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Multiplayer battle game-inspired optimizer for complex optimization problems

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Abstract

Various popular multiplayer battle royale games share a lot of common elements. Drawing from our observations, we summarized these shared characteristics and subsequently proposed a novel heuristic algorithm named multiplayer battle game-inspired optimizer (MBGO). The proposed MBGO streamlines mainstream multiplayer battle royale games into two discrete phases: movement and battle. Specifically, the movement phase incorporates the principles of commonly encountered “safe zones” to incentivize participants to relocate to areas with a higher survival potential. The battle phase simulates a range of strategies players adopt in various situations to enhance the diversity of the population. To evaluate and analyze the performance of the proposed MBGO, we executed it alongside ten other algorithms, including three classics and five latest ones, across multiple diverse dimensions within the CEC2017 and CEC2020 benchmark functions. In addition, we employed several industrial design problems to evaluate the scalability and practicality of the proposed MBGO. The statistical analysis results reveal that the novel MBGO demonstrates significant competitiveness, excelling in convergence speed and achieving high levels of convergence accuracy across both benchmark functions and real-world problems.

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Funding

No funding was received for conducting this study.

Author information

Authors and Affiliations

  1. School of Engineering, University of Fukui, Fukui, Japan

    Yuefeng Xu

  2. Graduate School of Information Science and Technology, Hokkaido University, Sapporo, Japan

    Rui Zhong

  3. Department of Engineering, University of Fukui, Fukui, Japan

    Chao Zhang

  4. Institute of Science and Technology, Niigata University, Niigata, Japan

    Jun Yu

Authors
  1. Yuefeng Xu
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  2. Rui Zhong
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  3. Chao Zhang
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  4. Jun Yu
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Contributions

YX implemented all the code and completed the first draft. RZ realized the visualization of experimental results and corrected the paper for the first time. CZ and JY designed the optimization framework together. All authors reviewed the manuscript again.

Corresponding author

Correspondence to Jun Yu.

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Appendix A

Appendix A

WBP Coello [57] proposed this benchmark problem, and the objective of the problem was to find the minimum manufacturing cost of a welded beam. The variables are weld thickness \(h(x_1)\), height \(l(x_2)\), length \(t(x_3)\) and bar thickness \(b(x_4)\). The mathematical expression of the objective function is:

$$\begin{aligned}&\text {minimize:}\\&f(X) = 1.10471x_1^{2}x_2 + 0.04811x_3x_4(14.0+x_2), \\&\text {subject to:}\\&g_1(X)=\tau (X)-\tau _{max}\le 0,\\&g_2(X)=\sigma (X)- \sigma _{max}\le 0,\\&g_3(X)=\delta (X)- \delta _{max}\le 0,\\&g_4(X)=x_1-x_4\le 0,\\&g_5(X)=P-P_c(X)\le 0,\\&g_6(X)=0.125-x_1\le 0,\\&g_7(X)=1.10471x_1^{2}+0.04811x_3x_4(14.0+x_2)-5.0\le 0,\\&\tau (X)=\sqrt{({\tau }')^{2}+2{\tau }'{\tau }''\frac{x_2}{2R}+ ({\tau }'')^{2} },\\&{\tau }'(X)=\frac{P}{\sqrt{2} x_1x_2} ,\\&{\tau }''(X)=\frac{MR}{J} ,\\&M=P(L+\frac{x_2}{2} ),\\&R=\sqrt{\frac{x_2^{2} }{4} +(\frac{x_1+x_3}{2} )^{3} } ,\\&J=2\left\{ \sqrt{2}x_1x_2\left[ \frac{x_2^{2} }{4}+(\frac{x_1+x_3}{2} )^{2} \right] \right\} ,\\&\sigma (\textbf{X} ) =\frac{6PL}{x_4x_3^{2} } ,\\&\delta (\textbf{X} )=\frac{6PL^{3} }{Ex_3^{2}x_4 } ,\\&P_c(\textbf{X})=\frac{4.013E\sqrt{x_3^{2}x_4^{6}/36 } }{L^{2} }(1-\frac{x_3}{2L}\sqrt{\frac{E}{4G} } ) ,\\&P=6000\ \text {lb},\\&L=14\ \text {in},\\&\delta _{max}=0.25\ \text {in},\\&E=30\times 10^{6} \ \text {psi},\\&G=12\times 10^{6} \ \text {psi},\\&\tau _{max}=13,600\ \text {psi},\\&\sigma _{max}=30,000\ \text {psi},\\&\text {variable range:} \\&x_1\ge 0.1,\\&x_2\ge 0.1,\\&x_3\le 10,\\&x_4\le 2. \end{aligned}$$

