Multi-objective Optimization of Construction Project Time-Cost-Carbon Emission Based on BIM Technology

. In the construction process, a large number of resources are consumed in a short period, leading to high intensity and concentration of carbon emissions. From the perspective of China’s Double Carbon Policy, a multi-objective optimization problem integrating BIM technology and intelligent optimization algorithm is proposed in this paper. Firstly, using BIM modeling software to construct a 3D model generates bills of quantities as the basis to obtain the initial research data. Then a mathematical model of time-cost-carbon emissions is constructed with the idea of multi-objective optimization and a double-layer encoding NSGA-II algorithm is designed to solve the optimal set of solutions for the combination of activity execution modes under the premise of satisfying the priority relationship. Finally, a construction project is used as a case to verify the feasibility and rationality of the problem proposed in this paper, and demonstrate that the method can e ﬀ ectively guide the optimization management of the construction process, reduce construction carbon emissions, and promote sustainability of the construction industry.


Introduction
Excessive emissions of greenhouse gases have led to global warming, with serious impacts on natural ecosystems and human development. The construction industry is not only a pillar industry of China's national economy but also one of the largest carbon emission sources in the country, consuming about 30% of the world's energy and emitting 25% of greenhouse gases [1]. The construction process consumes a large number of resources in a short period, leading to high intensity and concentration of carbon emissions. China urgently needs to take corresponding countermeasures to improve the engineering management level, reduce carbon emissions in the construction industry, and contribute to the mitigation of global warming [2]. In the context of China's Double Carbon Policy, the Ministry of Housing and Urban-Rural Development of the People's Republic of China has introduced a series of policies to mandate the calculation of construction carbon emissions to improve energy resource utilization and promote high-quality and green development of the construction industry.
It is critical to complete project planning with less time and cost [3]. However, reducing the time of the project causes extra costs because this requires the use of additional resources. Therefore, the time-cost tradeoff problem is a well-known project scheduling problem that has been extensively studied [4][5][6][7][8][9][10][11]. A project consists of a series of activities, each of which may have multiple execution modes. Activities require different resources to execute in different modes, resulting in different time and cost, etc. This makes the multi-mode resource-constrained project scheduling problem (MRCPSP) one of the central problems of project management [12][13][14][15].
Project managers should not only consider the time and cost of the project when making optimization decisions but should also focus on the environment [16][17][18][19][20]. The United Nations Intergovernmental Panel on Climate Change (IPCC) proposed the carbon emission factor method, which provides a basis for quantifying carbon emissions. Peng Bo [21] applied this method in expressway construction. Building Information Modelling (BIM) is a major change in the construction industry and has been found to be the most effective tool among many digital technologies to reduce the cost and improve the management of construction projects [22,23]. Nguyen [24] used BIM technology to build a 3D model to summarize the resources needed for the project, and based on this, he built a mathematical model, and finally a novel algorithm was designed to solve this model. This paper will solve the time-cost-carbon emission trade-off problem. The paper contributes to the relevant body of knowledge as follows. (1) A novel problem by integrating BIM technology and intelligent optimization algorithm is proposed to facilitate resource tradeoff in project scheduling. (2) Incorporates carbon emission objective into the optimization category and a mathematical model of time-cost-carbon emissions is constructed, enriching the application of multi-objective optimization theory. (3) A double-layer encoding NSGA-II algorithm is designed to provide the optimal solution set of activity execution modes combination. (4) BIM technology is introduced to generate bills of quantities to obtain initial research data for project resources. (5) Findings of this research give useful references for construction project managers to determine quickly and accurately schedules during project implementation.
The remainder of this paper is organized as follows: In the second part, the time-costcarbon emission multi objective trade off problem is described, the assumptions of this problem are presented, and the meanings of the variables are defined. In the third part, a multi objective optimization model is developed for the proposed problem. In the fourth part, a double layer encoding NSGA-II algorithm is designed to solve the optimal solution set for the combination of activity execution models. In the fifth part, a construction project is used as a case study to verify the feasibility and rationality of the problem proposed in this paper, and the efficiency of the algorithm designed for the problem is demonstrated. Finally, conclusions and future research are outlined.

Problem Description
A project consists of a set of activities. Each activity i(i = 1 · · · n) has to be executed in one of modes. The different resource requirements for each mode will result in corresponding changes in time, cost and carbon emissions. Therefore, the optimal combination of time, cost and carbon emissions can be achieved by selecting the mode for each activity. A series of activities of the project and their priority relationships can be represented by an activity-onnode (AON) network diagram, where S t and F in are two dummy activities that represent the beginning and end of the project, as shown in figure. 1.

