An Improved Rider Optimization Algorithm for Solving Engineering Optimization Problems

This paper proposed an improved optimization algorithm for engineering optimization problems. This improved optimization algorithm, namely, gravitational search strategy-assisted rider optimization algorithm (GSSROA), combines the gravitational search strategy (GSS) with the rider optimization algorithm (ROA). In the GSSROA, the search step can be adaptively adjusted in each iteration. The performance of the GSSROA has been verified with three benchmark engineering design problems. The results indicate that the proposed GSSROA has the feasibility and good robustness. Then, the GSSROA is applied to optimize the weight of the front axle of an automobile under the non-linear constraints. The results show that the GSSROA can achieve the optimal structure with a weight that is reduced by 16.82% compared with the initial design.


I. INTRODUCTION
Structural optimization is a challenging and critical research work. The aim of the structural optimization is to maximize profits under the certain conditions. Many optimization problems in real-world are the nonlinear constraint problem, which usually are solved using the classical optimization algorithms [1]- [3]. It is well-known that the classical approaches are often inefficient and need the strong math assumptions, which are usually difficult to establish the mathematical model in the practical problems. With increasing complexity of the problems, the traditional optimization algorithm needs higher computational cost to obtain the global optimal solution. To handle these problems, many metaheuristic algorithms have been proposed in past decades, such as genetic algorithms (GA) [4], particle swarm optimization (PSO) [5], pattern search (PS) [6], differential evolution (DE) [7], genetic programming (GP) [8], central force optimization (CFO) [9], ant colony optimization (ACO) [10] and rider optimization algorithm (ROA) [11].
Although these optimization algorithms have been successfully applied in the real-world problems, the high computational cost and accuracy are still the challenge work.
The associate editor coordinating the review of this manuscript and approving it for publication was Chung Shue Chen.
According to No Free Lunch (NFL) theorem [12], there is no one algorithm which can solve all the optimization problems. For addressing this challenge, many researchers have are focus on improving the performance of the algorithms using different optimization strategies, such as socio evolution and learning optimization (SELO) [13], adaptive genetic algorithms (AGA) [14] and fitness-distance-ratio-based PSO (FDR-PSO) [15], and so forth.
Among these meta-heuristic optimization algorithms, the ROA is a novel optimization algorithm, which is first proposed by Binu in 2017 [11], which considers a few rider groups. ROA has been used for fault diagnosis in analog circuits, the results show that ROA can attain the global optimal solutions. To speed up the convergence rate, the gravity search strategy is applied to the ROA in this paper, which considers the individual inter-relation. The performance of improve rider optimization algorithm (IROA) is verified on three benchmark engineering optimization problems. The results of IROA show that it has strong reliability and robust exploration performances. Then, IROA is applied to evaluate the parameters for automobile front axle. Results show that IROA is more competitive in the computational efficiency and accuracy.
The remainder of the paper is organized as follows. The review of rider optimization algorithm is described in Section 2. Section 3 develops the gravity search strategy and improved rider optimization algorithm. The benchmark engineering optimization problems are discussed in Section 4. The optimization of the automobile front axle is dealt with in section 5. Section 6 gives the concluding remarks.

