An interval particle swarm optimization method for interval nonlinear uncertain optimization problems

Abstract

For nonlinear optimization problems involving interval variables or parameters, the interval possibility degree or interval order relation is conventionally adopted to convert them into a deterministic corresponding counterpart and then can be solved with linear programing approaches, through which computational resources and costs have been preserved to a certain extent. However, some information and computational accuracy will be discounted during the model transformation. In this paper, an interval optimization algorithm based on particle swarm optimization is proposed aiming to cut down the loss of useful information by means of straight optimization instead of model deterministic converting so as to enhance calculation accuracy with few growths of computing expenses. The proposed method firstly employs an interval satisfaction value to cope with the constraints. Consequently, the particle swarm optimization method is used to seek the optimum solution sets without the model deterministic converting. And a criterion based on the satisfaction value model of interval possibility degree plays the role in individual selecting out of the above solution sets. Finally, the effectiveness of the proposed method is verified by investigating four examples.

Keywords

Uncertainty interval particle swarm optimization nonlinear programing interval satisfaction value interval preference criterion

Introduction

More often than not, the uncertainty exists in the practical engineering problems as aleatory or epistemic reasons.¹ Studies have shown that if the above mentioned uncertainties are ignored or simplified in engineering optimization, it will often lead to markable large errors and even get opposite conclusions.^2–4 Therefore, reducing the error and improving the reliability have become the research goals of some scholars.^5–7 In order to acquire more reliable optimization results, uncertainties in practical problems should be considered in sync with the corresponding uncertain optimization algorithms. So far, many savants have issued some corresponding optimization algorithms to cope with the uncertain parameters which can be mainly divided into three categories: stochastic programing, fuzzy programing and interval programing. Stochastic programing^8–11 transforms the optimization problem into a stochastic programing problem based on the hypothesis that its probability distribution function is acquirable in advance. Fuzzy programing^12,13 transforms the optimization problem into a corresponding fuzzy programing problem by assuming that its membership function is foreknown, thereby using a fuzzy set to represent its corresponding uncertainty constraints. Interval programing^14–19 usually obtains the appropriate parameter variation range by determining the upper and lower bounds of parameter variation and directly treating the interval as a whole to solve.

From an implementation perspective, although stochastic programing and fuzzy programing can describe uncertainty more accurately, their application is limited in some special cases because a large number of test samples are needed to construct probability density function or membership functions. Fortunately, interval programing merely needs to know the upper and lower bounds of the interval uncertain parameters instead of the specific value of the probability density of uncertain parameters. Not affecting the solution accuracy or causing over conservative results under some conditions, interval programing is more and more popular for engineers and technicians due to its advantages of lower test cost and excellent comprehensibility. Recent decades have seen studies on interval programing accompanied by a series of effective interval programing models and methods.^20–22 Among them, in the community of linear programing involving interval variables, Ishibuchi and Tanaka²³ proposed the interval order relation to transform the interval objective function maximization problem into a multi-objective optimization problem. Elishakoff et al.²⁴ proposed a structural optimization design method that describes the design problem as a two-level optimization problem with bounded uncertain loads. Elishakoff²⁵ compared the probabilistic modeling and convex modeling of uncertainty, and proposed combinatorial optimization and anti-optimization based on uncertainty. Huang et al.²⁶ developed a two-step method through directly conveying the uncertain factors into the linear programing process and then obtained the final decision. As for optimal solutions obtained by the two-step method may not be within the scope of decision space, Wang and Huang²⁷ improved the above mentioned method by appending additional constraints to the optimization model. Lai et al.²⁸defined the effective solution based on interval order relation and the method can use the non-inferior solution defined by the second-order relationship between intervals to deal with linear optimization problems. Elishakoff and Ohsaki²⁹took full advantage of anti-optimization through a combination of structural optimizations to produce optimal designs considering worst-case scenarios. Rivaz and Yaghoobi³⁰ proposed the minimal and maximal regret solution of the multi-objective linear programing problem with interval objective function coefficients. Santoro et al.³¹ introduced parametric interval analysis to solve interval equations by optimization and anti-optimization problems. The method proposed by the above scholars are more suitable for solving linear optimization problems. Nevertheless most practical engineering problems are nonlinear or even strong nonlinear. For those more general nonlinear programing problems, some productive works have been derived. Cao and Huang³²studied the first derivative of the objective function, constructed the interval expansion function with at least second-order convergence, and introduced the interval monotonicity rule to solve it. Qiu and Elishakoff³³proposed an inner layer optimization method based on interval analysis and laid the mathematical basis of convex models. Jiang et al.³⁴ proposed an optimization method for solving uncertain structural problems based on nonlinear interval number programing method and interval analysis method. Jiang³⁵ proposed two transformation models of nonlinear interval number optimization, which realized the transformation from uncertain optimization problem to deterministic one. Elishakoff and Miglis³⁶ performed deterministic analysis of uncertainty problems by describing variation of various inner properties or external influences on the system are parameterized using trigonometric functions. Li et al.³⁷ transformed the interval uncertain model into a nested deterministic model, and then solved the transformed deterministic model according to the adaptive interval multi-objective optimization method and non-dominated sorting genetic algorithm. Wu et al.³⁸ used high-order coefficients of approximate Taylor inclusion functions to solve the extension problem of interval algorithms. The algorithm uses interval algorithm to optimize inner loop. Hu et al.³⁹ proposed a new robustness index to deal with the bounded constraints of performance variation, and transformed an interval robust optimization problem into a deterministic optimization. Xie and Liao⁴⁰ developed an enhanced programing approach to decouple the optimization problem considering correlation between interval parameters. Wang et al.⁴¹ proposed an interval uncertain optimization method using genetic algorithm to optimize the inner layer and non-dominated sorting genetic algorithm to optimize the outer layer. Wang et al.⁴² proposed a Legendre polynomial expansion method combined with subinterval technique to optimize nonlinear interval uncertain problems. Rahman et al.⁴³ proposed the concept of type-2 interval order relation and type-2 interval valued function. Tang et al.⁴⁴ proposed interval sequence linear programing to solve nonlinear robust optimization problems. Wei and Li⁴⁵ proposed an interval optimization method based on adaptive binary decomposition algorithm.