Assumptions
To facilitate the construction and calculation of the time-cost-carbon emissions optimization model for projects, the following assumptions are made: (1) A single project consists of a series of activities.
(2) Each activity can only be started after all previous activities have been completed, and once started, it cannot be interrupted or changed mode during the implementation.
(3) The mode of the activities in the project is limited, and the time, cost and carbon emissions generated under different modes are different.
(4) The time, cost and carbon emission for all activities shall not exceed a limited amount. (5) Cost is considered only the sum of labor, materials and machines costs. (6) The management objective is to minimize time, cost and carbon emission.

Define the model variables
The symbols and meanings of the variables used in the model are shown in table. 1.

Description of the Mathematical Model
The first objective is to minimize the time. The time of activities is based on three factors: quantity of work, work efficiency and the number of workers.
The second objective is to minimize the cost. The cost of the activity is influenced by the market price of workers, materials and machines.
The third objective is to minimize carbon emission. The carbon emissions of the activities are calculated using the quantities of workers, materials and machines consumed and carbon emission factors. The number of workers in activity i in mode j S i j Work efficiency of activity i in mode j C i j Cost of activity i in mode j k i j w The number of workers w consumed by activity i in mode j q w The unit price of worker w k i j m The number of materials m consumed by activity i in mode j q m The unit price of material m k i j r The number of machines r consumed by activity i in mode j q r The unit price of machine r e i j Carbon emission of activity i in mode j F w Carbon emission factor for workers w Maximum cost E up Maximum carbon emission

Description of Constraints
Since only one mode can be selected for each activity, therefore, we use the constraint: The start time of each activity depends on the completion of its predecessor activities: For projects, it is also realistic to limit the time, cost and carbon emission of the project.

Multi-objective Optimization Model
Based on the previous description, the "time-cost-carbon emission" multi-objective model is constructed. s.t.

Double-layer Encoding NSGA-II
The time-cost-carbon emission tradeoff problem is an NP-hard problem, which is difficult to solve by the exact solution methods [25]. Thus, use the improved NSGA-II algorithm to solve the problem. NSGA-II algorithm is a fast non-dominated multi-objective optimization algorithm with an elite retention strategy, which has been widely studied and applied in solving multi-objective problems [26][27][28]. In this paper, we combine the algorithmic framework of NSGA-II algorithm and design a new chromosome encoding strategy, crossover operator and mutation operator.

Encoding Strategy
Activities must satisfy the priority constraints and consider the modes, so it is a very important part of the algorithm to design a chromosome encoding strategy. This paper proposes a double-layer encoding strategy. The first layer is the activity, which represents the priority relationship of activities. Firstly, the activities without the predecessor activities are placed in set 1, then all the activities whose predecessor activities belong to set 1 are placed in set 2, and so on. Note that the activities in each set can be arranged arbitrarily, but the sets must be arranged in the order of set 1 first and then set 2. The second layer is the mode, which is used to specify the execution mode for the activity. figure. 2. is an example of a chromosome for figure. 1.

Crossover and Mutation Operator
After performing the crossover and mutation operation, the chromosome still needs to satisfy the priority constraint and assign the mode. The crossover and mutation operation are designed according to the encoding strategy introduced in 4.1. Assume that the chromosomes performing the crossover operation are A 1 and A 2 and the new chromosomes generated after crossover are B 1 and B 2 . Two parameters q 1 and q 2 are randomly generated and represent the position of crossover of the activity layer and mode layer, respectively. The activity layer of B 1 consists of the set before q 1 +1 in the activity layer of A 1 and the set after q 1 in the activity layer of A 2 , and the mode layer of B 1 consists of the set before q 2 +1 in the mode layer of A 1 and the set after q 2 in the mode layer of A 2 . The activity layer of B 2 consists of the set before q 1 +1 in the activity layer of A 2 and the set after q 1 in the activity layer of A 1 and the mode layer of B 2 consists of the set before q 2 +1 in the mode layer of A 2 and the set after q 2 in the mode layer of A 1 . For example, when q 1 =5 and q 2 =4. The diagram of the crossover operation is shown in figure. 3. A 1 is the chromosome to be mutated. Randomly generated parameters r 1 and r 2 , represent the mutation positions of the activity and mode layers, respectively. For the activity layer, if there are two or more activities in the selected set, two points are randomly generated in the set and the positions are exchanged, otherwise, r 1 is updated and re-mutated. For the mode layer, all activities in the set r 2 randomly select modes to replace the original modes. For example, when r 1 =6, two points are randomly generated in set 6 g 1 =2, g 2 =4 and exchange the positions of both. When r 2 =4, replace the modes for the activities in set 4. The diagram of the mutation operation is shown in figure. 4.

