Research on the sustainability of emergency supply chain based on evolutionary game and catastrophe theory

. In the post-COVID-19 era, the market for emergency medical supplies has shrunk, and related production capacity is rapidly overcapacity.The supply chain is at risk of interruption. First, we analyzed the decision-making behavior of supply chain members from the perspective of evolutionary game, and used the probability density function to transform the game model into a sharp mutation model; secondly, we simulated the impact of factors such as cooperative income on catastrophe phenomena; finally, we introduced capacity reserves to advise on the sustainability of emergency supply chains. The results show that: ① There is a set of divergence points in the emergency supply chain system. When the parameter combination is located in the set sum, the system is prone to mutation phenomenon; ② Before the supply chain state changes abruptly, the cooperation income of the supply chain is increased, and the initial cooperation among members is improved. Probability is the main idea to ensure the sustainability of the supply chain; ③ Capacity reserves are conducive to stabilizing cooperation among members, but factors such as reserve scale should also be avoided in the collection of divergence points.


Introduction
At the beginning of 2020, the COVID-19 epidemic quickly swept the world, and the supply level of epidemic prevention materials directly affected the efficiency of epidemic control in various countries.After the epidemic was brought under control, the market demand decreased rapidly, and the production capacity has rapidly transitioned from a serious shortage to a relative surplus.Some enterprises are facing the dilemma of closing down factories in the face of reduced market revenue and increased production costs, which is not conducive to the continuous supply and reserve of emergency materials. .Whether it is the new crown epidemic or other emergency emergencies that may break out in the future, ensuring the continuous and stable supply of emergency supplies is an important means to improve response capabilities.
Relevant research on emergency supply chain focuses on the construction of emergency supply chain [1][2][3][4], or based on actual case studies on the coordination and cooperation between various entities in emergency supply chain [5][6], while research on the sustainability and Mutation phenomenon of emergency supply chain is less.In recent years, evolutionary games have been more used in decision-making research.Liu et al. [7] constructed an evolutionary game model between the government and medical device companies to study the impact of dynamic rewards and punishments on corporate recycling strategies.Zhang et al. [8] constructed a financing game model between agricultural product suppliers and urban residents, and put forward suggestions for the transformation of agricultural product supply chains.
Thom [9] established the catastrophe theory on the basis of singularity theory and stability theory to study discontinuous change phenomena.Xu et al. [10] used the probability density function to transform the game model into a catastrophe model, and for the first time tried to use the random catastrophe theory to analyze the nonlinear phenomenon in strategic alliances.Lin et al. [11] established a random cusp mutation model, and discussed the mutation phenomenon and generation mechanism in "Chinesestyle crossing the road".Wang et al. [12] established a railway system safety analysis method based on the cusp catastrophe model, described the discontinuous change process of railway system safety, and considered the suddenness of safety accidents.Song et al. [13] embedded a cusp disaster model into a relative agreement (RA) model and explored the impact of network structure and delivery price on employee opinion evolution and employee turnover.
To sum up, based on evolutionary game and catastrophe theory, this paper analyzes the evolution process and catastrophe phenomenon of emergency supply chain formation under the background of overcapacity.Firstly, the game model is transformed into a cusp mutation model by using the probability density function, and the conditions for the mutation of the emergency supply chain are analyzed; then the simulation is carried out from two aspects of structural mutation and perturbation mutation, and the influence of different parameters on the supply chain mutation phenomenon is analyzed; Finally, the government's two reserve means are introduced into the mutation model to make suggestions for the sustainability of the emergency supply chain.

Problem description and analysis
Supply chain interruption caused by excess capacity mainly has the following characteristics: ① Supply chain interruption is not a continuous change, but a sudden change in the decision-making behavior of its members under the influence of profit, cost and other factors; ② When a member chooses to withdraw from the supply chain, cooperation cracks are created, causing more members to withdraw from the supply chain due to the loss of benefits.According to the above phenomenon, the assumptions of the game model established in this paper are summarized: (1) Supply chain members have two choices: ① Choose to continue production and maintain cooperation; ② The impact of overcapacity leads to an increase in costs and a decrease in revenue, so they choose to operate independently and withdraw from cooperation.
(2) When the cooperation is maintained, supply chain members can obtain cooperation benefits, and at the same time generate continued production costs; after withdrawing from cooperation, supply chain members can obtain independent operating benefits by selling raw materials or equipment, and at the same time, idle costs are incurred due to idle equipment or employees.The member pays compensation for breach of contract to the member maintaining the cooperation.The parameter description of the game model is shown in Table 1.To sum up, the single-period game revenue matrix of supply chain members after "overcapacity" is obtained, as shown in Table 2.