Among the above mentioned methods for solving nonlinear interval optimization problems, some need to know the membership function or probability density function, some are only suitable for solving problems with low uncertainty levels, and most need to transform the uncertain optimization model into the deterministic optimization model. In this transformation process, some useful information of uncertain parameters will be lost, which will lead to the error between the optimal solution of the optimization model and the real solution without reflecting the uncertainty in practical problems. Therefore, some attempts try to solve the nonlinear interval optimization model directly without deterministic transformation recently and few research results have been presented in this community. Cheng et al.⁴⁶ proposed a constrained interval optimization model and a new concept of interval order relation based on the priority criterion of the degree of interval constraint violation, the design vectors were directly sorted by the interval and then a direct interval optimization implemented by means of a nested genetic algorithm and Kriging technique. Fu and Cao⁴⁷ proposed an interval differential evolution algorithm with adaptive subinterval decomposition analysis which eliminated the double-loop nesting optimization and improved the computational efficiency while ensuring accuracy, and used adaptive subinterval decomposition analysis to calculate the bounds of objective functions and constraints in the uncertain domain. However, the research on directly solving interval nonlinear uncertain optimization model is also in primary stage. There are remaining some problems such as: 1. Which optimization algorithm should be chosen as the solution tool to guarantee the computational efficiency? 2. What criteria is the proper one to select a more appropriate optimal solution? Particle swarm optimization (PSO) is a parallel structure, which determines the next position of the particle according to the current individual optimal position of each particle in the particle swarm, the current global optimal position, and the individual’s inertia. The algorithm is simple and easy to implement. Therefore, this paper proposes an interval optimization model based on PSO and an interval preference judgment criterion based on interval satisfaction value to deal with the above problems. In practical applications, the optimization of vehicle ride comfort and the stability of aerial camera in work are to improve the overall performance. Therefore, when solving the above problems, the accuracy and reliability of the obtained results are particularly important. As the above problems are nonlinear involving interval parameters, the direct method can be employed. In this paper, the proposed method is investigated in engineering examples including the eight-degree-of-freedom dynamic optimization model of the vehicle and the robustness optimization of the aerial camera. The remainder of the paper is organized as follows: Section 2 introduces the expression of the nonlinear interval optimization problem. Section 3 presents interval particle swarm optimization and describes the details of the proposed algorithm in detail. Numerical example and application verify the effectiveness of the proposed method in Section 4. Section 5 gives the conclusion of this paper.

Statement of the problem

Interval numbers are defined as a pair of ordered real numbers:

A^{I} = [A^{L}, A^{R}]

(1)

where $I$ is an interval, $L$ and $R$ are the lower and upper bounds of the interval. When $A^{L} = A^{R}$ , the interval degenerates into a real number. This interval is expressed as follows:

A^{I} = 〈 A^{C}, A^{W} 〉

(2)

where $A^{C}$ is the midpoint value of interval $A^{I}$ . $A^{W}$ is the radius value of interval $A^{I}$ .

A^{C} = \frac{A^{R} + A^{L}}{2}

(3)

A^{W} = \frac{A^{R} - A^{L}}{2}

(4)

The nonlinear constrained optimization problem is expressed as follows:

{\begin{matrix} min_{X} f (X) \\ \begin{matrix} s . t . \\ g_{j} (X) \leq 0, j = 1, 2 . . . p \end{matrix} \end{matrix}

(5)

where $f (X)$ is the objective function. $X = (X_{1}, X_{2} . . . X_{n}) \in R^{n}, i = 1, 2 . . . n$ is an n-dimensional design vector. $g_{j} (X)$ is the $j$ th constraint. $p$ is the number of the constraints.

In practical problems, $f (X)$ and $g_{j} (X)$ will have uncertain parameters. In general, the objective function and constraint conditions will contain uncertain parameters at the same time. The interval is used to describe the uncertain parameters, which are expressed as follows:

\begin{matrix} {\begin{matrix} \min_{X} f (X) \\ \begin{matrix} s . t . \\ g_{j}^{I} (X, U) \leq b_{j}^{I}, j = 1, 2 . . . p \end{matrix} \\ b_{j}^{I} = [b_{j}^{L}, b_{j}^{R}] \end{matrix} \end{matrix}

(6)

where $f (X, U)$ is the objective function. $U = (U_{1}, U_{2} . . . U_{m}) \in R^{m}$ is an m-dimensional uncertain vector. $b_{j}^{I}$ is the spatial range of the $j$ th constraint. $m$ is the number of uncertain parameters.

The interval order relation transformation model can solve the nonlinear interval optimization problems. The interval order relation transformation model is to extend the linear interval number optimization method based on order relation to general nonlinear problems and transforms the uncertain optimization problem into a deterministic multi-objective optimization problem.

For solving the maximum and minimum optimization problems, there are the following five forms⁴⁸: interval order relation of preference for upper and lower bounds $\leq_{LR}$ interval order relation of preference for interval midpoint and radius $\leq_{CW}$ interval order relation of preference for interval lower bound and midpoint $\leq_{LC}$ interval order relation of preference for interval lower bound $\leq_{L}$ interval order relation of preference for upper bound of the interval $\leq_{R}$ .

Select $\leq_{CW}$ to solve equation (6) as an example, where the interval order relation $\leq_{CW}$ is expressed as follows:

(1) Maximum optimization problem:

{\begin{matrix} A^{I} \leq_{CW} B^{I} Only if A^{C} \leq B^{C} and A^{W} \geq B^{W} \\ A^{I} <_{CW} B^{I} Only if A^{I} \leq_{CW} B^{I} and A^{I} \neq B^{I} \end{matrix}

(7)

(2) Minimize optimization problem:

{\begin{matrix} A^{I} \leq_{CW} B^{I} Only if A^{C} \geq B^{C} and A^{W} \geq B^{W} \\ A^{I} <_{CW} B^{I} Only if A^{I} \leq_{CW} B^{I} and A^{I} \neq B^{I} \end{matrix}

(8)

The value range of the objective function $f (X)$ of a specific design vector $X$ is expressed as follows:

f^{I} (X) = [f^{L} (X), f^{R} (X)] = 〈 f^{C} (X), f^{W} (X) 〉

(9)

{\begin{matrix} f^{C} (X) = \frac{f^{R} (X) + f^{L} (X)}{2} \\ f^{W} (X) = \frac{f^{R} (X) - f^{L} (X)}{2} \end{matrix}

