Packing While Traveling: Mixed Integer Programming for a Class of Nonlinear Knapsack Problems

TL;DR

This paper introduces a nonlinear knapsack problem related to packing along routes with travel time considerations, and proposes mixed integer programming solutions demonstrating near-optimal results efficiently.

Contribution

It formulates a new nonlinear knapsack problem in routing contexts and develops MIP-based methods, including an approximation approach, for effective solutions.

Findings

MIP solutions are effective for the problem.

Approximate MIP often yields near-optimal solutions.

Approximate approach reduces computation time significantly.

Abstract

Packing and vehicle routing problems play an important role in the area of supply chain management. In this paper, we introduce a non-linear knapsack problem that occurs when packing items along a fixed route and taking into account travel time. We investigate constrained and unconstrained versions of the problem and show that both are NP-hard. In order to solve the problems, we provide a pre-processing scheme as well as exact and approximate mixed integer programming (MIP) solutions. Our experimental results show the effectiveness of the MIP solutions and in particular point out that the approximate MIP approach often leads to near optimal results within far less computation time than the exact approach.