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0PHYSICS AND MATHEMATICS
ANALYSIS OF PRODUCTIVITY OF LEONTIEF LINEAR ECONOMIC MODEL UNDER INTERVAL
DATA UNCERTAINTIES
Esonboyeva M.I.
Student ofphysical and mathematical faculty of Navoi State Pedagogical institute, Uzbekistan
Abstract
In the given work the interval problem statement of the problem the inter-industry balance is considered, being the base of the many linear economic models. Here we refer to the analysis of productivity of models under interval uncertainty by Leontief. It is formulated and proved that the theorem of productivity of the interval linear models by Leontief is right.
Keywords: Leontief linear economic model, inter-industry balance equation, interval uncertainty, interval methods, interval matrix, eigenvalue of Frobenius, productivity analysis.
Recently, a large number of mathematical models have been successfully used in solving real problems and have a practical effect. First of all, they should include the class of linear programming models such as the cutting problem, the diet problem, the transport problem, etc., as well as the intersectoral balance scheme that has become the working tool of the planning bodies of the national economy.
The desire to bring economic and mathematical modeling to reality, to make models more "computable" in the context of non-determinism of data dictates the need to develop adequate methods. One of them is considered to be methods of interval analysis. This paper proposes an interval formulation of the intersectoral balance problem, which is the basis of many linear economic models. The analysis of the productivity of the Leontief model with interval uncertainties is given.
Suppose that each industry produces only single type of product and different industries produce different products. This means that in the production and economic system under consideration, n types of products are produced. Each industry in the production process of its type of product needs products from other industries.
In production planning for a period of time [T0, T] , the task is formulated as follows: for a given
vector C final consumption is required to find the vector X gross output
a
and
X
Ax - c, x > 0
(i)
Equation (1), called the linear equation of intersectoral economic balance, is the classical Leontief equation [1].
In practice, determining the coefficients a, for an
individual enterprise is not difficult, but it is very difficult to find them across the industry. As a rule, instead of the exact values of these coefficients, they are operated with their estimates obtained using one or another method. It is even reasonable to assume that the coefficients of direct production costs are known to us only with some uncertainty, which we assume to be interval.
In other words, let
A=(aij) = ([a.ij, a ]).
Similarly, the demand for a vector of final consumption C is also natural to formulate in an interval form: we are usually satisfied with the situation when real consumption will be maintained within a certain interval C. In the real case, solving the system (1) with respect to X allows you to predict the production volumes by industry, necessary for obtaining the planned final consumption C.
Below we assume familiarity with the basics of interval analysis [2] and the system of notation for interval objects adopted in [3]. In particular, the interval objects in the text will be highlighted in bold, indicating, if necessary, the lower and upper boundaries:
a = [a, a] , and also uses the concept of interval magnitude | a |= max {Ia|, |a| j, which for vectors is
understood component by component.
In the interval case, instead of (1) we have the equation
x = Ax + c, x > 0 (2)
with an interval vector of final consumption by branches c e IRn and an interval matrix
A = (fly ) e IR"X" - coefficients of direct production
costs, which were considered in [1, 4].
In the interval version (2), the question can be formulated as follows:
for what production volumes X for any values of
direct production costs a within a , will we still get
final consumption from a given interval C ?
Here the set of all real vectors X is included in the interval vector - production volumes by n branches, and forms an admissible set of solutions of
the interval linear system
(I-A)x = c [2], where
Ig R
nxn
is the identity matrix. If there is an inverse
matrix ( I — A) 1, then there is a solution to equation (2). In the case when the solution of system (2) exists for any non-negative vector C E C of final demand,
then the Leontief interval model (and the matrix A ) is
productive.
In mathematical terms, the productivity of the model under consideration is fully determined by the
eigenvalue of Frobenius XA of the matrix A . It is with
this assumption that the issue of productivity was investigated in [1] and the following was proved:
Theorem 1. An interval Leontief model is productive if and only if | < 1.
This paper addresses another criterion of productivity:
Theorem 2. If for a non-negative and indecomposable interval matrix A, r is the sum of the elements of each row (column), \v\ ^ 1 and at least for
one row (column) i model is productive.
Proof. Let p^ be a left Frobenius interval vector
for an interval
Rn
matrix
and
e = (l,l,...,l) eRM. Then
Ae = r = {rl,r2,...,rn) . From here we get
p/| = I|r |1(Pa)J < I|(Pa)J • Here we use the
¡=1 ¡=1
fact that for an indecomposable matrix p^ > 0 . On
n
the other hand,
PaT\ — |^a|l|(pa)i|, whence i—1
we
< 1 , then the Leontief interval
get that < 1. According to Theorem 1, the last
inequality gives the productivity of this model.
We will conduct numerical experiments confirming the practical significance of this result, which is expressed in the expansion of the considered classes of problems.
Numerical example. Table 1 contains data on the balance of the three industries for a certain period in the real case.
Table 1.
Balance data for the three industries in the real case.
