The management model of construction plane layout based on Pareto ant colony genetic algorithm

Abstract

The problem of plane layout in construction site is to optimize the use of construction site space in the case of meeting multiple conflicting or uniform arrangement objectives and site constraints, and place the construction of temporary facilities in the construction site of the effective space. Aiming at the problems of the construction site layout, the genetic algorithm and the new ant colony algorithm were fused to gain the optimal solution of the continuous space, and the Pareto ant colony genetic algorithm was developed to solve the multi-objective optimization problem, using mathematical model to optimize the layout of the construction site to solve the problem of safety management from the objective level. The safety model of the construction site layout and the corresponding research methods were proposed, in order to improve the optimization of genetic algorithm, this paper used the ant colony algorithm to improve the quality of initial solution of genetic algorithm, and then get better optimal solution, through the experimental data analysis, the larger the initial population size based on Pareto ant colony genetic algorithm, the more diverse and global the resulting solutions, it was very suitable for solving the problem of the plane layout of the construction site.

Keywords

Construction plane layout ant colony genetic algorithm Pareto multi-objective optimization

1 Introduction

The book “An ant colony system hybridized with a new local search for the sequential ordering problem” presented that Genetic algorithm and Ant Colony System (ACS) has such common characteristics as wide range of adaptation and general performance and others, it was widely used in discrete systems engineering optimization [1]. They both have their own advantages and disadvantages: the genetic algorithm has a strong global search capability, especially when the crossover probability is relatively large, it can generate a lot of new individuals, improving the global search range, and the convergence speed of genetic algorithm is slow, “A genetic algorithm for parallel simulated annealing” has adopted various methods to improve the convergence speed of genetic algorithm, their method can be divided into two types, one is the use of the specific issues related to the solution of the crossover operator, the other one is to fuse other algorithms in genetic algorithm, such as Newton optimization algorithm, simulated annealing algorithm, constituting a hybrid genetic algorithm [2]. “Ant colony algorithm with characteristics of sensation and consciousness” adopted the principle of positive feedback of information and integrates the heuristic search idea, strong local search ability and fast convergence speed and so on, but the global search capability was weaker than the genetic algorithm, an important method was to improve the global search capability of the Ant Colony Algorithm using the adaptive Ant Colony Algorithm, etc. [3]. “A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm” figured out that Genetic Algorithm and Ant Colony Algorithm were complementary, they were likely to be organically fused together, to overcome their shortcomings, playing their respective advantages [4]. “WSN path optimization based on fusion of improved ant colony algorithm and genetic algorithm” proposed that fusion strategy of Genetic Algorithm and Ant Colony Algorithm, according to their different status and advantages in the integration algorithm, which generally can be divided into two categories: one is based on Ant Colony Algorithm as the main hybrid Ant Colony Algorithm, the other one is a hybrid genetic algorithm based on genetic algorithm [5]. Figure 1 (a–d) are some pictures of the construction plan layout.

Fig.1

Display of layout chart of construction site.

Field layout can be considered as the second distribution problem, and the Ant Colony Optimization Algorithm was firstly applied to the traveling salesman problem (TSP), this kind of problem was a special case of QAP. “Ant colony system: a cooperative learning approach to the traveling salesman problem” presented that the performance of ACO was to solve the problem of TSP which was much better than that of GA and Simulated Annealing (SA) algorithm [6]. “Ant system for job-shop scheduling” presented that a lot of classical optimization algorithms were compared in the study, from ACO to solve the QAP problem of many classic examples, the performance was better than other algorithms [7]. “Ant colony optimization” proved that in the ACO algorithm, ACO was a heuristic algorithm to solve the QAP problem, and the best one of GA, SA and Tabu search [8]. “A multiagent system approach for image segmentation using genetic algorithms and extremal optimization heuristics” figured out the MAS-GA algorithm for the single objective optimization problem of the construction site layout, and proved the superiority of MMAS-GA through the results [9]. In addition, MMAS-GA has better convergence, which was better than the GA to get the optimization results earlier, the result is far better than GA. “A novel ACO–GA hybrid algorithm for feature selection in protein function prediction” proposed that ACO-GA could solve the problem of single objective optimization of the construction site layout, which wanted to apply the algorithm to the multi-objective optimization problem, it was necessary to make some improvement [10].

2 Description of the problem

The construction site layout problems can be divided into: Dynamic and static problems, equal area and non-equal area problems, single facility layout problems and multi-facility layout problems, fixed facility layout problems and non-fixed facility layout problems [11]. Dynamic problems and static problems in the construction site layout: The main difference is that: with the development of construction, taking into account the transport capacity of different construction facilities involved in different construction stages, the construction site layout should be prepared with different construction site layout plan [12]. The problems of equal area and non-equal area in the plane layout of construction site: Two assumptions are going to be done for equal-area site layout. Firstly, it’s the assumption of the shape of the facility. In the study, the shape of the facility is often assumed to be regular, usually rectangular, then, in the process of the arrangement, it is assumed that the space between the facilities and the site can be arranged between the space constraints, that is, any kind of facilities can be arranged in any place that can be arranged in a blank position, and any one of the space can accommodate any facilities, obviously, this is not consistent with the reality of the situation [13]. However, in the study of specific issues, this assumption is often used to simplify the layout process. Single facility layout problem and multi-facility layout problem in the plane layout of construction site: Single facility layout problem refers to some heavy construction machinery, such as the research on layout of tower crane in the construction site [14]. From a broad perspective, the road layout of the construction site should also be included in the scope of a single facility layout problem. At present, the single facility layout problem is mainly focused on the optimization of tower crane location selection. The facility layout problem is that the position of certain facilities is fixed to solve the problem of the site layout of other construction facilities [15].

3 State of the art

3.1 Principles of construction layout design

The main contents are as below: building above ground and underground on the general plan and the size of the above ground and underground building or structure, and the position and size of all kinds of pipelines, the moving route of the mobile crane as well as the vertical transport facilities, topographic contour, location of the earth, arrangement of temporary facilities for construction services, various materials, semi-finished products, components and industrial equipment, and other construction road layout and the introduction of railway, highway and waterway location, temporary water supply lines, power supply lines, steam and compressed air piping, etc., and the locations of all safety and fire protection facilities [16].

Scientifically plan the construction plane is used to reduce the temporary facilities so as to achieve the purpose of reducing the cost. Temporary road layout of achieving permanent, temporary combination, and set back to the lane, to ensure smooth transportation, pavement quality to achieve rain or shine. In the construction stage, the construction plan should be in line with the construction sequence of the arrangement in the construction plan. Materials should be taken into consideration in the transportation and the convenience of the use, and to minimize the number of times of the second handling. Even if we need to carry a short distance, there is no reverse transport. Temporary facilities’ construction standards should be based on the length of the construction period and the size of the project, striving for the lowest cost [17].

