Monte Carlo analysis for evacuation in multipurpose event spaces

Citation: Snohová, A.; Kuˇcera, P.; Pokorny, J.; Bernatik, A. Monte Carlo Analysis for Evacuation in Multipurpose Event Spaces. Fire 2024, 7, 178. https://doi.org/10.3390/ fire7060178 Academic Editors: Qingsheng Wang and Hans Pasman Received: 5 April 2024 Revised: 17 May 2024 Accepted: 21 May 2024 Published: 23 May 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). fire Article Monte Carlo Analysis for Evacuation in Multipurpose Event Spaces Adéla Snohová* , Petr Kuˇcera , Jiri Pokorny and Ales Bernatik Faculty of Safety Engineering, VSB Technical University of Ostrava, Lumírova 630/13, Výškovice, 700 30 Ostrava, Czech Republic; petr[email protected] (P.K.); [email protected] (J.P.); [email protected] (A.B.) *Correspondence: [email protected]; Tel.: +420-596-992-973 Abstract: This paper addresses the evacuation of people from multipurpose halls and introduces an innovative approach that uses a probabilistic model, specifically the Monte Carlo method, to analyse iterative evacuation processes. The aim is to explore how this modern technology can contribute to the development of effective and safe evacuation plans for mass events. The Monte Carlo method was applied to a specific example of a multipurpose hall that offers different configurations for events such as sports matches, concerts, or performances. The evacuation of people was analysed for two configurations: a hockey match with a capacity of 9500 people and a concert with a capacity of 11,000 people. In both cases, the total evacuation of people from the hall was analysed, and the evacuation time was evaluated when two parameters were changed: speed of movement and preference for door selection. The results of the simulations can provide valuable information for the design of effective safety measures in multipurpose halls and other similar multipurpose venues. This innovative approach to evacuation analysis allows for a comprehensive assessment of the evacuation process, identification of critical areas, and verification of the layout of the space. Keywords: evacuation of people; concert; hockey match; Monte Carlo method; multipurpose hall; Pathfinder 1. Introduction As Decker describes in his article [ 1 ], the first application of the Monte Carlo method dates back to 1777, when a probabilistic method for calculating the value of π was introduced. Later, at the beginning of the twentieth century, the Monte Carlo method was used to study the Boltzmann equation and to estimate the correlation coefficient for the t-distribution. Over time, the Monte Carlo method has been applied in other diverse fields such as computer science, engineering, architecture, and economics [1]. The Monte Carlo method has also been applied in some cases dealing with the problem of evacuation of people. In this context, the term “evacuation of people” refers to the movement of all people from areas threatened by an emergency to a place of safety. The procedure for applying the Monte Carlo method in the field of evacuation of people is described by Phillips and Fahy in Chapter 80 of the SFPE handbook [2]. Ternero et al. used the Monte Carlo method in their study to design fire evacuation plans for buildings in case of emergency [ 3 ]. In this study, two proposed evacuation strategies for a specific building (Faculty of Engineering—USACH) were evaluated using a combination of simulations and data analysis using the Monte Carlo method. The simulations were performed using Pathfinder software version 2023.2 and highlighted key factors that affect a successful evacuation strategy, such as the distribution of people, the exit capacity, and the time available for evacuation. The optimisation of the evacuation plan was also addressed by Li et al. in their study [ 4 ] in which the game theory based on the Monte Carlo analysis method was applied. The Monte Carlo method was used in a study of crowd evacuation dynamics [ 5 ] by Tamang and Sun. The stimulus for their study was the growing need to understand Fire 2024,7, 178. https://doi.org/10.3390/fire7060178 https://www.mdpi.com/journal/fire Fire 2024,7, 178 2 of 13 the processes by which people leave buildings. Their study found that the evacuation time increases with the initial density of people and decreases with increasing exit width. The simulations also showed that the crowd was grouped around the exits. This study aimed to develop a quantitative method to facilitate the design of people facilities and the management of people flows during an emergency to reduce the number of injured people or casualties during evacuation. Shariff et al. [ 6 ] used the Monte Carlo method to evaluate the safety and risk associated with the evacuation of people when emergency exits are blocked, considering the safety regulations and procedures established in Malaysia. Xie et al. applied this method in their study [ 7 ] to determine the optimal distance between exits, which is a key aspect in the design of building fire safety. The multipurpose hall that is the subject of this study is considered a place where people gather. In these types of buildings and spaces, it is important to pay close attention to proper fire safety design, especially to ensure the safe evacuation of people. In a study [ 8 ], Chen and Mao addressed the issue of safely evacuating people from public areas, which can also be considered assembly areas. Two key indicators, ASET and RSET, were compared to assess the level of safe evacuation. To quantify uncertainties and their application in engineering design, the probability of failure model was used for the evacuation of people, followed by the Monte Carlo method. Several other studies have addressed the issue of evacuating people from stadiums [9,10] . In their study [ 11 ], Gravit et al. compared the flow of people during evacuation from two selected arenas (a sports arena and an entertainment arena) using simulated pedestrian dynamics. This study evaluates the effect of the evacuation route design on evacuation times and highlights the need for effective evacuation planning, especially when stairs are involved. Minegishi [ 12 ] investigated the problem of crowd dynamics during emergencies by observing people—the author was part of the crowd—returning from football matches held in eight different stadiums. Contribution, Scope, and Limitations of This Study This study presents an innovative approach that uses a probabilistic model, specifically the Monte Carlo method, to