# Building Construction Design Based on Particle Swarm Optimization Algorithm

Authors: Wenxue Song
Journal: Journal of Control Science and Engineering
 (2022)

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## Abstract

In order to take a scientific risk control strategy to reduce the safety risk of construction projects, a construction safety risk decision-making method based on particle swarm optimization algorithm was proposed. Through the analysis of prefabricated building construction safety risk factors, the combination of the Markov Chain and Bayesian networks method was used to estimate the probability of risk factors. The relationship between the various risk factors was described by conditional probability, and a safety risk loss-control investment double objective optimization model was built. The corresponding algorithm was designed and the R language programming was used to solve the problem. The experimental results showed that by taking a high degree of control over the risk factors of the investment strategy, when the constraint cost was RMB 200,000, the global optimal risk loss and the global optimal control cost were RMB 1,400,500 and 19,600, respectively. When the constraint cost was 280,000 yuan, the global optimal risk loss and global optimal control cost were 1.046 million yuan and 278.5 million yuan, respectively. When the constraint cost was 320,000 yuan, the global optimal risk loss and global optimal control cost were 910,100 yuan and 317,300, yuan respectively. It was concluded that, considering the risk correlation optimization model, a reasonable allocation strategy was adopted, combined with the actual situation, which performed a promoting function in improving the assembly building construction safety risk decision-making.


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## Body

## 1. Introduction

China’s annual new building area is about 2 billion square meters, accounting for 50% of the world’s annual new area [1]. The construction industry has made a great contribution to the national economy by bringing GDP growth and stimulating employment. However, the traditional construction production mode has not been suitable for the rapid development of the construction industry. And there are disadvantages such as heavy workload, long night construction time, serious waste of materials and resources, too many workers, and difficult management [2]. Therefore, it is easy to cause engineering accidents. Statistics show that at least three people die in construction accidents every day in our country. However, in the UK, where the prefabrication rate is as high as 70%, at least one construction worker dies every week on average [3]. Not only that, the environmental pollution, the exhaustion of energy, and the serious haze are also worth our concern.In the face of such a severe situation, it is imperative to vigorously develop prefabricated buildings. It can provide people with residential products with high quality, energy saving property, and environmental protection, as well as human and natural harmonious coexistence. Prefabricated components of prefabricated buildings are completed in the component factory, which can control the quality and avoid the impact of external factors on the quality of components. Prefabricated components can reduce on-site construction work and effectively inhibit dust pollution, noise pollution, and other environmental problems. Prefabricated components are standard components, which can effectively control the amount of raw materials, reduce the loss caused by on-site construction and pouring, and save resources [4]. The production environment of the factory is relatively stable, and the safety factor is much higher than that of the on-site construction. Mechanized production not only reduces the labor force greatly but also improves the work efficiency and reduces the safety risks, thus reducing the risks brought by the construction operation. See Figure 1.Figure 1
Based on particle swarm optimization algorithm.


## 2. Literature Review

From the fuzzy qualitative concept in the risk evaluation of prefabricated building safety performance, Fang analyzed the key factors affecting prefabricated structure safety. The evaluation index system was constructed. Based on the analytic hierarchy process and the entropy weight method, a cloud model-based prefabricated construction safety performance evaluation method was proposed [5]. Sukamani proposed a supply chain cost model using time-driven activity-based costing to minimize the cost of multiskill resources in prefabricated buildings [6]. The cost and time required for cross-training multiskill resources of prefabricated buildings were included in resource planning calculations by Hire, and integer and probabilistic optimization models were used to minimize the cost of using multiskill resources in off-site construction [7]. In order to optimize the cost management of prefabricated buildings and reduce the cost of capital, Tayeh discussed the variables affecting the high capital cost of prefabricated buildings and developed FAEM for cost optimization [8]. Combining the characteristics of construction engineering with order strategy, Duan built an inventory management model of building materials and realized the optimal value of inventory cost of building materials by solving a genetic algorithm [9]. Based on the meta-network analysis, Wang proposed an effective way to express the complex interaction of various factors involved in a project and extended the analysis scope to multiple dimensions to form a comprehensive project network covering the relationship between various influencing factors [10].The above research results enrich the construction safety optimization and risk correlation theory of construction projects, but there are few relevant research on the introduction of risk correlation into the construction safety optimization of prefabricated buildings, especially for the construction safety risk expected loss and risk control investment optimization problems that need to be further investigated. In conclusion, based on the identification of safety risk factors of prefabricated building construction, the research draws the Bayesian network diagram of safety risk factors of prefabricated building construction, describes the correlation between risk factors with conditional probability, and gives the occurrence probability of risk factors with Monte Carlo simulation. On the basis of this research, the dual-objective optimization model is constructed with the minimum security risk loss of the system and the optimal risk control investment as the objective function. Multi-objective particle swarm optimization is properly improved to be applied to decision-making scheme optimization. Using the R language programming, a multiobjective particle swarm optimization algorithm is designed to solve the model, and reasonable risk control strategies are provided.