Specific operations can be as follows:

Temporary housing should make full use of the existing buildings near the site and the construction sector to demolished buildings. Temporary housing standards for the construction of a long period of time can be higher, this will not increase maintenance costs due to the long construction period of the project using the low standard, short construction period of the project with high standard and increase the cost of construction.

Temporary road standards are related to the length of the project and the location of the project and the road of the heavy truck. In the long construction period, the rainy season is long, and the heavy truck is on uphill road, the standard is higher, when necessary, we should apply hard road. In the south, the rainy season of construction is long, road transport earthwork standard, on the one hand, the impact of the speed of the car directly affects the duration, on the other hand, the car has a great impact on its own, if the wheel and wear increases the maintenance cost of the vehicle and thus increase the cost. Temporary housing and facilities are in compliance with the requirements of labor protection, technical safety and fire protection specifications. Materials, machinery and equipment warehouse and temporary housing location must be in a higher place, in order to prevent the flood [18].

Explosive materials should be put in the position of the library to keep people in safe distance away, not only to ensure the safety of people and construction, but also the shortest distance transportation. Production, life, safety, fire protection, environmental protection, health, labor protection city are complied with the provisions of the state.

3.2 Ant colony genetic algorithm

Genetic algorithm and ant colony algorithm are the two bionic optimization algorithm with their own advantages and disadvantages. The combination of the two can make the old and new contrast and complement each other. Ant Colony Algorithm is used to find the optimal space, and the Genetic algorithm is to find the best way of space, and overcome the genetic algorithm premature convergence to a certain range, and can not find the best solution of the drawbacks [19]. The matrix A represents a chromosome, each of which is a course arrangement. Row value represents the time P, the column value represents all the teaching tasks D. At the initial time, the matrix is set as 0. $A = (\begin{matrix} a_{11} & a_{12} & \dots \dots & a_{1 n} \\ a_{21} & a_{22} & \dots \dots & a_{2 n} \\ a_{31} & a_{32} & \dots \dots & a_{3 n} \\ \dots & \dots & \dots \dots & \dots \\ a_{m 1} & a_{m 2} & \dots \dots & a_{mn} \end{matrix})$ (1)

The value of the row and the number of classes is on behalf of the class to get the constraint formula: $\sum_{j = 0}^{N} A_{ij} \Leftarrow K_{i}$ (2)

Among them, A is for the scheduling program, K is for a class teaching tasks. Each teacher, each class in a class at a time point can only be on a course, getting the constraint formula: $\sum_{j = x}^{N} A_{ij} \Leftarrow 1$ (3)

The fitness function is the concept of genetic algorithm, and the whole algorithm is carried out under the guidance of the fitness function.

Firstly, the fitness value is divided into two categories:

The former is used to record the hard conflict, the greater the conflict, the greater the value; therefore, when this value is 0, there is no hard conflict, the chromosome can be used as a course to be selected; The latter is used to record the soft constraints, for example, a course in the two class time interval is longer, the better the class effect is, the greater the value is [20].

Therefore, the current person is 0, the latter is bigger, the row lesson plan more reasonable.

The selection operation is also called genetic, which is used to simulate the natural selection of the biological world. It selects a certain chromosome from the old population, and put it into the matching set, preparing for new species for the production of chromosome exchange and mutation. The higher the degree of adaptation, the greater the probability of being selected, the distribution of the genetic genes in the next generation is more complex, the number of children and grandchildren appearing in the next generation is more. There are many methods to choose, such as turntable method, sorting method and so on, here, the adaptive sorting method is used to obtain the optimal chromosome [21]. The number of chromosomes pop_genome_num in a generation population was defined as 200; The replication rate of eli_rate was 0.1f. High fitness is in the front row, and the first 10% of the chromosomes have been preserved without any change.

The rest of the (1 - eli_rate) * genome chromosome is free from the generation of all chromosomes. Although the copy operation can choose the best from the old population, but can not create a new chromosome, therefore, the pioneer of genetic algorithm is proposed to operate it simulates the reproduction phenomenon in the process of biological evolution, through the combination of the exchange of two chromosomes, to produce a new fine variety. Ordinary genetic algorithm directly select two chromosomes from the last generation for crossover, in this way, it is possible to get a better solution because of local convergence. So we introduce the Ant Colony Algorithm to divide the matrix into several spaces, “Spraying” in each space. In order to get the formula of pheromone and turntable: $P_{ij} = C * \sum_{i = 0}^{N} \sum_{i = 0}^{M} (F_{i} [j] + P_{(i - 1) j})$ (3)

Among them, C is a constant, P is divided into the interval on the pheromone, F is the fitness of each chromosome. $W_{ij} = \sum_{i = 0}^{N} \sum_{i = 0}^{M} (P_{ij} * F_{i} [j])$ (4)

Among them, W is the Wheel value of each chromosome, and P is the pheromone value in the interval of the chromosome, F is the fitness value of the chromosome. The chromosome turntable value, which is in direct proportion to the information element of the interval, and the adaptive degree of the chromosome can be seen that this is a positive feedback system. A strong degree of fitness of the chromosome, in the range of a strong pheromone; The probability of chromosome selection is large. In this way, the best chromosome can be obtained in the optimal interval [22]. The method of chromosome selection is adopted by the rotating disk method, which is different from the sorting method in the genetic selection step (Fig. 2). In this method, the individual’s choice probability is proportional to the rotary table. Its choice probability formula is as follows: $P_{i} = W_{i} {(\sum_{i = 1}^{N} W_{i})}^{- 1}$ (5)

In the formula: P_i is the choice of probability; I is chromosome number; W_i is the wheel turntable value N of individual i; I is chromosome population. To promote the evolution of chromosomes, the crossover operation is bound to conform certain rules. It is necessary to destroy the rationality of the teaching task if the two chromosomes are randomly exchanged. Using a row value cross does not destroy any data [23].

Fig.2

Flow chart of ant colony genetic algorithm.

