Post on 30-Oct-2014
52
Actual Three-WeekWeek Bicycle Sales Moving Average
1 82 103 94 11 (8 � 10 � 9)/3 � 95 10 (10 � 9 � 11)/3 � 106 13 (9 � 11 � 10)/3 � 107 — (11 � 10 � 13)/3 � 11Z\c
Alternative Example 5.2: Weighted moving average
Bower’s Bikes decides to forecast bicycle sales by weighting thepast 3 weeks as follows:
Weights Applied Period
3 Last week2 Two weeks ago1 Three weeks ago6 Sum of weights
A 3-week weighted moving average appears below.
ActualBicycle
Week Sales Three-Week Moving Average
1 82 103 94 11 [(3 � 9) � (2 � 10) � (1 � 8)]/6 � 9Z\n
5 10 [(3 � 11) � (2 � 9) � (1 � 10)]/6 � 10Z\n
6 13 [(3 � 10) � (2 � 11) � (1 � 9)]/6 � 10Z\n
7 — [(3 � 13) � (2 � 10) � (1 � 11)]/6 � 11X\c
Alternative Example 5.3: A firm uses simple exponentialsmoothing with a � 0.1 to forecast demand. The forecast for theweek of January 1 was 500 units, whereas actual demand turnedout to be 450 units. The demand forecasted for the week of Janu-ary 8 is calculated as follows.
Ft�1 � Ft � α(At � Ft)
� 500 � 0.1(450 � 500) � 495 units
�∑ (weight for period )(demand in period n n))
∑ weights
TEACHING SUGGESTIONS
Teaching Suggestion 5.1: Wide Use of Forecasting.Forecasting is one of the most important tools a student can masterbecause every firm needs to conduct forecasts. It’s useful to moti-vate students with the idea that obscure sounding techniques suchas exponential smoothing are actually widely used in business, anda good manager is expected to understand forecasting. Regressionis commonly accepted as a tool in economic and legal cases.
Teaching Suggestion 5.2: Forecasting as an Art and a Science.Forecasting is as much an art as a science. Students should under-stand that qualitative analysis ( judgmental modeling) plays an im-portant role in predicting the future since not every factor can bequantified. Sometimes the best forecast is done by seat-of-the-pants methods.
Teaching Suggestion 5.3: Use of Simple Models.Many managers want to know what goes on behind the forecast.They may feel uncomfortable with complex statistical models withtoo many variables. They also need to feel a part of the process.
Teaching Suggestion 5.4: Management Input to the ExponentialSmoothing Model.One of the strengths of exponential smoothing is that it allows de-cision makers to input constants that give weight to recent data.Most managers want to feel a part of the modeling process andappreciate the opportunity to provide input.
Teaching Suggestion 5.5: Wide Use of Adaptive Models.With today’s dominant use of computers in forecasting, it ispossible for a program to constantly track the accuracy of amodel’s forecast. It’s important to understand that a programcan automatically select the best alpha and beta weights inexponential smoothing. Even if a firm has 10,000 products, theconstants can be selected very quickly and easily without humanintervention.
ALTERNATIVE EXAMPLES
Alternative Example 5.1:
Moving average
Bicycle sales at Bower’s Bikes are shown in the middle column of thefollowing table. A 3-week moving average appears on the right.
= ∑ demand in previous periodsn
n
5C H A P T E R
Forecasting Models
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CHAPTER 5 FORECAST ING MODELS 53
Alternative Example 5.4: Exponential smoothing is used toforecast automobile battery sales. Two values of � are examined,� � 0.8 and � � 0.5. To evaluate the accuracy of each smoothingconstant, we can compute the absolute deviations and MADs.Assume that the forecast for January was 22 batteries.
On the basis of this analysis, a smoothing constant of � � 0.8 ispreferred to � � 0.5 because it has a smaller MAD.
Alternative Example 5.5: Use the sales data given below to de-termine: (a) the least squares trend line, (b) the predicted value for2000 sales.
Year Sales (Units)
1993 1001994 1101995 1221996 1301997 1391998 1521999 164
To minimize computations, transform the value of x (time) to sim-pler numbers. In this case, designate 1993 as year 1, 1994 as year2, and so on.