The cusp catastrophe model of behavior under "overcapacity"
Assuming that due to the impact of overcapacity, the proportion of supply chain members who choose to continue cooperation is x, and the proportion of members who choose to quit cooperation is 1-x.At this time, the members who choose to continue cooperation expect earnings E 1 , the members who choose to quit cooperation expect earnings E 2 and the supply chain members' average expected earnings E̅ are: Therefore, the dynamic equation can be obtained as follows: Set  as the time variable,  as the independent variable, 0 as the initial value of the system state, and note the probability density function of ( ) as Set the system potential function to (), the potential function reflects the mapping relationship between ( ) and the model control variables, The formula (4) can be transformed into: The cusp mutation model consists of a state variable z and two control variables u, v, and the potential function is as follows: Let F'(z,u,v)=0, the equilibrium surface form of the cusp catastrophe model can be obtained as: -z 3 +uz+v=0 ( 9) According to the above theoretical analysis, and further analyze the influence of game model parameters on the behavioral decision-making of supply chain members, the following theorems can be obtained: Theorem 1 When the dynamic equation satisfies H+2B-C-D≠0, that is, when the cooperative benefit minus the cooperative production cost is not equal to the independent benefit minus the idle cost, the system evolution conforms to the characteristics of the cusp catastrophe model. Proof.
It can be seen from Theorem 1 that when the evolution process of the game behavior of supply chain members satisfies certain conditions, their decision-making behavior will mutate, that is, the proportion of selected cooperation will discontinuously mutate from a certain stable state to another stable state.Further research on the critical conditions for the occurrence of mutations can lead to the following conclusions: Theorem 2 In the game process of supply chain members, when the following two conditions are met: ① the stochastic dynamics equation satisfies H+2B-C-D≠0; ② when the control parameters satisfy H+B+W-C-D=0 or W-B=0, the supply chain members decisions will change.
Proof.When H+B-C-D≠0, it can be seen from the catastrophe theory that the surface represented by equation ( 9) is the equilibrium surface of the standard cusp catastrophe model, and the critical condition for the occurrence of catastrophe is the critical surface of the equilibrium surface, which must satisfy the following conditions: which is -z 3 +uz+v=0, -3z 2 +u=0# (16) Equation ( 16) can be obtained by eliminating z Substitute equation ( 14) into equation ( 17) to get , combined with formula (13), we can get, When the control parameters satisfy H+B+W-C-D=0 or W-B=0, the decision-making of supply chain members will mutate.
Therefore, in the evolution process of supply chain members' decision-making, when the control parameters change continuously and are in the neighborhood of a critical point in the equilibrium surface, the decision-making of supply chain members will make great changes, and a large number of members choose to withdraw from cooperation, causing a catastrophe phenomenon produce.

Analysis of catastrophe phenomenon under "overcapacity"
From Theorem 1 and Theorem 2, it can be known that the random evolution of the decision-making behavior of supply chain members satisfies the cusp mutation model.In order to reflect the mutation behavior more intuitively and discuss reasonable solutions, this paper conducts numerical simulations on the control parameters and random disturbances of the system, and analyzes the conditions and paths of the mutation.
Let the cooperation benefit H=70, the independent benefit D=60, the default cost W=20, the idle cost B=30, the initial cooperation probability x=0.4,and the cooperation cost C gradually increase from 20 to 70, and the evolution process is shown in Figure 2 .The evolution path trend under certain conditions is obvious and monotonous.The change of cooperation cost C causes the "catastrophe" phenomenon of supply chain members' decision-making.When the production cost changes from 20 to 40, the cooperation probability of supply chain members increases with time.It is close to 1, which means that supply chain members all choose to cooperate when making decisions, and the supply chain system can maintain production; and when the cooperation cost exceeds 40, the evolution trajectory of the cooperation probability drops sharply from 1 to 0 with no transition process in between.At this time, all members of the supply chain choose to withdraw from the cooperation, and the supply chain system cannot maintain production.Under the background of overcapacity, it is difficult to guarantee the benefits of cooperation among members of the supply chain, so the cost of cooperative production is C=40, the independent benefit D=60, the cost of default W=20, the initial cooperation probability x=0.4, the idle cost B=30, and the cooperation The income H gradually decreases from 90 to 40, and the evolution process is shown in Figure 3. Similarly, the change of cooperation income also causes the phenomenon of "catastrophe".When the cooperation income decreases from 90 to 70, the cooperation probability of supply chain members tends to 1 with time, which means that the cooperative relationship between members is very strong and the supply chain system is stable When the cooperation income is lower than 70, the evolution trajectory drops sharply from 1 to 0 and there is no transition process in between.At this time, all supply chain members choose to withdraw from the cooperation, and the supply chain system cannot be maintained.
Keeping other parameters unchanged, the cooperation income H is gradually reduced from 90 to 40, and the initial cooperation probability x is set to 0.3 and 0.7, respectively.The evolution process is shown in Figures 4 and 5. Compared with Figure 3, when x=0.3, H ≤70, the evolution path of supply chain member decision-making begins to exit; when x=0.4,H≤60, the evolution path begins to exit; when x=0.7, H≤50, the evolution path begins to exit.The path just begins to trend toward the exit.This shows that the initial cooperation probability has a positive effect on the stability of the supply chain, and a higher initial cooperation probability is beneficial for supply chain members to maintain cooperation even when they obtain lower cooperation benefits.