№ Industry Consuption Gross issue
1 2 3
1 Extraction and processing of hydrocarbons 5 35 20 50
2 Power industry 10 10 20 100
3 Engineering 20 10 19 70
Now suppose that each branch contains interval data, i.e. has interval uncertainty (table 2):
Table 2.
Balance data for the three industries in the interval case.
№ Industry Consuption x Gross issue x
1 2 3
1 Extraction and processing of hydrocarbons [4,6] [33,37] [19,22] [40,52]
2 Power industry [9,11] [9,11] [18,23] [88,104]
3 Engineering [15,22] [9,11] [17,21] [67,75]
Determine the coefficients of the matrix of direct production costs:
'[0.0769, 0.1501] [0.3173, 0.4205] [0.2533, 0.3236] A
A — X-- / x ■ —
y J
[0.1730, 0.2751] [0.0865, 0.1251] [0.2399, 0.3383] [0.2884, 0.5501] [0.0865, 0.1251] [0.2266, 0.3089]
Multiplying the matrix A by the vector e, we obtain the sum of the elements r of each row:
'[0.6475, 0.8940] ^ Ae = r = [0.4996, 0.7383] [0.6016, 0.9839]
Left interval Frobenius vector pA =([40,52],[88,104],[67,75]). \pAr\ = |[106.0570, 197.0505] = 197.0505 = ¿| r | |(p^ ,
1
V 1y
Now calculate
е K^)| = 231.0000,from here = 197.0505/231.0000=0.8530<1.
Since the conditions of Theorem 1 are satisfied. Thus, a similar calculation for columns
1 = 0.83645623538672<1, also states
that the condition of Theorem 1 is satisfied.
Calculations for this problem were carried out on the interval package INTLAB [5], which works in the core of the computer mathematical system MatLab.
Concluding, we note that the real Leontief model reflects only the potential possibilities incorporated in the production technology. In (1) it is assumed that the production process takes place instantly - all intermediate products are produced by the time when there is a need for them. In contrast, model (2) includes both the results of the already completed and the future cycle,
and the intervalness a,
IJ
and Ci allows scrolling sim-
ultaneously the continual set of production cycles and consumption options.
References
1. Z.Kh.Yuldashev, A.A.Ibragimov About interval version of inter-industry balance equation // Computational technologies, Volume 7, №5, 2002. -pp. 8-9.
2. S.P. Shary Finite-Dimensional Interval Analysis. ICT SB RAS -Novosibirsk: Electronic Book (2018): http://www.nsc.ru/interval/Library/InteBooks.
3. R.B.Kearfott, M.T.Nakao, A.Neumaier, S.M.Rump, S.P.Shary, P. Hentenryck Standardized notation in interval analysis. // Computational technologies 2010. Vol.15, №1, pp.7-13.
4. M.E.Jerrell Interval Arithmetic for Input-output Models with Inexact Data // Computational Economics, Kluwer Academic Publishers.-1997. -Vol.10. -P.89-100.
5. S.M. Rump. INTLAB - INTerval LABoratory. In Tibor Csendes, editor, Developments in Reliable Computing, pages 77-104. Kluwer Academic Publishers, Dordrecht, 1999. http://www.ti3.tu-har-burg.de/rump/intlab/index.html.
МОДЕЛЬ МИКРОКРИСТАЛЛИЗАЦИИ РАВНОМОЛЯРНЫХ ДВУХКОМПОНЕНТНЫХ МЕТАЛЛИЧЕСКИХ РАСПЛАВОВ В ДИФФУЗИОННО-РЕЛАКСАЦИОННОМ РЕЖИМЕ
Байков Ю.А.
Российский химико-технологический университет имени Д. И. Менделеева, профессор кафедры физики, д. ф.- м. н., профессор.
Петров Н.И.
Российский химико-технологический университет имени Д. И. Менделеева,
доцент кафедры физики, к. ф.- м. н., доцент.
Антонова Т.Л.
Российский химико-технологический университет имени Д. И. Менделеева, доцент кафедры физической химии, к. х. н., доцент.
Тимошина М.И.
Московский технический университет связи и информатики, доцент кафедры физики, к. т. н., доцент.
Акимов Е.В.
Московский технический университет связи и информатики,
ассистент кафедры физики.
THE EQUAL-MOLAR BINARY METALLIC MELTS' MICRO CRYSTALLIZATION MODEL IN
THE DIFFUSIVE-RELAXATION PROCEDURE
Baikov Yu.A.,
D. Mendeleev University of Chemical Technology of Russia, Full Professor, Physics Department, Dr. Sci (Phys. -Math).
Petrov N.I.,
D. Mendeleev University of Chemical Technology of Russia, Associate Professor, Physics Department, Cand. Sci (Phys.-Math).
Antonova T.L.,
D. Mendeleev University of Chemical Technology of Russia, Associate Professor, physical chemistry Department, Cand. Sci (Chem).
Timoshina M.I.,
Moscow Technical University of Communications and Informatics, Associate Professor, Physics Department, Cand. Sci (Tech.).
Akimov E.V.
Moscow Technical University of Communications and Informatics,
Lecturer, Physics Department.
