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.
Problem setup.
We have available a forecast of product demand 
 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 incurred for each order and there is an inventory holding cost per item per period ( and can also vary with time if desired). The problem is how many units to order now to minimize the sum of setup cost and inventory cost. Let me denote inventory:
formula_1
The functional equation representing minimal cost policy is:
formula_2
Where H() is the Heaviside step function. Wagner and Whitin proved the following four theorems:
Planning Horizon Theorem.
The precedent theorems are used in the proof of the Planning Horizon Theorem. Let
formula_4
denote the minimal cost program for periods 1 to t. If at period t* the minimum in F(t) occurs for j = t** ≤ t*, then in periods t > t* it is sufficient to consider only t** ≤ j ≤ t. In particular, if t* = t**, then it is sufficient to consider programs such that > 0.
The algorithm.
Wagner and Whitin gave an algorithm for finding the optimal solution by dynamic programming. Start with t*=1:
Because this method was perceived by some as too complex, a number of authors also developed approximate heuristics (e.g., the Silver-Meal heuristic) for the problem. 