forecasting examples

OR-NotesJ E Beasley

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

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

Forecasting examples

Forecasting example 1996 UG exam

The demand for a product in each of the last five months is shown below.

Month 1 2 3 4 5Demand ('00s) 13 17 19 23 24

Use a two month moving average to generate a forecast for demand in month 6.Apply exponential smoothing with a smoothing constant of 0.9 to generate a forecast for demand for

demand in month 6.Which of these two forecasts do you prefer and why?

Solution

The two month moving average for months two to five is given by:

m2 = (13 + 17)/2 = 15.0m3 = (17 + 19)/2 = 18.0m4 = (19 + 23)/2 = 21.0m5 = (23 + 24)/2 = 23.5

The forecast for month six is just the moving average for the month before that i.e. the moving average formonth 5= m5 = 2350.

Applying exponential smoothing with a smoothing constant of 0.9 we get:

M1 = Y1 = 13M2 = 0.9Y2 + 0.1M1 = 0.9(17) + 0.1(13) = 16.60M3 = 0.9Y3 + 0.1M2 = 0.9(19) + 0.1(16.60) = 18.76M4 = 0.9Y4 + 0.1M3 = 0.9(23) + 0.1(18.76) = 22.58M5 = 0.9Y5 + 0.1M4 = 0.9(24) + 0.1(22.58) = 23.86

As before the forecast for month six is just the average for month 5= M5 = 2386

To compare the two forecasts we calculate the mean squared deviation (MSD). If we do this we find that for

Forecasting examples http://people.brunel.ac.uk/~mastjjb/jeb/or/formore.html

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the moving average

MSD = [(15 - 19)² + (18 - 23)²+ (21 - 24)²]/3 = 16.67

and for the exponentially smoothed average with a smoothing constant of 0.9

MSD = [(13 - 17)² + (16.60 - 19)² +(18.76 - 23)²+ (22.58 - 24)²]/4 = 10.44

Overall then we see that exponential smoothing appears to give the best one month ahead forecasts as it has alower MSD. Hence we prefer the forecast of 2386 that has been produced by exponential smoothing.


The table below shows the demand for a new aftershave in a shop for each of the last 7 months.

Month 1 2 3 4 5 6 7Demand 23 29 33 40 41 43 49

Calculate a two month moving average for months two to seven. What would be your forecast for thedemand in month eight?Apply exponential smoothing with a smoothing constant of 0.1 to derive a forecast for the demand inmonth eight.Which of the two forecasts for month eight do you prefer and why?The shop keeper believes that customers are switching to this new aftershave from other brands.Discuss how you might model this switching behaviour and indicate the data that you would require toconfirm whether this switching is occurring or not.

Solution

The two month moving average for months two to seven is given by:

m2 = (23 + 29)/2 = 26.0m3 = (29 + 33)/2 = 31.0m4 = (33 + 40)/2 = 36.5m5 = (40 + 41)/2 = 40.5m6 = (41 + 43)/2 = 42.0m7 = (43 + 49)/2 = 46.0

The forecast for month eight is just the moving average for the month before that i.e. the moving average formonth 7 = m7 = 46.


M1 = Y1 = 23M2 = 0.1Y2 + 0.9M1 = 0.1(29) + 0.9(23) = 23.60M3 = 0.1Y3 + 0.9M2 = 0.1(33) + 0.9(23.60) = 24.54M4 = 0.1Y4 + 0.9M3 = 0.1(40) + 0.9(24.54) = 26.09M5 = 0.1Y5 + 0.9M4 = 0.1(41) + 0.9(26.09) = 27.58M6 = 0.1Y6 + 0.9M5 = 0.1(43) + 0.9(27.58) = 29.12M7 = 0.1Y7 + 0.9M6 = 0.1(49) + 0.9(29.12) = 31.11


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As before the forecast for month eight is just the average for month 7 = M7 = 31.11 = 31 (as we cannot havefractional demand).

