Microsoft Word - 1080-Source Texts-5245-1-11-20220930.docx Adv Syst Sci Appl 2022; 03; 18-35 Published online at https://ijassa.ipu.ru. Models of Industrial Risk Control Systems Mikhail Geraskin1, Elena Rostova2* 1) Samara National Research University, Samara, Russia E-mail: innovation@ssau.ru 2) Samara National Research University, Samara, Russia E-mail: el_rostova@mail.ru Abstract: We investigate a problem of searching for Pareto equilibrium sets of an insurance rate and an industrial damage utilization price. We consider a system, which, in the case of industrial accidents, arises around an industrial firm. An industrial firm, a waste utilization firm, and an insurance company are considered as the system’s agents. We develop profit functions for the agents, and we determine compromise prices on waste utilization and insurance, which provide the system’s stability. We analyze the set of industrial risk control systems with a various number of the agents and the agent’s relations. A problem of determining an optimal solution is solved on the basis of maximizing agents’ profit functions. The sets of an equilibrium industrial damage utilization price and an equilibrium insurance rate are defined as Pareto equilibrium. A problem of determining the set of an insurance rate is solved taking into account constraints according to requirements of an industrial firm and an insurance company. A problem of determining the set of an industrial damage utilization price is solved taking into account constraints according to requirements of an industrial firm and a waste utilization firm. We consider the following models of industrial risk control systems: agents have a strong relation and a weak relation, additionally, one agent of each type and of many agents of the same type. Keywords: industrial risk, insurance, waste utilization, optimization, risk control 1. INTRODUCTION The industrial risk control is an important problem for every firm because the influence of different external market factors. The risk control problems cover a financial risk, human errors, a non-fulfillment of contracts, an industrial risk, an environmental risk, etc. These problems were solved by means of the following methods: the scenario method [17], the multi-agent systems [1, 7, 19], the multi-criteria models [6, 27, 31]. Additionally, this problem was analyzed on various levels: the world market risk [11], the regional economic system risk [20, 24, 26], the firm’s risk [2, 28], the technology operation risk [3, 8]. The risk control problems are related to various aspects, in particular, an assessment of the risk factors, a choice of the risk management method, a prediction of the damage, etc. Rasmussen and Svedung emphasized that «risk management can no longer be based on responses to past accidents and incidents, but must be increasingly proactive» [21]. Therefore, an importance of developing measures to prevent risks prevails over minimizing the damage from accidents. Wu, Olson, and Choi indicated that «optimization and risk minimization inherently run counter to each other» [30]. Consequently, a choice of the risk management method should be based on an assessment of the preventive measures economic efficiency. These problems were solved on the basis of the multicriteria decision making (MCDM) methods [6, 27, 31], and the biconvex models and algorithms [25]. * Corresponding author: first@ras.ru MODELS OF INDUSTRIAL RISK CONTROL SYSTEMS 19 Copyright ©2022 ASSA. Adv. in Systems Science and Appl. (2022) For MCDM, Heller [27] proposed using a pair wise comparison of the risk competing objectives. The following criteria were analyzed: a power outage, a fire, a flood, an earthquake, a hurricane, a destruction of buildings, network failures, etc. As a result, a matrix of the risk criteria assessments was formed, which is used in a qualitative risk analysis for three buildings that differ in qualitative features. Abla et al. [6] used MCDM to derive an aggregated risk score on the basis of the fuzzy logic. For assessing development scenarios of the risk situation, the decision-making process was considered under the following criteria: a technical and functional efficiency, an economic sustainability, a social sustainability, an institutional environmental sustainability, an overall sustainability. The risk management strategy was selected based on a weighted sum of these criteria. In the case of