3.3 A new ant colony algorithm for continuous space optimization

Continuous space discrete method: the use of binary encoding ant travel path. Set function n independent variable is X₁, ⋯, X_n, the range of the independent variable X_i is (a_i, b_i), the length is 1, the total length of encoding n is r (r = ln). Set binary to encoding for 011001-110001 (two independent variables), to the binary one of the vertical coordinates, the travel time of the ants is as a cross, there are 12 points: (1, 0), (2, 1), (3, 1), (4, 0), (5, 0), (6, 1), (7, 1), (8, 1), (9, 0), (10, 0), (11, 0) and (12, 1), the line connecting these points is composed of the ant travel path, which is expressed in Fig. 1; On the other hand, the travel path of similar Fig. 1 can also be expressed by binary. This constitutes a binary encoding and the path of the corresponding relationship. Article k is an ant’s travel path which may be expressed as:

$\begin{matrix} T (b_{1, i_{1}}^{k} b_{2, i_{2}}^{k} \dots b_{j, i_{j}}^{k} b_{j + 1, i_{j + 1}}^{k} \dots b_{r, i_{r}}^{k}), \\ (b_{j, i_{j}}^{k} = 0, 1; i_{j} = 0, 1) \end{matrix}$ (6)

Encoding in accordance with the binary decoding method of genetic algorithm: $X_{i} = \frac{(X_{i_code})_{2}}{(11 \dots 11)_{2}} \times (b_{i} - a_{i})$ (7)

The path is shown in Fig. 3 which can be converted to a closed path shown in Fig. 4.

Fig.3

Ant travel path.

Fig.4

Ant travel close trail.

Figure 4 is composed of two concentric circles, which are R0 and R1, respectively, which are concentric circles and two circles. The radius of the circle corresponds to all points in Fig. 3, all points of the coordinate of the circle corresponding to the radius of R1 in Fig. 3. Figure 4 is the closed path of ants travel.

The purpose of doing so is to initialize the Ant Colony Algorithm, that is to say using the initialization. The purpose of doing so is to initialize the ant colony algorithm, that’s to say, when the initialization is used in Fig. 4, and in the process of the ant colony algorithm iteration in Fig. 3. These two approaches can be converted to each other [24].

According to Fig. 4, an ant is set up as a path from the first b_{k,i_k} (i_k = 0, 1):

T (b_{k,i_k}b_{k+1,i_k+1} ⋯ b_{r,i_r}b_{l,i_l} ⋯ b_{k-1,i_k-1}b_{k,i_k}), the path is a closed path. A path that can be set from b_{l,i_l}, and the same as the original path, by shifting the shift $T^{'} (b_{l, i_{l}} b_{2, i_{2}} \dots b_{k - 1, i_{k - 1}} b_{k, i_{k}} \dots b_{r, i} b_{l, i_{r}} b_{l, i_{l}})$ .

Transformation method is to exchange the location of a part of the path [b_{k,i_k}b_{k+1,i_k+1} ⋯ b_{r,i_r}] and the other part [b_{l,i_l}b_{k-1,i_k-1} ⋯ b_{k,i_k}] of the path T, removing the same point b_{k,i_k} and connected to the expression as:

$\begin{matrix} T ([b_{k, i_{k}} b_{k + 1, i_{k + 1}} \dots b_{r, i_{r}}] [b_{l, i_{l}} \dots b_{k - 1, i_{k - 1}} b_{k, i_{k}}]) \\ \overset{T_{M}}{⟶} T' (b_{l, i_{l}} b_{2, i_{2}} \dots b_{k - 1, i_{k - 1}} b_{k, i_{k}} \dots b_{r, i} b_{l, i_{r}} b_{l, i_{l}}) \end{matrix}$ (8)

In the ant colony algorithm, each ant leaves a hormone on the side of the path it passes through, in the Ant colony algorithm, the operation of the hormone concentration update operation is aimed at one side.

Knot - j (i, j = 1 ⋯ r, i = 0, 1) expresses the coordinates for the (j, i) travel node. In addition to the last node, the rest of each node has two successor travel nodes Next - knot - 0, Next - knot - 1, its coordinates are (j + 1, 0) and (j + 1, 1), Knot-ji and Next - knot - 0, Next - knot - 1 are connected by the two sides respectively, expressed as Edge (j, i, 0), Edge (j, i, 1), abbreviated as edge (j, i, i_j), among them, i_j = 0, 1. Their correspondence is shown in Fig. 5. The two sides can also be expressed as $Edge (b_{j, i_{j}}^{k} b_{j + 1, i_{j + 1}}^{k})$ .

New ant colony algorithm: $τ_{j, i, i_{j}} (t + 1) = ρ τ_{j, i, i_{j}} (t) + Δ τ_{j, i, i_{j}}$ (9) $Δ τ_{j, i, i_{j}} = \sum_{k = 1}^{m} Δ τ_{j, i, i_{j}}^{k}$ (10)

Fig.5

Relationship between adjacent nodes.

3.4 Fusion of genetic algorithm and new ant colony algorithm fusion

According to genetic algorithm and ant colony algorithm, the position and advantage of the two are different in a certain integration algorithm, generally it can be divided into two categories: one is based on Ant Colony Algorithm as the main hybrid Ant Colony Algorithm, in the literature, the genetic algorithm is used to find the optimal combination of ρ, α, β in GA and ACS. An ant colony algorithm with mutation operator is proposed in the paper; the other one is a hybrid genetic algorithm based on genetic algorithm. The new Ant Colony Algorithm and the basic genetic algorithm are used to form two hybrid Ant colony algorithms based on genetic algorithm. The first algorithm is to use the Genetic Algorithm to compare the global search capability, in a large scale, looking for a rough set of solutions, and then to the rough solution of this group as the initial path of the Ant Colony Algorithm, Ant Colony Algorithm is to quickly find the optimal solution $X_{best}^{*}$ , the method is called GA - HACS - I Genetic Algorithm-Hybrid Ant Colony System-I; The other one is the crossover operation in the ant colony algorithm, in order to generate the new path of the ant colony algorithm, improving the global search capability of ant colony algorithm, the method is called GA - HACS - II Genetic Algorithm-Hybrid Ant Colony System-II [25, 26].

3.5 Algorithm GA-HACS-I

Algorithm description:

Step 1. Use the genetic algorithm to obtain a set of optimal solution $T_{Better}^{(GA)} (0)$ .

Step 2. Use the $T_{Better}^{(GA)} (0)$ as the initial path, and use the ant colony algorithm for the other accurate solution $X_{best}^{*} (t)$ .

3.6 Algorithm GA-HACS-II

The travel path representation of the ant colony algorithm is the same as that of the genetic algorithm, which laid the foundation for the two. Path crossover operation, recombining the existing paths to generate a new path, this is because the genetic algorithm has stronger ability to find new solutions. Renew hormone to get a new way.

