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

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 + α √ 10+p ) ) log T , where: (i) > 0 and T are constants; (ii) The maximum quantity supplied is qmax = O(n )qmin, for some p > 0, where qmin is the minimum quantity supplied; (iii) α > 0 is a constant such that the optimal number of vehicles is always at least √ 10 + p/α. The second algorithm has an approximation ratio of ' 46(1 + + (2+α) √ 10+p ) log T . Finally, the third algorithm has an approximation ratio of ' 11(1 + 2 ) 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.