a differential equation approach to fuzzy non-linear programming problems

E L S E V I E R Fuzzy Sets and Systems 93 (1998) 57-61

FUZIY sets and systems

A differential equation approach to fuzzy non-linear programming problems

F a t m a M. Al i

Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt

Abstract

This paper deals with non-linear programming problems with fuzzy parameters (FNLPP). A non-linear autonomous system is introduced as the base theory instead of the usual approaches for solving (FNLPP). The relation between critical points and local s-optima of the original fuzzy optimization problem is proved. The asymptotic stability of the critical points is also proved. Finally, a numerical example is given to illustrate the theory developed in this paper. © 1998 Elsevier Science B.V.

Keywords: Fuzzy non-linear programming problem; Fuzzy parameters; Fuzzy number; Differential equations

1. Introduction

A differential equation approach is presented as a new method for solving equality constrained non- linear programming problems in [4]. Here we ex- tend this result to equality or inequality constrained non-linear programming problems with fuzzy parameters. Dubois and Parde [1,2-] and Sakawa and Yano [-6 8], have characterized the fuzzy parameters by fuzzy numbers. A non-linear autonomous differential system - fundamental equations - is introduced as the base theory instead of the usual approaches for solving (FNLPP) in Section 3. In Section 4 we discuss the relation between constrained a-optimal points of the a-non- linear programming problem and asymptotically stable critical points of an autonomous differential system associated with the optimization problem. In fact, we prove a necessary and sufficient condition that the constrained stationary point is a critical point of the differential system of the associated

0165-0114/98/$19.00 (~, 1998 Elsevier Science B.V. All rights reserved PIt S 0 1 6 5 - 0 1 1 4 ( 9 6 ) 0 0 2 1 7 - 5

optimization problem and it was proved that the critical point of the differential system which is asymptotically stable is a local a-optimal point of the original fuzzy optimization problem. Finally, in Section 5 we present a numerical example to show the efficiency of this approach.

2. Problem formulation

Consider the following non-linear programming problem involving fuzzy parameters in the objective function in the form

(FNLPP): Minimize f ( X , ~ ) , X ~ M (2.1a)

subject to X ~ M

= { x e R": ~(x) ~< 0},

G = ( g l , g 2 . . . . . gin) E R m,

(2.1b)

58 F.M. Ali / Fuzzy Sets and Systems 93 (1998) 57-61

where 2 are fuzzy parameters involved in the objective function f Here the fuzzy parameters are as- sumed to be characterized by fuzzy numbers as introduced in [-6], for this a membership function /~(2) is defined for a real fuzzy number 2, where ~ is a convex continuous fuzzy subset of the real line [6], the functions f gl, i = 1, 2 , . . . ,m possess continuous first and second derivatives, i.e. f, g~ ~ C2, i = 1 , 2 ..... m.

For simplicity in the notation, we defined the following vectors:

= (7~1, 7~2 . . . . , 7~ ) ,

/~ = (21 , ~'2 . . . . , 2p) , p ~ /'/,

Definition. The c~-level set of the number ~. is defined as an ordinary set L,(2) for which the degree of their membership function exceeds the level ~;

L,(~) = {2 : /~ (2 , ) ~> ~ (r = 1,2 .... ,p}.

For a certain degree ~, problem (2.1) can be for- mulated as the following non-fuzzy ~-NLPP,

(e-NLPP): Min. f ( X , 2) (2.3a)

subject to X E M, (2.3b)

2 ~ L,(~). (2.3c)

Then the corresponding equality constrained non- fuzzy optimization problem can be rewritten in the form

Min. f ( X , 2)

subject to

X' ~ m ' = {X' ~ ~n+m: g i (X ) ~_ S 2

= 0, i = 1,2 .. . . . m}

,~ '~ h 0 : ) = m , ( ~ ) - C - ~ = 0,

(2.4a)

(2.4b)

r = 1,2 .. . . . p,

(2.4c)

where

X ¢ = ( x 1 , 2¢ 2 . . . . . Xn,S1,S 2 . . . . . Sn) E ~ n + m (2.4d)

3. Fundamental equations

To solve the problem (2.4), we introduce the following system of differential equations:

BX" + ATTt = - V T x ' f , (3.1a)

C~' + AT72 [7 T = - ~,f , (3.1b)

A13~ ' ' = - G , (3.1c)

A22' = - h , (3.1d)

where B is a symmetric (n + m) × (n + m) matrix, C is a symmetric (2p × 2p) matrix, At = Vx,G(X') , A2 = V~,h(2'), the matrices A1, A2 are of full rank, 7t is an m-dimensional vector and 72 is a p-dimensional vector.

