Next Article in Journal
Optimal Configuration and Scheduling Model of a Multi-Park Integrated Energy System Based on Sustainable Development
Previous Article in Journal
Underwater Acoustic Target Recognition Based on Data Augmentation and Residual CNN
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Cross-Departmental Collaboration Approach for Earthquake Emergency Response Based on Synchronous Intersection between Traditional and Logical Petri Nets

1
College of Intelligent Equipment, Shandong University of Science and Technology, Tai’an 271000, China
2
College of Continuing Education, Shandong University of Science and Technology, Tai’an 271000, China
3
College of Computer Science and Engineering, Shandong University of Science and Technology, Qingdao 266000, China
*
Author to whom correspondence should be addressed.
Electronics 2023, 12(5), 1207; https://doi.org/10.3390/electronics12051207
Submission received: 16 January 2023 / Revised: 27 February 2023 / Accepted: 28 February 2023 / Published: 2 March 2023
(This article belongs to the Section Networks)

Abstract

:
In order to reduce the harm of earthquakes to human society, all governments actively promote the construction and development of earthquake emergency rescue work. The earthquake emergency response involves many departments, multiple personnel and large rescue forces, which presents a great challenge to the ability to carry out cross-departmental rescue work in a collaborative and joint manner. A novel collaboration approach based on traditional and logical Petri nets is proposed to improve the cross-departmental collaboration in earthquake emergency response. The approach extends the synchronization of transitions in traditional Petri nets to that of traditional and logical transitions in traditional and logical Petri nets, and defines the intersection of related logical functions. The approach builds a model for the earthquake emergency response plans of various departments with the help of traditional and logical Petri net models, and then performs synchronous intersection operations on the two kinds of Petri nets and merges the two kinds of Petri nets into new logical Petri nets. Meanwhile, through realizing the collaboration ability of cross-departmental work, the approach improves the rescue efficiency and reduces the damage of the earthquake emergency.

1. Introduction

The threat of earthquakes is enormous to the safety of human society. In 2021, 13 earthquakes of magnitude 7.0 or higher occurred worldwide, which caused 2274 deaths and 13,261 injuries. In 2022, nine earthquakes of magnitude 7.0 or higher occurred worldwide, which caused 29 deaths and 761 injuries. In order to reduce the human casualties and economic losses caused by earthquakes and to lessen the impact on human society, earthquake emergency rescue work has become one of the three major work systems of governments for earthquake prevention and mitigation, which is one of the important means for human beings to respond directly to earthquake disasters.
The institutional reform of the Chinese State Council in 2018 integrated the earthquake emergency rescue responsibilities of the China Earthquake Administration with other industry emergency management functions to establish the Ministry of Emergency Management. It further standardized the rescue process. The China Earthquake Administration was changed from a component department of the State Council to a subordinate department of the Ministry of Emergency Management. This change will test its ability to coordinate across departments. In addition, the damage caused by the earthquake is enormous. After the earthquake, many rescue forces and departments need to get involved in the rescue. It also puts forward higher requirements for earthquake emergency departments to carry out cross-departmental collaboration in rescue work.
Therefore, this study argues that an efficient earthquake emergency management system should not only include earthquake prediction and emergency planning but also have efficient post-earthquake emergency rescue with cross-departmental coordination. Only in this way can it help the earthquake emergency departments carry out rescue work efficiently and minimize casualties and property damage.
How to improve cross-organizational and cross-departmental collaboration capabilities will be the focus of this study. This study introduces business process management, process mining and other related knowledge into the field of earthquake emergency management. It uses traditional and logical Petri nets to model and analyze the relevant earthquake emergency management processes. It can further improve cross-departmental collaboration capabilities of earthquake emergency departments.
The main work of this paper has the following significance:
(1)
Traditional and logical Petri nets with better representation capability are applied to the field of earthquake emergency management;
(2)
The synchronization composition of traditional Petri nets is extended from the synchronization of transitions to the synchronization between logical and traditional transitions in traditional and logical Petri nets, and the intersection operation of logical functions of corresponding transitions is presented. For the first time, the logical function is extracted for traditional transitions, and the definition of synchronous transition unit and the calculation method of its logical functions are proposed;
(3)
The synchronous intersection operation between traditional and logical Petri nets fills the research gap. It also solves the coordination problem of the earthquake emergency departments;
(4)
For the superior department, a new logical Petri net can be obtained by modeling the workflow of its subordinate units between traditional and logical Petri nets and performing synchronous intersection operations. Then, we can more directly observe the workflow of each department and judge which work needs the cooperation of the two departments.
The rest of this paper is organized as follows: Related work is introduced in Section 2. In Section 3, some basic concepts are proposed. Section 4 presents a synchronous intersection approach based on traditional and logical Petri nets. Simulation experiments for cross-departmental collaboration in earthquake response are given in Section 5. We conclude in Section 6.

2. Related Work

2.1. Business Process Management and Process Mining

Earthquake emergency management is essentially Business Process Management (BPM) for the rescue work and rescue process after an earthquake. BPM is a specific way of management science [1]. With the help of BPM, the rescue work can be efficiently promoted and the rescue process can become smooth. After the earthquake, the rescue work will involve many departments, such as public security, health, transportation, and hospitals. In the whole rescue work, multiple departments need to coordinate to complete the cross-departmental coordination emergency. Therefore, the high cross-departmental coordination systems and methods become particularly important. BPM can play an important role in solving this problem.
BPM focuses on the integration of information technology and management technology. With the help of the event log, the entire business process is simulated and monitored for better management and correction [2]. As a component of BPM, Process Mining (PM) plays an increasingly prominent role in the field of BPM, which mainly extracts data information about processes from event data. The role of BPM and PM in earthquake emergency management is mainly to abstract the process model according to the relevant earthquake emergency plan or previous rescue workflow. After the earthquake occurs again, the relevant earthquake emergency departments can quickly promote the rescue work according to the guidance of the process model. It can realize the collaboration between different departments, as well as analyze the established process model to achieve a more effective rescue process. PM relies on process models. Currently, there are many methods to establish process models, such as BPMN [3], WFN [4,5] and Petri nets [6,7], etc.
Business Process Modeling Notation (BPMN) is a process model modeling language, which has been widely used in the field of BPM. Many BPM software vendors support its use. WorkFlow net(WFN) is considered as a subclass of Petri nets. WFN contains a source place and a sink place. All nodes of WFN are on the path from the source place to the sink place. With the help of modeling languages such as BPMN, WFN and PN, the process model can be more intuitive and readable.

2.2. Traditional and Logical Petri Nets

In fact, a Petri net is a traditional Petri net, also known as a classical Petri net. The reason why Petri nets are called traditional Petri nets is to distinguish them from a series of Petri nets derived later.
Traditional Petri nets have the ability to handle concurrency, asynchronous and contain uncertain information. High-level Petri nets, time Petri nets [8,9,10], Petri nets with inhibitor arcs, and logical Petri nets are further developed on the basis of Petri nets and have been widely recognized [11,12,13,14,15,16,17].
Logical Petri nets are developed on the basis of Petri nets with inhibitor arcs and high-level Petri nets. Traditional Petri nets and logical Petri nets are two different types of Petri nets with different applications. Logical Petri nets can describe the batch processing functions and the passing value indeterminacy of the system with the help of logical functions. The passing value indeterminacy is divided into input and output indeterminacy, which can be described by logical input functions and logical output functions, respectively. Therefore, compared with traditional Petri nets, logical Petri nets have stronger representation ability. At the same time, the application of logical Petri nets has greatly reduced the number of places in the model, making the constructed model more concise and readable. Logical Petri nets have been widely used in e-commerce and medical fields [18]. However, before this paper, neither logical Petri nets nor traditional Petri nets were applied to the field of earthquake emergency.

2.3. Sharing Composition and Synchronous Composition

Although both traditional and logical Petri nets have strong representation capabilities, there are still difficulties in modeling on larger scale systems [19]. Therefore, when facing complex systems and larger organizations for modeling and analysis, the design idea of combination is usually used to compose and decompose the models, which greatly reduces the complexity of modeling and analysis of large systems.
The study belongs to the composition section. Composition is usually divided into synchronous composition and sharing composition. Synchronous composition targets transitions’ synchronization and can be used to achieve synchronous operation of traditional Petri nets. The collaboration problems of earthquake emergency departments studied in this paper belong to the synchronous composition content.
Many studies have been carried out for sharing composition and synchronous composition, but more for traditional Petri nets. In synchronous composition, a lot of research has been done on the process of synchronous composition and complete behavior invariant of traditional Petri nets, and some examples are given to illustrate the correctness and effectiveness of this method in large-scale system modeling and analysis. The synchronous composition of two Petri nets has been extended to the synchronization of multiple Petri nets.
In the context of sharing composition, the preservation of liveness problem, siphon computation and the confluence property of Petri nets systems have been studied [20,21]. In addition, the methods of hierarchical process mining and deadlock detection have greatly improved the quality of mining models and the accuracy of deadlock detection [22].
Logical Petri nets have stronger modeling capability, which also leads to more complex research on their synchronous composition and sharing composition, and less relevant research results and data. Luan Wenjing et al. [23,24] proposed interactive logical Petri nets, analyzed their liveness and boundedness, and proposed a logical Petri net composition method of cooperative systems. In addition, a vector computational method is proposed to verify the properties of the sharing composition of logical Petri nets such as liveness and boundedness, which filled the gap between the study of sharing composition of logical Petri nets [25].