(10)

According to equation (10), the uncertain optimization problem in equation (6) is transformed into a definite multi-objective optimization problem by judging the midpoint value and radius value of the objective function, which is expressed as follows:

min_{X} (f^{C} (X), f^{W} (X))

(11)

The interval of the uncertain objective function is solved by twice optimization:

f^{L} (X) = min_{U} f (X, U)

(12)

f^{R} (X) = max_{U} f (X, U)

(13)

Through the above processing, the converted model is determined as follows:

{\begin{matrix} min_{X} (f^{C} (X), f^{W} (X)) \\ \begin{matrix} s . t . \\ P (M_{i}^{I} \leq N_{i}^{I}) \geq λ_{i}, i = 1, 2 . . . k \end{matrix} \\ X \in Ω^{n} \end{matrix}

(14)

\begin{matrix} {\begin{matrix} f^{C} (X) = \frac{max_{U} f (X, U) + min_{U} f (X, U)}{2} \\ f^{W} (X) = \frac{max_{U} f (X, U) - min_{U} f (X, U)}{2} \\ U = {U | U_{i}^{L} \leq U_{i} \leq U_{i}^{R}, i = 1, 2 . . . q} \end{matrix} \end{matrix}

(15)

Optimization based on the above optimization method requires model transformation, which is easy to lose information. In order to reduce information loss and improve accuracy, an interval particle swarm optimization (IPSO) based on PSO is proposed.

Interval particle swarm optimization (IPSO)

Particle swarm optimization

PSO is inspired by the research on the simulation between the movement and the swarm in the process of birds foraging, to develop an algorithm to solve the optimization problem.⁴⁹ In this algorithm, the particle swarm are initialized randomly in the initialization range, including the initial position $X_{i}^{0}$ and the initial velocity $V_{ij}^{0}$ . The fitness value is obtained in the fitness function. The fitness value of each particle is compared with the fitness value of the best position to obtain the individual extremum value $P_{best}$ . Then, the individual extremum $P_{best}$ is compared with the fitness value of the best position obtained in the particle swarm or its corresponding neighborhood to obtain the global extremum value $G_{best}$ . Finally, the optimal solution is obtained by constantly updating the position $X_{i}$ , velocity $V_{ij}$ , individual extremum $P_{best}$ , and global extremum $G_{best}$ . The inertia weight coefficient $ω$ , learning factors $c_{1}$ and $c_{2}$ , and random number $rand (0, 1)$ are added to iteratively solve the individual extremum value $P_{best}$ and the global extremum value $G_{best}$ until the optimum is reached in the given range. The updating formula of velocity vector and position vector of the $i$ th particle is:

\begin{matrix} V_{ij}^{k + 1} = ω V_{ij}^{k} + c_{1} rand (0, 1) \times (P_{best, ij} - X_{ij}^{k}) \\ + c_{2} rand (0, 1) \times (G_{best, j} - X_{ij}^{k}) \\ X_{ij}^{k + 1} = X_{ij}^{k} + V_{ij}^{k + 1} \\ i = 1, 2 . . . NP \\ j = 1, 2 . . . NP \end{matrix}

(16)

IPSO

IPSO is an optimization algorithm for solving nonlinear interval optimization problems based on PSO. The algorithm uses interval satisfaction values to deal with interval uncertainty constraints. Based on the interval possibility satisfaction values model, it will establish an interval preference criterion to solve the nonlinear uncertainty optimization problem caused by interval uncertainty.

Interval satisfaction value and constraint treatment

In order to construct the interval possibility satisfaction model and realize the processing of interval constraints, a left truncated interval possibility model based on the reliability-based possibility degree of interval (RPDI) model⁵⁰ is given to quantitatively describe the degree of one interval greater than (or better than) another. For intervals $A^{I}$ and $B^{I}$ , the left truncated interval possibility degree model $P_{r}' (A^{I} \leq B^{I})$ is defined as:

P_{r}' (A^{I} \leq B^{I}) = max (\frac{B^{R} - A^{L}}{2 A^{W} + 2 B^{W}}, 0)

(17)

When the interval $B^{I}$ or $A^{I}$ degenerates into a real number, $P_{r}' (A^{I} \leq B^{I})$ has the following form:

P_{r}' (A^{I} \leq b) = max (\frac{b - A^{L}}{2 A^{W}}, 0), P_{r}' (a \leq B^{I}) = max (\frac{B^{R} - a}{2 B^{W}}, 0)

(18)

$P_{r}' (A^{I} \leq B^{I})$ has the following properties:

(1) $P_{r}' (A^{I} \leq B^{I}) \in [0, + \infty)$ .

(2) If $A^{R} \leq B^{L}$ , then $P_{r}' (A^{I} \leq B^{I}) \geq 1$ , $A^{I}$ is complete to the left of $B^{I}$ on the real number line.

(3) If $A^{L} \geq B^{R}$ , then $P_{r}' (A^{I} \leq B^{I}) \geq 1$ , $A^{I}$ is complete to the right of $B^{I}$ on the real number line.

Based on the left truncated interval possibility model, the concept of interval possibility satisfaction value is proposed below. For intervals $A^{I}$ and $B^{I}$ , interval possibility satisfaction value $R_{λ}$ is defined as:

R_{λ} (A^{I} \leq B^{I}) = {\begin{matrix} P_{r}' (A^{I} \leq B^{I}) / λ, if P_{r}' (A^{I} \leq B^{I}) \leq λ \\ 1, otherwise \end{matrix}

(19)

where $λ$ is the pre-given interval possibility level. The interval possibility satisfaction value $R_{λ}$ can be used to quantitatively describe whether the state of $A^{I} \leq B^{I}$ satisfies the degree of a given possibility level $λ$ , or it can be called the interval possibility satisfaction value at a level $λ$ . It has the following properties:

(1) $R_{λ} (A^{I} \leq B^{I}) \in [0, 1]$ .

(2) When $P_{r}' (A^{I} \leq B^{I})$ is less than or equal to $λ$ , (when the given level of possibility is not reached), $R_{λ} (A^{I} \leq B^{I})$ is set as the value of $P_{r}' (A^{I} \leq B^{I}) / λ$ .

