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Let $I$, $J$ and $K$ denote index sets. There are two kinds of variables $x_{j,k} \in [0, 1]$ and $y_{i,k} \in [0, 8760]$ and two kinds of constants $a_{i,j,k} \geq 0$ and $b_j \geq 0$. I'm considering a system of $(|I| \cdot |K| + 1) \cdot |J|$ equations \begin{align} &0 = x_{j,k} \cdot b_j \cdot y_{i,k} - a_{i,j,k}\\[2mm] &0 = \sum_{k \in K}{x_{j,k}} - 1 \end{align} in $(|I| + |J|) \cdot |K|$ unknowns. I assume $|I|, |J| \geq 2$ such that the system is always overdetermined for all $|K|$. As the system is rather simple I was wondering whether

  1. there exists a unique solution
  2. the system is analytically tractable
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1 Answer 1

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For the first part, for the case when $b_j =1, a_{i,j,k} = 0 \,\, \forall i,j,k$, we can have multiple solutions. Moreover, when $\exists i,j,k \,\,\,\, b_j = 0, a_{i,j,k} > 0$, there are no solutions.

As for tractability, we need to do a few cases

Case 1: If there exists a $j$ such that $b_j =0$ and $a_{i,k,j} > 0$ for any $i,k$ then there is no solution that satisfies all the constraints.

Case 2: In case $b_j =0$, we have $a_{i,j,k} = 0$ for all $i,k$. In this case, neglect the equations with $b_j=0$.

If there is a $j$ such that $\exists i,k \,\, a_{i,j,k}/b_j > 8760$, there is no solution.

Let $c_{i,j,k} = a_{i,j,k}/b_j$

Hence, we have that $x_{j,k} = c_{i,j,k}/y_{i,k}$ from the first set of constraints.

Now, we can use the second set of constraints to solve for $y_{i,k}$. The problem can be converted to to solving set of linear equations with variables in the range $[1, \infty)$

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  • $\begingroup$ Regarding your second to last sentence I would have come up with $y_{i,k} = c_{i,j,k} / (1- \sum_{\ell \in K \setminus \{k\}}{x_{j,\ell}}) = c_{i,j,k} / (1- \sum_{\ell \in K \setminus \{k\}}{c_{i,j,\ell} / y_{i,\ell}})$. But I don't know about the last sentence. Would you mind stating the set of linear equations? $\endgroup$
    – clueless
    Commented Sep 9, 2022 at 7:20
  • $\begingroup$ Let $z_{i,k} = 8760/y_{i,k}$. First set is, $c_{i,j,k} - c_{i,j',k} = 0$. Second is, $\sum_{k\in K} c_{i,j,k}z_{i,k} = 8760$ $\endgroup$
    – Bridgesign
    Commented Sep 9, 2022 at 15:51

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