## 3. Research Methods

### 3.1. Problem Description and Relevant Theoretical Preparation

#### 3.1.1. Problem Description

In the process of construction, the economic loss due to risk events or casualties can be obtained through statistical data intuitively. However, there are numerous risk factors involved in the process of fabricated construction. Risk factors influence and correlate with each other. So a scientific and reasonable evaluation method needs to be used to evaluate the possibility of risk factors. Traditional risk evaluation methods are mostly based on the subjective experience of experts to evaluate the probability of risk occurrence, such as the AHP method, the fuzzy logic method, etc. [11]. These methods are subjective and do not take into account the actual implementation effect of the construction site, so the evaluation results are often biased. The Bayesian network can well construct the causal dependence relationship between various variables and carry out probability analysis and simulation for each variable under different circumstances, which has a good advantage in analyzing the probability of risk occurrence. In addition, the construction safety risk factors compared with the project investment elasticity are different. Namely, the amount of risk reduction achieved is different for the same planned investment. If the risk factor is in a state of high elasticity, the corresponding control measures will have obvious effects. However, if the elasticity is low, even if the planned investment is large, it cannot guarantee that the risk level can be significantly controlled. Therefore, in the case of limited planned control resources, risk control should be invested in the link with greater flexibility, so that the total risk can be significantly reduced.For the optimization problem of construction safety risk lossV and risk control investment C, the abstract form of the risk decision planning model is defined as (1)–(5) [12]:(1)minV=fx1,x2,…,xn,(2)s.t.C≤Cmax,(3)minC=gx1,x2,…,xn,(4)V≤Vmax,(5)xi∈D,i=1,…,n.In the formulas,xi represents the decision variable of the i th risk source Ri. Cmax represents the upper limit of risk control investment. D represents the localized space of decision variable xi.


#### 3.1.2. Theoretical Preparation

(1) Bayesian Networks(1)
The Bayes’ rulePrior probability and posterior probability are relative to certain evidence [13]. X and Y are set as two random variables, X=x as a certain hypothesis, and Y=y as a group of evidence. Before considering the evidence, the probability PX=x of the event X=x is called the prior probability, and after considering evidence Y=y, PX=x|Y=y is called the a posteriori probability. The relationship between prior probability and posterior probability expressed by the Bayes’ rule is shown in Equation (6) [14].(6)PX=x/Y=y=PX=x,Y=yPY=y.(2)
The Bayesian networksBayesian Networks (BNs) are topological structures composed of a directed acyclic graph and related probability distribution functions, representing the joint probability distribution ofn random variables [15], which is composed of two parts, BNs=G,θ. G stands for a directed acyclic graph, whose nodes are random variables X1,X2,…,Xn. The directed edge is the qualitative representation of the joint probability distribution, which is used to indicate the conditional independent relationship between random variables. θ represents a set of parameters used to quantify the network.Bayesian network is used to describe the initial risk state relationship among risk sources. Assume thatR1 and R2 are two risk sources in a construction project, and the initial risk state of R1 will affect the initial risk state of R2. Figure 2 shows the relationship between the initial risk state of R1 and R2, that is, the initial state of R2 is affected by R1. The state space of risk grade I is set as low, medium, and high. Table 1 shows the conditional probability distribution column of initial risk state of risk source R2 [16].Figure 2
Bayesian example diagram.Table 1
Joint probability distribution column of risk sources R1 and R2.
R2 = LowR2 = MediumR2 = HighNotesR1 = LowP11P12P13P11+P12+P13=1R1 = MediumP21P22P23P21+P22+P23=1R1 = HighP31P32P33P31+P32+P33=1(2) The Markov Chain. Russian mathematician Andre Markov proposed Markov properties in 1906 [17]. Markov chains are random processes with Markov properties and state space in probability theory and mathematical statistics, which can be defined by the transfer matrix and transfer graph. The risk set of assembly building construction project is set as R=Ri,i=1,2,…,n. The risk level of Ri in different risk states will only remain unchanged or transfer to a higher level without any strategy. The risk level of the state space of risk factors is set for I=I1,I2,…,Im. In the transfer process, any level forms a random sequence IRit,t=0,1,2,…;IRit∈I. Assume that the future risk level of the risk source is only related to the current risk level, that is, there is no aftereffect of the grade state, so the transfer of the risk source level constitutes a Markov chain, which satisfies the following equation (7) [18]:(7)PIRit+1∣IRi0,IRi1,…,IRit=PIRit+1∣IRit.Further, it is assumed that the transfer between risk levels is a time-homogeneous Markov chain, and the one-step transfer probability ofRi transferring from risk level Ij to risk level Ik is expressed in formula (8) below:(8)pij,k=PIRit+1=Ik∣IRit=Ij,t=0,1,2,…In the state spaceI, the one-step transition probability matrix constituted can be expressed as formula (9):(9)pi1,1pi1,2⋯pi1,mpi2,1pi2,2⋯pi2,m⋮⋮⋱⋮pim,1pim,2⋯pim,m.In the formula,∑k=1mpij,k=1,j=1,2,…,m.(3) Joint Distribution of Accidents. Suppose that the expectation of joint accident distribution of risk source R1,R2,…,Rn is μ=μ1,μ2,…,μnT, that is, the accident rate of risk source Ri is μi. Then μi describes the average level of accident probability after Ri moves from initial risk state IRi0 to state IRi1 [19].


### 3.2. Construction Safety Risk Optimization Model of Prefabricated Building Construction

#### 3.2.1. Model Description

For risk sourceRi, its decision localization space D is divided into two categories, 0 and 1, D=0,1. That is, the degree of risk investment in Ri is divided into two different levels. xi=0 has a low level of investment. xi=0 represents a high level of investment. Suppose that the risk loss of Ri under the investment degree of xi is Li,xi, and the control investment is Ci,xi, then the relationship between the risk loss and the control investment under different investment degrees is as follows:(10)Li0>Li1,i=1,…,n,(11)Ci0<Ci1,i=1,…,n.Construction safety risk lossV is depicted by the overall expected risk loss value of the system, then V in (12) is defined as(12)fx1,x2,…,xn=∑i=1nμiLi,xi.In formula (13), the system overall control investment gx is defined as(13)gx1,x2,…,xn=∑i=1nCi,xi.


