Define the functions $g_j(t)(j=1.\mathrm{.}4)$ as follows:

$g_1(t)=t$

$g_2(t)=t-7$

$g_3(t)=\{\begin{array}{ccc}(t-5{)}^2& if& t5\end{array}$

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$g_4(t)=2$max$\{0,t-3\}$

Develop the optimal schedule. Show all your work!

Interpretations of $g_j(t)$ :

Completion time of job $1$

Lateness of job $2,$ where $d_2=7$

Squared tardiness of job $3,$ where $d_3=5$

Weighted tardiness of job $4,$ where $w_4=2$ and

$d_4=3$

Provide a Gantt chart of the optimal schedule, and report its optimal objective function value. Consider the following $1$ prec $$hmax$\{$giCi$\}$ problem instance with precedence constraints: $2->4$ and $2->3$