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New model with T-fuzzy variations in linear programming
FU|ZY
sets and systems ELSEVIER
Fuzzy Sets and Systems 78 (1996) 289 292
New model with T-fuzzy variations in linear programming Cao Bing-yuan Department of Mathematics, Changsha University of Electric Power, Hunan, 410077 China
Received December 1993; revised December 1994
Abstract A programming problem with T-fuzzy variations has been widely used in the realistic world because it contains more information than a classical programming. Under cone index J, the author in the paper turns T-fuzzy variations into non-T-fuzzification, so that he turns a linear programming with T-fuzzy variations into a parametric linear programming problem, depending on cone index J, before he can obtain many corresponding properties about a linear programming problem with T-fuzzy variations such as existence of an optimal solution, its determining condition, and duality property. With a numerical example, he verifies the effectiveness of this model and its algorithm.
Keywords: T-fuzzy variation; Non-T-fuzzification; Linear programming; Cone index; Parametric optimal solution
1. Introduction
mm
C3~
It is of great practical value to build various models with T-fuzzy variations since a T-fuzzy number contains more information than an ordinary real one. In [1, 2], the author built a regression and self-regression forecasting model. In [3] he first developed an i n p u t - o u t p u t model with T-fuzzy data and obtained a consumption coefficient after getting a solution to a linear programming (LP) with T-fuzzy data. In this paper, on the basis of some property on T-fuzzy numbers, he solves a p r o g r a m m i n g problem, by the ideal method mentioned in [3], builds an L P problem with T-fuzzy variations, and provides a new approach to its optimal and dual optimal solutions.
s.t.
A)~ ~< b,
2. The LP problem with T-fuzzy variations Definition 1. Let the fuzzy L P problem be
(1)
where C is a real 1 x n matrix, A a real m x n matrix, )~, 0 are two real fuzzy n-vectors, and b" is a real fuzzym-vector. If X, b and (~ are T-fuzzy data defined in [2, 3], i.e. X ~-(Xl,)~2 . . . . ,Xn) T, here Xi=(Xi,~i,~i)T, /~ = (1, i, 1--)T,0 = (0, 0, 0)T, (1) is called an L P problem with T-fuzzy variations, written (r').
Definition 2. Let ~p = (xpl,xp2 . . . . ,.¢:pN)T (p = 1, .... m). We partition the set of natural numbers { 1,2 . . . . . n} into two exhaustive, mutually exclusive subsets J( + ) and J( - ), one of which may be an empty set. Each partition associates a binary multi-
C. Bing-vuan / Fuzzy Sets and Systems 78 (1996) 289 292
290
index J = (J1,J2 . . . . . J,) defined by {~ Jr=
if p c J ( + ) , if p e J ( - ) .
Especially, J0 = (0,0 . . . . . 0), J l = (1, 1. . . . . 1).
Theorem 1. Let the L P problem be given from Tfuzzy variations like (P). Then (t)) is equivalent to (r)(J)) for a given cone index J, and (P(J)) has an optimal solution depending on cone index J, which is equivalent to ([)) with a.fuzzy optimal one.
F r o m the equivalence of (~') and (F'(J)), we k n o w that (F'(J)) has an optimal solution depending on cone index J, which is equivalent to (P) with an optimal T-fuzzy solution. [] T h e o r e m 1 shows us that ([)) can be turned into an ordinary parametric linear p r o g r a m m i n g (P(J)) depending on cone index J, where (P(J)) has m a n y solutions and its optimal solution can be found in any works on LP,
3. Dual problem Proof. Let ~v = (.~pl,ff.p2 . . . . . .~pn)T be a column T-fuzzy variation tallying with (P), where Xpi:(Xi,~pi,~pi)T (p= l ..... m ; i = 1. . . . . n), We classify vectors of the column by subscripts, and might as well let p = 1. . . . . M correspond to a smaller fluctuating variation, while the other variations correspond to p = M + 1. . . . ,3M. Then to i = 1. . . . . N and each p,
The LP p r o b l e m with T-fuzzy variations always has a dual L P p r o b l e m with T-fuzzy variations corresponding to it. Since a close connection exists between the prime problem and the dual one, we can find the latter m o r e easily than the former. Because (p(J))
Uvi = x, + (g_v, + Q,)/2;
rain i=1
toi=N+
~a~i(xi+
s.t.
