Optimization Routine for two equations, three unknowns

Hi,

I am trying to set up a basic optimization routine in matlab.

Equation 1: 3.6*x*y = 750

Equation 2: 19.8*x*y*z = 800000

x = 6:1:12

y = 20:1:40

z = 150:1:250

Is there a tool/method to solve/optimize this ?

Thank You.

Answer 1

[Walter Roberson]

>> [X,Y,Z] = ndgrid(6:12,20:40,100:150);
>> mask = (3.6.*X.*Y == 750) & (1.98.*X.*Y.*Z == 800);
>> nnz(mask)
ans =
     0

There are no solutions under those constraints.

[Walter Roberson]

Those equations require that z = 32/165 independently of x and y. Notice that 32/165 is not even close to the z range you indicated -- it is roughly 0.1939 .

[DIP]

Sorry about that, I missed out on a unit conversion. It should be 800000

[Walter Roberson]

Under that circumstance then z must be 6400/33 which is about 193.9393, which at least is in the range you give, but is not one of the choices 150:1:250 . z must be that value exactly for the equations to hold.

[DIP]

So I basically need to treat this as a 2 equation and 2 unknown and optimize for X and Y keeping Z 193.93. How is it done ?

[Walter Roberson]

No, just examine it algebraically.

>> syms x y z
>> Y = solve(3.6*x*y == 750,y)
Y =
625/(3*x)
>> Z = solve(subs(19.8*x*y*z == 800000,y,Y),z)
Z =
6400/33;

>> lbx = solve(Y==20, x)
lbx =
125/12
>> ubx = solve(Y==40, x)
ubx =
125/24

so x must be in the range 125/24 to 125/12 for y to be in the ranges you specified. That is 5.20833 to 10.4166 .

Your x limit starts at 6 and ends at 12. 6 is inside the y range and 12 is outside the y range. So for everything to work out, x must be 6 to 125/12.

>> double(subs(Y,x,6))
ans =
          34.7222222222222
>> double(subs(Y,x,125/12))
ans =
    20

which fits the given limits.

So given continuous values instead of the discrete values you had asked for, x is any value in the range 6 to 125/12, and y = 625/3 * x, and z is 6400/33

[DIP]

Thanks Walter. I appreciate the help.