Modified salp swarm algorithm for global optimisation

Abstract

Salp swarm algorithm (SSA) is a newly swarm-based metaheuristic algorithm that simulate the swimming and foraging behaviour of salps in oceans so to search for global optimum solution. Similarly to other metaheuristic algorithms, SSA suffers from poor convergence rate and stagnation in local optima. In this paper, three different improvements to the original population update process are proposed in order to enhance its exploitation and exploration capabilities. The first modification (MSSA1) introduces the concept of local best information to the followers salps update process allowing a better exploration of local search neighbourhood. The second improvement (MSSA2) provide two followers update process. The first is based on a differential evolution combined with a randomly selected local best position, and the second uses a local search in the global best neighbourhood which is triggered by a non-improvement in the corresponding local best. A third modification to the SSA algorithm (MSSA3) penalises a non-improvement of the local best solution by computing a new corresponding follower’s position based on a local jump in the local best neighbourhood for better exploitation. The performances of the proposed algorithms are tested on 27 CEC’15 test suite, and two real-world optimisation problems. A comparative study using nonparametric statistical tests of the obtained results is conducted against those of eight well-known metaheuristics, including the original SSA. The results indicate an overall distinctive performance of all three modification compared to the remaining algorithms, while MSSA1 scored generally better than MSSA2 and MSSA3.

Appendices

Appendix A

Four variables are considered in solving the welded beam structure problem, namely thickness of the weld (h), length of the attached bar (l), height of the bar (t) and thickness of the bar (b). The objective of the problem is a welded beam designed for minimum cost subject to constraints on shear stress ($\tau$), bending stress ($\theta$), end deflection of the beam ($\delta$), buckling load on the bar ($P_C$), and side constrains. The mathematical formulation of the problem is listed below:

Consider:

$$\begin{aligned} {\mathbf {x}} = \left[ x_1,x_2,x_3,x_4 \right] = \left[ h,l,t,b\right] \end{aligned}$$

Minimise:

$$\begin{aligned} F({\mathbf {x}}) = 1.10471 x_1^2x_2 + 0.04811 x_3x_4 (14.0+x_2) \end{aligned}$$

Subject to:

$$\begin{aligned} g_1({\mathbf {x}})&= \tau ({\mathbf {x}})-\tau _{max} \le 0 \\ g_2({\mathbf {x}})&= \sigma ({\mathbf {x}})-\sigma _{max} \le 0 \\ g_3({\mathbf {x}})&= \delta ({\mathbf {x}})-\delta _{max} \le 0 \\ g_4({\mathbf {x}})&= x_1-x_4 \le 0 \\ g_5({\mathbf {x}})&= P - P_C({\mathbf {x}}) \le 0 \\ g_6({\mathbf {x}})&= 0.125 -x_1 \le 0 \\ g_7({\mathbf {x}})&= 1.10471 x_1^2 + 0.04811x_3x_4 (14.0+x_2)-5.0 \le 0 \end{aligned}$$

where

$$\begin{aligned}&\tau ({\mathbf {x}})=\sqrt{(\tau ')^2+2\tau '\tau ''\frac{x_2}{2R}+(\tau '')^2} \\&\tau ' =\frac{P}{\sqrt{2}x_1x_2}, ~ \tau '' =\frac{MR}{J},~ M=P(L+\frac{x_2}{2}) \\&R = \sqrt{\frac{x_2^2}{4}+\left( \frac{x_1+x_3}{2} \right) ^2 } \\&J = 2\left( \sqrt{2}x_1x_2\left[ \frac{x_2^2}{4}+\left( \frac{x_1+x_3}{2}\right) ^2 \right] \right) \\&\sigma ({\mathbf {x}}) = \frac{6PL}{x_4x_3^2}, ~ \delta ({\mathbf {x}}) =\frac{6PL^3}{x_4x_3^3} \\&P_C({\mathbf {x}}) = \frac{4.013 \sqrt{\frac{x_3^2x_4^6}{36}}}{L^2}\left( 1-\frac{x_3}{2L}\sqrt{\frac{E}{4G}}\right) \\&P=6000~lb,~L=14~in, \delta _{max} = 0.25~in,\\&\quad E=30\times 10^6~psi,~G=12\times 10^6 psi \\&\tau _{max} = 13600~psi, ~\sigma _{max}=30000~psi \end{aligned}$$

with

$$\begin{aligned} 0.1 \le x_1,~ x_4 \le 2.0 ~and~ 0.1 \le x_2,~ x_3 \le 10.0 \end{aligned}$$

Appendix B

The tension/ compression spring design problem consists of minimising the volume V of a coil spring under a fixed tension/ compression load. Mathematically, the problem is illustrated as follows: Consider:

$$\begin{aligned} {\mathbf {x}} = \left[ x_1,x_2,x_3 \right] = \left[ d, D, N\right] \end{aligned}$$

Minimise:

$$\begin{aligned} F({\mathbf {x}}) = (x_3+2)x_2x_1^2 \end{aligned}$$

Subject to:

$$\begin{aligned} g_1({\mathbf {x}})&= 1- \frac{x_2^3 x_3}{71785 x_1^4} \le 0 \\ g_2({\mathbf {x}})&= \frac{4x_2^2 - x_1x_2}{12566 (x_2x_1^3-x_1^4)}+ \frac{1}{5108 x_1^2}-1 \le 0 \\ g_3({\mathbf {x}})&= 1- \frac{140.45 x_1}{x_2^2 x_3} \le 0 \\ g_4({\mathbf {x}})&= \frac{x_2+x_1}{1.5}-1 \le 0 \end{aligned}$$

with

$$\begin{aligned} 2 \le x_1 \le 15, ~ 0.25 \le x_2 \le 1.3, ~0.05 \le x_3 \le 2. \end{aligned}$$