Improved formulations of the joint order batching and picker routing problem

Order picking is the process of retrieving ordered products from storage locations in warehouses. In picker-to-parts order picking systems, two or more customer orders may be grouped and assigned to a single picker. Then routing decision regarding the visiting sequence of items during a picking tour must be made. (J.Won and S.Olafsson 2005) found that solving the integrated problem of batching and routing enables warehouse managers to organize order picking operations more efficiently compared with solving the two problems separately and sequentially. We therefore investigate the mathematical programming formulation of this integrated problem. We present several improved formulations for the problem based on the findings of (Valle, Beasley, and da Cunha 2017), that can significantly improve computational results. More specifically, we reconstruct the connectivity constraints and generate new cutting planes in our branch-and-cut framework. We also discuss some problem properties by studying the structure of the graphical representation, and we present two types of additional constraints. We also consider the no-reversal case of this problem. We present efficient formulations by building different auxiliary graphs. Finally, we present computational results for publicly available test problems for single-block and multiple-block warehouse configurationsComment: 37 pages, 11 figures, 7 table

Some dynamical properties of delayed weakly reversible mass-action systems

This paper focuses on the dynamical properties of delayed complex balanced systems. We first study the relationship between the stoichiometric compatibility classes of delayed and non-delayed systems. Using this relation we give another way to derive the existence of positive equilibrium in each stoichiometric compatibility class for delayed complex balanced systems. And if time delays are constant, the result can be generalized to weakly reversible networks. Also, by utilizing the Lyapunov-Krasovskii functional, we can obtain a long-time dynamical property about $\omega$-limit set of the complex balanced system with constant time delays. An example is also provided to support our results

Automatic Implementation of Neural Networks through Reaction Networks -- Part I: Circuit Design and Convergence Analysis

Information processing relying on biochemical interactions in the cellular environment is essential for biological organisms. The implementation of molecular computational systems holds significant interest and potential in the fields of synthetic biology and molecular computation. This two-part article aims to introduce a programmable biochemical reaction network (BCRN) system endowed with mass action kinetics that realizes the fully connected neural network (FCNN) and has the potential to act automatically in vivo. In part I, the feedforward propagation computation, the backpropagation component, and all bridging processes of FCNN are ingeniously designed as specific BCRN modules based on their dynamics. This approach addresses a design gap in the biochemical assignment module and judgment termination module and provides a novel precise and robust realization of bi-molecular reactions for the learning process. Through equilibrium approaching, we demonstrate that the designed BCRN system achieves FCNN functionality with exponential convergence to target computational results, thereby enhancing the theoretical support for such work. Finally, the performance of this construction is further evaluated on two typical logic classification problems

Capturing persistence of delayed complex balanced chemical reaction systems via decomposition of semilocking sets

With the increasing complexity of time-delayed systems, the diversification of boundary types of chemical reaction systems poses a challenge for persistence analysis. This paper focuses on delayed complex balanced mass action systems (DeCBMAS) and derives that some boundaries of a DeCBMAS can not contain an $\omega$-limit point of some trajectory with positive initial point by using the method of semilocking set decomposition and the property of the facet, further expanding the range of persistence of delayed complex balanced systems. These findings demonstrate the effectiveness of semilocking set decomposition to address the complex boundaries and offer insights into the persistence analysis of delayed chemical reaction network systems

Caustic analysis of partially coherent self-accelerating beams: Investigating self-healing property

We employed caustic theory to analyze the propagation dynamics of partially coherent self-accelerating beams such as self-healing of partially coherent Airy beams. Our findings revealed that as the spatial coherence decreases, the self-healing ability of beams increases. This result have been demonstrated both in simulation and experiment. This is an innovative application of the caustic theory to the field of partially coherent structured beams, and provides a comprehensive understanding of self-healing property. Our results have significant implications for practical applications of partially coherent beams in fields such as optical communication, encryption, and imaging.Comment: 9 pages, 4 figure

On Stability of Two Kinds of Delayed Chemical Reaction Networks

For the networks that are linear conjugate to complex balanced systems, the delayed version may include two classes of networks: one class is still linear conjugate to the delayed complex balanced network, the other is not. In this paper, we prove the existence of the first class of networks, and emphasize the local asymptotic stability relative to a certain defined invariant set. For the second class of systems, we define a special subclass and derive th

On the relation between omega-limit set and boundaries of mass-action chemical reaction networks

ω-limit set can be used to understand the long term behavior of a dynamical system. In this paper, we use the Lyapunov function PDEs method, developed in our previous work, to study the relation between ω-limit points and boundaries for chemical reaction networks equipped with mass-action kinetics. Using the solution of the PDEs, some new checkable criteria are proposed to diagnose non ω-limit points of the network system. These criteria are successfully applied to verify that non-semilocking boundary points and some semilocking boundary points are not ω-limit points. Further, we derive the ω-limit theorem that precludes the limit cycle of some biochemical network systems. The validity of the results are demonstrated through some abstract and practical examples of chemical reaction networks