Time SalesYear Period (Units) x2 xy
1993 1 100 1 1001994 2 110 4 2201995 3 122 9 3661996 4 130 16 5201997 5 139 25 6951998 6 152 36 9121999 17 164 149 1,148
�x � 28 �y � 917 �x2 � 140 �xy � 3,961
Absolute AbsoluteActual Forecast Deviation Forecast DeviationBattery with with with with
Month Sales � �� 0.8 � �� 0.8 � �� 0.5 � �� 0.5
January 20 22 2 22 2February 21 20.40 0.6 21 0March 15 20.880 5.88 21 6April 14 16.176 2.176 18 4May 13 14.435 1.435 16 3June 16 13.287 2.713 14.5 31.5
Sum of absolute deviations: 15 16.5
MAD: 2.46 2.75
Alternative Example 5.6: The rated power capacity (in hours/week) over the past 6 years has been:
Rated CapacityYear (hrs/wk)
1 1152 1203 1184 1245 1236 130
Here is an alternative way to recode years which simplifies themath since �X � 0.
Renumbered CapacityYear Year (x) (y) x2 xy
1 �2.5 115 6.25 �287.52 �1.5 120 2.25 �1803 �.5 118 0.25 �594 �.5 124 0.25 �625 �1.5 123 2.25 �184.56 �2.5 130 6.25 �325
�X � 0 �Y � 730 �X2 � 17.5 �XY � 45
y � 121.67 � 2.57X
Year 7 � 121.67 � (2.57)(3.5)
�131
Alternative Example 5.7: The forecast demand and actual de-mand for 10-foot fishing boats are shown below. We compute thetracking signal and MAD.
Tracking SignalRSFE
MAD MADs= = − = −24
11 72 1
..
MAD Forecast errors= ∑ = =
n
70
611 7.
aY
n= ∑ = =730
6121 67.
bXY
X= ∑
∑= =
2
45
17 52 57
..
yy
n= ∑ = =917
7131
bxy nxy
x nx= ∑ −
∑ −= −
−2 2
3 961 7 4 131
140 7 4
, ( )( )( )
( )( 22
293
2810 464
).= =
a y bx= − = − =131 10 46 4 89 14. ( ) .
xx
n= ∑ = =28
74
Therefore, the least squares trend equation is,
To project demand in 2000, we denote the year 2000 as x � 8,
Sales in 2000 � 89.14 � 10.464(8) � 172.85
ˆ . .y a bx x= + = +89 14 10 464
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54 CHAPTER 5 FORECAST ING MODELS
SOLUTIONS TO DISCUSSION QUESTIONS
AND PROBLEMS
5-1. The steps that are used to develop any forecasting systemare:
1. Determine the use of the forecast.
2. Select the items or quantities that are to be forecasted.
3. Determine the time horizon of the forecast.
4. Select the forecasting model.
5. Gather the necessary data.
6. Validate the forecasting model.
7. Make the forecast.
8. Implement the results.
5-2. A time-series forecasting model uses historical data to pre-dict future trends.
5-3. The only difference between causal models and time-series models is that causal models take into account any factorsthat may influence the quantity being forecasted. Causal modelsuse historical data as well. Time-series models use only historicaldata.
5-4. Qualitative models incorporate subjective factors into theforecasting model. Judgmental models are useful when subjectivefactors are important. When quantitative data are difficult to ob-tain, qualitative models are appropriate.
5-5. The disadvantages of the moving average forecastingmodel are that the averages always stay within past levels, and themoving averages do not consider seasonal variations.
5-6. When the smoothing value, �, is high, more weight is givento recent data. When � is low, more weight is given to past data.
5-7. The Delphi technique involves analyzing the predictionsthat a group of experts have made, then allowing the experts to re-view the data again. This process may be repeated several times.After the final analysis, the forecast is developed. The group of experts may be geographically dispersed.
5-8. MAD is a technique for determining the accuracy of aforecasting model by taking the average of the absolute deviations.
MAD is important because it can be used to help increase forecast-ing accuracy.
5-9. If a seasonal index equals 1, that season is just an averageseason. If the index is less than 1, that season tends to be lowerthan average. If the index is greater than 1, that season tends to behigher than average.
