Wagner-Whitin model
The Wagner-Whitin model is a model of dynamic lot size planning and order quantity planning . It was first formulated in 1958 by Harvey M. Wagner and Thomson M. Whitin .
Assumptions
The model is based on the following assumptions:
- Only one product considered (economy)
- Demand that changes over time and is satisfied at the beginning of each period.
- no limitation of storage or production capacity
- Fixed lot run costs / order costs
- linear variable storage costs / production costs
- The purchase prices / production costs are constant over time and are therefore irrelevant to the decision
- Shortages are not allowed
- finite planning period of T periods
The goal is now to decide when and how much should be ordered / produced in order to keep the sum of fixed and variable costs as low as possible.
Formulations
The model can be solved with dynamic programming . Alternatively, it can be formulated as a warehouse location problem and solved using specialized methods. It can also be formulated as a shortest path problem in a graph .
As a warehouse location problem
In the case of a formulation as a warehouse location problem, the issue of a lot / order corresponds to the potential locations for the location construction costs (fixed costs). The individual periods correspond to the customers who are to be supplied from the locations (lots).
As the shortest route problem
If the Wagner-Whitin model is formulated as the shortest path problem in a graph, the directed graph has the following form:
- The T + 1 nodes correspond to the individual periods t = 1,2, ... T. The node T + 1 is fictitious in nature.
- There are arrows to each subsequent period / to each node with a higher t than the starting node. The arrows correspond to the individual options for placing lots. In a model with 10 periods, you have two options in period 9: Place one lot each in periods 9 and 10 (arrows from node 9 to 10 and 10 to 11) or a common lot for the last two periods (arrow from 9 to 11)
- The weights (length) of the arrows correspond to the sum of the costs of the corresponding lot.
Procedure
With the Wagner-Whitin algorithm named after them, Wagner and Whitin have found a procedure that delivers optimal solutions. There are also some heuristics:
- Moving economic lot size ( least-unit-cost method )
- Piece compensation period ( part-period method )
- Silver-meal heuristic
- Procedure by Groff
Evaluation of various procedures
Various methods of dynamic lot size determination offer time advantages that are bought at the expense of quality. Leinz quantifies these in terms of competitive advantages through the use of dynamic storage models as follows:
| Procedure | k | RZ in ms |
| Wagner Whitin method (optimal solution) | 0% | 325.7855 |
| Procedure by Leinz / Bossert / habenicht | 0.70% | 1.7213 |
| Groff method | 1.88% | 0.1620 |
| Least Unit Cost rule | 7.31% | 0.3104 |
| Period adjustment | 3.27% | 0.2499 |
| Dynamic planning calculation (Freeland / Colley) | 25.49% | 0.1782 |
k = additional costs in% compared to the solution determined using the Wagner-Whitin algorithm
RZ = computing time
In percentage terms it looks like this:
| Procedure | % Deviation from the reference solution |
| Procedure by Leinz / Bossert / habenicht | 0% |
| Groff method | - 268% |
| Period adjustment | - 466% |
| Least Unit Cost rule | - 1041% |
| Dynamic planning calculation (Freeland / Colley) | - 3632% |
Depending on the problem, a decision should be made that weighs the available time with the necessary accuracy.
Extensions to the model
The Wagner-Whitin model has been expanded in many ways, for example through scarce storage or production capacities, variable production costs or volume discounts.
literature
- Wolfgang Domschke , Armin Scholl , Stefan Voss : Production planning. Process organizational aspects. 2nd, revised and expanded edition. Springer, Berlin a. a. 1997, ISBN 3-540-63560-2 .
Individual evidence
- ^ Harvey M. Wagner , Thomson M. Whitin : Dynamic version of the economic lot size model. In: Management Science . Vol. 5, 1958, pp. 89-96, 195.
- ^ Domschke, Scholl, Voss: production planning. 2nd, revised and expanded edition. 1997, p. 115.
- ^ Domschke, Scholl, Voss: production planning. 2nd, revised and expanded edition. 1997, p. 119 f.
- ^ Domschke, Scholl, Voss: production planning. 2nd, revised and expanded edition. 1997, p. 117 f.
- ^ Domschke, Scholl, Voss: production planning. 2nd, revised and expanded edition. 1997, pp. 115-128.
- ↑ Jürgen Leinz: Competitive advantages through the use of dynamic storage models . ( Memento of September 21, 2002 in the Internet Archive ).