how to define binary decision variable in multi objective optimization (gamultiobj) problem

8 Ansichten (letzte 30 Tage)
limon saha
limon saha am 3 Sep. 2019
Bearbeitet: Matt J am 4 Sep. 2019
Our purpose is to solve a multi objective optimization problem using genetic algorithm. The decision variables x(9) to x(20) and x(29) to x(36) in the objective functions are binary. My question is how can i define those decision variables as binary?
objective = @sssobjective;
nvars = 36;
lb = [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];
ub = [inf,inf,inf,inf,inf,inf,inf,inf,inf,1,1,1,1,1,1,1,1,1,1,1,1,inf,inf,inf,inf,inf,inf,inf,1,1,1,1,1,1,1,1];
A = [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1000000 0 0 0 0 0 0 0;
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1000000 0 0 0 0 0 0;
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1000000 0 0 0 0 0;
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1000000 0 0 0 0;
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1000000 0 0 0;
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1000000 0 0;
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1000000 0;
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1000000;
1 1 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
1 1 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
1 1 0 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 1 1 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 1 1 0 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 1 1 0 0 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 1 1 0 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
-1 -1 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
-1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
-1 -1 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
-1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
-1 -1 0 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
-1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 -1 -1 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 -1 -1 0 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 -1 -1 0 0 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 1000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0;
];
b = [62000;
62000;
78000;
78000;
80000;
80000;
77000;
77000;
0;
0;
0;
0;
0;
0;
0;
0;
1004000;
1004000;
1008000;
1008000;
2000000;
2000000;
1008000;
1008000;
1015000;
1015000;
2000000;
2000000;
1005000;
1005000;
1010000;
1010000;
2000000;
2000000;
1010000;
1010000;
1020000;
1020000;
2000000;
2000000;
1000000
1000000;
995999;
995999;
991999;
991999;
1000000;
1000000;
991999;
991999;
984999;
984999;
1000000;
1000000;
994999;
994999;
989999;
989999;
1000000;
1000000;
989999;
989999;
979999;
979999;
64000;
80000;
72000;
88000;
30400;
];
Aeq = [ 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 ;
0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 ;
0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 ;
0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 1 -1 0 0 0 0 0 0 0 0 ;
0.04 0 0 0 0.03 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0.04 0 0 0 0.03 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0.04 0 0 0 0.03 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0.04 0 0 0 0.03 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
];
beq = [ 64000;
80000;
72000;
88000;
3200;
4000;
3600;
4400;
1;
];
nonlcon = @ncons;
IntCon=[];
options = optimoptions(@gamultiobj,'PlotFcn',@gaplotpareto);
[x,fval] = gamultiobj(objective,nvars,A,b,Aeq,beq,lb,ub,nonlcon,options);
# The multi objective functions are defined as follows
function y = sssobjective(x)
y(1)= x(1)*7*1 + x(2)*7*1 +x(3)*8*1 + x(4)*8*1 + x(5)*8*1 + x(6)*8*1 + x(7)*10*1 + x(8)*10*1 + x(1)*7*x(9) + x(1)*6*x(10) + x(1)*5.5*x(11) + x(2)*7*x(9) + x(2)*6*x(10) + x(2)*5.5*x(11)+ x(3)*8*x(12)+ x(3)*7*x(13) + x(3)*6*x(14) + x(4)*8*x(12) + x(4)*7*x(13) + x(4)*6*x(14) + x(5)*8*x(15) + x(5)*7.5*x(16) + x(5)*6*x(17) + x(6)*8*x(15) + x(6)*7.5*x(16) + x(6)*6*x(17) + x(7)*10*x(18) + x(7)*9*x(19) + x(7)*8*x(20) + x(8)*10*x(18) + x(8)*9*x(19) + x(8)*8*x(20) + 70000*x(29) + 70000*x(30) + 85000*x(31) + 85000*x(32) + 100000*x(33) + 100000*x(34) + 90000*x(35) + 90000*x(36) + 6*x(21)+ 6*x(22)+ 7*x(23)+ 7*x(24)+ 172*x(1) + 172*x(2) + 172*x(3) + 172*x(4) + 126*x(5) + 126*x(6) + 126*x(7) + 126*x(8) +1.6*x(25) + 1.5*x(26) + 2*x(27) + 2.2*x(28) ;
y(2)= -(0.39421*x(1) +0.39421*x(2) +0.39421*x(3) +0.39421*x(4) + 0.28560*x(5)+ 0.28560*x(6)+ 0.28560*x(7)+ 0.28560*x(8));
y(3)= -(.244360*x(1) +.244360*x(2) +.244360*x(3) +2.24436*x(4) + .51067*x(5)+ .51067*x(6)+ .51067*x(7)+ .51067*x(8));
nonlinear constraints are defined as follows
function [c,ceq] = ncons(x)
c=[x(1)* x(9)*7+ x(1)*x(10)*6 + x(1)*x(11)*5.5 + x(2)*x(9)*7 + x(2)*x(10)*6 + x(2)*x(11)*5.5 + x(3)*x(12)*8 + x(3)*x(13)*7 + x(3)*x(14)*6 + x(4)*x(12)*8 + x(4)*x(13)*7 + x(4)*x(14)*6 + x(5)*x(15)*8 + x(5)*x(16)*7.5 +x(5)*x(17)*6 + x(6)*x(15)*8 + x(6)*x(16)*7.5 + x(6)*x(17)*6 + x(7)*x(18)*10 + x(7)*x(19)*9 + x(7)*x(20)*8 + x(8)*x(18)*10 + x(8)*x(19)*9 + x(8)*x(20)*8 -1000000 ];
ceq=[];
end

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Antworten (1)

Matt J
Matt J am 4 Sep. 2019
Bearbeitet: Matt J am 4 Sep. 2019
There must be a reason why integer constraints are not supported, but you could try to enforce binariness through non-linear equality constraints,
function [c,ceq] = ncons(x)
c=[x(1)* x(9)*7+ x(1)*x(10)*6 + x(1)*x(11)*5.5 + x(2)*x(9)*7 + x(2)*x(10)*6 + x(2)*x(11)*5.5 + x(3)*x(12)*8 + x(3)*x(13)*7 + x(3)*x(14)*6 + x(4)*x(12)*8 + x(4)*x(13)*7 + x(4)*x(14)*6 + x(5)*x(15)*8 + x(5)*x(16)*7.5 +x(5)*x(17)*6 + x(6)*x(15)*8 + x(6)*x(16)*7.5 + x(6)*x(17)*6 + x(7)*x(18)*10 + x(7)*x(19)*9 + x(7)*x(20)*8 + x(8)*x(18)*10 + x(8)*x(19)*9 + x(8)*x(20)*8 -1000000 ];
xbin=x([9:20,29:36]);
ceq=xbin.*(xbin-1);
end
I would never suggest this for a local optimizer, but since ga() is global, it might be worth a try.

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