Abstract
This study proposes a new selection method called trisection population for genetic algorithm selection operations. In this new algorithm, the highest fitness of 2N/3 parent individuals is genetically manipulated to reproduce offspring. This selection method ensures a high rate of effective population evolution and overcomes the tendency of population to fall into local optimal solutions. Rastrigin’s test function was selected to verify the superiority of the method. Based on characteristics of arc tangent function, a genetic algorithm crossover and mutation probability adaptive methods were proposed. This allows individuals close to the average fitness to be operated with a greater probability of crossover and mutation, while individuals close to the maximum fitness are not easily destroyed. This study also analyzed the equipment layout constraints and objective functions of deep-water semisubmersible drilling platforms. The improved genetic algorithm was used to solve the layout plan. Optimization results demonstrate the effectiveness of the improved algorithm and the fit of layout plans.
1. Introduction
GA (genetic algorithms) [1] are a type of optimization method that is based on biological evolution. Due to their strong robustness and global optimization ability, GA have been widely used in artificial intelligence, engineering optimization, machine learning, expert system, and many other fields [2–4]. However, the efficiency of standard genetic algorithm computation is low, and it is easy to fall into local optimal solutions. Many researchers have proposed strategies to improve these algorithms. These studies have investigated the design of encoding genetic algorithms [5, 6], genetic operations improvement [7, 8], combination with other optimization algorithms [9–11], and so forth.
Many studies have focused on layout problems. Such problems arise in a variety of situations, including pallet loading, container stuffing, and placement problems [12]. The drilling equipment layout of semisubmersible platforms represents a typical complex system layout problem, in which layout scheme significantly affects platform performance.
In recent years, genetic algorithms have been shown to be particularly effective for solving NP-hard layout problems. In this study, the selection operator of GA was improved based on existing genetic algorithm. This study proposed that a genetic algorithm crossover and mutation probability adaptive methods are based on the characteristics of arc tangent function. An improved GA was used to solve the layout problem of drilling equipment for deep-water, semisubmersible platforms. Calculation results confirm that the improved algorithm is quite efficient for solving the NP-hard layout problem.
In standard GA, offspring is generated by parent individuals through the selection operator, crossover operator, and mutation operator. The parent-offspring competition mechanism (POGA) is often used in GA to produce excellent offspring [13]. However, all parent individuals participate in the generation procedure of offspring. Therefore, this is not useful for excellent individuals’ evolution at rapid speed. The elitist genetic algorithm (EGA) was introduced [14, 15]. In EGA, the best offspring remains in the next generation without being destroyed. However, the EGA leads to locally optimal solutions with a higher possibility. Other common selection operators include area selection, group selection, selection based on population gender, and selection based on model extraction and compensation.
The remainder of the paper is organized as follows. Section 2 describes the improvement of the genetic operator. Section 3 describes improvement of adaptive crossover operator and mutation operator. Section 4 describes convergence performance of T-AAGA and simulation testing results are listed. Drilling equipment layout of semisubmersible drilling platforms is introduced in Sections 5 and 6 that cover conclusions.
2. Improvement of Genetic Operator
This work proposes the trisection population method selection operator (TGA). Excellent two-thirds parent individuals were selected to participate in the genetic operation and the lower fitness one-third offspring individuals were eliminated. The remaining individuals of parent and offspring generation constituted the next generation of population. This selection mechanism ensures that excellent individual genes are stable through next generation. It also guarantees the diversity of the species gene pool and prevents algorithms from becoming stuck in local optimal solution.
The specific process of trisection population method selection mechanism is as follows.
Let us define the population of generations:where is the number of generations and is the number of individuals. The fitness of is expressed by .
To ensure the superiority of offspring, the parent individuals whose fitness is higher were selected for operating to generate new individuals. Based on calculation of the fitness function, the fitness of individuals is ranked from highest to lowest. Rank results are expressed by
In the formula (2), .
Let us define where is the integral function.
