智能算法之蜣螂优化算法(DBO)，原理公式详解，附matlab代码

蜣螂优化算法（Dung beetle optimizer，DBO）是一种新型群智能优化算法，该算法是受蜣螂滚球、跳舞、觅食、偷窃和繁殖行为启发而提出的，具有进化能力强、搜索速度快、寻优能力强的特点。该成果于2022年发表在知名SCI期刊THE JOURNAL OF SUPERCOMPUTING上。目前谷歌学术上查询被引102次。

DBO算法通过滚球、跳舞、觅食、偷窃和繁殖的过程，五个主要操作模拟了蜣螂的生存行为，最后选取最优解。

算法原理

（1）滚球行为

蜣螂在滚动的过程中利用天体导航，蜣螂滚动路径受环境影响，环境的变化也影响着蜣螂位置的改变:

 式中：t为当前迭代次数；  表示第  只蜣螂在第  次迭代时的位置信息；  为（0,0.2）之间的常数，表示缺陷系数；  为（0,1）的常数；  为-1或1的自然系数；  为全局最差位置；∆x为环境变化。

（2）跳舞行为

当蜣螂遇到障碍无法继续前进时，通过跳舞行为变换方向，获得新的移动路线。使用切线函数得到新的滚动方向，在[0,π]区间值，确定新方向后，继续滚球行为： 如果角度为0，π/2，π时，蜣螂的位置不更新。

（3）繁殖行为

蜣螂将粪球滚动到安全区域后进行隐藏，合适的产卵地点对于蜣螂后代来说至关重要，提出一种边界选择策略模拟雌性蜣螂的产卵区域： 式中：  为当前局部最佳位置；  ,  分别为产卵区的下限和上限；  表示动态选择因子；  为最大迭代次数；  ,  分别为优化问题的下限和上限。

确定产卵区域后，每只雌性蜣螂每次产一枚卵。产卵的区域随迭代次数动态调整，卵球的位置也是动态变化的：  式中，  表示第  次迭代时第  个卵球的位置；  ,  均为1×D的独立随机向量；D为优化问题的维数。

（4）觅食行为

幼虫从卵中钻出后，需要限定最佳觅食区域引导小蜣螂觅食，最佳觅食区域的边界为：  式中：  ,  分别为最佳觅食区域的下限和上限；  为全局最佳位置。小蜣螂在觅食过程中位置的更新为： 式中：C1为服从正态分布的随机数；C2为（0,1）的随机数。

（5）偷窃行为

蜣螂种群中，偷窃作为一种竞争行为十分常见，偷窃蜣螂的位置更新为：  式中：S为常数；g为服从正态分布的1×D随机向量。

 结果展示

以为CEC2005函数集为例，进行结果展示：

 MATLAB核心代码

function [fMin , bestX, Convergence_curve ] = DBO(pop, M,c,d,dim,fobj  )
        
   P_percent = 0.2;    % The population size of producers accounts for "P_percent" percent of the total population size       
 
 
 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
pNum = round( pop *  P_percent );    % The population size of the producers   
 
 
 
 
lb= c.*ones( 1,dim );    % Lower limit/bounds/     a vector
ub= d.*ones( 1,dim );    % Upper limit/bounds/     a vector
%Initialization
for i = 1 : pop
    
    x( i, : ) = lb + (ub - lb) .* rand( 1, dim );  
    fit( i ) = fobj( x( i, : ) ) ;                       
end
 
 
pFit = fit;                       
pX = x; 
 XX=pX;    
[ fMin, bestI ] = min( fit );      % fMin denotes the global optimum fitness value
bestX = x( bestI, : );             % bestX denotes the global optimum position corresponding to fMin
 
 
 % Start updating the solutions.
for t = 1 : M    
       
        [fmax,B]=max(fit);
        worse= x(B,:);   
       r2=rand(1);
 
