A Mathematical Model for Continuous Expression of Urban Underground Space Resource Multi-Object Evaluation

Abstract

Urban underground space resource (UUSR) constitutes a critical natural resource and a vital component of the natural environment, whose rational utilization is essential for the sustainable development of cities. Mathematical models are indispensable for the multi-object evaluation of UUSR. However, in previous research based on traditional mathematical models, neither the continuous expression of evaluation results has been considered, nor has the comparability been addressed. In this paper, the interval continuous mathematical model (ICMM) is presented at the theoretical level for UUSR multi-object evaluation. By achieving the continuous distribution of quantitative values of evaluation indicators, removing the step-like output features of evaluation results, and eliminating predefined grade boundaries, the ICMM achieves continuous expression of the results and improves their comparability across different areas. The correlation analysis conducted on the UUSR evaluation indicators demonstrates a significant monotonic relationship between the indicator value and the evaluation result, regardless of the increasing trend or the decreasing trend. Finally, a numerical experiment clearly demonstrates that the ICMM is valid for evaluating different UUSR objects.

1. Introduction

With the construction and development of modern cities, the urban development space is gradually shifting from “extending from the ground to the top” to “extending from the ground to the bottom with high density”, which means that urban underground space needs to take on more and more urban functions [1,2,3,4,5,6]. Worldwide, subways, underground pipeline networks, underground corridors, underground storage and underground commercial facilities have become an important part of three-dimensional urban space development [7,8,9,10,11]. It is worth noting that underground space, being an important resource for water, space, ecosystem support, and geothermal energy, needs to be considered as a nonrenewable resource requiring frugal use [12,13,14,15,16,17,18]. By the end of 2024, the cumulative floor area of urban underground space in China had reached 3.5 billion m², with total investment expected to exceed 2.8 trillion RMB in 2025. As a crucial natural environment and an important natural resource, the urban underground space resource (UUSR), along with its development and utilization in the 21st century, is becoming an important spatial resource and an actual approach for future urban expansion, improvement of human settlements, optimization of urban structure, disaster mitigation, prevention and sustainable development [19,20,21,22,23].

In order to make better utilization of urban underground space resources, we need to evaluate its different application objects reasonably. As an important and critical work of the UUSR evaluation, a large number of mathematical models for UUSR multi-object evaluation have been presented from different objects, including those for the quality [24,25,26], capacity [27,28], suitability [29,30], sustainability [31,32], development potential and value [33,34,35], and security [36]. For evaluating the different application objects of UUSR, numerous mathematical models, including linear weighted summation, fuzzy synthesis theory, gray system, extension theory, and variable fuzzy sets, are available for application in UUSR multi-object evaluation. In detail, the multi-object linear weighted summation mathematical model was used to evaluate the comprehensive quality, suitability, and potential value of UUSR [37,38]. The fuzzy synthesis theory was applied to construct the membership function and evaluation mathematical model for suitability evaluation of UUSR [39,40,41]. The development potential of some cities has been evaluated by the gray system and the extension method [34], respectively. In addition, in order to effectively reduce the uncertainty of the artificial division of resource quality assessment levels, variable fuzzy sets are utilized for the quality evaluation of UUSR in Foshan city [42]. However, the key limitation of the multi-object UUSR evaluation results produced by the above models is their hierarchical, step-like, and qualitative nature. This precludes a quantitative, refined, and continuous expression of results, thus obstructing the representation of detailed differentials and spatial gradation among different application objects of UUSR. Consequently, the suitable evaluation mathematical model should fully integrate and build upon existing mathematical models developed for UUSR multi-object evaluation, as well as those from other relevant disciplines.

This research aims to apply the interval continuous mathematical model (ICMM) to the UUSR multi-object evaluation at the theoretical level, including three specific tasks: (1) to construct a universal indicator system for UUSR evaluation; (2) to apply the ICMM for continuous result expression; (3) to validate the model through correlation analysis and numerical experiments. In detail, in Section 2, two commonly used mathematical models are analyzed, in detail, from the perspective of principles, advantages, and disadvantages. The ICMM method is introduced from the aspects of method hypotheses and calculation principles. In Section 3, combined with the UUSR multi-object evaluation indicators and their weights, the correlation of indicators is discussed fully. In Section 4, the application of the ICMM method is discussed. Section 5 summarizes three conclusions of the research and proposes some expectations for future research.