the flood risk, this decision making technique was applied to select the most sustainable strategy under uncertainty. Yazdani et al. [31] proposed MCDM type for assessing risks of the cultivated areas’ flooding; they ranked various agricultural projects, which can mitigate the flood risks. Dudin M. N. et al. [5] investigated the risks of an industrial enterprise and calculated external factors influence weights and a likelihood of unforeseen events for political, macroeconomic, social, and technological factors. On the basis of these weights, the external risk average level of Russian industrial enterprises was calculated. Internal risks of the enterprise were assessed according to the following criteria: a fulfillment of the production plan, an economic security of current obligations, an economic security of supply contracts, a human factor, and a likelihood of success in innovations commercialization. The authors examined hedging and insurance methods for the risk management of an industrial enterprise, however, the insurers and other related organizations were not considered as separate agents, and their interactions were not investigated. Krokhina J.A. et al. [16] explored the environmental risks of industrial enterprises using a tree method and assessed the integral risk of an industrial facility as a result of its negative impact on an environment, a human health, and an enterprise’s economy. The authors applied the tree method to assessing the risk of an accident on the main pipeline and assessed an economic efficiency of the measures to reduce the environmental risk of industrial enterprises, but did not take into account an interaction of the enterprise with other economic agents. In contrast to the aforementioned literature, multi-agent systems (MAS) models expanded a range of the risk management techniques. MAS models were studied in production design and development systems, a production planning and management, and a supply chain management (SCM) [19]. In a set of agents, MAS models described the participants in the production process supply chain. In SCM, an influence of small and medium-sized enterprises in the interaction with large firms [7] and buyers with suppliers [1] on the risk level was considered. In the case of different-scale enterprises in SCM, Finch [7] emphasized varying degrees of the supply chain disruption risk, because large and small enterprises are subject to different degrees of the risks and their sensitivity to the risk factors is different. Ahn and Park [1] studied the information exchange processes between participants in the supply chain as MAS agents and assessed an impact of an agent’s awareness on SCM. In general, MAS model was applied for the risk management in systems, which consist of an industrial enterprise and its suppliers, i.e., the participants in the supply chain. Finally, it should be emphasized that the aforementioned studies were carried out for specific industries. For example, the problems of risk in the chemical and oil industries were solved [3, 8] and technical, human and organizational factors of the industrial facility risk were taken into account; on the basis of the fuzzy logic, the risk of the industrial facility in the oil and gas industry was analyzed using the criteria of frequency, detectability and damage value [30]. Therefore, the results of these studies cannot be applied to all industries. Thus, we demonstrate the following research gap in the problem framework of the industrial risk management. On the one hand, the industrial researchers pointed to a need for 20 M. GERASKIN, E. ROSTOVA Copyright ©0000 ASSA Adv. in Systems Science and Appl. (0000) the insurance and the technical measures to prevent or eliminate consequences of industrial accidents. On the other hand, they did not investigate the interaction mechanisms of an enterprise with insurers and waste utilization firms as the specific agents. Additionally, they did not generalize the results for the universal industrial enterprise; they were limited to the specific industry. At the same time, MAS researchers did not study the principles of applying MAS in the process of industrial risk management. Hence, we can formulate the following research question of the industrial risk management problem: to describe the industrial risk management process for the universal industrial enterprise within the system of interconnected economic agents and calculate the equilibrium prices for services that circulate within this system. Our study aims at calculating the price equilibrium in the system, which appears as a result of the preventive measures to minimize the consequences of technical accidents in the industry. 