RANDOMNESS AND FUZZINESS IN A LINEAR. PROGRAMMING PROBLEM. S.T. Wierzchon. Institute of Computer Sciences. Polish Academy of Sciences.
RANDOMNESS AND FUZZINESS IN A LINEAR PROGRAMMING PROBLEM
S.T. Wierzchon Institute of Computer Sciences Polish Academy of Sciences P.O. Box 22 00-901 Warszawa, PKIN POLAND
ABSTRACT: An LP (Linear Programing) problem is studied under sumption that the right hand sides of the contraint
the
as-
inequalities
independently distributed normal r.v.'s (random variables) with
are fuzzy
mean values and fuzzy standard deviations. A version of Charnes-Cooper's method is formulated
and
possible
extensions of the approach are suggested. KEYWORDS: Fuzzy Numbers, Fuzzy Normal Distribution. Stochastic LP.
1. INTRODUCTION
In this paper an LP problem is studied under the assumption the right hand sides of the constraint inequalities are
that
independently
distributed normal random variables with fuzzy mean values
and
fuzzy
standard deviations. Such a problem can be a model of a general situation when an agent knows that that an exogenous variable is a r.v. with its
J. Kacprzyk et al. (eds.), Combining Fuzzy Imprecision with Probabilistic Uncertainty in Decision Making © Springer-Verlag Berlin Heidelberg 1988
228
sufficient
distribution function of a certain type but due to lack of knowledge
he
is
not
distribution function.
able
to
estimate
the
of
parameters
In general, however, the agent
this
possesses
evidence enabling him to roughly assess these parameters.
Fuzzy
some sets
theory seems to be an effective tool for solving such problems. The paper is organized as follows. Section 2 is a brief survey to the notions used in later sections. In
Section
the
3
main
needed for deriving deterministic equivalents to the fuzzy LP problem are presented. Section 4 is devoted to
the
results
stochastic
main
problem,
i.e. to the solution of the fuzzy stochastic LP problem.
2. SOME CONCEPTS OF FUZZY SET THEORY Suppose a is a parameter and its
value
a
is
Suppose
unknown.
further that we have only an evidence that the value of a is ABOUT a o ' The term ABOUT is vague and, according to Zadeh, its represented by the so-called membership function
meaning
may
be
Z --) [0,1) where
~A:
Z is a universe of discourse and A is a fuzzy set induced by the vague term. We can treat A as an elastic constraint that may be aSSigned to a.
In this context
degree to which the constraint ABOUT a ~
0
acting
~A(z)
on
the
values
is interpreted as the
represented by the fuzzy set A
is satisfied when z is assigned to a, i.e. Poss(a=z)
(1)
where Poss is the possibility measure [10). For the practical purposes the fuzzy subsets of are classified as fuzzy numbers provided they
are
~,
the real line,
convex,
unimodal,
normalized and having upper semi-continuous membership functions Of special importance
from
a
practical
standpoint
are
(1).
fuzzy
numbers characterized by the membership function of the form L(r) R(r)
o
for
ad$ r $ a a $ r $ a g $ +00 otherwise -00 $
for
where L (resp. R) is a nondecreasing
(resp.
(2)
nonincreasing)
function
such that L(a d ) = R(a g ) = 0 and L(a) = R(a) 1. Here ad (resp. a g ) is said to be the lower (resp. upper) bound of A and a is referred to as the main value of the fuzzy set A. Of course ad $ a $ an'
229
When ad
= ag = a
If ad
0
we get a crisp (i.e. nonfuzzy) number a.
(resp. a g ~ 0) then the fuzzy number A is positive (resp. negative). ~
Employing the Extension Principle (cf e.g.
said
we
[1])
to
be
extend
any
arithmetic operation . to the fuzzy case, namely ~AOB(r)
= sup
r=z·v
min
(~A(z),
In the sequel we will
use
(3)
~B(v»
triangular
fuzzy
numbers
(TFN
for
short) characterized by the membership function r - ad LA(r)
~A(r)
I
a - ad a
RA(r)
a
9 9
- r - a
a
r
~
~
a
(4)
9
otherwise
0
We will write
for
A=(ad,a,ag)~
case, using Lemma 1 in p.42 of
to denote that A is a [1],
we
derive
TFN.
from
(3)
In
this
the
next
expressions for the addition of two TFN'S and the multiplication of
a
TFN by a crisp number : (5a)
r(ad,a,ag)~=
fl
(rad,ra,rag)~
when
r
~
0
(5b) (rag,ra,rad)~
when
r
~
0
Unfortunately it is not possible to obtain such
simple
formulae
the extended multiplication and division of the
TFN·s.
