Abstract
The outbreak of emergencies usually triggers the phenomenon of a lack of emergency resources, and the lack of resources may lead to untimely rescue affecting the social life order. How to guarantee a smooth and efficient emergency resource allocation process and improve the efficiency of emergency relief disposal is the key to ensuring the successful completion of disaster relief tasks. To this end, this paper focuses on the emergency resource allocation process and considers the recycling of emergency resources after disaster relief with the background of emergencies. An emergency resource allocation process model is constructed based on the generalized stochastic Petri net, and the activity, reachability, boundedness, and security of the model are analyzed. The homogeneous Markov chain is utilized for simulation, and the performance analysis is carried out by calculating the steady-state probability and transition utilization rate of each state using examples. Simulation results show that the database and the recovery process should be the key steps to optimizing the emergency resource allocation process. Therefore, improving the effectiveness of these critical links’ emergency disposal will further improve the effectiveness of emergency resource allocation.
1. Introduction
In recent years, floods, typhoons, forest fires, droughts, earthquakes, and other global emergencies have occurred frequently, causing direct risks and impacts on human health, life, and property. At this time, certain measures need to be taken to prevent the situation from getting worse [1]. These intervention methods form a process, that is, the emergency response process in the emergency plan. In this process, all emergency personnel are trying their best to manage the disaster to reduce or avoid the secondary impact of the disaster. How to generate better emergency resource allocation before the emergency response has become a problem that needs to be solved urgently. As an indispensable part of emergency management, the dispatch and management of emergency resources will directly affect the efficiency of emergency response. Through the study of emergency resources, the strength of emergency response capabilities is determined to a certain extent. Therefore, it is of theoretical significance and practical value to explore and study the management of emergency resources.
Petri net is an intuitive graphical model that visualizes the relationship between processes. At the same time, Petri nets have strong mathematical support. By using mathematical method, the process can be analyzed with static structure and dynamic performance. Therefore, it becomes a modeling tool for many researchers to analyze decision process [2]. Zhong et al. [3] studied the urban emergency response system using Petri net model and illustrated the effectiveness of the model. Yang et al. [4] introduced the stochastic Petri net modeling method to establish emergency system and performance analysis. Liu et al. [5] also modeled the emergency management process of energy emergencies based on generalized stochastic Petri net and proposed suggestions for optimizing the process. Deng and Tang [6] modeled the operational process of military emergency logistics. Qi and Luo [7] designed the precipitation state chain, disaster evolution chain, and behavior disposal chain and constructed the process model of large-scale flood disaster. Wang Bo et al. [8] analyzed and modeled the public safety emergency management process using generalized stochastic Petri nets.
Based on existing research, most studies apply Petri nets to modeling and simulation of entire emergency systems or identify bottlenecks in emergency response in a particular area. However, this paper only analyzes some small research from the angle of emergency resource scheduling. Zhong et al. [9] constructed urban emergency resource reserve process model and urban emergency resource allocation process model from two aspects of reserve and allocation. Huang et al. [10] applied stochastic Petri nets to agricultural drought emergency response to find the key links of drought emergency response. Tian et al. [11] established an early warning mechanism for the network public opinion of emergencies and used stochastic Petri nets with Markov chains to simulate and analyze the early warning mechanism, based on which to formulate the early warning activation rules. The main contribution of this paper is to combine the generalized stochastic Petri net with the Markov chain to model and analyze the emergency resource scheduling process and construct the reachability graph. According to the model, the performance of the actual case process is analyzed quantitatively to describe the state transition of emergency, identify bottleneck links in the emergency resource scheduling process, optimize workflow, and improve emergency efficiency.
The remainder of this paper is arranged as follows.
The first section of this paper briefly introduces the research background and significance before moving on to the main work. The second section focuses on the relevant theories of Petri nets and summarizes the research methods of this paper. The third section introduces the modeling process of the emergency resource scheduling process based on GSPN and studies the effectiveness of the model. The fourth section analyzes the specific performance of the GSPN model and designs the homogeneous Markov chains and performance indicators of the GSPN model for qualitative analysis. The fifth section establishes the research model’s superiority and viability through practical cases. The summary and prospect of the full text are presented in the sixth section.
