The paper describes two mathematical models of trade ows among the territories of a region or country. First model uses the approach of modeling the complex communication system to determine the most probable values of trade ows in a case of incomplete information about the system. Second one is based on multi-commodity network ow equilibrium approach. Transport costs between the territories are modeled within the framework of gravity model. The payment for transportation depends on the distance between the regions, this distance is estimated as the shortest way length in a given transport network or geographical distance. The mathematical formulation of the problems belongs to the class of convex mathematical programming problems and assumes the numerical solution of nonlinear optimization problem with linear constraints. The paper demonstrates the simulation of interregional freight tra c of the Russian Far East region.
Simulation of interregional ows was proposed by Wassily Leontief [
A.G. Wilson and others developed a more general entropy modeling approach [
Boyce and others [
In this section the mathematical model, program implementation and visualization of trade ows for the Far East of Russia Macro-region is given. 2.1
Consider the model of the economy of k regions in each of which there is an n products. Let zirjm is an unknown number of a product of r-th type, r = 1; : : : ; n delivered from the i-th region in the j-th, i; j = 1; : : : ; k, the mode of transport is de ned as m = 1; 2; : : : ; M where M is the number of di erent types of transport used for transportation. Here and below the superscripts correspond to the product type and mode of transport, subscripts correspond regions. Note that the ow zirim is not necessarily assumed to be zero. The case it is strictly positive corresponds the part of the region production that is consumed in the same region.
The total ow from region r to the j-th region (\total consumption" of the product M k r to the region j) by mode of transport m equals P P zirjm. The last sum also m=1 i=1 equals to Vjr that is a known cumulative import of product r to the region j given by o cial statistics. Total export of product r from the region i in other regions (\total production" of product r to the region i) transported by all modes of transport is M k P P zirjm is also known from o cial statistics and is de ned as Wir. m=1 j=1
The production and consumption of each product r = 1; : : : ; n de ned in the above way is subject to obvious balance equations
M k k X X X zirjm = which imposes an additional restriction on the given values of Vjr and Wir.
However, since the real system of regions may be not closed and we can observe the trade of the assumed regions with others, the aforementioned balance (1) formed on the statistical data would not be observed. This means that there is a ow of products between these k regions and other unknown \external" regions in relation to the considered system of regions. The problem is complicated by the fact that neither the total import or export of such \external" regions are known. Obviously in this case the model requires modi cation.
To solve this problem let's aggregate the \external" regions to (k + 1)-th region and let's consider additional ows zir mk+1 and zkrm+1 j which are unknown and moreover are unidenti ed. (1)
Trade ows are carried out by economic agents under the in uence of the transportation costs which principally depends on the geographical distance between regions. Consider the gravity model for transportation costs which can be represented by virjm = exp( drmTij ), where virjm is a priori de ned the ow of products from the i-th region in the j-th, Tij is an assessment of the geographical distance between regions i and j, and drm are the parameters that are responsible for the ow sensitivity to distance for the product r and the mode of transport m used to transport it. Parameters drm are non-negative which means that the higher the value of the distance between, the smaller an amount of ow between the regions i and j is. It is additionally assumed that virim = 0 for Tii = 0 and virjm = vjrim because of Tij = Tji.
Calibration of non-negative parameters drm with actual o cial statistics is a separate problem of applied statistics. This assessment is carried out by methods such as least squares (LS) applied to the regression model which is represented by a linear by parameters model ln virjm = drmTij + rm for all li; j = 1; : : : ; k and i > j where rm is a normally distributed residuals of the regression for all r and m.
In papers [
r=1 m=1 i;j=1;i6=j
The presence of such features makes it necessary to specify strictly positive trade ows zirjm which is modeled by specifying lower restrictions on ows by preassigned small parameter " > 0.
Then we solve a nonlinear optimization problem with already considered balance
equations as linear constraints and the objective function that is motivated by the most
probable ows approach in a case of incomplete information about the communication
system [
X r=1 m=1 i;j=1;i6=j zirjmln(zirjm=v^irjm) ! min ; fzirjmg where v^irjm = exp( d^rmTij ); d^rm are known estimates of the parameters and constraints of the problem are given further: for all j = 1; 2; : : : ; k and r = 1; 2; : : : ; n,
M k+1 X X zirjm = Vjr; for all i = 1; 2; : : : ; k and r = 1; 2; : : : ; n, (2) (3) (4) zirjm " > 0 (5) for all i; j = 1; 2; : : : ; k + 1; r = 1; 2; : : : ; n; m = 1; 2; : : : ; M . 2.2
Implemented computer software for interregional trade simulation is used to determine the equilibrium interregional freight tra c in the transport network of railway, road and sea transport of the Far Eastern regions of Russia. The data is used as input data of Rosstat o cial statistical handbooks of di erent years \The regions of Russia. Socioeconomic indicators of the interregional trade". The main products (commodities) are foodstu s, fuel, goods for technical purposes.