Path crossover operation:

Random selection of two paths $T_{1} (b_{1, i_{1}}^{k_{1}} \dots b_{j, i_{j}}^{k_{1}} \dots b_{r, i_{r}}^{k_{1}}), T_{2} (b_{1, i_{1}}^{k_{2}} \dots b_{j, i_{j}}^{k_{2}} \dots b_{r, i_{r}}^{k_{2}})$ and a cross location pos, according to the genetic algorithm in the single point cross train of thought, exchange data, getting two new paths $T_{1} (B_{1, i_{1}}^{k_{1}} \dots B_{j, i_{j}}^{k_{1}} \dots b_{r, i_{r}}^{k_{1}}), T_{2} (B_{1, i_{1}}^{k_{2}} \dots B_{j, i_{j}}^{k_{2}} \dots b_{r, i_{r}}^{k_{2}})$ , the principle of path crossing is shown in Fig. 6.

Fig.6

Path crossover operation.

Algorithm description:

GA- ACS- II Algorithm:

Step 1. Initialization parameters:

Parameters of Ant Colony Algorithm; Path crossover probability pc;

Step 2. While (t < ACS-cycle), execute ant colony algorithm New-ACS;

For k = 1 to m

Execute crossover procedure;

End For;

t ← t + 1;

End While (t>ACS-cycle)

Step 3. Output.

4 Methodology

4.1 Pareto ant colony genetic algorithm

This paper presents a Pareto Ant Colony Genetic Algorithm (Pareto-based ACO-GA), combining with the advantages of ant colony algorithm, and combined with the guiding ideology of Pareto optimization, finding the optimal solution of multi-objective function. This algorithm combines Pareto ant colony algorithm and genetic algorithm, the first half of this algorithm is Pareto ant colony algorithm, Pareto ant colony algorithm is used to solve the problem of continuous multi-objective optimization, this paper is to solve the discrete problems of multi-objective optimization of the construction site layout, so for the algorithm it makes some changes, in solving the problem of contact.

The final solution space is limited, so that it can be applied to the discrete problem; The latter half of the algorithm is genetic algorithm, as discussed in section GA on the performance improvement of ACO, GA is a kind of self-learning optimization algorithm, in the initial random generation of solutions, according to the applicable function to select the superior solution, its excellent performance of the next generation. It can be seen that the solution of each generation has a direct impact on the quality of the optimal solution, in order to improve the performance of GA, the quality of the initial solution can be improved. If the use of ACO is to find the optimal generation of initial solution, and then use the powerful search function of GA, based on the advantages of GA and ACO two algorithms, an improved ACO-GA algorithm is designed, which can improve the efficiency of solving the QAP problem. The operation process of Pareto ant colony genetic algorithm is as follows: The first step, randomly generated an ant colony POP of scale for the initial N. Initial ant colony POP scale N can be set by itself, assuming a construction site layout problem of n × n, n equipment will be placed into n positions, so that the facility is placed on the n! kinds of solutions, with the scale of N ≤ n !.

The second step is to calculate the objective function and the constraint conditions of each ant. In Ant colony algorithm, each ant represents a solution, the objective function and constraint conditions of each ant are calculated, the objective function and the constraint condition of each solution in the initial ant colony POP is calculated. The objective function and constraint conditions are set by the construction site layout problem [26, 27].

The third step, the formation of an external set BP. The external set BP is the non-dominated solution set in the initial ant colony POP.

The fourth step, setting the maximum number of cycles N_max. When the number of cycles of the algorithm reaches N_max, the output of the optimal solution set.

The fifth step, the optimization of the solution. In ant colony algorithm, ants determine the direction of foraging based on the number of pheromone on the path, and the pheromone on each path is determined by the length of the path. When solving the single objective problem, the length of the path corresponds to the advantages and disadvantages of the single objective solution, the advantages and disadvantages of the solution can be expressed by the objective function. In the multi-objective optimization problem, there is no absolute optimal solution, which determines the merits of the solution, in order to determine the relationship between the solution, which determines the emission of pheromone. In the process of searching, the ants should find the non-dominated solutions, optimization process is not only affected by the information left by the companions of the impact, to find a better solution, and taking into account the best experience of the whole ant colony, not to lose some of the better solutions. So the algorithm has the local pheromone communication optimization method and the overall situation of the most experienced guidance optimization method, according to the random number p (0 ≤ p ≤ 1) and set the p₀. Choosing different ways of optimizing:

Optimization of local pheromone communication:

In the optimization method of local pheromone communication, the definition of the pheromone concentration of each ant j with θ_j is expressed as: $θ_{j} = {\begin{matrix} λ_{1}, if x_{j} is a non - feasible solution \\ λ_{2}, if x_{j} is a feasible solution, and x_{i} < x_{j} \\ λ_{3}, if x_{j} is a feasible solution, and x_{i} and \\ x_{j} are non - constrained dominance \\ λ_{4}, if x_{j} is a feasible solution, and x_{i} < x_{j} \end{matrix}$ (11)

Among them, i, j = 1, 2, ⋯, N, i ≠ j, N is the ant colony size; x_i and x_j are the corresponding solutions of ant i, j, λ₁, λ₂, λ₃, λ₄ are the four parameter of pheromone concentration, these four parameters depend on the relationship between x_i and x_j, and the λ₄ > λ₃ > λ₂ > λ₁;

The x_i < x_j shows that x_j dominance of x_i; x_j < x_i shows that x_i dominance of x_j. From the formula it can be seen in the choice of moving direction of the ants i, comparing the Pareto dominance relationship with other ants in the ant colony, this relationship determines the concentration of the pheromone released by the companion in their position [27, 28]. Ant j in Ant colony, j = 1, 2, ⋯ N, N is the ant colony size, if the x_j is not feasible solution, then j is not helpful for the search of the optimizing, and release little information; If the x_j is a feasible solution and control x_i, then x_j is in favor of finding the optimal direction and moving forward in the direction of Pareto forward or feasible solution. At this time, the ant j is about to release a large amount of information, to attract the mother ant j to find the optimizing. In addition to the direction of the optimization of the mother ant and the pheromone concentration, but also should be related to the distance between them, the probability of random optimization is defined as: $P_{j} = \frac{θ_{j} δ_{ij}}{\sum_{j = 1}^{m} θ_{j} δ_{ij}}, j \neq i, j = 1, 2, \dots, N$ (12)

Among them, δ_ij = 1/ - d_ij is the distance between i and j, which can be seen from the formula, if the pheromone concentration is greater, the closer the distance from the current ant i, the greater the probability of being selected. Global most experienced guidance optimization method:

