Model and Algorithm of Industrial Risk Control at
Regional Level
Mikhail Geraskin Elena Rostova
Department of mathematical methods in Economics Department of mathematical methods in Economics
Samara National Research University Samara National Research University
Samara, Russia Samara, Russia
innovation@ssau.ru ORCID: 0000-0002-6432-6590
Abstract—The paper investigates the problem of risk control complex and the price pi does not depend on the production
in the regional industrial complex. We consider the risk volume Qi
distribution among the industrial firms, the insurance sector and
the recovering enterprises. We study the model of the interaction dp i
0 , i 1, n , (1)
in this multi-agent system. We develop the algorithm for the dQ i
choice of the number of the waste utilization firms and the
number of the insurers, which provide the minimum of the where n is the number of the firms in the industrial regional
industrial firm’s risk costs. complex.
Keywords— industrial risk control, insurance, waste utilization, The waste utilization firms and the insurance companies
Pareto equilibrium work in the monopolistic competitive market with a falling
inverse demand curve
I. INTRODUCTION
The regional economy includes the industrial firms, and dp Yj Tk Tk
0 , j 1, m , 0, 0 , k 1, l , (2)
often their number achieves tens of thousands. Each firm is a dY j X k Yk
source of the industrial risk for the environment, legal entities
and individuals, including firm’s employees. The effective risk where pYj is the utilization price of the conventional waste unit
control in the industrial firms is based on the correct risk in the j-th waste utilization firm (WUF), Yj is the external
assessment and the reasonable choice of the control methods, damage accepted for utilization by the j-th WUF, m is the
such as the insurance, the waste utilization and the self- number of WUFs, Tk is the insurance rate of the k-th insurance
insurance. company, l is the number of the insurance companies in the
region, X is the internal damage, Y is the external damage.
The risk control issues were considered in wide range of
studies [1-10]. The risk of industrial firms was analyzed by Assumption 2. The production growth leads to a
using various mathematical tools: the game theory [11], [12], decreasing in return:
the penalties mechanisms [13], the simulation modeling [14],
[15]. The industrial risk was investigated at various levels, C Q Q 0 , i 1, n , (3)
i i
including the regional level [16] – [18] and the firms’ level [1],
[3], [4] – [6]. where Ci is the value of the i-th firm’s costs.
The risk control in the regional industrial complex Assumption 3. An increase in the production volume Qi
combines the regional insurance sector, and the waste leads to an increasing in the possible internal damage Xi; the
utilization firms of the regional recovering sector. In the internal damage Xi is reduced with an increase in the voluntary
region, the industrial firms may interact with many waste risk costs; the internal damage Xi is limited from above due to
utilization companies and insurers. In turn, the regional technology features and production volume
recovering firms and the regional insurers may interact with X i X i max max
many industrial firms. 0, 0, X i (0, X i ], X i 0 , (4)
Q i fi
The number of the industrial firms, the insurance
companies and the waste utilization firms is quite great. where Xmax is the limit of the internal damage, fi is the
Consequently, we consider the problem of determining the voluntary risk costs (VRC) of the i-th firm.
interaction parameters in the big data framework. Further, on Assumption 4. The external damage Yi is proportional to
the basis of the mathematical methods and tools [19 - 22], we the internal damage Xi:
search for the solution of this problem.
