Constraint to state the relation between 2 binary variables

Question

I'm trying to deal with a process planning and machine layout allocation simultaneously.

I have the following variables:

• $X_p{_w}_{cj}=1$ if an operation $p$ is done by a machine $w$ with a configuration $c$ at process plan position $j$, and zero otherwise

• $T_w{_{w'}}_{,jj+1}=1$ if there is a change of machine $w$ between position $j$ and $j+1$, and zero otherwise.

• $C_w{_{cc'}}_{,jj+1}=1$ if for a given machine $w$ there is a change of configuration between position $j$ and $j+1$, and zero otherwise.

Since the variable $X_p{_w}_{cj}$ gives me the position of each machine and configuration on the process plan, I think I must establish a relation between this variable and $T_w{_{w'}}_{,jj+1}$ and $C_w{_{cc'}}_{,jj+1}$, respectively.

In order to do that, I created the following constraint:

$$\sum_{c}(X_p{_w}_{cj}+X_{p+1}{_{w'}}_{cj+1})\leqslant T_w{_{w'}}_{,jj+1} + 1$$

With this constraint I would like to state that for 2 followed positions on the process plan, the sum of variables $X_p{_w}_{cj}$ must be equal or less than the variable $T_w{_{w'}}_{,jj+1}+1$. In other words, this constraint states if there is a change of machine between $j$ and $j+1$.

Similarly, the following constraint states if there is a change of configuration for the same machine between $j$ and $j+1$.

$$X_p{_w}_{cj}+X_{p+1}{_w}_{{c'}j+1}\leqslant C_w{_{cc'}}_{,jj+1} + 1$$

I would like to know if it is correct or if there is a better way to express these relations. Could someone help me?

Answer 1

The following constraint should be correct:

$$\sum_{c}(X_p{_w}_{cj}+X_{p+1}{_{w'}}_{cj+1})\leq T_w{_{w'}}_{,jj+1} +1 \ \ \ \forall w,w',p,p+1,j,j+1$$

Because if in your machine layout allocation, the process flows from machine $w$ to machine $w'$ then $X_{pwcj}$ and $X_{p+1w'cj+1}$ both should be $1$. Otherwise, either $T_{ww',jj+1}=0$ that means one of the following situations:

1. your process is $\{...,w,...,w',...\}$ ($w$ is in position $j$ but $w'$ is not in position $j+1$)

2. your process is $\{...,w,...,w',...\}$ ($w$ is not in position $j$ but $w'$ is in position $j+1$)

3. your process is $\{...,w,...,w',...\}$ (neither $w$ is in position $j$ nor $w'$ is in position $j+1$)

The same logic procedure can be seen in your second constraint.