Then the individuals with lower fitness would be eliminated:
Individual species are produced by through the selection operator, crossover operator, mutation operators, and other genetic forms of manipulation. The number of individual is :
The total number of and is . Then all of the individuals are ranked by the fitness value, and the individuals with lower fitness are eliminated. The individuals with higher fitness are retained to constitute the next generation of population :
The iterative process of GA carries on according to the above method until the condition of convergence is satisfied.
In TGA, offspring individuals are generated from parent individuals with higher fitness through the trisection population method selection operator. TGA can quickly lead the solution procedure to the globally optimal solution.
For comparison with the parent-offspring competition mechanism (POGA) and the elitist genetic algorithm (EGA), this study used Rastrigin’s function to test the effectiveness of three methods.
The function expression of Rastrigin’s function is listed as:
Figure 1 shows the graph of Rastrigin’s function. Form Figure 1, we know that there are many local maximum points, with just one global maximum point. The global maximum point is point (0, 0), and the maximum is 0. The progress of the genetic algorithm is easy to fall into local maximum points when finding globe maximum of this function.
This study uses binary encoding for the encoding mechanism of the three GAs (POGA, EGA, and TGA). Specific parameters are set as follows: the number of bits in the chromosomes was 40, the number of individual species was 100, the number of evolution was 200, the crossover probability was 0.9, and the mutation probability was 0.01. Operation selection was performed using the roulette wheel. One hundred running times were used to obtain the reality effect of three GA methods. The threshold value was set as so that the algorithm was convergent when the convergences value was always larger than the threshold value.
All three algorithms are implemented using Matlab 2012a. The algorithms were run on the same Intel Core i5-2400CPU @3.0 GHz with 2.0 GB RAM in the Windows XP Operating System. Table 1 displays test results.
Figure 2 shows the change in the largest fitness value for Rastrigin’s function in three GA methods, in which the abscissa is evolutionary algebra and the ordinate is the maximum fitness .
Table 1 indicates that the convergence probability and average convergence algebra of the trisection population method selection operator genetic algorithm (TGA) proposed in this study was far superior to that of POGA and EGA. Figure 2 shows that the shape of the curve of TGA is rapid on its way to the global maximum of Rastrigin’s function. Therefore, we can conclude that TGA is feasible and predominant.
3. Improvement of Adaptive Crossover Operator and Mutation Operator
In genetic algorithms, the crossover operator plays a central role in global search ability. New individuals are generated by exchanging the structure of part of the parent individuals. The location of the cross is randomly selected from one or more genes in the parent individuals’ genes. This indicates that the search capability of GA can be improved tremendously. Genetic operations depend on the selection operator, the crossover operator, and mutation operator of the genetic operator. The values of crossover probability and mutation probability significantly affect performance of the algorithm. The crossover probability and mutation probability of basic genetic algorithm are fixed, while different optimization problems progress through trial and error in order to improve and . However, in the optimization process of GA, different and are needed for individuals with different fitness.
If the value of is too large, new individuals are generated quickly. At the same time, genetic models may be destroyed. However, if the value of is too small, the search process is too slow. Genetic algorithms are easy to degenerate into the pure random search method if the value of is too large. However, it is difficult to produce new individuals if the value of is too small. In light of this, many scholars have developed new adaptive genetic operators. The values of and are distributed along the fitness of different individuals though these adaptive genetic operators.
Srinivas and Patnaik [16] introduced a new method in which the values of crossover probability and mutation probability change automatically along the fitness of individuals (adaptive GA, AGA). However, the AGA cannot improve individual evolution in the early part of the process, and it easily becomes trapped in local optimal solution.
Shi et al. [17] designed a new method in which the values of crossover probability and mutation probability change automatically based on the cosine function (CAGA). In CAGA, and change nonlinearly, but the adaptive adjustment curve nears the straight line when differentials of and are too large.