2. PROBLEM FRAMEWORK In this paper, the risk is considered at the firm’s level, and it includes an internal damage and an external damage. The internal damage causes a reduction in the firm’s assets. The external damage is the property wastes of other firms, individuals and the environment. Additionally, the fiscal penalties (the ecology payment, the penalty for a damage to health, and a property of other firms and individuals, the compensation caused by the non-fulfillment contracts) depend on a value of the external damage. The industrial firm can reduce the internal/external damage by means of additional expenses on the risk reduction. These costs expenses are named the voluntary risk costs (VRC). We analyze the problem of the industrial risk control for a system with three agents: the industrial firm, the waste utilization firm, and the insurance company. These agents are in various relationships in the system. We consider the following problem: to search for a compromise price of waste utilization and a compromise insurance rate, which are compliant with all agents of the system. We assume that each participant in the system is intended to increase its profits, and he chooses the optimal price. If these optimal prices are different, then the participants may not agree to conclude a contract, then the industrial risk management will not be implemented. Therefore, in this case, we determine the set of possible values of the insurance rate and the price of waste utilization, at which the participants in the system will agree. We introduce the following assumptions, which determine the applicability limits of the model. Assumption 1. The product price is an exogenous constant, that is, the firm does not affect the price ,0 dQ dp (1) where p is the price of the production, Q is the production volume. The waste utilization firms and the insurance companies are in the monopolistic competition market, that is, the following conditions are fulfilled: ,0,0  U Y U Y dX dp dY dp (2) ,0,0       SS Y T X T (3) MODELS OF INDUSTRIAL RISK CONTROL SYSTEMS 21 Copyright ©2022 ASSA. Adv. in Systems Science and Appl. (2022) where pY is the price of the utilization of a conventional waste unit, T is the insurance rate, XU and YU are the internal and external utilized damage, XS and YS are the internal and external insured damage. Assumption 2. The production growth leads to a decreasing return: ,0QQC (4) where C is a value of the firm’s costs. Assumption 3. An increase in the production assets leads to an increasing in the possible damage; the internal damage and the external damage are reduced with an increase in VRC; the internal damage is limited from above due to technology features and the production volume 0],,0(,0,0 maxmax       XXX f X Q X . (5) where Xmax is the maximum possible internal damage, X is the internal damage, f is VRC. Assumption 4. The external damage Y is proportional to the internal damage X: 0   X Y . (6) Assumption 5. The voluntary combination insurance is considered, the wear is not included. The insurance indemnity W is proportional to the insured damage XS and YS, the indemnity does not exceed the damage: .,0,0 SS SS YXW Y W X W       (7) Assumption 6. The cost of the utilization of a conventional waste unit cY is a constant. cY = const. (8) Assumption 7. The firm’s external damage Y=YS+YU+Yres consists of the insured external damage YS= S Y, the utilized external damage YU= U Y, and the residual external damage Yres= res Y. The firm’s internal damage X=XS+XU+Xres consists of the insured external damage XS= S X, the utilized internal damage XU= U X, and the residual internal damage Xres= res X. 0,,,1  resUSresUS  , (9) 0,,,1  resUSresUS  . (10) The production costs function, according to assumption 2, has the following form [4], [29]. BQQСQ )( , 0],2,1(],,1( maxmax  B , (11) where B and β are the parameters of the production costs function, βmax is the maximum possible parameter value. The internal damage function satisfies assumption 3, and it has the following form: feQfQX   )(),( , ],1,0(],,0( maxmax   0)(  Q . (12) This function Х(Q) expresses an exponential distribution of the damage, which corresponds to man-made accidents, φ(Q) is the dependence of the damage on the production volume Q,  is the parameters of the internal damage function, max is the maximum possible parameter value. 