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Sandpiper optimization algorithm: a novel approach for solving real-life engineering problems

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Abstract

This paper presents a novel bio-inspired algorithm called Sandpiper Optimization Algorithm (SOA) and applies it to solve challenging real-life problems. The main inspiration behind this algorithm is the migration and attacking behaviour of sandpipers. These two steps are modeled and implemented computationally to emphasize intensification and diversification in the search space. The comparison of proposed SOA algorithm is performed with nine competing optimization algorithms over 44 benchmark functions. The analysis of computational complexity and convergence behaviors of the proposed algorithm have been evaluated. Further, SOA algorithm is hybridized with decision tree machine-learning algorithm to solve real-life applications. The experimental results demonstrated that the proposed algorithm is able to solve challenging constrained optimization problems and outperforms the other state-of-the-art optimization algorithms.

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Notes

  1. The term fitness value is defined as a process which evaluates the population and gives a score or fitness. Whereas, the process is a function which measures the quality of the represented solution.

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Authors and Affiliations

  1. Computer Science and Engineering Department, Thapar Institute of Engineering and Technology, Patiala, India

    Amandeep Kaur & Sushma Jain

  2. Department of Computer Science and Engineering, Bennett University, Noida, India

    Shivani Goel

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  1. Amandeep Kaur
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Appendix: Unimodal, multimodal, and fixed-dimension multimodal benchmark test functions

Appendix: Unimodal, multimodal, and fixed-dimension multimodal benchmark test functions

1.1 A.1 Unimodal benchmark test functions

1.1.1 A.1.1 Sphere model

$$ \begin{array}{@{}rcl@{}} &&F_1(z)= \sum\limits_{i=1}^{30} z_i^2\\ &&-100 \leq z_i \leq 100, \quad f_{min} = 0, \quad Dim = 30 \\ \end{array} $$

1.1.2 A.1.2 Schwefel’s problem 2.22

$$ \begin{array}{@{}rcl@{}} &&F_2(z)=\sum\limits_{i=1}^{30} |z_i| + \prod\limits_{i =1}^{30} |z_i|\\ &&-10 \leq z_i \leq 10, \quad f_{min} = 0, \quad Dim = 30 \end{array} $$

1.1.3 A.1.3 Schwefel’s problem 1.2

$$ \begin{array}{@{}rcl@{}} &&F_3(z)= \sum\limits_{i=1}^{30} \left( \sum\limits_{j=1}^i z_j\right)^2\\ &&-100 \leq z_i \leq 100, \quad f_{min} = 0, \quad Dim = 30 \end{array} $$

1.1.4 A.1.4 Schwefel’s problem 2.21

$$ \begin{array}{@{}rcl@{}} &&F_4(z)= max_i\{|z_i|, 1 \leq i \leq 30\}\\ &&-100 \leq z_i \leq 100, \quad f_{min} = 0, \quad Dim = 30 \end{array} $$

1.1.5 A.1.5 Generalized Rosenbrock’s function

$$ \begin{array}{@{}rcl@{}} F_5(z)= \sum\limits_{i=1}^{29}[100(z_{i+1} - z_i^2)^2 + (z_i - 1)^2]\\ -30 \leq z_i \leq 30, \quad f_{min} = 0, \quad Dim = 30 \end{array} $$

1.1.6 A.1.6 Step function

$$ \begin{array}{@{}rcl@{}} &&F_6(z)= \sum\limits_{i=1}^{30}(\lfloor{z_i + 0.5}\rfloor)^2\\ &&-100 \leq z_i \leq 100, \quad f_{min} = 0, \quad Dim = 30 \end{array} $$

1.1.7 A.1.7 Quartic function

$$ \begin{array}{@{}rcl@{}} &&F_7(z)= \sum\limits_{i=1}^{30} iz_i^4 + random[0,1]\\ &&-1.28 \leq z_i \leq 1.28, \quad f_{min} = 0, \quad Dim = 30 \end{array} $$

