A Hierarchical Generating Method with Feasible Decomposition

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A HIERARCHICAL GENERATING METHOD WITH FEASIBLE DECOMPOSITION D. Li and Y. Y. Haimes Dl'jJarlllll'lIl

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Abstract. This paper investigates l arge-scale mulliobjective systems in the conte x t of h ierarchical generating met hods ~hich consider the problem of ho~ to find the set of all noninferior solutions by decomposition and coordination. A hierarchical generating method ~ith feasible decomposition is developed for a general c l ass of multiobjective systems ~ith nonconvex objectives, nonlinear subsystems, and nonseparable interaction among subsystems. A multi level scheme i s proposed based on a feasible decomposition theorem, and the decomposition i s attained by fixing t he output vecto rs of the s ub sys tems. Lo~er - dime n sioned multiobjective s ubproblems are so l ved at the lower le ve l by the E-constraint approa ch. A way to map th e noninf er ior fronti ers of subsystems into the noninferior frontier of the overall sys tem is established by using trade-off information in subsystems . At the upper level, coordination is accomplished by using the envelope approach in order to identif y all non inferior solutions of the primal problem. Keywords. large - scale multiobjectiv e system, noninferior so lution, hierarchical generati~g methods, envelope coordination, feasible de com position. By introducing the Lagra ngian multipliers \ . , i = 1,2, . . . ,N, the followin g vector-valued Lagr~ n gian probl em is constructed:

INTRODUCTION The fundamental c haracteristic of large-scale systems is their inescapab l y multifarious nature : they ha ve a varying number of multiple objectives and th e ir s tructure may be mult i level or h i e rarchical, multi s tage or dy namical, etc. Theoretical and methodological results of hierar chical multiobj ec ti ve analysis for large sca l e sys tems have been developed in r ece nt years (E s t e r, Riedel, and Pesc hel, 1975; Geoffrion and Hogan, 1972; Li and Haime s, 1985, 1987; Nijkamp and Rietveld, 1981; Shimizu and Aiyoshi, 1981; Straszak, 1972; Takama and Loucks , 1981; Tarvainen, 1986; Tarvainen and Haimes , 1982; Tarv ai nen, Haimes , and Lefko~itz, 198 3 ). A pap er by Haimes and Li (1986) present s a detailed s ur vey of th e use of h ierarchical multiobje c t i ve anal ys i s for l arge - sca l e sy s tems. One bas i c task in large-sca le multiobje c ti ve systems is generating the set of noninferior solutions. The purpose of hierarchi ca l generating methods (Li, 1987; Li and Haimes , 1987) i s to find the set of non inf e rior so lutions of a large-scale multiobj ec ti ve sys tem by using decompo s ition and coordination. The hi erarc hical generating method developed by Li a nd Haimes (1987) i s base d on the duality theory of multiobjec t i ve programming. This scheme can be seen as an extension of the nonfea s ible method (La sdo n a nd schoeffler, 1965) used in the singleobjective cas e. In the sc heme presented by Li and Haim es (198 7 ), the follo~ing multiobjective optimization convex problem is cons i dered:

N i

fl = E f l (x.,u.,z . ) i=l 1 1 1 min f

= 1, 2 , ... ,N

= 1,2, ... ,N i

=

1,2, ... ,N

(1)

N

i

N

T

N

i

E f (x.,u.,z.)+ E \. [ z.- E D.x. ) i =l l 1 1 1 i =l j=l J J

min L=

s . t.

Xl

=

AiZi+ B i ll i+C i

g , ( Xi, Ui ,Z,l S; 0

1 ,2, .. . ,N

( 2)

1, 2 , ... ,N

Using \ ' i = 1, 2 , ... ,N, as the coordination var iabl es , Eq. (2) i s se parable with respect to (X~,U i ' Z,l , i = 1, 2, ... ,N. Thus, N multiobjecti ve s UDprobl ems are so l ve d a t the lo wer level for a fixed value of \ , i = 1,2, ... ,N. Let Ext ) F)D) denote the set of D- extreme points of F ( Yu, 1974). Thus, the se t of noninferior solutions of the vec tor Lagrangi a n problem in Eq. (2) ~it h fixed A ca n be expre sse d as Ext ! (L(x,u,zIA) } IRk ) . Th e following dualit y th eo rem provides the + th eoreti cal basis for coor dination at the upper le vel . Th e duality theorem: If ( ~, ~,i) is a noninferior solution of (1), then (3)

where .p(A) = Ext ! (L(x,u,zIA)j IR: )

