Optimal allocation of a collective use center Текст научной статьи по специальности «Математика»

UDC 539.3

Vestnik of St. Petersburg University. Serie 10. 2014. Issue 4

V. V. Karelin, V. M. Bure

OPTIMAL ALLOCATION OF A COLLECTIVE USE CENTER

St. Petersburg State University, 7/9, Universitetskaya embankment, St. Petersburg, 199034, Russian Federation

In the paper we propose a method to define a place where a collective use center can be constructed. The center is intended to serve residents of some region. It is assumed that the center should be located in such a way that provide the best access for residents of the region. It is supposed that it will be built during T years, and actively exploited during к years. The problem is to choose a site that provides the most convenient access for most people of the region in the respective age groups during the whole time interval [T,T + к]. The dynamics of the number of residents of the settlements of the region is defined by a special system of equations. The theory of exact penalties is applied to analysis of the problem, and necessary optimality conditions are obtained. Bibliogr. 11.

Keywords: collective use center, exact penalties, necessary optimality conditions.

В. В. Карелин, В. М. Буре

ОПТИМАЛЬНОЕ РАЗМЕЩЕНИЕ ЦЕНТРА КОЛЛЕКТИВНОГО ПОЛЬЗОВАНИЯ

Санкт-Петербургский государственный университет, Российская Федерация, 199034, Санкт-Петербург, Университетская наб., 7/9

В статье рассматривается задача выбора места расположения центра коллективного пользования, который предназначен для обслуживания населения в некотором регионе. Предполагается, что центр коллективного пользования должен быть расположен таким образом, чтобы обеспечить наилучший доступ для всех жителей региона. Центр коллективного пользования будет построен через T лет, после чего он будет активно эксплуатироваться к лет. Задача заключается в таком выборе места для строительства центра, чтобы обеспечить наиболее удобный доступ к нему для большинства жителей региона в соответствующих возрастных группах, в промежуток времени [T, T + к], представляющий собой период активной эксплуатации центра. Динамика численности жителей задается системой уравнений. Теория точных штрафов применена к исследованию поставленной задачи. Найдены необходимые условия оптимальности. Библиогр. 11 назв.

Ключевые слова: центр коллективного пользования, точные штрафы, необходимые условия оптимальности.

1. Introduction. In this contribution we study the problem of how to choose a location for a collective use center. The center is going to be used by people of some region. It should be placed in such a way that allows the best access to the residents of the region during a certain time period, say k years. The center may be an airport, a complex of sports facilities, or a large regional hospital. The non-profit nature is an essential characteristic of the center, that will be used by regional residents independent

Karelin Vladimir Vitalievich — candidate of physical and mathematical sciences, associate professor; e-mail: [email protected]

Bure Vladimir Mansurovich — doctor of technical sciences, professor; e-mail: [email protected] Карелин Владимир Витальевич — кандидат физико-математических наук, доцент; e-mail: [email protected].

Буре Владимир Мансурович — доктор технических наук, профессор; e-mail: vlb310154@gmail.

com

of their income level. This means that the construction should be financing by regional budget, mainly. By T we denote the construction time of the center.

It is assumed that there are l settlements in the region, and we know the population dynamics of the settlements during the construction period and the active period, T + k. The problem is to choose a place where such a center should be placed in order to provide residents from different age groups the most suitable access during the time period [T, T + k], that is the life time of the center. Recent surveys, where similar problems have been studied can be found in [1, 2].

