CC BY
34
MSC 47D06, 49J15
DOI: 10.14529/ mmp 150108
THE MATHEMATICAL MODELLING
OF THE PRODUCTION OF CONSTRUCTION MIXTURES WITH PRESCRIBED PROPERTIES
A.L. Shestakov, South Ural State University, Chelyabinsk, Russian Federation, [email protected],
G.A. Sviridyuk, South Ural State University, Chelyabinsk, Russian Federation, [email protected],
M.D. Butakova South Ural State University, Chelyabinsk, Russian Federation, [email protected]
We propose a method for the mathematical modelling of the preparation of construction mixes with prescribed properties. The method rests on the optimal control theory for Leontieff-type systems. Leontieff-type equations originally arose as generalizations of the well-known input-output model of economics taking supplies into account. Then they were used with success in dynamical measurements, therefore giving rise to the theory of optimal measurements.
In the introduction we describe the ideology of the proposed model. As an illustration, we use an example of preparing of simple concrete mixes. In the first section we model the production process of similar construction mixtures (for instance, concrete mixtures) depending on investments. As a result, we determine the price of a unit of the product. In the second section we lay the foundation for the forthcoming construction of numerical algorithms and software, as well as conduction of simulations. Apart from that, we explain the prescribed properties of construction mixes being optimal with respect to expenses.
Ключевые слова: Leontieff-type system; production of construction mixes.
Introduction
Take two square matrices L and M of size n, allowing det L = 0. The system of ordinary differential equations
Lx = Mx + u (1)
is called a Leontieff-type system. For the first time system (1) appeared as a generalization of Leontieff input-output economical model taking supplies into account [1]. It was used to calculate the municipal economy of the town of Emanzhelinsk in Chelyabinsk region [2]. Subsequently the Leontieff-type equations theory advanced [3]; numerical algorithms were developed [4], software was constructed [5]. Eventually the results merged into the optimal control theory of Leontieff-type models [6]. This theory received a new impetus for development when it was adapted to the needs of dynamical measurements theory [7]. The foundations of the resulting optimal measurements theory are laid in [8], the first survey of its results appeared in [9], and a direction for its further development is proposed in [10]. Both theories rest on the Sobolev-type equations theory, which addresses the equations of the form (1) in infinite-dimensional Banach spaces Some of the first publications concerning this theory were [11,12]. Presently the number of books dealing with this theory is snowballing; let us just mention [13-22]. In addition, the Sobolev-type equations theory is already extended from Banach spaces to Frechet spaces [23], and now it is being
carried over to quasi-Banach spaces [24]. Finally, note also the closely related theory of algebro-differential systems [25,26] with all its numerous offshoots.
Start with considering of an example of preparing a simple concrete mix. It requires: water xi, sand x2, gravel x3, concrete x4. In addition, we need electricity x5 to rotate the mixer as well as labor x6 to combine everything. We can express the first four components either in mass units (tonnes for instance) or in spatial units (cubic meters for instance). The last two components cannot be measured in those units, however, we can express all components in terms of their cost. Following the approach of Leontieff, we represent the expenses to make concrete as the system of equations
AX + Bx = u, (2)
where A and B are square matrices of size 6 whose entries characterize the capital and current expenses to produce six variants of some grade of concrete. For instance,
t aik x t + tbik xk = ui
k=i k=i
are the total expenses to produce some standard grade of concrete. To produce other variants of the same grade, we want to use different equipment and take twice as much water. We obtain
6 6 y^ &2kxk + 26nxi + bikxk = U2. k=i k=2 AB
of six virtual variants of some grade of concrete.