To compare the two forecasts we calculate the mean squared deviation (MSD). If we do this we find that forthe moving average

MSD = [(26.0 - 33)² + ... + (42.0 - 49)²]/5 = 41.1


MSD = [(23 - 29)² + ... + (29.12 - 49)²]/6 = 203.15

Overall then we see that the two month moving average appears to give the best one month ahead forecastsas it has a lower MSD. Hence we prefer the forecast of 46 that has been produced by the two month movingaverage.

To examine switching we would need to use a Markov process model, where states = brands and we wouldneed initial state information and customer switching probabilities (from surveys). We would need to run themodel on historical data to see if we have a fit between the model and historical behaviour.


The table below shows the demand for a particular brand of razor in a shop for each of the last nine months.

Month 1 2 3 4 5 6 7 8 9Demand 10 12 13 17 15 19 20 21 20

Calculate a three month moving average for months three to nine. What would be your forecast for thedemand in month ten?Apply exponential smoothing with a smoothing constant of 0.3 to derive a forecast for the demand inmonth ten.Which of the two forecasts for month ten do you prefer and why?

Solution

The three month moving average for months 3 to 9 is given by:

m3 = (10 + 12 + 13)/3 = 11.67m4 = (12 + 13 + 17)/3 = 14.00m5 = (13 + 17 + 15)/3 = 15.00m6 = (17 + 15 + 19)/3 = 17.00m7 = (15 + 19 + 20)/3 = 18.00m8 = (19 + 20 + 21)/3 = 20.00m9 = (20 + 21 + 20)/3 = 20.33

The forecast for month 10 is just the moving average for the month before that i.e. the moving average formonth 9 = m9 = 20.33.

Hence (as we cannot have fractional demand) the forecast for month 10 is 20.



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M1 = Y1 = 10M2 = 0.3Y2 + 0.7M1 = 0.3(12) + 0.7(10) = 10.60M3 = 0.3Y3 + 0.7M2 = 0.3(13) + 0.7(10.60) = 11.32M4 = 0.3Y4 + 0.7M3 = 0.3(17) + 0.7(11.32) = 13.02M5 = 0.3Y5 + 0.7M4 = 0.3(15) + 0.7(13.02) = 13.61M6 = 0.3Y6 + 0.7M5 = 0.3(19) + 0.7(13.61) = 15.23M7 = 0.3Y7 + 0.7M6 = 0.3(20) + 0.7(15.23) = 16.66M8 = 0.3Y8 + 0.7M7 = 0.3(21) + 0.7(16.66) = 17.96M9 = 0.3Y9 + 0.7M8 = 0.3(20) + 0.7(17.96) = 18.57

As before the forecast for month 10 is just the average for month 9 = M9 = 18.57 = 19 (as we cannot havefractional demand).


MSD = [(11.67 - 17)² + ... + (20.00 - 20)²]/6 = 10.57


MSD = [(10 - 12)² + ... + (17.96 - 20)²]/8 = 15.08

Overall then we see that the three month moving average appears to give the best one month ahead forecastsas it has a lower MSD. Hence we prefer the forecast of 20 that has been produced by the three month movingaverage.


The table below shows the demand for a particular brand of fax machine in a department store in each of thelast twelve months.

Month 1 2 3 4 5 6 7 8 9 10 11 12Demand 12 15 19 23 27 30 32 33 37 41 49 58

Calculate the four month moving average for months 4 to 12. What would be your forecast for thedemand in month 13?Apply exponential smoothing with a smoothing constant of 0.2 to derive a forecast for the demand inmonth 13.Which of the two forecasts for month 13 do you prefer and why?What other factors, not considered in the above calculations, might influence demand for the faxmachine in month 13?