PVP The objective is to minimize the total cost, including material, forming, and welding costs. The problem has four variables, including shell thickness Ts(x1), head thickness Th(x2), inner radius R(x3), and the length of the cylindrical portion of the vessel (excluding the head) L(x4). In addition, x1 and x2 are integer multiples of 0.0625 inches, while the other variables are continuous. The optimization problem can be expressed as follows:

$$\begin{aligned}&\text {minimize:}\\&f(X) = 0.6224x_1x_3x_4+1.7781x_2x_3^{2}+3.1661x_1^{2}x_4+19.84x_1^{2}x_3, \\&\text {subject to:}\\&g_1(X)=-x_1+0.0193x_3\le 0,\\&g_2(X)=-x_2+0.00954x_3\le 0,\\&g_3(X)=-\pi x_3^2x_4-\frac{4}{3}\pi x_3^3+1,296,000 \le 0,\\&g_4(X)=x_4-240\le 0,\\&\text {variable range:} \\&x_1,x_2\in \left\{ 1\times 0.0625,2\times 0.0625,3\times 0.0625,\dots 1600\times 0.0625 \right\} ,\\&x_3\ge 10,\\&x_4\le 200. \end{aligned}$$

TBTD This problem is an optimization problem for a three-bar planar truss structure. The volume of a statically loaded 3-rod truss is to be minimized with stress (\(\sigma\)) constraints on each truss member. The objective is to evaluate the optimum cross-sectional areas \(A_1(x_1)\) and \(A_2(x_2)\). The optimization problem can be expressed as follows:

$$\begin{aligned}&\text {minimize:}\\&f(X) = (2\sqrt{2}x_1+x_2 )\times l, \\&\text {subject to:}\\&g_1(X)=\frac{\sqrt{2}x_1+x_2 }{\sqrt{2}x_1^2+2x_1x_2 }P-\sigma \le 0,\\&g_2(X)=\frac{x_2 }{\sqrt{2}x_1^2+2x_1x_2 }P-\sigma \le 0,\\&g_3(X)=\frac{1}{\sqrt{2}x_2+x_1 }P-\sigma \le 0,\\&l=100 \ \text {cm}, \\&P=2 \ \text {kN/cm}^3,\\&\sigma =2 \ \text {kN/cm}^3, \\&\text {variable range:} \\&x_1 \ge 0,\\&x_2 \le 1. \end{aligned}$$

GTD GTD is an unconstrained discrete design problem proposed by Sandgren [58]. This benchmark task minimizes the gear ratio, defined as the ratio of the output shaft’s angular velocity to the input shaft’s angular velocity. Ratio to input shaft angular velocity. The number of teeth of the gears \(n_A(x_1)\), \(n_B(x_2)\), \(n_C(x_3)\) and \(n_D(x_4)\) are considered as the design variables for the problem. The mathematical formulation is as follows:

$$\begin{aligned}&\text {minimize:}\\&f(X) = (\frac{1}{6.931}-\frac{x_3x_2}{x_1x_4} )^{2}, \\&\text {variable range:} \\&x_1,x_2,x_3,x_4\in \left\{ 12,13,14\dots ,60 \right\} . \end{aligned}$$

CBD This problem is a good benchmark for validating the ability of optimization methods to solve continuous, discrete, and mixed-variable structural design problems. The objective of the problem is to minimize the volume of the beam. The design variables include segment widths (x1, x2, x3, x4, x5) and segment heights (x6, x7, x8, x9, x10). In addition to the bending stress limitations, a specific aspect ratio is specified, i.e., the height-to-width ratio of the beam segments must be less than 20. The problem is formulated as follows:

$$\begin{aligned}&\text {minimize:}\\&f(X) = D(x_1x_6l_1+x_2x_7l_2+x_3x_8l_3+x_4x_9l_4+x_5x_{10}l_5), \\&\text {subject to:}\\&g_1(X)=\frac{6Pl_5}{x_5x_{10}^2}-\sigma _d \le 0,\\&g_2(X)=\frac{6P(l_5+l_4)}{x_4x_9^2}-\sigma _d \le 0,\\&g_3(X)=\frac{6P(l_5+l_4+l_3)}{x_3x_8^2}-\sigma _d \le 0,\\&g_4(X)=\frac{6P(l_5+l_4+l_3+l_2)}{x_2x_7^2}-\sigma _d \le 0,\\&g_5(X)=\frac{6P(l_5+l_4+l_3+l_2+l_1)}{x_1x_6^2}-\sigma _d \le 0,\\&g_6(X)=\frac{Pl^3}{3E} (\frac{1}{I_5}+\frac{1}{I_4}+\frac{1}{I_3}+\frac{1}{I_2}+\frac{1}{I_1} )-\Delta _{max}\le 0,\\&g_7(X)=\frac{x_{10}}{x_5}-20 \le 0,\\&g_8(X)=\frac{x_{9}}{x_4}-20 \le 0,\\&g_9(X)=\frac{x_{8}}{x_3}-20 \le 0,\\&g_{10}(X)=\frac{x_{7}}{x_2}-20 \le 0,\\&g_{11}(X)=\frac{x_{6}}{x_1}-20 \le 0,\\&P=50000N,\\&\sigma _d=14,000\ \text { N/cm}^2,\\&E=2\times 10^7\ \text { N/cm}^2,\\&\Delta _{max}=2.7\text {cm},\\&D=1.0,\\&\text {variable range:} \\&x_1\in \left\{ 1,2,3,4,5\right\} ,\\&x_2,x_3\in \left\{ 2.4,2.6,2.8,3.1 \right\} ,\\&x_4\ge 1,\\&x_5\le 5,\\&x_6\in \left\{ 30,31,32,\dots ,65 \right\} ,\\&x_7,x_8\in \left\{ 45,50,55,60,65 \right\} ,\\&x_9\ge 30,\\&x_{10}\le 65. \end{aligned}$$

