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Municipal Civil Engineering Construction Based on Finite Element Differential Equations

Publié en ligne: 15 Jul 2022
Volume & Edition: AHEAD OF PRINT
Pages: -
Reçu: 06 Mar 2022
Accepté: 05 May 2022
Détails du magazine
License
Format
Magazine
eISSN
2444-8656
Première parution
01 Jan 2016
Périodicité
2 fois par an
Langues
Anglais
Introduction

Many safety accidents accompany the rapid development of the construction industry. As an external binding force, the lack of supervision, anomie, and inefficiency of management in the building safety supervision department have hindered building safety levels. Therefore, we should seek effective regulatory methods. At the same time, we make a set of scientific and reasonable construction safety supervision mechanisms to restrain the behavior of supervision departments and construction enterprises [1]. This approach will be the direction of future government efforts.

Many scholars have used game theory to build building safety supervision models in recent years. Scholars have explored the evolutionary path of the main body strategies of government and enterprises. At the same time, they sort out the safety behaviors of construction companies under different equilibrium scenarios and expand and analyze the long-term mechanism to effectively reduce construction safety accidents. However, most researchers are limited to standard classical game theory and evolutionary games. They did not consider that the unstable accident hidden danger contained in the production system of the construction site is the potential cause of “disturbance” in the structure, function, and state of the construction safety accident chain. They ignore that accidents are formed due to the existence, accumulation, superposition, and loss of control of accident hazards. This is different from the dynamics of actual construction safety production governance. As a result, the game model cannot truly reflect the process of the constant adjustment of the complementary strategies of both parties with the change of the hidden danger of the accident. Therefore, scholars need to adopt a more reasonable method to discuss the issue of building safety supervision [2]. The differential game is to extend the game theory to continuous-time and fully consider the influence of the dynamic accumulation process of the hidden accident quantity in the construction site on the game. We can reveal the causes of construction safety accidents from the microscopic dimension. This makes up for the shortcomings of traditional game research.

This paper introduces the building safety performance appraisal system. At the same time, this paper builds a differential game model between the government and construction enterprises based on the analysis of hidden dangers of accidents [3]. This paper attempts to analyze the dynamic law of the influence of government regulatory factor endowment on the safety behavior of construction enterprises through research. The results of this paper provide helpful inspiration for the government to improve the building safety supervision policy.

Basic Assumptions and Construction of Differential Game Models
Basic assumptions of the model

Assumption 1: For any time t ⊂ [0, ∞), when it is profitable, construction companies n will violate the regulations. The accidental hidden danger hi (t) generated by the construction unit i is related to the engineering quantity di (t). Assume hi (t) = ∂di (t), where (I > ∂ > 0) = is the accident hidden danger generation coefficient. Construction enterprises obtain benefits through construction production [4]. We use the function of hi (t) to express the construction benefits and costs of construction companies. The net income of the enterprise is the difference between the product utility and the construction cost, aihi(t)vi2hi(t)2 {a_i}{h_i}\left(t \right) - {{{v_i}} \over 2}{h_i}{\left(t \right)^2} . where ai (ai > 0) and vi (vi > 0) represent the utility coefficient and the cost coefficient, respectively. k[hi (t) − rhi (t)] represents the benefits that illegal construction brings to the construction company. k(k > 0) represents the benefit of the hidden accident per unit, rhi (t) represents the number of hidden accident hazards reduced by the i construction unit through safety management, 0 < r < 1.

Hypothesis 2: The losses caused by safety accidents to local governments and construction enterprises are represented by the linear function of hidden dangers [5]. Assume that the loss caused by the safety accident to the local government is lfM (t), such as the accountability of the higher-level government due to the accident, the damage to the image and credibility of the government after the accident. Where f (f > 0) is the government loss cost. M (t) is the amount of accident hazards at the construction site. The loss cost caused by this method to the enterprise is blM (t). b (b > 0) is the loss cost of the business. l (l ≥ 0) is the probability of a safety accident.