The cusp catastrophe model under productive capacity reserve strategy
In the fourth chapter, based on the cusp catastrophe model, this paper analyzes the sudden change process and conditions of the emergency material supply chain system under the background of overcapacity.The disintegration of the supply chain will reduce the ability of society to respond to emergency emergencies.In order to maintain the coping ability and strengthen the cooperation among the members of the supply chain, this chapter introduces the means of government purchasing and storage, establishes the cooperation between the government and enterprises, and jointly shares the risk of overcapacity, so as to realize the sustainable operation of the emergency supply chain.
After the impact of emergencies has weakened, the market size of emergency materials has decreased sharply, and the scale of the remaining production capacity of enterprises has also been reduced accordingly.Moreover, the maintenance and management costs of enterprise production capacity are relatively high, and more companies choose to give up maintaining the remaining production capacity.Therefore, the existing production capacity scale is difficult to meet the large-scale emergency material demand.In order to ensure the production capacity required to respond to emergencies, the government needs to participate in maintaining production capacity, provide certain subsidies for enterprises to reserve production capacity, and share costs and risks with enterprises.This section introduces the capacity reserve method with government participation, assuming that the government first determines the subsidy coefficient a, and then the supply chain members determine the capacity reserve scale n, and its benefits and costs are directly related to the capacity reserve scale n.The larger the enterprise capacity reserve scale, the higher the government subsidy level, the subsidy level A(n)=a*n, the independent income D(n)=d * n, at this time, the capacity reserve scale can still meet the current market size, cooperation The return remains H. Combined with the actual situation, under the background of overcapacity, the marginal cost of the enterprise increases, so the cooperation cost C(n)=c * n 2 , the idle cost B(n)=b * n 2 , the initial cooperation probability is still is x, and the return matrix at this time is: This leads to the replica dynamic equation: Let cooperation benefit H=40, government subsidy coefficient a=3, independent benefit coefficient d=6, default cost W=20, idle cost coefficient b=0.75, initial cooperation probability x=0.4,cooperation cost coefficient c=1, capacity reserve The scale n is gradually increased from 4 to 9, and the evolution process is obtained as shown in Figure 5.When the capacity reserve scale is lower than 7, supply chain members are more inclined to cooperate.When the capacity reserve scale is higher than 7, supply chain members are more inclined to withdraw, which indicates that maintaining a certain capacity reserve scale is conducive to cooperation among supply chain members However, due to the increasing marginal cost, the excessively high reserve scale will also cause the cost of the enterprise to increase rapidly, resulting in sudden changes in the decision-making of supply chain members, which is not conducive to the stability of the emergency material supply chain.Comparing the evolution process in Figure 4, when the cooperation income H is 40, the enterprises that did not participate in the production capacity reserve have already withdrawn from the supply chain, while the enterprises with reserved production capacity can still maintain cooperation when the cooperation income is low, which shows that the government subsidies Capacity reserve can effectively reduce the cost of enterprises and is conducive to the sustainable development of the emergency supply chain.To sum up, the capacity reserve under the government subsidy can effectively reduce the cost of the enterprise, but the excessive reserve scale will still cause sudden changes.Therefore, it is necessary to guard against the accumulation of parameters such as the reserve scale through the divergence points to avoid the occurrence of sudden changes.Managers can adjust the scale of production capacity reserves according to the size of the market and the development of emergency events, reduce production capacity and reduce costs while meeting existing social needs, which can not only improve the ability to respond to emergencies, but also help maintain the stability of the emergency supply chain.

Conclusion
In order to maintain the sustainable supply of emergency supplies, this paper uses a combination of evolutionary game and catastrophe theory to establish a cusp catastrophe model from the perspective of sudden changes in decision-making of supply chain members, and discusses the impact of sudden changes in emergency supply chains under the background of overcapacity.factors, critical conditions and evolutionary processes.Subsequently, the government's purchasing and storage methods were introduced to share the risk of overcapacity among enterprises, and to propose management strategies for the sustainability of the emergency supplies supply chain.Research indicates: (1) The emergency supply chain will produce structural catastrophe under the internal action of the system.Structural catastrophe is caused by the combination of related parameters passing through the set of divergence points.Therefore, the enterprise must first clarify the boundaries of the set of divergence points, and avoid the combination of related parameters passing through the set.
(2) Before the state of the supply chain is abruptly changed, it is the main idea to ensure the sustainability of the supply chain to increase the net benefit of cooperation in the supply chain and improve the initial cooperation probability among members.The initial cooperation probability and cooperation benefits have a positive effect on the cooperation among members, while the cooperation cost has a negative effect on the cooperation among members.However, under the background of excess capacity, the market size is shrinking and the marginal cost is increasing, which requires the government to establish a reserve mechanism., and share the risk of overcapacity with enterprises.

Fig. 1 .
Fig. 1.The equilibrium surface of the cusp catastrophe model.

Table 1 .
Parameter description of the game model.

Table 2 .
The game benefit matrix of supply chain members.

Table 3 .
Profit matrix of capacity reserve strategy.