  This paper introduces the Packing While Traveling problem as a new non-linear
knapsack problem. Given are a set of cities that have a set of items of
distinct profits and weights and a vehicle that may collect the items when
visiting all the cities in a fixed order. Each selected item contributes its
profit, but produces a transportation cost relative to its weight. The problem
asks to find a subset of the items such that the total gain is maximized. We
investigate constrained and unconstrained versions of the problem and show that
both are NP-hard. We propose a pre-processing scheme that decreases the size of
instances making them easier for computation. We provide lower and upper bounds
based on mixed-integer programming (MIP) adopting the ideas of piecewise linear
approximation. Furthermore, we introduce two exact approaches: one is based on
MIP employing linearization technique, and another is a branch-infer-and-bound
(BIB) hybrid approach that compounds the upper bound procedure with a
constraint programming model strengthened with customized constraints. Our
experimental results show the effectiveness of our exact and approximate
solutions in terms of solution quality and computational time.
