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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)

Koinos
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1 Answers1

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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 $0 \leq x_i \leq 1$.

LinAlg
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  • 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 ) – Koinos Oct 07 '18 at 13:08
  • 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. – LinAlg Oct 07 '18 at 13:11
  • 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 – Koinos Oct 07 '18 at 13:17
  • I think your are right, i changed the model. – Koinos Oct 07 '18 at 13:20
  • don't you get $+s_{j-1}$? – LinAlg Oct 07 '18 at 13:21
  • It's correct now ? – Koinos Oct 07 '18 at 13:23
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    hard to say without the full context, but it is not obviously wrong – LinAlg Oct 07 '18 at 13:28
  • Ps: i came back to read your first reply. Even if $s_{j}$ can assume real values, is this model an example of integer programming due to the fact that $x$ is a integer vector ? – Koinos Oct 07 '18 at 13:42
  • it is typically called mixed integer optimization – LinAlg Oct 07 '18 at 14:19