IBD The objective of this problem is to minimize the vertical deflection of the I-beam. The variables of this problem include flange width \(b(x_1)\), section height \(h(x_2)\), web thickness \(t_w(x_3)\) and flange thickness \(t_f(x_4)\). The maximum vertical deflection of the beam is \(f(x) = PL^3 /48EI\) when its length (L) and modulus of elasticity (E) are 5200 cm and \(523.104 \text {kN/cm}^2\), respectively. The objective function of the problem The objective function and constraints of the problem are formulated as follows:

$$\begin{aligned}&\text {minimize:}\\&f(X) = \frac{5000}{x_3(x_2-2x_4)^3/12+(x_1x_4^3/6)+2bx_4(x_2-x_4/2)^2} , \\&\text {subject to:}\\&g_1(X)=2x_1x_3+x_3(x_2-2x_4) \le 300,\\&g_2(X)=\frac{18x_2\times 10^4}{x_3(x_2-2x_4)^3+2x_1x_3(4x_4^2+3x_2(x_2-2x_4))}\\&\quad +\frac{15x_1\times 10^3}{(x_2-2x_4)x_3^2+2x_3x_1^3} \le 56,\\&\text {variable range:} \\&10\le x_1\le 50,\\&10\le x_2\le 80,\\&0.9\le x_3\le 5,\\&0.9\le x_4\le 5. \end{aligned}$$

TCD This problem is an example of designing a uniform column of tubular cross-section to carry compressive loads at minimum cost. The problem has two design variables: the mean diameter of column \(d(x_1)\) and tube t’s thickness \((x_2)\). The material yield stress of the tube column is \(\sigma _y =500\text {kgf/cm}^2\), and the modulus of elasticity is \(E =0.85 \times 10^6 \text {kgf/cm}^2\) The optimization model for this problem is as follows:

$$\begin{aligned}&\text {minimize:}\\&f(X) = 9.8x_1x_2+2x_1, \\&\text {subject to:}\\&g_1(X)=\frac{P}{\pi x_1x_2\sigma _y}-1 \le 0,\\&g_2(X)=\frac{8PL^2}{\pi ^3Ex_1x_2(x_1^2+x_2^2)}-1 \le 0,\\&g_3(X)=\frac{2.0}{x_1}-1\le 0,\\&g_4(X)=\frac{x_1}{14}-1\le 0,\\&g_5(X)=\frac{0.2}{x_2}-1\le 0,\\&g_6(X)=\frac{x_2}{8}-1\le 0,\\&\text {variable range:} \\&2\le x_1\le 14,\\&0.2\le x_2\le 0.8. \end{aligned}$$

PLD The main objective of the problem is to locate the piston parts \(H(x_1)\), \(B(x_2)\), \(D(x_3)\) and \(X(x_4)\) by minimizing the amount of oil as the piston rod rises from \(0^{\circ }\) to \(45^{\circ }\). The formula for this problem is as follows:

$$\begin{aligned}&\text {minimize:}\\&f(X) = \frac{1}{4}\pi x_3^2(L_2-L_1), \\&\text {subject to:}\\&g_1(X)=QL\cos \theta -R\times F \le 0,\\&g_2(X)=Q(L-x_4)-M_{max} \le 0,\\&g_3(X)=1.2(L_2-L_1)-L_1\le 0,\\&g_4(X)=\frac{x_3}{2}-x_2 \le 0,\\&\text {where:} \\&R=\frac{\left| -x_4(x_4\sin \theta +x_1)+x_1(x_2-x_4\cos \theta ) \right| }{\sqrt{(x_4-x_2)^2+x_1^2} },\\&\theta = 45^{\circ } ,\\&Q=10,000\ \text {lbs},\\&L=240\ \text {in},\\&M_{max}=1.8\times 10^6\ \text {lbs in},\\&P=1500 \text {psi},\\&\text {variable range:} \\&0.05\le x_1,x_2,x_4\le 500,\\&0.05\le x_2\le 120. \end{aligned}$$