2. Statement of the problem. Let m be the number of age groups, the union of which forms the population of the region. Denote by z(i) the number of residents living

(i s)

at time t in the settlement i, i = l,...,l; and by z(' ) the number of these residents that belong to the group s. Then, the following equality holds

m

z(i) = £ 4^.

s = 1

Denote by x(i) = (x^ ¡xX^) the geographical coordinates of this settlement, and by x(c) = (x1c),xr,c)) the coordinates of the collective use center. Let pci = p(x(c),x(i)) be the distance between the center and the settlement. Taking into account the size of the settlement during the period [T, T + k] we introduce a weight qi of the settlement that is defined as follows

T+ k

£ £ z(i,s)

qi =

t=T sEB

l T+k

£ £ £ z(i,s)

j=1t=TsEB

l

Here B is the set of age groups that will use the center. By definition qi > 0, and £ qi = 1.

i=1

Let D be the area where the collective use center may be constructed. This area is defined by limitations reflecting specific features of the region. Define the performance index as follows:

l

Fi(x(c) ) = Y,(p(x(c),x(0))qi f. (1)

0

Place, where the center should be constructed, is defined as a point that provides the performance index minimal value under constrains defined by D. We assume that during the interval [0, T], T > 0, the dynamics of the number of residents in settlements i = 1, ...,l is given by a system of equations see [3-8]. Divide the interval [0,T] by the points kh, where h > 0, and Kh = T. Let , n = 0,1,...N, be the number of residents of the settlement i in the age group (nh, (n +1)h). Suppose that after the time interval h residents of a certain age move to the next age group, and number of the residents decreases due to the death,

(i(n+1)) (in) _ (in) f (in) <T,(in)\ f0\

zk+1 - zk = Vk \zk , wk ). (2)

Here function ^^ describes the influence of the other age groups on the mortality function /j,kn. The number of residents in the age group [0, h) at the time instant kh + h

is computed by the formula

N-1

4+i = e (kn) (zkn\ *kn)), (3)

n=0

where the reproduction functions depend on the number of residents in the age group, and term * ^ reflects influence of the other age groups. Initially, the number of residents is given

4m) = In- (4)

In (2)-(4) it is supposed that

z{kn) > 0, n,k = 0,1,--- . (5)

There exists an upper age limit Nh,

zkiN) =0, k = 0,1,...,

and by the obvious reason

№ (z^, *km)) < 0, n,k = 0,1,.... If a group is empty, then the corresponding mortality is set to be zero,

^n) (0,

The birth process does not reduce the number of residents

(kn) (zim),*(kn)) >0, n,k=0,i,...

,(in) f0,*=0, n,k = 0,1,... .

but is impossible without parents

(0, CO =0, nk = 0,1,.... (6)

3. Penalty function method [6, 9, 10]. We arrive at the optimization problem

min F1(x (c)) x(c) e D

subject to conditions (2)-(6), where F1(x(c)) is that of (1). The system (2)-(4) can be written in the form

n = g(n),

where n = (,...,VQ )T = (4i0) ,-,4"), V^,-,^-1», C ,...,*KN-1))) '

0 ,...,zK 0 ,...,* K-1 0 K-1

T

and the vector-valued function g = (g1,...,gQ) is given by the respective right-hand sides of (2)-(4), Q = 3NK + N + K +1. We replace each equation of the system by two

inequalities, hj(j?) <0, j = 1, S, S = 9NK + 2N + 3K + 2. Here vector h is defined by the left hand side of equations (5), (6),

h _ I z(iN) z(iN) _z(i0) _z(iN)

h — 1 z0 i-- zK , z0 ■¡■■■J zK ,

$ (0, 4i0)) ,■■■, KN-1)) (0, *K(-Nf1H) , (0, '■■■'

-4N- 1)) (0, 1)0, -/#0) (z(i0), *k0)) ,■■■, -KN- 1 )) (4(N- 1 )), *K(— 1)) m0 (0, *0i0)( ,■■■, Ki-1)) (0, *K(-i-1))(, -M0i0) (0, *0i0)( ,■■■, -K1-1)) (0,