5-10. If the smoothing constant equals 0, then
Ft�1 � Ft � 0(At � Ft) � Ft
This means that the forecast never changes.If the smoothing constant equals 1, then
Ft�1 � Ft � 1(At � Ft) � At
This means that the forecast is always equal to the actual value inthe prior period.
5-11. A centered moving average (CMA) should be used iftrend is present in data. If an overall average is used rather than aCMA, variations due to trend will be interpreted as variations due toseasonal factors. Thus, the seasonal indices will not be accurate.
5-12.
ActualMonth Shed Sales Four-Month Moving Average
Jan. 10Feb. 12Mar. 13Apr. 16May 19 (10 � 12 � 13 � 16)/4 � 51/4 � 12.75June 23 (12 � 13 � 16 � 19)/4 � 60/4 � 15July 26 (13 � 16 � 19 � 23)/4 � 70/4 � 17.75Aug. 30 (16 � 19 � 23 � 26)/4 � 84/4 � 21Sept. 28 (19 � 23 � 26 � 30)/4 � 98/4 � 24.5Oct. 18 (23 � 26 � 30 � 28)/4 � 107/4 � 26.75Nov. 16 (26 � 30 � 28 � 18)/4 � 102/4 � 25.5Dec. 14 (30 � 28 � 18 � 16)/4 � 92/4 � 23
The MAD � 7.78
See solution to 5-13 for calculations.
Table for Alternate Example 5.7
Forecast Actual Forecast Cumulative TrackingYear Demand Demand Error RSFE Error Error MAD Signal
1 78 71 �7 �7 7 7 7.0 �1.02 75 80 5 �2 5 12 6.0 �0.33 83 101 18 16 18 30 10.0 �1.64 84 84 0 16 0 30 7.5 �2.15 88 60 �28 �12 28 58 11.6 �1.06 85 73 �12 �24 12 70 11.7 �2.1
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CHAPTER 5 FORECAST ING MODELS 55
5-14.
Three-Year Weighted Three-Year Three-Year Three-Year WeightedYear Demand Moving Averages Moving Averages Absolute Deviation Absolute Deviation
1 42 63 4 sum of the weights4 5 (4 � 6 � 4)/3 � 42⁄3 [(2 � 4) � 6 � 4]/4 � 41⁄2 0.34 0.555 10 (6 � 4 � 5)/3 � 5 [(2 � 5) � 4 � 6]/4 � 50 5.55 5.556 8 (4 � 5 � 10)/3 � 61⁄3 [(2 � 10) � 5 � 4]/4 � 71⁄4 1.67 0.757 7 (5 � 10 � 8)/3 � 72⁄3 [(2 � 8) � 10 � 5]/4 � 73⁄4 0.67 0.758 9 (10 � 8 � 7)/3 � 81⁄3 [(2 � 7) � 8 � 10]/4 � 80 0.67 1.559 12 (8 � 7 � 9)/3 � 8 [(2 � 9) � 7 � 8]/4 � 81⁄4 4.55 3.75
10 14 (7 � 9 � 12)/3 � 91⁄3 [(2 � 12) � 9 � 7]/4 � 10 4.67 4.5511 15 (9 � 12 � 14)/3 � 112⁄3 [(2 � 14) � 12 � 9]/4 � 121⁄4 3.34 2.75
Total absolute deviations: 20.36 18.5
MAD for 3-year average � 2.54
MAD for weighted 3-year average � 2.32
The weighted moving average appears to be slightly more accuratein its annual forecasts.
5-15. Using Excel or QM for Windows, the trend line is
Y � 2.22 � 1.05X
Where X � time period (1, 2, . . .) Y � demand
The 3-month moving average appears to be more accurate. How-ever, if weighted moving averages had been used, the resultsmight be different.
Three- Four-Three- Month Four- Month
Actual Month Absolute Month AbsoluteMonth Shed Sales Forecast Deviation Forecast Deviation
Jan. 10Feb. 12Mar. 13Apr. 16 11.66 4.34May 19 13.66 5.34 12.75 6.25June 23 16 7 15 8July 26 19.33 6.67 17.75 8.25Aug. 30 22.66 7.34 21 9Sept. 28 26.33 1.67 24.5 3.5Oct. 18 28 10 26.75 8.75Nov. 16 25.33 9.33 25.5 9.5Dec. 14 20.66 56.66 23 69.25
58.35 62.25
5-13.