#### 3.2.2. Algorithm Solution Design

The particle swarm optimization (PSO) algorithm is a kind of meta-heuristic algorithm based on swarm, which has the advantages of simple setting, fewer optimization parameters, and fast convergence. A large number of practical problems can be transformed into multiobjective optimization problems. The particle swarm optimization algorithm has been widely used to solve multiobjective optimization problems in recent years due to its excellent characteristics in intelligent optimization algorithms. Multiobjective optimization problems cannot compare a set of advantages and disadvantages of noninferiority solution sets, while the actual decision-making situations usually need to select one from noninferiority solutions or some solution as a final solution problem. The main task of solving the multiobjective optimization problem is to find the noninferiority solution set as much as possible or get the best noninferiority solution set. In the research, in order to minimize the goalV and C, a particle is set as S0. For any particle xi,xj∈S0. Then the objective function value is Vi, Ci and Vj, Cj, respectively. If Vi≤Vj and Ci≤Cj, then xi is superior to xj. If Vj≤Vi and Cj≤Ci,xj is superior than xi . In other cases, xi and xj are called noninferiority relations. x0 is called a noninferiority particle if ∃x0∈S0, for ∀xi∈S0, there is a noninferiority relationship between x0 and xi or x0 is superior than xi.The established optimization model is a 0–1 integer programming model, so it is improved on the basis of the multiobjective particle swarm optimization algorithm. And the value range of particles in each position dimension is limited to {0, 1}. The inertia weight W is converted by the linear decreasing weight (LDW) strategy formula proposed by Shi, and the particle velocity update formula S is converted by the form constraint transformation function sigmoid (v).t is the iteration cycle, k is the particle number, and i is the risk source number. Let xkt=xk1t,xk2t,…,xknt′ be the position vector of particle k in round t iteration, vkt=vk1t,vk2t,…,vknt′ be the position vector of particle k in round t iteration, ξkt=ξk1t,ξk2t,…,ξknt′ be the historical optimal position vector of particle k before round t iteration, and ξgt=ξg1t,ξg2t,…,ξgnt′ be the global optimal position vector before round t iteration. The specific implementation steps [20,21] are as follows:Step 1: The parameters and initialization of the particle swarm are determined. The number of particles ism, the number of iterations is T, the upper limit of the number of elements Smax, the acceleration factors φ1,φ2, and the inertial weight parameters vmax and vmin are determined. Initialized particle position is x10,x20,…,xn0. The probability of each dimension of particles are taken as 0 or 1. Initialized particle velocity is v10,v20,…,vn0. Random number of Uniform−1,1 distribution is taken as each dimension of each particle.Step 2: Takeξk0=xk0k=1,2,…,n and determine the noninferior particle set S=o1,o2,…,ol according to ξ10,ξ20,…,ξn0.Step 3: Updateξkk=1,2,…,n and the noninferior solution set. For particle k, if xkt is superior than ξkt, then ξkt+1=xkt. If ξkt and xkt are noninferior relationship, then ξkt or xkt are randomly chosen as ξkt+1 with equal probability. According to ξ01,ξ02,…,ξ0t+1, the temporary noninferior solution set S′ is determined. According to S and S′, the new noninferior particles are determined. In accordance with it, S. is updated.Step 4: Calculate the probability information of noninferior particles inS. The particle position space is a set of 2n points. Suppose the particles of the noninferior solution set cover h points H1,H2,…,Hh in space. Suppose the point Hb have lb particles ∑b=1hlb=l, then take the characteristic probability of the noninferior particle qk as(14)qk=1hb,ok∈Hb,k=1,2,…,m.Ifl>Smax, Smax elements are randomly selected from S according to the unequal probability sampling method as the noninferior solution, and the sampling probability is the calculation result of Equation (14). According to Equation (14), the characteristic probability of noninferior particles is redefined. ξg is randomly selected from S by the equal probability sampling method.Step 5: Updatev and particle swarm velocity and position. As shown in Formula (15)–(18)(15)vt=vmax−vmax−vminTt,(16)vkt+1=vtvkt+φ1r1ktξkt−xkt+φ2r2kt⋅ξgt−xkt,(17)sigmoidvkit+1=11+exp−vkit+1,k=1,2,…,m;i=1,2,…,n,(18)xkit+1=1,rkit+1<sigmoidvkit+10,others,k=1,2,…,m;i=1,2,…,n.In the formulas,r1kt,r2kt,…,rkt=rk1t,rk2t,…,rknt′ is an n dimensional vector, and the value of each dimension is a random number with Uniform (0,1) independence and distribution.Step 6: If the iteration is completed, the particle position with the minimum value ofV+C is selected from S as the global optimal position. Otherwise, return to Step 3.


## 3.1. Problem Description and Relevant Theoretical Preparation

### 3.1.1. Problem Description

In the process of construction, the economic loss due to risk events or casualties can be obtained through statistical data intuitively. However, there are numerous risk factors involved in the process of fabricated construction. Risk factors influence and correlate with each other. So a scientific and reasonable evaluation method needs to be used to evaluate the possibility of risk factors. Traditional risk evaluation methods are mostly based on the subjective experience of experts to evaluate the probability of risk occurrence, such as the AHP method, the fuzzy logic method, etc. [11]. These methods are subjective and do not take into account the actual implementation effect of the construction site, so the evaluation results are often biased. The Bayesian network can well construct the causal dependence relationship between various variables and carry out probability analysis and simulation for each variable under different circumstances, which has a good advantage in analyzing the probability of risk occurrence. In addition, the construction safety risk factors compared with the project investment elasticity are different. Namely, the amount of risk reduction achieved is different for the same planned investment. If the risk factor is in a state of high elasticity, the corresponding control measures will have obvious effects. However, if the elasticity is low, even if the planned investment is large, it cannot guarantee that the risk level can be significantly controlled. Therefore, in the case of limited planned control resources, risk control should be invested in the link with greater flexibility, so that the total risk can be significantly reduced.For the optimization problem of construction safety risk lossV and risk control investment C, the abstract form of the risk decision planning model is defined as (1)–(5) [12]:(1)minV=fx1,x2,…,xn,(2)s.t.C≤Cmax,(3)minC=gx1,x2,…,xn,(4)V≤Vmax,(5)xi∈D,i=1,…,n.In the formulas,xi represents the decision variable of the i th risk source Ri. Cmax represents the upper limit of risk control investment. D represents the localized space of decision variable xi.


