Mixed-Integer Linear Programming (Capacity Planning)

Question

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!

Answer 1

$I$: Number of products

$|P_i|$: Number of available processes for product $i$

$|p|$: Number of machines in each process $p$

You can define two new binary variables for each $k$ machine as follow:

$s_{tipk} = \left\{\begin{array}{l}1 & \text{if machine $k$ in time $t$ under process $p$ is working on product $i$}\\0 & \text{otherwise}\end{array}\right.$

$q_{pk} = \left\{\begin{array}{l}1 & \text{if machine $k$ is among the ones that are being used in process $p$}\\0 & \text{otherwise}\end{array}\right.$

Now you need to add the following constraints to the model:

1. $\sum_{i\in I} s_{tipk} =1 \ \ \ \forall \ t,k$ (each machine can produce only one type of product at each time period)

2. $\sum_{k} q_{pk} =|p|*u_{tip} \ \ \ \forall \ \ i,t$

3. $s_{tipk} \leq q_{pk} \ \ \ \forall \ \ t,k,i,p$

I believe this answer will give you at least some hints on how to model the problem (if it hasn't already covered all the necessary details).