2. Relevant Theories of Petri Nets
The materials and methods section should contain sufficient detail so that all procedures can be repeated. It may be divided into headed subsections if several methods are described. In the 1960s, Petri [12] invented Petri nets, which are used to represent discrete parallel systems and can be described and simulated. There is no time factor in the original Petri net. To analyze the delay process, Molloy, Florin, and Natkin combined each transition of the Petri net with delay, introduced the actual rate, and obtained the stochastic Petri net (SPN) with time, which can be associated with queuing theory, stochastic process, and other methods. The transition delay of Petri nets is considered to be a continuous random variable subject to the negative exponential distribution. When the problem continues to grow, the space of Petri nets will grow exponentially, and SPN becomes very difficult to deal with such problems.
Therefore, based on stochastic Petri nets, Marsan [13] supplemented and improved the generalized stochastic Petri nets theory, mainly dividing the transition into two categories. (i) Instantaneous transition is associated with random switches and the implementation delay is zero. (ii) Time change is associated with the implementation delay of exponential random distribution. Generalized stochastic Petri nets are defined as seven-tuple GSPN = (P, T, F, V, W, M0, λ) [1]. The elements are specified in Table 1.
To better solve the stability probability of each state of GSPN [14], first, there are the following assumptions: (i) the reachable set is finite; (ii) the actual rate of change is independent of time; (iii) the probability of any reachable identification returning to the initial identification is not zero. Without considering the effect of time, GSPN can be seen as embedding Markov chains and then calculating the stability probability of each state based on Markov chains.
There are two main methods here: (i) remove the disappearing state from the embedded Markov chain, and only calculate the transition probability between the existing states on the compressed embedded Markov chain and (ii) the conclusive equation formula is based on the theory of the Stochastic Discontinuous Finite State Markov Process (SDFSMP) [15]. In this paper, the second algorithm is used to solve the problem, as shown in equation (3).
3. GSPN-Based Emergency Resource Scheduling Process Modeling
Emergency resource scheduling modeling is a relatively complicated process, especially since this article considers reverse logistics based on emergency forward logistics. The forward process of emergency resource scheduling mainly includes the determination of emergency resource demand and supply information, the formulation of resource raising plans, and the scheduling of resource distribution vehicles. In this paper, Petri net is used to model the emergency resource scheduling process to find the key factors affecting the allocation of emergency resources and provide certain theoretical support for emergency rescue and resource utilization.
In the emergency dispatch process, the Markov chain is integrated into the Petri net, and its main characteristics are as follows.(i)The state of emergencies is dynamically developing and changing.(ii)The information of emergencies is constantly improved, from vague to clear, from incomplete to complete.(iii)The response strategy formulated under the condition of incomplete information can be adjusted in time when the information is complete [16].
3.1. Analysis of Emergency Resource Scheduling Process
According to the general mechanism of emergency events, when an emergency occurs, the emergency command center first receives disaster information. At the same time, it judges the level of emergencies, formulates emergency plans, and notifies the corresponding departments to prepare for action. Then, all media, the Internet, etc. will successively report and disseminate the relevant disaster. The current process of dispatching emergency resources in my country is as follows (take an earthquake emergency as an example). After classifying the events, the emergency command center carries out scenario analysis and emergency resource demand forecast, which can directly call the emergency resources in the local repository if within the controllable scope. When the level exceeds a certain range, the resources in the repository cannot meet the needs of the disaster area. Therefore, the government must contact the cooperating suppliers for market procurement and production, or directly requisition local resources and equipment, etc. At the same time, individuals and social organizations make donations under the propaganda of the Internet and other media. Through the active actions of all parties, all kinds of emergency resources are gathered at the emergency resource dispatching center. Then, they were dispatched to various distribution points and distributed to the people in the disaster area. Finally, the resource information is fed back to the emergency command center according to the local disaster relief situation.
After the disaster is controlled, it enters the reverse logistics stage of emergency resources. Emergency resources no longer have the characteristics of an emergency, but only need to be processed according to general resources. The reverse logistics process of emergency resources mainly includes collecting and summarizing the emergency resources after the disaster relief in the disaster area, and then the professionals will detect and classify the various emergency resources according to the standards. Generally, detection can be divided into three categories. (i) Nonrecyclable resources, which are no longer of value: to prevent pollution, green disposal will be carried out. (ii) Resources that cannot be directly recycled: such resources have been used but still have a certain value and will be processed by the cooperative supplier again. (iii) Resources that can be directly recycled: such resources have not been used and have complete value and will be stored in the local emergency storage warehouse. The process of resource scheduling after an emergency is shown in Figure 1.