analyse iterative evacuation processes. This research focusses on evaluating the change in evacuation time when the speed of movement is set according to different sources and on the effect of changing preferences for door selection on evacuation time. To properly evaluate the effect of changes on evacuation time, a convergence rate of iterations is performed for simulations. This ensures that the simulations are repeated until the results are acceptable. This study addresses the problem of how to evacuate a large number of people safely and effectively from a multipurpose hall that is considered an indoor assembly area. Evacuation models provide some representation of human behaviour. However, they cannot fully reflect reality because they do not consider key aspects such as fear, panic, or uncertainty. Nonetheless, these models can provide valuable information to help develop effective and safe evacuation plans. The remainder of this paper is structured as follows: Section 2briefly describes the multipurpose hall that was the subject of the case study. It also includes a description of Pathfinder software version 2023.3, detailing the input parameters used to simulate evacuation of people, and a description of the Monte Carlo method. Section 3presents the results of the analysis of evacuation times and space usage maps. Section 4discusses and evaluates the sensitivity analysis of the convergence rate. Finally, Section 5presents the conclusions and suggestions for future directions. 2. Methods 2.1. Case Study The aim is to investigate how modern technology (the Monte Carlo method) can contribute to the development of effective and safe evacuation plans for mass events. This Fire 2024,7, 178 3 of 13 study focusses on modelling the evacuation of people for two specific configurations of the multipurpose hall. These are a hockey match with a capacity of 9500 people and a concert with a capacity of 11,000 people. For both configurations, we investigated how the evacuation of people is affected by the input of speed of movement from different sources [ 13 – 15 ] and what effect changes in the preference for door selection have on the evacuation of people. The Pathfinder software version 2023.3, which supports the Monte Carlo method, was chosen to implement the simulation process and configure agents (people). The combination of the two tools was used to obtain data on evacuation times, means, standard deviations (SDs), and people concentrations in the selected area. The results of the simulations provided valuable information for the design of effective safety measures in multipurpose halls and other similar multipurpose venues. A multipurpose hall in the Czech Republic is the subject of this study. The hall has a total of 19 grandstands, which are marked with letters, and the individual rows are marked with Arabic numerals. The grandstands are located around the ice area. Matches, tournaments, and various performances are held in the ice area. The access to the grandstands is through stairs that lead from the entrance hall. Three entrances lead to the multipurpose hall and also serve as emergency exits in an emergency event. There are three entrance doors on the north and south sides, while the east side offers two entrance doors. All entrances to the multipurpose hall are wheelchair accessible. The multipurpose hall and its emergency exits are shown in Figure 1. Fire 2024, 7, x FOR PEER REVIEW 3 of 13 2. Methods 2.1. Case Study The aim is to investigate how modern technology (the Monte Carlo method) can contribute to the development of effective and safe evacuation plans for mass events. This study focusses on modelling the evacuation of people for two specific configurations of the multipurpose hall. These are a hockey match with a capacity of 9500 people and a concert with a capacity of 11,000 people. For both configurations, we investigated how the evacuation of people is affected by the input of speed of movement from different sources [13–15] and what effect changes in the preference for door selection have on the evacuation of people. The Pathfinder software version 2023.3, which supports the Monte Carlo method, was chosen to implement the simulation process and configure agents (people). The combination of the two tools was used to obtain data on evacuation times, means, standard deviations (SDs), and people concentrations in the selected area. The results of the simulations provided valuable information for the design of effective safety measures in multipurpose halls and other similar multipurpose venues. A multipurpose hall in the Czech Republic is the subject of this study. The hall has a total of 19 grandstands, which are marked with leers, and the individual rows are marked with Arabic numerals. The grandstands are located around the ice area. Matches, tournaments, and various performances are held in the ice area. The access to the grandstands is through stairs that lead from the entrance hall. Three entrances lead to the multipurpose hall and also serve as emergency exits in an emergency event. There are three entrance doors on the north and south sides, while the east side offers two entrance doors. All entrances to the multipurpose hall are wheelchair accessible. The multipurpose hall and its emergency exits are shown in Figure 1. Figure 1. Geometry of the multipurpose hall in Pathfinder. 2.2. Pathfinder Software As mentioned above, the Pathfinder software version 2023.3 was used for this study [16]. The software was developed by Thunderhead Engineering, which provides regular updates of this software. Pathfinder is now a widely used software, as evidenced by the Figure 1. Geometry of the multipurpose hall in Pathfinder. 