Clustered vehicle routing problems (CluVRPs) represent a complex class of combinatorial optimization problems with significant real-world relevance. They extend classic VRPs by introducing pre-specified customer clusters and requiring effective routing both between clusters and within each cluster. While numerous deep learning approaches have been developed to address the standard VRP, research on CluVRPs remains relatively limited, presenting opportunities and challenges for advancing solutions to more practical VRPs with cluster-related constraints. This paper offers a deep reinforcement learning (DRL) approach to solving CluVRPs. We propose a cluster-aware attention module in the encoder, along with inter-cluster and intra-cluster decoders to specialize the constructive policies within and between clusters. Symmetrical data augmentation is adopted in the training to improve the performance. Empirical results in different CluVRP variants manifest that the DRL method outperforms existing approaches, consistently offering advantages for various instances.

The authors regret an error in the abstract of the published article: the text ‘‘(i) the Schottky emission mechanism at a low reverse voltage (0–1 V) before the current is fully turned on.’’ should be changed to ‘‘(i) the Schottky emission mechanism at a low reverse voltage (0 to 1 V) before the current is fully turned on.’’ This change does not affect the main conclusions of the manuscript. The authors would like to apologize for any inconvenience caused. The Royal Society of Chemistry apologises for these errors and any consequent inconvenience to authors and readers.

Recent research focusing on wide-bandgap and two-dimensional materials with a Schottky junction has provided a new concept for ultraviolet photodetectors. However, the working mechanism of the Schottky junction-based detector varies depending on the photosensitive materials used and the device structure. We demonstrated a TiO₂/AlGaN/GaN heterostructure-based photodetector with a Schottky junction, integrating an ultraviolet photosensitive TiO₂ nanolayer, a two-dimensional electron gas (2DEG) field effect transistor, and a metal-semiconductor Schottky diode. The spectral response wavelength region of the detector is 200-365 nm and the peak responsivity is 37.396 A W⁻¹ at −5 V bias under 240 nm UV illumination, respectively. Meanwhile, a peak photo-to-dark current ratio (PDCR) of 5.1 × 10² at −2 V bias voltage was observed under 274 nm UV irradiation. This Schottky-based 2DEG heterostructure detector can realize three dominant working principles: (i) the Schottky emission mechanism at a low reverse voltage (0-1 V) before the current is fully turned on, (ii) the Poole-Frenkel emission mechanism at a medium reverse voltage (−1 to −2 V) with peak PDCR, and (iii) the Fowler-Nordheim tunneling mechanism at a high reverse voltage (>−2 V) with a high responsivity. Continuous cycle response measurement results indicate that the detectors have good response repeatability and reliability. The characteristics of response wavelength, responsivity, and stability show that the detector can be used for several commercial applications, including sunscreen UV monitoring and LED sterilization light source detection.