Metaheuristic Algorithms Based on Fuzzy Logic

Erik Cuevas; Daniel Zaldívar; Marco Pérez-Cisneros

doi:10.1007/978-3-319-89309-9_8

Advances in Metaheuristics Algorithms: Methods and Applications pp 167-218 | Cite as

Metaheuristic Algorithms Based on Fuzzy Logic

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Erik CuevasEmail author
Daniel Zaldívar
Marco Pérez-Cisneros

Chapter

First Online: 11 April 2018

Part of the Studies in Computational Intelligence book series (SCI, volume 775)

Abstract

Several systems are extremely complicated to be handled quantitatively. In spite of this humans undergo them by using simplistic rules that are obtained from their own experiences.

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Appendix: List of Benchmark Functions

See Tables 8.17, 8.18 and 8.19.

Table 8.17

Unimodal test functions used in the experimental study

Function	S	Dim	Minimum
\( f_{1} ({\mathbf{x}}) = \sum\nolimits_{i = 1}^{n} {x_{i}^{2} } \)	\( \left[ { - 100{,}100} \right]^{n} \)	\( n = 50 \) \( n = 100 \)	\( \begin{aligned} & {\mathbf{x}}^{} = (0, \ldots ,0); \\ & f({\mathbf{x}}^{} ) = 0 \end{aligned} \)
\( f_{2} ({\mathbf{x}}) = \sum\nolimits_{i = 1}^{n} {\left\| {x_{i} } \right\| + \prod\nolimits_{i = 1}^{n} {\left\| {x_{i} } \right\|} } \)	\( \left[ { - 10{,}10} \right]^{n} \)	\( n = 50 \) \( n = 100 \)	\( \begin{aligned} & {\mathbf{x}}^{} = (0, \ldots ,0); \\ & f({\mathbf{x}}^{} ) = 0 \end{aligned} \)
\( f_{3} ({\mathbf{x}}) = \sum\nolimits_{i = 1}^{n} {\left( {\sum\nolimits_{j = 1}^{i} {x_{j} } } \right)}^{2} \)	\( \left[ { - 100{,}100} \right]^{n} \)	\( n = 50 \) \( n = 100 \)	\( \begin{aligned} & {\mathbf{x}}^{} = (0, \ldots ,0); \\ & f({\mathbf{x}}^{} ) = 0 \end{aligned} \)
\( f_{4} ({\mathbf{x}}) = \mathop {\hbox{max} }\nolimits_{i} \left\{ {\left\| {x_{i} } \right\|, \, 1 \le i \le n} \right\} \)	\( \left[ { - 100{,}100} \right]^{n} \)	\( n = 50 \) \( n = 100 \)	\( \begin{aligned} & {\mathbf{x}}^{} = (0, \ldots ,0); \\ & f({\mathbf{x}}^{} ) = 0 \end{aligned} \)
\( f_{5} ({\mathbf{x}}) = \sum\nolimits_{i = 1}^{n - 1} {\left[ {100(x_{i + 1} - x_{i}^{2} )^{2} + (x_{i} - 1)^{2} } \right]} \)	\( \left[ { - 30{,}30} \right]^{n} \)	\( n = 50 \) \( n = 100 \)	\( \begin{aligned} & {\mathbf{x}}^{} = (1, \ldots ,1); \\ & f({\mathbf{x}}^{} ) = 0 \end{aligned} \)
\( f_{6} ({\mathbf{x}}) = \sum\nolimits_{i = 1}^{n} {\left( {\left\lfloor {x_{i} + 0.5} \right\rfloor } \right)^{2} } \)	\( \left[ { - 100{,}100} \right]^{n} \)	\( n = 50 \) \( n = 100 \)	\( \begin{aligned} & {\mathbf{x}}^{} = (0, \ldots ,0); \\ & f({\mathbf{x}}^{} ) = 0 \end{aligned} \)
\( f_{7} ({\mathbf{x}}) = \sum\nolimits_{i = 1}^{n} {ix_{i}^{4} + random\left( {0{,}1} \right)} \)	\( \left[ { - 1.28{,}1.28} \right]^{n} \)	\( n = 50 \) \( n = 100 \)	\( \begin{aligned} & {\mathbf{x}}^{} = (0, \ldots ,0); \\ & f({\mathbf{x}}^{} ) = 0 \end{aligned} \)

