I don't know what kinds of programming model it is with an indicator function in the constrained. Thanks for providing any keywords!
Maxmize $30R_1+20R_2+12R_3+15R_4$
Subject to: $0\leq R_{1} \leq 20\\ 0\leq R_{1}+R_{2} \leq 40\\ 0\leq R_{1}+R_{2}+R_{3} \leq 80\\ 0\leq R_{1}+R_{2}+R_{3}+R_{4} \leq 90\\ 20 \times g_{1}(20) \leq R_{1} \leq 20 \times g_{2}(20) \\ 30 \times g_{1}(40-R_1) \leq R_{2} \leq 30 \times g_{2}(40-R_1) \\ 50 \times g_{1}(80-R_1-R_2) \leq R_{3} \leq 50 \times g_{2}(80-R_1-R_2) \\ 40 \times g_{1}(90-R_1-R_2-R_3) \leq R_{4} \leq 40 \times g_{2}(90-R_1-R_2-R_3) \\$
$ g_{1}(S_i)=\begin{cases} 0.9&\text{, if } 30 < S_{i}\\ 0.5&\text{, if } S_{i} \leq 30\\ \end{cases} \qquad i=1,2,\dots,4 $
$ g_{2}(S_i)=\begin{cases} 1&\text{, if } 30 < S_{i}\\ 0.8&\text{, if } S_{i} \leq 30\\ \end{cases} \qquad i=1,2,\dots,4 $
You can solve it using mixed integer linear programming and big-M modelling.
For every use of the $g$ function (except the first one which is constant), you introduce two binary variables ($\delta$) to indicate which case you have. The function value will then be, e.g., $0.9\delta_1 +0.5 \delta_2$. As only one case can hold, you add $\delta_1+\delta_2=1$. To model the region in which the function is valid, you add, e.g., $30-M(1-\delta_1)\leq 40-R_1, 40-R_1\leq 30+M(1-\delta_2)$ where $M$ is a sufficiently large constant to make the constraint redundant when the region isn't active (thus the name big-M, although you should always make it as small as possible while being sufficiently large)