2. Methods

2.1. Linear Weighted Summation Model

The linear weighted summation model has the characteristics of a hierarchical structure, and it is similar to the hierarchical structure of the multi-object evaluation indicator system of UUSR. At the same time, the importance of the impact of the evaluation indicators and the degree of mutual interference can be comprehensively expressed through the functional relationship between the parameter quantification of the indicator system and the evaluation indicator weight system. The multi-object evaluation results of UUSR could be obtained using the following expression:

$$S={\displaystyle \sum _{h=1}^p}\left[{\displaystyle \sum _{j=1}^m}\left({\displaystyle \sum _{i=1}^n}{A}_i{B}_i\right)C_j\right]D_h$$

(1)

where S is the total score of the evaluation object of UUSR; $A_i$ is the quantized value of the indicator i, i = 1, 2, …, n; $B_i$ is the weight of the indicator i; $C_j$ is the weight of the sub-subject j, j = 1, 2, …, m; and $D_h$ is the weight of the subject h, h = 1, 2, …, p.

The main calculation process of the linear weighted summation model is subdivided into three steps, including determining the fixed quantized scores of evaluation indicators on the basis of the pre-set criteria, obtaining the total score of the evaluation object by combining the corresponding weights of the evaluation indicators, and classifying its rating by using the preset grade standard. When using the linear weighted summation model for UUSR multi-object evaluation, due to the division of evaluation indicator level intervals, the selection of attribute values of evaluation indicators is subjective and discrete, resulting in a step-like evaluation result.

2.2. Fuzzy Synthesis Theory

According to the weights of evaluation indicators and the evaluation membership matrix, the evaluation set of fuzzy synthesis theory is calculated using the following expression:

$$B_i=A·X=\left[a_1,a_2,\dots ,a_n\right]\left[\begin{array}{ccccc}\begin{array}{c}{x}_1^1\\ x_2^1\\ ⋮\\ x_n^1\end{array}& \begin{array}{c}{x}_1^2\\ x_2^2\\ ⋮\\ x_n^2\end{array}& \begin{array}{c}{x}_1^3\\ x_2^3\\ ⋮\\ x_n^3\end{array}& \begin{array}{c}{x}_1^4\\ x_2^4\\ ⋮\\ x_n^4\end{array}& \begin{array}{c}{x}_1^5\\ x_2^5\\ ⋮\\ x_n^5\end{array}\end{array}\right]=\left[b_{i1},b_{i2},b_{i3},b_{i4},b_{i5}\right]$$

(2)

where $B_i$ is the evaluation set of UUSR, $i=1,2,\dots ,m$; A is the evaluation indicator weight, ${\sum }_{i=1}^n{a}_i=1$; and X is the evaluation membership matrix.

If it is a two-level evaluation indicator system, its fuzzy synthesis evaluation model is expressed as

$$R=C·B=\left[c_1,c_2,\dots ,c_m\right]\left[\begin{array}{c}{B}_1\\ B_2\\ ⋮\\ B_m\end{array}\right]$$

(3)

where R is the two-level evaluation set of UUSR; C is the two-level evaluation indicator weight, ${\sum }_{i=1}^m{c}_i=1$; and B is the two-level evaluation membership matrix. If the evaluation indicator system is a three-level or above structure, its fuzzy synthesis evaluation model is recursive according to the above rules. Using fuzzy synthesis theory, the positive and negative properties of the UUSR multi-object evaluation indicators need to be taken into consideration. The boundary value b_i (i.e., the judgment domain value when determining the different ratings of the indicator value) needs to be analyzed when calculating the membership degree of the evaluation indicator. Additionally, the trapezoidal membership functions are usually applied for constructing the membership matrix of the positive or negative evaluation indicator on the five-level evaluation domain.

The main idea of fuzzy synthesis theory is to determine the quantized value of each evaluation indicator first, and then calculate the membership degree of each evaluation indicator for each level set in advance. Combining with the weight of the multi-object evaluation indicator of UUSR, the overall membership degree of each evaluation indicator for each level can be obtained. Finally, the rating of UUSR is determined according to the maximum membership degree criterion. When using the fuzzy synthesis theory model for UUSR multi-object evaluation, due to the high dependence of the membership degree of the evaluation indicators on the original data, the UUSR evaluation results are not comparable.

