=Paper=
{{Paper
|id=Vol-1623/papertl5
|storemode=property
|title=Interregional Transportation Modeling for the Far East of Russia Macro-region
|pdfUrl=https://ceur-ws.org/Vol-1623/papertl5.pdf
|volume=Vol-1623
|authors=Andrey Velichko
|dblpUrl=https://dblp.org/rec/conf/door/Velichko16
}}
==Interregional Transportation Modeling for the Far East of Russia Macro-region==
https://ceur-ws.org/Vol-1623/papertl5.pdf
Interregional Transportation Modeling
for the Far East of Russia Macro-region
Andrey Velichko
Institute for Automation and Control Processes FEB RAS,
5, Radio Str., Vladivostok, 690041, Russia
vandre@dvo.ru
http://iacp.dvo.ru
Abstract. The paper describes two mathematical models of trade flows 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 flows in a case of incomplete information about the system. Second one is
based on multi-commodity network flow 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 traffic of the
Russian Far East region.
Keywords: spatial, interregional, transportation, multi-commodity, multimodal,
gravity, entropy, network, flow, equilibrium, model
1 Introduction
Simulation of interregional flows was proposed by Wassily Leontief [1]. The researchers
of international trade and regional economy develop gravity modeling approach [2] to
explain the exports and imports flows of goods and services within a multiproduct
equilibrium flows on the transport network.
A.G. Wilson and others developed a more general entropy modeling approach [3] to
take into account the incompleteness of information in the application to an equilibrium
modeling for complex communication systems which is applied to simulate interregional
multi-product flows. In 1970s it was shown that gravity modeling approach and the
principle of entropy maximization [4] are interrelated in many ways.
Boyce and others [5]–[7] further applied Wilson’s approach for practical applications
of the transport system of the USA regarding the configuration of the transport network
and multimodal flows concerning various modes of transport. The entropy approach in
Copyright c by the paper’s authors. Copying permitted for private and academic purposes.
In: A. Kononov et al. (eds.): DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org
Interregional Transportation Modeling for the Far East of Russia 395
Soviet transportation science has been widely used for planning spatial development
of cities and industries. At present these approaches are now widely used for passenger
and freight traffic modeling in the transportation systems in Russia and abroad [8]–
[10]. Input data can be incomplete and possible modification assuming interval input
data can be done as discussed in [11].
2 Gravity model of trade flows
In this section the mathematical model, program implementation and visualization of
trade flows for the Far East of Russia Macro-region is given.
2.1 Mathematics of trade flows model
Consider the model of the economy of k regions in each of which there is an n prod-
rm
ucts. Let zij 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 defined as
m = 1, 2, . . . , M where M is the number of different types of transport used for trans-
portation. Here and below the superscripts correspond to the product type and mode
rm
of transport, subscripts correspond regions. Note that the flow zii 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 flow from region r to the j-th region (“total consumption” of the product
M Pk
rm
P
r to the region j) by mode of transport m equals zij . 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
official 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 P k
rm
is also known from official statistics and is defined as Wir .
P
zij
m=1 j=1
The production and consumption of each product r = 1, . . . , n defined in the above
way is subject to obvious balance equations
X k X
M X k X X
rm
zij = Vjr = Wir , (1)
m=1 i=1 j=1 j i
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 flow 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 modification.
To solve this problem let’s aggregate the “external” regions to (k + 1)-th region and
let’s consider additional flows zirm rm
k+1 and zk+1 j which are unknown and moreover are
unidentified.
396 A. Velichko
Trade flows are carried out by economic agents under the influence of the trans-
portation costs which principally depends on the geographical distance between re-
gions. Consider the gravity model for transportation costs which can be represented by
rm
vij = exp(−drm Tij ), where vij rm
is a priori defined the flow 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 flow sensitivity to dis-
tance 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 flow between the regions i and j is. It is additionally assumed
rm rm rm
that vii = 0 for Tii = 0 and vij = vji because of Tij = Tji .
Calibration of non-negative parameters drm with actual official statistics is a sepa-
rate 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
rm
parameters model ln vij = α − drm Tij + δ rm for all li, j = 1, . . . , k and i > j where
rm
δ is a normally distributed residuals of the regression for all r and m.
