Fractional Knapsack Problem with a Concave Penalized Objective Function and Position-Dependent Item Weights (Profits)

Fractional Knapsack Problem with Concave Penalties (FKPCP) is a variant of the knapsack problem in which items can be split at the expense of a concave penalty function depending on the fractional quantity. We consider the FKPCP with variable weights or profits of items, where the weight or profit of an item depends on its position in the packing sequence. Based on relevant properties of these problems, for FKPCP involving items with variable weights or profits, we propose pseudo-polynomial time exact algorithms and approximation algorithms, respectively.