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
- there exists a unique solution
- the system is analytically tractable
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 $[1, \infty)$