If B,C, A t B - t A t and A 2 C - ~ A 2 are non-singular, then the matrix

0

0 C 0 D = At 0 0

0 A 2 0

(3.2)

is non-singular and consequently (3.1) has a unique solution for X',,~',71 and 2:2 where

X ' = ck(X' ) = - P B - t Vx, f T _ PG, (3.3a)

2' = ~p(2') = -- K C - ~ Vz, f v - Kh , (3.3b)

= IZ T 7t ( A t B - 1 A ~ ) - ~ G - ( A t B - 1 A ~ ) - ~ A t B - t x , f ,

(3.3c)

72 = ( A z C - 1AzT)- lh

- ( A z C - ' A T z ) - I A e C - X V a , f T, (3.3d)

and

P = I - P1, (3.4a)

Pt = B - t A ~ ( A t B - I A ~ ) - tA1, (3.4b)

P = B - ~ A ~ ( A t B - tAXi ) - t , (3.4c)

K = I -- K1, (3.5a)

K t = C - 1Az(A 2 C - t aT ) - 1A2, (3.5b)

and

2 '=(21 ,22 . . . . . 2p,~1,~2 . . . . . iv), p<~n. (2.4e) / ~ = C - I iv -1 T - 1 A z ( A 2 C A z ) . (3.5c)

F.M. Ali / Fuzzy Sets and Systems 93 (1998) 57-61 59

Since ~b and ~ do not contain t explicity, the system (3.3) is au tonomous .

It is clear that P~ and K1 are idempotent , i.e. p2 = pa and K 2 = KI and consequently P , K are two projections which project any vector in ~R "+m, ~2p onto Ma and M2, respectively, where

M1 = {Z: A I ( X ) Z = 0, Z e 9t"+"}, (3.6a)

M2 = {v: Az( : )v = 0, v e ~]~2p}. (3.6b)

Let

M = M 1 @ M 2 ---= {(Z,v): A1Z = O,

A2v = 0, Z ~ ~n+ra and v e ~2p}. (3.6c)

Proof. If (X*, 2*) is a constrained s tat ionary point, then there exists ~* ~ 9U', y~ E 91 p such that

A~(X*)7~ + Vx , f T = 0 at G(X*) = O,

A~(X*)7~ + Vz,f T = 0 at h(2*) = 0.

Since D(X*,2*) is nonsingular, then (3.1) has a unique solution )f, ).', 71 and 72 which implies that ~b(X*) = 0 , ~0(2 '*)=0. On the other hand, if qS(X*) = 0, ~b(2*) = 0 then

AT(X*)7~ + Vx , f v = O, G (X*) = O,

AT(x*)7 * + Vz,f T = 0, h(2*) = 0.

and consequently (X*, 2*) is a constrained stationary point of f

4. Asymptotic properties of fundamental equations

In this section we discuss the relation between constrained a-opt imal points of ( a -NLPP) and an asymptotical ly stable critical point of the differential system [3, 5, 9]. Let us denote the solution of the system which passes through X' = t/i, 2' = q2 at t = 0 by nl(t/1, t) and ~ZZ(t/z,t ) and the whole trajectory by CI(t/1) and C2(q2) where

C1(/~1 ) ~- {H~(rh, t):tE T I ( r / 1 ) } ,

Ta(th) = (ta(ql), tb(t/,)),

C2(r /2) = {H2(r/2, t ) : t ~ T2(t/z)},

T2( r /2 ) = (ta(~/2), tb(r/2)),

where Tl(r/1), T2(r/2 ) are the maximal intervals of existence of the solutions.

