Abstract—It is known that statistical information on operating of the compound multisite system is often far from the description of actual state of the system and does not allow drawing any conclusions about the correctness of its operation. For example, from the world practice of operation of systems of water supply, water disposal, it is known that total measurements at consumers and at suppliers differ to 40 – 60%. It is connected with mathematical measure of inaccuracy as well as ineffective running of corresponding systems. Analysis of widely-distributed systems is more difficult, in which subjects, which are self-maintained in decision-making, carry out economic interaction in production, act of purchase and sale, resale and consumption. This work analyzed mathematical models of sellers, consumers, arbitragers and the models of their interaction in the provision of dispersed single- product market of perfect competition. On the basis of these models the methods, allowing estimating every subject’s operating options and systems as a whole are given.
Keywords—Dispersed systems, models, hydraulic network, algorithms.
I. INTRODUCTION
T is known that, proceeding of a perfect competition market subject is described by demand-and-supply curve, its equilibrium state [1]-[2] reaches in cusp of these curves.
Market is treated as focused, if the proceeding (behavior) of its all subjects (consumers and sellers) is described by one demand curve and one supply curve. The situation is more common, when the market subjects are overextended on the different points as local markets that are interconnected with trade-transportation lines. In this case new subjects are joined into the market (we will treat them as arbitrageurs). These new subjects are mediate layers between producers and consumers.
Purchase and sell processes are carried out between subjects of every point as well as between points of different points, the only difference is that in the latter case commodities must be conveyed by some trade-transportation lines. In this case, every point is characterized by its demand-and-supply curves;
every line of communication is characterized by cargo-
G.Kovalenko is with the Samara State University, Russia.
Ye. N. Amirgaliyev is with Institute of computer science problems and management, Kazakhstan.
A.U. Kalizhanova is with the Kazakh national technical university after K.I.Satpaev, Kazakhstan.
A. H.Kozbakova is with the Kazakh national technical university after K.I.Satpaev, Kazakhstan. ([email protected])
L.Sh. Balgabayeva is with the Kazakh national technical university after K.I.Satpaev, Kazakhstan.
Zh.S.Aitkulovis with the Kazakh national technical university after K.I.Satpaev, Kazakhstan.
carrying (transportation) curve, posing dependence of transfer volume between points from the difference of prices in incidental points. This kind of market is considered as dispersed market [2]-[3]. We treat dispersed market in its equilibrium state, when all the commodities are sold, bought, transferred, correspondingly to the correlations determined by the curves as well as the material balance equations of every point. For the characterization of dispersed market we will make use of machinery of hydraulic network theory [4] that uses cross network goal settings.
In economic-mathematical model building cross network economic goal settings are widely used. Transportation problem of linear programming in cross network goal settings is the most allied one to the hydraulic network theory tasks (problems) and reciprocal to the transportation problem [5].
Transportation network problem with convex cost function of transfer along the diagram’s semicircular arc without restrictions to fluencies place with the help of method of Lagrange multipliers is reduced to subproblem of power flow.
In this work the further buildup of the mentioned problems (tasks) is considered, that makes possible to outline dispersed market, and it is demonstrated that searching task of market’s equilibrium state is reduced to the task of power flow in the cross network with unfixed sortings [4].
Usually for feasible economic systems carriage inwards and outwards volumes, prices, determined in points, and transfer volume are established from statistical accounting, and as a rule, they don’t meet main balance comparators, demand-and-supply and cargo-carrying curves. As a rule demand-and-supply curves and cargo-carrying curves are indeterminate, except that exact these curves attract the greatest interest in economic researches. On this basis, it is possible to solve problems of investment prospects of points.
Thus, the curves’ identification problems arise.
From the mathematical point of view, consideration of real- life objects’ identification tasks equates to necessity for formulation and solving the inverse problem. Very often inverse problems are incorrectly formulated as well as they are difficult to be solved. Universal problems of identification of hydraulic networks and attitude of solving it are analyzed in this work [4].
One of these attitudes which has wide supplement in statistical methods of identification is attitude, which is based on ordinary least squares technique. In this work mentioned attitude is progressed according to economic-mathematical models being analyzed. The problem of minimization of sought quantities’ sums of squares’ excursion from posed on the assumptions is set. Network flows must comply with
Kovalenko A.G., Amirgaliyev Ye.N., Kalizhanova A.U., Balgabayeva L.Sh., Kozbakova A.H., Aitkulov Zh.S.
