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How to plot non-quadratic functions?

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Melda Hasiloglu
Melda Hasiloglu 2023-1-30
编辑： Torsten 2023-2-1
Hi everyone,
I have 3 non-quadratic/nonlinear profit functions (see below) in terms of 3 parameters which are alfa, F and delta.
To find for which alfa and F values which profit function is optimum, I need to compare these 3 profit functions and draw a plot based on alfa (x-axis) and F (y-axis) also alfa=F=[0, 1].
Manually I know how to solve it by I couldn't find a way to code it in Matlab.
I would appreciate if anyone can help me. Many thanks in advance!
totalprofit1=(8*F*(delta^3 - 3*delta^2 + 4*delta - 2))/((delta - 2)^2*(8*F*delta - 8*alfa - 4*delta - 12*F + 8*alfa*delta - 2*alfa*delta^2 - 4*alfa^2*delta + 4*alfa^2 + delta^2 + alfa^2*delta^2 + 4))
totalprofit2 =((delta - 1)^2*(- alfa^2 + 2*alfa + 4*F - 1)*(alfa^2 - 2*alfa - 4*F + 2*F*delta + 1)^2)/(4*(- 32*F^3*delta^3 + 112*F^3*delta^2 - 128*F^3*delta + 48*F^3 + 8*F^2*alfa^2*delta^3 - 56*F^2*alfa^2*delta^2 + 88*F^2*alfa^2*delta - 40*F^2*alfa^2 + 8*F^2*alfa*delta^4 - 40*F^2*alfa*delta^3 + 136*F^2*alfa*delta^2 - 184*F^2*alfa*delta + 80*F^2*alfa - 8*F^2*delta^4 + 32*F^2*delta^3 - 80*F^2*delta^2 + 96*F^2*delta - 40*F^2 + 6*F*alfa^4*delta^2 - 18*F*alfa^4*delta + 11*F*alfa^4 - 2*F*alfa^3*delta^4 + 12*F*alfa^3*delta^3 - 40*F*alfa^3*delta^2 + 78*F*alfa^3*delta - 44*F*alfa^3 + 5*F*alfa^2*delta^4 - 34*F*alfa^2*delta^3 + 83*F*alfa^2*delta^2 - 126*F*alfa^2*delta + 66*F*alfa^2 - 4*F*alfa*delta^4 + 32*F*alfa*delta^3 - 70*F*alfa*delta^2 + 90*F*alfa*delta - 44*F*alfa + F*delta^4 - 10*F*delta^3 + 21*F*delta^2 - 24*F*delta + 11*F + alfa^6*delta - alfa^6 - alfa^5*delta^3 + 2*alfa^5*delta^2 - 7*alfa^5*delta + 6*alfa^5 + 5*alfa^4*delta^3 - 10*alfa^4*delta^2 + 20*alfa^4*delta - 15*alfa^4 - 10*alfa^3*delta^3 + 20*alfa^3*delta^2 - 30*alfa^3*delta + 20*alfa^3 + 10*alfa^2*delta^3 - 20*alfa^2*delta^2 + 25*alfa^2*delta - 15*alfa^2 - 5*alfa*delta^3 + 10*alfa*delta^2 - 11*alfa*delta + 6*alfa + delta^3 - 2*delta^2 + 2*delta - 1))
totalprofit3 =-((delta - 2)^2*(2*F^2*delta^5 - 19*F^2*delta^4 + 72*F^2*delta^3 - 136*F^2*delta^2 + 128*F^2*delta - 48*F^2 + F*alfa^2*delta^4 - 10*F*alfa^2*delta^3 + 35*F*alfa^2*delta^2 - 52*F*alfa^2*delta + 28*F*alfa^2 - 2*F*alfa*delta^4 + 20*F*alfa*delta^3 - 70*F*alfa*delta^2 + 104*F*alfa*delta - 56*F*alfa + F*delta^4 - 14*F*delta^3 + 57*F*delta^2 - 92*F*delta + 52*F - alfa^4*delta^2 + 4*alfa^4*delta - 4*alfa^4 + 4*alfa^3*delta^2 - 16*alfa^3*delta + 16*alfa^3 - 6*alfa^2*delta^2 + 28*alfa^2*delta - 30*alfa^2 + 4*alfa*delta^2 - 24*alfa*delta + 28*alfa - delta^2 + 10*delta - 13))/(4*(20*F*delta - 8*alfa - 6*delta - 12*F + 8*alfa*delta - 11*F*delta^2 + 2*F*delta^3 - 2*alfa*delta^2 - 4*alfa^2*delta + 4*alfa^2 + delta^2 + alfa^2*delta^2 + 7)^2)

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采纳的回答

Torsten
Torsten 2023-1-30
编辑：Torsten 2023-1-30
Ran in:
I suggest you evaluate the three profit functions for a rectangular region [alpha_min,alpha_max] x [ F_min,F_max] and introduce a third variable profit_max that is 1 if the first profit function is maximum, 2 if the second profit function is maximum and 3 if the third profit function is maximum. Then use "contourf" to plot z in the region [alpha_min,alpha_max] x [ F_min,F_max].
