-The paper investigates the problem of risk control in the regional industrial complex. We consider the risk distribution among the industrial firms, the insurance sector and the recovering enterprises. We study the model of the interaction in this multi-agent system. We develop the algorithm for the choice of the number of the waste utilization firms and the number of the insurers, which provide the minimum of the industrial firm's risk costs.
The regional economy includes the industrial firms, and often their number achieves tens of thousands. Each firm is a source of the industrial risk for the environment, legal entities and individuals, including firm’s employees. The effective risk control in the industrial firms is based on the correct risk assessment and the reasonable choice of the control methods, such as the insurance, the waste utilization and the selfinsurance.
The risk control issues were considered in wide range of studies [1-10]. The risk of industrial firms was analyzed by using various mathematical tools: the game theory [11], [12], the penalties mechanisms [13], the simulation modeling [14], [15]. The industrial risk was investigated at various levels, including the regional level [16] – [18] and the firms’ level [1], [3], [4] – [6].
The risk control in the regional industrial complex combines the regional insurance sector, and the waste utilization firms of the regional recovering sector. In the region, the industrial firms may interact with many waste utilization companies and insurers. In turn, the regional recovering firms and the regional insurers may interact with many industrial firms.
The number of the industrial firms, the insurance companies and the waste utilization firms is quite great. Consequently, we consider the problem of determining the interaction parameters in the big data framework. Further, on the basis of the mathematical methods and tools [19 - 22], we search for the solution of this problem.
We introduce the following assumptions, which determine the applicability limits of our model.
Assumption 1. The industrial firms sell their products in
the perfect competitive markets. The product price pi is an
exogenous constant for the i-th firm in the industrial regional
(
The waste utilization firms and the insurance companies work in the monopolistic competitive market with a falling inverse demand curve dp Yj 0 , j 1, m , T k 0 , T k 0 , k 1, l , dY j X k Y k where pYj is the utilization price of the conventional waste unit in the j-th waste utilization firm (WUF), Yj is the external damage accepted for utilization by the j-th WUF, m is the number of WUFs, Tk is the insurance rate of the k-th insurance company, l is the number of the insurance companies in the region, X is the internal damage, Y is the external damage.
Assumption 2. The production growth leads to a decreasing in return:
C Q iQ i 0 , i 1, n , where Ci is the value of the i-th firm’s costs.
Assumption 3. An increase in the production volume Qi leads to an increasing in the possible internal damage Xi; the internal damage Xi is reduced with an increase in the voluntary risk costs; the internal damage Xi is limited from above due to technology features and production volume X i 0 , X i 0 , X i ( 0 , X imax ], X imax 0 , Q i f i (4) where Xmax is the limit of the internal damage, fi is the voluntary risk costs (VRC) of the i-th firm.
Assumption 4. The external damage Yi is proportional to the internal damage Xi: complex and the price pi does not depend on the production volume Qi Y i 0 , i 1, n .
X i
Assumption 5. The voluntary combination insurance is considered, the wear is not included. The insurance indemnity is proportional to the insured damage, the indemnity does not exceed the damage: W k 0 , W k 0 , W k X k Y k , k 1, l .
X k Y k
Assumption 6. The utilization cost of the conventional waste unit is constant.
cY = const.
Assumption 7. The external and internal damages of the i-th firm in the regional industrial complex consist of three elements:
If the regional industrial complex consists of n firms, the profit function of the industrial complex П I is
n n
П I (Q i p i W i ) (C Q i f i X ires V i H i Fi ) . (
The problem of searching for the optimal production volume vector Q*=(Q1*, Q2*, …Qn*) and the optimal VRC vector f*=(f1*, f2*, …fn*) is based on a maximization of the profit criterion { f *, Q *}
arg max f i A f ,Q i AQ
П I .
AQ {Q i R : Q i Q imax , Q imax 0} ,
A f { f i ( ) R : f i ( ) f imax , f imax ( 0, R i )} , (
According to the assumption 2, the production costs function has the following form [23, 24].
С Q i (Q ) B i Q i , i (1, imax ], imax (1, 2 ], B i 0 . (9) The internal damage function satisfies the assumption 3, and it has the following form:
X (Q i , f i ) (Q i ) e f i ,
( 0, max ], max ( 0, 1], (Q i ) 0 .
(
The function χ(Qi) expresses the relationship between the internal damage and the production volume. The parameter ξ characterizes the effectiveness of the measures to reducing in the internal damage The function Х(Q) expresses an exponential distribution of the damage, which corresponds to man-made accidents.
The external damage function satisfies the assumption 4:
Y ( X ) X , 0 .
The coefficient of the accident consequences expansion μ expresses the ratio of the external damage to the internal damage, taking into account the specifics of the regional industrial complex, the geographical features, etc.
The insurance indemnity satisfies the assumption 5:
W ( X , Y ) ( X Y ), 0 1 .
The penalty function has the following form: X i (Q i ) e f i , Y i X i , C Q i B i Q i , m m Fi Y i p Yj iUj X i p Yj iUj , j 1 j 1 H i aY ires , l l W i X i k iSk Y i k iSk , k 1 k 1 l l V i X i T k iSk Y i T k iSk , k 1 k 1 where R i is the limit value of VRC.
The vector f* is the solution of problem (
l l l K i k iSk k iSk iост T k iSk
k 1 k 1 k 1 l m m T k iSk a iост Yj iUj Yj iUj k 1 j 1 j 1 .