### 3.1.2. Theoretical Preparation

(1) Bayesian Networks(1)
The Bayes’ rulePrior probability and posterior probability are relative to certain evidence [13]. X and Y are set as two random variables, X=x as a certain hypothesis, and Y=y as a group of evidence. Before considering the evidence, the probability PX=x of the event X=x is called the prior probability, and after considering evidence Y=y, PX=x|Y=y is called the a posteriori probability. The relationship between prior probability and posterior probability expressed by the Bayes’ rule is shown in Equation (6) [14].(6)PX=x/Y=y=PX=x,Y=yPY=y.(2)
The Bayesian networksBayesian Networks (BNs) are topological structures composed of a directed acyclic graph and related probability distribution functions, representing the joint probability distribution ofn random variables [15], which is composed of two parts, BNs=G,θ. G stands for a directed acyclic graph, whose nodes are random variables X1,X2,…,Xn. The directed edge is the qualitative representation of the joint probability distribution, which is used to indicate the conditional independent relationship between random variables. θ represents a set of parameters used to quantify the network.Bayesian network is used to describe the initial risk state relationship among risk sources. Assume thatR1 and R2 are two risk sources in a construction project, and the initial risk state of R1 will affect the initial risk state of R2. Figure 2 shows the relationship between the initial risk state of R1 and R2, that is, the initial state of R2 is affected by R1. The state space of risk grade I is set as low, medium, and high. Table 1 shows the conditional probability distribution column of initial risk state of risk source R2 [16].Figure 2
Bayesian example diagram.Table 1
Joint probability distribution column of risk sources R1 and R2.
R2 = LowR2 = MediumR2 = HighNotesR1 = LowP11P12P13P11+P12+P13=1R1 = MediumP21P22P23P21+P22+P23=1R1 = HighP31P32P33P31+P32+P33=1(2) The Markov Chain. Russian mathematician Andre Markov proposed Markov properties in 1906 [17]. Markov chains are random processes with Markov properties and state space in probability theory and mathematical statistics, which can be defined by the transfer matrix and transfer graph. The risk set of assembly building construction project is set as R=Ri,i=1,2,…,n. The risk level of Ri in different risk states will only remain unchanged or transfer to a higher level without any strategy. The risk level of the state space of risk factors is set for I=I1,I2,…,Im. In the transfer process, any level forms a random sequence IRit,t=0,1,2,…;IRit∈I. Assume that the future risk level of the risk source is only related to the current risk level, that is, there is no aftereffect of the grade state, so the transfer of the risk source level constitutes a Markov chain, which satisfies the following equation (7) [18]:(7)PIRit+1∣IRi0,IRi1,…,IRit=PIRit+1∣IRit.Further, it is assumed that the transfer between risk levels is a time-homogeneous Markov chain, and the one-step transfer probability ofRi transferring from risk level Ij to risk level Ik is expressed in formula (8) below:(8)pij,k=PIRit+1=Ik∣IRit=Ij,t=0,1,2,…In the state spaceI, the one-step transition probability matrix constituted can be expressed as formula (9):(9)pi1,1pi1,2⋯pi1,mpi2,1pi2,2⋯pi2,m⋮⋮⋱⋮pim,1pim,2⋯pim,m.In the formula,∑k=1mpij,k=1,j=1,2,…,m.(3) Joint Distribution of Accidents. Suppose that the expectation of joint accident distribution of risk source R1,R2,…,Rn is μ=μ1,μ2,…,μnT, that is, the accident rate of risk source Ri is μi. Then μi describes the average level of accident probability after Ri moves from initial risk state IRi0 to state IRi1 [19].


## 3.1.1. Problem Description

In the process of construction, the economic loss due to risk events or casualties can be obtained through statistical data intuitively. However, there are numerous risk factors involved in the process of fabricated construction. Risk factors influence and correlate with each other. So a scientific and reasonable evaluation method needs to be used to evaluate the possibility of risk factors. Traditional risk evaluation methods are mostly based on the subjective experience of experts to evaluate the probability of risk occurrence, such as the AHP method, the fuzzy logic method, etc. [11]. These methods are subjective and do not take into account the actual implementation effect of the construction site, so the evaluation results are often biased. The Bayesian network can well construct the causal dependence relationship between various variables and carry out probability analysis and simulation for each variable under different circumstances, which has a good advantage in analyzing the probability of risk occurrence. In addition, the construction safety risk factors compared with the project investment elasticity are different. Namely, the amount of risk reduction achieved is different for the same planned investment. If the risk factor is in a state of high elasticity, the corresponding control measures will have obvious effects. However, if the elasticity is low, even if the planned investment is large, it cannot guarantee that the risk level can be significantly controlled. Therefore, in the case of limited planned control resources, risk control should be invested in the link with greater flexibility, so that the total risk can be significantly reduced.For the optimization problem of construction safety risk lossV and risk control investment C, the abstract form of the risk decision planning model is defined as (1)–(5) [12]:(1)minV=fx1,x2,…,xn,(2)s.t.C≤Cmax,(3)minC=gx1,x2,…,xn,(4)V≤Vmax,(5)xi∈D,i=1,…,n.In the formulas,xi represents the decision variable of the i th risk source Ri. Cmax represents the upper limit of risk control investment. D represents the localized space of decision variable xi.


