paper, we restrict our discussion within a
situation where the values of the CL are center point estimates.
[/ / i\ \ / I Y
5 15 20 30 35 45 0 5 10
Figure 2. Definitions of fuzzy numbers A, B and C
3.2. Two Usages of Fuzzy Simulation
In what follows, the notations in Table 1 will be used for -i1b,.l1 il In the proposed method, the fuzzy simulation
approach has two important roles: 1) calculation of CFfuSy of RUT and 2) estimation of MFconseq. We
discuss these two roles of the fuzzy simulation in the following two sections.
3.2.1. Calculation of CFfu~ y of RUT F
Since the uncertainty arising from the human reasoning process is easily represented by a rule associated with
CFepxprt, we introduced a way for emulating such processes by showing how fuzzy simulation can derive the confidence
factors from quantitative models. By doing this, we benefit from the comparison of the two rules in terms of their
CF values. However, since the CFexpert involves a -,1i, I live opinion, there is no theoretical formulation to derive
the CFfEzy whose value is exactly the same as the CFpert. Our solution is to define an equation in such a way
that its results agree with human intuition as much as possible. We used a weighted average method to create such
Let us define the CFfuzy using the weighted average method. Given a RUT let its two MFpremises
be A and B, where A and B are fuzzy subsets of a universe discourse X, and its MFconseq be C, where C is a fuzzy
subset of a universe discourse Y. Then we define the CFfEUy by the following equation:
E' 1=(IAG B(XJ) x X tc(yj))
CFfuzzy j= 1 (9)
where 0 denotes a fuzzy logic or arithmetic operator,
xj, = 1, 2,..., n, denote real values on the fuzzy set resulted from the operation of A 0 B,
yj, = 1, 2,..., n, denote real values on Y obtained from fuzzy simulation using xj.
Equation (9) can be divided into the following three steps for -ii! .!lif i,- its calculation: 1) perform the fuzzy
logic/arithmetic operation, 2) simulate using the fuzzy set obtained from the above step, and 3) calculate CFfuzzy
using the weighted average method. For example, given a RUT F IF Xi is A OR X2 is B THEN T is C,
with definitions of A, B and C as shown in Fig. 2, CFfuzzy for the RUT F can be calculated by performing
the following steps:
1. Perform the fuzzy OR operation for A and B. For each element x in X, the degree of membership of A OR
B, PA OR B(x), is obtained by (2). Fig. 3(a) shows the result of the operation.
2. Perform the fuzzy simulation on the fuzzy set of Fig. 3(a). The result is shown at Fig. 3(b).
3. Calculate CFfuzzy using the weighted average method.
(0.3 x 0.5) + (0.7 x 0.8) + (1.0 x 0.9) + (0.7 x 0.2)
CFf 33Y 0.3 + 0.7 + 1.0 + 0.7 + 1.0 + 0.7 + 0.3
The validity of calculating CFfuEy in this way can be easily shown as in Fig. 4. CFfu; y, using (9), is 1.0 and
0.0 for Fig. 4(a) and for Fig. 4(b) respectively. The results exactly match our intuition. When the CF falls into some
range between the two extreme cases above, we can intuitively - that each member in A supports the conclusion B
with a higher confidence, the greater CF we get. Using (9), we also get the results which support such an intuition.
A OR B A ORB I C
1.0 ----------.--- 1.0 --- --
0.7 ------- -- 0.7 _ __ 0.8 ---- -
0.3 -- -------- 0.2 -
x X m y
5 20 25 30 45 5 10 15 20 25 30 35 40 45 1 23 6 9 14
(a) The result of operation A ORB (b) The result offuzzy simulation on fuzzy set A OR B
Figure 3. Calculation of CFfuzy as an example
A B A B
1.0 -- 1.0 ---
I I I
0.5 ----- -- -I I 0.5 - - -
x y x
X1 X2 X3 X4 X5 Yl Yz Y3 Y4 Y5 X2 345 Y Y2 Y3 4
(a) All members of A support the conclusion B (b) None of the members of A support
the conclusion B
Figure 4. All members or none of members support the conclusion
3.2.2. Estimation of MFconseq
In the previous section, we used a fuzzy simulation to derive the CFfEzy when all definitions of the linguistic
terms in a rule are already known. Conversely, without knowing the definition of the linguistic term, particularly
the definition of the linguistic term in the consequence of the rule, we can use the fuzzy simulation to estimate its
Let's assume B is a ,ni i; triangular fuzzy number whose members are real numbers y. Knowing its center
point c and the width w of B, the degree of membership of 11;, real number, yl,Y2, y, Y can be obtained from the
2 x yi c
pB (Yi) = 1 (10)
where i = 1,2,...,m.
