A Diversity-Enhanced Constrained Particle Swarm Optimizer for Mixed Integer-Discrete-Continuous Engineering Design Problems

Abstract

Engineering optimization problems usually contain various constraints and mixed integer-discrete-continuous types of design variables. We propose an efficient particle swarm optimization (PSO) algorithm for such problems. First, we transform the constrained optimization problem into an unconstrained problem without introducing problem-dependent or user-defined parameters such as penalty factors or Lagrange multipliers (such parameters are usually required in general optimization algorithms). Then, we extend the above PSO method to handle integer, discrete, and continuous design variables in a simple manner with a high degree of precision. The proposed PSO scheme is fairly simple and therefore easy to implement. To demonstrate the effectiveness of our method, several mechanical design optimization problems are solved, and the numerical results are compared with results reported in the literature.

1. Introduction

Most engineering optimal design problems require optimization techniques that are capable of handling the problem of equality and inequality constraints. In recent years, the particle swarm optimization (PSO) algorithm [1], whose mechanism was inspired by the social and cooperative behavior of swarms such as a flock of birds or a group of people, has been applied to various optimization problems with some promising results. It is a population-based search algorithm where a randomly initialized particle of the swarm updates its state (position) in a search space based on the information of its own and other agents' historical behavior until an optimal state has been reached (see, e.g., [2 –4] and the references therein). However, the original PSO is naturally a stochastic unconstrained optimization method, and thus it cannot be directly applied to constrained optimization problems which is also the case for most swarm-inspired metaheuristic search technologies. This has motivated many researchers to investigate techniques for handling constrained optimization problems, and some methods have been proposed for example, (i) the modified feasible-based mechanism [5, 6], (ii) the penalty approach [7, 8], (iii) the augmented Lagrange multiplier approach [9, 10], (iv) the violated design points redirection [11, 12], (v) the fly-back mechanism method [13], (vi) the problem reformulation into an unconstrained multi-objective formulation [14, 15], and (vii) a series of sequential quadratic programming problems [16]. The distinctive features of conventional constraint handling schemes can be found in Aziz et al. [17], Kim et al. [18], and Jansen and Perez [10]. It is worth mentioning that many of these methods could not ensure a realization of strict constraint feasibility which would limit its possible use to practical applications [10]. Further, some of them require the adoption of problem-dependent or user-defined parameters such as penalty factors or Lagrange multipliers.

In this line of researches, the author recently proposed a considerably simple and efficient way to handle the constraint conditions in PSO when solving engineering optimization problems [18]. The key feature of the proposed constrained PSO method is that the given constrained optimization problem is transformed into an unconstrained problem without introducing problem-dependent or user-defined parameters such as penalty factors or Lagrange multipliers, although such parameters are usually required in general optimization algorithms. In this approach, particles of the swarm evolve as follows (see Kim et al. [18] for details). At the first stage of evolution, all particles try to move into a feasible region where constraint satisfaction is guaranteed. Thereafter, the particles try to explore the feasible region to find an optimal solution that satisfies the given constraint conditions. This constraint handling PSO scheme has been shown to outperform conventional methods for various continuous engineering optimization problems.

On the other hand, many types of engineering optimal design problems usually contain mixed integer-discrete-continuous- (MIDC-) type design variables together with the design constraints. They involve a high degree of nonlinearity due to the combinatorial aspect of the related integer-discrete variables. The development of global stochastic optimization techniques suitable for such optimization problems is a challenging and active research area in many engineering fields. However, a rigorous implementation of integer/discrete variables, even in the framework of unconstrained PSO schemes, still represents a challenge [29]. The authors discussed in Kim et al. [18] how the aforementioned constrained PSO scheme can be combined with conventional rounding-off techniques found in many studies (see, e.g., [2, 27, 38 –40]). The numerical example in this study shows that the proposed method could solve constrained optimization problems with integer, discrete, and continuous design variables in a simple manner. However, the constrained-PSO-based approach with rounding-off technique tested in Kim et al. [18] may not provide sufficient optimization reliability, in the sense that the probability of obtaining an optimal design variable is not sufficiently high. In fact, such a shortcoming may chiefly be due to the poor particle diversification characteristics which frequently causes the potential problem of premature convergence of the swarm to local optima in a practical implementation. Further, this unfavorable phenomenon becomes more persistent in the case of constrained optimizations with MIDC variables. This means that the particles' diversity is undoubtedly an indispensable issue in improving the performance of population-based search algorithms. Furthermore, without adding an additional efficient diversity enhancing mechanism to the constrained PSO, it may be very difficult to reach the true global optimal solution of such mixed variable optimization problems.