Three-month MAD
Four-month MAD = =62 25
87 78
..
= =58 35
96 48
..
a
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56 CHAPTER 5 FORECAST ING MODELS
5-17. � � 0.3. New forecast for year 2 is last period’s forecast ��(last period’s actual demand � last period’s forecast):
new forecast for year 2 � 5,000 � (0.3)(4,000 � 5,000)
� 5,000 � (0.3)(� 1,000)
� 5,000 � 300
� 4,700The calculations are:
Year Demand New Forecast
2 6,000 4,700 � 5,000 � (0.3)(4,000 � 5,000)3 4,000 5,090 � 4,700 � (0.3)(6,000 � 4,700)4 5,000 4,763 � 5,090 � (0.3)(4,000 � 5,090)5 10,000 4,834 � 4,763 � (0.3)(5,000 � 4,763)6 8,000 6,384 � 4,834 � (0.3)(10,000 � 4,834)7 7,000 6,869 � 6,384 � (0.3)(8,000 � 6,384)8 9,000 6,908 � 6,869 � (0.3)(7,000 � 6,869)9 12,000 7,536 � 6,908 � (0.3)(9,000 � 6,908)
10 14,000 8,875 � 7,536 � (0.3)(12,000 � 7,536)11 15,000 10,412 � 8,875 � (0.3)(14,000 � 8,875)
The mean absolute deviation (MAD) can be used to determinewhich forecasting method is more accurate.
WeightedMoving Absolute Absolute
Year Demand Average Deviation Exp. Sm. Deviation
1 4,000 5,000 1,0002 6,000 4,700 1,3003 4,000 5,090 1,0904 5,000 4,500 500 4,763 2375 10,000 5,000 5,000 4,834 5,1666 8,000 7,250 750 6,384 1,6167 7,000 7,750 750 6,869 1318 9,000 8,000 1,000 6,908 2,0929 12,000 8,250 3,750 7,536 4,464
10 14,000 10,000 4,000 8,875 5,12511 15,000 12,250 12,750 10,412 14,588
Total: 18,500 26,808Mean: 2,312.5 2,437
5-18.
5-19.
Year 1 2 3 4 5 6
Forecast 410.0 422.0 443.9 466.1 495.2 521.8
Year Sales Forecast Using � �� 0.6 Forecast Using � �� 0.9
1 4502 495 410 � (0.6) (450 � 410) � 434 410 � (0.9)(450 � 410) � 4463 518 434 � (0.6) (495 � 434) � 470.6 446 � (0.9)(495 � 446) � 490.14 563 470.6 � (0.6)(518 � 470.6) � 499.0 490.1 � (0.9)(518 � 490.1) � 515.215 584 499 � (0.6) (563 � 499) � 537.4 515.21 � (0.9)(563 � 515.21) � 558.26 ? 537.4 � (0.6)(584 � 537) � 565.6 558.221 � (0.9)(584 � 558.2) � 581.4
Thus, the 3-year weighted moving average model appears to bemore accurate.
5-16. Using the forecasts in the previous problems we obtain theabsolute deviations given in the table below.
3-Yr MA 3-Yr Wt. MA Trend line Year Demand |deviation| |deviation| |deviation|
11 14 — — 0.7312 16 — — 1.6713 14 — — 1.3814 15 0.33 0.50 1.4415 10 5.00 5.00 2.5116 18 1.67 0.75 0.5517 17 0.67 0.75 2.6018 19 0.67 1.00 1.6519 12 4.00 3.75 0.2910 14 4.67 4.00 1.2411 15 3.33 2.75 1.18
Total absolutedeviations � 20.33 18.50 15.24
MAD (3-year moving average) � 2.54MAD (3-year weighted moving average) � 2.31MAD (trend line) � 1.39The trend line is best because the MAD is lowest.
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CHAPTER 5 FORECAST ING MODELS 57
5-20.