Table 8.18

Multimodal test functions used in the experimental study

\( f_{8} ({\mathbf{x}}) = \sum\nolimits_{i = 1}^{n} { - x_{i} \sin \left( {\sqrt {\left\| {x_{i} } \right\|} } \right)} \)	\( \left[ { - 500{,}500} \right]^{n} \)	\( n = 50 \) \( n = 100 \)	\( \begin{aligned} & {\mathbf{x}}^{} = (420, \ldots ,420); \\ & f({\mathbf{x}}^{} ) = - 418.9829 \times n \end{aligned} \)
\( f_{9} ({\mathbf{x}}) = \sum\nolimits_{i = 1}^{n} {\left[ {x_{i}^{2} - 10\cos (2\pi x_{i} ) + 10} \right]} \)	\( \left[ { - 5.12{,}5.12} \right]^{n} \)	\( n = 50 \) \( n = 100 \)	\( \begin{aligned} & {\mathbf{x}}^{} = (0, \ldots ,0); \\ & f({\mathbf{x}}^{} ) = 0 \end{aligned} \)
\( \begin{aligned} f_{10} ({\mathbf{x}}) & = - 20\exp \left( { - 0.2\sqrt {\frac{1}{n}\sum\nolimits_{i = 1}^{n} {x_{i}^{2} } } } \right) \\ & \quad - \exp \left( {\frac{1}{n}\sum\nolimits_{i = 1}^{n} {\cos \left( {2\pi x_{i} } \right)} } \right) + 20 + \exp \end{aligned} \)	\( \left[ { - 32{,}32} \right]^{n} \)	\( n = 50 \) \( n = 100 \)	\( \begin{aligned} & {\mathbf{x}}^{} = (0, \ldots ,0); \\ & f({\mathbf{x}}^{} ) = 0 \end{aligned} \)
\( f_{11} \left( {\mathbf{x}} \right) = \frac{1}{4000}\sum\nolimits_{i = 1}^{n} {x_{i}^{2} } - \prod\nolimits_{i = 1}^{n} {\cos \left( {\frac{{x_{i} }}{\sqrt i }} \right) + 1} \)	\( \left[ { - 600{,}600} \right]^{n} \)	\( n = 50 \) \( n = 100 \)	\( \begin{aligned} & {\mathbf{x}}^{} = (0, \ldots ,0); \\ & f({\mathbf{x}}^{} ) = 0 \end{aligned} \)
\( \begin{aligned} f_{{12}} ({\mathbf{x}}) & = \frac{\pi }{n}\left\{ {10\sin (\pi y_{1} ) + \sum\nolimits_{{i = 1}}^{{n - 1}} {(y_{i} - 1)^{2} \left[ {1 + 10\sin ^{2} (\pi y_{{i + 1}} )} \right]} } \right. \\ & \quad \left. { + (y_{n} - 1)^{2} } \right\} + \sum\nolimits_{{i = 1}}^{n} {u(x_{i} ,10,100,4)} \\ \end{aligned} \) \( y_{i} = 1 + \frac{{\left( {x_{i} + 1} \right)}}{4} \) \( u\left( {x_{i} ,a,k,m} \right) = \left\{ {\begin{array}{{20}l} {k\left( {x_{i} - a} \right)^{m} } \\ 0 \\ {k\left( { - x_{i} - a} \right)^{m} } \\ \end{array} \begin{array}{{20}l} {x_{i} > a} \\ { - a \le x_{i} \le a} \\ {x_{i} < a} \\ \end{array} } \right. \)	\( \left[ { - 50{,}50} \right]^{n} \)	\( n = 50 \) \( n = 100 \)	\( \begin{aligned} & {\mathbf{x}}^{} = (0, \ldots ,0); \\ & f({\mathbf{x}}^{} ) = 0 \end{aligned} \)
\( \begin{aligned} f_{{13}} ({\mathbf{x}}) & = 0.1\left\{ {\sin ^{2} \left( {3\pi x_{1} } \right) + \sum\nolimits_{{i = 1}}^{n} {\left( {x_{i} - 1} \right)^{2} } } \right. \\ & \left. { \quad \left[ {1 + \sin ^{2} \left( {3\pi x_{i} + 1} \right)} \right] + \left( {x_{n} - 1} \right)^{2} \left[ {1 + \sin ^{2} \left( {2\pi x_{n} } \right)} \right]} \right\} \\ & \quad + \sum\nolimits_{{i = 1}}^{n} {u\left( {x_{i} ,5,100,4} \right)} ; \\ \end{aligned} \) \( u\left( {x_{i} ,a,k,m} \right) = \left\{ {\begin{array}{*{20}l} {k\left( {x_{i} - a} \right)^{m} } & {x_{i} > a} \\ 0 & { - a < x_{i} < a} \\ {k\left( { - x_{i} - a} \right)^{m} } & {x_{i} < - a} \\ \end{array} } \right. \)	\( \left[ { - 10{,}10} \right]^{n} \)	\( n = 50 \) \( n = 100 \)	\( \begin{aligned} & {\mathbf{x}}^{} = (1, \ldots ,1); \\ & f({\mathbf{x}}^{} ) = 0 \end{aligned} \)
\( f_{14} \left( x \right) = \frac{1}{2}\sum\nolimits_{i = 1}^{n} {\left( {x_{i}^{4} - 16x_{i}^{2} + 5x_{i} } \right)} \)	\( \left[ { - 5{,}5} \right]^{n} \)	\( n = 50 \) \( n = 100 \)	\( \begin{aligned} & {\mathbf{x}}^{} = ( - 2.90, \ldots , - 2.90); \\ & f({\mathbf{x}}^{} ) = - 39.16599 \times n \end{aligned} \)

