fuzzy and stochastic programming
TRANSCRIPT
Fuzzy Sets and Systems 22 (1987) 171-180 171 North-Holland
F U Z Z Y A N D STOCHASTIC P R O G R A M M I N G
A.V. YAZENIN Department of Algorithmic Languages, The University of Kalinin, Kalinin, USSR
Received December 1985
Fuzzy mathematical programming is considered on the basis of the concept of fuzzy variable [5] that is one of the most complete patterns of fuzziness. Statements of problems under consideration are analogous to those in stochastic programming. Equivalent precise analogues are derived for these problems. The relationship between fuzzy programming and stochastic programming is determined.
Keywords: Fuzzy mathematical programming, Stochastic programming, Fuzzy variable.
1. Introduction
Recently several publications have been dedicated to the problem of fuzzy mathematical programming. Every possible statement of problems of fuzzy mathematical programming and means of their analysis, working out the level of the problem on the whole, are rather fully described in [1, 2, 4, 6, 7, 9-11, 13, 14] and others.
However, it should be pointed out that all above-mentioned works have not got a single methodical basis. This is probably connected with the fact that there is no conventional mathematical model of fuzziness yet. One of the most complete models of fuzziness, to our mind, is proposed by Nahmias [5]. He introduced the concept of a fuzzy variable as a function X:F--*R, where F is a set with a nonnegative weight o defined on it and satisfying
(i) o(t~)= 0, o(F)= 1, and (ii) o(U~,~AA~) = sup~,~a o(A~) for every sub-family {A~; a~ • A} of P(r),
where P(F) is the set of all subsets of F. A membership function of a fuzzy variable X is determined in the following
way:
I~x(X) = o(X-l{x}) , Yx •R .
It is obvious that 0 ~< ~x(X) ~< 1 for every x • R and supx~x I~x(X) = 1. Normal fuzzy variables are of special interest. The main result of [5] is the
following. Let X1 . . . . . X. be mutually unrelated normal fuzzy variables with parameters (al, bl) . . . . , (a,, b,) and let c l , . . . , c ,~ :0 be scalars. Then Z = Y~'=lciX,- is a normal fuzzy variable with parameters a = P,'d=iciai and b = Y~.=x cib~. This result plays a basic part in the paper.
0165-0114/87/$3.50 ~) 1987, Elsevier Science Publishers B.V. (North-Holland)
172
2. Main results
A . V . Y a z e n i n
Consider the following problem of fuzzy mathematical programming:
o{fo(x, y) = 0} ~ max, (1)
o{fi(x, y )=O}>-oG i = 1 . . . . . m, x • X, (2)
where a~, • I = [0, 1] are given powers of a membership (of fuzziness), and y • F, X c R~_. We shall consider the situations:
(I) f ( x , V) = E'2=a aoxj - bi(y), where only bi is a fuzzy variable, i = 0 . . . . . m; (II) f/(x, y) = Ej%, a i j ( ] t ) x j - - b i ( ] l ) , where b i and a 0 are fuzzy variables. The problem (1), (2) is analogous in form to the stochastic programming
problem
P{fo(x, ~o) <. 0} ~ max, (3)
P{fii(x, ~o) <~ 0} >~Pi, i = 1 . . . . . m, x ~ X, (4)
where p~ • I and to is a state of nature that is a simple event of the probability space (g2, ~ , P). The problem (3), (4) is well studied [3], [12]. Consider the first situation. We have:
Theorem 1. Let membership functions #b, be nondecreasing. Then the problem (1), (2) has an equivalent precise analogy o f the following form:
aojxj---~ max, (5) ]=1
~ % x j > - r i , i = 1 . . . . . m, j=, (6) x • X ,
where ri = inf, {lAb,(t) t> 0~'i}.
Proof. From the determination of the fuzzy variable [5] it follows
o (x, = 0 } = o • = a,jxj j = l
= lAbi ai i • j = l i
In this case the problem (1), (2) is identical to
#bo ao~ --~ max, j = l
x,) lab, ai i >i oli, i = 1 , . . . , m, /'=1
x • X .
Fuzzy and stochastic programming 173
The membership functions under conditions of the theorem being nondecreasing, the last problem can be substitued by a more simple one:
~ aosXj---> max, j = l
~ aisxj >t ri, i = l . . . . . m, j = !
x E S .
This completes the proof.
Remark 1. If membership functions of fuzzy variables b i a r e strictly increasing, constants ri can be determined by solving the equations:
/Zb,(t)=tri, i = l , . . . , m .