Uvi
SXi -- ~pi, j p = O , xi + E.,,i, i,, = 1.
i=1
xi~>0
to i = 2N + 1. . . . . 3N and each p,
Uv i=
p=l
1. . . . . 2N and each p,
xi + %pi, .iv=O, xi -- ~_vi, .],,= 1.
~'vgl)~bk(J),
(2)
p~l
( i = 1. . . . . n ; k =
1,...,m),
where g'pi is + ~'vi or + > Substitute x'i = xi + Y~=l g'vi/l, and then we might as well let xi >1Y~= 1 ~.'pi/l, and turn (2) into --
--
%pi.
n
min Then, under a given cone index J, (P) is changed into
~ ci xl i=l
s.t.
~ aki x'i ~ &(J), i=1
(P(J))
min
s.t.
p=l
~ aki(~ i=1
xi/>0
Upjl
~ c, i=1
Upi/l)<.~ok(J)
p=l
(i=l ..... n;k=
rain
CX'
s.t.
AX' ~ or(J),
(3)
X'>~0.
(k = 1,2 . . . . . m),
while the dual form of (3) is
U~>0.
max
Yg(J)
s.t.
ATY >~C,
where U is n-vector and ~k(J) are numbers depending on index J.
1,2 . . . . . m).
Y~>O.
(4)
C. Bing-yuan / Fuzzy Sets and Systems 78 (1996) 289-292
Theorem 2. Suppose the L P problem (P) is deduced from T-fuzzy variations. Its dual form is max
17E
s.t.
A x I7 ~> ~,
(5)
291
Then ~o, f o are optimal T-fuzzy solutions of (P) and (5), respectively.
4. The numerical example Example. Find
and (15) has an optimal T-fuzzy solution which is equivalent to (5) with an optimal T-fuzzy one, and (15) has the same optimal T-fuzzy values as (5). Proof. As (15) can be changed into (15(J)) under a given cone index J, and the dual form of (15(J)) is equivalent to (4), (5) can be changed into (4) under cone index J above. Again, (i5) is known to be mutual dual with (5) because of the equivalence of (15) and (15(J)), (5) and (4), and the mutual duality of (15(J)) and (4). Again, (15(J)) and (4) are, respectively, an ordinary prime L P and a dual L P depending on the same cone index J. As for (15(J)) and (4), applying Theorem 2 in [4, Section 4.2], we know that if one of them has an optimal solution, so has the other and they contain the same optimal values, therefore the theorem holds from an arbitrary of cone index J. []
Theorem 3. Suppose that L P problem (P) is deduced from T-fuzzy variations. Then the necessary and sufficient condition that dual programmin9 (15) and (5) have optimal T-fuzzy solutions is that they have T-fuzzy feasible ones at the same time. Proof. Necessity is obvious and sufficiency is proved as follows. ([') can be changed into (['(J)), (5) into (4) under cone index J, and (P(J)) is equivalent to (4) under the same cone index J, so, in a way similar to Theorem 1 in [4, Section 4.2], we can prove necessary and sufficient condition that (P(J)) and (4) have feasible solutions depending on cone index J is that they have optimal solutions depending on cone index d. Again, according to the equivalence of ([') and (P(J)), (5) and (4), and the duality of (P) and (5), we know the theorem holds. [] Corollary. If .~o and 170 are feasible T-fuzzy solutions of (P) and (5), respectively, with ~ o = 17o~.
max
(33~1 -- "~2)
s.t.
2"~1 -- 3~2 ~ ~[,
where 2 = (2, O, O)x,
;1 "--<74,
where 74 = (4, O, O)r,
; 1 , ; z >~),
where 0 = (0, O, 0)i..
and give T-fuzzy data of a column xl"
1. (Xl,0.5,1.2)T
X2: 4.
(X2,0,0.4)T
2. (Xl,0.8,1)T
5. (X:,0.6,1)T
3. (Xl, 1, 1.4)T
6. (Xz, 1.5,0.9)T.