## 3.1.2. Theoretical Preparation

(1) Bayesian Networks(1)
The Bayes’ rulePrior probability and posterior probability are relative to certain evidence [13]. X and Y are set as two random variables, X=x as a certain hypothesis, and Y=y as a group of evidence. Before considering the evidence, the probability PX=x of the event X=x is called the prior probability, and after considering evidence Y=y, PX=x|Y=y is called the a posteriori probability. The relationship between prior probability and posterior probability expressed by the Bayes’ rule is shown in Equation (6) [14].(6)PX=x/Y=y=PX=x,Y=yPY=y.(2)
The Bayesian networksBayesian Networks (BNs) are topological structures composed of a directed acyclic graph and related probability distribution functions, representing the joint probability distribution ofn random variables [15], which is composed of two parts, BNs=G,θ. G stands for a directed acyclic graph, whose nodes are random variables X1,X2,…,Xn. The directed edge is the qualitative representation of the joint probability distribution, which is used to indicate the conditional independent relationship between random variables. θ represents a set of parameters used to quantify the network.Bayesian network is used to describe the initial risk state relationship among risk sources. Assume thatR1 and R2 are two risk sources in a construction project, and the initial risk state of R1 will affect the initial risk state of R2. Figure 2 shows the relationship between the initial risk state of R1 and R2, that is, the initial state of R2 is affected by R1. The state space of risk grade I is set as low, medium, and high. Table 1 shows the conditional probability distribution column of initial risk state of risk source R2 [16].Figure 2
Bayesian example diagram.Table 1
Joint probability distribution column of risk sources R1 and R2.
R2 = LowR2 = MediumR2 = HighNotesR1 = LowP11P12P13P11+P12+P13=1R1 = MediumP21P22P23P21+P22+P23=1R1 = HighP31P32P33P31+P32+P33=1(2) The Markov Chain. Russian mathematician Andre Markov proposed Markov properties in 1906 [17]. Markov chains are random processes with Markov properties and state space in probability theory and mathematical statistics, which can be defined by the transfer matrix and transfer graph. The risk set of assembly building construction project is set as R=Ri,i=1,2,…,n. The risk level of Ri in different risk states will only remain unchanged or transfer to a higher level without any strategy. The risk level of the state space of risk factors is set for I=I1,I2,…,Im. In the transfer process, any level forms a random sequence IRit,t=0,1,2,…;IRit∈I. Assume that the future risk level of the risk source is only related to the current risk level, that is, there is no aftereffect of the grade state, so the transfer of the risk source level constitutes a Markov chain, which satisfies the following equation (7) [18]:(7)PIRit+1∣IRi0,IRi1,…,IRit=PIRit+1∣IRit.Further, it is assumed that the transfer between risk levels is a time-homogeneous Markov chain, and the one-step transfer probability ofRi transferring from risk level Ij to risk level Ik is expressed in formula (8) below:(8)pij,k=PIRit+1=Ik∣IRit=Ij,t=0,1,2,…In the state spaceI, the one-step transition probability matrix constituted can be expressed as formula (9):(9)pi1,1pi1,2⋯pi1,mpi2,1pi2,2⋯pi2,m⋮⋮⋱⋮pim,1pim,2⋯pim,m.In the formula,∑k=1mpij,k=1,j=1,2,…,m.(3) Joint Distribution of Accidents. Suppose that the expectation of joint accident distribution of risk source R1,R2,…,Rn is μ=μ1,μ2,…,μnT, that is, the accident rate of risk source Ri is μi. Then μi describes the average level of accident probability after Ri moves from initial risk state IRi0 to state IRi1 [19].


## 3.2. Construction Safety Risk Optimization Model of Prefabricated Building Construction

### 3.2.1. Model Description

For risk sourceRi, its decision localization space D is divided into two categories, 0 and 1, D=0,1. That is, the degree of risk investment in Ri is divided into two different levels. xi=0 has a low level of investment. xi=0 represents a high level of investment. Suppose that the risk loss of Ri under the investment degree of xi is Li,xi, and the control investment is Ci,xi, then the relationship between the risk loss and the control investment under different investment degrees is as follows:(10)Li0>Li1,i=1,…,n,(11)Ci0<Ci1,i=1,…,n.Construction safety risk lossV is depicted by the overall expected risk loss value of the system, then V in (12) is defined as(12)fx1,x2,…,xn=∑i=1nμiLi,xi.In formula (13), the system overall control investment gx is defined as(13)gx1,x2,…,xn=∑i=1nCi,xi.