(4)

The di fferent va lu es of the coordination var iable A will generate a famil y of noninferior fro n tier s in the fun c tional s pa ce , and all noninferior solutio ns of the original problem given in Eq. (1) lie on the envelope of this family. In this paper, we consider a general class of systems that may have non convex object i ves , nonline ar s ub sys tem s, and nonseparable interaction among s ub sys tems. A hierarchical generating method is propo se d in ~hich de composi t ion i s attained by fi x in g the output vectors of the subsystems. In contrast ~ ith the non feasible ve rsion of the hierar chi cal generat ing method (Li and Haimes,

D. Li and Y. Y. Haimes

74

1987). the version presented here can be seen as an extension of the feasible method (Brosilow. Lasdon. and Pearson. 1965) used in the singleobjective case.

ft(xi'ui' z i )

f~( x i'ui'zi) min

PROBLEM STATEMENT

i

Let us consider a system consisting of N intercoupled subsystems. In each subsystem. 1.2 •.. . • N. let x . be the output vector of subsystem i . x . £Rni; let u . be the control vector of subsystem i: u . £RPi; ana let z . be the interaction input ' vector of subsy~tem i. zi£Rmi. The subsystem i s com pletely de scri bed by giving it s outputs as functions of its control and interaction input: =

1.2 ..... N

(5)

where the function Hi' i = 1.2 •...• N. differentiable with respect to u i and A model for the overall system i s obtained by adding to Eq. (5) a set of relations describing ho w the subsystems are interconnected: i = 1.2 ..... N

(6)

where the function C. • i = 1.2 •...• N. is differentiable with ~espect to x j ' j = 1.2 •...• N. Eq. (5) • which i s a nonlinear model. and Eq. (6). which describe s the nonlinear cou pling of the su bsys tem s of the model. cover a general class of systems. In addition. the assumption of nonlinear coupling gives us the flexibility to define the s ubsy s tems and enables us to do a coordinate transformation on the input - output relationship for s ome s ubsyste ms in order to s implify the treatment of the problem. The system's multiobjective function. whi ch is what we want to minimize. is assumed to be of the additively se parable form: N f1 f2

E

i=1 N E

i =1

i f 1 (xi' u i· z i)

f~( X i·ui· z i) (7)

The following assumption i s made in thi s multiobjecti ve optimization model: for each s ub sys tem. P . • the dimension of control u . • is greater than ' or equal to n i • the dimensio~ of Xi A FEASI BLE DECOMPOSITION THEOREM In a case where the syste m is large. it may be diffi cult to ta ckl e directl y. in a global way. the nonlinear multiobjective optimization problem given by Eqs. ( 5) - ( 7) . A way to avoid s uch diffi culties may be found among the hiera rc hi cal generating methods that solve the problem of optimization for large- sca le multiobjective sys tem s by decomposition and coordination. If we temporarily fix the va lues of xi' i = 1.2 •...• N. we can decompose the large-scale multiobjec ti ve optimization problem given in Eqs. (5) - ( 7) into the following N multiobjecti ve su bproblems :

fk(xi·ui·z i ) s. t.

xi

Hi (ui'zi)

zi

Ci (xl'x 2 ... · .x N)

(8)

In this feasible decomposition sc heme. at each s ub sys tem the cri teria are not changed; however. the output vector x. is fixed. as is the interaction input Z'. through Eq. (6) . ~e call Xi ' i = 1.2 •...• N. the coordination variables. For different coo rdination variables . we get a famil y of noninferior frontiers with parameters x in the functional space. ~e will prove in what follo ws that the envelope of this family gives us all the noninferior so lutions of the primal problem outlined in Eqs. (5) - (7). Theorem 1. Feasible Decompo sit ion Theorem

Ext [F(S) IR~ 1

(9)

where TT T TT T TT T x=[x 1 •··· .xNl • u=[u 1 .···.u NJ • z=[z1.··· .zNl S = {( x. u.z)l x i = Hi' zi = Ci • i = 1.2 •...• N) F(S) = X = (xl S(x) = f( S(x»

(f(x . u.z ) I (x.u .z )e:S) there exists u such that (x.u.z)£S) (x.u.z)l(x.u.z)£S. x is fixed and x£X) = (f( x.u. z)l(x.u .z )£S(x)}

Proof --necess it y:

Ass ume f( x. li .z )e: Ext[F(S)IR~I .

