Improved Barebones Particle Swarm Optimization with Neighborhood Search and Its Application on Ship Design

Abstract

Barebones particle swarm optimization (BPSO) is a new PSO variant, which has shown a good performance on many optimization problems. However, similar to the standard PSO, BPSO also suffers from premature convergence when solving complex optimization problems. In order to improve the performance of BPSO, this paper proposes a new BPSO variant called BPSO with neighborhood search (NSBPSO) to achieve a tradeoff between exploration and exploitation during the search process. Experiments are conducted on twelve benchmark functions and a real-world problem of ship design. Simulation results demonstrate that our approach outperforms the standard PSO, BPSO, and six other improved PSO algorithms.

1. Introduction

Particle swarm optimization (PSO), developed by Kennedy and Eberhart [1], is a new optimization technique inspired by swarm intelligence. Liker other evolutionary algorithms (EAs), PSO is also a population-based stochastic search algorithm, but it does not contain any crossover or mutation operator. During the search process, each particle adjusts its search behavior according to the search experiences of its previous best position (𝑝_best) and the global best position (𝑔_best). Due to its simplicity and easy implementation, PSO has been successfully applied to various practical optimization problems [2–5].

However, like other stochastic algorithms, PSO also suffers from premature convergence when handling complex multimodal problems. The main reason is that the attraction search pattern of PSO greatly depends on 𝑝_best and 𝑔_best. Once these best particles (𝑝_best and 𝑔_best) get stuck, all particles in the swarm will quickly converge to the trapped position. In order to enhance the performance of PSO, different versions of PSO have been proposed in the past decades. Shi and Eberhart [6] introduced an inertia weight 𝑤 into the original PSO to achieve a balance between the global and local search. Reported results show that a linearly decreased 𝑤 is a parameter setting. Parsopoulos and Vrahaits [7] proposed a unified PSO (UPSO) which is a hybrid algorithm by combining the global and local versions of PSO. In [8], a fully informed PSO, called FIPS, is proposed by employing a modified velocity model. van den Bergh and Engelbrecht [9] used a cooperative mechanism to improve the performance of PSO on multimodal optimization problems. In the standard PSO, particles are attracted by their corresponding previous best particles and the global best particle. This search pattern is a greedy method which may result in premature convergence. To tackle this problem, Liang et al. [10] proposed a comprehensive learning PSO (CLPSO), in which each particle can be attracted by different previous best positions. Computational results on a set of multimodal problems demonstrate the effectiveness of CLPSO. In [11], Wang et al. proposed a new PSO algorithm (NSPSO) to search the neighbors of particles. This can provide more chances of finding better candidate solutions. Simulation studies show that NSPSO outperforms UPSO, FIPS, CPSO-H, and CLPSO. In [12], another version of NSPSO is proposed by employing neighborhood search and diversity enhanced mechanism.

Similar to other EAs, the performance of PSO also greatly depends on its control parameters, 𝑤, 𝑐₁, and 𝑐₂. The first parameter is known as inertia weight, and the last two are acceleration coefficients. Slight differences of these parameters may result in significantly different performance. To tackle this problem, some PSO variants based on adaptive parameters have been proposed to minimize the dependency of these parameters [13, 14]. Compared to these adaptive PSO algorithms, Kennedy developed a novel PSO called barebones PSO (BPSO) [15], which eliminates the velocity term and does not contain the parameters 𝑤, 𝑐₁, and 𝑐₂. In BPSO, a Gaussian sampling is used to generate new positions of particles. Empirical studies demonstrate that the performance of BPSO is competitive to the standard PSO and some improved PSO algorithms. Inspired by the idea of BPSO, some new algorithms haven been proposed. In [16], Omran et al. combined BPSO with differential evolution (DE) and the proposed barebones DE (BBDE). The reported results show that BBDE outperforms the standard DE and BPSO and it also achieves promising solutions for unsupervised image classification. Krohling and Mendel [17] introduced Gaussian and Cauchy mutations into BPSO to improve its performance. Experimental studies on a suite of well-known multimodal benchmark functions demonstrate the effectiveness of this approach. Blackwell [18] presented a theoretical analysis of BPSO. A series of experimental trials confirmed that the BPSO situated at the edge of collapse is comparable to other PSO algorithms and that performance can be still further improved with the use of an adaptive distribution. In [19], Wang embedded opposition-based learning (OBL) into BPSO to solve constrained nonlinear optimization problems. In addition, a new boundary search strategy is utilized. Simulation studies on thirteen constrained benchmark functions show that the new approach outperforms PSO, BPSO, and six other improved PSO algorithms.