This work developed a new adaptive genetic operator based on arc tangent function (AAGA). We know that the arc tangent function has the following characteristics.(1)The function value of arc tangent function is between .(2)The arc tangent function is an odd function whose domain is .(3)The value of the arc tangent function’s derived function tends to zero when and tends to maximum when .
As shown in Figure 3 which displays graphs of the arc tangent function, the top and bottom of the graph are smooth, but the value of the function changes rapidly when constant grows. With the feature of the function, a new adaptive frame function of and is formed, so that individuals with higher fitness cross each other with smaller values of crossover probability, and individuals with fitness close to cross each other with larger values of crossover probability.
The expression of and in AAGA are as follows. The formulas are enumerated when finding the maximum, and they are similar when finding the minimum:where and are the top and bottom limitation of crossover probability, and are the top and bottom limitation of mutation probability, is the average fitness of the entire population, is the largest fitness of population individuals, is the bigger fitness of the two individuals that take part in the crossover, is the fitness of the mutation individuals, and is a constant that can control the changing rate when the value of the function is close to . Usually the value of is between 20 and 60. In this work, the value of is 40.
Figure 4 displays the curves of (8).
(a)
(b)
As shown in Figure 4, crossover probability and mutation probability change adaptively between the maximum adaptation degree and the average adaptation degree based on function. Compared to AGA and CAGA, individuals whose fitness degree is located as can be operated with bigger crossover probability and mutation probability , while individuals whose fitness degree is located at can be operated with the smaller crossover probability and mutation probability . This means that AAGA tends to the optimal solution quickly, ensuring that excellent individuals are not destroyed.
Now a new genetic algorithm (T-AAGA) is formed by combining trisection population method selection operator and adaptive genetic operator based on the arc tangent function. Figure 5 displays the flow diagram of T-AAGA.
4. Convergence Performance of T-AAGA and Simulation Testing
Form the improving process of T-AAGA, we can know that T-AAGA is a special case of EGA. Compared to EGA, more excellent individuals are retained to the next generation of population. Early in the 1990s, Eiben and Rudolph have proven that EGA can be convergent to globally optimal solution in a probability value of 100% [18, 19]. So we can draw a conclusion that T-AAGA is convergent.
To test the performance of the proposed IAFOA algorithm, we consider a total of 5 benchmark functions. The details of these 5 functions are found in Table 2.
Four widely used evolutionary algorithms are compared with the proposed T-AAGA approach: they are the particle swarm optimization (PSO) algorithm [20], the covariance matrix adaptation evolution strategy (CMAES) [21], the self-adaptive differential evolution (SaDE1) algorithm [22], and the self-adaptive differential evolution (SaDE2) algorithm [23]. The parameter settings and results of PSO, CMAES, SaDE1, and SaDE2 come from the literature [20–24]. The main parameter settings of T-AAGA are as follows: the number of bits in the chromosomes was 40, the number of individuals was 200, the number of evolution was 1000, , , and , . Stopping iterative criteria is that the maximum number of iterations equals 1000.
Owing to their stochastic nature, evolutionary algorithms may arrive at solutions that are better or worse than solutions they have previously reached during their search for new solutions. For this reason, it is beneficial to use statistical tools to compare the problem-solving success of one algorithm with that of another [24]. All five of these algorithms were run 50 times for 5 test functions.
The comparison of the results is reported in Table 3. The “Best” means the optimal objective function value, the “Mean” reflects the precision and rate of convergence, and the “Std.” reflects the stability and robustness of the algorithm. From Table 3, we can know that T-AAGA have best performance in “Best,” “Mean,” and “Std.” of , , and . Besides, T-AAGA have best performance in “Best” and “Std.” of and . It can be concluded that the proposed T-AAGA has made great improvement in astringency, accuracy, and stability in finding the optimal value.