Let

N = C 1 (t/1) ® C2(t/2 ) = {U(t/1, q2, t), t 6 T0/I , t/z)},

where

T(th, ~/2) = max {Tl(t/ ,) , T2(t/2)}.

Theorem 1. I f D(X*, 2*) is nonsingular, then a necessary and sufficient condition that (X*, 2*) is a constrained stationary points is that (X*, 2*) is a critical point of the system; qS(X*) -- 0, ~b(2*) = 0.

Corollary 2. I f at the critical point (X*,2*), the matrix

[ v Z , f ( X * , 2 *) + 22 VzZ, h(2 *) J

is a positive definite on tangent plane M(X*, 2*) oJ the constraint surface, i.e. Y T H Y > O, for any Y ~ M(X*, 2"), Y ~ 0 then (X*, 2*) is a strict local minimal point.

Theorem 3. Let (X*,2*) be a strict local minimal point, if there exist a neighbourhood U of (X*,2*) such that for any (X',2 ') ~ U~(G x h) the matrices B,C are positive definite on M, then (X*,2*) is asymptotically stable on (G × h).

Proofi For the solution (X'(t), 2(t)) ~ Uc~(G × h) we have

d f iX ' ,2 ' )

dt - - - V x , f ( X ' ) + Va,f(2')

= V x , f P B - ' Vx, f T + V z , f K C - ' 17;¢f T

= V a , f P P B - 1 V2, fT

+ V x , f K K C - 1 V2, fT

= V x , f B - 1pTBPB- 1 Vx, fT

+ V x , f C - 1KTCKC- a Vx,fT

= - ) [ 'TBX' + )~'C)(. (4.1)

60 F.M. Ali / Fuzzy Sets and Systems 93 (1998) 57-61

we have from the assumption that

d f d t < 0, (X', 2') # (X*, 2"), (4.2)

df(X*, 2*) dt

- 0 . (4.3)

On the other hand, from the strict minimality at (X*, 2*), there exists a neighbourhood V of (X*, 2*) such that for any (X', 2') • (G x h)

f (X' ,2 ' ) >f(S* ,2* ) , (X',2') ~ (S*,2*). (4.4)

Then, from the above properties, one can write

L(X', 2') = f (X', 2') - f (X*, 2"), (4.5)

which satisfies the following equations

L ( X ' , 2 ' ) ~ O, (X', 2') va (X*, 2"), (4.6a)

L(X*, 2*) = 0, (4.6b)

dL(X' , 2') < O, (X', 2') # (X*, 2*) (4.6c)

dt

dL(X*, 2*) = 0, for (X*, 2*) • Uc~ Vc~(G x h).

dt

(4.6d)

Thus the function L is a Liapunov function and (X*, 2*) is asymptotically stable on (G x h).

Lemma 4. A local minimal point (X*,2*) which satisfies the assumption of Theorem 3 is stable.

Proof. Suppose that (X*, 2*) is not stable, then for any neighbourhoods U and V of (X*,2*) there exists(Xo,20) Vsuch tha t + ' ' ' N (Xo,2;) c U, where

+ t ! t ~ t . ¢ t N (Xo ,2o)={H(Xo, ,~o , t ) . O < ~ t • T ( X o , 2 o ) } , and 0 is a closure of U. Let {(X;,,2;,)} be a sequence in V which converges to (X*, 2*) such that

+ t t N (Xk,2k) c U. Clearly, there exists t k > 0 such that FI(X'k,2'k,t) • U, 0 <~ t <~ t k, II(X'k,2'k,t) • OU, where (?U is a boundary of U. Since (?U is compact, the sequence {Yk}={II(X'k,2 'k, t)} must have a convergent subsequence converging to Y, since G(X*) = 0, h(2*) = 0 then G(X'k),h(2~) --* 0 and hence G x h(Y) = 0, then Y • G x h. Consequently,

FI(Y,-- t ) • Vc~Gxh for some t > 0 this contra- dicts the asymptotic stability of (X*, 2*) on G x h.

