Originally posted by: SystemAdmin

if x(i,j,k)=1 then A(i) >= B(j,k) ?
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Originally posted by: SystemAdmin

Let say x,A,B all are decision variables,
I need to write like this
if x(i,j,k) = 1 then A(i) >= B(j,k)

there is one possible way I could have a constraint like
A(i) >= B(j,k)* x(i,j,k),

But then that constraint would not be a liner it is quadratic. Do you know how to change quadratic constraints to liner constraints.
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Originally posted by: SystemAdmin

The best way is to just use an indicator constraint:

x(i,j,k) = 1 -> A(i) >= B(j,k)

CPLEX supports those logical implications and would transform it into a big-M constraint automatically if it thinks this is useful in terms of numerics and performance. If not, it just leaves it as a logical condition and deals with it by branching.

If you want to create the big-M constraint manually (which I do not recommend), then you have to know an upper bound for the term B(j,k) - A(i), for example as the upper bound of B(j,k) minus the lower bound of A(i). If M is an upper bound for B(j,k) - A(i), then the constraint looks as follows:

B(j,k) - A(i) + Mx(i,j,k) <= M

As you can see, if x = 0 then the constraint is redundant because

B(j,k) - A(i) <= M

is always satisfied (since M is an upper bound for the difference). On the other hand, if x = 1 then the constraint reads

B(j,k) - A(i) <= 0

which is exactly what you want.
Tobias
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Originally posted by: SystemAdmin

How do i pick the M?
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Originally posted by: SystemAdmin

As I said: M must be an upper bound for B(j,k) - A(i). For example, if you know that B(j,k) <= 1000 and A(i) >= 100, then M = 900 would be valid. If you do not have such an upper bound, then you cannot model this as a big-M constraint and have to rely on indicators.
But in any case, I still recommend to use indicators, even if you know some finite M.
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Originally posted by: SystemAdmin

But B(j,k) and A(i) are decsion variables,So how do I know the bounds?
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Originally posted by: SystemAdmin

indicators?
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Originally posted by: SystemAdmin

If you do not know what indicator constraints are, then please read the user's manual, for example here: http://publib.boulder.ibm.com/infocenter/cosinfoc/v12r3/index.jsp, and then navigate to CPLEX > User's Manual for CPLEX > Discrete optimization > Indicator constraints in optimization.
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Originally posted by: SystemAdmin

With "decision variables" you mean binary variables?
Then the bounds are of course 0 and 1, which means M = 1.
So, you get

B(j,k) - A(i) + x(i,j,k) <= 1

In this case, there is no difference with using indicators or big-M. CPLEX would convert an indicator constraint automatically into this "big-M" form.
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