Hypothesis 3: The superior government rewards and punishes the local government according to the construction safety performance [6]. The size of reward and punishment is represented by the linear function ϕα (t)[gz(t) − g0z0(t)] of building safety performance. Where ϕ(ϕ > 0) is the reward and punishment value. α(t) represents the importance of building safety performance appraisal. g0z0 (t) are the assessment standard values formulated by the higher-level government, respectively. When the product gz (t) of the local government's building safety supervision level and the degree of supervision effort exceeds the standard value g0z0(t), ϕα (t)[gz(t) − g0z0(t)] > 0. This means that the higher-level government rewards the local government; otherwise, it means punishment. At the same time, the discount rate of the government and the construction unit is the same and both are ρ.

Construction site safety risk level function

The dynamic accident potential amount M˙(t) \dot M\left(t \right) at the construction site is mainly composed of three parts. The total amount of hidden accident risks generated by the illegal behavior of n construction companies i1n[hi(t)rhi(t)] \sum\nolimits_{i - 1}^n {\left[{{h_i}\left(t \right) - r{h_i}\left(t \right)} \right]} , the amount of hidden accident risks eliminated by the government through supervision efforts z(t), and the amount of hidden accident risks destroyed by other factors on the construction site δ. xz (t) is the number of accident hazards eliminated by the government through safety supervision. x is the amount of accident hazards eliminated by local government unit supervision efforts, x > 0. δM (t) is the amount of accident hazards reduced by other factors δ on the construction site. δM (0) is the initial value of the accident hidden danger at the construction site, M (0) = 0. The level function of the safety risk of the construction site based on the above assumptions is: M˙(t)=i1n[hi(t)rhi(t)]xz(t)δM(t) \dot M\left(t \right) = \sum\nolimits_{i - 1}^n {\left[{{h_i}\left(t \right) - r{h_i}\left(t \right)} \right]} - xz\left(t \right) - \delta M\left(t \right)

Profit function of construction enterprises

The income of the construction unit can be divided into two parts: net income aihi(t)vi2hi(t)2 {a_i}{h_i}\left(t \right) - {{{v_i}} \over 2}{h_i}{\left(t \right)^2} from construction production and income k k[hi (t) − rhi (t)] from illegal construction.

The costs and expenses of the enterprise include: (1) The loss cost blM (t) caused by the safety accident to the enterprise. (2) The safety investment c12[rhi(t)]2 {{{c_1}} \over 2}{\left[{r{h_i}\left(t \right)} \right]^2} , c1 (c1 > 0) increased by the enterprise to reduce rhi (t) is the safety investment coefficient. (3) α (t)g[hi(t) − rhi (t)]ω means the construction company will be punished by the government after the illegal construction is discovered. α(t) g represents the probability of the company's illegal construction being verified [7]. Among them, g represents the government building safety supervision level. ω represents the fine paid per unit of potential accident risk when confirmed by the government. Therefore, the profit function of the construction enterprise is: 0t{aihi(t)vi2hi(t)2+k[hi(t)rhi(t)]blM(t)c12[rhi(t)]2α(t)g[hi(t)rhi(t)]ω}eρtdt \int_0^t {\left\{{{a_i}{h_i}\left(t \right) - {{{v_i}} \over 2}{h_i}{{\left(t \right)}^2} + k\left[{{h_i}\left(t \right) - r{h_i}\left(t \right)} \right] - blM\left(t \right) - {{{c_1}} \over 2}{{\left[{r{h_i}\left(t \right)} \right]}^2} - \alpha \left(t \right)g\left[{{h_i}\left(t \right) - r{h_i}\left(t \right)} \right]\omega} \right\}{e^{- \rho t}}dt}

Government revenue function

The government revenue is mainly composed of three parts. The economic benefits created by the construction enterprises for the society i1n[aihi(t)vi2hi(t)2] \sum\nolimits_{i - 1}^n {\left[{{a_i}{h_i}\left(t \right) - {{{v_i}} \over 2}{h_i}{{\left(t \right)}^2}} \right]} , the fines i1nα(t)y[hi(t)rhi(t)]ω \sum\nolimits_{i - 1}^n {\alpha \left(t \right)y\left[{{h_i}\left(t \right) - r{h_i}\left(t \right)} \right]\omega} paid by the enterprises after the illegal construction is discovered, and the rewards from the superior government after passing the assessment [8].