Let's assume another fuzzy number A whose members are real numbers x. Given an expert rule, IF / is A THEN
T is B, with its CFepert, performing a fuzzy simulation on A and ;, 11- 1 i ,_ the weighted average method to B yields
pCF (Pi A zi) X B (Yi))
Ei=l 1A (xXi)
where y, is a result of the fuzzy simulation on xi.
However, consider a situation where a fuzzy simulation is executed on A, but the width of B is unknown. Letting
the CFfuE, in (11) be equal to the CFexprt of the rule above, and substituting the right-hand side of (10) for the
A B B
10 ------ 10----------- 10 -
x O- y y
003 260270 290 320 230 270 310
(a) Fuzzy simulation with unknown width of B (b) Symmetric fuzzy membership function of B
Figure 5. Estimation of unknown width of B using fuzzy simulation
pIB(Yi) in (11), we get
Fe I(ltA(Xi) X (1 -2 c))
CFexpert Ei=1 fA( i)w (12)
From this equation, we can obtain the following equation to estimate the unknown width w of B.
2 x E' (tlA(Xi) X i c|)
w = = (13)
S Ep1-A(X) (CFxpert X E 1A [1 i(X))
Equation (13) has an important meaning: if we know an expert rule, its CFexpert, its MFpremise and the center
point of the fuzzy number in the consequence of the rule, then we can estimate the range of the fuzzy number with
an aid of fuzzy simulation.
For example, with the rule, IF T iA THEN T is B; CFeprt = 0.5; CLA = 0.03; CLB =27.0, and unknown
width w of B, suppose that the result of a fuzzy simulation is shown in Fig. 5(a). By ;,'l1 i;!- (13), a -- ''ir li;
triangular membership function for B can be obtained as shown in Fig. 5(b).
One constraint of;,' i, ii- (13) is that the CFxpert should not be equal to 1.0 (i.e., less than 1.0). Otherwise, the
value of the denominator in (13) would be zero. Even though such a case is currently a limitation of the equation, a
preliminary approach has been developed.
3.3. An algorithm
Once the expert's rules for a 1,1! -;, ,1 -. i! have been presented, and a relevant quantitative model has been found
during the model base search, we can apply the algorithm presented here for checking consistency between the two
models of knowledge. The algorithm generates the approximate definitions of fuzzy linguistic values by increasing
the ranges of fuzzy sets from their initial minimal width to fill in the expert knowledge. For such a process, two
confidence factors, CFexpert and CFfuzzy, are used to calculate local and global inconsistencies. These serve as the
quantitative closeness measures between the two different levels of knowledge. Two different usages of the fuzzy
simulation as discussed in section 3.3.1 and 3.3.2. are involved to help this process. When the algorithm reaches
a point where tuning membership functions does not improve the amount of closeness i- further, the algorithm
stops and returns the membership functions that have been tuned so far as an approximate set with which two levels
of knowledge match I.... .....I.., If the closeness is out of a reasonable range, human intervention is required for
resolving the inconsistencies: either the expert rules or the simulation components which show the inconsistency can
be reevaluated, or the definitions of the linguistic values generated by the algorithm can be changed interactively.
The algorithm presented here is also useful for this resolving phase, since the comparison results are quantitatively
calculated and visualized in response to the human interaction. When the goodness of fit reaches a reasonable point,
another fuzzy simulation with different values of fuzzy variables creates a more detailed level of rules than the level
of the expert's rules.
For this algorithm, we !!i1.. an iterative improvement method. This algorithm consists of the following three
Step 1 Hypothesize membership functions in RULEsimpl
Hypothesize MFpremises in RUEsimplexs -
Fuzzy simulation for each RULEsimplex
Obtam MFconseqs RULsimplexssmg equation (13)
Step 2 Apply hypothetical membership functions to RULE pounds
Fuzzy Logic/Arithmetic Operations for MFpremises RULEcompounds
Fuzzy Simulations for RULEcomponds
Calculation of CFfuzy for each RULEcompoundusng equation (9)
Calculation of LIs and Glfor RULEcompounds
Step 3 Improve hypothetical membership functions using RULEcompuds
Pick a RULEcompound which has the largest LI
Find all MFpremises causing the largest LI
Find a proper subset leading to the smallest GI
Compare this smallest GIto PI
GI < PI
Final approximate fuzzy membership functions
Figure 6. Three basic steps and their substeps of algorithm
1. Hypothesize membership functions in RUT F
2. Apply hypothetical membership functions to RUT I
3. Improve hypothetical membership functions using RUT I
Fig. 6 shows detailed substeps in each basic step. We explore them in the following three subsections.
