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The rapidly growing online food delivery (OFD) market presents substantial logistical challenges for last-mile delivery operations. Sidewalk delivery robots (SDRs) have emerged as a promising alternative to on-demand workers, as these compact, box-sized robots efficiently deliver food or groceries over short distances via sidewalks. We propose a two-stage stochastic optimization model for a single-depot SDR system with integrated battery-swapping operations. In the first stage, a continuous approximation (CA) method determines the optimal fleet size and the required number of additional swappable batteries. The second-stage solutions are critical to facilitate the first-stage method. These involve solving a routing problem that incorporates battery-swapping decisions and penalties for late arrivals. To address this, we develop a customized heuristic based on adaptive large neighborhood search (ALNS) to generate high-quality solutions for the second stage. The fitted CA model integrates key factors, including time windows, battery swapping, and pickup-delivery orders. Numerical examples highlight the proposed approach’s efficiency in reducing computational time while maintaining solution accuracy. A case study and sensitivity analysis conducted on Purdue University’s campus illustrate the practical impacts of fleet size and the number of swappable batteries. 

This paper considers off-street parking for the cruising vehicles of transportation network companies (TNCs) to reduce the traffic congestion. We propose a novel business that integrates the shared parking service into the TNC platform. In the proposed model, the platform (a) provides interfaces that connect passengers, drivers and garage operators (commercial or private garages); (b) determines the ride fare, driver payment, and parking rates; (c) matches passengers to TNC vehicles for ride-hailing services; and (d) matches vacant TNC vehicles to unoccupied parking garages to reduce the cruising cost. A queuing-theoretic model is proposed to capture the matching process of passengers, drivers, and parking garages. A market-equilibrium model is developed to capture the incentives of the passengers, drivers, and garage operators. An optimization-based model is formulated to capture the optimal pricing of the TNC platform. Through a realistic case study, we show that the proposed business model will offer a Pareto improvement that benefits all stakeholders, which leads to higher passenger surplus, higher drivers surplus, higher garage operator surplus, higher platform profit, and reduced traffic congestion. 

Abstract The nearby radio galaxy M87 is a prime target for studying black hole accretion and jet formation 1,2 . Event Horizon Telescope observations of M87 in 2017, at a wavelength of 1.3 mm, revealed a ring-like structure, which was interpreted as gravitationally lensed emission around a central black hole 3 . Here we report images of M87 obtained in 2018, at a wavelength of 3.5 mm, showing that the compact radio core is spatially resolved. High-resolution imaging shows a ring-like structure of $${8.4}_{-1.1}^{+0.5}$$ 8.4 − 1.1 + 0.5 Schwarzschild radii in diameter, approximately 50% larger than that seen at 1.3 mm. The outer edge at 3.5 mm is also larger than that at 1.3 mm. This larger and thicker ring indicates a substantial contribution from the accretion flow with absorption effects, in addition to the gravitationally lensed ring-like emission. The images show that the edge-brightened jet connects to the accretion flow of the black hole. Close to the black hole, the emission profile of the jet-launching region is wider than the expected profile of a black-hole-driven jet, suggesting the possible presence of a wind associated with the accretion flow.