Optimal control of thermal-neutron nuclear reactors

.,Vo1.27,~0.5,pp. 37-47,1987 U.S.S.R. Comput.Maths.Math.Phys Printed in Great Britain

0041-5553/87 $~O.OO+O.OO 01989 Pergamon Press plc

OPTIMAL CONTROL OF THERMAL-NEUTRONNUCLEAR REACTORS* V.S. NSRONOV

The problem of the optimal control of thermal-neutronnuclear reactors is considered. Questions regarding the existence of an optimal control and the derivation of the necessary conditions for the optimality of the control in the form of a maximum principle are investigated. Problems associated with the optimal control of thermal-neutronnuclear reactors as objects with distributed parameters are of considerable interest (for example, see /l-6/, etc.). A solution of the problem of the optimal control of nuclear reactors, including the solvability of the equations of the process, the existence of an optimal control and the derivation of the conditions for control optimality, is given in this paper. A non-linear integral functional is selected as the criterion of optimality. 1. Formulation of the problem. Let us consider a thermal neutron nuclear reactor with an active zone kR" bounded by a surface S. The non-stationaryprocesses in such reactors are described by the differential equations /7, 8/

(l.la)

Z.@fmerp(-B%) k hicitso, 2-I i-1,2...., m,

++e-ric,.

(5, t) E Q=QX (0, T) 7

(l.lb)

T
with initial and boundary conditions

X42,

@(I, O)=Wx),

C,(s, O)=C&),

@(x,t)=O.

(x,t)G=SX(O, T),

i=l,

2 ,...,m,

(1.2a) (1.2b)

where t is the time, x is a spatial variable, @=@(r, t) is the neutron flux density, Ci=Ci(x, t) is the concentration of precursors which delay neutrons of the i-th group, m is the number of groups of delayed neutrons, D is the effective coefficient of diffusion, & is the macroscopic cross-section for the absorption of neutrons,k=kcr +6k is the neutron breeding coefficient in an infinite medium, kc, is the critical neutron breeding coefficient, 6k is the excess reactivity, /% are the fractions of delayed neutrons of the i-th groups, V,Y is the velocity of the thermal neutrons, B’ is a material parameter, ? is the square of the moderation length, cp is the probability of avoiding the resonance absorption of neutrons during moderation, hc are the radioactive decay constants of the fission fragments and So is the density of the external neutron sources. To simplify the presentation we shall only consider the first group of delayed neutrons and the system (l.l), (1.2) will be represented in the form

-$-=div(agrady,)+

b,,yj+c,vy,+g,

(1.3a)

I-1 z-i --

b,yj+ww,

(x,t)= 0,

(1.3b)

k-1,2,

(1.4)

j-1 Yf(G

o)-Yo~(xh

Y‘k t)=O,

xq (I,0=X,

(1.5)

where y,-yl(s,t)-@(z, t),yr==yl(x, t)-C(z, t),a=v,D, b,,==vN&(kcr (I--B)sxp(-B'r)-I), b*r-nNcpexP(B*r)h,cl-v&( I-p)exp(-B’z), b,,pkcr Z.l% bn--A, Czsbrs, v--bk, Pvnh *Zh.vychisl.Mat.mat.Fiz.,27,9,1335-1348,1987 37

38

we select the excess reactivity &k~3~(~~Nu(~) as the control_and define a set,of permissible contxols o-(v[ti~v,v(QeG

a.** on

(~~~~~

in the space v=~tf% T), where a.%. means almost emwywhere and G is a convexI set in R'* We sepcify a functional (a criterion of the efficiency of the process)

closed, bounded

which is a Banach space with respect to the norm

for the functional spaces corresponds to that adapted in /9-U/ R6ma.rk * The notation where their properties are also described: conjunction, canvergence, etc.. me shall say that the function y(v) is a generalized solut5.onof Defini tfon 1. initial-boundaryvalue problem (l-3)-(1.5)if #(v)cy and, for almost all tai0, T) any (4 ~~~~~~(~~X~~(~~~ it satisfies the system of equations * Q/,(s) ah, Lc( "---GQ-~~= > (4

the

and

far

be satisfied. Than,

v&7, there exists a generalized solution y(V) of problem (Il.*3)-(1.5). far all If, in adclitionto (2.11,

