Dynamic lot-size model

The dynamic lot-size model in inventory theory, is a generalization of the economic order quantity model that takes into account that demand for the product varies over time. The model was introduced by Harvey M. Wagner and Thomson M. Whitin in 1958.

We have available a forecast of product demand d_t over a relevant time horizon t=1,2,...,N (for example we might know how many widgets will be needed each week for the next 52 weeks). There is a setup cost s_t incurred for each order and there is an inventory holding cost i_t per item per period (s_t and i_t can also vary with time if desired). The problem is how many units x_t to order now to minimize the sum of setup cost and inventory cost. Let me denote inventory:

${\displaystyle I=I_{0}+\sum _{j=1}^{t-1}x_{j}-\sum _{j=1}^{t-1}d_{j}\geq 0}$

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