2.2. Pathfinder Software As mentioned above, the Pathfinder software version 2023.3 was used for this study [ 16 ]. The software was developed by Thunderhead Engineering, which provides regular updates of this software. Pathfinder is now a widely used software, as evidenced by the number of studies on the evacuation of people in which the software has been applied [ 17 – 19 ]. Pathfinder version 2023.3 was used for the simulation processes. A people movement simulation software provides two simulation modes: the Society of Fire Protection Engineers (SFPE) mode and the steering mode. For this study, all simulations were performed in the steering mode. The control mode aims to mimic human Fire 2024,7, 178 4 of 13 behaviour and movements as closely as possible. This scheme is based on the autonomous behaviour of each agent. Agents move independently towards their goal while avoiding other people and obstacles [16]. In Pathfinder, people are replaced by agents, as mentioned above. By default, they are displayed as cylinders with a diameter of 45.58 cm and a height of 1.83 m or as 3D people figures. The dimensions of the agents used in the simulations and how they are represented in the simulation are listed in Table 1. Table 1. Type of people, their dimensions, and how to display them in Pathfinder. Type of People Dimensions of People Agents are Displayed in Pathfinder As Cylinders As People Men/women <30 years of age Men/women 30–50 years of age Men/women >50 years of age default 45.58 cm Pheasant [20]: men 42.0–46.5 cm; mean 43.8 cm, SD 28.0 cm women 35.5-43.5 cm; mean 39.5 cm, SD 24.0 cm Fire 2024, 7, x FOR PEER REVIEW 4 of 13 number of studies on the evacuation of people in which the software has been applied [17–19]. Pathfinder version 2023.3 was used for the simulation processes. A people movement simulation software provides two simulation modes: the Society of Fire Protection Engineers (SFPE) mode and the steering mode. For this study, all simulations were performed in the steering mode. The control mode aims to mimic human behaviour and movements as closely as possible. This scheme is based on the autonomous behaviour of each agent. Agents move independently towards their goal while avoiding other people and obstacles [16]. In Pathfinder, people are replaced by agents, as mentioned above. By default, they are displayed as cylinders with a diameter of 45.58 cm and a height of 1.83 m or as 3D people figures. The dimensions of the agents used in the simulations and how they are represented in the simulation are listed in Table 1. Table 1. Type of people, their dimensions, and how to display them in Pathfinder. Type of People Dimensions of People Agents are Displayed in Pathfinder As Cylinders As People Men/women <30 years of age Men/women 30–50 years of age Men/women >50 years of age default 45.58 cm Pheasant [20]: men 42.0–46.5 cm; mean 43.8 cm, SD 28.0 cm women 35.5-43.5 cm; mean 39.5 cm, SD 24.0 cm Men/women with disabilities default 45.58 cm Methodology [21]: 60.0–90.0 cm Men/women with disabilities − wheelchair + assistant 132.0 × 76.0 cm Four scenarios were developed to illustrate the evacuation of people from a multipurpose hall. Two (scenarios 1A and 1B) related to a hockey match and the other two (scenarios 2A and 2B) to a concert. Scenarios 1A and 2A show the evacuation of people without specifying the time before evacuation and without changing the preference for door selection, corresponding to a situation in which the agents are unfamiliar with the environment. These scenarios are called “original”. On the other hand, Scenarios 1B and 2B include agents who already have a set pre-evacuation time and their preference for door selection has also been adjusted. These scenarios are called “modified”. In the original scenarios, people were divided into two main categories: ordinary people and vulnerable people. These categories were further subdivided into several groups, which are detailed in Table 2. In the modified scenarios (1B and 2B), two additional profiles were added: “assistant men” and “assistant women”. The percentage distribution of the different groups within the scenarios is shown in Figure 2. Fire 2024, 7, x FOR PEER REVIEW 4 of 13 number of studies on the evacuation of people in which the software has been applied [17–19]. Pathfinder version 2023.3 was used for the simulation processes. A people movement simulation software provides two simulation modes: the Society of Fire Protection Engineers (SFPE) mode and the steering mode. For this study, all simulations were performed in the steering mode. The control mode aims to mimic human behaviour and movements as closely as possible. This scheme is based on the autonomous behaviour of each agent. Agents move independently towards their goal while avoiding other people and obstacles [16]. In Pathfinder, people are replaced by agents, as mentioned above. By default, they are displayed as cylinders with a diameter of 45.58 cm and a height of 1.83 m or as 3D people figures. The dimensions of the agents used in the simulations and how they are represented in the simulation are listed in Table 1. Table 1. Type of people, their dimensions, and how to display them in Pathfinder. Type of People Dimensions of People Agents are Displayed in Pathfinder As Cylinders As People Men/women <30 years of age Men/women 30–50 years of age Men/women >50 years of age default 45.58 cm Pheasant [20]: men 42.0–46.5 cm; mean 43.8 cm, SD 28.0 cm women 35.5-43.5 cm; mean 39.5 cm, SD 24.0 cm Men/women with disabilities default 45.58 cm Methodology [21]: 60.0–90.0 cm Men/women with disabilities − wheelchair + assistant 132.0 × 76.0 cm Four scenarios were developed to illustrate the evacuation of people from a multipurpose hall. Two (scenarios 1A and 1B) related to a hockey match and the other two (scenarios 2A and 2B) to a concert. Scenarios 1A and 2A show the evacuation of people without specifying the time before evacuation and without changing the preference for door selection, corresponding to a situation in which the agents are unfamiliar with the environment. These scenarios are called “original”. On the other hand, Scenarios 1B and 2B include agents who already have a set pre-evacuation time and their preference for door selection has also been adjusted. These scenarios are called “modified”. In the original scenarios, people were divided into two main categories: ordinary people and vulnerable people. These categories were further subdivided into several groups, which are detailed in Table 2. In the modified scenarios (1B and 2B), two additional profiles were added: “assistant men” and “assistant women”. The percentage distribution of the different groups within the scenarios is shown in Figure 2. Men/women with disabilities default 45.58 cm Methodology [21]: 60.0–90.0 cm Fire 2024, 7, x FOR PEER REVIEW 4 of 13 number of studies on the evacuation of people in which the software has been applied [17–19]. Pathfinder version 2023.3 was used for the simulation processes. A people movement simulation software provides two simulation modes: the Society of Fire Protection Engineers (SFPE) mode and the steering mode. For this study, all simulations were performed in the steering mode. The control mode aims to mimic human behaviour and movements as closely