Enhancing Fire Safety of Buildings’ Occupants: An Integrated Fire Risk Concept into Reliability-Based Evacuation Design Optimization Method Focused on At-Risk Groups

Fire safety is important for building occupants, especially the elderly and those with mobility or cognitive impartment, as they may face significant challenges during evacuation especially if they are alone [1]. Buildings must be designed and constructed so that people can evacuate safely during a fire. Performance-based design (PBD) and prescriptive design (PD) are two methods used to fulfil laws and regulations when considering fire safety during the design of the buildings [2, 3]. Not only the buildings should be safe in the event of a fire, but also people need to be able to evacuate safely. The two aims can differ in their approach. PD includes codes and standards that prescribe specific building fire safety measures [4]. These measures may include fire-resistant materials, fire-rated walls and doors, and sprinkler systems. The PD approach primarily concerns compliance with codes and standards. PBD adopts a performance-based approach, focusing on the specific fire safety needs of a building based on use and occupants. PBD allows for more flexibility in design and can lead to more cost-effective and sustainable solutions [5]. In general, a PBD can be implemented by considering a fire risk assessment [6]. That is why several scholars suggested the concept of risk-informed performance-based building codes [7‐9].

Generally, the purpose of PBD is to minimize fire risk or failure and its consequences for building occupants. It incorporates deterministic or probabilistic models and analyses [10]. However, in engineering problems, we face parameters and variables that are not always deterministic. They are inherently random, leading to uncertainties in assessing these problems. Due to these uncertainties, there is a pressing need for scientific and systematic methods to address this issue [11]. An alternative to the PBD method is the Reliability-Based Design (RBD) method, which involves designing systems to meet a specific level of safety over their expected service life [12]. This approach is based on statistical models and probabilistic analyses, accounting for design parameter uncertainties [13]. The difference between these two methods is their approach to ensuring the desired safety level. The PBD ensures that the system performs to a certain level under specified conditions and aims to meet or exceed performance criteria, such as strength, usability, durability, and serviceability. In contrast, RBD focuses on ensuring that the system has an acceptable probability of not failing during its whole intended life period.

So far, numerous researchers have used the RBD method to validate and enhance the fire safety of diverse structures and infrastructures, including roads, tunnels, and bridges [14‐16]. This approach has been adopted to evaluate and enhance the fire safety of buildings, e.g. Bjelland employed the reliability-based design concept to interpret safety margins in ASET/RSET (Available/Required Safe Egress Time) assessments within the Norwegian building industry [17] and McAllister et al. developed reliability-based resistance criteria for practical structural engineering approaches to fire safety assurance [18]. Frantzich et al. investigated the application of the reliability index (beta) method for building fire evacuation safety. They used the First Order Second Moment (FOSM) reliability method to derive the design values and expressed the safety level of the building against fire as a value of the reliability index [19]. In another study, Olsen & Frantzich proposed a simple risk-based method for determining fire safety engineering design values based on an acceptable risk value, utilising the FOSM reliability index [20]. They reviewed the results and discussed the method’s strengths and limitations, highlighting the importance of factors such as model uncertainty, occupant load, and response time. It can be concluded that the method can apply to fire evacuation scenarios involving a limited number of occupants. However, this method accounts only for the probability of failure, not the magnitude of the consequences. This limitation makes it inadequate for situations with high occupant loads, where consequences can vary significantly depending on the number of exposed individuals. Addressing uncertainty treatment in risk analysis across six defined levels of model detail and complexity [21], the FOSM reliability method is categorised as level 3, which relies on a “best estimate” or a central value (e.g., the mean, median, or mode) of the outcome (e.g., loss) distribution, typically derived from “best estimates” of different variables. To address high occupant load and its impact on evacuation design, a higher level of uncertainty treatment (level 4) is required, which relies on quantitative risk assessment (QRA) to obtain a distribution of the probabilities of different system states based on best estimates of models and parameter values. This approach allows for representing risk not as a single point estimate of outcomes or even as a distribution of system states but as a comprehensive distribution of potential losses. However, in the context of this study, because individuals within the target groups are assumed to reside alone in their accommodations, occupant load is not a relevant factor, and a level 3 uncertainty treatment is deemed sufficient.

Albrecht [22] critiques traditional code-compliant designs for complex architectures and introduces a probabilistic approach using risk analysis to quantify safety levels. The study employs a response surface method and sensitivity analysis to identify critical input parameters. In another study, Albrecht [23] extends this approach to analyse the impact of fire protection systems. Using event-tree analysis, the study not only models the failure probabilities of different systems but also measures their impact on overall safety, incorporating cost-benefit analysis for a comprehensive comparison of solutions. Van et al. [24] develop a computationally efficient methodology combining deterministic models with probabilistic techniques. This framework quantifies fire safety levels through failure probabilities and risk assessments, illustrated through a case study of a commercial shopping mall. These studies enhance the understanding of fire safety engineering by integrating probabilistic approaches into PBD and RBD, offering a more accurate measure of safety levels.