The coordinates of the vector Q* are the solution of the following equation p i B i i Q i * i 1
0 i 1, n . (Q i *) (Q i *)
n 1 П I * (Q i * p i B iQ * i f i * ) .
i 1 H i aY i a Х i , a 0 , i 1, n .
If the regional recovering sector includes m of WUFs, the profit function of this sector П II has the following form
m n П II [( p Y j cY j ) (Y iUj X iUj ) A j ] .
j 1 i 1
The problem of searching for the optimal price vector pY*=(pY1*, pY2*, …pYm*) is based on the recovering sector’s profit criterion p Y * arg max П II
p Y R p Y j p Y p n n
Y ( Y i iUj X i iUj ),
Y j i 1 i 1 n n Y i iUj X i iUj Y j . i 1 i 1 p Yj * cYj p 2
Y , j 1, m .
The vector pY* is the solution of problem (
For pY*, the value of the profit function ПII* is П I*I 1 m
(Y j ( p Y cYj ) 2 A j ) .
4 p Y j 1
If the regional insurance sector includes l of insurers, the profit function of this sector П III has the following form
The problem of searching for the optimal insurance rate vector T*=(Т1*, Т2*, …, Тl*) is based on the regional insurance sector’s profit criterion l П III (V k W k ) .
k 1 T * arg max
Tst k (
The optimal parameters of the regional risk-control system maximize the profits of the sectors (agents) individually. All agents interact in the process of the risk-control.
Further, we consider the problem of searching for the interaction parameters. We search for the Pareto equilibrium set of the compromise utilization prices in the following form pYcom arg max { П I , П II } ,
p Y G
G { pY | П I ( pY ) 0 П II ( pY ) 0} .
Additionally, we analyze the Pareto set of the compromise insurance rates in the following form
T com arg max { П I , П III } ,
T
{T | T k (
We investigate our model on the basis of the regional industrial complex of Volga Federal District, which includes 14 regions and republics of Russian Federation. In each region (or republic) of this District, tens of thousands industrial firms emit the waste (Table I).
The volumes of the waste in Volga Federal District is presented in table II. In these regions, as a rule, the volumes of the waste grow.
We calculate the interaction parameters of the insurances sector and WUF sector by using formulas (32) – (35).
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 X iUj Y iUj , and taking into account a minimum of the utilization expenses Fi . Similarly, the number of the insurers is chosen on the basis of the minimal insurance rate criterion
X iUj Y iUj Y j .
V i among insurers that meet the condition X iSk Y ikS 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 Among WUFs that meet this condition, we search for the best WUF according to the minimal waste utilization costs criterion. The choice of the insurers is organized in the same way.
If WUF satisfies the condition j 0 | X iUj0 Y iUj0 Y j0 , then the parameters of the interaction between the industrial firm and WUF correspond to the Pareto equilibrium set (Fig. 2).
Figure 2 indicates the solution of problem (32) – (33). Therefore, the compromise utilization price p com Y j belongs to the p Ycjom p Y cYj 1
2 2 empty if p Y c Yj 1 2 2
following ( p Y cYj ) 2 set 4 A j p Y
Y j ; p Y , that is not 2 ( p Y c Yj ) 2 4 A j p Y
Y j p Y . 2 industrial firm solves the problem of interaction with several WUFs. We consider the interaction of the industrial firm with two WUFs. The condition for waste utilization contracts is the existence of compromise prices pY1com, pY2com (Fig. 3).
In this situation, the price vector pYcom=(pY1com, pY2com) is the solution, where pYcjom pY cYj 1
2 2 and p Y c Yj 1 2 2
( pY cYj ) 2 ( p Y c Yj ) 2 4 A j pY
Y j 4 A j p Y
Y j ; p Y , j=1, 2, 2 p
2 vector pYcom=(pY1com, …, pYmcom), m>2, the solution of problem (32) – (33) is similar to the solution for m=2.
The problem of searching for the parameters of the interaction with the regional insurers is solved similar. The solution of problem (34) – (35) has the following form 1) for one insurer provided k 0 | X iSk 0 Y S the ,
According to our algorithm (Fig. 1), we calculate the minimum number of the insurers that are interconnected with one industrial firm in each region (Table III). In this case, we consider that the average cost of the conventional waste ton utilization is equal to 6 thousand rubbles.
Thus, our results allow us to determine the compromise waste utilization prices and the compromise insurance rates that meet the requirements of the industrial firms, the recovering enterprises and the insurance regional sector. In addition, we construct the firm-insuarer system, i.e., we calculate the number of insurers, which are nesessary to insure the firm’s damage. This solution includes the big data as input parameters that reflect the operating conditions of all agents in the regional industrial risk control system.
The developed models describe the functioning of the regional industrial risk control system on the basis of big data regarding to the industrial firms, the insurance companies and the waste utilization organizations. The number of agents in the system varies from region to region, but generally exceeds tens of thousands. Each firm of the industrial regional complex interacts with one or multiple agents of the environmental protection and the insurance regional sector. The formulated problems and the presented solutions allow us to determine the parameters of the agents’ interaction in the regional system based on Pareto equilibrium. Our results may be used by the industrial firms to determine the terms of waste utilization and insurance contracts. In the strategies designing, the simulation results may be useful for WUF and insurers to develop the requirements for the industrial firms.