An APX for the Maximum-Profit Routing Problem with Variable Supply

In this paper, we study the Maximum-Profit Routing Problem with Variable Supply (MPRP-VS). This is a more general version of the Maximum-Profit Public Transportation Route Planning Problem, or simply Maximum-Profit Routing Problem (MPRP), introduced in \cite{Armaselu-PETRA}. In this new version, the quantity $q_i(t)$ supplied at site $i$ is linearly increasing in time $t$, as opposed to \cite{Armaselu-PETRA}, where the quantity is constant in time. Our main result is a $5.5 \log{T} (1 + \epsilon) (1 + \frac{1}{1 + \sqrt{m}})^2$ approximation algorithm, where $T$ is the latest time window and $m$ is the number of vehicles used. In addition, we improve upon the MPRP algorithm in \cite{Armaselu-PETRA} under certain conditions.