DEVELOPMENT OF A FUZZY CALCULATOR FOR CONTINUOUS

FUNCTIONS OF NON-INTERACTIVE FUZZY VARIABLES

Karolien Scheerlinck, Hilde Vernieuwe and Bernard De Baets

Department of Applied Mathematics, Biometrics and Process Control, Ghent University, Coupure links 653, Ghent, Belgium

Keywords:

Extension principle, Particle swarm optimization, Fuzzy calculator, Non-interactive fuzzy variables.

Abstract:

The goal of this paper is to develop a Fuzzy Calculator, making it possible to calculate functions of fuzzy

intervals, as prescribed by the extension principle of Zadeh. The extension principle can be reversed, result-

ing in ﬁxed α-levels for which the minimum and the maximum of the function has to be determined. This

optimization problem can be tackled by different algorithms: Gradient Descent, SIMPSA, Particle Swarm

Optimization and Particle Swarm optimization in combination with Gradient Descent. Two approaches are

used to determine the number of α-levels: it is either ﬁxed to a predetermined value, or it is initially chosen

very small and subsequently expanded according to a suitable criterion. Both a non-parallel and a parallel im-

plementation of the Fuzzy Calculator are designed. In the parallel version, communication is used to optimize

the internal workings of PSO. The Fuzzy Calculator is applied to a number of test functions. The different

combinations of optimization algorithms are evaluated, both by the ﬁnal result and by the number of required

model evaluations. The results indicate that the parallel implementation of the Fuzzy Calculator starting with

a small number of α-levels and using PSO with Gradient Descent leads to the most accurate membership

function.

1 INTRODUCTION

The concept of fuzzy information plays a central role

in many engineering applications. This type of uncer-

tainty can be captured by fuzzy intervals. Calculat-

ing with fuzzy intervals is in general a complex pro-

cess, described by the extension principle of Zadeh.

It is possible to reverse the extension principle, and

to ﬁnd, for each membership degree α ∈]0,1], the

corresponding interval of the membership function

(Nguyen, 1978).

Several practical implementations of the exten-

sion principle based on the α-level concept are avail-

able for (locally) monotone continuous functions of

non-interactive fuzzy intervals. The vertex method

is developed for computing with monotone functions

of non-interactive fuzzy intervals. This method can

be extended to non-monotone functions by doing an

extreme value analysis (Dong and Shah, 1987; Otto

et al., 1993). However, this is not always possible,

for example when dealing with complex functions.

Another approach for (locally) monotone continuous

functions is working with gradual numbers (Fortin

et al., 2008; Dubois and Prade, 2008). However, for

general functions, an optimisation algorithm is

needed to determine the minimum and maximum of

the function at each value of α (Maskey et al., 2004;

Shrestha et al., 2007).

The objective of this paper is to develop a compu-

tationally efﬁcient Fuzzy Calculator to construct the

membership function of a fuzzy output interval that

depends on non-interactive fuzzy intervals, through a

general continuous function. In this paper four dif-

ferent optimisation algorithms are compared to de-

termine the minimum and maximum value of the

function at the different α-levels: (1) Gradient De-

scent based on Sequential Programming (GD) (Eaton,

2002; Nocedal and Wright, 1999), (2) the Simplex-

Simulated Annealing approach (SIMPSA) (Cardoso

et al., 1996), (3) Particle Swarm Optimisation (PSO)

(Kennedy and Eberhart, 1995) and (4) Particle Swarm

Optimisation in combination with Gradient Descent

(PSO

GD). The ﬁrst step is to determine the number

of α-levels. In this paper, two approaches are used.

Either we ﬁx the number of α-levels at the beginning

and kept constant throughout the algorithm or, alter-

natively, we start with only 3 α-levels and gradually

increase this number as required by a criterion based

on linear interpolation. The next step is to implement

the Fuzzy Calculator. Both a non-parallel and a par-

Scheerlinck K., Vernieuwe H. and De Baets B..

DEVELOPMENT OF A FUZZY CALCULATOR FOR CONTINUOUS FUNCTIONS OF NON-INTERACTIVE FUZZY VARIABLES.

