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In this paper, we study the Maximum Profit Pick-up Problem with Time Windowsand Capacity Constraint (MP-PPTWC). Our main results are 3 polynomial timealgorithms, all having constant approximation factors. The first algorithm hasan approximation ratio of $~46 (1 + (71/60 + \frac{\alpha}{\sqrt{10+p}})\epsilon) \log T$, where: (i) $\epsilon > 0$ and $T$ are constants; (ii) Themaximum quantity supplied is $q_{max} = O(n^p) q_{min}$, for some $p > 0$,where $q_{min}$ is the minimum quantity supplied; (iii) $\alpha > 0$ is aconstant such that the optimal number of vehicles is always at least $\sqrt{10+ p} / \alpha$. The second algorithm has an approximation ratio of $\simeq 46(1 + \epsilon + \frac{(2 + \alpha) \epsilon}{\sqrt{10 + p}}) \log T$. Finally,the third algorithm has an approximation ratio of $\simeq 11 (1 + 2 \epsilon)\log T$. While our algorithms may seem to have quite high approximation ratios,in practice they work well and, in the majority of cases, the profit obtainedis at least 1/2 of the optimum.

Approximation Algorithms for the Maximum Profit Pick-up Problem with Time Windows and Capacity Constraint