### 3.2.2. Algorithm Solution Design

The particle swarm optimization (PSO) algorithm is a kind of meta-heuristic algorithm based on swarm, which has the advantages of simple setting, fewer optimization parameters, and fast convergence. A large number of practical problems can be transformed into multiobjective optimization problems. The particle swarm optimization algorithm has been widely used to solve multiobjective optimization problems in recent years due to its excellent characteristics in intelligent optimization algorithms. Multiobjective optimization problems cannot compare a set of advantages and disadvantages of noninferiority solution sets, while the actual decision-making situations usually need to select one from noninferiority solutions or some solution as a final solution problem. The main task of solving the multiobjective optimization problem is to find the noninferiority solution set as much as possible or get the best noninferiority solution set. In the research, in order to minimize the goalV and C, a particle is set as S0. For any particle xi,xj∈S0. Then the objective function value is Vi, Ci and Vj, Cj, respectively. If Vi≤Vj and Ci≤Cj, then xi is superior to xj. If Vj≤Vi and Cj≤Ci,xj is superior than xi . In other cases, xi and xj are called noninferiority relations. x0 is called a noninferiority particle if ∃x0∈S0, for ∀xi∈S0, there is a noninferiority relationship between x0 and xi or x0 is superior than xi.The established optimization model is a 0–1 integer programming model, so it is improved on the basis of the multiobjective particle swarm optimization algorithm. And the value range of particles in each position dimension is limited to {0, 1}. The inertia weight W is converted by the linear decreasing weight (LDW) strategy formula proposed by Shi, and the particle velocity update formula S is converted by the form constraint transformation function sigmoid (v).t is the iteration cycle, k is the particle number, and i is the risk source number. Let xkt=xk1t,xk2t,…,xknt′ be the position vector of particle k in round t iteration, vkt=vk1t,vk2t,…,vknt′ be the position vector of particle k in round t iteration, ξkt=ξk1t,ξk2t,…,ξknt′ be the historical optimal position vector of particle k before round t iteration, and ξgt=ξg1t,ξg2t,…,ξgnt′ be the global optimal position vector before round t iteration. The specific implementation steps [20,21] are as follows:Step 1: The parameters and initialization of the particle swarm are determined. The number of particles ism, the number of iterations is T, the upper limit of the number of elements Smax, the acceleration factors φ1,φ2, and the inertial weight parameters vmax and vmin are determined. Initialized particle position is x10,x20,…,xn0. The probability of each dimension of particles are taken as 0 or 1. Initialized particle velocity is v10,v20,…,vn0. Random number of Uniform−1,1 distribution is taken as each dimension of each particle.Step 2: Takeξk0=xk0k=1,2,…,n and determine the noninferior particle set S=o1,o2,…,ol according to ξ10,ξ20,…,ξn0.Step 3: Updateξkk=1,2,…,n and the noninferior solution set. For particle k, if xkt is superior than ξkt, then ξkt+1=xkt. If ξkt and xkt are noninferior relationship, then ξkt or xkt are randomly chosen as ξkt+1 with equal probability. According to ξ01,ξ02,…,ξ0t+1, the temporary noninferior solution set S′ is determined. According to S and S′, the new noninferior particles are determined. In accordance with it, S. is updated.Step 4: Calculate the probability information of noninferior particles inS. The particle position space is a set of 2n points. Suppose the particles of the noninferior solution set cover h points H1,H2,…,Hh in space. Suppose the point Hb have lb particles ∑b=1hlb=l, then take the characteristic probability of the noninferior particle qk as(14)qk=1hb,ok∈Hb,k=1,2,…,m.Ifl>Smax, Smax elements are randomly selected from S according to the unequal probability sampling method as the noninferior solution, and the sampling probability is the calculation result of Equation (14). According to Equation (14), the characteristic probability of noninferior particles is redefined. ξg is randomly selected from S by the equal probability sampling method.Step 5: Updatev and particle swarm velocity and position. As shown in Formula (15)–(18)(15)vt=vmax−vmax−vminTt,(16)vkt+1=vtvkt+φ1r1ktξkt−xkt+φ2r2kt⋅ξgt−xkt,(17)sigmoidvkit+1=11+exp−vkit+1,k=1,2,…,m;i=1,2,…,n,(18)xkit+1=1,rkit+1<sigmoidvkit+10,others,k=1,2,…,m;i=1,2,…,n.In the formulas,r1kt,r2kt,…,rkt=rk1t,rk2t,…,rknt′ is an n dimensional vector, and the value of each dimension is a random number with Uniform (0,1) independence and distribution.Step 6: If the iteration is completed, the particle position with the minimum value ofV+C is selected from S as the global optimal position. Otherwise, return to Step 3.


## 3.2.1. Model Description

For risk sourceRi, its decision localization space D is divided into two categories, 0 and 1, D=0,1. That is, the degree of risk investment in Ri is divided into two different levels. xi=0 has a low level of investment. xi=0 represents a high level of investment. Suppose that the risk loss of Ri under the investment degree of xi is Li,xi, and the control investment is Ci,xi, then the relationship between the risk loss and the control investment under different investment degrees is as follows:(10)Li0>Li1,i=1,…,n,(11)Ci0<Ci1,i=1,…,n.Construction safety risk lossV is depicted by the overall expected risk loss value of the system, then V in (12) is defined as(12)fx1,x2,…,xn=∑i=1nμiLi,xi.In formula (13), the system overall control investment gx is defined as(13)gx1,x2,…,xn=∑i=1nCi,xi.


