clc, clear all % The code is Matlab code % Any one be so kind and help me, Thank

Question

clc, clear all

% The code is Matlab code
% Any one be so kind and help me, Thank alot !

% I have a 4x4 matrix X and I want to optimize and find the minimum
% element of the matrix so that the sum of all the min will be the
%optimum = 207.
% The min only alow to occur once, either in a column or in a row.
% In other words, the min are strictly not alow to be in the same row or
% column the optimum will be 207, but i run this code i got
% optimum = 205, which is not correct.

X=[54 54 51 53; 51 57 52 52; 50   53   54   56; 56   54   55   53];

for i=1:length(X)
for j=1:length(X)   
A(i,:)=min(X(i,:));   
Optimum=sum(A);   
end
end
A
Optimum

Explanation / Answer

X=[54 54 51 53; 51 57 52 52; 50 53 54 56; 56 54 55 53];

[row column]=size(X);

board = zeros(1,row);
chosen = zeros(1,row);

optimum = sum(sum(X));
for i = 1:row
board(1)=i;
[board optimum] = isSolution(X,board,2,optimum);
  
end
optimum

%% isSolution.m %%

function [board optimum] = isSolution(X,board,depth,optimum)
[row column]=size(X);
if depth == 5
for i = 1:column
r=board(i);
chosen(i)=X(r,i);
end
if sum(chosen) < optimum
optimum=sum(chosen);
end
return
end

for i = 1:row
if isSafe(board,depth,i)
board(depth)=i;
[board optimum] = isSolution(X,board,depth+1,optimum);
end
end
end

%% isSafe.m %%

function value = isSafe(board,depth,counter)

for i = 1:depth-1
if board(i) == counter
value=false;
return;
end
end
value=true;
return;
end

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