In papers [3], [4], [10] it is considered an approach of modeling flows in a com-
munication networks corresponding to the principle of the most likely values of the
distribution of flows in conditions of incomplete information when only some balance
equations for these flows are given. Adaptation of this approach for the model of in-
terregional trade flows makes it necessary to minimize the non-linear functions of the
n PM k+1
rm rm rm rm
P P
form zij ln(zij /νij ) on the set of unknown flows zij .
r=1 m=1 i,j=1,i6=j
The presence of such features makes it necessary to specify strictly positive trade
rm
flows zij which is modeled by specifying lower restrictions on flows by preassigned
small parameter ε > 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 flows approach in a case of incomplete information about the communication
system [3, 10, 11]:
X M
n X k+1
X
rm rm rm
zij ln(zij /v̂ij ) → min
rm
, (2)
{zij }
r=1 m=1 i,j=1,i6=j
rm
where v̂ij = exp(−dˆrm Tij ), dˆrm are known estimates of the parameters and constraints
of the problem are given further:
M k+1
X X
rm
zij = Vjr , (3)
m=1 i=1
for all j = 1, 2, . . . , k and r = 1, 2, . . . , n,
M k+1
X X
rm
zij = Wir (4)
m=1 j=1
for all i = 1, 2, . . . , k and r = 1, 2, . . . , n,
Interregional Transportation Modeling for the Far East of Russia 397
rm
zij ≥ε>0 (5)
for all i, j = 1, 2, . . . , k + 1, r = 1, 2, . . . , n, m = 1, 2, . . . , M .
2.2 The solution and visualization
Implemented computer software for interregional trade simulation is used to determine
the equilibrium interregional freight traffic 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 official statistical handbooks of different years “The regions of Russia. Socio-
economic indicators of the interregional trade”. The main products (commodities) are
foodstuffs, fuel, goods for technical purposes.
As the points of import and export products corresponding administrative cen-
ters 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.
Table 1. Estimates of the distances between the regions (Russian Far East), km
Regions 1 2 3 4 5 6 7 8 9
1 0 762 1410 938 3314 2490 990 2490 4490
2 762 0 779 176 2552 2534 1034 2534 4534
3 1410 779 0 603 2035 3182 1813 3182 5182
4 938 176 603 0 2376 2710 1210 2710 4710
5 3314 2552 2035 2376 0 1736 3586 2736 4736
6 2490 2534 3182 2710 1736 0 1500 1000 3000
7 990 1034 1813 1210 3586 1500 0 1500 3500
8 2490 2534 3182 2710 2736 1000 1500 0 2000
9 4490 4534 5182 4710 4736 3000 3500 2000 0
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.
398 A. Velichko
Table 2. Total outflow (production) from the regions (Russian Far East)
Goods Fuel, thous. Coal, Furniture, Meat and Cars, units
tons thous. tons thous. cub. poultry,
Regions m. tons
1 11.2 11.9 0.2 25.9 1175
2 507 263 47.7 431 0.01
3 0.01 495 2 2007 0.01
4 0.01 0.01 0.01 0.01 0.01
5 0.01 3514 10 0.01 0.01
6 0.2 81.1 0.01 0.01 0.01
7 0.01 61 0.01 0.01 0.01
8 0.01 0.01 0.01 0.01 0.01
9 0.01 129 0.01 0.01 0.01
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.
Table 3. Total inflow (consumption) of different goods (Russian Far East)
Goods Fuel, thous. Coal, Furniture, Meat and Cars, units
tons thous. tons thous. cub. poultry,
Region m. tons
1 459 2179 167 662 951
2 63.5 3386 6.6 2048 1424
3 138 693 0.01 154 921
4 45.2 377 0.01 2.4 2
5 42.4 81.4 5.4 485 661
6 0.8 323 0.01 5 33
7 21.6 0.01 0.01 5 564
8 37.2 202 0.01 9.5 27
9 0.2 77.8 0.1 0.01 0.01
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.
Table 4 shows the result of the coal traffic simulation as solution of the problem
to determine the most probable movement of goods in the system of the Far Eastern
regions and trade with other regions. External to the system of the regions in question
in the territory of the aggregated region that table shows the last row (column) under
the number “10”.