(i0) („(i0) lTf(i0)( ,,(i(N-1)) (z(i(N-1)) ,T,(i(N-1)^ xT

. (i0) / (i0) ^(¿0)\ i,(i(N-1)) ^z(i(N-1)) ^(i(N-1))\ \

M0 ^z0 , T0 J,■■■,^K-1 yzK-1 , TK-1 J J

The function _

u(n, x) =

Q _ _2 s

J2[vj - gj(n)] ^[Wj]2 + hj(n)

2

jj i hj (

j=1~ ' j=1"

is non-negative all over the space RM x RQ x Rs . The vectors n, X satisfy the equation

u(n,X) _ 0

We denote by

W

|(n, x) | n e Rq,x e Rs, u(n, x) _ 0|

the solution set of this equation. Consider the penalty function

F2(n,x,a)_ F1(n,X) + a u(n,X)■ (7)

Following [10], we prove the theorem.

Theorem. Point (n*,X*) is a solution of (7), if there exist vectors v* e RQ, w* e Rs, such that (n*,X*, v*, w*) satisfies the equations

ix*s}2 + hs(V*) = 0, * =

W*X* = Oj

Q f)„. s dh■ _

2 [bs V*s - cs] + v* - E v* $L(r,*) + E w; = 0, S = 1,Q,

j=1 u'ls j=1 U''s

Q da- s dh ■ _

" £ v* -§§^(V*) + E w* = 0, m = 1, M.

j=1 uum j=1 J uum

Proof. Function F1 (n, x) is differentiable in any direction, hence dF1 (n, x) _ {VF1 (n,x)}. Consider the differential properties of u(n,x). Suppose that u(n,x) > 0,

2

and choose a direction p = (q, r), q G RQ, r G RS, number a > 0 and point (n, x) G W. Then we compute the value of w at the point (n + aq, x + ar),

w(n + aq, x + ar) =

j=i

2 S

E nj + aqj - gj (n + aq) + E Ixj + arj ]2 + hj (n + aq)

j=i

1/2

Q

E

j=1

Vj - 9j(v)] +u\qj - E jSrls} + °(«2) J L s=i dns J

+

Since

then

where

+ E

j=1

iXjf + hjirj) 1 +a\2Xjrj + £ ^rqs} + o(a2)

s = 1

1/2

j=1

E nj - gj (n, ) = 0 and E [xj ] + hj (n) = 0

j=1

lim a

a|0 a

w(ap, n + aq,x + ar) - w(n, x)

Q

: E

j=i

■I' ~ >- 77^''-

s=1

S

Vj + E

j=1

Vj ~ 9jiv)

^Xjrj + E

s=1 dns

Wi

Q

\ E

m-gj (n)

S

+ Z j=i

[Xj ]2+hj (n)

[xj] + hj (n)

Q

\ = \ j = 1

nj-Sj (n)

+ E j=i

[Xj ]2+hj (n)

, j = i,Q,

j = s.

Now, the derivative of w in the direction p = (q, r) at (n, x) G W is equal to

w (n, x; p) = ^Z qi q* +

l = 1 s=1

where

QS

Ed9j . \ - ohj ——

j=1 dni j=1 dni

(8)

r* = 2XsWs, s = l,5. Now, consider the case when v(n,x) =0. Let v e RQ, and w e Rs, and observe that

(Q r 1 s r

Y.vi nj - gj (n) +Y.wi [Xj f + hj (n)

—: L J —: L

j=1 j=1

2

2

2

Vj =

2

W

j

2

2

s ' s

Choose a positive number a, a direction p = (q, r), q G RQ, r G RS, and a point (n, x) G W, then

Q {

u(n + aq, x + ar) = max < E Vj nj + aqj - gj (n + aqj)

||v||2 + ||w||2 = ll j=1 L

S r

+ £ wj [Xj + ar]2 + hj (n + aqj) j=i L

We obtain that in the case u(n,x) = 0, the following equality holds

+

Uj(r] + aq, X + ar) = a max < £ Vj qj - £ -pr^-qs + o{ot)

