A conceptual investigation of the effect of random numbers over the performance of metaheuristic algorithms

Abstract

A lot of research studies focus on the development of a new algorithm or the techniques which improve the performance of the original algorithm. Very few studies conduct the research on the effect of the initial population on the solution quality of algorithms. However, in these studies, one or two algorithms have been used, and a limited number of problems have been handled. To fill in the gap in the literature, this study presents a comprehensive analysis of the five algorithms on the effect of the initial population on their final results including both the numerical and real-world problems along with a wide variety of types of distributions. The study consisted of three rounds and followed the strategy for determining the candidate algorithms to be participated in the next rounds, supported by the statistical tests. Rather than using popular random numbers, fourteen different distributions are used to imitate the random numbers in the initial population generation mechanisms of the algorithms. Two different numerical benchmark sets along with nine real-world problems are used to evaluate the performance of the algorithms. The results are compared with the original ones and other distribution-integrated algorithms. Since knowledge of the appropriate random number source is not available a priori, this study could be a good foundation for future studies not only on the matter of the effect of several distributions on the performances of the algorithms but also introducing an alternative way in generating an initial population.

Appendices

Appendix 1: Benchmark set-1 used in this study

	Name	Equations	Dim	Range	Min
Unimodal	Sphere	$F_{1} \left( x \right) = \mathop \sum \limits_{i = 1}^{n} x_{i}^{2}$	$50$	$\left[ { - 100, 100} \right]$	$0$
	Schwefel 2.22	$F_{2} \left( x \right) = \mathop \sum \limits_{i = 1}^{n} \left| {x_{i} } \right| + \mathop \prod \limits_{i = 1}^{n} \left| {x_{i} } \right|$	$50$	$\left[ { - 10, 10} \right]$	$0$
	Schwefel 1.2	$F_{3} \left( x \right) = \mathop \sum \limits_{i = 1}^{n} \left\{ {\mathop \sum \limits_{j = 1}^{i} x_{j} } \right\}^{2}$	$50$	$\left[ { - 100, 100} \right]$	$0$
	Schwefel 2.21	$F_{4} \left( x \right) = \max_{i} \left\{ {\left| {x_{i} } \right|, 1 \le i \le n} \right\}$	$50$	$\left[ { - 100, 100} \right]$	$0$
	Rosenbrock	$F_{5} \left( x \right) = \mathop \sum \limits_{i = 1}^{n - 1} \left[ {100\left( {x_{i + 1} - x_{i}^{2} } \right)^{2} + \left( {x_{i} - 1} \right)^{2} } \right]$	$50$	$\left[ { - 30, 30} \right]$	$0$