To make up for the aforementioned deficiency, we propose a novel modified constrained PSO algorithm using a simple diversity enhancing mechanism with high applicability and reliability. Many diversity-guided PSO methods have recently been proposed (see, e.g., [41 –46] and the references therein). However, their effectiveness was examined only via continuous optimization problems with no constraint conditions; that is, their methods cannot be directly applied to solve constrained MIDC optimization problems. The proposed approach is designed to have particles of the swarm evolve as follows. At the beginning stage of evolution, all particles try to move into a feasible region where constraint satisfaction is guaranteed. Here, the individual particle first freely evolves in a continuous search space regardless of the variable type based on the diversity-guided velocity update law with three phases of attraction, repulsion, and a combination of attraction and repulsion (the so-called in-between phase) [43, 47]. It is worth mentioning that the PSO scheme proposed by Pant et al. [43, 47] originally targets the optimization problems with only continuous design variables and no constraint conditions. Then, for integer or discrete design variables, the corresponding continuous values of particles are rounded off to the nearest integer or to discrete values. Thereafter, the particles continue to evolve in the aforementioned manner and try to explore such a feasible region to find an optimal solution that satisfies the given constraint conditions. Note that although it may not be considered the best possible approach, the rounded-off technique has been used with promising results as demonstrated in many studies [2, 18, 24, 27, 28, 38 –40, 48, 49]. The introduced three-phase velocity control method has the following features (see Pant et al. [43, 47] for details):

if the population diversity is above an upper threshold, the particles attract each other,

if the diversity is below a lower threshold, the particles repel each other,

if the diversity is between upper and lower thresholds, then the individual particle is attracted by its own previous best position and is simultaneously repelled by the best known particle position.

This efficiently improves the population diversity in the framework of constrained PSO [18] and can reduce the probability of a premature convergence phenomenon (particles trapped in a local optimum due to the nature of mixed variables). Note that the procedure that transforms the given constrained optimization problem into an unconstrained problem introduces no problem-dependent extra parameters, such as penalty factors [23, 50] or Lagrange multipliers [51], which usually require time-consuming fine-tuning tasks. The proposed constrained PSO scheme, combined with the general integer and discrete variable handling technique and three phases of particle velocity control, can yield satisfactory optimal solution accuracy. We now demonstrate the scheme's distinctive features through several numerical examples.

2. Particle Swarm Optimizer for Constrained MIDC Optimization Problems

A novel PSO algorithm for constrained MIDC optimization problems with high practicality, simplicity, and implementability is proposed. Some of the advantages of the proposed method over conventional methods are discussed in detail.

2.1. Optimization Problem Description

In this study, we are interested in general constrained optimization problems that are mathematically formulated as follows:

\min_{x} f (x), x ∊ 𝔻 \subseteq ℝ^{n_{p}}

(1)

subject to

h_{ℓ} (x) \leq 0, h_{ℓ} : ℝ^{n_{p}} ⟶ ℝ, ℓ = 1,2, \dots, m,

(2)

where n_p is the number of independent design variables, m is the number of constraint conditions, x ∊ ℝ^{n
_p} is the design variable vector, 𝔻 is the search space, f(x) ∊ ℝ is the linear/nonlinear objective function, and h_ℓ(x) ∊ ℝ is the linear/nonlinear constraint function. It is assumed in general optimization problems that the design variable x ∊ ℝ^{n
_p} is continuously variable. However, some engineering optimization problems require treating the design variables as not only continuously varying quantities but also integer/discrete quantities. In such cases, the design variable vector x can be defined as follows:

x ∶ = {(x^{I}, x^{D}, x^{C})}^{T},

(3)

where the n_I-dimensional vector x^I includes integer variables, the n_D-dimensional vector x^D includes discrete variables, and the n_C-dimensional vector x^C includes continuous variables; that is,

\begin{matrix} x^{I} ∶ = {(x_{1}^{I}, x_{2}^{I}, \dots, x_{n_{I}}^{I})}^{T} ∊ ℝ^{n_{I}} : integer variables, \\ x^{D} ∶ = {(x_{1}^{D}, x_{2}^{D}, \dots, x_{n_{D}}^{D})}^{T} ∊ ℝ^{n_{D}} : discrete variables, \\ x^{C} ∶ = {(x_{1}^{C}, x_{2}^{C}, \dots, x_{n_{C}}^{C})}^{T} ∊ ℝ^{n_{C}} : continuous variables, \end{matrix}

(4)

where ℝ^{n
_I}, ℝ^{n
_D}, and ℝ^{n
_C} denote feasible subsets of integer, discrete, and continuous variables, respectively. The total number of design variables is n_p ∶ = n_I + n_D + n_C. Note that in order to guarantee particle evolution within the given search space, it is sometimes required that the search space information be incorporated into a form of boundary constraints such that $\underset{}{x} \leq x \leq \overset{—}{x}$ , where $\underset{}{x} ∊ ℝ^{n_{p}}$ and $\overset{—}{x} ∊ ℝ^{n_{p}}$ denote, respectively, the vectors of the lower and upper bounds of design variables and ≤ denotes the element-wise inequality. Thus, the considered general linear/nonlinear constrained MIDC optimization problem is defined as follows:

find x ∶ = {(x^{I}, x^{D}, x^{C})}^{T} ∊ 𝔻 \subseteq ℝ^{n_{p}}

(5)

to minimize f(x) subject to (2) and the following boundary constraints:

h_{m + 1} (x) ∶ = \underset{}{x} - x \leq 0, h_{m + 2} (x) ∶ = x - \overset{—}{x} \leq 0,

(6)

where 0 denotes a zero vector of an appropriate dimension. In Section 2.2, a novel PSO algorithm to find an optimal solution of the constrained MIDC problem defined in (5) is developed.

