I am currently working on a formulation for a linear program of a complex problem. At the moment I am facing to formulate the following logical condition:
There are two binary variables. Let's name them $\beta$ and $\delta$, there are existing for $ \forall t \in T$. $\beta$ should take the value 1 only if $\delta^{t-1} = 1$ and $\delta^t = 0$. For all other cases, it should be zero.
My first approach was the following inequality: $\delta^t - \delta^{t-1} + \beta^t = 0 \quad \forall t \in T$
It's working for the described case, but it forbids also the feasible solutions of $\delta^{t-1} = 0$ and $\delta^{t} = 1$. Also $\delta^{t-1}=1$ and $\delta^{t}=1$ is feasible. Is there any way of formulating additional or other (inequality) equations to model this?
I guess with an additional binary variable I can formulate this but of course, i would like to prevent this.
Thanks for your help!