A Hybridisation of Runner-Based and Seed-Based Plant Propagation Algorithms

Abstract

In this chapter we introduce a hybrid plant propagation algorithm which combines the standard PPA which uses runners as a means for search and SbPPA which uses seeds as a means for search. Runners are more suited for exploitation while seeds, when propagated by animals and birds, are more suited for exploration. Combining the two is a natural development to design an effective global optimisation algorithm. PPA and SbPPA will be recalled. The hybrid algorithm is then presented and comparative computational results are reported.

Appendices

Appendix

Constrained Global Optimization Problems

1.1 CP1

$$\begin{aligned} \begin{array}{ll} \mathrm{Min} &{} \quad f(x)=5{\sum }_{d=1}^{4}x_d-5{\sum }_{d=1}^{4}x_d^2-{\sum }_{d=5}^{13}x_d, \\ {\text {subject to}} &{}g_1(x)=2x_1+2x_2+x_{10}+x_{11}-10\le 0,\\ &{}\quad g_2(x)=2x_1+2x_3+x_{10}+x_{12}-10\le 0,\\ &{}\quad g_3(x)=2x_2+2x_3+x_{11}+x_{12}-10\le 0,\\ &{}\quad g_4(x)=-8x_1+x_{10}\le 0,\\ &{}\quad g_5(x)=-8x_2+x_{11}\le 0,\\ &{}\quad g_6(x)=-8x_3+x_{12}\le 0,\\ &{}\quad g_7(x)=-2x_4-x_5+x_{10}\le 0,\\ &{}\quad g_8(x)=-2x_6-x_7+x_{11}\le 0,\\ &{}\quad g_9(x)=-2x_8-x_9+x_{12}\le 0, \end{array} \end{aligned}$$

where bounds are $0 \le x_i \le 1 (i = 1, \ldots , 9, 13), 0 \le x_i \le 100 (i= 10, 11, 12)$. The global optimum is at $x^* = (1, 1, 1, 1, 1, 1, 1, 1, 1, , 3, 3, 3, 1), f(x^*)= -15$.

1.2 CP2

$$\begin{aligned} \begin{array}{ll} \mathrm{Min} &{}\quad f(x)=5.3578547x_2 +0.8356891x_1x_5+ 37.293239x_1 - 40792.141,\\ {\text {subject to}}&{} g_1(x)=85.334407+0.0056858x_2x_5+0.0006262x_1x_4-0.0022053x_3x_5-92\le 0,\\ &{}\quad g_2(x)=-85.334407-0.0056858x_2x_5-0.0006262x_1x_4+0.0022053x_3x_5\le 0,\\ &{}\quad g_3(x)=80.51249+0.0071317x2x5+0.0029955x1x2-0.0021813x_2-110\le 0,\\ &{}\quad g_4(x)=-80.51249-0.0071317x_2x_5+0.0029955x_1x_2-0.0021813x_2+90\le 0,\\ &{}\quad g_5(x)=9.300961-0.0047026x_3x_5-0.0012547x_1x_3-0.0019085x_3x_4-25\le 0,\\ &{}\quad g_6(x)=-9.300961-0.0047026x_3x_5-0.0012547x_1x_3-0.0019085x_3x_4+20\le 0, \end{array} \end{aligned}$$

where $78 \le x_1\le 102$, $33 \le x_2 \le 45$, $27 \le x_i\le 45$ $(i = 3, 4, 5)$. The optimum solution is $x^* = (78, 33, 29.995256025682, 45, 36.775812905788)$, where $f(x^*)= - 30665.539$. Constraints $g_1$ and $g_6$ are active.

1.3 CP3

$$\begin{aligned} \begin{array}{ll} \mathrm{Min} &{}\quad f(x)=(x_1-10)^3+(x_2-20)^3,\\ \text {subject to }&{} g_1(x)=-(x_1-5)^2-(x_2 -5)^2+100 \le 0,\\ &{}\quad g_2(x)=(x_1-6)^2 +(x^2-5)^2-82.81 \le 0, \end{array} \end{aligned}$$

where $13 \le x_1 \le 100$ and $0 \le x_2 \le 100$. The optimum solution is $x^* = (14.095, 0.84296)$ where $f(x^*)= -6961.81388$. Both constraints are active.