(i) Number the data by the aid of 1-6. Group the data into three parts from Definition 2: I, No. 1, 4; II, No. 2, 5;j2 = 0,j5 = 1; III. No. 3, 6;j~ = 1 , j 6 = 0. Here jp = 1 for odd numbers and jp = 0 for even numbers. (ii) Non-T-fuzzification. Let Xl,X2 be [-(X 1 -}- 0.85) q- (X 1 -- 0.8) + (X 1 nc 1.4)]/3
= xl + 0.483, [(x2 + 0.2) + (x2 + 1) + (x2 -- 1.5)]/3 = x2 -- 0.1. (iii) Obtain a programming problem corresponding to (2) as follows: max
(3Xl - x2 + 1.55)
s.t.
2xl - x2 + 1.07 ~ 2, Xa + 0.483 ~< 4,
Xl,X2 >~O. max
(3xi - x2 + 1.55)
s.t.
2xl - x2 ~< 0.93, xl ~< 3.52,
xl,x2 >~O. The optimal solution depending on cone index J is x l ( J ) = 3.52, x 2 ( J } = 6.11, and the optimal
292
C Bing-yuan / Fuzzy Sets and Systems 78 (1996) 289-292
value is 6.00. If xx (J) stands for expensive resource then x z ( J ) stands for cheap resource. Decrease x l ( J ) and increase x2(J) properly, and the optimal value we obtain is the same as with noncrisp case. Obviously it decreases its cost.
LP. Meanwhile, he extends the m e t h o d mentioned in this paper to a geometric programming, and this will be discussed in another paper.
References 5. Conclusion The a u t h o r builds theoretically a new L P model on the basis of T-fuzzy numbers, studies its dual form, n o n - T - fuzzifies it under cone index J, and turns an L P with T-fuzzy variations into an L P depending on cone index J. U n d e r such a theoretical state, he can transplant m a n y results for the
[1] B.Y. Cao, Study for a kind of regression forecasting model with fuzzy datums, J. Math. Statist. Appl. Probab. 4(2)(1989) 182 189. [2] B.Y. Cao, Study on non-distinct self-regression forecast model, Kexue Tongbao 34(17) (1989) 1291-1294. (Also in Chinese Sci. Bull. 35(13) (1990) 1057-1062.) [3] B.Y. Cao, Input-~output mathematical model with T-fuzzy data, Internat. J. Fuzzy Sets and Systems 59(1)(1993) 15-23. [4] M.G. Guan and H.D. Zheng, Linear Programming (Shandong Science and Technology Press, 1983).
sets and systems ELSEVIER
Fuzzy Sets and Systems 78 (1996) 289 292
New model with T-fuzzy variations in linear programming Cao Bing-yuan Department of Mathematics, Changsha University of Electric Power, Hunan, 410077 China
Received December 1993; revised December 1994
Abstract A programming problem with T-fuzzy variations has been widely used in the realistic world because it contains more information than a classical programming. Under cone index J, the author in the paper turns T-fuzzy variations into non-T-fuzzification, so that he turns a linear programming with T-fuzzy variations into a parametric linear programming problem, depending on cone index J, before he can obtain many corresponding properties about a linear programming problem with T-fuzzy variations such as existence of an optimal solution, its determining condition, and duality property. With a numerical example, he verifies the effectiveness of this model and its algorithm.
Keywords: T-fuzzy variation; Non-T-fuzzification; Linear programming; Cone index; Parametric optimal solution
1. Introduction
mm
C3~
It is of great practical value to build various models with T-fuzzy variations since a T-fuzzy number contains more information than an ordinary real one. In [1, 2], the author built a regression and self-regression forecasting model. In [3] he first developed an i n p u t - o u t p u t model with T-fuzzy data and obtained a consumption coefficient after getting a solution to a linear programming (LP) with T-fuzzy data. In this paper, on the basis of some property on T-fuzzy numbers, he solves a p r o g r a m m i n g problem, by the ideal method mentioned in [3], builds an L P problem with T-fuzzy variations, and provides a new approach to its optimal and dual optimal solutions.
s.t.
A)~ ~< b,
2. The LP problem with T-fuzzy variations Definition 1. Let the fuzzy L P problem be
(1)
where C is a real 1 x n matrix, A a real m x n matrix, )~, 0 are two real fuzzy n-vectors, and b" is a real fuzzym-vector. If X, b and (~ are T-fuzzy data defined in [2, 3], i.e. X ~-(Xl,)~2 . . . . ,Xn) T, here Xi=(Xi,~i,~i)T, /~ = (1, i, 1--)T,0 = (0, 0, 0)T, (1) is called an L P problem with T-fuzzy variations, written (r').