1.2 A.2 Multimodal benchmark test functions

1.2.1 A.2.1 Generalized Schwefel’s problem 2.26

$$ \begin{array}{@{}rcl@{}} &&F_8(z)= \sum\limits_{i=1}^{30} - z_i sin (\sqrt{|z_i|})\\ &&-500 \leq z_i \leq 500, \quad f_{min} = -12569.5, \quad Dim = 30 \end{array} $$

1.2.2 A.2.2 Generalized Rastrigin’s function

$$ \begin{array}{@{}rcl@{}} F_9(z)= \sum\limits_{i=1}^{30} [z_i^2 - 10cos(2\pi z_i) + 10]\\ -5.12 \leq z_i \leq 5.12, \quad f_{min} = 0, \quad Dim = 30 \end{array} $$

1.2.3 A.2.3 Ackley’s function

$$ \begin{array}{@{}rcl@{}} F_{10}(z)&=& -20exp \left( -0.2\sqrt{\frac{1}{30}\sum\limits_{i=1}^{30} z_i^2}\right) - exp\left( \frac{1}{30}\sum\limits_{i=1}^{30} cos(2\pi z_i)\right) \\&&+20 +e -32 \leq z_i \leq 32, \quad f_{min} = 0, \quad Dim = 30 \end{array} $$

1.2.4 A.2.4 Generalized Griewank function

$$ \begin{array}{@{}rcl@{}} &&F_{11}(z)= \frac{1}{4000} \sum\limits_{i=1}^{30} z_i^2 - \prod\limits_{i=1}^{30} cos \left( \frac{z_i}{\sqrt{i}}\right) + 1\\ &&-600 \leq z_i \leq 600, \quad f_{min} = 0, \quad Dim = 30 \end{array} $$

1.2.5 A.2.5 Generalized penalized functions

  • \(F_{12}(z)= \frac {\pi }{30}\{10sin(\pi x_1) + \sum \limits _{i=1}^{29}(x_i - 1)^2[1 + 10sin^2(\pi x_{i+1})] + (x_n - 1)^2 \} + \sum \limits _{i=1}^{30}u(z_i, 10, 100, 4)\)− 50 ≤ zi ≤ 50, fmin = 0, Dim = 30

  • \(F_{13}(z)= 0.1\{sin^2(3\pi z_1) + \sum \limits _{i=1}^{29}(z_i - 1)^2[1 + sin^2(3\pi z_i + 1)] + (z_n - 1)^2[1 + sin^2(2\pi z_{30})]\} + \sum \limits _{i=1}^Nu(z_i, 5, 100, 4)\)− 50 ≤ zi ≤ 50, fmin = 0, Dim = 30where, \( x_i = 1 + \frac {z_i+1}{4}\)

    \(u(z_i, a, k, m) = \left \{\begin {array}{ll} k(z_i - a)^m & z_i > a\\ 0 & -a<z_i<a\\ k(-z_i - a)^m & z_i<-a \end {array}\right .\)

1.3 A.3 Fixed-dimension multimodal benchmark test functions

1.3.1 A.3.1 Shekel’s foxholes function

$$ \begin{array}{@{}rcl@{}} &&F_{14}(z)= \left( \frac{1}{500} + \sum\limits_{j=1}^{25}\frac{1}{j+ \sum\limits_{i=1}^2(z_i - a_{ij})^6}\right)^{-1}\\ &&-65.536 \leq z_i \leq 65.536, \quad f_{min} \approx 1, \quad Dim = 2 \end{array} $$

1.3.2 A.3.2 Kowalik’s function

$$ \begin{array}{@{}rcl@{}} &&F_{15}(z)= \sum\limits_{i=1}^{11}\left[a_i - \frac{z_1(b_i^2+b_iz_2)}{b_i^2+b_iz_3+z_4}\right]^2\\ &&-5 \leq z_i \leq 5, \quad f_{min} \approx 0.0003075, \quad Dim = 4 \end{array} $$