DOI: 10.5220/0003053800140020

In Proceedings of the International Conference on Fuzzy Computation and 2nd International Conference on Neural Computation (ICFC-2010), pages

14-20

ISBN: 978-989-8425-32-4

 2010 SCITEPRESS (Science and Technology Publications, Lda.)

allel version are implemented. The latter application

is only important when PSO is employed, such that

several swarms simultaneously search at different α-

levels and moreover communicate with each other as

to locate the minimum and maximum of the function

more accurately.

This paper is organised as follows. Section 2 re-

minds the reader of the extension principle applied

to construct the membership function of a continuous

function of fuzzy intervals. Section 3 describes the

different optimisation algorithms used by the Fuzzy

Calculator. In Section 4, the implementation of the

Fuzzy Calculator and the different optimisation algo-

rithms are outlined in full detail. The test functions

used to validate the Fuzzy Calculator are presented in

Section 4. Section 5 describes the results and com-

pares them in great detail. Finally, Section 6 contains

our conclusion.

2 EXTENSION PRINCIPLE

The extension principle of Zadeh provides a general

method to transfer uncertainty described by fuzzy in-

tervals through a function. Concretely, the extension

principle of Zadeh determines the membership func-

tion µ

of a fuzzy output quantity Z that depends

on n fuzzy quantities X

,...,X

with known member-

ship functions µ

,...,µ

through a general function

Z = f(X

,...,X

) (Zadeh, 1975):

(z) = sup

z= f(x

,...,x

)

min(µ

),...,µ

)) (1)

In the case of a continuous function f and up-

per semi-continuous membership functions, i.e. all

α-cuts are closed: ∀α ∈]0, 1], [µ

]

is closed, with

a compact support S

, i.e. a bounded support, we

can reverse the extension principle. Instead of deter-

mining the membership degree α = π

(z) of a certain

value z = f(x

,...,x

), it is then possible to determine

the interval of values z ∈ [z

] which have a mem-

bership degree µ

(z) ≥ α (Nguyen, 1978). Thus, by

determining z

and z

for certain values of α, hence-

forth called α-levels, it is possible to reconstruct the

membership function π

(z). As the interval at α = 0

is not closed, we choose the ﬁrst α-level at a small

value α = δ > 0. The range of membership degrees

[δ,1] will be subdivided in m intervals of length 1/m,

with the m + 1 α-levels α

= j/m, j = δ,1,...,m, as

endpoints.

To simplify the search problem to determine the

membership function of the fuzzy output quantity, we

only use convex membership functions µ

(x):

∀(x

) ∈ R

,∀α ∈]0, 1] :

(αx

+ (1− α)x

) ≥ min(µ

),µ

))

(2)

Fuzzy quantities whose membership functions

satisfy the above described conditions of upper-semi

continuity and convexity and have a compact support

are henceforth called fuzzy intervals.

For functions without internal minima or maxima

for a certain α-level, the minimum z

and maximum

will be found on the boundary of the corresponding

α-cut of the input fuzzy intervals. For general func-

tions, however, the minimum z

and maximum z

can

either be located on the boundary or in the interior of

the corresponding α-cut of the input fuzzy intervals,

and optimisation algorithms will be necessary to try

to locate these points efﬁciently (Maskey et al., 2004;

Shrestha et al., 2007).

3 OPTIMISATION ALGORITHMS

As outlined in Section 2, the construction of the mem-

bership function of the fuzzy output interval of a gen-

eral function of non-interactive fuzzy input intervals

can be converted into a number of optimisation prob-

lems. Four different optimisation algorithms are ap-

plied to this problem: Gradient Descent based on

Sequential Quadratic Programming (GD), Simplex-

Simulated Annealing (SIMPSA), Particle Swarm Op-

timisation (PSO), and Particle Swarm Optimisation

based on Gradient Descent (PSO GD).

3.1 Gradient Descent based on

Sequential Quadratic Programming

Gradient Descent based on Sequential Quadratic Pro-

gramming (Eaton, 2002; Nocedal and Wright, 1999),

a local optimisation algorithm, is an extension of the

Quasi Newton method in order to handle constraints.