2.2. The Improved PSO for Constrained MIDC Problems

To solve the constrained MIDC optimization problem described by (5), we used the diversity enhanced particle evolution law that decreases the possibility of particles being trapped in a local optimum due to the mixed variable nature. Consider a swarm consisting of m_p particles: x₁, x₂, …, x_{m
_p}. Each particle x_i is, in effect, an n_p-dimensional vector:

x_{i} ∶ = {(x_{i}^{I}, x_{i}^{D}, x_{i}^{C})}^{T} = {(x_{i, 1}^{I}, \dots, x_{i, n_{I}}^{I}, x_{i, 1}^{D}, \dots, x_{i, n_{D}}^{D}, x_{i, 1}^{C}, \dots, x_{i, n_{C}}^{C})}^{T},

(7)

and its velocity v_i is also an n_p-dimensional vector denoted by

v_{i} ∶ = {(v_{i, 1}, v_{i, 2}, \dots, v_{i, n_{p}})}^{T} ∊ ℝ^{n_{p}},

(8)

where i ∊ [1, 2, …, m_p]. Then, to find an optimal solution for the constrained MIDC optimization problem in (5), each particle x_i ∊ ℝ^{n
_p} should evolve in the search space 𝔻. In this case, the evolution law of particles should efficiently improve the population diversity in the framework of constrained PSO [18] in order to reduce the probability of a premature convergence phenomenon (the particle being trapped in a local optimum due to the mixed variable nature). Therefore, we introduce a particle position updating algorithm using a combination of the constriction factor χ and the diversity-guided velocity update law with an attraction phase, repulsion phase, and in-between phase:

x_{i}^{k + 1} = x_{i}^{k} + v_{i}^{k + 1},

(9)

where $v_{i}^{k + 1}$ evolves as

v_{i}^{k + 1} = {\begin{matrix}   χ [v_{i}^{k} + c_{1} ζ_{i}^{k} (x_{i}^{best, k} - x_{i}^{k}) + c_{2} ξ_{i}^{k} (x_{swarm}^{best, k} - x_{i}^{k})], \\   if DIV (k) > D_{high}; \\   χ [v_{i}^{k} - c_{1} ζ_{i}^{k} (x_{i}^{best, k} - x_{i}^{k}) - c_{2} ξ_{i}^{k} (x_{swarm}^{best, k} - x_{i}^{k})], \\   if DIV (k) < D_{low}; \\   χ [v_{i}^{k} + c_{1} ζ_{i}^{k} (x_{i}^{best, k} - x_{i}^{k}) - c_{2} ξ_{i}^{k} (x_{swarm}^{best, k} - x_{i}^{k})], \\   otherwise, \end{matrix}

(10)

where c₁ is the cognitive scaling factor, c₂ is the social scaling factor, and χ is the constriction factor defined as

χ = \frac{2}{| 2 - φ - \sqrt{φ^{2} - 4 φ} |}, φ ∶ = c_{1} + c_{2} > 4,

(11)

where c₁ and c₂ are based on the authors' experience and were set at 2.02 for the example constrained MIDC optimization problems (c₁ and c₂ are usually set to 2.05 for optimization problems with only continuous design variables [18, 52]). The random numbers ζ_i^k and ξ_i^k are uniformly distributed in [0, 1] and represent the stochastic behavior of the PSO. $x_{i}^{best, k}$ denotes the best previously obtained position of the ith particle, and $x_{swarm}^{best, k}$ denotes the best position in the entire swarm at the current iteration k, whose mathematical formulation will be given subsequently. The term $c_{1} ζ_{i}^{k} (x_{i}^{best, k} - x_{i}^{k})$ is associated with cognition, since it takes into account only the best position of the particle's own experience. On the other hand, $c_{2} ξ_{i}^{k} (x_{swarm}^{best, k} - x_{i}^{k})$ represents the social interaction of all particles [51]. In (10), the first velocity update law means the attraction phase of particles, the second law means the repulsion phase, and the final law means the in-between phase, where diversity measure DIV (k) [43, 53] is defined as

DIV (k) = \frac{1}{m_{p}} \sum_{i = 1}^{m_{p}} \sqrt{\sum_{j = 1}^{n_{p}} {(x_{i, j}^{k} - {\hat{x}}_{j}^{k})}^{2}},

(12)

where

{\hat{x}}_{j}^{k} ∶ = \frac{(\sum_{i = 1}^{m_{p}} x_{i, j}^{k})}{m_{p}} .