as possible. This scheme is based on the autonomous behaviour of each agent. Agents move independently towards their goal while avoiding other people and obstacles [16]. In Pathfinder, people are replaced by agents, as mentioned above. By default, they are displayed as cylinders with a diameter of 45.58 cm and a height of 1.83 m or as 3D people figures. The dimensions of the agents used in the simulations and how they are represented in the simulation are listed in Table 1. Table 1. Type of people, their dimensions, and how to display them in Pathfinder. Type of People Dimensions of People Agents are Displayed in Pathfinder As Cylinders As People Men/women <30 years of age Men/women 30–50 years of age Men/women >50 years of age default 45.58 cm Pheasant [20]: men 42.0–46.5 cm; mean 43.8 cm, SD 28.0 cm women 35.5-43.5 cm; mean 39.5 cm, SD 24.0 cm Men/women with disabilities default 45.58 cm Methodology [21]: 60.0–90.0 cm Men/women with disabilities − wheelchair + assistant 132.0 × 76.0 cm Four scenarios were developed to illustrate the evacuation of people from a multipurpose hall. Two (scenarios 1A and 1B) related to a hockey match and the other two (scenarios 2A and 2B) to a concert. Scenarios 1A and 2A show the evacuation of people without specifying the time before evacuation and without changing the preference for door selection, corresponding to a situation in which the agents are unfamiliar with the environment. These scenarios are called “original”. On the other hand, Scenarios 1B and 2B include agents who already have a set pre-evacuation time and their preference for door selection has also been adjusted. These scenarios are called “modified”. In the original scenarios, people were divided into two main categories: ordinary people and vulnerable people. These categories were further subdivided into several groups, which are detailed in Table 2. In the modified scenarios (1B and 2B), two additional profiles were added: “assistant men” and “assistant women”. The percentage distribution of the different groups within the scenarios is shown in Figure 2. Fire 2024, 7, x FOR PEER REVIEW 4 of 13 number of studies on the evacuation of people in which the software has been applied [17–19]. Pathfinder version 2023.3 was used for the simulation processes. A people movement simulation software provides two simulation modes: the Society of Fire Protection Engineers (SFPE) mode and the steering mode. For this study, all simulations were performed in the steering mode. The control mode aims to mimic human behaviour and movements as closely as possible. This scheme is based on the autonomous behaviour of each agent. Agents move independently towards their goal while avoiding other people and obstacles [16]. In Pathfinder, people are replaced by agents, as mentioned above. By default, they are displayed as cylinders with a diameter of 45.58 cm and a height of 1.83 m or as 3D people figures. The dimensions of the agents used in the simulations and how they are represented in the simulation are listed in Table 1. Table 1. Type of people, their dimensions, and how to display them in Pathfinder. Type of People Dimensions of People Agents are Displayed in Pathfinder As Cylinders As People Men/women <30 years of age Men/women 30–50 years of age Men/women >50 years of age default 45.58 cm Pheasant [20]: men 42.0–46.5 cm; mean 43.8 cm, SD 28.0 cm women 35.5-43.5 cm; mean 39.5 cm, SD 24.0 cm Men/women with disabilities default 45.58 cm Methodology [21]: 60.0–90.0 cm Men/women with disabilities − wheelchair + assistant 132.0 × 76.0 cm Four scenarios were developed to illustrate the evacuation of people from a multipurpose hall. Two (scenarios 1A and 1B) related to a hockey match and the other two (scenarios 2A and 2B) to a concert. Scenarios 1A and 2A show the evacuation of people without specifying the time before evacuation and without changing the preference for door selection, corresponding to a situation in which the agents are unfamiliar with the environment. These scenarios are called “original”. On the other hand, Scenarios 1B and 2B include agents who already have a set pre-evacuation time and their preference for door selection has also been adjusted. These scenarios are called “modified”. In the original scenarios, people were divided into two main categories: ordinary people and vulnerable people. These categories were further subdivided into several groups, which are detailed in Table 2. In the modified scenarios (1B and 2B), two additional profiles were added: “assistant men” and “assistant women”. The percentage distribution of the different groups within the scenarios is shown in Figure 2. Men/women with disabilities − wheelchair + assistant 132.0 ×76.0 cm Fire 2024, 7, x FOR PEER REVIEW 4 of 13 number of studies on the evacuation of people in which the software has been applied [17–19]. Pathfinder version 2023.3 was used for the simulation processes. A people movement simulation software provides two simulation modes: the Society of Fire Protection Engineers (SFPE) mode and the steering mode. For this study, all simulations were performed in the steering mode. The control mode aims to mimic human behaviour and movements as closely as possible. This scheme is based on the autonomous behaviour of each agent. Agents move independently towards their goal while avoiding other people and obstacles [16]. In Pathfinder, people are replaced by agents, as mentioned above. By default, they are displayed as cylinders with a diameter of 45.58 cm and a height of 1.83 m or as 3D people figures. The dimensions of the agents used in the simulations and how they are represented in the simulation are listed in Table 1. Table 1. Type of people, their dimensions, and how to display them in Pathfinder. Type of People Dimensions of People Agents are Displayed in Pathfinder As Cylinders As People Men/women <30 years of age Men/women 30–50 years of age Men/women >50 years of age default 45.58 cm Pheasant [20]: men 42.0–46.5 cm; mean 43.8 cm, SD 28.0 cm women 35.5-43.5 cm; mean 39.5 cm, SD 24.0 cm Men/women with disabilities default 45.58 cm Methodology [21]: 60.0–90.0 cm Men/women with disabilities − wheelchair + assistant 132.0 × 76.0 cm Four scenarios were developed to illustrate the evacuation of people from a multipurpose hall. Two (scenarios 1A and 1B) related to a hockey match and the other two (scenarios 2A and 2B) to a concert. Scenarios 1A and 2A show the evacuation of people without specifying the time before evacuation and without