In these constraints $u_{g,t}^G$ is on at t=1, then $u_{g,t}^{G,start}$ should become 1 after $t_g^{G,start}$ time-steps then after it stays 1 for $t_g^{G,cr}$ time-steps then $u_{g,t}^{G,on}$ becomes 1. T is the complete time horizon. Is there any simple or alternate approach to these constraints?
$\sum\limits_{t \in{T} } {\left( {1 - u_{g,t}^{G,start}} \right)} = \left[ {\sum\limits_{t \in{T}} {\left( {1 - u_{g,t}^G} \right)} } \right] + t_g^{G,start},g \in {G}$, where $G=1,2,3,...N$
$\sum\limits_{t \in{T} } {\left( {1 - u_{g,t}^{G,on}} \right)} = \left[ {\sum\limits_{t \in{T}} {\left( {1 - u_{g,t}^G} \right)} } \right] + t_g^{G,start} + t_g^{G,cr},g \in{G} $ , where $G=1,2,3,...N$
For every g, there are two times, for example $t_g^{G,start}=2$ and $t_g^{G,cr}=3,$ and time $t=1,2,...,T$. For example in the MIP, at $t=3,$ the variable $u_{g,t}^G$ becomes $u_{g,t}^G=1,$ then $u_{g,t}^{G,start}$ becomes $u_{g,t}^{G,start}=1,$ at $t=5,$ and remains $1$ for the rest of the time interval $T$. After that $u_{g,t}^{G,on}$ becomes $u_{g,t}^{G,on}=1$ at $t=8$ and remains $1$ for the rest of the time interval $T$. I am trying to formulate a single constraint for this formulation.
If I understand correctly, you want to enforce a logical implication $x \implies \bigwedge_{i=1}^n y_i$, which you can do with linear constraints $x \le y_i$ or a single (weaker) aggregated linear constraint $n x \le \sum_{i=1}^n y_i$. The derivation via conjunctive normal form is: \begin{align} &x \implies \bigwedge_{i=1}^n y_i \\ &\lnot x \lor \bigwedge_{i=1}^n y_i \\ &\bigwedge_{i=1}^n (\lnot x \lor y_i) \\ &(1 - x) + y_i \ge 1 && \text{for $i \in \{1,\dots,n\}$} \\ &x \le y_i && \text{for $i \in \{1,\dots,n\}$} \\ \end{align}