Presently, public safety events are mainly divided into four categories, namely natural disasters, accidental disasters, public health events and social security incidents . Natural disasters mainly involve flood and drought, earthquake and meteorological, geological, marine and biological disasters while accidental disasters mainly include production and environmental disasters in mining, commerce and trade industries. Public health events mainly include infectious diseases, transmission of mass diseases of unknown cause to humans, animal epidemics and other events that seriously affect public health and life safety while social security incidents mainly involve aggressive events, economic security incidents, foreign emergencies, etc. The related content to the management of regional road network public security incidents studied in this paper belongs to the management of aggressive events in social security incidents. Since the ‘911’ event that caused great losses to the American society, public safety governance has gradually become the focus of researchers from all over the world, and domestic researchers have also begun to involve in studying emergencies and participate in the discussion of emergency problems. Then, after the SARS occurrence, domestic scholars have conducted  more research studies on public security incidents, and their results are also interesting. In recent years, with the development of science and technology, the level of public security governance has been constantly improved, and regional security incident governance has also become one of the research hotspots. However, factors like personnel congestion, open environment, rapid information transmission and others in regional security incidents also bring certain difficulties to public security event processing. As a result, once an aggressive public security event occurs, how to improve the emergency efficiency and to quickly and effectively allocate the rescue personnel and materials to the site is a key approach to reduce the loss [3, 4]. Currently, some relevant studies have been carried out centering on the site selection of public safety emergency facilities, among which the traditional site selection research studies of emergency facilities mostly focus on natural disasters, such as flood, hurricane and earthquake, and realise the layout of emergency facilities and the distribution of emergency supplies by virtue of the derived site selection-distribution model  and site selection-path model  besides solving by branch-bound, column generation, various heuristics or biological evolution algorithms . In recent years, the site selection of public safety emergency facilities has gradually attracted more attention. Unlike natural disasters, an attacker's attack strategy is adjusted according to the defender's defence deployment. Therefore, to characterise the game relationship between defenders and attackers, Berman and Gavious  constructed the emergency facility location problem as a Stackelberg game model, where after Berman et al.  took a further step to explore the Nash game location model under the not disclosed defender location selection information. Han et al.  supplemented the genetic algorithm for solving large-scale network problems. Chai et al.  considered the site selection of emergency facilities under the continuous attack. Xiang Yin and Wei Hang  studied the situation of realistic situation security resource allocation for attackers that is not completely rational and led to a new game model suitable for general scenarios. Meng et al.  studied the risk of interruption of emergency facilities. Xiang Yin  studied the optimal strategy for defenders’ location information in 2019.
All of the above studies provide effective countermeasures for defenders, which, nonetheless, leave room for improvement. Firstly, existing studies consider less about that the diffusion of the consequences of regional public safety event, the panic generated in groups and loss in the process of controlling public opinion that will produce joint losses. Therefore, the regional road network security resource facilities layout problem considering the diffusion of the loss needs to be solved urgently. Secondly, in the planning process, existing studies propose the assumption of unlimited facility capacity. The handling is in place at a time. However, materials often cannot be carried in place at once in reality. Accordingly, this paper considers the situation of multiple material handling into the material planning model. In contrast to the previous studies, in view of the layout of regional road network security resource facilities, this paper fully considers the diffusion of safety incidents and the limited material planning and handling and proposes a class of problems covering the safety resource facility allocation and planning optimisation, which enriches the site selection theory and algorithm of emergency facilities.