delta = 1;
totalprofit1=@(F,alfa)(8*F*(delta^3 - 3*delta^2 + 4*delta - 2))/((delta - 2)^2*(8*F*delta - 8*alfa - 4*delta - 12*F + 8*alfa*delta - 2*alfa*delta^2 - 4*alfa^2*delta + 4*alfa^2 + delta^2 + alfa^2*delta^2 + 4));
totalprofit2 =@(F,alfa)((delta - 1)^2*(- alfa^2 + 2*alfa + 4*F - 1)*(alfa^2 - 2*alfa - 4*F + 2*F*delta + 1)^2)/(4*(- 32*F^3*delta^3 + 112*F^3*delta^2 - 128*F^3*delta + 48*F^3 + 8*F^2*alfa^2*delta^3 - 56*F^2*alfa^2*delta^2 + 88*F^2*alfa^2*delta - 40*F^2*alfa^2 + 8*F^2*alfa*delta^4 - 40*F^2*alfa*delta^3 + 136*F^2*alfa*delta^2 - 184*F^2*alfa*delta + 80*F^2*alfa - 8*F^2*delta^4 + 32*F^2*delta^3 - 80*F^2*delta^2 + 96*F^2*delta - 40*F^2 + 6*F*alfa^4*delta^2 - 18*F*alfa^4*delta + 11*F*alfa^4 - 2*F*alfa^3*delta^4 + 12*F*alfa^3*delta^3 - 40*F*alfa^3*delta^2 + 78*F*alfa^3*delta - 44*F*alfa^3 + 5*F*alfa^2*delta^4 - 34*F*alfa^2*delta^3 + 83*F*alfa^2*delta^2 - 126*F*alfa^2*delta + 66*F*alfa^2 - 4*F*alfa*delta^4 + 32*F*alfa*delta^3 - 70*F*alfa*delta^2 + 90*F*alfa*delta - 44*F*alfa + F*delta^4 - 10*F*delta^3 + 21*F*delta^2 - 24*F*delta + 11*F + alfa^6*delta - alfa^6 - alfa^5*delta^3 + 2*alfa^5*delta^2 - 7*alfa^5*delta + 6*alfa^5 + 5*alfa^4*delta^3 - 10*alfa^4*delta^2 + 20*alfa^4*delta - 15*alfa^4 - 10*alfa^3*delta^3 + 20*alfa^3*delta^2 - 30*alfa^3*delta + 20*alfa^3 + 10*alfa^2*delta^3 - 20*alfa^2*delta^2 + 25*alfa^2*delta - 15*alfa^2 - 5*alfa*delta^3 + 10*alfa*delta^2 - 11*alfa*delta + 6*alfa + delta^3 - 2*delta^2 + 2*delta - 1));
totalprofit3 =@(F,alfa)-((delta - 2)^2*(2*F^2*delta^5 - 19*F^2*delta^4 + 72*F^2*delta^3 - 136*F^2*delta^2 + 128*F^2*delta - 48*F^2 + F*alfa^2*delta^4 - 10*F*alfa^2*delta^3 + 35*F*alfa^2*delta^2 - 52*F*alfa^2*delta + 28*F*alfa^2 - 2*F*alfa*delta^4 + 20*F*alfa*delta^3 - 70*F*alfa*delta^2 + 104*F*alfa*delta - 56*F*alfa + F*delta^4 - 14*F*delta^3 + 57*F*delta^2 - 92*F*delta + 52*F - alfa^4*delta^2 + 4*alfa^4*delta - 4*alfa^4 + 4*alfa^3*delta^2 - 16*alfa^3*delta + 16*alfa^3 - 6*alfa^2*delta^2 + 28*alfa^2*delta - 30*alfa^2 + 4*alfa*delta^2 - 24*alfa*delta + 28*alfa - delta^2 + 10*delta - 13))/(4*(20*F*delta - 8*alfa - 6*delta - 12*F + 8*alfa*delta - 11*F*delta^2 + 2*F*delta^3 - 2*alfa*delta^2 - 4*alfa^2*delta + 4*alfa^2 + delta^2 + alfa^2*delta^2 + 7)^2);