Solution

The four month moving average for months 4 to 12 is given by:

m4 = (23 + 19 + 15 + 12)/4 = 17.25m5 = (27 + 23 + 19 + 15)/4 = 21m6 = (30 + 27 + 23 + 19)/4 = 24.75m7 = (32 + 30 + 27 + 23)/4 = 28


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m8 = (33 + 32 + 30 + 27)/4 = 30.5m9 = (37 + 33 + 32 + 30)/4 = 33m10 = (41 + 37 + 33 + 32)/4 = 35.75m11 = (49 + 41 + 37 + 33)/4 = 40m12 = (58 + 49 + 41 + 37)/4 = 46.25




M1 = Y1 = 12M2 = 0.2Y2 + 0.8M1 = 0.2(15) + 0.8(12) = 12.600M3 = 0.2Y3 + 0.8M2 = 0.2(19) + 0.8(12.600) = 13.880M4 = 0.2Y4 + 0.8M3 = 0.2(23) + 0.8(13.880) = 15.704M5 = 0.2Y5 + 0.8M4 = 0.2(27) + 0.8(15.704) = 17.963M6 = 0.2Y6 + 0.8M5 = 0.2(30) + 0.8(17.963) = 20.370M7 = 0.2Y7 + 0.8M6 = 0.2(32) + 0.8(20.370) = 22.696M8 = 0.2Y8 + 0.8M7 = 0.2(33) + 0.8(22.696) = 24.757M9 = 0.2Y9 + 0.8M8 = 0.2(37) + 0.8(24.757) = 27.206M10 = 0.2Y10 + 0.8M9 = 0.2(41) + 0.8(27.206) = 29.965M11 = 0.2Y11 + 0.8M10 = 0.2(49) + 0.8(29.965) = 33.772M12 = 0.2Y12 + 0.8M11 = 0.2(58) + 0.8(33.772) = 38.618



MSD = [(17.25 - 27)² + ... + (40 - 58)²]/8 = 107.43


MSD = [(12 - 15)² + ... + (33.772 - 58)²]/11 = 176.05

Overall then we see that the four month moving average appears to give the best one month ahead forecastsas it has a lower MSD. Hence we prefer the forecast of 46 that has been produced by the four month movingaverage.

Other factors:

seasonal demandadvertisingprice changes, both this brand and other brandsgeneral economic situationnew technology


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The table below shows the demand for a particular brand of microwave oven in a department store in each ofthe last twelve months.

Month 1 2 3 4 5 6 7 8 9 10 11 12Demand 27 31 29 30 32 34 36 35 37 39 40 42

Calculate a six month moving average for each month. What would be your forecast for the demand inmonth 13?Apply exponential smoothing with a smoothing constant of 0.7 to derive a forecast for the demand inmonth 13.Which of the two forecasts for month 13 do you prefer and why?

Solution

Now we cannot calculate a six month moving average until we have at least 6 observations - i.e. we can onlycalculate such an average from month 6 onward. Hence we have:

m6 = (34 + 32 + 30 + 29 + 31 + 27)/6 = 30.50

m7 = (36 + 34 + 32 + 30 + 29 + 31)/6 = 32.00

m8 = (35 + 36 + 34 + 32 + 30 + 29)/6 = 32.67

m9 = (37 + 35 + 36 + 34 + 32 + 30)/6 = 34.00

m10 = (39 + 37 + 35 + 36 + 34 + 32)/6 = 35.50

m11 = (40 + 39 + 37 + 35 + 36 + 34)/6 = 36.83

m12 = (42 + 40 + 39 + 37 + 35 + 36)/6 = 38.17




M1 = Y1 = 27

M2 = 0.7Y2 + 0.3M1 = 0.7(31) + 0.3(27) = 29.80

M3 = 0.7Y3 + 0.3M2 = 0.7(29) + 0.3(29.80) = 29.24

M4 = 0.7Y4 + 0.3M3 = 0.7(30) + 0.3(29.24) = 29.77

M5 = 0.7Y5 + 0.3M4 = 0.7(32) + 0.3(29.77) = 31.33

M6 = 0.7Y6 + 0.3M5 = 0.7(34) + 0.3(31.33) = 33.20

M7 = 0.7Y7 + 0.3M6 = 0.7(36) + 0.3(33.20) = 35.16


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M8 = 0.7Y8 + 0.3M7 = 0.7(35) + 0.3(35.16) = 35.05

