A Simulation Analysis of the Coverage and Demand Suitability of the Firefighting Capacity in Complex Commercial Areas

Abstract

The initial firefighting capacity of complex commercial areas depends on the service level and the location of micro firefighting facilities. In response to the issue of coverage by micro firefighting facilities in complex commercial areas, a graded progressive coverage model is established. This model includes distance-progressive coverage and firefighting service level attenuation coverage. The former integrates fairness and efficiency in fire planning, while the latter considers the adaptability between the demand points and the fire service levels. The objectives include maximizing the coverage matching degree, effective coverage rate, medium- and high-risk coverage rate, and overall coverage rate of the fire service point and the demand points. A genetic algorithm is designed to solve the model, where the construction level and the number of micro fire stations are varied to analyze changes in various indicators. Central Street, characterized by complex buildings and high pedestrian traffic, is selected as a case for the experiment. The results show that simply adjusting the construction level of micro fire stations increases the effective coverage rate by 2.5%. The graded progressive coverage model shows a turning point in the effective coverage rate and the overall coverage rate when the number of new micro fire stations is 14 or 10, with coverage rates of 76.1% and 93.2%, respectively. The maximum progressive coverage model shows a turning point in the overall coverage rate when 9 new stations are added, which is 10.5% lower than that of the graded coverage model; when 10 stations are added, the overall coverage rate is 11.2% lower. These findings demonstrate the effectiveness of the graded progressive coverage model.

1. Introduction

With the development of urban economies, the construction scale of commercial areas (such as high-rise buildings, large shopping malls, and heritage buildings) has grown significantly, integrating production, commerce, transportation, and entertainment. These buildings are complex in structure, prone to collapse, have a high pedestrian density, and experience high electrical loads. In the event of a fire, these characteristics pose significant challenges for firefighting and rescue operations [1]. Heritage buildings possess immeasurable cultural value and irreplaceability. Due to their age and varying construction quality, they have a low ability to withstand fire risks. In recent years, frequent fire incidents have occurred at landmarks such as Notre-Dame Cathedral, the National Museum of Brazil, and the ancient city of Shangri-La [2]. High-rise building fires are accompanied by the chimney effect and vertical fire spread, commercial buildings have numerous electrical devices, and underground spaces are enclosed: all of these factors together contribute to an increased fire risks [3,4,5]. In the first half of 2024, there were 36,000 fires in high-rise buildings in China, and over the past decade, reports from foreign countries show that fires in high-rise and commercial buildings accounted for 11% of total fires [6]. For example, the collapse of the Plasco Building was caused by its complex structure, which delayed the arrival of firefighting forces and hindered early fire suppression, allowing the fire to quickly spread to store rooms and other areas [2]. Micro fire stations can provide initial firefighting and rescue services in complex commercial areas, and in special circumstances (especially when transportation and communication are interrupted), they can respond more quickly and organize evacuations faster than professional rescue teams [7]. Fire safety regulations require that businesses with a building area larger than 5000 m² must be equipped with micro fire stations. Modern commercial buildings are often operated by multiple merchants, and when each merchant’s area is less than 5000 m², but the total building area exceeds 5000 m², this can result in the omission of micro fire stations, reducing organizational and mobilization capabilities. For businesses with an area smaller than 5000 m² but a high fire risk, the lack of micro fire stations leads to insufficient coverage, which presents a significant flaw. Simulating the spread of fire in high-risk areas, it was found that adding micro fire stations at key locations reduced the time for controlling the fire spread to one-third of the original duration [8]. So, how should micro fire stations be added to cover a larger area?

The emergency facility coverage problem is typically addressed using the following four models: the P-center coverage model (P-CP), the P-median model (P-MP), the set covering model, and the maximum coverage model (MCLP). P-CP aims to minimize the total distance from points locations to demand points [9]. The objective of P-MP is to minimize the weighted total service distance within the coverage area, focusing on the efficiency of service provided from facilities to demand points [10]. P-MP typically allocates more firefighting resources to demand points with higher weights. References [11,12,13] describe the application of P-CP and P-MP in emergency facility location planning. The set covering model requires coverage of all demand points within the area, which is difficult to achieve in practice [14]. MCLP typically maximizes the area of the service region to ensure fairness in facility service [15]. References [16,17] describe the application of MCLP in fire station location planning. P-CP and P-MP focus on the distance between demand points and the nearest facility points, while the set covering model and MCLP focus on the number of demand points that are covered at least once by facility points. MCLP is modeled using a fixed radius method: a demand point is covered if it lies within the radius of a facility, and not covered otherwise. Karasakal et al. proposed the maximum progressive coverage model (MCLP-P), which improves upon the “covered-not covered” approach of MCLP [18]. In MCLP-P, the coverage degree between demand points and facilities decreases as the distance increases. The model adopts a maximum progressive coverage approach, where the service provided by a facility to a demand point is zero if the distance exceeds a certain maximum distance, while the service matching degree decreases with increasing distance. MCLP-P only considers fairness in the provision of services by facilities and does not consider the weight of demand points, thus avoiding the tilting of firefighting services. The modeling approach that integrates fairness and efficiency is usually achieved through multi-objective programming [19]. Multi-objective programming generates multiple Pareto solutions, which can hinder fire departments from making quick decisions [13].