Observe that for n = 6 system (2), up to changing notation and permutations, coincides with system (1), which we put down as the foundation of our mathematical model of production of construction mixes. This means, in particular, that we can increase without any limit the number of virtual construction mixtures, introducing additional components like, for instance, additives, catalyzers, inhibitors, and so on. Note also that the construction of matrices L and M is a problem of economics engineering, whose solution we leave outside the scope of this article. Henceforth we assume this problem to be solved; we emphasize only that necessarily det L = 0 since the last column of L, following the tradition going back to Leontief, corresponds to capital investments into labor and contains only zeroes.
Thus, system (1) is the basis of the mathematical model of construction mixes production. We consider solutions x = x(t) on the interval [0, t] with t E R+. Each solution is a vector function x = (xi,x2,..., xn); each component xk = xk (t) for k = 1, 2,... ,n and t E [0,t], corresponds to a component of the construction mix. The vector function u = u(t) of t E [0, t] in the right-hand side of (1) stands for the financial expenses to produce the construction mix, in particular, its component uk = uk(t) for k = 1, 2,... ,n and t E [0, t], stands for the expenses required to produce the (virtual) variant k of the construction mix. It may seem that as the parameter t E [0,t], which in our model corresponds to time, grows, the expenses uk = uk(t) can only increase. However, in reality this is not so. For instance, to speed up the process, at some moment of time ti E (0, t) we add a certain expensive catalyzer (for instance, gold) and remove it after some time at the moment ti + At E (0, t). Clearly, at the moment ti the expenses sharply increase, and at the moment ti + At they fall just as sharply.
We may assume that in the beginning of the production (that is, at t = 0) a number of components of the construction mix are already available (although this is not necessary). Thus, we should complement system (1) with the Cauchy initial conditions x(0) = x0 (actually, we can take xo = 0). However, it has been observed [27] that for Sobolev-type equations, and for Leontieff-type systems (1) as their particular case, the Showalter-Sidorov initial condition
is more natural, where P is a projector matrix constructed from L and M. Incidentally, condition (3) is preferable to the Cauchy condition [4, 6] in simulations as well. In the first section we present our results on the solvability of problem (1), (3) as rigorously as possible.
In the second part of this article we explain how we understand the prescribed properties of construction mixes. Since the proposed mathematical model comes from economics, by the prescribed properties we mean the optimization of financial expenses to produce the construction mixes. In other words, we are interested in the minimal cost we are ready to pay to obtain the construction mix of certain composition. We emphasize that in this article we just lay the foundation for a subsequent construction of numerical algorithms and software, as well as conduction of simulations. It is clear that this foundation requires powerful mathematics, which we borrow from both optimal control theory [6] and optimal measurements theory [9]. We are not planning to cover all possible situations immediately, but hope to have discussions with the experts in construction mixtures to improve our mathematical models.
The authors are grateful to A. V. Keller, A. A. Zamyshlyaeva, and M. A. Sagadeeva for their suggestions which helped to improve this article, and to S. A. Zagrebina and N. A. Manakova, whose work gave this article an almost ideal form. We ask the readers to address all praise to the first author, who initiated this study, and all critique to the other two authors.
1. Mathematical Modelling of the Production of Construction Mixtures
Take two square matrices ^d M of size n. The matrix M is called regular with respect to L (or briefly, L-regular) whenever there exists a number a E C such that det(aL — M) = 0. If M is L-regular then there exists at most n points ■ ■ ■, Hm} C C, with m < n,
such that det(^fcL — M) = 0 for k = 1, 2, ■■■, m. Refer to the set aL(M) = ■ ■ ■, Hm}
as an L-spectrum of M. Observe that if det L = 0 then the L-spectrum of M coincides with the spectra of both L-1M and ML-1. Assuming now that M is L-regular, choose the contour y = {^ E C : = r}, where r > max{|^1|, ■ ■ ■, \^m\}, and construct the matrices
P(x(0) - xo) = 0
(3)
Lemma 1.1. If M is an L-regular matrix then
(i) P2 = P and Q2 = Q,
(ii) LP = QL and MP = QM.
