International Journal of Fuzzy Systems, Vol. 15, No. 4, December 2013
488
Two Interpretations of Multidimensional RDM Interval Arithmetic Multiplication and Division
Andrzej Piegat and Marek Landowski
Abstract1
The paper presents two possible interpretations
and realization ways of interval multiplication and
division: the possibilistic, unconditional interpretation that is of great meaning for fuzzy arithmetic and
fuzzy systems, and the probabilistic, conditional interpretation that requires either knowledge of probability density distributions or assumptions concerning these distributions. The possibilistic interpretation has a great significance not only for fuzzy arithmetic but also for other sciences that use it such as
Computing with Words, Grey Systems, etc. These
two interpretations are explained in frame of a new,
multidimensional RDM interval-arithmetic. The possibility of realization of interval-arithmetic operations in two ways is an argument for reconciliation of
two competing scientific groups that propagate two
approaches to uncertainty modeling: the probabilistic
and possibilistic one. For many years Professor
Zadeh has been claiming in his publications that both
approaches are not contradictory but rather complementary.
Keywords: Computing with words, fuzzy arithmetic,
granular computing, interval arithmetic, interval equations, uncertainty theory.
1. Introduction
For many scientists and engineers interval arithmetic
seems to be a less important area of science. However,
its importance grows with the development of the uncertainty theory [1]. Scientists realize more and more that
taking into account the parameter and variable uncertainty in mathematical models is necessary. Data uncertainty is met everywhere. Let us consider as an example
Corresponding Author: Andrzej Piegat is with the Faculty of Computer Science and Information Technology, West Pomeranian University of Technology, Zolnierska 49, 71-210 Szczecin, Poland.
E-mail: [email protected]
Marek Landowski is with the Maritime University of Szczecin, Waly
Chrobrego 1-2, 70-500 Szczecin, Poland.
E-mail: [email protected]
Manuscript received 22 July 2013; revised 20 Nov. 2013; accepted 18
Dec. 2013.
the charging process of the car battery [16], where L0
[Ah] means the initial battery charge, L(t) [Ah] means
the charge after time t [h] of charging, and i [A] means
the charging current. The charging process can be described by (1).
(1)
L ( t )  L 0  it
If we want to determine the charging time T [h] of a
battery that is discharged, then (2) should be used.
(2)
T  L T  L 0  / i
In formula (2) LT means the required end charge of the
battery. However, in this problem we mostly do not
know precisely but only approximately the initial state
L0[ L 0 , L 0 ] as e.g. L0[4,7] Ah. Similarly, the charging current is not constant and varies during the charging
process. It can be approximated as i[ i, i ], e.g. i[3,4]
A. Therefore, to calculate the required charging time T
[h] we should not use (2) but rather the interval formula
(3)
[T , T ] 
LT  [ L 0 , L 0 ]
[i , i ]

