Linear Programming Based Decomposition Methods for Inventory Distribution Systems
By Sumit Kunnumkal, Huseyin Topaloglu
European Journal of Operational Research | June 2011
European Journal of Operational Research | June 2011
DOI
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Citation
Kunnumkal, Sumit., Topaloglu, Huseyin. Linear Programming Based Decomposition Methods for Inventory Distribution Systems European Journal of Operational Research pdf.sciencedirectassets.com/271700/1-s2.0-S0377221710X00224/1-s2.0-S0377221710008064/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEM7%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaCXVzLWVhc3QtMSJHMEUCIQDUxzDfByNF29LmaYJaJRQKsGGu7WoFNDl1wBYPO%2F0WlQIgRPqBDE73PjxtcP15KaN6fqYs%2B%2FVqPxbx1veKggoDw88qvAUItv%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARAFGgwwNTkwMDM1NDY4NjUiDEN8JogiZcaUy9PbViqQBesOCC6CpYYViC6e19ta34kb3zsh05x9B%2FR8eVM0S7pKctJZ3ZLmY4L2%2FsMPb9AoHMcxRCtsZfU1IEjGRgoOYfvYt1omJBcX1%2BVF2Q%2FxQDZwMd0kjZYWZEXKQYxxO16c8MDqXYrFguRGYPjxY0tbx0hpqOFYPvKliR54BI%2BuwPLWcr6gbt7zfyGTGrUYwjrYXE0nLaj9bPY27Jt7ok7yboa6faUOrw%2FruJ0rVig8IGMf%2BOSI8zD0d1fvUBHf%2B6w5bfn2BdulK04eqznVq8AycHr%2B9fCye7nm49GMB6bA%2FBQ5Xf52YV4Ujufqfp3KocjksuoQdfvFo783CRf8Jqcqr8YqImXt8RyvzcyRGZAR17WxOUWCgw0MyfEAl03oGpbeWW2jqKlQqTWR4rpZzGOR%2BMtrBi92GAAvqnJtacHuZJJNtwWRoK4hBK10FfeE0OrGVXCrn%2Fa%2Frbv60lyX2YCrN8vrqcxOwsSlCo7SdT4HxRjzHtUWSmgxLO6UzbAeZ5moRv8dU8UHTbpPgB5dZRMjcH1kAFnDwxdfdNx0K93JUpb%2FBxl3EIP2AKmDKHXBJv%2FeXLg0JnZfDCntYWSaywJQoN9LhGH7eozTYesK23ZHYEdCEISKyY6NMvADepcDYXs%2F5H5ZhMltSFtRMxSJrXQ2GTxBACjrYnUtARq0%2FC6DM1YKvG469nGvp84n%2BGGQ6CapdizDDB3CVGot2ho9mvODWRO7Ff%2BaUNdt58ipbKzIX6ZqTbD8x61%2FNVlTU2lbtq054eLmtVks94C2C2u9XP%2BTXq7uCVPedmDivtIjp5OnxHhMSHVrp0D03HfT%2FltBz7MvVPoZUls%2BZSlpBrJ%2BL32GIi8HAyA%2Fm6ect8g6IBC5RqCUMJb92KEGOrEBORBpcY3YhdWqes%2BfA8f8o3tj03%2B00mi6b0ExppZGEVFrEOUCSLL5kncmX1RPlEWzLXsvtgYbynHN%2Fnd%2FXOlldaQyvHV5hBnx7Hq4QkMa9nHuPe78s4%2FFVJp8K8a8QRHMYvIirSe4Ujvsw6rE59Pfm9ffDHXbHes7BNJbwVm25a%2BHxXcPfLXKuYiwfgnQZx8PIL1QC%2FPfnA%2B%2BVakLazqnsoamU7eE3D31KOCJm62YlYwV&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230412T062717Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTY4OHON5CP%2F20230412%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=8e21f6794e887f32b483632ce07882e41d6b219484d8b6b4e3d15c133ea6b5ba&hash=aa5d8038f6eb40a6b826e8d4a67ab328cf7781234da22218608a8370f55eb389&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=S0377221710008064&tid=spdf-fbdcf629-3a54-4fb5-ad7d-a9f8726fd426&sid=4b8ab7e650b9934cb81a29b60f3945aab730gxrqb&type=client&tsoh=d3d3LnNjaWVuY2VkaXJlY3QuY29t&ua=13085603575e535753055f&rr=7b69670f9f2d4af2&cc=in.
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European Journal of Operational Research, 2011
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Abstract
In this paper, we consider an inventory distribution system consisting of one warehouse and multiple
retailers. The retailers face random demand and are supplied by the warehouse. The warehouse
replenishes its stock from an external supplier. The objective is to minimize the total expected
replenishment, holding and backlogging cost over a ¯nite planning horizon. We begin by formulating
the problem as a dynamic program, which is di±cult to solve due to the high dimensionality of the
state variable. Nevertheless, this formulation allows us to observe that if the warehouse is allowed to
ship negative quantities to the retailers, then the problem decomposes by the locations. To exploit this
observation, we relax the constraints that ensure the nonnegativity of the shipments to the retailers
by associating Lagrange multipliers with them. This approach naturally raises the question of how to
choose a good set of Lagrange multipliers. In this paper, we propose two methods that can be used to
choose the Lagrange multipliers. The first method is based on a linear programming approximation to
the inventory distribution problem that is formulated under the assumption that the demand random
variables take on their expected values, whereas the second method is based on a linear programming
approximation that is formulated under the assumption that the demand realizations are known a
priori. Computational experiments indicate that the inventory replenishment policies obtained by
especially the second method outperform many standard benchmarks by significant margins.
retailers. The retailers face random demand and are supplied by the warehouse. The warehouse
replenishes its stock from an external supplier. The objective is to minimize the total expected
replenishment, holding and backlogging cost over a ¯nite planning horizon. We begin by formulating
the problem as a dynamic program, which is di±cult to solve due to the high dimensionality of the
state variable. Nevertheless, this formulation allows us to observe that if the warehouse is allowed to
ship negative quantities to the retailers, then the problem decomposes by the locations. To exploit this
observation, we relax the constraints that ensure the nonnegativity of the shipments to the retailers
by associating Lagrange multipliers with them. This approach naturally raises the question of how to
choose a good set of Lagrange multipliers. In this paper, we propose two methods that can be used to
choose the Lagrange multipliers. The first method is based on a linear programming approximation to
the inventory distribution problem that is formulated under the assumption that the demand random
variables take on their expected values, whereas the second method is based on a linear programming
approximation that is formulated under the assumption that the demand realizations are known a
priori. Computational experiments indicate that the inventory replenishment policies obtained by
especially the second method outperform many standard benchmarks by significant margins.
Sumit Kunnumkal is a Professor and Area Leader of Operations Management at the Indian School of Business (ISB). He holds a PhD in Operations Research from Cornell University. He received his MS in Transportation from the Massachusetts Institute of Technology and a B.Tech in Civil Engineering from the Indian Institute of Technology, Madras.
Professor Kunnumkal has previously taught at the Smith School of Business, Queen’s University, and has held visiting positions at the Singapore University of Technology and Design and Universitat Pompeu Fabra. His research interests lie in the areas of pricing and revenue management, retail operations, assortment planning, and approximate dynamic programming.
At ISB, he has taught in the PGP programme, the Fellow programme, and various Advanced Management and Executive Education programmes.
Sumit Kunnumkal