~e know x£X and f(x.li.z)£ E x t[f(S( x »IR~I .

Furthermore. f(x.li.z) must Ext[ (UE xt(f(S( x » IR~)J IR~J. lead to a contradiction of Proof--sufficiency: Assume

belong to otherwise. it will the assumption. f( x.li.z )£ E x t[(UE xt [f(S( x »IR~)JIR~I. If f(x.li.z)\: E xt [F( S)IR~ I . th en there exis t s ( x.li.x )£S such that f(x.li.z)£ f(x.u.z)+R~. If f(x.u.z)£ E xt [f(S(x»IR~I. thi s leads to a contradiction of the ass umption. If f(x.u.z)\: Ext[f(S(x»IR~I. then there exists ( x.G.z )£S( x) suc h that f(x.G.z)£ E xt [f(S( x» IR~ 1 and f(x.u.z)£ f(x.G.z)+R~. Thus. we ha ve f(x.li.z)£ f(x.G.z)+R~. which i s a contradiction. Q.E . D. It i s clear to concl ude from Theorem 1 that largesca l e nonlinear multiobjective optimi zation problems may be so l ved by multile ve l methods. At the l ower le ve l. the inner multiobjective optimization of Eq. (9). E x t[f(S( x » IR~ I. is dealt with; i.e .• each s ub syste m so l ves the problem given in Eq. (8) for a fixed va lue of x. At the upper le vel. if the family satisfies ce rtain condit ions (Li . 1987). then. the coordination may be accomplished by us ing the envelope approa ch (Li. 198 7). whi ch gives us the set of noninf erior so lutions of th e primal overall syste m. i.e .• Ext[F(S) IR:l. FEASIBLE DECOMPOSITION AND ENVELOP E COORDINATION ~e

do not make an as s umption of co nvex ity for the

75

A Hierarchial Generating Method problem given in Eqs. (5)-(7), and the weighting approach is not appropriate for some nonconvex cases due to the duality gap (Chankong and Haimes, 19B3). Because of these considerations, we adopt the £-constraint approach (Haimes, Lasdon, and Vismer, 1971) to deal with those multiobjective optimization problems that may be nonconvex. The Ith-objective £-constraint formulation for the problem in Eqs. (5)-(7) with a fixed value x is defined for some £ - [£" ... ,£,_"£,.,, ... ,£.I T as follows:

Remarks: 1) The assumption of regularity guarantees the existence of the Kuhn-Tucker multipliers. 2) The assumption of uniqueness guarantees that the Ith-objective e:-constraint formulation for each subsystem generates a noninferior solution of the problem given in Eq. (8). 3) To assure that [XT,U~T,Z~TIT is a regular point of p~(e:i';X), the co~dition' P.>n . +(k-1) must be satisfied. .- • Proof: Introduce variables £~, i - 1, ... ,N; j _ 1, ... ,k, j ;< 1, and rewrite P,(e:';x) as follows:

N

Pl(£;x): min fl

min fl -

-

N

f~(Xi,ui'Zi)

E

i-1 N

s. t. fj

E

i-1

,

Hi(ui,zi)

zi

Ci (x 1 ,x 2 ,··· ,x N),

-

1

(13)

1

1, ... ,k, j;
1, ... ,k, j ;<1

N

E

i-1

1,2, ... ,N

£~ *

- 1, ... , k, j;
J

1,2, ... ,N

For a given point [XT,U'T,Z'TI T, we use the symbol P (£';x) to represent the problem P,(£;x), where £' - £~ - f(x,u',z'), j - 1, ... ,k, j ~ 1. It f6110w~ frofu Theorem 4.2 in Chankong and Haimes (19B3) that i f [XT,U'T,Z'TI T is the unique solution of P,(£';x), then [XT,U'T,Z'TI T is a noninferior solution of the problem in Eqs. (5) (7) with a fixed value x.

i_ 1,2, ... ,N

Form the Lagrangian of the problem in Eq. (13): N

p~(£i;x): (11)

ft( Xi ,Ui,zi)

s. t. f~(x, ,Ui'Zi)~£~' Xi Hi(Ui'Zi)

1, ... ,k,

;< 1

.