Albertus Untadi proposed a dual objective site selection optimization method for fire stations based on the prediction of building socio-economic factors, maximizing regional coverage, and minimizing uncovered area distance [20]. Penjani Hopkins Nyimbili uses GIS-based fuzzy multi criteria method to evaluate the important factors affecting the location of fire stations and models them to study the location of urban fire stations [21]. Jing Yao conducted research on fire station site selection from two aspects: access points and service coverage, in order to optimize the fire rescue response time [19]. Ming Jinke proposed a collaborative coverage model for fire services based on MCLP-P, which needs to consider the scale of fire stations [22]. The fire risk of demand points is the main parameter in model design. Chen Jinyue combines static and dynamic indicators and uses Bayesian networks for historical building fire risk assessments [23]. Yu Wenhao studied the accessibility of fire stations and buildings at a fine scale, laying the foundation for the site selection and configuration of micro fire stations [24]. The above research provides new ideas for the location and layout of macro-level fire stations in cities. Complex commercial areas have diverse buildings and need to strengthen grassroots resilience. Therefore, from a micro perspective, the coverage capacity of fire protection facilities to demand points should be considered. The construction of micro fire stations has different capacity levels. Traditional coverage issues set fire service points to a fixed scale. However, when covering different risk points, the performance measurement standards for micro fire station coverage need to consider the adaptability between the construction level and the demand weight. In Lee’s generalized hierarchical coverage problem [25], it is required that the facilities providing services to demand points be facilities of equivalent or higher levels.

This article explores progressive coverage and graded attenuation coverage. Building a progressive coverage model, based on MCLP-P, divides the demand points with different risk weights into different ideal radii and maximum radii, which can integrate fairness and efficiency in fire planning into a single optimization objective. We further assume that the demand points should be covered by facilities of an equal or higher level and, when covered by facilities of a lower level, the coverage matching degree will decrease. We design a genetic algorithm that uses multivariate integer encoding for the location and the level of fire service points. Based on the data of fire facilities and fire risk demand points on Central Street in Harbin, we conduct a study on the coverage of fire station services in complex areas.

2. Materials and Methods

2.1. Suitability-Based Fire Service Coverage Model

This article proposes a hierarchical progressive coverage model, and the conceptual diagram is shown in Figure 1.

2.1.1. Variables and Parameters

i: the requirement point.

j: a demand point with firefighting capabilities, abbreviated as a firefighting service point.

I: a collection of demand points.

J: a collection of demand points with firefighting capabilities.

M: a set of demand points that do not have firefighting capabilities.

Number: the number of demand points for improving firefighting capabilities.

d_ij: the distance from i to j.

R_i: the ideal coverage radius for demand point i.

D: the maximum coverage radius.

ρ_r: the equivalent deterministic radius of range r.

B: the Fermi sensitivity parameter.

${\mathrm{S}}_{\mathrm{i}\mathrm{j}}^{\mathrm{b}}$: the coverage level of j for i is b.

P: the number of demand points for enhancing fire protection capabilities.

Q: the number of demand points with firefighting capabilities.

FM: the maximum asymptotic coverage total matching degree.

FG: the gradual coverage of graded total matching degree.

PM1: the effective coverage of the maximum progressive coverage model.

PM2: the overall coverage rate of the maximum progressive coverage model.

PM3: the coverage rate of medium- and high-risk points with maximum progressive coverage model.

PG1: the effective coverage rate of graded progressive coverage model.

PG2: the overall coverage rate of the graded progressive coverage model.

PG3: the coverage rate of medium- and high-risk points with graded progressive coverage model.

2.1.2. Problem Description

The model assumes that a complex area has a set of fire protection demand points, I, a set of demand points, M, without fire protection capabilities, and a set of demand points, J, with fire protection capabilities. Fire safety demand points refer to units that do not have their own fire safety capabilities but have fire safety needs; set M represents all the demand points that can enhance the fire protection capabilities, except for key fire prevention units; key fire prevention units have all improved their firefighting capabilities. Fire service points can provide fire services to demand points themselves and the surrounding demand points. The risk levels of the demand points are divided into high-risk, medium-risk, and low-risk. The fire protection capabilities of the fire service points are divided into Level 1, Level 2, and Level 3. When providing fire services, distance factors and adaptability requirements need to be considered comprehensively. The fire service point is located within the ideal radius of the demand point and can receive all services from the fire service point. The fire service point is located outside the ideal radius and within the maximum radius, and the demand point can obtain partial services from the fire service point. The adaptability requirement between fire service points and demand points refers to incomplete coverage, where low-fire-capacity service points cannot meet all the fire services required by high-risk points. The coverage levels of different fire protection level service points covering different risk-demand points are shown in Figure 2. Level A coverage is complete coverage, while levels B and C are incomplete coverage, with level C coverage being less than level B. Therefore, the decision-making problem is proposed as follows: how to improve the fire service point’s capability level or enhance the fire protection capability of the demand points without any fire protection capability under the premise of given resources in order to maximize the coverage rate within the research area. Based on the characteristics of the problem, the following assumptions are made: (1) the number and locations of key fire prevention units are known; (2) the higher the fire risk of a demand point, the greater its demand weight.

