How to optimise five equations and four unknowns?

Hi, I am looking to find the optimal power output for four generators given the equations and conditions:

0.31246x1 + 0.15x2 + 0.15x3 + 0.15x4 = 46,

0.15x1 + 0.3x2 + 0.15x3 + 0.15x4 = 40,

0.15x1 + 0.15x2 + 0.3x3 + 0.15x4 = 55,

0.15x1 + 0.15x2 + 0.15x3 + 0.3x4 = 50.5,

x1+x2+x3+x4=200

25<=x1<=100

5.5<=x2<=30

20<=x3<=130

25.5<=x4<=160

To maximise x1, x2, x3 and x4.

Please help.

[Torsten]

You can only maximize a single value, not four of them (here: x1,x2,x3,x4) simultaneously.

[Sophie Cai]

How do I maximise one single value? Do you have an example?

Answer 1

[Alan Weiss]

You have five equations in four unknowns, which means that you probably have no solution to your problem as stated. Consider solving the first four equations:

A = [0.31246 0.15 0.15 0.15
0.15 0.3 0.15 0.15 
0.15 0.15 0.3 0.15
0.15 0.15 0.15 0.3 ];
b = [46
40
55
50.5];
c = A\b

c =

   48.1346
   12.1330
  112.1330
   82.1330

sum(c)

ans =

254.5337

So you cannot satisfy all of your equations. What would you like to do? Find a least-squares solution? Something else?

Here is a least-squares solution:

A = [A;1 1 1 1];
b = [b;200];
c = A \ b

c =

   40.1980
   -1.1451
   98.8549
   68.8549

sum(c)
ans =

206.7627

Alan Weiss

MATLAB mathematical toolbox documentation

[Sophie Cai]

How do I maximise one single value? Do you have an example?