Evaluation of Parameters of Subject Models and Their Mutual Effects
I
sought quantities in equilibrium conditions. For the formulated problems necessary scenarios are set up, which are based on method of Lagrange multipliers. As the result, running, characterizing fluence movement as well as the Lagrange multipliers comply with the hydraulic flows movement in the network, and the problem converts to superimposition of two problems of power flow. This provides an opportunity to offer acquainted detailed mechanisms of power flow in hydraulic network with minor modification of them.
In conclusion of the work the analyze of suggested models and mechanisms is carried, the problems are considered, concerning their progress and attitude.
II. CLASSIFICATION OF MATHEMATICAL MODELS
For the characterization of models main determinations and designations from the theory of graphs are used. Assume that
H V E
G= , , − coherent resulting orgraph, Е − vertex set of diagram, V − multitude of semicircular arc, Н − mapping H:V→ Е × Е. For every semicircular arc v∈ V mapping H puts in compliance ordered dyad (h1(v),h2(v)) apexes from Е, h1(v) − the onset of the semicircular arc v, h2(v) − endpoint.
Let’s say that from apex i emerges semicircular arc v, if i=h1(v), and penetrates into apex j, if j=h2(v). Lots of semicircular arcs, penetrating into apex i, let us set V + (i), Lots of semicircular arcs, emerging from apex i, let us call V - (i).
Apexes i∈E are explanated as points− local markets of purchase and sale of some homogenous commodity. For every apex i∈Е in compliance assigned variables Рi ,Сi , explanating in compliance as price and volume of balance of payments of this material, Сi=Di − Si, Di − volume of demand, Si− volume of supply. If Сi<0, in this apex supply is dominant than demand and this apex is deposit of continuous production process, if Сi>0, demand is dominant than supply and this is drainage of continuous production process, Сi=0−intermediate vertex.
Semicircular arcs v∈V areexplanated as trading-transport systems, where traffic flow is carried. Let us denote Qv as flow volume, going along semicircular arc v∈V, semicircular arc’s direction indicates to positive direction of flow.
Transport dimension Qv through an arc v depends on price differences between incidental points and characterized by the dependence
Qv =ϕv(Ph2(v) − Ph1(v)), v∈V, (1) Qv increases with growth (Ph2(v) − Ph1(v)), if (Ph2(v) − Ph1(v))<0, subsequently Qv <0, and by contrast, if (Ph2(v) − Ph1(v))>0, subsequently Qv >0. This dependence really reflects supply of point h1(v) to point h2(v) at a difference of the prices (Ph2(v) − Ph1(v))≥0 , and by contrast, supply of point h2(v) to point h1(v)at a difference prices Ph2(v) − Ph1(v)<0 .
Let’s rewrite dependence (1) of the form
Ph2(v)=Ph1(v)+fv(Qv), v∈V, (2) That is the price Ph2(v) of the material (product) at the close of semicircular arc v equals to the product price in the onset of semicircular arc v plus supplemented price for transportation fv(Qv) , where fv , is inverse function for ϕv.
From the economic analysis of mentioned function it is possible to conclude that the price is incremental, uneven, where Qv>0 convex upwards. Analytically such function can be written down in the form
v v v v v
v Q AQ Q
f ( )= α , where ≥0
Av , 0<1+ ≤1 αv .
For every apexes i∈E material-balance equation must be fulfilled
i i V v
v i
V v
v Q C
Q −
∑
=∑
+ −∈
∈ () ()
, i∈E, (3)
where Сi− balance of payments. As it was mentioned above, if Сi<0, then i−source of flow, when a Сi>0, then sink point of flow, when a Сi= 0, then intermediate apex.
Let’s break up several apexes Е into two skew partitions Е1 and Е2. In apexes Е1 the prices of flow are prescribed (apexes with fixed pricesPi*)
*=0
− i
i P
P . i∈E1. (4) In apexes E2 the value of balance of payments is given (apexes with fixed balance of paymentsCi*)
0
*=
− i
i C
C , i∈E2. (5) Problem 1, described by (2) − (5), is the problem of load- flow in hydraulic network (further as problem of load-flow) and for its solution several effective mechanisms have been already made. Their characterization and survey are demonstrated in the investigation [4]. In the case, when several E1 consist of a single element, the problem 1 is called the problem with fixed selection.
