The one-warehouse multiretailer problem with an order-up-to level inventory policy

What is it about?

The problem is to jointly determine a lot sizing policy of the warehouse and retailers such that the total of inventory holding and order costs at both levels is minimized. The problem is not only important in its own right, but also arises as a subproblem in many involved settings such as variants with capacities over replenishment quantities and with multi-stop routing.

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Why is it important?

Mixed integer programming formulations for the problem, including a new combined transportation and shortest path based reformulation, and analyze the strength of their LP relaxations with respect to each other. We reveal that the new formulation is stronger than the transportation based one, which is stronger than the echelon stock formulation. It is contrary to the results in single-level lot sizing where shortest path and transportation based formulations are the LP equivalent (see Nemhauser and Wolsey 1988 for the ULS problem and Denizel et al. (2008) for the capacitated multi-item lot sizing problem with setups).

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