Site location of security resource emergency facilities can be described as below. In a given regional traffic network
The hypothesis on security resources path planning can be described as follows. On the basis of security resource emergency facilities site problem, we assume that the attack point
As it can be seen from the above description of the location of security resource emergency facilities, the defender's decision goal is to minimise the loss, while the attacker's is to maximise the perceived defender loss. The ultimate goal of path planning is to solve a point to rescue the vertex
Berman [8, 9] and Han et al.  discussed the facility site location model, where security information is that the attacker is completely rational, and abstracted the site selection problem into a Stackelberg game model. The defenders first set up k facility points and form the strategy space
In the relevant literature of Berman [8, 9] and Han et al. , the sum of defender disutility is defined as the sum of attack loss and facility construction costs, and attacker utility is defined as attack loss. Xiang Yin and Wei Hang  improved the utility function and obtained the defensive negative utility function in the general attack case. Based on the above research, considering the loss caused by the event diffusion of a road network in a public safety area, the paper sets the improved defensive negative utility function of
The utility formula of the attacker Eq. (2) lacks the item
Therefore, in the face of completely rational attackers, the integrated model formula of Eqs (1) and (2), Eq. (3) is written as follows:
In Eq. (3), there are three constraints. First, there is only one attack. Second, each vertex has an attack probability ≥ 0. And third, the attacker responds optimally to the defender's location strategy to maximise its utility function.
In allusion to the completely rational situation, the game problem becomes a single constraint linear planning problem. Assuming that
And the optimal solution of the defence's location strategy is written as follows:
Since this situation will hardly happen in reality, there is no literature focussing specifically on irrational attack game models. By the definition of irrationality through game theory, Xiang Yin and Wei Hang  described the location selection problem when the attacker is irrational as follows: in a regional traffic extension network
The site selection model in the irrational attack mode Eq. (6) is equivalent to the p-median model, where the attacker is completely irrational, and the attack probability is also given externally. Thus, the defensive location problem can be transformed into a 0–1 planning problem, namely a vertex placed facility point
Many real attackers are not completely rational; therefore, Hao et al.  and Xiang Yin [7,14] all considered the allocation model of incomplete rational attackers in the process of studying the security resource allocation problem. This paper learns from the introduction of a rational degree factor of modelling literature in the analysis of incomplete rational game situation from the documents of Xiang Yin and Wei Hang .
In the above two parts, the irrational attacker's attack probability is determined by exogenous factors, while the completely rational attacker's is completely determined by endogenous factors. Bringing the above reaction function into the Eq. (8), we can then work out the optimum solution of the only equilibrium solution
In existing research studies, the shortest path refers to the shortest path from the starting point and the end point and the smallest sum of the path weights. However, after the occurrence of the emergency, besides the shortest path to the demand point, the effective time of delivery materials and other restrictive condition are added. Comprehensively analysing, the so-called ‘optimal path of security resource scheduling path’ refers to the shortest one, ensuring the supplies to be delivered in order to reduce the more unnecessary losses incurred during the transportation process and arrived within the scope of time requirements when a security event occurs at a certain point in the road network of the park. Since there are more than one site in the campus for storing materials, it involves multiple material point, that is, the ‘optimal path’ of the campus security material scheduling. Therefore, it is necessary to propose the u-Dijkstra algorithm as shown in the following Eqs (10)–(12) on the basis of the existing improved Dijkstra algorithm proposed by Zhang Yajing et al. .
Through the illustration in the third section, completely rational security resource location model and irrational security resource location model are two extreme special cases when the incompletely rational model is in the attacker rational degree of 0 and 1, respectively. In the site selection algorithm design process, the Floyd (exhaustive attack method) algorithm can be employed to simulate. The Floyd algorithm refers to the method of enumerating all the elements in a set composed of finite elements one by one and studying. Its efficiency is not high and applicable to the cases with no obvious rules to follow. However, because the Floyd method will increase exponentially with the number of fixed points, the complexity of the calculation is too high and does not apply for strategic site selection. For instance, if there are 22 vertices and 3 points are selected as security resource points, we need to list
Genetic algorithm can greatly reduce the workload. First, the initial population composed of a certain number of site selection scheme is randomly generated. Then the biological genetic process is simulated to iterate and update the site selection scheme. The genetic algorithm adopts the parallel search method of multiple clues, and the calculation speed is fast.