	Step	$F_{6} \left( x \right) = \sum\limits_{i = 1}^{n} {\left\{ {\left[ {x_{i} + 0.5} \right]} \right\}}^{2}$	$50$	$\left[ { - 100, 100} \right]$	$0$
	Quartic	$F_{7} \left( x \right) = \mathop \sum \limits_{i = 1}^{n} ix_{i}^{4} + {\text{random}}\left[ {0,1} \right)$	$50$	$\left[ { - 1.28, 1.28} \right]$	$0$
Multimodal	Schwefel	$F_{8} \left( x \right) = \mathop \sum \limits_{i = 1}^{n} - x_{i} {\text{sin}}\left( {\sqrt {\left| {x_{i} } \right|} } \right)$	$50$	$\left[ { - 500, + 500} \right]$
	Rastrigin	$F_{9} \left( x \right) = \mathop \sum \limits_{i = 1}^{n} \left[ {x_{i}^{2} - 10\cos \left( {2\pi x_{i} } \right) + 10} \right]$	$50$	$\left[ { - 5.12, + 5.12} \right]$	$0$
	Ackley	$F_{10} \left( x \right) = - 20 \exp \left( { - 0.2\sqrt {\frac{1}{n}\mathop \sum \limits_{i = 1}^{n} x_{i}^{2} } } \right) - \exp \left( {\frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \cos \left( {2\pi x_{i} } \right)} \right) + 20 + e$	$50$	$\left[ { - 32, + 32} \right]$	$0$
	Griewank	$F_{11} \left( x \right) = \frac{1}{4000}\mathop \sum \limits_{i = 1}^{n} x_{i}^{2} - \mathop \prod \limits_{i = 1}^{n} \cos \left( {\frac{{x_{i} }}{\sqrt i }} \right) + 1$	$50$	$\left[ { - 600, + 600} \right]$	$0$
	Penalized-1	$F_{12} \left( x \right) = \frac{\pi }{n}\left\{ {10\sin \left( {\pi y_{1} } \right) + \mathop \sum \limits_{i = 1}^{n - 1} \left( {y_{i} - 1} \right)^{2} \left[ {1 + 10\sin^{2} \left( {\pi y_{i + 1} } \right)} \right] + \left( {y_{n} - 1} \right)^{2} } \right\} + \mathop \sum \limits_{i = 1}^{n} u\left( {x_{i} , 10, 100, 4} \right)$ $y_{i} = 1 + \frac{{x_{i} + 1}}{4} , u\left( {x_{i} , a, k, m} \right) = \left\{ {\begin{array}{*{20}c} {k(x_{i} - a)^{m} } & , & {x_{i} > a} \\ 0 & , & { - a < x_{i} < a} \\ {k( - x_{i} - a)^{m} } & , & {x_{i} < - a} \\ \end{array} } \right.$	$50$	$\left[ { - 50, + 50} \right]$	$0$
	Penalized-2	$F_{13} \left( x \right) = 0.1\left\{ {\sin^{2} \left( {3\pi x_{1} } \right) + \mathop \sum \limits_{i = 1}^{n} \left( {x_{i} - 1} \right)^{2} \left[ {1 + \sin^{2} \left( {3\pi x_{i} + 1} \right)} \right] + \left( {x_{n} - 1} \right)^{2} \left[ {1 + \sin^{2} \left( {2\pi x_{n} } \right)} \right]} \right\} + \mathop \sum \limits_{i = 1}^{n} u\left( {x_{i} , 5, 100, 4} \right)$	$50$	$\left[ { - 50, + 50} \right]$	$0$

Appendix 2: Benchmark set-2 used in this study (CEC2013)

	Name	Equation	${\varvec{F}}_{{\varvec{i}}}^{\user2{*}}$
Unimodal	Sphere	$F_{1} \left( x \right) = \mathop \sum \limits_{i = 1}^{D} z_{i}^{2} + F_{1}^{*} ,\;z = x - o$	− 1400
	Rotated High Conditioned Elliptic	$F_{2} \left( x \right) = \mathop \sum \limits_{i = 1}^{D} \left( {10^{6} } \right)^{{\frac{i - 1}{{D - 1}}}} z_{i}^{2} + F_{2}^{*} ,\;z = T_{osz} \left\{ {M_{1} \left( {x - o} \right)} \right\}$	− 1300
