The reasons of attracting each other among fireflies depend on two factors: their own brightness and attractiveness. The brightness of fireflies depends on the target of their location. The higher brightness indicates better location, which means the better target. Attractiveness is related to brightness. The brighter fireflies own higher attractiveness which can attract fireflies with weaker brightness to move in this direction within the scope of sight. If the brightness is identical, fireflies will move randomly. Brightness and attractiveness are inversely proportional to the distance between fireflies, both decrease with increasing distance. Mathematical description of the algorithm is as follows [17]:

Definition 1: The firefly relative fluorescence brightness is:

(3)

where: I₀– firefly maximum fluorescence brightness, the higher brightness indicates better location, which means the better target, more easy to attract other fireflies move to himself. γ– absorption coefficient of light intensity. It is usually set to be a constant, and represent a change of the fluorescence intensity in the communication process. r_ij– spatial distance between firefly i and j, It is determined by the initial position of fireflies, the greater the distance, the lower the attraction between fireflies.

Definition 2: The firefly's attractiveness is:

(4)

where: β₀– the maximum attractiveness, which is the attractiveness of light source. Fireflies were attracted by the one which has a better attractiveness, and other fireflies will move to it. β(r)- the attractiveness after moving.

Definition 3: The location update. Where the firefly i attracted by the firefly j moves is determined by the following equation:

(5)

where: x_i,x_j – the spatial location of firefly; α – step factor, which is usually a constant between [0, 1]; rand– random factor, which is uniformly distributed between 0 and 1.

Firefly algorithm has advantages and potential in searching for the global optimal solution and solving NP-hard problem, but the basic algorithm has defects, which are highly depended on the distribution of the initial solution, easy to stagnation in early stage, low solution accuracy, and slow convergence in late stage [18, 19].

In order to avoid stagnation existing in FA, reduce the possibility of falling into local optima in the initialization, cube mapping equation (6) was applied to produce the m individual fireflies. Then the equation (7) was applied to obtain the projection of the locations of m fireflies.

(6)

(7)

where: N_P – upper limits of the search space; N_P – lower limits of the search space; N_P – d^th dimension of the i^th firefly produced by equation (6); N_P – coordinate of the i^th firefly on the d^th dimension of the search space.

On this basis, dynamic population firefly algorithm was introduced. Firstly, set a threshold value MaxT, and then global search was done. When iterations reach MaxT, the equation (8) was used to get ps new individuals. In order to maintain the diversity of population, and strengthen the scope of the search, the original populations of ps individuals were replaced by some new generation individuals with a random way.

(8)

where: y_i,j – the location of the new generation of firefly individual i; ps– the number of new individuals generated; dim – dimension.

The equation (8) shows that the new firefly individuals scattered around the global optimum position. It can effectively increase the traction of the original population, greatly improve the convergence rate and make more opportunities for the algorithm to search more accurate solution. Finally, Gaussian disturbance was employed on the current optimal solutions of each generation generated based on equation (9), and a new global optimal solution was obtained by using equation (10).

(9)

where: NGbest – position after disturbance; Gbest*– position before disturbance

(10)

The framework of the algorithm was shown in Fig. (2).

Fig. (2). Framework of the improved algorithm.

TSP problem (Travelling Salesman Problem) is one of the famous combinatorial optimization problems with NP-hard. In order to verify the performance of the improved algorithm, three typical TSP problem (Berlin52, Pr107, D1198) were selected. Improved firefly algorithms and several typical basic algorithms (basic ACO, basic PSO, basic FA) were compared. The basic sparameters of our improved algorithm: Population size is 200 and the maximum evolution algebra is 600. The results were shown in Table 1.

As shown in Table 1, the basic ACO and basic PSO have slow convergence and low accuracy, and are easy to fall into local optima. The basic firefly algorithm is more successful, and it overcomes the premature defects to some extent, although there are some errors between the calculation results and the optimal value. The improved algorithm is better in local optimization, avoids the local stagnation effectively. The convergence rate and the number of iterations are better than basic firefly algorithm. The Fig. (3) shows four algorithms iterations and curve optimal value for D1998 test function.

Fig. (3). Four algorithms iterations and curve optimal value.