Actual � �� 0.3 Absolute � � 0.6 Absolute � � 0.9 AbsoluteYear Sales Forecast Deviation Forecast Deviation Forecast Deviation
1 450 410.0 40.0 410.0 40.0 410.0 40.02 495 422.0 73.0 434.0 61.0 446.0 49.03 518 443.9 74.1 470.6 47.4 490.1 27.94 563 466.1 96.9 499.0 64.0 515.2 47.85 584 495.2 88.8 537.4 46.6 558.2 25.86 ? 521.8 — 565.8 — 581.4 —
Total absolute deviation 372.8 259.0 190.5
MAD��0.3 � 372.8/5 � 74.56
MAD��0.6 � 259/5 � 51.8
MAD��0.9 � 190.5/5 � 38.1Because it has the lowest MAD, the smoothing constant � � 0.9gives the most accurate forecast.
5-21.
Year Sales Three-Year Moving Average
1 4502 4953 5184 563 (450 � 495 � 518)/3 � 487.6675 584 (495 � 518 � 563)/3 � 525.3336 ? (518 � 563 � 584)/3 � 555
5-22.
TimePeriod Sales
Year X Y X2 XY
1 1 450 1 4502 2 495 4 9903 3 518 9 15544 4 563 16 22525 5 2,584 125 2920
2,610 55 8166
b � 33.6
a � 421.2
Y � 421.2 � 33.6X
Projected sales in year 6,
Y � 421.2 � (33.6)(6)
� 622.8
5-23.
Three-Year Moving Time-SeriesYear Actual Sales Average Forecast Absolute Deviation Forecast Absolute Deviation
1 450 — — 454.8 4.82 495 — — 488.4 6.63 518 — — 522.0 4.04 563 487.7 75.3 555.6 7.45 584 525.3 58.7 589.2 5.26 ? 555.0 — 622.8 —
Total absolute deviation 134.0 28.0
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58 CHAPTER 5 FORECAST ING MODELS
MAD��0.3 � 74.56 (see Problem 5-20)
MADmoving average � 134/2 � 67
MADregression � 28/5 � 5.6Regression (trend line) is obviously the preferred method becauseof its low MAD.
5-24. To answer the discussion questions, two forecasting mod-els are required: a three-period moving average and a three-periodweighted moving average. Once the actual forecasts have beenmade, their accuracy can be compared using the mean average dif-ferences (MAD).
a, b.
Period Month Demand Average Weighted Average
4 Apr. 10 13.67 14.55 May 15 13.33 12.676 June 17 13.67 13.57 July 11 14 15.178 Aug. 14 14.33 13.679 Sept. 17 14 13.50
10 Oct. 12 14 1511 Nov. 14 14.33 1412 Dec. 16 14.33 13.8313 Jan. 11 14 14.6714 Feb. – 13.67 13.17
c. MAD for moving average is 2.2. MAD for weighted aver-age is 2.72. Moving average forecast for February is 13.6667.Weighted moving average forecast for February is 13.1667.
Because a three-period average forecasting method is used,forecasts start for period 4. As can be seen, the MAD for the mov-ing average is 2.2, and the MAD for the weighted moving averageis 2.7. Thus, based on this analysis, the moving average appears tobe more accurate. The forecast for February is about 14.
d. There are many other factors to consider, including sea-sonality and any underlying causal variables such as advertis-ing budget.
5-25. a.
Sum ofAbsolute
Actual Forecast ForecastWeek Miles (Ft) Error RSFE Errors MAD Track Signal
1 17 17.00 — — — — —2 21 17.00 �4.00 �4.00 4.00 4.00 13 19 17.80 �1.20 �5.20 5.20 2.60 24 23 18.04 �4.96 �10.16 10.16 3.39 35 18 19.03 �1.03 �9.13 11.19 2.80 3.36 16 18.83 �2.83 �6.30 14.02 2.80 2.257 20 18.26 �1.74 �8.04 15.76 2.63 3.058 18 18.61 �0.61 �7.43 16.37 2.34 3.179 22 18.49 �3.51 �10.94 19.88 2.49 4.21
10 20 19.19 �0.81 �11.75 20.69 2.30 5.1111 15 19.35 �4.35 �7.40 25.04 2.50 2.9612 22 18.48 �3.52 �10.92 28.56 2.60 4.20
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CHAPTER 5 FORECAST ING MODELS 59
b. The total MAD is 2.60.c. RSFE is consistently positive. Tracking signal exceeds 5MADs at week 10. This could indicate a problem.
