# On the bullwhip and inventory variance produced by an ordering policy

## Author

Listed:
• Disney, S. M.
• Towill, D. R.

## Abstract

The Bullwhip Problem in supply chains is first outlined. A discrete control theory model of a generic model of a replenishment rule is presented. From this model, an analytical expression for bullwhip is derived that is directly equivalent to the common statistical measure often used in simulation, statistical and empirical studies to quantify the bullwhip effect. This analytical expression clearly shows that bullwhip can be reduced by taking a fraction of the error in the inventory position and pipeline position, rather then account for all of the errors every time an ordering decision is made as is common in many scheduling systems. Furthermore, increasing the average age of the forecast reduces bullwhip, as does reducing the production lead-time. Next an analytical expression for the variance of the inventory position is derived and used together with the bullwhip expression to determine suitable ordering system designs that minimise both bullwhip and inventory variance for a range of weightings between the two variances. The relationship between our Bullwhip metric and the control theory metric Noise Bandwidth is highlighted. This contribution then derives and exploits an analytical expression for the Integral of Time * Absolute Error (ITAE) criterion often used to quantify inventory responsiveness in response to a deterministic demand. The results form a Decision Support System that can be used to design balanced supply chain ordering decisions and clearly shows how the Bullwhip Effect can be reduced, at the expense of increased inventory recovery times. Thus, this paper presents the general solution to the bullwhip problem on a sound mathematical basis.

## Suggested Citation

• Disney, S. M. & Towill, D. R., 2003. "On the bullwhip and inventory variance produced by an ordering policy," Omega, Elsevier, vol. 31(3), pages 157-167, June.
• Handle: RePEc:eee:jomega:v:31:y:2003:i:3:p:157-167
as

File URL: http://www.sciencedirect.com/science/article/pii/S0305-0483(03)00028-8

As the access to this document is restricted, you may want to search for a different version of it.

## References listed on IDEAS

as
1. Dejonckheere, J. & Disney, S. M. & Lambrecht, M. R. & Towill, D. R., 2003. "Measuring and avoiding the bullwhip effect: A control theoretic approach," European Journal of Operational Research, Elsevier, vol. 147(3), pages 567-590, June.
2. Hau L. Lee & V. Padmanabhan & Seungjin Whang, 1997. "Information Distortion in a Supply Chain: The Bullwhip Effect," Management Science, INFORMS, vol. 43(4), pages 546-558, April.
Full references (including those not matched with items on IDEAS)

### Keywords

Bullwhip effect Supply chains z-transforms ITAE APIOBPCS;

## Corrections

All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jomega:v:31:y:2003:i:3:p:157-167. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Haili He). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/375/description#description .

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.