Definition 2. Let ~p = (xpl,xp2 . . . . ,.¢:pN)T (p = 1, .... m). We partition the set of natural numbers { 1,2 . . . . . n} into two exhaustive, mutually exclusive subsets J( + ) and J( - ), one of which may be an empty set. Each partition associates a binary multi-
C. Bing-vuan / Fuzzy Sets and Systems 78 (1996) 289 292
290
index J = (J1,J2 . . . . . J,) defined by {~ Jr=
if p c J ( + ) , if p e J ( - ) .
Especially, J0 = (0,0 . . . . . 0), J l = (1, 1. . . . . 1).
Theorem 1. Let the L P problem be given from Tfuzzy variations like (P). Then (t)) is equivalent to (r)(J)) for a given cone index J, and (P(J)) has an optimal solution depending on cone index J, which is equivalent to ([)) with a.fuzzy optimal one.
F r o m the equivalence of (~') and (F'(J)), we k n o w that (F'(J)) has an optimal solution depending on cone index J, which is equivalent to (P) with an optimal T-fuzzy solution. [] T h e o r e m 1 shows us that ([)) can be turned into an ordinary parametric linear p r o g r a m m i n g (P(J)) depending on cone index J, where (P(J)) has m a n y solutions and its optimal solution can be found in any works on LP,
3. Dual problem Proof. Let ~v = (.~pl,ff.p2 . . . . . .~pn)T be a column T-fuzzy variation tallying with (P), where Xpi:(Xi,~pi,~pi)T (p= l ..... m ; i = 1. . . . . n), We classify vectors of the column by subscripts, and might as well let p = 1. . . . . M correspond to a smaller fluctuating variation, while the other variations correspond to p = M + 1. . . . ,3M. Then to i = 1. . . . . N and each p,
The LP p r o b l e m with T-fuzzy variations always has a dual L P p r o b l e m with T-fuzzy variations corresponding to it. Since a close connection exists between the prime problem and the dual one, we can find the latter m o r e easily than the former. Because (p(J))
Uvi = x, + (g_v, + Q,)/2;
rain i=1
toi=N+
~a~i(xi+
s.t.
Uvi
SXi -- ~pi, j p = O , xi + E.,,i, i,, = 1.
i=1
xi~>0
to i = 2N + 1. . . . . 3N and each p,
Uv i=
p=l
1. . . . . 2N and each p,
xi + %pi, .iv=O, xi -- ~_vi, .],,= 1.
~'vgl)~bk(J),
(2)
p~l
( i = 1. . . . . n ; k =
1,...,m),
where g'pi is + ~'vi or + > Substitute x'i = xi + Y~=l g'vi/l, and then we might as well let xi >1Y~= 1 ~.'pi/l, and turn (2) into --
--
%pi.
n
min Then, under a given cone index J, (P) is changed into
~ ci xl i=l
s.t.
~ aki x'i ~ &(J), i=1
(P(J))
min
s.t.
p=l
~ aki(~ i=1
xi/>0
Upjl
~ c, i=1
Upi/l)<.~ok(J)
p=l
(i=l ..... n;k=
rain
CX'
s.t.
AX' ~ or(J),
(3)
X'>~0.
(k = 1,2 . . . . . m),
while the dual form of (3) is
U~>0.
max
Yg(J)
s.t.
ATY >~C,
where U is n-vector and ~k(J) are numbers depending on index J.
1,2 . . . . . m).
Y~>O.
(4)
C. Bing-yuan / Fuzzy Sets and Systems 78 (1996) 289-292
Theorem 2. Suppose the L P problem (P) is deduced from T-fuzzy variations. Its dual form is max
17E
s.t.
A x I7 ~> ~,
(5)
291
Then ~o, f o are optimal T-fuzzy solutions of (P) and (5), respectively.