Government losses and costs include losses caused by security incidents to the government lfM (t). The second is the cost of government building safety supervision c22[xz(t)]2 {{{c_2}} \over 2}{\left[{xz\left(t \right)} \right]^2} .

c2 (c2 > 0) is the regulatory cost factor. Then the government revenue function in the game model is: 0t{i=1n[aihi(t)vi2hi(t)2]+i=1nα(t)g[hi(t)rhi(t)]ω+ϕα(t)[gu(t)g0u0(t)]lfM(t)c22[xz(t)]2}eρtdt \matrix{{\int_0^t {\left\{{\sum\limits_{i = 1}^n {\left[{{a_i}{h_i}\left(t \right) - {{{v_i}} \over 2}{h_i}{{\left(t \right)}^2}} \right] + \sum\limits_{i = 1}^n {\alpha \left(t \right)g\left[{{h_i}\left(t \right) - r{h_i}\left(t \right)} \right]\omega +}}} \right.}} \hfill \cr {\left. {\phi \alpha \left(t \right)\left[{gu\left(t \right) - {g_0}{u_0}\left(t \right)} \right] - lfM\left(t \right) - {{{c_2}} \over 2}{{\left[{xz\left(t \right)} \right]}^2}} \right\}{e^{- \rho t}}dt} \hfill \cr}

Differential game model solution and analysis
Model solution

We aim to obtain the continuous solution M (t) of equation (1). This paper first constructs the Hamilton Jacobi-Bellman (HJB) equation: ρVc(M)=maxhi(t)0{aihi(t)vi2hi(t)2+k[hi(t)rhi(t)]blM(t)c12[rhi(t)]2α(t)g[hi(t)rhi(t)]ω+Vc(M){i=1n[hi(t)rhi(t)]xz(t)δM(t)}} \matrix{{\rho {V_c}\left(M \right) = \mathop {\max}\limits_{{h_i}\left(t \right) \ge 0} \left\{{{a_i}{h_i}\left(t \right) - {{{v_i}} \over 2}{h_i}{{\left(t \right)}^2} + k\left[{{h_i}\left(t \right) - r{h_i}\left(t \right)} \right] -} \right.} \hfill \cr {blM\left(t \right) - {{{c_1}} \over 2}{{\left[{r{h_i}\left(t \right)} \right]}^2} - \alpha \left(t \right)g\left[{{h_i}\left(t \right) - r{h_i}\left(t \right)} \right]\omega + V_c^{'}\left(M \right)} \hfill \cr {\left\{{\left. {\sum\limits_{i = 1}^n {\left[{{h_i}\left(t \right) - r{h_i}\left(t \right)} \right]} - xz\left(t \right) - \delta M\left(t \right)} \right\}} \right\}} \hfill \cr} ρVg(M)=maxz(t)0{i=1naihi(t)vi2hi(t)2]+i=1nα(t)g[hi(t)rhi(t)]ω+ϕα(t)[gz(t)g0z0(t)]lfM(t)c22[xz(t)]2+Vg(M){i=1n[hi(t)rhi(t)]xz(t)δM(t)}} \matrix{{\rho {V_g}\left(M \right) = \mathop {\max}\limits_{z\left(t \right) \ge 0} \left\{{\sum\limits_{i = 1}^n {\left. {{a_i}{h_i}\left(t \right) - {{{v_i}} \over 2}{h_i}{{\left(t \right)}^2}} \right] +}} \right.} \hfill \cr {\sum\limits_{i = 1}^n {\alpha \left(t \right)g\left[{{h_i}\left(t \right) - r{h_i}\left(t \right)} \right]} \omega + \phi \alpha \left(t \right)\left[{gz\left(t \right) -} \right.} \hfill \cr {\left. {{g_0}{z_0}\left(t \right)} \right] - lfM\left(t \right) - {{{c_2}} \over 2}{{\left[{xz\left(t \right)} \right]}^2} + V_g^{'}\left(M \right)} \hfill \cr {\left\{{\left. {\sum\limits_{i = 1}^n {\left[{{h_i}\left(t \right) - r{h_i}\left(t \right)} \right]} - xz\left(t \right) - \delta M\left(t \right)} \right\}} \right\}} \hfill \cr}

