If you just want to know the magnitude of the overlap after optimization, then you can use the following formula for a given job $i$, if $\mbox{start}_i < 10800$:
$$\max\{0,\min\{\mbox{end}_i,10800\}-\max\{\mbox{start}_i,7200\}\} \tag{0}$$
If you want to use the overlap in your model as a variable, then you can proceed as follows:
Let $s,e$ denote the start and end time variables, respectively. Let $\ell, u$ denote the start and end times of the weekend, respectively.
Also, let $\delta_1 \in \{0,1\}$ be a binary variable that takes value $1$ if $\ell < s < u$, let $\delta_2 \in \{0,1\}$ be another binary variable that takes value $1$ if $\ell < e < u$, let $\delta_3 \in \{0,1\}$ be another binary variable that takes value $1$ if $e \le \ell$, and let $\delta_4 \in \{0,1\}$ be another binary variable that takes value $1$ if $s \ge u$.
And finally, let $O\in \mathbb{R}^+$ denote the magnitude of the overlap.
You want to model the following:
Or in algebraic terms:
\begin{align}
\ell < s < u \quad &\Longleftrightarrow \quad \delta_1 \tag{1} \\
\ell < e < u \quad &\Longleftrightarrow \quad \delta_2 \tag{2} \\
e \le \ell \quad &\Longleftrightarrow \quad \delta_3 \tag{3} \\
u \le s \quad &\Longleftrightarrow \quad \delta_4 \tag{4} \\
\delta_1 \wedge \delta_2 \quad &\Longrightarrow \quad O = e-s \tag{5} \\
\delta_1 \wedge \neg \delta_2 \quad &\Longrightarrow \quad O = u-s \tag{6} \\
\neg \delta_1 \wedge \delta_2 \quad &\Longrightarrow \quad O = e-\ell \tag{7} \\
\neg \delta_1 \wedge \neg \delta_2 \wedge (\delta_3 \vee \delta_4) \quad &\Longrightarrow \quad O = 0 \tag{8} \\
\neg \delta_1 \wedge \neg \delta_2 \wedge \neg \delta_3 \wedge \neg \delta_4 \quad &\Longrightarrow \quad O = u - \ell \tag{9}
\end{align}
For $(1)$, use the following big M constraints:
\begin{align}
\ell + \epsilon -M(1-\delta_1) \le s \le u - \epsilon + M(1-\delta_1) \tag{1a} \\
u -M(1-y_1) \le s \le \ell + M(1-y_2) \tag{1b} \\
\delta_1+y_1 +y_2 = 1 \tag{1c} \\
\delta_1,y_1, y_2 \in \{0,1\}
\end{align}
For $(2)$, use the following big M constraints:
\begin{align}
\ell + \epsilon-M(1-\delta_2) \le e \le u -\epsilon + M(1-\delta_2) \tag{2a}\\
u -M(1-y_3) \le e \le \ell + M(1-y_4) \tag{2b} \\
\delta_2+y_3 +y_4 = 1 \tag{2c} \\
\delta_2,y_3, y_4 \in \{0,1\}
\end{align}
For $(3)$, use the following big M constraints:
$$
\ell + \epsilon -M\delta_3 \le e \le \ell +M(1-\delta_3) \tag{3a}\\
\delta_3 \in \{0,1\}
$$
For $(4)$, use the following big M constraints:
$$
u - M(1-\delta_4)\le s \le u - \epsilon + M \delta_4 \tag{4a} \\
\delta_4 \in \{0,1\}
$$
For $(5)$, use the following big M constraints:
$$
e - s - M (2-\delta_1 - \delta_2) \le O \le e - s + M (2-\delta_1 - \delta_2) \tag{5a} \\
\delta_1,\delta_2 \in \{0,1\}
$$
For $(6)$, use the following big M constraints:
$$
u - s - M (1-\delta_1 + \delta_2) \le O \le u - s + M (1-\delta_1 + \delta_2) \tag{6a}\\
\delta_1,\delta_2 \in \{0,1\}
$$
For $(7)$, use the following big M constraints:
$$
e - \ell - M (1-\delta_2 + \delta_1) \le O \le e - \ell + M (1-\delta_2 + \delta_1) \tag{7a}\\
\delta_1,\delta_2 \in \{0,1\}
$$
For $(8)$, use the following big M constraints:
\begin{align}
0 - M (\delta_1 + \delta_2 + 1- \delta_3) \le O \le 0 + M (\delta_1 + \delta_2 + 1-\delta_3 ) \tag{8a} \\
0 - M (\delta_1 + \delta_2 + 1 -\delta_4) \le O \le 0 + M (\delta_1 + \delta_2 + 1-\delta_4 ) \tag{8b} \\
\delta_1,\delta_2,\delta_3,\delta_4 \in \{0,1\}
\end{align}
For $(9)$, use the following big M constraints:
$$
u - \ell - M (\delta_1 + \delta_2 + \delta_3 + \delta_4) \le O \le u - \ell + M (\delta_1 + \delta_2 + \delta_3 + \delta_4 ) \tag{9a}\\
\delta_1,\delta_2,\delta_3,\delta_4 \in \{0,1\}
$$
Note also that by definition $\delta_1 \; \Longrightarrow \; \neg \delta_3$, $\delta_1 \; \Longrightarrow \; \neg \delta_4$, $\delta_2 \; \Longrightarrow \; \neg \delta_3$, $\delta_2 \; \Longrightarrow \; \neg \delta_4$:
$$
\delta_1 \le 1- \delta_3 \\
\delta_1 \le 1- \delta_4 \\
\delta_2 \le 1- \delta_3 \\
\delta_2 \le 1- \delta_4 \\
$$
Note : Ideally, distinguish the different big $Ms$. Also, I would not be surprised if the above constraints could be simplified. In particular, I am not 100% sure if the double implication is required in constraints $(1)-(4)$.
Addendum
OP uses a formulation which linearizes equation $(0)$, without any binary variables. OP minimizes $O$ subject to
\begin{align}
O &\ge u - \ell - \alpha - \beta \\
\alpha &\ge s- \ell \\
\beta &\ge u -e \\
\alpha, \beta, O &\ge 0 \\
\end{align}
which is equivalent to
\begin{align}
O &\ge (u - \max\{u -e, 0\}) - (\ell + \max\{s- \ell, 0\})\\
O &\ge 0
\end{align}
or
\begin{align}
O &\ge \min\{u,e\} - \max\{s,\ell\}\\
O &\ge 0
\end{align}
which yields
$$
O = \max\{0,\min\{u,e\} - \max\{s,\ell\}\}
$$
Note however that this works because we are minimizing.
start[i]andend[i]are the variables for the start and end time for jobi, thenmax(end[i], 7200) - max(start[i], 7200)should give you the time of the job that is executed at the weekend. This assumes that jobs do terminate before Monday (as suggested by @CMichael). – Daniel Junglas Mar 31 '22 at 05:23