CBHD The problem aims to minimize the weight of the corrugated bulkheads of a chemical tanker [59], where the design variables include width \((x_1)\), depth \((x_2)\), length (x3) and plate thickness \((x_4)\). The mathematical model for this optimization problem is as follows:

$$\begin{aligned}&\text {minimize:}\\&f(X) = \frac{5.885x_4(x_1+x_3)}{x_1+\sqrt{\left| x_3^2-x_2^2 \right| }}, \\&\text {subject to:}\\&g_1(X)=-x_4x_2(0.4x_1+\frac{x_3}{6} )+8.94(x_1+\sqrt{\left| x_3^2-x_2^2 \right| } ) \le 0,\\&g_2(X)=-x_4x_2^2(0.2x_1+\frac{x_3}{12} )+2.2(8.94(x_1+\sqrt{\left| x_3^2-x_2^2 \right| } ) )^{4/3} \le 0,\\&g_3(X)=-x_4+0.0156x_1+0.15\le 0,\\&g_4(X)=-x_4+0.0156x_3+0.15\le 0,\\&g_5(X)=-x_4+1.05 \le 0,\\&g_6(X)=-x_3+x_2 \le 0,\\&\text {variable range:} \\&0\le x_1,x_2,x_3\le 100,\\&0\le x_4\le 5. \end{aligned}$$

RCB Amir and Hasegawa [60] presented a simplified design optimization problem for a reinforced concrete beam. The beam was assumed to be supported with a span of 30 ft. It was subjected to a live load of 2000 lbs. and a dead load of 1000 lbs., including the weight of the beam. The compressive strength of the concrete (\(\sigma _x\)) is 5 ksi, and the yield stress of the steel reinforcement (\(\sigma _y\)) is 50 ksi. The cost of the concrete is 0.02 dollars/sq.ft., and the cost of the steel is 1.0 dollars/sq.ft. The cost of the steel is 1.0 dollars/sq.ft. The cost of the concrete is 1.0 dollars/sq.ft. To minimize the total structural cost, the area of reinforcement A \(s(x_1)\), beam width \(b(x_2)\) and beam depth \(h(x_3)\) must be determined. According to ACI Building Code 318-77, the structure should be scaled to achieve the required strength as follows:

$$\begin{aligned}&M_u=0.9A_s\sigma _y(0.8h)(1.0-0.59\frac{A_s\sigma _y}{0.8bh\sigma _c} \ge 1.4M_d+1.7M_l),\\&\text {where}\, M_u, M_d \,\text {and}\, M_l \,\text {are the beam's flexural strength, dead weight, and live weight} \\&\text {bending moment, respectively. In this example,}\, M_d \,\text {is 1350 kip and}\, M_l \,\text {is 2700 kip,}\\&\text {and the depth-to-width ratio of the beam must be less than or equal to 4.}\\&\text {minimize:}\\&f(X) = 2.9x_1+0.6x_2x_3, \\&\text {subject to:}\\&g_1(X)=\frac{x_2}{x_3}-4 \le 0,\\&g_2(X)=180+7.375\frac{x_1^2}{x_3}-x_1x_2 \le 0,\\&\text {variable range:} \\&x_1\in \left\{ 6,6.16,6.32,6.6,7,7.11,7.2,7.8,7.9,8,8.4 \right\} ,\\&x_2\in \left\{ 28,29,30,\dots ,40 \right\} ,\\&5\le x_3\le 100. \end{aligned}$$

See Tables 3, 4, 5, 6, 7.

Table 3 The mean and standard deviation of the optimal solutions obtained from 30 trial runs for all algorithms are reported for the 10-D CEC2017 function functions
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Table 4 The mean and standard deviation of the optimal solutions obtained from 30 trial runs for all algorithms are reported for CEC2017-30D
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Table 5 The mean and standard deviation of the optimal solutions obtained from 30 trial runs for all algorithms are reported for the 50-D CEC2020 function functions
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Table 6 The mean and standard deviation of the optimal solutions obtained from 30 trial runs for all algorithms are reported for the 100-D CEC2020 function functions
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Table 7 The mean and standard deviation of the optimal solutions obtained from 30 trial runs for all algorithms are reported for 10 real-world problems
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Xu, Y., Zhong, R., Zhang, C. et al. Multiplayer battle game-inspired optimizer for complex optimization problems. Cluster Comput 27, 8307–8331 (2024). https://doi.org/10.1007/s10586-024-04448-w

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