L(Z f-1

holds

for

all v&,

b&,-f-w%, I..,

then #is

)G 0

vpw

solution is uniquia.

a.e. in

Q,

(2.2)

39

we shall prove the existence of a generalized solution of problem (1.3)-(1.5)by Proof. the Faedo-Galexkinmethod, following which, in the space H==H~'(QIxL(Q), we shall introduce a fundamental sequence (pi*, p~~),...,(Pt.. It~~f,...~ which possesses the following properties: lt (JL,~,IL,O=H for all i 2) for any s, the elements (~**,~~~),-..,(~,‘, PJ are linearly independent 3) the system of element8 (pith ~ZS) is complete in II,that is, the collection of finite linear combinations Le&rr,Ci&Z is dense in H. We shall seek an approximate solution ~‘=~.(~)-(~,.(~), &.(tffof problem (1.3)-(1.5)in the form

where the functions ft.(t) axe determined from the following conditions: (2.3a)

(2.4a) Yw+%~ in L%(Q) as ~-+m, i=l,2.

(2.4b)

By virtue of the linear independence of the elements P,~, i=l,2, system (2.3) is representable in the form

c

d&is

Air‘(t) Ei,*+Efd,s(t)

dt-

1

i=l,2,

j=1,2

,...,

s,

I=-’

where &j,(t) and &j,(t) axe known measurable functions. The resulting system of ordinary linear differential equations when augmented with the initial conditions Ss.=atj,, i=l, 2, j=i, 2,...,s, according to the well-known theorems on the existence of a solution (for example, sea /14, page 417/, is solvable in the interval [O,T]. Let us multiply each of the Eqs.CZ.3) by &t,(t)and sum over 7 from 1 to s. After this, by adding them to one another, we obtain ah.(t) a~dt) ah(t) yidW)+~(a~t--g-) = U at ’ , ,=-i 1-t

2

2

b,a,.(t)+ IXZ: i-, j-1

c,uy,.(t),

yt.(O > + (g,!/,.(t)).

Whence, we shall have

(2.5)

C(E&,~y,,(tf+civy,,~t),y,‘(~)) t-(&Yl60))7 1-1

where

/]*l\is a norm in

1-i

L(Q).

Allowing for conditions (2.1), the boundedness of the set of permissible controls U using Cauchy's inequality, we pass from the equality (2.5) to the inequality

and

where C is a certain positive constant. Now, by fntsgrating inequality (2.6) from 0 to t, we obtain

(2.7)

In particular, it follows from (2.7) that

By applying Gronwall's lenma to the resulting inequality and allowing for the fact that, by virtue of (2.41, Il~oi.~J~CoIIyo,~(, i--l, 2, where C, is a positive constant, we have

Ilyr,(t) II*4 const( ~llIlo~llz+Il~lltiai) 7

i=172.

Whence, by using conditions (2.1), we find that

IIYrr(~)Il-~st,

i=l, 2.

(2.8)

Then, from (2.7), we obtain i--l, 2,...,n. llaYlrlaZl/lLt(9~4const,

(2.9)

It follows from the estimates (2.8), (2.9) that one can separate out a subsequence {&} frcsnthe sequence (#.} such that, as k+m El*-ccpt*

&l-W Let us show that e@f=(P. BY putting s=k

weakly in ~~~O,~;~~(~)), i=l,Z, weakly in &(O, T, H*'(Q)).