We
can
for show
however the next proposition that will be used in the sequel Proposition 2.1 Let A = (ad,a,ag)~ and B = (bd,b,bg)~ be two TFN'S such that B is positive. Then C = AlB is a fuzzy number defined as follows: (i) If A is positive then
230
rb g
L(r)
for
a -ad + r(b -b)
b
9
R(r)
l
r
5
5
b
9
a
a
a g - rb d
~C(r)
a
ad
ad
for
a g -a + r(b-b d )
o
r
5
b
5
9
bd (6a)
otherwise
(ii> I f A is negative then
r
rb d - ad L(r)
a-ad - r(b-b d ) a
~C(r)
R(r)
rb g
9
9
for
b
9
5
b a
r
5
5
bg (6b)
otherwise
0
Proof.
bd
r
5
a
r(b -b)
a -a
a
ad for
Quite simple. We use the next results from [lJ: (i) (ii)
AlB = Ao (lIB) where lIB is the inverse of B. If A is negative then A.B = - (-A).B. then LH'lN (x*y)
(7a)
RH'lN (z*y)
(7b)
for any fuzzy numbers H,N and any increasing operation To end this section let us
mention
how
to
*.
•
compare
numbers. Let I denote any of the four relation .
the
fuzzy
Since
we
deal with imprecise data. it is not possible to make definite judgments; what we can do is to estimate an extent to which the statement
"A I 8" seems to be plausible or credible. This problem was
attempted
by Dubois and Prade [2J who proposed the following indices. Poss(A I B) = sup r
Cr(A I B)
min(~A(r),
~IBCr»
(8)
(9)
231
where Poss and Cr stand respectively. IB is a fuzzy
"possibility"
for set
of
numbers
and
"credibility" to B. More
I-related
precisely (10)
(11)
To get an intuitive meaning of these indices
notice
that
(a
simple
proof of these identities is left to the reader) Poss(A I B)
tiff
Cr(A I B) = t
t=sup
{O~v~l:
Av
t=l-sup {O~v~l: A
iff
~
v
IBv
0}
~
~ (IBc )
v
(12) ~ 0}
(13)
Here IBv= {reR: ~IB(r) ~ v} is the v-cut of IB and lBc is the complement of IB (recall that
~IBc(r)=
1-~IB(r)
for any r).
3. FUZZIFIED NORMAL DISTRIBUTION In this section we consider a model
leading
to
the
notion
of
fuzzy probability introduced by Zadeh [11]. Let X
~
N(m,s), i.e. X is a normal r.v. with
standard deviation s. Suppose that, due
to
the
mean lack
value of
m
and
sufficient
knowledge, both the parameters can be estimated by fuzzy numbers,
and
assume that m.
(md,m,mg)t.
(14)
s
(sd,S,Sg)t.
(15)
i.e. m. and s are finding Pr(X
~
triangular
fuzzy
numbers.
We
are
interested
in
a), the probability that X is not greater than a.
Assume for generality that a is a TFN of the form (16)
Following Yager [9] we can write Pr(X
~
a)
Pr(Y
a ~
m.
s
Prey
~
C)
F(C)
(17)
232 where C
=
(a-m)/s and Y
N(O,1).
F
probability distribution function of
stands
v.
for
According
the to
cumulative the
Extension
Principle, F(C) is a fuzzy number with the membership function
/-IF (C) (w)
f l
/-IC (F
-1
(w) )
o
otherwise
( 18)
Using (17) we derive Pr
(X
~
a)
1 - Pr(X :S a).
(19)
This last definition is quite reasonable. Denote namely by the t-cuts of the fuzzy numbers Pr(X :S
a)
and Pr(X
~
a)
P t and respectively.
The pair (Pt , P~) is regular in the sense of [6], i.e. for each PI P t there exists P2 in P~ such that PI + P2 = 1. Proposition 3.1
Let X
~
N(~s)
where m and s
are
TFN's
defined
in
by
(14) and (15). Let a be a TFN characterized by (16). Then Pr(X :S a) is
fuzzy number P with its membership function defined as follows: (i) If a-m
is a positive TFN, then
m
a
for
5
9
5
(20a)
m
a
for
o
otherwise.