M9 = 0.7Y9 + 0.3M8 = 0.7(37) + 0.3(35.05) = 36.42

M10 = 0.7Y10 + 0.3M9 = 0.7(39) + 0.3(36.42) = 38.23

M11 = 0.7Y11 + 0.3M10 = 0.7(40) + 0.3(38.23) = 39.47

M12 = 0.7Y12 + 0.3M11 = 0.7(42) + 0.3(39.47) = 41.24



MSD = [(30.50 - 36)² + ... + (36.83 - 42)²]/6 = 21.66


MSD = [(27 - 31)² + ... + (39.47 - 42)²]/11 = 5.25

a smoothing constant of 0.7 appears to give the best one month ahead forecasts as it has a lower MSD. Hencewe prefer the forecast of 41 that has been produced by exponential smoothing with a smoothing constant of0.7.


The table below shows the temperature (degrees C), at 11 p.m., over the last ten days:

Day 1 2 3 4 5 6 7 8 9 10Temperature 1.5 2.3 3.7 3.0 1.4 -1.3 -2.4 -3.7 -0.5 1.3

Calculate a three day moving average for each day.What would be your forecast for the temperature at 11 p.m. on day 11?Apply exponential smoothing with a smoothing constant of 0.8 to derive a forecast for the temperatureat 11 p.m. on day 11.Which of the two forecasts for the temperature at 11 p.m. on day 11 do you prefer and why?

Solution

Now we cannot calculate a 3 day moving average until we have at least 3 observations i.e. we can onlycalculate such an average from month 3 onward. Hence we have:

m3 = (1.5 + 2.3 + 3.7)/3 = 2.50

m4 = (2.3 + 3.7 + 3.0)/3 = 3.00

m5 = (3.7 + 3.0 + 1.4)/3 = 2.70

m6 = (3.0 + 1.4 - 1.3)/3 = 1.03


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m7 = (1.4 - 1.3 - 2.4)/3 = -0.77

m8 = (-1.3 - 2.4 - 3.7)/3 = -2.47

m9 = (-2.4 - 3.7 - 0.5)/3 = -2.20

m10 = (-3.7 - 0.5 + 1.3)/3 = -0.97

Hence the forecast for the temperature at 11 p.m. on day 11 is just m10 = -0.97.


M1 = Y1 = 1.5

M2 = 0.8Y2 + 0.2M1 = 0.8(2.3) + 0.2(1.5) = 2.14

M3 = 0.8Y3 + 0.2M2 = 0.8(3.7) + 0.2(2.14) = 3.39

M4 = 0.8Y4 + 0.2M3 = 0.8(3.0) + 0.2(3.39) = 3.08

M5 = 0.8Y5 + 0.2M4 = 0.8(1.4) + 0.2(3.08) = 1.74

M6 = 0.8Y6 + 0.2M5 = 0.8(-1.3) + 0.2(1.74) = -0.69

M7 = 0.8Y7 + 0.2M6 = 0.8(-2.4) + 0.2(-0.69) = -2.06

M8 = 0.8Y8 + 0.2M7 = 0.8(-3.7) + 0.2(-2.06) = -3.37

M9 = 0.8Y9 + 0.2M8 = 0.8(-0.5) + 0.2(-3.37) = -1.07

M10 = 0.8Y10 + 0.2M9 = 0.8(1.3) + 0.2(-1.07) = 0.83

Hence the forecast for the temperature at 11 p.m. on day 11 is just M10 = 0.83.

To compare the two forecasts we calculate the mean squared deviation (MSD). If we do this we find that forthe moving average MSD=7.90 and for the exponentially smoothed average with a smoothing constant of 0.8MSD=3.86.