Remark 2. If membership functions of fuzzy variables b~ are unimodal, it is not difficult to prove that the equivalent precise analogy of the problem (1), (2) takes the form:
l~bo aosxj --~ max, (7) j = l
ri<<-~aijx s<<-Ri, i = 1 . . . . . m, (8) j = l
where x EX,
r /= inf {/Xb~(t) >/O{i}, Ri = sup {/Xb,(t)/> 0¢i}. t t
Consider the second situation which is more common. If factors ais and bi are random values with normal distribution, one succeeds in constructing an equivalent determined analogy of the problem (3), (4). A similar result holds in fuzzy programming.
Theorem 2. Let aij, bi be unrelated normal f u z z y variables [5] with parameters (ais, c~is), (~i, c~i) respectively, i = 0 . . . . . m; j = 1 . . . . . n. Then the prob lem (1), (2) has the fo l lowing equivalent precise analogy:
(fo(x, ~,)-mo(x))tdo(x)(mo(x ) / do(X ) ) "---> max, (9)
r i <~ mi(x ) ld i ( x ) <~ R,, i = 1 . . . . . m, (10)
X E X , where
mi(x ) = ~ aijxj -- ~i, ]=1
di(x ) = ~ •ijx j dp 2i ~ const . > 0 , j=l
and ri, Ri are the least and the largest roots o f the equation e -'2 - o~i = O.
174 A.V. Yazenin
Proof. We have
o7,(x, r ) = o} = a{r • r:fi(x, r ) = o} = ~ x , r ~ ( 0 ) .
Since #c1,(x, r)( z ) = I~1,(x, r)( z / c ) it follows
~ ( x . ~ ) ( O ) = ~r,(x,~),~,(x)(O).
Introduce the fuzzy variable
q,(y) = ~ ( x , y ) - m , ( x ) ) / d , ( x ) .
In accordance with the results of the work [5] the fuzzy variable qi is normal. Its parameters are (0, 1). In fact
~(fii(x,y)_mi(x))/di(x)(Z ) = I~A(x,r)_m,(x)(zdi(x ) ) = o { y • F:fi i(x, y ) - m i ( x ) = z d i ( x ) }
= £r{~i(X , •) = zdi(x ) -t- mi(x)}
= #1,e~,r)(zdi(x) + m i ( x ) ) .
Since f/(x, y) is a normal fuzzy variable with parameters (m i ( x ) , d i (x ) ) , it follows
#i , (x .r)(zdi(x) + m i ( x ) ) = e x p ( - ( z d i ( x ) + me(x ) -- mi(x))2/d2i(x)) = exp(-z2).
The fuzzy variable qi being a normal fuzzy one, it is completely determined by its parameters, i.e. qi is independent upon x.
With this in mind, we get
O{fi(X, y ) = O } = o { fi(x' y ) - m i ( x ) - t - m i ( x ) ) - 0
= o{ fi(x' ~/) - mi(x ) mi(x)~ = - , u x ) J
= ~,~x,,)-m,<x)>,, , ,<x)(~ -- mi(x ) '~ di(x ) / "
Since
= ~,<.,,>.,<x)~,~,<x)(-mi(x)ldi(x)), it follows
o { f i ( x , Y) = O} = ~q,(r)(mi(x)/di(x)).
The restriction
l,tq,(r)(mi(x)/di(x)) >i Ol i
describes an oci-level subset of the fuzzy variable qi:
Q. , = {r • R I lZqi(r) ~ Oil}.
Fuzzy and stochastic programming 175
That is why it is an equivalent of the expression
ri <~ m, (x ) /d i (x ) <~ R,.
It follows that the problem (1), (2) has an equivalent precise analogy of the form
l-fifo(x, ~)-mo(x))/ao(x)(mo(x ) / do(x ) ) ~ max,
ri <<- mi(x) /d i (x) <<- Ri, i = 1 . . . . . m,
x e X .
This completes the proof.
The stochastic programming [12] covers as well the problem:
k---> min(max) (11)
subject to
P{fo(X, oJ) < =p0, P ~ ( x , to)~<O}~>pi, i = 1 . . . . . m, (12)
x E X .
Consider a problem of fuzzy mathematical programming which is analogous to the problem (11), (12):
k ~ min(max) (13)
under the restrictions
O{fo(x, y) = k} >I O~o,
a{fii(x, )') = 0} >~ a~i, i = 1 . . . . . m, (14)
x ~ X .