Let us set the demand curve byDi=Di(Pi), the supply curve by Si=Si(Pi), then the balance of payments curve will be as Сi=Сi(Pi)=Di(Рi)−Si(Pi). Let us break up a great deal of apexes Е into three noncrossing parts Е1, E2 and Е3. Every of these parts can be empty. In apexes from E1 the prices of flow are set (apexes with fixed prices) : Pi= Р*i, i∈E1. In apexes from Е2 let us set flow volume (apexes with fixed balance of payments) :
C
i= C
i*, i∈E2 . In apexes from Е3 flow volume equals to price coversine (apexes with unfixed balance of payments)Сi= Сi (Pi), i∈E3. (6)
Problem 2, described by (2)–(6) is the problem of load-flow in hydraulic network with unfixed selections and for its solution several effective mechanisms have been also made.
Let us remark that in hydraulic network for solution of the problems (2) – (6) additional assumptions can be superposed
Qv≥0, v∈V*, (7) where V*⊂V (a great number of V* can be empty). Into a good deal of V* penetrate semicircular arcs , interpretable as pumps, ported floats, etc.
Let us convert problem 2 to the problem 1. For this let us add additional apex ζ∉Е and assume Е= {ζ}∪Е. Let us set P(ζ) for the new apex, it will be the apex with fixed price.
In the beginning let us pool apex ζ with apexes i∈E3 additional semicircular arcs v, the onsets of that semicircular arcs is on the apex ζ, and endpoint in i. Let us suppose for these semicircular arcs v ( a great number of it we will denote by V−(ζ) )
Ph2(v)=Ph1(v)+fv(Qv), (8) where fv−1(Ph2(v)−Ph1(v)) =Si (Ph2(v)−Ph1(v))=Si(Pi).
These semicircular arcs will correspond to the supply apex.
Later let us join together apex ζ with apexesi∈E3 additional semicircular arcs v, endpoint of them are on the apex η , the onset is i. Let us suppose for these semicircular arcs v (a great number of it we will denote by V+(η) )
Ph2(v)=Ph1(v)+fv(Qv), (9) where fv−1(Ph2(v)−Ph1(v)) =Di (Ph1(v)−Ph2(v))=Di(Pi) . These semicircular arcs will correspond to the demand apex.
For these apexes i∈E3 let us assume that a Сi*=0, i.e..these apexes became apexes with trivial fixed balance of payment. Also let us rename a good deal of semicircular arcs, supposing V=V∪ V+(ζ)∪V−(ζ). It should be noted that
Qv≥0, v∈V+(ζ)∪V−(ζ) (10) As the result of these manipulations the problem 2 is reducible to the problem 1, in the process the in (10) are similarities to the in (7). Accordingly, further load flow problems we will analyze in scenario of the problem 1.
As it was mentioned above, in the analyzing process of real economic systems values of Qv, v∈V, Рi, Ci, i∈Е, are known from the statistical reporting and often they don’t measure up correlations (2), (3). As a rule, the parametric variables of curves fv are unknown, and exactly these curves are of practical interest. For the further analysis we need to define concretely function patterns. Let us consider them in the shape of
v v v v v
v Q AQ Q
f ( )= α , v∈V, (11)
Note, that for v∈ V+(ζ) ∪V− (ζ) Qv≥0, that’s why
+1
= v
v v v
vQ Q
Q α α , where for v∈V−(ζ) 0< (αv +1)≤1 , for v∈ V+(ζ) (αv +1)<0.
III. THE PROBLEM OF SEARCH FOR PARAMETERS OF SUBJECT
MODELS AND THEIR MUTUAL EFFECT IN THE CONDITIONS OF
DISPERSED MARKET
Let’s consider that values of variables Q*v, А*v, v∈V, С*i, P*i, i∈E, were set and, because of several reasons do not measure up equation systems (2), (3). When for some network element values of these variables are undeterminate, we set certain, reasonable values.