(3) When $P_{r}' (A^{I} \leq B^{I})$ is greater than $λ$ , that is, (when the given level of possibility is completely reached), $R_{λ} (A^{I} \leq B^{I})$ has a value of 1.

It should be pointed out that for the same set of intervals $A^{I}$ and $B^{I}$ , the values obtained under different possibility levels $R_{λ}$ may be different. The interval possibility satisfaction value in equation (9) can be used to deal with the interval uncertainty constraints. For any interval constraint $g_{s} (X, U) \leq b_{s}^{I}$ , the interval possibility satisfaction value $R_{λ_{s}}$ can be calculated:

R_{λ_{s}} (g_{s}^{I} (X) \leq b_{s}^{I}) = {\begin{matrix} P_{r}' (g_{s}^{I} (X) \leq b_{s}^{I}) / λ_{s}, if P_{r}' (g_{s}^{I} (X) \leq b_{s}^{I}) \leq λ_{s} \\ 1, otherwise \end{matrix}

(20)

where $λ_{s}$ is the interval possibility level of the pre-given $s$ th constraint. For multiple interval constraints in uncertain optimization problems, calculating the interval possibility satisfaction value of each constraint can obtain a global interval possibility satisfaction value $R_{t}$ :

R_{t} = \sum_{s = 1}^{l} R_{λ_{s}}

(21)

According to the overall interval possibility satisfaction value $R_{t}$ and the size of the interval constraint values $l$ , it is possible to determine whether a vector individual in the evolutionary population of IPSO satisfies the interval constraint conditions. The details are as follows:

(1) When $R_{t} < l$ , it means that at least one constraint cannot completely meet the requirement of the given interval possibility level, and the individual vector can be called an infeasible solution. At this point, the larger the value of $R_{t}$ , the greater the degree to which the vector individuals meet all the constraint possibility levels, and the individuals with a smaller value of $R_{t}$ should be preferentially retained in the next generation population.

(2) When $R_{t} = l$ , it means that all constraints completely meet the given interval possibility level requirements, at which point the individual vector is called a feasible solution.

Individual preference strategy based on interval preference criterion

Under the framework of IPSO, how to select excellent individuals from the target vector individuals to retain the next generation of the evolutionary population is directly related to the overall performance of IPSO. This paper establishes an interval preference criterion based on the above interval possibility satisfaction value model to realize individual preference, as shown in Figure 1. Assuming that $X_{1}$ and $X_{2}$ are any two bodies in the population of IPSO, the satisfaction values of the overall interval possibility degree are $R_{t} (X_{1})$ and $R_{t} (X_{2})$ , the midpoint values of the objective function are $f^{c} (X_{1})$ and $f^{c} (X_{2})$ , and the radius values of the objective function are $f^{w} (X_{1})$ and $f^{w} (X_{2})$ , respectively, the preferred process is as follows:

(1) When $R_{t} (X_{1}) < l$ and $R_{t} (X_{2}) = l$ , viable individual $X_{2}$ is selected and retained for the next generation population. If $R_{t} (X_{2}) < l$ and $R_{t} (X_{1}) = l$ , viable individual $X_{1}$ is reserved for the next generation population. This indicates that IPSO preferentially selects feasible solution.

(2) When $R_{t} (X_{1}) = R_{t} (X_{2}) = l$ , if $f^{c} (X_{1}) < f^{c} (X_{2})$ , select individual $X_{1}$ . Otherwise pick individual $X_{2}$ . This situation indicates that when two individuals are both feasible solutions, the individual with a smaller point value in the objective function should be selected to ensure that the nominal performance of the objective function value is better.

(3) When $R_{t} (X_{1}) = R_{t} (X_{2}) = l$ and $f^{c} (X_{1}) = f^{c} (X_{2})$ , if $f^{w} (X_{1}) < f^{w} (X_{2})$ , select individual $X_{1}$ . Otherwise pick individual $X_{2}$ . This situation indicates that when two individuals are both feasible solutions and the midpoint value of the objective function is equal, the individual with a smaller radius value of the objective function should be selected to ensure that robustness of the objective function is better.

(4) If $R_{t} (X_{1}) = R_{t} (X_{2}) = l$ , $f^{c} (X_{1}) = f^{c} (X_{2})$ and $f^{w} (X_{1}) = f^{w} (X_{2})$ , any individual is selected randomly and retained for the next generation.

(5) When $R_{t} (X_{1}) < l$ and $R_{t} (X_{2}) < l$ , if $R_{t} (X_{1}) > R_{t} (X_{2})$ , select individual $X_{1}$ . Otherwise pick individual $X_{2}$ . This shows that when the two bodies are infeasible individuals, the individual with a large overall interval possibility satisfaction value should be selected, because the larger the $R_{t}$ -value is, the greater degree that the vector individual satisfies all the constraint possibility levels.

Figure 1.

Individual preference strategy based on interval preference criterion.

Flow chart of IPSO

Based on interval satisfaction degree and interval preference criteria, this section proposes an algorithm to directly solve nonlinear uncertain optimization problems with interval parameters, namely, IPSO whose flow chart is shown in Figure 2. Detailed iteration steps are as follows:

Step 1 : An interval uncertain optimization problem with design vector space $X \in R^{n}$ and uncertain vector parameter $U \in R^{m}$ .

Step 2 : Set the initial parameter population size $NP$ , inertia weight coefficient $ω$ , learning factor $c_{1}$ , $c_{2}$ , interval possibility level $λ_{s}$ of the $s$ th constraint, maximum speed $V_{max}$ , minimum speed $V_{min}$ , maximum iteration number $K_{max}$ of IPSO.

Step 3 : Generate initial particle position $X_{i}^{k}$ and initial velocity $V_{i}^{k}$ of design variables.

Step 4 : In the current population range, calculate the boundary response values $f^{L} (X_{i}^{k})$ , $f^{R} (X_{i}^{k})$ of the objective function of design variable $X_{i}^{k}$ and the boundary response values $g_{j}^{L} (X_{i}^{k})$ , $g_{j}^{R} (X_{i}^{k})$ of the $j$ th constraint function.

Step 5 : Under the $j$ th constraint condition, calculates the possibility satisfaction value $R_{t} (X_{i}^{k})$ of the overall interval, the midpoint value $f^{c} (X_{i}^{k})$ and radius value $f^{w} (X_{i}^{k})$ of the corresponding objective function of the individual, and the midpoint value $g_{j}^{c} (X_{i}^{k})$ and radius value $g_{j}^{w} (X_{i}^{k})$ of the constraint function.