## 3.2.2. Algorithm Solution Design

The particle swarm optimization (PSO) algorithm is a kind of meta-heuristic algorithm based on swarm, which has the advantages of simple setting, fewer optimization parameters, and fast convergence. A large number of practical problems can be transformed into multiobjective optimization problems. The particle swarm optimization algorithm has been widely used to solve multiobjective optimization problems in recent years due to its excellent characteristics in intelligent optimization algorithms. Multiobjective optimization problems cannot compare a set of advantages and disadvantages of noninferiority solution sets, while the actual decision-making situations usually need to select one from noninferiority solutions or some solution as a final solution problem. The main task of solving the multiobjective optimization problem is to find the noninferiority solution set as much as possible or get the best noninferiority solution set. In the research, in order to minimize the goalV and C, a particle is set as S0. For any particle xi,xj∈S0. Then the objective function value is Vi, Ci and Vj, Cj, respectively. If Vi≤Vj and Ci≤Cj, then xi is superior to xj. If Vj≤Vi and Cj≤Ci,xj is superior than xi . In other cases, xi and xj are called noninferiority relations. x0 is called a noninferiority particle if ∃x0∈S0, for ∀xi∈S0, there is a noninferiority relationship between x0 and xi or x0 is superior than xi.The established optimization model is a 0–1 integer programming model, so it is improved on the basis of the multiobjective particle swarm optimization algorithm. And the value range of particles in each position dimension is limited to {0, 1}. The inertia weight W is converted by the linear decreasing weight (LDW) strategy formula proposed by Shi, and the particle velocity update formula S is converted by the form constraint transformation function sigmoid (v).t is the iteration cycle, k is the particle number, and i is the risk source number. Let xkt=xk1t,xk2t,…,xknt′ be the position vector of particle k in round t iteration, vkt=vk1t,vk2t,…,vknt′ be the position vector of particle k in round t iteration, ξkt=ξk1t,ξk2t,…,ξknt′ be the historical optimal position vector of particle k before round t iteration, and ξgt=ξg1t,ξg2t,…,ξgnt′ be the global optimal position vector before round t iteration. The specific implementation steps [20,21] are as follows:Step 1: The parameters and initialization of the particle swarm are determined. The number of particles ism, the number of iterations is T, the upper limit of the number of elements Smax, the acceleration factors φ1,φ2, and the inertial weight parameters vmax and vmin are determined. Initialized particle position is x10,x20,…,xn0. The probability of each dimension of particles are taken as 0 or 1. Initialized particle velocity is v10,v20,…,vn0. Random number of Uniform−1,1 distribution is taken as each dimension of each particle.Step 2: Takeξk0=xk0k=1,2,…,n and determine the noninferior particle set S=o1,o2,…,ol according to ξ10,ξ20,…,ξn0.Step 3: Updateξkk=1,2,…,n and the noninferior solution set. For particle k, if xkt is superior than ξkt, then ξkt+1=xkt. If ξkt and xkt are noninferior relationship, then ξkt or xkt are randomly chosen as ξkt+1 with equal probability. According to ξ01,ξ02,…,ξ0t+1, the temporary noninferior solution set S′ is determined. According to S and S′, the new noninferior particles are determined. In accordance with it, S. is updated.Step 4: Calculate the probability information of noninferior particles inS. The particle position space is a set of 2n points. Suppose the particles of the noninferior solution set cover h points H1,H2,…,Hh in space. Suppose the point Hb have lb particles ∑b=1hlb=l, then take the characteristic probability of the noninferior particle qk as(14)qk=1hb,ok∈Hb,k=1,2,…,m.Ifl>Smax, Smax elements are randomly selected from S according to the unequal probability sampling method as the noninferior solution, and the sampling probability is the calculation result of Equation (14). According to Equation (14), the characteristic probability of noninferior particles is redefined. ξg is randomly selected from S by the equal probability sampling method.Step 5: Updatev and particle swarm velocity and position. As shown in Formula (15)–(18)(15)vt=vmax−vmax−vminTt,(16)vkt+1=vtvkt+φ1r1ktξkt−xkt+φ2r2kt⋅ξgt−xkt,(17)sigmoidvkit+1=11+exp−vkit+1,k=1,2,…,m;i=1,2,…,n,(18)xkit+1=1,rkit+1<sigmoidvkit+10,others,k=1,2,…,m;i=1,2,…,n.In the formulas,r1kt,r2kt,…,rkt=rk1t,rk2t,…,rknt′ is an n dimensional vector, and the value of each dimension is a random number with Uniform (0,1) independence and distribution.Step 6: If the iteration is completed, the particle position with the minimum value ofV+C is selected from S as the global optimal position. Otherwise, return to Step 3.


## 4. Results Analysis

### 4.1. Security Risk Factor Analysis and Bayesian Networks Structure Construction

A prefabricated building project in a city was elected as a sample for example verification. On the basis of the existing literature review, combined with the principle of risk evaluation index selection, the project construction safety risk factors were mainly divided into 6 aspects, and these 6 aspects mainly involved 23 risk sources. Risk control cost and risk loss (the unit is ten thousand yuan) were obtained by referring to the relevant data of the project in the last three years and by asking relevant project personnel. The final results were averaged. Combined with the above risk factor analysis, the Bayesian network structure of the project was drawn [22], as shown in Figure 3.Figure 3
Diagram of Bayesian network risk factors.
(a)(b)(c)(d)(e)


### 4.2. Construction of State Risk Probability Table and Transition Matrix

Through collecting similar items for statistical analysis, the probability of occurrence of each risk factor in different states was determined.On the basis of the probability of each risk occurring in different original states and considering the characteristics of each risk factor, multiple state transition matrices were generated. Due to space limitation, the research took the state transition matrix of riskR1 as shown in Table 2 for example. When R1 was in a low-risk state, there were three risk transfer states. The probability of maintaining a low-risk state is 0.38, the probability of transferring to a medium-risk state 0.33, and the probability of transferring to a high-risk state is 0.29. When R1 was in the medium-risk state, it was the same for other situations except when it could not be transferred to the low-risk state [23]. When R1 was in the high-risk state, it could only keep the high-risk state and could not transfer to the low- or medium-risk state without adopting the strategy. See Table 2.Table 2
State transition matrix of riskR1.
R1LowMediumHighLow0.380.320.28Medium0.000.550.46High0.000.001.00


