ENCODING FUZZY DIAGNOSIS RULES AS

OPTIMISATION PROBLEMS

Antonio Sala, Alicia Esparza, Carlos Ari

no and Jose V. Roig

Systems Engineering and Control Dept., Univ. Polit

ecnica de Valencia

Cno. Vera s/n, 46022 Valencia, Spain

Keywords:

Fault detection and diagnosis, fuzzy mathematical programming, approximate reasoning, optimisation.

Abstract:

This paper discusses how to encode fuzzy knowledge bases for diagnostic tasks (i.e., list of symptoms pro-

duced by each fault, in linguistic terms described by fuzzy sets) as constrained optimisation problems. The

proposed setting allows more ﬂexibility than some fuzzy-logic inference rulebases in the speciﬁcation of the

diagnostic rules in a transparent, user-understandable way (in a ﬁrst approximation, rules map to zeros and

ones in a matrix), using widely-known techniques such as linear and quadratic programming.

1 INTRODUCTION

Many industrial activities depend on the correct op-

eration of complex technological processes. A fault

changes the behaviour ofa system in such a way that it

does no longer satisfy its nominal performance objec-

tives or even the system functionality is lost. The ob-

jective of diagnosis (process monitoring) is estimating

a vector of fault state parameters, f, from measure-

ments of the outputs of a dynamic system, whose tra-

jectories depend on f and on time, initial conditions,

external input variables, physical parameters, etc. De-

tecting faults with a gradation of severity from let’s

say zero to 100% may be advantageous in practice:

detecting faults in its early stages enables corrective

actions to be taken on time, if needed.

Different approaches to the problem appear in lit-

erature: data-based, knowledge-based or based on

differential-equation analytical models. A description

of many of these techniques appear in (Chiang et al.,

2001; Blanke et al., 2003). The broadest conception

of the problem may be set up in a probabilistic set-

ting. In that situation, a comprehensive approach to

the problem would involve estimation under nonlin-

ear stochastic dynamics (Timmer, 2000; Khalil, 2002)

as well as decision-theoretic criteria (Berger, 1985)

apart, of course, of the consideration of robustness of

the resulting results when subject to possibly signiﬁ-

cant modelling errors. The problem, as such, is quite

complex and possibly intractable. Hence, simpliﬁed

assumptions are often stated.

In quite a few practical cases, diagnostic-related

knowledge is available from experts, who express

it in linguistic terms (“fault F produces a pressure

in pipe 3 lower than normal”). The meaning of

some of those linguistic terms may be understood

as rules involving fuzzy concepts deﬁned on the nu-

meric range of the physical variables being measured.

This fact inspires the use of fuzzy logic in diag-

nosis (Angeli, 1999; Carrasco and et. al., 2004).

Other knowledge-based frameworks extend the ba-

sic logic reasoning schemes to uncertain reasoning

(Kruse et al., 1991; Shafer and Pearl, 1990), possi-

bilistic reasoning (Dubois and Prade, 2004; Yamada,

2004), Bayesian networks (Castillo et al., 1997; Rus-

sell and Norvig, 2003) or uncertain probability (Ky-

burg, 1988), or combine the approach with neural net-

works (Ayoubi and Isermann, 1997; Jie and Morris,

1996).

This paper presents an alternative approach: the ex-

pert knowledge stemming from some (partially sim-

pliﬁed) properties of a nonlinear dynamic system

is encoded as a constrained optimisation problem,

transcoding fuzzy assertions as approximate linear

equations in the linguistic domain. The idea of trans-

forming fuzzy statements into equations also appears

in (Juuso, 1999; Sala and Albertos, 2001) in a differ-

ent context of system modelling.

The proposed approach seems to possesses signi-

ﬁcative advantages with respect to a classical fuzzy

IF-THEN rulebase, particularly in multiple fault situ-

ations, while keeping the problem readable (reduced

Sala A., Esparza A., Ariño C. and V. Roig J. (2006).