In this paper, we also propose an improved barebones PSO called NSBPSO which employs a global and local neighborhood search strategies to make a balance between exploration and exploitation during the search process. In order to verify the performance of NSBPSO, twelve well-known benchmark functions and a real-world problem on ship design are used in the experiments. Computational results show that our approach outperforms PSO, BPSO, and several other improved PSO variants in terms of the quality of solutions.

The rest of the paper is organized as follows. The standard PSO and barebones PSO are given in Section 2. In Section 3, our approach NSBPSO is proposed. Experimental studies are presented in Section 4. Section 5 presents a real-world application on ship design. Finally, the work is summarized in Section 6.

2. Barebones Particle Swarm Optimization

PSO is a population-based stochastic search algorithm, which simulates the behaviors of fish schooling or birds flocking. Each particle has a velocity and a position vectors. During the search space, a particle dynamically adjusts its velocity to generate a new position as follows: 𝑉_𝑖(𝑡+1)=𝑤⋅𝑉_𝑖(𝑡)+𝑐₁⋅𝑟₁⋅𝑝_best𝑖−𝑋_𝑖(𝑡)+𝑐₂⋅𝑟₂⋅𝑔_best−𝑋_𝑖(𝑡),𝑋_𝑖(𝑡+1)=𝑋_𝑖(𝑡)+𝑉_𝑖(𝑡+1), (1) where 𝑋_𝑖 and 𝑉_𝑖 are the position and velocity vector for the 𝑖th particle, respectively. 𝑝_best𝑖 is the previous best particle of the 𝑖th particle and 𝑔_best is the global best particle. 𝑟₁ and 𝑟₂ are two independently generated random numbers within [0,1]. The parameter 𝑤 is known as inertia weight. 𝑐₁ and 𝑐₂ are acceleration coefficients.

A recent study [20] proved that the particles in PSO converge to the weighted position of 𝑝_best and 𝑔_best as follows: lim_𝑡→+∞𝑋_𝑖(𝑡)=𝑐₁⋅𝑝_best𝑖+𝑐₂⋅𝑔_best𝑐₁+𝑐₂. (2)

Based on the convergence characteristic of PSO, Kennedy [15] proposed a new PSO variant called barebones PSO (BPSO), in which each particle only has a position vector and eliminates the velocity vector. Therefore, BPSO does not contain the parameters 𝑤, 𝑐₁, and 𝑐₂. In BPSO, a new position is updated by Gaussian sampling as follows: 𝑋_𝑖()𝑔𝑡+1=𝑁^best+𝑝_best𝑖2,𝑔_best−𝑝_best𝑖, (3) where 𝑁(⋅) indicates a Gaussian distribution with mean (𝑔_best+𝑝_best𝑖)/2 and standard deviation |𝑔_best−𝑝_best𝑖|.

3. Barebones PSO with Neighborhood Search

Due to the intrinsic randomness, both PSO and EAs suffer from premature convergence when solving complex multimodal problems. Sometimes, the suboptima are near to the global optimum and the neighborhoods of trapped particles may contain the global optimum. For this case, searching the neighbors of particles is beneficial for finding better solutions. Based on this idea, some neighborhood search strategies have been successfully applied to various algorithms.

In [11], Wang et al. proposed a new PSO algorithm called PSO with neighborhood search strategies (NSPSO), which utilizes one local and two global neighborhood search strategies. The NSPSO includes two operations. First, for each particle, three trial particles are generated by the above neighborhood search strategies, respectively. Second, the best one among the three trial particles and the current particle is chosen as the new current particle. Simulation studies on twelve unimodal and multimodal benchmark problems show that NSPSO achieves better results than standard PSO and five other improved PSO algorithms.

Although NSPSO has shown good search abilities, its performance is still seriously influenced by its control parameters, 𝑤, 𝑐₁, and 𝑐₂. In [11], NSPSO used an empirical parameter settings, 𝑐₁=𝑐₂=1.49618 and 𝑤=0.72984. In order to minimize the effects of the control parameters on the performance of NSPSO, this paper proposes an improved PSO algorithm by combining barebones PSO and the neighborhood search strategies.

