With the increasingly complicated engineering problems during the past few years, many researchers devote themselves to researching new intelligent optimization algorithms. In 2011, a new heuristic optimization algorithm named fruit fly optimization algorithm (FOA) is proposed by Pan [1] who is inspired by the feeding behaviors of drosophila. FOA is easy to be understood, and it can deal with the optimization problems with fast speed and high accuracy, while, the results are influenced a lot by the initial solutions [2]. Based on the phototropic growth characteristics of plants, a new global optimization algorithm called plant growth simulation algorithm is proposed by Li et al., which is a kind of bionic random algorithm and suitable for largescale, multimodal and nonlinear integer programming [3], however, for its complex calculation theory, the algorithm is not widely applied in industry and scientific research. Artificial bee colony algorithm [4] is a new application of swarm intelligence, which simulates the social behaviors of bees, whose defects are slow convergence speed and easy to trap into local optimum [5].
Mirror is a common necessity, which plays an important role in daily life. Inspired by the optical function of mirror, a new algorithm called specular reflection algorithm (SRA) is raised by this paper. SRA, similar to genetic algorithm [6][8], particle swarm optimization [9][11], simulated annealing algorithm [12], [13], differential evolution algorithm [14], [15], etc, can be widely used in science and engineering. The SRA has many outstanding advantages, such as simple principle, easy programming, high precision and fast calculation speed, and its unique nonpopulation searching mode distinguishes itself from original swarm algorithm. Furthermore, the global searching ability is significantly improved by the specific acceptance criterion of the new solution. In order to verify above mentioned features of SRA, a great deal of comparative experiments are adopted in this paper. At last, the reliability based design and robust design are combined with the SRA, in order to evaluate the ability of SRA in reliability based robust optimization design.
2 SRA 2.1 Introduction of SRAMirror is a life necessity and a product of human civilization, which can change the direction of propagation of light. There are various kinds of mirrors, such as magnifying glass, microscope, etc. With the help of mirror, a great deal of stuff can be observed, even if they are out of the range of visibility. For example, the submarine soldier is able to catch sight of the object above the water by periscope. This reflection property of mirror is simulated by the SRA.
Object, suspected target, eyes and mirror are the four basic elements of specular reflection system.
Object is the objective function of optimization. Getting its exact coordinate is the purpose of the SRA. It is not involved in the optimization procedure for the location of the object is unpredictable.
Suspected target is the coordinate of the object observed by eyes, which is approximate to the optimal solution. There is an error between the suspected target and object, because the coordinate of the object observed by eyes is not accurate. The suspected target is located around the object, and it is the element nearest to the object.
Mirror can change the direction of propagation of light. The vision of eyes can be broaden by mirror. All the things that can reflect light (glass, water, etc.) are taken as mirror.
Eyes are the subject of the SRA, which can acquire the approximate coordinate of the object. And it is the element farthest from the object.
2.2 Definition$ \begin{align}\label{eq1} &\min f(X), \ X = (x^1, x^2, \ldots, x^N), \quad X \in \mathbb{R}^N \notag\\ & {\rm s.t.}\ \ g_j (x) = 0, \ \ j = 1, 2, \ldots, m \notag\\ &\qquad h_k (x) \le 0, \ \ k = 1, 2, \ldots, l. \end{align} $  (1) 
Taking the constrained optimization problem showed in (1) as an example, the definition of SRA will be drawn as following:
Set the specular reflection system as a
Searching the new coordinate: the coordinates of
$ \begin{align} \begin{cases} X_{\rm New1}^n = x_1^n + \xi (2{\rm rand}  1)(x_1^n  x_3^n ) \\[2mm] X_{\rm New2}^n = x_1^n + \xi (2{\rm rand}  1)(2x_1^n  x_2^n  x_3^n ) \end{cases} \end{align} $  (2) 
where
$ \begin{align} \label{eq3} \begin{cases} X_{\rm New} = X_{\rm New1}, f(X_{\rm New1} ) \leq f(X_{\rm New2} ) \\[2mm] X_{\rm New} = X_{\rm New2}, f(X_{\rm New1} ) \ge f(X_{\rm New2} ). \end{cases} \end{align} $  (3) 
Updating the specular reflection system: Once the coordinate of
The optimization steps of the SRA are shown as follows:
Step 1: Define the initial value
Step 2: If the precision or the maximum iteration number reaches the design requirements, the coordinate of
Step 3: Search the coordinate of
In conclusion, the optimization flow chart of the SRA is given by Fig. 2.