	Rotated bent cigar	$F_{3} \left( x \right) = z_{1}^{2} + 10^{6} \mathop \sum \limits_{i = 2}^{D} z_{i}^{2} + F_{3}^{*} ,\;z = M_{2} T_{asy}^{0.5} \left\{ {M_{1} \left( {x - o} \right)} \right\}$	− 1200
	Rotated discus	$F_{4} \left( x \right) = 10^{6} z_{1}^{2} + \mathop \sum \limits_{i = 2}^{D} z_{i}^{2} + F_{4}^{*} ,\;z = T_{{{\text{osz}}}} \left\{ {M_{1} \left( {x - o} \right)} \right\}$	− 1100
	Different powers	$F_{5} \left( x \right) = \sqrt {\mathop \sum \limits_{i = 1}^{D} \left| {z_{i} } \right|^{{2 + 4\frac{i - 1}{{D - 1}}}} } + F_{5}^{*} ,\;z = x - o$	− 1000
Multimodal	Rotated Rosenbrock's	$\begin{aligned} F_{6} \left( x \right) = & \;\mathop \sum \limits_{i = 1}^{D - 1} \left\{ {100\left( {z_{i}^{2} - z_{i - 1}^{2} } \right)^{2} + \left( {z_{i} - 1} \right)^{2} } \right\} + F_{6}^{*} , \\ z = & \;M_{1} \left( {\frac{{2.048\left( {x - o} \right)}}{100}} \right) + 1 \\ \end{aligned}$	− 900
	Rotated Schaffers F7	$\begin{aligned} F_{7} \left( x \right) = & \;\left\{ {\frac{1}{{D - 1}}\mathop \sum \limits_{{i = 1}}^{{D - 1}} \left[ {\sqrt {z_{i} } + \sqrt {z_{i} } \sin ^{2} \left( {50\sqrt[5]{{z_{i} }}} \right)} \right]} \right\}^{2} \\ & \; + F_{7}^{*} ,\;\begin{array}{*{20}c} {z_{i} = \sqrt {y_{i}^{2} + y_{{i + 1}}^{2} } ~~,~~i = 1, \ldots ,D} \\ {y = \Lambda ^{{10}} M_{2} T_{{{\text{asy}}}}^{{0.5}} \left\{ {M_{1} \left( {x - o} \right)} \right\}} \\ \end{array} \\ \end{aligned}$	− 800
	Rotated Ackley's	$\begin{aligned} F_{8} \left( x \right) = & \; - 20e^{{ - 0.2\sqrt {\frac{1}{D}\mathop \sum \limits_{i = 1}^{D} z_{i}^{2} } }} - e^{{\frac{1}{D}\mathop \sum \limits_{i = 1}^{D} \cos \left( {2\pi z_{i} } \right)}} + 20 + e + F_{8}^{*} , \\ z = & \;\Lambda^{10} M_{2} T_{asy}^{0.5} \left\{ {M_{1} \left( {x - o} \right)} \right\} \\ \end{aligned}$	− 700
	Rotated Weierstrass	$\begin{aligned} F_{9} \left( x \right) = & \;\mathop \sum \limits_{i = 1}^{D} \left\{ {\mathop \sum \limits_{k = 0}^{{k_{\max } }} \left[ {a^{k} \cos \left( {2\pi b^{k} \left( {z_{i} + 0.5} \right)} \right)} \right]} \right\} \\ & \; - D\mathop \sum \limits_{k = 0}^{{k_{\max } }} \left[ {a^{k} \cos \left( {\pi b^{k} } \right)} \right] + F_{9}^{*} ,\;\frac{{a = 0.5 , b = 3 , k_{\max } = 20}}{{z = \Lambda^{10} M_{2} T_{asy}^{0.5} \left\{ {M_{1} \frac{{0.5\left( {x - o} \right)}}{100}} \right\}}} \\ \end{aligned}$	− 600
	Rotated Griewank’s	$F_{10} \left( x \right) = \mathop \sum \limits_{i = 1}^{D} \frac{{z_{i}^{2} }}{4000} - \mathop \prod \limits_{i = 1}^{D} \cos \left( {\frac{{z_{i} }}{\sqrt i }} \right) + 1 + F_{10}^{*} ,\;z = \Lambda^{100} M_{1} \frac{{600\left( {x - o} \right)}}{100}$	− 500