(13)

Note that the constriction factor χ in PSO inherently enables particles to search different regions efficiently by avoiding premature convergence to local optima [38, 54]. Therefore, the introduced three-phase evolution method in (10) could be a better way to solve MIDC optimization problems than the approach in [43]. On the other hand, although the evolution law of particles described in (9) and (10) has an effectiveness of improving the population diversity that prevents the premature convergence phenomenon in PSO, its main drawback is the lack of fine-searching ability in the current search area. Note that the convergence of the particle swarm is necessary to refine the optimal solution [55]. Therefore, to overcome such a problem, we applied the three-phase velocity update law described in (10) and the following law in consecutive order (see Figure 1):

v_{i}^{k + 1} = χ [v_{i}^{k} + c_{1} ζ_{i}^{k} (x_{i}^{best, k} - x_{i}^{k}) + c_{2} ξ_{i}^{k} (x_{swarm}^{best, k} - x_{i}^{k})];

(14)

that is, the three-phase velocity update procedure from (10) was applied to an early stage of evolution of particles, and after a predetermined number of iterations (k_DIV), the conventional scheme described in (14) is introduced. We verified its effectiveness through several numerical examples, as discussed in Section 3.

Figure 1:

Flowchart of the proposed PSO scheme for constrained MIDC optimization problems.

Next, we considered how to handle the mixed integer-discrete-continuous design variables $x_{i} ∶ = {(x_{i}^{I}, x_{i}^{D}, x_{i}^{C})}^{T}$ , i = 1, 2, …, m_p, within the framework of the PSO update scheme. To achieve this objective, we combined the particle evolution law with rounding-off techniques. We note that although the preceding scheme seems a bit rough and led to unsatisfactory optimal solution accuracy, promising results were obtained in PSO frameworks as demonstrated in previous studies (see, e.g., [2, 18, 24, 27, 28, 38 –40, 48, 49]). The particle evolution procedure, which is a combination of the three-phase position update law described in (9) and (10) and rounding-off techniques, is described as follows.

When particles in the swarm are initialized, each particle first freely evolves in a continuous search space regardless of the variable type based on the position and velocity updating law in (9) and (10). Therefore, all design variables take real values (i.e., $x_{i}^{k} ∊ ℝ^{n_{p}}$ ) in this stage.

Next, for an integer variable $x_{i, j}^{I, k} ∊ x_{i}^{k}$ (j = 1, 2, …, n_I), we exploit the rounding-off method; namely, x_{i, j}^{I, k} ← INT (x_{i, j}^k), where INT (x_{i, j}^k) denotes that x_{i, j}^k ∊ ℝ is rounded off to the nearest integer. Similarly, for a discrete design variable $x_{i, n_{I} + j}^{D, k} ∊ x_{i}^{k}$ (j = 1, 2, …, n_D), x_{i, n_I + j}^{D, k} ← DIS (x_{i, n_I + j}^k), where DIS (x_{i, n_I + j}^k) denotes that x_{i, n_I + j}^k ∊ ℝ takes the nearest discrete value in a given set of equally and unequally spaced discrete values.

Once all particles in the swarm have moved to their new positions according to the aforementioned procedure, then the given objective function f(x), subject to various forms of constraints h_ℓ(x), ℓ = 1, 2, …, m, m + 1, m + 2, should be taken into account in any form to calculate the fitness value of each individual and to find $x_{i}^{best, k}$ and $x_{swarm}^{best, k}$ in (10). Thus, we incorporated the simple constraint handling method [56] using a virtual objective function f_v(x) with the particle evolution law. Note that f_v(x_i) could be a function that simultaneously satisfies the following two properties: (i) f_v(x_i) < 0 holds for any x_i satisfying h_ℓ(x_i) ≤ 0, and (ii) f_v(x_i) < f_v(x_j) holds whenever f(x_i) < f(x_j) is satisfied. One possible candidate for f_v(x_i) is f_v(x_i) ∶ = arctan{f(x_i)} – π/2 which was used to solve the numerical examples given in the next section (see [18, 56] for details and its distinctive features). Then, the original constrained MIDC optimization problem in (5) was modified into an unconstrained problem as follows.

\min_{x_{i} ∶ = {(x_{i}^{I}, x_{i}^{D}, x_{i}^{C})}^{T}} ℒ (x_{i}),

(15)

where h_max(x_i) ∶ = max[h₁(x_i), h₂(x_i), …, h_{m + 2}(x_i)] and

ℒ (x_{i}) ∶ = {\begin{cases} h_{\max} (x_{i}), & if h_{\max} (x_{i}) \geq 0, \\ f_{v} (x_{i}), & otherwise . \end{cases}

(16)

Note that the procedure transforming the given constrained optimization problem into an unconstrained problem introduces no problem-dependent extra parameter such as penalty factors [23, 50] or Lagrange multipliers [51] that usually require time-consuming fine-tuning tasks. Based on the definition of a modified objective function ℒ(x_i) in (16), rigorous mathematical formulations of $x_{i}^{best, k}$ and $x_{swarm}^{best, k}$ were obtained as follows:

\begin{matrix} x_{i}^{best, k} ∶ = \arg \min_{x_{i}^{j}} {ℒ (x_{i}^{j}), 0 \leq j \leq k}, \\ x_{swarm}^{best, k} ∶ = \arg \min_{x_{i}^{k}} {ℒ (x_{i}^{k}), i = 1,2, \dots, m_{p}} . \end{matrix}

(17)

From the preceding observation, the overall computational steps of the proposed PSO algorithm for constrained MIDC optimization problems are described as shown in Figure 1.

