We study the possibility of using drones to implement an automated picking system in a warehouse. We imagine a warehouse divided into two contiguous areas: in one area, the drone moves according to the Euclidean distance, while in the other area, the drone moves according to the Manhattan distance. For each customer-order (CO), the automated picking system is in charge of gathering the items requested in the CO to a predefined location where the cart of the drone is positioned. For each item of the order, the drone flies to the location where the item is stored, grasps it, and brings it back to its cart. Our goal is to find the position of the drone's cart that minimizes the sum of the distances traversed by the drone to pick-up all the items of the CO. We propose algorithms to find such a location when the items to be collected are in Euclidean and Manhattan areas. We can prove a √2-approximation factor for our solutions. Moreover, we compare the efficiency of the automated picking system employing a drone with that of a traditional picking system employing a worker that pushes a cart, and we find under which conditions the drone can be more efficient.

We introduce a drone delivery system for the "last-mile" logistics of small parcels. The system serves mixed delivery areas modeled as EMs. The shortest path between two destinations of an EM concatenates the Euclidean-and Manhattan-distance metrics. The drone's mission consists in one delivery for each destination of the grid, and, due to the strict payload constraint, the drone returns to the warehouse after each delivery. Our goal is to set the drone's warehouse in the delivery area so as the sum of the distances between the locations to be served and the warehouse is minimized. We exactly solve the problem proposing an algorithm that takes logarithmic time in the length of the Euclidean side of the EM. We also devise two approximate solutions that select the warehouse among a constant number of vertices of the EM. Such solutions are almost as good as the exact solution and we prove a √2-approximation bound in the worst case. Finally, we propose an exact solution for the two warehouse problem in a Manhattan grid, and an approximate solution for EMs.

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In the context of mesh-like, parallel processing computers for (i) approximating continuous space and (ii) analog simulation of the motion of objects and waves in continuous space, the present paper is concerned with which mesh-like interconnection of processors might be particularly suitable for the task and why. Processor interconnection schemes based on nearest neighbor connections in geometric lattices are presented along with motivation. Then two major threads are exploded regarding which lattices would be good: the regular lattices , for their symmetry and other properties in common with continuous space, and the well-known root lattices , for being, in a sense, the lattices required for physically natural basic algorithms for motion. The main theorem of the present paper implies that the well-known lattice A n is the regular lattice having the maximum number of nearest neighbors among the n -dimensional regular lattices. It is noted that the only n -dimensional lattices that are both regular and root are A n and Z n (Z n is the lattice of n -cubes. The remainder of the paper specifies other desirable properties of A n including other ways it is superior to Z n for our purposes.

An approach to analogical representation for objects and their motions in space is proposed. This approach involves lattice computer architectures and associated algorithms and is shown to be abstracted from the behaviour of human beings mentally solving spatial/kinematic puzzles. There is also discussion of where in this approach the modelling of human cognition leaves off and the engineering begins. The possible relevance of the approach to a number of issues in artificial intelligence is discussed. These issues include efficiency of sentential versus analogical representations, common sense reasoning, update propagation, learning performance tasks, diagrammatic representations, spatial reasoning, metaphor, human categorization, and pattern recognition. Lastly there is a discussion of the somewhat related approach involving cellular automata applied to computational physics.