22 M. GERASKIN, E. ROSTOVA Copyright ©0000 ASSA Adv. in Systems Science and Appl. (0000) The external damage function satisfies assumption 4: 0,)(  XXY . (13) The coefficient of the accident consequences expansion μ expresses the ratio of the external damage and the internal damage, taking into account the specifics of the industrial complex in a region, geographical features, etc. The insurance indemnity satisfies assumption 5: ,10),(),(   SSSS YXYXW (14) where α is the coefficient of the insurance indemnity. The penalty function has the following form: ,0,  aХaaYH  (15) where a is the parameter of a relationship between the penalty and the external damage. We consider the systems, which include the agents of three types: the 1st agent is the industrial firm, the 2nd agent is the waste utilization firm, and the 3rd agent is the insurance company. The industrial firm (the 1st agent) produces the production volume Q, and it sells the product at the price p. We introduce the following notation: CQ is the production costs function, X is the internal damage, Y is the external damage, f is VRC, H(Y) is the penalty function, F(Х,Y) is the value of waste utilization costs, V(X, Y) is the insurance premium, W(X, Y) is the insurance indemnity, Qmax is the maximum possible production volume, fmax is the maximum possible VRC. The revenue function of the 1st agent is WQpR  . (16) The total costs function of the 1st agent is FHVXfCС res Q   . (17) The profit function of the 1st agent is ПI=R – C∑. (18) We formulate the problem of the firm’s choice as follows: to search for the production volume and VRC function, which maximize the profit of the 1st agent, that is: .maxarg*}*,{ , I AQAf ПQf Qf   (19) }0,:{ maxmax   QQQRQAQ (20) )},0(,)(:)({ maxmax if RfffRfA   (21)                     ).( ),( , ),( , , ,)( SS SS res UU Y Q f YXTV YXW aYH XYpF BQC XY eQX         (22) The waste utilization firm (the 2nd agent) has the following parameters. The utilized damage UU XY   does not exceed the level Y , the price pY does not exceed the level Yp , where Yp is the maximum possible price, Y is the maximum possible waste utilization. Consequently, the inverse demand function of the 2nd agent, according to assumption 1, has the form: )( UUY YY XY Y p pp   . If 0Yp , then YXYYX UU  :, MODELS OF INDUSTRIAL RISK CONTROL SYSTEMS 23 Copyright ©2022 ASSA. Adv. in Systems Science and Appl. (2022) The profit function of the 2nd agent is ))(( UU YYII XYcpП   . (23) We formulate the problem of the choice of pY : to search for the price of the utilization of a conventional waste unit, which maximizes the profit of the 2nd agent, that is: II Rp Y Пp Y   maxarg* (24) ).( UUY YY XY Y p pp   (25) The insurance company (the 3rd agent) has the following parameters. The insurance premium depends on the insurance rate T and the insured damage SS XY   ; T is the maximum possible insurance rate, X is the maximum possible insured damage. Consequently, the inverse demand function of the 3rd agent, according to assumption 1, has the form: X T XYTT SS )(   . If 0T , then XXYYX SS  :, . The profit function of the 3rd agent is WVПIII  . (26) We formulate the problem of the choice of the insurance rate: to search for the insurance rate, which maximizes the profit of the 3rd agent, that is: III Tst ПТ )1,0( maxarg*   (27)             . ),( ,)( ),( XXY XYW X T XYTT XYTV SS SS SS SS     (28) We consider the following problem of the agent’s optimal control: to search for the pair (Q*, f*), which is optimal according to criterion (19), to search for the price pY*, which is optimal according to criterion (24), and to search for the rate T*, which is optimal according to criterion (27). This system consists of three agents, and, if we vary the parameters Q, f, pY, T, then three agents achieve maximums of their profits. We consider the system of the industrial damage control of the following types. The agents have a strong relation, if they have the collective criterion function, and their costs are not separable. The agents have a weak relation, if the costs are separable, and each agent has the individual criterion function. We investigate the types of the system in the following models. Model 1: the 1st agent