Theorem 1.1. If M is an L-regular matrix then there exist matrices L' and M' such that L'L = P, LL' = Q, M'M = - P, and MM' = In - Q.
Here and henceforth ^denotes the size n identity matrix. Proofs of both claims amount to finite-dimensional adaptations of the infinite-dimensional results of subsection 4.1 of [17], and so we omit them.
Furthermore, taking the contour y C C as above, construct the matrix functions
U = h Y RL(Mpt = ^ti Y LLM
where R%(M) = (pL - M)-1L is the right, and LL(M) = L(pL - M)-1 is the left L-M
C
ML
(i) U0 = P and F0 = Q,
(ii) LUt = FtL and MUt = FtM for all t E C, (Hi) Us+t = UsUt and Fs+t = FsFt for all s,t E C.
Proofs of these claims are also adaptations of the results of subsection 4.4 of [17] to our situation, and so we omit them. It is interesting to look at these results from the classical viewpoint, see Ch. 12 of [28].
ML A mid B of size n such that
(i) ALB = [Jni ,Jn2,..., Jnk, Ir}
(ii) AMB = (Ini,In2Ink}.
Here Jni for l = 1, 2,..., k are size nl Jordan biocks, Nr are size r square matrices,
k
y^ ni + r = n, 1=1
( , }
the results of Weierstrass, see [28], Ch. 12. Section 3. Hence,
B-1PB = — [ B-1UL - M)-lA-lALBdp =
2ni JY
2- I {(J-Irn (J-In2 (J-Ink Г\ -Nr )-1]d^ = {On-r , Ir } = P
'1
since
ni
(J - Ini) 1 = -Ini V3 Jni
3=1
for l = 1,2,..., k. ^Similarly, A-1QA = (On-r, Ir} = Q and A-lUtA (On-r ,etNr} = Ut = Fl. Here On-r is the zero size n - r square matrix and
e
j-к ]\тк
tNr V^ 1 Nr
к!
к=0
Furthermore, L' = B{Jn, Jn2,..., №, Ir }A. Indeed
ni > "U2' • • • ' nk
L'L = B J J.., JZ, Ir }AA-1{Jn 1 ,Jn2.., Jnk, Ir }B-1 = BPB-1 = P.
In exactly the same fashion we obtain LL' = Q. Similarly, M' = B{In-r, Or}A.
Consider now problem (1), (3). Refer to a vector function x E C([0,rj; Rn) H C 1((0,r); Rn) satisfying system (1) as a solution to this system (for some vector function u = u(t)). Call the solution x = x(t) to (1) a solution to (1), (3) whenever it also satisfies condition (3) for some vector xo E Rn.
Theorem 1.4. If M is an L-regular matrix then for all x0 E Rn and u = u(t) such that u0 = (In - Q)u E Cp+1([0,r]; Rn) and u1 = Qu E C([0,t]; Rn) there exists a unique solution x = x(t) to problem (1), (3), which, in addition, is given by
p r-1
x(t) = -Y^ Hq M'(In - Q)u0(q)(t) + U x +/ Ut-sL'Qu1(s)ds. (4)
q=o Jo
Here H = (In — P)M'L(In — P) and p = max{n1,U2,... ,nk}■ Since
B 1HB = {Jn1, Jn2 , ■ ■ ■ , Jnk , Or },
Hp of [17] or give it independently, using Theorem 1.3.
Remark 1.1. The components of the vector function x = x(t), t E [0, t], are meant to be dimensionless quantities characterizing the production of one of the (virtual) variants of construction mix depending on the investment u = u(t) at the moment of time t = [0, t]. Therefore, it is necessary to include into our mathematical model the system
y = Sx, (5)
where the vector function y = y(t), t E [0,t], expresses the quantity of construction mixtures produced in the mass (tonnes) or spatial (cubic meters) units per unit time t E [0, t]. Determination of the entries oft he matrix S is an engineering problem, which we leave outside the scope of this article. We assume that matrices L, M, and S are constructed as a result of experiments. However, we can use the vector function of expenses u = u(t) and the vector function of resulting const ruction mixes y = y(t) to obtain the
p = p( t)
function of prices is of the form pk(t) = ykfor k = 1, 2,..., n and t E [0, t]. Thus, at
uk (t)
the moment of time t = t, when the production is complete, the finite price of a unit of
variant k of the construction mixture is pk (t ) = yk ((T |, fo r k = 1, 2,... ,n.
