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Computers and Mathematics with Applications 37 (1999) 63-76

Linear Programming with Fuzzy Coefficients in Constraints S . - C . FANG* Operations Research and Industrial Engineering North Carolina State University Raleigh, NC 27695-7906, U.S.A.

C.-F. Hu Department of Applied Mathematics, I-Shou University Kaohsiung, Taiwan, R.O.C.

H.-F. WANG Department of Industrial Engineering National Tsing Hua University Hsinchu, Taiwan, R.O.C.

S.-Y. W u Department of Mathematics National Cheng Kung University Tainan, Taiwan, R.O.C.

(Received March 1998; revised and accepted September 1998) A b s t r a c t - - T h i s paper presents a new method for solving linearprogramming problems with fuzzy coefficientsin constraints. It is shown that such problems can be reduced to a linear semi-infinite programming problem. The relations between optimal solutions and extreme points of the linear semi-infinite program are established. A cutting plane algorithm is introduced with a convergence proof, and a numerical example is included to illustratethe solution procedure. (~) 1999 Elsevier Science Ltd. All rights reserved. Keywords--Fuzzy

mathematical programming, Linear semi-infiniteprogramming.

1. I N T R O D U C T I O N This paper studies a linear programming problem with fuzzy coefficients in constraints. To describe this problem, we consider the following linear program in the conventional form [1]: maximize

c T x,

s.t.

Ax [-8, 1, 3], [-1, 1, 1Ix3 >_ [-10, 1, 2],

s.t.

Xl~X2,X

O,

3 >

which is equivalent to the following linear semi-infiniteprogramming problem: min

(LSIP) s.t.

- Xl - 2 x 2 - 2 x 3 ,

~ -20t3'

0

- 2 + t~

-t30

-t40

xl

|-11

+t2

x3X2 _> \[-5-8-- 2t43t3 ' Vt, E [a, 1],

X l , X 2 , X 3 ~ O.

Given any a E [0, 1], say a = 0.6 in this example and an arbitrary starting point, say t 1 = (t~,'t2, t3,1 t ~ ) = (0.7, 0.8, 0.7, 0 . 8 ) , w e h a v e a regular l i n e a r p r o g r a m min (LP 1)

s.t.

01/( )

-- Xl -- 2X2 -- 2X3,

- 4 + 2t~

- 2 + tl

0

0

0

0

Xl x2

t2

-2

-t~

/

> -

=3

-11 + t~ -5

-8-

-

3t~

'

2t,'

X l , X 2 , X 3 > O.

Solving (LP 1) results in an optimal solution x 1 = (x~ , x2, 1 x3) 1 = (0, 6.3846, 8.5). Define

v~ (tl) = ( - 4 + 2tl) =~ + ( - 2 + t,) =~ - ( , 9 + tl) = 5.3846tl - 3.7692, v~ (t2) -- ( - 2 + t 2 ) x ~ - ( - 1 1 + t2) = 7 . 5 t 2 -- 6, V 2 (t3) = --2t3X ] -- ~3 xl -- (--5 -- 353)

------3.3846t3 +

5,

V2(t4) = --t4X~ -- (--8 -- 2t4) -------6.5t4 "4- 8.

The minimizers of v~(tl), v~(t2), v2(t3), v~(t4) over [c~,1] = [0.6, 1] are 0.6,0.6,1,1, respectively. Hence, we choose t2 = (tl,2 t2, 2 t3, 2 t4) 2 = (0.6, 0.6, 1, 1).

Linear Programming

75

Since v2(t 2) _

+ t~ 0

-tl

- 9 + t~ - 1 1 + t~ - 5 - 3t32 - 8 - 2t 2

2:l,X2,X 3 ~ O.

