My question is how to model a resource constraint with MIP model (how should I modify it?)
A factory is making blocks. If there is space left at that time, the block can be assembled. (If the factory has space, it can do multiple tasks at the same time.)
I'm asking this question because I don't know how to formulate it even though I've been thinking about it after getting feedback in the comments from prubin to my previous question: Space constraint in scheduling problem.
Here are my notations:
Index
$i,j$ is the block number
$f$ is factory number
$t$ is time
Parameters
$y_{i,f}=\begin{cases}1\quad\text{if}\,i\,\text{process at factory}\,f\\0\quad\text{otherwise}\end{cases}$
$sp_i$ is required area for each block
$c_f$ is available space of factory f
$P_i$ is procees time of each block
$M$ is a big number
Variables
$S_i$, $C_i$ the start time, completion time of block $i$ respectively
$y_{i,j,f}=\begin{cases}1\quad\text{if}\,i\,\text{process before}\,j\,\text{at factory}\,f\\0\quad\text{otherwise}\end{cases}$
$s_{i,f,t}=\begin{cases}1\quad\text{if}\,i\,\text{start at factory }\,f\text{ at time t }\quad\\0\quad\text{otherwise}\end{cases}$
$s2_{i,f,t}=\begin{cases}1\quad\text{if}\,i\,\text{process at factory }\,f\text{ at time t }\quad\\0\quad\text{otherwise}\end{cases}$
Constraints
$$ \sum_{f=1}\sum_{t=1} (y_{i,f} \cdot s_{i,f,t}) =1\quad\forall i \quad (1)$$
$$ \sum_{f=1}\sum_{t}(t \cdot s_{i,f,t}) = S_i \quad \forall i,f \quad (2)$$$$ \sum_{f=1}\sum_{t=1}(t \cdot s_{i,f,t}) = S_i \quad \forall i \quad (2)$$
$$ S_i+p_i = C_{i}\quad\quad \forall i \quad (3)$$
$$ S_i-M(1-y_{i,j,f}) \le S_{j} \quad\quad \forall i,j,f \quad (4)$$
I think this part is the problem because $t$ depends on variables $S_i$,$C_i$
I'm not sure why Problems running when running with gurobi. $$ \sum_{t=S_i}^{C_i} s2_{i,f,t} = p_{i}\quad\forall f\quad (5)$$
$$ \sum_{i} sp_i \cdot s2_{i,f,t} \le c_{f}\quad\forall f,t\quad (6)$$
thanks for your reading