is the customer of the waste utilization and the insurance, the 2nd agent is the contractor of the waste utilization, the 3rd agent is the insurer of the internal damage and the external damage of the 1st agent. This system has the weak relation type. (Fig. 1) 24 M. GERASKIN, E. ROSTOVA Copyright ©0000 ASSA Adv. in Systems Science and Appl. (0000) Fig. 1. The agent’s interaction schema in Model 1. The 1st agent pays the sum )( UU Y XYpF   to the 2nd agent, and the agents achieve the contract, if the price pY complies with everyone. We formulate the problem with two criteria to search for the compromise price com Yp , under which the system is stable. By using an analogy with the previous case, the 1st agent pays the sum )( SS XYTV   to the 3rd agent, and the agents achieve the contract, if the insurance rate T complies with everyone. We formulate the problem with two criteria to search for the compromise rate comT , under which the system is stable. Thus, the system of three agents is stable, if the compromise price com Yp and compromise rate comT are indicated in the contracts. Consequently, we formulate the following problems: to search for the compromise price com Yp and the compromise rate comT , which satisfy the following conditions: II Gp I Gp ПП YY   maxmax , (29) III T I T ПП  maxmax , (30) }0)(0)(|{  YIIYIY pПpПpG , (31) }0)(0)()1,0(|{  TПTПTT IIII . (32) In formulas (29), (30), the symbol of the conjunction “ “ means that the maximums are determined according to both criteria, taking into account the Pareto optimal principle. Model 2: the 1st agent and the 2nd agent have the strong relation; the 3rd agent is the insurer of the internal damage and the external damage of the 1st and the 2nd agents; the 3rd agent has the weak relation to other agents. (Fig. 2) MODELS OF INDUSTRIAL RISK CONTROL SYSTEMS 25 Copyright ©2022 ASSA. Adv. in Systems Science and Appl. (2022) Fig. 2. The agent’s interaction schema in Model 2. The aggregate profit function of the 1st and the 2nd agents in the system is resUU YQIIIIII XXYcHVfCWQpППП   )(, . (33) The problem of searching for the optimal production volume and VRC for the 1st agent, and, additionally, the optimal rate of the 3rd agent, has the following form: III AQAf ПQf Qf , , maxarg*}*,{   , (34)                     ).( ),( , , , , ,)( SS SS res UU Q f YXTV YXW aYH YXY BQC XY eQX         (35) These variables influence on the waste utilization, the insured damage, the penalties, and the insurance rate. We formulate the problem of searching for the compromise rate accounting to the following conditions: III T III T ПП 11 maxmax ,   , (36) }0)(0)()1,0(|{ ,1  TПTПTT IIIIII . (37) Model 3: two agents of the 1st type are customers of the waste utilization and the insurance; the 2nd agent is the contractor of the waste utilization; the 3rd agent is the insurer of the internal damage and the external damage of the 1st agent. This system has the weak relation. (Fig 3) 26 M. GERASKIN, E. ROSTOVA Copyright ©0000 ASSA Adv. in Systems Science and Appl. (0000) Fig. 3. The agent’s interaction schema in Model 3. The problem of searching for VRC and the optimal production volume of the 1st type agents is iii res iiiiQiiiiI FHVXfCWpQП   , 2,1i , (38) I AQAf ii Qifi Qf maxarg , *}*,{   , (39)                     ).( ),( , ),( , , ,)( S ii S iii S ii S iiii res iii U ii U iiYi iiiQ ii f iii YXTV YXW aYH XYpF QBC XY eQX i i         2,1i (40) where }2,1,{  iП IiI is the vector of criteria in model 3. The profit function (23) of the 2nd agent is the sum of the revenues, which are provided by two agents of the 1st type, and it has the following form:    2 1 )()( i U ii U iiYYII XYcpП  . (41) The problem of searching for the price pY* according to maximization of the 2nd agent’s profit function is II Rp Y Пp Y   maxarg* , (42) .2,1),(  iXY Y p pp U ii U ii Y YY  (43) By using an analogy with the previous case, the profit function (26) of the 3rd agent is    2 1 )( i iiIII WVП . (44) The problem of searching for the insurance rate T according to the maximization of the 3rd agent’s profit function is III Tst ПТ )1,0( maxarg*   (45) MODELS OF INDUSTRIAL RISK CONTROL SYSTEMS 27 Copyright ©2022 ASSA. Adv. in Systems Science and Appl. (2022)                .)