90  [ 4 , 7 ]
[ 3, 4 ]
(3)
The mathematical battery-model is simple. However,
in practical problems much more complicated models
occur, e.g. ship movement model, airplane, rocket,
space-ship movement models, economical, medical,
mechanical, environmental models, etc. In most of these
models occur uncertainties and taking them into account
is necessary if we want to get realistic results. This situation explains the increasing interest in uncertainty theory
[1], Grey Systems [8], Granular Computing [11], Computing with Words [21, 22], fuzzy systems [7, 12] in
science areas that allow for processing uncertain data
and modeling uncertain systems. Interval arithmetic is
the basic arithmetic of approximate data, because interval [ x, x ] is the simplest and a very convenient way of
elicitation and modeling uncertainty. This arithmetic is
also used as a basis by other sciences dealing with uncertainty. And so, all calculation results delivered by
fuzzy arithmetic [3, 6] have to be consistent with results
delivered by interval arithmetic in respect of their supports (it results from the -cut method). The same refers
to probabilistic arithmetic [4, 5, 19], Grey Systems, etc.
The so called “precise” models as e.g. battery model (1)
are at present called “academic” or “laboratory” models,
because they are not rather realistic. They make illusion
© 2013 TFSA
A. Piegat and M. Landowski: Two Interpretations of Multidimensional RDM Interval Arithmetic - Multiplication and Division
of accuracy that in reality is not attainable. In a way,
“precise” models deceive their user. The simplest models taking into account parameter- and variable-uncertainty are interval ones. Their using requires
knowledge of interval arithmetic. The origins and development of this arithmetic is mainly assigned to Moore
[9, 10, 11]. In scientific literature one can find many
examples of its applications, e.g. [2, 3, 8, 10, 11, 17, 18].
However, many faults or weak-points of this arithmetic
have been detected [2, 3, 18]. They are rather well
known: the excess width effect, the dependency problem,
difficulties of solving even the simplest interval equation,
interval equation’s right hand-side problem, absurd solutions and request to introduce negative entropy into the
system. Descriptions of certain faults can also be found
in Wikipedia. Thus they will not be discussed in this paper. In spite of clear faults of Moore arithmetic it is used
by scientists all the time, also in fuzzy arithmetic. The
examples can be found in [2, 3, 8, 10, 17]. It is probably
caused by the fact that Moore arithmetic is very intuitive
and easy to understand. The authors of this paper do not
maintain that Moore’s interval arithmetic generally is
incorrect. However, they are of the opinion that it only
allows for solution of relatively simple problems. Its
particular fault is a difficulty or impossibility of solving
even simple interval equations with unknowns [2]. This
perception was the reason for elaborating multidimensional interval- arithmetic [14, 15]. The concept author
of this arithmetic is Andrzej Piegat and its investigation
and development is made in collaboration with Marek
Landowski and other co-workers. Further on, shortly, the
concept of the multidimensional RDM-arithmetic will be
presented. In the author's opinion one-dimensional approach to interval-arithmetic operations that is proposed
by Moore-arithmetic generally is not correct and the
multidimensional approach is necessary because in
problems with uncertainty input-intervals create a multidimensional rectangular or cuboid granule and the calculation result is, in the general case, not a
one-dimensional interval but a multidimensional, irregular, non-cuboidal output granule. Thus, in case of interval arithmetic we have to do with calculations of the
Granular-Computing character [11]. The abbreviation
“RDM” comes from the notion “Relative- Distance-Measure”. If an unknown value of variable x is
approximated or precisiated by interval [ x, x ], where x
means the lower limit and x the upper limit of the interval, then a value x lying in the interval can be expressed with the use of the RDM-variable x[0,1], (4).
(4)
x  x   x ( x  x),  x  [0,1]
If e.g. x[5,7], then this information can be expressed
in form of (5).
(5)
x  5  2 x ,  x  [0,1]
489
The meaning of the RDM-variable x[0,1] is visualized in Figure 1.
Figure 1. Illustration of meaning of the RDM-variable x in
case of the normal interval [ x, x ], x  x .
Thanks to introducing the variable x, the interior of
interval [ x, x ] becomes not anonymous, achieves a
meaning and can take part in calculations. Why is RDM
arithmetic characterized as multidimensional? Because it
allows us to realize that each parameter-uncertainty of a
system increases its dimensionality. Let us consider once
more the car-battery charging process (1), L(t)=L0+it.
In this equation variables are time t and the
end-charge LT=L(T) of the battery, where T means the
charging time [h] and the equation is a linear one. If the
current i is known only approximately (i[3,4]A) and
similarly the initial battery charge L0[4,7], then these
parameters can be described with (6) and the linear battery charging model is transformed in the nonlinear
model (7).
x  4  3 L0 ,  L0  [0,1]
(6)
i  3   i ,  i  [0,1]
L(t )  (3   i )  (4  3 L0 )t  3   i  4t  3t L0
(7)
 i  [0,1],  L0  [0,1]
The linear battery-model (1) is defined in
2-dimensional space LT, but the non-linear
RDM-model (7) is defined in much larger,
4-dimensional space LT  i   L0 . This phenomenon
of increasing model-dimensionality and nonlinearity degree caused by parameter uncertainty is the reason of
great difficulties in solving problems with uncertainty.
However, many scientists do not realize that.
2. Possibilistic Version of Interval Multiplication
According to Moore-arithmetic multiplication of two
intervals [ a, a ][ b, b ]=[ x, x ] should be realized with (8).
[ a , a ][ b , b ]  [min( ab , a b , a b , ab ), max( ab , a b , a b , ab )]
 [ x, x]
(8)
Operations min and max are necessary if intervals are
not positive. If both intervals are positive then the multiplication can be realized according to the simplified
formula (9).
(9)
[ a , a ][ b , b ]  [ ab , ab ]  [ x , x ], a  0, b  0
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International Journal of Fuzzy Systems, Vol. 15, No. 4, December 2013
If e.g. [a]=[1,3] and [b]=[1,5/3] then the multiplication
is realized according to (10) and visualization of this operation is shown in Figure 2.
(10)
[ a , a ][ b , b ]  [1,3][1,5 / 3]  [ x , x ]
Figure 2. Illustration of the one-dimensional multiplication of
two not precisely known values a and b that have been approximated by intervals a  [1, 3] and b  [1, 5/3] with the
use of Moore-arithmetic.
According to Moore-arithmetic the multiplied values
a and b are treated fully anonymously. It has no meaning
whether they represent the same or different physical,
biological, economical, etc. variables. The multiplication-operation is always realized 1-dimensionally and
only interval borders have importance and take part in
calculations. Numbers contained inside of intervals have
no meaning. However, let us consider the interval multiplication [a][b]=[x] as a 3-dimensional operation. Figure
3 shows the 3-dimensional functional surface of multiplication ab=x. Figure 3 also presents contour lines of
constant results of the multiplication ab=x=const, e.g.
ab=4.
Figure 4. Projection of contour lines ab=x=const on 2D-space
AB.
The operation of interval multiplication [a][b]=
[1,3][1,5/3] that in Figure 2 has been realized one- dimensionally with Moore-arithmetic now will be realized
with the use of multidimensional RDM- arithmetic.
Values a and b will be modeled with the use of
RDM-variables a and b, formulas (11) and (12) respectively.
(11)
[ a ]  a   a ( a  a )  1  2 a ,  a  [ 0,1]
(12)
[b ]  b   b (b  b )  1  2 b / 3,  b  [ 0,1]
The multiplication operation [a][b] = [x] is determined
with (13).
[ a ][ b ]  [ a   a ( a  a )][ b   b ( b  b )]
(13)
 (1  2 a )(1  2 b / 3),  a  [ 0 ,1],  b  [ 0 ,1]
The multiplication operation of two positive intervals
is characterized by a monotonically rising functional
surface x=ab shown in Figure 3 and Figure 4. Thus, the
extremum values x and x lie on borders of the multiplication domain AB, that is shown in Table 1.
Table
1.
Multiplication results of two intervals
for border values of RDM-variables
a[0,1] and b[0,1].
[ a , a ][ b , b ]  [ x , x ]
a
b
Figure 3. 3-dimensional functional surface of multiplication
ab=x for a0 and b0.
The multiplication surface can be shown not only in
3D-space ABX but also as a projection on 2D-space
AB, Figure 4.
0
0
1
1
0
1
0
1
x
ab
ab
ab
ab
x
1
5/3
3
5
The solution of the interval multiplication with the use
of RDM-arithmetic is visualized in Figure 5.
The solution of the interval multiplication [a][b]=[x] is
a set of all subsets x=ab=const in the form of contour
lines that lie inside of the solution granule shown in Figure 5. Each of the contour lines represents one solution
subset. However, these subsets are not equal. The subset
x=ab=1 contains only one tuple (a,b)=(1,1). Similarly
the subset x=(a,b)=(3,5/3). Instead, remaining solution
A. Piegat and M. Landowski: Two Interpretations of Multidimensional RDM Interval Arithmetic - Multiplication and Division
subsets, e.g. x=ab=5/3, contain infinite number of tuples
(a,b) that satisfy the condition x=ab=const. Each tuple
(a,b) has a full membership to its solution subset
x(a,b)=1. This is presented in Figure 6.
AbsPoss ( x )  L ( x ), x  ab
(15)
An event possibility should not be mistaken with the
event probability. For discrete events this difference was
explained by Zadeh in [20] on the example of eggs that
John is able to eat for breakfast, see also [13, 23]. An
event possibility can be interpreted as easiness of the
event occurrence. Let us assume that John is able to eat
maximally 8 eggs. Then, the possibility distribution for n
eggs is as below.
1
Poss ( n )  
0
Figure 5. Multiplication of intervals [a][b]=[x] with use of
RDM-arithmetic in projection on the space AB.
Figure 6. Examples of membership functions x(a,b) of tuples
(a,b) to particular solution subsets x=ab=const.
The membership function x(a,b) is tantamount to the
characteristic function of the solution subset x=ab=const
because the membership function is a generalization of
the characteristic function of a classical set, and the classical set is a special case of fuzzy set [7, 12]. A measure
of each solution subset {(a,b)|ab=x} is the cardinality of
this subset Card{(a,b)|ab=x}, (14).
Card {( a , b ) | ab  x}   x (( a , b ) | a  x / b ) db (14)