Nowadays, the increasing presence of elderly individuals and those with reduced mobility or cognitive disabilities highlights the critical need for enhanced fire safety measures for these vulnerable populations. Despite this necessity, there is a gap in research regarding the RBD of buildings that accounts for the evacuation challenges these at-risk groups face. Due to their different behaviour in fire scenarios, these individuals represent a significant portion of fire fatalities, particularly in residential fires. To address this issue, this study integrates probabilistic risk concepts into an RBD method for fire evacuation problems alongside an optimization algorithm to incorporate the characteristics of the aforementioned at-risk groups, building geometry, fire properties, and related constraints. The goal is to determine the optimal design values for design variables related to building geometry and occupant characteristics for a fire egress time model. The primary objective is to ensure a specific level of safety compliance tailored to the unique needs of at-risk groups, thereby mitigating life loss during fire incidents and enhancing overall fire safety for building residents.

The Reliability-Based Design Optimization (RBDO) method is suitable for ongoing study as it incorporates both the probabilistic nature of design variables and optimisation techniques, offering a more holistic and adaptive solution than traditional methods like quantitative risk analysis (QRA) or Monte Carlo simulations. While QRA provides a comprehensive overview of potential risks and is beneficial for understanding event probabilities and consequences, it does not inherently optimise design parameters. Similarly, Monte Carlo simulations are excellent for modelling uncertainties but lack the built-in optimisation framework to find the best design solutions. RBDO, on the other hand, combines these strengths by not only accounting for uncertainties but also directly identifying optimal design parameters to ensure safety compliance and minimize risks specifically for at-risk groups. However, RBDO also has its drawbacks; it is computationally intensive and requires detailed statistical data on the design variables, which can be challenging to obtain. It also necessitates sophisticated modelling techniques and expert knowledge, making it less accessible for routine design tasks than to traditional methods. Despite these challenges, the ability of RBDO to provide tailored, optimal solutions for fire safety, makes it a compelling choice for this study.

Based on this introduction, the structure of the paper is as follows: Section 2 outlines the methodological approach for improving the safety of buildings occupied by individuals with mobility and cognitive impairments. Section 3 addresses the building fire evacuation problem, focusing on the egress timeline model for estimating the Available Safe Egress Time (ASET) and the Required Safe Egress Time (RSET). Section 4 details the methods for calculating the safety index, optimizing the reliability-based design, and establishing life safety or target safety criteria for the specified group. It also discusses individual risk to life and suggests criteria for the required safety level for individuals with reduced functionality by integrating probabilistic risk analysis and investigation of expected risk of life. Section 5 presents a case study that illustrates the proposed method, using data on the characteristics of people with mobility impairments derived from available datasets, including those provided by Fahy et al. [28], which cover evacuees’ premovement time and travel speed. Section 6 includes discussion and limitations, and Section 7 concludes with key findings and recommendations for future work.

In this section, we outline the methodology employed in our study, detailing the steps taken to address the research objectives. The first step is to present the building fire evacuation problem by defining a function that establishes the boundary between successful and unsuccessful evacuations during fire incidents. Based on the fire egress timeline model and data extracted from established pre-movement time datasets [25, 26], this study reviewing relevant literature on the pre-movement time of at-risk groups and updates previous findings related to this phase of evacuation [27, 28]. As presented earlier, individuals with cognitive and mobility impairments may experience unique challenges in fire evacuation beyond merely extended response times. These challenges include difficulties in preparation for evacuation, perceiving the fire threat, deciding on appropriate actions, and executing those actions, which collectively impact the pre-movement phase of evacuation [29, 30]. Additionally, their movement phase may require assistance from care staff, affecting overall evacuation efficiency. Although this study uses evacuation time as a comparative metric, the broader spectrum of behavioural and perceptual differences is acknowledged between individuals with and without mentioned disabilities.

Following this, presented methodology explains how the safety index, RBDO method, and criteria for life safety, which is the level of safety that should be fulfilled in building fire safety design, considering at-risk groups [31‐33] can be used. This allows for attaining the optimum values for the most critical variables in the fire evacuation model according to the defined safety level. The usability of this approach is illustrated by a case study via a fire evacuation problem for considering three different types of building occupants’ characteristics. Accordingly, the suggested target level of safety for the mentioned occupants considers fire properties, building geometry, and occupants’ characteristics. This approach will contribute to a better understanding of the fire evacuation problem using the RBD method to ensure that buildings are designed to meet a required level of safety.