( ),( ,)( ),( 2 1 XXY XYW X T XYTT XYTV i S ii S ii S ii S iiii S ii S ii S ii S iii     (46) We formulate the problem of searching for the compromise price com Yp and the compromise rate comT , accounting to the following conditions: II Gp I Gp I Gp ППП YYY 111 maxmaxmax 21   . (47) }0)(0)(0)(|{ 211  YIIYIYIY pПpПpПpG . (48) III T I T I T ППП 222 maxmaxmax 21   (49) }.0)(0)(0)()1,0(|{ 212  TПTПTПTT IIIII (50) 3. RESULTS Assertion 1. The function |)(|ln 1 * KQf    , U Y U Y resSSresSS aTTK   and the value Q*, which is calculated from the equation 0 )( )(1     Q Q QBp    , are the solution of the problem (19 – 22) for the continuously differentiable functions )( and under conditions 0))(())()1(()( 222   KeQKeQQBKeQ fff   and K > 0, 0)()()( 2  QQQ  . The maximal profit of the 1st agent is   1 ****  fBQpQПI . Proof. The profit function of the 1st agent (18) is   resresfSS I aYeQfBQYXTQpП   )())(( )( UU Y YXp   . We search for the partial derivatives of this function, which are equal to zero: 0)]())([()(1     UU Y resresSSfI paTeQ f П   0)]())([()(1     UU Y resresSSfI paTeQQBp Q П   We solve these equations as follows: 1)]())([()(  UU Y resresSSf paTeQ   28 M. GERASKIN, E. ROSTOVA Copyright ©0000 ASSA Adv. in Systems Science and Appl. (0000) We introduce the denotation )())(( UU Y resresSS paTK   , then we can write the optimal function VRI as follows: ))(ln( 1 * KQf    . We write the equation 0)(1   KeQQBp f  , and, for the aforementioned symbols K and f*, we get 0 )( )(1     Q Q QBp    . We solve this equation, and we search for Q*. We consider a function )( )( )( 1 Q Q QBpQh       . This function is continuous function QAQ . Due to the features of continuous functions, the equation h(Q)=0 has a solution, because 0)( Qh for )( )( | 1 Q Q QBpAQ Q       , 0)( Qh for )( )( | 1 Q Q QBpAQ Q       , and 0 )( )()()( )1()( 2 2 2     Q QQQ QBQh    if 0)()()( 2  QQQ  . We check a fulfillment of the maximum sufficient condition for the function ПI(Q, f) at f=f* and Q=Q*. For this purpose, we define the sign of 22 2 2 2 2                 Qf П Q П f П III . KeQ f П fI     )(2 2 2 , KeQQB Q П fI       )()1( 2 2 2 , KeQ Qf П fI     )( 2 . 0))(())()1(()( 222   KeQKeQQBKeQ fff   . If K>0, then 2 2 f П I   <0 and 2 2 Q ПI   <0, consequently, the pair (f*,Q*) is the maximum point according to Sylvester’s criterion. █ Assertion 2. The value 2 * YY Y pc p   is the solution to the problem (24 – 25) for the continuously differentiable function Y(pY). The maximal profit of the 2nd agent is ПII*=ПII( * Yp )= ))(( 4 1 2 YY Y cpY p  . Proof: We write the profit function of the 2nd agent (23), which is subjected to the condition (25): MODELS OF INDUSTRIAL RISK CONTROL SYSTEMS 29 Copyright ©2022 ASSA. Adv. in Systems Science and Appl. (2022) .))(( Y YYYYII p Y ppcpП  We transform this expression as follows: .2 YY Y Y Y YII cYc p Y Yp p Y pП              This expression is the second-order power function, and it has the parabola graph with the maximum at . 2 * YY Y cp p   . We search for the maximum of the profit function *)(* YIIII pПП  of the 2nd agent:                            YY Y YY Y YY II cYc p Y Y cp p Ycp П 22 * 2 2)( 4 YY Y cp p Y  .█ Assertion 3. The insurance rate 2 *   T T is the solution to the problem (27 – 28) for the continuously differentiable functions X(T) and Y(T). The maximal profit of the 3rd agent is 2)( 4 1 *)(*  TX T TПП IIIIII . Proof: The profit function of the 3rd agent is T TT XTXYTWVП SS III   )())((  . We transform this function as follows: X T X XT T X TП III               2 . This function has the parabola graph with the maximum point at 2 *   T T . We determine the maximum of the profit function of the 3rd agent as follows: 2 2 )( 422 *)(*                             T T X X T X X T T XT TПП IIIIII .█ Assertion 4. If 2 Y Y p c  , then     2 ; Y Y com Y p cp is the solution of the problem (29), (31) for the continuously differentiable functions )( , else com Yp Ø. Proof: The profit function of the 1st agent is )()( UU Y res QYI YXpHVXfCWQppП   The utilized damage UU YX   corresponds to the demand function: . Y YUU p p YYYX   Then the profit function of the 1st agent is 1 2 1 1)( KYp p Y p p p YpKpП Y Y Y Y Y YYI        , 30 M. GERASKIN, E. ROSTOVA Copyright ©0000 ASSA Adv. in Systems Science and Appl. (0000) where 1K is HVXfCWQp res Q   . The maximum point of this function is 2 Y Y p p  , and the maximum of the profit function of the 1st agent )( YI pП is 142 K Ypp П YY I       . The profit function ПII(pY) is analyzed in the proof of assertion 2. Fig. 4. Compromise set of the 1st and the 2nd agents The value 0 Yp is Yc . The figure 4 shows that the 1st agent prefers price     2 ,0 Y Y p p , the 2nd agent prefers price . 