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

Abstract

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 [2]. In this new version, the quantity q i (t) supplied at site i is linearly increasing in time t, as opposed to [2], where the quantity is constant in time. Our main result is a 5.5 log T (1+ )(1+ 1 1+ √ 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 [2] under certain conditions.

Maximum-Profit Routing Problem with Multiple Vehicles per Site

“…We know from [2] that an optimal MPRP algorithm run on S may collect a quantity at least 1 1+ as much as an optimal MPRP-VS algorithm run on S. Since for m = 1, an instance of MPRP-M also belongs to MPRP, and an instance of MPRP-MVS is also an instance of MPRP-VS, we get the following. Lemma 2.…”

“…q i (t) = q i (e i ) t−s i e i −s i . We adapt the algorithm in [2] for solving MPRP-VS, to work in our case, by using a similar reduction as the one from MPRP-M to MPRP described in the previous section, to reduce MPRP-MVS to MPRP-VS.…”

“…Given an instance I = (S = {S i : 1 ≤ i ≤ n}, D = (x 0 , y 0 ), d = {d(i, j) : 0 ≤ i < j ≤ n}, T, Q, m) of MPRP-MVS, we apply the reduction in [2] to obtain an instance I ∈ M P RP − M , and then solve MPRP-M using the algorithm in Section 2. Again, this is not straightforward since the approach in [2] depends on single vehicle assignment which is not the case here. Thus, we need a more insightful construction of I than directly applying the algorithm in [2].…”

“…Again, this is not straightforward since the approach in [2] depends on single vehicle assignment which is not the case here. Thus, we need a more insightful construction of I than directly applying the algorithm in [2].…”

“…For m = 1, an optimal MPRP-M algorithm when run on S may collect a quantity at least 1 1+ as high as an optimal MPRP-MVS algorithm when run on S. Lemma 3. [2] Let A be an algorithm for a MPRP (resp., MPRP-M) and let P (I) be the profit obtained by A when run on an instance I with one vehicle. Then, on an instance I with m vehicles, the profit obtained by A is P (I ) ≥…”