Step 6 : Determine whether the overall interval possibility satisfaction value $R_{t}$ is less than the interval constraint value $l$ , satisfies the condition, then updates the particle velocity and position, and jumps back to Step 3 for circulation. If the conditions are not satisfied, $X_{i}^{k}$ is stored in the feasible solution alternative library, an iteration number is added, and whether the iteration number is less than or equal to the given maximum iteration number is judged. If the iteration number is greater than or equal to the given maximum iteration number, the speed and position of the particles are updated, and the loop is skipped back to Step 3 . If the iteration number is less than or equal to the given maximum iteration number, all the fitness values in the feasible solution alternative library are calculated, and the optimal feasible solution $X_{optimum}^{*}$ is selected based on the individual preference strategy of the interval preference criterion.

Step 7 : Output the optimal design vector and finish running.

Figure 2.

Flow chart of interval particle swarm optimization.

Examples and applications

Numerical example 1

The following constrained interval optimization problem³⁵ is used as an example:

\begin{matrix} {\begin{matrix} min_{X} f (X, U) = 130 - U_{1}^{2} (X_{1} + 2) - U_{2} X_{2}^{2} - U_{3}^{2} X_{3}^{2} \\ s . t . \\ g_{1} (X, U) = U_{1} X_{1}^{2} - U_{2}^{2} X_{2} + U_{3} X_{3} = [6.2, 7.3] \\ g_{2} (X, U) = U_{1} X_{1} + U_{2} X_{2} + U_{3}^{2} X_{3}^{2} = [15.0, 20.0] \\ - 1 \leq X_{1} \leq 5 \\ - 3 \leq X_{2} \leq 6 \\ - 2 \leq X_{3} \leq 7 \\ U_{1} \in [1.0, 1.3] \\ U_{2} \in [0.9, 1.1] \\ U_{3} \in [1.2, 1.4] \end{matrix} \end{matrix}

(22)

For this example, the parameters are set as follows: population size $NP = 30$ , inertia weight coefficient $ω$ , learning factor $c_{1}$ , $c_{2}$ , interval possibility level of constraint $λ_{s}$ , maximum number of iterations $K_{\max} = 30$ .

IPSO proposed in Section 3 is used to solve the above interval optimization problem. The maximum number of iterations is given as the stopping criterion. The optimization results under different interval possibility satisfaction values are shown in Figure 3 As can be seen from Figure 4, the optimal target value converges at the 11th generation. This proves that the proposed algorithm can efficiently solve the interval optimization problem with small uncertainty. The optimal solution is $X = (1.83, 6.00, - 1.72)$ , and the corresponding objective function is $f (X, U) = 〈 67.09, 9.13 〉$ . According to the interval possibility satisfaction value and the interval preference criterion, the two constraints $g_{1}$ and $g_{2}$ in equation (22) are satisfied when $X = (1.83, 6.00, - 1.72)$ , so when the satisfaction level is 1, $X = (1.83, 6.00, - 1.72)$ is the optimal solution.

Figure 3.

Optimization results under different interval possibility satisfaction values.

Figure 4.

Convergence curve of example 1with $R_{λ} = 1$ .

The optimal solution of equation (22) is $X^{*} = (- 0.13, 0.10, 5.24)$ when the acceptable level of a constraint is 1 by an indirect algorithm that transforms an interval model into a deterministic model. As can be seen from Table 1, the optimal solutions positioned by the proposed algorithm and Jiang’s³⁵ method both meet the constraints. However, the optimal solution obtained by this algorithm is better than that of Jiang’s³⁵ method because its corresponding target value is smaller than that of Jiang’s³⁵ method. Therefore, the direct interval optimization algorithm proposed in this paper is superior to Jiang’s³⁵ method.

Table 1.

Comparison of optimization results.

Algorithm	Optimal solution	Objection f
IPSO	(1.83, 6.00, −1.72)	(67.09, 9.13)
Jiang’s³⁵ method	(−0.13, 0.10, 5.24)	(80.80, 7.79)

Numerical example 2

The following constrained interval optimization problem⁵¹ is used as an example:

{\begin{matrix} min_{X} f (X, U) = U_{1} {(X_{1} - 2)}^{2} + U_{2}^{2} {(X_{2} - 1)}^{2} + U_{3}^{2} X_{3}^{2} - U_{1} U_{2} X_{1} \\ s . t . \\ g (X, U) = U_{1}^{2} X_{1}^{2} + U_{1} U_{2} X_{1} - U_{2}^{2} X_{2}^{2} + U_{2} U_{3} X_{2} \\ + U_{1} U_{3} X_{3}^{2} + U_{3}^{2} X_{3} \geq [65, 70] \\ - 2 \leq X_{1} \leq 6 \\ - 4 \leq X_{2} \leq 7 \\ - 3 \leq X_{3} \leq 8 \\ U_{1} \in [1.5, 1.8] \\ U_{2} \in [1.2, 1.5] \\ U_{3} \in [1.6, 2.0] \end{matrix}

(23)

Due to the random errors in IPSO calculation, the mean of 20 times of calculation is taken as the final result. Results of 20 calculations are shown in Tables 2 and 3. The optimization results under different interval possibility satisfaction values are shown in Figure 5 When the interval possibility satisfaction value is 1.0, $X = (5.11, 0.91, 0.16)$ is the optimal solution, and the corresponding objective function is $f (X, U) = 〈 3.17, 7.03 〉$ . The convergence curves under different $R_{λ}$ are shown in Figure 6. The optimal solution is obtained in the 32nd generation, which shows that the IPSO is fast.

Table 2.

The calculation results $X$ under different interval possibility satisfaction values.