Overall Procedure of The Presented Method
step1. Set the population size, the maximum number of iterations, the crossover probability and the mutation probability.
step2. Generate the initial population using the encoding strategy presented in Section 4.1.
step3. Calculate non-dominated rank and distance of crowding. step4. Using the binary tournament selection method, two chromosomes in the population are selected randomly based on the non-dominance rank and crowding degree.
step5. Using the crossover operator and mutation operator proposed in section 4.2 to generate offspring populations.
step6. Merge the parent and offspring populations and generate a new population. step7. If is reached, step8 is performed, otherwise, step3 is returned. step8. Output the solution set.

Description of Case Problem
A construction project in the Sichuan Province of China is selected as the case study. BIM modeling software is used to obtain bills of quantities for 11 activities of this project, and the priority relationship between activities is shown in figure. 2. During the project implementation, construction activities can be studied in normal mode, crash mode and saving mode. The time, cost and carbon emission indicators of the project are calculated separately by the method proposed in the previous section, and these indicators are used as the initial data of the multi-objective optimization model, as shown in table. 2. In this case, if all activities are executed in normal mode, the time, cost and carbon emissions are (307, 69076, 33338). Therefore, the limits of the optimization objectives are set as follows: T up =307, C up =67076, E up =33338.

Result of Case Problem
Set the population size N to 150, the maximum number of iterations G to 200, the crossover probability P c to 0.8, and the mutation probability M u to 0.05. The double-layer encoding NSGA-II algorithm is programmed in Matlab to solve the multi-objective model, and 95 optimal solutions are finally obtained. table. 4 shows the five solutions in the Pareto solution set. It can be seen that the activities in each solution are satisfying the priority constraint and are assigned an execution mode. One objective value of a solution is better, but the other two objective values are at a disadvantage.

Discussion
In traditional engineering management generally discusses time, cost and quality together [15] [19], the introduction of carbon emission objectives is a new expansion in the field of multi-objective optimization of engineering management. Analyzing the existing research results, expert estimation method is mostly used for quantification of carbon emissions [18] [24], while the carbon emission factor method used in this paper can calculate the carbon emission of the construction process more scientifically and accurately. In addition, the double-layer encoding NSGA-II algorithm is more efficient than the traditional genetic algorithm (GA) in that it uses a non-dominated ranking procedure to simplify the multiobjective to an adaptation function, which can solve any number of objective problems using this method. The crossover and mutation operation are designed to avoid infeasible solutions.
The optimization results show that the NSGA-II algorithm is able to provide the optimal set of solutions for the combination of activity execution modes under the premise of satisfying the priority relationship. The resulting offers useful information for project implementation and Project managers can evaluate the cons and pros of each potential solution to achieve the best project schedule based on the actual situation of the project.

Conclusion and Further Work
In this paper, a multi-objective optimization problem that combines BIM technology with intelligent optimization algorithms is proposed with activity execution mode as the decision variable. BIM modeling software to construct a 3D model, which generates bills of quantities. By aggregating the types and quantities of resource consumption of each activity under different execution modes, the time, cost and carbon emission of the project are calculated, and the mathematical model of time-cost-carbon emission is established with the idea of multiobjective optimization. For the specificity of the proposed problem, a double-layer encoding NSGA-II algorithm is designed to solve the optimal set of solutions for the combination of activity execution modes under the premise of satisfying the priority relationship. The feasibility and rationality of the problem proposed in this paper are verified by a case study, which proves that the method can effectively guide the optimal management of the construction process, reduce construction carbon emissions, and promote the high-quality and green development of the construction industry. Future research in this area has four aspects: calculating carbon emissions is predicated on carbon emission factors, so we need to compile a unified and comprehensive library of carbon emission factors. Secondly, practical problems are often complex and need to consider more constraints and more objective dimensions. In order to solve NP-hard problems, more efficient heuristic algorithms need to be designed. There are more aspects of BIM technology being used in the optimization field that can be explored.