Theorem 1: The constraint optimization problem presented in (1) can converge to the global extremum with 100 % probability by the SRA.
Proof: Provided that
First, get the feasible initial solutions
Second, the new solutions
$ \begin{align} p^k =&\ \int\nolimits_{X_{\rm Object}  \varepsilon }^{X_{\rm Object} + \varepsilon } \frac{1}{X_{\max }^k  X_{\min }^k }dX = \frac{2\varepsilon }{X_{\max }^k  X_{\min }^k } \nonumber\\[2mm] \ge&\ \frac{2\varepsilon }{X_{\max }  X_{\min } } > 0 \end{align} $  (4) 
where
The probability that the feasible solution
$ \begin{align} \begin{cases} P^1 = P\{X_{\rm Suspect}^0 \subseteq [X_{\rm Object}\varepsilon, X_{\rm Object} + \varepsilon]\} \\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\mbox{ }P^1 = P \\ Q^1 = P\{X_{\rm Suspect}^0 \not\subset [X_{\rm Object}\varepsilon, X_{\rm Object} + \varepsilon]\} \\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\mbox{ }Q^1 = P \end{cases} \end{align} $  (5) 
where
The probability that the feasible solution gotten for the second time still failing to be the optimal value is:
$ \begin{align} Q^2=Q^1(1P)=(1P)^2. \end{align} $  (6) 
So, the probability that the solution is optimal is:
$ \begin{align} P^2=1(1P)^2. \end{align} $  (7) 
After
$ \begin{align}\label{2} P^n& = 1  (1  P)^n = 1  \prod _{i = 1}^n \left( {1  \frac{2\varepsilon }{X_{\max }^i  X_{\min }^i }} \right) \nonumber\\[1mm] &\ge 1  \left( {1  \frac{2\varepsilon }{X_{\max }  X{ }_{\min }}} \right) ^n. \end{align} $  (8) 
Calculate the extreme value of (8):
$ \begin{align} \lim _{n \to \infty } P^n& = \lim\limits_{n \to \infty } \left[{1 \prod _{i = 1}^n \left( {1\frac{2\varepsilon }{X_{\max }^i X_{\min }^i }} \right)} \right] \nonumber\\ &\ge \lim _{n \to \infty } \left[{1\left( {1 \frac{2\varepsilon }{X_{\max }X_{\min } }} \right)^n} \right] = 1. \end{align} $  (9) 
With the iterations going on, it is more and more likely to achieve the optimum solution. When
The control parameter is closely related to the space complexity of optimized target, which has an effect on the capability of algorithm. The control parameters of classical optimization algorithm are gotten by experience or experiment, such as the learning parameter
$ \begin{align} f (x_1, x_2, \ldots, x_N)=\sum\limits_{j=1}^N j\times x_j^2. \end{align} $  (10) 
The test function is illustrated by (10), and its threedimension diagram is shown in Fig. 3. The global minimum value in theory of this function is 0
As shown in Table Ⅰ, all the results fall in between 10
According to the Table Ⅰ, the conclusions can be drawn as follows: when
$ \begin{align} \xi=\frac{2.15}{N}+0.84. \end{align} $  (11) 
To verify the global optimization ability of SRA, four numerical test functions in [10] are used, each test function is listed in Table Ⅱ in detail. The total iteration time is set as 2000. The SRA will be executed 50 times, and the average values are listed in Table Ⅲ, other results are references from [10], Figs. 47 show the iteration curves of the objective functions of each test function respectively.
