Modified forensic-based investigation algorithm for global optimization

Abstract

Forensic-based investigation (FBI) is recently developed metaheuristic algorithm inspired by the suspect investigation–location–pursuit operations of police officers. This study focuses on the search processes of the FBI algorithm, called Step A and Step B, to improve and increase its performance. For this purpose, opposition-based learning is adopted to Step A to enhance diversity, while Cauchy-based mutation is integrated with Step B to guide the search to different regions and to jump out of local minima. To show the effectiveness of these improvements, the proposed algorithm has been tested with two different benchmark sets. To verify the performance of the new modified algorithm, the statistical test is carried out on numerical functions. This study also investigates the application of the proposed algorithm to a set of six real-world problems. The proposed and adapted/integrated methods appear to have a significant impact on the FBI algorithm, which augments its performance, resulting in better solutions than the compared algorithms in most of the functions and real-world problems.

Appendix

Equations	Dim	Range	Min
$f_{1} \left( x \right) = \sum\nolimits_{i = 1}^{n} {x_{i}^{2} }$	$50$	$\left[ { - 100, + 100} \right]$	$0$
$f_{2} \left( x \right) = \sum\nolimits_{i = 1}^{n} {\left| {x_{i} } \right|} + \Pi_{i = 1}^{n} \left| {x_{i} } \right|$	$50$	$\left[ { - 10, + 10} \right]$	$0$
$f_{3} \left( x \right) = \sum\nolimits_{i = 1}^{n} {\left( {\sum\nolimits_{j - 1}^{i} {x_{j} } } \right)^{2} }$	$50$	$\left[ { - 100, + 100} \right]$	$0$
$f_{4} \left( x \right) = {\text{max}}_{i} \left\{ {\left| {x_{i} } \right|, 1 \le i \le n} \right\}$	$50$	$\left[ { - 100, + 100} \right]$	$0$
$f_{5} \left( x \right) = \sum\nolimits_{i = 1}^{n - 1} {\left[ {100\left( {x_{i + 1} - x_{i}^{2} } \right)^{2} + \left( {x_{i} - 1} \right)^{2} } \right]}$	$50$	$\left[ { - 30, + 30} \right]$	$0$
$f_{6} \left( x \right) = \sum\nolimits_{i = 1}^{n} {\left( {\left[ {x_{i} + 0.5} \right]} \right)^{2} }$	$50$	$\left[ { - 100, + 100} \right]$	$0$
$f_{7} \left( x \right) = \sum\nolimits_{i = 1}^{n} {ix_{i}^{4} + {\text{random}}\left[ {0,1} \right)}$	$50$	$\left[ { - 1.28, + 1.28} \right]$	$0$
$f_{8} \left( x \right) = \sum\nolimits_{i = 1}^{n} { - x_{i} {\text{sin}}\left( {\sqrt {\left| {x_{i} } \right|} } \right)}$	$50$	$\left[ { - 500, + 500} \right]$	$- 418.9829X{\text{dim}}$
$f_{9} \left( x \right) = \sum\nolimits_{i = 1}^{n} {\left[ {x_{i}^{2} - 10\cos \left( {2\pi x_{i} } \right) + 10} \right]}$	$50$	$\left[ { - 5.12, + 5.12} \right]$	$0$
$f_{10} \left( x \right) = - 20 {\text{exp}}\left( { - 0.2\sqrt {\frac{1}{n}\sum\nolimits_{i = 1}^{n} {x_{i}^{2} } } } \right) - {\text{exp}}\left( {\frac{1}{n}\sum\nolimits_{i = 1}^{n} {\cos \left( {2\pi x_{i} } \right)} } \right) + 20 + e$	$50$	$\left[ { - 32, + 32} \right]$	$0$
$f_{11} \left( x \right) = \frac{1}{4000}\sum\nolimits_{i = 1}^{n} {x_{i}^{2} } - \prod_{i = 1}^{n} \cos \left( {\frac{{x_{i} }}{\sqrt i }} \right) + 1$	$50$	$\left[ { - 600, + 600} \right]$	$0$