2.1.3. Grid-Based Path Planning

In complex areas, there are many obstacles (such as walls and fences) between the fire demand point and the fire service point. When firefighters are deployed, they need to avoid these obstacles and find the fastest path to the fire site. Based on the real-time situation of the fire, they must also predict future fire trends in a short period of time; consider changes in the accident location, hazardous areas, and obstacle routes; and plan the path accordingly [26]. The model constructs objects in the complex environment that may affect path planning, where objects that are far away or invisible in the environment are ignored, and only the filtered objects are considered. Scene objects are represented by a series of polyline segments, as shown in Figure 3. The construction steps are as follows:

1. Create a spatial grid with the size of the scene, with a grid resolution of m.

2. Divide the polyline into multiple segments. When both endpoints of the starting segment lie on the grid boundary, the grid records the ID; otherwise, it does not record it. The same method is applied to the ending segment.

In this paper, walls and fences are represented by a series of polyline segments in the scene, which can either be closed or open. The shortest path between the fire demand point and the fire service point is used as the distance for further research. The shortest path calculation must avoid grids in the spatial grid that have been marked with an ID.

2.1.4. Multi-Level Progressive Coverage Model

The traditional maximum coverage model models fire service points using a fixed radius approach. There are only two situations for the service level of the fire service points to the demand points: coverage or non coverage, which is called 0–1 type coverage. That is, if the demand point is located within the fixed radius range of the fire service point, it is called covered and the coverage matching degree is 1. Otherwise, the demand point is not covered and the coverage matching degree is 0. The maximum progressive coverage model proposes that when the demand point is located outside the fixed radius range of the fire service point, the coverage matching degree is 0; when the demand point is located within a fixed radius of the fire service point, the matching degree decreases as the distance between the demand point and the fire service point gradually increases. This approach only considers the fairness of the fire service. Fair distribution refers to the evaluation of the degree of return received by participants [27]. This article assigns different ideal radii and the same maximum radius to demand points with different weights and studies the fairness and the efficiency of comprehensive fire protection services. The objective function is to find the maximum value of the coverage matching degree of all the demand points. On the one hand, setting the maximum coverage radius, D, can serve more demand points. Each demand point is treated equally within the maximum coverage radius to ensure fairness in service. On the other hand, we could set the ideal coverage radius. The higher the risk of the demand point, the smaller the ideal coverage radius. This enables the demand points with higher weights to receive more firefighting resources, ensuring service efficiency. In summary, the matching degree of the demand points within the ideal coverage radius is 1. Outside the ideal coverage radius and within the maximum radius, the matching degree of demand point coverage decreases with increasing distance.yj

• Assumptions and Preparations

The model defines two coverage radii, Ri (ideal coverage radius) and D (maximum coverage radius), for each demand point. There are the following situations: (1) the fire service point is located within the ideal coverage radius R, and the demand point receives all services from the fire service point, with a coverage matching degree of 1; (2) the fire service point is located outside the maximum coverage radius D, and the demand point cannot receive services from the fire service point. The coverage matching degree is 0; (3) the fire service point is located between the ideal coverage radius R and the maximum coverage radius D, and the demand point receives partial services from the fire demand point. The coverage matching degree is given by f_ij(d_ij). In this study, the Fermi function is used to formulate the multi-level progressive coverage model, where the coverage level of fire service point j for demand point i is expressed as

$${\mathrm{f}}_{\mathrm{ij}}\left({\mathrm{d}}_{\mathrm{ij}}\right)=\left\{\begin{array}{c}1, {\mathrm{d}}_{\mathrm{ij}}\le {\mathrm{R}}_{\mathrm{i}}\\ {\displaystyle \frac{1}{1+{10}^{\left[\right({\mathrm{d}}_{\mathrm{ij}}-{\mathrm{R}}_{\mathrm{i}})/(\mathsf{\rho }-{\mathrm{R}}_{\mathrm{i}})-1]/\mathrm{b}}}}, {\mathrm{R}}_{\mathrm{i}}<{\mathrm{d}}_{\mathrm{ij}}\le \mathrm{D}\\ 0, \mathrm{otherwise}.\end{array}\right.$$

(1)

where ρ (ρ > R_i) represents the deterministic coverage radius at which the coverage probability is equal to 0.5, and b is a “sensitivity” parameter. Figure 4 illustrates the Fermi function for different values of b, with R_i = 0.2 km, D = 0.8 km, and ρ = 0.5. When b = 0, the Fermi function model approximates a deterministic coverage model with a radius of ρ. The variable d_ij represents the actual shortest path distance from demand point i to facility point j, and R_i is the ideal coverage radius for demand point I, based on its fire risk level. Within the ideal coverage radius, the demand point receives all the fire services provided by the fire service point, and a value of 1 indicates that the coverage matching degree between the demand point and the fire service point is 100%. The degree of matching is determined by both the distance from the demand point to the facility point and the demand point’s weight.

2.

