Energy-Constrained Interval Task Scheduling

An autonomous rover needs to select a set of non-overlapping tasks to execute in order to maximize its total profit.

You are given:

• tasks: A list of objects, where each task $i$ has:
  • start: The start time $S_i$ of the task ($S_i \ge 0$).
  • end: The finish time $F_i$ of the task ($F_i > S_i$).
  • profit: The profit $P_i$ earned upon completing the task ($P_i > 0$).
  • energy_cost: The energy $C_i$ consumed by executing the task ($C_i \ge 0$).
• E_max: An integer representing the rover's maximum energy capacity.
• R: An integer representing the energy recharge rate per unit of time during idle periods.

Rules:

1. Execution: The rover can execute at most one task at any time. If task $i$ is chosen, it runs continuously from $S_i$ to $F_i$. Two tasks $i$ and $j$ do not overlap if $F_i \le S_j$ or $F_j \le S_i$.
2. Energy Capacity & Start State: The rover starts at time $t = 0$ with full energy $E_{max}$. Its energy level can never exceed $E_{max}$ or drop below $0$.
3. Task Energy Consumption: To execute task $i$, the rover must have at least $C_i$ energy at time $S_i$. Immediately after the task completes at time $F_i$, its energy decreases by $C_i$.
4. Idle Energy Recharge: During any idle period from $t_1$ to $t_2$ (where no task is running), the rover recharges energy at rate $R$. Energy at $t_2$ becomes $\min(E_{max}, e_{t_1} + R \times (t_2 - t_1))$.

Return the maximum total profit obtainable by selecting a valid sequence of tasks.