The target of the problem 3 is finding values of variables Av, Qv, v ∈ V, Сi, Рi, i∈E, measuring up equation and in equation systems (2), (3), (10) , and minimizing functional
⎟⎠
⎜ ⎞
⎝
⎛ − + − + − + −
= ∑ ∑ ∑ ∑
∈
∈
∈
∈ iE
i i i E
i
i i i V
v
v v v V
v
v v
vQ Q a A A c C C p P P
q
F ( *)2 ( *)2 ( *)2 ( *)2 2
1
, where qv, av, v∈V, ci, pi, i∈E − positive coefficients, formulating additives to dimensionless form and defining its certainty. The bigger coefficient value, the more certainty of corresponded additives, For network elements with certain, reasonable values Qv, Av, Ci, Pi, conformable vales qv, av, ci, pi must be sufficiently small, for network elements Qv, Av, Ci, Pi which values are beyond questions (for example, Ci = 0 for intermediate vertexes) conformable values qv, av, ci, pi must be sufficiently great.
The methods of a solution of the problem. It was mentioned that the methods of a solution of the problems 1 and 2 were analyzed in details in the work [4], that’s why we will dwell on an issue of methods of solution of the problem 3. Lets generate Lagrange function, the function of minimization of functional F in the setting of assumptions (2), (3)
( )
∑ ∑ ∑ ∑
∈ ∈ ∈ ∈ ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ − −
+
−
− +
=
+ −
V
v iE vV i
i i V v
v v
i v v v h v h
vP P f Q y Q Q C
x F L
)
( ()
) ( 1 ) (
2 ( )
Further the Lagrange multipliers xv, v∈V, will be interpreted as Lagrange flows, multipliers уi ,i∈E as Lagrange prices. Variables Qv, v∈V, Рi, i∈E, will be interpreted as flows and prices. The meaning of these interpretations will be obvious from the further datas.
Necessary criterions of minimum of functional F in the setting of assumptions (2), (3) are further correlations:
. , 0 , 0 , 0
, , 0 , 0 , 0
E y i
L P
L C
L
V x v
L A
L Q
L
i i
i
v v
v
∈
∂ =
= ∂
∂
= ∂
∂
∂
∈
∂ =
= ∂
∂
= ∂
∂
∂
In the case, if the solution of this equation system also complies with in equations Qv>0, v∈V+(η)∪V−(ζ), then the
solution will measure up necessary criterions minimums of the problem 3.
With light development of these correlations, further comes out.
v v v v v h v
h P AQ Q
P2()= 1()+ α , v∈V, (12)
∑
+∑
−∈ ∈
+
=
−
)
( 1
* ) ( i
V v
i i i V v
v
v c
C y Q Q
, i∈ E, (13) Q v
A x
Q Q q y
yh2(v)= h1(v)− v( v− v*)+ v(αv+1) v α , v∈V, (14) )
( *
) ( ) (
i i i i V v
v i
V v
v x p P P
x −
∑
=− −∑
+ −∈
∈ i∈E, (15)
v v v v v
v A xQ Q a
A = *+ αv/ v∈V. (16) Summarizing correlations (13) in accordance with i∈E, further comes out (this comes out very easily from the equation system analysis (15) in matrix form [4])
∑
∑
∈ ∈−
=
E
i i
i E
i
i c
C* y . (17)
Analogously summing correlations (15) in accordance with i∈E , we obtain
∑
∑
∈ ∈=
E i
i i E
i i
iP p P
p * (18) Thus, the problem of searching of minimum of function F is brought to solution of equation systems (12) − (18). It is obvious, that under Lagrange flows xv , prices y i and values A v equation systems (12), (13), (18) are represented load- flow problem of productive flows. And conversely under the fixed productive flows Q v , prices P i and values A v , equation systems (14), (15), (17) are represented load-flow problems of Lagrange flows of the same type as the load-flow of productive flows, Lagrange flows xv appears in the equation (14) in linear fashion, that makes easier the solution.
A little uncommon view have some correlations (17), (18).
Resulting structure of equation defines methods of solution of the problem.
The key methods of solution of the problem of load-flow are divided into nodal and contour line interrelationship.
However, correlations (17), (18) make difficult usage of methods of nodal interrelationship to solve the resulting problems pf load-flow. Let’s demonstrate how to use contour line interrelationship to solve this problem.