ENCODING FUZZY DIAGNOSIS RULES AS OPTIMISATION PROBLEMS.

In Proceedings of the Third International Conference on Informatics in Control, Automation and Robotics, pages 34-39

DOI: 10.5220/0001203100340039

 SciTePress

number of “rules”) and computationally tractable (ef-

ﬁcient off-the-shelf linear programming (LP) soft-

ware exists able to deal with hundreds, even thou-

sands, of constraints and variables, and quadratic pro-

gramming (QP) routines are also widely available).

LP is a widely known tool (Sierksma, 2001; Gass,

2003), taught in many undergraduate disciplines so

that user understanding of both the rules and the infer-

ence tools implies that the approach might be useful

in practical applications.

The structure of the paper is as follows: a prelimi-

nary section will justify some approximate additivity

properties of systems based on linearisation. Section

3 will discuss how a fuzzy rulebase may be encoded

as equations. Section 4 will discuss the available al-

gorithms to solve them and how they should be mod-

iﬁed for the problem in consideration. A conclusion

section closes the paper.

2 PRELIMINARIES

Dynamic systems. Let us assume that a system

to be diagnosed is governed by equations ˙x =

ψ(x, u, f, θ, t) where x is the system’s dynamical

state, u is a set of known input variables, f is the fail-

ure state to be estimated and θ are a set of system pa-

rameters, also assumed to be approximately known,

and t is the time variable (Khalil, 2002).

First, it will be assumed that the initial conditions

x(t

) and inputs of standarised tests are known be-

forehand, as well as system-dependent parameters θ.

The result of such tests will be a ﬁnite collection of

measurements at some time instants:

y = ψ(x(t

), θ, u, t

, t, f)

where, by assumption, only f is unknown. Let us de-

note as q = (x(t

), θ, u, t

, t) the set of known vari-

ables; they will be denoted as the experiment context

variables. The diagnostic problem may then be cast

as estimating f from y(q, f), i.e., obtaining the im-

plicit function, if it exists f = γ(q, y). In general,

the analytic expression for γ cannot be obtained ex-

cept in the simplest of the cases. If enough sensors

exist,and pre-classiﬁed data from the system (y,q,f)

were available, a functional approximator such as a

neural network might be used in order to try to learn

γ by example, or at least to learn the normal behaviour

and generate some residuals in case of faults. Appli-

cations of the approach are reported in (Chow et al.,

1993; Jie and Morris, 1996). However, learning for

successful fault isolation in a complex system may

require an impractical number of data. The approach

will not be pursued further in this paper.

The diagnosis problem becomes easier if some sim-

pliﬁcations and assumptions are made. Taking a Tay-

lor series expansion of ψ on the variable f , around

f = 0, which will be deﬁned as the “normal” situa-

tion, the result is an afﬁne model:

y(q, f) = ψ(q, 0) +

∂ψ

∂f

(q, 0) ∗ f + o(f

) (1)

This Taylor series expansion justiﬁes that at least ap-

proximately a linearity assumption holds:

y(q, f) ≈ ψ(q) + C

(q) ∗ f (2)

where C

is a linear operator (a matrix whose ele-

ments depend on the known information q). Usu-

ally, a parameterised analytical model of the system

is not available so ψ(q) and C

(q) cannot be calcu-

lated. However, human operators possess knowledge

that may be encoded in terms of fuzzy sets. Express-

ing such a knowledge in the form (2) is discussed in

next section.

3 FUZZY DIAGNOSTIC MODELS

In the above discussed context, a collection of fuzzy

sets mapping y to the interval [0, 1] are assumed to

deﬁned by an expert on the system to be diagnosed,

µ : R

→ [0, 1]

where p is the number of measure-

ments and k is the number of fuzzy concepts. Usu-

ally, those sets are denoted by user-deﬁned linguistic

labels.