	Rastrigin’s	$\begin{aligned} F_{11} \left( x \right) = & \;\mathop \sum \limits_{i = 1}^{D} \left\{ {z_{i}^{2} - 10\cos \left( {2\pi z_{i} } \right) + 10} \right\} + F_{11}^{*} , \\ z = & \;\Lambda^{10} T_{{{\text{asy}}}}^{0.2} \left\{ {T_{{{\text{osz}}}} \left( {\frac{{5.12\left( {x - o} \right)}}{100}} \right)} \right\} \\ \end{aligned}$	− 4000
	Rotated Rastrigin’s	$\begin{aligned} F_{12} \left( x \right) = & \;\mathop \sum \limits_{i = 1}^{D} \left\{ {z_{i}^{2} - 10\cos \left( {2\pi z_{i} } \right) + 10} \right\} + F_{12}^{*} , \\ z = & \;M_{1} \Lambda^{10} M_{2} T_{{{\text{asy}}}}^{0.2} \left\{ {T_{{{\text{osz}}}} \left( {M_{1} \frac{{5.12\left( {x - o} \right)}}{100}} \right)} \right\} \\ \end{aligned}$	− 300
	Non-Continuous Rotated Rastrigin's	$F_{13} \left( x \right) = \mathop \sum \limits_{i = 1}^{D} \left\{ {z_{i}^{2} - 10\cos \left( {2\pi z_{i} } \right) + 10} \right\} + F_{13}^{*}$ $\begin{aligned} \hat{x} = & \;M_{1} \frac{{5.12\left( {x - o} \right)}}{100} \\ y_{i} = & \;\left\{ {\begin{array}{*{20}l} {\hat{x}_{i} \quad } \hfill & {\left| {\hat{x}_{i} } \right| \le 0.5} \hfill \\ {{\text{round}}\left( {2\hat{x}_{i} } \right)/2\quad } \hfill & {\left| {\hat{x}_{i} } \right| > 0.5} \hfill \\ \end{array} } \right., \\ z = & \;M_{1} \Lambda^{10} M_{2} T_{{{\text{asy}}}}^{0.2} \left\{ {T_{{{\text{osz}}}} \left( y \right)} \right\} \\ \end{aligned}$	− 200
	Schwefel's	$\begin{aligned} F_{14} \left( z \right) = & \;418.9829D - \mathop \sum \limits_{i = 1}^{D} g\left( {z_{i} } \right) + F_{14}^{*} , \\ z = & \;{\Lambda }^{10} \left( {\frac{{1000\left( {x - o} \right)}}{100}} \right) + 420.9687462275036 \\ \end{aligned}$ $g\left( {z_{i} } \right) = \left\{ {\begin{array}{*{20}l} {\left\{ {\bmod \left( {\left| {z_{i} } \right|,500} \right) - 500} \right\}\sin \left( {\sqrt {\left| {\bmod \left( {\left| {z_{i} } \right|,500} \right) - 500} \right|} } \right) - \frac{{\left( {z_{i} + 500} \right)^{2} }}{10000D},\quad } \hfill & {z_{i} < - 500} \hfill \\ {z_{i} \sin \left( {\sqrt {\left| {z_{i} } \right|} } \right),\quad } \hfill & {\left| {z_{i} } \right| \le 500} \hfill \\ {\left\{ {500 - \bmod \left( {\left| {z_{i} } \right|,500} \right)} \right\}\sin \left( {\sqrt {\left| {500 - \bmod \left( {\left| {z_{i} } \right|,500} \right)} \right|} } \right) - \frac{{\left( {z_{i} - 500} \right)^{2} }}{10000D},\quad } \hfill & {\left| {z_{i} } \right| > 500} \hfill \\ \end{array} } \right.$	− 100