In the following sections, the effectiveness of the proposed PSO scheme for constrained MIDC optimization problems is demonstrated via several engineering optimal design example problems (it may be difficult to guarantee the performance of our method theoretically due to the probabilistic nature of PSO).

3. Numerical Experimentation

Several engineering optimal design problems were investigated to evaluate the effectiveness of the proposed PSO technique described in Section 2. The best results obtained by the proposed method are compared to problems reported in the literature that were solved by various evolutionary computation and other optimization algorithms. In the following examples, all computations were performed on the MATLAB platform.

3.1. Engineering Optimization Problems

Problem 1 (design of a pressure vessel described in [19 –22]).

A cylindrical vessel is capped at both ends by hemispherical heads as shown in Figure 2. The objective is to minimize the total cost including the cost of the material, forming, and welding. There are four design variables: T_s (the thickness of the shell), T_h (the thickness of the head), R (the inner radius), and L (the length of the cylindrical section of the vessel not including the head). T_s and T_h are integer multiples of 0.0625 (the available thickness of rolled steel plates), and R and L are continuous. These design variables are specified as

x = {(x_{1}, x_{2}, x_{3}, x_{4})}^{T} ∶ = {(T_{s}, T_{h}, R, L)}^{T} .

(18)

The optimization problem is formulated as follows:

\min_{x} f (x) ∶ = 0.6224 x_{1} x_{3} x_{4} + 1.7781 x_{2} x_{3}^{2} + 3.1661 x_{1}^{2} x_{4} + 19.84 x_{1}^{2} x_{3}

(19)

subject to

h_{1} (x) ∶ = 0.0193 x_{3} - x_{1} \leq 0,

(20)

h_{2} (x) ∶ = 0.00954 x_{3} - x_{2} \leq 0,

(21)

h_{3} (x) ∶ = 1296000 - π x_{3}^{2} x_{4} - \frac{4}{3} π x_{3}^{3} \leq 0,

(22)

h_{4} (x) ∶ = x_{4} - 240 \leq 0,

(23)

h_{5} (x) ∶ = 1.1 - x_{1} \leq 0,

(24)

h_{6} (x) ∶ = 0.6 - x_{2} \leq 0,

(25)

where the search space was set as 𝔻 ∶ = {x ∊ ℝ⁴: (1.1, 0.6, 10, 10)^T ≤ x ≤ (10, 10, 150, 150)^T}.

Figure 2:

Configuration of a pressure vessel.

Problem 2 (design of a pressure vessel described in [23 –27]).

For the cylindrical pressure vessel shown in Figure 2, the constrained optimization problem formulated as (19) subject to (20)–(23) was considered in [23 –27]. Note that the constraint conditions described in (24) and (25) are not included in this case. Design variables x₁ (∶ = T_s) and x₂ (∶ = T_h) are integer multiples of 0.0625, and x₃ (∶ = R) and x₄ (∶ = L) are continuous. The search space was set as 𝔻 ∶ = {x ∊ ℝ⁴: (0, 0, 10, 10)^T ≤ x ≤ (99, 99, 200, 200)^T}.

Problem 3 (design of a speed reducer described in [24, 26]).

This example deals with a design problem of the speed reducer shown in Figure 3. The design parameters are as follows: b (the face width), m (the module of teeth), n (the number of teeth on the pinion), l₁ (the length of shaft 1 between bearings), l₂ (the length of shaft 2 between bearings), d₁ (the diameter of shaft 1), and d₂ (the diameter of shaft 2). The design variables are specified as

x = {(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7})}^{T} ∶ = {(b, m, n, l_{1}, l_{2}, d_{1}, d_{2})}^{T},

(26)

where x₃ (∶ = n) is integer and the rest of x are continuous. The objective is to minimize the total weight of a speed reducer. The constraints include limitations on the bending stress and surface stress of the gear teeth, transverse deflections of shafts 1 and 2 due to the transmitted force, and stresses in shaft 1 and 2. The problem is formulated as follows:

\min_{x} f (x) ∶ = 0.7854 x_{1} x_{2}^{2} (3.3333 x_{3}^{2} + 14.9334 x_{3} - 43.0934) - 1.508 x_{1} (x_{6}^{2} + x_{7}^{2}) + 7.4777 (x_{6}^{3} + x_{7}^{3}) + 0.7854 (x_{4} x_{6}^{2} + x_{5} x_{7}^{2})