Then, the expert also knows which effects each

fault has in the measurements. This knowledge is usu-

ally expressed in terms of rules:

If the isolated fault F

occurs, then abnormal

symptoms S

, . . . , S

should be observed, be-

ing the rest normal

Those rules, in a fuzzy context, may be basically un-

derstood as

If the severity of fault F

is f

, 0 ≤ f

≤ 1, and it

is the only fault occurring in the system, then the

intensity of symptoms S

, . . . , S

is approxi-

mately f

, and the intensity of the rest of them is

zero.

where f

= 0 denotes fault not occurring and f

1 denotes the fault occurring at a very signiﬁcative

severity level requiring user attention

, and the inten-

sity of the symptoms is the membership function of

the suitably deﬁned fuzzy concepts.

In a multiple-fault situation, the system is assumed

to verify an expression such as (2). If the observed

outputs are the fuzziﬁed ones, it will be assumed that

It is up to the user to ﬁx a maximum value of f

(usually

1) in the optimisation procedures to be discussed, or to set

= 1 as a “landmark” point but considering higher values

(more severe) possible.

ENCODING FUZZY DIAGNOSIS RULES AS OPTIMISATION PROBLEMS

the system veriﬁes the following linear equation in the

domain [0, 1]

of logic values:

(y) ≈ C(q)f + D(q) (3)

where C and D are a known function of the system’s

parameters, input variables and initial conditions of

the experiment. The notation µ

indicates that even

the deﬁnition of the membership function may de-

pend on a priori information, such as historical data,

system-dependent parameters, etc. C(q) will be sim-

ilar to

∂µ

∂y

(q). Note that a differentiable system in-

deed does verify such an equation

for small values

of f based in (2). The basic assumption here is that,

with suitably deﬁned fuzzy sets, the range in which

such an equation fulﬁlls is large enough to be useful

for diagnosis.

Fuzzy sets may be deﬁned on the difference be-

tween the observed readings and those in a normal

situation, as an alternative to the absolute reference

frame for concepts implicit in equation (3), so that:

(y − D(q)) ≈ C(q)f (4)

In a practical situation, both types of fuzzy sets (3)

or (4) may be used. In summary, the following linear

equation must be solved in the diagnosis process:

µ(y, q) = C(q)f (5)

Other authors also pose matrix representations of the

relationship between fuzzy faults and symptoms (for

instance, (Yao and Yao, 2001) uses a fuzzy rela-

tion approach). The assumption of linearity in the

logic domain, with suitable deﬁnitions of membership

functions is also used, in a control context, in (Juuso,

1999).

3.1 Rule Encoding

The proposed expert rules are encoded in the format

required in (3) in a simple way. Let us discuss several

situations which will be clariﬁed by examples.

First, the simplest setting would imply that, under

a standarised test (q is ﬁxed for all diagnostic experi-

ments), a fuzzy set denoting an “abnormal” situation

is deﬁned for every measured variable. In that case,

the knowledge:

causes abnormality in y

, y

, . . .

will deﬁne a column of matrix C, where elements at

rows 1, 3, . . . will be 1 and the rest of elements not

explicitly enumerated at the above assertion will be

set as zero. The usual fuzzy negation operator may be

used if some variables have a fuzzy set deﬁning the

“normal” value of a variable.

Formally, left and right derivatives may be needed, but

details are not relevant.

Example 1 Let us have an industrial boiler where a

fault f

causes: no variation on a temperature mea-

surement t

, and increasing of temperature t

. An-

other fault, f

, causes increases in both temperatures.