Fire safety analysis aims to confirm the possibility of a safe evacuation from a burning building to a safe area. Developing and accessing models that account for fire propagation within structures and human reactions during evacuations is crucial. These models are applicable in fire safety engineering to evaluate the safety conditions of various buildings and infrastructures [34]. Among these models, the egress timeline model provides a simplified representation of the evacuation process, as illustrated in Fig. 1 [34]. This model encompasses the detection and awareness time (DAT), pre-movement time (PT), which is the duration between normal activities among occupants and the initiation of purposeful evacuation movements, and the movement phase (MOT) itself, which denotes the time required to reach a safe location [34]. The cumulative of the mentioned time intervals is denoted as RSET (Required Safe Egress Time), representing the duration of time required for all occupants within a building to safely evacuate in the event of a fire and reach a secure area. Conversely, ASET (Available Safe Egress Time) considers various factors, including the spread of fire and smoke and specific building characteristics. ASET indicates the point in time when conditions within the building become untenable, potentially leading to people being trapped and injured within the fire-affected area [34].

A critical aspect to consider is the time difference between RSET and ASET, particularly before the conditions become life-threatening. A more significant time gap between these two values signifies a larger margin of safety, as depicted in Fig. 1. The Q(x) represents the function that shows the boundary for a secure evacuation process, also known as the Limit State Function (LSF), signifying three potential evacuation modes: failure (Q(X) < 0), on time (Q(X) = 0), and with a margin of safety (Q(X) > 0). Further details are discussed in this section.

This section addresses fire safety for building occupants by introducing a safety index to evaluate and improve measure and discussing the RBDO method to efficiently achieve safety goals. To define the safety index of an occupant or the probability of being trapped in a fire, let x denotes basic variables of the fire safety limit state function, such as fire properties, the geometry of buildings, and people’s behaviour based on given fire scenarios. This vector may be modelled by a random vector X, and where \(\:{f}_{x}\left(x\right)\) is its probability density function (PDF). Also, the failure state is denoted by a limit state function Q, and conventionally with the following properties [31]:

\(\:Q:\:{R}^{n}\:\to\:\:R\:;\:\:\:\:\:\:X\to\:\:Q\left(X\right)\) such that:

The random variable defined by Z = Q(X) is also called the safety margin of the building evacuation. According to the definition of Q, the evacuation process fails when the safety margin is lower or equal to zero. Therefore, the failure domain is defined by \(\:\xi\:=\:\{x\in\:{R}^{n},\:Q(x)\le\:0\}\) (considering that the border δ (\(\:\xi\:\)) is also a failure set) the probability of unsuccessful evacuation or being trapped by the fire, \(\:{f}_{P}\) is given by Eqs. 7,

Generally, solving Eq. 7 is challenging because the border of \(\:\xi\:\) is non-linear, nor is the vector X a linear function of a standard Gaussian vector (i.e., the components of vector X are not Gaussian, or they may be correlated or have different types of probability distribution function) [58]. To solve Eq. 7, the main approaches are direct integration of PDF on the failure domain, which is usually done by using Monte-Carlo simulations [58], first and second-order reliability methods [59], and discrete approximation [60]. Besides these methods, Hasofer and Lind proposed a safety index β that can be used to calculate the probability of failure of an event. This index contains information about the margin of safety in the LSF and the uncertainty of the variables. It provides a general overview of a system’s safety level and can be considered a criterion for decision-making in rehabilitation and promotion of system status. The β safety index is proposed in a Λ space, where the vector components are Gaussian standard. To calculate the reliability index and probability of failure, let U be any vector of this space, Γ the n-dimensional surface defined by the limit state function Q(x) in the physical variables space \(\:\varPsi\:\), and \(\:\varSigma\:=T\left(\varGamma\:\right)\) its image in the standard Gaussian space \(\:\varLambda\:\).