Let us distinguish certain edge G′= E,V′,H in the diagram G. Semicircular arcs u∈V\V′, which will be treated as bisecants, in combination with apexes and arcs the edges
define fundamental system of cycles Gu = Eu,Vu,H . Let’s define sense of rotation coinciding with direction of the arc и for every cycle. Also let’s put in function sgnu (v), presented on the multitudes V for every cycle
G
u:⎪⎩
⎪⎨
⎧
∈
∈
−
∈
=
u
u u u
V V v
u arc ctionofthe iththedire coincidesw V
v fthearc directiono
u arc ctionofthe iththedire coincidesw V
v fthearc directiono v
sgn
\ 0 1 1 ) (
It is obvious, that
(Рi2−Рi1)+ (Рi3−Рi2)+ (Pi4−Pi3)+... +
+ (Рik−Рi1) =0,u∈V\V', (19) where i1, i2, i3, ... , ik− apexes of the cycle Си, pointed out in the direction of the arc и. Analogously for the Lagrange pressures
(yi2−yi1)+ (yi3−yi2) + (yi4−yi3)+... +
+ (yik−yi1)=0,u∈V\V'. (20) For every arc v∈V let us denote ΔРv= Ph2(v)−Ph1(v), Δyv= yh2(v)−yh1(v).substituting in (12), (14), (19), (20), we get
v
v v v
v AQ Q
P = α
Δ , v∈V, (21)
v v v v v v v v
v q Q Q x a A Q
y =− ( − *)+ ( −1) α
Δ , v∈V, (22)
∑
∈= Δ
Vu
v
v
u v P
sgn ( ) 0 , (23)
∑
∈= Δ
Vu
v
v
u v y
sgn ( ) 0. (24)
Equations (23) represents Kirchhoff second productive flows law, equations (24) represents Kirchhoff second Lagrange flows law. If these laws hold for the system of fundamental cycles, further they work for any line cycle [4].
Let us fix variable values Av, v∈V и уi, i∈E, in the system of equations (13), (21), (23), after that with any mechanism of contour interrelationship [2] it is possible to get variable values ΔРv, Qv, v∈V out of it. Let us show how it is possible to get values of variables Рi, i∈E with the help of them. Let’s fix in some apexes k∈E certain values Рk. Then system of equations
) ( 1 ) (
2v h v
h
v P P
P = ′ − ′
Δ , v∈V',
( on an edge ) identically define the value of Р'i in all other apexes i∈E. However, these values not always measure up a correlation (18). Let us assume that Pi=P'i+C, i∈E, where.
∑
∑
∈
∈
− ′
=
E i
i E i
i i i
p P P p C
) ( *
It is obvious, that for every i,j∈E the equation Pi−Pj=P'i−P'j is correctly, it means that for the values of variables Рi, obtained with the same method, the equations (12) are fulfilled as well as the equations (18), i.e. we obtain solution of system of equations (12), (13), (18).
Analogously for a system (14), (15), (17), having fixed values of Аv , Q v , v∈V, P i , i ∈ E, with any mechanism of contour interrelationship we get values of variables хv, Δуv, v∈V. Fixing in certain apex k∈E certain value y'i , from a system of equationsΔуv= y'h2(v) −y'h1(v), v∈V,
We find value of y'i, in all apexes. Assuming yi=y'i+c, i∈E, where
∑
∑ ∑
∈
∈
∈
′+
−
=
E i
i E i
i
E
i i
i
c C c
c y
/ 1
*
, we get solution of a system (14),
(15), (17).
IV. CHARACTERIZATION OF ALGORITHM OF SOLVING THE
PROBLEM OF SEARCHING OF PARAMETER MODELS
To organize algorithm of solution of system of equations (12) −(18) it is necessary to organize recomputation of values of Av in formulas Av=Av*+xvQvQvαv/av,v∈V.
This method is characterized in the further algorithm. Let’s introduce further designations:
* AG(Av, xv,v∈V; уi,i∈E) − an algorithm, making possible to define values of Qv, v∈V, Рi, i∈E with the help of values of Av, xv ,v∈V, уi , i∈E
* AL(Av, Qv, v∈V; Рi, i∈E) − an algorithm, making possible to define values of xv, v∈V, yi, i∈E with the help of values of Av, Qv, v∈V, Рi,.
Characterizations of the methods which must be entered into these algorithms are demonstrated above. After that algorithm of solution of system of equations (12) − (18) can be presented as the further iteration process.
Algorithm.
Step 0. Let us assume Av(0)= A*v, xv(0)=0, v∈V, yi(0)=0, i∈E.
Step k ( k =1,2,3,...)
1. With algorithm AG(Av(k−1), xv(k−1) ,v∈V; уi(k−1), i∈E) we define values of Qv(k), v∈V, Рi(k), i∈E.
2. We assume v
k v k v k v v k
v A x Q Q a
A()= *+ (−1) () ()αv/
, v∈V.