II. REVIEW OF RIDER OPTIMIZATION ALGORITHM
The inspiration of ROA is a group of riders, including the bypass rider, follower, overtaker, and attacker. In ROA, all the riders should follow the predefined strategy, which can be described as follows: 1) The bypass rider can reach the target with the leading path. The group riders initialization can be sated as follows: where R is the number of the riders, Q denotes the dimension of the optimization problem, t denotes the time instant, X t is the position of ith rider at time t. For the bypass rider, it implies that he is the rider in the leading position, and the position can be calculated as follows: where δ is a random within [0,1], η denotes the a random number within [1, R], ξ is a number which can select from 1 to R. In addition, β represents a random within [0,1], but of size 1 × Q.
2) The follower arrives at the destination by following the bypass rider. The position of follower is relevant to bypass rider, the main reason for this is that the walk path of the follower depends on the bypass rider. Thus, the position of follower can be updated using Eq. (3).
where k represents the coordinate selector, X L denotes the position of bypass rider, L is the index of bypass rider, T t i,k denotes the steering angle of the ith rider in the kth coordinate, and d t i is the distance to be traveled of the ith rider.
3) The overtaker not only follows his own position, but also the information are gathered according to the bypass rider. The position of overtaker is in dependence on three factors, including the coordinate selector, relative success rate and direction indicator. The main reason for this is that the overtaker needs to gather the information of the bypass rider and his own. Therefore, the overtaker's position can be updated by Eq. (4). where X t (i, k) represents the position of the ith rider in the kth coordinate, D I t (i) denotes the direction indicator of the ith rider. The direction indicator can be calculated as follows: where S R t (i) represents the ith rider's success rate. 4) The attacker uses the maximum speed to reach the target point. The purpose of the attacker is to obtain the position of the leader, and the travelling strategy is similar to the follower. Notably, all the attackers' position are updated in this strategy, rather than the selected individual. The position of the attacker can be sated as follows: where X L (L, j) represents the leader's position, T t i,j denotes the steering angle of the ith rider in the jth coordinate.
The pseudo-code of the ROA is shown in Fig. 1.

III. AN IMPROVED RIDER OPTIMIZATION ALGORITHM A. GRAVITATIONAL SEARCH STRATEGY (GSS)
In this section, the gravitational search strategy (GSS) is introduced. Gravitation search algorithm is first proposed by Rashedi et al. in 2009 [16], which has been widely used to resolve the optimization problems in the literature [17]- [21]. The basis of gravitation search algorithm is the Newton gravitational law, i.e., each individual attracts every other individual with a force. In gravitation search algorithm, each individual consist of four parts, including the position of the individual, its inertial mass, its active gravitational mass and passive mass. Correspondingly, the gravitational and inertial mass can be consider as the fitness functions, and the position of the mass is supposed to the solution of the optimization problem. Suppose that mass j acts mass i, the gravitational force can be stated as follows: where M i and M j are the mass of the ith and jth individual, respectively. G denotes the gravitational constant. ε is the a small constant. R ij denotes the distance of the between ith and jth individual, which can be calculated as follows: The total force acting on the ith individual can be stated as follows: where rand represents a random within [0, 1]. kbest is the set of the first individuals with the best function values. According to the Newton's second law, the acceleration of the inertial mass M ii can be calculated as follows:

B. IMPROVED RIDER OPTIMIZATION ALGORITHM
An improved optimization algorithm is proposed in this section, which is named the gravitational search strategy based on ROA (GSSROA). In GSSROA, the correlation between rider was taken into account. To clearly describe the search process, the position of the rider can be defined by Eq. (11).
where x d i is the position of the ith rider in the dth dimension. The acting force can be given by: where m i and m j are the best fitness of the ith and jth rider. g(t) denotes the search constant at the iteration t. l ij is the distance of the between ith and jth rider. The total acting force and acceleration of the ith rider can be calculated as follows: Therefore, the position of bypass rider, follower, overtaker, and attacker can be updated as follows: where a B , a F , a O and a A denote the acceleration of the bypass rider, follower, overtaker, and attacker, respectively. The pseudo-code of the GSSROA is shown in Fig. 2.

IV. ENGINEERING DESIGN OPTIMIZATION PROBLEMS
To verify the performance of the GSSROA, three engineering optimization problems are adopted, including the multiple disc clutch brake [22]- [23], three-bar truss design [24] and hydrodynamic thrust bearing [22]- [23]. The benchmark optimization problems are described in detail in the following sub-sections.

