Improved Rao algorithm: a simple and effective algorithm for constrained mechanical design optimization problems

Abstract

This paper proposes a simple and effective optimization algorithm named as "Improved Rao (I-Rao) algorithm". The I-Rao algorithm consists of two phases: the local exploitation phase and the global exploration phase. The local exploitation phase improves the exploitation of the search process of Rao algorithms and enhances the algorithm's convergence speed. The global exploration phase helps the algorithm to get away from the local optimum solutions and enhances the exploration ability of the search process of the algorithm. Like Rao algorithms, the I-Rao algorithm also does not need algorithm-specific control parameters. The effectiveness of the I-Rao algorithm is tested on 25 unconstrained benchmark functions, 45 complex CEC test functions, and 19 real-world constrained mechanical design optimization problems of CEC2020. The comparison of results reveals the effectiveness of the I-Rao algorithm over many state-of-the-art advanced optimization algorithms such as GWO, SCA, PSO, WOA, Jaya, IJAYA, MJAYA, PGJAYA, EJAYA, IUDE, εMAgES, iLSHADEε, and COLSHADE. The Friedman test is also carried out to check the overall performance of the I-Rao algorithm as compared to other advanced optimization algorithms considered. The convergence plots illustrate the convergence speed of the I-Rao algorithm.

Appendix A: Demonstration of the proposed I-Rao algorithm

For demonstration of the proposed algorithm, the three-bar truss design problem (Kumar et al. 2020) is considered. Let, the population size (NP) is 6. Table

Table 8 Initial random population

8 shows the initial random population. Here $\overline{v }$ is the mean constraint violation (Kumar et al. 2020). The candidate solutions are ranked based on values of $\overline{v }$ and f(x).

In 1st iteration, the phase change probability (P_PC) is assumed as 0.3. As P_PC is less than 0.5, the local exploitation phase is carried out in 1^st iteration. The mean vector $\overline{M}_{{{\text{best}}\_{\text{pop}}}}$ is generated using best candidates 5, 6 and 2. The mean vector $\overline{M}_{{{\text{worst}}\_{\text{pop}}}}$ is generated using worst candidates 1, 3 and 4. Then, the local best-mean vector (LBM) and the local worst-mean vector (LWM) is calculated using Eqs. (4–5) and listed in Table

Table 9 Local best-mean vector and the local worst-mean vector in 1st iteration

9. Here, r₁ = 0.35 and r₂ = 0.50.

The old population (X) listed in Table 8 is then updated using Eq. (6) and listed in Table

Table 10 Updated population in 1st iteration (local exploitation)

10. Here, r₃ = 0.15 and r₄ = 0.65.

Then the old population (X) in Table 8 is compared with new population (X_new) in Table 10 based on values of $\overline{v }$ and f(x), and best candidates are listed in Table

Table 11 Population at the end of 1st iteration

11. The 1st iteration is now completed.

In 2nd iteration, the phase change probability (P_PC) is assumed as 0.7. As P_PC is greater than 0.5, the global exploration phase is carried out in 2nd iteration. The global population (X_G) is generated using Eqs. (7–8) and listed in Table

Table 12 Global population (X_G) in 2nd iteration

12.

The old population (X) listed in Table 11 is then updated using Eq. (9) and listed in Table

Table 13 Updated population in 2nd iteration (global exploration)

13. Here, rn = 1.23.

Then the old population (X) in Table 11 is compared with new population (X_new) in Table 13 based on values of $\overline{v }$ and f(x), and best candidates are listed in Table

Table 14 Population at the end of 2nd iteration

14. The 2nd iteration is now completed. This procedure is to be repeated until the termination criteria is satisfied. It is to be noted that the values of P_PC, r₁, r₂, r₃, r₄ and rn are to be generated randomly during execution of the algorithm.