n = 100;
alfa = linspace(0,1,n);
F = linspace(0,1,n);
[F,alfa] = meshgrid(F,alfa);
for i=1:n
for j=1:n
f = F(i,j);
a = alfa(i,j);
z1 = totalprofit1(f,a);
z2 = totalprofit2(f,a);
z3 = totalprofit3(f,a);
[~,index] = max([z1,z2,z3]);
profit_max(i,j) = index;
end
end
contourf(F,alfa,profit_max)
colorbar
  4 个评论
显示 2更早的评论
Melda Hasiloglu
Melda Hasiloglu 2023-2-1
Hi Torsten,
I add some conditions for each profit function but it doesn't work can you please have a look at?
My aim is: For example, if totalprofit1 is maximum and if it meets the condition 1 then color it blue, if doesn't meet than color it black.
totalprofit1=@(alfa,F)(8*F*(delta^3 - 3*delta^2 + 4*delta - 2))/((delta - 2)^2*(8*F*delta - 8*alfa - 4*delta - 12*F + 8*alfa*delta - 2*alfa*delta^2 - 4*alfa^2*delta + 4*alfa^2 + delta^2 + alfa^2*delta^2 + 4));
totalprofit2 =@(alfa,F)((delta - 1)^2*(- alfa^2 + 2*alfa + 4*F - 1)*(alfa^2 - 2*alfa - 4*F + 2*F*delta + 1)^2)/(4*(- 32*F^3*delta^3 + 112*F^3*delta^2 - 128*F^3*delta + 48*F^3 + 8*F^2*alfa^2*delta^3 - 56*F^2*alfa^2*delta^2 + 88*F^2*alfa^2*delta - 40*F^2*alfa^2 + 8*F^2*alfa*delta^4 - 40*F^2*alfa*delta^3 + 136*F^2*alfa*delta^2 - 184*F^2*alfa*delta + 80*F^2*alfa - 8*F^2*delta^4 + 32*F^2*delta^3 - 80*F^2*delta^2 + 96*F^2*delta - 40*F^2 + 6*F*alfa^4*delta^2 - 18*F*alfa^4*delta + 11*F*alfa^4 - 2*F*alfa^3*delta^4 + 12*F*alfa^3*delta^3 - 40*F*alfa^3*delta^2 + 78*F*alfa^3*delta - 44*F*alfa^3 + 5*F*alfa^2*delta^4 - 34*F*alfa^2*delta^3 + 83*F*alfa^2*delta^2 - 126*F*alfa^2*delta + 66*F*alfa^2 - 4*F*alfa*delta^4 + 32*F*alfa*delta^3 - 70*F*alfa*delta^2 + 90*F*alfa*delta - 44*F*alfa + F*delta^4 - 10*F*delta^3 + 21*F*delta^2 - 24*F*delta + 11*F + alfa^6*delta - alfa^6 - alfa^5*delta^3 + 2*alfa^5*delta^2 - 7*alfa^5*delta + 6*alfa^5 + 5*alfa^4*delta^3 - 10*alfa^4*delta^2 + 20*alfa^4*delta - 15*alfa^4 - 10*alfa^3*delta^3 + 20*alfa^3*delta^2 - 30*alfa^3*delta + 20*alfa^3 + 10*alfa^2*delta^3 - 20*alfa^2*delta^2 + 25*alfa^2*delta - 15*alfa^2 - 5*alfa*delta^3 + 10*alfa*delta^2 - 11*alfa*delta + 6*alfa + delta^3 - 2*delta^2 + 2*delta - 1));
