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Until now i formulated some Linear Programming problems with integer constraints and some with continuous constraints. Now, i've written a linear programming model that both contains variables with integer constraints and with continuous constraints. Is it possible ?

$max \sum_{i=1}^{n} p_{i}x_{i}$

$ \sum_{i=1}^{n} a_{ij}x_{i}=b_{j}-s_{j}+s_{j-1}$ , $j=1..T$

$x_{i}\in \left \{ 0,1 \right \}$ , $i=1..n$

$s_{j}>=0$ , $j=1..T$

Is this a valid linear programming model ? (x and s are variables)

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    $\begingroup$ This is a community. Please go over your previous question and indicate if the person's answer was helpful to you. $\endgroup$ Commented Oct 7, 2018 at 13:05
  • $\begingroup$ You are right, i completely forgot. I'm very sorry $\endgroup$ Commented Oct 7, 2018 at 13:06

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Yes, that's perfectly possible. The criterion is whether it is a linear optimization problem if you replace $x_i \in \{0,1\}$ with $0 \leq x_i \leq 1$.

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  • $\begingroup$ Thanks so much. I tried to model that "the budget available in each year is equal to a fixed rate of the remaining budget of previous periods"; do you think is a correct model ? (a is the cost and p is the profit ) $\endgroup$ Commented Oct 7, 2018 at 13:08
  • $\begingroup$ I do not see why you get $b_j+s_j + s_{j+1}$ (both the $b_j$ and the two plus signs), and where the "fixed rate" kicks in. $\endgroup$ Commented Oct 7, 2018 at 13:11
  • $\begingroup$ I think your are right, i changed the model. ps:the $b_{j}$ is the fixed budget, $s_{j-1}$ is the surplus of j-1-th equation, and $s_{j}$ is the surplus at j-th equation $\endgroup$ Commented Oct 7, 2018 at 13:17
  • $\begingroup$ I think your are right, i changed the model. $\endgroup$ Commented Oct 7, 2018 at 13:20
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    $\begingroup$ hard to say without the full context, but it is not obviously wrong $\endgroup$ Commented Oct 7, 2018 at 13:28

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