As shown in Fig. 3, the Hasofer–Lind reliability index β, is defined as \(\:\beta\:=min-\left(d\right(O,M\left)\right)\), where \(\:O\) is the center (origin) of space \(\:\varLambda\:\), and M evolves on the surface \(\:\varSigma\:\). The transformation \(\:{T}_{i}\left(X\right)=U\), may be obtained by \(\:{T}_{i}\left(X\right)={U}_{i}(i=1,\dots\:,n)\) [31], . The main aim is to calculate β in space \(\:\varLambda\:\). To achieve this, one must solve a constrained optimization problem that is:

The solution of Eq. 8, \(\:{u}^{\text{*}}\) is called the design point and enables the calculation of the safety index as \(\:\beta\:=\left|\right|{u}^{\text{*}}\left|\right|.\) Then, the obtained safety index can be utilized to define the probability of failure (being trapped in fire) by computing the inverse cumulative normal distribution of the safety index value, \(\:{f}_{P}=\varPhi\:(-\beta\:)\) or \(\:\beta\:={\varPhi\:}^{-1}(1-{f}_{P})\).

Optimising the designs in performance-based design in engineering disciplines is common, and considering uncertainties is vital for the best outcomes [32]. The RBDO aims to create strong designs by considering existing uncertainties [61]. When tackling the optimisation design of a system, there are two main design principles to consider. The first is deterministic optimisation design, which treats known or unknown quantities as non-statistical. The second is RBDO, where some or all the design problems are treated as random variables [13]. The fire safety design of buildings involves a wide range of inherently indeterministic variables, which can be addressed through Reliability-Based Design Optimization (RBDO) during the design process.

Figure 4 illustrates a schematic view of the RBDO method process for finding a solution to a 2-dimensional problem. The red region represents the failure zone, signifying that solutions within this area fail to satisfy the problem design constraints, specifically the limit state functions \(\:{Q}_{1}\left(U\right)\) and \(\:{Q}_{2}\left(U\right)\). The white region in the first quadrant of the coordinate system signifies the space from which potential solutions can be chosen, provided they adhere to RBDO problem constraints, including target safety level constraints and design constraints related to design variables and problem parameters [61]. As the number of variables increases to find optimal values, the boundary between the safe region and the failure region transforms into a hyperplane, increasing the complexity of the problem. The process begins with a randomly selected initial design point, progressively advancing to meet the specified constraints at each step. Alternatively, deterministic design points can be employed as starting points, representing intersections of limit state functions. A standard RBDO model can be formulated as follows:

where d is the deterministic design vector, the mean value \(\:{\mu\:}_{x}\) of a random vector, X is an uncertain design vector, \(\:f\left(d,\:{\mu\:}_{x}\right)\) is the objective function, \(\:{Q}_{i}\left(d,x\right)\le\:0\) is used to represent the failure, \(\:{P}_{i}^{T}\)is the target failure probability, \(\:{\beta\:}_{i}^{T}\) is the target index of safety, \(\:{C}_{j}\left(d\right)\le\:0\) is the deterministic constraint, Pr[.] is the probability operator, n is the number of uncertain constraints, and m is the number of deterministic constraints [12]. Figure 5 gives a schematic diagram of the RBDO problem. It involves nested optimisation layers, such as the reliability analysis loop within the outer optimisation loop. Solving the inner layer’s reliability analysis, which determines failure probability, involves complex high-dimensional integration due to the non-linear behaviour of the failure domain. However, direct integration is challenging due to the complicated probability density function and integration space.

The reliability index approach (RIA) and performance measurement approach (PMA) are the most common practical approaches for solving RBDO problems. For further insights into these methods, read the details provided in reference [12]. As shown above (see Eq. 9), one of the essential aspects of RBDO problems is defining the desired level of safety in terms of target failure probability or target safety index, which directly impacts the results obtained for design variables using this method. This issue has been addressed in the following section for reliability-based evacuation design in this study, which focuses on at-risk groups.

A case study illustrates the proposed method. The aim is to determine an optimal combination of values for the design variables associated with fire evacuation in residential incidents, specifically for considered at-risk groups. This determination is guided by a defined and tailored target level of safety against fire risk. It is essential to consider assumptions and constraints related to fire properties, building geometry, and occupant characteristics to achieve this aim.

As RBDO is the chosen approach, certain preparations are required. First, target safety level for the groups under in the study must be established in each scenario. Next, the objective function and constraints that must be satisfied during the RBDO process must be defined. The constraints can be in either equality or inequality forms. Applied statistical information related to the design variables of the fire evacuation model has been presented based on residential fire (see Table 4). Sensitivity analysis is necessary to define the objective function and to identify the most important or sensitive parameters influencing occupant safety. Once these variables and objective functions are identified, it will be possible to apply RBDO to determine the design variables’ values that fulfil the desired level of safety.