	Rotated Schwefel's	$\begin{aligned} F_{15} \left( z \right) = & \;418.9829D - \mathop \sum \limits_{i = 1}^{D} g\left( {z_{i} } \right) + F_{15}^{*} , \\ z = & \;\Lambda^{10} M_{1} \left( {\frac{{1000\left( {x - o} \right)}}{100}} \right) + 420.9687462275036 \\ \end{aligned}$ $g\left( {z_{i} } \right) = \left\{ {\begin{array}{*{20}c} {\left\{ {\bmod \left( {\left| {z_{i} } \right|,500} \right) - 500} \right\}\sin \left( {\sqrt {\left| {\bmod \left( {\left| {z_{i} } \right|,500} \right) - 500} \right|} } \right) + \frac{{\left( {z_{i} + 500} \right)^{2} }}{10000D}} & , & {z_{i} < - 500} \\ {z_{i} \sin \left( {\sqrt {\left| {z_{i} } \right|} } \right)} & , & {\left| {z_{i} } \right| \le 500} \\ {\left\{ {500 - \bmod \left( {\left| {z_{i} } \right|,500} \right)} \right\}\sin \left( {\sqrt {\left| {500 - \bmod \left( {\left| {z_{i} } \right|,500} \right)} \right|} } \right) + \frac{{\left( {z_{i} + 500} \right)^{2} }}{10000D}} & , & {\left| {z_{i} } \right| > 500} \\ \end{array} } \right.$	100
	Rotated Katsuura	$\begin{aligned} F_{16} \left( x \right) = & \;\frac{10}{{D^{2} }}\mathop \prod \limits_{i = 1}^{D} \left\{ {1 + i\mathop \sum \limits_{j = 1}^{32} \frac{{\left| {2^{j} z_{i} - {\text{round}}\left( {2^{j} z_{i} } \right)} \right|}}{{2^{j} }}} \right\}^{{\frac{10}{{D^{1.2} }}}} - \frac{10}{{D^{2} }} + F_{16}^{*} , \\ z = & \;M_{2} \Lambda^{100} \left( {M_{1} \frac{{5\left( {x - o} \right)}}{100}} \right) \\ \end{aligned}$	200
	Lunacek Bi_Rastrigin	$\begin{aligned} F_{17} \left( x \right) = & \;\min \left\{ {\mathop \sum \limits_{i = 1}^{D} \left( {\hat{x}_{i} - \mu_{0} } \right)^{2} ,dD + s\mathop \sum \limits_{i = 1}^{D} \left( {\hat{x}_{i} - \mu_{1} } \right)^{2} } \right\} \\ & \; + 10\left\{ {D - \mathop \sum \limits_{i = 1}^{D} \cos \left( {2\pi \hat{z}_{i} } \right)} \right\} + F_{17}^{*} \\ \end{aligned}$ $\begin{aligned} \mu_{0} = & \;2.5,\;s = 1 - \frac{1}{{2\sqrt {D + 20} - 8.2}} \\ d = & \;1,\;\mu_{1} = - \sqrt {\frac{{\mu_{0}^{2} - d}}{s}} \\ \end{aligned}$ $y = \frac{{10\left( {x - o} \right)}}{100},\;\hat{x}_{i} = 2sign\left( {x_{i}^{*} } \right)y_{i} + \mu_{0} ,\;z = \Lambda^{100} \left( {\hat{x} - \mu_{0} } \right)$	300
	Rotated Lunacek Bi_Rastrigin	$\begin{aligned} F_{18} \left( x \right) = & \;\min \left\{ {\mathop \sum \limits_{i = 1}^{D} \left( {\hat{x}_{i} - \mu_{0} } \right)^{2} ,{\text{d}}D + s\mathop \sum \limits_{i = 1}^{D} \left( {\hat{x}_{i} - \mu_{1} } \right)^{2} } \right\} \\ & \; + 10\left\{ {D - \mathop \sum \limits_{i = 1}^{D} \cos \left( {2\pi \hat{z}_{i} } \right)} \right\} + F_{18}^{*} \\ \end{aligned}$ $\mu_{0} = 2.5,\;s = 1 - \frac{1}{{2\sqrt {D + 20} - 8.2}},\;d = 1,\;\mu_{1} = - \sqrt {\frac{{\mu_{0}^{2} - d}}{s}}$ $y = \frac{{10\left( {x - o} \right)}}{100},\;\hat{x}_{i} = 2\;{\text{sign}}\;\left( {y_{i}^{*} } \right)y_{i} + \mu_{0} ,\;z = M_{2} \Lambda^{100} \left( {M_{1} \left( {\hat{x} - \mu_{0} } \right)} \right)$	400