Times	0.90	1.00	1.10	1.20	1.30
1	(4.89, 0.93, 0.16)	(5.13, 0.88, 0.16)	(5.40, 0.91, 0.17)	(5.56, 0.92, 0.18)	(5.68, 1.09, 0.38)
2	(4.90, 0.93, 0.15)	(5.11, 0.86, 0.14)	(5.42, 0.90, 0.19)	(5.58, 0.94, 0.19)	(5.71, 1.15, 0.44)
3	(4.93, 0.90, 0.17)	(5.08, 0.88, 0.13)	(5.41, 0.93, 0.18)	(5.54, 0.94, 0.20)	(5.73, 1.11, 0.38)
4	(4.93, 0.96, 0.19)	(5.12, 0.91, 0.15)	(5.44, 0.95, 0.17)	(5.57, 0.98, 0.21)	(5.70, 1.17, 0.39)
5	(4.92, 0.93, 0.15)	(5.10, 0.90, 0.13)	(5.40, 0.91, 0.19)	(5.56, 0.96, 0.20)	(5.68, 1.11, 0.40)
6	(4.89, 0.90, 0.18)	(5.13, 0.90, 0.14)	(5.45, 0.94, 0.18)	(5.54, 0.92, 0.23)	(5.74, 1.09, 0.42)
7	(4.92, 0.93, 0.19)	(5.11, 0.86, 0.15)	(5.44, 0.90, 0.17)	(5.59, 0.96, 0.21)	(5.68, 1.17, 0.38)
8	(4.91, 0.92, 0.19)	(5.09, 0.91, 0.13)	(5.48, 0.93, 0.18)	(5.58, 0.95, 0.22)	(5.73, 1.13, 0.41)
9	(4.88, 0.92, 0.17)	(5.14, 0.96, 0.17)	(5.41, 0.92, 0.18)	(5.57, 0.95, 0.20)	(5.76, 1.11, 0.41)
10	(4.88, 0.94, 0.15)	(5.11, 0.92, 0.18)	(5.42, 0.89, 0.19)	(5.56, 0.92, 0.24)	(5.74, 1.13, 0.39)
11	(4.94, 0.94, 0.17)	(5.09, 0.89, 0.19)	(5.41, 0.91, 0.17)	(5.58, 0.95, 0.27)	(5.70, 1.09, 0.44)
12	(4.88, 0.92, 0.17)	(5.11, 0.96, 0.19)	(5.40, 0.94, 0.19)	(5.59, 0.98, 0.28)	(5.72, 1.12, 0.42)
13	(4.94, 0.94, 0.14)	(5.14, 0.91, 0.17)	(5.46, 0.93, 0.18)	(5.54, 0.94, 0.30)	(5.76, 1.15, 0.42)
14	(4.93, 0.92, 0.18)	(5.10, 0.91, 0.16)	(5.48, 0.91, 0.19)	(5.56, 0.92, 0.27)	(5.72, 1.13, 0.44)
15	(4.92, 0.92, 0.19)	(5.11, 0.92, 0.17)	(5.44, 0.92, 0.17)	(5.59, 0.95, 0.28)	(5.71, 1.17, 0.38)
16	(4.89, 0.94, 0.16)	(5.08, 0.94, 0.17)	(5.48, 0.91, 0.18)	(5.58, 0.98, 0.29)	(5.76, 1.09, 0.42)
17	(4.93, 0.93, 0.16)	(5.12, 0.93, 0.18)	(5.46, 0.93, 0.19)	(5.58, 0.98, 0.28)	(5.71, 1.17, 0.41)
18	(4.93, 0.96, 0.18)	(5.11, 0.91, 0.19)	(5.45, 0.92, 0.17)	(5.57, 0.95, 0.26)	(5.72, 1.15, 0.41)
19	(4.91, 0.94, 0.15)	(5.11, 0.91, 0.16)	(5.44, 0.92, 0.18)	(5.57, 0.95, 0.25)	(5.73, 1.13, 0.44)
20	(4.94, 0.93, 0.20)	(5.11, 0.94, 0.15)	(5.45, 0.93, 0.18)	(5.56, 0.96, 0.24)	(5.72, 1.14, 0.40)
Mean	(4.91, 0.93, 0.17)	(5.11, 0.91, 0.16)	(5.44, 0.92, 0.18)	(5.57, 0.95, 0.24)	(5.72, 1.13, 0.41)
Standard deviation	(0.02, 0.02, 0.02)	(0.02, 0.03, 0.02)	(0.03, 0.01, 0.01)	(0.02, 0.02, 0.04)	(0.02, 0.03, 0.02)

Table 3.

The calculation results $f (X, U)$ under different interval possibility satisfaction values.

Times	0.90	1.00	1.10	1.20	1.30
1	(2.47, 5.10)	(3.16, 7.01)	(4.20, 8.16)	(5.74, 9.86)	(6.30, 11.02)
2	(2.48, 5.11)	(3.14, 7.02)	(4.23, 8.17)	(5.76, 9.88)	(6.31, 11.01)
3	(2.51, 5.11)	(3.16, 7.05)	(4.22, 8.17)	(5.76, 9.86)	(6.30, 10.98)
4	(2.51, 5.14)	(3.16, 7.05)	(4.24, 8.16)	(5.77, 9.91)	(6.33, 10.99)
5	(2.49, 5.10)	(3.16, 7.05)	(4.25, 8.18)	(5.76, 9.92)	(6.32, 10.98)
6	(2.48, 5.13)	(3.15, 7.04)	(4.20, 8.15)	(5.76, 9.89)	(6.32, 10.98)
7	(2.47, 5.14)	(3.17, 7.03)	(4.21, 8.15)	(5.79, 9.94)	(6.34, 11.02)
8	(2.49, 5.12)	(3.20, 7.04)	(4.22, 8.15)	(5.80, 9.94)	(6.33, 11.03)
9	(2.48, 5.13)	(3.15, 7.01)	(4.19, 8.17)	(5.78, 9.90)	(6.32, 11.05)
10	(2.51, 5.14)	(3.17, 7.01)	(4.27, 8.14)	(5.80, 9.90)	(6.30, 11.06)
11	(2.49, 5.10)	(3.14, 7.04)	(4.23, 8.16)	(5.82, 9.88)	(6.35, 11.06)
12	(2.49, 5.12)	(3.18, 7.02)	(4.20, 8.16)	(5.78, 9.88)	(6.34, 11.04)
13	(2.48, 5.11)	(3.19, 7.03)	(4.24, 8.14)	(5.80, 9.90)	(6.36, 11.06)
14	(2.49, 5.13)	(3.17, 7.02)	(4.24, 8.18)	(5.78, 9.92)	(6.38, 11.02)
15	(2.47, 5.12)	(3.18, 7.05)	(4.22, 8.13)	(5.80, 9.91)	(6.36, 11.02)
16	(2.51, 5.10)	(3.18, 7.03)	(4.22, 8.16)	(5.78, 9.89)	(6.38, 11.00)
17	(2.49, 5.14)	(3.20, 7.02)	(4.20, 8.16)	(5.77, 9.89)	(6.38, 11.01)
18	(2.47, 5.14)	(3.17, 7.01)	(4.21, 8.19)	(5.78, 9.92)	(6.36, 11.03)
19	(2.51, 5.10)	(3.19, 7.04)	(4.24, 8.17)	(5.79, 9.90)	(6.37, 11.03)
20	(2.47, 5.12)	(3.18, 7.03)	(4.17, 8.15)	(5.78, 9.91)	(6.35, 11.01)
Mean	(2.49, 5.12)	(3.17, 7.03)	(4.22, 8.16)	(5.78, 9.90)	(6.34, 11.02)
Standard deviation	(0.01, 0.02)	(0.02, 0.01)	(0.02, 0.01)	(0.02, 0.02)	(0.03, 0.03)