L - E

i-1

+

Similarly, the Ith-objective £-constraint formulation for the subsystem i in Eq.(B) with a fixed x is defined for some £i _ [£~, ... , £~_" £~." ... , £~IT as follows:

min

1

(10)

fj(Xi'Ui'Zi)~£j ,

xi

. fl'(x. ,u. ,z.)

E

i-1

ft(x. ,u. ,z . ) 1

k

1

1

N +

E

i-1

A.

E

j-l jU

J

(14) where >{., i - 1, ... ,N, j - 1, ... ,k, j ~ 1, are Kuhn-Tucker multipliers, and A.(j _ 1, ... ,k, j ~ 1), IJ(i - 1,2, ... ,N), and cr . d - 1, ... ,N) are Lagra~gian multipliers. Ve ~ay get the firstorder necessary conditions for the optimal solutions of Eq. (13) as follows: j - 1, ... , k , j;
For a given point [x:,U:T,Z:TI T, we use the symbol p~(£i';X) to represent the problem p~(£i;X), where

i- 1 , 2, ... ,N; j - 1 , .. . ,k, j;
£~ - e:r - f~(xi'u:,z:>, j - 1, ... ,k, j ~ 1. And we know that if [x:' u: T,
Theorem 2. Assume tha t [~, u: T, Z:T IT is the unique solution of p~(£i';X) and that it is a regular point, i - 1,2, ... ,N. If [x~, U;T ,Z;T; ... ; ~,U~T ,Z~T I T solves the problem P,(e:';x), where e:

*

E e: i*

N

i-l

~j(e:")

~ j ( e:") - ... -

j-1, ...

T dH i (ui'zi) IJi dUi dL dZ i

df~(Xi,ui,zi) dZi

0

Ai' j (e:

N

')

(12)

,k,j~1

where >{j is the optimal Kuhn-Tucker multiplier of problem p~(e:i';X) with respect to the jth £ constraint.

1, ... ,N k

+

E

j-1 jU

T dH i (ui'zi) IJi + cri dZ i

i - 1, ... ,N;

x i - Hi(ui,zi) - 0

then we have

(16)

df;(Xi'Ui,Zi) dUi

noninferior solution of the problem in Eq. (B) with a fixed value x. In the following, we are going to investigate the relationship between the noninferior solutions of the problem given in Eqs. (5)-(7) with a fixed value x and the noninferior solutions of N subsystems, which are expressed by Eq. (B) with the fixed value X.

(15)

AL

-

0

(17)

af~ (xi' ui' zi) dZ i

i

-

1, ... ,N

1, ... ,k, j

~

1

i - 1, ... ,N

o

(1B)

(20)

1, ... ,N (21)

If we choose e: i equal to £i', i _ 1, ... , N, j _

1, ... ,k, j ;< l~ Eq. (15) i1s satisfied. Equations (17)-(21) are the first-order necessary conditions for [XT,U'T,Z~TIT to be the optimal solution of p~(e:if;X). The necessary condition in

I).

7(;

Eq. (12) in Theorem 2 (16). O.E.D .

follo~s

Li alld Y. Y. Haimt:s

directly from Eq.

In the follo~ing, ~e consider the case ~ here the noninferior frontier for each s ub syste m in Eq. (8) ~ith a fixed x, i = 1,2, . . . ,N, is of k-1 dimension and the optimal Kuhn - Tucker multiplier A{ j' i = 1, . . . , N, j = 1, ... , k, j t 1, is a s tr i c t ly monotonic function of e: i , j = 1, . . . ,k, j t 1. The scheme to map the no ninterior frontier s of N s ubsystems into the noninferior frontier of the overall system is established by usi ng the tradeoff information in the s ubsystems in accordance Id th Theorem 2.

fl

f1 (e:~;x)

f2

f2(e:~;x )

af 1

~

af 2

ax - as ax £2

a &~

For

(24)

af 1

k~3,

[ V. L. O. G. , take

f1

f 1 ( £~ , . .. , £k; x)

f2

f 2 ( £~ , . .. , £k; x)