2 ,0       YY YY pc pp If 2 0 Y Y p p  then     2 ,0 Y Y com Y p pp else         2 ,0 2 ,0 YYY Y ppc p Ø █ The value com Yp is the solution of the problem (29), (31), and it enables us to establish the price Yp , which complies with the 1st and the 2nd agents. If one of the agents changes this price, then the profit of other agent decreases, therefore, this is Pareto optimal equilibrium set for 1st and the 2nd agents’ prices according to profit function (18) as the criterion of the 1st agent and profit function (23) as the criterion of the 2nd agent. Assertion 5. If 2 T  , then        2 ; T T com  is the solution of the problem (30), (32) for the continuously differentiable functions )( , else comT Ø. Proof: The profit function 1st agent is )()( SSres QI YXTFHXfCWQpTП   . MODELS OF INDUSTRIAL RISK CONTROL SYSTEMS 31 Copyright ©2022 ASSA. Adv. in Systems Science and Appl. (2022) The insured damage SS YX   corresponds to the demand function in the insurance market: . T T XXYX SS   We transform the profit function of 1st agent as follows: .1)( 2 2 2 KXT T X T T T XTKTПI        The value 2K is HFXfCWQp res Q   . The minimum point of profit function )(TПI is 2 T T  and the minimum of the function is 242 K XTT ПI       . The profit function ПIII(T) is analyzed in the proof of assertion 3. The graphs of the 1st and the 3rd agents’ profits for problem (30), (32) are demonstrated in Fig. 5. Fig. 5. Compromise set of the 1st and the 3rd agents The acceptable set comT is interval       2 ; T . These values enable us to transact of the insurance contract, because the value comT complies with the 1st and the 3rd agents. The deviation of the insurance rate relative to comT leads to a decrease in the profit of one of the agents. Thus, we identify the set of the possible values of the insurance rate comT and the price of waste utilization com Yp , at which the system agents interact, and the process of the industrial risk management is implemented. Similar to the conclusion from assertion 4, we establish the Pareto optimal equilibrium set for the 1st and the 3rd agents’ prices according to profit function (18) as the criterion of the 1st agent and profit function (26) as the criterion of the 3rd agent. 32 M. GERASKIN, E. ROSTOVA Copyright ©0000 ASSA Adv. in Systems Science and Appl. (0000) 4. DISCUSSION The currently accepted risk management model is based on the standards [13] - [15] and, in fact, implements the intra-firm management [3], [4], [6], [8], [16], [27], [31]. In comparison with the aforementioned literature, we investigate the process of the industrial risk management in the system of several organizations interconnected within the framework of this process. Our system consists of three agents (the industrial firm, the insurer, and the waste utilization firm) and provides a comprehensive examination of the risk management process. In contrast to use of MAS for SCM, we consider the MAS model, which includes the agents that are not linked in a supply chain. We add the insurers and the waste utilization firm to this model; therefore, our results extend the studies [18], [23]. In addition, our results are industry invariant; consequently, it can be used for any industrial production. This generality of results distinguishes our study from authors who considered risks for a particular industry, for example [3], [8]. We derive an analytical form of the preventive risk costs function for an industrial enterprise and prove that these costs depend logarithmically on the production volume of an enterprise. This pattern means that the preventive risk costs increase more slowly in comparison with a growth of the production. This feature encourages enterprises to take the preventive measures and develops the concept of the proactive risk management [21]. Therefore, we prove that the proactive risk management strategy is optimal, i.e., it corresponds to the minimum costs. We calculate the optimal values of the waste utilization price and the insurance rate, which maximize the objective functions of the waste utilization firm and the insurer. In comparison with the approach [4], [16], which is based on the exogenously specified insurance rate, in our model, the insurance rate is determined endogenously. In other words, our model is based on such values of the waste utilization price and the insurance