A mixed‐integer linear model for the multiple heterogeneous knapsack problem with realistic container loading constraints and bins' priority

Abstract

We propose a mixed-integer linear model that solves the multiple heterogeneous knapsack problem, minimising the wasted space of the bins, taking into account their priority and considering the following constraints: non-overlapping, boundaries and positioning (both constrained and free), rotations (around the vertical axis), orthogonal displacement, weight limit, static stability, weight distribution in a pyramidal region and load bearing considering items' arbitrary centre of mass (CoM). In the experiments, the priority parameter is set to prioritise smaller bins, but the model is generic to cover other cases. We study the trade-off of adding more constraints to make the problem more realistic and the complexity of finding a solution. We introduce new metrics that facilitate the comparison of datasets used in experiments. These metrics allow the export of theoretical results in the industry. We also propose a constructive heuristic named weight first best fit to handle large-scale instances in a reasonable time.

Physical Internet deployment in industry: literature review and research opportunities

“…Considering features and requirements of the FMCG industry, Landschützer et al (2015) provided and discussed the design methodology for PI-containers from aspects of sizing, design and loading. To support the utilization maximization in the PI, the mathematical models for determining the appropriate number (Lin et al , 2014) and priority of PI-containers (Deplano et al , 2019), as well as wireless sensor networks designed for PI-containers to easily monitor their position (Tran-Dang et al , 2017), have been proposed and discussed.…”

“…Some research mentioned above aimed to solve the theoretical problem in the PI without experiments or case studies involving specific industries (Lin et al , 2014; Żuchowski, 2018; Deplano et al , 2019). Simulation scenarios were also used for experiments and simulation in some research (Sallez et al , 2015; Tran-Dang et al , 2017).…”

“…Innovative inventory-control models have been proposed in terms of the interconnected PI vision (Pan et al, 2015;Yang et al, 2015) and have been iteratively assessed to optimize performance (Yang et al, 2017a, b). Some research mentioned above aimed to solve the theoretical problem in the PI without experiments or case studies involving specific industries (Lin et al, 2014;_ Zuchowski, 2018;Deplano et al, 2019). Simulation scenarios were also used for experiments and simulation in some research (Sallez et al, 2015;Tran-Dang et al, 2017).…”

Balancing the profit and capacity under uncertainties: a target‐based distributionally robust knapsack problem

“…Roca et al (2016) applied the knapsack formulation to find a locally optimal solution for error minimization in solar hydrogen production control. Deplano et al (2021) proposed a mixed-integer linear programming (MILP) model to solve the heterogeneous knapsack problem, which minimizes the wasted space of the bins under realistic container loading constraints. All the abovementioned knapsack problems are deterministic.…”

“…In order to further elaborate how our work fill the gap in the existing literature, we present a comparison of this study and the existing works in Table 1. As shown, Deplano et al (2021) investigated a multiple heterogeneous knapsack problem, which was solved by an MILP. Caserta and VoB (2018) addressed a multiple-choice MKP (MMKP), and Robert and Heerde (2015) studied a robust retail assortment optimization problem.…”