0

1J

Each subsystem i solves Eq. (8) ~ith the fixed x independently by using the £-constraint approach . The parametric form of the set of non inferior s olutions of subsystem i can be expressed with t he corresponding trade-off values in the follo~ing " ay:

(25) af 2

ax

i i [' 1 = £1

+. . •

afk

ax

0

(26)

EXAMPLES The set of noninferior solutions of the problem in Eqs. (5)-(7) ~ith thi s fi xed x can be obtained by s umming the objec ti ves of the correspo nding points in efficient frontiers of each subsystem that are satisfied by the necess ary condition, Eq . (12). Since there are N' (k - 1) unknowns in the (N - l) ' (k - 1) tl'ad e - off equations of (1 2), it is possible to express e:', i = 1, ... ,N, i t s , j = 1, ... ,k, j t 1, as th~ fun ction of e:; , e:; , ... , e:~ _ 1' e:~+ l' . . . , e:;, where 1~s~N. Thus , th e parametric form of the set of noninf er ior so lut io ns of the problem in Eqs. (5) - (7) with tixed x can be expressed as

Example 1. Application to multiobjective discrete dynamic systems Consider the follo~ing multiobjective discrete dynamic system : T-1 f1 fZ min

1:

t=O T-1 1:

t =O

T- 1 1:

t=O

(23)

s.t.

For different values of coordination variable x, ~e get a family of noninferior frontier s as ex pre s sed in Eq. ( 23 ). Bas ed on Theorem 1 and the envelope approach (Li, 1987), the coo rdination algorithm is des igned to identify the set of non inferior s olutions of the primal probl em given in Eq s . (5) - (7) by us ing the envelore approa ch. In fact, the union of Ext[f(S(x»IR.J comprises a family, and the multiobjective optimization problem, Ex t [ {UExt[f(S(x»IRkJl IRkJ , may be solved by identifying the envelope ;f thi s family based on th e envelope approach (Li, 19B7). The envelope of the family defined by Eq. (23), ~ hich has parameters x, can be obtained by the follo~ing formulas : For k=2 [ V.L.O.G., take 1

1J

fi(x(t),u(t),t)

f~(x(t),u(t),t)

f~(x(t),u(t),t)

x(t +1) h( x(t),u(t» x(O) = xo' x(T) = xT

(27)

The coordination va riable z(t) is defined as z(t-1)

=

(28)

x(t)

Thu s

the subproblem t, t ~ritten dire c tly as fj(x(t),u(t),

=

1, ... ,T- 2, could

be

t)

f~(x(t),u(t),t) min

f~(x(t),U(t),t) s.t.

z(t) = h(x(t),u(t» xCt) = z(t-1) for given z(t-1) and z(t)

To take into account the initial

(29)

state, the first

i7

A Hierarchial Generating l\Iet!Joc!

2

2 '32 xl - f.3 x 2

1 - '3 gl1

f~(xo'u(O),O)

u 12

1 1 2 1 2 1 1 '3 gl - '3 xl + '3 xl - '3 x 2

fZ(xo'u(O),O)

A21

+

'3 xl

+

2 '32 [x 1- x 22 - gl1 + xl]

1

min

1

u 11

subproblem becomes:

(35) (30)

z(O) = h(xo'u(O»

s. t.

The condition on the final subproblem T-1 be:

+

for given z(O) state

requires that

with Subsystem 2: 1 g2 + x 2 - 1Z- x + 1Z- x x 2 1 2 2 Z- 1 1 1 g2 + x 2 + 1 Z- x 2 - Z- xl x 2 2 Z- 1 1 2 3 g2 3x 2 - 1Z-x 2 +Z-x 1x 2 Z- 1

fI- 1 (x(T-1),U(T-1),T-1)

f~-1(x(T-1),U(T-1),T-1) min

g2 1

fI- 1 (x(T-1),U(T-1),T-1) xT = h(x(T-1),u(T-1» (31) x(T-1) = z(T-2) for given z(T-2)

s. t.

As we stated earlier, the hierarchical generating method with feasible decomposition is only applicable if the dimension of u ~ the dimension of x + (k - 1). Example 2. Consider the following optimization problem:

multiobjective

min

s. t.

xl

u11 + u12 + zl

x2

- u 21 + u 22 + z2

zl

2 x2

z2

xl· x 2

This system may be decomposed into for some given xl and x 2 .

