The Packing While Traveling Problem

TL;DR

This paper defines the Packing While Traveling problem, a new non-linear knapsack variant involving maximizing profit while considering transportation costs, and proposes exact and approximate solution methods with demonstrated effectiveness.

Contribution

It introduces the novel Packing While Traveling problem, analyzes its complexity, and develops new solution techniques including MIP-based bounds and hybrid algorithms.

Findings

Problem is NP-hard.

Pre-processing reduces instance size.

Exact and approximate methods are effective.

Abstract

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.