If only using the local optimization of the pheromone between ants to find the best, it will affect the search time of ants, and the search time of the algorithm will become longer, the superiority of the solution is sometimes ignored, and it is not easy to maintain the diversity of the solution. In the algorithm, a method is presented to find the optimal method under the guidance of global optimal experience, setting up an external set BP to preserve all the non-dominated solutions found by the entire ant colony, searching for the most sparse non dominated solutions in the set BP, its position is the current direction of the ants [28, 29]. If there are P non-dominated solutions x = (x₁, x₂, ⋯, x_p) in the current set BP, the distance from each solution to the other solutions is: $d_{ij} = \sqrt{\sum_{i = 1}^{k} (f_{i} (x_{i}) - f_{i} (x_{j}))^{2}}$ (13)

Among them, i = 1, 2, ⋯, p, j = 1, 2, ⋯, p, i ≠ j, then the shared function value is: $S (d_{ij}) = {\begin{matrix} 1 - d_{ij} / σ_{share}, & d_{ij} < σ_{share} \\ 0, & d_{ij} \geq σ_{share} \end{matrix}$ (14)

Where σ_share is the niche radius of non-dominated solution for its niche number is: $niche (i) = \sum_{j = 1}^{p} S (d_{ij}), j = 1, 2, \dots, p, i \neq j$ (15)

The location of the non-dominated solutions i of Niche count niche (i) is the current Ant Optimization method. The sixth step, update the external set BP. According to the results after optimization, it finds non-dominated solutions in the solution space, forming a new external set BP. The seventh step is to calculate the fitness function of the solution. When the number of cycles reaches N_max, then ends the ant colony algorithm, and uses the operation of the algorithm to solve the input. The fitness function is selected as the objective function of the plane layout of the construction site, according to the fitness function, selection, crossover and mutation operation of genetic manipulation. The eighth step: genetic algorithm’s genetic operation of selection, crossover and mutation operation of genetic manipulation. In this algorithm, the adaptive proportion is chosen, and the expected number of each individual is related to their fitness value and the proportion of the average fitness of the population [29, 30]; The pseudo code for the crossover and mutation operation is shown in Tables 1 and 2.

Table 1

Single point crossover operation of the pseudo code

Step	Operation
1	Determine the maximum number N of cycles of crossover operation
2	Select cross monomer
3	If p_rand < p_c, p_rand is a real number between 0 and 1, p_c is the crossover rate, then ...
4	Determine the intersection point
5	Single point operation for left and right exchange
6	Complete the crossover operation in this cycle
7	If the number of cycles is reached, then the output results
8	If the number of cycles cannot reach N, then go on the second step

Table 2

Variation operation of the pseudo code

Step	Operation
1	For each individual of a generation
2	If p_rand < p_m, p_rand is a real number between 0 and 1, p_m is the rate of variation, then ...
3	Determine the intersection point
4	Determine the variance of m_value
5	Change the chromosome of the mutation according to the variation
6	Output results

The ninth step: the output of Pareto optimal solution. When the program reaches the end conditions, you can output the optimal solution of Pareto. The termination condition of the algorithm is to determine a maximum of the evolutionary algebra, in the evolution of this generation, the algorithm automatically terminate the operation. In this paper, we mainly study the plane layout of the construction site based on the safety angle. In order to solve the safety problem of the construction site layout, this paper set the multi-objective function. For the multi-objective optimization problem, through the calculation of the optimal Pareto ant colony genetic algorithm (Pareto-based Aco-ga), we can get multiple Pareto optimal solutions, namely multiple conforms to the objective function requirements of construction site layout scheme. If in the candidate solution, one of the closest to the ideal solution, at the same time away from the negative ideal solution, the program is the best candidate in the program. The ideal solution and negative ideal solution for Euclidean distance in two-dimentional space are solved. The ideal solution contains all the optimal solutions that meet the properties, and the negative ideal solution contains the worst solution that meets all the properties [30, 31]. A formula is expressed as: $A^{+} = (x_{1}^{+}, x_{2}^{+}, \dots, x_{j}^{+}, \dots, x_{n}^{+})$ (16) $A^{-} = (x_{1}^{-}, x_{2}^{-}, \dots, x_{j}^{-}, \dots, x_{n}^{-})$ (17)

The Euclidean distance to measure each solution near degree for the ideal solution and the negative ideal solution, the n of the existing evaluation schemes and M attribute decision problem hypothesis and operation steps of Entropy TOPSIS are as follows:

The first step, multi objective decision making matrix D is:

$\begin{matrix} \begin{matrix} a_{1} & a_{2} & \dots & a_{m} \end{matrix} \\ D = \begin{matrix} y_{1} \\ y_{2} \\ \dots \\ y_{n} \end{matrix} [\begin{matrix} r_{11} & r_{12} & \dots & r_{1 m} \\ r_{21} & r_{22} & \dots & r_{2 m} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ r_{n 1} & r_{n 2} & \dots & r_{nm} \end{matrix}] \end{matrix}$ (18)

$W = [\begin{matrix} w_{1} & w_{2} & \dots & w_{m} \end{matrix}]$ (19)

Among them, D and W are decision matrix and weight vector; Y = [y₁, y₂, ⋯, y_n] is a candidate evaluation scheme for N; A = [a₁, a₂, ⋯, a_m] is the evaluation of the properties of M; so, r_ij, (i = 1, 2, ⋯, m) is a candidate program x_i for the evaluation of attribute a_j; w_j (j = 1, 2, ⋯, m) is the weight of the attribute a_j, using information entropy to solve.

The second step is to construct the canonical decision matrix $\bar{D}$ . $\bar{D} = [\begin{matrix} z_{11} & z_{12} & \dots & z_{1 m} \\ z_{21} & z_{22} & \dots & z_{2 m} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ z_{n 1} & z_{n 2} & \dots & z_{nm} \end{matrix}]$ (20) $\begin{matrix} z_{ij} = r_{ij} / \sqrt{\sum_{i = 1}^{n} r_{ij}^{2}}, i = 1, 2, \dots, n; \\ j = 1, 2, \dots, m \end{matrix}$ (21)

In the decision matrix D, if the original attribute value is used, the dimension of each attribute is different, which will not be conducive to analysis, and may have a significant difference in the numerical. So, the value of each attribute should be normalized and unified in the [0,1] range. The third step is to construct a weighted normalized decision matrix $\bar{X}$ . $\bar{X} = [\begin{matrix} x_{11} & x_{12} & \dots & x_{1 m} \\ x_{21} & x_{22} & \dots & x_{2 m} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ x_{n 1} & x_{n 2} & \dots & x_{nm} \end{matrix}]$ (22) $x_{ij} = w_{j} z_{ij}, i = 1, \dots, n; j = 1, \dots, m$ (23)