Yi
II. METHODS 0 , i 1, n . (5)
X i
We introduce the following assumptions, which determine
the applicability limits of our model. Assumption 5. The voluntary combination insurance is
considered, the wear is not included. The insurance indemnity
Assumption 1. The industrial firms sell their products in is proportional to the insured damage, the indemnity does not
the perfect competitive markets. The product price pi is an exceed the damage:
exogenous constant for the i-th firm in the industrial regional
Copyright © 2020 for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0)
Data Science
W k W k If the regional industrial complex consists of n firms, the
0, 0 , W k X k Y k , k 1, l . (6)
X k Yk profit function of the industrial complex П I is
Assumption 6. The utilization cost of the conventional n n
res
П I (Q i p i W i ) (C Q f i X i V i H i F i ) . (14)
waste unit is constant. i 1 i 1
i
cY = const. (7) The problem of searching for the optimal production
Assumption 7. The external and internal damages of the volume vector Q*=(Q1*, Q2*, …Qn*) and the optimal VRC
i-th firm in the regional industrial complex consist of three vector f*=(f1*, f2*, …fn*) is based on a maximization of the
elements: profit criterion
l m { f *, Q *} arg max ПI. (15)
S U res
ik ij i 1, f i A f ,Q i AQ
k 1 j 1
max max
l m AQ {Q i R : Qi Qi ,Qi 0} , (16)
S U res
ik ij i 1, i 1, n , (8)
k 1 j 1 max max
A f { f i ( ) R : f i ( ) f i , fi ( 0 , R i )} , (17)
where ikS ( ikS ) are the fractions of the external or internal
fi
damages, which are insured in the k-th insurance company, X i (Q i ) e ,
U
ij ( ijS ) are the fractions of the external or internal damages, Y i X i ,
C Q B iQ i ,
which are accepted for utilization by the j-th WUF, ires i
m m
U U
( ires ) are the rest fractions of the external or internal F Y i p Yj ij X i p Yj ij ,
i
j 1 j 1 (18)
damages, which are rectified by the i-th firm.
H i aY i
res
,
According to the assumption 2, the production costs l l
function has the following form [23, 24]. W i X i k ikS Y i k ikS ,
k 1 k 1
i l l
С Q (Q ) B iQ , i (1, imax ], imax (1, 2 ], B i 0 . (9) S
V i X i T k ik Y i T k ik ,
S
i
k 1 k 1
The internal damage function satisfies the assumption 3,
and it has the following form: where R i is the limit value of VRC.
X (Q i , f i ) (Q i ) e
fi
, The vector f* is the solution of problem (14)-(18), and it
has the following coordinates:
max max
(0, ], ( 0 , 1 ], ( Q i ) 0 . (10) 1
fi * ln | ( Q i ) K i | , (19)
The function χ(Qi) expresses the relationship between the
internal damage and the production volume. The parameter ξ where
characterizes the effectiveness of the measures to reducing in
the internal damage The function Х(Q) expresses an l
S
l
S ост
l
S
K i k ik k ik i T k ik
exponential distribution of the damage, which corresponds to
k 1 k 1 k 1
man-made accidents. .
l m m
S ост U U
T k ik a i Yj ij Yj ij
The external damage function satisfies the assumption 4: k 1 j 1 j 1
Y ( X ) X , 0 . (11)
The coordinates of the vector Q* are the solution of the
The coefficient of the accident consequences expansion μ following equation
expresses the ratio of the external damage to the internal i 1 ( Q i *)
damage, taking into account the specifics of the regional pi B i iQ i * 0 i 1, n . (20)
( Q i *)
industrial complex, the geographical features, etc.
The insurance indemnity satisfies the assumption 5: For f*, Q*, the value ПI* is
W ( X , Y ) ( X Y ), 0 1 . (12) n
i 1
П I * (Q i * p i B iQ * fi * ). (21)
i 1
The penalty function has the following form:
If the regional recovering sector includes m of WUFs, the
H i aY i a Х i , a 0 , i 1, n . (13) profit function of this sector П II has the following form
VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 6
Data Science
m n
U U
Further, we consider the problem of searching for the
П II [( p Y c Y ) ( Y ij X ij ) A j ] . (22) interaction parameters. We search for the Pareto equilibrium
j j
j 1 i 1
set of the compromise utilization prices in the following form
The problem of searching for the optimal price vector pYcom arg max { П I , П II } , (32)
pY*=(pY1*, pY2*, …pYm*) is based on the recovering sector’s pY G
profit criterion
G { p Y | П I ( p Y ) 0 П II ( p Y ) 0 } . (33)
p Y * arg max П II (23)
pY R Additionally, we analyze the Pareto set of the compromise
insurance rates in the following form
pY n
U
n
U
pY j pY ( Y i ij X i ij ),
arg max { П I , П III } ,
com
Y j i 1 i 1
T (34)
(24) T
n n
U U
Y i ij X i ij Y j .