$\begin{gathered} f_{12} \left( x \right) = \frac{\pi }{n}\left\{ {10\sin \left( {\pi y_{1} } \right) + \sum\nolimits_{i = 1}^{n - 1} {\left( {y_{i} - 1} \right)^{2} } \left[ {1 + 10\sin^{2} \left( {\pi y_{i + 1} } \right)} \right] + y_{n} - 1^{2} } \right\} \hfill \\ + \sum\nolimits_{i = 1}^{n} {u\left( {x_{i} , 10, 100, 4} \right)} \hfill \\ \end{gathered}$ $y_{i} = 1 + \frac{{x_{i} + 1}}{4}u\left( {x_{i} , a, k, m} \right) = \left\{ {\begin{array}{*{20}c} {k(x_{i} - a)^{m} x_{i} > a} \\ {0 - a < x_{i} < a} \\ {k( - x_{i} - a)^{m} x_{i} < - a} \\ \end{array} } \right.$	$50$	$\left[ { - 50, + 50} \right]$	$0$
$\begin{gathered} f_{13} \left( x \right) = 0.1\left\{ {\sin^{2} \left( {3\pi x_{1} } \right) + \sum\nolimits_{i = 1}^{n} {\left( {x_{i} - 1} \right)^{2} } \left[ {1 + \sin^{2} \left( {3\pi x_{i} + 1} \right)} \right] + \left( {x_{n} - 1} \right)^{2} \left[ {1 + \sin^{2} \left( {2\pi x_{n} } \right)} \right]} \right\} \hfill \\ + \sum\nolimits_{i = 1}^{n} {u\left( {x_{i} , 5, 100, 4} \right)} \hfill \\ \end{gathered}$	$50$	$\left[ { - 50, + 50} \right]$	$0$
$f_{14} \left( x \right) = \left( {\frac{1}{500} + \sum\nolimits_{j = 1}^{25} {\frac{1}{{j + \mathop \sum \nolimits_{i = 1}^{2} \left( {x_{i} - a_{ij} } \right)^{6} }}} } \right)^{ - 1}$	2	$\left[ { - 65.536, + 65.536} \right]$	1
$f_{15} \left( x \right) = \sum\nolimits_{i = 1}^{11} {\left| {a_{i} - \frac{{x_{{1\left( {b_{i}^{2} + b_{i} x_{2} } \right)}} }}{{b_{i}^{2} + b_{i} x_{3} + x_{4} }}} \right|^{2} }$	4	$\left[ { - 5, + 5} \right]$	0.0003
$f_{16} \left( x \right) = 4x_{1}^{2} - 2.1x_{1}^{4} + \frac{1}{3}x_{1}^{6} + x_{1} x_{2} - 4x_{2}^{2} + 4x_{2}^{4}$	2	$\left[ { - 5, + 5} \right]$	− 1.0316
$f_{17} \left( x \right) = \left( {x_{2} - \frac{5.1}{{4\pi^{2} }}x_{1}^{2} + \frac{5}{\pi }x_{1} - 6} \right)^{2} + 10\left( {1 - \frac{1}{8\pi }} \right)\cos x_{1} + 10$	2	$\left[ { - 5, + 0} \right]$ $\left[ {10,15} \right]$	0.398
$f_{18} \left( x \right) = \left[ {1 + \left( {x_{1} + x_{2} + 1} \right)^{2} \left( {19 - 14x_{1} + 3x_{1}^{2} - 14x_{2} + 6x_{1} x_{2} + 3x_{2}^{2} } \right)} \right] \times \left[ {30 + \left( {2x_{1} - 3x_{2} } \right)^{2} \times \left( {18 - 32x_{1} + 12x_{1}^{2} + 48x_{2} - 36x_{1} x_{2} + 27x_{2}^{2} } \right)} \right]$	2	$\left[ { - 2, + 2} \right]$	3
$f_{19} \left( x \right) = - \sum\nolimits_{i = 1}^{4} {c_{i } {\text{exp}}\left( { - \mathop \sum \limits_{j = 1}^{3} a_{ij} (x_{j} - p_{ij} )^{2} } \right)}$	3	$\left[ {0, 1} \right]$	− 3.86
$f_{20} \left( x \right) = - \sum\nolimits_{i = 1}^{4} {c_{i } {\text{exp}}\left( { - \mathop \sum \limits_{j = 1}^{6} a_{ij} (x_{j} - p_{ij} )^{2} } \right)}$	6	$\left[ {0, 10} \right]$	− 3.32
$f_{21} \left( x \right) = - \sum\nolimits_{i = 1}^{5} {\left[ {\left( {X - a_{i} } \right)\left( {X - a_{i} } \right)^{T} + c_{i} } \right]^{ - 1} }$	4	$\left[ {0, 10} \right]$	− 10.1532
$f_{22} \left( x \right) = - \sum\nolimits_{i = 1}^{7} {\left[ {\left( {X - a_{i} } \right)\left( {X - a_{i} } \right)^{T} + c_{i} } \right]^{ - 1} }$	4	$\left[ {0, 10} \right]$	− 10.4028
$f_{23} \left( x \right) = - \sum\nolimits_{i = 1}^{10} {\left[ {\left( {X - a_{i} } \right)\left( {X - a_{i} } \right)^{T} + c_{i} } \right]^{ - 1} }$	4	$\left[ {0, 10} \right]$	− 10.5363