Interregional Transportation Modeling for the Far East of Russia 399
Table 4. The simulation results. Transportation of coal, thous. tons
Regions 1 2 3 4 5 6 7 8 9 10
1 0 5.53 2.29 0.78 0.72 1.45 0 0.78 0.36 0
2 100.69 0 44.63 5.18 19.6 52.13 0 27.98 12.79 0
3 172.49 184.94 0 16.44 14.47 60.61 0 32.53 13.53 0
4 0 0 0 0 0 0 0 0 0 0
5 1133 1693.23 301.64 180.99 0 92.41 0.01 78.16 34.55 0.01
6 20.95 41.37 11.61 5.08 0.85 0 0 0.7 0.54 0
7 12.86 26.08 10.21 3.5 2.71 3.03 0 1.63 0.97 0
8 0 0.01 0 0 0 0 0 0 0 0
9 33.11 64.88 16.57 7.74 2.03 3.44 0 1.23 0 0
10 705.91 1369.96 306.05 157.29 41.03 109.91 0 58.99 15.06 0
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.
Fig. 1. The simulation results. Transportation of coal, thous. tons
400 A. Velichko
3 Network flow model of interregional trade
In this section the mathematical model within the more general network flow equilib-
rium framework is discussed and the visualization of simulated trade flows for the Far
East of Russia Macro-region is given.
3.1 Mathematical model
Total network equilibrium problem takes the form of the following nonlinear optimiza-
tion problem
M X Lm Z flm
(m)
X
cl (y) dy → min . (6)
m=1 l=1 0
Here
(m)
cl (y) is the cost of flow y moving on arc l by m-th type of transportation, further
called “mode”,
M is a number of modes of transport used for transportation,
Lm is a number of network arcs for m-th mode,
Pm
flm = dm m
P
lp hp is a flow along the arc l of transport network graph for m-th type of
p=1
transportation, where Pm is a number of all possible paths between any pair of different
regions on the m-th mode. Numbers dm lp equals to one if for fixed m the arc l is a part
of path p. For fixed m the elements {dmlp } form a matrix of dimension Lm × Pm .
Then let’s define unknown variables hm p that is a total flow along the path p for
m
mode m and zij is an unknown trade flow from the region i to the region j by mode
m.
Then we have such constraints:
Pm
ij
X
hm m
p = zij , (7)
p=1
where Pijm is a number of all possible paths between the region i and j for mode m
which can be represented as
XPm
apij hm m
p = zij , (8)
p=1
where apij = 1, if the path p connects region i and j.
Balance constraints for transportation between the regions have the form
M X
X N M X
X N
m m
zij = Vj , zij = Wi . (9)
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 official statistics.
Interregional Transportation Modeling for the Far East of Russia 401
Solving the problem (6) under constraints (8) and (9) equilibrium distribution of
flows over the network expressed complementary slackness conditions for flows in a
form
XLm
hm
p ( cm m m m
l (fl )dlp − uij ) = 0, (10)
l
which reflects the principle of Wardrop for network equilibrium that, firstly, if the flow
hm m
p along the path p is not equal to zero i.e. hp > 0, then the total cost of flow moving
L m
cm cm m m m
P
p = l (fl )dlp on all the paths p = 1, Pij are equal to the equilibrium value costs
l=1
umij 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. cm m
p > uij , then
m
all hp = 0. All these mean that none of the unloaded paths do not have a lower cost
for transportation than cmp .
Fig. 2. Aggregated transportation network of Far East of Russia
402 A. Velichko
3.2 The visualization of a solution
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 (Ma-
gadan region), 7 - Yuzhno-Sakhalinsk (Sakhalin region), 8 - Petropavlovsk (Kam-
chatka 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 flow model for aggregated is
shown in Fig. 3.
Fig. 3. The simulation result. Aggregated transportation flows
4 Conclusion
The paper sketches two mathematical models of trade flows among the territories of a
region or country. They are based on a most probable values of flows in a case of in-
complete information about the communication system and multi-commodity network
flow equilibrium approach. The mathematics of the models assumes nonlinear convex
optimization problem with linear constraints.
Interregional Transportation Modeling for the Far East of Russia 403
The paper demonstrates the simulation of interregional freight traffic 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 flows 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 includ-
ing parallel ones which could be efficient in a case of a huge number of constraints in
the considered mathematical problems and therefore high-performance computations
would be needed.
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