Linear constraint formulation (OR-statement)

Question

I have the decision variable $X_{iz}$

And I have two parameters $T_i\in\{0,1\}$ and $IT_z\in\{0,1,2\}$. I can only assign $i$ to $z$ if the following holds:

for $T_i=0$, $IT_z$ needs to be $0$ or $2$
for $T_i=1$, $IT_z$ needs to be $1$ or $2$
So for every value of $T_i$, a value of $2$ for $IT_z$ must satisfy the constraint
Or $T_i$ needs to be equal to $IT_z$

I cannot seem to figure out how to make a valid constraint for this problem, any tips?

Do you have 2 $T$ and 3 $IT$ or do you have one $T$ and one $IT$ where $T$ can be ${0, 1}$ and $IT$ can be ${0, 1, 2}$. If the latter, then you don't need the indices. Just say $T \in {0,1}$ and $IT \in {0,1,2}$ — EhsanK, Nov 22 '19 at 14:50
No I have multiple $i$'s and multiple $z$'s, each has their own value of $T\in{0,1}$ and $IT\in{0,1,2}$. — Harry van t Kamp, Nov 22 '19 at 14:53

score 5 · Accepted Answer · answered Nov 22 '19 at 20:09

5

For the new problem description, it seems like you just want to fix $X_{i,z}=0$ for some disallowed $(i,z)$ pairs. An even better approach is to avoid defining $X_{i,z}$ in those cases.

answered Nov 22 '19 at 20:09

RobPratt

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RobPratt · Answer 2 · 2019-11-22T15:04:35.047

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Introduce a new binary variable $z_i$ and linear constraints: \begin{align} -z_i \le \text{IT}_i - \text{T}_i &\le 2 z_i \\ \text{IT}_i &\ge 2 z_i \end{align} If $z_i=0$ then $\text{IT}_i = \text{T}_i$. If $z_i=1$ then $\text{IT}_i \ge 2$, hence $=2$.

edited Nov 22 '19 at 15:04

answered Nov 22 '19 at 13:46

RobPratt

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I think this is not correct, because when $T_i=0$ and $IT_i=1$ the constraint also satisfies. And this should not be the case. That is what I am struggling with. – Harry van t Kamp Nov 22 '19 at 14:25
Sorry, I misread $0$ or $2$ as $0$ to $2$. – RobPratt Nov 22 '19 at 14:54
Corrected just now. – RobPratt Nov 22 '19 at 15:04
Note: all your indices are of type $i$, but actually $IT$ has index $z$. Is the $z$ (you introduced) of type $i$ or of type $z$? I tried some things now, but the model seems infeasible – Harry van t Kamp Nov 22 '19 at 16:44
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Hmm, you changed the question after I posted my answer, and I don't quite understand the new question. My binary variable $z_i$ has nothing to do with your new index $z$. Is $X_{i,z}$ a binary decision variable? Are $\text{T}i$ and $\text{IT}{z}$ now decision variables or constants? – RobPratt Nov 22 '19 at 18:18
$X_{i,z}$ is a binary decision variable and $IT_z$ and $T_i$ are constants. I am sorry for the confusion. – Harry van t Kamp Nov 22 '19 at 18:21
Let us continue this discussion in chat. – Harry van t Kamp Nov 22 '19 at 18:25
Was this meant as an alternative to the other answer? – SecretAgentMan Nov 22 '19 at 22:24
No, it was the first answer, which answered the original problem that has now been lost. – RobPratt Nov 22 '19 at 22:37

Linear constraint formulation (OR-statement)

2 Answers2