The paper develops a know mathematical model of multiregional ows of goods. We determine the most probable spatial distribution of ows. Transportation costs depend on the distance between the regions within the \gravity model" approach. We use the shortest way length in a transportation network as a measure of these distances. The underlying mathematics of the model is a convex mathematical programming. The problem is represented as nonlinear minimization of a function with linear constraints which is solved numerically. Developed software is implemented for interactive modeling and visualization of the problem solution. The code is implemented on a high-performance cloudserver platform and consist of modules for simulation, visualization and control. Asynchronous http-queries are used for interaction between the platforms. For the data exchange between these modules a declarative model in the JSON format is implemented. The paper demonstrates a practical problem solution for the gasoline freight transportation for Paci c Russia region and the simulation software usage.
EnModeling multi-regional freight transportation ows for a national economy is the important problem for studying a level of economic integration among regions of the country. In a market economy such ows are formed in accordance with the economic agents own free interests. We assume that such ows are selforganized in a transportation network. In a case of incomplete statistics about all the bilateral ows among regions it is possible to use the equilibrium network approach which allows to get the most probable spatial distribution of ows. The power of such equilibrium models is that they can be used for simulation Copyright c by the paper's authors. Copying permitted for private and academic purposes.
In: S. Belim et al. (eds.): OPTA-SCL 2018, Omsk, Russia, published at http://ceur-ws.org the ows distribution among regions in a `comparative statics' approach when technical and economic parameters of the model are changed.
It seems that the issue of multi-regional ows simulation was pioneered by
Leont'ev [
The paper is organized as follows. First we describe a mathematical model for solving the problem mentioned above. Then a web-based computer service for more convenient analysis and interactive simulation of the problem is presented. And afterwards we provide an example of the proposed mathematical model and software for the multi-regional ows modeling for Paci c Russia. 2 \Gravity" Principle in the Transportation
This section sketches the model, solution for the correspondent mathematical problem and gives its visualization for gasoline freight transportation for Paci c Russia region. 2.1
The model consider the economy of k regions producing and consuming a product. Let zij is an unknown volume of a product delivered from region i to the region j, i; j = 1; : : : ; k. The ow zii is not necessarily assumed to be zero. Strictly positive values of zii corresponds the part of the production of a region that is consumed in this region.
The total ow from region i to j can be interpreted as the total consumption k of the product in the region j which equals P zij . We de ne Vj as a known i=1 in ow (import) of a product to the region j given by o cial statistics. Obviously k Vj = P zij . Total out ow (export) of a product from the region i to other regions i=1 k is the total production of a product in the region i. The value P zij is assumed j=1 to be known from o cial statistics, and let it be de ned as Wi.
The production and consumption of a product de ned above is subject to evident balance equations k k X Vj = X Wi = X X zij : j i i=1 j=1 (1) Equations (1) put an additional restriction on the given values of Wi and Vj so the total ows should be balanced in the considered system of regions.
However since the system of regions cannot be closed and we can observe ows of the assumed regions with others, the aforementioned balance (1) based on statistics would not be observed. This means that for the system of regions j = 1; : : : ; k there are ows of a product between these regions and other unde ned \external" regions. The problem is complicated by the fact that neither the total in ow or out ow 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 zi k+1 and zk+1 j which are unknown but moreover can be identi ed.
Freight transportation is carried out by economic agents under the in uence of the transportation costs which depend on the distances between regions. These distances are estimated as the shortest way length in a transportation network by available modes of transport. Consider the gravity model for transportation costs which can be represented by vij = exp( d Tij ), where vij is `a priori' de ned the ow of products from region i to j, Tij is an assessment of the distance between regions i and j, and d is the parameter that are responsible for the ow sensitivity to distance transportation for a product. Parameter d is non-negative which means that the higher the value of the distance, the smaller an amount of ow between the regions i and j is. It is additionally assumed that vii = 0 for Tii = 0 and vij = vji because we assume that Tij = Tji.
Calibration of such non-negative parameter d with actual o cial statistics is a separate problem of applied statistics. This assessment is carried out by methods such as least squares applied to the linear regression model ln vij = d Tij + "ij for all li; j = 1; : : : ; k and i > j where "ij is a normally distributed residuals of the regression for all i and j.
Paper [
The presence of such features makes it necessary to specify strictly positive freight transportation zij 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 i;j=1;i6=j zij ln(zij =v^ij ) ! min ;
fzijg k+1 X zij = Vj ; j = 1; 2; : : : ; k; i=1 k+1 X zij = Wi; i = 1; 2; : : : ; k; j=1 zij
" > 0; i; j = 1; 2; : : : ; k + 1: 2.2
Implemented computer software for multi-regional freight transportation simulation is used to determine the equilibrium multi-regional freight tra c in the transportation network of railway, road and marine transport of the Paci c Russia regions. The data is used as input data of Rosstat o cial statistical handbooks of di erent years \The regions of Russia. Socio-economic indicators of the multi-regional trade". The main products (commodities) are foodstu , fuel, goods for technical purposes. 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.