There are various population topologies, such as ring, wheel, star, Von Neumann, and random. A recent study shows that the complexity of population topology affects the performance of PSO. A population topology with few connections (low complexity) may perform well on multimodal problems, while a highly interconnected population topology may perform well on unimodal problems. In this paper, a ring topology is used by the suggestions of [11].

The ring topology assumes that particles are organized as a ring. In [21], a special ring topology is proposed by connecting the indices of particles. For example, the fourth particle 𝑃₄ is connected by the third one 𝑃₃ and the fifth one 𝑃₅. In other words, 𝑃₃ and 𝑃₅ are two immediate neighbors of 𝑃₄. Figure 1 shows the employed ring topology. Based on the ring topology, a 𝑘-neighborhood radius is defined, where 𝑘 is a predefined integer number. For each particle 𝑃_𝑖, its 𝑘-neighborhood radius consists 2𝑘+1 particles (include itself), which are 𝑃_𝑖−𝑘,…,𝑃_𝑖−1,𝑃_𝑖,𝑃_𝑖+1,…,𝑃_𝑖+𝑘. It is obvious that the parameter 𝑘 satisfies 0≤𝑘≤(𝑁−1)/2. Figure 1 shows the 2-neighborhood radius of 𝑃₄, where 5 particles are covered by the neighborhood. By the suggestions of [11], 𝑘=2 is used in this paper.

Figure 1 

The ring topology and 2-neighborhood radius.

Based on the 𝑘-neighborhood radius, a local neighborhood search strategy is proposed. For each particle 𝑃_𝑖, a local particle 𝐿_𝑖 is generated as follows [11]: 𝐿𝑋_𝑖(𝑡+1)=𝑎₁⋅𝑋_𝑖(𝑡)+𝑎₂⋅𝑝_best𝑖+𝑎₃⋅𝑋_𝑖1(𝑡)−𝑋_𝑖2(𝑡), (4) where 𝑋_𝑖1 and 𝑋_𝑖2 are the position vectors of two particles, 𝑃_𝑖1 and 𝑃_𝑖2, randomly selected from the 𝑘-neighborhood neighborhood, 𝑖1,𝑖2∈[𝑖−𝑘,𝑖+𝑘]∧𝑖1≠𝑖2≠𝑖, 𝑎₁, 𝑎₂, 𝑎₃ are three random numbers within (0,1), and 𝑎₁+𝑎₂+𝑎₃=1. In [11], the velocity of 𝐿_𝑖 keeps the same with 𝑃_𝑖. Although the velocity mechanism is simple, it may not beneficial for the next flight of 𝐿_𝑖. Therefore, we use a similar method to generate 𝐿𝑉_𝑖: 𝐿𝑉_𝑖(𝑡+1)=𝑎₁⋅𝑉_𝑖(𝑡)+𝑎₂⋅𝑝_best𝑉𝑖+𝑎₃⋅𝑉_𝑖1(𝑡)−𝑉_𝑖2(𝑡), (5) where 𝑝_best𝑉𝑖 is the velocity vector of 𝑝_best𝑖 and 𝑉_𝑖1, 𝑉_𝑖2 are the velocity vectors of 𝑃_𝑖1 and 𝑃_𝑖2, respectively.

Beside the local neighbor strategy, a global neighborhood search strategy is proposed. For each particle 𝑃_𝑖, a global particle 𝐺_𝑖 is generated as follows [11]: 𝐺𝑋_𝑖(𝑡+1)=𝑎₁⋅𝑋_𝑖(𝑡)+𝑎₂⋅𝑔_best+𝑎₃⋅𝑋_𝑖3(𝑡)−𝑋_𝑖4(𝑡), (6) where 𝑋_𝑖3 and 𝑋_𝑖4 are the position vectors of two particles, 𝑃_𝑖3 and 𝑃_𝑖4, randomly selected from the current swarm, 𝑖3,𝑖4∈[1,𝑁]∧𝑖3≠𝑖4≠𝑖, 𝑎₁, 𝑎₂, 𝑎₃ are three random numbers within (0,1), and 𝑎₁+𝑎₂+𝑎₃=1. In [11], the velocities of 𝐺_𝑖 keeps the same with 𝑃_𝑖. So, the velocity of 𝐿_𝑖, 𝐺_𝑖, and 𝑃_𝑖 are the same, but 𝐿_𝑖 (local) and 𝐺_𝑖 (global) are two different types of particles. Like (5), this paper uses a new method to generate 𝐺𝑉_𝑖: 𝐿𝑉_𝑖(𝑡+1)=𝑎₁⋅𝑉_𝑖(𝑡)+𝑎₂⋅𝑔_best𝑉+𝑎₃⋅𝑉_𝑖3(𝑡)−𝑉_𝑖4(𝑡), (7) where 𝑔_best𝑉 is the velocity vector of 𝑔_best and 𝑉_𝑖3, 𝑉_𝑖4 are the velocity vectors of 𝑃_𝑖3 and 𝑃_𝑖4, respectively.