By maximizing the right-hand sides of equations (4) and (5) with respect to hi (t), z (t), respectively, we can obtain: hi(t)=ai+k(1r)ωα(t)g(1r)+Vc(M)(1r)nvi+c1r2 {h_i}\left(t \right) = {{{a_i} + k\left({1 - r} \right) - \omega \alpha \left(t \right)g\left({1 - r} \right) + V_c^{'}\left(M \right)\left({1 - r} \right)n} \over {{v_i} + {c_1}{r^2}}} z(t)=ϕα(t)gxVg(M)c2x2 z\left(t \right) = {{\phi \alpha \left(t \right)g - xV_g^{'}\left(M \right)} \over {{c_2}{x^2}}}

Our purpose here is to understand the value function Vc (M), Vg (M). Suppose Vc (M) = ε1 + π1 M (t), Vg (M) = ε2 + π2 M (t), and ε1, ε2, π1, π2 are both constants. Then substitute Vg(M)=π1 V_g^{'}\left(M \right) = {\pi _1} , Vg(M)=π2 V_g^{'}\left(M \right) = {\pi _2} into formulas (4) and (5) respectively to get: ρ[ε1+π1M(t)]=aihi(t)vi2hi(t)2+k[hi(t)rhi(t)]blM(t)c12[rhi(t)]2α(t)g[hi(t)rhi(t)]ω+π1{i=1n[hi(t)rhi(t)]xz(t)δM(t)}ρ[ε2+π2M(t)]=i=1n[aihi(t)vi2hi(t)2]+i=1nα(t)g[hi(t)rhi(t)]ω+ϕα(t)[gz(t)g0z0(t)]lfM(t)c22[xz(t)]2+π2{i=1n[hi(t)rhi(t)]xz(t)δM(t)} \matrix{{\rho \left[{{\varepsilon _1} + {\pi _1}M\left(t \right)} \right] = {a_i}{h_i}\left(t \right) - {{{v_i}} \over 2}{h_i}{{\left(t \right)}^2} + k\left[{{h_i}\left(t \right) - r{h_i}\left(t \right)} \right]} \hfill \cr {- blM\left(t \right) - {{{c_1}} \over 2}{{\left[{r{h_i}\left(t \right)} \right]}^2} - \alpha \left(t \right)g\left[{{h_i}\left(t \right) - r{h_i}\left(t \right)} \right]\omega + {\pi _1}} \hfill \cr {\left\{{\sum\limits_{i = 1}^n {\left[{{h_i}\left(t \right) - r{h_i}\left(t \right)} \right]} - xz\left(t \right) - \delta M\left(t \right)} \right\}} \hfill \cr {\rho \left[{{\varepsilon _2} + {\pi _2}M\left(t \right)} \right] = \sum\limits_{i = 1}^n {\left[{{a_i}{h_i}\left(t \right) - {{{v_i}} \over 2}{h_i}{{\left(t \right)}^2}} \right] +}} \hfill \cr {\sum\limits_{i = 1}^n {\alpha \left(t \right)g\left[{{h_i}\left(t \right) - r{h_i}\left(t \right)} \right]\omega + \phi \alpha \left(t \right)\left[{gz\left(t \right) - {g_0}{z_0}\left(t \right)} \right]} - lfM\left(t \right)} \hfill \cr {- {{{c_2}} \over 2}{{\left[{xz\left(t \right)} \right]}^2} + {\pi _2}\left\{{\sum\limits_{i = 1}^n {\left[{{h_i}\left(t \right) - r{h_i}\left(t \right)} \right]} - xz\left(t \right) - \delta M\left(t \right)} \right\}} \hfill \cr}