Solving (LP 2) results in an optimal solution x2 = (xl,2 x2,2 x 2) = (0,6.0001,7.4287). Define

v~ (t~) = (-4 + 2t,) =~ + (-2 + t,) =~ - (-9 + t,) = 5.0001t, - 3.0003, v 3 (t2) = ( - 2 + t2) x32 - ( - 1 1 + t2) = 6.4287t2 - 3.8575,

v~ (ts) = - 2 t 3 = ~ - t~=~ - ( - 5 - 3t~) = - 3 . 0 0 0 1 t ~ + 5, v~ (t4) = - t 4 = ~ - ( - 8 - 2t4) = - 5 4287t4 + 8. T h e minimizers of v3(tl), v3(t2), v3(t3), v3(t4) over [0.6,1] are 0.6,0.6,1,1, respectively. Hence, we choose t = (t 3, t2,t3,3 3 t43) = (0.6, 0.6, 1, 1). Now, since v3(t~) >_ O, va(t 3) >_ O, v3(t~) > O, v3(t 3) >_ O, the algorithm stops and returns an o p t i m a l solution x* = x 2 = (0, 6.0001, 7.4287) to t h e fuzzy linear p r o g r a m (20) with (~ = 0.6.

7. C O N C L U D I N G

REMARKS

In this paper, a linear p r o g r a m m i n g p r o b l e m with fuzzy coefficients in A and b is studied. Based on the specific ranking of fuzzy numbers, we have shown t h a t such p r o b l e m s can be reduced to a linear semi-infinite p r o g r a m m i n g problem. T h e relationship between the o p t i m a l solutions and e x t r e m e points of (LSIP) are established. A cutting plane algorithm is proposed for solving a linear p r o g r a m m i n g p r o b l e m with fuzzy coefficients in t e r m s of linear semi-infinite p r o g r a m m i n g . O n l y those constraints which b e c o m e binding are generated and used in t h e solution procedure.

REFERENCES 1. S.-C. Fang and S.C. Puthenpura, Linear Optimization and Extensions: Theory and Algorithms, Prentice Hall, Englewood Cliffs, N J, (1993). 2. H.-J. Zimmermann, Fuzzy Set Theory and Its Applications, Kluwer Academic, Norwell, MA, (1991). 3. Y.-J. Lai and C.-L. Hwang, T~uzzyMathematical Programming: Methods and Applications, Springer-Verlag, Heidelberg, (1992). 4. H. Tanaka, H. Ichihashi and K. Asai, A formulation of fuzzy linear programming problems based on comparison of fuzzy numbers, Control and Cybernet 13, 185-194 (1984). 5. J. Ramfk and J. Rfm~nek, Inequality relation between fuzzy numbers and its use in fuzzy optimization, F~zzy Sets and Systems 16, 123-138 (1985). 6. M. Delgado, J.L. Verdegay and M.A. Vila, A general model for fuzzy linear programming, Fuzzy Sets and Systems 29, 21-29 (1989). 7. S.-J. Chen and C.-L. Hwang, Fuzzy Attribute Decision Making, Springer-Verlag, New York, (1992). 8. J.M. Adamo, Fuzzy decision trees, Fuzzy Sets and Systems 4, 207-219 (1980). 9. J.J. Buckley, A fast method of ranking alternatives using fuzzy numbers, Fuzzy Sets and Systems 30, 337-338 (1989). 10. K.S. Kretschmer, Programs in paired spaces, Canadian Journal of Mathematics 13, 221-238 (1961).

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11. E.J. Anderson, A review of duality theory for linear programming over topological vector spaces, Journal of Mathematical Analysis and Applications 97, 380-392 (1983). 12. E.J. Anderson and P. Nash, Linear Programming in Infinite-Dimensional Spaces: Theory and Applications, John Wiley & Sons, Great Britain, (1987). 13. A. Hettich and K. Kortanek, Semi-infinite programming: Theory, method and applications, SIAM Review 35, 380-429 (1993). 14. S.-C. Fang, J.R. Rajasekera and H.-S.J. Tsao, Entropy Optimization and Mathematical Programming, Kluwer Academic, Norwell, MA, (1997).