B
In respect of value cardinality of the subset
{(a,b)|ab=x} is surface area of the membership function
{x(a,b)|ab=x} that is presented in Figure 6. Because
membership of each tuple (a,b) to one of the subsets is
full, therefore the cardinality is equal and tantamount to
the length of a contour line ab=x=const shown in Figure
5 and Figure 6. This length L(x) can also be interpreted
as the measure of the absolute possibility AbsPoss(x)
of the event occurring ab=x, (15).
491
for
n  {0 ,1, 2 ,3, 4 ,5, 6 ,7 ,8}
for
n8
However, the fact that John potentially can eat 8 eggs
does not mean that he eats 8 eggs for breakfast every day.
He usually eats 1-3 eggs because the bigger quantity
would be too many. Thus, the probability distribution of
the egg number eaten for breakfast, determined on the
basis of statistical observations, for John can be as below.
prob ( 0 )  0 .1, prob (1)  0 .2 , prob ( 2 )  0 .6 ,
prob ( 3)  0 .1, prob ( n  3)  0
An event possibility is of potential character and is
more often connected with mental analyses and a problem understanding, and seldom with experiments. An
event probability is rather of real character and results
from observations and experiments. However, both notions are not contrary because the increase in an event
possibility is mostly accompanied by the increase of the
event probability. This phenomenon is described by
Zadeh’s consistency principle of possibility and probability [20]. Length L(x) of contour lines lying in each of
3 zones shown in Figure 5 (for a  0 , b  0 and
ab  ab ) can be calculated from (16).
x / a