uk (t )
2. Mathematical Modelling of the Production of n Construction Mixes with Prescribed Properties
Let us continue the constructing the mathematical model for production of construction mixtures with prescribed properties. To explain our understanding of
prescribed properties of construction mixes, introduce the space of production states X = {x,x E L2((0,t),Rn)} for some fixed t E R+. As we indicated in the introduction, the components of the vector functions x E X are dimensionless, and their relation to the vector function u = u(t) of production expenses is expressed by equation (1), which, in turn, is constructed experimentally. Only upon finding (1), verifying that M is an L-regular matrix, and determining the number p E {0} U N, we can define the space of financial expenses U = {u,u(p+1 E L2((0,t),Rra)^. Refer to a vector function x E X satisfying almost everywhere on (0, t) system (1) for some u E U as a strong solution to this system. Call the strong solution x = x(t) to (1) a strong solution to problem (1), (3) for some x0 E Rn whenever it also satisfies the Showalter-Sidorov condition (3). By the
x E X
continuous on [0,t]; therefore, the Showalter-Sidorov condition is well-posed in this case.
Theorem 2.1. If M is an L-regular matrix then for all x0 E Rn and u E U there exists a unique strong solution x = x(t), for t E [0, t], to problem (1), (3), which, in addition, is given by (4)-
A proof of Theorem 2.1 follows directly from Theorem 1.4. Moreover, it is not difficult to give an independent proof. We will continue construction of the mathematical model, but let us firstly explain what we mean by the prescribed properties of construction mixes. Unfortunately, in the framework of our model we cannot account for the full variety of properties of construction mixes like, for instance, strength, water resistance, temperature resistance, and so forth, separately. We have only a single integral characteristic, the cost (per unit) of the product at almost every moment of time. Certainly, this characteristic is sufficiently universal and can indirectly represent every combination of prescribed properties. Nevertheless, we are ready to have discussions with the experts in construction mixes aiming to make our mathematical model more adequate to their requirements. Therefore, we introduce the main detail of our mathematical model, the penalty functional
Here pk = pk (t) is the fc-th component of the vector-function of prices on each (virtual) construction mix, pk = — (see Remark 1 for details). We find the vector-function of prices
P = (p1,P2,---,Pn) in the process of production. In contrast to it, the vector-function of prices p° = (pi,p°,... ,рП) is specified at the outset and reflects (in an indirect and integrated way) our prescribed properties of construction mixtures. We obtain the vector-function y = (y1, y2,... ,yn) from (4) and (5); each of its components is the quantity of construction mix produced by the moment t E [0, т] expressed in the spatial or mass units. The square matrices Rk for k = 0,1,..., ж are symmetric and nonnegative by defined. Over all, the second term in the penalty functional J is to control the increase of finances during production. Here we control not only the amounts received, but also the rate, acceleration, and the derivatives up to order p +1. We introduce a such strict control here only for the
yk
completeness of the picture; while conducting simulations, we can replace some Rk by the zero matrices (that is, Rk = On for some k = 0,1, ■ ■ ■, «)■ In the fast term of J we also take into account not only the changes in the states x = (x\,x2, ■ ■ ■, xn) of production, but also their rates. However, here we cannot neglect the rates as yet due to the mathematical properties of the proposed model. Finally, we determine the normalization parameters a E (0,3 =1 — a in the course of simulations, and (•, •) stands for the Euclidean inner product on Rn.