totalprofit3 =@(alfa,F)-((delta - 2)^2*(2*F^2*delta^5 - 19*F^2*delta^4 + 72*F^2*delta^3 - 136*F^2*delta^2 + 128*F^2*delta - 48*F^2 + F*alfa^2*delta^4 - 10*F*alfa^2*delta^3 + 35*F*alfa^2*delta^2 - 52*F*alfa^2*delta + 28*F*alfa^2 - 2*F*alfa*delta^4 + 20*F*alfa*delta^3 - 70*F*alfa*delta^2 + 104*F*alfa*delta - 56*F*alfa + F*delta^4 - 14*F*delta^3 + 57*F*delta^2 - 92*F*delta + 52*F - alfa^4*delta^2 + 4*alfa^4*delta - 4*alfa^4 + 4*alfa^3*delta^2 - 16*alfa^3*delta + 16*alfa^3 - 6*alfa^2*delta^2 + 28*alfa^2*delta - 30*alfa^2 + 4*alfa*delta^2 - 24*alfa*delta + 28*alfa - delta^2 + 10*delta - 13))/(4*(20*F*delta - 8*alfa - 6*delta - 12*F + 8*alfa*delta - 11*F*delta^2 + 2*F*delta^3 - 2*alfa*delta^2 - 4*alfa^2*delta + 4*alfa^2 + delta^2 + alfa^2*delta^2 + 7)^2);
alfa = 0:0.005:1;
f = 0:0.005:1;
delta=0.8
[Alfa,F] = meshgrid(alfa,f);
[~,idx] = arrayfun(@(Alfa,F)max([totalprofit1(Alfa,F),totalprofit2(Alfa,F),totalprofit3(Alfa,F)]),Alfa,F);
if max([totalprofit1(Alfa,F),totalprofit2(Alfa,F),totalprofit3(Alfa,F)])==totalprofit1(Alfa,F)
%%condition 1
if - (Alfa - 1)^2 - (2*F*(4*delta - 6))/(delta - 2)^2>0 &((Alfa - 1)*(2*N - 4*delta - 4*Alfa*delta + 6))/(delta + Alfa*delta - 2)^2<0 & - (Alfa - 1)^2 - (2*F*(Alfa - 1)*(2*Alfa - 4*delta - 4*Alfa*delta + 6))/(delta + Alfa*delta - 2)^2 >0
contourf(Alfa,F,idx, 'b')
else
contourf(Alfa,F,idx, 'k')
end
end
if max([totalprofit1(Alfa,F),totalprofit2(Alfa,F),totalprofit3(Alfa,F)])==totalprofit2(Alfa,F)
%%condition 2
if 4*F>(1-Alfa)^2 & delta <(0.5*(4.0*F + Alfa - 1.0*(16.0*F^2 - 8.0*F*Alfa + 8.0*F + 4.0*Alfa^3 - 11.0*Alfa^2 + 10.0*Alfa - 3.0)^(1/2) - 1.0))/(Alfa - 1.0) & delta< (1.0*(F*Alfa - 2.0*Alfa - 4.0*F + Alfa^2 + 1.0))/(2.0*F*Alfa - 2.0*Alfa - 4.0*F + Alfa^2 + 1.0)
contourf(Alfa,F,idx, 'g')
else
contourf(Alfa,F,idx, 'k')
end
end
if max([totalprofit1(Alfa,F),totalprofit2(Alfa,F),totalprofit3(Alfa,F)])==totalprofit3(Alfa,F)
%%condition 3
if -(4*(Alfa^2 - 2*Alfa - 3*F + 2*F*delta + 1))/(delta - 2)^2>0 & Alfa <1 & (3*F+F*Alfa-2*F*delta+Alfa*delta^2-delta^2-2*F*Alfa*delta)>0
contourf(Alfa,F,idx, 'b')
else
contourf(Alfa,F,idx, 'k')
end
end
colorbar
Torsten
Torsten 2023-2-1
编辑：Torsten 2023-2-1
Ran in:
Maybe you can even choose the color for a certain value of Fun. Ad hoc, I can't.