1  x 2 / b 4 db
 b

b
 1  x 2 / b 4 db
L( x)  
b
 b

1  x 2 / b 4 db
x / a
0




for
x  [ ab , a b )
for
x  [a b, a b )
for
x  [ a b , ab )
for
others
Figure 7 shows a diagram of the absolute
AbsPoss( x)  L( x) of the event occurrence
Figure 8 shows a diagram of the relative
RelPoss( x)   ( x)  L( x) / max L( x) , i.e. of the
normalized to max L(x)=1.
(16)
possibility
ab=x and
possibility
possibility
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International Journal of Fuzzy Systems, Vol. 15, No. 4, December 2013
pdf ( a )  1 /( a  a ), a  [ a , a ]  [1,3 ]
pdf ( b )  1 /( b  b ), b  [ b , b ]  [1,5 / 3 ]
(17)
Distributions pdf(a) and pdf(b) are shown in Figure 9.
Figure 7. Distribution of the absolute possibility AbsPoss(x) of
the solution occurrence x=ab in interval multiplication
[a][b]=[1,3][1,5/3]=[x] with the use of multidimensional
RDM-arithmetic, AbsPoss( a b )=AbsPoss(5/3)0.9528, AbsPoss( a b )=AbsPoss(3)1.3832.
Figure 9. Uniform distributions of probability density pdf(a)
and pdf(b) of multiplied intervals [a] and [b].
As a result of interval multiplication [a][b]=[x] a solution granule consisting of an infinitive number of solutions ab=x=const shown in Figure 10 is achieved. Figure
10 presents only 4 of all solutions x=const shown in
form of contour lines.
Figure 8. Distribution of the relative possibility RelPoss(x)=
(x), normalized to $\max (x)=1, of the solution occurrence
x=ab=[1,3][1,5/3] in interval multiplication with the use of
multidimensional, RDM-arithmetic.
The possibilistic solution of interval multiplication
[a][b]=[x] is unconditional: it was achieved without initial assumptions.
3. Probabilistic Version of Interval Multiplication
The very knowledge that value a  [a, a] and
b [b, b] is not connected with any distributions of
probability density (pdf). This knowledge only means
that the limits a , a and b , b of intervals that approximate the precisely unknown values a and b are
known. However, in certain problems experts can possess knowledge concerning probability density distributions pdf(a) and pdf(b). Then, this knowledge can be
used in the interval multiplication and the multiplication
result will also have the form of a probability density
distribution pdf(x). Let us assume, that according to the
expert knowledge of the problem, both pdf(a) and pdf(b)
are uniform distributions (17).
Figure 10. Solution granule of interval multiplication [a][b]=[x]
with 4 contour lines representing 4 of all possible solutions
ab=x=const and with 3 solution zones.
If pdf(a) and pdf(b) are uniform and variables a and b
are independent then also the joint distribution
pdf(a,b)=const and the total probability mass is uniformly distributed in domain AB of the interval multiplication, Figure 10. The cumulated probability of the
event occurrence p(abx|pdf(a,b)=const) is then proportional to the area A1(x) under the contour line x, Figure
11.
Figure 11 helps to construct (18) for the area A1(x)
below the contour line x.
x/a
A1 ( x ) 