3.3.1. Step 1: Hypothesize membership functions in RUT I
The purpose of this step is to 1i i.!. -- .. each MFpremise in RUT F and to obtain its corresponding 1i- Ii-
thetical MFconsq using (13).
In the first substep, two cases should be handled differently. That is, when the algorithm initially starts, we
construct an initial hypothetical MFpr.emise so that its range is 2Ad with the center point, where Ad is a optimal
resolution size for simulation execution. Ad can be determined by experts or simulationists. For other case, this
substep modifies MFpremise by increasing its range by Ad on either sides. After executing the last substep, we obtain
a hypothetical pair of MFpre,,is and MFconseq for each RUT F which satisfies CFfu,,,,, CFexpert.
3.3.2. Step 2 : Apply hypothetical membership functions to RUT F
The obtained MFpremises and MFconseqs from the previous step are consistent only for the RUT F in a
sense that CFfuzzy ,, CFexpert for each RUT F Our claim is that if those membership functions are really
consistent, then this also should be the case with the all RUT F Thus, the purpose of this step is to apply
these hypothetical membership functions to the RUT F to check their validities.
For each RUT F we define its local ....... .... LI, as
LI = CFfuzzy CF 2pert (14)
Then, using the LI, we define the global inconsistency for all RUT TF GI, as
GI = LI, (15)
m = total number of RUT F
Searching for the largest LI enables us to identify the most inconsistent RUT : between two different
knowledge sources. Moreover, the GI calculated in this way allows us to measure the total amount of inconsistency.
3.3.3. Step 3 : Improve hypothetical membership functions using RUT F
The purpose of this step is to reduce the GI by picking up a RUT F : which has the largest LI and in. 1i f- i!-, a
proper subset of the MFpremises among all subsets of MFpremis, which caused that LI. We can find the proper subset
by searching for a combination of the MFpremises which leads the GI to the smallest value among all combinations
of the MFpremises which caused the largest LI. Notice that we should not regard the MFpremises that can reduce
the largest LI into the smallest amount as the proper subset. The reason is that if ;,i- such MFprem,ise is used
also for other rules, then the modification of this definition could make other LIs in those rules worse than before,
possibly causing increased GI as a whole. For this reason we introduce the GI instead of the LI as a p. .' f ...... ...
index (PI). Therefore, we need to find a subset of MFpre-mis, which improves the GI by the greatest amount by
executing the step 1 and step 2 for each subset of MFpremises. When we eventually reach the smallest GI after
incrementally reducing the inconsistencies, we can regard the hypothetical set of the MFpremises and the MFcon,, qs
as the final approximate fuzzy set with which the expert's rule-based model matches maximally the quantitative
4. EXAMPLE: BOILING WATER MODEL
To illustrate the application of proposed method, we have chosen boiling water12 as a simple example. Due to the
limited length of the paper, we showed only inputs and outputs of the consistency checking phase. We used a total of
sixteen rules (eight RUT F and eight RUT F ) and their center point estimates as shown in Fig. 7(a)
as the expert's inputs to describe the water temperatures depending on the on and off position of the knob over time.
Using these expert rules as inputs and 1,1~' ii- fuzzy simulations to the quantitative simulation model12 for this
-1 in we obtained sixteen rules and an approximate set of fuzzy membership functions for those rules as shown in
Notice that each CFexpert in Fig. 7(a) is closely equal to the corresponding CFfuzy in Fig. 7(b). Moreover,
PI turns out to be 0.229479, which can be regarded as f',,'. ., consistent. Therefore, the expert's rule-based model
and the quantitative simulation model for this particular boiling water problem can be considered to be consistent
without processing an additional resolving inconsistency phase. By executing another fuzzy simulation with different
values of the fuzzy variables, we got more detailed rules (8 x 8 = 64 rules) as a li- i"..1! -;- of the expert's deep
knowledge. Table 2 shows a part of such knowledge.
