Let's assume there is a demand node with demand $D$ which this demand should be covered within a 24 hours window. The objective is to find the best delivery times to maximize the amount of covered demand volume within a time window $T$ where $T$ can be defined as follows: $$ T \in \{ [t_1,t_2],[t_3,t_4]\} $$ where $t_k$ denote a time of a day (e.g., $t_1$ is 10 AM). We have following constraints:
- $2 \le t_2-t_1 \le 5$
- $2 \le t_4-t_3 \le 5$
- $t_3-t_2 \ge 1$
- $t_2-t_1+t_4-t_3 \le 8$
(1) and (2) make sure that each time range length should be minimum 2 hours and maximum 5 hours and also (3) indicates that at least one hour gap should be established between the two ranges, and finally (4) makes sure that the length of the two ranges won't exceed 8 hours. There is also a constraint with respect to delivery rate which limits max delivery throughput within each hour to be a constant value $U \in \mathbb{Z}^+$.
I was thinking to approach this problem as a Maximum coverage problem and generate all possible time ranges and set set each generated time range as a binary decision variable e.g.,$x_1$ is to cover within [5AM,9AM] and $y_1$ is the amount that is covered within this time range, but not sure if this is the best approach to formulate this problem.
Thanks in advance for any insights.