As the points of import and export products corresponding administrative centers of 9 Far East regions are considered: Primorsky Krai (Vladivostok), Khabarovsk (Khabarovsk), the Amur Region (Blagoveshchensk), the Jewish Autonomous Region (Birobidzhan), the Republic of Sakha - Yakutia (Yakutsk), Magadan region (Magadan), Sakhalin region (Yuzhno-Sakhalinsk), Kamchatka region (Petropavlovsk-Kamchatsky), Chukotka Autonomous Okrug (Anadyr).
Estimates of the distances between regions are shown in Table 1 which correspond to the shortest paths between the administrative centers of the regions in question in the transport network of railway, road and sea transport of the Far East of Russia. Primorskiy kray - 1, Khabarovskiy kray - 2, Amurskaya oblast' - 3, Evreyskaya avtonomnaya oblast' - 4, Respublika Sakha (Yakutiya) - 5, Magadanskaya oblast' - 6, Sakhalinskaya oblast' - 7, Kamchatskiy kray - 8, Chukotskiy avtonomnyy okrug - 9, Other regions - 10.
The visual presentation of the data from Tab. 4 is shown in Fig. 1.
In this section the mathematical model within the more general network ow equilibrium framework is discussed and the visualization of simulated trade ows for the Far East of Russia Macro-region is given. 3.1
Total network equilibrium problem takes the form of the following nonlinear optimization problem
M XLm Z flm X where Pimj is a number of all possible paths between the region i and j for mode m which can be represented as where aipj = 1, if the path p connects region i and j.
Balance constraints for transportation between the regions have the form M N M N X X zimj = Vj ; X X zimj = Wi: m=1 i=1 m=1 j=1
Value Vj is a known cumulative import to the region j and total export from the region i in other regions is also known as Wi both given by o cial statistics.
Solving the problem (6) under constraints (8) and (9) equilibrium distribution of ows over the network expressed complementary slackness conditions for ows in a form uimj ) = 0; (10) which re ects the principle of Wardrop for network equilibrium that, rstly, if the ow hpm along the path p is not equal to zero i.e. hpm > 0, then the total cost of ow moving
Lm cpm = P clm(flm)dlmp on all the paths p = 1; Pimj are equal to the equilibrium value costs l=1 uimj which is independent of the path. Secondly, if for some way between the regions i and j total expenses is strictly greater than equilibrium value costs, i.e. cpm > uimj , then all hm = 0. All these mean that none of the unloaded paths do not have a lower cost p for transportation than cpm.
We consider 12 cities and administrative centers of corresponding regions of the Far East of Russia as a nodes of transport network: 1 - Vladivostok (Primorye), 2 Khabarovsk (Khabarovsk Territory), 3 - Birobidzhan (Jewish autonomous region), 4 - Blagoveshchensk (Amur region), 5 - Yakutsk (Sakha-Yakutia), 6 - Magadan (Magadan region), 7 - Yuzhno-Sakhalinsk (Sakhalin region), 8 - Petropavlovsk (Kamchatka region), 9 - Anadyr (Chukotka Autonomous region), 10 - Komsomolsk-on-Amur (Khabarovsk Territory), 11 - Sovetskaya Gavan (Khabarovsk Territory), 12 - Tynda (Amur region).
Aggergated transportation network for all modes is represented on Fig. 2.
The visual presentation of the simulation for network ow model for aggregated is shown in Fig. 3. The paper sketches two mathematical models of trade ows among the territories of a region or country. They are based on a most probable values of ows in a case of incomplete information about the communication system and multi-commodity network ow equilibrium approach. The mathematics of the models assumes nonlinear convex optimization problem with linear constraints.
The paper demonstrates the simulation of interregional freight tra c of the Russian Far East region. The implemented software package is designed for professionals from various ministries and departments dealing with the problem of optimizing the planning and interregional ows industries products based on their multi-product in multimodal transport networks, and can be used for interregional trade simulation of other set of regions in Russia and the world.
Further research could be developed in a way of numerical algorithms design including parallel ones which could be e cient in a case of a huge number of constraints in the considered mathematical problems and therefore high-performance computations would be needed.