Don't forget to set N to the correct value.
delta=0.8;
N = 1;
totalprofit1=@(alfa,F)(8*F*(delta^3 - 3*delta^2 + 4*delta - 2))/((delta - 2)^2*(8*F*delta - 8*alfa - 4*delta - 12*F + 8*alfa*delta - 2*alfa*delta^2 - 4*alfa^2*delta + 4*alfa^2 + delta^2 + alfa^2*delta^2 + 4));
totalprofit2 =@(alfa,F)((delta - 1)^2*(- alfa^2 + 2*alfa + 4*F - 1)*(alfa^2 - 2*alfa - 4*F + 2*F*delta + 1)^2)/(4*(- 32*F^3*delta^3 + 112*F^3*delta^2 - 128*F^3*delta + 48*F^3 + 8*F^2*alfa^2*delta^3 - 56*F^2*alfa^2*delta^2 + 88*F^2*alfa^2*delta - 40*F^2*alfa^2 + 8*F^2*alfa*delta^4 - 40*F^2*alfa*delta^3 + 136*F^2*alfa*delta^2 - 184*F^2*alfa*delta + 80*F^2*alfa - 8*F^2*delta^4 + 32*F^2*delta^3 - 80*F^2*delta^2 + 96*F^2*delta - 40*F^2 + 6*F*alfa^4*delta^2 - 18*F*alfa^4*delta + 11*F*alfa^4 - 2*F*alfa^3*delta^4 + 12*F*alfa^3*delta^3 - 40*F*alfa^3*delta^2 + 78*F*alfa^3*delta - 44*F*alfa^3 + 5*F*alfa^2*delta^4 - 34*F*alfa^2*delta^3 + 83*F*alfa^2*delta^2 - 126*F*alfa^2*delta + 66*F*alfa^2 - 4*F*alfa*delta^4 + 32*F*alfa*delta^3 - 70*F*alfa*delta^2 + 90*F*alfa*delta - 44*F*alfa + F*delta^4 - 10*F*delta^3 + 21*F*delta^2 - 24*F*delta + 11*F + alfa^6*delta - alfa^6 - alfa^5*delta^3 + 2*alfa^5*delta^2 - 7*alfa^5*delta + 6*alfa^5 + 5*alfa^4*delta^3 - 10*alfa^4*delta^2 + 20*alfa^4*delta - 15*alfa^4 - 10*alfa^3*delta^3 + 20*alfa^3*delta^2 - 30*alfa^3*delta + 20*alfa^3 + 10*alfa^2*delta^3 - 20*alfa^2*delta^2 + 25*alfa^2*delta - 15*alfa^2 - 5*alfa*delta^3 + 10*alfa*delta^2 - 11*alfa*delta + 6*alfa + delta^3 - 2*delta^2 + 2*delta - 1));
totalprofit3 =@(alfa,F)-((delta - 2)^2*(2*F^2*delta^5 - 19*F^2*delta^4 + 72*F^2*delta^3 - 136*F^2*delta^2 + 128*F^2*delta - 48*F^2 + F*alfa^2*delta^4 - 10*F*alfa^2*delta^3 + 35*F*alfa^2*delta^2 - 52*F*alfa^2*delta + 28*F*alfa^2 - 2*F*alfa*delta^4 + 20*F*alfa*delta^3 - 70*F*alfa*delta^2 + 104*F*alfa*delta - 56*F*alfa + F*delta^4 - 14*F*delta^3 + 57*F*delta^2 - 92*F*delta + 52*F - alfa^4*delta^2 + 4*alfa^4*delta - 4*alfa^4 + 4*alfa^3*delta^2 - 16*alfa^3*delta + 16*alfa^3 - 6*alfa^2*delta^2 + 28*alfa^2*delta - 30*alfa^2 + 4*alfa*delta^2 - 24*alfa*delta + 28*alfa - delta^2 + 10*delta - 13))/(4*(20*F*delta - 8*alfa - 6*delta - 12*F + 8*alfa*delta - 11*F*delta^2 + 2*F*delta^3 - 2*alfa*delta^2 - 4*alfa^2*delta + 4*alfa^2 + delta^2 + alfa^2*delta^2 + 7)^2);