Figure 5.

Optimization results under different interval possibility satisfaction values.

Figure 6.

Convergence curves for example 2 under different $R_{λ}$ .

In order to verify the accuracy and effectiveness of the proposed method, a comparison is made with Fu et al.’s method in.⁵¹ The comparison results are shown in Table 4. The relative errors of the IPSO’s optimal solutions are respectively 0.87%, 2.15%, and 5.89%. Therefore, IPSO has better accuracy. IPSO calls only $3.6 \times 10^{4}$ functions, whereas Fu et al.′s method in⁵¹ requires $4.2 \times 10^{4}$ . This means that IPSO improves computing efficiency.

Table 4.

Comparison results for example 2 under interval possibility satisfaction value 1.

Algorithm	Optimal solution (relative errors from the precise optimum)	Objection f (relative errors from the precise bounds)
IPSO	(5.11, 0.91, 0.16) (0.78%, 2.15%, 5.89%)	(3.17, 7.03) (6.21%, 2.48%)
Fu et al.’s⁵¹ method	(5.15, 0.93, 0.17)	(3.38, 6.86)

Application of eight degrees of freedom vehicle body vibration model

The vehicle with four-wheel independent suspension is simplified as a spatial eight-degree-of-freedom dynamic model shown in Figure 7 Eight-degrees-freedom dynamic model. The parameters are selected as follows: the front and rear tire mass $m_{1} = 40.5 kg$ , $m_{2} = 40.5 kg$ , body mass $m_{3} = 690 kg$ , pitching moment of inertia $I = 1222 kg \cdot m^{2}$ , the quality of the driver and the chair $m_{5} = 75 kg$ , the distance from the front and rear wheels to the center of mass of the body $a = 1.25 m, b = 1.51 m$ , the distance from the driver’s seat to the center of mass $l = 0.57 m$ , the front and rear wheel stiffness $k_{1} = 192000 kN / m$ , $k_{2} = 192000 kN / m$ , seat stiffness $k_{5} = 7500 kN / m$ , seat damping $c_{5} = 450 kNs / m$ , front and rear suspension damping $c_{1} = 1500 k Ns / m$ , $c_{2} = 1500 kNs / m$ , front and rear suspension stiffness $k_{3}, k_{4}$ . The acceleration root mean square value of the body mass center is taken as the objective function, and the design variable is $X = (k_{3}, k_{4})$ .

Figure 7.

Eight-degrees-freedom dynamic model of a vehicle.

The following optimization models can be established:

\begin{matrix} {\begin{matrix} min_{X} f (X) = σ_{\overset{\cdot\cdot}{z}} (X) \\ s . t . \\ g_{1} (X) = \frac{m_{3 i} g}{0.3} - k_{i}^{I} \leq 0 \\ g_{2} (X) = k_{i}^{I} - \frac{m_{3 i} g}{0.3} \leq 0.1 \\ g_{3} (X) = \frac{- C^{I}}{2 \sqrt{k^{I} m_{3 i}}} \leq - 0.2 \\ g_{4} (X) = \frac{C^{I}}{2 \sqrt{k^{I} m_{3 i}}} \leq 0.4 \\ g_{5} (X) = - \frac{1}{2 π} \sqrt{\frac{k_{i}^{I}}{m_{3 i}}} \leq - 1.0 \\ \begin{matrix} g_{6} (X) = \frac{1}{2 π} \sqrt{\frac{k_{i}^{I}}{m_{3 i}}} \leq 1.5 \\ X = [k_{3}, k_{4}] \end{matrix} \end{matrix} \end{matrix}

(24)

In the formula: $i = 1, 2$ represent the front and rear suspension $m_{33} = \frac{m_{3} b}{2 (a + b)}, m_{34} = \frac{m_{3} a}{2 (a + b)}$ .

The optimization results are shown in Figure 8 and Table 5. It can be concluded from the optimization results that the upper and lower boundary values and the midpoint values of the objective function also increase with the increase of the interval possibility satisfaction value. When the interval possibility satisfaction value is 1, the optimal solution $X = (15024, 14106)$ , $k_{3} = 15024 N / m, k_{4} = 14106 N / m$ , is obtained. As can be seen from Figure 9, the optimal target value converges at the 26th generation. Under this design scheme, the root means the square value of the acceleration of the center of mass of the body is the smallest, and $σ_{\overset{\cdot\cdot}{z}} (X) = 1.036$ , at this time, the optimized vehicle ride comfort is also good.

Figure 8.

Optimization results under different interval possibility satisfaction values.

Table 5.

Interval optimization results under different interval possibility satisfaction values.

Interval possibility satisfaction values	K ₃ ( $N / m$ )	K ₄ ( $N / m$ )	Interval of objective function ( $m / s^{2}$ )	The midpoint of objective function ( $m / s^{2}$ )
0.9	15,024	14,106	(0.986, 1.086)	1.036
1.0	15,526	14,473	(0.989, 1.089)	1.039
1.1	16,630	15,539	(0.991, 1.091)	1.041
1.2	15,762	14,638	(0.995, 1.095)	1.045
1.3	16,920	15,237	(1.001, 1.101)	1.051

Figure 9.

