This is getting somewhat ugly. I would use indicator constraints; see section 14.2.4 in the Language Reference for more information.

p(i) = 1  →  a(i,j) <= C(i,j)*H(j)*(1-p(j)) + C(i,j)*p(j)

p(j) = 1  →  a(i,j) <= C(i,j)*H(i)*(1-p(i)) + C(i,j)*p(i)

p(i) = 0  →  a(i,j) <= C(i,j) * (H(i) + H(j))/2 * (1-p(j)) + M*p(j)

Here M is a big number (“big M”). The left part of each of these three constraints represents the activating condition.

Please note that indicator constraints are only supported by CPLEX and Gurobi.

Thanks very much!

That’s indeed a good idea to try with indicator constraints. I haven’t thought about it. I will give it a try!

Dear staff,

I would like to ask you further help for a potential extension of the constraint mentioned in the previous messages. (I don’t know if I should open a separate discussion for this, but I try to continue here for now).

Within a graph, I define arcs (i-j) where “i” and “j” are nodes. For each arc and node, I also have the following variables and parameters (basically, there is a new index “y” compared to the previous case):

p(i,y) binary variable

p(j,y) binary variable

a(i,j,y) continuous variable

C(i,j) parameter

H(i,y) parameter

H(j,y) parameter

I would like to write a constraint expressing the following conditions:

For all y,

if sum(y1|((ord(y1)<=ord(y)), p(i,y1)) = 1 and sum(y1|((ord(y1)<=ord(y)), p(j,y1)) = 0, then a(i,j,y) <= C(i,j)*H(j,y)

if sum(y1|((ord(y1)<=ord(y)), p(i,y1)) = 0 and sum(y1|((ord(y1)<=ord(y)), p(j,y1)) = 1, then a(i,j,y) <= C(i,j)*H(i,y)

if sum(y1|((ord(y1)<=ord(y)), p(i,y1)) = 0 and sum(y1|((ord(y1)<=ord(y)), p(j,y1)) = 0, then  a(i,j,y) <= C(i,j) * (H(i,y) + H(j,y))/2

if sum(y1|((ord(y1)<=ord(y)), p(i,y1)) = 1 and sum(y1|((ord(y1)<=ord(y)), p(j,y1)) = 1, then a(i,j,y) <= C(i,j)

If you have any suggestions on how to write the above extended conditions within a set of constraint, I will be glad. I am not sure about the possibility of including the summation together with the indicator constraints.

Thank you,

Hi @Chibo, i and j are indices of same set. So, create a new variable varSum(i, y) like below 

sumsy1|(ord(y1)<= ord(y)), p(i, y1)] = varSum(i, y)

sumsy1|(ord(y1)<= ord(y)), p(j, y1)] = varSum(j, y)

varSum can be binary as your post suggests that the above sums are either 0 or 1. 

Create another auxillary binary variable auxVar(y) with below constraints. forall (i, y)

auxVar(y) >= varSum(i, y)

auxVar(y) >= varSum(j, y)

auxVar(y) <= varSum(i, y) + varSum(j, y)

Now, for the constraint you would like to write - forall (i, j, y)

a(i, j, y) <= C(i, j) * )H(i, y)/2 + H(j, y)/2

- varSum(i)*H(i, y)/2 - varSum(j)*H(j, y)/2 

+ varSum(i, y) + varSum(j, y) - auxVar(y)] 

As H and C are parameters, there is no multiplication of variables here. the 4 cases in your constraint will evaluate as 

1. varSum(i, y) = 1, varSum(j, y) = 0 -> auxVar(y) = 1 & a(i, j, y) <= C(i, j) * H(j,y)/2

2. varSum(i, y) = 0, varSum(j, y) = 1 -> auxVar(y) = 1 & a(i, j, y) <= C(i, j) * H(i,y)/2

3. varSum(i, y) = 0, varSum(j, y) = 0 -> auxVar(y) = 0 & a(i, j, y) <= C(i, j) * (H(i,y)/2 + H(j, y)/2)

4. varSum(i, y) = 1, varSum(j, y) = 1 -> auxVar(y) = 1 & a(i, j, y) <= C(i, j) * 1
 

It might have gotten some of the indices wrong but this should work for your case. Using varSum is not necessary, you can directly type in the sum ..] expression but you will need the additional auxVar constraint. 

Thank you very much for your precious inputs.

I will try to implement them