3. With algorithm AL(Av(k), Qv(k),v∈V;Рi(k),i∈E) we define values of xv(k), v∈V, yi(k), i∈E.
4. We assume Av(k)=Av*+xv(k)Qv(k)Qv(k)αv/av, v∈V.
5. Let’s pass on to step (k+1)
Multiprocessing of further in equations can be treated as stop criterion
)
1( Δ < ε
∑
∈Vu vv
u
v P
sgn
,∑ ( ) Δ < ε
2∈Vu v
v
u
v y
sgn
.For all
u ∈ V \ V ′
, where ε1,ε2 — maximum allowed value of disparity of the sum of changes in price conforming to productive and Lagrange flows in the system of fundamental cycles.As the result of processing of algorithm, besides values of productive and Lagrange flows and prices, we obtain:
1. Networkwide general estimator F of resulting solution, 2. Through an arc:
− values of parametersAv =Av*+xvQvQvαv/av, and their excursions
v v v
vQ Q a
x αv/ , from set values
A
v*, v∈V;− values of productive flows Qv and their excursions
|Qv−Q*v| from set values Q*v, i∈E;
3. Through apexes of the network:
- value of balance of payments Ci =C*i+yi/ci and its excursions |yi/ci| from set value C*i, i∈E,
- values of prices Pi and their excursions |Pi −P*i | from set values P*i , i∈E.
V. CONCLUSION
In this work for problem of analysis of single-product market:
The cognition of focused and dispersed market is distinguished. The problem of defining the equilibrium state of market is set, which is characterized by the load-flow problem in hydraulic network. The problem of defining parameters of demand-and-supply curves is analyzed as well as the parameters of transportation curves through acquainted values of flows on network arcs and prices on apexes. For this purpose the problem of minimization of sum of squared deviations of sought quantities is set, deviations from measured under the conditions, which must measure up productive flows in network.
It is illustrated, that demanded conditions of minimum of problem being analyzed represent superimposition of load- flow problems:
1. productive flows, 2. Lagrange flows.
Algorithm of solution of set problem was offered, which represents interleaved array of algorithm modification of contour line interrelationship.
Analyzed models and offered algorithms are theoretical base of further analysis of dispersed market. Most likely, further researches must be based on simulation experiments, because for hydraulic network theory nowadays basic results of the algorithm experiments are based upon experiences of solution of dozens of practical problems. It is impossible to identically carry over results of hydraulic network theory into models being analyzed, at minimum the movement of flows are differentiated. For dispersed market on the basis of offered model new problems are arised. New problems count in
classical roadblocks of focused market, concerning characterization of subjects’ behavior under the disequilibrium conditions. As it was mentioned above, algorithms (mechanisms) of solving load-flow problems were based on contour and nodal interrelationship.
Investigation of algorithms of searching of dispersed market’s equilibrium state and characterization of subjects’
behavior of the market will make possible to set its dynamic model. It will allow to explain the possibilities of passage from oligopoly to monopoly and vice versa as well as to estimate influence of distance between points in event of a passage, and it will allow to develop measures, preventing formation of monopoly.
Significance of identification of model parameters was mentioned above. Let’s see that identification was carried out through parameters, which linearly enter into model dependence, and was not carried out through parameters αv, v∈V, which nonlinearly enter, although exactly they define the function elasticity fv(Qv), v∈V. In the theory of hydraulic network most likely αv+1=2 (square flow pattern of liquid), in the models of dispersed market to estimate the parameters αv, v∈V additional investigations are required. Verbally, it was possible to add them to method of search for parameters of problem 3, but in this case after setting the demanded conditions more difficult problem will arise.
REFERENCES
[1] Nikaido H. Convex Structures and Mathematical Economics. − М.,
“Мir”, 1972, p. 514.
[2] Kovalenko A.G. On mathematical modelling of a dispersed market //
Economics and mathematical methods. – 1999. – Т. 35. – № 3. – p. 108 – 115.
[3] Kovalenko A.G. Mathematical models of interindustry balance under the conditions of dispersed market // Economics and mathematical methods.
– 2001. – Т. 37. – № 2. – p. 92–106.
[4] Merenkov A.P., HasilevV. Theory of hydraulic network −М.,Nauka, 1985, p.278.
[5] Golyshtein E.G., Yudin D.B. Problems of linear programming of transportation type.−М., Nauka, 1969, p.382.