After generating two new particles 𝐿_𝑖 and 𝐺_𝑖, a greedy selection mechanism is used. Among 𝑃_𝑖, 𝐿_𝑖, and 𝐺_𝑖, we select the best one as the new 𝑃_𝑖.

In our approach NSBPSO, it embeds the local and global neighborhood search strategies into barebones PSO. The neighborhood search strategies focus on searching the neighbors of particles and provide different search behaviors. The BPSO concentrates on minimizing the dependency of the control parameters (without 𝑤, 𝑐₁, and 𝑐₂). By hybridization of BPSO and the neighborhood strategies, NSBPSO is almost a parameter-free algorithm (except for the probability of the neighborhood search), which achieves a tradeoff between exploration and exploitation.

The main steps of NSBPSO are listed as follows.

Step 1. Randomly initialize the swarm, and evaluate the fitness values of all particles.

Step 2. Initialize 𝑝_best and 𝑔_best.

Step 3. For each particle 𝑃_𝑖, calculate its new position vector 𝑋_𝑖 according to (3). Evaluate the fitness value of 𝑃_𝑖. If needed, update 𝑝_best𝑖 and 𝑔_best.

Step 4. For each particle 𝑃_𝑖, if 𝑟 and (0,1)<𝑃_ns, where 𝑟 and (0,1) is a random number within [0,1] and 𝑃_ns is the probability of conducting neighborhood search, then go to Step 5; otherwise go to Step 6.

Step 5. Generate a local particle 𝐿_𝑖 according to (4) and (5). Generate a local particle 𝐺_𝑖 according to (6) and (7). Evaluate the fitness values of 𝐿_𝑖 and 𝐺_𝑖. Among 𝑃_𝑖, 𝐿_𝑖, and 𝐺_𝑖, we select the best one as the new 𝑃_𝑖. If needed, update 𝑝_best𝑖 and 𝑔_best.

Step 6. If the stop criterion is satisfied, then stop the algorithm and output the results; otherwise go to Step 3.

4. Experimental Study

4.1. Test Problems

In order to verify the performance of our approach, there are twelve well-known benchmark problems used in the following experiments [10]. According to the properties of these problems, they are divided into two three types: unimodal problems (𝑓₁–𝑓₂), unrotated multimodal problems (𝑓₃–𝑓₈), and rotated multimodal problems (𝑓₉–𝑓₁₂). For rotated problems, the original variable 𝑥 is left multiplied by the orthogonal matrix 𝐌 to get the new rotated variable 𝑦=𝐌∗𝑥. For all test problems, they are to be minimized and their global optima are zero. The specific descriptions of these problems are presented in Table 1.

Table 1 

The twelve benchmark problems.

4.2. Effects of the Parameter 𝑃_ns

The main contribution of this paper is to minimize the effects of the control parameters and improve the performance of BPSO. Although NSBPSO eliminates the control parameters, 𝑤, 𝑐₁, and 𝑐₂, it introduces two new parameters 𝑘 and 𝑃_ns. The parameter 𝑘 is the size of neighborhood radius. The ring population topology used in this paper assumes that particles are connected by their indices. Although 𝑃₂ and 𝑃₃ are two neighbors of 𝑃₁, they may not be the nearest one to 𝑃₁ (Euclidean distance). So, the size of the neighborhood radius does not affect the selection of particles in the local neighborhood search. Our empirical studies also confirm it (here we do not list the results of NSBSO with different 𝑘-neighborhood radius). According to the suggestions of [11], 𝑘 is set to 2 in this paper.