{T | T k ( 0 , 1 ) П I ( T ) 0 П III ( T ) 0 }. (35)
i 1 i 1
The vector pY* is the solution of problem (22) - (24), and This problems and theirs solutions enable us to determine
the coordinates of this vector are the interaction parameters of the regional risk-control system
for a variety of the regional industrial firms, the regional
c Yj p Y insurers and WUFs in the regional recovering sector.
p Yj * , j 1, m . (25)
2
III. RESULTS AND DISCUSSION
For pY*, the value of the profit function ПII* is We investigate our model on the basis of the regional
1 m industrial complex of Volga Federal District, which includes
* 2
П II ( Y j ( p Y c Yj ) Aj) . (26) 14 regions and republics of Russian Federation. In each region
4 pY j 1 (or republic) of this District, tens of thousands industrial firms
If the regional insurance sector includes l of insurers, the emit the waste (Table I).
profit function of this sector П III has the following form TABLE I. NUMBER OF INDUSTRIAL FIRMS AND INSURERS IN VOLGA
FEDERAL DISTRICT [25]
l
П III (V k W k ) . (27)
Region Number of Firms Number of Insurers
k 1
The problem of searching for the optimal insurance rate Republic of Bashkortostan 128025 85
vector T*=(Т1*, Т2*, …, Тl*) is based on the regional Republic Of Mari El 20299 63
insurance sector’s profit criterion
Republic of Mordovia 20462 65
T * arg max П III (28) Republic of Tatarstan 160009 93
Tst k ( 0 , 1 )
Udmurt Republic 57393 75
S
n
S
n
Chuvash Republic 45665 68
V k T k ( X i ik Y i ik ),
i 1 i 1
Perm region 102906 73
n
S
n
S T
T k T ( X i ik Y i ik ) , (29) Kirov region 48160 76
i 1 i 1 Xk
n n Nizhny Novgorod region 128867 84
S S
W k k ( X i ik Y i ik ).
Orenburg region 57584 68
i 1 i 1
The problem (27)-(29) has the solution Penza region 45395 67
T*=(Т1*, Т2*, …, Тl*), where T*k is Samara region 135063 80
T k Saratov region 75304 71
Tk * , k 1, l (30)
2 Ulyanovsk region 43765 71
For Tk*, the value of the insurance regional sector profit is The volumes of the waste in Volga Federal District is
1 n presented in table II. In these regions, as a rule, the volumes of
2
П III * X k (T k ) . (31) the waste grow.
4T k 1
We calculate the interaction parameters of the insurances
The optimal parameters of the regional risk-control system sector and WUF sector by using formulas (32) – (35).
maximize the profits of the sectors (agents) individually. All
agents interact in the process of the risk-control.
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TABLE II. VOLUMES OF WASTE IN REGIONS OF VOLGA FEDERAL
DISTRICT (THOUSAND TONS )[25]
Region 2013 2014 2015 2016 2017
Republic of
448942 459365 434914 460888 417781
Bashkortostan
Republic Of Mari El 26869 24619 22348 36437 34993
Republic of Mordovia 36298 34964 31761 40538 53849
Republic of Tatarstan 298102 293675 293594 338227 285914
Udmurt Republic 171910 175820 147945 146845 139201
Chuvash Republic 29428 35878 26870 25341 42818
Perm region 367988 312486 298597 308912 310841
Kirov region 103339 114908 96093 98636 98081
Nizhny Novgorod
125909 125647 132661 149689 150340
region
Orenburg region 512809 410574 490210 512068 475103
Penza region 28401 33478 38856 44483 37388
Samara region 261000 266394 261143 253250 251274
Saratov region 98808 119924 118198 109971 122586
Ulyanovsk region 38102 34182 33195 32619 34028
Next, we analyze the number of WUFs and the number of
the insurers, which provide the minimum of the industrial
firm’s risk costs. The number of WUFs is determined on the
basis of the WUF’s capability Y j for the waste volume Fig. 1. Algorithm of WUFs number selection.
U U
X ij Y ij , and taking into account a minimum of the
utilization expenses F i . Similarly, the number of the insurers
is chosen on the basis of the minimal insurance rate criterion
S S
V i among insurers that meet the condition X ik Y ik X k .
