集成在PlatEMOv2.9上的PSP代码

PSP（Pareto Sets Proximity）反映了得到的PSs与真实PSs之间的相似性

$$PSP=\frac{CR}{IGDX}$$

其中$CR$为覆盖率，$IGDX$为决策空间的反转世代距离。

CR公式如下：
$${\left(\prod _{i=0}^n{\sigma }_l\right)}^{1/2n}$$

• 其中${\sigma }_l$ 如下：
  $${\sigma }_l\{\begin{array}{ll}1& V_l^{max}=V_l^{min}\\ 0& v_l^{min}\ge V_l^{max}\left|\right|v_l^{max}\le V_l^{min}\\ {\left(\frac{min(v_l^{max},V_l^{max})-max(v_l^{min},V_l^{min})}{V_l^{max}-V_l^{max}}\right)}^2& otherwise\end{array}$$
  在这个式中式中$n$为决策空间的维数；$v_l^{max}$和 $v_l^{min}$分别为第$l$个变量得到的$PS$的最大值和最小值；$V_l^{max}$和$V_l^{min}$是第$l$个变量的真$PS$的最大值和最小值。

PSP不仅能反映得到PS的收敛性，还能表示真PS与得到PS的重叠率。更大的PSP值是可取的。

下面是matlab代码

function Score=PSP(Parameter)
% <metric> <max>
% PSP: Calculate the Pareto Sets Proximity
%% Input:
%                              Dimension                                                                    Description
%      obtained_ps     population_size x n_var                                                 Obtained Pareto set     
%      reference_ps    num_of_solutions_in_reference_ps x n_var                     Reference Pareto set
%      fname——the type of the test problem

%% Output:
%                           Description
%      Score             Pareto Sets Proximity
    IGDX_Score = IGDX(Parameter);
    CR_Score = CR_calculation(Parameter);
    Score = CR_Score / IGDX_Score;
end
function CR_Score=CR_calculation(Parameter)
% CR_calculation: Calculate the cover rate of the obtained Pareto set

% Dimension: n_var --- dimensions of decision space

%% Input:
%                              Dimension                                                                    Description
%      PopDec     population_size x n_var                                                 Obtained Pareto set     
%      PS    num_of_solutions_in_reference_ps x n_var                     Reference Pareto set

%% Output:
%                       Description
%      CR             Cover rate of the obtained Pareto set
[~,~,PopDec,PS,fname,~] = deal(Parameter{:});
n_var=size(PS,2);
%find the maximum and minimum in each dimension of obtained Pareto set
obtained_min=min(PopDec);
obtained_max=max(PopDec);
%find the maximum and minimum in each dimension of reference Pareto set
reference_min=min(PS);
reference_max=max(PS);
for i=1:n_var
    if reference_max(i)==reference_min(i)||(strcmp(fname,'MMF9')&&i==2)||(strcmp(fname,'MMF10')&&i==2)...
            ||(strcmp(fname,'MMF11')&&i==2)||(strcmp(fname,'MMF12')&&i==2)||(strcmp(fname,'MMF20')&&i==2)||(strcmp(fname,'MMF14')&&i==3)||(strcmp(fname,'MMF15')&&i==3)
        kesi(i)=1;% if the i_th value of reference Pareto set are all the same, ignore the i_th dimension.
    elseif obtained_min(i)>=reference_max(i)||reference_min(i)>=obtained_max(i)
         kesi(i)=0;
    else
        kesi(i)=((min(obtained_max(i),reference_max(i))-max(obtained_min(i),reference_min(i)))/...
        (reference_max(i)-reference_min(i)))^2;
    end
end
CR_Score=nthroot(prod(kesi),2*n_var);
end