(2.1Oa) (2.lGbf

q?=(cp,, cpt)is a generalized solution of problem (1.3)-(1.51,that is, in

(2.3) and fixing T
(2.11b)

Accordfng to (2.10), as

k-m

Q/u, PLq)-t(WI Pd’

weakly in

L(O, T),

i--1,2,

and, consequently (for example, see /11, page 26/I, (2.12)

Next, by virtue of (2.10)‘ we find that, as

k-r= (2.13a)

f2.13b) *-weakly in L,(O,T),i--l, BY passing, in (2.121, to the limit as we obtain

2. k+-

and taking account of (2.12) and (2.13)t

(2.14a)

41

The equalities (2.14) hold for any fixed 7. Then, by virtue of the denseness of the set of finite linear combinations of the elements {~,,,n,,) in the space H, the validity of the equalities " 1.

follows, from which it follows that the function 9 satisfies the system of equations (1.3) and the boundary condition (1.5). Let us now establish that cp~y. We find from the system of Eqs. f1.3), by virtue of the inclusions (2.15) CP,=L,(O, T; w=2)m(o, 2-i few), tpz~~~~o, T; ~3) and conditions

(2.11,

that c%p,lc%=L, (0, T; H-’ ($2)) ,

@hlatEL,

(2.16)

(0, T; L, (52) ) .

when conditions (2.15) and (2.16) are satisfied, the inclusions

(2.17)

i=l,2,

cpi=c(Io, Tl;L*(Q)), hold

by

virtue of Theorem 3.1 from /9, page 33/. Finally, let us show that the function cp satisfies the initial conditions (1.4). According to (2.171, we have, as k-tmr weakly in i=l,2, IrdO) -W(O) G(Q),

and since, by virtue of (2.41, ~~(O~~~Oi

strongly

in

L,(Q),

i=l,2,

then tp~(O)=J/0<. We will now prove the uniqueness of the generalized solution. Let there exist two different soltuions 61"and y' of problem (l-3)-(1.5)for the same initial data. Then, their difference cp=y"--@' will satisfy the system of equations

(2.18b)

wtth the initial and boundary conditions cp,(z,O)=O,r=n, i=l,2, Multiplying each of Eqs.(2.18) by integration, we shall have 2

rp,l.=O.

(P&, t), We add them to one another. Then, after

f n

whence, allowing for conditions (2.1) and (2.2), we obtain the inequality

from which it follows that q-0

and, consequently, that y"=y'. The theorem is proved.

3, The existence of optimal controls. Let us introduce a set of optimal controls U,-(ulud7, Z,~Z(u)=infZ(v,y(uff}, .lU Theorem 2. s--R*

tur : @w-R,

inequalities:

w:s2X(O, T)X Let the conditions of Theorem 1 be satisfied. The integrants satisfy the Caratheodory conditions and the following lv. : (0, T) x2-R

42

Iwk t, u)l~~b.

d5, t, y),-D

t)+clsrl:

for almost all (r,t)EQ and for any y=R'; IW(r*Y(T))IB~r(s)fC,ly(T)I', W(5,y(T))a-D7 for almost all rEs1

and for any Y(T)ER~; Iw.(t, U)I+(t)+C.IuI*,

for almost all

tE(O, T)

w.(t,u)>-D. where (P'L(C?),$&L(Q),

and for any ue:G,

$.=L(O,T),C,CT,C.>O,D,

D,, D.=const. Moreover,

the function W(X,t, 611, .) is convex on R for almost all (x,t)=Q and for any y,=RE WT(X,.) is convex on R’ for almost all x=9 and w.(t,.)is convex on G for almost all tE(O,T). Then, tne set of optimal controls UD+@ and any minimizing sequence {v,}Cu converges weakly to U, in V.

Proof. Let

{u.)Cu

be a certain minimizing sequence, that is, limI(u.)=infI(v)= I,. .-b" 0-u

(3.1)

In order to establish the continuity of the mapping v+Y(v), let us consider the process (1.3)-(1.5)in the presence of a control u.EU: P F=

div(agrady.,)f z buy.&c,v.y.,+g, I-1 in b,,y.r+sv,g.,

yai(x, O)=ydx),

x4,

Q;

i=l, 2,

(3.2a)

(3.2b) y.llz=O,

where y.~=blr(v,), i=l,2. By multiplying each of the Eqs.(3.2) by

1(,(, after a certain amount of manipulation we

have

(3.3) 2

I

h/.,(t)+ UC i-i

cw/.,(t),

y.,(t)) +(g, y.,(t)).