Lagrange multipliers are used to incorporate (non-

linear and linear) equality constraints. In addition, in-

equality constraints are allowed. At the starting point,

the objective function is approximated by a quadratic

function, obtained through a Taylor expansion. The

algorithm exactly determines the minimum or max-

imum of that quadratic function, which is the new

starting point and the procedure is repeated until a

convergence criterion is fulﬁlled. A possible conver-

gence criterion is that the gradient is approximately

zero, which is the mathematical condition for a local

extremum (Eaton, 2002; Nocedal and Wright, 1999).

DEVELOPMENT OF A FUZZY CALCULATOR FOR CONTINUOUS FUNCTIONS OF NON-INTERACTIVE FUZZY

VARIABLES

3.2 Simplex-simulated Annealing

The Simplex-Simulated Annealing (SIMPSA) algo-

rithm (Cardoso et al., 1996) is an optimisation al-

gorithm based on a combination of the non-linear

Simplex algorithm (Nelder and Mead, 1965) and the

Simulated Annealing algorithm (Kirkpatrick et al.,

1983). The non-linear Simplex algorithm starts with

a randomly chosen starting point~x = (x

,...,x

), for

which a simplex, i.e. a polytope determined by n + 1

n-dimensional vertices, is created. In a next step, the

function values at the vertices of the simplex are com-

pared. The objective is to move away from the worst

point. However, in order to be able to ﬁnd global op-

tima, wrong-way movements must sometimes be al-

lowed. This is regulated by combining the non-linear

simplex algorithm with the Simulated Annealing al-

gorithm. Simulated Annealing is a global optimisa-

tion algorithm based on the physical thermal process

of annealing (Kirkpatrick et al., 1983). New conﬁg-

urations are accepted by the Simulated Annealing al-

gorithm with a certain probability p, depending on the

ﬁtness of the solution and the current system temper-

ature. In combination with the non-linear simplex al-

gorithm, the possible conﬁgurations are represented

by the vertices of the simplex. As for all heuris-

tic optimisation algorithms, the performance of the

SIMPSA algorithm largely depends on the choice of

the parameter values inherent to the algorithm (Car-

doso et al., 1996).

3.3 Particle Swarm Optimisation

Particle Swarm Optimisation (PSO) (Kennedy and

Eberhart, 1995), a population-based global optimisa-

tion algorithm, starts with the initialisation of a pop-

ulation of N particles, numbered i = 1,...,N, having

randomly chosen position vectors~x

and velocity vec-

tors ~v

. The position of each particle corresponds to

a candidate solution of the optimisation problem. In

each iteration, the particle’s position vector is trans-

ported over its velocity. The velocity vector is redi-

rected toward the particle’s personal best position and

local best position. The contributions of the particle’s

personal and local best position are weighted through

the stochastic variables c

and c

. The positive

constants c

and c

are the cognitive and social pa-

rameters, the factors r

and r

are random numbers

between 0 and 1, and are regenerated in each iteration

step. When a particle is positioned outside the search

space, it is repositioned on the boundary it crossed

and its velocity in that direction is set to zero. An in-

ertia weight w is used to decelerate the particle’s ve-

locity if a suitable solution is found. As with the other

Figure 1: Linear interpolation of α

through α

and α

optimisation algorithms, the performance of the algo-

rithm largely depends on the choice of the parameter

values.

4 METHODOLOGY

4.1 Subdivision in α-levels

The construction of the membership function of the

output variable of a continuous function of non-

interactive fuzzy variables described by fuzzy inter-

vals can be handled through the application of the ex-

tension principle and the subdivision in α-levels. The

number of α-levels that have to be determined is im-

portant, as for a ﬁnite number of m+ 1 α-levels, only

an approximation for the true membership function

(z) will be obtained. Increasing the number of α-

levels will improve the approximation.

