index constraint

回答済み

Rayan SAAD

2023年06月13日 13:18

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Helloo,

I have 2 binary variables Z[i,j,e] and u[i,j,e],

I want to express this constraint :

Z[i,j]+u[i,j] <= 1 (regardless of e)

I dont know how to express it in Gurobipy,

Thank you !!

関連

• Modeling

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  Riley Clement

  • Gurobi Staff

  2023年06月14日 02:42 編集済み

  Hi Rayan,

  Let's say i, j, e, come from sets I, J and E respectively.   If I understand correctly you are wanting to model  constraints of the form

  \[z_{ije} + u_{ije'} \leq 1, \quad \forall i \in I, j \in J, e \in E, e' \in E\]

  If you have defined z and u like so

  z = model.addVars(I,J,E, ...)
  u = model.addVars(I,J,E, ...)

  then the you can add the constraints like so

  model.addConstrs(
      z[i,j,e] + u[i,j,e2] <= 1
      for i in I
      for j in J
      for e in E
      for e2 in E
  )

  However you can likely make this formulation stronger if both z and u variables are binary.  Consider a set of  binary variables, x, and a set of binary variables y and the constraints

  \[x_m + y_n \leq 1, \quad \forall m \in M, n \in N\]

  If you think of each variable as representing a node in a graph, and an edge exists between nodes if they linked by the above constraints, then what you have is a complete bipartite graph.  One set of variables in a solution must always be zero.  The moment any x variable has value 1, all y variables must be 0.  The moment any y variable has value 1, all x variables must be zero.

  We can therefore introduce a binary variable w which is used to decide which set will be 0.  We can then replace the constraints above with the following

  \begin{align}
  x_m \leq w, \quad \forall m \in M\\
  y_n \leq 1-w, \quad \forall n \in N
  \end{align}

  which is likely to run much faster.

  You have this exact situation for each specific i and j, and so you can introduce binary variables w

  w = model.addVars(I,J, vtype="B")

  and use the following constraints instead of the ones above

  model.addConstrs(
     z[i,j,e] <= w[i,j]
     for i in I
     for j in J
     for e in E
  )
  model.addConstrs(
    u[i,j,e] + w[i,j] <= 1
    for i in I
    for j in J
    for e in E
  )

  If your z and u variables are not binary then this is all irrelevant though.

  - Riley

   

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  Rayan SAAD

  2023年06月20日 14:42

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  Thank youu !!

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