3.2. Construction of the GSPN Model for Emergency Resource Scheduling Process
According to the emergency resource scheduling process shown in Figure 1 and the related principles of Petri nets mentioned above, the emergency resource scheduling process model of generalized stochastic Petri nets is constructed. The model includes two stages of forwarding logistics and reverse logistics in the emergency resource scheduling process, as shown in Figure 2. In the beginning, P1 and P3 each have 1 logo. In the model, there are 18 places P0∼P17, and there are 18 transitions T0∼T17. The meanings of specific places and transitions are shown in Table 2.
3.3. Validity Analysis of the GSPN Model
The validity analysis of the model in this paper is the same as the judgment method of the general SNP model, which can be analyzed by the structural characteristics of three aspects: reachability, boundedness, and activity [15]. Generally, it is judged by calculating the invariants of the GSPN model, including (i) P-invariant representing the flow path of the emergency resource allocation process; (ii) T− invariant representing the effect of model transition. In this paper, the validity of the model is judged by calculating T− invariant. The T− invariant can be obtained by the incidence matrix method, and its solution can be obtained by formula (1):
A is the incidence matrix of the GSPN model, A = A+ − A−, X is a T− invariant set of GSPN model, and X is a nonnegative integer vector. T− invariant can be obtained by calculation:
Each T− invariant represents the order in which the model transitions in different environments. Element 1 represents the transition here, and 0 represents the nontransition. The above calculation results show that all the transitions in the GSPN model are triggered more than once, and there is no deadlock phenomenon, so the model can be judged to be active. According to the definition of the boundedness of the generalized stochastic Petri net, there is no resource overflow phenomenon in the emergency resource scheduling process model. The sufficient and necessary condition of the structural bounded net is that there is a positive integer vector X such that AX ≤ 0, as well as T− invariants such that the condition holds; that is, the model is bounded. At the same time, it is proved that the reachable graph of the model can be obtained according to the T− invariant, so the GSPN model network is reachable. In summary, the GSPN model of the emergency resource allocation process in this paper is effective.
4. Performance Analysis of the GSPN Model
4.1. Qualitative Analysis of the GSPN Model
It can be seen from Figure 2 that the GSPN model of the emergency resource scheduling process has the structure of sequence, concurrency, and cycle. T2, T3, T4 and T5, T6, T7, T8 and T9, T14, T15, and T16 are concurrent structural relationships, T11 and T1 are cyclic structural relationships, and their relationships are sequential structural relationships. At the same time, in the operation of the process model, all transitions have their input and output places, representing that each emergency resource scheduling task has its conditions. And the model does not exist with dead tasks. That is, the entire emergency resource scheduling process can be implemented, and emergency resource scheduling tasks can be completed.
4.2. GSPN Model Isomorphic Markov Chain and Performance Index Design
After qualitatively analyzing the GSPN model of the emergency resource scheduling process, this section conducts a quantitative analysis based on the Markov chain. Quantitative analysis can be the following steps. (i) Find the reachable set of the model and construct the reachable graph. (ii) Construct isomorphic Markov chain (MC) corresponding to the GSPN model. (iii) Calculate the stable probability of each state based on MC theory. (iv) Calculate various performance indicators of the model based on the stable probability of each state.
According to the established GSPN model of the emergency resource scheduling process, the model reachable graph as shown in Figure 3 is obtained by using the Petri net modeling software—PIPE (Platform Independent Petri net Editor). Then, the Ti transition on each directed arc is represented by the corresponding λi. That is, there are embedded Markov chains with isomorphic models, as shown in Figure 4. There are 19 states in a Markov chain, and their initial identification is M0= (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), which means that there is a token in P0 and P2 in the model. For the sake of simplicity, it is abbreviated as M0 = (0,2). By analogy according to the process, the following state sets are obtained: M0 = (0,2); M1 = (1,2); M2 = (3); M3 = (4); M4 = (5); M5 = (6); M6 = (7); M7 = (8); M8 = (9); M9 = (2,10); M10 = (2, 11); M11 = (2, 12, 13, 14); M12 = (2, 13, 14, 15); M13 = (2, 14, 15, 16); M14 = (2, 15, 16, 17); M15 = (2, 13, 15, 17); M16 = (2, 12, 16); M17 = (2, 12, 16, 17); M18 = (2, 12, 13, 17).
According to the isomorphic Markov chain stochastic process, the equation system is as follows [15]:
Π is the set of probabilities when each state of the Markov process is stable, namely, (P(M0), P(M1), ..., P(M18)); Q is the set of rates of transition from state Mi to state Mj. If the state Mi is connected to the state Mj by a directed arc, the element qij = λi(i≠j) of the matrix Q; otherwise, qij = 0, where the diagonal element qij = −∑qij(i = j).