In addition, these key parameters (i.e., the gray correlation degree in the gray system, the degree of conformity in extension theory, and the relative membership degree in variable fuzzy sets) are quite similar to the membership degree in fuzzy synthesis theory. Therefore, these methods, including gray system, extension theory, and variable fuzzy sets, are of the same type of evaluation mathematical model as fuzzy synthesis theory for UUSR evaluation. These mathematical models have similar construction ideas, which result in similar shortcomings in the UUSR evaluation results.

In summary, the two kinds of evaluation models described above exhibit significant shortcomings in UUSR multi-object evaluation, including discrete values of evaluation indicators, step-like output of evaluation results, lack of comparability among evaluation results, and reliance on predefined grade boundaries. To address these issues, we propose the ICMM for UUSR multi-object evaluation.

2.3. Interval Continuous Mathematical Model

Through constructing the benchmark intervals of evaluation indicators and five conversion mathematical models, the interval continuous mathematical model (ICMM) has been first presented for the quality classification of slope rock mass. On the basis of similar application conditions, the ICMM can also be utilized for the multi-object evaluation of UUSR. The calculation steps of the ICMM include the determination of benchmark intervals of evaluation indicators, the calculation of relative quantized values, and the combination with a linear weighted function [43].

(1)

Determination of the benchmark intervals of indicators

For qualitative indicators, the initial values ($x_{ij}$) could be chosen in the fixed interval [0, 5]. For quantitative indicators, the initial values ($x_{ij}$) could be selected in the dynamic interval [$V_l$, $V_r$].

(2)

Calculation of the relative quantized value ($a_{ij}$) of indicators

The calculation rules of $a_{ij}$ are divided into five conversion models. Equations (4), (6), (8), (10) and (12) are used for the positive indicators, and Equations (5), (7), (9), (11) and (13) are utilized for the negative indicators, as shown concretely below.

• Linear model, which can be obtained as:

$$a_{ij}={\displaystyle \frac{x_{ij}-V_l}{V_r-x_{ij}}}$$

(4)

$$a_{ij}=-{\displaystyle \frac{x_{ij}-V_l}{V_r-x_{ij}}}+1$$

(5)

b.

Square model, which is described as:

$$a_{ij}={\left({\displaystyle \frac{x_{ij}-V_l}{V_r-x_{ij}}}\right)}^2$$

(6)

$$a_{ij}={-\left({\displaystyle \frac{x_{ij}-V_l}{V_r-x_{ij}}}\right)}^2+1$$

(7)

c.

Square root model, which can be expressed as:

$$a_{ij}={\left({\displaystyle \frac{x_{ij}-V_l}{V_r-x_{ij}}}\right)}^{{\displaystyle \frac{1}{2}}}$$

(8)

$$a_{ij}={-\left({\displaystyle \frac{x_{ij}-V_l}{V_r-x_{ij}}}\right)}^{{\displaystyle \frac{1}{2}}}+1$$

(9)

d.

Cubic model, which is described as:

$$a_{ij}={\left({\displaystyle \frac{x_{ij}-V_l}{V_r-x_{ij}}}\right)}^3$$

(10)

$$a_{ij}={-\left({\displaystyle \frac{x_{ij}-V_l}{V_r-x_{ij}}}\right)}^3+1$$

(11)

e.

Cubic root model, which can be calculated as:

$$a_{ij}={\left({\displaystyle \frac{x_{ij}-V_l}{V_r-x_{ij}}}\right)}^{{\displaystyle \frac{1}{3}}}$$

(12)

$$a_{ij}={-\left({\displaystyle \frac{x_{ij}-V_l}{V_r-x_{ij}}}\right)}^{{\displaystyle \frac{1}{3}}}+1$$

(13)

(3)

Combination with a linear weighted function

Combining with the reasonable weights $W_j$, the final evaluation values can be calculated. Its expression is:

$$L={\displaystyle \sum _{j=1}^n}{W}_j{a}_{ij}$$

(14)

Among these five conversion models, the linear model exhibits constant change, the square and cubic models show change that accelerates with larger indicator values (a positive correlation), and the square root and cubic root models display change that decelerates as indicator values increase (a negative correlation). Therefore, we can reasonably select a kind of conversion model for UUSR multi-object evaluation based on the data structure and characteristics of the real data.