Hence overall prefer the exponentially smoothed forecast as that seems to give the best one day aheadforecasts as it has a smaller MSD.


The table below shows the sales of a toy robot over the last 11 months.

Month 1 2 3 4 5 6 7 8 9 10 11Sales 3651 4015 3874 3501 3307 3105 2986 3100 3209 3450 3507

Calculate a four month moving average for each month. What would be your forecast for the sales inmonth 12?Apply exponential smoothing with a smoothing constant of 0.9 to derive a forecast for the sales inmonth 12.


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Which of the two forecasts for month 12 do you prefer and why?

Solution

Now we cannot calculate a 4 month moving average until we have at least 4 observations - i.e. we can onlycalculate such an average from month 4 onward. Hence we have:

m4 = (3651 + 4015 + 3874 + 3501)/4 = 3760.25

m5 = (4015 + 3874 + 3501 + 3307)/4 = 3674.25

m6 = (3874 + 3501 + 3307 + 3105)/4 = 3446.75

m7 = (3501 + 3307 + 3105 + 2986)/4 = 3224.75

m8 = (3307 + 3105 + 2986 + 3100)/4 = 3124.50

m9 = (3105 + 2986 + 3100 + 3209)/4 = 3100.00

m10 = (2986 + 3100 + 3209 + 3450)/4 = 3186.25

m11 = (3100 + 3209 + 3450 + 3507)/4 = 3316.50

The forecast for month 12 is just the moving average for the month before that i.e. the moving average formonth 11 = m11 = 3316.50


M1 = Y1 = 3651

M2 = 0.9Y2 + 0.1M1 = 0.9(4015) + 0.1(3651) = 3978.60

M3 = 0.9Y3 + 0.1M2 = 0.9(3874) + 0.1(3978.60) = 3884.46

M4 = 0.9Y4 + 0.1M3 = 0.9(3501) + 0.1(3884.46) = 3539.35

M5 = 0.9Y5 + 0.1M4 = 0.9(3307) + 0.1(3539.35) = 3330.24

M6 = 0.9Y6 + 0.1M5 = 0.9(3105) + 0.1(3330.24) = 3127.52

M7 = 0.9Y7 + 0.1M6 = 0.9(2986) + 0.1(3127.52) = 3000.15

M8 = 0.9Y8 + 0.1M7 = 0.9(3100) + 0.1(3000.15) = 3090.02

M9 = 0.9Y9 + 0.1M8 = 0.9(3209) + 0.1(3090.02) = 3197.10

M10= 0.9Y10 + 0.1M9 = 0.9(3450) + 0.1(3197.10) = 3424.71

M11 = 0.9Y11 + 0.1M10 = 0.9(3507) + 0.1(3424.71) = 3498.77

As before the forecast for month 12 is just the average for month 11 = M11 = 3498.77.


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MSD= [(3760.25 - 3307)² + ... + (3186.25 - 3507)²]/7 = 141407.9


MSD= [(3651 - 4015)² + ... + (3424.71 - 3507)²]/10 = 51008.3

Overall then we see that exponential smoothing with a smoothing constant of 0.9 appears to give the best onemonth ahead forecasts as it has a lower MSD. Hence we prefer the forecast of 3498.77 that has beenproduced by exponential smoothing with a smoothing constant of 0.9.

Forecasting example

The table below shows the movement of the price of a commodity over 12 months.

Month 1 2 3 4 5 6 7 8 9 10 11 12Price 25 30 32 33 32 31 30 29 28 28 29 31

Calculate a 6 month moving average for each month. What is the forecast for month 13?Apply exponential smoothing with smoothing constants of 0.7 and 0.8 to derive forecasts for month 13.Which of the two forecasts based on exponential smoothing for month 13 do you prefer and why?

