Modeling XNOR in Pyomo - linear-programming

Question

I am writing a nurse patient matching algorithm, and I want to incorporate something into the objective function that measures how well the new patient-nurse assignments match the patient-nurse assignments from the previous day. I've introduced a new binary variable model.MATCHES_PREVIOUS which should be equal to the XNOR operation of model.Prev_Assignments and model.Assignments (if both are 0 or both are 1, then model.MATCHES_PREVIOUS is 1, otherwise it is 0).
#Sets
model.PatientIDs = {0, 1, 2}
model.NurseIDs = {a, b}
Indexed parameter (indexed by (patient, nurse))
model.Prev_Assignments = {(0, a): 1, (0, b): 0, (1, a) : 0, (1, b) : 1, (2, a) : 1, (2, b) : 0}
Indexed variable (indexed by (patient, nurse))
model.Assignments = {(0, a): 1, (0, b): 0, (1, a) : 0, (1, b) : 1, (2, a) : 0, (2, b) : 1}
model.Matches_Previous = {(0, a): 1, (0, b): 1, (1, a) : 1, (1, b) : 1, (2, a) : 0, (2, b) : 0}
Currently, I'm trying to implement this with the below constraint, which I thought was the proper way to translate XNOR into a linear expression:
def matches_previous(self, patient, nurse):
return model.Matches_Previous[patient, nurse] == (model.Assignments[patient, nurse] * model.Prev_Assignments[patient, nurse]) + (1 - model.Assignments[patient, nurse]) * (1 - model.Prev_Assignments[patient, nurse])
In the objective function, I include (-model.Matches_Previous) along with the other components (since I am minimizing the objective, this would maximize model.Matches_Previous).
However, this isn't giving the behavior I want. Note that there are other aspects of the model which would motivate it to produce assignments that are different from the previous assignments (e.g. changing patient workloads), but I want it to match the previous assignments as best as possible.
Any idea how to implement this better? I looked into Pyomo.GDP and LogicalConstriants, but I haven't been able to get this to work and the documentation is lacking for these modeling extensions.

Answer 1

We always can write
z = x xnor y
as
z >= 1-x-y
z <= 1-x+y
z <= 1+x-y
z >= x+y-1
where x,y,x are binary variables. This is rigorous (derivation is here) and tight (we can even relax z to be continuous between 0 and 1).
Sometimes we can drop the <= or >= inequalities because of how the objective (or constraint) works.

Answer 2

So here is a strategy you might try. Note your attempt above makes the problem non-linear by multiplying variables, which is probably NOT desired.
You can code the XNOR variable in linear fashion, assuming there is some "pressure" on it... Let me explain.
In pseudocode:
Let X = binary variable
Let X_prev = other binary variable
Let P = binary variable {1: if X, X_prev are not equal, else 0}
So, we intend to make P a penalty that can be indexed (optionally) and summed up, then introduce 2 constraints
P >= X - X_prev
P >= X_prev - X
And in your objective function, use P to induce a penalty not a reward and it should work... something like:
Obj = <some other components> + p_weight * sum(P) ; minimized