I-I

Allowing for the boundedness of the set of permissible controls and carrying out manipulations of equality (3.3) as in the proof of Theorem 1, it can be shown that

where c,and G are positive constants which are independent of s. A subsequence (u~,~~} may then be separated out from the sequence {V.,Sr.1such that, as k+=, V*‘U

weakly in V,

UaJ,

(3.4a)

yr1*ycl - weakly in L,(O, T; L(n)),

i-i,

2,

(3.4b)

VA*,+Y~weakly in L,(O,T;&YB)),

(3.4c)

&-+$

(3.46)

at

-*at at

weakly in L,(O,T;H-'(Q)),

*-weaklyin

L, (0, T; L (W ).

(3.4e)

We note that the inclusion u=U holds by virtue of the weak compactness of the set Cl. Let us show that y=(y,, y?) is a generalized solution of problem (1.3)-(1.5)corresponding to a control U. By virtue of conditions (3-4), for any (AI, h8)EH0'(P)XLI(Q),

$ (l/r,, h)

d

+--(Yl,h) dt

weakly in L,(O,T),

(3.5a)

43

$ (Y*z,w * z-1

d

(Y,,

z

w

*-weakly in L, (O,T),

~~,~)+~(a-$,$) ,-I

*

(3.5b)

(3.5c)

weakly in L,(O,T).

,

*-weakly in L,(O,T), i=l.2. ,=l Furthermore, by virtue of the continuity of the functions t+y~: [O, T]+&(Q),

(3.5d) we have

Yki(O)+Yi(O) weakly in J%(Q), i=l,2. Now, in order to establish that Y'Y(U). it is necessary to substantiate the passages to the limit (Way,,, hi)-+(CiUY,, h,), Lemma 1. is compact.

i=l,2.

The embedding of the space L*(O,T; H,‘(Q))nH’(O,

(3.6) T; H-‘(Q))

in

L,(O, T; L,(Q))=L,(Q)

Proof. According to Rellich's theorem, the embedding Ho’(Q)+L,(Q) is compact and, furthermore, we have H,‘(Q)cL,(Q)cH-‘(52). The assertion of Lemma 1 then follows from Theorem 5.1 in /11, p.70/. By applying Lemma 1 to conditions (3.51, we obtain, in particular, that, as k-m, Ykl+Ylstrongly in L,(Q).

(3.7)

The passages to the limit (3.6) will then hold (for example, see /lo, p.94/) in the space

a'(07 T) (1.5).

and, consequently, y=Y(u)

actually is a generalized solution of problem (1.3)-

Let us now prove that u is an optimal control.

beacaratheodory function which Lemma 2. (see /15, p.694/). Let o :EX(RmXR”) +R satisfies the inequality 0(&p, 9)2-c for almost all e=.8 and for any (p,q)E R”XR”, where and C=const and, furthermore, the function 0(.&p;) is convex on R” for almost all @s for any p=Rm. Let (p.) be a sequence of measurable functions which converges strongly to P in (&(E))m and {q.} be a sequence of measurable functions which converges weakly to g in (L2(9))“. Then,

, we obtain according to Lemma 2 that By using conditions (3.4)-(3.7) (3.8)

liminfI(v,)>Z(u). )I-NOW,

for conditions (3.1) and (3.8), we shall have I,=limI(u,)=liminfZ(u~)~Z(u). 1-m Il-0,

(3.9)

On the other hand, from the definition of the lower edge of the functional we have (3.10)

Z0GI(u).