5-26. a, b. See the accompanying table for a comparison ofthe calculations for the exponentially smoothed forecasts usingconstants of 0.1 and 0.6.
c. Students should note how stable the smoothed values forthe 0.1 smoothing constant are. When compared to actualweek 25 calls of 85, the 0.6 smoothing constant appears to doa better job. On the basis of the forecast error, the 0.6 con-stant is better also. However, other smoothing constants needto be examined.
Actual Smoothed SmoothedWeek, Value, Value, Forecast Value, Forecast
t At Ft (� �� 0.1) Error Ft (� �� 0.6) Error
1 50 50 — —2 35 50 �15 50 �153 25 48 �23 41 �164 40 46 �6 31 �85 45 45 0 37 �96 35 45 �10 42 �77 20 44 �24 38 �188 30 42 �12 27 �39 35 41 �6 29 �6
10 20 40 �20 32 �1211 15 38 �23 25 �1012 40 36 �4 19 �2113 55 36 �19 32 �2314 35 38 �3 46 �1115 25 38 �13 39 �1416 55 37 �18 31 �2417 55 38 �16 45 �1018 40 40 0 51 �1219 35 40 �5 44 �1020 60 40 �20 39 �2121 75 42 �33 51 �2322 50 45 �5 66 �1623 40 45 �5 56 �1624 65 45 �20 46 �1825 47 58
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60 CHAPTER 5 FORECAST ING MODELS
5-27. Using data from Problem 5-26, with � � 0.9
Actual SmoothedValue Value Forecast
Week At Ft Error
1 50 50 —2 35 50 �153 25 36 �114 40 26 145 45 39 66 35 44 �97 20 36 �168 30 22 89 35 29 6
10 20 34 �1411 15 21 �612 40 16 2413 55 38 1714 35 53 �1815 25 37 �1216 55 26 2917 55 52 318 40 55 �1519 35 41 �620 60 36 2421 75 58 1722 50 73 �2323 40 52 �1224 65 41 2425 62
MAD � 14.48
Note that in this problem, the initial forecast (for the first period) wasnot used in computing the MAD. Either approach is considered valid.
5-28. Exponential smoothing with � � 0.1
Month Income Forecast Error
Feb. 70.0 65.0 —March 68.5 65.0 � 0.1 (70 � 65) � 65.5 3.0April 64.8 65.5 � 0.1(68.5 � 65.5) � 65.8 �1.0May 71.7 65.8 � 0.1(64.8 � 65.8) � 65.7 6.0June 71.3 65.7 � 0.1(71.7 � 65.7) � 66.3 5.0July 72.8 66.3 � 0.1(71.3 � 66.3) � 66.8 6.0Aug. 66.8 � 0.1(72.8 � 66.8) � 67.4
MAD � 4.20
Note that in this problem, the initial forecast (for the first period) wasnot used in computing the MAD. Either approach is considered valid.
5-29. Exponential smoothing with � � 0.3
Month Income Forecast Error
Feb. 70.0 65.0 —March 68.5 66.5 2.0April 64.8 67.1 �2.3May 71.7 66.4 5.3June 71.3 68.0 3.3July 72.8 69.0 3.8Aug. 70.1
MAD � 3.34
Based on MAD, � � 0.3 produces a better forecast than � � 0.1(of Problem 5-28).
Note that in this problem, the initial forecast (for the first period) wasnot used in computing the MAD. Either approach is considered valid.
5-30. Using QM for Windows, we select Forecasting - TimeSeries and multiplicative decomposition. Then specify CenteredMoving Average and we have the following results:
a. Quarter 1 index � 0.8825; Quarter 2 index � 0.9816;Quarter 3 index � 0.9712; Quarter 4 index � 1.1569
b. The trendline is Y � 237.7478 � 3.6658X
c. Quarter 1: Y � 237.7478 � 3.6658(17) � 300.0662
Quarter 2: Y � 237.7478 � 3.6658(18) � 303.7320
Quarter 3: Y � 237.7478 � 3.6658(19) � 307.3978
Quarter 4: Y � 237.7478 � 3.6658(20) � 311.0636
d. Quarter 1: 300.0662(0.8825) � 264.7938
Quarter 2: 303.7320(0.9816) � 298.1579
Quarter 3: 307.3978(0.9712) � 298.5336
Quarter 4: 311.0636(1.1569) � 359.8719
5-31. Letting
t � time period (1, 2, 3, . . . , 16)
Q1 � 1 if quarter 1, 0 otherwise
Q2 � 1 if quarter 2, 0 otherwise
Q3 � 1 if quarter 3, 0 otherwise
Note: if Q1 � Q2 � Q3 � 0, then it is quarter 4.