2.4. Emergency Management and Cross-Organizational Collaboration

In response to emergencies, an efficient emergency management system not only serves to maintain social stability and reduce the loss of life and property, but also enables various departments to respond quickly and advance in an integrated manner. Currently, great progress has been made in using traditional and colored Petri nets to simulate and model the urban emergency linkage systems and focusing on cross-organizational task collaboration. Some scholars and researchers have studied the way of cross-organizational [26,27] and cross-modal [28,29] collaborative emergency response processes and the privacy problem [30,31] involved. A method of automatically extracting the BPMN model of the emergency response process is also applied. These research results not only greatly promote the development of cross-departmental collaborative emergency response systems, but also point out the direction for the development of the industrial emergency response system in a specific field.
Due to the complexity of earthquake emergency management research, the modeling and cross-organizational collaboration of earthquake emergency departments are still in a blank state. Compared with the general cross-departmental emergency response collaboration, the cross-departmental collaboration in emergency response is more complex. In general, the cross-departmental emergency response is relatively simple and can be realized by traditional Petri nets. Moreover, this part has been studied and is relatively mature. However, the situation is more complicated in the face of cross-departmental collaboration in an earthquake emergency, which is determined by many factors, such as the large number of departments and personnel involved in the earthquake rescue work, and the need for collaboration among multiple departments in various rescue work. Traditional Petri nets can not meet the basic modeling requirements in some cases. Therefore, this paper uses traditional and logical Petri nets for modeling to meet the needs of different situations.
In summary, it can be concluded that there are several problems with the current study as follows.
(1)
At present, the main model of emergency management is constructed using traditional Petri nets, which leads to many limitations in modeling complex situations.
(2)
The application of logical Petri nets in earthquake emergency management is still a gap, and its value has not been fully utilized.
(3)
The research on cross-departmental collaboration is still more dependent on traditional Petri nets than on logical Petri nets.
(4)
In the composition of logical Petri nets, due to its complexity, the current research focus is mainly on the sharing composition. Therefore, the synchronous composition of logical Petri nets, which has the significance of cross-organizational, cross-departmental collaboration research, has not achieved research results.

2.5. Solution

In order to solve the above problems, this paper introduces the application of traditional and logical Petri nets to the field of earthquake emergency response. In this paper, the emergency plans or workflows of each earthquake emergency department are transformed into traditional or logical Petri nets models. Since there is no research on synchronous composition of logical Petri nets, there is no way to directly apply traditional and logical Petri nets across departments. In order to solve the problem of cross-departmental collaboration of traditional and logical Petri nets, this study researches the logical Petri nets and proposes the synchronous intersection operation for the two kinds of Petri nets to realize the synchronous composition. Then, it can realize the cross-departmental collaboration for traditional and logical Petri nets in order to realize the collaboration of earthquake emergency departments.

3. Preliminaries

Before studying the synchronous intersection operations of traditional and logical Petri nets to achieve cross-departmental collaboration, it is necessary to acquire knowledge about traditional and logical Petri nets. For example, the definitions of traditional Petri nets, logical Petri nets and the related meanings of transition and place are as follows:
Definition 1.
(traditional Petri nets) Let A be a set of activities, and let PN be a five-tuple traditional Petri nets over the set of activities A , P N = P , T ; F , M , α . P is a finite set of places on PN; T is a finite set of transitions on PN; F P × T T × P is a finite set of directed arcs on PN; M : P Ν + is a marking function on PN, M p represents the number of token in p; α : T A is a mapping from a set of transitions to a set of activities. The transition firing rules for traditional Petri net PN are as follows:
(1)
For t i T , if p i t i , M p i 1 , ti is enabled at M, then ti can be fired, it can be denoted by M[ti>.
(2)
When ti is fired under the marking M, a new marking M′ is created, and it can be denoted by M[ti>M′. If p i t i , and p i t i , then M p i 1 = M p i ; if p i t i , and p i t i , then M p i + 1 = M p i ; else, M p i = M p i .
For t i T , t i = { p i | p i , t i F } , t i is called a preset of t i . For t i T , t i = { p i | t i , p i F } , t i is called a postset of t i .
The main difference between traditional and logical Petri nets is the constraint of the logical function of the transitions to the places. It enhances the representation capability of logical Petri nets.
Definition 2.
(Logical Petri nets) Let A be a set of activities, logical Petri net is a seven-tuple on the set of activities A , can be denoted by LPN, L P N = P , T ; F , I , O , M , α , where
(1)
P is a finite set of places on LPN;
(2)
T is a finite set of transitions on LPN, T = T D T I T O , where
(a)
T D   denotes the set of traditional transitions of LPN, whose firing rules are consistent with those of the transitions in traditional Petri nets, with specific reference to the contents of Definition 1;
(b)
T I   denotes the set of logical input transitions of LPN; for t i T I , the firing of ti is constrained by the logical input function f I t i of ti for its place p i , and p i t i . For t i T I , if f I t i | M = . T . , ti is enabled at M, then ti can be fired, it can be denoted by M[ti>; When ti is fired under the marking M, a new marking M′ is created, and it can be denoted by M[ti>M′. For p t i , M p = 0 and M p = 1 ; for p t i t i , M p = M p ;
(c)
T O   denotes the set of logical output transitions of LPN; for t i T O , the firing result is constrained by the logical output function f O t i of ti for its place p i , and p i t i . For t i T O , if p i t i and M p i 1 , t i is enabled at M, then t i can be fired, it can be denoted by M[ti>; when ti is fired under the marking M a new marking M′ is created, and it can be denoted by M[ti>M′. For p t i , M p = 1 and M p = 0 ; for p t i t i , M p = M p ;
(3)
F P × T T × P   is a finite set of directed arcs on LPN;
(4)
I is a mapping from a set of logical input transitions to a logical input function, for t i T I , I t i = f I t i ;
(5)
O is a mapping from a set of logical output transitions to a logical output function, for t i T O , O t i = f O t i ;
(6)
α : T A   is a mapping from a set of transitions to a set of activities;
(7)
M : P Ν +   is a marking function on LPN, M p represents the number of tokens in p.
This paper uses some symbols , , .   T   .   and the reachable marking functions M = M + R T W T X . Please refer to [11] for the use of these symbols and functions. In this paper, "*" represents the number 0, 1, or −1. Please refer to [11] for its use and calculation method.
Figure 1 gives an example of logical Petri nets, denoted by LPN. For LPN, P = p 1 , p 2 , p 3 , p 4 , p 5 , p 6 , p 7 , T = T D T I T O , T D = t 1 , T I = t 2 , T O = t 3 , M = 1 , 1 , 0 , 0 , 0 , 0 , 0 T . t 1 is a traditional transition; t 2 is a logical input transition, and I t 2 = f I t 2 = p 2 p 4 p 3 ; t 3 is a logical output transition, and O t 3 = f O t 3 = p 6 p 7 . Under the marking M , both p1 and p2 contain a token; at this time, t1 is enabled. After t1 is fired, p3 contains a token; when t2 is fired, p5 contains a token; p2 and p4 do not contain any tokens at the same time.
Since both p2 and p3 contain a token, and p4 does not contain a token, t2 is enabled. When t2 is fired, p5 contains a token. At this point, t3 is enabled. When t3 is fired, there will be two kinds of cases: (1) p6 contains a token; (2) p7 contains a token. Assuming that case (1) occurs, p6 contains a token and p7 does not contain a token.
Each logical function is transformed into a disjunctive normal form with the help of discrete mathematical knowledge, and that is unique. Each disjunctive clause in the disjunctive normal form is a conjunction of all the positive or negative places associated with a logical transition. Each conjunct in the disjunctive can be converted to a vector. Then, we calculate the reachable marking of the logical Petri nets; e.g., for I t 2 = f I t 2 = p 2 p 4 p 3 , it can be transformed into a disjunctive normal form: p 2 ¬ p 4 p 3 ¬ p 2 p 4 p 3 . Calculating the reachable marking of the logical Petri nets requires a lot of other knowledge, e.g., LPN conjunct vector, logical transition vector sets, the logical incidence matrix, the logical function matrix, reachable marking functions, and so on, which can be found in the related literature [11] and will not be discussed in this paper.
Using L P N = P , T ; F , I , O , M , α in Figure 1 as an example, the logical incidence matrix of LPN is constructed:
R = 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0
The logical function matrix of the LPN is as follows:
W = 0 0 0 0 0 0 0 ( * , * ) T ( 1 , 0 ) T ( 1 , 1 ) T ( 0 , 1 ) T ( * , * ) T ( * , * ) T ( * , * ) T ( * , * ) T ( * , * ) T ( * , * ) T ( * , * ) T ( * , * ) T ( 1 , 0 ) T ( 0 , 1 ) T
According to the firing process of LPN, a final marking M = 0 , 0 , 0 , 0 , 0 , 1 , 0 T and a vector X = 1 , 1 , 0 , 0 , 1 T are obtained.
Finally, the reachable marking functions of LPN is as follows: M = M + R T W T X = 0 , 0 , 0 , 0 , 0 , 1 , 0 T .