2

'3

1 4 2 4 2 1 2 2 r1 + '3 xl + xl + x 2 - '3 gl X1 - '3 v1 xl 2 r1 2 2 3 2 2 2 v1 x 2 + '3 xl - '3 xl x 2 - 2x 1x 2

(32)

two subsystems

+

(36)

2 2 322 2 2 4 3 2 3 + 4" (gl) + 3x 2 + 4" x2 + 4" x 1x 2 + 3£1 x 2 3 2 3 1 2 3 1 2X Z- gl 2 - Z- gl xl x 2 + x 2 - x 1x 2 - Z- x 1x 2

wi th By Theorem 2, the corresponding points on the efficient frontiers of subsystems 1 and 2 are satisfied by Eq. (12). Thus, we have the relation 1 2 1 2 2 2 '3 [x 1 - x 2 - gl1 + xlI = - l2 g21 - 3x 2 - Z- x 2+ Z- x 1x 2 3 3 gl = + 9 2 + xl2 + x + 7Z- x 2 + 4" i. e. , x 2- 4" xl x 2 4" £1 1 2 1 (37) Substituting Eq. (37) into Eq. (35) and summing the objectives of corresponding points, we get the parametric form of the set of noninferior solutions of a problem with parameters x, and x,: 13 2 z;-- £1 + xl2 + xl + 7Z- x 22 + 4"3 x 2 - 4"3 x 1x 2 (38) 39 (r2)2 r2 (39 2 13 13) 16 v1 + v1 4 x 2 - S- x 1x 2 + S- x 2

Subsystem 1:

min

s. t.

2 xl - u11 + 2u 12 [ 2 2u 11 + u 212 xl zl

u 11 + u 1 2 + zl x2 2

The envelope of the family defined by Eq. (38) with parameters xl and x 2 can be obtained by Eq. (24) as follows: (33)

2

3

rz-

2x2 + u 21 + u22 + u 221 + 2u 222 (34)

Using the second-objective g-constraint formulation, we get the parametric forms of the set of noninferior solutions for both subsystems and 2 with given x, and x,. Subsystem 1:

7

2 2 125 4 13 3 15 2 x 2 + z;-- x 2 + 16 x 2 + '3 xl + 13 3 77 2 15 2 2 (39) + 16 x 1x 2 - z;-- x 1x 2 - 24 x 1x 2 117 2 39 £2 39 2 13 2 13 3 x1x 2 S1 + z;-- gl X1 + S- £l x 2 + z;-- x 2 + 8

2

s. t.

2

39 (r2)2 2 39 2 13 16 v1 + gl (z;-- x 2 - S- x 1x 2

Subsystem 2: min

13 g2

z;-- 1 + xl + xl + Z- x 2 + 4" x 2

13 13 - ~ xl + S- x 2 =0 13 x 1x 22

=

°

Table 1 gives some points on this envelope that are noninferior solutions of Eq. (32).

D. Li and Y. Y. Haimes

78 Table 1

A Sample of Noninferior Solutions 2

f1

3

4

5

-1 -6.390934 -15.147470 -32.531120 -170.7199

0 1. 618811 f2 1 1 1 -1. 390934 f1 = £1 f1 0 1.490825 2 2 2 -5 f 1= £1 -2 f2 0 0.127986 2 xl 1 0.929036