The optimal WUFs number selection procedure is presented as
the algorithm in Figure 1. The iteration procedure allows us to
calculate the number m according to a fulfillment of the
condition X ijU Y ijU Y j . Among WUFs that meet this
condition, we search for the best WUF according to the
minimal waste utilization costs criterion. The choice of the Fig. 2. Agreement of industrial firm and WUF.
insurers is organized in the same way.
If there is no WUF, i.e. j 1, m | X ijU Y ijU Y j , than the
If WUF satisfies the condition j 0 | X ijU Y
U
ij 0
Yj
0
, then
0 industrial firm solves the problem of interaction with several
the parameters of the interaction between the industrial firm WUFs. We consider the interaction of the industrial firm with
and WUF correspond to the Pareto equilibrium set (Fig. 2). two WUFs. The condition for waste utilization contracts is the
existence of compromise prices pY1com, pY2com (Fig. 3).
Figure 2 indicates the solution of problem (32) – (33). In this situation, the price vector pYcom=(pY1com, pY2com) is the
Therefore, the compromise utilization price p Ycomj belongs to solution, where
the following set p c 1 4 A j pY pY
, j=1, 2,
com Y Yj 2
p Yj ( p Y c Yj ) ;
p c 4 A j pY pY 2 2 Yj 2
com Y Yj 1 2
p Yj ( p Y c Yj ) ; , that is not
2 2 Yj 2 p Y c Yj
1 2
4 A j pY pY
and ( p Y c Yj ) . For price
p Y c Yj 1 4 A j pY pY 2 2 Y j 2
empty if ( p Y c Yj )
2
.
2 2 Y j 2 vector pYcom=(pY1com, …, pYmcom), m>2, the solution of problem
(32) – (33) is similar to the solution for m=2.
VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 8
Data Science
The problem of searching for the parameters of the 2) for multiple insurers provided
interaction with the regional insurers is solved similar. The k
S
1, l | X ik
S
Y ik Xk the solution is
solution of problem (34) – (35) has the following form
T T
S S Тcom=(Т1com, …, Тlcom), where T kcom k ; for k .
1) for one insurer provided k0 | X Y Xk the 2
ik 0 ik 0 0 2
T T
solution is T kcom k 0 ; for k 0 ,
0
2 2
Fig. 3. Agreement of industrial firm and two WUFs.
According to our algorithm (Fig. 1), we calculate the Thus, our results allow us to determine the compromise
minimum number of the insurers that are interconnected with waste utilization prices and the compromise insurance rates
one industrial firm in each region (Table III). In this case, we that meet the requirements of the industrial firms, the
consider that the average cost of the conventional waste ton recovering enterprises and the insurance regional sector. In
utilization is equal to 6 thousand rubbles. addition, we construct the firm-insuarer system, i.e., we
calculate the number of insurers, which are nesessary to insure
TABLE III. CALCULATION MINIMUM NUMBER OF INSURERS the firm’s damage. This solution includes the big data as input
Average Estimate parameters that reflect the operating conditions of all agents in
Average the regional industrial risk control system.
Insured Minimum
Waste per
Damage per Number of
Region Firm, IV. CONCLUSION
Insurers, Insurers per
thousand
million Firm
tons
rubbles The developed models describe the functioning of the
Republic of regional industrial risk control system on the basis of big data
3.26 29 490.42 3 regarding to the industrial firms, the insurance companies and
Bashkortostan
Republic Of
1.72 3 332.67 1 the waste utilization organizations. The number of agents in the
Mari El system varies from region to region, but generally exceeds tens
Republic of of thousands. Each firm of the industrial regional complex
2.63 4 970.68 1
Mordovia
Republic of interacts with one or multiple agents of the environmental
1.9 18 446.06 2 protection and the insurance regional sector. The formulated
Tatarstan
Udmurt problems and the presented solutions allow us to determine the
2.43 11 136.08 1
Republic parameters of the agents’ interaction in the regional system
Chuvash based on Pareto equilibrium. Our results may be used by the
0.94 3 778.06 1
Republic
industrial firms to determine the terms of waste utilization and
Perm region 3.02 25 548.58 3 insurance contracts. In the strategies designing, the simulation
Kirov region 2.04 7 743.24 1 results may be useful for WUF and insurers to develop the
Nizhny
requirements for the industrial firms.
Novgorod 1.17 10 738.57 1
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