Establishment of the Multi-level Progressive Coverage Model

In this study, a multi-level progressive coverage model is employed, which considers the variable ideal coverage radius for demand points. Figure 5 illustrates the hierarchical diagram of the ideal coverage radius for demand points. Each demand point, i is assigned an ideal coverage radius, R_j, and a maximum coverage radius, D, based on its risk level. Building on the Fermi function model from Section 2.2.1, the decision-maker selects an equivalent deterministic radius, ρ_r (ρ_r = (R_i + D)/2), for each demand point to apply the progressive coverage principle. In Equation (1), ρ is replaced by ρ_r to obtain f_ij(d_ij), thus relaxing the Fermi function model to accommodate varying coverage conditions. Figure 5 is a schematic diagram of the ideal coverage radius hierarchy for the demand points. Demand points are classified into their ideal radius, R_j, and the maximum coverage radius, D, based on their own risk level. Assuming the demand point is a high-risk point and fire services cannot be obtained within the ideal coverage radius, R₁, partial services of j₁ and j₂ can be obtained. Assuming the demand point is a medium-risk point and fire services cannot be obtained within the ideal coverage radius, R₂, all the services of j₁ and partial services of j₂ can be obtained.

The time from alarm reception to deployment for firefighters is 1 min. The rescue speed of firefighters in densely populated complex areas is 1.5 m/s. Firefighting regulations require that the response time for micro fire stations be within 3 min. The ideal coverage radii for high-risk, medium-risk, and low-risk demand points are set at 0.2 km, 0.3 km, and 0.4 km, respectively. These parameter values are set both to verify the effectiveness of the model and because no unified reference standard has been proposed for the construction of urban fire service points. Figure 6 shows the coverage matching rate variation curve for different ideal coverage radii when D = 0.8 km and b = 0.5. As shown in the figure, the smaller the ideal coverage radius, the shorter the distance required to achieve the same matching (d₁ < d₂ < d₃), indicating the prioritization of rescue efforts.

2.1.5. Fire Service Hierarchical Coverage Model

The hierarchical coverage model for fire service point to demand points introduces a loss function based on graded coverage. The selection of the loss function fully considers the fire evolution process. As shown in Figure 7a, a fire progresses through four distinct phases: incubation, outbreak, peak, and decay. During these phases, it evolves from initial ignition, through flashover and intense combustion, until it finally extinguishes. Based on Figure 7a, the fire loss curve is shown in Figure 7b. At first, the loss rate increases slowly. After the flashover occurs, it escalates exponentially and reaches its maximum at the peak temperature. As the fire subsides, the total losses continue to accumulate, but the rate of increase drops sharply. The primary role of the micro fire stations is to respond to fires in the early stages. The main function of a micro fire station is to respond to early-stage fires, and only by reaching the demand point before the onset of a flashover can they significantly delay the spread of the fire, thereby minimizing the potential damage.

Based on the analysis of the four stages of fire loss, the fire loss curve is fitted using a Logistic function. By examining variations in key parameters, the critical factors influencing fire loss are explored, and a practical micro fire station location model is developed. To investigate the variation pattern of the fire loss rate, the higher-order derivatives of Equation (2) are computed.

$$\mathrm{y}={\displaystyle \frac{\mathrm{m}}{1+\mathrm{aexp}\left(\text{-}\mathrm{bt}\right)}}$$

(2)

$${\displaystyle \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{d}}^2\mathrm{t}}}={\displaystyle \frac{\mathrm{mabexp}\left(\text{-}\mathrm{bt}\right)\left(\mathrm{abexp}\right(\text{-}\mathrm{bt}\left)\text{-}\mathrm{b}\right)}{{(1+\mathrm{aexp}(\text{-}\mathrm{bt}\left)\right)}^3}}$$

(3)

$${\displaystyle \frac{{\mathrm{d}}^3\mathrm{y}}{{\mathrm{d}}^3\mathrm{t}}}={\displaystyle \frac{{6\mathrm{ma}}^3{\mathrm{b}}^3\exp \left(\text{-}3\mathrm{bt}\right)}{{\left(\mathrm{aexp}\right(\text{-}\mathrm{bt})+1)}^4}}\text{-}{\displaystyle \frac{{6\mathrm{ma}}^2{\mathrm{b}}^3\exp \left(\text{-}2\mathrm{bt}\right)}{{\left(\mathrm{aexp}\right(\text{-}\mathrm{bt})+1)}^3}}+{\displaystyle \frac{{\mathrm{mab}}^3\exp \left(\text{-}\mathrm{bt}\right)}{{\left(\mathrm{aexp}\right(\text{-}\mathrm{bt})+1)}^2}}$$

(4)

When the second derivative is equal to 0, the value of t at this point is

$${\mathrm{t}}_0={\displaystyle \frac{{\ln }^{\mathrm{a}}}{\mathrm{b}}}$$

(5)

When the third derivative is equal to 0, the value of t at this point is

$${\mathrm{t}}_1={\displaystyle \frac{{\ln }^{\mathrm{a}}-1.317}{\mathrm{b}}}$$

(6)

$${\mathrm{t}}_2={\displaystyle \frac{{\ln }^{\mathrm{a}}+1.317}{\mathrm{b}}}$$

(7)

When the time is 0, the fire loss is 0. The Logistic function at t = 0 is y = m/(1 + a). The further fitting of Equation (2) gives the resulting loss function as

$$\mathrm{y}\left(\mathrm{t}\right)={\displaystyle \frac{\mathrm{m}}{1+\mathrm{aexp}(-\mathrm{bt})}}-{\displaystyle \frac{\mathrm{m}}{1+\mathrm{a}}}$$

(8)

