Regarding robust optimisation

[sh.mojta...@gmail.com]

Dear all,

I have a LP model here as follow:

Min = .42*x1 + .56*x2 + .70*x3;

S.t.

x1 + x2 + x3 = 900;

x1 <= 400 * y1;
x2 <= 700 * y2;
x3 <= 600 * y3;

30*x1 <= 12500;
40*x2 <= 20000;
50*x3 <=15000;

.15*x1 + .2*x2 +.15*x3 >= 100;
.2*x1 + .05*x2 + .2*x3 >= 100;
.25*x1 + .15*x2+ .05*x3 >= 150;

y1+y2+y3 = 2;

xi>=0,

yi=0, if x=o
yi=1, if x>=o

The constraints

.15*x1 + .2*x2 +.15*x3 >= 100;
.2*x1 + .05*x2 + .2*x3 >= 100;
.25*x1 + .15*x2+ .05*x3 >= 150;

have uncertainties in x1, x2, and x3 coefficients. I want to know how
can I make a robust optimisation model for this LP model?

for example, if we know that all the coefficients have variations
about 30%.

Thank you,
Shab

[Peter Spellucci]

In article <3b9efb53-6447-4755...@r15g2000prd.googlegroups.com>,

sh.mojta...@gmail.com writes:
>Dear all,
>
>I have a LP model here as follow:

no, this is a MILP and even with stochastic part.
I know of no ready to use software which combines these three features
in one code, but maybe there exists some.

But if this indeed your problem, not only a model,
then, with only three 0-1-variables you could simply try all possible
combinations. for the stochastic part, you could try stochastic LP
or, if you require strict bounds, then an interval arithmetic based
LP (with the 0-1-variables fixed in different combinations)
an overview on codes is obtainable from
http://plato.asu.edu/guide.html

there is an ambiguity in your model:
if x (you mean: x_i?) is zero, you allow y_i =0 or y_i =1 ?

hth
peter

[Johan Löfberg]

Assuming you want it to be feasible in the worst-case, you can
explicitly derive it

sum (a_i+d_i*delta_i)*xi >b where |delta_i|<1

You have a model where d_i = 0.3*a_i

write as

sum a_i*xi + sum 0.3*a_i*delta_i*xi > b where |delta_i|<1

worst case is when delta_i = -sign(a_i*x_i) and the explicit expression is

sum a_i*xi - sum 0.3*abs(a_i*xi) > b

This can be handled by introducing a new variable t_i to model the
absolute value

sum a_i*xi - sum 0.3*t_i > b
t_i > a_i*x_i > -t_i

FYI, these worst-case reformulations are done automatically in the
robust optimization framework in the MATLAB toolbox YALMIP.

/johan