The Dynamic Lot Sizing Problem is a fundamental model in inventory management where a decision-maker must determine the optimal order quantities for a single item over a finite planning horizon.

Problem Definition

A manager decides how much inventory to order in each period $t$ to satisfy demand while minimizing total costs, which typically include ordering costs and holding costs.

Key Assumptions

  1. Single-Item, Single-Level: We consider only the end product, ignoring raw materials or multi-echelon interactions.
  2. Finite Planning Horizon: The decision-making process spans a discrete, finite timeline $t = 1, 2, \dots, T$.
  3. Known Dynamic Demand: The demand $d_t$ for each period varies over time (hence “dynamic”) but is known in advance for the entire sequence $d_1, \dots, d_T$ (deterministic).
  4. Periodic Review: Inventory levels are reviewed, and ordering decisions are made at the beginning of each period.
  5. Unconstrained Capacity: There are no limits on the order quantity or inventory storage (infinite warehouse assumption).
  6. No Backorders: Demand must be fully met in the period it occurs; shortages are not permitted.

Input variables:

  • $d_t$: demand in period $t$
  • $c_o$: ordering cost
  • $c_h$: holding cost

Calcium Imaging

given y_1, …, y_T, want to decide s_1, …, s_T and c_1,.. c_T c_t: current inventory level. s_t: order quantity in period t. perishable good: c_t = s_t + \gamma c_(t-1) the holding cost is 1/2(y_t-c_t)^2.