# 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]}

## Contents

## Problem setup[edit]

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 me denote inventory:

The functional equation representing minimal cost policy is:

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

- There exists an optimal program such that I
`x`_{t}=0; ∀t - There exists an optimal program such that ∀t: either
`x`_{t}=0 or 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

## Planning Horizon Theorem[edit]

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 `x`_{t*} > 0.

## The algorithm[edit]

Wagner and Whitin gave an algorithm for finding the optimal solution by dynamic programming.^{[1]} Start with t*=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 - 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 - From these t* alternatives, select the minimum cost policy for periods 1 through t*
- 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.

## See also[edit]

- 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

## References[edit]

- ^
^{a}^{b}^{c}^{d}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

## Further reading[edit]

- 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