OR-Notes

J E Beasley

OR-Notes are a series of introductory notes on topics that fall under the broad heading of the field of operations research (OR). They were originally used by me in an introductory OR course I give at Imperial College. They are now available for use by any students and teachers interested in OR subject to the following conditions.

A full list of the topics available in OR-Notes can be found here.

Forecasting tutorial solution

• Five week moving average, MSD 11.576, forecast 23
• Exponential smoothing with 0.8 MSD 9.588, forecast 22.57, i.e. 23 when rounded to the nearest whole number
• Best exponential smoothing has MSD of 8.620, forecast 22.73, i.e. 23 when rounded to the nearest whole number, obtained with a smoothing constant of 0.47

• Single exponential smoothing, MSD 93422, forecast 2182
• Single exponential smoothing with trend, MSD 93422, forecast 2182
• Double exponential smoothing, MSD 96370, forecast 2170
• Double exponential smoothing with trend, MSD 97833, forecast 2177

Hence best result is an MSD of 93422 associated with a forecast of 2182

However, consider the January figures, do you believe this forecast?

Taking the Holt-Winters additive model with a seasonal cycle of 12 (months) gives an MSD of 82921 and a forecast of 1955 which seems much more reasonable given the previous January figures.

The graph of the data for the J785 sales can be seen below. You will note that there appears to be a linear trend (straight line) to the data and no visible seasonal effect (such as sales always rising in summer and dropping in winter). This illustrates that even simple visual inspection of data can give you useful information.

This would probably mean that a forecasting method involving a linear trend would give us a better forecast than other methods. You will see from the detailed results given below that this is so (e.g. double exponential smoothing with trend is better than double exponential smoothing).

The figures for the various forecasting methods are as follows:

Method                                             MSD    Forecast
Simple average                                  545675    1189
Weighted moving average        
    3 month average                              15376    2367
    6 month average                              35169    2260
    12 month average                            111280    2133 
Moving average with linear trend
    3 month average                               14833   2418 
    6 month average                                8754   2492
   12 month average                                7234   2428 
Single exponential smoothing                      10876   2407 
Single exponential smoothing with trend            6409   2416 
Double exponential smoothing                      10840   2406
Double exponential smoothing with trend            6792   2449 
Linear regression                                  5014   2456 
Holt-Winter's additive model
    seasonal cycle of 12                          18951   2316
    no seasonal cycle specified                    6408   2417
Holt-Winter's multiplicative model
    seasonal cycle of 12                          49971   2442
    no seasonal cycle specified                    6327   2419 

The lowest MSD is for linear regression with MSD=5014 and a forecast for the next month of 2456. The next lowest MSD is for Holt-Winter's multiplicative model with no seasonal cycle specified (MSD=6327) and a forecast of 2419 for the next month. However single exponential smoothing with trend with MSD=6409 and a forecast of 2416 for the next month is also a reasonable choice.

The graph of the linear regression, together with the actual data, is shown below.