We have available a forecast of product demanddt 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 costst incurred for each order and there is an inventory holding costit 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 us 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.[1] 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.
↑ 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
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
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