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.^[1][2]

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 us denote inventory:

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

The functional equation representing minimal cost policy is:

${\displaystyle f_{t}(I)={\underset {x_{t}\geq 0 \atop I+x_{t}\geq d_{t}}{\min }}\left[i_{t-1}I+H(x_{t})s_{t}+f_{t+1}\left(I+x_{t}-d_{t}\right)\right]}$

Where H() is the Heaviside step function. Wagner and Whitin^[1] proved the following four theorems:

• There exists an optimal program such that Ix_t=0; ∀t
• There exists an optimal program such that ∀t: either x_t=0 or ${\displaystyle x_{t}=\textstyle \sum _{j=t}^{k}d_{j}}$ for some k (t≤k≤N)
• There exists an optimal program such that if d_t* is satisfied by some x_t**, t**<t*, then d_t, t=t**+1,...,t*-1, is also satisfied by x_t**
• Given that I = 0 for period t, it is optimal to consider periods 1 through t - 1 by themselves

The precedent theorems are used in the proof of the Planning Horizon Theorem.^[1] Let

${\displaystyle F(t)=\min \left[{{\underset {1\leq j<t}{\min }}\left[s_{j}+\sum _{h=j}^{t-1}\sum _{k=h+1}^{t}i_{h}d_{k}+F(j-1)\right] \atop s_{t}+F(t-1)}\right]}$

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 x_t* > 0.

Wagner and Whitin gave an algorithm for finding the optimal solution by dynamic programming.^[1] Start with t*=1:

1. Consider the policies of ordering at period t**, t** = 1, 2, ... , t*, and filling demands d_t , t = t**, t** + 1, ... , t*, by this order
2. Add H(x_t**)s_t**+i_t**I_t** to the costs of acting optimally for periods 1 to t**-1 determined in the previous iteration of the algorithm
3. From these t* alternatives, select the minimum cost policy for periods 1 through t*
4. Proceed to period t*+1 (or stop if t*=N)

Because this method was perceived by some as too complex, a number of authors also developed approximate heuristics (e.g., the Silver-Meal heuristic^[3]) for the problem.

• 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
• Base stock model

• 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

• Solving the Lot Sizing Problem using the Wagner-Whitin Algorithm
• Dynamic lot size model
• Python implementation of the Wagner-Whitin algorithm.