changing the preference for door selection, corresponding to a situation in which the agents are unfamiliar with the environment. These scenarios are called “original”. On the other hand, Scenarios 1B and 2B include agents who already have a set pre-evacuation time and their preference for door selection has also been adjusted. These scenarios are called “modified”. In the original scenarios, people were divided into two main categories: ordinary people and vulnerable people. These categories were further subdivided into several groups, which are detailed in Table 2. In the modified scenarios (1B and 2B), two additional profiles were added: “assistant men” and “assistant women”. The percentage distribution of the different groups within the scenarios is shown in Figure 2. Fire 2024, 7, x FOR PEER REVIEW 4 of 13 number of studies on the evacuation of people in which the software has been applied [17–19]. Pathfinder version 2023.3 was used for the simulation processes. A people movement simulation software provides two simulation modes: the Society of Fire Protection Engineers (SFPE) mode and the steering mode. For this study, all simulations were performed in the steering mode. The control mode aims to mimic human behaviour and movements as closely as possible. This scheme is based on the autonomous behaviour of each agent. Agents move independently towards their goal while avoiding other people and obstacles [16]. In Pathfinder, people are replaced by agents, as mentioned above. By default, they are displayed as cylinders with a diameter of 45.58 cm and a height of 1.83 m or as 3D people figures. The dimensions of the agents used in the simulations and how they are represented in the simulation are listed in Table 1. Table 1. Type of people, their dimensions, and how to display them in Pathfinder. Type of People Dimensions of People Agents are Displayed in Pathfinder As Cylinders As People Men/women <30 years of age Men/women 30–50 years of age Men/women >50 years of age default 45.58 cm Pheasant [20]: men 42.0–46.5 cm; mean 43.8 cm, SD 28.0 cm women 35.5-43.5 cm; mean 39.5 cm, SD 24.0 cm Men/women with disabilities default 45.58 cm Methodology [21]: 60.0–90.0 cm Men/women with disabilities − wheelchair + assistant 132.0 × 76.0 cm Four scenarios were developed to illustrate the evacuation of people from a multipurpose hall. Two (scenarios 1A and 1B) related to a hockey match and the other two (scenarios 2A and 2B) to a concert. Scenarios 1A and 2A show the evacuation of people without specifying the time before evacuation and without changing the preference for door selection, corresponding to a situation in which the agents are unfamiliar with the environment. These scenarios are called “original”. On the other hand, Scenarios 1B and 2B include agents who already have a set pre-evacuation time and their preference for door selection has also been adjusted. These scenarios are called “modified”. In the original scenarios, people were divided into two main categories: ordinary people and vulnerable people. These categories were further subdivided into several groups, which are detailed in Table 2. In the modified scenarios (1B and 2B), two additional profiles were added: “assistant men” and “assistant women”. The percentage distribution of the different groups within the scenarios is shown in Figure 2. Four scenarios were developed to illustrate the evacuation of people from a multipurpose hall. Two (scenarios 1A and 1B) related to a hockey match and the other two (scenarios 2A and 2B) to a concert. Scenarios 1A and 2A show the evacuation of people without specifying the time before evacuation and without changing the preference for door selection, corresponding to a situation in which the agents are unfamiliar with the environment. These scenarios are called “original”. On the other hand, Scenarios 1B and 2B include agents who already have a set pre-evacuation time and their preference for door selection has also been adjusted. These scenarios are called “modified”. In the original scenarios, people were divided into two main categories: ordinary people and vulnerable people. These categories were further subdivided into several groups, which are detailed in Table 2. In the modified scenarios (1B and 2B), two additional profiles were added: “assistant men” and “assistant women”. The percentage distribution of the different groups within the scenarios is shown in Figure 2. In all four scenarios, the speeds of movement from different sources were successively entered, as shown in Table 2. The aim was to find out how a change in the speed of movement set according to various sources (according to different authors) can affect the evacuation of people. In scenarios in which the speed of movement was set according to Fruin [ 13 ], this speed of movement was set with a normal distribution with a mean value of 1.2 m/s and a standard deviation (SD) of 0.2 m/s. In scenarios in which the speed of movement was set according to Weidmann [ 15 ], the effect of changing the dimensions of people (change in shoulder width) was also investigated. A comparison was made between the default dimensions (45.58 cm for all equally) and the dimensions according to Pheasant [20] with a normal distribution. As stated previously, Scenarios 1B and 2B involved setting a time before evacuation. This period was determined based on data from the publication [ 22 ], which specifies a pre-evacuation time between 41 and 60 s for assembly areas with an auditorium. This range was applied to all people. In Scenarios 1B and 2B, the door selection parameters were also modified, as shown in Table 3. In both scenarios, agents consider their distance to the door in the room but can change their choice during the simulation. Scenario 1B assumes Fire 2024,7, 178 5 of 13 that half of the visitors to the multipurpose hall are very familiar with the environment because they regularly attend hockey matches. On the contrary, in Scenario 2B, the setting was based on the idea that most visitors are not local and are, therefore, unfamiliar with the area. Table 2. Division of people into categories and groups and values of the speed of movement of people according to Fruin [13], International Maritime Organisation (IMO) [14], and Weidmann [15]. Category Group v (m/s) Fruin v (m/s) IMO v (m/s) Weidmann min max min max min max Normal people Men < 30 years of age 0.6 1.8 1.11 1.85 0.580 1.610 Women < 30 years of age 0.6 1.8 0.93 1.55 0.516 1.433 Men 30–50 years of age 0.6 1.8 