As the centers of production and consumption corresponding administrative centers of 9 Paci c Russia regions are considered: Primorskiy krai (Vladivostok), Khabarovskiy krai (Khabarovsk), Amurskaya oblast' (Blagoveshchensk), Evreyskaya avtonomnaya oblast' (Birobidzhan), Respublika Sakha (Yakutsk), Magadanskaya oblast' (Magadan), Sakhalin region (Yuzhno-Sakhalinsk), Kamchatskiy krai (Petropavlovsk-Kamchatsky), Chukotskiy avtonomnyi okrug (Anadyr).
Estimates of the distances between regions are shown in Table 2 which correspond to the shortest paths between the administrative centers of the regions in aggregated transportation network of railway, road and marine transport of Paci c Russia.
Table 3 shows the result for the gasoline freight transportation simulation as solution of the problem described above as the spatial distribution of the most probable ows in the system of the Paci c Russia regions and other \external" regions. These \external" regions are aggregated in one additional region that is added in Tab.3 in the last row under the number \10".
Consider the transportation of motor gasoline between the regions of Paci c Russia (Figure 2, Table 4). From the simulation result shown in Tab.3 it is possible we can conclude that rather large quantities of gasoline is transported from Khabarovskiy krai to Primorsky krai, smaller quantities are transported to Amurskaya oblast', and a very little amounts are transported to Respublika Sakha (Yakutiya) and Kamchatskiy krai, as well as to Evreyskaya avtonomnaya oblast' and Sakhalinskaya oblast'. Second conclusion is that there is a huge amount of gasoline that is needed to be transported from other \external" regions that are not regions of the Paci c Russia. Gasoline was transported mainly to Primorsky Kray, at less extent to Khabarovskiy krai, Amurskaya oblast' and Evreyskaya avtonomnaya oblast'.
The visual presentation of the data from Tab. 3 is shown in Fig. 2. Software that implements the described above mathematical model is realized. It is used to nd the equilibrium multi-regional freight tra c in the transportation network system of regions. An indicator of \connectedness" of the regions for a 2(zij+zji) speci c industry or aggregated for all products is de ned as Lij = Ei+Ej+Ii+Ij where zij is an out ow from the region of i to the region j and Em is a total export of a product from the region m to all other regions, and Im is total imports to the region m from all other regions.
Output results of the program are calculated equilibrium values of the freight tra c between transportation network nodes, the matrix of multi-regional ows and \connectedness' matrix of regions.
Model implementation is carried out in the MPL language [
The control module uses the http-protocol and asynchronous requests. The visualization module sends an asynchronous http-request to the control module for getting input data from the simulation module. The control module initiates an http-request for getting or calculating data about a state of freight transportation. The simulation module forms data in the declarative representation and then sends an answer to the visualization module using the control module. Then the visualization module builds a graph of freight transportation by obtained data. The control module is di erent from existing methods for multi-regional ows because it provides interaction between two environments (environment of calculations and environment of visualization) which are on the di erent platforms. 3.2
The visualization module gives a possibility to represent complicated matrix data about freight ows visually (graphically) in form of oriented graph. Arcs of that graph are freight ows and vertexes are points of destination of these ows.
The visualized transport net allows users to see results of the modeling and to change them interactively, by editing parameters of vertexes and arcs by a program interface.
The visualization module displays di erent variants of the oriented graph in
depend of needed parameters. For this on base of the declarative model there are
formed automatically control elements for the graph of freight transportation [
Interactive editing of the graph supposes a change of the parameters vertexes and arcs of the graph. (Fig. 5). In accordance with user modi cation the dynamic asynchronous request is sent to the simulation module server via the control module and as a results a new declarative model of trade ows will be received and graph visualization will be changed.
The parametric sampling and the visualization of only part of information matters a lot for clarity of the received results especially if parts of these results are entirely (or partially) are independent from each other. Addition graphical parameters are used for the visualization of the transport ows graph: a color and thickness of arcs. The color characterizes a type of communication of ows. The thickness characterizes a volume of ows' loads.
Implementation of the visualization module as a cloud service gives a convenient way to use and display the results of mathematical modeling for many users via the Internet. 4
The paper sketches the \gravitation" model for multi-regional trade. The underlying mathematical problem of the model is based on the approach of nding the most probable spatial distribution of ows in a case of incomplete information about the communication system. The nonlinear convex optimization problem with linear constraints is needed to be solved. The paper demonstrates the simulation of multi-regional freight transportation of gasoline for the Paci c Russia.
The software package is a cloud service, and it consists of three main modules. First, simulation module of trade ows based on the mathematics of the model. Second, the control module, and the third, module for visualization. Modules for control and visualization are implemented on the cloud platform using a multi-agent approach. The interaction between the cloud platform and the high-performance computing platform is organized with dynamic asynchronous requests using a http-protocol.
The software is designed for professionals dealing with the problem of analyzing multi-regional ows of products and planning the strategical plans of regional economic development. Overall methodology and conclusions can be used for multi-regional trade simulation for other regions and municipalities.
Due to a huge dimension of a problem and high practical computational complexity further research could be managed in a way of special numerical algorithms design including parallel ones. The use of a high-performance computing platform could be e cient in a case of a huge number of constraints in the considered mathematical problems.