Theorem 5. A local minimal point (X*, 2*) which satisfies the assumption of Theorem 3 is asymptotically stable.

Proof. From the stability of (X*, 2"), then for any neighbourhood U of (X*, 2*) there exists a neighbourhood V of (X*,2*) such that N+(X' ,2 ') c U for any (X', 2') • V. Therefore the positive limit set F + ( X ' , 2 ' ) c U , but F+(X ' ,2 ' ) cOG×h, where Uo×h = (G x h)c~Uo. Then we can choose U such that F + (Y) = (X*, 2*) for any Y • Oa ×h. If we take Y • F+ (X',2') c fJG×h, then F+ (Y) = F+ (X',2), F + (X', 2') = (X*, 2*), and hence (X*, 2*) is asymptotically stable.

5. An illustrative example

In this section .we provide a simple numerical example to clarify the theory developed in the paper. The problem under consideration is a non- linear programming problem involving a fuzzy parameter in the objective function.

(FNLPP) Minimize f(X,-2)

subject to x + y ~ < l

where

f(X,-2) = x 2 + i 2y 2.

Let the fuzzy parameter be characterized by the following fuzzy number ~ = (17,19.5, 20.9, 22.5). Assume that the membership function for each fuzzy number ~ in (FNLPP) problem is defined by

~:(2)=

0, 2~<pl , ( 2 ~ p 2 ) 2, pl<~,~<~p2,

1 - ~ - P 2 /

1, P2 ~<2 ~<P3, 1 - ( L ~ P 3 ~ 2, p 3 ~ . ~ p 4 ,

\P3 -- P4./ 0, 2 ~>P4.

Consider the a-level set or a-cut of the fuzzy numbers given by #4,(2) ~> 0.9 and the non-fuzzy a-non-

F.M. Ali / Fuzzy Sets and Systems 93 (1998) 57-61 61

linear problem (e-NLPP) with equality constraints can be written as follows:

(e-NLPP): Minimize f (X, 2) -- x 2 + 2y 2

subject to x + y + s 2 - 1 = 0 ,

~L~(/~) - - ~ 2 __ 0.9 = 0

using the method of the fourth-order Runge-Kutta or Euler method with initial point (x°,y°,s °, 2°,¢ °) = (0.1,0.1 0.8, 18,0) for solving the differential autonomous system associated with the above problem, we get the a-optimal point (0,0,1,19, 53,0).

Acknowledgements

The author gratefully acknowledges the sugges- tions and comments made by Prof. Mohamed S.A. Osman, Higher Technological Institute, Ramadan 10th City, Cairo, Egypt and the Journal referees.

References

[1] D. Dubois and H. Parde, Operations on fuzzy numbers, lnternat. J. Systems Sci. 9(6) (1978) 613-626.

[2] D. Dubois and H. Parde, Fuzzy Sets and Systems, Theory and Applications (Academic Press, New York, 1980).

[3] W. Hahn, Stability of motion (Springer, Berlin, 1967). [4] Hiroshi Yamashita, A differential equation approach to

non-linear programming, Math. Programmino 18 (1980) 155-168.

[5] J. Lasalle and S. Lefschetz, Stability Theory by Liapunov's Direct Method (Academic Press, London, 1961).

[6] M. Sakawa and H. Yano, Interactive decision making for multiobjective nonlinear programming problems with fuzzy parameters, Fuzzy Sets and Systems 29 (1989) 315-326.

[7] M. Sakawa and H. Yano, An interactive fuzzy satisficing method for generalized multiobjective linear programming problems with fuzzy parameters, Fuzzy Sets and Systems 35 (1990) 125-142.

[8] M. Sakawa and H. Yano, Feasibility and Pareto optimality for multiobjective non-linear programming problems with fuzzy parameters, Fuzzy Sets Systems 43 (1991) 1-16.

[91 T. Yoshizawa, Stability Theory by Liapunov's Second Method (Mathematical Society of Japan, Tokyo, 1966).

a differential equation approach to fuzzy non-linear programming problems

Documents