Mdthbs

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 $iin I$ 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:

$$x_{tip} le u_{tip}cdot M quad quad forall t in T,iin I, p in P_i$$
$$u_{tip} = u_{t+1ip} quad quad forall t in T,iin I, p in P_i$$

$$sum_{p∈P_i} u_{tip} le 1 quad quad forall t in T,iin I$$
$$u_{tip} in {0,1}quad quad forall t in T,iin I, p in P_i$$

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 $P_1in{1,3}$. 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!

$I$: 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. $sum_{iin I} s_{tipk} =1 forall t,k$ (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).