4. Synchronous Intersection of Traditional and Logical Petri Nets

4.1. Definition of Synchronous Intersection

The relevant rescue work of the earthquake emergency department often needs the collaboration between two or more departments, which has the characteristics of multi- department collaboration. When solving the problem of collaboration between two departments, the problem of synchronous composition between traditional and logical Petri nets often arises. In essence, it is to solve the problem of synchronous composition between traditional and logical Petri nets. Therefore, we propose a synchronous intersection operation for the two kinds of Petri nets. It is mainly an extension on the basis of transition synchronization in the synchronous composition of traditional Petri nets, so that the synchronous composition of traditional and logical Petri nets includes the synchronization between traditional and logical transitions and the intersection of logical functions. The synchronous intersection operation is actually a synchronous intersection operation of some transitions and logical functions in the two kinds of Petri nets. Therefore, on the basis of logical Petri nets, this paper further generalizes the meaning of transitions and the logical functions. Firstly, the corresponding logical functions are extracted for the traditional transitions in traditional Petri nets. Then, the definition and calculation approach of synchronous transition units and the logical functions of synchronous transition units are proposed. It needs to start with two transitions with the same meaning, i.e., the mapping meaning of these transitions on the set of activities is consistent. Finally, the logical functions of synchronous transition units are obtained, and then the synchronous intersection operations between traditional and logical Petri nets are realized.
Definition 3.
(Synchronous transition unit) Let A 1 and A 2 be two sets of activities, and A 1 A 2 φ . P N 1 = P 1 , T 1 ; F 1 , M 1 , α 1 is a traditional Petri net based on the activities set A 1 , and L P N 2 = P 2 , T 2 ; F 2 , I 2 , O 2 , M 2 , α 2 is a logical Petri net based on the activities set A 2 . For a i A 1 , and t i T 1 , α 1 t i = a i ; for a j A 2 , and t j T 2 , α 2 t j = a ; if a i = a j , i.e., α 1 t i = α 2 t j , synchronous transition unit can be denoted by < t i , t j > .
We define the meaning of the synchronous transition unit < t i , t j > as a transition, which is approximately similar to what is represented by transitions ti, tj and has all the characteristics of transitions ti, tj. In fact, < t i , t j > is obtained by performing synchronous mergence and synchronous intersection operations on the transitions ti, tj. Therefore, the synchronous transition unit is essentially a transition. Its nature, characteristics and meaning are the same as those of a transition, but the denotation approach is different. Its purpose is mainly to replace the original transition in the appropriate scene, to make a unified symbolic representation and to distinguish it from the traditional and logical transitions.
In this paper, there are three cases considered for the synchronous intersection operation between traditional and logical Petri nets: (1) the synchronous intersection operation between a pair of traditional transitions, and the synchronous transition unit is regarded as a traditional transition; (2) the synchronous intersection operation between a traditional transition and a logical input transition, and the synchronous transition unit is regarded as a logical input transition; (3) the synchronous intersection operation between a traditional transition and a logical output transition, and the synchronous transition unit is regarded as a logical output transition.
Both synchronous transition unit, traditional transition and logical transition all have input and output. The logical input transition and the logical output transition have related logical function to constrain their firing. The relevant logical functions can also be extracted for the traditional transition, and the logical function of the traditional transition is often used in performing synchronous intersection operations.
Definition 4.
(Extraction of traditional transition logical functions) Let L P N = P , T ; F , I , O , M , α be a logical Petri net. For t i T , a logical input function f I t i and a logical output function f O t i can be extracted, where
  • For t i T D T O , the logical input function is f I t i = p i t i p i ;
  • For t i T D T I , the logical output function is f O t i = p i t i p i ;
  • For t i T in traditional Petri nets, its logical functions are extracted in the same way as the way for t i T D in logical Petri nets. In addition, obtaining functions for the synchronous transition unit can be done by intersecting the logical functions of the two transitions that have not been merged before.
Definition 5.
(Synchronous transition unit logical functions) Let P N 1 = P 1 , T 1 ; F 1 , M 1 , α 1 be a traditional Petri net based on A 1 , let L P N 2 = P 2 , T 2 ; F 2 , I 2 , O 2 , M 2 , α 2 be a logical Petri net based on A 2 , and A 1 A 2 φ , there exists a synchronous transition unit < t i , t j > in P N 1 and L P N 2 . f I < t i , t j > is called a logical input function of < t i , t j > , and f O < t i , t j > is called a logical output function of < t i , t j > , where (1) f I < t i , t j > = f I t i f I t j ; (2) f O < t i , t j > = f O t i f O t j .

4.2. Computation of Synchronous Intersection

The synchronous intersection of traditional and logical Petri nets is actually the synchronous intersection of some transitions, which enables some transitions to be synchronized and merged in order to realize the collaboration between two departments in some work. In the two kinds of Petri nets, the transitions are divided into the traditional transitions, the logical input transitions, and the logical output transitions. Therefore, there are many cases included in the synchronous intersection operation of traditional and logical Petri nets. In this paper, we mainly consider three cases of synchronous intersection operations: (1) The traditional transitions and the traditional transitions; (2) The traditional transitions and the logical input transitions; (3) The traditional transitions and the logical output transitions.
Let P N 1 = P 1 , T 1 ; F 1 , M 1 , α 1 be a traditional Petri net based on A 1 , let L P N 2 = P 2 , T 2 ; F 2 , I 2 , O 2 , M 2 , α 2 be a logical Petri net based on A 2 , and A 1 A 2 φ . For LPN2, T 2 = T 2 , D T 2 , I T 2 , O . The three cases of synchronous intersection operations for traditional and logical Petri nets are discussed separately.

4.2.1. Synchronous Intersection Operation of Traditional Transitions

For t i T 1 and t j T 2 , D , α 1 t i = a i = α 2 t j = a j , a i A 1 , a j A 2 , at this time, t i and t j can perform synchronous intersection operations to make t i and t j merge together; it can be recorded as < t i , t j > , and replace t i and t j . A new logical Petri net L P N 3 = P 3 , T 3 ; F 3 , I 3 , O 3 , M 3 , α 3 is obtained by PN1 and LPN2 after the synchronous intersection operation. The acquisition of the new net consists of several aspects: the set of activities composition, the set of places composition, the set of transitions composition, the set of directed arcs composition, mapping I composition, mapping O composition, marking composition, and mapping α composition.
(1)
The set of activities composition
A 3 = A 1 A 2 a j . Composing the set of activities A 1 , A 2 of PN1 and LPN2. Since the activities mapped in the set of activities A 1 and A 2 are the same for transitions ti and tj, i.e., a i = a j , it needs to delete one of the activities a i or a j . Then preforming the union operation on A 1 and A 2 , A 3 can be obtained;
(2)
The set of places composition
P 3 = P 1 P 2 . Since the place deletion is not included in the synchronous intersection operation, preforming the union operation on P 1 and P 2 , P 3 can be obtained;
(3)
The set of transitions composition
T 3 , D = T 1 t i T 2 , D t j < t i , t j > . Since ti, tj and < t i , t j > can be regarded as a traditional transition and belong to the set of traditional transitions, hence, it is necessary to delete ti, tj from T 1 and T 2 , D , respectively. Then preforming the union operation on T 1 , T 2 , D and < t i , t j > , T 3 , D can be obtained;
T 3 , I = T 2 , I , T 3 , O = T 2 , O . Since this part performs the synchronous intersection operation of two traditional transitions, and PN1 is a traditional Petri net and LPN2 is a logical Petri net, so assign T 2 , I to T 3 , I and T 2 , O to T 3 , O to obtain the set of logical input transitions and the set of logical output transitions of the new logical Petri net LPN3;
(4)
The set of directed arcs composition
F 3 = F 1 ( t i , p j p j t i ) ( p i p i t i , t i ) . Firstly, delete all arcs from F1 that contain ti, F3 can be obtained: F 3 = F 3 ( F 2 ( t j , p j p j t j ) ( p i p i t j , t j ) ) ; delete all arcs from F2 that contain tj, and then preform the union operation on F3 and F2;
F 3 = F 3 ( < t i , t j > , p j p j t i ) ( p i p i t i , < t i , t j ) ; add the new directed arcs to F3 by replacing ti in the previously deleted arcs with < t i , t j > ; F 3 = F 3 ( < t i , t j > , p j p j t j ) ( p i p i t j , < t i , t j > ) ; add the new directed arcs to F3 by replacing tj in the previously deleted arcs with < t i , t j > ;
(5)
Mapping I composition
I3 = I2. Because this part is the synchronous intersection of two traditional transitions, it does not include mapping I composition. Assign I2 to I3 in order to get the mapping I of LPN3;
(6)
Mapping O composition
O3 = O2. Assign O2 to O3 in order to get the mapping O of LPN3;
(7)
Marking composition
M 3 = M 1 M 2 . Since the place deletion is not included in the synchronous intersection operation, marking composition is the same as the set of places composition approach. Preforming the union operation on M1 and M2, M3 can be obtained;
(8)
Mapping composition
α 3 = α 1 α 1 t i α 2 α 2 t j . Delete α 1 t i , α 2 t j from α 1 , α 2 . Then, preforming the union operation on α 1 and α 2 , α 3 can be obtained;
α 3 = α 3 α 3 < t i , t j > = a i ; add the mapping α 3 < t i , t j > = a i of < t i , t j > for α 3 .
Note: Since the synchronous transition unit obtained by the synchronous intersection operation of two traditional transitions is still a traditional transition, the extraction of logical functions and the union operation are not involved. In addition, there has been much research on the synchronous intersection operation of traditional transitions. For specific examples, please refer to relevant literature, and this paper will not give any examples.