To determine the number of α-levels, two ap-

proaches are followed. Firstly, the number of α-levels

is set to a ﬁxed number m + 1, determined at the be-

ginning of the algorithm. The corresponding α-levels

are equidistantly distributed at values α

= j/m for

j = δ,...,m. Secondly, we start with m + 1 = 3 and

gradually increase the number of α-levels according

to a criterion based on linear interpolation. More

speciﬁcally, we compare α with a value

α that is cal-

culated by linear interpolation. We choose to compare

α with its linear interpolation

α based on z

, instead

of the other way around, since α is always in the in-

terval ]0,1] and it is thus possible to work with an

absolute tolerance level ε independent of the problem

at hand. This is illustrated in Figure 1.

Since the membership function µ

(z) should be

upper semi-continuous and convex, it is impossible

that z

j−1

> z

or that z

j−1

< z

. Therefore, it is

not possible to determine the different α-levels inde-

pendently and we have to correct the endpoints of the

α-levels when this situations occurs. There are two

ways to correct for these inconsistencies. If for ex-

ample z

j+1

> z

, z

can be reset to z

j+1

. Another

possibility is to discard the old result at α-level α

and

ICFC 2010 - International Conference on Fuzzy Computation

recalculate z

. If the optimisation algorithm accepts

a starting point, then providing the location x

j+1

will

ensure that the inconsistency is solved.

4.2 Non-parallel versus Parallel

Implementation

The Fuzzy Calculator is implemented in the program-

ming environment Octave (Eaton, 2002) and accepts

a general n-ary function as input. Both a non-parallel,

and a parallel version were designed using the Mes-

sage Passing Interface (MPI) of Octave.

Non-parallel. As mentioned above, in the α-level

approach the number of α-levels is ﬁxed to m + 1.

The non-parallel version of the algorithm then starts

with searching the optima (the minimum and maxi-

mum) of the function f at the level α

= 1. As this

is the smallest interval, chances are higher to ﬁnd the

correct optima z

and z

. The algorithm then con-

tinues with the determination of the optima for the

other α-levels, for decreasing values of α. When al-

lowed by the optimisation algorithm, the optima of

the function at the previous α-level can be provided

as starting point. With this approach, the inconsis-

tency mentioned in the last paragraph of Section 4.1

cannot occur.

Parallel. The parallel version of the implementation

is based on a master-slave conﬁguration. For the ﬁxed

number of m+ 1 α-levels, we used 2m + 3 processes:

one master and 2(m+ 1) slaves for the determination

of the left (right) sides z

), for j = δ,...,m. The

master sends the input intervalsbelongingto the m+1

α-levels to the slaves and the slaves optimise the func-

tion for these intervals and send these optima back to

the master. When the master receives the optima for

all α-levels, these values are corrected for inconsis-

tencies if needed. In this part, the ﬁrst correction ap-

proach is applied.

In the second α-level approach, we start from

m + 1 = 3 α-levels and expand this number through

linear interpolation when necessary. This algorithm

starts with searching the optima of the function at

the α-levels 1, 0.5 and δ. The obtained values are

compared and corrected if inconsistencies should oc-

cur. Next, a linear interpolation is applied to examine

whether an even ﬁner sampling (additional α-levels)

is required. Each time the master receives new re-

sults, it checks for possible inconsistencies and sub-

sequently examines whether it has to request the cal-

culation of additional α-levels. The algorithm stops

when the convergence criterion of the linear interpo-

lation is fulﬁlled for all α-levels.

4.3 Optimisation Algorithms

Gradient Descent based on Sequential Quadratic Pro-

gramming (GD) is a standard function of Octave,

namely

sqp

(Eaton, 2002). This function has no extra

parameters that have to be determined and can thus be

applied directly to our optimisation problem.