From Equations (5) to (7), it is evident that m does not affect the fire loss rate. The further examination of parameters a and b (Figure 8) reveals that when b is held constant, changes in a produce minimal effects on the fire rate, whereas fixing a makes variations in b substantially more impactful. Thus, b becomes the key factor determining fire loss dynamics. As b increases, the fire loss rate increases, resulting in greater loss within the same time frame. In the case of graded fire service coverage, the higher the coverage level, the more reduced the fire loss rate, leading to less fire loss within the same time frame. Therefore, the variation in b reflects the influence of graded coverage on fire loss: b = 1 denotes Grade A coverage, b = 2 denotes Grade B coverage, and b = 3 denotes Grade C coverage. $S_{ij}^b$ indicates that the coverage level of facility point j to demand point i is b. The speed of the firefighter is constant at v, and the shortest path distance from facility point j to demand point i is d_ij. Based on the above analysis, the loss function based on the graded coverage of fire facilities is formulated as follows:

$$\mathrm{y}({\mathrm{d}}_{\mathrm{ij}},{\mathrm{S}}_{\mathrm{ij}}^{\mathrm{b}})={\displaystyle \frac{\mathrm{m}}{1+\mathrm{aexp}(-{\mathrm{S}}_{\mathrm{ij}}^{\mathrm{b}}\frac{{\mathrm{d}}_{\mathrm{ij}}}{\mathrm{v}})}}-{\displaystyle \frac{\mathrm{m}}{1+\mathrm{a}}}$$

(9)

2.1.6. The Construction of the Objective Function Model

The Fermi function is applied to implement the progressive coverage between the fire service point and the demand points, based on distance. On this foundation, the Logistic loss function is employed to achieve the graded coverage of fire services, integrating both distance-related factors and adaptability criteria into a unified, comprehensive coverage model. The coverage matching formula is provided in Equation (10).

$${\mathrm{F}}_{\mathrm{ij}}\left({\mathrm{d}}_{\mathrm{ij}}, {\mathrm{S}}_{\mathrm{ij}}^{\mathrm{b}}\right)={\overline{\mathrm{f}}}_{\mathrm{ij}}\left({\mathrm{d}}_{\mathrm{ij}}\right)\ast \exp \left[\mathrm{y}\left(d_{ij}, 1\right)-\mathrm{y}\left(d_{ij}, {\mathrm{S}}_{\mathrm{ij}}^{\mathrm{b}}\right)\right]$$

(10)

The final objective function is constructed as follows:

$$\mathrm{Z}=\max {\displaystyle \sum _{\mathrm{i}\in \mathrm{I}}}{\displaystyle \sum _{\mathrm{j}\in \mathrm{J}}}{\mathrm{F}}_{\mathrm{ij}}({\mathrm{d}}_{\mathrm{ij}}, {\mathrm{S}}_{\mathrm{ij}}^{\mathrm{b}}){\mathrm{z}}_{\mathrm{i}}$$

(11)

Constraints:

$${\displaystyle \sum _{\mathrm{j}\in \mathrm{M}}}{\mathrm{y}}_{\mathrm{j}}=\mathrm{P}$$

(12)

$${\displaystyle \sum _{\mathrm{j}\in \mathrm{J}}}{\mathrm{t}}_{\mathrm{j}}=\mathrm{Q}$$

(13)

$${\displaystyle {\sum }_{\mathrm{j}=1}^{\mathrm{n}}}{\mathrm{x}}_{\mathrm{ij}}\le 1, \forall \mathrm{i}\in \mathrm{I}$$

(14)

$${\mathrm{x}}_{\mathrm{ij}}\le {\mathrm{y}}_{\mathrm{j}}, \forall \mathrm{i}\in \mathrm{I} \forall \mathrm{i}\in \mathrm{I}$$

(15)

$${\mathrm{s}1}_{\mathrm{j}}\le {\mathrm{s}2}_{\mathrm{j}}, \forall \mathrm{j}\in \mathrm{J} \forall \mathrm{s}1,\mathrm{s}2\in \{1,2,3\}$$

(16)

$${\mathrm{x}}_{\mathrm{ij}}=\left\{\begin{array}{c}1, {\mathrm{d}}_{\mathrm{ij}}\le {\mathrm{R}}_{\mathrm{i}}\\ 0, \mathrm{otherwise}.\end{array}\right.$$

(17)

$${\mathrm{y}}_{\mathrm{j}}=\left\{\begin{array}{c}1, \mathrm{if} \mathrm{point} \mathrm{j} \mathrm{has} \mathrm{firefighting} \mathrm{capability}.\\ 0, \mathrm{otherwise}.\end{array}\right.$$

(18)

$${\mathrm{z}}_{\mathrm{i}}=\left\{\begin{array}{c}1, \mathrm{if} \mathrm{facility} \mathrm{j} \mathrm{is} \mathrm{sited} \mathrm{at} \mathrm{j} \mathrm{and} {\mathrm{d}}_{\mathrm{ij}}\le \mathrm{D}.\\ 0, \mathrm{otherwise}.\end{array}\right.$$

(19)

Among them, the objective function (11) is to maximize the sum of the coverage of the matching degrees of the demand points. Constraint (12) represents the proposed improvement of fire protection capabilities for P demand points. Constraint (13) represents selecting Q fire service points to change the fire protection capability level. Constraint (14) indicates that within the ideal coverage radius, a maximum of one fire service point is allowed to exist that can meet all the fire services of the demand point. Constraint (15) indicates that only fire service points can provide services to demand points. Constraint (16) indicates that the level of the fire service point can only be increased and cannot be decreased. s1j and s2j, respectively, represent the level of the fire service point before and after changing its fire protection capability. Constraints (17)–(19) represent the decision variables x_ij, yj, and z_i as 0 and 1, respectively. When x_ij is 1, the j that meets the required fire protection services for i is located within the ideal coverage radius of i, otherwise it is not; y_j indicates that point j has some fire protection capability, otherwise it does not. When z_i is 1, i within the maximum coverage radius of i has some firefighting capability.

