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

TL;DR

This paper introduces three polynomial-time approximation algorithms for the Maximum Profit Pick-up Problem with Time Windows and Capacity Constraints, achieving constant approximation ratios and practical effectiveness.

Contribution

The paper presents new approximation algorithms with proven constant ratios for MP-PPTWC, improving solution quality for this complex logistics problem.

Findings

Algorithms achieve constant approximation ratios.

Practical results often reach at least half of the optimal profit.

Algorithms perform well despite high theoretical approximation bounds.

Abstract

In this paper, we study the Maximum Profit Pick-up Problem with Time Windows and Capacity Constraint (MP-PPTWC). Our main results are 3 polynomial time algorithms, all having constant approximation factors. The first algorithm has an approximation ratio of $~46 (1 + (71/60 + \frac{\alpha}{\sqrt{10+p}}) \epsilon) \log T$, where: (i) $\epsilon > 0$ and $T$ are constants; (ii) The maximum 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 a constant 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 obtained is at least 1/2 of the optimum.