3. Results

3.1. Evaluation Indicators

Generally, the multi-object evaluation of UUSR is mainly affected by factors such as geological medium, rock and soil condition, economic condition, social condition, geographical condition, construction condition, policy condition, and geological defect in different geological types, different evaluation scales, and different evaluation time states. The universal UUSR multi-object evaluation indicators are constructed by taking the urban geological types, the scale effect of the evaluation indicators, the temporal change in the indicators, and the accessibility of the indicator values into consideration, as seen in

Table 1. The indicators for UUSR multi-object evaluation.

Subject Layer	Indicators	Characteristics
Geological medium	Ground surface slope X1	Quantitative
	Soft soil thickness X2	Quantitative
	Liquefaction index of sandy soil X3	Quantitative
	Corrosiveness of groundwater X4	Quantitative
	Phreatic water depth X5	Quantitative
Rock and soil condition	Cohesion stress X6	Quantitative
	Internal friction angle X7	Quantitative
	Bearing capacity X8	Quantitative
	Permeability coefficient X9	Quantitative
Economic condition	GDP per capita X10	Quantitative
	Income per capita X11	Quantitative
	Expenditure per capita X12	Quantitative
Social condition	Population density X13	Quantitative
	Benchmark land price X14	Quantitative
	Urbanization rate X15	Quantitative
Construction condition	Ground facility types X16	Qualitative
	Underground facility types X17	Qualitative
Geographic location condition	Distance from downtown X18	Quantitative
Policy condition	National policy X19	Qualitative
Geological defect	Fault X20	Qualitative
	Karst X21	Qualitative

:

3.2. Evaluation Weights

In general, all subjective and objective methods, such as the analytic hierarchy process (AHP) [44,45,46,47], Delphi [34], and the entropy weight method [42], could be utilized for calculating the weights of evaluation indicators in different disciplines. Due to the hypothetical data used for UUSR multi-object evaluation in this paper, weighting methods such as Delphi and AHP that rely on expert experience are not applicable. Considering the need to solve the problems of large amounts of information and difficult quantification of UUSR multi-object evaluation indicators, and to express the amount of information contained in the indicators to the greatest extent, the entropy weight method [48] is selected for determining the weights of the indicators above. Based on the corresponding series of hypothetical data of UUSR multi-object evaluation indicators (

Table 2. The data used for the calculation of the weights evaluation indicators.

Data	X1	X2	X3	X4	X5	X6	X7	X8	X9	X10	X11
A1	1	3.5	6	6	10	100	10	100	1	10,000	5000
A2	1.2	3.3	5	6.2	15	150	11	150	1.1	15,000	5300
A3	2.1	3	4	6.5	20	200	15	200	1.2	20,000	5500
A4	2.9	2.5	3	6.7	25	250	25	250	1.3	25,000	6000
A5	3.6	2.3	2	6.9	30	350	30	400	1.4	30,000	6500
A6	4.1	2	1	7	50	800	40	800	1.5	60,000	7000
Data	X12	X13	X14	X15	X16	X17	X18	X19	X20	X21
A1	3000	1000	4000	50	1	1	4000	0	1.5	0.5
A2	3500	1200	4500	52	1.1	1.3	3500	1	2.1	1.2
A3	4000	1500	5000	56	1.3	1.5	3000	2	2.6	1.4
A4	4200	1800	5500	58	1.6	1.7	2500	3	3.2	1.5
A5	4500	1900	6500	59	1.9	1.8	1500	4	3.6	1.7
A6	5000	2000	7000	60	2	2	1000	5	4	2

), the weights of UUSR multi-object evaluation indicators are calculated, respectively, as seen in Figure 1.

3.3. Correlation Analysis

Aiming at the different effects of the changes in indicators on evaluation results, the correlation analysis of the UUSR multi-object evaluation indicators is carried out before UUSR evaluation. The influence of evaluation indicators is analyzed in turn by changing each of the variables for the check calculation. The UUSR multi-object evaluation indicators are divided into hard factors, soft factors, and control factors. In detail, the hard factors contain two categories, including geological medium, rock and soil condition. The soft factors contain four parts, including economic condition, social condition, construction condition, and geographic location condition. In addition, the control factors contain two aspects, including policy conditions and geological defects.

