Suppose there are $n$ resources which can do some work. Each resource has a number of time windows: $tw_{i,k}=\{start_{i, k},stop_{i, k}\}$, such that the resource can perform its functions only within these time windows.
There is a vehicle, which needs to drive to each resource in a fixed order and it takes $d_i$ time to drive from resource $i$ to resource $i+1$. Each resource requires time $s_i$ to do its work.
If the vehicle arrives earlier than resource's time window start, it has to wait for $w_i$ units of time.
The vehicle starts at time $t_0$, drives for $d_0$ units of time to the first resource, waits for $w_0$, then the first resource performs some work for $s_0$ units and the the vehicle drives to the next resource.
Given that the order of resources is fixed and $d_i, s_i, tw_{i, k}$ are constants I need to find vehicle's start time $t_0$, which minimizes the total waiting time $w=\sum _n w_i$, where $w_i=\min_k(\max\{0, tw_{i,k}|start-t_i\})$ and $(t_i+w_i,t_n+w_n+s_n)\subseteq tw_{i,k}$
Is there an efficient algorithm for this problem? Has it been studied in the literature?