1.4 CP4

$$\begin{aligned} \begin{array}{ll} \mathrm{Min} &{} f(x)=x_1^2+x_2^2+x_1x_2-14x_1-16x_2+(x_3-10)^2+ 4(x_4 - 5)^2 + (x_5 - 3)^2 \\ &{}\qquad +2(x_6 -1)^2+ 5x_7^2 + 7(x_8 - 11)^2+ 2(x_9 - 10)^2 + (x_{10}-7)^2 + 45,\\ {\text {subject to}} &{} g_1(x)=-105+4x_1+5x_2-3x_7+9x_8 \le 0,\\ &{}\quad g_2(x)=10x_1 - 8x_2 - 17x_7 + 2x_8 \le 0,\\ &{}\quad g_3(x)=-8x_1 + 2x_2 + 5x_9 - 2x_{10} - 12 \le 0,\\ &{}\quad g_4(x)=3(x_1 - 2)^2 + 4(x_2 - 3)^2 + 2x_3^2 - 7x_4 - 120 \le 0,\\ &{}\quad g_5(x)=5x_1^2 + 8x_2 + (x_3 - 6)^2 - 2x^4 - 40 \le 0,\\ &{}\quad g_6(x)=x_1^2 + 2(x_2 - 2)^2 - 2x_1x_2 + 14x_5 - 6x_6 \le 0,\\ &{}\quad g_7(x)=0.5(x_1 - 8)^2 + 2(x_2 - 4)^2 + 3x_5^2 - x_6 - 30 \le 0,\\ &{}\quad g_8(x)=-3x_1 + 6x_2 + 12(x_9 - 8)^2 - 7x_{10} \le 0,\\ \end{array} \end{aligned}$$

where $-10 \le x_i \le 10$ $(i = 1,\ldots , 10)$. The global optimum is $x^* = (2.171996, 2.363683, 8.773926, 5.095984, 0.9906548, 1.430574, 1.321644, 9.828726, 8.280092, 8.375927)$, where $f(x^*) = 24.3062091$. Constraints $g_1, g_2, g_3, g_4, g_5$ and $g_6$ are active.

1.5 CP5

$$\begin{aligned} \begin{array}{ll} \mathrm{Min}&{}f(x)=x_1^2 + (x_2 - 1)^2\\ {\text {subject to}}&{}g_1(x)=x_2 - x_1^2=0, \end{array} \end{aligned}$$

where $1 \le x_1 \le 1$, $1 \le x_2 \le 1$. The optimum solution is $x^*= (\pm 1/\sqrt{(2)}, 1/2)$,

where $f(x^*) = 0.7499$.

1.6 Welded Beam Design Optimisation

The welded beam design is a standard test problem for constrained design optimisation, [6, 27]. There are four design variables: the width w and length L of the welded area, the depth d and thickness h of the main beam. The objective is to minimise the overall fabrication cost, under the appropriate constraints of shear stress $\tau$, bending stress $\sigma$, buckling load P and maximum end deflection $\delta$. The optimization model is summarized as follows, where $x^T=(w,L,d,h).$

$$\begin{aligned} Minimise\quad f(x) = 1.10471w^2L + 0.04811dh(14.0 + L), \end{aligned}$$

(11)

subject to

$$\begin{aligned} \begin{aligned}&g_{1}(x) = w-h \le 0, \\&g_{2}(x) = \delta (x)-0.25 \le 0, \\&g_{3}(x) = \tau (x)-13,600 \le 0, \\&g_{4}(x) = \sigma (x)-30,000\le 0, \\&g_{5}(x) = 1.10471w^2 + 0.04811dh(14.0 + L)-5.0 \le 0, \\&g_{6}(x) = 0.125-w\le 0, \\&g_{7}(x) = 6000-P(x) \le 0,\\ \end{aligned} \end{aligned}$$

(12)

where

$$\begin{aligned} \begin{aligned}&\sigma (x) =\frac{504,000}{hd^2},\\&D = \frac{1}{2}\sqrt{L^2+(w+d)^2},\\&\delta =\frac{65,856}{30,000hd^3},\\&\alpha = \frac{6000}{\sqrt{2}wL},\\&P =0.61423 \times 10^6 \frac{dh^3}{6}\left( 1-\frac{\root d \of {\frac{30}{48}}}{28}\right) .\\ \end{aligned} \begin{aligned}&Q =6000\left( 14+\frac{L}{2}\right) ,\\&J = \sqrt{2}w L \left( \frac{L^2}{6}+\frac{(w+d)^2}{2}\right) ,\\&\beta = \frac{QD}{J},\\&\tau (x) =\sqrt{\alpha ^2+\frac{\alpha \beta L}{D}+\beta ^2},\\ \end{aligned} \end{aligned}$$