x/a
( a ( b )  a ) db 
b
 ( x ln | b |  ab )
 ( x / b  a ) db
b
x/a
|b
 x ln | x / ab |  x  ab
 x ln | x / x1 |  x  x1
(18)
A. Piegat and M. Landowski: Two Interpretations of Multidimensional RDM Interval Arithmetic - Multiplication and Division
493
Figure 11. Illustration of the calculation way of cumulated
probability p(abx|pdf(a,b)=const) of the product occurrence
ab=x, x[x1,x2], x1= ab , x2= a b , for a b  a b .
Figure 12. Illustration of the calculation of the cumulated
probability density p(abx|pdf(a,b)=const) for Zone 2,
x  [ x 2 , x 3 ]  [ a b , a b ] , for a b  a b .
The cumulated probability p(abx|pdf(a,b)=const) is
equal to the participation ratio of the area A1(x) in the
area A of the full solution granule, (19), where
A  (a  a)(b  b) .
Figure 13 illustrates a calculation way of the cumulative probability of the event occurrence ab=x for Zone 3
of the solution domain.
p ( ab  x | pdf ( a , b )  const )  ( x ln | x / x1 |  x  x1 ) / A
(19)
Probability density pdf(x|pdf(a,b)=const) can be calculated from (20).
pdf ( x | pdf ( a , b )  const )
d
p ( ab  x | pdf ( a , b )  const )  (ln | x / x1 |) / A

dx
x  [ x1 , x 2 ], x1  ab , x 2  a b
(20)
Figure 12 illustrates the calculation of the cumulated
probability density p(abx|pdf(a,b)=const) for Zone 2,
x  [ x 2 , x3 ]  [ab, ab] .
The cumulative probability is proportional to the area
A2(x) below the contour line ab=x. It can be calculated
from (21).
b
A2 ( x ) 

b
x  [ a b , ab ]  [ x 3 , x 4 ] .
b
( a ( b )  a ) db 
 ( x / b  a ) db
(21)
b
 x ln | b / b |  a ( b  b )
The cumulative probability is equal to the quotient
A2(x)/A, where A  (a  a)(b  b) , (22).
pdf ( ab  x | pdf ( a , b )  const )
 ( x ln | b / b |  a (b  b )) / A
(22)
The probability density pdf(x|pdf(a,b)=const) can be
calculated from (23).
pdf ( x | pdf ( a , b )  const )
d
p ( ab  x | pdf ( a , b )  const )
dx
 (ln | b / b |) / A  (ln | x 2 / x1 |) / A

Figure 13. Illustration of the calculation way of the cumulative
probability of the event occurrence ab=x for Zone 3:
(23)
In Zone 3 the cumulative probability p(ab
x|pdf(a,b)=const) is expressed by (24), where
A  (a  a)(b  b) .
pdf ( ab  x | pdf ( a , b )  const )
b




 A3 / A   A  ( a  a (b )) db  / A


x/a


b




  A  ( a  x / b ) db  / A


x/a


b

 / A
  A  ( ab  x ln | b |) |
x/a 




(24)

 A  x ln | ab / x |  ab  x / A
The probability density (pdf(x|pdf(a,b)=const) is ex-
International Journal of Fuzzy Systems, Vol. 15, No. 4, December 2013
494
pdf ( x | pdf ( a , b )  const )  k N ln | ab / x |
pressed by (25).
pdf ( x | pdf ( a , b )  const )
d

p ( ab  x | pdf ( a , b )  const )
dx
 (ln | ab / x |) / A  (ln | x 4 / x |) / A
(28)
where k N  1 / ab .
(25)
The distribution of probability density of the result x
of interval multiplication in case of [a][b]=[x]0 is presented by (26) and in Figure 14.
pdf ( x | pdf ( a , b )  const )
 (ln | x / x1 |) / A
 (ln | x / x |) / A

