I'm currently developing a small capacity planning problem and right now I am struggling with the "activation" of a subset. Needless to say I am not an expert in this kind of things.
I have a set of $i\in I$ products. Each product $i$ can be produced by $p∈P_i$ processes. Each process $p$ requires a set of different machines $w∈W_p$. Decision variable $x_{tip}$ represents the quantity of product $i$ produced in period $t$ through process $p$. The quantity is subject to change in each period to satisfy the demand $D_{ti}$. However, the company must stick to the process selected in $t=1$ for the whole planning period $T$. This is indicated by the binary variable $u_{tip}$. I already formulated the relevant constraints to force this behavior:
$$x_{tip} \le u_{tip}\cdot M \quad \quad \forall t \in T,i\in I, p \in P_i$$ $$u_{tip} = u_{t+1ip} \quad \quad \forall t \in T,i\in I, p \in P_i$$
$$\sum_{p∈P_i} u_{tip} \le 1 \quad \quad \forall t \in T,i\in I$$ $$u_{tip} \in \{0,1\}\quad \quad \forall t \in T,i\in I, p \in P_i$$
So now I want to use $u_{tip}$ to activate all machines that are required for the selected process $p$. However, I am at my wit's end and just have no clue how to implement this behavior linearly.
Example: Assume we have a product $i=1$ that can be produced by processes $P_1\in\{1,3\}$. Disregarding the other products, process $1$ is selected for product $1$, so $u_{t11}=1$. Process $1$ requires machines $W_1 \in \{2,3\}$. Is there a way to use a new indicator variable, e.g. $y_{tiw}$, to "activate" all machines $w \in W_1$ for product $1$?
As I try to minimize my set-up costs, I need to know whether a machine $w$ is used by a product $i$ in period $t$. I tried it with $$u_{tip} \le y_{tiw} \quad \quad \forall t \in T, i \in I, p \in P_i, w \in W_i$$ but after toying around I don't think this works. Note that every process $p$ may have a different no. of required machines $w$.
As I said I am currently struggling to find a way to make this work linearly. I tried to find similar papers in the operations management literature but that wasn't successful either. I would gladly appreciate any help or hints/references on similar work. Thank you!