(27)

subject to

\begin{matrix} h_{1} (x) ∶ = 27 x_{1}^{- 1} x_{2}^{- 2} x_{3}^{- 1} - 1 \leq 0, \\ h_{2} (x) ∶ = 397.5 x_{1}^{- 1} x_{2}^{- 2} x_{3}^{- 2} - 1 \leq 0, \\ h_{3} (x) ∶ = 1.93 x_{2}^{- 1} x_{3}^{- 1} x_{4}^{3} x_{6}^{- 4} - 1 \leq 0, \\ h_{4} (x) ∶ = 1.93 x_{2}^{- 1} x_{3}^{- 1} x_{5}^{3} x_{7}^{- 4} - 1 \leq 0, \\ h_{5} (x) ∶ = \frac{\sqrt{{(745 x_{4} / x_{2} x_{3})}^{2} + 16.9 \times 1 0^{6}}}{0.1 x_{6}^{3}} - 1100 \leq 0, \\ h_{6} (x) ∶ = \frac{\sqrt{{(745 x_{5} / x_{2} x_{3})}^{2} + 157.5 \times 1 0^{6}}}{0.1 x_{7}^{3}} - 850 \leq 0, \\ h_{7} (x) ∶ = x_{2} x_{3} - 40 \leq 0, \\ h_{8} (x) ∶ = 5 - (\frac{x_{1}}{x_{2}}) \leq 0, \\ h_{9} (x) ∶ = \frac{x_{1}}{x_{2}} - 12 \leq 0, \\ h_{10} (x) ∶ = (1.5 x_{6} + 1.9) x_{4}^{- 1} - 1 \leq 0, \\ h_{11} (x) ∶ = (1.1 x_{7} + 1.9) x_{5}^{- 1} - 1 \leq 0, \end{matrix}

(28)

where 2.6 ≤ x₁ ≤ 3.6, 0.7 ≤ x₂ ≤ 0.8, 17 ≤ x₃ ≤ 28, 7.3 ≤ x₄ ≤ 8.3, 7.8 ≤ x₅ ≤ 8.3, 2.9 ≤ x₆ ≤ 3.9, 5.0 ≤ x₇ ≤ 5.5.

Figure 3:

Configuration of a speed reducer.

Problem 4 (design of a speed reducer described in [22, 32 –34]).

It is assumed that for the same optimal design problem of the speed reducer presented in Problem 5, x₃ is an integer variable; x₁, x₂, x₄, and x₅ are integer multiples of 0.1; and x₆ and x₇ are integer multiples of 0.01. In the cited studies, the minimization problem given in (27) was modified to be

\min_{x} f (x) ∶ = 0.7854 x_{1} x_{2}^{2} (3.3333 x_{3}^{2} + 14.9334 x_{3} - 43.0934) - 1.508 x_{1} (x_{6}^{2} + x_{7}^{2}) + 7.477 (x_{6}^{3} + x_{7}^{3}) + 0.7854 (x_{4} x_{6}^{2} + x_{5} x_{7}^{2}) .

(29)

Note also that 7.3 ≤ x₅ ≤ 8.3 was considered in this case.

Problem 5 (design of a tension/compression coil spring described in [21, 22, 27, 35, 36, 57]). In this optimal design problem for tension/compression coil spring shown in Figure 4, x₁ (∶ = N), x₂ (∶ = D), and x₃ (∶ = d) were considered as integer, continuous, and discrete variables, respectively, as follows:

\begin{matrix} x_{1} = 5 + κ, where κ = 0,1, 2, \dots, 15, \\ 1.0 \leq x_{2} \leq 3.0, \\ x_{3} ∊ {0.207,0.225,0.244,0.263,0.283,0.307,0.331, 0.362,0.394,0.4375,0.50} . \end{matrix}

(30)

The following optimization problem was investigated:

\min_{x} f (x) ∶ = \frac{π^{2} x_{2} x_{3}^{2} (x_{1} + 2)}{4}

(31)

subject to

\begin{matrix} h_{1} (x) ∶ = S - \frac{8 K_{s} P_{\max} x_{2}}{π x_{3}^{3}} \geq 0, \\ h_{2} (x) ∶ = l_{\max} - (δ + 1.05 (x_{1} + 2) x_{3}) \geq 0, \\ h_{3} (x) ∶ = x_{3} - d_{\min} \geq 0, \\ h_{4} (x) ∶ = D_{\max} - x_{2} \geq 0, \\ h_{5} (x) ∶ = C - 3 \geq 0, \\ h_{6} (x) ∶ = δ_{p m} - δ \geq 0, \\ h_{7} (x) ∶ = l_{f} - δ - \frac{P_{\max} - P_{L}}{K} - 1.05 (x_{1} + 2) x_{3} \geq 0, \\ h_{8} (x) ∶ = \frac{P_{\max} - P_{L}}{K} - δ_{w} \geq 0, \end{matrix}

(32)

with

\begin{matrix} C = \frac{x_{2}}{x_{3}}, K_{s} = \frac{4 C - 1}{4 C - 4} + \frac{0.615}{C}, \\ δ = \frac{8 P_{\max} x_{2}^{3} x_{1}}{G x_{3}^{4}}, K = \frac{G x_{3}^{4}}{8 x_{2}^{3} x_{1}}, \end{matrix}

(33)

where the maximum work load P_max = 1000. 0 lb, the maximum free length l_max = 14.0 in, the minimum wire diameter d_min = 0.2 in, the allowable maximum shear stress S = 189000. 0 psi, the maximum outside diameter of the spring D_max = 3.0 in, the preload compression force P_L = 300. 0 lb, the allowable maximum deflection under preload δ_pm = 6.0 in, the deflection from preload position to maximum load position δ_w = 1.25 in, and the shear modulus of the material G = 11. 5 × 10⁶ psi.