The single-story detached home of the target group is simulated as a rectangular compartment with independent exit. The compartment has a fixed height of 2.7 m and a total area of 208 square meters. The available space for smoke propagation, which depends on the fire’s location and the status (open or closed) of interior doors, was simulated as a compartment that has been divided into two different sections. This division splits the total available area of the home into a space where smoke can spread and another space that is free of smoke. It has been assumed that this simulation reflects the situation where the interior space for smoke propagation varies based on the status of interior doors (open or closed). The fire source is a sofa with a height of 0.5 m. The ambient temperature is 25˚C. The fire detection system is a smoke detector with an Response Time Index (RTI) of 25, sensitivity to a temperature rise of 15˚C per minute, detector spacing of 4.5 m, and a radial distance of 3 m from the fire plume axis. The process of attaining the desired results is outlined step by step following this introduction.

First step: Possible scenarios for fire in the building are developed. Figure 8 shows the possible scenarios. For calculating the probability of occurrence for each scenario, required information such as ignition time and probability of basic events can be extracted from statistical analysis and experimental research concerning the operational reliability of fire protection systems [75].

Table 2 shows the assumed probability for the operation reliability of the fire protection system considered in the case study for the mentioned events. Then, the probability of occurrence of each scenario can be calculated, assuming that all the mentioned events are independent from a statistical point of view.

Based on the obtained results from the event tree of Fig. 8 for the probability of occurrence of each scenario and determined \(\:ERL\) for occupants with different characteristics, the target failure probability of a safe evacuation \(\:\left({P}_{{f}_{i}}\right)\) can be calculated (refer to Sect. 4.1).

In the defined scenarios, the worst-case situations involve a fire occurring at nighttime when occupants are not awake (Scenario 3), and especially when the smoke and alarm systems fail to activate (Scenarios 4 and 5). In such cases, occupants would need to wake up and become alert due to the alarm sound (Scenario 3), or fire detection would rely on the occupant’s ability to independently recognize and perceive fire cues, such as flickering light, smell, and auditory cues like crackling and shuffling (Scenarios 4 and 5).

A study conducted on unimpaired adults with self-reported normal hearing and sleep patterns examined their response times to the mentioned fire cues during the night while sleeping [77]. The study indicated that among all participants, the lowest sensitivity and reaction to fire cues were from the light of fire and smoke odour, with only 48% ad 58% o occupants, respectively, waking up and responding to the fire. Reaction to auditory cues was better, with more than 80% o participants responded to fire cues during sleep. Also, it has been shown that around 50% of unimpaired occupants will independently respond to fire cues at nighttime [77].

Based on the presented data related to response times to the two mentioned fire cues, the response time in second to the light follows a lognormal distribution with the statistical properties (shape: 18.6, location: 30, and scale: (\(\:6.8\times\:{10}^{-6}\)). The response time to the smoke odour has a mean of 101 s and a standard deviation of 56 s, with a range between 45 and 205 s, suggesting that it also follows a lognormal distribution with the statistical properties (shape: 13, location: 50, and scale: 0.15) [77]. As the study was conducted on occupants without disabilities, its results cannot be directly generalized to individuals with cognitive disabilities. However, the proposed values for independent fire detection time might be applicable to occupants with mobility disabilities, as fire recognition and detection are primarily related to cognitive perception rather than mobility functionality. Given the unknown response times to fire cues for individuals with cognitive disabilities, scenarios 4 and 5 are excluded from the analysis for this group, as calculating their safe evacuation probability under such conditions is not feasible. Furthermore, Scenario 5 is also excluded for occupants without disabilities and those with mobility disabilities. This is because it assumes a non-activated fire alarm, and in a nighttime fire, a significant portion (e.g., 50%) of occupants are assumed unable to manually distinguish fire cues, rendering independent evacuation within the available safe egress time (ASET) infeasible, particularly when other fire protection systems like sprinklers are not considered.

According to Eq. 13, if a fire scenario is more likely to happen, the target failure probability for a safe evacuation must be lower. In other words, the more probable a fire scenario is, the stricter the safety requirements become. Table 3 shows the results for the likelihood of the fire scenarios for three different groups of occupants. As the obtained ERL for the occupants without two considered mobility (Group 1) in this study was more than the recommended value by the design code [66, 67], it has been set to the acceptable individual risk to the life by design code \(\:{(10}^{-6})\). For occupants with mobility disability (Group2) and cognitive disability (Group3), the obtained values for the observed ERL from the fire statistics have been set as individual risk to the life or ERL, which are \(\:6.15\times\:{10}^{-7}\) and \(\:1.28\times\:{10}^{-7}\), respectively.

Table 3 shows achieved \(\:{P}_{{f}_{i}}^{T}\) values for the considered scenarios and corresponding safety indexes to these target probabilities. The probability of being trapped in a fire for a person with these specific characteristics should not exceed \(\:{P}_{{f}_{i}}^{T}\) for the given scenarios (Fig. 9).