4.2.2. Synchronous Intersection Operation of Traditional and Logical-Input Transitions

For t i T 1 and t j T 2 , I , α 1 t i = a i = α 2 t j = a j , a i A 1 , a j A 2 ; at this time, t i and t j can perform synchronous intersection operations to make t i and t j merge together; it can be recorded as < t i , t j > , and replace t i and t j . A new logical Petri net L P N 4 = P 4 , T 4 ; F 4 , I 4 , O 4 , M 4 , α 4 is obtained by PN1 and LPN2 after a synchronous intersection operation. The acquisition of the new net consists of several aspects: the set of activities composition, the set of places composition, the set of transitions composition, the set of directed arcs composition, the logical input functions composition, mapping I composition, mapping O composition, marking composition, and mapping α composition.
Some of the steps and approaches in the synchronous intersection operation of traditional transitions and logical input transitions are consistent with the synchronous intersection operation of traditional transitions. The same aspects will not be introduced here; only the differences will be introduced. The same aspects mainly include the set of activities composition, the set of places composition, the set of directed arcs composition, marking composition, and mapping α composition. The different aspects are as follows:
(1)
The set of transitions composition
T 4 , D = T 1 t i T 2 , D , T 4 , I = T 2 , I t j t i , t j . Since < t i , t j > , obtained after synchronous intersection operation of t i and t j , can be regarded as a logical input transition and belongs to the set of logical input transitions, therefore it is necessary to delete ti from T 1 . Then, preforming the union operation on T 1 and T 2 , D , T 4 , D can be obtained. Finally, deleting tj from T 2 , I and preforming the union operation on T 2 , I and < t i , t j > , T 4 , I can be obtained.
T 4 , O = T 2 , O . Because this part is the synchronous intersection of a traditional transition and a logical input transition, it does not include the logical output transitions composition. Assign T 2 , O to T 4 , O , and then get the set of logical output transitions of LPN4.
(2)
The logical input functions composition
f I < t i , t j > = f I t i f I t j , obtain the logical input function f I t = p i t i p i for ti; then, intersect f I t i with f I t j to obtain the logical input function f I < t i , t j > of < t i , t j > .
(3)
Mapping I composition
I 4 = I 2 I 2 t j ; since PN1 is a traditional Petri net, it does not involve mapping I. After deleting mapping I 2 t j = f I t j , I2 is assigned to I4 in order to get the mapping I4 of LPN4.
I 4 = I 4 I 4 < t i , t j > = f I < t i , t j > ; since < t i , t j > can be regarded as a logical input transition, the mapping from < t i , t j > to f I < t i , t j > needs to be added to I4 in order to finally obtain the mapping I4 in LPN4.
(4)
Mapping O composition
O4=O2, because this part is the synchronous intersection of a traditional transition and a logical input transition, it does not include mapping O composition. Assign O2 to O4 in order to get the mapping O of LPN4.
An example of a traditional Petri net is given in Figure 2a, denoted by PN5. An example of a logical Petri net is given in Figure 2b, denoted by LPN6. P N 5 = P 5 , T 5 ; F 5 , M 5 , α 5 is based on A5, L P N 6 = P 6 , T 6 ; F 6 , I 6 , O 6 , M 6 , α 6 is based on A6, and A 5 A 6 φ . For P N 5 , T 5 = t 1 , P 5 = p 1 , p 2 , M 5 = 1 , 0 T . For L P N 6 , T 6 , D = φ , T 6 , I = t 1 , T 6 , O = φ , P 6 = p 1 , p 2 , p 3 , p 4 , I 6 t 1 = f I t 1 = p 1 p 3 p 2 , O = φ , M 6 = 1 , 1 , 0 , 0 T .
(1)
Reachable marking functions about PN5
Construct the logical incidence matrix of PN5: R 5 = 1 1 , since PN5 is a traditional Petri net, it is not necessary to construct the logical function matrix. t1 is enabled under the marking M 5 = 1 , 0 T , and it can be fired. Then, a vector X 5 = 1 is obtained.
Finally, the reachable marking functions of PN5 is as follows: M 5 = M 5 + R 5 T X 5 = 0 , 1 T , the final marking M 5 = 0 , 1 T is obtained. It means that p2 contains a token, whereas p1 does not.
(2)
Reachable marking functions about LPN6
Construct the logical incidence matrix of LPN6: R 6 = 0 0 0 1 ; construct the logical function matrix of LPN6: W 6 = ( 1 , 0 ) T ( 1 , 1 ) T ( 0 , 1 ) T ( * , * ) T . t 1 is enabled at M 6 = 1 , 1 , 0 , 0 T , and it can be fired. Let p1′∧p2′ be fired under the marking M 6 = 1 , 1 , 0 , 0 T , then a vector X 6 = ( 1 , 0 ) T is obtained.
Finally, the reachable marking functions of LPN6 is as follows: M 6 = M 6 + R 6 T W 6 T X 6 = 0 , 0 , 0 , 1 T , the final marking M 6 = 0 , 0 , 0 , 1 T is obtained. It means that p4′ contains a token, whereas p1′, p2′, and p3′ do not.
(3)
Synchronous intersection operation of PN5 and LPN6
By the synchronous intersection operation of PN5 and LPN6, LPN7 can be obtained. The specific structure of LPN7 is given in Figure 2c. The steps are as follows:
Step 1 The set of places composition: P 7 = P 5 P 6 = p 1 , p 2 , p 1 , p 2 , p 3 , p 4 ;
Step 2 The set of transitions composition:
T 7 , D = T 5 t 1 T 6 , D = φ , T 7 , I = T 6 , I t 1 < t 1 , t 1 > , T 7 , O = T 6 , O = φ ;
Step 3 The set of directed arcs composition:
F 7 = F 5 ( t 1 , p j p j t 1 ) ( p i p i t 1 , t 1 ) ,   F 7 = F 7 ( F 6 ( t 1 , p j p j t 1 ) ) , F 7 = F 7 ( F 6 ( p i t 1 , t 1 ) ) , F 7 = F 7 ( < t 1 , t 1 > , p j p j t 1 . ) , F 7 = F 7 ( p i p i t 1 , < t 1 , t 1 > ) , F 7 = F 7 ( < t 1 , t 1 > , p j p j t 1 . ) , F 7 = F 7 ( p i p i t 1 , < t 1 , t 1 > ) ;
Step 4 The logical input functions composition:
Obtain the logical input function f I t 1 = p i t 1 p i = p 1 for t1, then intersect f I t 1 . with f I t 1 of t 1 to obtain the logical input function f I < t 1 , t 1 > = f I t 1 f I t 1 = p 1 p 1 p 3 p 2 of < t 1 , t 1 > ;
Step 5 Mapping I composition:
I 7 = I 6 I 6 t 1 , since PN5 is a traditional Petri net, it does not involve mapping I. After deleting mapping I 6 t 1 = f I t , I6 is assigned to I7 in order to get the mapping I7 of LPN7.
I 7 = I 7 I 7 < t 1 , t 1 > = f I < t 1 , t 1 > , the mapping from < t 1 , t 1 > to f I < t 1 , t 1 > needs to be added to I7 in order to finally obtain the mapping I7 in LPN7;
Step 6 Mapping O composition: O7 = O6;
Step 7 Marking composition: M 7 = M 5 M 6 = 1 , 0 , 1 , 1 , 0 , 0 T ;
Construct the logical incidence matrix of LPN7: R 7 = 0 1 0 0 0 1 ; Construct the logical function matrix of LPN7: W 7 = [ ( 1 , 1 ) T , ( * , * ) T , ( 1 , 0 ) T , ( 1 , 1 ) T , ( 0 , 1 ) T , * , * ) T .
< t 1 , t 1 > is enabled at M 7 = 1 , 01 , 1 , 0 , 0 T , and it can be fired. Let p1p1′∧p2′ be fired under the marking M 7 = 1 , 01 , 1 , 0 , 0 T , then a vector X 7 = ( 1 , 0 ) T is obtained.
Finally, the reachable marking functions of the LPN7 is as follows: M 7 = M + R 7 T W 7 T X 7 = 0 , 1 , 0 , 0 , 0 , 1 T , the final marking M 7 = 0 , 1 , 0 , 0 , 0 , 1 T is obtained. It means that p2 and p4′ each contain a token, whereas p1, p1′, p2′, p3′ do not.