alfa = 0:0.005:1;
f = 0:0.005:1;
for i = 1:numel(alfa)
Alfa = alfa(i);
for j = 1:numel(f)
F = f(j);
t1 = totalprofit1(Alfa,F);
t2 = totalprofit2(Alfa,F);
t3 = totalprofit3(Alfa,F);
[tmax,idx] = max([t1,t2,t3]);
if idx==1
if - (Alfa - 1)^2 - (2*F*(4*delta - 6))/(delta - 2)^2>0 &((Alfa - 1)*(2*N - 4*delta - 4*Alfa*delta + 6))/(delta + Alfa*delta - 2)^2<0 & - (Alfa - 1)^2 - (2*F*(Alfa - 1)*(2*Alfa - 4*delta - 4*Alfa*delta + 6))/(delta + Alfa*delta - 2)^2 >0
Fun(i,j) = 1;
else
Fun(i,j) = 2;
end
end
if idx==2
if 4*F>(1-Alfa)^2 & delta <(0.5*(4.0*F + Alfa - 1.0*(16.0*F^2 - 8.0*F*Alfa + 8.0*F + 4.0*Alfa^3 - 11.0*Alfa^2 + 10.0*Alfa - 3.0)^(1/2) - 1.0))/(Alfa - 1.0) & delta< (1.0*(F*Alfa - 2.0*Alfa - 4.0*F + Alfa^2 + 1.0))/(2.0*F*Alfa - 2.0*Alfa - 4.0*F + Alfa^2 + 1.0)
Fun(i,j) = 3;
else
Fun(i,j) = 4;
end
end
if idx==3
if -(4*(Alfa^2 - 2*Alfa - 3*F + 2*F*delta + 1))/(delta - 2)^2>0 & Alfa <1 & (3*F+F*Alfa-2*F*delta+Alfa*delta^2-delta^2-2*F*Alfa*delta)>0
Fun(i,j) = 5;
else
Fun(i,j) = 6;
end
end
end
end
colormap(jet(6))
contourf(alfa,f,Fun)
colorbar('Ticks',1:6,'TickLabels',["1" "2" "3" "4" "5" "6" " "]);

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更多回答（1 个）

Alan Weiss
Alan Weiss 2023-1-30
This sounds like a multiobjective optimization problem. See Generate and Plot Pareto Front and, if you have Global Optimization Toolbox, Multiobjective Optimization.
Alan Weiss
MATLAB mathematical toolbox documentation
  2 个评论
Melda Hasiloglu
Melda Hasiloglu 2023-1-30
Hi Alan,
Thanks for your reply. Did you mean that I can solve this by using the same approach as multiobjective optimization? Because my problem isn't multiobjective and the provided profit functions are already the optimal functions which I obtained them after several calculations. My current goal is to compare these 3 scenarios based on parameters, alfa and F.
Alan Weiss
Alan Weiss 2023-1-30
I do not understand your problem. You say that you have three objective functions. But you say that you do not have a multiobjective problem. So what are you trying to do? I do not understand what "compare these 3 scenarios based on parameters, alfa and F" means.
Alan Weiss
MATLAB mathematical toolbox documentation

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