0.97 1.62 1.410 1.514 Women 30–50 years of age 0.6 1.8 0.71 1.19 1.255 1.371 Men assistant 0.6 1.8 0.97 1.62 1.410 1.514 Women assistant 0.6 1.8 0.71 1.19 1.255 1.371 Vulnerable people Men with disabilities—wheelchair 0.6 1.8 0.55 0.91 0.460 0.760 Men with disabilities 0.6 1.8 0.55 0.91 0.460 0.760 Women with disabilities—wheelchair 0.6 1.8 0.37 0.61 0.409 0.676 Women with disabilities 0.6 1.8 0.37 0.61 0.409 0.676 Men > 50 years of age 0.6 1.8 0.64 1.06 0.671 1.392 Women > 50 years of age 0.6 1.8 0.43 0.71 0.605 1.255 Fire 2024, 7, x FOR PEER REVIEW 5 of 13 (a) (b) Figure 2. Percentage distribution of people: (a) Scenarios 1A and 2A, (b) Scenarios 1B and 2B. In all four scenarios, the speeds of movement from different sources were successively entered, as shown in Table 2. The aim was to find out how a change in the speed of movement set according to various sources (according to different authors) can affect the evacuation of people. In scenarios in which the speed of movement was set according to Fruin [13], this speed of movement was set with a normal distribution with a mean value of 1.2 m/s and a standard deviation (SD) of 0.2 m/s. In scenarios in which the speed of movement was set according to Weidmann [15], the effect of changing the dimensions of people (change in shoulder width) was also investigated. A comparison was made between the default dimensions (45.58 cm for all equally) and the dimensions according to Pheasant [20] with a normal distribution. Table 2. Division of people into categories and groups and values of the speed of movement of people according to Fruin [13], International Maritime Organisation (IMO) [14], and Weidmann [15]. Category Group v (m/s) Fruin v (m/s) IMO v (m/s) Weidmann min max min max min max Normal people Men < 30 years of age 0.6 1.8 1.11 1.85 0.580 1.610 Women < 30 years of age 0.6 1.8 0.93 1.55 0.516 1.433 Men 30–50 years of age 0.6 1.8 0.97 1.62 1.410 1.514 Women 30–50 years of age 0.6 1.8 0.71 1.19 1.255 1.371 Men assistant 0.6 1.8 0.97 1.62 1.410 1.514 Women assistant 0.6 1.8 0.71 1.19 1.255 1.371 Vulnerable people Men with disabilities—wheelchair 0.6 1.8 0.55 0.91 0.460 0.760 Men with disabilities 0.6 1.8 0.55 0.91 0.460 0.760 Women with disabilities—wheelchair 0.6 1.8 0.37 0.61 0.409 0.676 Women with disabilities 0.6 1.8 0.37 0.61 0.409 0.676 Men > 50 years of age 0.6 1.8 0.64 1.06 0.671 1.392 Women > 50 years of age 0.6 1.8 0.43 0.71 0.605 1.255 As stated previously, Scenarios 1B and 2B involved seing a time before evacuation. This period was determined based on data from the publication [22], which specifies a pre-evacuation time between 41 and 60 s for assembly areas with an auditorium. This range was applied to all people. In Scenarios 1B and 2B, the door selection parameters were also modified, as shown in Table 3. In both scenarios, agents consider their distance to the door in the room but can change their choice during the simulation. Scenario 1B assumes that half of the visitors to the multipurpose hall are very familiar with the Figure 2. Percentage distribution of people: (a) Scenarios 1A and 2A, (b) Scenarios 1B and 2B. Table 3. Setting up decision processes—preference for door selection. CRTT CRQT GTT CDP CRDP Hockey match—Scenario 1A Normal and vulnerable people 1 1 1 35 35 Hockey match—Scenario 1B Normal people + people > 50 years of age + people with disabilities 5 5 1 50 35 People with disabilities—wheelchair + assistant 10 0 1 70 35 Fire 2024,7, 178 6 of 13 Table 3. Cont. CRTT CRQT GTT CDP CRDP Concert—Scenario 2A Normal and vulnerable people 1 1 1 35 35 Concert—Scenario 2B Normal people + people > 50 years of age + people with disabilities 2 2 1 50 35 People with disabilities—wheelchair + assistant 10 0 1 50 35 Explanatory notes: CRTT (current room travel time) ( − ) is the agent’s travel time to the door, regardless of other agents. The higher the value, the more likely the agent will select the closest door. CRQT (current room queue time) ( − ) is the time the agent spends in the queue at the door according to their current position in the queue and the current calculated flow (per/s) of the door. The higher the value CRQT, the more likely that agent can change the preference for door selection, leave the queue, and go to another door. GTT (global travel time) ( − ) is the agent’s travel time until they reach the exit of the object, regardless of other agents. The higher the value, the more likely the agent will select the closest exit from the object. CDP (current door preference) (%) is the percentage of compliance with the selected door in the room. The higher the value, the smaller the change in the selection of doors in the room in question. CRDP (current room distance penalty) (m) is a parameter that simulates agent fatigue, that is, as the distance increases, the agent selects a closer exit. 2.3. Using the Monte Carlo Method The Monte Carlo method is an effective tool for investigating patterns of behaviour. It is applied to determine the ideal number of simulations in evacuation models such as Pathfinder; the results are then subjected to convergence analysis (sensitivity analysis). The aim is to check whether the results are sufficiently accurate in relation to the expected values. The process involves a series of repeated simulations that generate average total evacuation time (TET) values that converge to the average evacuation time course. Analysis of the average TET values provides information on the time required for a safe escape. To determine the convergence rate of the TET, the method proposed by Ronchi in collaboration with Reneke and Peacock [ 23 ] is used. First, the arithmetic mean of the TET for the j-runs must be determined. This can be expressed as TETavj and is calculated using Equation (1). TETavj = 1 j j ∑ i=1 TETi(1) Then, the convergence rate of two consecutive average evacuation times TET avj (TET av1 , TET av2 , etc.) is determined. The convergence rate according to [ 23 ] is obtained by calculating TET convj using Equation (2). The TET convj value is expressed as a percentage (%) as the absolute value of the difference between two consecutive average TET values divided by the last average evacuation time [ 23 ]. The last value of TET convj corresponds to all runs and can be referred to as TETconvFIN. TETconvj =      TETavj −TETavj−1 TETavj      (2) The convergence rate values obtained by TET convj (including TET convFIN ) are then compared with the TR TET threshold. The determination of the threshold value according to [ 23 ] depends on various factors, such as the evacuation scenario or the model used. The potential value in the context of fire safety engineering can be the potential value in the context of modelling according to [23], TRTET = 0.5%. TETconvj <TRTET (3) The aim of this study is to achieve consistent results in terms of the average evacuation times of people. Using the Monte Carlo method, it is possible to generate the desired number of variations (iterations) with random agent positions and profiles for all agents [ 16 ]. The agent profile includes parameters such as speed, shape (diameter, height), and pref- Fire 2024,7, 178 7 of 13 erence for door selection. In this study, 30 repetitions of each simulation were performed. This approach avoids interpretation of the study topic solely based on individual behaviour. The actual implementation of the Monte Carlo method involved the following steps: 1. Define the problem to be solved. 