Table 8.19

Hybrid test functions used in the experimental study

Function	S	Dim	Minimum
\( f_{15} ({\mathbf{x}}) = f_{1} ({\mathbf{x}}) + f_{2} ({\mathbf{x}}) + f_{9} ({\mathbf{x}}) \)	\( \left[ { - 100{,}100} \right]^{n} \)	\( n = 50 \) \( n = 100 \)	\( \begin{aligned} & {\mathbf{x}}^{} = (0, \ldots ,0); \\ & f({\mathbf{x}}^{} ) = 0 \end{aligned} \)
\( f_{16} ({\mathbf{x}}) = f_{9} ({\mathbf{x}}) + f_{5} ({\mathbf{x}}) + f_{11} ({\mathbf{x}}) \)	\( \left[ { - 100{,}100} \right]^{n} \)	\( n = 50 \) \( n = 100 \)	\( \begin{aligned} & {\mathbf{x}}^{} = (0, \ldots ,0); \\ & f({\mathbf{x}}^{} ) = n - 1 \end{aligned} \)
\( f_{17} ({\mathbf{x}}) = f_{3} ({\mathbf{x}}) + f_{5} ({\mathbf{x}}) + f_{10} ({\mathbf{x}}) + f_{13} ({\mathbf{x}}) \)	\( \left[ { - 100{,}100} \right]^{n} \)	\( n = 50 \) \( n = 100 \)	\( \begin{aligned} & {\mathbf{x}}^{} = (0, \ldots ,0); \\ & f({\mathbf{x}}^{} ) = (1.1 \times n) - 1 \end{aligned} \)
\( f_{18} ({\mathbf{x}}) = f_{2} ({\mathbf{x}}) + f_{5} ({\mathbf{x}}) + f_{9} ({\mathbf{x}}) + f_{10} ({\mathbf{x}}) + f_{11} ({\mathbf{x}}) \)	\( \left[ { - 100{,}100} \right]^{n} \)	\( n = 50 \) \( n = 100 \)	\( \begin{aligned} & {\mathbf{x}}^{} = (0, \ldots ,0); \\ & f({\mathbf{x}}^{} ) = n - 1 \end{aligned} \)
\( f_{19} ({\mathbf{x}}) = f_{1} ({\mathbf{x}}) + f_{2} ({\mathbf{x}}) + f_{8} ({\mathbf{x}}) + f_{10} ({\mathbf{x}}) + f_{12} ({\mathbf{x}}) \)	\( \left[ { - 100{,}100} \right]^{n} \)	\( n = 50 \) \( n = 100 \)	\( \begin{aligned} &{\mathbf{x}}^{} = (1, \ldots ,1); \\ & f({\mathbf{x}}^{} ) = 0 \end{aligned} \)

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Erik Cuevas
- 1
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Daniel Zaldívar
- 1
Marco Pérez-Cisneros
- 1

1.CUCEIUniversidad de GuadalajaraGuadalajaraMexico

Metaheuristic Algorithms Based on Fuzzy Logic

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