They both cause an increase of pressure p. Deﬁning

abnormally high temperatures with fuzzy sets denoted

as “T

abnormal”, µ

and “T

abnormal”, µ

, and “nor-

mal pressure” with another fuzzy set (such as a trian-

gular one), µ

, the basic diagnosis equation would be:

)

1 − µ

(p)

0 1

1 1

Alternatively, if the nominal normal pressure de-

pends on a variable q, denoted as P

(q), (for instance,

q might be the load regime of the boiler), a fuzzy set

may be deﬁned on the pressure increment so that the

diagnosis equation would be written as:

)

1 − µ

(p − P

(q))

0 1

1 1

Model errors and sensor faults. Another situation

takes into account the approximate nature of (5) al-

lowing for an instrumental “fault” variable associ-

ated to each of the measurements. That instrumen-

tal fault variable encompasses both sensor faults and

modelling errors (inaccuracy in the deﬁnition of the

membership functions).

Example 2 In the example being considered, the fault

vector may be extended as:

)

1 − µ

(p)

0 1 1 0 0

1 1 0 1 0

1 1 0 0 1

∗

(6)

where f

∗

are three instrumental fault variables and f

denotes the “primary” faults.

In other situations, different faults have opposite ef-

fects on a particular variable so that its simultaneous

occurrence does not deviate its measurements from

the normal condition. The following example clari-

ﬁes how to encode such a knowledge.

Example 3 If, in example 1 fault 1 decreases T1, be-

ing the rest of symptoms the same as previously de-

scribed, with concepts ”abnormally high”, µ

, and ”ab-

normally low”, µ

, deﬁned for T

, then the rulebase

should be encoded as:

)

1 − µ

(p)

−1 1

1 −1

1 1

ICINCO 2006 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

In the above example, depending on the units and

membership deﬁnitions, the −1 terms may be a dif-

ferent negative number. Note that some combinations

of faults may yield negative membership values; this

is not a problem if inequality restrictions are consid-

ered, as discussed in section 4.1.

In any of the examples, a coefﬁcient in matrix

C lower (higher) than 1 would indicate a milder

(stronger) effect of the fault on the symptom, as de-

ﬁned by the membership function. Modifying the co-

efﬁcients might be needed in a ﬁne-tuning phase be-

cause not all the faults inﬂuence with the same in-

tensity a particular variable. However, taking into ac-

count “sensor” errors f

∗

, an initial setting with mostly

0 and 1 coefﬁcients in C may be enough in order to

achieve a reasonable output.

4 INFERENCE

A fault estimation f is consistent with the observed

symptoms if it is a solution of the basic diagnosis

equation (5). Hence, fuzzy diagnosis amounts to ﬁnd-

ing the set of solutions of (5). However, some consid-

erations need to be made: indeed, if sensor faults and

modelling errors, f

∗

, are considered, the solution set

includes any conceivable primary fault f , as sensor

faults can accommodate any reading. For instance,

in (6) a value of f

∗

can be calculated for any sensor

reading and any value of f . However, if sensor faults

are not considered, with less faults than sensors the

set of consistent faults will be usually empty, as (5)

will have no solution due to modelling errors. So, the

above idea must be reﬁned for practical usability.

Inference as optimisation. As above discussed,

sensor faults must be considered in practice. Then,

(5) are equality restrictions and a criterion should be

used in order to rank all the feasible solutions. Note

also that inequality restrictions f

> 0 implicitly ap-

ply, unless the user casts a meaning for negative fault

severities.

A possible criterion to be chosen is minimising the

norm of the “instrumental” fault component, i.e., giv-

ing as the diagnosis solution the one that achieves less

discrepancy between the measurements and the pre-

dictions C(q)f. The discrepancy µ − C(q)f will be

denoted as inference error.