1.3.3 A.3.3 Six-Hump camel-back function

$$ \begin{array}{@{}rcl@{}} &&F_{16}(z)= 4z_1^2 - 2.1z_1^4 + \frac{1}{3}z_1^6 + z_1z_2 - 4z_2^2 + 4z_2^4\\ &&-5 \leq z_i \leq 5, \quad f_{min} = -1.0316285, \quad Dim = 2 \end{array} $$

1.3.4 A.3.4 Branin function

$$ \begin{array}{@{}rcl@{}} F_{17}(z)&=& \left( z_2 - \frac{5.1}{4\pi^2}z_1^2 + \frac{5}{\pi}z_1 - 6 \right)^2 + 10\left( 1 - \frac{1}{8\pi}\right)cosz_1 + 10\\ &&-5 \leq z_1 \leq 10, \quad 0 \leq z_2 \leq 15, \quad f_{min} = 0.398, \quad Dim = 2 \end{array} $$

1.3.5 A.3.5 Goldstein-price function

$$ \begin{array}{@{}rcl@{}} F_{18}(z)&=& [1 + (z_1 + z_2 + 1)^2(19 - 14z_1 + 3z_1^2 - 14z_2 + 6z_1z_2 + 3z_2^2)] \\ &&\times [30 + (2z_1 - 3z_2)^2 \times (18 - 32z_1 + 12z_1^2 + 48z_2 - 36z_1z_2 \\&&+ 27z_2^2)] -2 \leq z_i \leq 2, \quad f_{min} = 3, \quad Dim = 2 \end{array} $$

1.3.6 A.3.6 Hartman’s family

  • \(F_{19}(z)= -\sum \limits _{i=1}^4c_i exp(-\sum \limits _{j=1}^3 a_{ij}(z_j - p_{ij})^2)\) 0 ≤ zj ≤ 1, fmin = − 3.86, Dim = 3

  • \(F_{20}(z)= -\sum \limits _{i=1}^4c_i exp(-\sum \limits _{j=1}^6 a_{ij}(z_j - p_{ij})^2)\) 0 ≤ zj ≤ 1, fmin = − 3.32, Dim = 6

1.3.7 A.3.7 Shekel’s foxholes function

  • \(F_{21}(z)= -\sum \limits _{i=1}^5[(X - a_i)(X - a_i)^T + c_i]^{-1} \) 0 ≤ zi ≤ 10, fmin = − 10.1532, Dim = 4

  • \(F_{22}(z)= -\sum \limits _{i=1}^7[(X - a_i)(X - a_i)^T + c_i]^{-1}\) 0 ≤ zi ≤ 10, fmin = − 10.4028, Dim = 4

  • \(F_{23}(z)= -\sum \limits _{i=1}^{10}[(X - a_i)(X - a_i)^T + c_i]^{-1}\) 0 ≤ zi ≤ 10, fmin = − 10.536, Dim = 4

Table 29 Shekel’s foxholes function F14
Full size table
Table 30 Hartman function F19
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Table 31 Shekel foxholes functions F21, F22, F23
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Table 32 Hartman function F20
Full size table

1.4 A.4 Basic composite benchmark test functions

1.4.1 A.4.1 Weierstrass Function

$$ \begin{array}{@{}rcl@{}} F(z)&=&\sum\limits_{i=1}^{30} \left( \sum\limits_{k=0}^{20} [0.5^k cos(2\pi 3^k(z_i+0.5))] \right)\\ &&-30\sum\limits_{k=0}^{20}[0.5^k cos(2\pi 3^k \times 0.5)] \end{array} $$

Note that the Sphere, Rastrigin’s, Griewank’s, and Ackley’s functions in composite benchmark suite are same as above mentioned F1, F9, F11, and F10 benchmark test functions.

Table 33 Composite benchmark test functions
Full size table
Table 34 CEC 2015 benchmark test functions
Full size table

1.5 A.5 Basic CEC 2015 benchmark test functions

1.5.1 A.5.1 Bent cigar function

$$ \begin{array}{@{}rcl@{}} F(z)= z_1^2+ 10^6\sum\limits_{i=2}^{30}z_i^2 \end{array} $$

1.5.2 A.5.2 Discus function

$$ \begin{array}{@{}rcl@{}} F(z)= 10^6z_1^2+\sum\limits_{i=2}^{30}z_i^2 \end{array} $$

1.5.3 A.5.3 Modified Schwefel’s function

$$ \begin{array}{@{}rcl@{}} F(z)& = & 418.9829 \times 30 - \sum\limits_{i=1}^{30}g(y_i), \quad y_i=z_i\!+4.209687462275036e+002\\ g(y_i) & = & \left\{\begin{array}{l} y_i sin(|y_i|^{1/2}) \\ \quad \textit{where, if} |y_i| \leq 500,\\ (500-mod(y_i,500))sin(\sqrt{|500-mod(y_i,500)|}) - \frac{(y_i-500)^2}{10000\times 30}\\ \quad \textit{where, if} y_i > 500,\\ (mod(|y_i|,500) - 500)sin(\sqrt{|mod(|y_i|,500) - 500|}) - \frac{(y_i + 500)^2}{10000\!\times\! 30}\\ \quad \textit{where, if} y_i < -500 \end{array}\right. \end{array} $$

1.5.4 A.5.4 Katsuura function

$$ \begin{array}{@{}rcl@{}} F(z)=\frac{10}{30^2} \prod\limits_{i=1}^{30}\left( 1+i\sum\limits_{j=1}^{32}\frac{|2^jz_i-round(2^jz_i)|}{2^j}\right)^{\frac{10}{30^{1.2}}}-\frac{10}{30^2} \end{array} $$

1.5.5 A.5.5 HappyCat function

$$ \begin{array}{@{}rcl@{}} F(z)= \left| \sum\limits_{i=1}^{30}z_i^2 - 30\right|^{1/4} + \left( 0.5 \sum\limits_{i=1}^{30}z_i^2 + \sum\limits_{i=1}^{30}z_i\right)/30+0.5 \end{array} $$

1.5.6 A.5.6 HGBat function

$$ \begin{array}{@{}rcl@{}} F(z)&=& \left| \left( \sum\limits_{i=1}^{30}z_i^2\right)^2 - \left( \sum\limits_{i=1}^{30}z_i\right)^2\right|^{1/2}\\ &&+ \left( 0.5 \sum\limits_{i=1}^{30}z_i^2 + \sum\limits_{i=1}^{30}z_i\right)/30+0.5 \end{array} $$

1.5.7 A.5.7 Expanded Griewank’s plus Rosenbrock’s function

$$ \begin{array}{@{}rcl@{}} F(z)&=& F_{39}(F_{38}(z_1,z_2))\\ &&+F_{39}(F_{38}(z_2,z_3))\\ &&+\ldots+F_{39}(F_{38}(z_{30},z_1)) \end{array} $$

1.5.8 A.5.8 Expanded Scaffer’s F6 function

$$ \begin{array}{@{}rcl@{}} &&\text{Scaffer's} F6 \text{Function:}\\ &&g(z,x)=0.5+\frac{(sin^2(\sqrt{z^2+x^2})-0.5)}{(1+0.001(z^2+x^2))^2}\\ &&F(z)= g(z_1,z_2) + g(z_2,z_3) + {\ldots} + g(z_{30},z_1) \end{array} $$

1.5.9 A.5.9 High conditioned elliptic function

$$ \begin{array}{@{}rcl@{}} F(z)= \sum\limits_{i=1}^{30}(10^6)^{\frac{i-1}{30-1}}z_i^2 \end{array} $$

Note that the Weierstrass, Rosenbrock’s, Griewank’s, Rastrigin’s, and Ackley’s functions in CEC 2015 benchmark test suite are same as above mentioned Weierstrass, F5, F11, F9, and F10 benchmark test functions.

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Kaur, A., Jain, S. & Goel, S. Sandpiper optimization algorithm: a novel approach for solving real-life engineering problems. Appl Intell 50, 582–619 (2020). https://doi.org/10.1007/s10489-019-01507-3

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