We need a closed convex subset Uad C U, the set of admissible financial expenses. For instance,
Uad = {u E U : u(t) > 0,t E [0,r]},
that is, the cone of nonnegative vector-functions in U. (Recall that a vector-function u : [0, t] ^ Rn is called nonnegative whenever all its components are nonnegative functions.) Let us now state the problem of finding the minima of the penalty functional J on Uad, that is,
J(v) = min J(u). (6)
ueUad
Theorem 2.2. The penalty functional J has a unique minimum point on the set Uad of admissible financial expenses.
Indeed, the set Uad is convex and closed; consequently, it is weakly closed. The J
point v E Uad such that (6) holds. Insert this vector function v = v(t), for t E [0,t], into (4) instead of u(t), and then insert the result into (5). This yields n (virtual) construction mixes whose properties are close to those prescribed.
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Received October 14, 2014
УДК 517.9 DOI: 10.14529/mmpl50108
МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ СОСТАВА СТРОИТЕЛЬНЫХ СМЕСЕЙ С ЗАДАННЫМИ СВОЙСТВАМИ
А.Л. Шестаков, Г.А. Свиридюк, М.Д. Бутакова
Предложен метод математического моделирования состава строительных смесей с заданными свойствами. В основе метода лежит теория оптимального управления системами уравнений леонтьевского типа. Уравнения леонтьевского типа первоначально возникли как обобщения известной экономической модели В. Леонтьева «затраты - выпуск» с учетом запасов. Затем они с успехом были использованы в динамических измерениях, породив тем самым теорию оптимальных измерений. Во введении на описательном уровне обсуждается идеология предлагаемой модели. Для иллюстрации использован пример составления простейших бетонных смесей. В первом параграфе моделируется процесс производства однотипных строительных смесей (например, бетонных смесей) в зависимости от финансовых вложений. В результате определяется цена единицы произведенной продукции. Во втором параграфе закладывается основа для будущего построения численных алгоритмов, конструирования комплексов программ и проведения вычислительных экспериментов. Помимо этого дается объяснение заданных свойств строительных смесей как оптимальных по затратам.
Ключевые слова: системы леонтьевского типа; производство строительных смесей.
Литература
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2. Брычев, C.B. Исследование математической модели экономики коммунального хозяй-
...
3. Свиридюк, Г.А. О сходимости численного решения задач оптимального управления для систем уравнений леонтьевского типа / Г.А. Свиридюк, A.B. Келлер // Вестник Самарского государственного технического университета. Серия: Физико-математические науки. - 2011. - №2. - С. 24-33.
4. Келлер, A.B. Задача оптимального измерения: численное решение, алгоритм программы / A.B. Келлер, Е.И. Назарова // Известия Иркутского государственного университета. Серия: Математика. - 2011. - №3. - С. 74-82.
5. Келлер, A.B. Численное решение задачи оптимального управления вырожденной линейной системой уравнений с начальными условиями Шоуолтера - Сидорова / A.B. Келлер // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. -2008. - № 27 (127), вып. 2. - С. 50-56.
6. Келлер, A.B. Численное исследование задач оптимального управления для моделей леонтьевского типа: дис. .. .д-ра физ.-мат. наук / A.B. Келлер. - Челябинск, 2011.
I ()(ц Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
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Александр Леонидович Шестаков, доктор технических наук, профессор, кафедра «Информационно-измерительная техника:», Южно-Уральский государственный уни-вврситбт (г. Челябинск, Российская Федерация), [email protected].
Георгий Анатольевич Свиридюк, доктор физико-математических наук, профессор, кафедра «Уравнения математической физики», Южно-Уральский государствен-ныи университет (г. Челябинск, Российская Федерация), [email protected].
Марина Дмитриевна Бутакова, кандидат технических наук, доцент, кафедра «Строительные материалы», Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), [email protected].
Поступила в редакцию Ц октября 2014 г.
