In the field of e-commerce warehousing, maximizing the utilization of packing bins is a fundamental goal for all major logistics enterprises. However, determining the appropriate size of packing bins poses a practical challenge for many logistics companies. Given the limited research on the open-size 3D bin packing problem as well as the high complexity and lengthy computation time of existing models, this study focuses on optimizing multiple-bin sizes within the e-commerce context. Building upon existing research, we propose a hybrid integer programming model, denoted as the three dimensional multiple option dimensional rectangular packing problem (3D-MODRPP), to address the multiple-bin size 3D bin packing problem. Additionally, we leverage well-established hardware and software technologies to propose a 3D bin packing system capable of accommodating multiple bin types with open dimensions. To reduce the complexity of the model and the number of constraints, we introduce a novel assumption method for 0–1 integer variables in the overlap and rotation constraints. By applying this approach, we significantly streamline the computational complexity associated with the model calculations. Furthermore, we refine the dataset by developing a customized version based on the classical Three-Dimensional One-Size Dependent Rectangular Packing Problem (3D-ODRPP) dataset, leading to improved outcomes. Through comprehensive analysis of the research results, our model exhibits remarkable advancements in addressing the strong heterogeneous bin packing problem, the weak heterogeneous bin packing problem, the actual bin packing problem, and the bin packing problem with multiple bin types and open sizes. Specifically, it significantly reduces model complexity and computation time and increases space utilization. The system designed in this study paves the way for practical applications based on the proposed model, providing researchers with broader research prospects and directions to expand the scope of investigation in the field of 3D bin packing. Consequently, this system contributes to solving complex 3D packing problems, reducing space waste, and enhancing transportation efficiency.

In this paper, we consider the following class of Kirchhoff-type equation with doubly critical Hartree-type nonlinearities $ \left\{ \begin{align}& -(a+b\int _{\mathbb{R}^N}|\nabla u|^2\text{d}x)\Delta u+V(x)u=(I_{\alpha}*|u|^p)|u|^{p-2}u+(I_{\alpha}*|u|^q)|u|^{q-2}u,\,\, \text{in}\,\,\mathbb{R}^N,\\& u\,\in\,H^1(\mathbb{R}^N), \end{align} \right. $ where $ a>0 $, $ b\geq0 $, $ N\geq 3 $, $ \alpha\,\in\,(N-2,N) $, $ p=\frac{N+\alpha }{N-2} $, $ q=\frac{N+\alpha }{N} $, $ V:\,\mathbb{R}^N \rightarrow \mathbb{R} $ is a potential function and $ I_{\alpha} $ is a Riesz potential of order $ \alpha\,\in\,(N-2,N) $. Under certain assumptions on potential function $ V(x) $, we prove that the equation has at least a ground state solution by variational methods.

In this paper, we consider the following nonlinear magnetic Kirchhoff Choquard type equation $ \begin{align*} &[a+b\int _{ \mathbb{R}^N}(|\nabla_A u|^2+\lambda V(x)|u|^2)\text{d}x](-\Delta_A u+\lambda V(x)u)\\=&(I_{\alpha}*F(|u|))\frac{f(|u|)}{|u|}u, \, \, \text{in}\, \, \mathbb{R}^N, \end{align*} $ where $ u: \mathbb{R}^N\rightarrow \mathbb{C} $, $ A: \mathbb{R}^N\rightarrow \mathbb{R}^N $ is a vector potential, $ N\geq 3 $, $ a>0 $, $ b>0 $, $ \alpha\, \in\, (N-2, N] $, $ V:\, \mathbb{R}^N \rightarrow \mathbb{R} $ is a scalar potential function and $ I_{\alpha} $ is a Riesz potential of order $ \alpha\, \in\, (N-2, N] $. Under certain assumptions on $ A(x) $, $ V(x) $ and $ f(t) $, we prove that the equation has at least one ground state solution by variational methods and investigate the asymptotic behavior of solutions.