Convergence curve of example 3 with $R_{λ} = 1$ .

Application for an aerial camera system

Aerial camera system is composed of unmanned aerial vehicle and aerial camera.⁵² And the aerial camera is the most important remote sensing sensor in aerial camera system, which needs to maintain stable and reliable performance at work. In the process of its robust optimization design model involving interval variables established in Tang et al.,⁴⁴ the method proposed in this paper might be adopted as the solving approach as well. The finite element model of the lens, the motherboard, the filter and the housing are shown in Figure 10. By assigning different boundary conditions, the finite element analysis results of temperature and displacement under high temperature conditions and the dynamic analysis results under strong vibration conditions are shown in Figure 11. Considering the two conditions of temperature and vibration, the robust optimization of aerial camera can be obtained as:

\begin{matrix} {\begin{matrix} min_{X} f (X^{C}, U^{C}) = - 0.9914 {(X_{1}^{C})}^{2} - 3.761 X_{1}^{C} + 0.01086 {(X_{2}^{C})}^{2} - 0.6826 X_{2}^{C} + 0.6346 X_{1}^{C} X_{2}^{C} + 20.55 \\ s . t . \\ pr (g_{j} (X, U) \leq b_{j}^{I}) \geq λ_{j}, j = 1, 2, 3, 4, 5 \\ \begin{matrix} g_{1} (X, U) = 2.109 U_{1}^{2} - 1.9520 U_{1} + 2.559 X_{1}^{2} + 0.0745 X_{1} X_{2} - 18.58 X_{1} + 0.003174 X_{2}^{2} \\ - 0.4111 X_{2} + 93.41 \end{matrix} \\ \begin{matrix} g_{2} (X, U) = - 1.525 \times 10^{- 7} U_{2}^{2} + 7.98 \times 10^{- 5} U_{2} + 3.366 \times 10^{- 3} U_{3} + 5.150 \times 10^{- 3} (X_{1}^{C})^{2} \end{matrix} \\ - 7.857 \times 10^{- 4} X_{1}^{C} X_{2}^{C} + 1.233 \times 10^{- 2} X_{1}^{C} + 1.563 \times 10^{- 2} (X_{2}^{C})^{2} - 1.125 X_{2}^{C} + 1.603 \\ g_{3} (X, U) = f (X, U) - f (X^{C}, U^{C}) \\ g_{4} (X, U) = f (X^{C}, U^{C}) - f (X, U) \\ 1 \leq X_{1}^{C} \leq 3 \\ 30 \leq X_{2}^{C} \leq 40 \\ X_{1}^{W} = 0.2 \\ X_{2}^{W} = 0.4 \\ X_{i} \in [X_{i}^{C} - X_{i}^{W}, X_{i}^{C} + X_{i}^{W}], i = 1, 2 \\ U_{1} \in U_{1}^{I} = [1.8, 2.2] \\ U_{2} \in U_{2}^{I} = [10000, 12000] \\ b_{1}^{I} = [66, 66] \\ \begin{matrix} b_{2}^{I} = [1, 1] \\ b_{3}^{I} = b_{4}^{I} = [3.0, 3.5] \end{matrix} \end{matrix} \end{matrix}

(25)

where the nominated value of design vector is $X^{C}$ , and its radius vector is $X^{W}$ , $U \in U^{I}$ is the uncertain parameter vector, $g_{1}$ , $g_{2}$ , $g_{3}$ , and $g_{4}$ are constraints.

Figure 10.

The finite element model of the aerial camera:⁴⁴ (a) profile view and (b) sectional view.

Figure 11.

The finite element analysis results of the aerial camera:⁴⁴ (a) the high-temperature condition and (b) the vibration condition.

The interval optimization results with different interval possibility satisfaction values are shown in Figure 12 and Table 6. The convergence curves under different $R_{λ}$ are shown in Figure 13. When the interval possibility satisfaction value is 1, the optimal solution is $X^{C} = (2.43, 33.60)$ , $f (X^{C}, U^{C}) = 46.73$ . The optimization results obtained by IPSO are compared with these obtained by the double-loop nested optimization method, as shown in Table 7. The results of two algorithms are basically consistent. But the optimal solution of IPSO can be obtained with 96 times of interval analysis calls. Compared with 324 times of interval analysis calls of the double-loop nested optimization, IPSO has better efficiency and accuracy.

Figure 12.

Optimization results under different interval possibility satisfaction values for example 4.

Table 6.

Interval optimization results under different interval possibility satisfaction values.

Interval possibility satisfaction values	$X_{1}^{C}$ ( $mm$ )	$X_{2}^{C}$ ( $mm$ )	The midpoint of objective function ( $kg$ )
0.9	1.71	31.59	45.72
1.0	2.43	33.60	46.73
1.1	3.16	34.61	47.51
1.2	3.84	35.20	48.84
1.3	4.27	35.94	49.85

Figure 13.

Curves of convergence for example 4 under different $R_{λ}$ .

Table 7.

Comparison of optimization results.

Algorithm	Optimal solution	Objection f
IPSO	(2.43, 33.60)	46.73
Double-loop	(2.43, 33.60)	46.76

Conclusion

In order to solve the problem involving the loss of some information and accuracy in the process of converting a nonlinear optimization model containing interval variables or parameters into a deterministic model using interval possibility or interval order relationship. In this paper, an interval optimization algorithm based on PSO is proposed, which aims to reduce the loss of useful information through straight optimization instead of deterministic transformation for the interval model so as to improve computational accuracy with a small increasing in computational cost. The method firstly uses interval satisfaction value to deal with constraints and consequently a particle swarm optimization algorithm is proposed to find the optimal solution set without deterministic conveying for the nonlinear interval optimization model. The individual selection in the above solution set is implemented by means of the criteria based on the interval possibility satisfaction value model. Finally, the effectiveness of the proposed method is verified by four examples. The results show that the algorithm has advantages in solving constrained interval optimization problems with arbitrary uncertainty levels, and the result is valid.

Footnotes

Handling Editor: Chenhui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the Hunan Provincial Natural Science Foundation of China (Grant No. 2021JJ40205), the China Postdoctoral Science Foundation (Grant No. 2021M690988), and the National Natural Science Foundation of China (Grant No. 51905165).

ORCID iD

Huichao Xie

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