The parameter 𝑃_ns controls the probability of conducting neighborhood search. A larger 𝑃_ns will result in more neighborhood search operations, while a smaller 𝑃_ns will have less. This may affect the performance of NSBPSO. To investigate the effects of 𝑃_ns, this section presents an experimental study. In the experiment, the 𝑃_ns is set to 0.0, 0.1, 0.3, 0.5, 0.7, and 1.0, respectively. The performance of NSBPSO with different 𝑃_ns is compared.

For other parameters of NSBPSO, we use the following settings by the suggestions of [10]. The population size 𝑁 is set to 40. When the number of fitness evaluations (FEs) reaches the maximum value MAX_FEs, the algorithm stops running. In the experiment, MAX_FEs is set to 2.0𝑒+05. For each test problem, NSBPSO is run 30 times and the mean fitness error values are reported.

Table 2 presents the computational results of NSBPSO under different 𝑃_ns, where “Mean” represents the mean fitness error values. The best results among the comparison are shown in boldface. As seen, the performance of NSBPSO is not sensitive to the parameter 𝑃_ns. A smaller (𝑃_ns<0.3) or larger (𝑃_ns>0.5) value of 𝑃_ns almost achieves similar results. For 𝑃_ns=0.0, NSBPSO is equal to the original BPSO, because the neighborhood search operations are not conducted. For this case, the algorithm shows poor performance and falls into local minima on most test functions. When 𝑃_ns=0.1, NSBPSO significantly outperforms NSBPSO with 𝑃_ns=0.0. It demonstrates that the neighborhood search strategies are very effective. Even if we use a small 𝑃_ns, NSBPSO can also obtain promising results.

Table 2 

Results achieved by NSBPSO with different 𝑃_ns.

The value of 𝑃_ns does not affect the performance of NSBPSO, and 𝑃_ns>0 is applicable for all test problems. In this paper, 𝑃_ns=0.3 is used in the following experiments.

Figure 2 presents the convergence processes of NSBPSO with different 𝑃_ns. Although different 𝑃_ns of NSBPSO can find the global optimum on the majority of test functions, they show different convergence characteristics. For problem 𝑓₁, larger 𝑃_ns converges faster than smaller 𝑃_ns. For problem 𝑓₂, 𝑃_ns=0.1 converges fastest than other values. For 𝑓₈, 𝑃_ns=0.3 shows the fastest convergence speed.

(a) 𝑓1

(b) 𝑓2

(c) 𝑓3

(d) 𝑓4

(e) 𝑓8

(f) 𝑓9

(g) 𝑓10

(h) 𝑓11

Figure 2 

The convergence curves of NSBPSO with different 𝑃ns.

4.3. Comparison of NSBPSO with Other PSO Algorithms

In this section, experiments are conducted to compare nine PSO algorithms including the proposed NSBPSO on the 12 test problems. The involved algorithms are listed as follows. (1)standard PSO,(2)barebones PSO (BPSO),(3)unified PSO (UPSO) [7],(4)fully informed PSO (FIPS) [8],(5)cooperative PSO (CPSO-H) [9],(6)comprehensive learning PSO (CLPSO) [10],(7)adaptive learning PSO (APSO) [13],(8)pSO with neighborhood search (NSPSO) [11],(9)our approach (NSBPSO).

For the sake of fair comparison, we use the same settings for the same parameters. For all algorithms, the population size 𝑁 is set to 40, and the maximum number of fitness evaluations (MAX_FEs) is set to 2.0𝑒+05. For standard PSO, 𝑤 is linearly decreased from 0.9 to 0.4, and 𝑐₁=𝑐₂=1.49618. For NSPSO, the probability of neighborhood search 𝑃_ns is set to 0.3. The parameter settings of UPSO, CPSO-H, FIPS, and CLPSO are described in [10]. For NSPSO and APSO, the same parameters are used by the literature [11, 13], respectively. For each test problem, each algorithm is conducted 30 times and the mean fitness error values are reported.

Table 3 lists the comparison results of NSBPSO with other eight PSO algorithms, where “Mean” represents the mean fitness error values. The best results are shown in boldface. From the results, it can be seen that NSBPSO outperforms PSO, BPSO, and FIPS on all test problems. UPSO and APSO perform better than NSBPSO on 𝑓₂, while NSBPSO achieves better results for the rest 11 problems. CPSO-H obtains better solution than NSBPSO on 𝑓₂, while NSBPSO outperforms CPSO-H on 10 problems. For problem 𝑓₆, both NSBPSO and CPSO-H can find the global optimum. NSPSO performs better than NSBPSO on 𝑓₂, while NSBPSO outperforms NSPSO on 4 problems. Both of them can converge to the global optimum on 7 problems.