4.2.3. Synchronous Intersection Operation of Traditional and Logical-Output Transitions

For t i T 1 and t j T 2 , O , α 1 t i = a i = α 2 t j = a j , a i A 1 , a j A 2 , at this time, ti and tj can perform synchronous intersection operation to make ti and tj merge together, it can be recorded as < t i , t j > , and replace ti and tj. A new logical Petri net L P N 8 = P 8 , T 8 ; F 8 , I 8 , O 8 , M 8 , α 8 is obtained by PN1 and LPN2 after synchronous intersection operation. The acquisition of the new net consists of several aspects: the set of activities composition, the set of places composition, the set of transitions composition, the set of directed arcs composition, mapping I composition, the logical output functions composition, mapping O composition, marking composition, mapping α composition.
Some of the steps and approaches in the synchronous intersection operation of traditional transition and logical output transition are consistent with the synchronous intersection operation of traditional transitions. The same aspects will not be introduced here, only the differences will be introduced. The same aspects mainly include the set of activities composition, the set of places composition, the set of directed arcs composition, marking composition, and mapping composition. The different aspects are as follows:
(1)
The set of transitions composition
T 8 , D = T 1 t i T 2 , D , T 8 , O = T 2 , O t j < t i , t j > . Since < t i , t j > obtained after synchronous intersection operation of ti and tj can be regarded as a logical output transition and belongs to the set of logical output transitions, it is necessary to delete ti from T1. Then, preforming the union operation on T1 and T2,D, T8,D can be obtained. Finally, deleting tj from T2,O, and then preforming the union operation on T2,O and < t i , t j > , T 8 , O can be obtained;
T8,I = T2,I. Because this part is the synchronous intersection of a traditional transition and a logical output transition, it does not include the logical input transitions composition. Assign T2,I to T8,I in order to get the set of logical input transitions of LPN8;
(2)
Mapping I composition
I8 = I2. Because this part is the synchronous intersection of a traditional transition and a logical output transition, it does not include mapping I composition. Assign I2 to I8 in order to get the mapping I of LPN8;
(3)
The logical output functions composition
f O < t i , t j > = f O t i f O t j . Firstly, obtain f O t i = p i t i .   p i for ti; then, intersect f O t i with f O t j of tj to obtain f O < t i , t j > of < t i , t j > ;
(4)
Mapping O composition
O 8 = O 2 O 2 t j . Since PN1 is a traditional Petri net, it does not involve mapping O. After deleting mapping O 2 t j = f O t j , O2 is assigned to O8 in order to obtain the mapping O8 of LPN8:
O 8 = O 8 O 8 < t i , t j > = f O < t i , t j > . Since < t i , t j > can be regarded as a logical output transition, the mapping of < t i , t j > to the logical output function f O < t i , t j > needs to be added to O8, which eventually obtains the mapping O8 in LPN8.
An example of a traditional Petri net is shown in Figure 3a, denoted by PN9. An example of a logical Petri net is shown in Figure 3b, denoted by LPN10. P N 9 = P 9 , T 9 ; F 9 , M 9 , α 9 is based on A9; L P N 10 = P 10 , T 10 ; F 10 , I 10 , O 10 , M 10 , α 10 is based on A10, and A 9 A 10 φ . For PN9, T 9 = t 1 , P 9 = p 1 , p 2 , and M 9 = 1 , 0 T . For LPN10, T 10 , D = φ , T 10 , I = φ , T 10 , O = t 1 , P 10 = p 1 , p 2 , p 3 , p 4 , I 10 = φ , O 10 t 1 = f O t 1 = p 2 p 3 p 4 , and M 10 = 1 , 0 , 0 , 0 T .
(1)
Reachable marking functions about PN9
The reachable marking functions steps and results are the same as the reachable marking functions of PN5 in Figure 2a, and the final marking M 9 = 0 , 1 T is obtained. It means that p2 contains a token, whereas p1 does not.
(2)
Reachable marking functions about LPN10
Construct the logical incidence matrix of LPN10: R 10 = 1 0 0 0 ; construct the logical function matrix of LPN10: W 10 = [ ( * , * ) T , ( 1 , 0 ) T , ( 0 , 1 ) T , 1 , 1 ) T . t1′ is enabled at M 10 = 1 , 0 , 0 , 0 T , and it can be fired. Let p2′∧p4′ be fired under the marking M 10 = 1 , 0 , 0 , 0 T , then a vector X 10 = ( 1 , 0 ) T is obtained.
Finally, the reachable marking functions of the LPN10 is as follows: M 10 = M 10 + R 10 T W 10 T X 10 = 0 , 1 , 0 , 1 T , and the final marking M 10 = 0 , 1 , 0 , 1 T is obtained. It means that both p2′ and p4′ contain a token, whereas p1′ and p3′ do not.
(3)
Synchronous intersection operation of PN9 and LPN10
By the synchronous intersection operation of PN9 and LPN10, LPN11 can be obtained. The specific structure of LPN11 is given in Figure 3c. The steps are as follows:
Step 1 The set of places composition:
P 11 = P 9 P 10 = p 1 , p 2 , p 1 , p 2 , p 3 , p 4 ;
Step 2 The set of transitions composition:
T 11 , D = T 9 t 1 T 10 , D = φ , T 11 , I = T 10 , I = φ , T 11 , O = T 10 , O t 1 < t 1 , t 1 > ;
Step 3 The set of directed arcs composition:
F 11 = F 9 ( t 1 , p j p j t 1 ) ( p i p i t 1 , t 1 ) , F 11 = F 11 ( F 10 ( t 1 , p j p j t 1 ) ( p i p i t 1 , t 1 ) ) , F 11 = F 11 ( < t 1 , t 1 > , p j p j t 1 ) ( p i p i t 1 , < t 1 , t 1 > ) , F 11 = F 11 ( < t 1 , t 1 > , p j p j t 1 ) ( p i p i t 1 , < t 1 , t 1 > ) ;
Step 4 The mapping I composition: I11 = I10;
Step 5 The logical output functions composition:
Obtain f O t 1 = p i t 1 p i = p 2 for t1, then intersect f O t 1 with f O t 1 of t1′ to obtain the logical output function f O < t 1 , t > = f O t 1 f O t 1 = p 2 p 2 p 3 p 4 of < t 1 , t 1 > ;
Step 6 Mapping O composition:
O 11 = O 10 O 10 t 1 . Since PN9 is a traditional Petri net, it does not involve mapping O. After deleting mapping O 10 t 1 = f O t 1 , O10 is assigned to O11 in order to get the mapping O11 of LPN11;
O 11 = O 11 O 11 < t 1 , t 1 > = f O < t 1 , t 1 > . Since < t 1 , t 1 > can be regarded as a logical output transition, the mapping from < t 1 , t 1 > to f O < t i , t j > needs to be added to O11, which eventually obtains O11 in LPN11;
Step 7 The marking composition:
M 11 = M 9 M 10 = 1 , 0 , 1 , 0 , 0 , 0 T ;
Construct the logical incidence matrix of LPN11: R 11 = 1 , 0 , 1 , 0 , 0 , 0 ; construct the logical function matrix of LPN11: W 11 = [ ( , ) T , ( 1 , 1 ) T , ( , ) T , ( 1 , 0 ) T , ( 0 , 1 ) T , 1 , 1 ) T ;
< t 1 , t 1 > is enabled at M 11 = 1 , 01 , 0 , 0 , 0 T , and it can be fired. Let p2p2′∧p4′ be fired under the marking M 11 = 1 , 01 , 0 , 0 , 0 T , then a vector X 11 = ( 1 , 0 ) T is obtained.
Finally, the reachable marking functions of LPN11 is as follows: M 11 = M 11 + R 11 T W 11 T X 11 = 0 , 1 , 0 , 1 , 0 , 1 T , and the final marking M 11 = 0 , 1 , 0 , 1 , 0 , 1 T is obtained. It means that p2, p2′,and p4′ each contain a token, whereas p1, p1′,and p3′ do not.

5. Simulation Experiments

In Section 4, three cases of synchronous intersection operation of traditional and logical Petri nets are introduced and proved by examples. This chapter begins to take the emergency plan of a Provincial Earthquake Administration as an example to verify and simulate the synchronous intersection operation.
According to the information of the emergency plan, the subordinate units of the Provincial Earthquake Administration include the Bureau Office, the Emergency Rescue Department, the Earthquake Monitoring and Forecasting Center, the Earthquake Damage Defense Department, the Earthquake Disaster Defense Center, the Science and Technology Department (STD), the institutional service center, the development and finance department, and the Earth Observation Institute (EOI) affiliated with the National Earthquake Administration (NEA). These departments have corresponding work arrangements and a complete set of emergency response processes after an earthquake. Once the earthquake occurs, the above-mentioned departments will quickly start emergency response according to the established plan and complete a series of rescue work in the shortest time in order to shorten rescue time, improve rescue efficiency and reduce loss of life and property.
Since modeling experiments and simulations for the complete emergency plan and all departments in this part of the work would lead to a huge workload and a long article, this paper selects one of them, the EOI and the STD, as an example for modeling verification. In this way, it not only reduces the workload but also verifies the feasibility of synchronous intersection operations.
In this section, a traditional Petri net and a logical Petri net model are modeled for the EOI and the STD, respectively. Then, according to Section 4.2, synchronous intersection operation between traditional and logical Petri net model is conducted, and finally a complete, cross-departmental logical Petri net model is obtained. It also includes a series of emergency work completed by the EOI and the STD after the earthquake.

5.1. Emergency Planning Restatement

Through careful analysis of the emergency plan, this paper extracts tasks from the EOI and the STD, mainly including the following tasks and processes:
The EOI: after the earthquake, the EOI starts the emergency response and then mainly performs two tasks: (1) sending a team to the site to set up a mobile monitoring center station in the earthquake area, obtaining the mobile monitoring center station data in the earthquake area and reporting it to the NEA, and then obtaining the approval of the NEA; (2) starting the emergency response for strong earthquakes in key areas in collaboration with the STD, organizing relevant experts from the institute to work with the staff of the STD and combine with the approval of the NEA to come up with emergency recommendations for strong earthquakes in key areas, and reporting the recommendations to the NEA.
The STD: After the earthquake, the STD initiates emergency response for strong earthquakes in key areas in collaboration with the EOI, organizes experts in reservoirs, key buildings, strong vibration and other industries to conduct research, invites gravity experts to participate in research when necessary, and finally proposes emergency response recommendations for strong earthquakes in key areas in collaboration with the EOI and implements all emergency recommendations.

