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We consider the placement, delivery promise, and fulfillment decisions of an online retailer. We have a set of products with given numbers of units to be placed at capacitated fulfillment centers. Once we make the placement decisions, we face demands for the products arriving from different demand regions randomly over time. In response to each demand, we pick a delivery promise to offer, which determines the probability that the demand converts into sales as well as choose a fulfillment center to use to serve the demand. Our goal is to decide where to place the units at the beginning of the selling horizon and to find a policy to make delivery promise and fulfillment decisions over the selling horizon so that we maximize the total expected profit. We give an approximation strategy to obtain solutions with performance guarantees for this joint placement, delivery promise, and fulfillment problem. In our approximation strategy, we construct a bounding function that upper bounds the total expected profit from the delivery promise and fulfillment policy when viewed as a function of the placement decisions. To make the placement decisions, we maximize the bounding function subject to the capacity constraints at the fulfillment centers. To make the delivery promise and fulfillment decisions, we construct a policy that obtains a constant fraction of the bounding function. Using our approximation strategy with appropriate bounding functions, we obtain a solution with a constant factor performance guarantee, but if the size of the system, measured by the numbers of units that we need to place and capacities of the fulfillment centers, is large, then we get an asymptotically optimal solution. We compare our approximation strategy with approaches that ignore the interactions between the placement, delivery promise, and fulfillment decisions as well as heuristics that are based on Lagrangian relaxation, demonstrating that our approximation strategy compares quite favorably.
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Spectral gaps on complete Riemannian manifolds
In this short note, we survey some basic results related to the New Weyl criterion for the essential spectrum. We then use the language of Gromov-Hausdorff convergence to prove a spectral gap theorem.
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- Award ID(s):
- 1908513
- PAR ID:
- 10273519
- Date Published:
- Journal Name:
- Contemporary mathematics
- Volume:
- 756
- ISSN:
- 2705-1064
- Page Range / eLocation ID:
- 57-67
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/conm/756