2. Model the system and identify key parameters. 3. Repeat the simulation of the system. 4. Perform statistical analysis. 5. Interpret the results obtained. 3. Results 3.1. Analysis of Evacuation Times The analysis of the evacuation time of people for both situations (hockey—Scenario 1A, 1B; concert—Scenario 2A, 2B) was carried out based on the tables of values generated by the Pathfinder software version 2023.3. During the analysis, investigations were carried out on how the evacuation time changes and how its course depends on changes in the input parameters of the speed of movement. The graphs in Figure 3were generated for the average values obtained using the Monte Carlo method for 30 iterations of each scenario. These graphs express the relationship between the number of people remaining in the model (space) and the evacuation time in seconds. This analysis allows a better understanding of the factors that affect the effective evacuation process and highlights the need to improve the safety of people in an emergency event. In Figure 3, the greatest increase in evacuation time occurs when the speed of movement is set according to the International Maritime Organisation (IMO) test. For Scenarios 1A and 1B, the total evacuation time increased by approximately 300 s. For Scenarios 2A and 2B, it extended by 500 s compared to the Fruin speed of movement and by 300 s compared to the Weidmann speed of movement. It is also important to warn about the difference between the evacuation time at the speed of movement according to Fruin and the evacuation time at the speed of movement according to Weidmann. For Scenarios 1A and 1B, this difference was approximately 30 s, whereas for Scenarios 2A and 2B, the difference was approximately 200 s. As part of this study, the effect of changing the width of the agents’ shoulders on the evacuation time was also monitored when entering the speed of movement according to Weidmann. It can be seen in Figure 3that changing the width of the shoulders from the default software setting (yellow curve) to the dimensions according to Pheasant (grey curve) did not have a significant effect on the evacuation time and its smooth course. The time difference between the evacuation time for the speed of movement according to Weidmann and the shoulder width according to Pheasant and the evacuation time for the speed of movement according to Weidmann and the shoulder width according to the default setting was about 30 s. In Figure 3, the difference in the time that the first agent leaves the multipurpose hall can also be observed. If a pre-evacuation time is set, the evacuation time will be longer. This setting makes the situation of the model more realistic. It takes into account the time for decision-making and preparation for evacuation. It is unlikely that everyone will start evacuating at the same time without first deciding on the next steps. During the evaluation of the results, it was found that the change in preference for the door selection set for Scenario 1B and Scenario 2B affected the evacuation time. This can be seen in Figure 3. In both cases, preferences for door selection resulted in an increase in evacuation time. The increase in evacuation time is due to the pre-evacuation time setting (in the Section 2.2), which was not specified in the original scenarios (Scenarios 1A and 2A). Although Scenario 1B assumed knowledge of the facility for about half of the visitors, the evacuation time was longer than that in Scenario 1A with the original preferences for door selection. This finding is unexpected, as the evacuation time was expected to be shorter or the same as in Scenario 1A. A greater difference in evacuation times was observed between Scenarios 2A and 2B, where the preference for door selection was only changed from a Fire 2024,7, 178 8 of 13 value of 1 to a value of 2. These results show that even a small change in preferences for door selection can have a significant impact on overall evacuation times. Fire 2024, 7, x FOR PEER REVIEW 8 of 13 (a) (b) (c) (d) Figure 3. Dependence of the remaining number of people in the model (space) on the time in seconds: (a) Scenario 1A, (b) Scenario 1B, (c) Scenario 2A, and (d) Scenario 2B. In Figure 3, the difference in the time that the first agent leaves the multipurpose hall can also be observed. If a pre-evacuation time is set, the evacuation time will be longer. This seing makes the situation of the model more realistic. It takes into account the time for decision-making and preparation for evacuation. It is unlikely that everyone will start evacuating at the same time without first deciding on the next steps. During the evaluation of the results, it was found that the change in preference for the door selection set for Scenario 1B and Scenario 2B affected the evacuation time. This can be seen in Figure 3. In both cases, preferences for door selection resulted in an increase in evacuation time. The increase in evacuation time is due to the pre-evacuation time setting (in the Section 2.2), which was not specified in the original scenarios (Scenarios 1A and 2A). Although Scenario 1B assumed knowledge of the facility for about half of the visitors, the evacuation time was longer than that in Scenario 1A with the original preferences for door selection. This finding is unexpected, as the evacuation time was expected to be shorter or the same as in Scenario 1A. A greater difference in evacuation times was observed between Scenarios 2A and 2B, where the preference for door selection was only changed from a value of 1 to a value of 2. These results show that even a small change in preferences for door selection can have a significant impact on overall evacuation times. 