5.2. EOI Model and Its Reachable Markings

Based on the EOI emergency response workflow in Section 5.1, a set of activities can be extracted. A12 = {Activate emergency, Dispatch on-site team, Set up stations, Obtain station data, Report to the NEA, Get approval, Activate strong earthquake response, Organize Institute experts, Make a suggestion, Presented to the NEA}; a traditional Petri net model P N 12 = P 12 , T 12 ; F 12 , M 12 , α 12 about the EOI can be developed based on this and in conjunction with the emergency response workflow of the EOI. For P N 12 = P 12 , T 12 ; F 12 , M 12 , α 12 , P 12 = p 1 , p 2 , p 3 , p 4 , p 5 , p 6 , p 7 , p 8 , p 9 , p 10 , p 11 , p 12 , T 12 = t 1 , t 2 , t 3 , t 4 , t 5 , t 6 , t 7 , t 8 , t 9 , t 10 , and M 12 = 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 T , Figure 4 shows the specific structure of the PN12 model.
The emergency response workflow for the EOI can be calculated for its marking. Firstly, construct the logical incidence matrix of PN12:
R 12 = 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1
Since PN12 is a traditional Petri net, it is not necessary to construct the logical function matrix.
Combining the emergency response workflow of the EOI in Section 5.1 and Figure 4, it can be seen that t1 is enabled at M 12 , and the new marking M 12 1 = 0 , 1 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 T is obtained after t1 is fired. At this time, t2 and t7 are enabled, then t2 and t7 are fired to obtain the new marking M 12 2 = 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 T . Next, t3, t4, t5, t6, and t8 are fired in turn to obtain the marking M 12 3 = 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 1 , 0 , 0 T .
Then, t9 is enabled at M 12 3 , and new marking M 12 4 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 T is obtained after t9 is fired. t10 is enabled at M 12 4 , and the final marking M 12 5 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 T is obtained after t10 is fired. At this point, the relevant emergency response workflow of the EOI has been completed. Finally, p12 contains a token, whereas p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, and p11 do not.
Accordingly, X 12 = 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 T can be constructed. Finally, the reachable marking function of PN12 is as follows: M 12 = M 12 + R 12 T X 12 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 T ; by marking the calculation result, M 12 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 T can be judged to be consistent with the final marking M 12 5 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 T .
According to the calculation results, it can be judged that M 12 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 T is consistent with the final marking M 12 5 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 T .

5.3. STD Model and Its Reachable Markings

Based on the STD emergency response workflow in Section 5.1, a set of activities can be extracted. A13 = {Initiate emergency, Activate strong earthquake response, Organize reservoir experts, Organize construction experts, Organize vibration experts, Organize gravity experts, Make a suggestion, Implementing recommendations}, a logical Petri net model L P N 13 = P 13 , T 13 ; F 13 , I 13 , O 13 , M 13 , α 13 about the STD can be developed based on this and in conjunction with the emergency response workflow of the STD. For L P N 13 = P 13 , T 13 ; F 13 , I 13 , O 13 , M 13 , α 13 , P 13 = p 1 , p 2 , p 3 , p 4 , p 5 , p 6 , p 7 , p 8 , p 9 , p 10 , p 11 , p 12 , T 13 = T 13 , I T 13 , O T 13 , D , T 13 , I = t 7 , T 13 , O = t 2 , T 13 , D = t 1 , t 3 , t 4 , t 5 , t 6 , t 8 , O 13 t 2 = f O t 2 = p 3 p 5 p 7 p 9 , I 13 t 7 = f I t 7 = p 4 p 6 p 8 p 10 , and M 13 = 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 T . Figure 5 shows the specific structure of the PN13 model.
The emergency response workflow for the STD can be calculated for Its marking. Firstly, construct the logical incidence matrix of LPN13:
R 13 = 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1
Construct the logical function matrix of LPN13:
W 13 = 0 ( * , * ) 0 0 0 0 ( * , * ) 0 0 ( * , * ) 0 0 0 0 ( * , * ) 0 0 ( 1 , 1 ) 0 0 0 0 ( * , * ) 0 0 ( * , * ) 0 0 0 0 ( 1 , 1 ) 0 0 ( 1 , 1 ) 0 0 0 0 ( * , * ) 0 0 ( * , * ) 0 0 0 0 ( 1 , 1 ) 0 0 ( 1 , 1 ) 0 0 0 0 ( * , * ) 0 0 ( * , * ) 0 0 0 0 ( 1 , 1 ) 0 0 ( 1 , 0 ) 0 0 0 0 ( * , * ) 0 0 ( * , * ) 0 0 0 0 ( 1 , 0 ) 0 0 ( * , * ) 0 0 0 0 ( * , * ) 0 0 ( * , * ) 0 0 0 0 ( * , * ) 0 T
Combining the emergency response workflow of the STD in Section 5.1 and Figure 5, it can be seen that t1′ is enabled at M 13 , and the new marking M 13 1 = 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 T is obtained after t1′ is fired. At this time, t2′ is enabled. Let p3′∧p5′∧p7′∧¬p9′ be fired under the marking M 13 1 , and t2′ are fired to get the new marking M 13 2 = 0 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 0 , 0 , 0 , 0 T . Next, t1′, t3′, t4′, and t5′ are fired in turn to get the marking M 13 3 = 0 , 0 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 0 , 0 , 0 T . The logical input function p4′∧p6′∧p8′∧¬p10′ for t7′ is fired under M 13 3 , whereas t7′ is enabled. Then, the new marking M 13 4 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 T is obtained after t7′ is fired. At this time, t8′ is enabled at M 13 4 , and the final marking M 13 5 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 T is obtained after t8′ is fired. At this point, the relevant emergency response workflow of the STD has been completed. Finally, p12′ contains a token, whereas p1′, p2′, p3′, p4′, p5′, p6′, p7′,p8′, p9′, p10′, and p11′ do not.
Accordingly, a vector X 13 = 1 , ( 0 , 1 ) T , 1 , 1 , 1 , 0 , ( 0 , 1 ) T , 1 T can be constructed. Finally, the reachable marking functions of LPN13 is as follows: M 13 = M 13 + R 13 T W 13 T X 13 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 T .
According to the calculation results, it can be judged that M 13 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 T is consistent with the final marking M 13 5 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 T .