If the chosen norm is the Euclidean one, fuzzy in-

ference is equivalent to a least squares problem. Let

us consider an equation µ = Cf where sensor faults

are also members of f. This is a linear system of

equations with more unknowns than equations, which

can be solved in the following sense:

The feasible solution f that minimises the squared

Euclidean norm of W f, where W is a diagonal

weight matrix, is given by the pseudo-inverse formula

(Meyer, 2001):

f = W

−2

(CW

−2

)

−1

µ (7)

In a practical setting, a high penalisation in W must

be speciﬁed for the sensor fault components f

∗

. For

invertibility of W , small positive weights in the pri-

mary faults need to be introduced

If the chosen norm is the 1-norm of W f (sum of

absolute value of the components), then inference can

be carried out in a linear programming setting. The

LP framework needs to introduce dummy variables

for positive and negative sensor errors f

∗

= f

−f

−

> 0, f

−

> 0, to calculate the 1-norm as the sum

of f

+ f

−

. This change of variables is standard in

LP textbooks.

Note that LS algorithms produce an “intermediate”

point as a result (not a vertex of the feasible region),

sharing the error between all the equations, as small

errors are not signiﬁcant (because of the squaring) so

LS tries to reduce big errors. On the contrary, LP pro-

duce a result in a vertex of the feasible solution space,

and increments from either small or big errors weight

the same.

4.1 Constrained Optimisation

Under the proposed settings, it is implicitly assumed

that a reasonable diagnostic should verify f

≥ 0 in

all components of the primary faults.

Also, fuzzy concepts saturate in [0, 1]. Hence, si-

multaneous faults yielding the same symptom cannot

fulﬁll, for instance, 1 = f

+ f

if they are fully ac-

tive. However, two easy options are available (or a

combination of them) in that case:

• When a fuzziﬁed sensor reading is saturated, re-

place the equality constraint in the diagnosis equa-

tion by an inequality (1 ≤ C

f, 0 ≥ C

f, where

denotes the i-th row of C).

• Translate an ordered fuzzy partition on a domain

into numerical values (for instance, {very low, low,

normal, high, very high} into {−2, −1, 0, 1, 2}) in

the spirit of the so-called linguistic equation (Juuso,

1999; Jarvensivu et al., 2001). In this way, the basic

diagnosis equation (5) may involve sensor values

ranging more than [0, 1], but somehow keeping the

linguistic meaning.

Algorithms. LP algorithms incorporate linear in-

equality restrictions seamlessly. However, the least

squares formula (7) must then be discarded and

quadratic programming (QP) routines used instead.

Solving y = Cf by standard least squares, f =

−1

µ, is equivalent to the proposed approach

when the primary-fault weights tend to zero and sensor ones

are equal to the same constant.

ENCODING FUZZY DIAGNOSIS RULES AS OPTIMISATION PROBLEMS

For instance, if the sensor reading in example (4) had

been {.2, .18, 0} the output of the LS formula would

have been f

= 0.197, f

= −0.003, out of the

constraint space so QP or non-negative least squares

would have been needed.

As the involved restrictions are linear, efﬁcient

code exists for both LP and QP settings. For brevity,

mostly linear programming settings will be consid-

ered in the sequel, although a similar version posed as

QP would produce comparable results. For instance,

commercially available LP software is able to efﬁ-

ciently deal with hundreds (even thousands) of vari-

ables and restrictions, allowing for large-scale imple-

mentation of the ideas in this work.

Binary faults. Some faults may be only either 0

or 1, without intermediate values (for instance, circuit

breaker ON vs. OFF). To carry out optimisation, ex-

plicit enumeration of all the involved binary (or inte-

ger) variables and solving for each of them the optimi-

sation on the remaining real variables may be an ap-

proach. Alternatively, mixed linear integer program-

ming or branch-and-bound methodologies may also

be applied (Sierksma, 2001).

Managing non-unique solutions. The optimisa-

tion routines stop at an approximately optimal point.

However, there might be other optimal points or, at

least, which may have a very similar value of the

cost index. The situation is particularly frequent in

the case of missing measurements, which amounts to

deleting the corresponding row of C.

In order to choose between possible nonunique so-

lutions, the weights of the different fault components

should penalise each fault according to its probabil-

ity: in that way, the solution would tend to be the most

likely fault consistent with the sensor measurements.

