Drawing Relation between two set of variables
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Greetings,
In my modeling, I have two set of variables. X[0..n] and Y[0..n]. X is independent of Y. But I want Y to be dependent on X. Using some optimality constraints, X [0..n] is getting populated. Now, i need to populate Y based on additional constrains, only for those cells, for which X[i] is not equal to zero.
Is the following right way ?
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Hi Sesha,
Do you like to implement the constraint below?
\[\mbox{if} ~~ X_i \neq 0, \mbox{then}~~ X_i Y_i \geq 0.001, ~~ i=0, \cdots, n\]Assuming \(X\) and \(Y\) are continuous variables, you would need to introduce a binary variable \(Z\) to model the following:
- \[\mbox{if} ~~ X_i \neq 0, \mbox{then}~~Z_i = 1, ~~ i=0, \cdots, n\]
- \[\mbox{if} ~~ Z_i = 1, \mbox{then}~~ X_i Y_i \geq 0.001, ~~ i=0, \cdots, n\]
The first indicator constraint can be modelled using the following linear constraints where \(M\) is a sufficiently large value representing the domain of the \(X\) variables.
\(-M Z_i \leq X_i \leq M Z_i, ~~ i=0, \cdots, n\): This gurantees that if \(X_i \neq 0\), then \(Z_i\) has to be 1.
The second indicator constraint can then be modelled as below:
\( X_i Y_i \geq 0.001 - M (1 -Z_i), ~~ i=0, \cdots, n\): This gurantees that if \(Z_i = 1\) (i.e., \(X_i \neq 0\)), then \(X_i Y_i \geq 0.001\).
Best regards,
Maliheh
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Thank you Maliheh for you reply. Unfortunately, that is not the constraint that I want to implemt.
I have variables X[0..n] which are getting populated based on some constraints. Some of those can be zero. Now, I want another set of variables Y[0..n] that should be populated with optimized values again based on some other constraints. I want only those variables Y[i] to be populated with NZ values if X[i] is NZ.
Thanks,
Shesha.
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Unfortunately, that is not the constraint that I want to implemt.
Could you please write the mathematical formulation of the constraint you would like to implement? The previous post implements the constraint below:
\[\mbox{if} ~~ X_i \neq 0, \mbox{then}~~ X_i Y_i \geq 0.001, ~~ i=0, \cdots, n\]How does the constraint you intend to implement differ from the above constraint?
Maliheh
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Hi Maliheh ,
I see your point. I am new to gurobi. How do I program indicator constraints in Python? In my example X is a continuous variable in the range of {0..1}.
Regards,
Shesha.0 -
This is my attempt to model it.
M = 1
Z = [[None for j in range(NCONTAINERS)] for i in range (NVARS)]
EPSILON = 0.001
for i in range (NVARS):
for j in range (NCONTAINERS):
varname = "Z_%02d_%02d" % (i, j)
Z[i][j] = opt_mod.addVar(vtype=gp.GRB.BINARY, name=varname)
varname = "GT_%02d_%02d" % (i, j)
opt_mod.addConstr(X[i][j] >= -M * Z[i][j], name = varname)
varname = "LT_%02d_%02d" % (i, j)
opt_mod.addConstr(X[i][j] <= M * Z[i][j], name = varname)
varname = "SAMEBLK_%02d_%02d" % (i, j)
opt_mod.addConstr(X[i][j] * Y[i][j] >= EPSILON - M * (1 - Z[i][j]), name = varname)However, I am not seeing the correct results :(
X_00_00 0.0 Y_00_00 32.0
X_01_00 0.0 Y_01_00 16.0
X_02_00 0.0 Y_02_00 16.0
X_03_00 0.0 Y_03_00 32.0
X_04_00 0.0 Y_04_00 32.0
X_05_00 0.0 Y_05_00 32.0
X_06_00 0.0 Y_06_00 32.0
X_07_00 0.0 Y_07_00 32.0
X_08_00 0.0 Y_08_00 6.0
X_09_00 0.0 Y_09_00 32.0
X_10_00 0.0 Y_10_00 48.0
X_11_00 0.333 Y_11_00 76.0
X_00_01 0.0 Y_00_01 32.0
X_01_01 0.0 Y_01_01 16.0
X_02_01 0.0 Y_02_01 16.0
X_03_01 0.0 Y_03_01 32.0
X_04_01 0.0 Y_04_01 32.0
X_05_01 0.0 Y_05_01 32.00 -
Hi Sesha,
The results look consistent with the constraints implemented. The variable \(X_{11,00}\) equals a non-zero value (\(0.333\)) and its associated \(Y_{11,00}\) variable equals \(76\) such that \(0.333 \times 76 = 25.308 \geq 0.001\). What else would you like to see? Would you like to further enforce \(Y\) equals 0 if \(X\) equals 0?
Maliheh
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Yes, exactly. :)
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You need to add the following constraints where \(M\) is a sufficiently large value covering the domain of the \(Y\) variables.
- \(Y_i \leq X_i + MZ_i\)
- \(Y_i \geq X_i - MZ_i\)
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Thanks a lot for answering all my queries. Sincerely appreciate it.
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