We will show that an equivalent precise analogy can be built for this problem too.
Theorem 3. Let
f,-(x, 7) = ~ ai/(7)xi - bi(7) i = 1
with aij, bi normal unrelated f u z zy variables. Then the problem (13), (14) has an equivalent precise analogy o f the following form:
k ~ min(max) (15)
subject to
k - mo(x) ro <~ ~ Ro, do(x)
mi(x) r i<<-~<<-Ri , i = 1 . . . . . m,
x e X .
(16)
176 A. V. Yazenin
P r o o f . We have
, ,go(X, r ) = !,} = o{fo(x, r ) - / , = 0}.
As before denote the parameters of the normal fuzzy variable f~(x, y) by mi(x) and di(x) respectively. It is obvious that
Ittfo(x,r)-k)/do(x)(O) = tZA(x,r)-k(O), do(x) >i const. > O.
Then we follow the proof of the Theorem 2. Introduce a fuzzy variable
fi(x, y) - mi(x) qi(~') - , di(x) >! const. > 0.
d,(x)
This fuzzy variable is normal and its parameters are (0, 1) for every x, i.e. it is independent on x. Therefore
a~f0(x, r ) - k = o} = ~y0(x, ._k(0)
= UOro(X, y)-mo(x)+mo(x)-k)/do(x)(O)
"~ ~.L(fo(x. y)_mo(x))/do(x)( k -~ m ° ( x ) ~ do(x) /"
Determine values ro, Ro:
ro = inf {#qo(t) = Cro}, Ro = sup {#qo(t) = ~ro}. t t
The restriction #qo(mo(x)/do(x))>t oco describes an C~o-level subset by a fuzzy variable qo:
Q~o = {r e g ] #qo(r) ~ O~0}.
That is why it is an equivalent of the inequality
ro <- (k - mo(x)) /do(X) <~ Ro.
The rest of restrictions are done as with the proof of the Theorem 2. This completes the proof.
Consider a problem which is somewhat unlike the problem (13), (14):
k--+ min(max) (17)
subject to
o{fo(x, r ) = k} = O~o,
a{fi(x, y) = 0} >i oq, i = 1 . . . . . m, (18)
x E X .
Using the procedure of the proof of the Theorem 3 we get:
O(fo(X ' )t) -- k = 0 } = [~fo(x,y)(k) = [.Lqo((k - mo(x) ) /do(x ) ) = a:o.
Fuzzy and stochastic programming
Since the fuzzy variable is a normal one with parameters (0, 1), we have
t~qo((k - mo(X)) /do(x) ) = exp( (k - mo(x))2] = d~(x) / Olo.
177
Solving the latter equation, we obtain two expressions for k:
k l ( x ) = mo(X) + rodo(x) and k2(x) = too(X) + Rodo(x),
where to, R0 are the roots of the equation e -'2 - a~o -- 0, r0 < Ro. In this case the procedure of the selection of the optimum solution is as shown. Solve problems
kl(x)--~ min(max) and kE(X)--~ min(max)
subject to
ri <~ mi ( x ) /d i ( x ) ~ Ri, i = 1 . . . . . m,
x E X .
Let i , ~ be solutions of these problems. Then an optimum solution is x* • {i, ~} such that
x* = arg min [min{km(x), k2(x)}] X
o r
x* = arg max [max{kl(x), k2(x)}]. X
Remark 3. In the problem (15), (16), k, x ~ , . . . , xn are variable parameters.
3. Comparative analysis of models of fuzzy programming and stochastic programming
In this section we shall show an analogy between fuzzy programming and stochastic programming. Using results of the above section and known results of stochastic programming [3, 12], we can give Table 1 which yields a connection between fuzzy and stochastic programming.
Table 1 shows that the main difference between problems of fuzzy program- ming and stochastic programming follows from the fact that:
(1) the variance of a random value fi(x, to) is a quadratic function and the fuzziness characteristic of a normal variable f~(x, y) is a linear function;
(2) a distribution function is nondecreasing and a membership function is not necessarily so.
If a set X is defined by simultaneous linear restrictions, the problems (1), (2) and (3), (4) have linear equivalent 'determined' analogues in the first case. In the second case the problem (1), (2) is an equivalent of a problem of a nonlinear programming; the problem (3), (4) is an equivalent of a quadratic programming problem. The problems (11), (12) and (13), (14) have problems of quadratic programming and linear programming as their 'determined' equivalent analogy respectively. One can draw the inference that in terms of numerical realization models, fuzzy mathematical programming is more preferable.