In other words, the safety level of these persons should not be lower than the corresponding value of the target safety index. Figure 10 illustrates the results obtained for the target safety index \(\:{\beta\:}_{T}\) corresponding to the target probability of safety for three different groups of building occupants.

Comparing the failure probabilities of safe evacuation and the corresponding safety indexes for different occupant types provides crucial insights into the effectiveness of evacuation measures in building fires. For example, in scenario 3, for occupants with cognitive disabilities (Group 3), \(\:{P}_{{f}_{i}}^{T}\) is set at a maximum failure probability of \(\:1.50\times\:{10}^{-4}\). This means that the actual failure probability of evacuation should not exceed this threshold to meet the desired safety level. Additionally, the safety index for this scenario is 3.61, indicating a high level of safety compared to other groups (Group 1:3.04, Group 2: 3.18). These findings suggest that the safety level to be considered in evacuation design for building occupants is highly dependent on their individual characteristics and varies among them.

Second Step: This step involves defining the corresponding limit state function (LSF), constraints, and objective function for the critical scenario. Subsequently, a sensitivity analysis is conducted on LSF variables to identify those with the highest impact on the probability of occupants being trapped in a fire. According to Fig. 8, Scenario 4 for groups 1 and 2, and Scenario 3 among all considered groups are the critical scenarios, as the fire occurred during the night and the fire alarm system had poor performance in alerting the occupants to the fire risk. Therefore, the general form of the corresponding Limit State Functions (LSFs) for Scenarios 3 and 4 among the three different groups should be as follows.

The only difference among these scenarios lies in the approach for defining appropriate values for the fire detection time \(\:DAT\), pre-movement time \(\:PT\), and rate of movement \(\:V\) during MOT phase. In all scenarios, it is assumed that the occupants are located at the farthest point in the building from the exit door to consider the worst case in terms of the movement phase. The fire detection time in Scenario 3 can be calculated based on the method presented in Sect. 3.3.1, while for Scenario 4 it can be predicted based on the suggested values for response times to fire cues by occupants during the night [77]. Regarding the pre-movement time \(\:PT\), it can be predicted based on the suggested values presented in Table 1. To perform sensitivity analysis effectively on design variables of limit state function, having statistical properties of the other variables is crucial. The selected statistical distributions for these variables should accurately reflect real-world conditions. In this study, representative statistical distributions have been obtained from the fire safety literature, as presented in Table 4.

A sensitivity analysis of LSF depicts which variables are most sensitive regarding the failure of safe evacuation in that scenario. It would be beneficial to define the fitness function for the RBDO phase. A recent study has used a fire evacuation model that considers various factors, including fire properties, building geometry, and people characteristics [31]. The most sensitive variables identified are the building’s area, the length of escape routes, and the premovement time. Increasing the length of escape routes and premovement time raises the probability of failure in a safe evacuation. Conversely, increasing the building’s area reduces the probability of an incomplete evacuation. This is because more space in a home allows smoke to spread out, providing more time to safely exit the building before smoke fills the area [31].

Figure 11 also shows a sensitivity analysis of available variables in considered LSF corresponding to the scenario 3 conditions for occupants with and without mobility or cognitive impairment.

Bild vergrößern

As seen in Fig. 11, the sensitivity analysis of LSF variables for building occupants with different characteristics shows that pre-movement time (\(\:PT\)) is the most influential variable on the probability of safe evacuation, playing a pivotal role and significantly impacting fire safety during the evacuation process compared to other variables of the time-based evacuation’s LSF. The area of the building, length of the escape route, fire growth rate, and manual fire cue detection by occupants are among the important variables that impact the failure probability of safe evacuation. The impact of other variables, such as, the width of escape doors, the height of the building’s ceiling, and speed of movement, are negligible compared to mentioned impactful variables. Regarding the fire growth rate and manual detection of fire cues by occupants (DAT (manual)), these are treated as random variables based on the proposed statistical properties and probability density functions (PDFs) in these studies [44, 78]. These factors depend on the burning items and the response time of occupants to fire cues during sleep at home and are not considered design variables.