(1)

Analysis of hard factors

According to a change in amplitude of 10%, for the geological medium, the quantized values of ground surface slope, soft soil thickness, liquefaction index of sandy soil, corrosiveness of groundwater, and phreatic water depth change in the numerical range of 1 to 4.1, 2 to 3.5, 1 to 6, 6 to 7, and 10 to 50, respectively. For rock and soil condition, the quantized values of cohesion stress, internal friction angle, bearing capacity, and permeability coefficient change in the numerical range of 100 to 800, 10 to 40, 100 to 800, and 1 to 1.5, respectively. On the basis of the five conversion models, the UUSR evaluation results are obtained as shown in Figure 2.

Figure 2 indicates that the evaluation results show an increasing trend with the value increase of seven indicators (i.e., X₁, X₄, X₅, X₆, X₇, X₈, and X₉), which can be called positive indicators. In detail, the change rate in the linear model is always constant; the change rate in square model and cubic model increases with the increase in indicator value, that is, the greater the indicator value, the greater the impact; and the change rate in square root model and cubic root model decreases with the increase in indicator value, that is, the greater the indicator value, the smaller the impact. On the contrary, the change trend of indicators X₂ and X₃ decreased with the increase in indicator value, which is opposite to that of positive indicators and can be called negative indicators. Interestingly, their patterns of change rate in each conversion model are the same as those of positive indicators.

(2)

Analysis of soft factors

According to a change in amplitude of 10%, for economic condition, the quantized values of GDP per capita, income per capita, and expenditure per capita change in the numerical range of 10,000 to 60,000, 5000 to 7000, and 3000 to 5000, respectively. For social conditions, the quantized values of population density, benchmark land price, and urbanization rate change in the numerical range of 1000 to 2000, 4000 to 7000, and 50 to 60, respectively. For construction conditions, the quantized values of ground facility types and underground facility types change in the same numerical range of 1 to 2. For the geographic location condition, the distance from downtown changes in the numerical range of 1000 to 4000. On the basis of the five conversion models, the UUSR evaluation results are obtained as shown in Figure 3.

Figure 3 indicates that the evaluation results show an increasing trend with the value increase of eight indicators (i.e., X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, and X₁₇), which can be called positive indicators. In detail, the change rate in the linear model is always constant; the change rate in square model and cubic model increases with the increase in indicator value, that is, the greater the indicator value, the greater the impact; and the change rate in square root model and cubic root model decreases with the increase in indicator value, that is, the greater the indicator value, the smaller the impact. On the contrary, the change trend of indicator X₁₈ decreased with the increase in indicator value, which is opposite to that of positive indicators and can be called a negative indicator. Interestingly, their patterns of change rate in each conversion model are the same as those of positive indicators.

(3)

Analysis of control factors

According to a change in amplitude of 10%, for the policy condition, the quantized value of national policy changes in the numerical range of 0 to 5. For geological defects, the quantized values of fault and karst change in the numerical range of 1.5 to 4, and 0.5 to 2, respectively. On the basis of the five conversion models, the UUSR evaluation results are obtained as shown in Figure 4.

Figure 4 indicates that the evaluation results show an increasing trend with the value increase in all three indicators (i.e., X₁₉, X₂₀, and X₂₁), which can be called positive indicators. In detail, the change rate in the linear model is always constant; the change rate in square model and cubic model increases with the increase in indicator value, that is, the greater the indicator value, the greater the impact; and the change rate in square root model and cubic root model decreases with the increase in indicator value, that is, the greater the indicator value, the smaller the impact.

Finally, the correlation analysis indicates that the UUSR multi-object evaluation results show an increasing trend with the increase in the values of most of the indicators, except for X₂, X₃, and X_18, which show a decreasing trend. In addition, the correlation of evaluation indicators is also consistent with their positive and negative characteristics.