(13)

The simple limit or bounds are $0.1 \le L,d \le 10$ and $0.1 \le w,h \le 2.0.$

1.7 Spring Design Optimisation

The main objective of this problem, [4, 5] is to minimize the weight of a tension/compression string, subject to constraints of minimum deflection, shear stress, surge frequency, and limits on outside diameter and on design variables. There are three design variables: the wire diameter $x_1$, the mean coil diameter $x_2$, and the number of active coils $x_3$, [6]. The mathematical formulation of this problem, where $x^T=(x_1,x_2,x_3)$, is as follows.

$$\begin{aligned} Minimise\quad f(x)= (x_3+2)x_2x_1^2, \end{aligned}$$

(14)

subject to

$$\begin{aligned} \begin{aligned}&g_{1}(x) = 1-\frac{x_2^3x_3}{7,178x_1^4}\le 0, \\&g_{2}(x) = \frac{4x_2^2-x_1x_2}{12,566(x_2x_1^3)-x_1^4}+\frac{1}{5,108x_1^2}-1 \le 0, \\&g_{3}(x) = 1-\frac{140.45x_1}{x_2^2x_3} \le 0, \\&g_{4}(x) = \frac{x_2+x_1}{1.5}-1\le 0, \\ \end{aligned} \end{aligned}$$

(15)

The simple limits on the design variables are $0.05\le x_1\le 2.0$, $0.25 \le x_2 \le 1.3$ and $2.0\le x_3\le 15.0.$

1.8 Speed Reducer Design Optimization

The problem of designing speed reducer, [11], is a standard test problem, it consists of the design variables as: face width $x_1$, module of teeth $x_2$, number of teeth on pinion $x_3$, length of the first shaft between bearings $x_4$, length of the second shaft between bearings $x_5$, diameter of the first shaft $x_6$, and diameter of the first shaft $x_7$ (all variables continuous except $x_3$ that is integer). The weight of the speed reducer is to be minimized subject to constraints on bending stress of the gear teeth, surface stress, transverse deflections of the shafts and stresses in the shaft, [6]. The mathematical formulation of the problem, where $x^T=(x_1,x_2,x_3,x_4,x_5,x_6,x_7)$, is as follows.

$$\begin{aligned} Minimise \quad f(x) =&\,0.7854x_1x_2^2(3.3333x_3^2 + 14.9334x_{3}43.0934)\nonumber \\&-1.508x_1(x_6^2 + x_7^3 ) + 7.4777(x_6^3 + x_7^3 )+ 0.7854(x_4x_6^2 + x_5x_7^2 ), \end{aligned}$$

(16)

subject to

$$\begin{aligned} \begin{aligned}&g_{1}(x) = \frac{27}{x_1x_2^2x_3}-1\le 0, \\&g_{2}(x) = \frac{397.5}{x_1x_2^2x_3^2}-1 \le 0, \\&g_{3}(x) = \frac{1.93x_4^3}{x_2x_3x_6^4}-1 \le 0, \\&g_{4}(x) = \frac{1.93x_5^3}{x_2x_3x_7^4}-1\le 0, \\&g_{5}(x) = \frac{1.0}{110x_6^3}\sqrt{\left( \frac{745.0x_4}{x_2x_3}\right) ^2+16.9\times 10^6}-1 \le 0, \\&g_{6}(x) = \frac{1.0}{85x_7^3}\sqrt{\left( \frac{745.0x_5}{x_2x_3}\right) ^2+157.5\times 10^6}-1 \le 0, \\&g_{7}(x) = \frac{x_2x_3}{40}-1 \le 0,\\&g_{8}(x) = \frac{5x_2}{x_1}-1\le 0, \\&g_{9}(x) = \frac{x_1}{12x_2}-1 \le 0, \\&g_{10}(x) = \frac{1.5x_6+1.9}{x_4}-1\le 0, \\&g_{11}(x) = \frac{1.1x_7+1.9}{x_5}-1 \le 0,\\ \end{aligned} \end{aligned}$$

(17)

The simple limits on the design variables are $2.6\le x_1\le 3.6$, $0.7 \le x_2 \le 0.8$, $17\le x_3\le 28$, $7.3\le x_4\le 8.3$, $7.8 \le x_5 \le 8.3$, $2.9\le x_6\le 3.9$ and $5.0\le x_7\le 5.5$.