5.4. Synchronous Intersection of PN12 and LPN13

For PN12 and LPN13, t 7 , t 9 P N 12 , t 2 , t 7 L P N 13 , α 12 t 7 = α 13 t 2 = Activate   strong   earthquake   response and α 12 t 9 = α 13 t 7 = Make   a   suggestion . At this time, < t 7 , t 2 > and < t 9 , t 7 > can be obtained by performing synchronous intersection operations of t7 and t2′, t9 and t7′. Since t7 and t9 belong to PN12, and t2′ and t7′ belong to LPN13, the synchronous intersection operations of t7 and t2′ and the synchronous intersection operations of t9 and t7′ can be performed in parallel. A new logical Petri net LPN14 is obtained by synchronous intersection operations of PN12 and LPN13, and L P N 14 = P 14 , T 14 ; F 14 , I 14 , O 14 , M 14 , α 14 .
Step 1 The set of places composition: P14 = P12P13 = {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p1′, p2′, p3′, p4′, p5′, p6′, p7′, p8′, p9′, p10′, p11′, p12′};
Step 2 The set of transitions composition:
T 14 , D = T 12 t 7 t 9 T 13 , D = t 1 , t 2 , t 3 , t 4 , t 5 , t 6 , t 8 , t 10 , t 1 , t 3 , t 4 , t 5 , t 6 , t 8 , T 14 , I = T 13 , I t 7 < t 9 , t 7 > , T 14 , O = T 13 , O t 2 < t 7 , t 2 > , T 14 = T 14 , D T 14 , I T 14 , O ;
Since t7 is a traditional transition and t2′ is a logical output transition, < t 7 , t 2 > can be regarded as a logical output transition. Since t9 is a traditional transition and t7′ is a logical input transition, < t 9 , t 7 > can be regarded as a logical input transition.
Step 3 The set of directed arcs composition:
F 14 = F 12 ( t 7 , p i p i t 7 ) ( p j p j t 7 , t 7 ) , F 14 = F 14 ( < t 7 , t 2 > , p i p i t 7 ) ( p j p j t 7 , < t 7 , t 2 > ) ,
F 14 = F 14 ( F 13 ( t 2 , p i p i t 2 ) ( p j p j t 2 , t 2 ) ) , F 14 = F 14 ( < t 7 , t 2 > , p i p i t 2 ) ( p j p j t 2 , < t 7 , t 2 > ) ,
F 14 = F 14 ( F 12 ( t 9 , p i p i t 9 ) ( p j p j t 9 , t 9 ) ) , F 14 = F 14 ( < t 9 , t 7 > , p i p i t 9 ) ( p j p j t 9 , < t 9 , t 7 > ) ,
F 14 = F 14 ( F 13 ( t 7 , p i p i t 7 ) ( p j p j t 7 , t 7 ) ) , F 14 = F 14 ( < t 9 , t 7 > , p i p i t 7 ) ( p j p j t 7 , < t 9 , t 7 > ) ;
After the above operation the final set of directed arcs of the logical Petri net LPN14 is obtained. The specific structure of LPN14 is given in Figure 6.
Step 4 The logical output functions composition:
< t 7 , t 2 > can be regarded as a logical output transition. Therefore, it is necessary to obtain the logical output function f O t 7 = p i t 7 p i = p 9 for t7, then intersect f O t 7 with f O t 2 = p 3 p 5 p 7 p 9 of t2′ to obtain the logical output function f O < t 7 , t 2 > = f O t 7 f O t 2 = p 9 p 3 p 5 p 7 p 9 of < t 7 , t 2 > ;
Step 5 Mapping O composition:
O 14 = O 13 O 13 t 2 ; since PN12 is a traditional Petri net, it does not involve mapping O. After deleting the mapping O 13 t 2 = f O t 2 , O13 is assigned to O14 in order to get the mapping O14 of LPN14;
O 14 = O 14 O 14 < t 7 , t 2 > = f O < t 7 , t 2 > ; since < t 7 , t 2 > can be regarded as a logical output transition, the mapping from < t 7 , t 2 > to f O < t 7 , t 2 > needs to be added to O14, which eventually obtains O14 in LPN14;
Step 6 The logical input functions composition:
< t 9 , t 7 > can be regarded as a logical input transition. Therefore, it is necessary to obtain the logical input function f I t 9 = p i t 9 p i = p 7 p 10 for t9, then intersect f I t 9 with f I t 7 = p 4 p 6 p 8 p 10 of t7′ to obtain the logical input function f I < t 9 , t 7 > = f I t 9 f I t 7 = p 7 p 10 p 4 p 6 p 8 p 10 of < t 9 , t 7 > ;
Step 7 Mapping I composition:
I 14 = I 13 I 13 t 7 ; since PN12 is a traditional Petri net, it does not involve mapping I. After deleting the mapping I 13 t 7 = f I t 7 , I13 is assigned to I14 in order to get the mapping I14 of LPN14;
I 14 = I 14 I 14 < t 9 , t 7 > = f I < t 9 , t 7 > ; since < t 9 , t 7 > can be regarded as a logical input transition, the mapping from < t 9 , t 7 > to f I < t 9 , t 7 > needs to be added to I14, which eventually obtains I14 in LPN14;
Step 8 The marking composition:
M 14 = M 12 M 13 = 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 T ;
Construct the logical incidence matrix of LPN14: Since the logical incidence matrix R 14 of LPN14 is large, it is split into multiple row vectors.
r t 1 = 1 , 1 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
r t 2 = 0 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
r t 3 = 0 , 0 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
r t 4 = 0 , 0 , 0 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
r t 5 = 0 , 0 , 0 , 0 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
r t 6 = 0 , 0 , 0 , 0 , 0 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
r < t 7 , t 2 > = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
r t 8 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
r < t 9 , t 7 > = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 ,
r t 10 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
r t 1 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
r t 3 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
r t 4 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 ,
r t 5 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 0 , 0 , 0 , 0 ,
r t 6 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 0 , 0 ,
r t 8 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 ;
The final logical incidence matrix of LPN14 can be obtained: R14 = (r[t1]T,r[t2]T,r[t3]T, r[t4]T,r[t5]T,r[t6]T,r[<t7,t2>]T,r[t8]T,r[<t9,t7>]T,r[t10]T,r[t1′]T,r[t3′]T,r[t4′]T,r[t5′]T,r[t6′]T,r[t8′]T);
Construct the logical function matrix of LPN14: Since the logical function matrix W 14 of LPN14 is large, it is split into multiple row vectors.
w t 1 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
w t 2 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
w t 3 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
w t 4 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
w t 5 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
w t 6 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
w[<t7,t2′>] = ((*,*)T,(*,*)T,(*,*)T,(*,*)T,(*,*)T,(*,*)T,(*,*)T,(*,*)T,(1,1)T,(*,*)T,(*,*)T,(*,*)T,(*,*)T,(*,*)T,(1,1)T,(*,*)T,(1,1)T,(*,*)T,(1,1)T,(*,*)T,(1,0)T,(*,*)T,(*,*)T,(*,*)T),
w t 8 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
w[<t9,t7′>] = ((*,*)T,(*,*)T,(*,*)T,(*,*)T,(*,*)T,(*,*)T,(*,*)T,(*,*)T,(*,*)T,(−1,−1)T,(*,*)T,(*,*)T, (*,*)T,(*,*)T,(−1,−1)T,(*,*)T,(−1,−1)T,(*,*)T,(−1,−1)T,(*,*)T,(−1,0)T,(*,*)T,(*,*)T,(*,*)T),
w t 10 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
w t 1 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
w t 3 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
w t 4 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
w t 5 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
w t 6 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
w t 8 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ;
The final logical function matrix of LPN14 can be obtained: W14 = (w[t1]T,w[t2]T,w[t3]T, w[t4]T,w[t5]T,w[t6]T,w[<t7,t2>]T,w[t8]T,w[<t9,t7>]T,w[t10]T,w[t1′]T,w[t3′]T,w[t4′]T,w[t5′]T,w[t6′]T,w[t8′]T).
Based on the vector X 12 = 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 T constructed from the emergency response workflow of the EOI and the vector X 13 = 1 , ( 0 , 1 ) T , 1 , 1 , 1 , 0 , ( 0 , 1 ) T , 1 T constructed from the emergency response workflow of the STD, X 14 = 1 , 1 , 1 , 1 , 1 , 1 , ( 0 , 1 ) T , 1 , ( 0 , 1 ) T , 1 , 1 , 1 , 1 , 1 , 0 , 1 T constructed from the emergency response workflow of LPN14 can be obtained.
The reachable marking functions of LPN14 is as follows: M 14 = M 14 + R 14 T W 14 T X 14 , and M 14 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 T , which means that both p12 and p12′ contain a token, whereas p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p1′, p2′, p3′, p4′, p5′, p6′, p7′, p8′, p9′, p10′, and p11′ do not. The marking M 14 is consistent with the final marking M 12 = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 T and M 13 = ( 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 ) T of PN12 and LPN13. Thus, LPN14 is fully capable of representing all the reachable marking states of PN12 and LPN13, and synchronization of t7, t2′ and t9, t7′ is achieved.
In the synchronous intersection operation of PN12 and LPN13, two synchronous transition units < t 7 , t 2 > and < t 9 , t 7 > are obtained by synchronous intersection operations of the transitions t7, t2′ and t9, t7′. The synchronous intersection operation of PN12 and LPN13 achieves the collaboration between the EOI and the STD for the tasks of “Activate strong earthquake response” and “Make a suggestion”, which solves the problem of cross-departmental collaboration.
The above examples can illustrate that after the synchronous intersection of the original Petri nets and the original logical Petri nets, the new logical Petri nets can still represent the original Petri nets and the original logical Petri nets in each of the reachable marking states. It also realizes the cross-departmental collaboration, the synchronization of logical transition and traditional transition, and the intersection operation of related logical functions; meanwhile, it provides an approach for the collaboration between traditional and logical Petri nets for the relevant earthquake emergency departments.

6. Conclusions

This study proposes a synchronous intersection operation for traditional and logical Petri nets to solve the problem of the collaboration between different departments. It realizes the mergence and synchronization of transitions in the two kinds of Petri nets, and effectively solves the cooperation problem of earthquake emergency departments in practical applications.
Firstly, this paper introduces the related knowledge of traditional and logical Petri nets. Secondly, the definition of synchronous transition unit, logical functions of synchronous transition unit, and the approaches to extraction logical functions of traditional transition are proposed. Then, three cases of synchronous intersection operations in traditional and logical Petri nets are introduced and exemplified, respectively.
Finally, this paper applies the modeling approaches of traditional and logical Petri nets to the field of earthquake emergency response. Through the model extraction of the earthquake emergency plan and the verification of relevant examples, it can be concluded that the synchronous intersection operation of traditional and logical Petri nets can solve the synchronization problem with the two kinds of Petri nets in order to realize the collaboration between two departments, especially the earthquake emergency department.
At present, there are still a lot of research issues for the synchronous intersection operation of traditional and logical Petri nets. The future research mainly includes several aspects as follows: (1) it is meaningful to extend the synchronous intersection operation to synchronize multiple traditional and logical Petri nets; (2) the synchronous intersection operation among the logical Petri nets is further considered; (3) the approaches to compute the reachable markings lay the foundation for the development of synchronous intersection operations of the logical-input and logical-output transitions; (4) in order to verify the availability and soundness of the approach proposed in this paper, we will prove the reachability, boundedness, safety, liveness and fairness of the composition models in the future.

