Skip to main content
Log in

Use of individual and group seasonal indices in subaggregate demand forecasting

  • Theoretical Paper
  • Published:
Journal of the Operational Research Society

Abstract

An individual seasonal indices (ISI) method and two group seasonal indices (GSI) methods proposed in the literature are compared, based on two models. Rules have been established to choose between these methods and insights are gained on the conditions under which one method outperforms the others. Simulation findings confirm that using the rules improves forecasting accuracy against universal application of these methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Brown RG (1959). Statistical Forecasting for Inventory Control. McGraw-Hill: New York.

    Google Scholar 

  • Brown RG (1963a). Estimating aggregate inventory standards. Nav Res Logist Quart 10: 55–71.

    Article  Google Scholar 

  • Brown RG (1963b). Smoothing, Forecasting and Prediction of Discrete Time Series. Prentice-Hall: Englewood Cliffs, NJ.

    Google Scholar 

  • Brown RG (1967). Decision Rules for Inventory Management. Holt, Rinehart and Winston: Austin, TX.

    Google Scholar 

  • Bunn DW and Vassilopoulos AI (1993). Using group seasonal indices in multi-item short-term forecasting. Int J Forecasting 9: 517–526.

    Article  Google Scholar 

  • Bunn DW and Vassilopoulos AI (1999). Comparison of seasonal estimation methods in multi-item short-term forecasting. Int J Forecasting 15: 431–443.

    Article  Google Scholar 

  • Chatfield C (2004). The Analysis of Time Series, 6th edition. Chapman & Hall/CRC: London.

    Google Scholar 

  • Chen H (2005). Comparing individual with group seasonal indices to forecast subaggregate demand. PhD thesis, Buckinghamshire Chilterns University College, Brunel University, UK.

  • Dalhart G (1974). Class seasonality—a new approach. Published in Forecasting, 2nd edition. American Production and Inventory Control Society, Washington DC, pp. 11–16.

  • Dangerfield B and Morris JS (1988). An empirical evaluation of top-down and bottom-up forecasting strategies. Proceedings of the 1988 Meeting of The Western Decision Sciences Institute, pp. 322–324.

  • Dangerfield BJ and Morris JS (1992). Top-down or bottom-up: Aggregate versus disaggregate extrapolations. Int J Forecasting 8: 233–241.

    Article  Google Scholar 

  • Dekker M, van Donselaar K and Ouwehand P (2004). How to use aggregation and combined forecasting to improve seasonal demand forecasts. Int J Prod Econom 90: 151–167.

    Article  Google Scholar 

  • Duncan G, Gorr W and Szczypula J (1993). Bayesian forecasting for seemingly unrelated time series: Application to local government revenue forecasting. Mngt Sci 39: 275–293.

    Article  Google Scholar 

  • Duncan G, Gorr W and Szczypula J (1998). Forecasting analogous time series. Working Paper 1998-4, Heinz School, Carnegie Mellon University.

  • Gilchrist W (1976). Statistical Forecasting. Wiley: London.

    Google Scholar 

  • Johnston FR, Taylor SJ and Oliveria RMMC (1988). Setting company stock levels. J Opl Res Soc 39: 15–21.

    Article  Google Scholar 

  • Kendall MG, Stuart A and Ord JK and Arnold SF (eds) (1998). Advanced Theory of Statistics: Classical Inference, Vol. 2A. Charles Griffin & Company Limited: London.

  • Ouwehand P, van Donselaar KH and de Kok AG (2005). The impact of the forecasting horizon when forecasting with group seasonal indices. Working Paper, Department of Technology Management, Eindhoven University of Technology.

  • Schwarzkopf AB, Tersine RJ and Morris JS (1988). Top-down versus bottom-up forecasting strategies. Int J Prod Res 26: 1833–1843.

    Article  Google Scholar 

  • Shlifer E and Wolff RW (1979). Aggregation and proration in forecasting. Mngt Sci. 25: 594–603.

    Article  Google Scholar 

  • Stevens CF (1974). On the variability of demand for families of items. Op Res Quart 25: 411–419.

    Article  Google Scholar 

  • Wharton F (1975). On estimating aggregate inventory characteristics. Op Res Quart 26: 543–551.

    Article  Google Scholar 

  • Withycombe R (1989). Forecasting with combined seasonal indices. Int J Forecasting 5: 547–552.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to H Chen.

Appendices

Appendix A. Rule for the mixed model

The forecast for the ith series, hth season, in year r+1 using ISI is

MSE is

The same forecast using DGSI is

MSE is

MSEISI i > MSEDGSI i if and only if

The forecast of hth season in year r+1 for the ith item using WGSI is

MSE is

MSEISI i > MSEWGSI i if and only if

Appendix B. Rule for the additive model

The additive model is specified as

Forecast for item i, the hth season in year r+1 using ISI is

MSE of the forecast is

The GSI estimator is given as

Therefore, the forecast of item i, the hth season in year r+1 is

MSE of the forecast is as follows:

The difference in MSE for the forecasts using ISI and GSI is:

MSEGSI i < MSEISI i if and only if

Appendix C. Maximum likelihood estimators for the mixed model

where ∑q h=1 S h =q, and ε i,th iid N(0,σ i 2).

Hence, differentiating with respect to μ i and S h (h=1,…,q) and with respect to σ i 2,

If the q equations ∂logL/∂S h =0 (h=1,…,q) are all satisfied, then:

And so, summing all of these equations:

This gives the same condition as the first equation ∂logL/∂μ i =0

So, if all of the q equations are satisfied, the first equation will also be satisfied. Solving these q equations gives:

Summing, ∑r t=1q h=1 Y ith =r μ i q h=1 S h

Since the mixed model assumes that the seasonal indices sum to q (the number of seasons):

And the estimators for S h (h=1,…,q) are given by:

And the final condition gives:

We checked that all the criteria for maximum likelihood estimators have been satisfied. The maximum likelihood estimators for the aggregate mixed model and for the additive model can be obtained by the same method.

Appendix D. Bias properties of ISI, DGSI, and WGSI

Expectation of ISI:

Use Taylor series to approximate E(Y i,1h +Y i,2h +⋯+Y i,rh /∑r t=1q h=1 Y i,th )

Let X 1=Y i,1h +Y i,2h +⋯+Y i,rh ,

Without loss of generality, suppose h=1,

Then for any h,

Therefore,

Expectation of DGSI:

Expectation of WGSI:

Both ISI and WGSI are maximum likelihood estimators, one at the individual level and one at the aggregate level. E(ISI i,h )=S h +(S h −1)σ i 2/qr μ i 2 and, therefore, E(WGSI h )=S h +σ A 2(S h −1)/qr μ A 2.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, H., Boylan, J. Use of individual and group seasonal indices in subaggregate demand forecasting. J Oper Res Soc 58, 1660–1671 (2007). https://doi.org/10.1057/palgrave.jors.2602310

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1057/palgrave.jors.2602310

Keywords

Navigation