Use index of decision variable in max_ constraint
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I have a set of decision variables to denote time, \(\texttt{T[n,i,k]}\), and for each \(\texttt{k}\), I want to get the maximum value of all the decision variables irrespective of \(\texttt{n,i}\).
For that, I introduced a new decision variable \(\texttt{V[k]}\) and use the \(\texttt{max_}\) constraint:
for each k in K:
m.addConstrs((V[k] == gp.max_(list(T.select('*','*',k)))),"Max_value")
Thus far it seems straightforward, and I get the constraint as desired.
The issue is that I need to do some follow up computations with \(\texttt{V[k]}\) for each \(\texttt{k}\), and for that I also require the index/value for \(\texttt{i}\) where \(\texttt{T[n,i,k]}\) is maximal.
Is it possible to extract this somehow? Maybe by adding another decision variable? I also looked if I could make the variable with two indices, \(\texttt{U[i,k]}\), but I couldn't figure out how to get the index information from the max_ constraint.
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You are on the right track. Here is one way to do this, though this formulation can certainly be modified a few different ways. I'll assume your index sets are \( N \), \( I \), and \( K \).
First, we introduce a variable \( y_{ik} \) for all \( i \in I \) and \( k \in K\). We set \( y_{ik} \) equal to \( \max_{n \in N} T_{nik} \):
y = m.addVars(I, K, name='y')
m.addConstrs((y[i, k] == gp.max_(T.select('*', i, k)) for i in I for k in K), name='max_value')Then, we add your \( V_k \) variables and set \( V_k \geq \max_{n \in N, i \in I} T_{nik} \) for all \( k \in K \):
V = m.addVars(K, name='V')
m.addConstrs((V[k] >= y[i, k] for i in I for k in K), name='v_con')Next, we introduce binary variables \( u_{ik} \) for all \( i \in I \) and \( k \in K \). For each \( k \), exactly one \( u_{ik} \) is \( 1 \). Additionally, \( u_{ik} \) is equal to \( 1 \) only if \( T_{nik} \) is "maximal" for some \( n \in N \). If multiple indices \( i \in I \) are maximal, then \( u_{ik} \) is \( 1 \) for only one of them.
u = m.addVars(I, K, vtype=gp.GRB.BINARY, name='u')
m.addConstrs((u.sum('*', k) == 1 for k in K), name='sum_to_one')Finally, for all \( i \in I \) and \( k \in K \), we add indicator constraints that set \( y_{ik} \geq V_k \) if \( u_{ik} = 1 \):
m.addConstrs(((u[i, k] == 1) >> (y[i, k] >= V[k]) for i in I for k in K), name='indicator')
Let's see if this formulation makes sense. Consider a fixed \( k \in K \). Let \( i \in I\) satisfy \( u_{ik} = 1 \). The indicator constraint gives us \( y_{ik} \geq V_{k} \). By our other constraints, \( \max_{n \in N} T_{nik} = y_{ik} \leq V_k \). Altogether, this gives us \( u_{ik} = 1 \implies \max_{n \in N} T_{nik} = V_k \), as desired.
Note that we moved the max_ constraints to the new \( y_{ik} \) variables instead of the \( V_k \) variables. We no longer need these constraints on the \( V_k \) variables. To see this, consider a fixed \( k \in K \). By the indicator constraints, there exists \( \hat{i} \in I \) such that \( \max_{n \in N} T_{n\hat{i}k} \geq V_k \). Together with the constraints \( V_k \geq \max_{n \in N} T_{nik} \) for all \( i \in I, \) this implies \( V_k = \max_{n \in N, i \in I} T_{nik} \).
Now, for a fixed \( k \in K \), we can use the expression \( \sum_{i \in I} i u_{ik} \) to obtain an \( i \in I \) satisfying \( \max_{n \in N} T_{nik} = V_k \).
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Hi, I got the same problem but I´m doing it on Python, do you know how can I write this:
T.select('*', i, k)on python?
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The code is written in Python. It assumes \( \texttt{T} \) is a tupledict object (see tupledict.select()), which is the type of object returned by Model.addVars(). For example:
>>> x = m.addVars(3, 4, 5, name='x')
>>> type(x)
<class 'gurobipy.tupledict'>
>>> m.update()
>>> x.select('*', 1, 2)
[<gurobi.Var x[0,1,2]>, <gurobi.Var x[1,1,2]>, <gurobi.Var x[2,1,2]>]0 -
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