A. MULTIPLE DISC CLUTCH BRAKE
The multiple disc clutch brake optimization design problem is to find the minimize the power loss [22], [23]. As shown in Fig. 3, the optimization problem consists of five decision variables, including inner radius, outer radius, thickness of discs, actuating force, and number of friction surfaces. The mathematical formulation of hydrodynamic thrust bearing can be described as follows: where   Table 1 reports the results of the GSSROA and the results from Refs. [22], [25], [26]. Clearly, the GSSROA can obtain the optimal solution with 81859.74, whose optimal value identical to these of the ABC [22] and TLBO [22]. However, the mean values and worst values are different. The main reason for this is that these optimization algorithms have different convergence rate, so that the robustness of the algorithms are different. The performances of WCA [25] and MBA [26] are worse than the GSSROA. Therefore, we can conclude that GSSROA has a good robustness.

B. THE THREE-BAR TRUSS DESIGN
The three-bar truss design optimization problem is chosen from the Ref. [24], which is a widely used benchmark engineering design problem. The purpose of the three-bar truss design is to minimize the weight. The objective function is simple, and it subjects to the highly non-linear constraint, including the stress, deflection, and buckling constraints. The configuration of the three-bar plane framework is shown in Fig. 4.
The mathematical formulation of three-bar truss design problem can be described as follows: The optimized results of these popular algorithms for the three-bar truss design are reported in Table 2. Interestingly, these algorithms can obtain the optimal solution, the main reason for this is that the three-bar truss design optimization problem is simple and only consist of two design variables. In view of the results of five algorithms, it can be concluded that the GSSROA has the feasibility and good robustness.

C. HYDRODYNAMIC THRUST BEARING
The objective of hydrodynamic thrust bearing is find the minimize the power loss. The hydrodynamic thrust bearing is shown in Fig. 5. This optimization case has four decision variables, including the bearing step radius (R), recess radius (R 0 ), oil viscosity (µ) and flow rate (Q). This engineering optimization problem consist of seven non-linear constraints. The mathematical formulation of the hydrodynamic thrust bearing problem can be described as follows:  The optimization results of GSSROA are compared with the results from Ref. [22], which are listed in Table 3. Obviously, the GSSROA can obtain the optimal solution with 1625.44, whose optimal value identical to these of the ABC [22] and TLBO [22]. By comparing the results of these optimization algorithms, it can be concluded that the GSSROA demonstrates a satisfactory exploration ability and can be used to solve the engineering optimization problems.

V. FRONT AXLE OF AN AUTOMOBILE OPTIMIZATION PROBLEM
The front axle of an automobile is a typical optimization problem in real life [33]. As shown in Fig. 6, this case is a I-beam structure, which has been widely used in the modern engineering designs because of its high bending strength and light weight. The purpose of the front axle of an automobile is to find the minimum weight.
where a, b, c and h are shown in Fig. 6(b); M 0 and T 0 are the moment and torque; and t is the time parameter within [0,1]. The strength of the front axle can be given by Eq. (26): where σ s is the limit yield stress. The optimization results of the front axle model are listed in the Table 4. Notably, ''+'' (''−'') denotes the increase (decrease). As seen from Table 4, it is clearly to note that the weight of the automobile front axle decreases from 51.95 Kg to 43.21 Kg, 16.82% lower than the initial value. These results further demonstrate that the GSSROA has a satisfactory exploration ability, and is an efficient algorithm to solve this engineering optimization problem.

VI. CONCLUSION
As no single meta-heuristic algorithm can obtain the global optimal solutions for all optimization problems. To address the challenge, this paper proposes a new optimization algorithm, namely gravitational search strategy assisted rider optimization algorithm (GSSROA). The performance of GSSROA is firstly evaluated on three engineering optimization problems are adopted, including the multiple disc clutch brake, three-bar truss design and hydrodynamic thrust bearing. By comparing the results of these optimization algorithms, it can be concluded that the GSSROA demonstrates a satisfactory exploration ability.
In addition, GSSROA is applied to optimize the front axle of an automobile. Results show that GSSROA is more competitive in the computational efficiency and accuracy. In the future work, GSSROA can be applied to multi-objective optimization problems.