The fourth step: to determine the ideal solution and the negative ideal solution. $A^{+} = {(max_{j} x_{ij}), (min_{j} x_{ij}), j = 1, 2, \dots, m | i = 1, 2, \dots, n} = {x_{1}^{+}, x_{2}^{+}, \dots, x_{j}^{+}, \dots, x_{n}^{+}}$ (24) $A^{-} = {(min_{j} x_{ij}), (max_{j} x_{ij}), j = 1, 2, \dots, m | i = 1, 2, \dots, n} = {x_{1}^{-}, x_{2}^{-}, \dots, x_{j}^{-}, \dots, x_{m}^{-}}$ (25)

Among them, max x_ij is the benefit type goal, min x_ij is the cost target. The fifth step is calculated for each solution to the ideal solution and the negative ideal solution of the Euclidean distance between E⁺ and E^-. $E^{+} = \sqrt{\sum_{i = 1}^{m} (x_{ij} - x^{+})^{2}}, i = 1, 2, \dots, n$ (26) $E^{-} = \sqrt{\sum_{i = 1}^{m} (x_{ij} - x^{-})^{2}}, i = 1, 2, \dots, n$ (27)

The sixth step is to calculate the relative close distance of each solution to the ideal solution, the negative ideal solution, and proximity factor C_i. $C_{i} = E_{i}^{-} / (E_{i}^{+} + E_{i}^{-}), i = 1, 2, \dots, n$ (28)

The seventh step: to sort the program. To sort according to the size of the C_i, In the first place, it should be used.

4.2 Total construction site management

General layout of the construction, in order to fully guarantee the key points of the construction stage, for the purpose of ensuring the smooth implementation of the construction, before the construction of the project, it’s need to make detailed use of equipment, program, production, processing, stacking and transportation of major materials and working materials, the various types of construction team into the exit adjustment plan, and formulate specific measures for the specific implementation of the program, with the method, the combination of mandatory guidance, strict implementation of scientific and civilized construction management level. Construction plan management system: the project manager is responsible for the management of the general plan, coordinate and adjust the use of the general plan according to the project schedule and construction, the daily management of the general plan is in charge of the engineering department [31, 32].

Plan of plan management:

The key to scientific management of construction plane is scientific planning and detailed planning, in the progress of the project network plan based on the formation of materials, machinery, labor and field. In order to ensure the progress of the project, the full balance of the use of the plane as the goal, to formulate a plan for the implementation of the plan in line with the actual situation and dynamic control management.

Implementation of plan management: According to the implementation of the project schedule adjustment, phased release plan implementation plan, including time schedule, responsible person, the implementation of standards, standards of reward and punishment, in order to fully protect the stage of construction, ensuring the smooth implementation of the progress plan. In the plan, the scheduling will be held regularly, fully coordinated. After research, publishing the plan adjustment book. The engineering department is responsible for organizing the periodical and non-periodical inspection and supervision, ensuring the implementation of plan management plan. Key projects in the underground construction stage: The orderly arrangement of the use of the road in the field. It refers to field inside and outside the environmental sanitation and safe use of electricity. Key projects in the construction phase of the ground: vertical transport safety management, material placement tool Farfield orderly adjustment, management, materials, machinery for entering and exiting. General layout of construction site, the plan and use should be carried out in a letter to the general layout, according to the specific requirements of the city standard site management, on the field, the presence of the equipment, material stacking, security facilities, such as field capacity management, by the full-time administrator of the construction of the full implementation of dynamic management, speed up the site turnover, the site formation of field capacity of roads, in order to clean working environment [32, 33].

4.3 Total construction site management

General principles of planar Management: Planning and using the general plan of the construction site should strictly implement the principle of unified management. Implementation of dynamic management according to the construction content of schedule plan, and set up a special person for planning, coordination, setting up a civilized construction team to supervise and inspect. The use of the site after the sub-contracting units should comply with the provisions of the site civilized construction. General layout of the construction, in order to fully guarantee the key points of the construction stage, for the purpose of ensuring the smooth implementation of the construction, before the construction of the project, it’s need to make detailed use of equipment, program, production, processing, stacking and transportation of major materials and working materials, the various types of construction team into the exit adjustment plan, and formulate specific measures for the specific implementation of the program, with the method, the combination of mandatory guidance, strict implementation of scientific and civilized construction management level. Construction plan management system: the project manager is responsible for the management of the general plan, coordinate and adjust the use of the general plan according to the project schedule and construction, the daily management of the general plan is in charge of the engineering department.

Plan of plan management: The key to scientific management of construction plane is scientific planning and detailed planning, in the progress of the project network plan based on the formation of materials, machinery, labor and field. In order to ensure the progress of the project, the full balance of the use of the plane as the goal, to formulate a plan for the implementation of the plan in line with the actual situation and dynamic control management. Implementation of plan management: According to the implementation of the project schedule adjustment, phased release plan implementation plan, including time schedule, responsible person, the implementation of standards, standards of reward and punishment, in order to fully protect the stage of construction, ensuring the smooth implementation of the progress plan. In the plan, the scheduling will be held regularly, fully coordinated. After research, publishing the plan adjustment book. The engineering department is responsible for organizing the periodical and non-periodical inspection and supervision, ensuring the implementation of plan management plan [33, 34].

4.4 The establishment of the safety model of the plane layout of the construction site

The construction period of the infrastructure construction is long, the construction site is constantly changing, the construction site is difficult to define, involving a single temporary facility, resulting in the construction site layout of the safety impact of the limited. In this paper, the construction site layout security model includes four parts: input, optimization, evaluation and output section, as shown in Fig. 7.

Fig.7

Safety model structure of construction site layout.

Input part of the identification of the construction site needs to be arranged temporary facilities, some of which temporary facilities, for example, the tower crane is usually not needed to optimize the layout, because the tower crane layout needs from the building shape, around the building construction site conditions, lifting process, crane and other objects in the safe distance to consider, belonging to the single facility optimization problem. In this paper it can be regarded as fixed facilities. The guard is usually arranged at the construction site of the main traffic crossing location, it can also be regarded as a fixed facility. Other temporary facilities that have little effect on the construction production, for example, the toilet, smoking room does not belong to the scope of the temporary facilities. Optimization is the core of the model, the efficiency of the algorithm has a direct impact on the effectiveness and feasibility of the model. The optimization of the model uses Pareto ant colony genetic algorithm, this algorithm is proposed by using ant colony algorithm to solve the continuous multi-objective optimization problem, and through the ant colony algorithm to optimize the initial solution to improve the performance of genetic algorithm optimization. The research goal of this paper is to design the construction site plan to improve the safety level of the construction site, which can not only improve the safety of the construction site, but also can effectively control the cost of the safety level. In the three elements of the project management, the cost of the project managers always need to consider, without a reasonable cost budget, the project management objectives cannot be achieved [34, 35].