The results in Table Ⅳ indicate that: when
According to the reliability design theory, the reliability can be calculated by (12):
$ \begin{align} R=\int_{g(X)} f_x(X) dX \end{align} $  (12) 
where
$ \begin{align} \begin{cases} g(X)\leq 0, &{\rm failure}\\[2mm] g(X)>0, &{\rm safe.} \end{cases} \end{align} $  (13) 
The basic random variables
Robust design is a modern design technique that can improve the efficiency and quality and reduce the cost of products [20], [21]. The robust design of mechanical products can make the products insensitive to the changes of design parameters. The product which is designed by robust design method has the characteristic of stability. Even if there is an error in the designed parameters, the product still has excellent performance. Reliability is a kind of design method to eliminate the weaknesses, failure modes and guard against malfunction. The reliability robust optimization design is a new method by combining the robust design and reliability design, which possess all the merits of the two methods. The products designed by the reliability robust optimization design method are reliable and have robustness.
$ \begin{align} &\min f(X)=\omega_1 f_1(X)+\omega_2 f_2(X)\notag\\ & {\rm s.t.} \ \ R\geq R_0\notag\\ &\qquad p_i(X)\geq 0, \ i=1, 2, \ldots, l\notag\\ &\qquad q_j(X)\geq 0, \ j=1, 2, \ldots, m \end{align} $  (14) 
where
$ \begin{align} f_2 (X) = \sqrt {\sum\limits_{i = 1}^n \left( {\frac{\partial R}{\partial X_i }} \right)^2} \end{align} $  (15) 
where
$ \begin{align} \begin{cases} \omega _1 = \dfrac{f_2 (X^{1\ast })  f_2 (X^{2\ast })}{[f_1 (X^{2\ast }) f_1 (X^{1\ast })] + [f_2 (X^{1\ast })f_2 (X^{2\ast })]} \\[4mm] \omega _2 = \dfrac{f_1 (X^{2\ast })  f_1 (X^{1\ast })}{[f_1 (X^{2\ast }) f_1 (X^{1\ast })] + [f_2 (X^{1\ast })f_2 (X^{2\ast })]} \end{cases} \end{align} $  (16) 
where
The bridge crane is taken as an example to verify the capability of the SRA in solving the engineering problems. The SRA is adopted to design the structure with optimized design, reliability optimization design and robust reliability optimization design, and the results are listed in Table Ⅲ together with the results calculated by PSO and FOA, which are used for analysing the performance of the SRA.
4.1 Design ParametersThe mechanical model of the bridge crane is shown in Fig. 8, the uniform load
The parameters
Objective function: According to the characteristics of the structural optimization problem, the objective function can be defined as shown in (17).
$ \begin{align} {\rm min} f(x_1, x_2, x_3, x_4, x_5)=2x_1x_5+2x_2x_4. \end{align} $  (17) 
Constraint condition: Strength, stiffness and stability are the three basic failure modes of bridge crane. Therefore, the constraint condition can be defined as following:
1) Strength Constraint: The maximum stress of dangerous point in midspan section must be smaller than the ultimate stress
$ \begin{align} &h_1(x_1, x_2, x_3, x_4, x_5)=f_{rd}\sigma\notag\\ &\qquad =f_{rd}\frac{qS^2+2FS}{8I_Z}\left(\frac{x_4}{2}+x_1\right) \end{align} $  (18) 
where
2) Stiffness Constraint: The maximum deflection of the structure must be smaller than the allowable value
$ \begin{align} &h_2(x_1, x_2, x_3, x_4, x_5)=\gamma_0\gamma\notag\\ &\qquad =\gamma_0\left(\frac{5qS^4}{384EI_Z}+\frac{FS^3}{48EI_Z}\right). \end{align} $  (19) 
3) Stability Constraint: The depthwidth ratio of Section 2.1 must be smaller than 3.
$ \begin{align} h_3(x_1, x_2, x_3, x_4, x_5)=3\frac{x_4+2x_1}{x_3+2x_2}. \end{align} $  (20) 
In conclusion, the optimization model of the bridge crane can be built as (21).