Table 3 

Results achieved by the nine PSO algorithms.

From the comparison of BPSO and PSO, BPSO outperforms PSO on 6 problems, while PSO achieves better results than BPSO for the rest 6 problems. The results demonstrate that the performance of BPSO is similar to PSO on these problems. Compared to PSO, BPSO is more competitive, because BPSO does not contain any control parameter (except for the population size), while PSO employs empirical parameter settings. By hybridization of BPSO (or PSO) and the neighborhood search, NSBPSO (or NSPSO) achieves significantly improvements on the performance of BPSO (or PSO). Compared to NSPSO, NSBPSO not only achieves better results, but also has less control parameters.

In order to compare the performance differences between NSBPSO and the other eight PSO algorithms, we conduct the Wilcoxon signed-rank test by the suggestions of [22]. Table 4 shows the 𝑃-values achieved by the Wilcoxon test. The 𝑃 values below 0.05 are shown in boldface. As shown, NSBPSO is significantly better than all other algorithms except for CLPSO and NSPSO. Though NSBPSO is not significantly better than them, it outperforms them in the majority of test problems.

Table 4 

Results of Wilcoxon signed-rank test between NSBPSO and other eight PSO algorithms.

5. Application on Ship Design

5.1. Problem Description

This section investigates the performance of our approach NSBPSO for a conceptual ship design. The original optimization statements are presented in [23, 24]. The ship design optimization problem used in this paper has six design variables, three objectives, and 9 inequality constraints. The design variables are length (𝐿), beam (𝐵), depth (𝐷), draft (𝑇), block coefficient (𝐶_𝐵), and speed in knots (𝑉_𝑘). The ship design problem aims to minimize transportation cost (𝑇_𝑐) and lightship weight (𝐿_𝑠) and maximize annual cargo (𝐴_𝑐) [24]: Minimize𝑇_𝑐,𝐿_𝑠Maximize𝐴_𝑐, (8) where 𝑇_𝑐=𝐶_𝐴/𝐴_𝑐, 𝐴_𝑐=(DWT−𝐹_𝑐−DWT_𝑀)⋅RTPA, and 𝐿_𝑠=𝑊_𝑠+𝑊_𝑜+𝑊_𝑚. The specific model definition is described in Table 5 [25]. The search ranges of the six variables 𝐿, 𝐵, 𝐷, 𝑇, 𝐶_𝐵, and 𝑉_𝑘 are listed in Table 6.

Table 5 

Model definition of the ship design problem.

Table 6 

Search ranges of the six design variables.

There are 9 inequality constraints listed as follows: 𝐿6−𝐵𝐿≤0,𝐷𝐿−15≤0,𝑇−19≤0,𝑇−0.45DWT^0.31𝐹≤0,𝑇−0.7𝐷+0.7≤0,DWT−500000≤0,25000−DWT≤0,_𝑛−0.32≤0,0.07𝐵−𝐾𝐵−𝐵𝑀𝑇+𝐾𝐺≤0. (9)

5.2. Constraint Handling

In order to deal with the constraints, an adaptive penalty method is employed by the suggestions of [19]. Let 𝑓(𝑥) be the objective function (𝑇_𝑐, 𝐿_𝑠, or 𝐴_𝑐). The fitness evaluation function 𝐹(𝑥) is defined by 𝐹(𝑥)=𝑓(𝑥),if𝑥isfeasible,𝑓(𝑥)+∑^𝑚_𝑗=1𝑘_𝑗⋅𝐶𝑉_𝑗,otherwise, (10) where 𝑚 is the number of inequality constraints, 𝐶𝑉_𝑗=min{0,𝑐_𝑗(𝑥)} is the constraint violation for the 𝑗th constraint, 𝑐_𝑗(𝑥) is the 𝑗th constraint, 𝑓(𝑥) is the mean objective function value in the current swarm, and 𝑘_𝑗 is a penalty coefficient defined as follows: 𝑘_𝑗=𝑓(𝑥)⋅𝐶𝑉_𝑗(𝑥)∑^𝑚_𝑖=1𝐶𝑉_𝑖(𝑥)², (11) where 𝐶𝑉_𝑗(𝑥) is the average violation of the 𝑗th constraint for all particles in the swarm.