Using computer software we get
Y � 281.6 � 3.7t � 75.7Q1 � 48.9Q2 � 52.1Q3
The forecasts for the next 4 quarters are:
Y � 281.6 � 3.7(17) � 75.7(1) � 48.9(0) � 52.1(0) � 268.7
Y � 281.6 � 3.7(18) � 75.7(0) � 48.9(1) � 52.1(0) � 299.2
Y � 281.6 � 3.7(19) � 75.7(0) � 48.9(0) � 52.1(1) � 299.7
Y � 281.6 � 3.7(20) � 75.7(0) � 48.9(0) � 52.1(0) � 355.4
5-32. For a smoothing constant of 0.2, the forecast for year 11is 6.489.
Year Rate Forecast |Error|
1 7.2 7.2 02 7 7.2 0.23 6.2 7.16 0.964 5.5 6.968 1.4685 5.3 6.674 1.3746 5.5 6.400 0.9007 6.7 6.220 0.4808 7.4 6.316 1.0849 6.8 6.533 0.267
10 6.1 6.586 0.48611 6.489
MAD = 0.722For a smoothing constant of 0.4, the forecast for year 11 is 6.458.
Year Rate Forecast |Error|
1 7.2 7.2 02 7 7.2 0.23 6.2 7.12 0.924 5.5 6.752 1.2525 5.3 6.251 0.9516 5.5 5.871 0.3717 6.7 5.722 0.9788 7.4 6.113 1.2879 6.8 6.628 0.172
10 6.1 6.697 0.59711 6.458
MAD = 0.673
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CHAPTER 5 FORECAST ING MODELS 61
For a smoothing constant of 0.6, the forecast for year 11 is 6.401.
Year Rate Forecast |Error|
1 7.2 7.2 02 7 7.2 0.23 6.2 7.08 0.884 5.5 6.552 1.0525 5.3 5.921 0.6216 5.5 5.548 0.0487 6.7 5.519 1.1818 7.4 6.228 1.1729 6.8 6.931 0.131
10 6.1 6.852 0.75211 6.401
MAD = 0.604For a smoothing constant of 0.8, the forecast for year 11 is 6.256.
Year Rate Forecast |Error|
1 7.2 7.2 02 7 7.2 0.23 6.2 7.04 0.844 5.5 6.368 0.8685 5.3 5.674 0.3746 5.5 5.375 0.1257 6.7 5.475 1.2258 7.4 6.455 0.9459 6.8 7.211 0.411
10 6.1 6.882 0.78211 6.256
MAD = 0.577The lowest MAD is 0.577 for a smoothing constant of 0.8.
5-33. To compute a seasonalized or adjusted sales forecast, wejust multiply each seasonal index by the appropriate trend forecast.
Y � seasonal index � Ytrend forecast
Hence for:
Quarter I: YI � (1.30)($100,000) � $130,000
Quarter II: YII � (0.90)($120,000) � $108,000
Quarter III: YIII � (0.70)($140,000) � $98,000
Quarter IV: YIV � (1.10)($160,000) � $176,000
5-34.
(Average demandfor season)
Overall averagedemand
= ×1 200
4
,season index
Year 3 demandnew annual demand
4=
Season index(average for season)
overall av=
eerage demand
sum of all values= ( )
8
(year 1 demand) +�
((year 2 demand)
2
Solution Table for Problem 5-34
AverageYear 1 Year 2 (Average Year 1- Season Season Year 3
Season Demand Demand Year 2 Demand) Demand Index Demand
Fall 200 250 225.0 250 0.90 270Winter 350 300 325.0 250 1.30 390Spring 150 165 157.5 250 0.63 189Summer 300 285 292.5 250 1.17 351
5-35. Using Excel, the trend equation is Y � 1582.61 � 612.37X.