PREMISES CONSEQUENCES CF
KNOB ON KNOB OFF TEMPERATURE e
veryveryshort cold 0 8
veryshort cool 05
short tepid 0 7
slightly moderate warm 0 8
moderate hot 0 6
slightlylong veryhot 0 3
long veryhot 0 6
verylong veryhot 0 8
slightly moderate very veryshort warm 0 8
slightly moderate long tepid 0 9
moderate short hot 0 5
short verylong cool 0 4
very_long veryvery_short veryhot 08
moderate verylong tepid 0 6
veryshort verylong cool 07
veryveryshort verylong cold 09
CENTER POINT ESTIMATE( CL)
very veryshort 0 0 cold 20 0
veryshort 0 03 cool 27 0
short 0 06 tepid 37 0
slightly moderate 0 09 warm 44 0
moderate 0 13 hot 520
slightlylong 0 17 veryhot 700
verylong 0 23
20 27 37 44 52
(a) Expert's rules and center point estimates
(b) Rules extracted from fuzzy simulations
and final approximate fuzzy membership functions
Figure 7. The inputs presented by expert and the outputs of the proposed method
5. DISCUSSION AND FUTURE RESEARCH
The proposed algorithm is an iterative improvement algorithm !l!.- i ,1 the gradient descent method, because it
executes a loop that continually moves in the direction of decreasing GI. It keeps track of only the current states,
and does not look ahead '1, -..il the immediate neighbors of that state. Its solution ii! be a local minima. This
local minima problem can be cured if we choose all paths whose GIs are better than PI, instead of choosing the path
which has the best GI. ('!. I this solution costs more in terms of simulation time and memory than before, but
we can better avoid the local minima problem. Alternatively, we can take a middle position between these extreme
strategies. For example, when the problem space is too large to adopt the latter -li i -, we can choose two or three
best paths at every iteration.
We showed the proposed algorithm can deal with central point estimates. However, in order to handle the various
levels of uncertainties arising from linguistic vagueness, we will extend the method to cover the other cases as well.
Specifically, we will enhance our method to cover the situations where experts present various levels of confidence on
the linguistic terms in the following ways:
central point estimates
* interval estimates which represent the possible ranges of fuzzy sets
PREMISES CONSEQUENCES CF
KNOB ON KNOB OFF TEMPERATURE
veryveryshort cold 0 8
veryshort cool 05
short tepid 0 7
slightly moderate warm 0 8
moderate hot 06
slightly long veryhot 0 3
long veryhot 0 6
verylong veryhot 0 8
slightly moderate very veryshort warm 0 766667
slightly moderate long tepid 0 892000
moderate short hot 0553164
short very long cool 0410000
very_long veryvery_short veryhot 0 840909
moderate verylong tepid 0 684073
veryshort verylong cool 0 700000
veryveryshort verylong cold 0 900000
Table 2. A part of detailed rules extracted from fuzzy simulation
KNOB_ON KNOB_OFF TEMP. CFfu3 y
moderate very_very short hot (I 1.1! 1 li
moderate veryshort hot 0.603797
moderate short hot 0.553164
moderate slightlymoderate warm 0.403333
moderate moderate warm I I, _'. ;I
moderate slightlylong warm 0.552000
moderate long tepid 0.556800
moderate verylong tepid 0.684073
approximate fuzzy membership functions such as triangular or trapezoid fuzzy numbers
fuzzy membership functions with complete definitions
combinations of the above forms.
The motivation for this work lies with the problem of resolving the difference between qualitative and quantitative
forms of knowledge about '1,! -, ,1 -I. ,- The fuzzy simulation method introduced here bridges the gaps between
the two .l!I!. i. Ii levels of knowledge. We showed how two itl[!. i. Ii extreme levels of knowledge can be directly
compared and maintained in a -- -I. i i,,i manner. Since the uncertainty arising from the human reasoning process
is easily represented by rules associated with confidence factors, we devised a way for emulating such processes by
showing how fuzzy simulation can derive the confidence factors from quantitative models. For handling another
form of uncertainty arising from linguistic vagueness, we assumed that central point estimates were presented by
experts. Although this form of estimation is a very limited form of uncertainty representation, we assert that the
presented method serves as a stepping stone for developing a more robust method which can capture the other forms
of expert's confidence levels in the future. By devising a method of integrated qualitative and quantitative dynamical
S-. in knowledge refinement, we hope to provide a way to obtain better models in general about i'! -i. 1 -'. n,-
by exploiting knowledge at all levels, whether qualitative or quantitative.
ACKNOWLEDGE EN IENTS
We would like to thank the following funding sources that have contributed towards our study of modeling and
implementation of a multimodeling simulation environment for ;,i i,1 -- and planning: (1) Rome Laboratory, (; ItIi--
Air Force Base, New York under contract 1 ;i.1 i 2-95-C-0267 and grant 1 ;1.1 1 2-95-1-0031; (2) Department of the
Interior under grant 14-45-0009-1544-154 and the (3) National Science Foundation Engineering Research Center
(. i:l ) in Particle Science and T. 1!1 i!.1 at the U i. i i of Florida (with Industrial Partners of the 1. i 1:) under
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