Imprecise measurements. Some sensors may be

imprecise, in the sense that a small deviation from the

expected values is frequent and acceptable to assume

when producing a diagnosis. In a sense, quadratic

cost indices naturally take that fact into account, but

LP settings need a straightforward change of variable

to do that

. This is easily carried out, by express-

ing each “sensor fault” by a sum of two sub-faults,

bounding the maximum value of one of them (with

a small weight in the inference cost index) and set-

ting a much larger penalisation on the deviations of

the non-limited one. If the small sensor errors have

zero weight, the setting is equivalent to interval mea-

surements:

∗

= f

+ f

− f

−

− f

−

(8)

0 ≤ f

≤ l

, 0 ≤ f

−

≤ l

−

, 0 ≤ f

−

(9)

Heavy cost in f

and f

−

, no penalisation in f

, f

−

results in an interval sensor reading allowing, with no

The same change of variable may be used in QP for-

mulations to ﬁne-tune the cost index formula, if so wished.

cost, diagnostics involving the actual reading, say σ,

plus or minus the desired bound: [σ − l

−

, σ + l

Example 4 Let us consider a knowledge base:

Fault 1 produces S1, S2. Fault 2 produces S2,

and a fuzziﬁed sensor reading were {0.2,0.43,0.15}.

The setting for minimal squared inference error

(weighting by 0.5 the “conﬁdence” on the accuracy of

equation 2, because the addition of individual faults

may not result in the exact addition of the results due

to possible system nonlinearity) would be carried out

by the following Matlab code:

C=[1 0;1 1;0 1];C2=[C eye(3)];

j=diag([0.01 0.01 1 .5 1]);

cw=C2

inv(j);

f=inv(j)

pinv(cw)

[.2 .43 .15]’

The result is f

= 0.21, f

= 0.16.

Alternatively, the Matlab code for the above problem

minimising the 1-norm of the inference error via linear

programming is:

C=[1 0;1 1;0 1];

C2=[C eye(3) -eye(3)];

j=[0.01 0.01 1 .5 1 1 .5 1];

x=linprog(j,[],[],C2,

[.2 .43.15],zeros(8,1));

which produces f

= 0.2, f

= 0.15, apportioning all

error to the less reliable sensor 2. Vector j contains

the weights for f

, f

and the positive and negative

components of the 3 possible sensor/modelling faults.

The Matlab manual explains the meaning of the argu-

ments to linprog.

5 CONCLUSIONS

This paper presents a methodology for approximately

translating expert diagnostic knowledge into mathe-

matical programming problems (constrained optimi-

sation). The knowledge is a series of statements about

the list of symptoms caused by the occurrence of a

particular fault, expressed via linguistic statements in-

volving fuzzy sets.

The approach deals naturally with multiple faults

approximately following a linear equation in the lin-

guistic domain. The diagnostic procedure operates

satisfactorily with missing measurements or a limited

number of faulty sensors.

The proposed approach can be thought of as an in-

termediate between IF-THEN rulebases and diagnosis

based on a full mathematical model. It keeps a lin-

guistic interpretation while allowing for many combi-

nations of requirements difﬁcult to be taken into ac-

count in a pure logic framework. Marginal possible

intervals of fault severities are also easily calculated

in the case of non-unique solutions of the optimisation

problem. The problem is computationally tractable

ICINCO 2006 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

even in a large-scale framework as efﬁcient software

exists for the proposed optimisation techniques.

The presented framework has been discussed on a

theoretical level with simple academic examples. De-

tailed comparative analysis and application to realis-

tic diagnostic environments with a large number of

“rules” is under research at this moment.

REFERENCES

Angeli, C. (1999). Online expert system for fault diagnosis

in hydraulic systems. Expert Systems, 16:115–120.

Ayoubi, M. and Isermann, R. (1997). Neuro-fuzzy systems

for diagnosis. Fuzzy Sets and Systems, 89(3):289–307.