Figure 4:

Configuration of a tension/compression coil spring.

3.2. Numerical Experimental Results and Discussion

The performance of the proposed PSO algorithm was investigated using the five MIDC optimization problems. For brevity, the swarm size m_p is 100, the maximum number of iterations is 3000, the number of experiments is 50, and c₁ = c₂ = 2.02 for all the examples. The maximum number of iterations for applying the three phases velocity update procedure described in (10) was set to k_DIV = 2000. Fifty runs were performed for each of the five optimization problems, and the best solutions are summarized in Tables 1 to 7. The optimal design variables and the corresponding optimal objective function values are indicated in these tables. To further assess the effectiveness of the proposed technique, the results are compared with the best optima results available in the published literature. A comparison of the optimal objective function values in Table 1 (Problem 1) and Table 4 (Problem 3) demonstrates the superiority of the proposed PSO algorithm over conventional techniques. Also, Table 2 (Problem 2), Table 6 (Problem 4), and Table 7 (Problem 5) prove our method's ability to obtain results that coincide with results found in the literature. A statistical comparison of the solution quality analysis given in Table 3 (Problem 2), Table 5 (Problem 3), and Table 8 (Problem 5) demonstrates the remarkable reliability of the proposed PSO method. Note that the statistical analysis results for Problems 1, 4, and 5 show a nearly 100% confidence level (i.e., the standard deviations are all approximately zero); these are omitted here for the sake of brevity. In addition, the presented optimal design variables are all feasible solutions (i.e., they guarantee h_ℓ(x) ≤ 0, ∀ ℓ). In Table 3, the worst value of the objective function is not particularly good, but the obtained mean value indicates that our PSO scheme is comparable to the method by Kim et al. [18] or is more robust than any conventional technique. The worst value appears only one time in the results set of 50 experiments which is confirmed in Figure 5. Overall, our experimental results demonstrate that the proposed PSO strategy for handling mixed variables in optimization problems is more likely to find the global optimum solution with greater accuracy compared to existing methods.

Table 1:

Optimal design of the pressure vessel—Problem 1.

Quantity	x₁ (discrete)	x₂ (discrete)	x₃ (continuous)	x₄ (continuous)	f(x)

Shih and Lai [19]	1.125	0.625	47.448	119.98	8160.80
Fu et al. [20]	1.125	0.625	48.3807	111.7449	8048.619
Sandgren [21]	1.123	0.625	48.97	106.72	7982.5
Rao and Xiong [22]	1.1875	0.625	61.4483	27.4037	7284.02
Proposed method	1.125	0.625	58.29015544	43.69265624	7198.00542037

Table 2:

Optimal design of the pressure vessel—Problem 2.

Quantity (N/A: not available)	x₁ (discrete)	x₂ (discrete)	x₃ (continuous)	x₄ (continuous)	f(x)

Parsopoulos and Vrahatis [23]	N/A	N/A	N/A	N/A	6154.7
Akhtar et al. [24]	0.8125	0.4375	41.9768	182.2845	6171.0
Coello and Mezura-Montes [25]	0.8125	0.4375	40.097398	176.654047	6059.946341
Mezura-Montes et al. [26]	0.8125	0.4375	42.098370	176.637146	6059.714355
He et al. [27]	0.8125	0.4375	42.0984456	176.63659584	6059.714335
Cagnina et al. [28]	0.8125	0.4375	42.098445	176.636595	6059.714335
Tomassetti [29]	0.8125	0.4375	42.098446	176.636596	6059.714337
Proposed method	0.8125	0.4375	42.0984456	176.63659584	6059.714335

Table 3:

Statistical comparison for the pressure vessel design problem—Problem 2.

Quantity f(x)	Best	Mean	Worst	Standard deviation

Parsopoulos and Vrahatis [23]	6154.70	8016.37	9387.77	745.869
Mezura-Montes et al. [26]	6059.714355	6355.343115	6846.628418	256.0437950
Mezura-Montes and Coello [30]	6059.7143	6379.9380	N/A	210.0
Cagnina et al. [28]	6059.714355	6092.0498	N/A	12.1725
Aguirre et al. [31]	6059.714355	6071.0133	N/A	15.1011
Tomassetti [29]	6059.714337	6086.9	6411.4	39.2722
Kim et al. [18]	6059.714335	6059.727721	6060.074434	0.065870503
Proposed method	6059.714335	6060.33057699	6090.52620169	4.35745530

Table 4:

Optimal design of the speed reducer—Problem 3.