$ \begin{align} & \min f(x_1, x_2, x_3, x_4, x_5) \notag\\ & {\rm s.t.} \ \ h_k(x_1, x_2, x_3, x_4, x_5)\geq 0, \quad k=1, 2, 3\notag \\ &\qquad 6\leq x_1, \ x_2\leq 30\notag \\ &\qquad 50\leq x_3, \ x_4\leq 5000. \end{align} $  (21) 
The reliability constraint of structure is added to (21) to achieve the reliability optimization design. The failure of any mode will result in the failure of the structure, so the reliability
$ \begin{align} R_v=\prod\limits_{k=1}^3 R_k %(h_k\geq 0) \end{align} $  (22) 
where
$ \begin{align} & \min f(x_1, x_2, x_3, x_4, x_5)\notag \\ & {\rm s.t.}\ \ h_k(x_1, x_2, x_3, x_4, x_5)\geq 0, \quad k=1, 2, 3\notag\\ &\qquad 6\leq x_1, \ x_2\leq 30\notag\\ &\qquad 50\leq x_3, \ x_4\leq 5000\notag\\ &\qquad R_vR_0\geq 0. \end{align} $  (23) 
According to the robust reliability optimization design model which is shown in (14), the index of reliability and robustness are taken into account, the multiobjective optimization model is built by (24).
$ \begin{align} & \min \omega_1\times f(x_1, x_2, x_3, x_4, x_5)+w_2\times f'(x)\notag \\ & {\rm s.t.} \ \ h_k(x_1, x_2, x_3, x_4, x_5)\geq 0, \quad k=1, 2, 3\notag\\ &\qquad 6\leq x_1, \ x_2\leq 30\notag\\ &\qquad 50\leq x_3, \ x_4\leq 5000\notag\\ &\qquad R_vR_0\geq 0 \end{align} $  (24) 
where
The three optimization models shown in (21), (23) and (24) are calculated by the SRA, PSO and FOA, respectively. And the results are presented in Table Ⅲ, from which the conclusions can be drawn as follows:
1) For structural optimization, the results obtained by the three algorithms are 10 704, 11 544 and 11 328, the optimum among the three is 10 704 which is calculated by the SRA, which proves the ability of SRA is higher than PSO and FOA. The reliability results of the three groups of parameters are 0.5071, 0.5314 and 0.5132 respectively, which are unable to meet the requirement of reliability design for the reliability constraint is ignored.
2) The reliability of the structure can be ensured and the robustness can be improved after reliability optimization design. However, the areas of Section 2.1 are increased to 11 652, 12 334 and 16 576 at the same time, and the best result is also calculated by SRA.
3) With the requirements of the robustness, the reliability sensitivity index of design variables are significantly reduced, and the robustness of structure is improved notably.
5 ConclusionsIn this paper, a new optimization algorithm — specular reflection algorithm (SRA) is proposed, which is inspired by the optical property of the mirror. The SRA has a particular searching strategy which is different from the swarm intelligence optimization algorithms. The convergence ability of the SRA is verified by the traditional mathematical method, it converges to the global optimum value with the probability of 100 %. The reasonable values of the control parameters are analysed, and their computational formula is deduced by the method of data fitting, so that the control parameters will vary with the different problems and thus the adaptation and the operability of the SRA will be improved. Four classical numerical test functions are analysed by the SRA, and the results indicate that the ability of the SRA is better than the traditional intelligent optimization algorithms. Then, the theories of the reliability optimization and robust design are combined to establish the mathematical models of the optimization design, reliability optimization design and robust reliability optimization design for the bridge crane as an example system, which are calculated by the SRA and other two optimization methods (PSO and FOA). The conclusions are drawn after the simulation, that the structure designed by the SRA is reliable and robust. The results calculated by the SRA are superior to the PSO and the FOA. All in all, the SRA is the latest research in the area of intelligent optimization, which has the better calculation capability than other optimization algorithms, and the ability for the structure design is verified in this paper. SRA can be widely applied in other fields and create more value.
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