	Rotated Expanded Griewank’s plus Rosenbrock's	$F_{19} \left( x \right) = g_{1} \left\{ {g_{2} \left( {z_{1} ,z_{2} } \right)} \right\} + g_{1} \left\{ {g_{2} \left( {z_{2} ,z_{3} } \right)} \right\} + \ldots + g_{1} \left\{ {g_{2} \left( {z_{D - 1} ,z_{D} } \right)} \right\} + g_{1} \left\{ {g_{2} \left( {z_{D} ,z_{1} } \right)} \right\} + F_{19}^{*}$ $\begin{array}{*{20}c} {g_{1} \left( x \right) = \mathop \sum \limits_{i = 1}^{D} \frac{{x_{i}^{2} }}{4000} - \mathop \prod \limits_{i = 1}^{D} \cos \left( {\frac{{x_{i} }}{\sqrt i }} \right) + 1} & , & {g_{2} \left( x \right) = \mathop \sum \limits_{i = 1}^{D - 1} \left\{ {100\left( {x_{i}^{2} - x_{i + 1} } \right)^{2} + \left( {x_{i} } \right)^{2} } \right\}} \\ \end{array}$ $z = M_{1} \left( {\frac{{5\left( {x - o} \right)}}{100}} \right) + 1$	500
	Rotated Expanded Schaffers F6	$F_{20} \left( x \right) = g\left( {z_{1} ,z_{2} } \right) + g\left( {z_{2} ,z_{3} } \right) + \ldots + g\left( {z_{D - 1} ,z_{D} } \right) + g\left( {z_{D} ,z_{1} } \right) + F_{19}^{*}$ $g\left( {x,y} \right) = 0.5 + \frac{{\sin^{2} \left( {\sqrt {x^{2} + y^{2} } } \right) - 0.5}}{{\left\{ {1 + 0.001\left( {x^{2} + y^{2} } \right)} \right\}^{2} }},\;z = M_{2} T_{asy}^{0.5} \left( {M_{1} \left( {x - o} \right)} \right)$	600
Descriptions
	$D$: Dimension, Minimization problem: $\min \left\{ {F\left( x \right)} \right\}$, $x = \left[ {x_{1} ,x_{2} , \ldots ,x_{D} } \right]^{T}$, $i = 1,2, \ldots ,D$, $F_{i}^{*} = F_{i} \left( {x^{*} } \right)$ $o = \left[ {o_{1} ,o_{2} , \ldots ,o_{D} } \right]^{T}$: The shifted global optimum, $M_{1} ,M_{2} , \ldots ,M_{10}$: Orthogonal (rotation) matrix ${\Lambda }^{\alpha }$: Diagonal matrix in $D$ dimensions with the $i$ th diagonal element as $\lambda_{ii} = \alpha^{{\frac{i - 1}{{2\left( {D - 1} \right)}}}}$, $T_{{{\text{asy}}}}^{\beta }$: $\begin{array}{*{20}c} {x_{i} > 0 \Rightarrow x_{i} = x_{i}^{{1 + \beta \frac{i - 1}{{D - 1}}\sqrt {x_{i} } }} } \\ \end{array}$ $T_{osz}$:$x_{i} = {\text{sign}}\left( {x_{i} } \right)e^{{\left\{ {\hat{x}_{i} + 0.049\left[ {\sin \left( {c_{1} \hat{x}_{i} } \right) + \sin \left( {c_{2} \hat{x}_{i} } \right)} \right]} \right\}}} ,\;\left\{ {\begin{array}{*{20}l} {{\text{sign}}\left( {x_{i} } \right) = \left\{ {\begin{array}{*{20}l} { - 1} \hfill & {x_{i} < 0} \hfill \\ 0 \hfill & {x_{i} = 0} \hfill \\ 1 \hfill & {x_{i} > 0} \hfill \\ \end{array} ,\quad } \right.} \hfill & {\hat{x}_{i} = \left\{ {\begin{array}{*{20}l} {\log \left( {\left| {x_{i} } \right|} \right)} \hfill & {x_{i} \ne 0} \hfill \\ 0 \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right.