5. Drilling Equipment Layout of Semisubmersible Drilling Platforms
This section describes how T-AAGA was used to examine the drilling equipment layout of semisubmersible drilling platforms. Drilling equipment layout is an important part of the general design of drilling rig systems. It has important effects on operation performance and drilling efficacy and forms the main basis of further detailed designs of drilling rig systems. An effective layout ensures smooth drilling efficiency and considers effects on the overall performance of the drilling platform. A reasonable layout scheme retains drilling platform stability, security, reliability, and other indicators in a better state. This paper briefly discusses the layout mathematical model of drilling equipment in semisubmersible platforms.
5.1. Mathematical Model
In real-world engineering, factors such as operation cost, operation efficiency, and stability of the platform are the main focus, with the optimization goal to minimize the centroid transverse deviator of the platform. The centroid transverse deviator refers to the distance between the actual barycenter position of all equipment and the center of the centroid. To improve the stability of the platform, the centroid transverse deviator should be as small as possible, with the unit in meters:where is the weight (unit: KG) of the th equipment and and , respectively, are the barycentric abscissa and ordinate.
The layout of drilling equipment is limited by many constraint conditions. In this work, the significant constraints include the boundary constraint, interference constraint, and process constraint. The layout mathematical model can be expressed as follows:where is the expression of the boundary constraint, is the expression of the interference constraint, and , , , and are the expression of process constraints. All meanings of mathematical quantities are found in the literature [25].
5.2. Numerical Examples
To verify the effectiveness of T-AAGA, a specific example is adopted; the experimental data is covered in the literature [25]. In this example, the length of the deck was 78 meters, and the width of deck was 70 meters. The details on size and weight of allocated objects are listed in Table 4. It is notable that the location of the drill floor area center must be coincident with the drilling well location.
The improved genetic algorithm is used to solve numerical examples. For comparing results, the algorithm parameter is similar to parameter in the literature [25]. The details are as follows: the population size is 100; the number of iterations is 600; the top limitation and bottom limitation of crossover probability, respectively, are 0.70 and 0.35; top limitation and the bottom limitation of crossover probability, respectively, are 0.10 and 0.05.
The concrete solving process is implemented using Matlab 2012a for 300 times. Figure 6 displays performance tracking of the objective function value and the average value of the population.
As shown in Figure 6, simulation results demonstrate that the algorithm has fast convergence performance and high accuracy. A special layout case shown in Figure 7 was selected from the computational results.
In Figure 7, the coordinate planes represent layout plane of platform, and coordinate center represents the drilling center.
The centroid transverse deviator of the special case in Figure 7 is 0.128 m, and the allowable value of centroid transverse deviator is 0.7 m. Compared with [26] that uses hybrid genetic algorithms (HCGA), we know that 0.128 meters is far less than the results 0.392 m, 0.687 m, and 0.495 m in [26]. The genetic annealing algorithm used for solving the value of centroid transverse deviator and the results 0.5845 m and 0.4398 m are larger than 0.128 m. Therefore, we conclude that the improved genetic algorithm T-AAGA is more efficient.
6. Conclusions
(1)This study introduced a selection operator and proposed a trisection population method. Using the competition test function, this study compared the elitist genetic algorithm and the parent-offspring competition mechanism. Results show that the improved genetic algorithm (T-AAGA) is superior in convergence probability and convergence speed. At the same time, the iterative process of T-AAGA is relatively smooth and steady.(2)Compared with AGA and CAGA, a new genetic operator based on arc tangent function can change the values of crossover probability and mutation probability adaptively. Therefore, individuals with different fitness levels show different crossover probability and mutation probability, and the convergence rate of the population is guaranteed.(3)This study used the improved genetic algorithm T-AAGA to solve layouts for deep-water, semisubmersible platform drilling equipment. Combined with numerical examples, the results show considerable improvement compared with references. The centroid transverse deviator is just 0.128 m, which is far less than the allowable value.
Conflict of Interests
The authors declare that they have no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported in part by the project foundation of China Ministry of Industry and Information Technology “Research of Gordian Technique of Deep-Water Semi-Submersible platforms,” and the National 863 Plan Project Foundation of China “Key technology research of automated processing of deep-water drilling rig and string” (no. 2012AA09A203).