2.2. Genetic Algorithm for Model Solution

Since the progressive graded coverage model based on matching degrees constructed in this study is an extension of the maximum coverage model and belongs to the NP-Hard problem, it is difficult to use exact algorithms to solve such problems, especially for large-scale solving cases. Therefore, heuristic algorithms are used to calculate approximate results. In this study, a genetic algorithm is used to solve the model, primarily because of its inherent parallel computation capability and the universality of its encoding feasible solutions.

2.2.1. Encoding

The model includes two factors, the location and the level of fire service points, and adopts a multivariate integer encoding method. During the encoding process of the genetic individuals, the fire service points, (built_n) 1~k (the location of the demand points with firefighting capabilities); the locations of the demand points that need to improve their firefighting capabilities, (candidate_n) 1~m; the construction level of the fire service points, (b_grade_n) 1~3; and the construction level of the demand points that need to improve their fire services, (c_grade_n) 1~3, are assigned values. Each encoded gene represents a specific solution. We set the chromosome length as the sum of the number of fire service points, Q, and the proposed increase in fire service points, P, assuming k = 100, m = 100, P = 10, and Q = 5. As shown in Figure 9, it is a chromosome generated by encoding.

$$\mathrm{X}=[\left({\mathrm{built}}_1,{{\mathrm{b}}_{\mathrm{grade}}}_1\right),\dots ,\left({\mathrm{built}}_{\mathrm{k}},{{\mathrm{b}}_{\mathrm{grade}}}_{\mathrm{k}}\right),\left({\mathrm{candidate}}_1,{{\mathrm{c}}_{\mathrm{grade}}}_1\right),\dots ,\left({\mathrm{candidate}}_{\mathrm{m}},{{\mathrm{c}}_{\mathrm{grade}}}_{\mathrm{m}}\right)]$$

(20)

where X represents a chromosome, indicating a set of feasible solutions.

2.2.2. Initial Population Generation

The initial population is generated randomly, with the population size defined as M = [2(k/Q) + 2(m/P)]

2.2.3. Fitness Calculation

The fitness value of each chromosome is computed using the objective function (11). The specific process is as follows:

• Obtain the demand weight of demand point i, determine the risk level, and partition the ideal coverage radius.
• Calculate the shortest path from demand point i to each builtn and candn gene, and compute the fire service graded coverage loss for each b_graden and c_graden gene. The coverage matching degree of demand point i is then derived using Equation (10).
• Repeat steps (a) and (b) until all the demand points have been computed. The total coverage matching degree of all the demand points for this chromosome is then summed to obtain the fitness value of the chromosome.

2.2.4. Selection

A roulette wheel selection method is employed to choose the parent generation. The probability of selecting each individual is proportional to its fitness value, and the calculation formula is as follows:

$${\mathrm{P}}_{\mathrm{k}}={\displaystyle \frac{\mathrm{F}\left({\mathrm{X}}_{\mathrm{k}}\right)}{{\displaystyle {\sum }_{\mathrm{k}=1}^{\mathrm{M}}}\mathrm{F}({\mathrm{X}}_{\mathrm{k}})}}$$

(21)

P_k is the probability of each chromosome being selected, and F(X_k) is the fitness value of each chromosome.

2.2.5. Crossover

A crossover operator θ_c is defined, and a random number ε within the real interval (0, 1) is selected. If ε exceeds θ_c, a crossover occurs. This process is repeated until M offspring are generated. The crossover operation adopts a two-point overall cross operation, using a random generator to generate [(k + m)/2] positions. The genes at the corresponding positions are then exchanged between the two selected parent chromosomes, producing two offspring individuals. The crossover must ensure that the fire service point position genes exchanged are not repeated.

$${\mathsf{\theta }}_{\mathrm{c}}=\left\{\begin{array}{cc}0.8,\hfill & \mathrm{iter}\le \mathrm{maxiter}/4\\ \hspace{1em}0.5,& \mathrm{maxiter}/2<\mathrm{iter}<3\mathrm{maxiter}/4\\ 0.2,\hfill & 3 \mathrm{maxiter}/4\le \mathrm{iter}\end{array}\right.$$

(22)

2.2.6. Mutation

A mutation operator, θ_v, is set, and a random number, ω, within the real interval (0, 1) is selected. If ω exceeds θ_v, a mutation occurs. This process is repeated M times. The mutation operation is divided into three types: fire service point level mutation (b_grade_i), the location of demand points requiring enhanced fire service undergoes mutation (candidatei), and the fire service level of demand points requiring improvement undergoes mutation (c_grade_i). For b_grade_i, a random selection of a level higher than itself replaces b_grade_i; for candidate_i, a shift mutation is applied by randomly selecting a child’s candidate facility position level to replace candidate_i; for c_grade_i, the same mutation method as for candidate_i is applied.