### 4.3. Optimization Model Solution

The parameters are set as follows:ωmax=0.9, ωmax=0.4, T=5000, φ1=φ2=1.496, particle population size of 100, external memory capacity of 200.In order to reflect the specific relationship between the expected risk loss and the risk control investment, it is assumed that the control investment and the overall risk loss value of the system are unconstrained under the condition of sufficient capital. The calculation results are shown in Figure4, from which it can be seen that the risk loss value presents a nonlinear relationship with the risk control investment. With the increase of risk control investment, the overall risk loss value of the system gradually decreases. When the control investment approaches the minimum value, the system risk loss is the maximum. When the risk control investment approaches 400,000 yuan, all risks are controlled to the maximum extent.Figure 4
Diagram of V–C without constraints.However, due to limited funds, it is impossible to control all risks without limiting the actual projects, so risks should be controlled selectively according to the actual investment budget. In the research, two different constraint conditions are set. Constraint 1: the upper limit of overall risk lossVmax=2 million yuan, as well as the upper limit of total risk control investment Cmax=200,000 yuan. Constraint 2: Vmax=2 million yuan, Cmax=240,000 yuan. Noninferior solution sets are obtained under different constraints, as shown in Figures 5(a) and 5(b), respectively. It can be seen from Figure 5 that V–C is approximately nonlinear under different constraints. Noninferior solution sets are shown in Figure 5(a) and Figure 5(b), while the noninferior points in Figure 5(a) are relatively dispersed.Figure 5
Comparison of noninferior solution sets under different constraints. (a) The cost constraint is 200,000 yuan. (b) The cost constraint is 240,000 yuan.
(a)(b)The optimization control effects under different constraints are analyzed. When the constraint cost is RMB 200,000, the risk factors for adopting high degree of control investment strategy areR1,R5,R6,R7,R14,R19,R20,R22. Then the global optimal risk loss and the global optimal control cost are 144. 500,000 yuan and 199,600 yuan, respectively [24].When the constraint cost is 240,000 yuan, the risk factors for adopting high degree of control investment strategy areR1,R2,R5,R6,R7,R14,R15,R19,R20,R21,R22,R23. Then the global optimal risk loss and global optimal control cost are 1,194,100 yuan and 239,400 yuan, respectively [25].


## 4.1. Security Risk Factor Analysis and Bayesian Networks Structure Construction

A prefabricated building project in a city was elected as a sample for example verification. On the basis of the existing literature review, combined with the principle of risk evaluation index selection, the project construction safety risk factors were mainly divided into 6 aspects, and these 6 aspects mainly involved 23 risk sources. Risk control cost and risk loss (the unit is ten thousand yuan) were obtained by referring to the relevant data of the project in the last three years and by asking relevant project personnel. The final results were averaged. Combined with the above risk factor analysis, the Bayesian network structure of the project was drawn [22], as shown in Figure 3.Figure 3
Diagram of Bayesian network risk factors.
(a)(b)(c)(d)(e)


## 4.2. Construction of State Risk Probability Table and Transition Matrix

Through collecting similar items for statistical analysis, the probability of occurrence of each risk factor in different states was determined.On the basis of the probability of each risk occurring in different original states and considering the characteristics of each risk factor, multiple state transition matrices were generated. Due to space limitation, the research took the state transition matrix of riskR1 as shown in Table 2 for example. When R1 was in a low-risk state, there were three risk transfer states. The probability of maintaining a low-risk state is 0.38, the probability of transferring to a medium-risk state 0.33, and the probability of transferring to a high-risk state is 0.29. When R1 was in the medium-risk state, it was the same for other situations except when it could not be transferred to the low-risk state [23]. When R1 was in the high-risk state, it could only keep the high-risk state and could not transfer to the low- or medium-risk state without adopting the strategy. See Table 2.Table 2
State transition matrix of riskR1.
R1LowMediumHighLow0.380.320.28Medium0.000.550.46High0.000.001.00


## 4.3. Optimization Model Solution

The parameters are set as follows:ωmax=0.9, ωmax=0.4, T=5000, φ1=φ2=1.496, particle population size of 100, external memory capacity of 200.In order to reflect the specific relationship between the expected risk loss and the risk control investment, it is assumed that the control investment and the overall risk loss value of the system are unconstrained under the condition of sufficient capital. The calculation results are shown in Figure4, from which it can be seen that the risk loss value presents a nonlinear relationship with the risk control investment. With the increase of risk control investment, the overall risk loss value of the system gradually decreases. When the control investment approaches the minimum value, the system risk loss is the maximum. When the risk control investment approaches 400,000 yuan, all risks are controlled to the maximum extent.Figure 4
Diagram of V–C without constraints.However, due to limited funds, it is impossible to control all risks without limiting the actual projects, so risks should be controlled selectively according to the actual investment budget. In the research, two different constraint conditions are set. Constraint 1: the upper limit of overall risk lossVmax=2 million yuan, as well as the upper limit of total risk control investment Cmax=200,000 yuan. Constraint 2: Vmax=2 million yuan, Cmax=240,000 yuan. Noninferior solution sets are obtained under different constraints, as shown in Figures 5(a) and 5(b), respectively. It can be seen from Figure 5 that V–C is approximately nonlinear under different constraints. Noninferior solution sets are shown in Figure 5(a) and Figure 5(b), while the noninferior points in Figure 5(a) are relatively dispersed.Figure 5
Comparison of noninferior solution sets under different constraints. (a) The cost constraint is 200,000 yuan. (b) The cost constraint is 240,000 yuan.
(a)(b)The optimization control effects under different constraints are analyzed. When the constraint cost is RMB 200,000, the risk factors for adopting high degree of control investment strategy areR1,R5,R6,R7,R14,R19,R20,R22. Then the global optimal risk loss and the global optimal control cost are 144. 500,000 yuan and 199,600 yuan, respectively [24].When the constraint cost is 240,000 yuan, the risk factors for adopting high degree of control investment strategy areR1,R2,R5,R6,R7,R14,R15,R19,R20,R21,R22,R23. Then the global optimal risk loss and global optimal control cost are 1,194,100 yuan and 239,400 yuan, respectively [25].


## 5. Conclusions

Through the analysis of prefabricated building construction safety risk factors, the combination of the Markov Chain and Bayesian networks methods was used to estimate the probability of risk factors. The relationship between the various risk factors was described by conditional probability, and a safety risk loss-control investment double objective optimization model was built. The corresponding algorithm was designed and the R language programming was used to solve the problem. In the model, different control strategies can be adopted according to the available risk control investment, which provides a new idea for assembly building construction safety risk management.For the control intensity of each risk factor, only two degrees of comparison were considered. The control degree of each factor was set as a continuous variable, and the control degree was expressed by a function model. A variety of algorithms were used to verify the model.



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*Source: 7139230-2022-06-29.xml*