These prerequisites steps enable the formulation of the Reliability-Based Design Optimization (RBDO) problem using available information, variables, and constraints. Initially, an objective function should be developed and defined to consider effective variables for a safe evacuation. The optimisation process must find solutions that minimise the length of the escape route and pre-movement time, maximise the area of the building, and at the same time satisfy the fire properties, building layout, and people characteristics constraints. The objective function and mentioned constraints can be defined as follows:

where \(\:f\left(d,\:{w}_{i}\right)\) is the objective function, \(\:{d}_{i}\) are the most sensitive design variables, and \(\:{w}_{i}\) is the weight coefficients of these variables obtained from the sensitivity analysis. Here, based on the sensitivity analysis of the critical LSF variables, the objective function can be formulated as follows:

where \(\:A\) is the available area for smoke spread, \(\:L\) is the length of the escape route and \(\:PT\) is the pre-movement time. Obtained values for \(\:{w}_{i}\) according to the sensitivity analysis of LSF for the concern groups are W_{Group 3} = (w₁, w₂, w₃ ) = (0.88,0.025, –0.05), W_{Group 2} = (w₁, w₂, w₃ ) = (0.98,0.0029, –0.004), and W_{Group 1} = (w₁, w₂, w₃ ) = (0.98,0.0027, –0.0027). The maximum resulted from experiments (see Table 5), maximum \(\:L\) is based on design code recommendation and the maximum \(\:A\) is the maximum area of a building where smoke can spread. The corresponding target failure probability of safe evacuation \(\:{P}_{{f}_{i}}^{T}\) or target safety index \(\:{\beta\:}_{T}\:\) for building’s occupants can be chosen from the obtained values in Table 3. As mentioned in the RBDO for the fire evacuation problem, solutions should satisfy various constraints. For instance, in the considered case study, given the building and concerned group of occupants, the following constraint have been defined based on the fire scenario. The first constraints that should be satisfied and has been developed based on the egress timeline model of building evacuation is the limit state function presented in Eq. 14. The first indication of smoke at any time or the smoke layer interface height, which is based on the unsteady and t-square fire behaviour, should satisfy the following constraints [40] :

For example, for Scenario 3, the RBDO of building fire evacuation can be formulated for occupants with cognitive disability as follows:

\(\:{C}_{4}:\:L<18\:\) in residential buildings for disabled people

\(\:{C}_{5}:\:\sqrt{2.26A}\le\:L\le\:18\:for\:Num.\:of\:Exit\ge\:2\).

Table 6 shows the obtained results for the considered design variables in RBDO of fire evacuation design problems for occupants with cognitive impairment, mobility impairment, and general occupants with typical body functionality. Additionally, the obtained values for PT, L, and A for Groups 1 and 2 in Scenario 4 are presented in this table, based on the considered constraints and the developed LSF, similar to Scenario 3.

As seen in Scenario 3, in a home with the same layout and fire scenario, individuals with mobility impairments (Group 2) have the shortest available pre-movement time for evacuation compared to those with cognitive impairments and able-bodied individuals, as they need to spend most of their evacuation on MOT phase compared to the other groups. Increasing the target safety index for building occupants with cognitive disability reduces the available time for the pre-movement time phase of an evacuation for this group compared to Group1 without disability. Figure 12 illustrates the relationship between the fire safety index and pre-movement time (PT) for three distinct groups of building occupants in Scenario 3.

The black points on each group’s curve indicate the target level of safety index and corresponding pre-movement time that must be satisfied to ensure safe evacuation conditions. In other word, the occupants of each group will be in margin of safety for fire evacuation if their spend time for this phase of evacuation being lower that defined Pre-movement time. This consideration is based on the fact that the maximum length of the escape route should not exceed the constraints for the length of the travel distance by an evacuee. Therefore, any possible design solution that maintains the same value for the available area for smoke spread while considering the farthest evacuee to the exit should account for a lower pre-movement time to satisfy the specified target level of safety. The length of the escape route is the distance an evacuee at the farthest point from the fire exit must cover in the worst-case scenario. If the distance between an evacuee and the nearest usable exit due to the fire exceeds 18 m, a new exit must be considered to ensure safe evacuation and compliance with the safety level defined by the fire safety index.

Regarding Scenario 4, the fire cues detection time and pre-movement time depend on the occupants’ ability to detect fire cues, perceive the associated risk, and respond appropriately. To ensure that evacuees have a sufficient safety margin, the total time spent on manual fire detection and pre-movement activities should not exceed 89 s for Group 1 and 82 s for Group 2 when smoke is able to spread throughout the entire building. If these thresholds are exceeded, additional fire protection systems become essential for life safety. The shorter available time for safe evacuation for Group 2, compared to Group 1, is due to their mobility impairments, which result in a longer movement phase (See Table 6).

When considering the available area for smoke propagation, obstacles such as walls and doors can hinder or slow down smoke spread. In such cases, the available area should be defined as the usable space for smoke propagation. Figure 13 illustrates the effect of available space for smoke propagation on premovement time. As shown in the figure, a decrease in the available area for smoke propagation leads to a reduction in the available manual fire detection and pre-movement time for evacuees. This adjustment ensures that the specified safety level is maintained. Figure 14 shows the schematic view of simulated conditions of the considered case study when the available space for smoke propagation changes based on open/closed doors, which is simulated by a divider wall.