$${\mathsf{\theta }}_{\mathrm{v}}=\left\{\begin{array}{cc}0.2,\hfill & \mathrm{iter}\le \mathrm{maxiter}/4\\ \hspace{1em}0.5,& \mathrm{maxiter}/2<\mathrm{iter}<3 \mathrm{maxiter}/4\\ 0.8,\hfill & 3 \mathrm{maxiter}/4\le \mathrm{iter}\end{array}\right.$$

(23)

where iter is the current iteration count and maxiter is the maximum iteration count, the crossover and mutation operations are both functions related to the algorithm’s iteration count. Early in the iterations, a high crossover probability can increase the diversity of the population, and a low mutation probability helps avoid damaging high-fitness individuals; in later iterations, a low crossover probability avoids the destruction of elite individuals. At this time, the individual fitness values of the population are relatively high. The chromosomes’ excellent gene segments are largely similar, which can lead the algorithm into local optima. The later high mutation probability helps the algorithm escape local optima and achieve a global optimum.

2.2.7. Termination Criteria

The termination condition is that the algorithm reaches the maximum number of iterations, such as 500 iterations. After the iterations are completed, the individual with the highest fitness value is considered the optimal solution to the model.

3. Results

3.1. Data Sources

Central Street is a prominent commercial district located in Harbin, with numerous historical buildings, municipal heritage sites, century-old Western restaurants, and multi-story shopping malls. The streets are lined with food stalls and various shops, with an average daily foot traffic of approximately 1 million and up to 1.5 million during holidays. This area is considered a complex building zone, with a high fire risk due to the nature of the buildings. By the end of 2023, 20 key fire protection units in Central Street had established micro fire stations, as shown in Figure 10a. Previous studies have indicated that the fire service demand points in the area can be represented by POIs [24,28,29].

In this study, key data were retrieved, sorted, and filtered using Amap, resulting in a total of 4210 POI points. These POIs were divided into 14 categories, as shown in

Table 1. Classification of fire risk factors.

Fire Risk Factor	POI Type
High Population Density	Hotels, Public Entertainment Venues, Transportation Hubs, Shopping Centers
Key Protection	Historical Buildings, Municipal Protected Sites
Easily Flammable	Restaurants, Cosmetics Warehouses
General Commodity	Convenience Stores, Clothing Stores, Household Stores, Financial Institutions, Bookstores, Pharmacies

. Using nationwide fire data from 2009 to 2022, the fire occurrence probability for each of the 14 POI categories was calculated; that is, the proportion of fire occurrences in each category relative to the total number of fires nationwide. The weights of the four types of risk factors were preliminarily proposed to be 0.4, 0.4, 0.6, and 0.2, respectively. Using ArcGIS, the Central Street area was discretized into 815 grids of 3 m by 3 m, and the fire occurrence probabilities for the four risk factors and 14 POI categories were weighted to calculate the fire risk value for each grid. These risk values were categorized into three levels, as shown in Figure 10b: the colors red, yellow, and green represent high, medium, and low fire risk values, respectively.

3.2. Testing Problem

There are 771 demand points in the research area, of which, 20 have fire protection capabilities. The fire protection capability standards are uniformly classified as level two. After excluding locations like roads and obstacles, 190 points for fire protection improvement were identified. This case study aimed to change the service level of the fire service points and improve the demand points that do not have fire protection capabilities. We analyzed the changes in four indicators of the model: the total matching degree, effective coverage rate, coverage rate for high-risk and medium-risk points, and overall coverage rate. To compare the graded progressive coverage method proposed here with the maximum progressive coverage method developed by Karasakal [18], the maximum progressive coverage model and the graded progressive coverage model were tested separately. The ideal coverage radius, Ri, for the demand points in each risk category were set 0.2 km, 0.3 km, and 0.4 km, respectively, with a maximum coverage radius of 0.8 km. The maximum progressive coverage method was set to have a complete coverage radius of 0.6 km and a maximum coverage radius of 0.8 km.

4. Discussion

Table 2. The coverage rate changes under the two models.

Number	PM1	PM2	PM3	PG1	PG2	PG3
0	0.453	0.501	0.483	0.478	0.524	0.558
1	0.475	0.560	0.545	0.504	0.574	0.605
2	0.496	0.597	0.582	0.532	0.612	0.652
3	0.513	0.629	0.620	0.566	0.656	0.706
4	0.530	0.666	0.661	0.584	0.699	0.754
5	0.545	0.692	0.697	0.607	0.735	0.799
6	0.564	0.735	0.740	0.628	0.761	0.838
7	0.583	0.772	0.770	0.645	0.799	0.870
8	0.593	0.802	0.803	0.667	0.825	0.897
9	0.603	0.816	0.813	0.690	0.840	0.921
10	0.614	0.821	0.819	0.703	0.850	0.933
11	0.620	0.825	0.825	0.724	0.856	0.940
12	0.625	0.830	0.831	0.741	0.860	0.945
13	0.629	0.834	0.836	0.751	0.865	0.949
14	0.631	0.838	0.840	0.761	0.871	0.953
15	0.634	0.844	0.845	0.763	0.873	0.956
16	0.636	0.848	0.848	0.765	0.879	0.960
17	0.639	0.852	0.853	0.767	0.885	0.964
18	0.640	0.855	0.856	0.769	0.889	0.968
19	0.642	0.859	0.860	0.771	0.890	0.972
20	0.645	0.862	0.865	0.772	0.893	0.974