In: ρπ1M(t)=blM(t)π1δM(t)ρπ2M(t)=lfM(t)π2δM(t) \matrix{{\rho {\pi _1}M\left(t \right) = - blM\left(t \right) - {\pi _1}\delta M\left(t \right)} \hfill \cr {\rho {\pi _2}M\left(t \right) = - lfM\left(t \right) - {\pi _2}\delta M\left(t \right)} \hfill \cr} .

So: π1=bρ+δlπ2=lρ+δf \matrix{{{\pi _1} = {{- b} \over {\rho + \delta}}l\,} \hfill & {{\pi _2} = {{- l} \over {\rho + \delta}}f} \hfill \cr}

Then there are: Vc(M)=π1=bρ+δlVg(M)=π2=lρ+δf V_c^{'}\left(M \right) = {\pi _1} = {{- b} \over {\rho + \delta}}lV_g^{'}\left(M \right) = {\pi _2} = {{- l} \over {\rho + \delta}}f

We can obtain the feedback Nash equilibrium strategy hi*(t) h_i^*\left(t \right) , z* (t) by substituting the above parameters into formulas (6) and (7), respectively: hi*(t)=αi+k(1r)ωα(t)g(1r)bnlρ+δ(1r)vi+c1r2 h_i^*\left(t \right) = {{{\alpha _i} + k\left({1 - r} \right) - \omega \alpha \left(t \right)g\left({1 - r} \right) - {{bnl} \over {\rho + \delta}}\left({1 - r} \right)} \over {{v_i} + {c_1}{r^2}}} z*(t)=ϕα(t)g+lxfρ+δc2x2 {z^*}\left(t \right) = {{\phi \alpha \left(t \right)g + {{lxf} \over {\rho + \delta}}} \over {{c_2}{x^2}}}

Analysis of Relevant Influencing Factors in Differential Game Equilibrium

From equations (8) and (9), it can be known that hi*(t) h_i^*\left(t \right) z* (t) is interfered by many factors such as α (t), ω, ϕ, k. The safety risk status of construction projects is related to factors such as the importance of the safety performance assessment, the degree of supervision effort, the supervision cost, and the government supervision department [9]. We take the derivative of α (t), ω, ϕ and other parameters through hi*(t) h_i^*\left(t \right) , z* (t) respectively. The article compares and analyzes the derivation results. Therefore, the following relationship holds. See Table 1 for details.

Influencing factors in differential game equilibrium

Equilibrium strategy Parameter Influence trend Explain
hi*(t) h_i^*\left(t \right) α (t) hi*(t)α(t)=ωg(1r)vi+c1r2<0 {{\partial h_i^*\left(t \right)} \over {\partial \alpha \left(t \right)}} = - {{\omega g\left({1 - r} \right)} \over {{v_i} + {c_1}{r^2}}} < 0 Increasing the proportion of building safety in the performance appraisal and reducing the investment in the management of hidden dangers of accidents will reduce the number of hidden risks of accidents on the construction site
ω hi*(t)ω=α(t)g(1r)vi+c1r2<0 {{\partial h_i^*\left(t \right)} \over {\partial \omega}} = - {{\alpha \left(t \right)g\left({1 - r} \right)} \over {{v_i} + {c_1}{r^2}}} < 0 Increasing the intensity of punishment for illegal construction companies will stimulate the decline in the number of hidden accidents on the construction site
k hi*(t)k=1rvi+c1r2>0 {{\partial h_i^*\left(t \right)} \over {\partial k}} = {{1 - r} \over {{v_i} + {c_1}{r^2}}} > 0 The increase of illegal construction profits of construction enterprises will stimulate the rise of hidden dangers of accidents
z* (t) α (t) z*(t)α(t)=ϕgc2x2>0 {{\partial {z^*}\left(t \right)} \over {\partial \alpha \left(t \right)}} = - {{\phi g} \over {{c_2}{x^2}}} > 0 The increased importance of building safety in performance appraisal will increase the level of government safety supervision efforts
c2 z*(t)c2=ϕα(t)g+x1ρ+δfc22x2<0 {{\partial {z^*}\left(t \right)} \over {\partial {c_2}}} = - {{\phi \alpha \left(t \right)g + x{1 \over {\rho + \delta}}f} \over {c_2^2{x^2}}} < 0 Rising costs of building safety regulation will attenuate government safety regulation efforts
ϕ z*(t)c2=ϕα(t)g+x1ρ+δfc22x2<0 {{\partial {z^*}\left(t \right)} \over {\partial {c_2}}} = - {{\phi \alpha \left(t \right)g + x{1 \over {\rho + \delta}}f} \over {c_2^2{x^2}}} < 0 The increase of reward and punishment coefficient will stimulate the enthusiasm of government safety supervision
Numerical Simulation and Analysis