Solution

Now we cannot calculate a 6 month moving average until we have at least 6 observations - i.e. we can onlycalculate such an average from month 6 onward. Hence we have:

m6 = (25 + 30 + 32 + 33 + 32 + 31)/6 = 30.50

m7 = (30 + 32 + 33 + 32 + 31 + 30)/6 = 31.33

m8 = (32 + 33 + 32 + 31 + 30 + 29)/6 = 31.17

m9 = (33 + 32 + 31 + 30 + 29 + 28)/6 = 30.50

m10 = (32 + 31 + 30 + 29 + 28 + 28)/6 = 29.67

m11 = (31 + 30 + 29 + 28 + 28 + 29)/6 = 29.17

m12 = (30 + 29 + 28 + 28 + 29 + 31)/6 = 29.17



M1 = Y1 = 25

M2 = 0.7Y2 + 0.3M1 = 0.7(30) + 0.3(25) = 28.50


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M3 = 0.7Y3 + 0.3M2 = 0.7(32) + 0.3(28.50) = 30.95

M4 = 0.7Y4 + 0.3M3 = 0.7(33) + 0.3(30.95) = 32.39

M5 = 0.7Y5 + 0.3M4 = 0.7(32) + 0.3(32.39) = 32.12

M6 = 0.7Y6 + 0.3M5 = 0.7(31) + 0.3(32.12) = 31.34

M7 = 0.7Y7 + 0.3M6 = 0.7(30) + 0.3(31.34) = 30.40

M8 = 0.7Y8 + 0.3M7 = 0.7(29) + 0.3(30.40) = 29.42

M9 = 0.7Y9 + 0.3M8 = 0.7(28) + 0.3(29.42) = 28.43

M10 = 0.7Y10 + 0.3M9 = 0.7(28) + 0.3(28.43) = 28.13

M11 = 0.7Y11 + 0.3M10 = 0.7(29) + 0.3(28.13) = 28.74

M12 = 0.7Y12 + 0.3M11 = 0.7(31) + 0.3(28.74) = 30.32



M1 = Y1 = 25

M2 = 0.8Y2 + 0.2M1 = 0.8(30) + 0.2(25) = 29.00

M3 = 0.8Y3 + 0.2M2 = 0.8(32) + 0.2(29.00) = 31.40

M4 = 0.8Y4 + 0.2M3 = 0.8(33) + 0.2(31.40) = 32.68

M5 = 0.8Y5 + 0.2M4 = 0.8(32) + 0.2(32.68) = 32.14

M6 = 0.8Y6 + 0.2M5 = 0.8(31) + 0.2(32.14) = 31.23

M7 = 0.8Y7 + 0.2M6 = 0.8(30) + 0.2(31.23) = 30.25

M8 = 0.8Y8 + 0.2M7 = 0.8(29) + 0.2(30.25) = 29.25

M9 = 0.8Y9 + 0.2M8 = 0.8(28) + 0.2(29.25) = 28.25

M10 = 0.8Y10 + 0.2M9 = 0.8(28) + 0.2(28.25) = 28.05

M11 = 0.8Y11 + 0.2M10 = 0.8(29) + 0.2(28.05) = 28.81

M12 = 0.8Y12 + 0.2M11 = 0.8(31) + 0.2(28.81) = 30.56


To decide which of the two forecasts based on exponential smoothing we prefer we calculate the MSD for thetwo exponentially smoothed averages. The resulting figures represent the historical accuracy of the twoforecasting procedures with respect to one month ahead forecasts. Knowing this accuracy tells us which of


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the two exponentially smoothed forecasts for month 13 we prefer.

Performing the calculations we find that for exponential smoothing with a smoothing constant of 0.7MSD=4.97 whilst for exponential smoothing with a smoothing constant of 0.8 MSD=4.43.

Overall then we see that exponential smoothing with a smoothing constant of 0.8 appears to give the best onemonth ahead forecasts over the last 11 months. Hence we prefer the forecast of 30.56 for month 13 that hasbeen produced by exponential smoothing with a smoothing constant of 0.8.


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