IMr + IMr = 1 j=1 —1

Q r £ dg3

vj Qj-T,

1 s = 1

Q dhj

1 dns

+

+ £ wj 2Xjrj + £ -jP-qs + o(a)

j=1 L s = 1 dns

and

Therefore,

w(9 + ap,n + aq, x + ar) u{ri,X',P) = lim-■

a|0 a

w'(n,x; P) = „ max 2 ^ y^qt q* + V

II ir II 2 II w II 2 = 1 '

|| V|| ^+11 w

l=1

s=1

(9)

Expressions (8) and (9) imply that the derivative of function w in the direction of

p = (q, r), q G RQ, r G Rs is equal to

w'(n,x; P)= max ,\v,p),

v G 8w(n,x)s

here

dw(n,x) = {v = (p*, q*, r*)

r* = 2xsws, s = 1,5.

Q dgj dhj

j=1 j=1

rj rj t ■

where (v,w)G W(d,n,x) C RQ+S■

Due to the fact that functions F1 and w are Lipschitz, there exists a positive constant a0, such that for any a > a0 the set of minimum points of F1 on the set W coincides with the set of minimum points of function F2 on the whole space [10]. It is known [11] that the following condition:

0 G 3F2(V*,x*,a*), is necessary for the minimum of F2. Here

3F2(V*,x*,a*) = 3F1(V* ,x*) + a*dw(V*,x*).

We finally arrive at the conclusion that for a minimum point (n* ,X*) there is a pair (v, w) g Wi(n*,x*), that satisfies the equalities

Q f)„. s dh ■ _

2 [M:-cs]+OTs-«Ev3 f(»)Y) + «Ew3 = s = l,Q,

j=1 j=l arls

awsX*s=0, s = l,S.

To end the proof we simply introduce vectors v* = av, and w* = aw.

4. Conclusion. In this contribution we show how the problem of location of a collective use center can be presented in the form of unconstrained optimization problem. For this optimization problem we derive necessary optimality conditions. Global minimum of the original conditional minimization problem can be fined with the exact penalty function approach.

References

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Литература

1. Bayram Armagan, Solak Senay, Johnson Michael. Stochastic models for strategic resource allocation in nonproit foreclosed housing acquisitions // European Journal of Operational Research. 2014. Vol. 233, issue 1. P. 246-262.

2. Prodhon Caroline, Prins Christian. A survey of recent research on location-routing problems // European Journal of Operational Research. 2014. Vol. 238, issue 1. P. 1-17.

3. Hallam T. G., Ma Z. Persistence in population models with demographic fluctuations // J. Math. Biol. 1986. Vol. 24. P. 327-340.

4. Маценко В. Г. Анализ некоторых непрерывных моделей динамики возрастной структуры популяций. M.: Наука, 1981. 326 с.

5. Douglas J. Jr., Milner F. A. Numerical methods for a model of population dynamics // Calcolo. 1987. Vol. 24. P. 247-254.

6. Swick K. E. A model of single species population dynamics SIAM //J. Math. Anal. 1977. Vol. 32. P. 484-498.

7. Busenberg S., Iannelli M. Separable models in age-dependent population dynamics // J. Math. Biol. 1985. Vol. 22, N 2. P. 145-173.

8. Tuljaprukar S. Population dynamics in a variable environment // Theor. Pop. Biol. 1982. Vol. 21. P. 141-165.

9. Иарелин В. В. Штрафные функции в одной задаче управления // Автоматика и телемеханика. 2004. № 3. С. 137-147.

10. Demyanov V. F., Giannessi F., Karelin V. V. Optimal control problems via exact penalty functions // J. of Global Optimiz. 1998. Vol. 12, N 3. P. 215-223.

11. Демьянов В. Ф., Рубинов A. M. Основы негладкого анализа и квазидифференциальное исчисление. M.: Наука, 1990. 431 с.

The article is received by the editorial office on June 26, 2014.

Статья поступила в редакцию 26 июня 2014 г.