Author Contributions

Conceptualization, Y.T. and X.P.; methodology, Y.T. and X.P.; software, Y.S.; validation, Y.S. and D.H.; formal analysis, D.H.; investigation, D.H., Y.T. and X.P.; resources, Y.T. and Y.D.; data curation, Y.T. and X.P.; writing—original draft preparation, D.H., Y.T. and X.P.; writing—review and editing, Y.S., Y.T. and Y.D.; supervision, Y.T.; project administration, Y.T. and D.H.; funding acquisition, Y.T. and D.H. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported in the part by the National Natural Science Foundation of China under Grant 61973180 and Grant 72101137; in part by the Natural Science Foundation of Shandong Province under Grant ZR2019MF033, Grant ZR2020MF033 and Grant ZR2021MF117; in part by the Education Ministry Humanities and Social Science Research Youth Fund Project of China under Grant 20YJCZH159 and Grant 21YJCZH150; in part by the Shandong Key R&D Program (Soft Science) Project under Grant 2022RKY02009; and in part by the “Taishan Scholar” Construction Program of Shandong Province.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data used to support the findings of the study are available within the article.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Reijers, H.A. Business Process Management: The evolution of a discipline. Comput. Ind. 2021, 126, 103404. [Google Scholar] [CrossRef]
  2. Song, W.; Xia, X.; Jacobsen, H.A.; Zhang, P.C.; Hu, H. Efficient alignment between event logs and process models. IEEE Trans. Serv. Comput. 2017, 10, 136–1149. [Google Scholar] [CrossRef]
  3. Rodríguez, A.; Caro, A.; Cappiello, C.; Cabalero, I. A BPMN extension for including data quality requirements in business process modeling. In International Workshop on Business Process Modeling Notation; Springer: Berlin/Heidelberg, Germany, 2012; pp. 116–125. [Google Scholar]
  4. Xiang, D.M.; Liu, G.J.; Yan, C.G.; Jiang, C.J. A guard-driven analysis approach of workflow net with data. IEEE Trans. Serv. Comput. 2021, 14, 1650–1661. [Google Scholar] [CrossRef]
  5. Zhao, F.; Xiang, D.; Liu, G.; Jiang, C. A New Method for Measuring the Behavioral Consistency Degree of WF-Net Systems. IEEE Trans. Comput. Soc. Syst. 2022, 9, 480–493. [Google Scholar] [CrossRef]
  6. Liu, C.; Zeng, Q.; Duan, H.; Zhou, M.; Lu, F.; Cheng, J. E-net modeling and analysis of emergency response processes constrained by resources and uncertain durations. IEEE Trans. Syst. Man Cybern. Syst. 2015, 45, 84–86. [Google Scholar] [CrossRef]
  7. Qi, L.; Zhou, M.; Luan, W. A two-level traffic light control strategy for preventing incident-based urban traffic congestion. IEEE Trans. Intell. Transp. Syst. 2018, 19, 13–24. [Google Scholar] [CrossRef]
  8. Li, L.; Basile, F.; Li, Z. An Approach to Improve Permissiveness of Supervisors for GMECs in Time Petri Net Systems. IEEE Trans. Autom. Control 2020, 65, 237–251. [Google Scholar] [CrossRef]
  9. He, Z.; Ma, Z.; Li, Z.; Giua, A. Parametric transformation of timed weighted marked graphs: Applications in optimal resource allocation. IEEE/CAA J. Autom. Sin. 2021, 8, 179–188. [Google Scholar] [CrossRef]
  10. Li, L.; Basile, F.; Li, Z. Closed-Loop Deadlock-Free Supervision for GMECs in Time Petri Net Systems. IEEE Trans. Autom. Control 2021, 66, 5326–5341. [Google Scholar] [CrossRef]
  11. Du, Y.; Qi, L.; Zhou, M. A vector matching method for analysing logic Petri nets. Enterp. Inf. Syst. 2011, 5, 449–468. [Google Scholar] [CrossRef]
  12. Pang, S. Modeling and Verification of Workflow Based on Resource Constraint. Acta Electron. Sin. 2012, 40, 1497–1502. [Google Scholar]
  13. He, H.; Pang, S.; Zhao, Z. Dynamic scalable stochastic petri net: A novel model for designing and analysis of resource scheduling in cloud computing. Sci. Program. 2016, 13, 1–13. [Google Scholar] [CrossRef] [Green Version]
  14. Du, Y.Y.; Wang, L.; Qi, M. Constructing Service Clusters Based on Service Space. Int. J. Parallel Program. 2017, 45, 982–1000. [Google Scholar] [CrossRef]
  15. Teng, Y.; Du, Y.; Qi, L.; Luan, W. A Logic Petri Net-Based Method for Repairing Process Models With Concurrent Blocks. IEEE Access 2019, 7, 8266–8282. [Google Scholar] [CrossRef]
  16. Li, J.; Yang, R.; Ding, Z.; Pan, M. A Method for Learning a Petri Net Model Based on Region Theory. Comput. Inf. 2020, 39, 174–192. [Google Scholar] [CrossRef]
  17. He, H.; Zhao, Y.; Pang, S. Stochastic modeling and performance analysis of energy-aware cloud data center based on dynamic scalable stochastic petri net. Comput. Inf. 2020, 39, 28–50. [Google Scholar] [CrossRef]
  18. Li, H.; Bu, Z.; Wang, Z.; Cao, J. Dynamical clustering in electronic commerce systems via optimization and leadership expansion. IEEE Trans. Ind. Inf. 2020, 16, 5327–5334. [Google Scholar] [CrossRef]
  19. Hu, L.; Pan, X.; Tan, Z.; Luo, X. A Fast Fuzzy Clustering Algorithm for Complex Networks via a Generalized Momentum Method. IEEE Trans. Fuzzy Syst. 2021, 30, 3473–3485. [Google Scholar] [CrossRef]
  20. Wang, Z.; Luan, W.; Du, Y.; Qi, L. Composition and application of extended colored logic Petri nets to E-commerce systems. IEEE Access 2020, 8, 36386–36397. [Google Scholar] [CrossRef]
  21. Duo, W.; Jiang, X.; Karoui, O.; Guo, X.; You, D.; Wang, S.; Ruan, Y. A deadlock prevention policy for a class of multi-threaded software. IEEE Access 2020, 8, 16676–16688. [Google Scholar] [CrossRef]
  22. Liu, G. Complexity of the deadlock problem for Petri nets modeling resource allocation systems. Inf. Sci. 2016, 363, 190–197. [Google Scholar] [CrossRef]
  23. Luan, W.; Qi, L.; Du, Y. Composition of logical Petri nets and compatibility analysis. IEEE Access 2017, 5, 9152–9162. [Google Scholar] [CrossRef]
  24. Luan, W.; Qi, L.; Zhao, Z.; Liu, J.; Du, Y. Logic Petri net synthesis for cooperative systems. IEEE Access 2019, 7, 161937–161948. [Google Scholar] [CrossRef]
  25. Qi, L.; Luan, W.; Lu, X.S.; Guo, X. Shared P-type logic Petri net composition and property analysis: A vector computational method. IEEE Access 2020, 8, 34644–34653. [Google Scholar] [CrossRef]
  26. Liu, C.; Duan, H.; Zeng, Q.; Zhou, M.; Lu, F.; Cheng, J. Towards comprehensive support for privacy preservation cross-organization business process mining. IEEE Trans. Serv. Comput. 2019, 12, 639–653. [Google Scholar] [CrossRef]
  27. Duan, H.; Liu, C.; Zeng, Q.; Zhou, M. Refinement-based hierarchical modeling and correctness verification of cross-organization collaborative emergency response processes. IEEE Trans. Syst. Man Cybern. Syst. 2020, 50, 2845–2859. [Google Scholar] [CrossRef]
  28. Zhen, L.; Hu, P.; Peng, X.; Goh, R.; Zhou, J.T. Deep Multimodal Transfer Learning for Cross-Modal Retrieval. IEEE Trans. Neural Netw. Learn. Syst. 2020, 33, 798–810. [Google Scholar] [CrossRef]
  29. Wang, Y.; Chen, Z.; Luo, X.; Li, R.; Xu, X. Fast Cross-Modal Hashing with Global and Local Similarity Embedding. IEEE Trans. Cybern. 2021, 52, 10064–10077. [Google Scholar] [CrossRef]
  30. Liu, C.; Zeng, Q.; Cheng, L.; Duan, H.; Zhou, M.; Cheng, J. Privacy-preserving behavioral correctness verification of cross-organizational workflow with task synchronization patterns. IEEE Trans. Autom. Sci. Eng. 2021, 18, 1037–1048. [Google Scholar] [CrossRef]
  31. Chen, W.; Liu, L.; Liu, G. Privacy-Preserving Distributed Economic Dispatch of Microgrids: A Dynamic Quantization Based Consensus Scheme with Homomorphic Encryption. IEEE Transf. Smart Grid 2022, 14, 701–713. [Google Scholar] [CrossRef]
Figure 1. LPN model.
Figure 1. LPN model.
Electronics 12 01207 g001
Figure 2. Synchronous intersection of a traditional transition and a logical input transition. (a) The traditional Petri net model PN5. (b) The logical Petri net model LPN6. (c) The logical Petri net model LPN7.
Figure 2. Synchronous intersection of a traditional transition and a logical input transition. (a) The traditional Petri net model PN5. (b) The logical Petri net model LPN6. (c) The logical Petri net model LPN7.
Electronics 12 01207 g002
Figure 3. Synchronous intersection of a traditional transition and a logical output transition. (a) The traditional Petri net model PN9. (b) The logical Petri net model LPN10. (c) The logical Petri net model LPN11.
Figure 3. Synchronous intersection of a traditional transition and a logical output transition. (a) The traditional Petri net model PN9. (b) The logical Petri net model LPN10. (c) The logical Petri net model LPN11.
Electronics 12 01207 g003
Figure 4. The traditional Petri net model PN12 structure of the EOI.
Figure 4. The traditional Petri net model PN12 structure of the EOI.
Electronics 12 01207 g004
Figure 5. The logical Petri net model LPN13 structure of the STD.
Figure 5. The logical Petri net model LPN13 structure of the STD.
Electronics 12 01207 g005
Figure 6. The logical Petri net model LPN14.
Figure 6. The logical Petri net model LPN14.
Electronics 12 01207 g006
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Tian, Y.; Pang, X.; Su, Y.; Han, D.; Du, Y. Cross-Departmental Collaboration Approach for Earthquake Emergency Response Based on Synchronous Intersection between Traditional and Logical Petri Nets. Electronics 2023, 12, 1207. https://doi.org/10.3390/electronics12051207

AMA Style

Tian Y, Pang X, Su Y, Han D, Du Y. Cross-Departmental Collaboration Approach for Earthquake Emergency Response Based on Synchronous Intersection between Traditional and Logical Petri Nets. Electronics. 2023; 12(5):1207. https://doi.org/10.3390/electronics12051207

Chicago/Turabian Style

Tian, Yinhua, Xiaowen Pang, Yan Su, Dong Han, and Yuyue Du. 2023. "Cross-Departmental Collaboration Approach for Earthquake Emergency Response Based on Synchronous Intersection between Traditional and Logical Petri Nets" Electronics 12, no. 5: 1207. https://doi.org/10.3390/electronics12051207

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop