SP-J48: a novel optimization and machine-learning-based approach for solving complex problems: special application in software engineering for detecting code smells

Abstract

This paper presents a novel hybrid algorithm based on optimization and machine-learning approaches for solving real-life complex problems. The optimization algorithm is inspired from the searching and attacking behaviors of sandpipers, called as Sandpiper Optimization Algorithm (SPOA). These two behaviors are modeled and implemented computationally to emphasize intensification and diversification in the search space. A comparison of the proposed SPOA algorithm is performed with nine competing optimization algorithms over 23 benchmark test functions. The proposed SPOA is further hybridized with B-J48 pruned machine-learning approach for efficiently detecting the code smells from the data set. The results reveal that the proposed technique is able to solve challenging problems and outperforms the other well-known approaches.

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

1.1 Unimodal benchmark test functions

1.1.1 Sphere model

$$\begin{aligned} F_1(z) \,=\, & {} \sum _{i=1}^{30} z_i^2\\ -100\le & {} z_i \le 100, \quad f_{\min } = 0, \quad {\mathrm{Dim}} = 30 \end{aligned}$$

1.1.2 Schwefel’s Problem 2.22

$$\begin{aligned}&F_2(z)= \sum _{i=1}^{30} |z_i| + \prod _{i=1}^{30} |z_i|\\&-10 \le z_i \le 10, \quad f_{\min } = 0, \quad {\mathrm{Dim}} = 30 \end{aligned}$$

1.1.3 Schwefel’s Problem 1.2

$$\begin{aligned}&F_3(z)= \sum _{i=1}^{30} \Big (\sum _{j=1}^i z_j\Big )^2\\&-100 \le z_i \le 100, \quad f_{\min } = 0, \quad {\mathrm{Dim}} = 30 \end{aligned}$$

1.1.4 Schwefel’s Problem 2.21

$$\begin{aligned}&F_4(z)= \max _i\{|z_i|, 1 \le i \le 30\}\\&-100 \le z_i \le 100, \quad f_{\min } = 0, \quad {\mathrm{Dim}} = 30 \end{aligned}$$

1.1.5 Generalized Rosenbrock’s function

$$\begin{aligned}&F_5(z)= \sum _{i=1}^{29}\left[ 100\left( z_{i+1} - z_i^2\right) ^2 + (z_i - 1)^2\right] \\&-30 \le z_i \le 30, \quad f_{\min } = 0, \quad {\mathrm{Dim}} = 30 \end{aligned}$$

1.1.6 Step function

$$\begin{aligned}&F_6(z)= \sum _{i=1}^{30}(\lfloor {z_i + 0.5}\rfloor )^2\\&-100 \le z_i \le 100, \quad f_{\min } = 0, \quad {\mathrm{Dim}} = 30 \end{aligned}$$

1.1.7 Quartic function

$$\begin{aligned}&F_7(z)= \sum _{i=1}^{30} iz_i^4 + {\mathrm{random}}[0,1]\\&-1.28 \le z_i \le 1.28, \quad f_{\min } = 0, \quad {\mathrm{Dim}} = 30 \end{aligned}$$

1.2 Multimodal benchmark test functions

1.2.1 Generalized Schwefel’s Problem 2.26

$$\begin{aligned}&F_8(z)= \sum _{i=1}^{30} - z_i \sin (\sqrt{|z_i|})\\&-500 \le z_i \le 500, \quad f_{\min } = -12569.5, \quad {\mathrm{Dim}} = 30 \end{aligned}$$

1.2.2 Generalized Rastrigin’s function

$$\begin{aligned}&F_9(z)= \sum _{i=1}^{30} [z_i^2 - 10\cos (2\pi z_i) + 10]\\&-5.12 \le z_i \le 5.12, \quad f_{\min } = 0, \quad {\mathrm{Dim}} = 30 \end{aligned}$$

1.2.3 Ackley’s function

$$\begin{aligned}F_{10}(z)&= -20\exp \bigg (-0.2\sqrt{\dfrac{1}{30}\sum \nolimits _{i=1}^{30} z_i^2}\bigg ) \\&\quad - \exp \bigg (\dfrac{1}{30}\sum _{i=1}^{30} \cos (2\pi z_i)\bigg ) +20 +e\\&\quad -32 \le z_i \le 32, \quad f_{\min } = 0, \quad {\mathrm{Dim}} = 30 \end{aligned}$$

1.2.4 Generalized Griewank function

$$\begin{aligned}&F_{11}(z)\\&= \dfrac{1}{4000} \sum _{i=1}^{30} z_i^2 - \prod _{i=1}^{30} \cos \bigg (\dfrac{z_i}{\sqrt{i}}\bigg ) + 1\\&-600 \le z_i \le 600, \quad f_{\min } = 0, \quad {\mathrm{Dim}} = 30 \end{aligned}$$