The implementation of the Simplex-Simulated

Annealing (SIMPSA) algorithm was taken from

(Donckels, 2009; Donckels et al., 2009) with the au-

thor’s permission. As mentioned in Section 3.2, the

maximal number of iterations, the cooling ratio, the

freezing temperature and the tolerance of the conver-

gence criterion haveto be determined. After some test

simulations we decided to set the maximal number of

iterations for each simulated annealing cycle to 2500,

the cooling ratio to 0.7 and the freezing temperature

to 1. The tolerance level for convergence was set to

−6

The implementation of Particle Swarm Optimi-

sation (PSO) was taken from previous work of the

present authors (Scheerlinck et al., 2009) and appro-

priately modiﬁed. This algorithm requires the deter-

mination of a number of parameters inherent to the

algorithm. After some test simulations, we decided to

work with ﬁxed parameter values c

= 1, c

= 1.5 and

w = 0.7, while different population sizes of N = 10,

N = 15 and N = 20 were used. The convergence cri-

terion used requires that half of the population has

the same position (with tolerance level 10

−6

). Ex-

plicitly, the algorithm stops if the mean distance be-

tween the particles of the best half of the population

is smaller than 10

−6

. The parallel Fuzzy Calcula-

tor using PSO can be interpreted as several swarms

that search on different α-levels at the same time and

are thus able to communicate about candidate solu-

tions. We have modiﬁed the PSO algorithm such that

each swarm broadcasts its current global best position

to the other running PSO processes. When a swarm

receives a global best position located in its search

space and which is better than its own global best po-

sition, the swarm will change its global best position.

When communication occurs, the remaining particles

are reinitialised. This is necessary to prevent slow

convergence, for example when the current swarm is

already close to converging to a local optimum (far)

away from the newly obtained optimum. In this way,

a new parameter is introduced, namely the frequency

of communication. We varied this parameter by run-

ning instances in which the swarms communicate at

every 2, 5 or 10 iterations.

DEVELOPMENT OF A FUZZY CALCULATOR FOR CONTINUOUS FUNCTIONS OF NON-INTERACTIVE FUZZY

VARIABLES

As it is not certain that PSO will converge to a lo-

cal/global optimum (Engelbrecht, 2006), it may be

recommended to combine PSO with a local optimi-

sation algorithm such as GD. We performed GD at

several positions in the algorithm, in order to have a

ﬁnal best solution which is assured to be a local op-

timum. First, we performed GD on the initial parti-

cles. As after this, all the particles will be positioned

in a local optimum, we only kept the particle with the

best position in the population. The other particles of

the population are repositioned at their original posi-

tion received during the initialisation of the algorithm.

Then, we performedGD each time the swarm changes

its global best position through communication. Af-

ter convergence takes place, we perform GD on the

global best position of the swarm one last time.

4.4 Test Functions

In order to compare the different optimisation algo-

rithms, we made use of 9 different test functions, such

as the cosine function in one dimension, an arbitrary

function in 2 dimensions with multiple minima and

maxima, the alpine function in 2 dimensions for dif-

ferent intervals and the alpine function in 3, 4 and 5

dimensions.

The membership functions of the non-interactive

fuzzy input intervals are chosen to be normal, i.e.

∃x ∈ R such that µ

(x) = 1, and have a trapezoidal

shape. The interval [x

i,δ

] at α = δ, corresponds

to the intervals of the test functions. At α = 1, this

interval is reduced to [x

i,1

] with

i,1

i,δ

+ x

i,δ

−



i,δ

− x

i,δ



i,1

i,δ

+ x

i,δ



i,δ

− x

i,δ



In general, our implementation can deal with any

upper semi-continuous membership functions. When

the fuzzy intervals X

are interactive, i.e. a joint mem-

bership function can be speciﬁed, however, using the

Fuzzy Calculator for such type of membership func-

tion will beharder, as the search region at each α-level

is no longer hyper-rectangular.

5 RESULTS

The membership functions of the fuzzy output in-

tervals of the test functions are constructed with the

Fuzzy Calculator using the optimisation algorithms

discussed above. As for each α-level, the error on the

determination of z

and z

will depend on the opti-

misation algorithm, the lowest minimum and highest

maximum found by any of the optimisation algorithm

can be used as reference. We can thus use the area un-

der the membership function as a global quality mea-

sure. The Fuzzy Calculators are also compared on

the level of number of function evaluations. To allow

for a statistical comparison between the Fuzzy Calcu-

lators using different optimisation algorithms or be-

tween the non-parallel and parallel implementation,

each algorithm is repeated 50 times. To compare the

different Fuzzy Calculators, a mixed ANOVA model

with the test functions as random effectand a Satterth-

waite correction for unequal variances is used (Neter

et al., 2004).