The algorithm used in this article is shown in the following figure:
Based on the simulation planning and analysis of the traffic topology network in a university campus, we mainly study the corresponding facility layout strategy, the applicable situation and practical value of path planning under the condition that the attacker is not fully rational.
In this paper, Xinjiang Normal University is treated as an example. The campus map is shown in Figure 1a. Campus network number map and important campus places are abstract as random numbers, and roads between important sites are edges. The park network map is shown in Figure 1b. The map is composed of 44 nodes and 64 edges.
In this paper, considering the real data and importance data on the campus security, data demensitisation processing, select the vertex 1–22 importance value as shown in Table 1, the actual test distance between vertices, exhaustive calculation, and then according to a certain proportion, get the shortest path distance of each vertex to other vertices, as shown in Figure 2 below:
Campus node importance assignment
First, in order to intuitively observe the impact of the number of parameter security resource facility points m on the loss of defenders, this paper uses mathematical modelling software MATLAB to experiment. The results are shown in Figure 3, and the number of security facility points increases after the end of the second genetic algorithm.
The results shown in Figure 5 show that in the fixed number of fixed emergency facilities,
Secondly, in Figure 5, when the degree of rationality is fixed, the defensive loss gradually decreases as the number of facilities increases, but obviously, when the number of facilities increases to a certain extent, increasing the number of facilities has little impact on the defensive loss. And by changing the rationality of the attacker, the relationship between the number of facilities and the defensive utility remains.
In this paper, the simulation is conducted at p=0.5, p = 0.5, m=3 and m=5. According to the results of the site selection strategy, the blue route planning is shown in Figures 6 and 7. After calculating the verification of the weights of the path and the shortest path in the table, the results prove that the planning path is accurate.
The real-world regional public security management problems with the attackers are not completely rational attackers. If the existing classic site selection model and game model for emergency facilities site selection decision are used, it is easy to obtain the inefficient site selection scheme and make the security event spread persistent negative effects, thus increasing the loss caused by the attack. Considering the degree of incomplete rationality, the number of security resources and facilities, the diffusion characteristics of security events and other realistic situations, this paper fuses a game idea, putting forward a new hybrid site selection model, conducting path planning for the attack point of security resources scheduling and combining with a university case to analyse the impact of the damage point relief loss, thus providing scientific decision support for regional road network public security event management.
In the subsequent work, there is still a lot of research space for regional public safety management. In view of the proliferation of security incidents in the regional road network and the incomplete rational characteristics of the attackers, further research in the future can involve the following content:
First, conduct the related study of the dynamic game security resource site selection model. In reality, many security events also have continuity characteristics, and the multi-round game is more suitable for this scenario. Therefore, in the future research work, it can be more efficient to build multi-party game, multi-round game and multi-layer game models.
Second, generalise the regional road network security resource site selection model. We can further optimise the results of larger data and make the effect of the algorithm model more suitable for more scenarios.
Third, use the attacker's not completely rational characteristics to solve practical problems. From the fifth summary conclusion, it can be concluded that the incomplete rational characteristics can be used in the governance process of the public security events. On the one hand, the rational degree of the defence can be strengthened by further studying the algorithm. On the other hand, the rational degree of the attacker can be reduced by changing strategies and hiding information.
Fourth, for other research content, the model scenario assumption can be used in the field of public safety and can be employed in disaster governance, public opinion control, disease control and other fields as an attempt.
Campus node importance assignment
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Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training Nonlinear Channel Estimation for Internet of Vehicles Some Necessary Conditions for Feedback Functions of de Bruijn Sequences The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model An empirical research on economic growth from industrial structure optimisation in the Three Gorges Reservoir area The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Study of agricultural finance policy information extraction based on ELECTRA-BiLSTM-CRF Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Innovative research of vertical video creation under the background of mobile communication Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model Genetic algorithm-based congestion control optimisation for mobile data network