The article intuitively expounds on the evolution process of the differential game equilibrium strategy between the government and construction enterprises [10]. We calculate and analyze by assigning values to each variable. Where n = 3, ai = 20, vi = 1, k = 10, ω = 3, α (t) = 0.7, g = 0.8, l = 0.1, b = 2, ϕ = 0.01, c1 = 0.2, c2 = 0.04, f = 0.8, δ = 0.1, ρ = 0.004, x = 0.3, r = 0.3. We can obtain hi*(t)=21.4 h_i^*\left(t \right) = 21.4 , z* (t) = 65.66 by substituting the above parameters into equations (8) and (9). We replace the feedback Nash equilibrium strategy hi*(t) h_i^*\left(t \right) , z* (t) into Equation (1) to know: M = 252.42−10 e−0.1t. From the dynamic equation of accident hidden danger, we can obtain the change of accident hidden danger quantity at the construction site, as shown in Figure 1. It can be seen from Figure 1 that as time t continues, the hidden accident risk amount M (t) at the construction site increases continuously from 242.42.

Figure 1

The M(t) curve of the hidden accident quantity at the construction site

Figure 2 shows the change in the number of hidden dangers of accidents in construction enterprises. The curve h (t1) represents the number of hidden accidents caused by the construction of enterprises when the importance of building safety performance assessment changes. The curve h (t2) represents the change of hidden dangers under the superposition of the investment cost c1 required by the construction unit to control the hidden risks of accidents and the importance of assessing building safety performance. h (t3) represents the evolution of accident potential under the combined action of variable α (t) and punishment ω. It can be seen from Figure 2 that the output of hidden accident hazards of construction units is negatively correlated with the importance of building safety performance assessment [11]. This shows that the safety production performance assessment significantly impacts the government's safety production governance effect. If we reduce the cost of accident hidden danger management c1 simultaneously, the hidden accident quantity will drop to 19.52 at a faster rate. The curve h (t4) shows a positive correlation between the equilibrium accident hidden dangers of construction enterprises and the illegal construction income k of construction enterprises. The increase of illicit revenue construction k will lead to a decrease in the probability of safe construction of enterprises. The hidden accident quantity hi*(t) h_i^*\left(t \right) increases the possibility of building safety accidents.

Figure 2

hi (t) curve of hidden accident amount of construction enterprises

Figure 3 shows the dynamic change of government building safety supervision efforts. The curve z (t1) represents the development trend of the government's building safety supervision efforts with the shift in building safety performance assessment. z (t2) represents the degree of government building safety supervision efforts under the combined effect of the importance of building safety performance assessment and the strength of rewards and punishments [12]. The curve z (t3) shows the dynamic relationship between the government's building safety regulatory efforts and regulatory costs. It can be seen from Figure 3 that the government's building safety supervision efforts z (t) are positively correlated with the importance α (t) of building safety performance assessment and are negatively correlated with the supervision cost c2. This shows that the lower the cost of the government to verify whether construction companies have illegal construction, the higher the importance of building safety performance assessment. We compare curves z (t1) and curves z (t2). It is found that the slope of the curve z (t1) is much smaller than the slope of the curve z (t2). This shows that under the coupling effect of building safety performance assessment and reward and punishment incentives, the government's enthusiasm to implement building safety supervision is demonstrated.