  
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    for i = 1 : pNum    
        if(r2<0.8)
            r1=rand(1);
          a=rand(1,1);
          if (a>0.1)
           a=1;
          else
           a=-1;
          end
    x( i , : ) =  pX(  i , :)+0.3*abs(pX(i , : )-worse)+a*0.1*(XX( i , :)); % Equation (1)
       else
            
           aaa= randperm(180,1);
           if ( aaa==0 ||aaa==90 ||aaa==180 )
            x(  i , : ) = pX(  i , :);   
           end
         theta= aaa*pi/180;   
       
       x(  i , : ) = pX(  i , :)+tan(theta).*abs(pX(i , : )-XX( i , :));    % Equation (2)      
 
 
        end
      
        x(  i , : ) = Bounds( x(i , : ), lb, ub );    
        fit(  i  ) = fobj( x(i , : ) );
    end 
 [ fMMin, bestII ] = min( fit );      % fMin denotes the current optimum fitness value
  bestXX = x( bestII, : );             % bestXX denotes the current optimum position 
 
 
 R=1-t/M;                           %
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 Xnew1 = bestXX.*(1-R); 
     Xnew2 =bestXX.*(1+R);                    %%% Equation (3)
   Xnew1= Bounds( Xnew1, lb, ub );
   Xnew2 = Bounds( Xnew2, lb, ub );
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     Xnew11 = bestX.*(1-R); 
     Xnew22 =bestX.*(1+R);                     %%% Equation (5)
   Xnew11= Bounds( Xnew11, lb, ub );
    Xnew22 = Bounds( Xnew22, lb, ub );
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
    for i = ( pNum + 1 ) :12                  % Equation (4)
     x( i, : )=bestXX+((rand(1,dim)).*(pX( i , : )-Xnew1)+(rand(1,dim)).*(pX( i , : )-Xnew2));
   x(i, : ) = Bounds( x(i, : ), Xnew1, Xnew2 );
  fit(i ) = fobj(  x(i,:) ) ;
   end
   
  for i = 13: 19                  % Equation (6)
 
 
       x( i, : )=pX( i , : )+((randn(1)).*(pX( i , : )-Xnew11)+((rand(1,dim)).*(pX( i , : )-Xnew22)));
       x(i, : ) = Bounds( x(i, : ),lb, ub);
       fit(i ) = fobj(  x(i,:) ) ;
  
  end
  
  for j = 20 : pop                 % Equation (7)
       x( j,: )=bestX+randn(1,dim).*((abs(( pX(j,:  )-bestXX)))+(abs(( pX(j,:  )-bestX))))./2;
      x(j, : ) = Bounds( x(j, : ), lb, ub );
      fit(j ) = fobj(  x(j,:) ) ;
  end
   % Update the individual's best fitness vlaue and the global best fitness value
     XX=pX;
    for i = 1 : pop 
        if ( fit( i ) < pFit( i ) )
            pFit( i ) = fit( i );
            pX( i, : ) = x( i, : );
        end
        
        if( pFit( i ) < fMin )
           % fMin= pFit( i );
           fMin= pFit( i );
            bestX = pX( i, : );
          %  a(i)=fMin;
            
        end
    end
  
     Convergence_curve(t)=fMin;
  
    
  
end
% Application of simple limits/bounds
function s = Bounds( s, Lb, Ub)
  % Apply the lower bound vector
  temp = s;
  I = temp < Lb;
  temp(I) = Lb(I);
  
  % Apply the upper bound vector 
  J = temp > Ub;
  temp(J) = Ub(J);
  % Update this new move 
  s = temp;
function S = Boundss( SS, LLb, UUb)
  % Apply the lower bound vector
  temp = SS;
  I = temp < LLb;
  temp(I) = LLb(I);
  
  % Apply the upper bound vector 
  J = temp > UUb;
  temp(J) = UUb(J);
  % Update this new move 
  S = temp;

参考文献

[1] Xue J, Shen B. Dung beetle optimizer: A new meta-heuristic algorithm for global optimization[J]. The Journal of Supercomputing, 2023, 79(7): 7305-7336.

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