5.1 Fixed Number of α-levels

In this section, the number of α-levels is ﬁxed at a

value of m+ 1 = 11.

5.1.1 Non-parallel Implementation

This paragraph examines the capability of GD,

SIMPSA, PSO with a population of 10, 15 and 20 par-

ticles and PSO GD with a population of 10, 15 and 20

particles to construct the membership functions of the

fuzzy output intervals of the 9 test functions. Firstly, a

mixed ANOVA model with the different test functions

as random effect is applied on the data of the area un-

der the membership function composed by the Fuzzy

Calculator using these optimisation algorithms. This

test indicates that the application of the Fuzzy Calcu-

lator using the algorithms PSO GD with a population

size of 15 and 20 particles and SIMPSA are signif-

icantly better in constructing the membership func-

tions than the Fuzzy Calculator using the other opti-

misation algorithms.

The number of function evaluations needed to

construct the membership function is a measure for

the computational cost. The number of function eval-

uations of the Fuzzy Calculator using PSO GD with

a population size of 15 particles (28551) and 20 parti-

cles (37662) and SIMPSA (72390) are signiﬁcantly

different. As the Fuzzy Calculator using SIMPSA

needs a very high mean number of function evalua-

tions, we can conclude that this algorithm is computa-

tionally inefﬁcient. Therefore, the SIMPSA algorithm

will not be used in the parallel Fuzzy Calculator.

5.1.2 Parallel Implementation

For the parallel Fuzzy Calculator, we restrict to the

optimisation algorithms PSO and PSO GD.

Again, a mixed ANOVA model with the test func-

tions as random effect is performed. Since the factors

population and communication are available in our

ICFC 2010 - International Conference on Fuzzy Computation

two optimisation algorithms, we can put them in this

model as nested factors. This leads to a more correct

estimate of the p-values in our model. The results of

the mixed ANOVA model indicate that in all possible

combinations of population size and communication

strategy, PSO GD is signiﬁcantly better than PSO.

Furthermore, communication at every 5 iterations and

a population size of 20 particles is signiﬁcantly better

than the other parameter combinations. This leads to

the conclusion that for the parallel Fuzzy Calculator

using PSO GD with a population size of 20 particles

and communication at every 5 iterations is the best

algorithm to construct the membership functions of

the test functions. For the number of function evalu-

ations, more frequent communication or a larger pop-

ulation size needs signiﬁcantly more function evalua-

tions.

The last step is the comparison of the parallel

results with the non-parallel results. We compared

the results of the non-parallel Fuzzy Calculator using

PSO GD with a population size of 15 and 20 particles

and the parallel Fuzzy Calculator using PSO GD with

a population size of 20 particles and communication

at every 5 iterations. The mean area obtained with the

parallel Fuzzy Calculator using PSO GD with a pop-

ulation size of 20 particles is signiﬁcantly higher than

the mean areas obtained with the non-parallel Fuzzy

Calculator using PSO GD with a population size of

15 and 20 particles, which leads to the conclusion

that our parallel implementation gives the best results.

However, the number of function evaluations of the

parallel Fuzzy Calculator using PSO GD with com-

munication at every 5 iterations is signiﬁcantly higher

than the non-parallel Fuzzy Calculator using PSO GD

with a population size of 15 and 20 particles.

5.2 Starting from 3 α-levels

A disadvantage of working with a ﬁxed number of

α-levels is that this number has to be determined in

advance. Consequently, it is possible that too many

α-levels are used to determine the membership func-

tion of the fuzzy output interval of a simple func-

tion whereas too few α-levels are used for more dif-

ﬁcult functions. A solution to this problem is starting

with 3 α-levels and expanding this number through

linear interpolation when necessary. For this linear

interpolation, a convergence criterion has to be de-

termined. As convergence criterion, we decided to

set a tolerance of 0.01 between α and the guess of

calculated through linear interpolation. With the use

of the mixed ANOVA model with the different test

functions as random effect, we compared the parallel

Fuzzy Calculator starting with 3 α-levels with the pa-

rallel Fuzzy Calculator using the ﬁxed number of 11

α-levels. The difference between the mean area un-

der the membership function for the Fuzzy Calculator

starting with 3 α-levels and the Fuzzy Calculator us-

ing 11 α-levels is signiﬁcant, which leads to the con-

clusion that the Fuzzy Calculator starting with 3 α-

levels and expanding this number through linear inter-

polation gives a more accurate membership function

than the Fuzzy Calculator using 11 α-levels.