To find out the factors affecting the emergency resource scheduling process more effectively from the model, according to the characteristics of the emergency resource scheduling process, the following two performance indicators are set for discussion and analysis.
4.2.1. Probability When the Place Pi Is Busy
The probability density function for each library. That is, in the stable state of the Markov chain, the probability of the number of Tokens contained in each library. Here, P[M(Pi) = i] is set as the probability that the process information in the place Pi has passed, and the probability when the place Pi is busy is obtained according to the following formula:P[M(Pi) = i] is the sum of the steady-state probabilities of all the identifiers in the process information of a place.
4.2.2. Utilization Ratio of Transition Ti
Transition utilization is the sum of the steady-state probabilities of each state in which a transition can occur. Find the time-consuming process in the process through its height, and explore the factors that have the greatest impact on the configuration process, which is calculated by the following formula [17]:where M represents the set of all reachable identifiers that make transition T implementable.
5. Case and Performance Analysis
At 7 : 49 on April 14, 2010, the Yushu earthquake reached 7.1. This is the most devastating, widest, and most disaster-relief earthquake disaster ever recorded in Yushu. Involving 19 townships in 6 counties of Yushu, 2698 people were killed and 270 disappeared. After the earthquake, the government quickly formulated the emergency disaster relief plan and cooperated with multiple departments to carry out disaster relief activities. Civil society organizations have also joined the disaster relief team. Emergency resources mobilized by all parties are pooled at the Disaster Emergency Resource Dispatch Centre and then transported uniformly to distribution points and sent to the people in the affected areas.
It is assumed here that the implementation rate λi of emergency resource scheduling process model transition is (8, 3, 3, 5, 6, 8, 6, 9, 8, 9, 10, 4, 2, 2, 3, 2, 1). Supposing the stable probability of each state of the Markov chain is , according to (1), the probability (6) of the stable state of each state of the stochastic process can be obtained:
The stable probability of each state of the Markov chain of the GSPN model is calculated from the above equations, as shown in Table 3.
Based on the stability probability of each state in Table 3, the busy probability and transition utilization rate of the place are calculated by (4) and (5), respectively. The results are shown in Tables 4 and 5.
The results of Tables 4 and 5 show that the probability of P2 and P12 being busy is the greatest. That is to say, in the process of emergency resource scheduling, the forward logistics P2 (emergency command center database) and reverse logistics P12 (nonrecyclable resources) are easy to accumulate information. In particular, the emergency command center needs to control the overall situation and continuously obtain the latest real-time data from the disaster area, command and dispatch personnel, vehicles, and other resources, which can easily cause information congestion, and subsequent work will be affected. Therefore, to increase the speed of the overall process, this link must be the focus of optimization.
In the process of reverse logistics, in addition to P12, with the trigger of T12 (collecting and summarizing post-disaster emergency resources), T14 (green processing), T15 (supplier reprocessing), and T16 (transporting to the local repository), the busy probabilities of P11 (waste emergency resources), P13 (nonrecyclable emergency resources), P15 (fuel and fertilizer), P16 (other raw resources and products), and P17 (reserve resources) are also large. It shows that the reverse logistics of emergency resources after disaster relief is also a process that cannot be ignored, which is easy to cause damage to the ecological environment and still requires more manpower and material resources.
6. Conclusion
The emergency resource allocation process has a certain stage. It comprehensively considers the predictive analysis of the emergency resources in the early stage and the forward and reverse logistics in the allocation process. In the process of restoring earthquake emergency management, the model simulation and performance analysis of the Yushu earthquake in Qinghai are carried out by using GSPN. The results show that the demand information of the emergency command center database and the recovery process of nonrecyclable resources should be the focus of process optimization. The emergency command center needs to attach great importance to the accuracy of the acquisition of emergency resource demand forecast information, improve the degree of informatization construction of the emergency command center, and meet the various and messy needs after the disaster.
At the same time, to avoid pollution and damage to the environment, scientific detection and treatment plans should be formulated for the recovery of different emergency resources. Finally, from the perspective of the transition process, the feedback link of the emergency configuration process is also very important, and the post-disaster summary evaluation should be strengthened. This paper still has many deficiencies in the analysis of the GSPN-based emergency resource scheduling process and the application of the research model. There are more uncertainties in the process of emergencies, which cannot be classified, integrated, and analyzed comprehensively and effectively. The uncertainties in interpreting the emergency resource allocation process are not addressed due to spatial and individual capabilities and need further optimization in future research.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors do not have any possible conflicts of interest.