Quantity	Akhtar et al. [24]	Mezura-Montes et al. [26]	Cagnina et al. [28]	Tomassetti [29]	Proposed method

x₁ (continuous)	3.503122	3.506163	3.5	3.5	3.499993
x₂ (continuous)	0.700006	0.700831	0.7	0.7	0.7
x₃ (integer)	17	17	17	17	17
x₄ (continuous)	7.549126	7.460181	7.3	7.3	7.3
x₅ (continuous)	7.859330	7.962143	7.8	7.8	7.8
x₆ (continuous)	3.365576	3.362900	3.350214	3.350215	3.35021467
x₇ (continuous)	5.289773	5.308949	5.286683	5.286683	5.28668323
f(x)	3008.08	3025.00513	2996.348165	2996.348165	2996.34541555

Table 5:

Statistical comparison for the speed reducer design problem—Problem 3.

Quantity f(x)	Best	Mean	Worst	Standard deviation

Akhtar et al. [24]	3008.08	3012.12	3028.28	N/A
Mezura-Montes et al. [26]	3025.005127	3088.777816	3226.248291	47.361890
Cagnina et al. [28]	2996.348165	2296.3482	N/A	0.0000
Aguirre et al. [31]	2996.372448	2996.4085	N/A	0.0286
Tomassetti [29]	2996.348165	2.9965E + 03	3.0020E + 03	0.9121
Proposed method	2996.34541555	2996.34541555	2996.34541556	2.46561597E – 09

Table 6:

Optimal design of the speed reducer—Problem 4.

Quantity	Li and Azarm [32]	Li and Papalambros [33]	Kuang et al. [34]	Rao and Xiong [22]	Proposed method

x₁ (discrete)	3.5	3.5	3.6	3.5	3.5
x₂ (discrete)	0.7	0.7	0.7	0.7	0.7
x₃ (integer)	17	20	17	17	17
x₄ (discrete)	7.3	7.3	7.3	7.3	7.3
x₅ (discrete)	7.71	7.3	7.8	7.8	7.8
x₆ (discrete)	3.35	3.35	3.4	3.36	3.36
x₇ (discrete)	5.29	5.29	5.0	5.29	5.29
f(x)	2996.3 (infeasible)	2985.22 (infeasible)	2876.12 (infeasible)	3000.83	3000.82953716

Table 7:

Optimal design of the tension/compression coil spring—Problem 5.

Quantity	x₁ (integer)	x₂ (continuous)	x₃ (discrete)	f(x)

Deb and Goyal [35]	9	1.226	0.283	2.665
Sandgren [21]	10	1.180701	0.283	2.7995
Rao and Xiong [22]	9	1.225280	0.283	2.66342
Lampinen and Zelinka [36]	9	1.223041	0.283	2.65856
He et al. [27]	9	1.223041	0.283	2.65856
Proposed method	9	1.22304101	0.283	2.65855917

Table 8:

Statistical comparison for the tension/compression coil spring—Problem 5.

Quantity f(x)	Best	Mean	Worst	Standard deviation

He et al. [27]	2.65856	2.738024	N/A	0.107061
Gandomi et al. [37]	2.658575665	4.3835958	7.8162919	4.6076313
Proposed method	2.65855917	2.68803204	2.69949371	0.01856616

Figure 5:

Objective function values of Problem 2 in 50 experiments.

4. Conclusion

We presented a novel modified constrained PSO algorithm using a simple diversity enhancing mechanism with high applicability and reliability for finding optimal mixed integer-discrete-continuous design variables. The proposed approach causes the particles of the swarm to evolve as follows. At the beginning stage of evolution, all particles try to move into a feasible region where constraint satisfaction is guaranteed. Here, individual particles first freely evolve in a continuous search space regardless of the variable type based on the diversity-guided velocity update law that includes three phases: attraction, repulsion, and a combination of attraction and repulsion. The introduced three-phase velocity control method improves the population diversity in the framework of constrained PSO efficiently and thus can reduce the probability of a premature convergence phenomenon, particles being trapped in a local optimum due to the mixed variable nature. For integer or discrete design variables, the corresponding continuous values of particles are rounded off to the nearest integer or discrete values. Thereafter, the particles continue to evolve in the aforementioned manner and try to explore the feasible region to find an optimal solution that satisfies the given constraint conditions. Note that the procedure transforming the given constrained optimization problem into an unconstrained problem introduces no problem-dependent extra parameters such as penalty factors or Lagrange multipliers which usually require time-consuming fine-tuning tasks. Our numerical experiment verified that the proposed constrained PSO scheme, combined with a general integer and discrete variable handling technique and three phases of particle velocity control law, can yield satisfactory optimal solution accuracy.

Footnotes

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2012-012295) and the Chung-Ang University excellent freshman scholarship grants in 2013.

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