1.2.5 Generalized penalized functions

• $$\begin{aligned}&F_{12}(z)= \dfrac{\pi }{30}\{10\sin (\pi x_1) + \sum _{i=1}^{29}(x_i - 1)^2[1 + 10\sin ^2(\pi x_{i+1})] \\&\quad + \,(x_n - 1)^2 \} + \sum _{i=1}^{30}u(z_i, 10, 100, 4)\\&\quad -\,50 \le z_i \le 50, \quad f_{\min } = 0, \quad {\mathrm{Dim}} = 30 \\ \end{aligned}$$
• $$\begin{aligned}&F_{13}(z)= 0.1\{\sin ^2(3\pi z_1) + \sum _{i=1}^{29}(z_i - 1)^2[1 + \sin ^2(3\pi z_i + 1)]\\&\quad + \,(z_n - 1)^2[1 + \sin ^2(2\pi z_{30})]\} + \sum _{i=1}^Nu(z_i, 5, 100, 4)\\&\quad -50 \le z_i \le 50, \quad f_{\min } = 0, \quad {\mathrm{Dim}} = 30 \end{aligned}$$

where $x_i = 1 + \dfrac{z_i+1}{4}$

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

1.3 Fixed-dimension multimodal benchmark test functions

1.3.1 Shekel’s Foxholes function

See Table 10.

$$\begin{aligned}F_{14}(z)=& \bigg (\dfrac{1}{500} + \sum _{j=1}^{25}\dfrac{1}{j+\sum _{i=1}^2(z_i - a_{ij})^6}\bigg )^{-1}\\&-65.536 \le z_i \le 65.536, \quad f_{\min } \approx 1, \quad {\mathrm{Dim}} = 2 \end{aligned}$$

Table 10 Shekel’s Foxholes function $F_{14}$

1.3.2 Kowalik’s function

$$\begin{aligned}F_{15}(z)=& \sum _{i=1}^{11}\bigg [a_i - \dfrac{z_1(b_i^2+b_iz_2)}{b_i^2+b_iz_3+z_4}\bigg ]^2\\&\quad -5 \le z_i \le 5, \quad f_{\min } \approx 0.0003075, \quad {\mathrm{Dim}} = 4 \end{aligned}$$

1.3.3 Six-hump camel-back function

$$\begin{aligned}F_{16}(z)&= 4z_1^2 - 2.1z_1^4 + \dfrac{1}{3}z_1^6 + z_1z_2 - 4z_2^2 + 4z_2^4\\&\quad -5 \le z_i \le 5, \quad f_{\min } = -1.0316285, \quad {\mathrm{Dim}} = 2 \end{aligned}$$

1.3.4 Branin function

$$\begin{aligned}F_{17}(z)=& \bigg (z_2 - \dfrac{5.1}{4\pi ^2}z_1^2 + \dfrac{5}{\pi }z_1 - 6 \bigg )^2 \\&\quad +\, 10\bigg (1 - \dfrac{1}{8\pi }\bigg )\cos z_1 + 10\\&\quad -5 \le z_1 \le 10, \quad 0 \le z_2 \le 15, \quad f_{\min } = 0.398, \quad {\mathrm{Dim}} = 2 \end{aligned}$$

1.3.5 Goldstein–Price function

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

1.3.6 Hartman’s family

See Table 11.

• $$\begin{aligned}&F_{19}(z)= -\sum _{i=1}^4c_i \exp \left( -\sum _{j=1}^3 a_{ij}(z_j - p_{ij})^2\right) \\&0 \le z_j \le 1, \quad f_{\min } = -3.86, \quad {\mathrm{Dim}} = 3 \end{aligned}$$
• $$\begin{aligned}&F_{20}(z)= -\sum _{i=1}^4c_i \exp \left( -\sum _{j=1}^6 a_{ij}(z_j - p_{ij})^2\right) \\&0 \le z_j \le 1, \quad f_{\min } = -3.32, \quad {\mathrm{Dim}} = 6 \end{aligned}$$

Table 11 Hartman function $F_{19}$

1.3.7 Shekel’s Foxholes function

See Tables 12 and 13.

• $$\begin{aligned}&F_{21}(z)= -\sum _{i=1}^5\left[ (X - a_i)(X - a_i)^{\mathrm{T}} + c_i\right] ^{-1}\\&0 \le z_i \le 10, \quad f_{\min } = -10.1532, \quad {\mathrm{Dim}} = 4 \end{aligned}$$
• $$\begin{aligned}&F_{22}(z)= -\sum _{i=1}^7\left[ (X - a_i)(X - a_i)^{\mathrm{T}} + c_i\right] ^{-1}\\&0 \le z_i \le 10, \quad f_{\min } = -10.4028, \quad {\mathrm{Dim}} = 4 \end{aligned}$$
• $$\begin{aligned}&F_{23}(z)= -\sum _{i=1}^{10}\left[ (X - a_i)(X - a_i)^{\mathrm{T}} + c_i\right] ^{-1}\\&0 \le z_i \le 10, \quad f_{\min } = -10.536, \quad {\mathrm{Dim}} = 4 \end{aligned}$$

Table 12 Shekel Foxholes functions $F_{21}, F_{22}, F_{23}$

Table 13 Hartman function $F_{20}$