4. Discussion

In view of the application of the ICMM method, the benchmark intervals of the UUSR multi-object evaluation indicators are necessarily given. According to the construction principle 1 of ICMM (i.e., determination of the benchmark intervals of indicators), it stipulates that for qualitative indicators, the initial values ($x_{ij}$) could be chosen in the fixed interval [0, 5]; and for quantitative indicators, the initial values ($x_{ij}$) could be selected in the dynamic interval [$V_l$, $V_r$]. On the basis of the correlation features and extant indicators used for the UUSR multi-object evaluation, all the left endpoint values $V_l$ of indicators are assigned as 0. In addition, the right endpoint values for qualitative indicators (i.e., X₁₆, X₁₇, X₁₉, X₂₀, and X₂₁) are determined as 5. On the basis of the range of statistical values of the quantitative indicators (i.e., X₁–X₁₅, X₁₈), their right endpoint values are determined as the numerical vector. Taking the internal friction angle (X₇) as an example, the internal friction angle of most intact rocks in nature is between 25° and 50°, and extremely hard intact rocks can briefly exceed 50°; the right endpoint value is taken as 60° for calculation based on expert experience. Finally, the right endpoint values ($V_r$) of the indicators are determined as a numerical vector V = (30, 10, 10, 7, 60, 800, 60, 800, 1.5, 10,000, 8000, 6000, 4000, 12,000, 85, 5, 5, 5000, 5, 5, 5) in order from X₁ to X₂₁. Then, combined with the above evaluation indicators, the entropy weight method, and the interval continuous mathematical model, the multi-object evaluation results of UUSR are obtained as follows (Figure 5), based on the basic hypothetical data used for the weight calculation of evaluation indicators.

As shown in Figure 5, the evaluation results of UUSR have been calculated by using the five conversion models of the ICMM method. In detail, when utilizing the data in the same area (e.g., A₁), the evaluation results using the five different conversion models of ICMM have the following relationships from largest to smallest, arranged in order as cubic root model, square root model, linear model, square model, and cubic model. The result clearly demonstrates that the ICMM method possesses strong evaluation consistency, indicating its high reliability across different conversion models. It consistently yields concordant rankings, making the method reliable for UUSR evaluation. On the other hand, when utilizing the same conversion model (e.g., cubic root model), the evaluation results in different areas always have the following relationships from large to small, arranged in order as A₆, A₅, A₄, A₃, A₂, and A₁. This finding confirms that the ICMM method exhibits strong evaluation stability, producing consistent and reliable results across different conditions. Compared to the linear weighted summation model, the ICMM uses nonlinear models, especially root and power functions, which can capture the decrease or increase in marginal revenue. Meanwhile, due to the difference in the rate of variation in the five conversion models when using the ICMM method for the UUSR multi-object evaluation, the evaluation results are discrete and different, which means that the five different conversion models have different sensitivity to the data of evaluation indicators. In other words, it also shows that the ICMM method can be adapted to any type of evaluation indicator data. In addition, the evaluation results are continuously increasing, and all fall into the interval of 0 to 1, which means the ICMM method successfully realizes the continuous expression and improves the comparability of the evaluation results in the UUSR multi-object evaluation across different evaluation areas.

5. Conclusions

As a suitable and reasonable mathematical method, the interval continuous mathematical model (ICMM) is applied for UUSR multi-object evaluation based on five different conversion models. This paper presents three main conclusions. Firstly, a universal UUSR multi-object evaluation indicator system is constructed by taking the urban geological type, the scale effect of the indicators, the temporal change in the indicators, and the accessibility of the indicator values into consideration, which is essential for the UUSR evaluation. Secondly, the correlation analysis of UUSR evaluation indicators is carried out with the five conversion models of ICMM, which indicates that the UUSR evaluation results exhibit a monotonic trend, regardless of the increasing trend or the decreasing trend. Thirdly, the ICMM method can realize the continuous expression of UUSR multi-object evaluation results by removing the step-like features, and can improve the comparability of the results in different areas. This study can promote the rational utilization of UUSR, which is essential for the sustainable development of cities, particularly in future urban expansion, urban planning, and urban investment.

The present study focuses on the theoretical development and numerical validation of the ICMM. However, there are several points in this study that need further improvement: lack of empirical validation against ground real data, absence of comparative benchmarking with other mathematical models, and lack of principles in the selection of conversion models. In view of this, future applied research will apply the proposed model to real-world urban underground space cases with GIS, integrating actual geological, economic, and social data to evaluate development potential, suitability, and resource efficiency. In addition, the sensitivity of the model will be analyzed based on actual data from real study areas in future applied research. The limitations of the conversion models should be systematically analyzed in conjunction with the actual conditions of study areas in different types of cities. Additionally, other assumptions made in this paper require further validation or refinement through future research.