5.3. Computational Results

The ship design problem is a multiobjective optimization problem which has three objectives. By the suggestions of [24, 25], this paper only considers single objective optimization. Therefore, the whole problem is divided into three single objective optimization problems: (1) minimize transportation cost (𝑇_𝑐), (2) minimize lightship weight (𝐿_𝑠), and (3) maximize annual cargo (𝐴_𝑐).

In this section, we conduct three series of experiments for the three single optimization problems. In order to verify the performance of our approach NSBPSO, we compare it with four other algorithms. The involved algorithms are listed as follows:(1)parsons and Scott’s method [24],(2)standard PSO,(3)barebones PSO (BPSO),(4)PSO with neighborhood search (NSPSO),(5)our approach (NSBPSO).

To have a fair comparison, the same parameter settings are used for common parameters. For all algorithms, the population size and the maximum number of fitness evaluations (MAX_FEs) are set to 100 and 1.0𝑒+06. For standard PSO, 𝑤 is linearly decreased from 0.9 to 0.4 and 𝑐₁=𝑐₂=1.49618. For NSBPSO and NSPSO, 𝑃_ns is set to 0.3. For each optimization problem, each algorithm is run 10 times and the best results among these runs are presented.

Tables 7–9 show the computational results for the three problems. For Table 7, NSBPSO achieves the minimal transportation cost among the five algorithms; but it also obtains the minimal value of annual cargo. For the objective of 𝑇_𝑐, NSBPSO is the best among the five algorithms, however, it could not obtain the best results for all three objectives. For annual cargo 𝐴_𝑐, Parsons and Scott’s [24] method is the best. Tables 8 and 9 can also get similar conclusions. The results demonstrate that NSBPSO shows better performance than the other three algorithms for single objective optimization problem of the ship design. When considering all objectives, we cannot conclude which algorithm is the best. To perfectly solve this problem, we may use multiobjective optimization algorithms. Figures 3, 4, and 5 present the convergence curves of PSO, BPSO, NSPSO, and NSBPSO for the three single objective problems. For minimizing transportation cost, NSBPSO shows faster convergence speed at the last stage of the evolution. For minimizing lightship weight, NSBPSO converges faster than other three PSO algorithms. For maximizing annual cargo, both NSBPSO and NSPSO show similar convergence characteristics.

Table 7 

Computational results for minimizing transportation cost.

Table 8 

Computational results for minimizing lightship weight.

Table 9 

Computational results for maximizing annual cargo.

Figure 3 

The convergence curves of PSO, BPSO, NSPSO and NSBPSO for minimizing transportation cost.

Figure 4 

The convergence curves of PSO, BPSO, NSPSO, and NSBPSO for minimizing lightship weight.

Figure 5 

The convergence curves of PSO, BPSO, NSPSO, and NSBPSO for maximizing annual cargo.

6. Conclusions

Barebones PSO (BPSO) is a new variant of PSO which eliminates the velocity term. Although some reported results show that BPSO is better than PSO, it still gets stuck when solving complex multimodal problems. In order to enhance the performance of BPSO, this paper proposes an improved version called BPSO with neighborhood search (NSBPSO). The new approach embeds one local and one global neighborhood search strategies into the original BPSO to achieve a tradeoff between exploration and exploitation. Compared to other improved PSO algorithms, NSBPSO is almost parameter-free algorithm except for the probability of neighborhood search (𝑃_ns).

Experimental studies are conducted on twelve well-known benchmark problems, including unimodal, multimodal, and rotated multimodal problems. Computational results show that the parameter 𝑃_ns does not affect the performance of NSBPSO. When 𝑃_ns>0, NSBPSO can obtain good performance. Another comparison demonstrates that NSBPSO performs better than, or at least comparable to, several other state-of-the-art PSO algorithms. Compared to PSO with neighborhood search (NSPSO), our approach NSBPSO not only achieves better results, but also has less control parameters.

For the ship design problem, NSBPSO performs better than other three algorithms when optimizing a single objective. When considering all three objectives, we cannot determine which algorithm is the best. Because one algorithm only achieves better results than other algorithms on one or two objectives. To tackle this problem, we can use multiobjective optimization algorithms to optimize the three objectives at the same time. This will be investigated in the future work.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (no. 51209057).