4. The numerical example Example. Find
and (15) has an optimal T-fuzzy solution which is equivalent to (5) with an optimal T-fuzzy one, and (15) has the same optimal T-fuzzy values as (5). Proof. As (15) can be changed into (15(J)) under a given cone index J, and the dual form of (15(J)) is equivalent to (4), (5) can be changed into (4) under cone index J above. Again, (i5) is known to be mutual dual with (5) because of the equivalence of (15) and (15(J)), (5) and (4), and the mutual duality of (15(J)) and (4). Again, (15(J)) and (4) are, respectively, an ordinary prime L P and a dual L P depending on the same cone index J. As for (15(J)) and (4), applying Theorem 2 in [4, Section 4.2], we know that if one of them has an optimal solution, so has the other and they contain the same optimal values, therefore the theorem holds from an arbitrary of cone index J. []
Theorem 3. Suppose that L P problem (P) is deduced from T-fuzzy variations. Then the necessary and sufficient condition that dual programmin9 (15) and (5) have optimal T-fuzzy solutions is that they have T-fuzzy feasible ones at the same time. Proof. Necessity is obvious and sufficiency is proved as follows. ([') can be changed into (['(J)), (5) into (4) under cone index J, and (P(J)) is equivalent to (4) under the same cone index J, so, in a way similar to Theorem 1 in [4, Section 4.2], we can prove necessary and sufficient condition that (P(J)) and (4) have feasible solutions depending on cone index J is that they have optimal solutions depending on cone index d. Again, according to the equivalence of ([') and (P(J)), (5) and (4), and the duality of (P) and (5), we know the theorem holds. [] Corollary. If .~o and 170 are feasible T-fuzzy solutions of (P) and (5), respectively, with ~ o = 17o~.
max
(33~1 -- "~2)
s.t.
2"~1 -- 3~2 ~ ~[,
where 2 = (2, O, O)x,
;1 "--<74,
where 74 = (4, O, O)r,
; 1 , ; z >~),
where 0 = (0, O, 0)i..
and give T-fuzzy data of a column xl"
1. (Xl,0.5,1.2)T
X2: 4.
(X2,0,0.4)T
2. (Xl,0.8,1)T
5. (X:,0.6,1)T
3. (Xl, 1, 1.4)T
6. (Xz, 1.5,0.9)T.
(i) Number the data by the aid of 1-6. Group the data into three parts from Definition 2: I, No. 1, 4; II, No. 2, 5;j2 = 0,j5 = 1; III. No. 3, 6;j~ = 1 , j 6 = 0. Here jp = 1 for odd numbers and jp = 0 for even numbers. (ii) Non-T-fuzzification. Let Xl,X2 be [-(X 1 -}- 0.85) q- (X 1 -- 0.8) + (X 1 nc 1.4)]/3
= xl + 0.483, [(x2 + 0.2) + (x2 + 1) + (x2 -- 1.5)]/3 = x2 -- 0.1. (iii) Obtain a programming problem corresponding to (2) as follows: max
(3Xl - x2 + 1.55)
s.t.
2xl - x2 + 1.07 ~ 2, Xa + 0.483 ~< 4,
Xl,X2 >~O. max
(3xi - x2 + 1.55)
s.t.
2xl - x2 ~< 0.93, xl ~< 3.52,
xl,x2 >~O. The optimal solution depending on cone index J is x l ( J ) = 3.52, x 2 ( J } = 6.11, and the optimal
292
C Bing-yuan / Fuzzy Sets and Systems 78 (1996) 289-292
value is 6.00. If xx (J) stands for expensive resource then x z ( J ) stands for cheap resource. Decrease x l ( J ) and increase x2(J) properly, and the optimal value we obtain is the same as with noncrisp case. Obviously it decreases its cost.
LP. Meanwhile, he extends the m e t h o d mentioned in this paper to a geometric programming, and this will be discussed in another paper.
References 5. Conclusion The a u t h o r builds theoretically a new L P model on the basis of T-fuzzy numbers, studies its dual form, n o n - T - fuzzifies it under cone index J, and turns an L P with T-fuzzy variations into an L P depending on cone index J. U n d e r such a theoretical state, he can transplant m a n y results for the
[1] B.Y. Cao, Study for a kind of regression forecasting model with fuzzy datums, J. Math. Statist. Appl. Probab. 4(2)(1989) 182 189. [2] B.Y. Cao, Study on non-distinct self-regression forecast model, Kexue Tongbao 34(17) (1989) 1291-1294. (Also in Chinese Sci. Bull. 35(13) (1990) 1057-1062.) [3] B.Y. Cao, Input-~output mathematical model with T-fuzzy data, Internat. J. Fuzzy Sets and Systems 59(1)(1993) 15-23. [4] M.G. Guan and H.D. Zheng, Linear Programming (Shandong Science and Technology Press, 1983).