For 2008, X � 19; Y � 1582.61 � 612.37(19) � 13217.6
For 2009, X � 20; Y � 1582.61 � 612.37(20) � 13830.0
For 2010, X � 21; Y � 1582.61 � 612.37(21) � 14442.4
The MSE from the Excel output is 1654334.7.
5-36. a. With a smoothing constant of 0.3, the forecast for 2008is 11211.2 with MSE � 3246841.
b. Using QM for Windows, the best smoothing constant is1.0. This gives the lowest MSE of 1443842.
5-37. Using Excel, the trend equation is Y � 1.1940 � 0.0095X.
For January of 2007, X � 13; Y � 1.1940 � 0.0095(13) � 1.318.
For February of 2007, X � 14; Y � 1.1940 � 0.0095(14) � 1.327.
5-38. The forecast for January 2007 would be 1.286.
The MSE with the trend equation is 0.0003. The MSE with thisexponential smoothing model is 0.0010.
5-39. With a � 0.4, forecast for 2004 � 10,339 and MAD �837. With a � 0.6, forecast for 2004 � 10,698 and MAD � 612. 5-40. Using Excel, the trend line is: GDP � 6142.7 �
441.4(time). For 2004 (time � 12) the forecast is GDP � 6142.7 �441.4(12) � 11,439.5.5-41. The trend line found using Excel is: Patients � 29.73 �
3.28(time). Note these coefficients are rounded. For the next3 years (time � 11, 12, and 13) the forecasts for the number ofpatients are:
Patients � 29.73 � 3.28(11) � 65.8 Patients � 29.73 � 3.28(12) � 69.1 Patients � 29.73 � 3.28(13) � 72.4
The coefficient of determination is 0.85, so the model is a fairmodel.
SOLUTIONS TO INTERNET HOMEWORK PROBLEMS
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62 CHAPTER 5 FORECAST ING MODELS
Deposits and GSP over Time
0
20
40
60
80
100
1950 1960 1970 1980 1990 2000 2010
Time
DEPOSITS
GSP
5-42. The trend line found using Excel is: Crime Rate � 51.98� 6.09(time). Note these coefficients are rounded. For thenext 3 years (time � 11, 12, and 13) the forecasts for the crimerates are:
Crime Rate � 51.98 � 6.09(11) � 118.97Crime Rate � 51.98 � 6.09(12) � 125.06Crime Rate � 51.98 � 6.09(13) � 131.15
The coefficient of determination is 0.96, so this is a very goodmodel.
5-43. The regression equation (from Excel) is: Patients � 1.23 �0.54(crime rate). Note these coefficients are rounded. If the crimerate is 131.2, the forecast number of patients is:
Patients � 1.23 � 0.54(131.2) � 72.1
If the crime rate is 90.6, the forecast number of patients is:
Patients � 1.23 � 0.54(90.6) � 50.2
The coefficient of determination is 0.90, so this is a good model.
5-44. With a � 0.6, forecast for 2003 � 86.2 and MAD �3.42. With a � 0.2, forecast for 2003 � 63.87 and MAD � 7.23.The model with a � 0.6 is better since it has a lower MAD.
5-45. With a � 0.6, forecast for 2003 � 4.86 and MAD �0.23. With a � 0.2, forecast for 2003 � 4.52 and MAD � 0.48.The model with a � 0.6 is better since it has a lower MAD.
5-46. The trend line (coefficients from Excel are rounded) fordeposits is:
Deposits � �18.968 � 1.638(time)For 2003, 2004, and 2005, time � 45, 46, and 47 respectively. Theforecasts are:
Deposits � �18.968 � 1.638(45) � 54.7Deposits � �18.968 � 1.638(46) � 56.4Deposits � �18.968 � 1.638(47) � 58.0
The trend line (coefficients from Excel are rounded) for GSP is:GSP � 0.090 � 0.112(time). The forecasts are:GSP � 0.090 � 0.112(45) � 5.1GSP � 0.090 � 0.112(46) � 5.2GSP � 0.090 � 0.112(47) � 5.4
5-47. The regression equation from Excel is Deposits � �17.64 � 13.59(GSP)
In the scatterplot of this data that follows, the pattern appears tochange around 1985. There are definitely different relationshipsbefore 1985 and after 1985, so perhaps the model should be devel-oped with 1985 as the first year of data.
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