11.155450

55.427220

-5.147474

-12.531120 -70.71992

1606.168

10.282440

51.107530

-10

-20

1481. 335 -100

0.873010

4.319690

124.8334

0.878465

0.815300

0.566455

x2

1 1.510584

2.105128

2.955837

6.573870

zl

1 2.281864

4.431564

8.736972

43.215767

z2

1 1. 403387

1. 849281

2.409894

3.7238015

u ll

0 -0.15054

-0.39569

- 0.88250

-4.752612

u 12

0 -1.20229

-3. 15742

-7.03917

-37.89671

u 21

0 - 0.27173

-0.69636

-1. 53600

-8.209264

u 22

0 -0.16454

-0 .44051

-0.99006

-5.359195

A2l

0

1. 57738

3.51611

18.927650

0.60081

CONCLUSIONS A feasible version of a hierarchical generating method has been proposed in this paper. A theoretical basis and methodological grounding for decomposition and coordination are developed. Unlike the nonfeasible version of hierarchical generating methods (Li and Haimes, 1987), this feasible version can be applied to hierarchical multiobjective systems with non convex objecti ves , nonlinear subsystems, and nonseparable interactions among subsystems. Also, the local multiobjecti ve optimization problems at the lower level seem to be simpler than in the nonfea sib le vers ion (Li and Haimes, 1987), s ince the vector objective function for each subsystem is not modified and all subsystem's outputs x. (and thus all interaction inputs, z . ) are speCified at the upper level. However, this feasible version has a severe applicability condition: the dimension of u . must be larger than or equal to the dimension ol x . + (k - 1), i = 1,2, ... ,N, in order to guar~~tee that (xI,U:T,Z;T)T is a regular point of p'(£' ;x). In addltion, the way to get the c~rresponding noninferior frontier of the overall system from the noninferior frontiers of the subsystems is much more complex than in the nonfea s ible version (Li and Haimes, 1987). REFERENCES Brosilow, C., L. S. Lasdon, and J. D. Pears on (1965). Feasible optimization methods for interconnected systems. Proceedings of the Joint Automatic Control Conference, June 2225, 1965, Troy, NY, pp. 79-84. Chankong, V., and Y. Y. Haimes (1983) . Multiobjective Decision Making: Theory and Methodology. North-Holland, Amsterdam. Ester, J., C. Riedel, and M. Peschel (1975). Polioptimal approach to the control of hierarchical multilevel systems. Proceedings of the 6th Vorld Congress of IFAC, Boston, 1975. Geoffrion, A. M. , and V. V. Hogan (1972). Coordination of two-level organizations with mUltiple objectives. In A. V. Balakrishnan (Ed.), Techniques of Optimization, Academic Press, New York. Haimes, Y. Y., L. Lasdon, and D. Vismer (1971). On a bicriterion formulation of the problems of

integrated systems identification and system optimization. IEEE-SMC, SMC-l, 296-297. Haimes, Y. Y., and D. Li (1986). Hierarchical multiobjective analysis for large-scale systems: Current status. The 4th IFAC/IFORS Symposium on Large-Scale Systems, Zurich, August 26-29, 1986. Lasdon, L. S., and J. D. Schoeffler (1965). A multilevel technique for optimization. Proceedings of the Joint Automatic Control Conference, June 22-25, 1965, Troy, NY., pp. 85-92. Li, D. (1987). Optimization of large-scale hierarchical multiobjective systems: The envelope approach. Ph. D. thesis, Case Vestern Reserve University, Cleveland, Ohio, USA. Li, D., and Y. Y. Haimes (1987). A hierarchical generating method for large-scale multiobjective systems. To appear in J. Optimization Theory & Appl., 54 (2), 1987. Li, D., and Y. Y. Haimes (1985). The envelope approach for multiobjective optimization problems. Proceedings of 1985 IEEE International Conference on Systems, Man & Cybernetics, Tucson, Arizona, November 12-15, 1985, pp. 1039-1046. Nijkamp, P., and P. Rietveld (1981). Multiobjective multi-level policy models: An application to regional and environmental planning. Eur. Econ. Rev., 15, 63-89. Shimizu, K., and E. Aiyoshi (1981). Hierarchical multiobjective decision systems for general resource allocation problems. J. Optimization Theory & Appl., 35 (4). Straszak, A. (1972).-rolioptimization of large scale systems with multi-level control structures. Proceedings of the 5th Vorld Congress of IFAC, Paris, 1972. Takama, N., and D. P. Loucks (1981) . A multilevel model and algorithm for some multiobjective problems. Vater Resources Bull., 17 (3). Tarvainen, K. (1986). On the generating of Pareto optimal alternatives in large-scale systems. The 4th IFAC/IFORS Symposium on Large-Scale Systems, Zurich, August 26-29, 1986. Tarvainen, K., and Y. Y. Haimes (1982). Coordination of hierarchical-mutiobjective systems: Theory and methodology. IEEE-SMC, SMC-12 (6). Tarvainen, K. , Y. Y. Haimes, and I. Lefkowitz (1983). Decomposition method in multiobjective discrete - time dynamic problems. Automatica, 19 (1). Yu, ~ L. (1974). Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives. J. Optimization Theory & Appl., 14, 319-377.