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MOAVOA: a new multi-objective artificial vultures optimization algorithm

Abstract

This paper presents a multi-objective version of the artificial vultures optimization algorithm (AVOA) for a multi-objective optimization problem called a multi-objective AVOA (MOAVOA). The inspirational concept of the AVOA is based on African vultures' lifestyles. Archive, grid, and leader selection mechanisms are used for developing the MOAVOA. The proposed MOAVOA algorithm is tested oneight real-world engineering design problems and seventeen unconstrained and constrained mathematical optimization problems to investigates its appropriateness in estimating Pareto optimal solutions. Multi-objective particle swarm optimization, multi-objective ant lion optimization, multi-objective multi-verse optimization, multi-objective genetic algorithms, multi-objective salp swarm algorithm, and multi-objective grey wolf optimizer are compared with MOAVOA using generational distance, inverted generational distance, maximum spread, and spacing performance indicators. This paper demonstrates that MOAVOA is capable of outranking the other approaches. It is concluded that the proposed MOAVOA has merits in solving challenging multi-objective problems.

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Appendix 1: constrained multi-objective test problems

Appendix 1: constrained multi-objective test problems

CONSTR

There are two constraints and two design variables in this problem, which have a convex Pareto front.

$$\begin{array}{*{20}l} {{\text{Minimize}}:} \hfill & {f_{1} \left( x \right) = x_{1} } \hfill & {\left( {{\text{A}}.1} \right)} \hfill \\ {{\text{Minimize}}:~} \hfill & {f_{2} \left( x \right) = \left( {1 + x_{2} } \right)/x_{1} } \hfill & {\left( {{\text{A}}.2} \right)} \hfill \\ {{\text{where}}:} \hfill & {g_{1} \left( x \right)~ = 6 - \left( {x_{2} + 9x_{1} } \right)} \hfill & {\left( {{\text{A}}.3} \right)} \hfill \\ {} \hfill & {g_{2} \left( x \right)~ = \left( {1 + x_{2} - 9x_{1} } \right)} \hfill & {\left( {{\text{A}}.4} \right)} \hfill \\ {} \hfill & {0.1 \le x_{1} \le 1,0 \le x_{2} \le 5} \hfill & {} \hfill \\ \end{array}$$

SRN

Srinivas and Deb [67] suggested a continuous Pareto optimal front for the next problem as follows:

$$\begin{array}{*{20}l} {{\text{Minimize}}:} \hfill & {f_{1} \left( x \right)~ = ~2~ + ~\left( {x_{1} ~ - ~2} \right)^{2} + ~\left( {x_{2} ~ - ~1} \right)^{2} } \hfill & {\left( {A.5} \right)} \hfill \\ {{\text{Minimize}}:} \hfill & {f_{2} \left( x \right)~ = ~9x_{1} ~ - ~\left( {x_{2} ~ - ~1} \right)^{2} } \hfill & {\left( {A.6} \right)} \hfill \\ {{\text{where}}:} \hfill & {g_{1} \left( x \right)~ = x_{1}^{2} + x_{2}^{2} - 255} \hfill & {\left( {A.7} \right)} \hfill \\ {} \hfill & {g_{2} \left( x \right)~ = x_{1} - 3x_{2} + 10} \hfill & {\left( {A.8} \right)} \hfill \\ {} \hfill & { - 20 \le x_{1} \le 20, - 20 \le x_{2} \le 20} \hfill & {} \hfill \\ \end{array}$$

BNH

Binh and Korn [68] were the first to propose this problem as follows:

$$\begin{array}{*{20}l} {{\text{Minimize}}:} \hfill & {f_{1} \left( x \right) = 4x_{1}^{2} + 4x_{2}^{2} } \hfill & {\left( {{\text{A}}.9} \right)} \hfill \\ {{\text{Minimize}}:} \hfill & {f_{2} \left( x \right) = (x_{1} - 5)^{2} + (x_{2} - 5)^{2} } \hfill & {\left( {{\text{A}}.10} \right)} \hfill \\ {{\text{where}}:} \hfill & {g_{1} \left( x \right) = (x_{1} - 5)^{2} + x_{2}^{2} - 25} \hfill & {\left( {{\text{A}}.11} \right)} \hfill \\ {} \hfill & {g_{2} \left( x \right) = 7.7 - (x_{1} - 8)^{2} - (x_{2} + 3)^{2} } \hfill & {\left( {{\text{A}}.12} \right)} \hfill \\ {} \hfill & {0 \le x_{1} \le 5,0 \le x_{2} \le 3} \hfill & {} \hfill \\ \end{array}$$

OSY

Osyczka and Kundu [69] proposed five distinct regions for the OSY test issue. There are also six constraints and six design variables to consider as below:

Minimize: \(\begin{array}{*{20}l} {{\text{Minimize:}}} \hfill & {f_{1} \left( x \right) = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} + x_{5}^{2} + x_{6}^{2} } \hfill & {\left( {{\text{A}}.13} \right)} \hfill \\ {{\text{Minimize}}:} \hfill & {f_{2} \left( x \right) = \left[ {25(x_{1} - 2)^{2} + (x_{2} - 1)^{2} + \left( {x_{3} - 1} \right) + (x_{4} - 4)^{2} + \left( {x_{5} - 1^{2} } \right)~} \right]} \hfill & {\left( {{\text{A}}.14} \right)} \hfill \\ {{\text{Where:}}} \hfill & {g_{1} \left( x \right)~ = \left( {2 - x_{1} - x_{2} } \right)} \hfill & {\left( {{\text{A}}.15} \right)} \hfill \\ {} \hfill & {g_{2} \left( x \right) = - 6 + x_{1} + x_{2} } \hfill & {\left( {{\text{A}}.16} \right)} \hfill \\ {} \hfill & {g_{3} \left( x \right) = - 2 - x_{1} + x_{2} } \hfill & {\left( {{\text{A}}.17} \right)} \hfill \\ {} \hfill & {g_{4} \left( x \right) = - 2 + x_{1} - 3x_{2} } \hfill & {\left( {{\text{A}}.18} \right)} \hfill \\ {} \hfill & {g_{5} \left( x \right) = - 4 + x_{4} + (x_{3} - 3)^{2} } \hfill & {\left( {{\text{A}}.19} \right)} \hfill \\ {} \hfill & {g_{6} \left( x \right) = 4 - x_{6} - (x_{5} - 3)^{2} } \hfill & {\left( {{\text{A}}.20} \right)} \hfill \\ {} \hfill & {0 \le x_{1} \le 10,~~~~~0 \le x_{2} \le 10,~~~~~1 \le x_{3} \le 5~} \hfill & {\left( {{\text{A}}.21} \right)} \hfill \\ {} \hfill & {0 \le x_{4} \le 6,~\quad 1 \le x_{5} \le 5,\quad 0 \le x_{6} \le 10} \hfill & {} \hfill \\ \end{array}\)

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Khodadadi, N., Soleimanian Gharehchopogh, F. & Mirjalili, S. MOAVOA: a new multi-objective artificial vultures optimization algorithm. Neural Comput & Applic (2022). https://doi.org/10.1007/s00521-022-07557-y

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Keywords

  • Multi-objective problem
  • Artificial vultures optimization algorithm
  • Pareto optimal solution
  • Optimization
  • Algorithm
  • Particle Swarm Optimization
  • Performance indicator