178 A.V. Yazenin
Table 1
Stochastic programming Fuzzy programming
The problem (3), (4)
(I) f/(x, to) = ~ aOx i - bi(~o ). 1=1
An equivalent determined analogy [3]:
~_. aojx j --~ rain, j = l
~_ .a~;~r i , i = l . . . . . m, /=1
x EX,
where r i = sup {/-//(t) ~< 1 -p i} ,
H i is a distribution function of random value by
(n ) fi(x, co) = ~ a,j(~o)xj - b,(o~). j = l
Factors %, bi are independent normal random values.
An equivalent determined analogy [3]:
m°( x ) --* min ,
~ < r ~ , . . . , m, i 1,
x E X ,
mi(x), di(x ) are mathematical expectation and variance of random value f/(x, ¢o).
The problem (11), (12). An equivalent determined analogy [12]:
k = mo - ro doV~-)-'-~ min(max)
~ < = r i , i = 1 , m, Vd~(x) . . . . .
x ~ X .
The problem (1), (2)
(I) f/(x, y) = ~ ai;xj - bi(Y). j=l
Conditions of Theorem 1 are satisfied. An equivalent precise analogy:
~_~ ao/xf--~ max, j=1
~ ai/x/ >~ ri, i = I . . . . . m, 1=1
x e X ,
where ri = inf {/~bi(t) I> ~i}, t
#b~ is a membership function of fuzzy variable b i.
n
(II) fi(x, y) = ~] %(y)xj - bi(y). 1=1
Factors %, b i are unrelated normal fuzzy variables.
An equivalent precise analogy:
/ mo(x ) ~__, l~ . , r )_ .o (x ) ) /ao . )~d- -~] max,
_< mi(x) r i ~ d ~ R i , i = 1 . . . . . m,
x E X ,
mi(x ), di(x) are modal value and fuzziness char- acteristic of fuzzy variable f/(x, y).
The problem (13), (14). An equivalent precise analogy:
k ~ min(max),
k - mo(x ) _< u ro <~ do(x ) ~-,o,
mAx) ~ r i ~ d - ~ R i , i= 1 . . . . . m,
x E X .
4. Numerical example
We shall illustrate one of the problems of fuzzy mathematical programming with a numerical example. Let us consider
k---~ max,
Fuzzy and stochastic programming 179
cr{aolxl + ao.zX2 = k} >I 13,
o{allxl + a12x2 = bl} >I ~,
Xl, x2 I> 0.6.
Coefficients ao~, ao2, al~, a12, bl are normal fuzzy variables with parameters (1, 2), (2, 3), (1, ½), (1, ½), (2, ½) respectively. Therefore
too(X) = Xl + 2x2, do(x) = 2Xl + 3x2,
m l ( x ) = Xl + x 2 - 2, d l ( x ) 1 1 1 = 2X1 ÷ 2 X 2 ÷ 2-
Determine values ro, R0, rl, R~ by solving the equations:
e- := 13, e - := l . As a result we have
r0 = -1.04, Ro= 1.04, rl = -1.18, RI= 1.18.
In accordance with Theorem 3 the precise analogy consideration has the following form:
k---> max,
k -x~ - 2x2< --1.04 ~ 1.04,
2Xl + 3x2
-1.18~<- x-!1 + x 2 - 2 ~< 1.18, + ½x2 + ½
X l , X 2 ~ 0 . 6 .
Carrying out pertinent transformations, we programming:
k--* max subject to
0~<k + 1.08xl + 1.12x2,
O~ < - k + 3.08Xl + 5.12x2, 1.41 ~< 1.59x~ + 1.59X2,
0.41x~ + 0.41x2 ~< 2.59,
Xl , X 2 ~ 0.6.
The optimal solution of this problem is
k =31.11, x~ =0.6, X2=5.71.
of the problem under
have the problem of linear
5. Conclusion
In the paper some statements of the problems of fuzzy mathematical programming have been considered. A mathematical basis of their investigation is
180 A. V. Yazenin
the model of fuzziness introduced by [5]. A comparative analysis of models of fuzzy programming and stochastic programming has been carried out, revealing an analogy between them. The suitability of applying a stochastic model or a fuzzy model might be determined in the practical solution of a concrete applied problem in the process of revealing the nature of uncertainty.
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