rate that induce these firms to participate in contracts with the industrial enterprise. Consequently, in these conditions, the decentralized decision-making system is configured according to the mechanism of the centralized system. In addition, we find the conditions of the compromise domain, i.e., the ranges of the waste utilization price and the insurance rate, within which the system participants are interested in concluding contracts. Therefore, we expand the MAS approach [1, 7, 19] related to the revenue sharing, and reformulated it into the mechanism of the price sharing contract. Next, we prove that the contract prices within the specified ranges are Pareto efficient. Therefore, when the price varies within the compromise domain, the profit of one participant grows, and the profit of the counterparty decreases, i.e., the price sharing contract corresponds to the profit sharing contract. This is an important advantage of our approach: our model not only allows us to estimate the damage from an industrial accident, but also to calculate the economic effects of all participants in the risk management process and choose the preventive costs sum that corresponds to the optimal solutions for all participants. The results of this article can be used by industrial enterprises, insurance companies, and waste utilization firms to determine the insurance rate and the utilization price. The resulting compromise values of the rate and the price demonstrate the set of acceptable values at which a contract will be concluded. Finally, we briefly outline the directions for further research within our version of the MAS model. Our results are obtained under certain restrictions on the type of a market (assumption 1), the production function (assumption 2), and the damage function (assumption 3). In the future, we plan to expand the study to consider other types of markets and other production and damage functions. MODELS OF INDUSTRIAL RISK CONTROL SYSTEMS 33 Copyright ©2022 ASSA. Adv. in Systems Science and Appl. (2022) 5. CONCLUSION The optimization problems of searching for the firms’ risk prevention costs were investigated in our previous articles [9], [22]. In particular, the optimal VRC function in the case of the internal damage prevention according to the condition of the firm’s profit maximization was derived. The problem of the external damage control was analyzed, and the optimal VRC function with regard to fiscal penalties for the environmental damage and the civil penalties for individuals’ property damages was proved. The problem of searching for the optimal risk costs, taking into account the reinvestment of the firm’s profit, was considered, and the optimal VRC function for a firm’s activity in the consequent periods was obtained. In this paper, the problem of the industrial risk control in different system’s structures is investigated. We consider a risk control system of an industrial firm, which includes the insurance company and the waste utilization firm. This system enables us to determine the conditions of the insurance contract and the waste utilization contract. The problem of the industrial risk control is a problem of the agents’ interests congruence. The calculated values of the insurance rate and the waste utilization price are determined as the Pareto equilibrium set for price and insurance rate. We obtain the following results. The problem of the firm’s choice of the product volume and VRC function in the system, which includes an industrial firm, a waste utilization firm, and an insurance company, is solved in assertion 1. The problem of searching for the price of the utilization, which maximizes the profit of a waste utilization firm, is solved in assertion 2. The problem of searching for the insurance rate, which maximizes the profit of an insurance company, is solved in assertion 3. The problem of searching for the compromise utilization price and the compromise insurance rate, taking into account Pareto optimal principle for this variable, is solved in assertion 5. The final result provides the set of acceptable values at which a contract will be concluded. REFERENCES 1. Ahn, H.J., Park, S.J. (2003) Modeling of a Multi-agent System for Coordination of Supply Chains with Complexity and Uncertainty. In: Lee J., Barley M. (eds) Intelligent Agents and Multi-Agent Systems. PRIMA 2003. 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