Space constraint in scheduling problem

Question

I'm studying an MIP model with a scheduling problem and I'm wondering if the space constraint is correct.

If there is space in the factory at that time, the block can be processed, where

• $i$ is the block number

• $S_i$, $C_i$ and $P_i$ are the strat time, completion time and processing time of block $i$ respectively

• $y_{i,j,f}=\begin{cases}1\quad\text{if}\,i\,\text{process before}\,j\,\text{at factory}\,f\\0\quad\text{otherwise}\end{cases}$

• $y_{i,f}=\begin{cases}1\quad\text{if}\,i\,\text{process at factory}\,f\\0\quad\text{otherwise}\end{cases}$

• $M$ is a big number

• $sp_i$ is the required space of block $i$

• $c_{f,t}$ is the available space of factory $f$ at time $t$

\begin{align}S_i -M(1-y_{i,j,f}) &\le S_j\quad\forall i,j,f\\S_i + P_i &= C_i\quad \forall i\end{align}

The space constraint is $$sp_i \cdot y_{i,f} \le c_{f,t}\quad\forall i,j\quad\text{and}\quad t=S_i,\cdots,C_i.$$

Answer 1

No, your space constraint is incorrect. There are multiple problems, including the fact that the range of the index $t$ depends on variables $S_i$ and $C_i$. Note that your current space constraint would not stop two jobs from using the same capacity simultaneously. (Your precedence constraint forces one job to start before another, but does not force the second job to wait until the first job has completed.) Also, your precedence constraint should be $\le$, not $\lt$.