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 dt 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 st incurred for each order and there is an inventory holding cost it per item per period (st and it can also vary with time if desired). The problem is how many units xt to order now to minimize the sum of setup cost and inventory cost. Let me denote inventory:
The functional equation representing minimal cost policy is:
- There exists an optimal program such that Ixt=0; ∀t
- There exists an optimal program such that ∀t: either xt=0 or for some k (t≤k≤N)
- There exists an optimal program such that if dt* is satisfied by some xt**, t**<t*, then dt, t=t**+1,...,t*-1, is also satisfied by xt**
- Given that I = 0 for period t, it is optimal to consider periods 1 through t - 1 by themselves
Planning Horizon Theorem
The precedent theorems are used in the proof of the Planning Horizon Theorem. Let
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 xt* > 0.
- Consider the policies of ordering at period t**, t** = 1, 2, ... , t*, and filling demands dt , t = t**, t** + 1, ... , t*, by this order
- Add H(xt**)st**+it**It** to the costs of acting optimally for periods 1 to t**-1 determined in the previous iteration of the algorithm
- From these t* alternatives, select the minimum cost policy for periods 1 through t*
- Proceed to period t*+1 (or stop if t*=N)
- Infinite fill rate for the part being produced: Economic order quantity
- Constant fill rate for the part being produced: Economic production quantity
- Demand is random: classical Newsvendor model
- Several products produced on the same machine: Economic lot scheduling problem
- Reorder point
- Harvey M. Wagner and Thomson M. Whitin, "Dynamic version of the economic lot size model," Management Science, Vol. 5, pp. 89–96, 1958
- Wagelmans, Albert, Stan Van Hoesel, and Antoon Kolen. "Economic lot sizing: an O (n log n) algorithm that runs in linear time in the Wagner-Whitin case." Operations Research 40.1-Supplement - 1 (1992): S145-S156.
- EA Silver, HC Meal, A heuristic for selecting lot size quantities for the case of a deterministic time-varying demand rate and discrete opportunities for replenishment, Production and inventory management, 1973
- Lee, Chung-Yee, Sila Çetinkaya, and Albert PM Wagelmans. "A dynamic lot-sizing model with demand time windows." Management Science 47.10 (2001): 1384-1395.
- Federgruen, Awi, and Michal Tzur. "A simple forward algorithm to solve general dynamic lot sizing models with n periods in 0 (n log n) or 0 (n) time." Management Science 37.8 (1991): 909-925.
- Jans, Raf, and Zeger Degraeve. "Meta-heuristics for dynamic lot sizing: a review and comparison of solution approaches." European Journal of Operational Research 177.3 (2007): 1855-1875.
- H.M. Wagner and T. Whitin, "Dynamic version of the economic lot size model," Management Science, Vol. 5, pp. 89–96, 1958
- H.M. Wagner: "Comments on Dynamic version of the economic lot size model", Management Science, Vol. 50 No. 12 Suppl., December 2004