By comparing (3.9) and (3.101,we obtain f(n)=lo. Hence it follows that U,#@ and any element u which isaweak limit of any subsequence of the sequence (v.} belongs to Uo,that is, converges weakly to Ua in V. The theorem is proved. W 4. The necessary conditions of optimality. Let us introduce the conjugate state p=p(s, t)-(p,(z, problem 2 s

+ div(a grad pt) + r,

j-l

t))

as a solution of the

dw(u)

(b,r+c,u)~, = -

_$+xb,,p,=!ff-$-

ah

a.e.

in

Q,

a.e.

in

Q,

(4.la)

’

(4.lb)

.*

I-1

p,(z, T)--b(u)/8y,(T) PI@, t) =0

t), p,(z,

a.e.

on

8,

i--i,

2;

(4.lc) (4.ld)

where w(u)=w(z, t,y(s,t;u)), WT(U)=WT(~,Y(z,T; U)). is a generalized solution of the Definition 2. We shall say that the function p-p(u) and it satisfies the system of equations conjugate system (4.1) if p(u)EY

44 x

;(P*,k,)- U i-l

with the limited conditions Pc(T)=-awT(u)&/<(T) for almost all t=(O,T) Theorem 3.

a.e. in

i--l, 2.

Q,

and for any (h,,~)~~~'(~)XL*(~).

Let the conditions of Theorem 2 be satisfied and let 2. a~~(~)/aY~(~)~~~(~), i--1,

aw(n)/ay&(Q),

Then, for all u=Uo, there exists a unique generalized solution of the conjugate system (4.1). U&D,

Proof. When the conditions of Theorem 2 are satisfied, the set of optimal controls and, consequently, the derivatives

ae4/aihv),

aw(u)/ay,

id,

2,

can be considered in the corresponding region Q and 6;1to be functions of the arguments .rand t and, then, the existence and uniqueness of the generalized solution of the conjugate problem is proved using the scheme for the proof of Theorem 1. In order to formulate the conditions for the optimality of the control, let us introduce the function

Ra+R, &&y,: Theorem 4. Let the conditions of Theorem 3 be satisfied; aw/0y,(:eX(O,T)X QXRe+R becarathecdory functions and suppose constants L,Lr>O exist such that, for almost all (z,t)EQ and for any (y",y')~ R’XR’, the inequalities z

I

@w(x,3 lP1 aYf

are satisfied and, for almost all

I a4dwah(t)

-

aw(~,~,~') --I =sL t: Iyf-y{l, ayr I-1

are satisfied. Then, an optimal control ueUo

(4.2)

(y”(T), y’(T))eR*XR”,

r&-J and for any

awr;;;;;;T”

i-&2,

1 G L,$yj”(T,-y,‘(T),,

the inequalities

+=-1,2

(4.3)

satisfies the condition of a maximum

aEptt,v(t;~),p(t;u),u(t))=maxl(t,y(t;u),p(t;u),q) (I*(i

(4.4)

almost everywhere (a.e.) on (0,T). Proof. By using (4.1), we represent an increment of the functional I in the form

A~--j~~(t.y(t;u),p(t;U),~(t))mt,yk

(4.5)

uf, P(t;u),u(t))ldtfq;BO vv=u,

where Ar~Z(v)-Z(u),

rl””

U ,..I

U i-i

aww+ ah

4111

----

awwm

~w~(Y(T;u)+~AY(T))

&h(T) I

aye ’

Ay, >

_aw&V;u)) MY,

+

fit(P)

, AydT)) -

45 and the increments Ay,=Ahy& t) are determined from the system of equations

* BAY,

at-div(agredAY3)f

x WY,+C,[W~(~)- aY~(s)l, j-9

bzjAYf+Cz[aY*(a) - aYt(a) I ,-1

in

Q

(4.6e)

(4.6b)

with initial and boundary conditions Ay,(z, O)=O, z=B, i=l,Z,

y,lr=O.

Let us introduce Thc(O,T), intervals of straight lines with a centre at the point t. into the treatment. It is assumed that T,+tcT, and, as k-+-y mes T&-+0.