According to the ultimate aims of construction system reliability, which are shortest duration and the lowest cost, best quality and best safety performance, the objective function was established, as shown in equation (1). Constraints were set to meet the goals of the project, which are minimum duration, minimum costs, minimum schedule reliability, minimum cost reliability, minimum quality reliability and minimum safety reliability, as shown in equation (2). Calculation model of construction system reliability was established. Dynamic firefly algorithm was applied to optimize the function. Specific steps were shown as follows.

Step 1: Initialize the firefly size N, the number of firefly m, maximum attractiveness β₀ of each firefly, absorption coefficient γ, random step size α and the iterations MaxT.

Step 2: Initialize the position of fireflies: select the optimum firefly as an initial individual in feasible space.

Step 3: Update the brightness, attractiveness and position of fireflies according to equations (3), (4) and (5) respectively.

Step 4: If the number of iterations is equal to MaxT, then,

Apply equation (6) to produce ps new individuals.

Replace ps individuals of original population by the new individuals in the random way.

For each replaced individual firefly, calculate the target value f(t_i, t_i,c_i,q_i,s_i) corresponding to its new position.

Record the current optimum, go to step 7.

Step 5: Employ Gaussian disturbance to obtain current optimal value based on equation (9) and obtain new global optimum value through equation (10).

Step 6: If the algorithm stopping criterion is satisfied, then stop. Otherwise, go to step 3.

Step 7: Output the optimum value Gbest.

A sub-project of a large construction project in Handan City, Hebei Province, China was selected as an example, and its construction reliability was analysed. The sub-project is a residential building of thirty-two-story frame shear wall structure. Construction area is 30240 square meters, the completion period stipulated is 428 days, and the estimated total project cost is 42088622 yuan. The network plan was divided into thirteen partitioned projects and each partitioned project of engineering was regarded as unit of work. The construction reliability relationship is shown in Fig. (4).

Fig. (4). Construction reliability relationship.

The raw data were shown in Table 2.

Here, four aspects of unit of work, which are work time, cost, quality and safety, were taken into consideration. Based on the data shown in Table 2, Monte Carlo simulation and Fault Tree method were used to acquire probability distribution histogram of the total duration of the completed project (Fig. 5), probability distribution histogram of the total cost of the completed project (Fig. 6), quality reliability and safety reliability (Table 3), and cost - time, quality - time, safety - time relationship model (Table 4).

Through the statistics, the probability density function of the total duration approximately obeys normal distribution N (413.79,11.442²). The duration reliability of 428 days (the completion period stipulated) is R_T =0.8365.

And the probability density function of the total cost approximately obeys normal distribution N (415.32,13.952²). The cost reliability of 42088622 yuan (the estimated total project cost) in 428 days is R_T =0.7357.

Fig. (5). Probability distribution of the total duration of the completed project.

Fig. (6). Probability distribution of the total cost of the completed project.

The objective function:

(11)

where: CP– key line of the network. α_i – unified taken 2100yuan / day.

(12)

The improved Firefly algorithm was used to optimize the model. And the parameters are: the size of population N = 500, maximum evolution algebra MaxT = 1000, crossover probability pc = 0.55, mutation probability pm = 0.03. MATLAB Programming was used to get the optimal solution after 200 cycles. Each optimal solution represents a decision combination of time, cost, quality and safety of the 13 units of work. Optimal results obtained are shown in Table 5.

From the above data, we can know that the reliability of each subsystem is in a "qualified" status, the construction period and costs are below expectations. Thus, construction feasibility of the project is verified.

In order to solve the problems of multi-objective control, objective quantification difficulties and computational complexity existing in the construction management. The process of construction project was divided into several units of work based on system decomposition theory, and the reliability of each unit of work was calculated. Then, the characteristics of chaos and dynamic population were employed to improve the initial solution and the scope of search, and employed the improved firefly algorithm to obtain construction system reliability. Engineering practice shows that the improved algorithm has higher convergence speed and accuracy of optimization. Thus, we can draw the following conclusions:

• (1)  The reliability relationship between each unit of work in project construction system are analyzed, specifically, the construction project network diagram is build and the construction system reliability analysis of network graph model is established. It completed the multi-objective collaborative optimization.
• (2)  Firefly algorithm is introduced and improved for its weakness. What’s more, engineering practice shows that the improved firefly algorithm has higher convergence speed and accuracy of optimization.
• (3)  The study provides a new way for effective construction management, and it has an important theory and practical significance to improve construction management, at the same time, it also expand the application area of firefly algorithm.