Figure 3

z (t) curve of government building safety supervision efforts

Conclusion

This paper uses the differential game model of construction safety supervision to describe the behavioral characteristics, strategy selection, and equilibrium of the two main actors of the government and construction enterprises. Through numerical simulation, this paper makes the difference of the supervision effect under different parameters and obtains the following results. The hidden accident quantity hi*(t) h_i^*\left(t \right) of construction enterprises is negatively correlated with the importance a(t) of building safety performance assessment. It is negatively related to the government's punishment intensity ω. It is proportional to its own illegal construction income k. The government's balanced supervision effort z* (t) is an increasing function of the importance a (t) of building safety performance appraisal and is negatively correlated with its own supervision cost c2. It is positively related to the rewards and punishments of the higher-level government.

Figure 1

The M(t) curve of the hidden accident quantity at the construction site
The M(t) curve of the hidden accident quantity at the construction site

Figure 2

hi (t) curve of hidden accident amount of construction enterprises
hi (t) curve of hidden accident amount of construction enterprises

Figure 3

z (t) curve of government building safety supervision efforts
z (t) curve of government building safety supervision efforts

Influencing factors in differential game equilibrium

Equilibrium strategy Parameter Influence trend Explain
hi*(t) h_i^*\left(t \right) α (t) hi*(t)α(t)=ωg(1r)vi+c1r2<0 {{\partial h_i^*\left(t \right)} \over {\partial \alpha \left(t \right)}} = - {{\omega g\left({1 - r} \right)} \over {{v_i} + {c_1}{r^2}}} < 0 Increasing the proportion of building safety in the performance appraisal and reducing the investment in the management of hidden dangers of accidents will reduce the number of hidden risks of accidents on the construction site
ω hi*(t)ω=α(t)g(1r)vi+c1r2<0 {{\partial h_i^*\left(t \right)} \over {\partial \omega}} = - {{\alpha \left(t \right)g\left({1 - r} \right)} \over {{v_i} + {c_1}{r^2}}} < 0 Increasing the intensity of punishment for illegal construction companies will stimulate the decline in the number of hidden accidents on the construction site
k hi*(t)k=1rvi+c1r2>0 {{\partial h_i^*\left(t \right)} \over {\partial k}} = {{1 - r} \over {{v_i} + {c_1}{r^2}}} > 0 The increase of illegal construction profits of construction enterprises will stimulate the rise of hidden dangers of accidents
z* (t) α (t) z*(t)α(t)=ϕgc2x2>0 {{\partial {z^*}\left(t \right)} \over {\partial \alpha \left(t \right)}} = - {{\phi g} \over {{c_2}{x^2}}} > 0 The increased importance of building safety in performance appraisal will increase the level of government safety supervision efforts
c2 z*(t)c2=ϕα(t)g+x1ρ+δfc22x2<0 {{\partial {z^*}\left(t \right)} \over {\partial {c_2}}} = - {{\phi \alpha \left(t \right)g + x{1 \over {\rho + \delta}}f} \over {c_2^2{x^2}}} < 0 Rising costs of building safety regulation will attenuate government safety regulation efforts
ϕ z*(t)c2=ϕα(t)g+x1ρ+δfc22x2<0 {{\partial {z^*}\left(t \right)} \over {\partial {c_2}}} = - {{\phi \alpha \left(t \right)g + x{1 \over {\rho + \delta}}f} \over {c_2^2{x^2}}} < 0 The increase of reward and punishment coefficient will stimulate the enthusiasm of government safety supervision

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