(4.7)

Let us consider the permissible controls

u,=(f-~~)a+~~g, Q=G, where xh are the characteristic functions of the sets T,. By putting v=vk in the formula for the increment of the functional (4.5) and dividing we shall have it by mesTI>O,

Y&k!

[~(t,Y(u),p(u),v,)-~(t,Y(u),p(u),a)ldt+~,~O,

(4.8)

x

where nr=n/mesTh. tit us show that nr+O as k+m. We multiply each of the Eqs.(4.6) by AYr and add them to one another. Then, by integrating the resulting equality over the domain 9, we obtain

(4.9)

By using conditions (2.1), the Cauchy inequalities and the boundedness of the set of permissible controls U, we shall have from (4.9)

Here and henceforth, C are positiveconstants. Hence, with the aid of Gronwall's lemma and Hkilder'sinequality, we find that

Allowing for the fact that Y,(u)EL,(O,T; L(Q)) obtain from (4.10)

and applying Hijlder'sinequality, we

(tAY,Ilcc:o.rl;r,c~,, ~Ctlv--u!~~ ,o.~, i , , llA~,ll‘,co,t;~~,.,,, i-l,&

(4.11)

lls-+llr--l, 19sc2. Since IlAY~lt~~,o,=; L~,P~~~CI~Y,~IC~,O.~.~; WQ)I~ i-L 2, we Shall have from (4.11)

By putting V-v*

i--l, 2, l
i==l, 2,

1<8<2.

(4.12)

46 We will now estimate % Using conditions (2.11, (4.2) and (4.3) and Cauchy's Hijlder'sinequalities,we obtain I *

and

Whence, taking account of the fact that the functions t+j)p,(t; u))), i--i, 2 belong to bounded subsets of the space L,(O,T) and applying Hijlder'sinequality, we shall have

(4.13)

Substituting the estimates (4.12) into inequality (4.13) and taking account of the boundedness of the set of permissible controls U, we find (qrl
TA”+(mes

TAoI,

where a=Z/s-l>O, p=l/h+l/s-00. According to (4.71, it follows from (4.14) that k+m. ?j*'O as Let T. be the collection of points (0, T)

(4.14)

(4.15)

which are Lebesgue points for the function

t-(n(t, g(t;u),p(t;u),0,)-z@@,y(t;u),p(t;u),u(t))). Then, for any fixed point

t.=T.,we have from (4.8) taking account of (4.7) and (4.15)