} \hfill \\ {c_{1} = \left\{ {\begin{array}{*{20}l} {10} \hfill & {x_{i} > 0} \hfill \\ {5.5} \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right.,\quad } \hfill & {c_{2} = \left\{ {\begin{array}{*{20}l} {7.9} \hfill & {x_{i} > 0} \hfill \\ {3.1} \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right.} \hfill \\ \end{array} } \right.$

Appendix 3: Real-world problems used in this study

Problem	Objective function	Constraints/Related equations
Parameter estimation of FM sound waves	$F\left( {\vec{X}} \right) = \mathop \sum \limits_{t = 0}^{100} \left\{ {y\left( t \right) - y_{0} \left( t \right)} \right\}^{2}$	$y\left( t \right) = a_{1} \sin \left\{ {\omega_{1} t\theta + a_{2} \sin \left( {\omega_{2} t\theta + a_{3} \sin \left( {\omega_{3} t\theta } \right)} \right)} \right\}$ $y_{0} \left( t \right) = \sin \left\{ {5t\theta - 1.5\sin \left( {4.8t\theta + 2\sin \left( {4.9t\theta } \right)} \right)} \right\}$
Welded beam design problem	$F\left( x \right) = 1.10471x_{1}^{2} x_{2} + 0.04811x_{3} x_{4} \left( {x_{2} + 14} \right)$	$g\left( 1 \right) = \tau \left( x \right) - \tau_{\max } \le 0$ $g\left( 2 \right) = \sigma \left( x \right) - \sigma_{\max } \le 0$ $g\left( 3 \right) = x_{1} - x_{4} \le 0$ $g\left( 4 \right) = 1.10471x_{1}^{2} + 0.04811x_{3} x_{4} \left( {x_{2} + 14} \right) - 5 \le 0$ $g\left( 5 \right) = 0.125 - x_{1} \le 0$ $g\left( 6 \right) = \delta \left( x \right) - \delta_{\max } \le 0$ $g\left( 7 \right) = P - Pc\left( x \right) \le 0$
Dynamic economic load dispatch problems	$\mathop \sum \limits_{t = 1}^{n} \mathop \sum \limits_{i = 1}^{N} F_{i} \left( {P_{it} } \right) + c_{1} \left\{ {\mathop \sum \limits_{t = 1}^{n} \mathop \sum \limits_{i = 1}^{N} P_{it} - P_{Dt} } \right\}^{2} + c_{2} \left\{ {\mathop \sum \limits_{t = 1}^{n} \mathop \sum \limits_{i = 1}^{N} P_{it} - P_{r\lim } } \right\}^{2}$	$P_{i}^{\min } \le P_{it} \le P_{i}^{\max }$ $\max \left( {P_{i}^{\min } ,UR_{i} - P_{i} } \right) \le P_{i} \le \min \left( {P_{i}^{\max } ,P_{i}^{t - 1} - DR_{i} } \right)$
Static economic load dispatch problems	Objective functions without valve point effect $F_{i} \left( {P_{i} } \right) = a_{i} P_{i}^{2} + b_{i} P_{i} + c_{i}$ Objective functions with valve point effect $F_{i} \left( {P_{i} } \right) = a_{i} P_{i}^{2} + b_{i} P_{i} + c_{i} + \left| {e_{i} \sin \left\{ {f_{i} \left( {P_{i}^{\min } - P_{i} } \right)} \right\}} \right|$	$P_{i}^{\min } \le P_{i} \le P_{i}^{\max }$ $P_{i} \le \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{P}^{pz} \;{\text{and}}\;P_{i} \ge \hat{P}^{pz}$