Berger, J. (1985). Statistical Decision Theory and Bayesian

Analysis. Springer-Verlag, London.

Blanke, M., Kinnaert, M., Lunze, J., and Staroswiecki, M.,

editors (2003). Diagnosis and Fault-Tolerant Control.

Springer, London.

Carrasco, E. and et. al., J. R. (2004). Diagnosis of acidiﬁca-

tion states in an anaerobic wastewater treatment plant

using a fuzzy-based expert system. Control Engineer-

ing Practice, 12(1):59–64.

Castillo, E., Gutierrez, J., and Hadi, A. (1997). Expert

Systems and Probabilistic Network Models. Springer,

London.

Chiang, L., Russell, E., and Braatz, R. (2001). Fault Detec-

tion and Diagnosis in Industrial Systems. Springer-

Verlag.

Chow, M., Sharpe, R., and Hung, J. (1993). On the ap-

plication and design consideration of artiﬁcial neural

network fault detectors. IEEE Trans. Industrial Elec-

tronics, 40:181–198.

Dubois, D. and Prade, H. (2004). Possibilistic logic: a ret-

rospective and prospective view. Fuzzy Sets and Sys-

tems, 144(1):3–23.

Gass, S. (2003). Linear Programming: methods and appli-

cations. Dover, 5th edition.

Jarvensivu, M., Juuso, E., and Ahavac, O. (2001). Intelli-

gent control of a rotary kiln ﬁred with producer gas

generated from biomass. Eng. Applic. Artif. Intelli-

gence, 14:629–653.

Jie, Z. and Morris, J. (1996). Process modelling and fault

diagnosis using fuzzy neural networks. Fuzzy Sets and

Systems, 79(1):127–140.

Juuso, E. (1999). Fuzzy control in process industry: the

linguistic equation approach. In Verbruggen, H., Zim-

mermann, H.-J., and Babuska, R., editors, Fuzzy Algo-

rithms for Control, pages 243–300. Kluwer, Boston.

Khalil, H. (2002). Nonlinear Systems. Prentice Hall, third

edition.

Kruse, R., Schwecke, E., and Heinsohn, J., editors (1991).

Uncertainty and vagueness in Knowledge Based Sys-

tems: Numerical Methods (Artiﬁcial Intelligence).

Springer-Verlag.

Kyburg, H. (1988). Higher order probabilities and intervals.

Int. J. Approximate Reasoning, 2:195–209.

Meyer, C. (2001). Matrix Analysis and Applied Linear Al-

gebra. Society for Industrial & Applied Mathematics

(SIAM).

Russell, S. and Norvig, P. (2003). Artiﬁcial Intelligence: a

modern approach. Prentice-Hall, 2nd edition.

Sala, A. and Albertos, P. (2001). Inference error minimisa-

tion: Fuzzy modelling of ambiguous functions. Fuzzy

Sets and Systems, 121(1):95–111.

Shafer, G. and Pearl, J., editors (1990). Readings in uncer-

tain reasoning. Morgan Kauffman, San Mateo (CA),

USA.

Sierksma, G. (2001). Linear and Integer Programming:

Theory and Practice. Marcel Dekker Pub., New York,

2nd edition.

Timmer, J. (2000). Parameter estimation in nonlinear sto-

chastic differential equations. Chaos, Solitons and

Fractals, 11(15):2571–2578.

Yamada, K. (2004). Diagnosis under compound effects

and multiple causes by means of the conditional

causal possibility approach. Fuzzy Sets and Systems,

145:183–212.

Yao, J. and Yao, J. (2001). Fuzzy decision making for

medical diagnosis based on fuzzy number and com-

positional rule of inference. Fuzzy Sets and Systems,

120:351–366.

ENCODING FUZZY DIAGNOSIS RULES AS OPTIMISATION PROBLEMS