The number of function evaluations is signiﬁ-

cantly higher for the Fuzzy Calculator starting with

3 α-levels (145074 versus 45119). An explanation is

that much more α-levels are optimised when starting

with 3 α-levels and expanding this number through

linear interpolation (Table 1).

Table 1: Mean number of α-levels needed to construct the

left (# α

min

levels) and right (#α

max

levels) side of the mem-

bership function of the fuzzy output interval for the different

test functions with the Fuzzy Calculator starting with 3-α-

levels.

Test function # α

min

levels #α

max

levels

1 21 3

2 6.2 37.6

3 33.6 40.72

4 31.16 19.88

5 21.12 43.56

6 40.28 46.24

7 44.32 62.6

8 45.8 64.16

9 49.64 72.4

As mentioned in Section 4.2, there are two ways

to correct for inconsistencies between the α-levels.

In the previous results, the optima of all α-levels are

compared and replaced by the optima of a higher α-

level if necessary. The other approach is to recalcu-

late the optima of the inconsistent α-levels with as

starting point the location of the optima of the above

lying α-level. We used this last approach in combi-

nation with the Fuzzy calculator starting from 3 α-

levels and expanding this number through linear inter-

polation when necessary. We compared these results

to those of the Fuzzy Calculator starting with 3 α-

levels, using the ﬁrst approach to correct for inconsis-

tencies between the α-levels. The difference between

the mean area under the possibility distribution, ob-

tained with the two approaches, is very small and not

signiﬁcant (p-value > 0.05).

The difference between the number of function

evaluations, however, is signiﬁcant (p-value < 0.05).

The second approach for dealing with inconsistent

α-levels needs signiﬁcantly less function evaluations

(102101 versus 145074). This is caused by the fact

DEVELOPMENT OF A FUZZY CALCULATOR FOR CONTINUOUS FUNCTIONS OF NON-INTERACTIVE FUZZY

VARIABLES

that in general less α-levels are needed when using

the second approach for dealing with inconsistencies

between α-levels (Table 2).

Table 2: Mean number of α-levels needed to constructing

the left (# α

min

levels) and right (#α

max

levels) side of the

membership function of the fuzzy output interval for the

different test functions with the Fuzzy Calculator starting

with an expanding number of α-levels with recalculating

incorrectly found optima.

Test function # α

min

levels #α

max

levels

1 21 3

2 4.9 34.76

3 26.26 38.38

4 19.54 19.22

5 10.24 45.56

6 51.92 54.78

7 25.58 42.82

8 23.4 43.36

9 22.28 46.64

6 CONCLUSIONS

The results indicate that the parallel Fuzzy Calcula-

tor is the best way to construct the membership func-

tion of the fuzzy output interval of a continuous func-

tion of non-interactive fuzzy intervals. The best ap-

proach is an expanding number of α-levels, with Par-

ticle Swarm Optimisation in combination with Gradi-

ent Descent as optimisation algorithm, using a popu-

lation size of 20 particles and communication at ev-

ery 5 iterations, and by recalculating inconsistent α-

levels. The number of function evaluations, however,

can be quite high, depending on the number of α-

levels that will be constructed. This can be regulated

by the tolerance level in the criterion that determines

the insertion of additional α-levels. In addition, as the

implementation is parallel and several processors can

be used, an elevated number of function evaluations

will not pose a major problem for most applications if

a high performance facility is available.

ACKNOWLEDGEMENTS

This work was supported by the Special Research

Fund of Ghent University and the Belgian Science

Policy (STEREO-project SR/00/100).

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