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 200 400 600 800 1000 1200 Number of people Time in seconds Total evacuation time: Fruin (default) 1088.22 s IMO test (default) 1377.92 s Weidmann (Pheasant) 1045.96 s Weidmann (default) 1071.31 s 10,000 0 2000 4000 6000 8000 10000 12000 0 250 500 750 1000 1250 1500 1750 Number of people Time in seconds Total evacuation time: Fruin (default) 1316.89 s IMO test (default) 1807.56 s Weidmann (Pheasant) 1539.47 s Weidmann (default) 1505.88 s 10,000 12,000 0 2000 4000 6000 8000 10000 12000 0 250 500 750 1000 1250 1500 1750 2000 Number of people Time in seconds Total evacuation time: Fruin (default) 1528.48 s IMO test (default) 2080.25 s Weidmann (Pheasant) 1714.59 s Weidmann (default) 1687.37 s 10,000 12,000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 200 400 600 800 1000 1200 Number of people Time in seconds Total evacuation time: Fruin (default) 927.86 s IMO test (default) 1231.57 s Weidmann (Pheasant) 956.66 s Weidmann (default) 971.61 s 10,000 Figure 3. Dependence of the remaining number of people in the model (space) on the time in seconds: (a) Scenario 1A, (b) Scenario 1B, (c) Scenario 2A, and (d) Scenario 2B. 3.2. Analysis of Space Use Maps Space usage maps are a useful tool to represent the evacuation process. These maps allow you to identify areas in which queues and congestion form during evacuations. Furthermore, they also allow one to identify areas with a high density of people and thus help to understand and reduce the risks associated with evacuation. For better clarity, the individual areas are distinguished by colour. The areas with the highest loads are marked in dark red, indicating areas with a high risk of injury or death. Identifying these locations can make the evacuation process more efficient. Figure 4shows space usage maps for all four scenarios in this study and provides a 3D view of two key areas within the building: (i) Grandstand B and East exit and (ii) Grandstand C and North exit. Although evacuation times vary from scenario to scenario, critical areas with high concentrations of people and queues appear in the same locations in all scenarios. This includes all the stairs in the building, the corridors, and exits of the multipurpose hall. The main difference can be seen when comparing the situation of a hockey match with a concert. During the concert (Scenarios 2A and 2B), people can Fire 2024,7, 178 9 of 13 only leave the ice area by the North exit. Barriers are placed at the South exit to prevent escape in that direction. The space behind these barriers contains the technical and sound equipment needed for the concert to run smoothly. During the concert, a very critical area with a high concentration of people is the area by Grandstand C towards the North exit. Fire 2024, 7, x FOR PEER REVIEW 9 of 13 3.2. Analysis of Space Use Maps Space usage maps are a useful tool to represent the evacuation process. These maps allow you to identify areas in which queues and congestion form during evacuations. Furthermore, they also allow one to identify areas with a high density of people and thus help to understand and reduce the risks associated with evacuation. For beer clarity, the individual areas are distinguished by colour. The areas with the highest loads are marked in dark red, indicating areas with a high risk of injury or death. Identifying these locations can make the evacuation process more efficient. Figure 4 shows space usage maps for all four scenarios in this study and provides a 3D view of two key areas within the building: (i) Grandstand B and East exit and (ii) Grandstand C and North exit. Although evacuation times vary from scenario to scenario, critical areas with high concentrations of people and queues appear in the same locations in all scenarios. This includes all the stairs in the building, the corridors, and exits of the multipurpose hall. The main difference can be seen when comparing the situation of a hockey match with a concert. During the concert (Scenarios 2A and 2B), people can only leave the ice area by the North exit. Barriers are placed at the South exit to prevent escape in that direction. The space behind these barriers contains the technical and sound equipment needed for the concert to run smoothly. During the concert, a very critical area with a high concentration of people is the area by Grandstand C towards the North exit. Grandstand B and East exit Grandstand C and North exit Scenario 1A Scenario 1B Scenario 2A Fire 2024, 7, x FOR PEER REVIEW 10 of 13 Scenario 2B Figure 4. Space usage maps for the different scenarios. 4. Discussion This study proposes the use of a probabilistic model, specifically the Monte Carlo method, combined with simulation software to develop an effective and safe evacuation plan for a multipurpose hall designed to host mass events. For the model to be as realistic as possible, the agents’ parameters must be set correctly. In this case study, the movement speeds for the different agent profiles in Pathfinder were set sequentially according to different sources. First, the minimum and maximum velocity values were set between 0.6 m/s and 1.8 m/s [13] with a normal distribution. The minimum and maximum velocity values were then set between 0.37 m/s and 1.85 m/s [14]. Finally, the minimum and maximum velocity values were set between 0.409 m/s and 1.610 m/s [15]. As part of this study, a convergence rate analysis of the total evacuation time (TET) of people was carried out. The method based on the search for convergence is shown in the graphs in Figure 5. The best results were achieved in Scenario 1A. After five runs, the deviation of the average evacuation time did not exceed 0.5%. In Scenario 1B, the deviation fell below the TRTET 0.5% after the twelfth run. When comparing Scenarios 2A and 2B, beer results were obtained in Scenario 2B in which the deviation of the average evacuation time exceeded 0.5% for the last time in the eighth run and then stayed below this value. For Scenario 2A, the deviation fell to below 0.5% after the fifteenth run. Convergence is thus demonstrated. (a) (b) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 TETconvj (%) Number of runs (n) TRTET = 0.5 % 0 0.5 1 1.5 2 2.5 3 3.5 0 5 10 15 20 25 30 TETconvj (%) Number of runs (n) TRTET = 0.5 % Figure 4. Space usage maps for the different scenarios.