that

14&J tl(t,Y(t;u),P(t;u),u,)-

*--

L

~(t,y(t;u),p(t;u),u(t))ldt-llk gg(L, g(L; u),p(L; u),q)--wt., vq=G. NOW,since

1= Y(k u),p(C u), u(G)~O

mes[ (O,T)\T.]=O,we obtain n(t, y(t;u),p(t;u),u(t))-a(t,y(t;u),P(t;u),4)20 VqEG

a.e. on

(O,T),

which is the condition of a maximum to be proved. The theorem is proved. REFERENCES 1. CHOLJDHURIS.P., Distributed optimal control in a nuclear reactor, 5th Asilomar Conference Circuits and Sys. Pacific Grove, Calif., 1971, Rec. North Hollywood, Calif., 205-209,1972. 2. TZAFESTAS S., Distributed-parameternuclear reactor optimal control, An. System.et.Orientat. NOUV. Colloq. IRIA, Versailles-Rocquencourt,1976, Textes Communs. Rocquencourt S.A., 33-62, 1976. 3. KURODA Y. and MAKINO A., Some problems arising in distributed parameter reactor systems, Lect. Notes Control Inform. Sci., 2, 92-101, 1977. 4. EGOROV A.I., Optimal Control of Thermal and Diffusion Processes, Nauka, Moscow, 1978. 5. NIYEVA R. and CHRISTENSEN G.S., Optimal control of distributed nuclear reactors using functional analysis, J. Optimizat. Theory and Appl., 34, 3, 455-458, 1981. 6. NERONOV V.S., Optimal control of nuclear reactors on heat neutrons, Progr. Internat. COnf. on Power Plant Simulation, Mexico Cuernavaca, Morelos, 19, 1984. 7. DEMBNT'YEV B.A., Kinetics and the Regulation of Nuclear Reactors (Kinetika i regulirovaniye yadernykh reaktorov), Atomizdat, Moscow, 1973. 8. HETRICK D.L., Dynamics of Nuclear Reactors, /Russian translation/,Atomizdat, Moscow, 1975. 9. LIONS J.L. and MARGENES E., Non-homogeneousBoundary Value Problems and Applications, /Russian translation/,Mir, Moscow, 1971. 10. LIONS J.L., Optimal Control of Systems governed by Partial Differential Equations, /Russian translation/$Mir, Moscow, 1972. 11. LIONS J.L., Quelques Methodes de R&solution des Probli?mesaux Limites Non-Linbaires, /Russian translation/, Mir, Moscow, 1972. 12. GAYEVSKII KH., GREGER K. and EACHARIAS K., Non-linear Operator Equations and Operator Differential Equation, /Russian translation/,Mir, Moscow, 1978. 13. LITVINOV V.G., Motion of a Non-Linearly Viscous Liquid, Nauka, Moscow, 1982.

4-l

14. VASIL'YEV F.P., Numerical Methods of Solving Extremal Problems, Nauka, Moscow, 1980. 15. PICTNIKOV V.I., Theorems on the existence of optimizing functions for optimal systems with distributed parameters, Izv. Akad. Nauk SSSR, Ser. Matem., 34, 3, 689-711, 1970.

Translated by E.L.S.

U.S.S.R. Comput.Maths.Math.Phys Printed in Great Britain

.,Vo1.27,No.5,pp.47-54,1987

CC41-5553/87 $~O.CC+O.CC 01989 Pergamon Press plc

GENERATION OF COVERING AND PARTITION TEST PROBLEMS*

R.D. BABAYEV

An algorithm is proposed to generate test problems in two well-known classes of Boolean programming problems weighted covering and partition problems. The problems thus generated have a unique integer optimum, which is easily determined during the actual construction of the problem. None of the test problems are such as can be solved as linear programming problems.

Introduction. Consider the weighted covering problem

(1)

(2)

(3) where

(4) and the weighted partition problem, obtained from (l)-(4) when inequalities (2) are replaced by an equality. In the special case when c,-1 for all j, these problems are known as covering andpartitionproblems, respectively. In this paper we consider weighted covering and partition problems (for brevity, the word "weighted" will be omitted henceforth), a class of combinatorial optimization problems which has numerous applications /l-3/. In view of these applications, considerable efforts have been made to work out methods for solving them. In the overwhelming majority of cases, the efficiency of methods of solution and algorithms for covering and partition problems is evaluated statistically,by processing the results of computatfonal experiments. The performance of such experiments requires test problems whose optimum solutions are known in advance. The present author /4, 5/ has developed test-problem generators for Boolean programming with an arbitrary given matrix of constraints. In these, however, the right-hand sides of the constraints are defined during the actual construction of the test problem and in general cannot be assigned a priori. Hence, they are useless for the generation of covering and partition problems, in which the right-hand sides of the constraints are by definition equal to 1. Two computationallydifficult test problems for the covering problem were given in /6/. In this paper we propose a method of generating test problems in the class of covering and partition problems, with the following properties. 1) The optimum solution of the test problem is unique and known. 2) The solution of the linear relaxation of the test problem does not consistoflntegers. Property 2 is necessary in order to exclude trivial test problems, which are solvable as linear progranrningproblems, or, following /4/, to ensure the essential discreteness of the test problem. Let A denote the constraint matrix of problem (l)-(4): A-=(ac,}. 1. x.at

Construction of the constraint matrix k>l

be some integer that divides m, i.e.,

*Zh.vychfal.Mdt.mat.Ffz.,27,9,1349-1359,1987

A