5 Results analysis and discussion

Genetic algorithm single peak value function verification: The convergence speed and calculation accuracy of GA-HACS-II and GA-HACS-I are verified by using the single peak value function. Rosenbrock function F1: $\begin{matrix} min (x_{1}, x_{2}) = 100 f (x_{1}^{2}, x_{2})^{2} + (1 - x_{1})^{2} \\ - 2.048 \leq x_{1} \leq 2.048 i = 1, 2 \end{matrix}$

Experimental parameters: m = 50, ACS - cycle = 50, n = 2, l = 8, m = 10, d = 0.8, Q = 1, T = 1, U = 2 Genetic algorithm experimental parameters: chromosome population size $\begin{matrix} P = m, GA - cycle = 20, p_{c} = 0.6, \\ p_{m} = 0.1; p_{T_{c}} = 0.6 \end{matrix}$

Figure 8(a) is the initial distribution of all algorithms x1, x2; Fig. 8(b) is the algorithm GA-HACS-I twentieth times the loop variable x2, x1 distribution map, Fig. 8(c) is the algorithm GA-HACS-II twentieth times the loop variable x2, x1 distribution map.

Fig.8

Variables value initial distribution.

The minimum value of f(1.002, 1.002) = 0.0004.

Tables 3 and 4 are GA, GAHACS-I and GA-HACS-II’s calculation speed and calculation accuracy (a total of 50 experiments).

Table 3

Convergence speed of three algorithms

Algorithm name	Mean convergence rate
SGA	36.5
GA-ACS-I	23
GA-ACS-II	19

Table 4

Computation precision distribution of three algorithms SGA,GA-HACS-I, and GA-HACS-II

Algorithm name	Statistical distribution of optimal values
	f < 0.005	0.005 < f<0.01	0.01 < f < 0.1	f > 0.1
SGA	58%	28%	8%	6%
GA-ACS-I	66%	14%	12%	8%
GA-ACS-II	72%	14%	10%	4%

Convergence rate is used for mean convergence.

Using genetic algorithm to solve the binary encoding ant colony algorithm and solve the discretization of the continuous function. According to the advantages and disadvantages of genetic algorithm and ant colony algorithm, two hybrid ant colony algorithms GA-HACS-I and GA-HACS-II are constructed. Algorithms make full use of the crossover operator in genetic algorithm, which can generate new individual characteristics to generate the new path of the ant colony algorithm, so as to achieve the combination of genetic algorithm and ant colony algorithm. The model established in this paper is based on the Pareto ant colony genetic algorithm, genetic algorithm uses some genetic operators, continuously update the initial solution to obtain the final optimal solution, so the quality of the initial solution directly affects the performance of the algorithm. In order to improve the quality of the initial solution, the ant colony algorithm is used to improve the initial solution of genetic algorithm, so the optimization performance of ant colony algorithm is very important. The important parameter setting of ant colony algorithm is shown in Table 5.

Table 5

Parameters of ant colony algorithm

Parameter	Settings
Initial ant colony size	100/200
λ ₁	0.01
λ ₂	2
λ ₃	8
λ ₄	32

In the operation process, the effect of initial population size N on the results was tested, when λ₁, λ₂, λ₃ and λ₄ are under the same conditions, changing the initial population size of 100 and 200, the results of the calculation is shown in Table 6.

Table 6

Numerical value of Pareto optimal solution

Parameter setting	Objective function value of Pareto optimal solution		Original layout objective function value		Optimization of objective function (%)
	Safety cost	Safety level	Safety cost	Safety level	Safety cost optimization rate	Safety level optimization rate
N = 100	430540	918245	472098	799259	8.8	13.0
	435599	928110	472098	799259	7.7	13.9
	414752	881034	472098	799259	12.1	9.3
	423416	913944	472098	799259	10.3	12.5
	411152	852460	472098	799259	12.9	6.2
	414168	854358	472098	799259	12.3	6.4
	417145	887621	472098	799259	11.6	10.0
N = 200	424862	912937	472098	799259	10.0	12.5
	411653	872423	472098	799259	12.8	8.4
	408720	856488	472098	799259	13.4	6.7
	411050	864614	472098	799259	12.9	7.6
	418410	907491	472098	799259	11.4	11.9
	429122	955330	472098	799259	9.1	16.3

From the above data, we can see that the initial population size of N has a great impact on the results. The optimal solution of N = 200 in Table 6 is better than that of N = 100. The greater the initial population size is, the greater the diversity and the global nature of the solutions are, the greater the probability of generating the optimal solution in the initial population. If you want to get a better solution, you can set the size of the initial population. λ₁, λ₂, λ₃, λ₄ are the four parameter of pheromone concentration, these four parameters depend on the relationship between x_i and x_j in the population, according to the Pareto dominance relationship of ants, this relationship determines the concentration of the pheromone released by the companion in their position, and λ₄ > λ₃ > λ₂ > λ₁, the greater the difference between λ₁, λ₂, λ₃, λ₄, the faster the convergence rate, the larger the initial population size, the more sensitive to the setting of λ₁, λ₂, λ₃, λ₄. And for this problem, the initial size of the set, the effect of λ₁, λ₂, λ₃, λ₄ on the results is not obvious. In Pareto ant colony genetic algorithm, the performance of the algorithm is greatly affected by the initial population size N. The results of N = 200 in this paper are better than N = 100. The larger the initial population size, the resulting solutions are more diverse and global, so the possibility of generating the optimal solution in the initial population is also greater. If you want to get a better solution, you should try to set a larger initial population size.

6 Conclusions

The types of construction site layout problems are more and more complex, in the planning and use of the general plan of the construction site, the principle of unified management should be strictly implemented, the construction site layout problem can be regarded as the two distribution optimization problem in mathematics. Genetic algorithm and ant colony algorithm are two bionic optimization algorithms, which have their own advantages and disadvantages. The article combines the two together, and puts forward the method based on Pareto ant colony genetic algorithm, becoming a major innovation point of this paper. The article also according to the general principles of the construction plane management, based on this algorithm, the security model of the construction site layout is established, which can effectively solve the problem of multi objective optimization of the construction site.

