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Ladybug Beetle Optimization algorithm: application for real-world problems

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Abstract

In this paper, a novel optimization algorithm is proposed, called the Ladybug Beetle Optimization (LBO) algorithm, which is inspired by the behavior of ladybugs in nature when they search for a warm place in winter. The new proposed algorithm consists of three main parts: (1) determine the heat value in the position of each ladybug, (2) update the position of ladybugs, and (3) ignore the annihilated ladybug(s). The main innovations of LBO are related to both updating the position of the population, which is done in two separate ways, and ignoring the worst members, which leads to an increase in the search speed. Also, LBO algorithm is performed to optimize 78 well-known benchmark functions. The proposed algorithm has reached the optimal values of 73.3% of the benchmark functions and is the only algorithm that achieved the best solution of 20.5% of them. These results prove that LBO is substantially the best algorithm among other well-known optimization methods. In addition, two fundamentally different real-world optimization problems include the Economic-Environmental Dispatch Problem (EEDP) as an engineering problem and the Covid-19 pandemic modeling problem as an estimation and forecasting problem. The EEDP results illustrate that the proposed algorithm has obtained the best values in either the cost of production or the emission or even both, and the use of LBO for Covid-19 pandemic modeling problem leads to the least error compared to others.

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Data availability

The MATLAB and python source code of the LBO algorithm that support the findings of this study are available in https://github.com/Saadat-Safiri/LBO-algorithm-matlab-code and https://github.com/Saadat-Safiri/LBO-algorithm-python-code, respectively.

Abbreviations

\(i\) :

Member of the population that is being updated

\(j\) :

Member of the population that is used to update the \(i\)th member

\(k\) :

Iteration

\(k_{\max }\) :

Maximum iteration that used to terminate the optimization algorithm

\(t\) :

Index of summation

\({\text{rand}}\) :

A uniformly distributed random number between 0 and 1

\(N\left( 0 \right)\) :

Number of the initial population

\(N\left( k \right)\) :

Number of the population in the \(k\)th iteration

\(N_{\min }\) :

Minimum number of the population during the algorithm process

\({\text{NFE}}\) :

Number of function evaluation

\({\text{NFE}}_{\max }\) :

The maximum number of function evaluation, which used to terminate the optimization algorithm

\(x_{i} \left( k \right), x_{j} \left( k \right)\) :

Position of the \(i\)th and \(j\)th members in the search space

\(D\) :

Dimensions of the decision vector

\(f\left( {x_{i} \left( k \right)} \right)\) :

The value of the cost function for the \(i\)th member in the \(k\)th iteration

\(f_{{{\text{worst}}}}\) :

The worst value of the cost function up to the current iteration during the algorithm process

\(f_{{{\text{opt}}}}\) :

The optimal global value for the cost function

\(C_{i}\) :

The ratio of the \(i\)th member cost to total members cost

\(\overrightarrow {{r_{1} }} , \overrightarrow {{r_{2} }} , {\text{and}}\; \overrightarrow {{r_{3} }}\) :

Three vectors that are used to update the \(i\)th member of the population

\(P\) :

Generated vector for the population in Roulette-wheel selection method

\(\beta\) :

Pressure coefficient in Roulette-wheel selection method

\(i\) :

The power unit in the grid

\(N_{{\text{G}}}\) :

The number of power unit in the grid

\(P_{i}\) :

The value of power generation in the \(i\)th unit

\(P_{i}^{\min } ,P_{i}^{\max }\) :

The lower and upper bound of power generation in the \(i\)th unit

\(F_{{P_{i} }} \left( {P_{i} } \right)\) :

The value of generation cost for the \(i\)th unit to generate \(P_{i}\) MW of power

\(a_{i} , b_{i} , c_{i} , g_{i} , {\text{and}}\;h_{i}\) :

The constant parameters for calculating generation cost in the \(i\)th unit

\(F_{{{\text{E}}_{i} }} \left( {P_{i} } \right)\) :

The value of emission for the \(i\)th unit to generate \(P_{i}\) MW of power

\(\alpha_{i} , \beta_{i} ,\gamma_{i} ,\eta_{i} , {\text{and}}\;\delta_{i}\) :

The constant parameters for calculating emission in the \(i\)th unit

\(F\) :

The total cost function

\(p\) :

Penalty coefficient

\(D\) :

Power demand in MW

\(L_{{\text{P}}}\) :

The value of power losses during the transmission

\(B\) :

The constant matrix for calculating power losses

\(S\left( t \right)\) :

Susceptible individuals

\(I\left( t \right)\) :

Infected individuals

\(D\left( t \right)\) :

Diagnosed individuals

\(A\left( t \right)\) :

Ailing individuals

\(R\left( t \right)\) :

Recognized individuals

\(T\left( t \right)\) :

Threatened individuals

\(H\left( t \right)\) :

Recovered individuals

\(E\left( t \right)\) :

Death cases

\(\alpha\) :

Transmission rate from the infected to the susceptible individual

\(\beta\) :

Transmission rate from the diagnosed to the susceptible individual

\(\gamma\) :

Transmission rate from the recognized to the susceptible individual

\(\delta\) :

Transmission rate from the ailing to the susceptible individual

\(\varepsilon\) :

Detection rate of the individual with no symptoms

\(\theta\) :

Detection rate of the individual with symptoms

\(\zeta\) :

The probability that the infected individual knows that they are infected

\(\eta\) :

The probability that the infected individual does not know that they are infected

\(\mu\) :

The probability of developing life-threatening symptoms

\(\nu\) :

The probability of developing life-threatening symptoms for a detected case

\(\tau\) :

Death rate

\(\lambda\) :

The recovery rate

\(\kappa\) :

The recovery rate

\(\xi\) :

The recovery rate

\(\rho\) :

The recovery rate

\(\sigma\) :

The recovery rate

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  1. Faculty of Electrical Engineering, K. N. Toosi University of Technology, Tehran, Iran

    Saadat Safiri & Amirhossein Nikoofard

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Safiri, S., Nikoofard, A. Ladybug Beetle Optimization algorithm: application for real-world problems. J Supercomput 79, 3511–3560 (2023). https://doi.org/10.1007/s11227-022-04755-2

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