shows the coverage of the optimal solutions for two types of models. Figure 11 shows the relationship between the changes in various indicators under two models when adjusting the level of the fire service points and the number of demand points with improved fire protection capabilities. Each point in the graph represents an optimal solution. The performances of the optimal solutions of the two types of models were evaluated for different indicators, including the total matching degree, effective coverage rate, medium/high-risk coverage rate, and total coverage rate. The total matching degree refers to the sum of the coverage matching degrees between the fire service points and the demand points under the proposed scheme of the model.

The total matching degree refers to the sum of the coverage matching degrees between the fire service points and the demand points under the proposed scheme of the model. The total matching degree in Figure 11a increases linearly with the number of improved fire capacity points. The effective coverage rate refers to the proportion of demand points with a coverage matching degree exceeding 1. In Figure 11a, point A represents that adjusting only the level of fire service points increases the effective coverage rate by 2.5%; when the fire protection capability is increased from 0 to 14, the effective coverage rate increases by 38.3%. From 14 to 20 stations, the effective coverage rate increases by 1.1%. The reason is that the medium- and high-risk points are concentrated in the central area of the region. When the number of demand points for improving fire protection capabilities increases by 14, most of the medium- and high-risk points are covered, resulting in a significant increase in the effective coverage rate. As the number of micro fire stations continues to increase, they will cover more low-risk demand points or provide overlapping coverage for medium- and high-risk points. These areas contribute less to the improvement of effective coverage, so the growth rate of effective coverage will decrease. Covering other demand points requires more firefighting resources. The coverage rate for the high-risk and medium-risk points, representing the probability that these points are covered, and the overall coverage rate, representing the proportion of demand points covered by fair service points, reflect the overall coverage performance of the models. Figure 11b shows that both indicators follow a “rapid increase followed by gradual flattening” trend. In the graded coverage model, the inflection point for both the high-risk coverage rate and the total coverage rate occurs when the number of fire service capability improvement points reaches 10. The high-risk coverage rate is 8.2% higher than the total coverage rate, indicating that the graded coverage model prioritizes covering higher-risk demand points. The turning point of the total coverage rate in the maximum progressive coverage model occurs when the number of fire service capability improvement points reaches 9, with a total coverage rate of 81.3%. The inflection point of the effective coverage rate in the graded coverage model occurs when the number of fire service capability improvement points reaches 14, with an effective coverage rate of 76.1%. The results suggest that, by combining the construction level of micro fire stations with the risk weight of demand points, the graded coverage model requires more fire resources to achieve the desired effective coverage rate.

Currently, the fire facility configuration in complex commercial areas follows a unified standard, without resource prioritization for demand points of with varying risk levels. In the case study set in this paper, if the fire coverage area is considered without accounting for the fire risk weight of each demand point, it may result in the over-allocation of fire resources to low-risk areas, while high-risk areas remain inadequately protected, thus affecting the rationality and the effectiveness of fire planning. The model proposed in this article can be combined with existing fire safety regulations. Firstly, design a localized alarm system to collect the locations of fire risk demand points, fire service points, and the matching degree between the demand points and the fire service points. When a fire occurs, call the fire service point that matches the demand point the most. Meanwhile, the localized alarm system can monitor the situation where the fire demand points are covered by the fire service points. Secondly, it is necessary to strengthen the construction of firefighting facilities in high-risk areas and provide specialized training for personnel. In response to the complexity of commercial area construction and the firefighting characteristics of micro fire stations, the alarm can be accurately located at specific locations (such as a door or a side of a building), in order to quickly respond to firefighting needs and improve the firefighting capabilities of commercial areas.

5. Conclusions

Based on the different risk weights of different demand points, different ideal coverage radii were divided for different demand points. The method of dividing the ideal coverage radius adopted the “high risk level small coverage radius” approach to ensure the efficiency of the service. We unified the maximum coverage radius for all the demand points to ensure fairness in service. We constructed a multi-level progressive coverage model for the distance between the fire service points and the demand points and proposed a coverage matching index. The coverage matching degree within the ideal coverage radius was 1, while the coverage matching degree outside the ideal coverage radius decreased with increasing distance, achieving fairness and efficiency in fire services.

When the high-risk demand points were covered by the same or higher-level fire service points, it was possible to meet all the fire services required by the demand points by matching the different levels of the fire service points with the differently weighted demand points. Otherwise, the coverage matching degree between the demand points and the fire service points decreased. We explored the impact of the fire service level on fire losses at the demand points, fitted a fire service grading coverage loss function, and achieved the requirement of fire service adaptability.

The graded progressive coverage model that was studied combines the multi-level progressive coverage and the graded decay coverage of fire services. Designing a genetic algorithm to solve the model has achieved adaptability, efficiency, and fairness in fire services, providing a basis for fire planning guidance. The objective function is to maximize the coverage matching degree. Future research directions can study the accessibility of fire service points and demand points from a micro perspective, calculate response times more accurately, and improve fire rescue efficiency.