5
233 (ii) If a-m is a negative TFN, then
a-ad + mg-m -
(s-Sd)F
-1
(w)
a - m ad - mg 1 for ------------ S F- (w)S s (20b)
a-a
9
+
m- md -
(s 9 -s) F- 1 (w)
for
o
otherwise
•
The proof follows from definition (18) and Proposition 2.1. The result derived above, although far from a
general
statement
is quite sufficient for applications. Having determined the fuzzy probability we may be
interested
in
the determination of the conditions that should be imposed on the
TFN
a to fulfil the requirement P I p where p is a prespecified value
and
I ~ { }. As we argued earlier, the comparison of P with must be done in the sense of the indices (8) or (9). Hence we have Proposition 3.2
Let X
~
N(m,s) be a normal r.v. with
the
p
parameters
given by the TFN's. Suppose a is a TFN such that a-m is positive. Then (i)
Poss{Pr(X S a) 2 p} 2 t
iff
(21a) (ii)
Poss{Pr(X 2 a) 2 p} 2 t i f f ad+t(a-a d )
s
m -t(m -m) + (s -t(s -s» g
9
(iii) Cr{Pr(X S a) 2 p} 2 t
9
9
Cr{Pr(X 2 a) 2 p} 2 t
-1
(l-p)
(22a)
iff
a-tea-ad) 2 m+t(mg-m) + (s+t(s -s» 9 (iv)
F
iff
F- 1 (p)
(21b)
234
a+t (a
9
-a)
To prove (i) -
S;
(22b)
m -
(iv) it suffices to notice that when P
Poss(P
p)
~
when p
o
Cr (P
~
p)
otherwise
l-L p (p)
< P S P when P d-
1
when Pg
0
otherwise
and (when necessary) to employ p
=
Pr(X S a)
=
s P
(P d ' P,
P
~
definition
Pg)
a
fuzzy
Here we have denoted
(19) •
number
of
type
The
(2).
•
membership function of this fuzzy number is defined in (20a). Proceeding in the same way and assuming that a-m
is
a
negative
TFN we state for instance that
(23a)
and Cr{Pr(X
a)
~
~
p}
tiff
~
a + t(a -a) S m
+ t(s
9
9
F- 1 (l-p)
-s»
(23b)
Comparing (23a) with (22a) we state Let p
Corollar:l! 3.1
U) Cr{Pr(X
~
0.5. Then
The conditions ~
(i i )
a)
~
p}
~
t
Poss{Pr(X
The conditions
Cr{Pr(X S a)
~
p}
~
t
Part (ii) of this
~
a)
~
p}
can be satisfied iff Poss {Pr(X S a)
~
can be satisfied iff corollary
can
be
~
t
and is a negative
a-m.
p}
~
a-m.
seen
t
TFN.
and is a positive TFN ••
after
deriving
counterparts of (21) for the membership function defined in (20b).
the
235 Cr(P
Corollary 3.2
~
p) ) 0 implies
Poss(P
This property shows that the truth
~
p)
•
1.
quantification
performed
by
using the Cr index is much more restrictive than that done by the Poss index. To be more illustrative notice that Poss(P
= tiff
p)
~
t = sup{ 0
v
~
~
(24)
1:
When t
"w
each w E [0, vol, i.e. it is possible that ~
pl. Equating to the
unity
amount of belief (concerning the possible location of p Cr(P
follows
0 for each v in [0.1]. Suppose that p is in [P d , P], are the lower bound and the main value of P,
~
p)
=1
- Poss(P
pl. When
~
p
our
with p
max v(Pr (aix S b i ) ~ pi) i x ~ 0, 0 < P , t S 1
c x
i
where v stands for Poss or Cr, and t
(26)
1, •••• I
is a degree of truth to which all
the chance constraints should be satisfied. In
practice
we
aspire
to
find
a
constraint with a high value of pi. Thus
solution to
satisfying
each
derive
a
deterministic
equivalent of (26) we should assume that aix - mi is
a
negative
(cf Corollary 3.1). Taking into account the equations (23a) and
TFN (23b)
we immediately obtain Proposi t i on 4. 1
When v
Poss then the deterministic equivalent of
(26) is c x ---> max i i i ad x + tea - ad) x
(27)
and when v
Cr then (26) is
c x ---> max i i a x + tea g - a i ) x
x
~
0, 0
Cr{Pr(a 1 x Poss{Pr(a i
:::
b1)
5 x
max
5
bi )
pi}
:::
i, P •
:::
t
:::
t, i
0,
Assume for simplicity that t=0.6 and Pl = P2 = P3 = P i.e. F- 1 (p) = 2.5. Under these assumptions our initial
0.994, problem takes
the form 2x 1 + }:2 ---> max 5 13.7 2.6x 1 + 6.2x 2 5 37.4 3.6x 1 + 2.6>:2 5 21.3 2.6x 1 + 1.6x 2
x 1 ,x 2
:::
One verifies that Xo =
0 (5.26~,
0) is a solution
to
this
problem.
Applying (20b) we can find pI, the probability that the i-th
constra-
2~
int is "violated". For instance 2F- 1 (w) + 22.23
~p
$
Fd
-1 5.77 - 2F (w)
F- 1 (w)
F
$
~Pr(a1x ~b1) (w)
1
a
10.59+6F- 1 (w)
F
2F- 1 (W) - 5.77
F- 1 (w)
$
F
$
g
where Fd = -10.94, F A = -4.1155, Fg = -1.78. In ather wards PI e (0, 0.03754] and the mast plausible value
of
-1
P1 is F (~4.1155) = 0.00003. Moreover one can verify that -1 Cr(P1 $ 1-F (2.5» = 0.6. Proceeding in the same way we find that P2 = (F- 1 (-10.4), F- 1