To determine the reliability of the contributions, possible concerns regarding the utilised studies for prediction of pre-movement times, and the suggested RBDO method for improving the fire safety level are discussed. The dataset presented in Table 1 originates from actual fire incidents and fire drills in residential buildings. Premovement times of evacuees with movement disabilities might be overestimated based on the datasets, as the additional time required to retrieve and use assistive tools, such as wheelchairs and canes, is regarded as premovement activities. Additionally, all these datasets were represented recently in a peer-reviewed paper [55].

The suggested statistical properties for pre-movement time and criteria for life safety and individual risk to life, are based on the available datasets for pre-movement time and reports about the number of fatalities. The proposed values of this study should be updated when new datasets or reports become available. Additionally, obtained values for the safety level of building occupants with reduced functionality (mobility and cognitive) are more conservative than the current used target safety level. It should be noted that the calculated safety level is based on U.S. fire fatality data among people with disability. Other countries should implement their own safety levels based on their own fire fatality statistical data. The value obtained from the RBDO method for pre-movement time for occupants with mobility disability is significantly lower than the average observed pre-movement times for mobility impaired during fire drills (See Table 1). This imposes a limitation for design engineers when calculating the RSET of people with mobility disability in fire safety design. It is important to note that the obtained values for pre-movement time for people with cognitive disabilities are specific to particular types of cognitive disability, namely learning difficulties and intellectual impairment (with mild or moderate levels) and may not apply to other types of cognitive impairments. Furthermore, the amount of data collected for this group is relatively small. Currently, only data from two unannounced fire drills in residential buildings with occupants with learning and intellectual impairments are available, and where approximately 60% of residents responded to the fire alarm.

The proposed flow chart in Fig. 7 calculates the target level of safety, and the optimum values for the most sensitive or important design variables in the evacuation process is currently sufficient for attaining this study’s purposes. However, obtaining more classified data regarding fire fatalities among different types of occupants with various reduced functionalities and those with multiple impairments could potentially improve the accuracy and generality of the proposed approach for the target safety level. Nevertheless, access to this classified data on disability is not enough. It also needs to consider the characteristics of these new groups of occupants with disability in the model, such as their rate of movement and the pre-movement time required based on their impairments during the evacuation.

The specific case presented in Sect. 5 is based on statistical data obtained for a building fire evacuation model in a single-story building, a common building type with a smoke detector and alarm system but without a sprinkler system. However, it is important to note that in other cases such as buildings with multiple floors and fire extinguisher systems additional factors must be considered, including the vertical movement of evacuees and the impact of the extinguisher system on fire and smoke propagation. Therefore, the extent to which this particular case is typical or applicable to other cases depends on the specific characteristics and features of those cases, particularly in terms of building design and evacuation dynamics. The major hindrance to the broader use of the proposed approach is the lack of data regarding fire fatalities for different types of building occupants based on their characteristics. The availability of comprehensive and reliable data is crucial for accurately calibrating and validating the approach and assessing its effectiveness in various scenarios. Without this data, it can be challenging to confidently apply it to different building types and occupant profiles. This study is a step towards illustrating the need for better-granulated data to consider individual safety already in the design phase of buildings.

This study underscores the necessity of addressing fire-related threats for building occupants, particularly at-risk groups with mobility and cognitive impairments. Employing a probabilistic risk analysis approach, this study determined that higher target safety levels were required for mobility, and cognitively disabled occupants than those with typical physical functionality. This study emphasises the need for customised safety solutions that consider individual characteristics, ensuring accuracy in safety assessments. This study suggests that by integrating reliability-based design optimisation methods into fire evacuation design models is an effective approach for enhancing evacuation reliability. The results obtained using this method can be utilised by design engineers and policy decision-makers to implement essential actions that increase building safety for occupants.This may include equipping buildings with new safety facilities or planning better-organized emergency evacuation training courses. Individuals with impairments often rely on care staff during hazardous situations, making their role critical in the pre-movement phase of evacuation. Additionally, factors such as time of fire incident, the performance of the alarm system, the cause of evacuation (whether a real fire or a fire drill), and occupant characteristics such as gender can potentially influence pre-movement time during fire evacuation. Future research will evaluate the impact of these factors on pre-movement time using data from drills or actual fire incidents as it becomes available. This will contribute to more accurate predictions of pre-movement time in timeline-based fire egress models.