References

Gambardella

L.M.

and Dorigo

, An ant colony system hybridized with a new local search for the sequential ordering problem, INFORMS Journal on Computing12(3) (2000), 237–255.

Mahfoud

S.W.

, A genetic algorithm for parallel simulated annealing, Parallel Problem Solving from Nature2 (1992), 301–310.

Chen

, Qin

, Chen

, et al., Ant colony algorithm with characteristics of sensation and consciousness, Acta Simulata Systematica Sinica10 (2003), 019.

Karaboga

and Basturk

, A powerful and efficient algorithm for numerical function optimization: Artificial bee colony (ABC) algorithm, Journal of Global Optimization39(3) (2007), 459–471.

Sun

and Tian

, WSN path optimization based on fusion of improved ant colony algorithm and genetic algorithm, Journal of Computational Information Systems6(5) (2010), 1591–1599.

Dorigo

and Gambardella

L.M.

, Ant colony system: A cooperative learning approach to the traveling salesman problem, Evolutionary Computation, IEEE Transactions on1(1) (1997), 53–66.

Colorni

, Dorigo

, Maniezzo

, et al., Ant system for job-shop scheduling, Belgian Journal of Operations Research, Statistics and Computer Science34(1) (1994), 39–53.

Dorigo

, Birattari

and Stützle

, Ant colony optimization, Computational Intelligence Magazine, IEEE1(4) (2006), 28–39.

Melkemi

K.E.

, Batouche

and Foufou

, A multiagent system approach for image segmentation using genetic algorithms and extremal optimization heuristics, Pattern Recognition Letters27(11) (2006), 1230–1238.

10.

Nemati

, Basiri

M.E.

, Ghasem-Aghaee

, et al., A novel ACO–GA hybrid algorithm for feature selection in protein function prediction, Expert Systems with Applications36(10) (2009), 12086–12094.

11.

Zhan

Z.H.

, Zhang

, Li

, et al., An efficient ant colony system based on receding horizon control for the aircraft arrival sequencing and scheduling problem, Intelligent Transportation Systems, IEEE Transactions on11(2) (2010), 399–412.

12.

Socha

and Dorigo

, Ant colony optimization for continuous domains, European Journal of Operational Research185(3) (2008), 1155–1173.

13.

Gómez

, Khodr

H.M.

, De Oliveira

P.M.

, et al., Ant colony system algorithm for the planning of primary distribution circuits, Power Systems, IEEE Transactions on19(2) (2004), 996–1004.

14.

Kaveh

, Hassani

, Shojaee

, et al., Structural topology optimization using ant colony methodology, Engineering Structures30(9) (2008), 2559–2565.

15.

Singh

S.P.

and Sharma

R.R.K.

, A review of different approaches to the facility layout problems, The International Journal of Advanced Manufacturing Technology30(5-6) (2006), 425–433.

16.

Le Riche

and Haftka

R.T.

, Optimization of laminate stacking sequence for buckling load maximization by genetic algorithm, AIAA Journal31(5) (1993), 951–956.

17.

Chaharsooghi

S.K.

and Kermani

A.H.M.

, An effective ant colony optimization algorithm (ACO) for multi-objective resource allocation problem (MORAP), Applied Mathematics and Computation200(1) (2008), 167–177.

18.

Drira

, Pierreval

and Hajri-Gabouj

, Facility layout problems: A survey, Annual Reviews in Control31(2) (2007), 255–267.

19.

McKendall

A.R.

and Shang

, Hybrid ant systems for the dynamic facility layout problem, Computers & Operations Research33(3) (2006), 790–803.

20.

McKendall

A.R.

and Shang

, Hybrid ant systems for the dynamic facility layout problem, Computers & Operations Research33(3) (2006), 790–803.

21.

B.H.

, Zhang

, Wang

S.L.

, et al., Cloud manufacturing: A new service-oriented networked manufacturing model, Computer Integrated Manufacturing Systems16(1) (2010), 1–7.

22.

Mohan

B.C.

and Baskaran

, A survey: Ant Colony Optimization based recent research and implementation on several engineering domain, Expert Systems with Applications39(4) (2012), 4618–4627.

23.

Gambardella

L.M.

and Dorigo

, An ant colony system hybridized with a new local search for the sequential ordering problem, INFORMS Journal on Computing12(3) (2000), 237–255.

24.

Wei

G.A.O.

and Yingren

, Ant colony algorithm and its application into optimization of construction order for underground house groups, Chinese Journal of Rock Mechanics and Engineering4 (2002), 002.

25.

, Duan

and Liu

, Improved ant colony algorithm for global optimal trajectory planning of UAV under complex environment, IJCSA4(3) (2007), 57–68.

26.

Yang

, Yu

and Cheng

, A parallel ant colony algorithm for bus network optimization, Computer-Aided Civil and Infrastructure Engineering22(1) (2007), 44–55.

27.

Maniezzo

and Colorni

, The ant system applied to the quadratic assignment problem, Knowledge and Data Engineering, IEEE Transactions on11(5) (1999), 769–778.

28.

Uğur

and Aydin

, An interactive simulation and analysis software for solving TSP using Ant Colony Optimization algorithms, Advances in Engineering software40(5) (2009), 341–349.

29.

Dowsland

K.A.

and Thompson

J.M.

, Ant colony optimization for the examination scheduling problem, Journal of the Operational Research Society56(4) (2005), 426–438.

30.

Hing

, Jin

and Gao

, Multi-layer feedforward neural network based on ant colony system, Journal of Harbin Institute of Technology7 (2003), 016.

31.

Baykasoglu

, Ozbakir

and Tapkan

, Artificial bee colony algorithm and its application to generalized assignment problem, Swarm Intelligence: Focus on Ant and Particle Swarm Optimization (2007), 113–144.

32.

Kaveh

, Azar

B.F.

and Talatahari

, Ant colony optimization for design of space trusses, International Journal of Space Structures23(3) (2008), 167–181.

33.

Kulturel-Konak

and Konak

, Unequal area flexible bay facility layout using ant colony optimisation, International Journal of Production Research49(7) (2011), 1877–1902.

34.

Ruiz

and Stützle

, A simple and effective iterated greedy algorithm for the permutation flowshop scheduling problem, European Journal of Operational Research177(3) (2007), 2033–2049.

35.

and Xiao

R.B.

, Ant colony algorithm for layout optimization with equilibrium constraints, Control and Decision23(1) (2008), 25.