2
1

 (ln | x 4 / x |) / A
 0
for
x1  x  x 2
for
x 2  x  x3
for
x3  x  x4
for
others
Figure 15. Conditional distribution of probability density
pdf(x|pdf(a,b)=const) of multiplication result x of two intervals
with zero-lower-limits [ 0 , a ][ 0 , b ]  [ 0 , 2 ][ 0 ,3]  [ x ] .
where A  ( a  a )( b  b ), x1  min( ab , ab ),
x 2  min( a b , a b ), x 3  max( a b , a b ), x 4  max( ab , ab )
4. Interval Division
(26)
Interval division [a, a] /[b, b] can be replaced by interval multiplication. If e.g. both divided intervals are
positive, then (29) can be used.
[ a , a ] /[ b , b ]  [ a , a ]  [1 / b ,1 / b ]  [ a / b , a / b ], a  0, b  0
(29)
In the general case, when intervals are not positive,
(30) should be used.
[ a , a ] /[ b , b ]  [min( a / b , a / b , a / b , a / b ),
(30)
max( a / b , a / b , a / b , a / b )]
Figure 14. The normalized distribution pdf(x|pdf(a,b)=const)
of probability density of the result x of the interval multiplication [a]=[1,3] and [b]=[1,5/3] with the use of RDM-arithmetic
for case a b  a b .
The distribution of probability density of interval multiplication in case [a][b]=[x]<0 is presented with (27).
pdf ( x | pdf ( a , b )  const )
 (ln | x / x1 |) / A
 (ln | x / x |) / A

2
1

(ln
|
x
/
x
|) / A
4

 0
for
x 2  x  x1
for
for
x3  x  x2
x 4  x  x3
for
others
where A  ( a  a )( b  b ), x1  max( a b , a b ),
x 2  max( ab , ab ), x 3  min( ab , ab ), x 4  min( a b , a b )
(27)
In other cases of interval multiplication [a][b]=[x] the
distribution of probability density is a combination of
(26) and (27).
Figure 14 concerns the case x1  ab  0 . For example,
if both lower limits of the multiplied intervals [a] and [b]
are equal to zero ( [a, a]  [0, a] and [b, b]  [0, b] ) then the
distribution of probability density of result x is determined by (28) and illustrated by Figure 15.
Division by an interval containing zero is not defined.
5. Conclusion
The paper has shown that 2 interpretations of interval
multiplication are possible. The first interpretation is a
possibilistic one and is of unconditional character, i.e.
this interpretation does not require assumptions concerning distributions of variables inside the intervals.
However, if a problem-expert knowledge about the
probability density distributions exists then it can be
used in frame of the probabilistic version of interval
multiplication. Figure 16 shows the comparison of results of interval multiplications [a][b], where
[a, a]  [1,3] and [b, b]  [1,5 / 3] achieved with the use of
the probabilistic and possibilistic interpretations.
Of course, the probabilistic result (Figure 16b) is correct only if assumptions about uniform probability density distributions inside the intervals are satisfied. If, according to the problem-expert knowledge, the distributions are of other forms, then the resulting distribution
can also be calculated with the use of the
PACAL-program [4, 5]. The achieved distribution will
then have another form than that shown in Figure 16b.
Instead, the possibilistic result (Figure 16a) is always of
A. Piegat and M. Landowski: Two Interpretations of Multidimensional RDM Interval Arithmetic - Multiplication and Division
the same and constant form because it was calculated
without any assumptions concerning interval interiors.
Thus, the possibilistic interval-multiplication version has
a great practical meaning, because in real problems we
frequently possess no knowledge about distributions of
probability density. Mostly, only the knowledge of interval limits as approximation of unknown parameters or
variable values can be delivered by problem-experts.
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
Figure 16. Comparison of multiplication results of two intervals [ a , a ]  [1,3] and [b , b ]  [1,5 / 3] achieved with the use
of the unconditional, possibilistic version (Figure 16a) and of
conditional, probabilistic version of RDM-interval- arithmetic
at assumption pdf(a)=const and pdf(b)=const (Figure 16b).
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Andrzej Piegat received his Ph.D. degree in 1979 in modeling and control of
production systems from the Technical
University of Szczecin, Poland, the DSc
degree in control of underwater vehicles
from the University of Rostock, Germany, in 1998, and the professor title in
2001. At present he is professor at the
West Pomeranian University of Technology, Poland. His current research is focused on uncertainty
theory, fuzzy logic, computing with words and info-gap theory.
Marek Landowski received the MSc
degree in Mathematics from the
Szczecin University, Poland, in 2002, the
MSc eng degree and the Ph.D. degree in
Computer Science from the West Pomeranian University of Technology, Poland, in 2004 and 2009 respectively.
Currently he is assistant professor at the
Maritime University of Szczecin, Poland.
At present his research interests are focused on fuzzy arithmetic, uncertainty theory, and computing with words.